An authoritative reference on the new generation of VSC-FACTS and VSC-HVDC systems and their applicability within curren

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*Table of contents : CoverTitle PageCopyrightContentsPrefaceAbout the BookAcknowledgementsAbout the Companion WebsiteChapter 1 Flexible Electrical Energy Systems 1.1 Introduction 1.2 Classification of Flexible Transmission System Equipment 1.2.1 SVC 1.2.2 STATCOM 1.2.3 SSSC 1.2.4 Compound VSC Equipment for AC Applications 1.2.5 CSC-HVDC Links 1.2.6 VSC-HVDC 1.3 Flexible Systems Vs Conventional Systems 1.3.1 Transmission 1.3.1.1 HVAC Vs HVDC Power Transmission for Increased Power Throughputs 1.3.1.2 VAR Compensation 1.3.1.3 Frequency Compensation 1.3.2 Generation 1.3.2.1 Wind Power Generation 1.3.2.2 Solar Power Generation 1.3.3 Distribution 1.3.3.1 Load Compensation 1.3.3.2 Dynamic Voltage Support 1.3.3.3 Flexible Reconfigurations 1.3.3.4 AC-DC Distribution Systems 1.3.3.5 DC Power Grids with Multiple Voltage Levels 1.3.3.6 Smart Grids 1.4 Phasor Measurement Units 1.5 Future Developments and Challenges 1.5.1 Generation 1.5.2 Transmission 1.5.3 Distribution ReferencesChapter 2 Power Electronics for VSC-Based Bridges 2.1 Introduction 2.2 Power Semiconductor Switches 2.2.1 The Diode 2.2.2 The Thyristor 2.2.3 The Bipolar Junction Transistor 2.2.4 The Metal-Oxide-Semiconductor Field-Effect Transistor 2.2.5 The Insulated-Gate Bipolar Transistor 2.2.6 The Gate Turn-Off Thyristor 2.2.7 The MOS-Controlled Thyristor 2.2.8 Considerations for the Switch Selection Process 2.3 Voltage Source Converters 2.3.1 Basic Concepts of Pulse Width Modulated-Output Schemes and Half-Bridge VSC 2.3.2 Single-Phase Full-Bridge VSC 2.3.2.1 PWM with Bipolar Switching 2.3.2.2 PWM with Unipolar Switching 2.3.2.3 Square-Wave Mode 2.3.2.4 Phase-Shift Control Operation 2.3.3 Three-Phase VSC 2.3.4 Three-Phase Multilevel VSC 2.3.4.1 The Multilevel NPC VSC 2.3.4.2 The Multilevel FC VSC 2.3.4.3 The Cascaded H-Bridge VSC 2.3.4.4 PWM Techniques for Multilevel VSCs 2.3.4.5 An Alternative Multilevel Converter Topology 2.4 HVDC Systems Based on VSC 2.5 Conclusions ReferencesChapter 3 Power Flows 3.1 Introduction 3.2 Power Network Modelling 3.2.1 Transmission Lines Modelling 3.2.2 Conventional Transformers Modelling 3.2.3 LTC Transformers Modelling 3.2.4 Phase-Shifting Transformers Modelling 3.2.5 Compound Transformers Modelling 3.2.6 Series and Shunt Compensation Modelling 3.2.7 Load Modelling 3.2.8 Network Nodal Admittance 3.3 Peculiarities of the Power Flow Formulation 3.4 The Nodal Power Flow Equations 3.5 The Newton-Raphson Method in Rectangular Coordinates 3.5.1 The Linearized Equations 3.5.2 Convergence Characteristics of the Newton-Raphson Method 3.5.3 Initialization of Newton-Raphson Power Flow Solutions 3.5.4 Incorporation of PMU Information in Newton-Raphson Power Flow Solutions 3.6 The Voltage Source Converter Model 3.6.1 VSC Nodal Admittance Matrix Representation 3.6.2 Full VSC Station Model 3.6.3 VSC Nodal Power Equations 3.6.4 VSC Linearized System of Equations 3.6.5 Non-Regulated Power Flow Solutions 3.6.6 Practical Implementations 3.6.6.1 Control Strategy 3.6.6.2 Initial Parameters and Limits 3.6.7 VSC Numerical Examples 3.7 The STATCOM Model 3.7.1 STATCOM Numerical Examples 3.8 VSC-HVDC Systems Modelling 3.8.1 VSC-HVDC Nodal Power Equations 3.8.2 VSC-HVDC Linearized Equations 3.8.3 Back-to-Back VSC-HVDC Systems Modelling 3.8.4 VSC-HVDC Numerical Examples 3.9 Three-Terminal VSC-HVDC System Model 3.9.1 VSC Types 3.9.2 Power Mismatches 3.9.3 Linearized System of Equations 3.10 Multi-Terminal VSC-HVDC System Model 3.10.1 Multi-Terminal VSC-HVDC System with Common DC Bus Model 3.10.2 Unified Solutions of AC-DC Networks 3.10.3 Unified vs Quasi-Unified Power Flow Solutions 3.10.4 Test Case 9 3.11 Conclusions ReferencesChapter 4 Optimal Power Flows 4.1 Introduction 4.2 Power Flows in Polar Coordinates 4.3 Optimal Power Flow Formulation 4.4 The Lagrangian Methods 4.4.1 Necessary Optimality Conditions (Karush-Kuhn-Tucker Conditions) 4.5 AC OPF Formulation 4.5.1 Objective Function 4.5.2 Linearized System of Equations 4.5.3 Augmented Lagrangian Function 4.5.4 Selecting the OPF Solution Algorithm 4.5.5 Control Enforcement in the OPF Algorithm 4.5.6 Handling Limits of State Variables 4.5.7 Handling Limits of Functions 4.5.8 A Simple Network Model 4.5.8.1 Step One-Identifying State and Control Variables 4.5.8.2 Step Two-Identifying Constraints 4.5.8.3 Step Three-Forming the Lagrangian Function 4.5.8.4 Step Four-Linearized System of Equations 4.5.8.5 Step Five-Implementation of the Augmented Lagrangian 4.5.9 Recent Extensions in the OPF Problem 4.5.10 Test Case: IEEE 30-Bus System 4.5.10.1 Test System 4.5.10.2 Problem Formulation 4.5.10.3 OPF Test Cases 4.5.10.4 Benchmark Test Case (With No Voltage Control) 4.5.10.5 Test Case with Voltage Control Using Variable Transformers Taps (Case I) 4.5.10.6 Test Case with Nodal Voltage Regulation (Case II) 4.5.10.7 Test Case with Nodal Voltage Regulation (Case III) 4.5.10.8 A Summary of Results 4.6 Generalization of the OPF Formulation for AC-DC Networks 4.7 Inclusion of the VSC Model in OPF 4.7.1 VSC Power Balance Equations 4.7.2 VSC Control Considerations 4.7.3 VSC Linearized System of Equations 4.8 The Point-to-Point and Back-to-Back VSC-HVDC Links Models in OPF 4.8.1 VSC-HVDC Link Power Balance Formulation 4.8.2 VSC-HVDC Link Control 4.8.3 VSC-HVDC Full Set of Equality Constraints 4.8.4 Linearized System of Equations 4.9 Multi-Terminal VSC-HVDC Systems in OPF 4.9.1 The Expanded, General Formulation 4.9.2 Multi-Terminal VSC-HVDC Test Case 4.9.2.1 DC Network 4.9.2.2 AC Network 4.9.2.3 Objective Function 4.9.2.4 Summary of OPF Results 4.9.2.5 Converter Outputs-No Converter Losses 4.9.2.6 Converter Outputs-With Converter Losses 4.9.2.7 Power Flows in AC Transmission Lines-With No Converter Losses 4.9.2.8 Power Flows in AC Transmission Lines-With Converter Losses 4.10 Conclusion ReferencesChapter 5 State Estimation 5.1 Introduction 5.2 State Estimation of Electrical Networks 5.3 Network Model and Measurement System 5.3.1 Topological Processing 5.3.2 Network Model 5.3.3 The Measurements System Model 5.4 Calculation of the Estimated State 5.4.1 Solution by the Normal Equations 5.4.2 Equality-Constrained WLS 5.4.3 Observability Analysis and Reference Phase 5.4.4 Weighted Least Squares State Estimator (WLS-SE) Using Matlab Code 5.5 Bad Data Identification 5.5.1 Bad Data 5.5.2 The Largest Normalized Residual Test 5.5.3 Bad Data Identification Using WLS-SE 5.6 FACTS Device State Estimation Modelling in Electrical Power Grids 5.6.1 Incorporation of New Models in State Estimation 5.6.2 Voltage Source Converters 5.6.3 STATCOM 5.6.4 STATCOM Model in WLS-SE 5.6.5 Unified Power Flow Controller 5.6.6 The UPFC Model in WLS-SE 5.6.7 High Voltage Direct Current Based on Voltage Source Converters 5.6.8 VSC-HVDC Model in WLS-SE 5.6.9 Multi-terminal HVDC 5.6.10 MT-VSC-HVDC Model in WLS-SE 5.7 Incorporation of Measurements Furnished by PMUs 5.7.1 Incorporation of Synchrophasors in State Estimation 5.7.2 Synchrophasors Formulations 5.7.3 Phase Reference 5.7.4 PMU Outputs in WLS-SE 5.A.1 Input Data and Output Results in WLS-SE 5.A.1.1 Input Data 5.A.1.2 Network Data 5.A.1.3 Measurements Data 5.A.1.4 State Estimator Configuration 5.A.2 Output Results ReferencesChapter 6 Dynamic Simulations of Power Systems 6.1 Introduction 6.2 Modelling of Conventional Power System Components 6.2.1 Modelling of Synchronous Generators 6.2.2 Synchronous Generator Controllers 6.2.2.1 Speed Governors 6.2.2.2 Steam Turbine and Hydro Turbine 6.2.2.3 Automatic Voltage Regulator 6.2.2.4 Transmission Line Model 6.2.2.5 Load Model 6.3 Time Domain Solution Philosophy 6.3.1 Numerical Solution Technique 6.3.2 Benchmark Numerical Example 6.4 Modelling of the STATCOM for Dynamic Simulations 6.4.1 Discretization and Linearization of the STATCOM Differential Equations 6.4.2 Numerical Example with STATCOMs 6.5 Modelling of VSC-HVDC Links for Dynamic Simulations 6.5.1 Discretization and Linearization of the Differential Equations of the VSC-HVDC 6.5.2 Validation of the VSC-HVDC Link Model 6.5.3 Numerical Example with an Embedded VSC-HVDC Link 6.5.4 Dynamic Model of the VSC-HVDC Link with Frequency Regulation Capabilities 6.5.4.1 Linearization of the Equations of the VSC-HVDC Model with Frequency Regulation Capabilities 6.5.4.2 Validation of the VSC-HVDC Link Model Providing Frequency Support 6.5.4.3 Numerical Example with a VSC-HVDC Link Model Providing Frequency Support 6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations 6.6.1 Three-terminal VSC-HVDC Dynamic Model 6.6.2 Validation of the Three-Terminal VSC-HVDC Dynamic Model 6.6.3 Multi-Terminal VSC-HVDC Dynamic Model 6.6.4 Numerical Example with a Six-Terminal VSC-HVDC Link Forming a DC Ring 6.6.4.1 Disconnection of a DC Transmission Line 6.6.4.2 Three-Phase Fault Applied to AC3 6.7 Conclusion ReferencesChapter 7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment 7.1 Introduction 7.2 The STATCOM Case 7.3 STATCOM Based on Multilevel VSC 7.4 Example of HVDC based on Multilevel FC Converter 7.5 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters 7.6 Conclusions ReferencesIndexEULA*

VSC-FACTS-HVDC

VSC-FACTS-HVDC Analysis, Modelling and Simulation in Power Grids

Professor Dr Enrique Acha Laboratory of Electrical Energy Engineering Tampere University Tampere, Finland

Dr Pedro Roncero-Sánchez Department of Electronics Electrical Engineering and Control Systems University of Castilla-La Mancha, Spain

Dr Antonio de la Villa Jaén Department of Electrical Engineering University of Seville, Spain

Dr Luis M. Castro Faculty of Engineering National University of Mexico (UNAM) Mexico City, Mexico

Dr Behzad Kazemtabrizi School of Engineering Durham University, UK

This edition ﬁrst published 2019 © 2019 John Wiley & Sons Ltd All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Professor Dr Enrique Acha, Dr Pedro Roncero-Sánchez, Dr Antonio de la Villa Jaén, Dr Luis M. Castro and Dr Behzad Kazemtabrizi to be identiﬁed as the authors of this work has been asserted in accordance with law. Registered Oﬃces John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Oﬃce The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK For details of our global editorial oﬃces, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best eﬀorts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and speciﬁcally disclaim all warranties, including without limitation any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

® ®

MATLAB is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This work’s use or discussion of MATLAB software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB software. Library of Congress Cataloging-in-Publication Data Names: Acha, Enrique, author. Title: VSC-FACTS-HVDC : analysis, modelling and simulation in power grids / Professor Dr Enrique Acha, Tampere University, Tampere, Finland, Dr Pedro Roncero-Sanchez, Universidad de Castilla-La Mancha, Ciudad Real, Espana Dr Antonio de la Villa Jan, Universidad de Sevilla, Sevilla, Espana, Dr Luis M Castro, Universidad Nacional Autonoma de Mexico (UNAM), Mexico City, Mexico, Dr Behzad Kazemtabrizi, Durham University, Durham, England. Description: First edition. | Hoboken, NJ : John Wiley & Sons Ltd, 2019. | Includes bibliographical references and index. | Identiﬁers: LCCN 2018051883 (print) | LCCN 2018055480 (ebook) | ISBN 9781118965801 (Adobe PDF) | ISBN 9781118965849 (ePub) | ISBN 9781119973980 (hardcover) Subjects: LCSH: Smart power grids. | Flexible AC transmission systems. | Electric power transmission–Direct current. Classiﬁcation: LCC TK3105 (ebook) | LCC TK3105 .A25 2019 (print) | DDC 621.319–dc23 LC record available at https://lccn.loc.gov/2018051883 Cover Design: Wiley Cover Images: Background: © Teka77/iStock.com, Diagram: Courtesy of Enrique Acha Set in 10/12pt WarnockPro by SPi Global, Chennai, India Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY 10 9 8 7 6 5 4 3 2 1

®

To the memory of Jos Arrillaga, the one who wrote the most and best about HVDC transmission.

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Contents Preface xiii About the Book xvii Acknowledgements xxi About the Companion Website xxiii 1

Flexible Electrical Energy Systems 1

1.1 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.3 1.3.1 1.3.1.1 1.3.1.2 1.3.1.3 1.3.2 1.3.2.1 1.3.2.2 1.3.3 1.3.3.1 1.3.3.2 1.3.3.3 1.3.3.4 1.3.3.5 1.3.3.6 1.4 1.5 1.5.1 1.5.2 1.5.3

Introduction 1 Classiﬁcation of Flexible Transmission System Equipment 5 SVC 6 STATCOM 7 SSSC 9 Compound VSC Equipment for AC Applications 10 CSC-HVDC Links 12 VSC-HVDC 13 Flexible Systems Vs Conventional Systems 15 Transmission 16 HVAC Vs HVDC Power Transmission for Increased Power Throughputs VAR Compensation 19 Frequency Compensation 24 Generation 27 Wind Power Generation 28 Solar Power Generation 30 Distribution 33 Load Compensation 35 Dynamic Voltage Support 35 Flexible Reconﬁgurations 36 AC-DC Distribution Systems 37 DC Power Grids with Multiple Voltage Levels 40 Smart Grids 40 Phasor Measurement Units 43 Future Developments and Challenges 46 Generation 46 Transmission 47 Distribution 48 References 49

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2

Power Electronics for VSC-Based Bridges 53

2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.3 2.3.1

Introduction 53 Power Semiconductor Switches 53 The Diode 55 The Thyristor 56 The Bipolar Junction Transistor 57 The Metal-Oxide-Semiconductor Field-Eﬀect Transistor 59 The Insulated-Gate Bipolar Transistor 59 The Gate Turn-Oﬀ Thyristor 59 The MOS-Controlled Thyristor 60 Considerations for the Switch Selection Process 61 Voltage Source Converters 61 Basic Concepts of Pulse Width Modulated-Output Schemes and Half-Bridge VSC 62 Single-Phase Full-Bridge VSC 66 PWM with Bipolar Switching 67 PWM with Unipolar Switching 69 Square-Wave Mode 69 Phase-Shift Control Operation 69 Three-Phase VSC 72 Three-Phase Multilevel VSC 74 The Multilevel NPC VSC 76 The Multilevel FC VSC 80 The Cascaded H-Bridge VSC 81 PWM Techniques for Multilevel VSCs 85 An Alternative Multilevel Converter Topology 85 HVDC Systems Based on VSC 88 Conclusions 94 References 95

2.3.2 2.3.2.1 2.3.2.2 2.3.2.3 2.3.2.4 2.3.3 2.3.4 2.3.4.1 2.3.4.2 2.3.4.3 2.3.4.4 2.3.4.5 2.4 2.5

3

Power Flows 99

3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.2.8 3.3 3.4 3.5 3.5.1 3.5.2 3.5.3

Introduction 99 Power Network Modelling 100 Transmission Lines Modelling 100 Conventional Transformers Modelling 100 LTC Transformers Modelling 101 Phase-Shifting Transformers Modelling 101 Compound Transformers Modelling 102 Series and Shunt Compensation Modelling 102 Load Modelling 102 Network Nodal Admittance 102 Peculiarities of the Power Flow Formulation 103 The Nodal Power Flow Equations 105 The Newton-Raphson Method in Rectangular Coordinates 106 The Linearized Equations 107 Convergence Characteristics of the Newton-Raphson Method 108 Initialization of Newton-Raphson Power Flow Solutions 109

Contents

3.5.4 3.6 3.6.1 3.6.2 3.6.3 3.6.4 3.6.5 3.6.6 3.6.6.1 3.6.6.2 3.6.7 3.7 3.7.1 3.8 3.8.1 3.8.2 3.8.3 3.8.4 3.9 3.9.1 3.9.2 3.9.3 3.10 3.10.1 3.10.2 3.10.3 3.10.4 3.11

Incorporation of PMU Information in Newton-Raphson Power Flow Solutions 111 The Voltage Source Converter Model 112 VSC Nodal Admittance Matrix Representation 113 Full VSC Station Model 115 VSC Nodal Power Equations 117 VSC Linearized System of Equations 117 Non-Regulated Power Flow Solutions 119 Practical Implementations 120 Control Strategy 120 Initial Parameters and Limits 120 VSC Numerical Examples 121 The STATCOM Model 125 STATCOM Numerical Examples 127 VSC-HVDC Systems Modelling 129 VSC-HVDC Nodal Power Equations 131 VSC-HVDC Linearized Equations 133 Back-to-Back VSC-HVDC Systems Modelling 135 VSC-HVDC Numerical Examples 135 Three-Terminal VSC-HVDC System Model 139 VSC Types 142 Power Mismatches 142 Linearized System of Equations 143 Multi-Terminal VSC-HVDC System Model 146 Multi-Terminal VSC-HVDC System with Common DC Bus Model 147 Uniﬁed Solutions of AC-DC Networks 148 Uniﬁed vs Quasi-Uniﬁed Power Flow Solutions 148 Test Case 9 150 Conclusions 153 References 153 3.A Appendix 154 3.B Appendix 156

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Optimal Power Flows 159

4.1 4.2 4.3 4.4 4.4.1 4.5 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6 4.5.7

Introduction 159 Power Flows in Polar Coordinates 160 Optimal Power Flow Formulation 161 The Lagrangian Methods 162 Necessary Optimality Conditions (Karush-Kuhn-Tucker Conditions) 163 AC OPF Formulation 164 Objective Function 165 Linearized System of Equations 165 Augmented Lagrangian Function 167 Selecting the OPF Solution Algorithm 168 Control Enforcement in the OPF Algorithm 168 Handling Limits of State Variables 169 Handling Limits of Functions 169

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4.5.8 4.5.8.1 4.5.8.2 4.5.8.3 4.5.8.4 4.5.8.5 4.5.9 4.5.10 4.5.10.1 4.5.10.2 4.5.10.3 4.5.10.4 4.5.10.5

A Simple Network Model 170 Step One – Identifying State and Control Variables 170 Step Two – Identifying Constraints 170 Step Three – Forming the Lagrangian Function 171 Step Four – Linearized System of Equations 172 Step Five – Implementation of the Augmented Lagrangian 172 Recent Extensions in the OPF Problem 173 Test Case: IEEE 30-Bus System 173 Test System 173 Problem Formulation 173 OPF Test Cases 174 Benchmark Test Case (With No Voltage Control) 175 Test Case with Voltage Control Using Variable Transformers Taps (Case I) 176 4.5.10.6 Test Case with Nodal Voltage Regulation (Case II) 176 4.5.10.7 Test Case with Nodal Voltage Regulation (Case III) 177 4.5.10.8 A Summary of Results 177 4.6 Generalization of the OPF Formulation for AC-DC Networks 179 4.7 Inclusion of the VSC Model in OPF 181 4.7.1 VSC Power Balance Equations 181 4.7.2 VSC Control Considerations 183 4.7.3 VSC Linearized System of Equations 184 4.8 The Point-to-Point and Back-to-Back VSC-HVDC Links Models in OPF 184 4.8.1 VSC-HVDC Link Power Balance Formulation 185 4.8.2 VSC-HVDC Link Control 187 4.8.3 VSC-HVDC Full Set of Equality Constraints 188 4.8.4 Linearized System of Equations 189 4.9 Multi-Terminal VSC-HVDC Systems in OPF 191 4.9.1 The Expanded, General Formulation 192 4.9.2 Multi-Terminal VSC-HVDC Test Case 193 4.9.2.1 DC Network 193 4.9.2.2 AC Network 194 4.9.2.3 Objective Function 194 4.9.2.4 Summary of OPF Results 195 DC Network 196 4.9.2.5 Converter Outputs – No Converter Losses 196 4.9.2.6 Converter Outputs – With Converter Losses 197 AC Network 199 4.9.2.7 Power Flows in AC Transmission Lines – With No Converter Losses 199 4.9.2.8 Power Flows in AC Transmission Lines – With Converter Losses 200 4.10 Conclusion 200 References 201 5

State Estimation 203

5.1 5.2

Introduction 203 State Estimation of Electrical Networks

204

Contents

5.3 5.3.1 5.3.2 5.3.3 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.5 5.5.1 5.5.2 5.5.3 5.6 5.6.1 5.6.2 5.6.3 5.6.4 5.6.5 5.6.6 5.6.7 5.6.8 5.6.9 5.6.10 5.7 5.7.1 5.7.2 5.7.3 5.7.4 5.A 5.A.1 5.A.1.1 5.A.1.2 5.A.1.3 5.A.1.4 5.A.2

Network Model and Measurement System 206 Topological Processing 206 Network Model 206 The Measurements System Model 208 Calculation of the Estimated State 210 Solution by the Normal Equations 210 Equality-Constrained WLS 212 Observability Analysis and Reference Phase 213 Weighted Least Squares State Estimator (WLS-SE) Using Matlab Code 215 Bad Data Identiﬁcation 217 Bad Data 217 The Largest Normalized Residual Test 218 Bad Data Identiﬁcation Using WLS-SE 219 FACTS Device State Estimation Modelling in Electrical Power Grids 220 Incorporation of New Models in State Estimation 220 Voltage Source Converters 221 STATCOM 224 STATCOM Model in WLS-SE 225 Uniﬁed Power Flow Controller 227 The UPFC Model in WLS-SE 228 High Voltage Direct Current Based on Voltage Source Converters 230 VSC-HVDC Model in WLS-SE 231 Multi-terminal HVDC 233 MT-VSC-HVDC Model in WLS-SE 235 Incorporation of Measurements Furnished by PMUs 236 Incorporation of Synchrophasors in State Estimation 236 Synchrophasors Formulations 237 Phase Reference 239 PMU Outputs in WLS-SE 239 Appendix 240 Input Data and Output Results in WLS-SE 240 Input Data 240 Network Data 240 Measurements Data 242 State Estimator Conﬁguration 243 Output Results 243 References 244

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Dynamic Simulations of Power Systems 247

6.1 6.2 6.2.1 6.2.2 6.2.2.1 6.2.2.2 6.2.2.3 6.2.2.4

Introduction 247 Modelling of Conventional Power System Components Modelling of Synchronous Generators 248 Synchronous Generator Controllers 250 Speed Governors 250 Steam Turbine and Hydro Turbine 251 Automatic Voltage Regulator 252 Transmission Line Model 253

248

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6.2.2.5 6.3 6.3.1 6.3.2 6.4 6.4.1 6.4.2 6.5 6.5.1 6.5.2 6.5.3 6.5.4 6.5.4.1 6.5.4.2 6.5.4.3 6.6 6.6.1 6.6.2 6.6.3 6.6.4 6.6.4.1 6.6.4.2 6.7

Load Model 253 Time Domain Solution Philosophy 254 Numerical Solution Technique 254 Benchmark Numerical Example 257 Modelling of the STATCOM for Dynamic Simulations 261 Discretization and Linearization of the STATCOM Diﬀerential Equations 264 Numerical Example with STATCOMs 266 Modelling of VSC-HVDC Links for Dynamic Simulations 272 Discretization and Linearization of the Diﬀerential Equations of the VSC-HVDC 276 Validation of the VSC-HVDC Link Model 280 Numerical Example with an Embedded VSC-HVDC Link 283 Dynamic Model of the VSC-HVDC Link with Frequency Regulation Capabilities 289 Linearization of the Equations of the VSC-HVDC Model with Frequency Regulation Capabilities 291 Validation of the VSC-HVDC Link Model Providing Frequency Support 292 Numerical Example with a VSC-HVDC Link Model Providing Frequency Support 294 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations 298 Three-terminal VSC-HVDC Dynamic Model 299 Validation of the Three-Terminal VSC-HVDC Dynamic Model 307 Multi-Terminal VSC-HVDC Dynamic Model 310 Numerical Example with a Six-Terminal VSC-HVDC Link Forming a DC Ring 314 Disconnection of a DC Transmission Line 314 Three-Phase Fault Applied to AC3 314 Conclusion 317 References 318

7

Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment 321

7.1 7.2 7.3 7.4 7.5

Introduction 321 The STATCOM Case 322 STATCOM Based on Multilevel VSC 336 Example of HVDC based on Multilevel FC Converter 347 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters 358 Conclusions 375 References 375

7.6

Index 377

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Preface Electrical power transmission using high voltage direct current (HVDC) is a wellestablished practice. There is common agreement that the world’s ﬁrst commercial HVDC link was the Gotland link, built in 1954, designed to carry undersea power from the east coast of Sweden to the Island of Gotland, some 90 km away. The original design was rated at 20 MW, 100 kV and used mercury-arc valve converters. Its power and voltage ratings were increased in 1970 to 30 MW and 150 kV, respectively. Solid-state electronic valves were used for the ﬁrst time in the upgrade, with the new type of valve, termed silicon-controlled rectiﬁer (SCR) or thyristor, being connected in series with the mercury-arc valves. This kind of HVDC link, and ancillary technology, has been magniﬁcently described in earlier treatises by Adamson and Hingorani, Kimbark, Uhlmann and Arrillaga. By the turn of the second millennium, there had been 56 HVDC links of various topologies and capacities built around the world: 22 in North and South America, 14 in Europe, 2 in Africa and 18 in Australasia. They ranged from the small, 25 MW, Corsica tapping of the Sardinia–Italy HVDC link to the large, 6300 MW HVDC link, a part of the awesome Itaipu hydro-electric development on the Brazil–Paraguay border. At the time, three other large capacity HVDC links were at the planning/construction stage in China, to transport hydro-electric power from the Three Gorges to the east and southeast of the country, each spanning distances of around 1000 km, rated at 3000 MW – the Three Gorges is a gigantic hydro resource in central China, with an estimated power capacity of 22 MW. Heretofore all the HVDC links in the world had employed either mercury-arc rectiﬁer or thyristor bridges and phase control to enable the rectiﬁcation/inversion process. These converters are said to be line commutated and when applied to HVDC transmission are termed LCC-HVDC converters. The LCC-HVDC technology has continued its upward trend and ﬁve other high-power, high-voltage, long-distance DC links have been built in China since 2010. The most recent LCC-HVDC in operation was commissioned in 2012; it is the Jinping-Sunan link in East China, rated at 7200 MW and ±800 kV, spanning a distance of 2100 km. In LCC-HVDC systems the current is unidirectional, ﬂowing from the rectiﬁer to the inverter stations. Such a fundamental physical constraint in thyristor-based converters limits applicability to the following HVDC system topologies: point-to-point, back-to-back and radial, multi-terminal links. In this context, the conventional, or classical, HVDC transmission technology is not a meshed grid maker; rather, its role has been to interconnect AC systems where an AC interconnection is deemed too expensive or technically infeasible.

xiv

Preface

However, one has to bear in mind that nowadays, in many situations, robust AC interconnections may be achieved more economically using one or more of the options aﬀorded by the Flexible Alternating Current Transmission Systems (FACTS) technology, an array of power electronics-based equipment and control methods which became commercially available in around 1990. It is widely acknowledged that N.G. Hingorani and L. Gyugyi stand out prominently as the intellectual driving force behind the development of the FACTS technology. The main aim of the FACTS technology is to enable almost instantaneous control of the nodal voltages and power ﬂows in the vicinity of where the FACTS equipment has been installed. We should not forget that power ﬂows over an AC line can be manipulated very eﬀectively by controlling the line impedance, or the phase angles, or the voltages, or a combination of these parameters up to the thermal rating of the equipment. A key element of the FACTS technology is the so-called static compensator (STATCOM), which, in the parlance of a power electronics engineer, is a voltage source converter (VSC) and serves the purpose of injecting/absorbing reactive power to enable tight voltage magnitude regulation at its point of connection with the AC power grid. The advent of the STATCOM in the mid-1990s was made possible by the development of power semiconductor valves with forced turn-oﬀ capabilities, like the gate turn-oﬀ (GTO) ﬁrst and the insulated gate bi-polar transistor (IGBT) soon afterwards. GTOs are like thyristors, which can be turned on by a positive gate pulse when the anode– cathode voltage is positive, and, unlike thyristors, can be turned oﬀ by a negative gate pulse. This turn-oﬀ feature led to new circuit concepts and methods such as selfcommutated, pulse-width-modulated, soft-switching, voltage-driven and multi-level converters. These circuits may be made to operate at higher internal switching frequencies than the fundamental level, at several hundreds of hertz, which, in turn, reduces low-order harmonics and allows operation at unity and leading power factors. This contrasts sharply with what can be achieved with the normal thyristors. Advances in the design of the power GTO and its applications in Japan and the USA continued apace by virtue of strategic collaborative R&D projects funded by utilities, manufacturers and governments. In Japan there was a target to develop 300 MW GTO converters for back-to-back HVDC interconnections, while in the USA a 100 MVAR GTO-STATCOM was commissioned in 1996 for the Tennessee Valley Authority. Meanwhile, similar eﬀorts were conducted in Europe in the design of the power IGBT. It is reported that on 10 March 1997, power was ﬁrst transmitted between Hellsjön and Grängesberg in central Sweden using an HVDC link employing IGBT converters driven by pulse-width-modulation (PWM) control. The link is 10 km long, rated at 3 MW, 10 kV and is used to test new components for HVDC. In spite of the great many technical advantages and operational ﬂexibility of the VSC compared with the thyristor bridge, the GTO-based converters did not make inroads into HVDC applications because of the much higher power losses and cost of GTOs compared with thyristors. A further reason is that the ratings of GTOs are low compared with those of thyristors. All this conspired to make VSC-HVDC installations expensive. The impasse was broken with the use of IGBT valves, which exhibit lower switching losses than GTO valves, and decreasing manufacturing costs. Three years after the commissioning of the Hellsjön-Grängesberg, four other VSC-HVDC links had been commissioned in very distant parts of the world: a 50 MVA DC link in the emblematic Island of Gotland to evacuate wind power, an 8 MVA DC link in West Denmark to

Preface

link an oﬀshore wind farm, the 180 MVA Directlink or Terranora project in Australia for power export from New South Wales into Southern Queensland, and a 36 MVA DC link for system interconnection on the Mexican–Texan border. The undersea Estlink 1, linking the Estonian and Finnish power grids, was commissioned in 2006, rated at 350 MW and using VSC stations. Intriguingly, the Estlink 2, rated at 650 MW and commissioned in 2014, uses the classical thyristor-converter technology. It should be noted that all the VSC stations used in HVDC projects until 2010 had been of the so-called two- and three-level power converters. In around 2008, a new breed of VSCs was introduced into the market, the modular multilevel converters (MMCs), which switch at low frequencies, yield minimum harmonic production and have power losses just above those of the classical thyristor-based HVDC converters. Equally important is the fact that it has been possible to increase the capacity of VSC-HVDC links using MMC, by a very considerable margin, say 1000 MW per circuit, such as in the INELFE DC link between Baixas, France, and Santa Llogaia, Spain. Two identical circuits make up for a transmission capacity of 2000 MW. The link was commissioned at the end of 2013. Note that this application comes into the realm of bulk power transmission and is already eating into the niche area of classical thyristor-based HVDC technology, namely asynchronous bulk power transmission, an area until recently thought to be unassailable. The Trans Bay Cable link was the ﬁrst MMC VSC-HVDC, commissioned in 2010, transmitting up to 400 MW of power from Pittsburg in the East Bay to Potrero Hill in the centre of San Francisco, California. Furthermore, there are new application areas in which VSC-HVDC transmission does not seem to have a competitor in sight – the connection of wind sites lying more than 70 km away from the shore is one of the most obvious applications, but there are a few others. For instance, the connection of microgrids with insuﬃcient local generation and little or no inertia (inertia-less power grids), the electricity supply of oil and gas rigs in deep waters, the infeed of densely populated urban centres with power grids already experiencing high short-circuit ratios. Moreover, the unassailable characteristic of the HVDC transmission using the VSC technology is that it is a natural enabler of meshed DC power grids, with such a high level of operational ﬂexibility, reliability and eﬃciency that one day may surpass that of the meshed AC power grids. To get to this point, though, further technological breakthroughs are still awaited in the ancillary areas of DC circuit breaker technology and high-temperature superconductor cables and circuit breakers, as well as more aﬀordable VSCs.

xv

xvii

About the Book The purpose of this book is to facilitate the study of technology that has emerged over the past 15 years in the area of ﬂexible alternating current transmission systems (FACTS) and its technological convergence with the long-standing application of high voltage direct current (HVDC) but now using voltage source converters (VSC). This includes the back-to-back, point-to-point and multi-terminal VSC-HVDC applications. The subject is addressed from a modern perspective, including the latest development in the power systems industry that will extend the applicability of the VSC-FACTS-HVDC technology. Contrary to FACTS Modelling and Simulation of Power Networks, published by the leading author and colleagues in 2004, which was limited to material on FACTS power ﬂows and optimal power ﬂows (OPFs), this book will address new FACTS power system application areas which have received much attention from industry over the past 12 years. These areas include FACTS state estimation, FACTS-constrained OPF, studies of FACTS dynamic performance and control, and the all-important topic of electromagnetic transients. These applications areas coincide with research areas, which the authors have developed over the past 15 years and have published widely in the top journals and presented in their research work at international forums. The book is aimed at a very wide sector of the power engineering community, encompassing utility and equipment manufacturing engineers, researchers, university professors and PhD and MSc students, and undergraduate students in their ﬁnal year. The reader is expected to have a sound knowledge of electrical and electronic circuits, algebra and numerical methods, and a working knowledge of electrical power and control engineering – an undergraduate student embarking on their ﬁnal year in an electrical and electronics degree course should be well qualiﬁed to read the book. It goes without saying that utility engineers and managers with a background in electrical power would also take to the book like a ‘duck to water’. Students conducting research in any of the topics covered in the book will ﬁnd useful the modelling approach adopted by the authors which has resulted in ﬂexible and comprehensive FACTS and HVDC-VSC models with which to carry out a wide range of power network-wide simulation studies, ranging from steady-state to dynamic and transient studies. Chapter 1 gives an overview of the role that the VSC plays in the area of power systems VAR compensation. To qualify its prowess in this arena, a qualitative comparison is carried out against the long-enduring static var compensator. The VSC is a rather ﬂexible piece of equipment which may be connected either in shunt or in series with the AC system, according to requirement. Two or more of them may be made to combine to give

xviii

About the Book

rise to compound equipment or systems, such as the UPFC and the various ﬂavours of VSC-HVDC systems. Moreover, it is shown that the VSC combines well with the DC-DC converters and plays a pivotal role in enabling the grid connection of the renewable sources of electricity and energy storage systems. Equally important is the fact that the VSC technology is a builder of multi-terminal HVDC systems, with a DC network which may be a single node, a radial system or a meshed system. This is in stark contrast to the classical CSC-HVDC technology, whose ﬂexibility is very limited as far as multi-terminal schemes is concerned. This chapter illustrates that a strategic feature of the VSC technology is to enable the conversion of AC transmission systems into DC transmission systems with an unassailable power transfer capacity and having, at least, an equal level of operational ﬂexibility. Future developments in distribution systems seem to lie squarely in the incorporation of VSCs to enable greater operational ﬂexibility, greater power throughputs and the incorporation of renewable electricity sources and storage. Chapter 2 presents the theory of power electronics, which is essential to a good understanding of modern power converters topologies. The most popular semiconductor valves are presented ﬁrst, followed by the classical two-level, single-phase and three-phase power converters. This forms the preamble of the study of multi-level converters, which is the technology used today by industrial vendors. The chapter presents a comparison between the HVDC systems based on voltage source converters and those that employ current source converters. A comprehensive list of current VSC-HVDC installations around the world is given at the end of the chapter. Chapter 3 addresses the theory of power ﬂows. The chapter may be seen as consisting of two parts: (i) the conventional power ﬂow theory, including a Newton-Raphson power ﬂow method in rectangular coordinates used to solve the set of nodal power equations describing the power grid during steady-state operation; and (ii) the power ﬂow equations of the VSC model, which are derived from ﬁrst principles and then extended to establish the power ﬂow models of the STATCOM, VSC-HVDC links and generalized multi-terminal VSC-HVDC systems. The algebraic, non-linear equations describing the steady-state performance of the VSC, the STATCOM and VSC-HVDC systems are solved using the Newton-Raphson method in rectangular coordinates. The ensuing solutions fulﬁl the quadratic convergence characteristic which is the hallmark of the Newton-Raphson method. The VSC is the basic building block with which all the VSC-FACTS and VSC-HVDC equipment is assembled; hence, a Newton-Raphson power ﬂow computer program written in Matlab, with the model of the VSC included, is available at www.wiley.com/go/acha_vsc_facts for the user to gain hands-on experience. Chapter 4 introduces the topic of optimal power ﬂow (OPF) used by transmission system operators for optimal economic and security assessments of their power grids. The chapter is divided into three main sections: (i) a general overview of the OPF problem and its applications in power systems operational planning; (ii) the introduction of the OPF problem as a non-linear optimization problem and possible solution methods; and (iii) an extension to the OPF formulation to incorporate hybrid AC-DC networks using VSC-HVDC systems. The OPF methodology introduced in this chapter uses the VSC model developed in Chapter 3 to formulate versatile models of hybrid AC-DC networks suitable for minimum-cost assessments of power systems subject to realistic operational constraints in both the AC and the DC grids. The overall solution algorithm adopted for

About the Book

solving the non-linear system of equations is the de facto industry’s standard Newton’s method. Concerning the introduction of models of VSC-based equipment, the chapter follows a similar line of development as in Chapter 3, starting with the VSC and progressing to develop models of the various kinds of VSC-HVDC systems. However, in this chapter the complex nodal voltages are represented in polar form as opposed to rectangular form because the former is widely employed in the OPF literature. The models are formulated and solved using a general-purpose mathematical solver package called AIMMS (Advanced Interactive Multidimensional Modelling System), employing its nonlinear augmented Lagrangian solver. However, the reader may implement these models using any equivalent general-purpose simulation platform. Alternatively, the more advanced readers may wish to write their own OPF computer program using MATLAB scripting. In any case, a free academic licence of AIMMS can be obtained for academic research purposes. Chapter 5 presents the theory of power systems state estimation. The chapter is divided into two main parts, addressing the following issues: (i) the classical power systems state estimation theory using the weighted least squares method as the solution algorithm; and (ii) the state estimation models of the VSC, the STATCOM, UPFC, VSC-HVDC links and generalized multi-terminal VSC-HVDC systems. The timely topic of PMUs in power systems state estimation is addressed in this chapter. Each major topic in this chapter is accompanied by a set of well-designed numerical exercises using a MATLAB environment (WLS-SE) for the user to gain hands-on experience. Chapter 6 is dedicated to the study of power systems dynamics in time domain. It uses a similar outline to the previous four chapters. In the ﬁrst part, it introduces the theory of conventional power systems dynamics, where the synchronous generators and their controls are the only equipment that exhibit a dynamic behaviour following a disturbance in the power grid. The transmission lines, transformers and loads are taken to exhibit a static behaviour, although provisions are made for the models of these equipment to have a voltage and frequency dependency. In this chapter the interest is the study of power system dynamic phenomena which exhibit a relatively low variation in time. Hence, the dynamics of the power grid is described well by a set of algebraic-diﬀerential equations which are discretized and linearized in order to carry out the solution by iteration, which is valid for a single point in time. The solution algorithm used is an implicit simultaneous method employing the Newton-Raphson method. The resulting mathematical model is coded in software and applied to assess the dynamic behaviour of a test system, in order to illustrate the usefulness of the overall dynamic model. In the second part of the chapter, the dynamic model of the VSC-STATCOM, receives a similar treatment to the synchronous generator but having a detailed representation of the dynamics of its DC bus. This dynamic model is then suitably extended to encompass the dynamic models of the back-to-back, point-to-point and multi-terminal VSC-HVDC system. The VSC-HVDC model is applied to study the timely issues of frequency support in power grids with near-zero inertia and supplied by a VSC-HVDC link. Chapter 7 is devoted to the simulation of the transient responses of various FACTS and HVDC systems using PSCAD/EMTDC, a commercial software package for electromagnetic transient analysis, which is widely used in industry and academia. Four diﬀerent systems are simulated: (i) a STATCOM based on a conventional two-level voltage source converter; (ii) an extension of the STATCOM using a three-level ﬂying capacitor converter as an example of a multilevel converter; (iii) a two-terminal

xix

xx

About the Book

HVDC system based on a multilevel voltage source converter topology; and (iv) a multi-terminal HVDC system which also employs multilevel VSCs. Furthermore, the control schemes of the diﬀerent power systems are comprehensively explained and control design speciﬁcations are provided.

xxi

Acknowledgements Bringing this book project to a close has been an endeavour made possible only with the support of colleagues and institutions from across the world, having started in the research laboratories of the University of Glasgow, Scotland, in 2008 and completing today, when the authors work in the following universities: Tampere University, Finland, Universidad de Castilla-La Mancha, Spain, Universidad de Sevilla, Spain, Universidad Nacional Autónoma de México, Mexico, and Durham University, England. Our appreciation goes foremost to the University of Glasgow and our respective home universities for the time that allowed us to bring this project to fruition. We would like to thank Dr Rodrigo Garcia Valle from Ørsted, Denmark, and Dr Luigi Vanfreti from Rensselaer Polytechnic Institute, USA, for their early contribution to the book project. We would like to thank our respective families for the time that we were lovingly spared throughout the project. Enrique Acha would like to thank Antonio Gómez Expósito, Jose Maria Maza-Ortega and Sigridt Garcia for having written the following award-winning paper: J.M. MazaOrtega, E. Acha, S. Garcia, A. Gomez-Exposito, ‘Overview of power electronics technology and applications in power generation, transmission and distribution’, J. Mod. Power Syst. Clean Energy – Springer (2017) 5(4):499–514, which provided the inspiration for Chapter 1. Luis Miguel Castro and Enrique Acha would like to acknowledge the ﬁnancial assistance of Consejo Nacional de Ciencia y Technología (CONACYT), México, and Professor Pertti Järventausta from the Tampere University, Finland, through the SGEM project, to conduct fundamental research on the modelling and simulation of multi-terminal Voltage Source Converter High-Voltage Direct Current (VSC-HVDC) systems. This research forms the basis of Chapters 3 and 6. Behzad Kazemtabrizi would like to thank Ahmad Asrul Bin-Ibrahim of Durham University, England, for his help in producing and verifying the results for the AC/DC optimal power ﬂow (OPF) test case used in Chapter 4. Antonio de la Villa would like to thank the Spanish Ministry of Economy and Competitiveness (MINECO) under grants ENE 2010-18867, which provided the facilities for the work of Chapter 5. Thanks are also expressed to the following faculty staﬀ of Universidad de Sevilla: Antonio Gómez Expósito, Esther Romero Ramos and Pedro Cruz Romero, for their useful suggestions during the preparation of this chapter. Pedro Roncero would like to thank MINECO, whose ﬁnancial support, at various stages in the preparation of the book, proved instrumental in seeing its completion.

xxii

Acknowledgements

The large number of simulations in the book were enabled by the use of a wide range of open source and commercial software (educational versions): Matlab, Simulink, MATPOWER, the Advanced Interactive Multidimensional Modelling System (AIMMS) and Power System Computer Aided Design/Electromagnetic Transient Direct Current (PSCAD/EMTDC). We would like to extend our most ample gratitude to all the owners and developers of such powerful simulation platforms. We are grateful to the staﬀ of John Wiley & Sons for their utmost patience and continuous encouragement throughout the preparation of the manuscript.

xxiii

About the Companion Website This book is accompanied by a companion website:

www.wiley.com/go/acha_vsc_facts The website includes software ﬁles associated to Chapters 3, 5 and 7: • Matlab ﬁles corresponding to Chapters 3 and 5. • Two PSCAD ﬁles of Cases 1 and 4 corresponding to Chapter 7. Scan this QR code to visit the companion website.

1

1 Flexible Electrical Energy Systems 1.1 Introduction Following a sustained programme of expansion of high-voltage power grids in the 1960s and their widespread interconnection in the 1970s, by the end of that decade the expansion programmes of many utilities had become thwarted by a variety of well-founded, environmental, land-use and regulatory pressures, preventing the licensing and building of new transmission lines and electricity generating plants. This was in the face of sustained global demand for electricity. An in-depth analysis of the options available for increasing power throughputs with high levels of reliability and stability pointed towards the use of modern power electronics equipment, control techniques and methods [1]. Such a far-reaching work was carried out ﬁrst at the Electric Power Research Institute (EPRI) in Palo Alto, CA, under the leadership of N.G. Hingorani. The result was an integrated philosophy for AC network reinforcement using electronics principles, endowing AC transmissions lines with a degree of operational ﬂexibility and power-carrying capacity that had not been possible before. Flexible alternating current transmission systems (FACTS) was the name given to the family of power electronic-based equipment, control techniques and methods emanating from this initiative [2]. In the same time span, electricity distribution companies were experiencing a marked increase in the deployment of end-user equipment which was highly sensitive to poor-quality electricity supply. Several large industrial users reported experiencing signiﬁcant ﬁnancial losses as a result of even minor lapses in the quality of electricity supply. A great many eﬀorts were made to remedy the situation, with solutions based on the use of the latest power electronic technology of the time [3]. A range of custom-made equipment and solution techniques was put to the fore, with the key ideas emanating from EPRI. This initiative, aimed at ameliorating adverse power quality phenomena at the interface between the low-voltage distribution power grid and the industrial user, was given the name ‘custom power’ by its creator, N.G. Hingorani. Indeed, custom power technology was announced as the low-voltage counterpart of the FACTS technology, aimed at high-voltage power transmission applications and emerging as a credible solution to many of the problems relating to continuity of supply at the end-user level [2]. It is fair to say that many of the ideas upon which the foundation of FACTS rests evolved over a period of several decades, building on the experience gained in the areas VSC-FACTS-HVDC: Analysis, Modelling and Simulation in Power Grids, First Edition. Enrique Acha, Pedro Roncero-Sánchez, Antonio de la Villa Jaén, Luis M. Castro and Behzad Kazemtabrizi. © 2019 John Wiley & Sons Ltd. Published 2019 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/acha_vsc_facts

2

1 Flexible Electrical Energy Systems

of high-voltage direct current (HVDC) transmission and reactive power compensation equipment, methods and operational experiences [4, 5]. Nevertheless, there is widespread agreement that FACTS, as an integrated philosophy, was a novel concept brought to fruition at EPRI in the 1980s. Since those early days of the FACTS technology, a great many breakthroughs have taken place in the area of power electronics, encompassing new valves, control methods and converter topologies [6]. To a greater or lesser extent, the recent technological developments have all been incorporated into the ﬁelds of FACTS and HVDC, giving rise to a new generation of power transmission equipment in either AC or DC, with unrivalled operational ﬂexibility [7]. The original boundaries between HVDC and FACTS were drawn along the type of solid-state converters employed and their control [1], but these boundaries became blurred with the arrival of newer technology. For instance, the static compensator (STATCOM), which is essentially a voltage source converter (VSC), is a product of the FACTS technology used to provide reactive power support [8]. Two such devices connected in series on their DC sides results in the modern expression of an HVDC transmission system. This has been designated VSC-HVDC to distinguish it from the classical HVDC transmission using thyristor-based bridges and phase control [9]. The largest vendors of power electronics equipment, ABB, Siemens and Alstom, have proprietary equipment termed HVDC Light, HVDC Plus and MaxSine HVDC, respectively. It is documented that the use of a VSC in a utility-level application was in the form of a STATCOM, which falls squarely within the realm of the FACTS technology. The VSC application in HVDC transmission came next, which, it may be argued, is an application comprising two STATCOMs connected back-to-back or through a DC cable. Of course, such an argument is more diﬃcult to sustain when we progress into the realm of multi-terminal VSC-HVDC [10]. From a traditional perspective, artiﬁcial lines have been drawn between the FACTS and the HVDC technologies. It is argued here that these lines be removed and that, instead, the focus should be on ﬂexible transmission systems (FTS), a unifying concept bridging the FACTS and HVDC technologies – the aim being to enable the best-of-breed solutions underpinning the new power-carrying structures that the smart grids demand [11]. The breakthroughs in power electronics impacted not only the transmission and distribution sectors of the electrical energy industry but also the generation sector, particularly the renewable generation and energy storage technologies [12]. The use of advanced power electronic converters enabled the wind power equipment manufacturers to transit from the ﬁrst generation of ﬁxed-speed wind turbines to the second generation of variable-speed wind turbines, which are larger, more eﬃcient and fully compliant with modern grid codes [13]. The use of advanced power electronic converters also led to the proliferation, on a global scale, of grid-connected photo-voltaic generators, with full compliance to modern grid codes [14]. More recently, with the widespread availability of aﬀordable lithium-ion batteries suitable for power applications, battery energy storage systems (BESS) are becoming oﬀ-the-shelf products [15]. It is very likely that, once BESS prices decrease further, this equipment will become ubiquitous in the power grid since it has a potentially major role to play in electrical energy retailing. In a more ample technological sense than FTS or ﬂexible power generation, a wide range of enabling technologies has become cost-eﬀective, such as extruded cables, smart

1.1 Introduction

meters, phasor measurement unit (PMU), advanced protection systems, accessible satellite communications and distributed energy resources such as EV charging stations [16]. Equally important is the fact that information and communication technologies (ICTs), the internet, the web and distributed computing, have become even more powerful and popular than, say, only one decade ago. The widespread availability of all these technological developments has been seized on by the proponents of the smart grid philosophy, who argue that the coming together, in an all-encompassing manner, of these technologies should provide a solid foundation on which to build new, smarter energy grids, now that a large portion of the existing infrastructure in many countries is ageing and up for renewal [17, 18]. Key drivers of the smart grid technology are enhanced security of supply and self-healing properties, its market-oriented philosophy, progressive demand-side response (DSR) and demand-side management (DSM) policies [19]. The available technologies and drivers of the smart grid concept are voltage-level independent and network structure independent. Hence, it is argued here that it should be feasible to talk about smart grids at either the low-voltage distribution system level or the high-voltage transmission system level. Admittedly, each has its own peculiarities but with a great many common objectives and interrelated technology issues yet to be resolved. The power network is expected to incorporate increasing amounts of wind and solar power, leading to new challenges in its operation owing to the intermittent nature of these two new forms of renewable power generation. Frequency oscillations, resulting from temporary power imbalances, may become more common. Also, voltage control may become a signiﬁcant problem if no suitable FTS equipment is in place, such as static var compensators (SVCs) or VSCs in the form of STATCOMs, BESSs or as part of VSC-HVDC systems [7]. Moreover, the power-carrying structures of AC low-voltage smart grids are likely to use mainly underground cables – which is already happening in Denmark – and inductive reactive power compensation equipment may be required [20]. Alternatively, smart grids using AC and DC microgrids may reach the commercial stage, following on from the experience gained with current prototypes deployed by some of the distribution companies in Finland [21]. Indeed, structures based on multi-terminal VSC-HVDC systems are just the kind of transmission structures that are likely to be used by the next generation of smart grids, both at the transmission level and at the distribution level [7]. This may be a system within a larger AC system, as exempliﬁed in the Figure 1.1. In this one-line diagram, an AC power system incorporating a fair amount of FACTS controllers and a multi-terminal VSC-HVDC system is also equipped with synchronized measurement systems at key points of the power grid using PMU and satellite communications. The deployment of such equipment would go a long way towards establishing a high-voltage smart grid, particularly if the FTS equipment is ﬁtted with processor agents, sensors and a ﬁbre optic network – or some other means of fast communications – linking all the FTS equipment, so that a coordinated action takes place at the system level as opposed to the local, individual level [22]. Such arguments would also apply to low-voltage, low-power microgrids where the core system could be a multi-terminal VSC-HVDC system interconnecting an arbitrary number of AC systems – this point is elaborated further in Section 1.3.

3

15 kV

PMU

1000 km

Off-shore wind farm

400 kV

Island with “dead” load

PMU

CSP plant

Multiterminal CSCHVDC Link

100 km DFIG SVC

132 kV

CSP plant

STATCOM

Mainland Multi-terminal VSC-HVDC grid

PMU 400 km

TCSC

Figure 1.1 Flexible transmission system with renewable energy sources.

50 km

Off-shore wind farm

1.2 Classiﬁcation of Flexible Transmission System Equipment

1.2 Classiﬁcation of Flexible Transmission System Equipment Many of the ideas upon which the foundation of FACTS rests evolved over a period of many decades. Chieﬂy among them is: – experience gained with HVDC transmission technology [4] – experience gained with SVC technology [5] – experience gained with electric drives for motor control [23]. In a nutshell, FACTS and HVDC equipment uses power semiconductor devices and advanced power electronics control techniques and methods to fulﬁl its task in a matter of a few milliseconds [24]. The wide range of modern equipment available is bundled together in this book under the umbrella title of FTS equipment. The range of functions that this technology can fulﬁl is very wide but it is equipment-dependent. Table 1.1 lists the equipment comprising the FTS technology and the respective areas of power systems application. Various classiﬁcations of the FTS equipment are possible. It can be classiﬁed in terms of the power systems application, as outlined in Table 1.1: (i) voltage control, (ii) reactive power control, (iii) active power control, (iv) frequency control, and (v) AC systems interconnection. This list is not exhaustive by any means and other applications exist, such as loop ﬂow control. Alternative classiﬁcations may be drawn according to the number of power converters used or the way in which the equipment connects to the AC power grid or grids, Table 1.1 FTS equipment and the respective areas of power systems applications.

Equipment

Voltage control

Reactive power control

SVC

✓

✓

STATCOM

✓

✓

BESS

✓

✓

SSTC

✓

Active power control

Frequency control

AC systems interconnection

✓

TSSC

✓

TCSC

✓ ✓

SSPS ✓

SSSC

✓

✓

✓

✓

✓

VFT

✓

✓

✓

CSC-HVDC

✓

✓

✓

✓

✓

✓

UPFC

✓

✓

IPFC

VSC-HVDC

✓

✓

SVC: static var compensator; STATCOM: static compensator; BESS: battery energy storage system; SSTC: solid-state tap changer; TSSC: thyristor switched series capacitor; TCSC: thyristor-controlled series compensator; SSPS: solid-state phase shifter; SSSC: solid-state series compensator; UPFC: uniﬁed power ﬂow controller; IPFC: interphase power ﬂow controller; VFT: variable frequency transformer; CSC-HVDC: current source converter high-voltage direct current; VSC-HVDC: voltage source converter high-voltage direct current.

5

6

1 Flexible Electrical Energy Systems

Table 1.2 Classiﬁcation of FTS equipment according to converter type. Converter type

Equipment

Thyristor and phase control

SVC, SSTC, TSSC, TCSC, SSPS, CSC-HVDC

IGBT or GTO and PWM control

STATCOM, BESS, SSSC, UPFC, IPFC, VSC-HVDC, VFT

namely, shunt, series, cascade and multi-terminal. The equipment can also be classiﬁed according to the type of semiconductor valves and switching control that the converters use: (i) thyristor valves and phase control, and (ii) insulated gate bipolar transistor (IGBT) or gate turn-oﬀ (GTO) valves and pulse width modulation (PWM) control [2]. This classiﬁcation is shown in Table 1.2. For any practical purpose, an IGBT valve outperforms a GTO valve in terms of its speed of response, better power loss performance and improved reliability; they have become the standard forced commutated valves used in converters aimed at power systems applications. They are driven by PWM control of various kinds [24]. In this book, the application of IGBT-based converters driven by sinusoidal-PWM control is given priority. In particular, the ubiquitous STATCOM and the VSC-HVDC. Other equipment which uses the STATCOM as the basic building block also receives attention in this chapter, such as the BESS, the solid-state series compensator (SSSC), the uniﬁed power ﬂow controller (UPFC) and the interphase power ﬂow controller (IPFC). As a means of emphasizing the much-increased functionality that the STATCOM has brought into the arena of electrical power systems, in general, the SVC and the current source converter high-voltage direct current (CSC-HVDC) are also covered in suﬃcient detail. 1.2.1

SVC

The SVC appeared on the power systems scene at least two decades before the FACTS initiative was put forward [5]. In some respects, the SVC may be considered the forebear of some of the equipment developed under the auspices of the FACTS initiative [1]. It comprises a bank of thyristor-controlled reactors (TCRs) in parallel with a bank of thyristor switched capacitors (TSCs). The one-line schematic representation of the SVC is shown in Figure 1.2. The SVC is connected in shunt with the AC system through a step-up transformer. Its main function is to supply/absorb reactive power to support a speciﬁed voltage magnitude at the high-voltage side of its connecting transformer. At the construction level, a criticism levelled at the SVC is its rather large footprint [8]. The inductor, in Figure 1.2a, is a bulky air-core reactor, to prevent saturation; the capacitor banks are also quite sizable and so are the tuned ﬁlters. As suggested in Figure 1.2b, the SVC performs like a variable susceptance. The TCR consumes variable reactive power up to its design limit, governed by ﬁring angle control, 𝛼, in the range: 𝜋/2 ≤ 𝛼 ≤ 𝜋. It achieves its fundamental frequency operating point at the expense of generating harmonic currents, which is an undesirable side eﬀect. Hence, the three-phase TCR is connected in delta to prevent the triple harmonics from reaching the power system. In addition, passive ﬁltering is required

1.2 Classiﬁcation of Flexible Transmission System Equipment HV compensator bus t

PT

Ik MV compensator bus

ISVC Controller

jBSVC

Vref TSC

TSC

Vk

jXt VSVC ISVC = jBSVC (α)VSVC πXL (α) – XC [2(π – α) + sin 2α] BSVC (α) = πXLXC α1 = α2 = α

Permanently connected filters

TCR

(a)

(b)

Figure 1.2 (a) An SVC (©MPCE, 2017) and (b) its equivalent circuit representation.

HV bus t

Ik Vk

PT MV bus

jXt VVSC

Smoothing inductor

jXVSC VAC =

Controller

3 · mae jφEDC 8

ma EDC

Vref Figure 1.3 (a) A STATCOM (©MPCE, 2017) and (b) its equivalent circuit representation.

to mitigate the 6 k ± 1 and 12 k ± 1 harmonics generated by the 6-pulse and 12-pulse converter topologies, respectively, with k = 1,2,3… being the harmonic order. The TSC generates reactive power in a variable, discrete manner, with the thyristor pairs operating as switches (on/oﬀ ); hence, during steady-state operation, no harmonic distortion is produced by the TSC [5]. 1.2.2

STATCOM

The STATCOM is the modern counterpart of the SVC [8]. Coincidentally, its main operational task is to inject regulated volt-ampere reactives (VAR) to provide voltage support at the high-voltage side of its connecting transformer, but much more eﬀectively. As illustrated in Figure 1.3, it connects in shunt with the AC power system. Its main elements are the VSC, the smoothing inductor, the interfacing transformer and the PWM control system.

7

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1 Flexible Electrical Energy Systems

Table 1.3 STATCOM operating modes. Voltages

Operating mode

Functionality

V k > V AC

Consuming vars

Standard function

V k < V AC

Injecting vars

Standard function

−𝛿

Consuming watts

Normal operation

+𝛿

Injecting watts

Possible only with DC storage

Contrary to the SVC, the STATCOM does not use bulky inductors and banks of capacitors to absorb and to generate reactive power, respectively. This can be appreciated from Figure 1.3a. The reactive power production process is carried out entirely by the electronic processing of the voltage and current waveforms within the valves, to enable either leading or lagging VAR production to satisfy operational requirements [2]. The smoothing inductor laying between the VSC and the step-up transformer is used to eliminate the high-order harmonics produced by the action of the PWM control. The DC capacitor is an extruded capacitor of a small rating, employed only to support and stabilize the DC voltage to enable the converter operation – it does not play a signiﬁcant part in the VAR generation process. The DC current in the capacitor in Figure 1.3a is taken to be zero during steady-state operation, as an indication that the capacitor is fully charged. During dynamic and transient operation, I DC will diﬀer from zero. The STATCOM’s operational behaviour is superior to that of the SVC because its operation mimics a variable voltage source as opposed to a variable susceptance [8]. This is illustrated in Figure 1.3b. In fact, the STATCOM’s operational performance is closer to that of a rotating synchronous condenser but with a faster speed of response because it has no moving parts. The basic operating principles of the STATCOM may be explained with reference to the complex voltages V AC = VAC ∠𝛿 and V k = Vk ∠0 in Figure 1.3b, where the amplitude and the phase angle of the voltage drop across the reactances X SVC and X t can be controlled, to deﬁne the amount and direction of both active and reactive power ﬂows. For leading and lagging VARs, the STATCOM’s active and reactive power ﬂows are deﬁned by the following fundamental expressions: Vk VAC • sin 𝛿 XVSC + Xt Vk2 Vk VAC • cos 𝛿 − Q= XVSC + Xt XVSC + Xt P=

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(1.1)

To a large extent, the equation set (1.1) deﬁnes the STATCOM operating modes, which are summarized in Table 1.3. The static voltage–current characteristics shown in Figure 1.4 correspond to the SVC and the STATCOM. The thick line is for the SVC (capacitor and thyristor-controlled inductor) at rated values; only one value of capacitor has been considered and the broken lines on the inductor side would be for various ﬁring angle values of the TCR. As seen

1.2 Classiﬁcation of Flexible Transmission System Equipment

VT (p.u.) 1.0 0.9 0.8 Transient Inductive Rating for SVC and STATCOM

0.7

Transient capacitive rating for STATCOM

0.6 0.5 0.4 0.3 0.2 0.1

IC (p.u.)

–1

0

1

IL (p.u.)

Figure 1.4 Static V-I characteristics of the VSC and the STATCOM.

from the ﬁgure, the characteristics of a STATCOM and a SVC of comparable ratings will coincide only at their rated values. It can be seen from this characteristic that the SVC yields little capacitive current at low voltages. In contrast, the STATCOM is able to produce its full range of current, both inductive and capacitive, even when the voltage has dropped to about 10% of its nominal value. From the operational vantage, one of the main criticisms levelled at the SVC is that its ability to contribute reactive power becomes severely impaired in the presence of low system voltages in its vicinity, for instance in cases of voltage collapse. Conversely, the STATCOM’s reactive power provision is system voltage-independent. In spite of its superior technical performance, the STATCOM technology still carries a higher price tag than the SVC technology of comparable rating. Equipment manufacturer Alstom, Finland, has recently patented a new piece of VAR compensation equipment selecting the best attributes of the SVC and the STATCOM [25]. 1.2.3

SSSC

This equipment may be seen as a series-connected STATCOM and serves the purpose of injecting a variable, controllable voltage to an incoming voltage to achieve a range of purposes, one of which is active power ﬂow control through the power line and the injection of controlled reactive power at one of the two nodes [2]. An SSSC is normally a piece of equipment of small rating, with full control of the magnitude, phase and polarity of the injected series voltage. One may think of a small STATCOM which is connected to the secondary winding of an interfacing transformer whose primary winding is connected in series with the AC power grid, as illustrated schematically in Figure 1.5. Note that the SSSC should be designed to carry the full line current, with the rated voltage being only a fraction of the rated line voltage.

9

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1 Flexible Electrical Energy Systems

Figure 1.5 SSSC schematic representation (©MPCE, 2017).

P δcI Controller

Pref

maI

1.2.4

Compound VSC Equipment for AC Applications

The VSC, which forms the kernel of the STATCOM and the SSSC, is a rather ﬂexible device; not only does it connect easily in shunt or in series with the AC power system, it also combines rather well in any number and in any combination to suit a speciﬁc power systems application. This is illustrated in Figure 1.6. The schematic diagrams of the four most popular compound FACTS devices – the UPFC, the IPFC, the back-to-back VSC-HVDC and the BESS – are shown in Figure 1.6. Their main salient characteristics are outlined below, according to the time at which they seem to have been conceptualized or emerged on the applications scene. • The UPFC employs two VSCs, one connected in shunt and the other in series with the AC system. The two VSCs are connected back-to-back on their DC sides and have a uniﬁed control structure, giving rise to the UPFC, which has great operational functionality. Its schematic diagram is shown in Figure 1.6c. In this application, the shunt-connected VSC is rated at full capacity whereas the series-connected VSC is rated to carry only a fraction of the line current and have a fraction of the line voltage. Note that this contrasts with the SSSC, which is rated to carry the full line current [26]. • The UPFC combines the operational control capabilities of the STATCOM and the SSSC. It is capable of regulating, simultaneously, voltage magnitude at the high-voltage node of the shunt-connected VSC, active power ﬂow arriving at the receiving node of the series-connected transformer – the opposite node to that where the shunt converter is connected – and the injection of reactive power at that node. Regulation of these parameters is limited by the ratings of the shunt and series converters. The UPFC plays the role, in any combination, of a STATCOM, a TCSC (thyristor-controlled series compensator), an SSSC, a phase-shifting transformer and a tap-changing transformer, simultaneously and in any combination. For all its operational ﬂexibility and great expectations when it was conceptualized, the UPFC has not been a commercial success. Only two prototypes are known to exist in the world, in the USA [2] and Korea [27]. • If two or more transmission lines connecting to the same AC bus are ﬁtted each with a SSSC then these can be made to share the same DC capacitor and to have a coordinated control system, giving rise to the IPFC [28]. Its schematic diagram is shown in Figure 1.6d. It has been designed to regulate active power ﬂows between the various transmission lines by exchanging power through the common DC bus. No IPFC installation is known to exist at present but this is likely to change as more commercially-driven energy transactions take place between neighbouring transmission companies.

1.2 Classiﬁcation of Flexible Transmission System Equipment

δI

maI Pref δ1

fref

Controller

δ2

Controller

Vref

P&f

VSI

Qref

(a) PT (f)

Battery pack

Boost converter

Vref

Shunt

(b)

maI Pref

Shunt

(e) δI Inverter controller

P&Q

Qref

Shunt

PT

PT

maI

δI

Controller

maI Controller

PT

PR & QR m aR δR Rectifier controller

11

Series

(c)

PI & QI

maR δR

maI δI

Unified control system

(d) P2 & Q2

P1 & Q1 ma1 δ1

P3 & Q3 ma3

δ3 ma2

δ2

IPFC controller Pref 1 Qref 1 Pref 2 Qref 2 Pref 3 Qref 3

Figure 1.6 The most popular VSC-based equipment, placed, clockwise, in the order in which they appeared in the open literature: (a) STATCOM; (b) SSSC; (c) UPFC; (d) IPFC; (e) back-to-back VSC-HVDC: (f ) BESS.

• Two VSCs connected in tandem form a back-to-back VSC-HVDC link, as shown in Figure 1.6e. The VSCs are connected in shunt at their respective AC systems and have a common DC bus [9]. There is some resemblance between the structures of the UPFC and the back-to-back VSC-HVDC. It may be argued that the operational functionality of the two controllers is comparable but the back-to-back VSC-HVDC achieves this at the expense of using two fully rated converters. Nevertheless, a key attribute of the VSC-HVDC link that the UPFC lacks is the ability to connect, in an asynchronous manner, two otherwise independent AC systems, which may have the same or different operating frequencies. Furthermore, the two VSCs of the HVDC link do not need to be connected back-to-back but instead may be linked by a cable and used to transport electrical power in DC form with less power loss than an AC transmission line of comparable rating and distance. Among all the VSC-based equipment, the VSC-HVDC link is the technology that has experienced the highest rate of growth

PI & QI

12

1 Flexible Electrical Energy Systems

since its introduction in 2000. Examples of VSC-HVDC installations in the world are given in Section 2.4 of Chapter 2. • The VSC combines quite naturally with modern battery packs, such as lithium-ion batteries. The combined system is termed BESS, having the structure shown in Figure 1.6f. It should be noted that the battery pack connects to the VSC through a DC-DC converter to enable the smooth operation of the battery. Provided there is suﬃcient energy stored in the battery pack, the BESS is capable of injecting active power into the AC system, in a matter of milliseconds, to provide inertial and primary frequency support in the presence of synchronous generators frequency oscillations. Furthermore, the associated VSC acts as a source of VARs to enable eﬀective voltage regulation at the AC bus. Indeed, not only can a modern BESS emulate the operation of a synchronous generator, it can also provide additional ﬂexibility, such as adaptive time-varying behaviour during faults or perturbations. It is very likely that once BESS prices decrease further, this equipment will become ubiquitous in the power grid since it has a potentially major role to play in electrical energy retailing. BESSs commissioned in Chile and California [29, 30] in recent years are two good examples of where MW-size BESS have been installed.

1.2.5

CSC-HVDC Links

HVDC transmission using the classical six-pulse Graetz bridge, which uses thyristor valves and phase control, has been available, on a commercial basis, for more than half a century. Its main application has been in the area of bulk power transmission over long distances. The earliest schemes were point-to-point of the monopolar kind with ground return. The back-to-back and the point-to-point bipolar conﬁgurations were developed soon afterwards [4]. Today’s classical HVDC transmission uses 12-pulse converters as opposed to 6-pulse converters. This type of HVDC technology is termed CSC-HVDC to distinguish it from the new HVDC technology which uses forced commutated valves and PWM control, namely VSC-HVDC [24]. The rectiﬁcation and inversion processes are imperfect and the characteristic harmonics are produced in both the DC circuit and the AC circuits of the HVDC links. Hence, passive ﬁltering is required to mitigate the 6 k ± 1 and 12 k ± 1 AC harmonic currents generated by the 6-pulse and 12-pulse converter topologies, respectively, with k = 1,2,3… being the harmonic order. On the DC side, ﬁlters may be required for the 6 k order. The rectiﬁcation process is achieved with thyristor ﬁring angles in the range 0 ≤ 𝛼 < 𝜋/2 and the inversion process in the range 𝜋/2 < 𝛼 ≤ 𝜋, although some margin would need to be left to avoid commutation failures. Moreover, the rectiﬁcation/inversion process requires the provision of reactive power, which needs to be supplied locally to enable suitable operation of the link, with at least a part of this requirement being met by the passive harmonic ﬁlters in the installation. Contrary to the VSC, the Graetz bridge does not have VAR production capability and is unable to regulate AC voltage. Hence, connection to both AC grids is carried out through tap-changing transformers to enable AC voltage control. Cases of a monopolar, back-to-back and bipolar, point-to-point HVDC schemes are exempliﬁed in Figures 1.7a,b, respectively. For a full set of CSC-HVDC topologies, refer to [4].

1.2 Classiﬁcation of Flexible Transmission System Equipment

IDC

(a) IDC RDC + or – VDC

– or + VDC RDC

(b)

IDC

Figure 1.7 Two CSC-HVDC links: (a) back-to-back, monopolar HVDC; (b) point-to-point, bipolar HVDC.

Notice that the current ﬂow is from the rectiﬁer towards the inverter and so is the power ﬂow when the voltage polarity is positive. Alternatively, the power follows an opposite direction to the current when the voltage polarity reverses, an operational characteristic achieved through ﬁring angle control. The two units of a back-to-back HVDC link have equal rating. They seem to be economical for voltage ratings as low as 50 kV. The bipolar link may be seen to comprise two monopolar links, one at positive and one at negative polarity with respect to ground. It is plausible to operate both monopolar links independently, each having its own ground return, but it is more eﬀective to operate them together because their currents, being equal, cancel each other’s ground return to zero. Indeed, the ground path is a valuable resource for cases when one pole is out of service due to a planned or an unplanned event [4]. This HVDC technology seems to have hit an intrinsic limitation when applied to multi-terminal HVDC systems because CSC-HVDC is based on current balances. Hence, only series, multi-terminal HVDC schemes seem to be realizable using this technology. 1.2.6

VSC-HVDC

In contrast to the UPFC, the two VSCs of an HVDC link are not constrained to be housed in the same substation [9]. They can be hundreds of kilometres apart and linked together by an overhead DC line or an underground or submarine DC cable, or a combination of these, to satisfy geographical, economical, technical and aesthetic requirements. Such

13

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1 Flexible Electrical Energy Systems

IDC RDC

RDC IDC Figure 1.8 Point-to-point, bipolar VSC-HVDC system.

a VSC-HVDC link is termed point-to-point to diﬀerentiate it from the back-to-back VSC-HVDC link shown in Figure 1.6e. Since the point-to-point is aimed at bulk power transmission applications, it is normal to build it as a bipolar system as opposed to a monopolar one, as shown in Figure 1.8. Note that the rectiﬁer’s AC system and the inverter’s AC system do not necessarily need to have the same power frequency. The VSCs forming the bipole will be multi-level VSCs as opposed to the simpler two-level VSCs. Owing to the fast voltage-regulating capabilities aﬀorded by the VSCs, it is likely that the connecting transformers will not need to have tap-changing facilities, saving on costs. Normally the rectiﬁer is set to regulate power ﬂow and the inverter is set to regulate DC voltage. It is clear that the symmetrical bipole carries twice the rated power of a monopole HVDC link, with zero ground return. In the event of one of the poles being out of service because of maintenance or due to a contingency event, the link will remain in operation at 50% capacity. Tapping along the length of the DC line to pick up generation or to supply infeed points is carried out with ease. It requires only one extra converter station at each additional point. This is illustrated in the circuit diagram shown in Figure 1.9. In some respects, the VSC-HVDC link in Figure 1.9 can be classiﬁed as a multiterminal VSC-HVDC system [10], albeit of the radial type. As a matter of fact, the recent developments in VSC-HVDC technology are in the arena of multi-terminal VSC-HVDC systems; in particular, the kinds of multi-terminal systems which form meshed DC power grids. This is exempliﬁed by the case of four VSCs interconnected in their DC sides through cables to make up a four-terminal VSC-HVDC system, as shown in Figure 1.10. This is a generic concept that may be expanded to comprise n VSCs to link n AC systems of varying sizes, topologies and operational complexity. The DC cables may be overhead, underground or submarine, according to practical requirements. The DC network may even comprise a single node – a common DC bus where the n VSCs would be sharing a DC capacitor, i.e. multi-terminal back-to-back conﬁguration. Just as in meshed AC transmission systems there are transmission lines with non-regulated and regulated power ﬂows, power ﬂows sharing between neighbouring transmission lines, nodes with regulated and non-regulated voltages and so on, a meshed DC transmission system will have similar operational capabilities, as well as

1.3 Flexible Systems Vs Conventional Systems

Tapping 1

Tapping 2

Figure 1.9 Radial, bipolar VSC-HVDC system with tappings.

EDC,R1

PT

P maR1

RDC1

δvR1

Controller

maI1 RDC4

RDC3

EDC,I2

P

maI2

δvI2

δcI1

P

PT

P

PT

Controller VrefI1 PrefI1

Vref R1 PrefR1

PT

EDC,I1

RDC2

EDC,R2

maR2

δcR2

Controller

Controller

VrefI2 PrefI2

VrefR2 PrefR2

Figure 1.10 Four-terminal VSC-HVDC system.

suitable provisions for incorporating DC generation and DC loads directly. This will involve the use of DC-DC converters.

1.3 Flexible Systems Vs Conventional Systems Conventional electrical energy systems have traditionally been divided into generation systems, transmission systems and distribution systems. The energy ﬂows from the

15

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1 Flexible Electrical Energy Systems

generation systems towards the distribution systems. It is assumed that generation exists neither in the transmission system nor in the distribution system. The three-phase voltage and currents waveforms are largely sinusoidal and symmetrical, with a current frequency of either 50 Hz or 60 Hz. In these AC systems, a transmission line is not normally loaded up to its thermal limit since angular stability limits take place at much lower values than the thermal limits. Furthermore, transmission lines longer than 300 km will normally require series and shunt VAR compensation to be able to operate in steady-state. Power electronics valves, converter topologies switching techniques and control methods embedded in software have permeated all three sectors of the electricity supply industry, namely, generation, transmission and distribution. Power electronics has enabled the transformation of large, inﬂexible, ineﬃcient, failure-prone and environmentally unfriendly power systems into ﬂexible, eﬃcient, reliable and environmentally benign power systems which may be large in size or a concatenation of microgrids. 1.3.1

Transmission

The large blocks of electrical energy which are moved from the large generating power plants to the cities and factories are transported at high voltages over long or extra-long distances. Three-phase AC transmission lines and HVDC transmission lines are employed to carry out this task. In particular, the latter option serves the purpose of transporting large amounts of electrical energy over extra-long distances, with lower power losses than a comparable high-voltage AC transmission line (HVAC) [4]. Further applications where HVDC outperforms the HVAC option, regardless of whether or not it uses FACTS upgrades, are submarine power transmission longer than 70 km and the interconnection of AC power grids exhibiting diﬀerent operating frequencies or a substantial diﬀerence in network strength, i.e. connection of a weak network to a strong network [9]. It should be noted that the FACTS concept is based on the incorporation of power electronic devices and methods into the high-voltage side of the AC network to increase the control of power ﬂows in the high-voltage side of the network during steady-state and transient conditions [31]. From the outset, the developers of the FACTS initiative emphasized that this was not intended to be a direct competitor to HVDC transmission but, rather, an initiative able to provide technical solutions to speciﬁc power transmission problems at a lower cost [1, 2], particularly when the AC transmission corridor already existed. In any case, the aim is to apply the FTS solution that carries the best technical performance and the best value for money for a speciﬁc power transmission problem [7]. By way of example, the following technical issues call for the application of FTS equipment and methods, either in AC form or in DC form: – Higher power throughputs using the same right-of-way – VAR compensation – Frequency compensation/virtual inertia 1.3.1.1

HVAC Vs HVDC Power Transmission for Increased Power Throughputs

It is entirely feasible to have reinforced transmission systems using modern technology which perform their intended operating tasks in a rather smooth manner;

1.3 Flexible Systems Vs Conventional Systems

one which does not require more right-of-way than the original AC transmission system while increasing the power throughputs very substantially. Several options are available using FTS technology, one of which is to transport bulk power in DC form as opposed to AC form. The example presented in Figure 1.11 aims to illustrate this point. Converting existing AC transmission systems to carry power in DC form, when there is a technical and economical justiﬁcation for it, has been a longstanding aim [4, 32]. However, the technology associated with the DC transmission is still more expensive than the AC technology of comparable rating and the former, until quite recently, lacked the operational ﬂexibility of the latter, although this paradigm is changing rapidly. The attractiveness of the conversion from AC to DC transmission is that at least a threefold increase in power-carrying can be achieved and with only a fraction of the investment that would otherwise be required. Certainly, this would be met with no need to widen the existing right-of-way, which nowadays is a major obstacle for building or expanding on existing transmission circuits. As explained in [4], the insulation used in a standard AC transmission line to sustain the peak value of voltage to earth is entirely suitable to sustain a DC voltage to earth, particularly if atmospheric pollution is not a concern. Hence, the AC conductors become DC poles in the converted transmission system which may be operated up to their thermal rating. This contrasts with the AC conductors, where the transmission limiting factor is not the thermal rating of the cable but rather the stability limits associated with voltage phase angle. Accordingly, AC transmission limits have a great deal of slack capacity as far as their power-carrying potential is concerned. In connection with the transmission systems shown in Figure 1.11, a conventional power transmission corridor is shown in Figure 1.11a, comprising three AC circuits running in parallel and delivering 800 MW of power, supplied mainly from hydro-power stations located 431 km away. There is intermediate and local generation and load points. An example of what was possible to achieve with the DC technology available in the early 1970s, the hybrid AC-DC transmission circuit, is shown in Figure 1.11b [32]. Two of the three-phase AC circuits are converted into two point-to-point HVDC links running in parallel. Each DC circuit is a bipole, one using two conductors per pole and the other using one conductor per pole, commensurate with the six conductors of the two three-phase circuits singled out for conversion. The power converters use thyristor valves and phase control since that was the standard DC technology available until the ﬁrst decade of this millennium. The CSC-HVDC technology does not allow for the tapping of intermediate load/ generation points. Hence, the third AC circuit in Figure 1.1a is left untouched and used instead to supply the intermediate loads and to pick up the intermediate generation. The conversion of the two AC circuits into two bipolar DC circuits would increase almost fourfold the power-carrying capacity of the transmission scheme and with no compromises on operational ﬂexibility. This is achieved at the expense of extra investment in power converters, additional transformers, compensation equipment, ﬁlters and switchgear, and protection equipment. Note that the rectiﬁcation and inversion processes are imperfect and the characteristic harmonics are produced in both the DC circuit and the AC circuits of the HVDC links. Hence, AC ﬁlters are required for correct operation of the link and perhaps DC ﬁlters, if problems like interference in communications circuits are a cause for concern.

17

18

1 Flexible Electrical Energy Systems 2 × 350 mm2 ACSR conductors 220 kV

330 kV

330 kV

123 km

800 MW

Hydro-generation

Local generation and compensation

98 km

210 km C

D

(a)

B

A

220 kV

330 kV

330 kV

(b) 600 kVDC

2200 MW

Hydro-generation

Local generation and compensation 431 km

8 × 270 MW thyristor bridges

3600 A

8 × 270 MW thyristor bridges

330 kV

220 kV 210 km

123 km

98 km

600 kVDC 4 × 180 MW VSCs

4 × 180 MW VSCs

2500 MW

2400 A

(c)

600 kVDC

Hydro-generation

Local generation and compensation 8 × 270 MW thyristor bridges

431 km 2400 A

8 × 270 MW thyristor bridges

Figure 1.11 AC power transmission reinforcement/conversion using DC technology: (a) conventional AC transmission system [4]; (b) two AC circuits converted for DC transmission [4]; (c) three AC circuits converted for DC transmission.

1.3 Flexible Systems Vs Conventional Systems

A very important fact, from where we stand at this point in time, is that no extra land would be required for the upgraded right-of-way. Moreover, DC transmission systems do not contribute to fault level and this is normally considered to be an advantage, particularly in power grids where high levels of short-circuit ratio already have a problem. It should be noted that in bipolar HVDC systems, the poles may be operated independently when the need arises. Reﬂecting on what is possible today, with the availability of a new generation of HVDC technology based on forced-commutated valves and modern switching valve techniques, it is possible to convert the AC transmission line in Figure 1.11b to transport electricity in only DC form. This requires the use of VSCs as opposed to CSCs. Contrary to the early HVDC technology, the new technology does allow for the tapping of intermediate load/generation points by installing as many inverter stations as required. In the all-HVDC transmission system shown in Figure 1.11c, the three-phase AC circuit has been converted into an HVDC bipole. Note that this is not a point-to-point scheme but rather a multi-terminal VSC-HVDC scheme, albeit of the radial form. Notice that one of the existing CSC-HVDC bipoles has been modiﬁed to include only three conductors as opposed to four. The fourth conductor is used to make up the new VSC-HVDC bipole. The key point to bear in mind is that this upgrade fully meets the operational ﬂexibility of the AC transmission system in Figure 1.11a while increasing more than fourfold the power-carrying capacity of the transmission corridor. Also, the use of VSCs increases the availability of dynamic reactive compensation at both the rectiﬁer and the inverter sides of the hybrid HVDC transmission system, enabling a higher degree of system controllability. Equally important is the fact that the use of VSCs enables the supply of AC loads with no local synchronous generation, i.e. passive loads, such as the one connected to node C in Figure 1.11a. 1.3.1.2

VAR Compensation

Concerning shunt VAR compensation and the operational inﬂexibility associated with ﬁxed capacitor banks and reactors, this issue has long been a concern for the power transmission industry and equipment manufacturers [5]. In the early days of AC power transmission, the requirements for dynamic reactive compensations were met with synchronous condensers, i.e. synchronous generators with no prime mover, which were a rather expensive and bulky option to provide dynamic VARs with. In the late 1970s, with the availability of aﬀordable and reliable thyristor valve technology, TSC and TCR became commercially available to enable thyristor-ﬁred VAR compensators, which had no moving parts, i.e. static, and exhibit dynamic reactive power capabilities, with an acceptable degree of operational performance. SVC designs improved within a decade and their prices dropped, displacing the synchronous condenser as the preferred option for providing dynamic reactive power support, except for very specialized applications. The use of shunt dynamic compensation became pervasive in high-voltage AC power transmission; it would be diﬃcult to ﬁnd a transmission system that does not have one or more of these compensators installed. Towards the end of the millennium, a more powerful and versatile static VAR compensator emerged, which has been termed STATCOM. As discussed in Section 1.2, the STATCOM is essentially a VSC [8].

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Steady-State Performance of Shunt VAR Compensation The steady-state power transfer

concept, applied to the symmetrical transmission system shown in Figure 1.12, may be used to good eﬀect to illustrate the eﬀectiveness of adding a shunt compensator with dynamic capabilities, at the electrical midpoint [5]. In theory, the power transfer capability of a transmission line would double if an idealized shunt compensator were to be used at midpoint, as given by the following relationship: 𝛿 (1.2) 2 where Pmax is the ratio of the voltage magnitudes product at both ends of the transmission line, and its total inductive reactance. The angle 𝛿 is the phase angle diﬀerence of the two voltages at both ends of the line. Having said that, some practical limitations should be borne in mind which decrease the eﬀectiveness of the dynamic shunt controller. For instance, the compensator cannot respond instantaneously; the small positive slope in the characteristic of Figure 1.4, which accounts for the droop, and the inherent response delay in the controller combine to reduce the available area in the power-angle characteristic, as illustrated in Figure 1.13. Notice that the lost area is diﬀerent for the SVC and the STATCOM, since the SVC will act as a ﬁxed capacitor at low voltage. Psh = 2Pmax • sin

Transient Performance of Shunt VAR Compensation By way of example, the simpliﬁed

dynamic loops of the SVC and the STATCOM are shown in Figure 1.14 together with their simpliﬁed, schematic representation. The former takes the approach of modulating an equivalent susceptance in response to a voltage mismatch whereas the latter modulates directly the output AC voltage. The SVC and the STATCOM will perform similarly when operating at around their design ratings. This statement can be extended to cover the dynamic range of operation. To show the eﬀectiveness of the STATCOM to provide fault ride-through capability, its

E′, jX′d

jXtl

jXtl

jXtl

jXtl

jXt

jXt CB1

E′, jX′d

CB2 CB3

SVC/STATCOM Figure 1.12 Symmetrical transmission system with midpoint dynamic shunt compensation.

1.3 Flexible Systems Vs Conventional Systems

2.0 No shunt compensation line Practical shunt compensation line Point at which the SVC starts Idealized shunt compensation line to act as a fixed capacitor Limits imposed by limited capacitive current

Power (p.u.)

1.5

Area gained with an SVC

Area lost due to non-ideal compensation Additional area gained with a STATCOM

1.0

Area with no shunt compensation

0.5

0

1.5708 Angle (rad)

3.1416

Figure 1.13 Power-angle characteristic for the transmission system shown in Figure 1.12.

ΔVmax

VT ∠ θT

ISVC

Vref

jBSVC

+ –

ΔV VT

ΔBSVC 1 1 + sTSVC

KSVC ΔVmin

(a)

VAC + jδ EDC k1mae –

(b) ΔVmax

jXSTAT ISTAT + VT ∠ θT –

(c)

BSVC

Vref

+ –

ΔV K STAT VT

ΔVSTAT 1 1 + sTSTAT

VT

ΔVmin (d)

Figure 1.14 Dynamic SVC and STATCOM simpliﬁed representations: (a) schematic representation of the SVC; (b) dynamic control loop of the SVC; (c) schematic representation of the STATCOM; (d) dynamic control loop of the STATCOM.

voltage response to a three-phase fault applied to the electrical midpoint of the system shown in Figure 1.12 is given in Figure 1.15. To visualize better the marked improvement that controllable-shunt compensation has on power systems dynamics, the classical equal area criterion for transient stability assessment is used below, as applied to the symmetrical transmission system with midpoint compensation, shown in Figure 1.12.

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1 Flexible Electrical Energy Systems

Voltage performance 1.4 Without STATCOM With STATCOM

V [pu]

1.2 1 0.8 0.6 0.4

0

0.5

1

1.5 Time [s] (a)

2

2.5

3

Reactive power injected by the STATCOM

250 200 Qgen [MVAr]

22

150 100 50 0 –50

0

0.5

1

1.5 Time [s] (b)

2

2.5

3

Figure 1.15 Fault ride-through capability of the STATCOM: (a) voltage performance at the electrical midpoint of the system shown in Figure 1.12; (b) reactive power generated by the STATCOM.

Let us consider ﬁrst the case when CB3 is in open position and a short-circuit fault takes place between CBs 1 and 2; the fault is cleared by the opening of CBs 1 and 2. The three power-angle curves associated with the three diﬀerent states of the transmission system are shown in Figure 1.16a. The top curve corresponds to the healthy system – before the fault occurred. The bottom curve is for the period when the fault is in place, whereas the middle curve illustrates the situation when the faulted transmission line has been removed by the opening of CBs 1 and 2. To ensure that the overall system remains stable, these CBs should open at a time termed critical clearance time, tc , which carries an associated critical clearance angle, dc . In connection with Figure 1.16a, due to the occurrence of the fault, the energy enclosed by area A1 accelerate the rotors of the synchronous machines. Notice that before the occurrence of the fault, the system was operating in a stable equilibrium (P1 , d1 ). Removal of the faulted transmission line results in rotor speed and angle

1.3 Flexible Systems Vs Conventional Systems

Pmax

Pre-fault characteristics Post-fault characteristics Characteristics during fault

A2 P1

A1

0

2Pmax

d1

dc

1.5708 Angle (rad) (a)

Pre-fault characteristics Post-fault characteristics Characteristics during the fault Idealized shunt characteristics

1.5Pmax

dmax

3.1416

Point at which the SVC starts to act as a capacitor

Pmax Area margin for SVC A2

P1 A1 0

d1

dc

dmax

1.5708 Angle (rad)

3.1416

(b) Figure 1.16 An application of the equal area criterion used to illustrate the stability margins gained by using midpoint shunt compensation in connection with the symmetrical transmission system of Figure 1.12: (a) power-angle characteristics with no compensator; (b) power-angle characteristics with dynamic shunt compensator (SVC/STATCOM) of large rating, suitable for power transmission applications.

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oscillations. To ensure overall system stability, the area A2 enclosed by the post-fault electrical characteristic and the unchanged, constant mechanical power between the angle limits dc and dmax ought to be equal to area A1 . Otherwise, if area A1 is greater than area A2 , the system would become unstable. Let us consider now the previous operating scenario but with CB3 in closed position. The corresponding three power-angle curves associated with the three diﬀerent states of the transmission system but with the dynamic shunt compensator in place are shown in Figure 1.16b. The top curve corresponds to the pre-fault state. Notice that all three curves incorporate the limiting boundaries associated with the operation of a realistic dynamic shunt compensator, introduced in Figure 1.13. This applies to both the SVC and the STATCOM. Let us examine now the case with dynamic shunt compensation, for the same pre-fault power as for the case with no compensation. Hence, area A1 should be equal to area A2 in order to ensure a stable system. It is observed from Figure 1.16b that this criterion is fulﬁlled rather handsomely; the available decelerating area enlarges as a result of the shunt compensator and is only partially used, increasing the transient stability margin (or limit) by a signiﬁcant margin before the SVC starts behaving as a ﬁxed capacitor. This means that the compensated transmission system could be made to work harder during steady-state operation than the system with no compensation, at least in principle, up to the point in which all the decelerating area is used up. Notice that the slack area is larger for the STATCOM than it is for the SVC. Having said that, one has to bear in mind the rather simplifying nature of the equal area criterion for stability, such as the assumption of constant mechanical power and constant internal voltage of the synchronous generators. Detailed transient stability studies should be conducted using any of the industry-grade transient stability computer programs available. 1.3.1.3

Frequency Compensation

Large increases in wind and solar penetration in a power system present new challenges in its operation. This is intermittent generation, which may aﬀect adversely the operation of the high-voltage transmission system in a number of ways. For one, the system-level power balance would become more diﬃcult to attain, but also voltage control may become a signiﬁcant problem under certain conditions [7]. Hence, the measures required to ameliorate the problems brought about by power ﬂuctuations are worth investigating. For instance, the MW-level application of battery-BESSs is a case in point. It would be fair to say that interest in this application has been longstanding but the BESS technology had not matured suﬃciently to warrant much interest from transmission system operators. However, this is changing rapidly with the advent of a new generation of BESS using Li-ion batteries and state-of-the-art power electronic converter topologies and control techniques [15]. Reports on operational experiences in two of the newest installations, using state-of-the-art technology, are rather encouraging [29, 30]. The structure of a modern BESS is shown in Figure 1.17, comprising several parts: (i) battery pack; (ii) DC-DC boost converter; (iii) VSC; (iv) step-up power transformer; (v) controls. The DC-DC converter is responsible for active power control. It is designed to have bidirectional capabilities, with large enough capacitors to keep the ripple in its output

1.3 Flexible Systems Vs Conventional Systems

Pref

fref

P&f

Crtl δI

δE

Crtl

EB

VSC1 VvR

EDC ma1

2EDC

VSC2 Battery pack

Boost converter

EDC

PT ma2

Crtl Vref

Figure 1.17 Structure of a BESS.

voltage as small as possible and to keep the voltage between the switches constant as it transfers power between the batteries and the VSC. The VSC injects/draws reactive power by controlling the output DC voltage of the inverter. Apart from enabling the connection of the battery, through the DC-DC converter, to the AC system, the main control objective of the VSC is to regulate the output voltage of the inverter and with it the reactive power ﬂow. In order to control the active power exchange between the storage system and the grid, the duty cycles of the DC-DC converter switches are adjusted. One switch controls the power drawn from the battery and another switch controls the power injected into the battery. Needless to say, the two switches cannot be closed at the same time. The switches also control the terminal voltage of the battery [15]. When the terminal voltage is higher or lower than the internal voltage of the battery, the converter injects or draws power from the battery, respectively. It is well known that synchronous generators regulate frequency by adjusting the amount of power injected into the power grid. Following a system load change, all generators’ rotors accelerate/decelerate to compensate the momentary power imbalance between the electrical and mechanical powers. This diﬀerence in power will be made up by the rotating mass (kinetic energy) of each generator. This is followed by the action of the speed-governing system, which acts to reduce the error between the nominal and the actual generator’s speed by changing the input power to the turbine. This control system is termed load-frequency control [33]. It has been observed that a BESS possesses similar operating characteristics to those of a conventional power plant, even though it has no rotating parts. This makes the BESS much quicker to respond. The energy stored in the battery has the characteristics of inertia by acting as a stored kinetic energy. When the system load exceeds the power delivered by the synchronous generators, the BESS injects power into the grid, hence the battery’s state of charge (SOC) decreases, and vice versa. To emphasize further the operational resemblance between the BESS and a conventional power plant, an analogy between a BESS and a hydropower generator presented in [15] is reproduced in Figure 1.18.

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Volumetric flow rate Q at pressure P Reservoir Turbine (water capacity)

Torque T at speed S

Generator

Power PAC at voltage VAC (a)

Governor

AC voltage controller

∑ Reference speed

Battery (chemical)

Current I at voltage VDC

DC/DC converter

δ1,δ2

∑ Reference AC voltage

Power PDC at voltage VDC

VSC

Power PAC at voltage VAC (b)

Duty cycle controller

∑ Reference DC voltage

ma

AC voltage controller

∑ Reference AC voltage

Figure 1.18 (a) A conventional turbine-governor control of a synchronous generator; (b) the BESS voltage and power controls (©IEEE, 2016).

The governor controls the inlet of water into the turbine (with the reservoir being discharged) and the duty cycle controller manages the rate of discharge of the battery. Note that the case of battery charging in this analogy would correspond to the case of hydro-pumped storage. The AC voltage controller in both cases is a regulated gain. At the outputs of the reservoir and battery blocks, the volumetric ﬂow rate Q is akin to current I and pressure P is akin to voltage V DC . Likewise, at the output of the turbine and DC-DC converter blocks, torque and speed are akin to power and AC voltage, respectively. In order to illustrate the eﬀectiveness of the BESS to impact on system frequency, we use the contrived test system in Figure 1.19. However, notice that this system is a simpliﬁed reﬂection of the practical BESS installation in Antofagasta in northern Chile [30]. The system responses to a load increase and a load decrease are presented with BESS and with no BESS in Figures 1.20 and 1.21, respectively. The case of 100 MW load increase with BESS and with no BESS is discussed ﬁrst. The positive contribution of the 22 MW BESS is self-evident in the system frequency plot in Figure 1.20, showing the result when the BESS is connected and when it is not. At a sudden increase in load, the frequency drop is smaller when the CB is in closed position. The BESS not only improves the inertial response (i.e. ﬁrst swing), it also reduces frequency oscillations and restores the system frequency to its 60 Hz nominal frequency (i.e. it provides secondary frequency support). As expected, the case of 100 MW load decrease would also beneﬁt from the use of the BESS, in very much the same manner as the case for the load increase, even though

1.3 Flexible Systems Vs Conventional Systems

1000 MVA 13.8/230 kV

Double-circuit transmission line 200 km

800 MW

1000 MW 60 Hz CB

30 MVA 33/230 kV

BESS 22 MW Figure 1.19 Test circuit to assess the frequency response of a BESS.

Frequency of generators Frequency of generator without BESS Frequency of generator with BESS

1.01

Frequency (p.u.)

1.0 0.99 0.98 0.97 0.96 0.95

0

50

100

150

Time Figure 1.20 Frequency response of the BESS in the transmission circuit in Figure 1.19 – due to a load increase.

slightly more frequency oscillations are experienced in the BESS response, as seen from the frequency plots in Figure 1.21. 1.3.2

Generation

The vast majority of the electrical energy consumed today is produced by large, three-phase synchronous generators which use, as primary energy resources, the energy stored in huge water reservoirs, in fossil fuels and in nuclear materials [33].

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1 Flexible Electrical Energy Systems

Frequency of generators Frequency of generator without BESS Frequency of generator with BESS

1.05 1.04 1.03 Frequency (p.u.)

28

1.02 1.01 1 0.99 0.98

0

50

100

150

Time Figure 1.21 Frequency response of the BESS in the transmission circuit in Figure 1.19 – due to a load reduction.

However, there is widespread consensus that burning fossil fuels at the current rate is having adverse consequences for the global climate on a scale that poses a threat to many regions of the world. A large reduction in greenhouse gas emissions seems not only desirable but essential in order to reign in global warming. For the last decade, there have been calls from many governments around the world to have in place a 60–80% cut in emissions before 2050. However, reducing global emissions is one of the greatest challenges facing mankind because environmental wellbeing and economic growth are heavily inter-linked. Cost-eﬀective, low-carbon electricity generation sources, together with eﬀective demand-side and energy-saving measures, are pre-conditions for a low-carbon electric energy sector. Growth in wind and solar generation continues to outstrip all other forms of renewable generation, driven by concerns over climate change and energy diversity and by technological break-through in equipment and methods, and a better understanding of the wind and solar resources. 1.3.2.1

Wind Power Generation

The early commercial wind turbines were very basic in terms of their construction and their operation. They used a squirrel-cage induction generator (SCIG) driven above the synchronous speed by coupling the low-rotating shaft of the wind turbine and the high-speed shaft of the induction generator, using a gearbox [13]. This is illustrated in Figure 1.22a. They run at almost constant speed (1–2% above synchronous speed) and that is the reason why they are known as ﬁxed-speed induction generators (FSIGs). On the electrical side, the stator winding is connected to the secondary side of a step-up transformer, with the primary winding of the transformer connected to the AC power grid.

1.3 Flexible Systems Vs Conventional Systems

Soft-starter Gearbox

SC IG

(a)

Capacitor bank Crowbar

Gearbox

PM SG

DF IG

(b)

(c)

Figure 1.22 Schematic description of (a) FSIG, (b) DFIG and (c) PMSG with fully rated converters.

The reactive power required to ﬂux the air-gap ﬁeld of the induction machine was imported from the power grid or ﬁxed bank of capacitors were installed at the wind farm for improved overall performance. Notwithstanding this, the following technical issues started to raise serious concerns among power system operators when the wind farms began to grow in capacity: (i) steady-state voltage control, (ii) reactive power support, (iii) transient/dynamic stability, (iv) fault ride-through capability and (v) power quality [34]. The reason for these technical issues was that the most likely location for an on-shore wind farm was in rural areas where the network was poorly developed (it had a low X/R ratio); hence, ﬂuctuations in the wind speed induced power output changes which, in turn, led to ﬂuctuations in the network voltage. Connection to a weak network also caused problems for the wind farm and network during transient conditions, which led to instability. These early wind generators were prone to instability following a network fault due to excessive reactive power absorption at high speeds. Nowadays, wind generators are required to have fault ride-through capability. From the power quality perspective, ﬂickering was a concern. Aiming to overcome these issues, power electronic-based compensators, such as the SVC and the STATCOM, were employed to provide dynamic reactive power to the wind farm. Particularly with the use of the STATCOM, those ﬁrst-generation wind farms went a long way towards satisfying the more stringent low-voltage ride-through capability imposed by new grid codes [7].

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Meanwhile, a new generation of wind turbines became available which were larger in size, able to capture the available wind resource in an optimal manner and with a much more enhanced ride-through capability, whose performance rivalled that of the conventional synchronous generators for any practical purpose relating to grid codes requirements. This was possible through the incorporation of power electronics technology. Suitable use of a back-to-back VSC-HVDC link enabled the electrical generators to have variable-speed capabilities. Two quite distinct technologies emerged: the doubly-fed induction generator (DFIG) and the permanent magnet synchronous generator (PMSG). Both topologies are illustrated schematically in Figure 1.22b,c, respectively. In the former, a controlled ﬂux is superimposed onto the rotor ﬂux using a fractional VSC and slip rings. In the latter, a controlled voltage is injected directly into the stator winding through a fully rated VSC. In both cases, it has the eﬀect of varying, in a ﬁnely regulated manner, the speed of rotation of the machines aiming at matching, in an optimal manner, the available wind resource. The DFIG employs an induction generator with a wound rotor whereas the PMSG employs multi-pole designs with permanent magnets to provide the required rotor excitation to enable the rotor to rotate at synchronous speed. However, the speed of rotation of the PMSG is very low; in fact, the speed of rotation is the same as that of the blades, hence no gear-box is required, as illustrated in Figure 1.22c. It follows that its current frequency is very low compared with the current frequency of the power grid and its connection to it is carried out through a back-to-back HVDC link, not restricted to a VSC-HVDC link. Sometimes, on economic grounds, a hybrid HVDC link has been suggested. It may be argued that a simple diode bridge rectiﬁer could be used instead of the VSC connected to the stator winding of the PMSG in Figure 1.22c. The induction generators of the kind used in wind power normally employ four poles and their speed of rotation must be quite high to produce a current frequency that matches that of the power grid; hence, the use of gear-boxes becomes mandatory. The FSIG, illustrated in Figure 1.15a, is driven above the synchronous speed to be able to perform as a generator. However, the DFIG, illustrated in Figure 1.15b, operates under a range of controllable speeds around the synchronous speed. In the presence of short-circuit faults in the vicinity of the installation, the crowbar quickly short-circuits the rotor windings to prevent large fault currents from entering into the rotor winding and causing terminal damage; the DFIG eﬀectively becomes a FSIG for the duration of the short-circuit fault. This increases the ride-through fault capability of the DFIG technology, as shown in Figure 1.23. The DFIG is less expensive than the PMSG of comparable rating because its machine-side converter (rotor converter) is only a fraction of the machine-side converter (stator converter) of the PMSG. Moreover, its electrical generator uses standard rotating machinery designs compared with the PMSG designs employed in wind power applications. 1.3.2.2

Solar Power Generation

Solar power generation using the PV phenomenon is a growing industry around the world, particularly the case of grid-connected PV installations [14]. Since a PV generator produces voltage and current in DC form, connection to the AC power grid requires a power inverter. Early PV designs were able to track only the maximum amount of solar power available while maintaining a unit power factor at the connection point with the AC power grid. Modern PV designs incorporate a DC-DC boost converter,

Active power [MW]

40 30 20 10 0 –10

0

100 ms 120 ms 1 2 Time [s]

140 ms 3

Reactive power [MVAR]

1.3 Flexible Systems Vs Conventional Systems

20 0 –20 –40 –60 –80

0

100 ms 120 ms 1 2 Time [s]

140 ms 3

100

50

0 0.5

1

120 ms 140 ms 160 ms 180 ms 200 ms 1.5

Reactive power [MVAR]

Active power [MW]

(a)

20 0 –20 –40 –60 –80

0.5

1

120 ms 140 ms 160 ms 180 ms 200 ms 1.5

Time [s]

Time [s] (b)

Figure 1.23 Fault ride-through capability tested for a (a) FSIG-based wind farm and (b) DFIG-based wind farm, for diﬀerent time-fault durations.

which sits between the PV panels and the power inverter. This has enabled the new generation of PV installations to improve operational performance – they can perform voltage regulation at the point of common coupling with the AC power grid and provide frequency support, albeit this would be achieved at the expense of spilling out some of the solar resource available [35]. A schematic structure is shown in Figure 1.24. To enable an even more powerful PV-based system, which would rival fully the operational ﬂexibility of conventional, large synchronous generators, including the much after-sought characteristic of ﬁrm generation, the PV installation could be combined with local energy storage. In particular, BESS and superconductor magnetic energy storage (SMES) systems are most suitable because they have a relatively small footprint and use power electronics similar to those of the modern PV generator. The scheme, shown in Figure 1.25, would capture and store all the solar resource available, regardless of the grid-support function that it is required to perform by the grid operators [36]. Provided there is suﬃcient energy in the storage system, the associated controllers can quickly react by injecting active and reactive power into the AC system to provide frequency and voltage support in the event of power system disturbances. One area of concern is that as renewable power continues to displace larger blocks of conventional generation, AC power grids are losing inertia and synchronizing power, with the ensuing detrimental eﬀects on power system stability. To overcome

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Pref

fref

P&f

Crtl δI

IPV

δE

Crtl

VSC1 VvR

EDC ma1

2EDC

VSC2 PV stack

EDC

Unidirectional boost converter

PT ma2

Crtl Vref

Figure 1.24 Structure of a PV generator.

Pref

fref

P&f

Crtl δI

δE

Crtl

ES

2EDC

VSC1 EDC

Storage

VvR

Bidirectional boost converter

ma1 VSC2 EDC

IPV

Unidirectional boost converter

PT ma2

2EDC

Crtl Vref

PV stack Figure 1.25 Structure of a PV generator with storage.

this undesirable side eﬀect of increasing the generation share of the badly needed renewable generation, the power industry is engaged in the development of controllers for renewable generators and storage systems, to enable them to assist in primary frequency support by contributing synthetic inertia [7]. In order to illustrate the eﬀectiveness of the BESS, to support the power output of a PV generator undergoing irradiance and temperature variations, the output powers shown in Figure 1.26 are self-evident. They show the case with no BESS and with BESS, respectively.

1.3 Flexible Systems Vs Conventional Systems

100

Power (KW)

80 60 40 20 0 –20 0

0.5

1

2

1.5

2.5

Time (secs) (a) 120 100 Power (KW)

80 60 40 20 0 –20 0

1

2

3

4 Time (secs) (b)

5

6

7

8

Figure 1.26 Power responses of (a) PV generator with varying solar irradiance conditions and (b) PV generator with BESS with varying solar irradiance conditions [36].

1.3.3

Distribution

It is widely acknowledged that the distribution system load is unbalanced and non-linear, in stark contrast to the bulk load points at the transmission level, which are taken to be balanced and linear. To limit the adverse eﬀects that such loads may introduce into the distribution system, the industry has developed stringent grid codes aimed at maintaining high power-quality standards to increasingly demanding end users [7]. Moreover, the past two decades had seen a marked increase in the deployment of end-user equipment that was highly sensitive to poor power quality supply. In the early 1990s, several large industrial users were reported to have experienced signiﬁcant ﬁnancial losses as a result of even minor lapses in the quality of electricity supply [3]. A great many eﬀorts were made to remedy the situation, where solutions based on the use of the latest power electronics technology ﬁgured prominently.

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1 Flexible Electrical Energy Systems

Indeed, custom power electronic technology, the low-voltage counterpart of the more widely known FACTS technology, aimed at high-voltage power transmission applications, emerged as a credible solution to solve many of the problems relating to continuity of supply at the end-user level. Both the original FACTS and custom power concepts were developed at EPRI [1, 3]. Several power electronics controllers have been developed as part of the custom power applications, such as the STATCOM for low-voltage applications, known as D-STATCOM, the dynamic voltage restorer (DVR) and the solid-state transfer switch (SSTS). These are system-level pieces of equipment which lies at the interface between the distribution system and industrial users, aiming at mitigating upstream disturbances [37], as indicated in Figure 1.27. The D-STATCOM is aimed at load compensation whereas the DVR and SSTS provide dynamic voltage support. To a greater or lesser extent, the short-circuit faults cause voltage sags in the neighbouring areas where they take place; in turn, voltage sags may impair industrial processes. Even though events like capacitor switching and voltage sags due to motor starting do play an important role in equipment tripping, by and large equipment tripping is due to short-circuit faults in the distribution system. A number of mitigation actions could be applied to reduce the number and severity of such unforeseen events. For instance: (i) to improve the power system design so that it reﬂects in less severe and frequent events; (ii) to increase the immunity of equipment; (iii) to use mitigation equipment at the interface between the utility company and users with sensitive equipment [37]. The Custom Power equipment, installed at the interface between a utility company and an industrial user with sensitive loads, is commercially available from several vendors. The following are some of the potential beneﬁts of the Custom Power technology to end users with critical loads: practically no power interruptions, magnitude and duration of voltage dips/swells within speciﬁed, stringent limits, low harmonic distortion, low phase unbalance, acceptance in the neighbourhood of ﬂuctuating, non-linear and low power factor loads [3]. Actions: Distribution system

Improve system design

PQ disturbance Interface

Mitigate disturbances using Custom Power solutions

Ameliorated PQ disturbance Industrial equipment

Improve equipment design/protect individual equipment

Figure 1.27 Upstream PQ disturbances and actions.

1.3 Flexible Systems Vs Conventional Systems

1.3.3.1

Load Compensation

The D-STATCOM is the conventional solution used to overcome power quality (PQ) phenomena such as voltage imbalances, poor power factor and harmonic distortion. If the D-STATCOM’s control possesses harmonic cancellation capabilities, it is termed active power ﬁlter (APF). If the D-STATCOM is ﬁtted with an energy source on its DC bus, such as a battery pack, it is termed BESS. Concerning the amelioration of the above PQ phenomena, these devices inject a set of suitable compensating currents, which restore the operating currents to their ideal, balanced and sinusoidal form. 1.3.3.2

Dynamic Voltage Support

Voltage sags/swells have the potential to impair industrial processes, with entire production lines shutting downs [3]. To a larger or greater extent, these power quality disturbances are the result of upstream short-circuit faults. Design improvements on the power grid and intelligent equipment maintenance programs are preconditions for reducing the number and severity of short-circuit faults; however, it is highly unlikely that these will be eradicated for good [37]. In view of this, a practical alternative for industrial users with sensitive equipment and processes, such as paper mills, is to shield their installations against the adverse voltage phenomena resulting from upstream short-circuit faults, namely voltage sags/swells. Possible options to protect their loads include using a DVR or an SSTS. The DVR injects a series voltage in phase with the incoming voltage to restore the voltage waveform to what it was prior to the fault. The SSTS ensures continuous high-quality power supply to sensitive loads by transferring, within milliseconds, the load from the faulted bus to an auxiliary healthy bus [3]. The test system shown in Figure 1.28 contains a sensitive load connected to Node 2. Nodes 1 and 2 are linked through the primary winding of a series-connected transformer. A DVR connects to the secondary winding of the transformer through a switch S1 . Note that this test system is an abstraction of Figure 8 in reference [38]. Two scenarios are considered below, one with no DVR and the other with DVR. The former scenario assumes that a three-phase-to-ground short-circuit through a fault resistance occurs at Node 1 in the time period 300–600 ms of the simulation. The RMS voltage proﬁle in Figure 1.29a shows the expected behaviour of the voltage as measured Node 2

Node 1

Sensitive load Supply 3 ϕ S.C. fault through a fault resistance of 0.66 Ω

S1 DVR

Figure 1.28 Equivalent circuit of a shielded sensitive load against upstream short-circuit faults.

35

1 Flexible Electrical Energy Systems

1.2

Vrms (pu)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.1

0.2

0.3

0.4 0.5 Time (sec) (a)

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4 0.5 Time (sec) (b)

0.6

0.7

0.8

0.9

1.2 1.0 Vrms (pu)

36

0.8 0.6 0.4 0.2 0.0 0.0

Figure 1.29 RMS voltage response at Node 2 of Figure 1.28 when a three-phase-to-ground short-circuit fault takes place at Node 1 through a fault resistance of a 0.66 Ω, (a) with no DVR and (b) with DVR (©IEEE, 2002).

at Node 2. The voltage sag at the load point is 50% with respect to the reference voltage. The simulation experiment is repeated but now with switch S1 connected during the time of the fault. The new RMS voltage proﬁle is shown in Figure 1.29b. The last simulation result shows the eﬀectiveness of the DVR to provide dynamic voltage support. 1.3.3.3

Flexible Reconﬁgurations

Distribution systems may be classiﬁed as either radial or meshed networks, according to their structure. However, they are normally operated as if they were radial networks, owing to the simpler operation and cheaper protection equipment of radial networks. Heretofore, this practice has served well the aims and objectives of the electrical power utilities worldwide; however, the widespread installation of small-scale generators of various kinds at the customers’ premises is challenging this long-standing practice. This array of new sources of electrical energy is termed distributed generation (DG) and is said to be turning the otherwise passive distribution systems into active

1.3 Flexible Systems Vs Conventional Systems

distribution systems. One distinct characteristic of the passive distribution systems is that the power ﬂows are unidirectional, i.e. from the utility company towards the customers, whereas in the active distribution systems the power ﬂows are, at least in principle, bidirectional [7]. With the advent of reliable and aﬀordable power converters for low-voltage, low-power applications, this technology is ﬁnding an increasing number of uses. For instance, the concept of switching centres has been put forward in [39]. This concept is illustrated with reference to Figure 1.30, where a four-terminal switching centre is installed at the far end of four three-phase, AC distribution feeders. It should be noticed that in all respects the switching centre is a back-to-back, multi-terminal VSC-HVDC system. It enables controlled power exchanges between the feeders through a common DC bus. The DC bus is also a useful resource for incorporating a wide range of storage and electrical energy sources, which produce power in DC form. It is argued in [39] that key advantages of a ﬂexible bridge, with respect to mechanical switches, are: • The DC link is capable of carrying out scheduled exchanges of active power through the common DC bus, in a continuous fashion. • Controlled injections of reactive power at the far end of the feeders are carried out by their respective VSCs. This is particularly useful when the switching centre is located at the far end of the feeder since the voltage drops will be more pronounced at such points in passive feeders. The opposite may occur in feeders containing distributed generation where the risk of over-voltages is high. • The VSC technology is very eﬀective in overcoming adverse PQ environments, i.e. poor voltage regulation and power factor, voltage imbalances, low-order harmonics. • Anomalous phenomena such as short-circuit currents are not passed on between feeders owing to the almost instantaneous speed of response of the VSCs, which act to block the ﬂow of anomalous currents. One concern with the use of this technology that springs to mind is the high investment costs required, compared with those of an electromechanically based switching centre. However, it is conjectured that this technology may be cost eﬀective when accompanied by the substantial integration of DERs [40]. 1.3.3.4

AC-DC Distribution Systems

A conventional three-phase AC distribution feeder may be converted to carry DC current instead of AC current, with the use of suitable power electronic converters, as exempliﬁed by the AC and DC distribution systems shown in Figure 1.31a,b, respectively. In the DC system, the conventional three-phase loads may be served in a more controllable manner while enabling the direct connection of DC distributed energy resources (DERs), such as battery packs, fuel cell stacks, PV stacks and EV recharging stations. Furthermore, if the amount of energy produced by the in-situ generation and storage is suﬃcient to meet the local demand plus the associated power losses, then the DC feeders may act as an independent microgrid, isolated from the HV utility supply, with the AC connection points taking the character of backup supply points. It is stated that key concerns in today’s distribution network development are cost eﬀectiveness and system reliability [21]. In Finland, the distribution voltage levels were 20/0.4 kV until the beneﬁts of an intermediate 1 kV voltage level was established, i.e.

37

LV

LV CB

LV

SW

SW

SW

LV

LV

SW

SW PV stack

CB SW HV

SW

SW

Unidirectional boost converter

SW

MV LV

LV LV

LV CB

LV

DC bus

LV

LV

SW

SW

SW

LV

LV

SW

SW

Fuel cell stack Unidirectional boost converter

CB SW HV

SW

SW

SW

MV LV

LV

LV

LV

Battery pack Bidirectional boost converter

Switching centre

Figure 1.30 Distribution feeders with enhanced functionality with the use of power electronic converters and DC DERs.

1 kV

CB

SW 0.4 kV

SW 0.4 kV

SW 0.4 kV

SW 0.4 kV

SW 0.4 kV

SW 0.4 kV

SW 0.4 kV

SW 0.4 kV

0.4 kV

0.4 kV

20 kV

CB

(a) PV stack

FC stack

0.4 kV 0.4 kV

0.4 kV

1kV 1.5 kV DC

0.4 kV 0.4 kV

0.4 kV 0.4 kV

20 kV

1.5 kV DC

0.4 kV

0.4 kV (b)

0.4 kV

Figure 1.31 (a) Example of 1 kV AC distribution feeders and (b) LVDC bipoles incorporating DC DERs.

Battery pack

Extended DC bus for battery recharging

40

1 Flexible Electrical Energy Systems

20/1/0.4 kV. This has proven to be a proﬁtable solution but with application possibilities limited to small transmission powers and short transmission distances which, nonetheless, are suitable for the power distribution conditions prevailing in Finland. However, this is beginning to change as the challenges imposed by the introduction of DG are given due consideration [41]. A solution which has been put forward within Finnish academic circles is to transform the 20/1/0.4 kV AC distribution system into a distribution system operating at 1.5 kV DC. The study carried out in [21] for transforming an AC distribution system into a DC distribution system explores a number of monopolar and bipolar DC topologies, aiming at reusing as many AC conductors as possible. A rather interesting result shows that for a distance of 500 m and a 6% voltage drop, the maximum transmitted power in a 400 V AC system is 20.5 kW. In contrast, for the same distance and voltage drop, the maximum transmitted power in a unipolar 1500 V DC system is 270.5 kW. The same comparison for the bipolar ±1500 V DC system gives 345 kW of transmitted power. The authors conclude that for the Finnish distribution system, the low-voltage DC distribution system enables higher transmission powers and longer transmission distances compared with those of their traditional low-voltage system. Customers’ voltage quality improves when voltage dips, and ﬂuctuations and short time voltage drops can be eliminated using power electronic devices. They foresee more economical and reliable distribution networks than those we have today. However, for this to happen, power electronic converters prices should decrease even further and functionality should increase. 1.3.3.5

DC Power Grids with Multiple Voltage Levels

Power transformers have been used systematically in AC systems to suitably interconnect power circuits with diﬀerent voltage levels. As a matter of fact, these devices were instrumental in the early developments of the AC power systems, vastly surpassing the applicability of the DC power systems at the time. However, over the past two decades, the DC technology has developed at an unprecedented pace and the AC power system has to integrate more and more DC subsystems and components, employing a variety of voltage levels, from LV to UHV. This has created a need for eﬃciently performing voltage level conversion at the points of interconnection using DC-DC converters. However, the kind of converters used in low power applications [42] may not be suitable for the handling of high power and high voltages. A suitable solution involves the use of DC-AC converters employing two-level or multilevel VSCs, as illustrated in Figure 1.32 [7]. Note that the frequency on the AC transformer can be much higher than the power frequency, aiming at reducing the transformer size and, consequently, that of the DC-AC converters. 1.3.3.6

Smart Grids

Besides the incorporation of power electronic-based solutions to ameliorate PQ phenomena [3], to achieve higher power throughputs [6] and to enhance the operational ﬂexibility of distribution power grids [39], the electricity industry is embarking on unprecedented changes to be able to cope with major challenges arising from an ageing infrastructure, market liberalization and the incorporation of renewable generation, aﬀordable storage and smart metering [18]. It is said that tomorrow’s power

1.3 Flexible Systems Vs Conventional Systems

+ EDC1 –

+ EDC2 – maR

δvR

maI

δcI

Controller Pref Figure 1.32 A DC-DC converter based on VSCs (©MPCE, 2017).

grids must ensure secure and sustainable electricity supplies with low energy losses and low CO2 emissions. In Europe, these power grids should comply with new policy imperatives and changing business frameworks, and should incorporate the state-of-the-art information technology, communications technology and the latest generation of electrical equipment. The conﬂuence of these technologies and policies has given rise to the smart grid concept [19]. Paramount in this array of technologies is the ubiquitous power electronic converter [43]. In the near future, this would pave the way for the existence of AC-DC smart grids as opposed to today’s AC smart grids [17]. AC-DC smart grids would be amenable to higher energy yields than AC smart grids, reducing very considerably carbon footprints and using fewer material resources. This would be rather signiﬁcant in the face of ever escalating prices of metals such as copper and iron, which are a major cost factor in equipment manufacturing and in the electricity supply industry. The design and operation of future AC-DC smart grids call for the development of models, methods and control techniques embedded in software. These developments in software would be invaluable in making informed recommendations concerning the transformation of traditionally driven, energy-intensive processes such as in the mining industry into a more energy-eﬃcient, reliable operation with a much-reduced carbon footprint. It is expected that the trend of CO2 emissions reduction will accelerate when a full deployment of the smart grid technology takes place in earnest. It should be noted that the smart grid technology is likely to have a signiﬁcant inﬂuence on the next top contributor of CO2 emissions, namely the transport sector – which is reported to contribute to as much as 22% of total global emissions. Additional key drivers of the smart grid technology are its user-centric approach; electricity networks renewal due to ageing infrastructure; enhanced security of supply owing to the underground nature of its power cabling system and self-healing properties; the liberalized markets philosophy; the interoperability of European electricity networks due to standardization; the distributed generation and renewable energy sources (RES) approach; central generation environmental issues; the demand response and DSM policies; European regulatory and social aspects [19]. The multi-terminal VSC-HVDC systems (MT-VSC-HVDC) are very well placed to be the transmission structures that will be used in the next generation of smart grids, both

41

DS2 PEV charging station

DS1

DC cable Comms cable Processor agent

VSC2

VSC1 v2 DC ring

v1

v3

Micro-grid 3

VSC3 VSC5 v5

Micro-grid 2

VSC4

Micro-grid 1 v4

Figure 1.33 Structure of a DC microgrid (©PSCC, 2016).

1.4 Phasor Measurement Units

at the transmission level and at the distribution level [44]. This may be a system within a larger AC system, as exempliﬁed in Figure 1.1. Alternatively, the core system may be a MT-VSC-HVDC system interconnecting a number of AC systems, such as low-voltage, low-power microgrids, as illustrated by Figure 1.33. It was suggested earlier in the chapter that we should call the equipment of FACTS and HVDC, collectively, FTS equipment. Heretofore, FTS equipment has been used to solve only one or more problems at the local level as opposed to the system level; FTS equipment has been endowed neither with intelligence nor with advanced ICT capabilities. The current challenge is to make FTS equipment have a system reach within the power grid and to act either alone or in a coordinated fashion with other FTS equipment to ameliorate disturbances arising anywhere in the power grid, in an optimal, smart fashion. The challenge is in devising mechanisms that would provide FTS equipment with the required intelligence that would enable smart grids to be fault-tolerant [7] and, in the longer term, ‘self-healing’ [19]. To such an end, as illustrated in Figure 1.33, the individual equipment would be ﬁtted with embedded processing agents [22], which would be linked by a dedicated communications network running in parallel with the AC-DC power grid.

1.4 Phasor Measurement Units The developments of the PMUs are strongly linked to the development of distance relay using symmetrical components algorithms in the early 1970s [45]. By the early 1980s, GPS (Global Positioning System) satellites had been deployed in suﬃcient numbers and the great potential of digital relays to provide an instantaneous picture of the state of the power system had been realized. The basic idea was to use GPS time signals as inputs to the sampling clocks in the measurement system of the digital relays. More recently, the concept has evolved to encompass the calculation of phasors in real-time synchronized to an absolute time reference – the term ‘synchrophasor’ has been coined to describe such a concept. Phasors have been used to describe AC power circuit voltages and currents since their introduction by Charles P. Steinmetz in 1893 using complex quantities [46]. In theory, phasors are applicable only to AC circuits operating in a ‘true’ steady-state condition, but in practice they have been used to good eﬀect to describe the behaviour of AC power systems when their operations depart from the steady-state. For instance, when a power system undergoes electro-mechanical oscillations, the voltage and current waveforms will not observe constant amplitudes over the whole range of the oscillation period and the power system frequency itself will depart from its nominal operating value. Nevertheless, since the voltage and current variations are relatively slow, it makes sense to group a number of waveform cycles and to treat those as a steady-state condition corresponding to a given time interval; hence, the whole range of the oscillation period may be treated as a series of connected, steady-state conditions. Indeed, it may be argued that the limiting point of such an approach would be when the ‘steady-state’ is made up of one full cycle of the voltage and current waveforms. In practice, there are applications, such as relaying and power electronic control, where it is not uncommon to use phasors of voltage and current over a half-cycle observation window [47].

43

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1 Flexible Electrical Energy Systems

In a digital measuring system, samples of the waveform are collected for one full period of the fundamental frequency, starting at t = 0. If the interest is the fundamental frequency component in the Fourier series, then the following relation, derived from the Discrete Fourier Transform (DFT), may be used: √ N j2k𝜋 2∑ X= Xk •e N (1.3) N k=1 where N is the number of samples in one period, X is the phasor and X k is the waveform samples. The phasor expression (1.3) yields the correct value of the fundamental frequency component even in the presence of transient components. As expected, if the incoming signal frequency diﬀers from the nominal frequency, the magnitude and phase angle of the phasor will incur some error. However, as it turns out, the phasor error may be used to determine the frequency of the incoming signal [48]. Other sources of error may not be as useful and ought to be minimized as much as practicable. The processing is based on the use of the DFT and frequencies that are not harmonics of the fundamental frequency will introduce error in the phasor calculation. Moreover, any frequency term above the Nyquist rate (Nf s /2) will not be processed correctly by the DFT – this being the so-called aliasing phenomenon – which requires suitable data ﬁltering to bring it down to acceptable levels. It was recognized very early on in the area of synchrophasor applications that all the voltages and currents measured in a power system ought to be measured at exactly the same point in time to ensure the existence of a common reference for all the measured quantities. This represented a formidable challenge in network-wide applications, which was resolved satisfactorily only when fully ﬂedged satellite systems were deployed and opened to business for civilian applications. More than one option is available but the preferred system is the Navstar GPS satellite transmission – a system originally designed for navigational purposes which is capable of providing a common-access timing pulse at any location on earth with an accuracy of up to 1μs. The Navstar system is made up of a constellation of 24 satellites orbiting the earth at 10 000 miles. For accurate acquisition of the timing pulse, only one of the satellites needs to be visible to the antenna which would normally be mounted on the roof of a substation control house. Nevertheless, with the current array of 24 satellites, any point on earth is visible by more than one satellite all the time. Satellites systems are not the only option open to phasors synchronization. In fact, most of the communication systems normally available in electrical utilities – namely, leased lines, microwave transmission, AM radio broadcasts and ﬁbre optic links – have been used in the past with varying degrees of success. Apart from the dedicated ﬁbre optic links, the other non-satellite communications means have proved too coarse to be of any practical use in this application. Even multiplexed ﬁbre channels may not be suitable – they are reported to incur errors of up to 100 μs. Other satellite systems, such as the Geostationary Operational Environmental Satellite (GOES), have also been used for the purpose of phasors synchronization but their performance has not been suﬃciently accurate [48].

1.4 Phasor Measurement Units

A PMU may take many forms but it is widely agreed that its origins may be traced back to the ﬁeld of computer relaying and, in particular, to the pioneering work of Phadke and Thorp at the Virginia Polytechnic Institute in the early 1970s. Indeed, evidence of the earliest PMU prototypes that used GPS transmission to synchronize their sampling clocks can be found in the Power Systems Research Laboratory at Virginia Tech. It is reported that in the early 1980s the GPS receivers were quite expensive because GPS satellites were few in number and crystal clocks were used as part of the receivers’ circuitry to keep time accurately until the next satellite came into viewing. A full deployment of the satellite system and its opening to commercial use solved the logistical problems of having to allocate dedicated ﬁbre optic links to distribute timing pulses of suﬃcient accuracy and to rely on expensive crystal clocks inside the GPS receivers to keep accurate timing – the chip set of today’s GPS receivers is quite aﬀordable. A modern PMU uses one-pulse-per-second signals provided by the GPS receiver fed to a phase-locked oscillator (PLO) which is tasked to generate a sequence of high-speed timing pulses to sample waveforms from which synchrophasor information is extracted. A functional block diagram of the PMU and associated equipment is shown in Figure 1.34. The GPS receiver’s function is twofold: to provide the one-pulse-per-second signal to the PLO and to provide a time-tag to each extracted synchrophasor. The time-tag comprises the year, day, hour, minute and second, which may be provided with reference to the local time where the PMU is located or in Universal Time Coordinated (UTC) units. The PLO’s input from the GPS receiver is split up into a predeﬁned sequence of high-speed timing pulses for sampling of the analogue signals. These signals, which are derived from the secondary outputs of potential and current transformers, are conditioned with anti-aliasing and surge ﬁltering to remove from the incoming waveforms any trace of frequency components above the Nyquist rate. The microprocessor executes all the required DFT phasor calculations, and the timing message from the GPS receiver together with the sample number at the start of a data window is assigned to the calculated phasor as its identifying tag – a process that gives rise to a synchrophasor. A string of positive-sequence synchrophasors is transmitted to one or more remote collection

GPS receiver Analog inputs

Anti-aliasing filters

Phase-locked oscillator

16-bit A/D conv

Modems

Phasor microprocessor

Figure 1.34 Functional block diagram of the elements in a phasor measurement unit. The general structure is similar to many power system relays and digital fault recorders (©IEEE, 2006).

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1 Flexible Electrical Energy Systems

devices known as phasor data concentrators (PDC) over a dedicated communication line using modems. Note that for a purely sinusoidal waveform, the phasor expression (1.3) has the following representation: X X = √m •ej𝜑 2

(1.4)

where X is now termed synchrophasor, X m is the waveform amplitude and 𝜑 is its argument. Alternatively, the polar expression (1.4) may be expressed in rectangular form: X = X • cos 𝜑 + jX • sin 𝜑 (1.5) √ where X = Xm ∕ 2 is the rms value of the synchrophasor X. Owing to the growing use of synchrophasors in power systems state estimation, the most popular PMU measurements are synchrophasors of nodal voltages, branch currents and nodal current injections. This is elaborated on in Chapter 5.

1.5 Future Developments and Challenges Current research eﬀorts in the power electronics area concentrate on the development of modular, multilevel power electronic converters, SiC semiconductor valves, self-monitoring and fault-tolerant converters, converters with embedded processor agents and ICTs to add intelligence [7]. This would result in more eﬃcient, scalable, reliable and inexpensive converters with longer lifetimes and improved performances and, in the fullness of time, would produce transformer-less, grid-connected power converters. It is not diﬃcult to agree that current and future developments in the power electronics industry will drive, to a large extent, the direction and structure of future electric energy grids. These will impact all sectors of the electrical energy industry, namely generation, transmission and distribution. 1.5.1

Generation

Sustained growth of wind and PV solar installations is expected over the next decades. This include both MW-level installations and micro-generation for domestic and commercial applications. Wind power plants are likely to be large oﬀshore installations connected to mainland AC power grids in an asynchronous manner using VSC-HVDC links. The individual wind turbines may reach 10 MW capacity and incorporate superconductor materials. PV solar generators naturally produce electrical power in DC form and require a VSC for connection to the AC power grid. In any case, this is intermittent generation, much of it installed at the distribution system level, which may aﬀect adversely the operation of the high-voltage transmission system in a number of ways: for one, the system level power balance would become more diﬃcult to attain but

1.5 Future Developments and Challenges

also voltage control may become a signiﬁcant problem. Both issues are raising concerns among power system operators, since power exchange between countries ought to be free of frequency deviations. Hence, the search for measures to ameliorate present and future problems caused by intermittent power generation points in the direction of MW-level BESSs. As with PV generators, the BESSs connect to the AC power grid through a VSC. It is also likely that the conventional synchronous generators, those driven by steam and waterfalls, will be replaced by permanent magnet synchronous generators and connected to the power grid through VSCs. This would open the possibility of operating the power grid at frequencies higher than the customary 50 Hz or 60 Hz power frequencies, leading to more compact power transmission grids. A higher share of renewables in the generation mix calls for more advanced controllers than those we have at present, controllers able to mimic or even improve the operational behaviour of conventional power plants [49]. The very large penetration of distributed and renewable resources would require a mammoth eﬀort to develop new standards, addressing communication and protection issues, as well as the grid codes, imposing stringent connection requirements [50]. All this is poised to increase the share of non-synchronous generation in AC power grids, bringing new challenges to power system operators and power systems application software developers. It is in this tenor that this book is aiming to contribute, presenting new models, methods and control techniques embedded in software, with which to plan and operate the future electrical power systems, which will integrate more and more DC subsystems and components, connected to a diversity of voltage levels, from LV to UHV. 1.5.2

Transmission

It is quite clear that the current major challenges in power transmission are in the area of DC transmission as opposed to AC transmission, such as increasing the capacity of VSC-HVDC to enable higher voltage and power levels. The VSC-HVDC option stands at 2000 MW and ± 380 kV, as exempliﬁed by the INELFE and Ultranet VSC-HVDC transmission systems on the Spain–France border and in Germany, respectively – see Tables 2.3 and 2.4. As a yardstick, the classical HVDC option is currently rated at 7200 MW and ± 800 kV, as exempliﬁed by the Jinping-Sunan HVDC link in East China, spanning a distance of 2100 km – see Table 2.5. However, current limitations in the VSC-HVDC technology seem to be in making further progress in power electronic valves and converters as opposed to any lack of fundamental knowledge of the VSC-HVDC technology, at least as far as the point-to-point VSC-HVDC links are concerned. Indeed, the VSC-HVDC technology has reached such a good level of development that converting some of the existing long-distance, AC transmission corridors into point-to-point VSC-HVDC links may be a proﬁtable option, from the economic and technical vantages. A diﬀerent proposition is the multi-terminal HVDC technology, where the classical HVDC option does not present a challenge to the VSC-HVDC technology, owing to the intrinsic ﬂexibility of the former. However, practical problems still need to be resolved before fully ﬂedged multi-terminal VSC-HVDC technology becomes available [51].

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1 Flexible Electrical Energy Systems

For instance, a simple and cost-eﬀective DC circuit breaker has yet to be developed [52, 53]. In point-to-point HVDC links, short-circuit faults are cleared by AC switches. However, AC switches cannot be applied in multi-terminal VSC-HVDC systems without compromising the integrity of the neighbouring AC systems. Also, suitable mechanisms of power ﬂow control in meshed DC power grids still need to be realized. Having said that, it is likely that an option similar to the IPFC, designed for power ﬂow control in AC power grids, may be devised for power ﬂow control in DC power grids. Furthermore, as the DC power grid develops further, it is likely that diﬀerent voltage levels will co-exist and suitable DC power transformers, capable of handling large power throughputs, would need to be developed. The investigation of some of these outstanding issues in DC transmission may beneﬁt from using, as a springboard, the theory presented in this book. The material in the various chapters is entirely relevant to gain insight into the current and future issues that are likely to dominate the scene of DC power transmission circuits for the foreseeable future. One key technological development pertaining to power transmission is high temperature superconductivity (HTSC). It is thought that when this technology is fully developed, it will have a profound impact on both AC and DC transmission systems. In particular, DC circuits stand to beneﬁt from HTSC not only because of the near-zero power loss that HTSC cables incur but also because these cables may have an embedded current-blocking characteristic, which is temperature-triggered, hence resolving the outstanding issue of lack of DC circuit breakers. 1.5.3

Distribution

It is thought that low-voltage distribution networks will continue to experience vast improvements in a number of areas as a result of the widespread use of power electronics controllers and storage. Power quality is an area which is clearly beneﬁting from the availability of advanced power electronic equipment and control methods [2, 3]. However, there is more to come: the massive penetration of new agents with power electronic components, such as electric vehicles, distributed generation and storage, will lead to a massive number of distributed energy resources which can be controlled in a coordinated fashion to optimize the operation of utilities and large users [16]. It is highly likely, once power electronic converter prices decrease further, that power distribution systems which currently are three-phase radial AC systems will migrate to become multi-terminal, bipolar, VSC-DC systems [21, 54]. This would be due to higher energy throughputs, lower energy losses and smaller footprints of the DC equipment and cables [55]. On the DC side of the system, the conventional three-phase loads may be served in a more controllable manner while enabling the direct connection of DC DERs, such as PV battery packs, fuel cell stacks, PV stacks and EV recharging stations. Furthermore, if the amount of energy produced by the in-situ generation and storage is suﬃcient to meet the local demand plus the associated power losses, then the DC feeders may act as an independent microgrid, isolated from the HV utility supply, with the AC connection points taking the character of backup supply points [39]. In the meantime, an issue worth investigating, particularly in urban areas, is the suitability of current AC infrastructure for hosting DC power grids in order to reduce operating costs and to combat the growing problem of high short-circuit levels in AC power grids.

References

References 1 Hingorani, N.G. (1988). High power electronics and ﬂexible AC transmission sys-

tems. IEEE Power Engineering Review 8 (7): 3–4. 2 Hingorani, N.G. and Gyugyi, N. (1999). Understanding FACTS: Concepts and Tech3 4 5 6 7

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nology of Flexible AC Transmission Systems. Wiley–IEEE Press. Hingorani, N.G. (1995). Introducing custom power. IEEE Spectrum 32 (6): 41–48. Arrillaga, J. (1998). High Voltage Direct Current Transmission. IET. Miller, T.J.E. (1982). Reactive Power Control in Electric Systems. Wiley. Hingorani, N.G. (1993). Flexible AC transmission. IEEE Spectrum 30 (4): 44–45. Maza-Ortega, J.M., Acha, E., Garcia, S., and Gomez-Exposito, A. (2017). Overview of power electronics technology and applications in power generation, transmission and distribution. Journal of Modern Power Systems and Clean Energy 5, (4): 499–514. Gyugyi, L. (1994). Dynamic compensation of AC transmission lines by solid-state synchronous voltage sources. IEEE Transactions on Power Delivery 9 (2): 904–911. Asplund, G. (2000). Application of HVDC light to power systems enhancement. IEEE PES Winter Meeting 4: 2498–2503. Acha, E. and Castro, L.M. (2016). A generalized frame of reference for the incorporation of multi-terminal VSC-HVDC systems in power ﬂow solutions. Electric Power Systems Research 136: 415–424. Ekanayake, J.B., Jenkins, N., Liyanage, K. et al. (2012). Smart Grid: Technology and Applications. Wiley. Ribeiro, P., Johnson, B.K., Crow, M.L. et al. (2001). Energy storage systems for advanced power applications. Proceedings of the IEEE 89 (12): 1744–1756. Anaya-Lara, O., Jenkins, N., Ekanayake, J. et al. (2009). Wind Energy Generation: Modelling and Control. Wiley. Xiao, W. (2017). Photovoltaic Power Systems: Modelling, Design and Control. Wiley. J. Servotte, The Impact of a Distributed Battery Energy Storage System on Transmission and Distribution Power Grids, MSc Thesis, Tampere University of Technology, 2013. Acha, S. (2013). Modelling Distributed Energy Resources in Energy Service Networks. London: IET. P. Järventausta, P. Verho, J. Partanen and D. Kronman, “Finnish Smart Grids – A Migration From Version One to the Next Generation ”, 21st Int. Conf. & Exhib. on Electricity Distribution, CIRED, Frankfurt, Germany, 6–9 June 2011. Momoh, J. (2012). Smart Grid: Fundamentals of Design and Analysis. Wiley–IEEE Press. European Commission, “Smart Grids: Vision and Strategy for Europe’s Electricity Networks of the Future”, Directorate-General for Research, Sustainable Energy Systems, 2006. Z.K. Rather, Z. Chen and P. Th𝜙rgersen, “Challenges of Danish Power System and Their Solutions”, IEEE Int. Conf. on Power Systems Technology (POWERCON), 30 October–2 November 2012, Auckland, New Zealand. T. Kaipia, P. Salonen, J. Lassila and J. Partanen, “Possibilities of the Low Voltage DC Distribution Systems”, Nordic Distribution Automation Conference (NORDAC), Stockholm, August 2006.

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22 S.M. Amin and B.F. Wollenberg, “Towards a Smart Grid”, IEEE Power & Energy

Magazine, pp. 34–41, September–October 2005. 23 Miller, T.J.E. (1989). Brushless Permanent Magnet and Reluctance Motor Drives,

Monographs in Electrical Engineering. Oxford Press. 24 Acha, E., Agelidis, V.G., Anaya-Lara, O., and Miller, T.J.E. (2001). Power Electronic

Control in Electrical Systems, Newnes Power Engineering Series. Elsevier. 25 O. Törhönen, Beneﬁts of Main Reactor-Based SVC in Utility Applications, MSc The-

sis, Tampere University of Technology, 2016. 26 Gyugyi, L., Rietman, T.R., Edris, A. et al. (1995). The uniﬁed power ﬂow controller:

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33 34 35 36 37 38 39

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a new approach to power transmission control. IEEE Transactions on Power Delivery 10 (2): 1085–1097. Y.S. Han, I.Y. Suh, J.M. Kim, H.S. Lee, J.B. Choo and B.H. Chang, “Commissioning and Testing of the KangJin UPFC in Korea”. Paper presented at the 2004 CIGRE Conference, Paris, session B4–211, August 2004. Gyugyi, L., Kalyan, K., and Schauder, C.D. (1999). The interline power ﬂow controller concept: a new approach to power ﬂow management in transmission systems. IEEE Transactions on Power Delivery 14 (3): 1115–1123. Miller, N.W., Zrebiec, R.S., Hunt, G., and Deimerico, R.W. (1996). Design and commissioning of a 5 MVA, 2.5 MW-h battery energy storage system. In: Proc. of IEEE Transmission and Distribution Conference, 339–345. “AES Combines Advanced Battery-Based Energy Storage with a Traditional Power Plant”, Available: https://www.businesswire.com/news/home/20120503005472/en/ AES-Combines-Advanced-Battery-Based-Energy-Storage-Traditional 25.11.2018. Acha, E., Fuerte-Esquivel, C.R., Ambriz-Perez, H., and Angeles-Camacho, C. (2004). FACTS Modelling and Simulation in Power Networks. Wiley. Jones, K.M. and Kennedy, M.W. (1973). Existing AC transmission facilities converted for use with DC. In: IEE Conf. on High-Voltage DC and/or AC Power Transmission, 253–260. London. Kundur, P. (1993). Power System Stability and Control. McGraw-Hill. Jenkins, N., Allan, R., Crossley, P. et al. (2000). Embedded Generation. IET. A. Pazynych, A study of the harmonic content of distribution power grids with distributed PV systems, MSc Thesis, Tampere University of Technology, 2014. M.S. Rahman, Modelling and Simulation of Combined PV-BESS systems, MSc Thesis, Tampere University of Technology, 2017. Bollen, M.H. (1999). Understanding Power Quality Problems: Voltage Sags and Interruptions. Wiley–IEEE Press. Anaya-Lara, O. and Acha, E. (2002). Modeling and analysis of custom power systems by PSCAD/EMTDC. IEEE Transactions on Power Delivery 17 (1): 266–272. Romero-Ramos, E., Gomez-Exposito, A., Marano-Marcolini, A. et al. (2011). Assessing the load ability of active distribution networks in the presence of DC controllable links. IET Generation, Transmission & Distribution 5 (11): 1105–1113. Maza-Ortega, J.M., Gomez-Exposito, A., Barragan-Villarejo, M. et al. (2012). Voltage source converter-based topologies to further integrate renewable energy sources in distribution systems. IET Renewable Power Generation 6 (6): 435–445. J. Lohjala, T. Kaipia, J. Lassila, J. Partanen, P. Järventausta and P. Verho, “Potentiality and Eﬀects of the 1 kV Low Voltage Distribution System”, IEEE Future Power Systems Conference, 18 November 2005, Amsterdam, Netherlands.

References

42 Mohan, N., Undeland, T.M., and Robbins, W.P. (2003). Power Electronics, Converters,

Applications, and Design. Wiley. 43 Suntio, T., Messo, T., and Puukko, J. (2018). Power Electronic Converters: Dynamics

and Control in Conventional and Renewable Energy Applications. Wiley–VCH. 44 E. Acha, T. Rubbrecht and L.M. Castro, “Power Flow Solutions of AC/DC

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Micro-grid Structures”, 19th Power Systems Computation Conference (PSCC’16), Genoa, Italy, 19–24 June 2016. A.G. Phadke and J.S. Thorp, “History and Applications of Phasor Measurements”, Power Systems Conference and Exposition, 29 October–1 November 2006, Atlanta, GA. Cayemitee, F.I. (2000). Contribution of the Theory of Complex Numbers to the Field of Electrical Engineering Education and Practice. NY, USA: Columbia University. Phadke, A.G. and Thorp, J.S. (2017). Synchronized Phasor Measurements and Their Applications. Springer International Publishing AG. Phadke, A.G. (1993). Synchronized phasor measurements in power systems. IEEE Computer Applications in Power 102 (2): 10–15. M. Seyedi and M. Bollen, “The Utilization of Synthetic Inertia From Wind Farms and Its Impact on Existing Speed Governors and System Performance”, Vindforsk Project Report V-369, Part 2. ELFORSK, Stockholm, Sweden, 2013. Mohseni, M. and Islam, S.M. (2012). Review of international grid codes for wind power integration: diversity, technology and a case for global standard. Renewable Sustainable Energy Review 16 (6): 3876–3389. P. Rodriguez and K. Rouzbehi, “Multi-terminal DC grids: challenges and prospects”, Journal of Modern Power Systems and Clean Energy, Springer Link, Open Access, 8 July 2017. Y.L. Li, X.J. Shi and F. Wang, “DC Fault Protection of Multi-terminal VSC-HVDC System With Hybrid DC Circuit Breaker”, Proc. 2016 IEEE Energy Conversion Congress and Exposition (ECCE’16), Milwaukee, WI, USA, 18–22 September 2016. Liu, X., Wang, P., and Loh, P.C. (2011). A hybrid AC/DC micro-grid and its coordination control. IEEE Transactions on Smart Grid 2 (2): 278–286. Antoniou, D., Tzimas, A., and Rowland, S.M. (2015). Transition from alternating current to direct current low voltage distribution networks. IET Generation, Transmission & Distribution 9 (12): 1391–1401. Justo, J.J., Mwasilu, F., Lee, J., and Jung, J.W. (2013). ACM-microgrids versus DC-microgrids with distributed energy resources: a review. Renewable and Sustainable Energy Reviews 24 (C): 387–405.

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2 Power Electronics for VSC-Based Bridges 2.1 Introduction Over the past decade there have been huge developments in power semiconductors. The main features of such developments are an increase in power capability, easier control, higher switching frequencies, reduced switching losses, improved packaging and cost reduction. These technology improvements have resulted in power semiconductors that resemble, more and more, ideal switches. One consequence is that more simpliﬁed models and approaches are justiﬁed when studying power semiconductor circuits, particularly if they are aimed at power grid applications. New converter topologies and applications have emerged, such as voltage source converters (VSCs) of various kinds, which are already ubiquitous in electrical power networks. They have turned out to be key elements in the harnessing of oﬀshore wind power, solar power and the enablement of energy storage with fast speed of response. The ﬁrst part of this chapter deals with the basic power semiconductors and their main features, while the fundamentals of VSCs and the most popular conﬁgurations are explained in the second part. The various VSC multilevel topologies and their advantages, compared to the traditional two-level VSCs, are presented here. The last part of the chapter is devoted to the application of VSC topologies to HVDC systems. Furthermore, the chapter contrasts the main advantages of the VSC-HVDC technology and the classical, CSC-HVDC technology, and gives a comprehensive list of the current VSC-HVDC installations around the world, their ratings, year of installation and type of converters.

2.2 Power Semiconductor Switches The technology of semiconductor materials intended for use in high-current high-power applications continues to make steady progress in terms of developing a variety of switches that are able to withstand larger breakdown voltages, lower on-state losses, faster turn-on and turn-oﬀ capabilities and to handle larger amounts of power. However, these properties tend to be mutually exclusive as opposed to mutually inclusive, and the various kinds of switches are endowed with only a varying degree of such physical

VSC-FACTS-HVDC: Analysis, Modelling and Simulation in Power Grids, First Edition. Enrique Acha, Pedro Roncero-Sánchez, Antonio de la Villa Jaén, Luis M. Castro and Behzad Kazemtabrizi. © 2019 John Wiley & Sons Ltd. Published 2019 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/acha_vsc_facts

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Anode (A)

Cathode (K)

Anode (A)

Cathode (K) Thyristor

Diode

Gate (G) Anode (A)

Cathode (K)

Gate (G) Anode (A)

Cathode (K)

Gate turn-off (GTO) thyristor

Triac Collector (C)

Collector (C)

Gate (G) Gate (G) Emitter (E) Insulated-gate bipolar transistor (IGBT)

Emitter (E) Metal-oxide field effect transistor (MOSFET)

Figure 2.1 Electronic symbols of the most popular semiconductor switches.

attributes – for all the progress made, there is no single semiconductor device that dominates and is ﬁt for all applications. We are thus in the realm of the elusive ideal semiconductor switch. The main semiconductor switches used in power electronics applications and their symbols are shown in Figure 2.1. In general, there is a trade-oﬀ between break-down voltages and on-state losses, and in the bipolar-type devices there is also a trade-oﬀ between on-state losses and switching speeds. The choice of a particular power semiconductor device thus tends to be application-dependent, and more often than not a number of individual semiconductor switches of the same kind are combined to make up series and parallel combinations in order to meet speciﬁed voltage and current levels for the bridge. The exception rather than the rule is the diode, which combines rather well with a number of the switching valves either to provide some form of protection or to aid the process of commutation. The combined symbols shown in Figure 2.2 are quite popular in the power electronics literature. The semiconductor valves can be classiﬁed according to their switching characteristics: 1. Uncontrolled. The on and oﬀ states in these devices are governed by the external circuit operating conditions. It is possible that the diode is the only device in this category. The diode is a two-terminal device, as indicated by its electronic symbol in Figure 2.1, in which these terminals are termed as anode (A) and cathode (K), respectively. 2. Semi-controlled. If provisions are made for a third terminal in the device, which is termed as a gate (G), then it becomes possible to use this terminal to inject an

2.2 Power Semiconductor Switches

C G A

K

G

E (a)

(b)

Figure 2.2 Compound symbols. (a) GTO with series diode. One limitation of the GTO thyristor is its reduced reverse break-down voltage in comparison with a conventional thyristor, a drawback that can be improved on by using a series diode. (b) IGBT with anti-parallel diode.

avalanche of electrons at predetermined time periods in order to accurately control the turning on of the device. The avalanche of electrons through the gate takes the form of electric pulses of current or light and in many ways may be thought of as a catalytic process. As with the diode, no such control exists for the turning oﬀ of the device, which is governed by the external circuit operating conditions. The thyristor and the triac belong to this category. 3. Fully controlled. An extended gate circuit may be used not only to inject positive pulses of current to control the turning on of the device but also to enable the injection of negative pulses of currents in order to turn the device oﬀ before it reaches its natural turning-oﬀ point. The gate turn-oﬀ (GTO) thyristor embodies such a description rather well, and at the macroscopic level it can be viewed as a thyristor with added functionality. This is an area in which power semiconductor technology has made great progress over the last quarter of a century, during which a great many devices have been improved as regards their design and operational capabilities, although new devices have also emerged. In addition to the current driven devices, such as the GTO, voltage driven devices are widely available in the form of power transistors, such as the bipolar junction transistor (BJT) and the metal-oxide-semiconductor ﬁeld-eﬀect transistor (MOSFET). In the quest for the ideal semiconductor device, attention has turned to hybrid devices, such as the insulated-gate bipolar transistor (IGBT) and the MOS-controlled thyristor (MCT).

2.2.1

The Diode

The diode is the most basic electronic switch whose on- and oﬀ-states are determined by the power circuit. The electrical symbol of the diode is shown in Figure 2.3a, while its ideal steady-state i-v characteristic is plotted in Figure 2.3b. The diode is forward-biased when the applied voltage vAK is zero and the anode current is positive. When the voltage vAK becomes negative, the diode does not conduct current through it and exhibits open circuit behaviour. Nevertheless, the actual i-v characteristics depicted in Figure 2.3c show that a positive voltage across the diode (the forward voltage drop) is necessary if the diode is to begin to conduct.

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A

iA iA

iA On

+ vAK

Off vAK

–

vAK

K (a)

(b)

(c)

Figure 2.3 Diode (© Wiley, 2002), (a) electrical symbol; (b) ideal i-v characteristics; and (c) i-v characteristics. Figure 2.4 Diode reverse-recovery time (© Wiley, 2002).

iA On

Off t trr

The switch-on and switch-oﬀ processes can be considered as instantaneous if the ideal model is used. However, when the real model is considered, the diode has a dynamic response at switch-oﬀ: the current becomes negative for a time interval, known as reverse-recovery time t rr , before becoming completely zero, as shown in Figure 2.4. During the reverse-recovery time, the current can produce overvoltages in inductive circuits [1]. The reverse-recovery time is a key factor in high-frequency applications where fast-recovery diodes, which guarantee a small t rr , are normally used. Schottky diodes are used in applications which require a low forward voltage drop: these diodes typically have a voltage drop of 0.3 V when forward-biased [2]. Line-frequency diodes have larger reverse-recovery times, signifying that they are not suitable for high-frequency applications and are normally used in grid applications, which require only a frequency of 50 Hz or 60 Hz. 2.2.2

The Thyristor

The thyristor is an electronic switch whose turn-on state is controlled, but the turn-oﬀ is determined by the power circuit to which the thyristor is connected. It is, therefore, a semi-controlled device which has three terminals: anode (A), cathode (K) and gate (G). Although the most well-known thyristor is the silicon-controlled rectiﬁer (SCR), other

2.2 Power Semiconductor Switches

A

iA iA

iA

On +

On to Off

Off

vAK

G

vAK

–

vAK

K (a)

(b)

(c)

Figure 2.5 Thyristor (a) electrical symbol of the SCR; (b) ideal i-v characteristics; and (c) actual i-v characteristics.

electronic devices are included in the group of thyristors, such as the GTO thyristor or the MCT, among others. Figure 2.5a shows the electrical symbol of an SCR, while Figures 2.5b,c show the ideal and real i-v characteristics, respectively. The SCR starts to conduct if the voltage vAK is positive, and a positive pulse of current is simultaneously applied to the gate. Once the SCR has begun to conduct, the current pulse applied to the gate is no longer necessary. When the anode current attempts to become negative, the SCR turns oﬀ and the current is zero. In order to switch the SCR on again, it is necessary to repeat the process of applying a positive current pulse to the gate. The SCR undergoes a forward voltage drop when it starts to conduct, as shown in Figure 2.5c. The value of this voltage typically varies between 1 V and 3 V [1]. Unlike the diode, the SCR is able to block forward and reverse voltages vAK , and the rating values of these voltages are normally the same. Several types of thyristors exist depending on the diﬀerent applications. The two best known types are: • Phase-control thyristors. These are normally used as rectiﬁers of the grid voltage and in the control of large DC motors. They are slow devices and do not therefore support large voltage- or current-time derivatives. They are manufactured to conduct an average current of up to 4 kA and to block voltages of up to 7 kV [1]. • Light-triggered thyristors. Rather than applying a current pulse to the gate in order to start to conduct, these thyristors are triggered by means of a light pulse that is transmitted by means of optic ﬁbre. The main advantages of these thyristors are: (i) galvanic isolation between the control circuit and the power circuit to which the thyristor is connected; and (ii) control-circuit immunity to electromagnetic interferences (EMI) since light signals are not aﬀected by them [3]. This galvanic isolation permits the series connection of the various thyristors required in high-voltage applications. As the potential of each thyristor increases with the numbers of devices, it is diﬃcult to trigger these devices when using conventional thyristors. 2.2.3

The Bipolar Junction Transistor

The BJT is a current-controlled device which can operate in three diﬀerent modes: cut-oﬀ mode, active mode and saturation mode. The active mode is used only if the BJT

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C

iC iC + vCE

iB

B

On

– Off vCE

E (a)

(b) C

iC iB4 I

iB3

B

iB2 iB = 0

vCE(SAT) (c)

iB1 vCE

E (d)

Figure 2.6 Bipolar transistor (NPN) (© Wiley, 2002) (a) electrical symbol; (b) ideal i-v characteristics; (c) actual i-v characteristics; and (d) Darlington topology.

is used as a linear device, i.e. ampliﬁer applications, whereas the cut-oﬀ and saturation modes are used for switching applications [4]. Figure 2.6a shows the electrical symbol of an NPN BJT: the three terminals of the transistor are the emitter (E), the collector (C) and the base (B). The ideal and actual i-v characteristics are plotted in Figure 2.6b,c, respectively. The oﬀ-state is achieved when the BJT operates in the cut-oﬀ mode, which implies that the base current is zero. When the base current is suﬃciently large, the transistor operates in the saturation mode, which determines the on-state. The condition needed to saturate the transistor is: I IB > C (2.1) 𝛽 where the parameter 𝛽 is the DC current gain whose value is normally between 5 and 10 for high-power transistors, unlike a signal transistor whose DC current gain is usually 100. The transistor undergoes a drop voltage vCE(SAT) when conducting, as shown in Figure 2.6c, which typically varies between 1 V and 2 V. The dissipated power during the on-state is therefore small. In order to increase the DC current gain, Darlington conﬁgurations using two or more transistors can be employed, as shown in Figure 2.6d, and can be built either using a discrete transistor or as an integrated device. This avoids the need to use large values of the current base to saturate the transistor. In the past, power transistors were frequently used in power applications. However, increased power capabilities and the ease of control of new devices signify that they have gradually been substituted by MOSFETs and IGBTs.

2.2 Power Semiconductor Switches

2.2.4

The Metal-Oxide-Semiconductor Field-Eﬀect Transistor

The MOSFET is a voltage-controlled transistor device that is capable of operating at high switching frequencies, i.e. up to the range of megahertz. It cannot handle as much power as the BJT and is therefore used in low-power applications. Its voltage ratings are in the range of 1000–1500 V for small current values and current ratings of up to 600 A for small voltage values. Figure 2.7a shows the electrical symbol of an N-channel MOSFET. It contains three terminals: the gate (G), the drain (D) and the source (S). The ideal i-v characteristics of the MOSFET are shown in Figure 2.7b, whereas the actual i-v characteristics are depicted in Figure 2.7c. The oﬀ-state is determined by a gate-to-source voltage vGS that is lower than a threshold value, and it is necessary to continuously apply a voltage vGS that is higher than an appropriate value in order to achieve the on-state. When the MOSFET is in the on-state, it can be modelled as a small resistance between the drain and the source rDS(ON) of a few milliohms. As the gate impedance is almost inﬁnite, the power required to control the on- and oﬀ-states is very small. 2.2.5

The Insulated-Gate Bipolar Transistor

The IGBT combines some of the advantages of the MOSFET and the BJT. On the one hand, it is a voltage-controlled device and has as high a gate impedance as the MOSFET, signifying that it is easy to implement the control circuit of the switch and it requires little power. The IGBT has, on the other hand, similar on-state characteristics to those of the BJT, with a small forward drop voltage. The electrical symbol of the IGBT is shown in Figure 2.8a: the three terminals are the gate (G), the collector (C) and the emitter (E). Figure 2.8b shows the ideal i-v characteristics: unlike the BJT or the MOSFET, the IGBT can be designed to block negative voltages. The typical i-v characteristics are plotted in Figure 2.8c. BJTs have progressively been replaced by IGBTs in applications such as AC motor drives. They can block voltages of up to 3.3 kV and their current ratings are as large as 1200 A. [5]. Commercial IGBTs can operate at switching frequencies of up to 80 kHz [6]. 2.2.6

The Gate Turn-Oﬀ Thyristor

Unlike the conventional SCR in which only the on-state can be controlled, the GTO thyristor is a fully controlled device. The symbol of the GTO is shown in D

iD

iD iC + vDS

G

vGS3 On

vGS2

– +

vGS

–

vGS1 Off

(a)

vGS = 0

vDS

S

(b)

vDS

(c)

Figure 2.7 N-channel MOSFET (©Wiley, 2002) (a) electrical symbol; (b) ideal i-v characteristics; and (c) actual i-v characteristics.

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C

iC iC

G

iC On

+ vCE

+ vGS

vGE4 vGE3 vGE2 vGE1

Off vCE

–

vCE

– E (a)

(b)

(c)

Figure 2.8 An IGBT (© Wiley, 2002) (a) electrical symbol; (b) ideal i-v characteristics; and (c) actual i-v characteristics.

A

iA iA

+

On to Off and Off to On

iA

vAK

G

vAK

– K (a)

(b)

vAK

(c)

Figure 2.9 A GTO (© Wiley, 2002) (a) electrical symbol; (b) ideal i-v characteristics; and (c) actual i-v characteristics.

Figure 2.9a, whereas the ideal and actual i-v characteristics are plotted in Figure 2.9b,c, respectively. As in the case of the SCR, the on-state of the GTO is obtained by applying a positive current pulse to the gate when the voltage vAK is positive. Furthermore, the GTO can be switched oﬀ with a negative gate current for a short period of time (the turn-oﬀ time of the GTO, which is typically a few microseconds). However, the amplitude of the gate current needed to turn oﬀ the GTO can be up to one third of the anode current and complex gate control circuits are therefore required to process this current value. The GTOs can block negative voltages and are slow devices in comparison with other switches such as IGBTs or MOSFETs, i.e. the switching frequency is lower than 1 kHz [3], while the maximum voltage and current ratings for commercial GTOs are 6.5 kV and 4.5 kA. 2.2.7

The MOS-Controlled Thyristor

The MCT is a device which combines features of MOS and thyristor technologies. It is a voltage-controlled switch whose electrical symbols are shown in Figure 2.10a,b for an N-channel MCT and a P-channel MCT, respectively. The i-v characteristics are very similar to those of the GTO, as can be seen in Figure 2.10b,c, with a small voltage drop in the on-state. Nonetheless, the on- and

2.3 Voltage Source Converters

A

A

N-MCT

G

P-MCT

K (b)

K (a)

iA

G

On to Off and Off to On

iA

vAK

(c)

vAK

(d)

Figure 2.10 An MCT (© Wiley, 2002) (a) N-channel MCT electrical symbol; (b) P-channel MCT electrical symbol; (c) ideal i-v characteristics; and (d) actual i-v characteristics.

oﬀ-states are controlled by means of short voltage pulses that are applied to the gate, unlike the GTO which is controlled by the gate current. Moreover, the MCT is capable of operating at higher switching frequencies than the GTO. These are the main advantages of the MCT over the GTO. Commercial MCTs have voltage ratings of 1500 V at current ratings in the range of 50 A and a few hundred amperes [1]. 2.2.8

Considerations for the Switch Selection Process

In the selection process of a switch, the device chosen depends on the type of application and the main factors to take into consideration are the voltages and currents required, necessary switching frequency, maximum acceptable value of power losses, device price and degree of controllability.

2.3 Voltage Source Converters HVDC systems use power electronic converters to control the electrical energy ﬂow between the DC side and the AC side in a bidirectional manner. Two technologies can be employed [7]: 1. Current source converters use a DC current source as their input source. The load current can be controlled and the load voltage depends on the nature of the load. Mercury-arc rectiﬁers and thyristors have been employed in this type of converter using line-commutated control schemes.

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2. In voltage source converters, the input source is a voltage source, the voltage across the load can be controlled and the current through the load depends on its nature. The power switches that are most frequently used in VSCs are IGBTs and GTOs. These converters are self-commutated. CSCs have been used in HVDC systems since the mid-1950s; they represent a mature technology [8]. With the recent advances in power semiconductors, VSCs are on the ascendency over the CSC technology. The following sections explain the working principles of the VSC. According to [1], it is possible to classify VSCs as follows: 1. Pulse width-modulated VSCs. In these converters, the output voltage is controlled by using a PWM scheme in order to obtain an output voltage that is close to a sinusoidal waveform with a small distortion, keeping the voltage constant on the DC side. The switching frequency used in these converters is much higher than the fundamental frequency of the output voltage. 2. Square-wave VSCs. The VSC can be operated in square-wave mode. In this mode, the switching frequency is equal to the fundamental harmonic of the output voltage and the magnitude of this voltage is controlled by modifying the value of the DC-side voltage. In these converters, the output voltage is a square waveform with odd harmonic components and all the switches on the converter have a duty ratio of 50%. 3. Single-phase VSCs operated by phase-shift control. These can be operated in squarewave mode and the output voltage is controlled by overlapping the two leg voltages at a given angle. As in the previous case, since the converter works in square-wave mode, only odd harmonics are present in the output voltage and the switching frequency is equal to the fundamental frequency of the output voltage.

2.3.1 Basic Concepts of Pulse Width Modulated-Output Schemes and Half-Bridge VSC PWM provides a method with which to control the amplitude of the output voltage in a VSC, thus maintaining a low distortion of that voltage. In order to fully understand the PWM process, some concepts will be deﬁned ﬁrst. Any PWM scheme uses two waveforms to operate: a carrier signal, which is usually a triangular waveform in the case of the VSC control, whose frequency is the switching frequency of the converter, f sw , and a modulating signal, whose frequency f 1 is the fundamental frequency of the VSC output voltage. In the case of VSCs, the modulating signal is a sinusoidal waveform. ̂mod and V ̂tri be the amplitudes of the modulating signal and the carrier, respecLet V tively. The amplitude modulation index is deﬁned as: ma =

̂mod V ̂tri V

(2.2)

while the frequency modulation index is: mf =

fsw f1

(2.3)

2.3 Voltage Source Converters

Figure 2.11 Single-phase half-bridge VSC.

+ Vdc 2 –

+ Vdc –

T1 C+

2 –

A

0

c + Vdc

D1

T2 C–

D2

Let us consider a basic topology of the VSC: the single-phase half-bridge VSC, which is plotted in Figure 2.11, in which two capacitors are connected in series and split the DC voltage of the input in half. The VSC also contains a leg with two switches, T 1 and T 2 . The output voltage vo is measured between the terminal A and the midpoint of the two series capacitors. The on- and oﬀ-states of the two switches are determined by comparing a modulating signal vmod (sinusoidal) and a carrier signal vtri (triangular). Bearing in mind that the two switches of the leg cannot be simultaneously on since this would cause short-circuits, and they cannot be oﬀ since the output voltage must be deﬁned with regard to the VSC V circuit, the valve T 1 will be on when vmod > vtri , with vo = 2dc , and the valve T 2 will be V switched on when vmod < vtri , and therefore vo = − 2dc . When the PWM process is applied to control the output voltage of the VSC, the waveforms plotted in Figure 2.12 are obtained. Figure 2.12a shows the sinusoidal signal and the carrier waveform when ma = 0.8 and mf = 15: the frequency of the carrier signal is the switching frequency of the valves T 1 and T 2 . It should be noted that if ma ≤ 1, then the amplitude of the fundamental harmonic of the output voltage is [1]: Vdc (2.4) 2 This can be seen in Figure 2.12b. Furthermore, if the frequency modulation index is suﬃciently high (typically mf ≥ 9), the frequencies f h of the harmonic components of the output voltage are distributed around the switching frequency and its integer multiples, as shown in Figure 2.12c, as follows: ̂o1 = ma V

fh = (mf ± k)f1

(2.5)

with k = 1, 3, 5, … when mf = 0, 2, 4, … k = 2, 4, 6, … when mf = 1, 3, 5, …

(2.6)

What is more, the amplitudes of the diﬀerent harmonic components are almost independent of mf . If the frequency modulation index is chosen to be an odd integer number, the resulting output voltage will have odd symmetry and no even harmonic will be present. When ma ≤ 1, the PWM process has a linear behaviour according to (2.4), as shown in Figure 2.12b. Nevertheless, the amplitude modulation index can be greater than 1.0, and

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2 Power Electronics for VSC-Based Bridges

vmod

Vˆ tri

(a)

0

–Vˆ tri 1

Fundamental harmonic

(b)

Vdc/2

vo

0

–Vdc/2

(c)

Time fsw

0

T1

Time

1 0.8

Vˆ o 0.6 Vdc/2 0.4 fsw – 2f1 fsw + 2f1

0.2 0

f1

fsw – 4f1

fsw

fsw + 4f1

2fsw – f1 2fsw + f1

2fsw – 3f1

2fsw + 3f1

Frequency Figure 2.12 PWM method for a half-bridge VSC (© Wiley, 2002), (a) modulating and carrier waveforms for ma = 0.8 and mf = 15; (b) output voltage of the VSC v o (−) and the fundamental harmonic (− −); and (c) normalized harmonic components of the output voltage.

the PWM operates in the so-called overmodulation region. In this case, the amplitude of the fundamental harmonic does not vary linearly with ma and low-frequency components are present in the output voltage. Figure 2.13 shows the results obtained when using a PWM scheme with ma = 2 and mf = 15. As shown in Figure 2.13a, the amplitude of the modulating signal is twice the amplitude of the carrier waveform. Nevertheless, the amplitude of the fundamental harmonic does not vary linearly, as Figures 2.13b,c show. Furthermore, the harmonic spectrum does not satisfy the relationship (2.5) previously explained, and low-order odd harmonics are now present in the output voltage.

2.3 Voltage Source Converters

(a)

2

vmod

vtri

1 0 –1 –2

1 (b)

Time

fsw

Fundamental harmonic Vdc/2

vo

0

–Vdc/2 0

T1

Time

(c) 1 Vˆ o Vdc/2

0.5

0

1

3

5

7

9

11 13 15 17 Harmonic order h

19

21

23

25

27

Figure 2.13 PWM scheme operating in the overmodulation region, (a) modulating and carrier waveforms for ma = 2 and mf = 15; (b) output voltage of the VSC v o (−) and the fundamental harmonic (− −); and (c) normalized harmonic components of the output voltage (© Wiley, 2002)

If the value of the amplitude modulation index continues to increase, the number of intersections between the modulating signal and the carrier waveform is reduced. These intersections are eventually reduced to the zero crossing instants of the modulating signal, with a maximum normalized value of the amplitude of the fundamental component equal to 4∕𝜋 . In this situation, the VSC is working in the square-wave operation mode and each switch of the leg is turned on the 50% of the fundamental period of the output voltage. Figure 2.14 shows the evolution of the normalized amplitude of the fundamental

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2 Power Electronics for VSC-Based Bridges

4 π 1

Vˆo

1

Vdc/2

0

1

0 Linear region

3.24 Overmodulation region ma

Square wave

Figure 2.14 Normalized amplitude of fundamental component of the output voltage vs. ma , for mf = 15.

component of the output voltage with ma . This evolution has been divided into three zones: the linear region, the overmodulation region and the square-wave operation. When the VSC operates using the square-wave scheme, the output frequency is the fundamental frequency and the output voltage is a square wave with a duty cycle of 50%, as shown in Figure 2.15a. The harmonic spectrum can be obtaining by using Fourier analysis and the amplitudes of the diﬀerent components are: ̂oh ̂o1 V 4 V 1 = = , h = 1, 3, 5, 7, … ; Vdc∕2 π V h ̂o1

(2.7)

The diﬀerent harmonic components of the output voltage can be seen in Figure 2.15b. When the VSC works using the square-wave scheme, the switching frequency is the fundamental frequency of the output voltage, which may be an advantage in high-power applications in which the power devices cannot commutate at high frequencies. 2.3.2

Single-Phase Full-Bridge VSC

The scheme of the single-phase full-bridge VSC is depicted in Figure 2.16. In this case, the VSC comprises two legs (A and B) and the output voltage is obtained as vo = vAN − vBN . The full-bridge VSC allows a maximum output voltage equal to the voltage of the DC side, unlike the half-bridge VSC whose maximum output voltage is half of the DC voltage. There are two PWM alternatives with which to control the full-bridge VSC: PWM with bipolar switching and PWM with unipolar switching.

2.3 Voltage Source Converters

(a) Fundamental harmonic Vdc/2 vo

0 –Vdc/2 0

(b)

T1

Time

1.5

1 Vˆ o Vdc/2 0.5

0

0

1

3

5

7

9 11 13 Harmonic order h

15

17

19

21

Figure 2.15 Square-wave operation of the VSC, (a) output voltage of the VSC vo (−) and the fundamental harmonic (− −); and (b) normalized harmonic components of the output voltage. Figure 2.16 Single-phase full-bridge VSC.

T1

T3 D1

D3

A

B

+ Vdc –

c T4

T2 D4

D2 N

2.3.2.1

PWM with Bipolar Switching

In this scheme, the switches are gathered in pairs (T 1 ,T 2 ) and (T 3 ,T 4 ) and their onand oﬀ-states are controlled by means of the intersection of one modulating signal with a carrier signal. When vmod > vtri , both members of the pair (T 1 ,T 2 ) are on while both members of the pair (T 3 ,T 4 ) are oﬀ, resulting in vAN = V dc , vBN = 0 and vo = V dc . On the contrary, when vmod < vtri , the switches (T 1 ,T 2 ) are both oﬀ and the devices (T 3 ,T 4 ) are on and, therefore, vAN = 0, vBN = V dc and vo = − V dc . Figure 2.17a shows the sinusoidal signal and the carrier waveform when ma = 0.8 and mf = 15. Since ma ≤ 1, the VSC operates in the linear region and the amplitude of the

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2 Power Electronics for VSC-Based Bridges

(a)

vmod

Vˆ tri

0

–Vˆ tri 1

Time fsw Fundamental harmonic

(b)

Vdc

vo

0

–Vdc 0 (c)

T1

Time

1 0.8

Vˆ o 0.6 Vdc 0.4 fsw – 2f1

0.2 0

f1

fsw – 4f1

fsw

fsw + 2f1

fsw + 4f1

2fsw – f1 2fsw + f1

2fsw – 3f1

2fsw + 3f1

Frequency Figure 2.17 PWM with bipolar switching for a full-bridge VSC (© Wiley, 2002), (a) modulating and carrier waveforms for ma = 0.8 and mf = 15; (b) output voltage of the VSC v o (−) and the fundamental harmonic (− −); and (c) normalized harmonic components of the output voltage.

fundamental harmonic of the output voltage is: ̂o1 = ma Vdc V

(2.8)

Figure 2.17b shows the output voltage and the fundamental component of this voltage and it will be noted that this component is proportional to ma . In the bipolar switching scheme, the output is switched from V dc to −V dc , as shown in Figure 2.17b. The harmonic components are plotted in Figure 2.17c: the distribution of these components

2.3 Voltage Source Converters

also fulﬁls (2.5) and (2.6), as in the case of the half-bridge VSC operated with PWM in the linear region. 2.3.2.2

PWM with Unipolar Switching

If the unipolar PWM method is used, the voltage output is switched from either V dc to zero or −V dc to zero. This scheme is therefore called PWM with unipolar switching. In this scheme, there are two modulating signals,vmod and −vmod , with which to individually control the switches of the legs A and B, signifying that the modulating signals are shifted 180∘ with regard to each other. Recall that the switches of the pairs (T 1 ,T 4 ) and (T 3 ,T 2 ) cannot be simultaneously on, and that when vmod > vtri the switch T 1 is on and vAN = V dc and when vmod < vtri the switch T 4 is on, and therefore vAN = 0. Similarly, when −vmod > vtri the switch T 3 is on with vBN = V dc , and when −vmod < vtri the switch T 2 is on, and therefore vBN = 0. The comparison process between the modulating signals and the carrier signal is shown in Figure 2.18a for ma = 0.8 and mf = 15. The output voltage is shown in Figure 2.18b: since ma < 1, the amplitude of the fundamental component of the output voltage again fulﬁls (2.8). The main advantage of this PWM scheme is that the eﬀective switching frequency is twice the actual switching frequency. This is owing to the phase-shift of 180∘ of the modulating signals [3]. The harmonic spectrum of the output voltage shown in Figure 2.18c exhibits the fundamental component, and the switching components around twice the switching frequency and its even multiples. 2.3.2.3

Square-Wave Mode

As in the case of the half-bridge VSC, the full-bridge VSC can be operated in order to obtain a square waveform with odd harmonic components in which all the switches on the VSC have a duty ratio of 50%. The magnitude of the ﬁrst harmonic of the output voltage is: ̂o1 = 4 Vdc (2.9) V π whereas the magnitudes of the remaining harmonic components can be obtained as: ̂oh V 1 = , h = 1, 3, 5, 7, … h ̂ Vo1

(2.10)

where h is the harmonic order. 2.3.2.4

Phase-Shift Control Operation

In this mode, the converter is operated in square-wave mode and the output voltage is controlled by overlapping the two leg voltages vAN and vBN at an angle of 𝛼, as shown in Figure 2.19. When the two leg voltages are overlapped, the output voltage is zero since either switches T 1 and T 3 are on or switches T 2 and T 4 are on. Since the converter operates in square-wave mode, as Figures 2.19a,b show, only odd harmonics are present in the output voltage. This can also be observed in Figure 2.19c in which the output-voltage waveform is an odd function and hence contains only odd components. Let 𝛽 be deﬁned as: 𝜋−𝛼 𝛽= (2.11) 2

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2 Power Electronics for VSC-Based Bridges

vmod

Vˆ tri

(a)

0

–Vˆ tri

–vmod

Time 1

(b)

fsw

Fundamental harmonic

Vdc

vo

0

–Vdc 0 (c)

T1

Time

0 0.75

Vˆ o 0.5 Vdc 0.25 0

2fsw – 3f1 f1

2fsw – f1

2fsw + 3f1

2fsw + f1 Frequency

4fsw

Figure 2.18 PWM with unipolar switching for a full-bridge VSC (© Wiley, 2002), (a) modulating and carrier waveforms for ma = 0.8 and mf = 15; (b) output voltage of the VSC v o (−) and the fundamental harmonic (− −); and (c) normalized harmonic components of the output voltage.

The magnitude of the harmonic of order h of the output voltage can be easily obtained by using Fourier analysis as in [1]: | 𝜋∕2 | | | | 2 || 𝛽 4 | ̂oh = 2 || V vo cos(h𝜏)d𝜏 || = | vo cos(h𝜏)d𝜏 | = (2.12) V sin(h𝛽) | 𝜋h dc 𝜋 |∫− 𝜋∕2 | 𝜋 ||∫−𝛽 | | |

2.3 Voltage Source Converters

(a)

Vdc

vAN

180°

0 α

(b)

Time

Vdc

vBN

180°

0 Time (c)

Vdc 180°–α vo

0

–Vdc Time Figure 2.19 Waveforms obtained in the VSC using the phase-shift control (© Wiley, 2002), (a) v AN ; (b) v BN ; and (c) output voltage v o .

The value required for the ﬁrst harmonic component can therefore be obtained by simply varying the overlap angle 𝛼. Figure 2.20 shows the amplitudes of the diﬀerent harmonic components as a function of the angle 𝛼. The results show that the angle 𝛼 can be chosen either to control the amplitude of the fundamental frequency or to cancel a particular harmonic component (e.g. if 𝛼 = 60∘ , the third harmonic is not present in the output voltage).

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2 Power Electronics for VSC-Based Bridges

1

0.8 h=1 0.6 Vˆ oh Vˆ o1

04 h=3 h=5

0.2

0

h=7

h=9 0

30

90 α (degrees)

60

120

150

180

Figure 2.20 Normalized harmonic components of the output voltage (from h = 1to h = 9) as a function of the overlap angle 𝛼 (© Wiley, 2002).

2.3.3

Three-Phase VSC

In high-power applications, three-phase VSCs are normally used rather than the single-phase conﬁguration. The best-known topology of the three-phase VSC is depicted in Figure 2.21, in which the VSC consists of three independent legs (A, B and C), with two switches per leg. As in the case of the full-bridge VSC, the three-phase conﬁguration allows a maximum output voltage per leg equal to the DC voltage to be obtained, and both switches on the same legs cannot be either on or oﬀ simultaneously. The three-phase VSC can be controlled using a PWM scheme. In this case, three sinusoidal waveforms vmod A , vmod B and vmod C , i.e. the modulating signals with a phase-shift of 120∘ with regard to each other, are compared with the triangular signal vtri in order to produce the three independent voltages vAN , vBN and vCN , as shown in Figure 2.22a

T1

T3

T5

D1

D3

D5

A

B

C

+ Vdc –

c T4

T6 D4

T2 D6

N

Figure 2.21 Three-phase two-level VSC.

D2

2.3 Voltage Source Converters

vmod A

Vˆ tri

(a)

vmod B

vmod C

0

–Vˆ tri 1 (b)

Time fsw

Vdc vAN

0

(c)

0

Time

T1

0

Time

T1

Vdc vBN

0

(d)

vAB

0

–Vdc (e)

Fundamental harmonic

Vdc

0

T1

Time

1 0.8

Vˆ AB 0.6 Vdc 0.4

2fsw + f1

0.2 0

f1

fsw – 2f1 fsw + 2f1

2fsw – f1

3fsw + 2f1 3fsw – 2f1

Frequency Figure 2.22 PWM process for a three-phase VSC (© Wiley, 2002), (a) modulating and carrier waveforms for ma = 0.8 and mf = 15; (b) output voltage v AN ; (c) output voltage v BN ; (d) line-to-line output voltage v AB (−) and the fundamental harmonic (− −); and (e) normalized harmonic components of the line-to-line output voltage.

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2 Power Electronics for VSC-Based Bridges

for ma = 0.8 and mf = 15. When vmod A > vtri , the switch T 1 is on and therefore vAN = V dc . On the contrary, if vmod A < vtri , the valve T 4 is on and therefore vAN = 0. The process is repeated in a similar way in the other two legs involving the modulating signals vmod B and vmod C , and the resulting output voltages are plotted in Figures 2.22b,c for the vAN and vBN , respectively. It should be noted that the output voltage of each leg has a DC component which is exactly the same for the three legs, signifying that the line-to-line output voltage does not contain any DC component, as shown in Figure 2.22d. The harmonic components of the line-to-line voltage are plotted in Figure 2.22e: in the linear ̂AB is [1]: modulation region, the fundamental component amplitude V 1 √ 3 ̂AB = V (2.13) m V 1 2 a dc With regard to the remaining harmonic components, they are odd components if mf is chosen to be uneven and they are placed at the sidebands of the switching frequency f sw and its multiples. Furthermore, owing to the 120∘ phase shift between the modulating signals, if mf is chosen as an odd integer multiple of 3, the harmonic components at frequencies nf sw , n = 1, 3, 5… are not present [1]. The maximum value of the fundamental-harmonic amplitude is obtained when ma = 1 in (2.13). This value can be increased if the VSC operates in the overmodulation region at the expense of generating low-frequency harmonics and obtaining a nonlinear control of the amplitude of the fundamental harmonic. This nonlinear behaviour is shown in Figure 2.23, in which it will be observed that the maximum amplitude is obtained at the square-wave region and is: √ 2 3 ̂AB = (2.14) Vdc V 1 𝜋 When the VSC operates in the square-wave mode, all switches have a 50% duty cycle and the period of the output voltage is the fundamental period. Figures 2.24a–c show the voltages of the three legs, while the line-to-line voltage of the VSC is shown in Figure 2.24d. The resulting harmonic spectrum of the line-to-line voltage is plotted in Figure 2.24e, in which it will be noted that only odd harmonics that are not multiples of 3 are present. The amplitudes of the harmonics are inversely proportional at the harmonic order: ̂AB V 1 h = , h = 1, 5, 7, 11, 13, … (2.15) h ̂AB V 1

2.3.4

Three-Phase Multilevel VSC

Power electronic converters are very often connected to a medium-voltage grid, and the use of a conventional two-level VSC is not appropriate owing to the high voltages that the devices valves must block. Multilevel VSC topologies have been developed as an alternative to two-level VSCs, aimed at high-power applications. They have a low harmonic distortion and are well suited for applications such as reactive power compensation and HVDC transmission. Furthermore, as the various levels allow the switching frequency to be reduced, the switching losses are also reduced.

2.3 Voltage Source Converters

2 3 π 3 2 VˆAB

1

Vdc

0

0

1 Linear region

3.24 Overmodulation region ma

Square wave

Figure 2.23 Normalized fundamental-component amplitude of the line-to-line output voltage vs. ma , for mf = 15.

(a)

Vdc vAN

0

(b)

0

Time

0

Time

T1

Vdc vBN

0

Figure 2.24 Square-wave operation of a three-phase VSC, (a) output voltage vAN ; (b) output voltage v BN ; (c) output voltage v CN ; (d) line-to-line output voltage v AB (−) and the fundamental harmonic (− −); and (e) normalized harmonic components of the line-to-line output voltage.

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2 Power Electronics for VSC-Based Bridges

(c)

Vdc vCN

0 0

Time Fundamental harmonic

(d)

Vdc vAB 0

–Vdc 0

Time

(e) 1 Vˆ AB Vdc 0.5

0

0

1

5

7

11 13 Harmonic order h

17

19

Figure 2.24 (Continued)

Various multilevel VSC topologies exist, but the most well-known are the neutralpoint-clamped (NPC), the ﬂying capacitor (FC), the H-bridge converter, the modular multilevel converter, and the hybrid multilevel converter. 2.3.4.1

The Multilevel NPC VSC

Figure 2.25 shows the simplest multilevel NPC topology for a three-phase VSC: the three-level NPC conﬁguration in which each leg comprises four switches and four diodes. In order to explain the operation of the converter, the DC voltage has been split in half by means of two capacitors and the output voltage of each leg is measured with regard to the midpoint of these two capacitors, e.g. vA0 . A positive voltage is obtained when the two upper switches are on, i.e. vA0 = V dc /2 for leg A, while a negative voltage is produced if the two lower switches are on, i.e. vA0 = − V dc /2 for leg A. Besides these two voltage levels, this topology can also generate zero voltage (the third voltage level) when the switches T x2 and T x3 are on simultaneously, where the subscript x stands for the A, B and C legs and the devices T x1 and T x4 are oﬀ; the diodes Dcx1 and Dcx2 allow the current to ﬂow in both directions when the converter works in this zero-voltage region: for positive current value, the devices T x2 and Dcx1 conduct, whereas if the current is

2.3 Voltage Source Converters

Ta1

Tb1 Da1

+ Vdc 2 –

Dc1

C Dca1

Ta2

Dcb1

Da2

Tb2 Db2

A

0

Tc2 Dc2 C

Tb3 Da3

Dca2

Dcc1

B

Ta3 + Vdc 2 –

Tc1 Db1

Tc3 Db3

Dcb2

Dc3

Dcc2

C Ta4

Tb4 Da4

Tc4 Db4

Dc4

N Figure 2.25 Three-level three-phase NPC VSC.

negative, the semiconductors T x3 and Dcx2 are on. In order to avoid short-circuits, it is obvious that when the switches T x1 and T x2 are on, the devices T x3 and T x4 must be oﬀ, and vice versa. These restrictions imply that the ﬁring signals of both switches T x1 and T x3 and switches T x2 and T x4 must be complementary. The resulting output voltage for the on- and oﬀ-states of the diﬀerent devices is summarized in Table 2.1. The operation of the converter can be carried out by using either a square-wave scheme or a PWM process. The PWM operation will be explained later, as it can be applied in a similar way to other multilevel topologies. The square-wave scheme is identical to the phase-shift control method applied to the full-bridge single-phase VSC: the fundamental harmonic and the harmonic content of the output voltage can be controlled by varying the angle 𝛼. Figure 2.26 shows the resulting output voltage for a speciﬁed angle 𝛼: the output voltage of each leg exhibits the three levels explained above, as shown in Figures 2.26a,b. Nonetheless, the line-to-line output voltage vAB has ﬁve levels (two additional levels), which implies a lower harmonic content. Table 2.1 Three-level NPC VSC: output voltage as a function of the switch states. Switch state T x1

T x2

T x3

T x4

Output voltage

On

On

Oﬀ

Oﬀ

V dc /2

Oﬀ

Oﬀ

On

On

−V dc /2

Oﬀ

On

On

Oﬀ

0

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2 Power Electronics for VSC-Based Bridges

(a) Vdc/2 α vA0

0

–Vdc/2

T1/2

T1

0

T1/2

T1

0

T1/2

T1

0

(b) Vdc/2 α vB0

0

–Vdc/2

(c)

Vdc Vdc/2

vAB

0

–Vdc/2 –Vdc

Figure 2.26 Output voltage of a three-level NPC VSC using square-wave operation, (a) output voltage v A0 ; (b) output voltage v B0 ; and (c) line-to-line output voltage v AB .

The number of levels can be increased in order to reduce not only the harmonic content but also the time derivative of the voltage. If the number of levels of a VSC is m, then the converter will require m − 1 capacitors on the DC side and 2 × (m − 1) controllable switches per leg. Figure 2.27 shows the DC side and one leg of a ﬁve-level NPC VSC, where it can be seen that there are four capacitors on the DC side and eight switches per leg. Six

2.3 Voltage Source Converters

Ta1 Da1 + Vdc

C

4 –

Dca1

Da2

Ta3

Dca2

+ Vdc

Ta2

Da3 C

4 –

Ta4

Dca3

Da4 0

A Ta5 Da5

Dca4 + Vdc 4 –

C

Ta6 Da6 Dca5

+ Vdc 4 –

Ta7 Da7 C

Dca6

Ta8 Da8 N Figure 2.27 Five-level NPC conﬁguration.

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2 Power Electronics for VSC-Based Bridges

clamping diodes are also required per leg: for given m levels. Although the controllable switches block a voltage equal to V dc /(m − 1), the clamping diodes must block a diﬀerent voltage [9] and the number of necessary clamping diodes is almost proportional to the square of the number of levels [10]. The main drawbacks of the NPC conﬁguration are the great diﬃculties present in keeping the voltages of the DC capacitors balanced and the unequal power rating of the diﬀerent switches. 2.3.4.2

The Multilevel FC VSC

Rather than using diodes, the multilevel FC VSC uses capacitors to clamp the switch voltage to the capacitor voltage level [11]. The most basic structure of an FC converter is the three-level FC VSC, shown in Figure 2.28: the voltage of the capacitor C 1 must be vC1 = V dc /2 and the converter provides three output-voltage levels: V dc /2, 0 and −V dc /2. As in the case of the three-level NPC VSC, if the switches T x1 and T x2 are on, the output voltage of the leg x is vx0 = V dc /2, whereas when the devices T x3 and T x4 are on, the output voltage is vx0 = − V dc /2. In order to obtain zero output voltage, either switches T x1 and T x3 or switches T x2 and T x4 must be on. The ﬂying capacitor C 1 is charged when the pair T x1 - T x3 is on, while it is discharged when the pair T x2 -T x4 is on. Table 2.2 shows the diﬀerent states of each switch and the corresponding output voltage. When T x1 and T x2 are on, the devices T x3 and T x4 must be oﬀ, and vice versa. These restrictions imply that the ﬁring signals of both switches T x1 and T x3 are complementary as well as the ﬁring signals of switches T x2 and T x4 . As in the case of the multilevel NPC VSC, the multilevel FC converter can be operated using either PWM techniques or a phase-shift control scheme. Ta1 + Vdc 2 –

Tb1 Da1

Db1

Ta2

Tb2

+ 0

vC1 –

2 –

Dc1

C Da2

+ Vdc

Tc1

Tc2 Db2

+ A

C1 Ta3

vC1

Dc2 +

B

C1 Tb3

– Da3

vC1 –

C

C1 Tc3

Db3

Dc3

C Ta4

Tb4 Da4

Db4 N

Figure 2.28 Three-level three-phase FC VSC.

Tc4 Dc4

2.3 Voltage Source Converters

Table 2.2 Three-level FC VSC: output voltage as a function of the switch states. Switch state T x1

T x2

T x3

T x4

Output voltage

On

On

Oﬀ

Oﬀ

V dc /2

Oﬀ

Oﬀ

On

On

−V dc /2

On

Oﬀ

On

Oﬀ

0

Oﬀ

On

Oﬀ

On

0

One of the main drawbacks of this topology is the large number of ﬂying capacitors when the number of voltage levels increases. For example, Figure 2.29 shows a ﬁve-level FC topology for the leg A, where the voltages of the diﬀerent capacitors must be vC1 = 3V dc /4, vC2 = V dc /2 and vC3 = V dc /4. As in the NPC case, the ﬁnal number of ﬂying capacitors is almost proportional to the square of the number of voltage levels [10]. Furthermore, the ratings of the ﬂying capacitors must be high since the full load current ﬂows through them, and although the ﬂying capacitor voltages are, in theory, balanced, in practice there are many factors that lead to asymmetrical operation, involving voltage imbalances in the ﬂying capacitors. A method with which to balance the voltages of the ﬂying capacitors is therefore required [12]. The main advantage of this topology is that it provides more ﬂexibility in the choice of switching combinations, allowing accurate control of the ﬂying capacitor voltages. Although it requires a large number of capacitors, the implementation of a three-level FC converter is less onerous than the NPC topology [13]. In this multilevel topology, all the devices have the same switching frequency. 2.3.4.3

The Cascaded H-Bridge VSC

This topology uses several single-phase full-bridge converters with independent DC voltage sources to build a multilevel VSC. Each leg of the multilevel VSC is obtained by connecting the diﬀerent single-phase converters in series, as shown in Figure 2.30. The resulting output voltage is the sum of the individual output voltage of each single-phase converter: vAN = vVSC1 + vVSC2 + vVSC3 . In general, for an n given number of full-bridge converters, the resulting number of levels in the output voltage is m = 2n + 1. Each individual converter can be controlled by using the phase-shift control technique, and the harmonic content of the output voltage vAN can be controlled by adjusting the diﬀerent angle 𝛼 of each single-phase converter. Figure 2.31 shows the voltage waveforms obtained by applying the phase-shift control technique to the VSC in Figure 2.30: the output voltages of each single-phase converter are plotted in Figures 2.31a–c, respectively; all these output voltages have the same three levels (V dc , 0 and −V dc ). The resulting output voltage vAN is a staircase waveform with a low distortion, which is highly sinusoidal and has seven diﬀerent voltage levels: 3V dc , 2V dc , V dc , 0, −V dc , −2V dc and −3V dc , as shown in Figure 2.31d.

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Ta1 Da1

Ta2 Da2

Ta3 + Vdc 2 –

Da3 C Ta4 Da4 + 0

vC1

+ C1

– + Vdc 2 –

vC2

+ C2

–

vC3 –

A

C3 Ta5

Da5 C Ta6 Da6

Ta7 Da7

Ta8 Da8 N Figure 2.29 Five-level FC conﬁguration.

2.3 Voltage Source Converters

VSC1

+ Vdc

A +

c

vVSC1 –

–

+

VSC2

+ Vdc

c

–

+ vVSC2 –

vAN

+ vVSC3 –

– N

VSC3

+ Vdc

c

–

Figure 2.30 Phase A of a seven-level cascaded H-bridge VSC.

The use of PWM techniques is very suitable for the case of the H-bridge topology as it allows more harmonics to be cancelled than other multilevel topologies [3]. Until not so long ago, it was considered that the drawbacks of this topology were the large number of individual VSCs and the number of isolated DC voltage sources required as the number of levels increases [14]. However, in today’s context, this seems to be a very interesting technology, which may be considered the forebear of modern multilevel converters presented in Section 2.3.4.5.

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(a)

Vdc α1

vVSC1a 0

–Vdc 0 (b)

T1/2

T1

Vdc α2

vVSC2a 0

–Vdc 0 (c)

T1/2

T1

Vdc α3

vVSC3a 0

–Vdc 0

T1/2

T1

Fundamental harmonic (d)

3Vdc 2Vdc Vdc vAN 0 –Vdc –2Vdc –3Vdc 0

T1/2

Figure 2.31 Output voltage of a seven-level cascaded H-bridge VSC using the phase-shift control technique, (a) output voltage of VSC1; (b) output voltage of VSC2; (c) output voltage of VSC3; and (d) resulting output voltage v AN for the phase A (−) and its fundamental harmonic (− −).

T1

2.3 Voltage Source Converters

2.3.4.4

PWM Techniques for Multilevel VSCs

The previous sections have presented multilevel topologies that use fundamental switching techniques, such as phase-shift control, in order to control the multilevel VSC. If a high switching frequency is required, one of the most frequently used techniques is that of the phase-shifted SPWM (sinusoidal pulse width modulation) switching method: for a given number of levels m and the number of individual VSC cell voltages n necessary to generate the m levels, this scheme employs a sinusoidal modulating signal which is compared with a number of triangular carrier signals equal to n [9]. These triangular signals are shifted to an angle 𝜃 = 2𝜋/n and the results of these comparisons are used to turn the diﬀerent switches on and oﬀ. One of the advantages of this technique is that the output voltage has an equivalent switching frequency of n times the frequency of each carrier signal f sw , i.e. the switching frequency of each individual cell. This allows the frequency of the carrier signals to be decreased, thus reducing the switching losses [15]. The three-level, three-phase FC VSC of Figure 2.28 has been operated by using the phase-shifted SPWM technique. In this case, only two carrier signals are necessary and the resultant angle 𝜃 is 𝜋 radians. The two carrier signals and the three modulating waveforms are plotted in Figure 2.32a. The output voltage vA0 has three levels, as shown in Figure 2.32b, and according to Figure 2.32c, the main harmonic components are placed around 2f sw . The line-to-line voltage vAB contains ﬁve levels: V dc , V dc /2, 0, −V dc /2 and −V dc , as can be observed in Figure 2.32d. Finally, the harmonic content of the line-to-line voltage shows that the eﬀective switching frequency is twice the frequency of the carrier signals. The phase-shifted SPWM switching technique is applied not only to multilevel VSCs but also to multiconverter applications [16] and DC-DC interleaved converters [17]. 2.3.4.5

An Alternative Multilevel Converter Topology

The need to improve on the voltage waveform and reduce switching losses has led to the development of multilevel converters. The classical multilevel VSC conﬁgurations – the NPC, the ﬂying capacitor and the H-bridge – have been described in the sections above. Each topology presents its advantages and disadvantages with respect to the others, but the current consensus is that in spite their many advantages over the two-level VSCs for low- and medium-power applications, they are diﬃcult to escalate for more than three or ﬁve voltage levels and hence are rendered unsuitable for high-voltage and high-power applications [18]. Some of these drawbacks have been stated at the end of Sections 2.3.4.1, 2.3.4.2 and 2.3.4.3. Aiming at reducing design circuit complexity, increasing modularity, lowering switching losses and achieving a more sinusoidal waveform of the output voltage, the modular multilevel converter (MMC) was put forward in [19]. Depending on the number of modules (levels−1) used, the MMC may synthesize a near-sinusoidal voltage waveform. This is achieved by the incremental switching of each module in turn to shape up as much as possible a sinusoidal waveform, with frequencies in the range of 100–150 Hz. This contrasts with the switching frequencies employed by two-level and three-level VSCs, which are in the range of 1–2 kHz. One outcome is that MMCs incur power losses of around 1% as opposed to 2% incurred by the two-level and three-level VSCs. This is a rather signiﬁcant achievement, but it should be noted that the losses incurred by MMCs are still higher than the losses incurred by CSCs, which stand at around 0.7%.

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There are currently three vendors of the MMC VSC technology: ABB, Siemens and Alstom. Each has its own proprietary designs and trademark names: HVDC Light, HVDC Plus and MaxSine HVDC, respectively. The HVDC Light and the HVDC Plus are very similar and based on the design put forward in [19]. Interestingly, the HVDC Light has also been referred to as a cascaded two-level converter [20]. Let us say that these two proprietary designs are pure MMC technologies. Meanwhile, MaxSine HVDC uses a hybrid VSC-HVDC converter concept, involving a two-level converter as the main switching component with a low switching frequency and a multilevel converter of small rating to cancel out, in an active fashion, the harmonics produced by the main, two-level converter. The MMC is built by cascading a large number of half-bridge submodules (SMs), as illustrated in Figure 2.33, where two such SMs are connected in series. Appropriate switching of S1 and S2 in each SM will enable to add up the two capacitances, or to bypass both of them, or to use either of them, to achieve the target output voltage level. In each SM, when switch S1 is on, switch S2 is oﬀ, and the other way around. When both switches S1 are on, the output voltage is V out = V sm1 + V sm2 . When both switches S2 are on the output, voltage is nil, i.e. V out = 0. These switching operations already represent two levels in the converter operation and a third level arises when S1 in either SM is on, e.g. V out = V sm1 . The schematic representation of a three-phase MMC is shown in Figure 2.34 where 2n SMs are employed to make up a phase leg, as illustrated in the ﬁgure. vcar1 Vˆ tri

(a)

vcar2 vmod A

vmod B

vmod C

0

–Vˆ tri 1 Vdc 2

(b)

vA0

–

Time fsw

0 Vdc 2 0

Time

T1

Figure 2.32 SPWM for the three-level three-phase FC VSC for ma = 0.8 and mf = 15, (a) modulating signals and carrier waveforms shifted 𝜋 radians; (b) output voltage vA0 (−) and its fundamental harmonic (− −); (c) normalized harmonic components of vA0 ; (d) line-to-line output voltage v AB (−) and its fundamental harmonic (− −); and (e) normalized harmonic components of vAB for the phase A (−) and its fundamental harmonic (− −).

2.3 Voltage Source Converters

(c)

0.8

0.6 Vˆ A0 0.4 Vdc 2 0.2 0 (d)

Vdc Vdc 2 vAB 0 V – dc 2 –Vdc

2fsw + f1 2fsw – 3f1 f1

0

2fsw + 3f1

2fsw – f1 Frequency

T1

Time

(e) Vˆ AB Vdc

0.5 2fsw + f1

0.25 0

f1

2fsw – f1 Frequency

Figure 2.32 (Continued)

Notice that the phase legs are connected in parallel in the DC side and ideally, each phase leg contributes one third to the DC current. Nonetheless, balancing or circulating current will ﬂow between the phase legs since a small natural unbalance exists between the voltages of the three phase legs. The function of the phase reactors is to limit such circulating currents, but they also limit the current rise in cases of DC faults, to enable the IGBTs to be turned oﬀ relatively safely. More speciﬁcally, the circulating currents are suppressed using a parallel resonant ﬁlter tuned at the second harmonic and in the case of the ABB’s proprietary design, placed at the centre of the phase reactors. The ﬁlter also serves to eliminate the third harmonic [18]. The two companies seem to take very diﬀerent approaches when it comes to the number of SMs employed in their MMCs. For instance, in the Trans Bay Cable installation, Siemens used 200 SMs in each phase arm of the converter. This VSC-HVDC link is rated at ±200 kV [21]. In contrast, ABB employs only 38 SMs per phase arm for a DC link of ±320 kV [20]. Owing to the rather large number of SMs, the former design requires no ﬁlter whereas the latter design employs a small ﬁlter.

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The hybrid VSC is shown schematically in Figure 2.35. It comprises an H-bridge connected in parallel with a wave-shaping circuit in each phase. In turn, a wave-shaping circuit comprises half-bridge SMs – see Figure 2.33. In this conﬁguration the three phases are connected in series at their DC sides; so, there are no circulating currents and no need for the use of inductors as is the case in the MMC topology. We shall use the acronym HMC for this converter topology. Another key advantage of this hybrid topology is that the wave-shaping circuit lies outside the main current path; hence, the number of SMs is very much reduced. The working principle of the H-bridge has already been explained in Section 2.3.2. The function of the wave-shaping circuit on the DC side is to produce a rectiﬁed voltage sinewave, which is transferred onto the AC side, resulting in a sinusoidal waveform with little distortion, which is a function of the number of SMs used in the wave-shaping circuit. An undesirable byproduct of this rectiﬁcation process is the appearance of a large sixth harmonic component which has to be removed using ﬁlters or an active ﬁltering technique [18], [22].

2.4 HVDC Systems Based on VSC In Section 1.2 it was illustrated that two power converters connected in tandem may be used to interconnect two AC systems to exchange electrical power in DC form. This involves one converter acting as a rectiﬁer and the other as an inverter, according to a pre-speciﬁed command. In principle, any pair of the VSC types presented in Section 2.3 may be used to such an aim, giving rise to single-phase and three-phase HVDC links. It should be noted that diode and thyristor-based bridges may also be used to form HVDC links or may even be combined with VSCs to form hybrid HVDC links. Moreover, we may have two-level and multilevel HVDC links of various kinds. In bulk power transmission applications, only three-phase HVDC links are employed owing to reasons of energy eﬃciency and costs. Ism2 +

+ Vsm2 S1

+ Vc2 –

SM2

S2 D2

D1 – Vout

Ism1 + Vsm1 S1

+ Vc1 – D1

–

SM1

S2 D2

–

Figure 2.33 Two SMs connected in series.

2.4 HVDC Systems Based on VSC

IDC IDC /3

IDC /3

+

IDC /3

SMn

SMn

SMn

SM2

SM2

SM2

+

VDC /2 SM1 IACa /2

SM1 IACb /2

SM1 IACc /2

Module

VA

–

VB +

VC Phase arm IACa /2

SM1

IACb /2

SM1

VDC

Phase leg IACc /2

SM1 VDC /2

SM2

SM2

SM2

SMn

SMn

SMn –

–

Figure 2.34 Schematic representation of a three-phase MMC.

Reputedly, the ﬁrst commercial HVDC facility was installed to connect the island of Götland and mainland Sweden in 1954 [23]. Technology has evolved since then, and modern power semiconductors such as IGBTs have substituted the old mercury arc valves and the classical thyristors in that installation. Classical HVDC systems employ line-commutated CSC thyristor bridges. This mature technology enables the transmission of power in the region of 8 GW and perhaps more if the requirement were to arise [24] – note that in 2012, the Jinping-Sunan link in

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East China came into operation, rated at 7200 MW and ±800 kV, spanning a distance of 2100 km. These HVDC links may be classiﬁed, depending on the function and location of the CSCs, in back-to-back HVDC systems, monopolar HVDC systems, bipolar HVDC systems and series, multi-terminal HVDC systems. A comprehensive description of these conﬁgurations can be found in [25]. In contrast, current development of power semiconductors permits the use of IGBT-based VSCs for power levels up to 1000 MW per circuit, as in the case of the 2000 MW INELFE DC link between Baixas, France and Santa Llogaia, Spain, commissioned in 2015. Figure 1.8 shows an HVDC system based on the VSC technology. There are two VSC stations, which can be of the following kinds: the basic two-level, the multilevel or the more advanced modular multilevel. In any case, the active power, in DC form, can ﬂow through the DC link in either direction, according to command, whereas the respective

+

+VDC /2

Phase A

Phase B VDC

Phase C

–

–VDC /2

Figure 2.35 Schematic representation of a three-phase hybrid VSC.

2.4 HVDC Systems Based on VSC

reactive powers can be controlled to ﬂow in and out of both VSCs, or even to operate at unity power factors. Note that no reactive power exists in a DC circuit. The use of an HVDC system based on the VSC technology compared with the CSC technology exhibits several advantages, some of which are listed below: • VSCs are four-quadrant operation converters in terms of their active and reactive powers, as opposed to two-quadrant operation, which is the case with CSC converters [7, 26]. • Unlike CSCs, there are no commutation failures in VSCs, which incidentally cause voltage sags and waveform distortion in the AC power grid. • VSC-HVDC links may be used to interconnect strong and weak grids (those exhibiting low short-circuit ratios) [26]. This option is very much curtailed with CSC-HVDC links. • VSC-HVDC links are ideally suited to connect wind and solar power parks to the main grid – in particular, oﬀshore wind parks – as well as to supply oil and gas rigs from the main power grid [27]. • VSC-HVDC links, like CSC-HVDC systems, are economically feasible solutions if the transmission distance exceeds a threshold value, termed the ‘break even distance’ [26], [18]. However, there are many instances where the distance does not even come into consideration, such as the back-to-back (or zero-distance) schemes and submarine applications, the latter being an application where the VSC-HVDC outperforms the CSC-HVDC. • Unlike CSCs, VSCs exhibit very fast control of the active and reactive powers owing to the high PWM switching frequency. • Since the power semiconductors used in VSCs are fully controlled, there is no need for bulky transformers to assist in the commutation process as is the case in CSCs [7]. • CSC-HVDCs produce characteristic harmonics, which are low-order harmonics. In contrast, two-level VSC-HVDCs may produce no low-order harmonics but at the expense of substantial converter power losses (switching losses) [26]. However, the new breed of MMC VSCs have much more reduced power losses than the two-level converters, which are in the order of those incurred by the CSC-HVDCs, and a much-improved harmonic proﬁle. In fact, the MMC produces only the second and third harmonics, which may necessitate ﬁltering out and the use of a transformer with its secondary winding connected in delta. However, if the number of SMs per phase arm is large enough (e.g. 200), such as in the case of the Siemens design, then the second harmonic is low in value and does not require ﬁltering. The hybrid multilevel design produces a large sixth harmonic which needs to be ﬁltered out. The high-order harmonics are negligibly low in the three MMC designs presented in Section 2.3.4.5. • Each terminal of a VSC-HVDC system can operate as the STATCOM when the active power in the DC link is zero. Such a feature is not an attribute of the CSC-HVDC technology. • VSC-HVDCs, but not CSC-HVDCs, have the capability to start up in the case of a dead network [28]. • Unlike CSC-HVDC, VSC-HVDC enables meshed DC power grids. However, no multi-terminal VSC-HVDC systems has yet been built, although it is thought that one would enable the electriﬁcation of the North Sea, bringing many technical and economic advantages to the whole of Northern Europe [29].

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Table 2.3 VSC-HVDC installations in operation. Installation name

Installed

Rated power (MW)

Rated voltage (kV)

Manufacturer

Converter topology

Hälsjön

1997

3

±10

ABB

2-level

Gotland

1999

50

±80

ABB

2-level

Tjaereborg

2000

7

±9

ABB

2-level

Terranora (Directlink)

2000

180

±80

ABB

2-level

Eagle Pass

2000

36

±15.9

ABB

2-level

Murraylink

2002

220

±150

ABB

3-level

Cross Sound Cable

2002

330

±150

ABB

3-level

Troll A 1&2

2005

88

±60

ABB

2-level

Estlink1

2006

350

±150

ABB

2-level

BorWin1

2009

400

±150

ABB

2-level

Trans Bay

2010

400

±200

Siemens

MMC

Caprivi Link

2010

300

+350

ABB

2-level

Valhall

2011

78

+150

ABB

2-level

East West Interconnector

2013

500

±200

ABB

2-level

INELFE

2013

2000

±320

Siemens

MMC

Mackinac

2014

200

±71

ABB

2-level

Skagerrak4

2014

700

+500

ABB

MMC MMC

BorWin2

2015

800

±300

Siemens

HelWin1

2015

576

±250

Siemens

MMC

HelWin2

2015

690

±320

Siemens

MMC

Troll A 3&4

2015

100

±60

ABB

2-level

Åland

2015

100

±80

ABB

2-level MMC

Nordbalt

2015

700

±300

ABB

DolWin1

2015

800

±320

ABB

MMC

DolWin2

2017

900

±320

ABB

MMC

EWIC

2017

500

±200

ABB

2-level

Caithness-Moray

2018

1200

±320

ABB

MMC

Maritime Link

2018

500

±200

ABB

MMC

Given that the VSC-HVDC technology is barely 20 years old, having started in 1997 with the Hälsjön prototype in Sweden, a healthy number of installations are already in operation around the world, as shown in Table 2.3. The names, ratings, year of commissioning, converter manufacturer and converter topology are presented for the most relevant VSC-HVDC installations. Table 2.4 shows similar information but for VSC-HVDC installations that are in the construction stage and due for commissioning in the period 2019–2020. The tables show that at the time of writing there are three major vendors of the VSC-HVDC technology: ABB, Siemens and Alstom.

2.4 HVDC Systems Based on VSC

Table 2.4 VSC-HVDC installation due for commissioning in the period 2019–2020. Rated power (MW)

Rated voltage (kV)

Manufacturer

Converter topology

South-West Link

1440

±300

Alstom

HMC

Italy–France link

1200

±320

Alstom

HMC HMC

Installation name

900

±320

Alstom

NEMO

DolWin3

1000

±400

Siemens

MMC

SylWin

864

±320

Siemens

MMC

Cobra Cable

700

±320

Siemens

MMC

BorWin3

900

±320

Siemens

MMC

Ultranet

2000

±380

Siemens

MMC

As illustrated in both tables and except for the speciﬁc low-power applications aimed at the power supply of oil and gas rigs, the upward trend is the use of VSC-HVDC systems of higher ratings, a task for which the multilevel topologies are very well suited, as described previously. The reason is that multilevel topologies extend the advantages of medium-power VSCs to high-power applications, which necessarily calls for higher voltages. Furthermore, the use of multilevel converters reduces not only the total harmonic voltage distortion but also the switching losses, hence reducing ﬁltering to a bare minimum – normally, there is no need to ﬁlter out any of the characteristic harmonics, 5th, 7th, 11th, 13th and above. Currently, the total power losses of two-level VSCs and MMC are in the region of 2% and 1%, respectively. Note that they are still higher than those of thyristor-based converters, which are in the region of 0.7% [18]. The following are some of the main applications of the VSC-HVDC technology: • Grid connection of oﬀshore wind farms, with oﬀshore wind power projects dominating the scene in Europe. There are suﬃcient technical and economic justiﬁcations for oﬀshore wind parks to be connected to the main power grid using a VSC-HVDC link as opposed to an AC cable when they lie more than 70 km away from the shore [18]. The former option helps to improve the grid transient stability since the VSCs are natural sources of fast reactive power control [26, 30]. The layout of the submarine cable is also simpler with the former option since there is no need for shunt reactors to be connected at regular intervals along the length of the cable, as would be the case with the latter option. Also, power losses will be less using a VSC-HVDC submarine link compared with a submarine AC cable of comparable length. • Power distribution in densely populated areas. It is highly likely that multi-terminal VSC-HVDC systems will be used in future distribution applications in large cities [8]. A generic four-terminal VSC-HVDC system is shown in Figure 1.10. One of the key advantages of multi-terminal topologies is the ability to connect a number of power grids in an asynchronous manner; they can therefore contribute towards delivering the electricity produced by diﬀerent kinds of distributed power sources to customers, with no increase in the troublesome short-circuit ratio. Moreover, they would help to improve on the power quality [31]. • VSC-HVDC systems for bulk power transmission. Unidirectional bulk power transmission has been one of the main applications of HVDC systems [26]. If a CSC

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Table 2.5 The 10 largest CSC-HVDC installations in the world. Rated power (MW)

Rated voltage (kV)

Installed

Distance (km)

Country

Jinping-Sunan

7200

±800

2012

2090

China

Xiangjiaba-Shanghai

6400

±800

2010

1980

China

Xiluodo-Guangdong

6400

±500

2014

1223

China

Installation name

Yunnan-Guangdong

5000

±800

2010

1418

China

Ningxia-Shandong

4000

±660

2010

1348

China

Itaipu 2

3150

±600

1984

805

Brazil

Itaipu 1

3150

±600

1984

785

Brazil

Paciﬁc DC Intertie

3100

±500

1970

1362

USA

Three Gorges-Changzhou

3000

±500

2003

860

China

Guizhou-Guangdong

3000

±500

2004

980

China

solution is used, power ratings above 6.4 MW can be achieved with no diﬃculty, as illustrated in Table 2.5, containing the names, ratings, distances and year of commissioning of the 10 largest CSC-HVDC links in the world. At present, as Tables 2.3 and 2.4 show, the VSC technology reaches values of 1000 MW (2000 MW using two circuits), such as the case of the INELFE and Ultranet links. Some existing HVAC systems are under consideration to be replaced or reconverted to HVDC systems using the VSC option. Furthermore, hybrid solutions combining CSC-HVDC and VSC-HVDC already exist in practice, as in the case of the Skagerrak 4. In fact, the Skagerrak HVDC transmission system comprises four HVDC links, with a total combined capacity of 1700 MW. It goes from Kristiansand in southern Norway to Tjele on Denmark’s Jutland peninsula. Skagerrak 1 and 2 rated 500 MW are CSC-HVDC links commissioned in 1976–1977 and so is Skagerrak 3, commissioned in 1993. Skagerrak 4, commissioned in 2014, is a VSC-HVDC link and operates in a bipole mode with Skagerrak 3, which is a CSC-HVDC link. The latter required a control system upgrade to enable this mode of operation.

2.5 Conclusions The evolution that has taken place in the power semiconductor ﬁeld over the past two decades has enabled new developments in power-electronics applications. Although the power semiconductor with the highest power capability is the thyristor, the emergence of the GTO ﬁrst and the IGBT later on, as fully controlled semiconductors, with reasonable high switching frequencies and medium voltage and medium power capabilities, has made possible their increasing use in power electronics applications. This is particularly the case with the IGBT, which outperforms the GTO in terms of power losses and exhibits a much higher switching frequency range. A clear example of its growing popularity in power grid applications is the ubiquitousness of VSCs that employ IGBTs. They are used in the generation, transmission,

References

distribution and industrial sectors. Modern control switching techniques allow multilevel VSCs to be operated with a very low output voltage distortion and power losses of around 1%. The majority of the early commercial solutions were based on the two-level conﬁguration. However, there is now a variety of well-developed multilevel topologies such as the NPC and the FC converters, which are key elements in high-voltage applications. The number and rating of installations using VSC systems which employ the IGBT technology have risen since 2013, with the majority using multilevel converter topologies. The current facilities are in the range of several hundred megawatts [32]. VAR compensation systems based on VSC technology such as the STATCOM have several advantages over the classical static VAR compensators using thyristor-switched capacitors and inductor-controlled reactors. Among them is the faster speed of response, smaller footprint and the ability to contribute reactive power when the power system is undergoing a voltage collapse. A further, more general feature of the VSC is its four-quadrant operation, which enables the connection of a DC energy source or an energy storage system onto its DC bus. This feature is extended even further when two or more VSCs are employed to make up an HVDC system, representing a newer form of DC power transmission that exhibits numerous advantages over the conventional CSC-HVDC systems. For instance, VSC-HVDC enables the connection of strong and weak AC power grids; each VSC converter is able to regulate voltage magnitude at the respective AC nodes and exerts fast control of active and reactive powers, according to a pre-speciﬁed control operating philosophy. An unrivalled feature of the VSC-HVDC technology over the CSC-HVDC technology is its ability to form multi-terminal HVDC systems of a generic nature. All these attributes of the VSC-HVDC technology have encouraged their use in an ever-growing number of applications. Further to the well-known long-distance power transmission application, they look unassailable in the grid connection of oﬀshore wind generators, in energy supply from the mainland to oﬀshore oil and gas rigs, and in underground power distribution in the city centres of large conurbations.

References 1 Mohan, N., Undeland, T.M., and Robbins, W.P. (2002). Power Electronics: Converters,

Applications, and Design, 3e. Wiley. 2 Hart, D.W. (2011). Power Electronics. McGraw-Hill. 3 Acha, E., Agelidis, V.G., Anaya-Lara, O., and Miller, T.J.E. (2002). Power Electronic

Control in Electrical Systems. Newness. 4 Sedra, A.S. and Smith, K.C. (1998). Microelectronic Circuits. Oxford University Press. 5 Mohan, N. (2012). Power Electronics. A First Course. Wiley. 6 Hu, A.P. (2009). Wireless/Contactless Power Supply – Inductively Coupled Resonant

Converter Solutions. VDM Verlag. 7 Flourentzou, N., Agelidis, V.G., and Demetriades, G.D. (2009). VSC-based HVDC

power transmission systems: an overview. IEEE Transactions on Power Electronics 592–602. 8 Sood, V.K. (2004). HVDC and FACTS Controllers. Applications of Static Converters in Power Systems. Boston, MA: Kluwer Academic Publishers.

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9 Rodríguez, J., Lai, J.-S., and Peng, F.Z. (2002). Multilevel inverters: a survey of

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topologies, controls, and applications. IEEE Transactions on Industrial Electronics 49 (4): 724–738. Arrillaga, J., Liu, Y.H., and Watson, N.R. (2007). Flexible Power Transmission. The HVDC Options. Wiley. Lai, J.-S. and Peng, F.Z. (1996). Multilevel converters – a new breed of power. IEEE Transactions on Industry Applications 32 (3): 509–517. Xu, L. and Agelidis, V.G. (May 2004). Active capacitor voltage control of ﬂying capacitor multilevel converters. IEEE Proceedings on Electric Power Applications 151 (3): 313–320. Feng, C., Liang, J., and Agelidis, V.G. (2007). Modiﬁed phase-shifted pwm control for ﬂying capacitor multilevel converters. IEEE Transactions on Power Electronics 22 (1): 178–185. Roncero-Sánchez, P. and Acha, E. (2009). Dynamic voltage restorer based on ﬂying capacitor multilevel converters operated by repetitive control. IEEE Transactions on Power Delivery 24 (2): 951–960. Liang, Y. and Nwankpa, C.O. (2000). A power-line conditioner based on ﬂying-capacitor multilevel voltage-source converter with phase-shift SPWM. IEEE Transactions on Industry Applications 36 (4): 965–971. Mwinyiwiwa, B., Ooi, B.-T., and Wolanski, Z. (1998). UPFC using multiconverter operated by phase-shifted triangle carrier SPWM strategy. IEEE Transactions on Industry Applications 34 (3): 495–500. Choe, G.-Y., Kim, J.-S., Kang, H.-S., and Lee, B.-K. (2010). An optimal design methodology of an interleaved boost converter for fuel cell applications. Journal of Electrical Engineering & Technology 5 (2): 319–328. J. Glasdam, J. Hjerrild, L.H. Kocewiak and C.L. Bak, “Review on multi-level voltage source converter based HVDC technologies for grid connection of large oﬀshore wind farms,” in Proceedings of the IEEE International Conference on Power System Technology (POWERCON), 2012, Auckland, 2012. A. Lesnicar and R. Marquardt, “An innovative modular multilevel converter topology suitable for a wide power range” in 2003 IEEE Bologna Power Tech Conference Proceedings, Bologna, 2003. B. Jacobson, P. Karlsson, G. Asplund and L. Harnefors, “VSC-HVDC transmission with cascaded two-level converters”, in Proceedings CIGRE Conference, Paris, 2010. H.J. Knaak, “Modular multilevel converters and HVDC/FACTS: A success story”, in Proceedings of the 201 I-14th European Conference on Power Electronics and Applications (EPE 2011), Birmingham, 2006. C.C. Davidson and D.R. Trainer, “Innovative concepts for hybrid multi-level converters for HVDC power transmission”, in 9th IET International Conference on AC and DC Power Transmission (ACDC 2010), London, 2010. Jacobson, B., Jiang-Härner, Y., Rey, P. et al. (2006). HVDC with voltage source converters and extruded cables for up to +/−300 kV and 1000 MW. In: Cigre Session. Canada: Montreal,. Hingorani, N.G. (1996). High-voltage DC transmission: a power electronics workhorse. IEEE Spectrum 33 (4): 63–72. Arrillaga, J. (1998). High Voltage Direct Current Transmission, 2e. London: The Institution of Electrical Engineers.

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Proceedings of the IEEE Power & Energy Society General Meeting, 2009. PES ’09, Calgary, 2009. Z.R. Zhang, Z.D. Yin and F.X. Hu, “Research of Multi-Farms Transmission of Distributed Generation Based on HVDC Light”, in Proceedings of the International Conference on Power System Technology, 2006. PowerCon 2006, Chongqing, 2006. W. Pan, Y. Chang and H. Chen, “Hybrid multi-terminal HVDC system for large scale wind power”, in Proceedings of the IEEE PES Power Systems Conference and Exposition, 2006, Atlanta, 2006. Van Hertem, D. and Ghandhari, M. (2010). Multi-terminal VSC HVDC for the European supergrid: obstacles. Renewable and Sustainable Energy Reviews 14 (9): 3156–3163. Haileselassie, T.M. (2008). Control of Multi-Terminal VSC-HVDC, Master of Science in Energy and Environment. Norwegian University of Science and Technology. C. Zhao and Y. Sun, “Study on control strategies to improve the stability of multi-infeed HVDC systems applying VSC-HVDC”, in Canadian Conference on Electrical and Computer Engineering. CCECE ’06, Ottawa, 2006. T.K. Vrana, “Review of HVDC component ratings: XLPE cables and VSC converters”, in Proceedings of the 2016 IEEE International Energy Conference (ENERGYCON), Leuven, Belgium, 2016.

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3 Power Flows 3.1 Introduction The main objective of a power ﬂow study is to assess the steady-state operating condition of an electrical power network operating under a ﬁxed set of generation and load pattern. Such assessment may conveniently be carried out by determining the voltage magnitudes and phase angles in all buses of the network; the ﬂow of active and reactive powers in all branches of the network; the active and reactive powers contributed by each generator; and the total network active and reactive power losses [1]. The planning and daily operation of modern power systems call for numerous power ﬂow studies. If the study indicates that there are voltage magnitudes outside bounds at certain points of the network, then appropriate control actions become necessary in order to regulate voltage magnitude. Similarly, if the study predicts that the power ﬂow in a given transmission line or transformer is beyond the power-carrying capability of the line or transformer, then a suitable control action is required. Voltage magnitude regulation is achieved by controlling the amount of reactive power generated/absorbed at key points of the network, as well as by controlling the ﬂow of reactive power throughout the network. In this application it is common practice to represent the power network using only positive-sequence parameters, based on the assumption that the power network is perfectly balanced in terms of both network impedance and generator voltage excitation [2]. From the mathematical modelling point of view, the power ﬂow exercise consists in solving a set of nonlinear, algebraic equations that describe the power network under steady-state operating conditions. Over the years, many approaches have been used to solve the nodal power ﬂow equations; the early methods employed the Gauss-Seidel technique and nodal impedance matrix concepts. These were followed by applications of the Newton-Raphson method, with the nodal complex voltages expressed in either polar coordinates or rectangular coordinates. Simpliﬁed formulations of the Newton-Raphson method that exhibit a constant Jacobian matrix, such as the fast decoupled and second-order Newton-Raphson methods, were put forward and used with varying degrees of success. At the present time, with the beneﬁt of modern computers with practically unlimited storage capacity, the power ﬂow Newton-Raphson method may be considered the de facto standard in conventional power ﬂow computations [3].

VSC-FACTS-HVDC: Analysis, Modelling and Simulation in Power Grids, First Edition. Enrique Acha, Pedro Roncero-Sánchez, Antonio de la Villa Jaén, Luis M. Castro and Behzad Kazemtabrizi. © 2019 John Wiley & Sons Ltd. Published 2019 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/acha_vsc_facts

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3.2 Power Network Modelling Conventional power ﬂow formulations take the approach of balancing to zero the complex powers leaving and entering a given node of the electrical power network. To be commensurate with this principle, which stems from the fundamental law of energy conservation, all the network elements are taken to be connected to one or more nodes, depending on their nature. Power generators are represented as power injections into the node to which they are connected and load points are represented as power drains at the node to which they are connected. The latter may be made to reﬂect a degree of voltage dependency to widen the representation of electrical power-consuming equipment, such as electric motors of various kinds, electric lighting, heating and power electronics-based appliances. For the purpose of electrical power transmission, the main pieces of equipment are overhead transmission lines and underground cables, power transformers, and series and shunt volt ampere reactive (VAR) compensating equipment of various kinds. Early VAR compensation equipment used iron-core inductors and switchable banks of capacitors with mechanical switches. Modern compensation equipment uses power electronic switches and its functionality has been extended to exert active power control as opposed to only reactive power control. This comes under the realm of the power technology area known as ﬂexible alternating current transmission systems (FACTSs) [4]. 3.2.1

Transmission Lines Modelling

In general, the magnetic and electric ﬁelds associated with an AC current-carrying overhead transmission lines and underground cables are suitably represented by series inductances and shunt capacitances which, in turn, may be represented by series impedances and shunt admittances at a given operating frequency. For the purpose of fundamental-frequency, power system application studies, the relationships between the nodal currents and nodal voltages at the sending end and the receiving end of these elements are well accommodated in a nodal transfer admittance matrix, as shown in (3.1): ][ ] [ ] [ −1 −1 Ik Zseries + Y shunt ∕2 −Zseries Vk = (3.1) −1 −1 Im V −Zseries Zseries + Y shunt ∕2 m where Zseries = R + j𝜔L is the series combination of the cable’s resistance and the inductive reactance at the fundamental frequency: 𝜔 = 2𝜋f and, say, f = 50 Hz. The shunt admittance accounts for the cable’s dielectric losses and capacitive eﬀects: Y shunt = G + j𝜔C. Notice that when the cable is an overhead conductor with no external insulation, the dielectric losses are practically zero, hence Y shunt contains only the imaginary part. 3.2.2

Conventional Transformers Modelling

The transformer’s operation at power frequencies is well represented by a series impedance linking the primary and secondary windings’ nodes: k and m. This involves

3.2 Power Network Modelling

a per-unit representation of the primary and secondary windings’ impedances with, say, the secondary winding impedance referred to as the transformer’s primary side. In nodal admittance form, this representation takes the following form: ][ ] [ ] [ Ik Y SC −Y SC V k = (3.2) Im −Y SC Y SC Vm −1

where Y SC = Z SC = 1∕(Rt + jXSC ) is the inverse of the series combination of the transformer’s resistance and the short-circuit reactance at the fundamental frequency, derived from standard short-circuit tests. 3.2.3

LTC Transformers Modelling

Power transformers may be made to have a degree of voltage-regulating capabilities by ﬁtting one of the main windings with an auxiliary winding having variable turns ratio capability; the auxiliary winding is placed in series with the main winding. It is normal to place the auxiliary winding in series with the primary winding since the current is lower in this winding. The tap-changing transformer, with the tap T placed on the primary winding, is illustrated in Figure 3.1. The following basic relationships may be established with reference to the circuit of Figure 3.1: V p = TV m TI k = −I m V p = V k − I k ZSC and suitable combination of these equations yields the following nodal transfer admittance matrix: ][ ] [ ] [ Ik Y SC −TY SC V k = (3.3) Im −TY SC T 2 Y SC V m 3.2.4

Phase-Shifting Transformers Modelling

If the tap T is complex, with magnitude and phase angle of 1 and 𝜑, respectively, the nodal admittance matrix would correspond to that of a phase-shifting transformer: [ ] [ ][ ] Ik Y SC −Y SC ⋅ (cos 𝜑 + jsin𝜑) V k = (3.4) Im Y SC Vm −Y SC ⋅ (cos 𝜑 − jsin𝜑)

Ik

Vk

Rt

T:1

XSC Vp

Vm

Im

Figure 3.1 LTC representation, with the per-unit impedance in the primary side of the transformer.

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3.2.5

Compound Transformers Modelling

It should be noted that the system of equations is asymmetrical and that the ensuing equivalent circuit is a non-reciprocal one, hence a passive circuit representation is not possible. In a circuit representation of a transformer where the tap is complex and the magnitude is not 1 but T, the nodal admittance matrix would be as follows: [ ] [ ][ ] Ik Y SC −Y SC ⋅ T(cos 𝜑 + jsin𝜑) V k = (3.5) Im T 2 Y SC Vm −Y SC ⋅ T(cos 𝜑 − jsin𝜑) 3.2.6

Series and Shunt Compensation Modelling

For completeness, a series component such as an inductor or a capacitor has the following nodal transfer admittance representation: [ ] [ ][ ] Ik Y series −Y series V k = (3.6) Im −Y series Y series Vm where Y series = 1∕j𝜔L or Y series = j𝜔C for the series reactor and series capacitor, respectively. Likewise, a shunt inductor or capacitor has the following nodal admittance representation: [I k ] = [Y shunt ][V k ]

(3.7)

where Y shunt = 1∕j𝜔L or Y shunt = j𝜔C for the shunt reactor and shunt capacitor, respectively. 3.2.7

Load Modelling

A composite power load point in the network, say load point k, may comprise a combination of three kinds of power loads, in various amounts. They are suitably represented by the power expressions given in (3.8) and (3.9): ] [ ( ( ) ) Vk 2 Vk PLk = P0k ⋅ ap ⋅ (3.8) + bp ⋅ + cp V0k V0k ] [ ( ( ) ) Vk 2 Vk QLk = Q0k ⋅ aq ⋅ (3.9) + bq ⋅ + cq V0k V0k The ﬁrst, second and third terms represent those portions of the load that exhibit constant impedance, constant current and constant power behaviours, respectively. Notice that diﬀerent coeﬃcients are used for active power and reactive power. 3.2.8

Network Nodal Admittance

The various nodal transfer admittance matrices representing the individual components – Eqs. (3.1)–(3.9) – are easily assembled to represent an electrical power network

3.3 Peculiarities of the Power Flow Formulation

of practically any size. The result would be the nodal admittance matrix of the entire electrical power system having the following structure for a network of n buses: ⎡I 1 ⎤ ⎡Y 11 ⎢I ⎥ ⎢Y ⎢ 2 ⎥ ⎢ 21 ⎢⋮⎥ = ⎢ ⋮ ⎢I k ⎥ ⎢Y k1 ⎢⋮⎥ ⎢ ⋮ ⎢ ⎥ ⎢ ⎣I n ⎦ ⎣Y n1

Y 12 Y 22 ⋮ Y k2 ⋮ Y n2

··· ··· ⋱ ··· ⋱ ···

Y 1k Y 2k ⋮ Y kk ⋮ Y nk

··· ··· ⋱ ··· ⋱ ···

Y 1n ⎤ ⎡V 1 ⎤ Y 2n ⎥ ⎢V 2 ⎥ ⎥⎢ ⎥ ⋮ ⎥⎢ ⋮ ⎥ Y kn ⎥ ⎢V k ⎥ ⋮ ⎥⎢ ⋮ ⎥ ⎥⎢ ⎥ Y nn ⎦ ⎣V n ⎦

(3.10)

It should be noted that inclusion of the powers Pk and Qk in expressions (3.8)–(3.9) requires conversion into admittances. This is done by dividing these powers by the square of the voltage magnitude at node k. Moreover, the power expression (3.9) is negated. The entries along the main diagonal, (Y 11 , Y 22 , Y kk , Y nn ), are termed self-elements of the nodal admittance matrix and the remaining entries are termed mutual terms. Most entries in the matrix equation (3.10) are complex and will comprise a real part and an imaginary part, say, Y kk = Gkk + jBkk and Y kn = Gkn + jBkn . In general, any self-element of the nodal admittance matrix in (3.10) corresponds to the sum of all the branch admittances that connect to that node. Moreover, a mutual term of the nodal admittance matrix corresponds to the negative value of the branch admittance connecting two nodes, say, branch-linking nodes k and n will have the negative of its admittance value placed in locations kn and nk of the matrix. If no direct connection exists between two nodes, its corresponding entry in the matrix will be nil. Sometimes it is convenient to represent the nodal admittance matrix equation (3.10) using real quantities as opposed to complex quantities, hence: ⎡ IRe,1 ⎤ ⎡G11 −B11 ⎢IIm,1 ⎥ ⎢ B11 G11 ⎥ ⎢ ⎢I ⎢ Re,2 ⎥ ⎢G21 −B21 ⎢IIm,2 ⎥ ⎢ B21 G21 ⎢ ⋮ ⎥ ⎢ ⋮ ⋮ ⎥ = ⎢G −B ⎢I k1 ⎢ Re,k ⎥ ⎢ k1 ⎢IIm,k ⎥ ⎢ Bk1 Gk1 ⎢ ⋮ ⎥ ⎢ ⋮ ⋮ ⎥ ⎢G −B ⎢I n1 ⎢ Re,n ⎥ ⎢ n1 ⎣IIm,n ⎦ ⎣ Bn1 Gn1

G12 −B12 B12 G12 G22 −B22 B22 G22 ⋮ ⋮ Gk2 −Bk2 Bk2 Gk2 ⋮ ⋮ Gn2 −Bn2 Bn2 Gn2

··· ··· ··· ··· ⋱ ··· ··· ⋱ ··· ···

G1k −B1k B1k G1k G2k −B2k B2k G2k ⋮ ⋮ Gkk −Bkk Bkk Gkk ⋮ ⋮ Gnk −Bnk Bnk Gnk

··· ··· ··· ··· ⋱ ··· ··· ⋱ ··· ···

G1n −B1n ⎤ ⎡ VRe,1 ⎤ B1n G1n ⎥ ⎢VIm,1 ⎥ G2n −B2n ⎥⎥ ⎢⎢ VRe,2 ⎥⎥ B2n G2n ⎥ ⎢VIm,2 ⎥ ⋮ ⋮ ⎥⎢ ⋮ ⎥ (3.11) Gkn −Bkn ⎥ ⎢ VRe,k ⎥ ⎥⎢ ⎥ Bkn Gkn ⎥ ⎢VIm,k ⎥ ⋮ ⋮ ⎥⎢ ⋮ ⎥ Gnn −Bnn ⎥ ⎢ VRe,n ⎥ ⎥⎢ ⎥ Bnn Gnn ⎦ ⎣VIm,n ⎦

In the above, note the following complex numbers equivalence: I = Y V ⇔ IRe + jIIm = (G + jB)(VRe + jVIm ) [ ] [ ][ ] IRe G −B VRe ⇔ = IIm B G VIm

(3.12)

3.3 Peculiarities of the Power Flow Formulation In conventional AC power ﬂow studies, buses can be of three diﬀerent types: voltagecontrolled, load and reference. If synchronous generation is available at the bus with

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suﬃcient reactive power provisions, the nodal voltage magnitude may be regulated at a speciﬁed value and the bus will be of the voltage-controlled type; if no synchronous generation exists at the bus, voltage magnitude cannot be regulated and the node will be of the load type. On the other hand, one synchronous generator bus in the network must be selected to act as reference for all the other buses in the network. In the parlance of power systems engineers the reference bus is termed slack bus or swing bus and its main function is to provide for transmission power losses and any shortfall in generation that the other generators in the network are unable to meet. In the power ﬂow application, the power injected at a bus is well handled by using the concept of net power, which is the diﬀerence between the scheduled generation (positive) and the total load (negative) at the bus. If the bus contains neither generation nor load, its net nodal power carries a zero value. Each bus in the network has associated two power equations and four nodal quantities: voltage magnitude, voltage phase angle, net active power and net reactive power. Two of these quantities are speciﬁed a priori and the other two are calculated using the two available power equations – one is for active power and the other is for reactive power. In a voltage-controlled node, the two speciﬁed quantities are the voltage magnitude and the net active power injected into the bus; in a load node, the two speciﬁed quantities are the net active power injected into the bus and the reactive power injected into the bus; in the reference node, the two speciﬁed quantities are the voltage magnitude and the voltage phase angle. As a complement of the speciﬁed nodal variables, the calculated variables are the net reactive power injected into the bus and the voltage phase angle in a voltage-controlled node; the voltage magnitude at the bus and the voltage phase angle in a load node; the net active power injected into the bus and the net reactive power injected into the bus in the slack node. Because of the nature of the speciﬁed variables in the voltage-controlled bus and the load bus, they are also known as PV-type bus and PQ-type bus, respectively. Notice that the nodal power injections may be either positive or negative. Three main control systems directly aﬀect the turbine-generator set: the boiler’s ﬁring control, the governor control and the excitation system control. The excitation system consists of the exciter and the automatic voltage regulator (AVR). The latter regulates the generator terminal voltage by controlling the amount of current supplied to the ﬁeld winding by the exciter. For the purpose of steady-state analysis, it is assumed that the three control systems act in an idealized manner, enabling the synchronous generator to produce constant power output, to run at synchronous speed and to regulate voltage magnitude at the generator’s terminal with no delay and up to its reactive power design limits. These are the reasons why in a voltage-controlled bus the nodal voltage magnitude and the active power are selected to be known quantities (and speciﬁed a priori) in the power ﬂow solution. On the other hand, in a load-type bus, the active power and the reactive power are taken to be available quantities derived from measurements, say supervisory control and data acquisition (SCADA) measurements. In network expansion studies, the active and reactive powers in load type buses may correspond to forecasted values determined from load growth assessments. In the slack bus, the voltage magnitude is speciﬁed because in essence this is a generator-type bus or the strongest point in the network (with the largest reserve of reactive power) and its phase voltage angle provides a reference for the phase voltage angles of all the other

3.4 The Nodal Power Flow Equations

buses in the network – it is quite normal to select a value of zero for the phase voltage angle of the slack bus but practically any other value may be selected instead. It should be noted that within the context of power ﬂows in DC power circuits, no complex variables exist, i.e. voltages and currents are real quantities and reactive power does not exist. Apart from this simpliﬁcation in variables, no other diﬀerence seems to exist between the power ﬂow solutions of an AC power circuit and a DC power circuit. As a matter of fact, the power ﬂow solution of DC circuits may be carried out with no difﬁculty using a conventional power ﬂow computer program which is normally intended for solving AC power circuits. The results furnished by an AC power ﬂow algorithm when solving a DC power circuit show nodal voltages with zero phase angle voltages at all nodes and zero reactive power ﬂows throughout the DC power circuit. It is quite clear that for this to happen, the slack’s phase angle voltage ought to be initialized at zero value. Moreover, it is common sense to interpret such complex voltages (provided by the AC power ﬂow algorithm) with phase angle voltages of zero to be real quantities, as one would expect to see in DC power circuits. Combined AC and DC power circuits employ frequency converter equipment of various kinds at the interface between AC and DC parts of the power circuit. These power converters require suitable modelling representation within the combined AC-DC power ﬂow solution. Strictly speaking, the frequency does not enter as state variable in conventional power ﬂow solutions, but it is clear that a HVDC link decouples, frequency-wise, the AC power circuits connected to the rectiﬁer and the AC power circuit connected to the inverter. As a consequence of this frequency decoupling, each AC circuit has its own slack node and no relationship exists between the nodal phase angle voltages of the two AC power circuits [5]. The above points apply also to multi-terminal HVDC systems where each AC power circuit connected to a terminal of the HVDC link will require its own slack node. In even more specialized power ﬂow studies, such as the voltage source converter VSC-HVDC-infeed of micro-grids where conventional synchronous generation is not readily available, the inverter’s AC internal angle plays the role of slack node. This concept also applies to the case of oﬀshore wind farms where the energy is transferred to the mainland AC system using VSC-HVDC transmission.

3.4 The Nodal Power Flow Equations The equation describing the complex power injection at bus l is the starting point for deriving nodal active and reactive power ﬂow equations suitable for the Newton-Raphson power ﬂow algorithm. This may be established by appealing to Kirchhoﬀ ’s Current Law (KCL), where the current balance at bus l is exempliﬁed in Figure 3.2a and its extension to complex powers balance is exempliﬁed in Figure 3.2b. The current balance in Figure 3.2a may be expressed as follows: Il =

n ∑ m=1

I lm =

n ∑

Y lm V m

(3.13)

m=1

Extending this to include all n nodes of the network yields the nodal admittance matrix expression (3.10), which in compact form may be written as follows: I = YV

(3.14)

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3 Power Flows

Vi

Vi

Ii1

Ii

Iik

Pi1 + jQi1

Pi + jQi

Pik + jQik

Iin

Pin + jQin

(a)

(b)

Figure 3.2 Current and power balances in bus l.

The nodal complex voltages V may be expressed in either rectangular coordinates, i.e. V = e + jf, or polar coordinates, i.e. V = Vej𝛉 . Similarly, the admittances may be expressed in either rectangular coordinates, i.e. Y = G + jB, or polar coordinates, i.e. Y = Yej𝛄 . Using rectangular notation as suggested in (3.12), Eq. (3.14) transforms into the following one: ][ ] [ ] [ G −B e Ire = (3.15) Iim B G f Moreover, the nodal complex power equations in rectangular form may be obtained by the matrix product of the nodal complex voltages and the complex conjugate of the nodal currents: ] [ ] [ ] [ ]∗ [ eIre + fIim P e −f Ire = (3.16) = fIre − eIim Q f e Iim In a more detailed form: ] [ ] [ ] ([ ] [ ])∗ [ e(Ge − Bf) + f(Be + Gf) P e −f G −B e = = ⋅ f(Ge − Bf) − e(Be + Gf) Q f e B G f

(3.17)

3.5 The Newton-Raphson Method in Rectangular Coordinates The nonlinear set of algebraic equations in (3.17) may be used to calculate the nodal voltages that exist at all nodes of the power network, corresponding to a given set of generation and load patterns. These calculated powers will match the speciﬁed net powers up to a speciﬁed tolerance. In turns, the net active power and net reactive power at a given node correspond to the diﬀerence between the generated power and the consumed power at that node. This gives rise to the nodal power mismatches which are solved by iteration using any suitable numerical method, such as the Newton-Raphson: 𝚫P = (Pgen − Pload ) − Pcal = Pnet − Pcal

(3.18)

𝚫Q = (Qgen − Qload ) − Qcal = Qnet − Qcal

(3.19)

gen

gen

load

load

where P , Q , P , Q are vectors of speciﬁed active and reactive powers at all nodes of the network; Pnet , Qnet are the corresponding diﬀerence vectors between nodal

3.5 The Newton-Raphson Method in Rectangular Coordinates

generation and demand; and Pcal , Qcal are vectors of calculated powers as determined by (3.17). It is very unlikely that the nodal voltages used in (3.17) in the ﬁrst instance to calculate Pcal , Qcal will yield suﬃcient accuracy to fulﬁl the following condition: 𝚫P ≤ 10−12 and 𝚫Q ≤ 10−12 . Hence, successive approximations (iterations) are normally required. 3.5.1

The Linearized Equations

The expansion of (3.18)–(3.19) in a Taylor series form and retaining only the ﬁrst-order derivative terms, expressed in compact matrix notation, yields: [ ] [ ][ ] 𝚫P 𝜕𝚫P∕𝜕e 𝜕𝚫P∕𝜕f 𝚫e =− (3.20) 𝚫Q 𝜕𝚫Q∕𝜕e 𝜕𝚫Q∕𝜕f 𝚫f The linearized system of Eq. (3.20) is solved for the vector of nodal voltage incre[ ]t ments 𝚫e 𝚫f and with it the nodal voltages from the previous iteration, say (r−1), are updated: e(r) = e(r−1) + 𝚫e(r) f

(r)

=f

(r−1)

+ 𝚫f

(r)

(3.21) (3.22)

where r is the iteration counter. The partial derivatives may be obtained in a number of ways. For instance, the form developed below makes for rather clean implementations and straightforward solutions. Starting from the voltage derivatives of the product voltage times the conjugate of the current: ] } [ ]∗ ) [ [ net ( { ] e −f I 𝚫P P − Pcal 1 1 1 =− × Re ⋅ ⋅ ⋅ DV = IIm f e 𝜕e, 𝜕f 𝚫Q 𝜕e, 𝜕f Qnet − Qcal 𝜕e, 𝜕f (3.23) where Pnet and Qnet are constant entries; DV {⋅} is a block-diagonal matrix of nodal voltages and the current vector accommodates contiguous values of the real and imaginary parts of the current of a given node. Elaborating further the expression (3.23): { { } [ ]∗ } [ ] e −f e −f I 𝚫P 1 1 × Re − DV ⋅ ⋅ DV =− f e f e IIm 𝜕e, 𝜕f 𝚫Q 𝜕e, 𝜕f [ ]∗ I 1 × ⋅ Re 𝜕e, 𝜕f IIm { }[ { } { } ] 𝟏 𝟎 IRe IIm e −f 𝜕e∕𝜕e = −DU − DV × DI ∗ −IIm IRe 𝟎 𝟏 f e 𝜕f∕𝜕f ([ ]∗ { }∗ ) G −B e −f 1 × × ⋅ DV ∗ B G f e 𝜕e, 𝜕f { } { } IRe IIm e −f = −DI ∗ − DV −IIm IRe f e [ ] { }[ ] G B 𝟏 𝟎 𝜕e∕𝜕e × × DU ∗ −B G 𝟎 −𝟏 𝜕f∕𝜕f { } { } [ ] IRe IIm e −f G −B = −DI ∗ − DV × (3.24) −IIm IRe f e −B −G

107

108

3 Power Flows

The following slightly modiﬁed result is preferred for implementation because it makes use of the already available nodal admittance matrix in (3.11): { } { }[ ] [ ] IRe IIm e f G −B 𝜕𝚫P∕𝜕e 𝜕𝚫P∕𝜕f −D (3.25) = −DI ∗ −IIm IRe f −e B G 𝜕𝚫∕𝜕𝚫e 𝜕𝚫Q∕𝜕f Further to the block-diagonal matrix of nodal voltages, DV {⋅} in (3.24), the conjugated nodal currents and voltages are also arranged in the same format, DI* {⋅} and DV* {⋅}, respectively. Notice also that the diﬀerentiation process in (3.24) yields a block-diagonal unit matrix and a block-diagonal matrix resulting from the derivative of the conjugated voltage, DU {⋅} and DU* {⋅}, respectively. The Jacobian matrix of the Newton-Raphson power ﬂow method in (3.25) involves the matrix product of a band-diagonal matrix of voltages and the nodal admittance matrix added to a band-diagonal matrix of nodal currents. Notice that the Jacobian matrix yields a negative term which with the negative term of linearized expression (3.20) becomes an overall positive Jacobian term. Notice also that the arrangement of the band-diagonal matrix of nodal voltages, D{⋅}, in (3.25) diﬀers from both DV {⋅} and DV* {⋅}. Eqs. (3.17)–(3.22) and (3.25) are valid for cases when all n−1 buses in the network are load type – it should be remembered that one bus is selected to be the slack bus. Voltage-controlled buses, where the generator has not reached its reactive power ceiling, require diﬀerent equations from those of load-type buses. For a given voltage-controlled node, say node l, the mismatch voltage ΔV l in (3.26) replaces the entry corresponding to reactive power mismatch ΔQl in (3.20): √ spec ΔUl = Ul − e2l + fl2 (3.26) spec

where Ul is the speciﬁed voltage to be attained, and el and f l are the real and imaginary components of the calculated voltage at a given iteration (r). The corresponding Jacobian entries for the voltage-controlled buses are: 𝜕Ul e = −√ l 𝜕el e2l + fl2

(3.27)

𝜕Ul f = −√ l 𝜕fl e2l + fl2

(3.28)

The various equations used to program the Newton-Raphson power ﬂow algorithm in rectangular coordinates may be summed up by the ﬂow diagram presented in Figure 3.3. Owing to the nature of these voltage equations, all locations in the Jacobian matrix corresponding to the row where ΔU l has replaced ΔQl will yield zero entries except the two self-locations corresponding to el and f l . 3.5.2

Convergence Characteristics of the Newton-Raphson Method

The convergence characteristics of the Newton-Raphson power ﬂow method in either rectangular or polar coordinates are known to be very strong, particularly if the problem at hand is suitably initialized – it is said to yield quadratic convergence. The reason for such a ﬂawless performance may be better appreciated in connection with the

3.5 The Newton-Raphson Method in Rectangular Coordinates

Read network data and initial conditions

Calculate active and reactive powers – use (3.17)

Calculate power mismatches – use (3.18) and (3.19) or (3.26) in cases of PV nodes

Are all power mismatch values smaller than a specified tolerance (e.g. 10−12)?

Yes

STOP – Power flow converged successfully in r iterations

No Evaluate Jacobian elements − use (4.25) or (4.27)−(4.28) in cases of PV nodes r=r+1 Solve the linearized system of equations – use (4.20)

Update nodal voltages – use (4.21) and (4.22)

r > max. number of iterations

No

Yes

STOP – Power flow did not converge in rmax iterations to the specified tolerance

Figure 3.3 Flow diagram for the Newton-Raphson power ﬂow in rectangular coordinates.

performance towards the convergence of a single variable function, say f (x), as shown in Figure 3.4. However, if the iterative solution is poorly initialized, convergence may be impaired and the method may even diverge. 3.5.3

Initialization of Newton-Raphson Power Flow Solutions

Owing to practical reasons, the electrical power network is designed to have very different rated voltages at the generation, transmission, distribution and utilization systems – the various voltage levels in the overall network are separated by step-up and step-down power transformers, according to requirement. Power engineers have wisely used the per-unit system of units. In essence, the per-unit system is a normalized system of units where the rated values of the interface transformers may be taken to be the base voltage values against which all voltage values in their respective zone of inﬂuence are normalized – so, in a power network with no load attached, neglecting charging currents

109

110

3 Power Flows

Quadratic convergence, e.g.: f(x)

Δf (0) = 1

Δx(1) = –Δf (0)/fʹ (0)

Δf (1) = 0.1

Δx(2) = –Δf (1)/fʹ (1)

f ʹ (0)

Δf (2) = 10−2

Δx(3) = –Δf (2)/f ʹ (2)

Δf (3) = 10−4

Δx(4) = –Δf (3)/fʹ (3) Δf (0)

Δf (4) = 10−8 Δf (5) = 10−16

f ʹ (1)

Δf (1)

f ʹ (2) x(3)

x(2)

Δx(3)

x(1) Δx(2)

x(0)

x

Δx(1)

Figure 3.4 Convergence characteristic of the Newton-Raphson method.

in equipment and with all synchronous generators operating at their rated voltages, the voltage proﬁle throughout the network will be one per-unit. When the power network is operating under normal loading conditions, the nodal voltages will depart from their rated 1 p.u. values to some degree but healthy operation of the power systems will call for maximum deviations of 0.06 around the rated 1 p.u. value. Concerning power, which is the other standard base value to be speciﬁed, a judicious single value of power is selected for the whole power system under analysis – in academic circles, a base power of 100 MVA is normally selected for high-voltage power networks whereas a base value of 100 kVA looks more amenable for low-voltage distribution systems. The working practice of using per-unit values in a number of power system applications is rather fortuitous for cases when the Newton-Raphson method is used to solve the associated nonlinear system of equations – which is carried out by iteration. It has been observed that the Newton-Raphson is at its best when the state variables are within the same order of magnitude and when they are initialized to be within close range of the ﬁnal solution. Thanks to the per-unit practice, all nodal voltage magnitudes that enter the Newton-Raphson formulation as variables may be initialized at 1 p.u. Notice that there will be some nodal voltage magnitudes that will be entered as control variables and that they may not necessarily be speciﬁed to keep a 1 p.u. value – generator buses may be cases in point. Phase angle voltages are handled in radians and it is customary to set them to zero at the start of the iterative solution. In power systems applications where a reference bus is mandatory – for instance, the slack bus in a power ﬂow study – its phase angle is normally taken to be zero, a value that does not change throughout the iterative solution because the phase angle of the slack bus is only a reference against which all other phase angle voltages in the network are measured. It should be remarked that the zero phase angle at the slack is only a convenience practice and that any other value might work just as well – upon convergence to the same tolerance, the only discernible diﬀerence

3.5 The Newton-Raphson Method in Rectangular Coordinates

between two solutions that use diﬀerent phase angle voltages at the slack bus would be the phase angle voltages throughout the network. However, their relative values, with respect to the slack bus, would still coincide. The discussion above impinges directly onto state variable initialization of the Newton-Raphson method in rectangular coordinates – where the real and the imaginary parts of most nodal voltages are set to 1 and 0 p.u. at the start. Such initialization, known as ﬂat voltage proﬁle, may cause the Newton-Raphson method in rectangular coordinates to crash during the ﬁrst iteration if all the branch elements connecting to a bus of the network contain no resistance – the reason is that one or more entries in the diagonal of the Jacobian will become zero. However, if this is a problem, the Jacobian and vector of incremental state variable values in the right-hand-side of Eq. (3.20) may be arranged diﬀerently to prevent this possibility. The voltage magnitude and phase angle of the series voltage sources, such as the one found in the uniﬁed power ﬂow controller (UPFC), requires special care at the initialization stage – its magnitude and phase angle values are likely to be around 0.1 p.u. and π/2 rad, respectively. Initializing at 1 p.u. and 0 rad would be seen by the Newton-Raphson solutions as being too far away from the ﬁnal solution; at best, impairing its quadratic convergence and at worst, causing the algorithm to diverge. 3.5.4 Incorporation of PMU Information in Newton-Raphson Power Flow Solutions The widespread adoption of the phasor measurement unit (PMU) technology may change the practice of initializing nodal phase angle voltages at zero values. The reason is that a PMU is designed to measure the voltage magnitude and the phase angle as well as the complex current at a point in the network – this is actual information, as it exists at the point in time at which the measurement was carried out [6]. It has been observed that the measured phase angles are normally very far away from zero – perhaps in the region of −π/2 rad. Values measured by PMU would remain constant throughout the iterative power ﬂow solution, with the remaining phase angle voltages being pulled in their direction. Hence, customary initialization of the non-PMU phase angle voltages may be seen as being too far away from the ﬁnal solution values, resulting in an impaired convergence. However, it should be borne in mind that the phase angle diﬀerence between adjacent nodes remains relatively small, just as in the conventional power ﬂow solution. Thus, it makes common sense to initialize the phase angle voltages of nodes with no PMW measurements at values close to those neighbouring nodes with available PMU measurements. Overall, the new information calls for a slight reformulation of the linearized system of Eq. (3.20), which is explicitly illustrated with speciﬁc reference to Eqs. (3.29) and (3.30). The ﬁrst equation shows the case when an n-node system comprises the slack node (taken to be node 1), PQ-type nodes and PV -type nodes. The slack node is essentially a PV -type bus but one where the voltage phase angle is known a priori. Hence the voltage magnitude and phase angle in this node remain constant from iteration to iteration, i.e. the real and imaginary part of the nodal voltage. This node would normally contain the largest reserve of reactive power in the system and in conventional power ﬂow solutions is singled out to cater for the network power losses. We shall refer to this slack node as a ‘physical’ slack node.

111

112

3 Power Flows

If PMU information becomes available, say at node k, then V k and 𝜃 k are known a priori and ek and f k remain constant throughout the iterative solution. Hence, node k changes its status from being a PQ-type node to being a ‘mathematical’ slack node where the voltage magnitude and phase angle remain ﬁxed as given by the PMU measurement; the corresponding implementation is shown in Eq. (3.30). ⎡ ΔP1 ⎤ ⎡1 ⎢ΔQ1 ⎥ ⎢0 ⎢ ⋮ ⎥ ⎢⋮ ⎥ ⎢ ⎢ ΔP ⎢ k ⎥ = − ⎢0 ⎢ΔQk ⎥ ⎢0 ⎢ ⋮ ⎥ ⎢⋮ ⎥ ⎢ ⎢ ΔP ⎢ n⎥ ⎢0 ⎣ΔVn ⎦ ⎣0

0 1 ⋮ 0 0 ⋮ 0 0

··· 0 0 ··· 0 0 ⋮ ⋮ ⋱ · · · 𝜕ΔPk ∕𝜕ek 𝜕ΔPk ∕𝜕fk · · · 𝜕ΔQk ∕𝜕ek 𝜕ΔQk ∕𝜕fk ⋮ ⋮ ⋱ · · · 𝜕ΔPn ∕𝜕ek 𝜕ΔPn ∕𝜕fk ··· 0 0

⎡Δe1 = 0 ⎤ ⎢ Δf1 = 0 ⎥ ⎢ ⋮ ⎥ ⎥ ⎢ Δek ⎥ ×⎢ ⎢ Δfk ⎥ ⎢ ⋮ ⎥ ⎥ ⎢ ⎢ Δen ⎥ ⎣ Δfn ⎦ 1 ΔP ⎤ ⎡ ⎡ 0 ··· 0 ⎢ΔQ1 ⎥ ⎢0 1 · · · 0 ⎢ ⋮ ⎥ ⎢⋮ ⋮ ⋱ ⋮ ⎥ ⎢ ⎢ ΔP ⎢ k ⎥ = − ⎢0 0 · · · 1 ⎢ΔQk ⎥ ⎢0 0 · · · 0 ⎢ ⋮ ⎥ ⎢⋮ ⋮ ⋱ ⋮ ⎥ ⎢ ⎢ ΔP ⎢ n⎥ ⎢0 0 · · · 0 ⎣ΔVn ⎦ ⎣0 0 · · · 0

··· ··· ⋱ ··· ··· ⋱ ··· ···

0 0 ⋮ 𝜕ΔPk ∕𝜕en 𝜕ΔQk ∕𝜕en ⋮ 𝜕ΔPn ∕𝜕en 𝜕ΔVn ∕𝜕en

0 ⎤ 0 ⎥ ⎥ ⋮ ⎥ 𝜕ΔPk ∕𝜕fn ⎥ 𝜕ΔQk ∕𝜕fn ⎥ ⎥ ⋮ ⎥ 𝜕ΔPn ∕𝜕fn ⎥ 𝜕ΔVn ∕𝜕fn ⎦

(3.29)

0 0 ⋮ 0 1 ⋮ 0 0

··· 0 ··· 0 ⋮ ⋱ ··· 0 ··· 0 ⋮ ⋱ · · · 𝜕ΔPn ∕𝜕en · · · 𝜕ΔVn ∕𝜕en

0 ⎤ ⎡Δe1 = 0 ⎤ 0 ⎥ ⎢ Δf1 = 0 ⎥ ⎥⎢ ⋮ ⎥ ⋮ ⎥⎢ ⎥ 0 ⎥ ⎢Δek = 0 ⎥ 0 ⎥ ⎢ Δfk = 0 ⎥ ⎥⎢ ⋮ ⎥ ⋮ ⎥⎢ ⎥ 𝜕ΔPn ∕𝜕fn ⎥ ⎢ Δen ⎥ 𝜕ΔVn ∕𝜕fn ⎦ ⎣ Δfn ⎦

(3.30)

If PMU information becomes available in more than one bus, we repeat the procedure carried out in Eq. (3.30), with the ensuing system of linearized equations containing the actual or physical slack node and as many mathematical slack nodes as there are PMU voltage measurements in the system. It should be noted that some of the newly created slack nodes may not even contain synchronous generation, such as the PQ-type buses. In these buses, steps must be taken to suitably constrain the net reactive power. Meanwhile, most PV -type buses will contain generation and the phasor voltages furnished by the PMU measurement at the corresponding bus will correctly account for the net power injected at that bus.

3.6 The Voltage Source Converter Model The STATCOM model comprises the series connection of a VSC and a transformer whose primary winding is shunt-connected with the AC power network [7]. Physically, the VSC is built as a two-level or a multilevel inverter that uses a converter bridge made up of self-commutating switches driven by is pulse width modulation (PWM) control

3.6 The Voltage Source Converter Model

+ EDC −

VʹvR

CDC

φ

ma

(a) DC circuit

AC circuit

I0 = 0 V0 Iʺ2 + EDC −

I2

φ

1:mʹa

Iʹ2 Gsw

V1 I 1

jX1

R1

VʹvR IʹvR

Iʹ1 jBeq

VALVE SET

(b) Figure 3.5 (a) VSC schematic representation, (b) VSC equivalent circuit. Source: ©IEEE, 2013.

[8]. It uses a small capacitor bank on its DC side to support and stabilize the DC voltage to enable converter operation. The converter keeps the capacitor charged to the required voltage level by making its output voltage lag the AC system voltage by a small phase angle. The DC capacitor bank of value C DC is shown schematically in Figure 3.5a. It should be stated that C DC is not used per se in the VAR generation/absorption process. Instead, this process is carried out by action of the PWM control, which shifts the voltage and current waveforms within the VSC to yield either leading or lagging VAR operation to satisfy operational requirements. It is noted that the VSC has practically no inertia, it does not signiﬁcantly alter the existing system impedance and it can internally generate reactive (both capacitive and inductive) power. For the purpose of fundamental frequency analysis, the VSC’s electronic processing of the voltage and current waveforms is well synthesized by the notional variable susceptance, Beq , which connects to the ac bus of the ideal complex tap-changing transformer – see Figure 3.5b. Note that the notional Beq is responsible for the whole of the reactive power production in the valve set of the VSC. 3.6.1

VSC Nodal Admittance Matrix Representation

The fundamental frequency operation of the VSC shown schematically in Figure 3.5a may be modelled by means of electric circuit components, as shown in Figure 3.5b.

113

114

3 Power Flows

From the conceptual point of view, the central component of this VSC model is the ideal tap-changing transformer with complex tap, which, in the absence of switching losses, may be seen to act as a nullator that constrains the source current to zero, with the source being the capacitor C DC [9]. In this context, the associated norator is the variable susceptance Beq . Indeed, in steady-state operation the DC capacitor may be thought of as a battery that yields voltage EDC and draws no current. The dynamic behaviour of the STATCOM’s DC capacitor is covered in Chapter 6. Notice that the winding connected to node 1 is an AC node internal to the VSC and that the winding connected to node 0 is a notional DC node. Two elements connect to the VSC’s DC bus: the source, EDC , and the current-dependent resistor, Gsw . Hence, the ideal tap-changing transformer is the element that provides the interface for the VSC’s AC and DC circuits, as illustrated in Figure 3.4b. It should be emphasized that no reactive power ﬂows through it, only real power which is akin to DC power. The following expression, which is the root mean square (RMS) counterpart of Eq. (2.13) in Chapter 2, is the cornerstone of the fundamental-frequency, positive-sequence STATCOM model, developed in this chapter: V 1 = m′ a ej𝜑 EDC

(3.31)

where V 1 corresponds to the fundamental-frequency component of the VSC’s output line-to-line voltage in RMS form. In this expression, the angle 𝜑 is the phase angle of the complex voltage V 1 relative to the system phase reference; EDC is the DC bus voltage and m′ a = k 1 ma is the inverter’s amplitude modulation coeﬃcient, where in the linear range of modulation, the amplitude modulation index ma takes values within bounds: 0 < m√ For the case of a two-level, three-phase VSC, the constant k 1 takes the form a < 1. √ LL LL ), where VB,AC and EB, DC are the base voltages in the AC and k1 = 3∕2 2 ⋅ (EB,DC ∕VB,AC DC sides, respectively. In the remainder of this chapter and the book, the base voltages of the AC side and the DC side will be considered to have the same values. Hence, in √ 3∕8. This value is explained in terms of this chapter the constant k takes the value 1 √ 1∕ 2 being the relationship between an instantaneous quantity and the RMS value of √ a sinusoidal quantity; 3 being the relationship between a phase-like quantity and a line-to-line quantity and 1/2 representing the fact that only half of the peak-to-peak AC voltage induced by a DC voltage is useful for the purpose of extracting the RMS value. Note that this √ diﬀers from the case in [5, 7] where the following base voltages are used: LL VB,AC and 2EB,DC . Another element of the electric circuit shown in Figure 3.45b is the series impedance, which is connected to the ideal transformer’s AC side. The series reactance X 1 represents the VSC’s interface magnetics. The series resistor R1 accounts for the ohmic losses which are proportional to the AC terminal current squared. Note that the secondary winding current I 2 = IDC , which is always a real quantity, splits into I ′ 2 and I ′′ 2 . The latter current is always zero during steady-state operation. As one would expect, the complex power conservation property of the ideal transformer in Figure 3.5b stands, but note that there is no reactive power ﬂowing through it, since all the reactive power requirements of the VSC model (generation/absorption) are met by the shunt branch Beq connected at node 1. The power relationships between

3.6 The Voltage Source Converter Model

nodes 1 and 0, which account for the full VSC model, are: ,

∗

EDC IDC = V 1 (I 1 − I 1 )∗ = V 1 I 1 + jBeq V12

⇒

∗

EDC IDC = Re{V 1 I 1 }

(3.32)

The switching loss model corresponds to a constant resistance (conductance) G0 , which under the presence of constant DC voltage and constant load current would yield constant power loss for a given switching frequency of the PWM converter. Admittedly, the constant resistance characteristic may be inaccurate because although the DC voltage is kept largely constant, the load current will vary according to the prevailing operating condition. Hence, it is proposed that the resistance characteristic derived at rated voltage and current be corrected by the quadratic ratio of the actual current to the nominal current, ( act )2 I1 ⇒ Gsw (3.33) G0 ⋅ nom I1 where Gsw would be a resistive term exhibiting a degree of power behaviour. Expressing the fundamental equation (3.31) as a voltage ratio and using this information into (3.32) yields the following two relationships: k1 m a ∠ − 𝜑 k m ∠𝜑 IDC V1 = 1 a and , = EDC 1 1 (I 1 − I 1 )

(3.34)

which corresponds well with the voltage and current ratios which are the hallmark of an ideal, complex tap-changing transformer. The current through the admittance connected between nodes vR and 1 is: ,

,

,

I1 = Y 1 (V vR − V 1 ) = Y 1 V vR − k1 ma ∠𝜑 Y 1 EDC = I vR

(3.35)

where Y 1 = 1∕(R1 + jX1 ). At node 0, the following relationship holds: I 0 = −I2 + I2,

,

= −k1 ma ∠ − 𝜑 (I 1 − I 1 ) + Gsw EDC ,

= −k1 ma ∠ − 𝜑 Y 1 V vR + Gsw EDC + k12 m2a (Y 1 + jBeq )EDC

(3.36)

Combining (3.35) and (3.36) and incorporating constraints from the electric circuit in Figure 3.5b: ( , ) ( )( , ) Y1 −k1 ma ∠𝜑 Y 1 I vR V vR = (3.37) EDC −k1 ma ∠ − 𝜑 Y 1 Gsw + k12 m2a (Y 1 + jBeq ) I0 = 0 Notice that this expression represents the VSC equivalent circuit in Figure 3.3b in steady-state, with the capacitor eﬀect represented by the DC voltage EDC . 3.6.2

Full VSC Station Model

As suggested in [10] and shown in Figure 3.6 for a generic VSC station, i connected to node k using a load tap changer (LTC) transformer. A complete VSC station comprises

115

116

3 Power Flows

I0i = 0 IʹvRi = 0

0i

+ EDCi −

vRiʹ

CDCi

I1i mai φi

VʹvRi Yreac

IvRi = 0

T

vRi VvRi

k Ik

YT

Vk

Yfilter

Figure 3.6 VSC station i with ancillary elements connected to node k through a LTC transformer. Source: ©ELSEVIER, 2016.

the AC-DC converter with its AC and DC buses explicitly represented, phase reactors and AC ﬁlters. The connecting LTC transformer is added for the sake of generality, but its role will be discussed in Section 3.7. The smoothing line reactor and the shunt ﬁlter are essential elements of the VSC station in order to enable harmonic-free currents and voltages at the secondary winding of the transformer. It should be noted that the resistive components of these elements are rather small, but they are represented here as admittances Y reac and Y ﬁlter for the sake of generality. The smoothing line reactor and the shunt ﬁlter are added below to the basic model of the VSC given by Eq. (3.37). Owing to the modular philosophy of the modelling approach, this can be done in two diﬀerent ways: (i) by simply extending the nodal admittance matrix to include one more node in the model, namely vR; (ii) by developing a more compact model where node vR′ is eliminated mathematically. Developing these two points and dropping the subscript i in Figure 3.6: −Y reac 0 ⎞ ⎛V vR ⎞ ⎛I vR ⎞ ⎛Y reac + Y ﬁlter ⎟ ⎜V , ⎟ ⎜I , ⎟ = ⎜ −Y Y + Y −k m ∠𝜑 Y reac reac 1 1 a 1 ⎟ ⎜ vR ⎟ ⎜ vR ⎟ ⎜ ⎝ I0 ⎠ ⎝ 0 −k1 ma ∠ − 𝜑 Y 1 k12 m2a (Y 1 + jBeq ) + Gsw ⎠ ⎝EDC ⎠

(3.38)

Moreover, since the external injected current at node vR′ is nil, the more compact representation of Eq. (3.39) is arrived at: )( ) ( ) ( Y vRvR −Y vR0 ∠𝜑 I vR V vR = (3.39) EDC −Y vR0 ∠ − 𝜑 (Gsw + jk12 m2a Beq ) + Y 00 I0 where Y vRvR = GvRvR + jBvRvR = Y ﬁlter + Y 1 Y reac ∕(Y 1 + Y reac ) Y vR0 = GvR0 + jBvR0 = k1 ma Y 1 Y reac ∕(Y 1 + Y reac ) Y 00 = G00 + jB00 = k12 m2a ⋅ Y 1 Y reac ∕(Y 1 + Y reac ) Both representations, Eq. (3.38) and Eq. (3.39), correctly account for all the elements existing in the electric circuit of Figure 3.5. However, it should be noted that the latter option is preferred with the use of the Newton-Raphson algorithm because voltage regulation using ma will be applied at a point of the VSC where the voltage signal has already been cleaned by action of the smoothing reactor and ﬁlter, namely node vR.

3.6 The Voltage Source Converter Model

3.6.3

VSC Nodal Power Equations

The complex power model is derived from the nodal admittance matrix: (

SvR S0

)

(

)( ∗ ) I vR = ∗ I0 ( ) (evR + jfvR ) 0 = 0 EDC {( )( )} (GvRvR − jBvRvR ) −(cos 𝜑 − jsin𝜑)(GvR0 − jBvR0 ) (evR − jfvR ) × −(cos 𝜑 + jsin𝜑)(GvR0 − jBvR0 ) (Gsw − jk12 m2a Beq ) + (G00 − jB00 ) EDC V vR 0 0 EDC

(3.40)

Following some arduous algebra, the following nodal active and reactive power expressions are arrived at: 2 PvR = GvRvR (e2vR + fvR ) − EDC (evR [GvR0 cos 𝜑 − BvR0 sin 𝜑]+

fvR [GvR0 sin 𝜑 + BvR0 cos 𝜑]) 2 ) − EDC (−evR [GvR0 sin 𝜑 + BvR0 cos 𝜑]+ QvR = −BvRvR (e2vR + fvR

fvR [GvR0 cos 𝜑 − BvR0 sin 𝜑])

(3.41)

and 2 P0 = (Gsw + G00 )EDC − EDC (evR [GvR0 cos 𝜑 + BvR0 sin 𝜑]

+ fvR [GvR0 sin 𝜑 − BvR0 cos 𝜑]) 2 − EDC (evR [GvR0 sin 𝜑 − BvR0 cos 𝜑] Q0 = −(k12 m2a Beq + B00 )EDC

− fvR [GvR0 cos 𝜑 + BvR0 sin 𝜑]

(3.42)

where GvRvR = Gﬁlter + ΔG and BvRvR = Bﬁlter + ΔB GvR0 = k1 ma ΔG and BvR0 = k1 ma ΔB G00 = k12 m2a ΔG and B00 = k12 m2a ΔB (G G − B1 Breac )(G1 + Greac ) + (G1 Breac + B1 Greac )(B1 + Breac ) ΔG = 1 reac (G1 + Greac )2 + (B1 + Breac )2 (G B + B1 Greac )(G1 + Greac ) − (G1 Greac − B1 Breac )(B1 + Breac ) ΔB = 1 reac (G1 + Greac )2 + (B1 + Breac )2 Note that GvR0 , BvR0 , G00 and B00 are functions of ma .

3.6.4

VSC Linearized System of Equations

These equations are nonlinear and their solution, for a predeﬁned set of generation and load pattern (PvR, net , QvR, net , P0, net , Q0, net ), may be carried out using the Newton-Raphson method. This involves repeated linearization of the nodal power equations. Their initial evaluation requires an informed guess of the state variable

117

118

3 Power Flows (0) (0) values (e(0) , f (0) , B(0) eq , ma , 𝜑 ), when the aim is to regulate voltage magnitude at bus vR vR vR using the VSC’s amplitude modulation ratio ma and to keep EDC at a constant value. In practice, the latter is possible due to the DC capacitor’s action. The linearized system of equations is:

0 𝜕ΔPvR ∕𝜕𝜑 ⎤ ⎡ ΔPvR ⎤ ⎡ 𝜕ΔPvR ∕𝜕evR 𝜕ΔPvR ∕𝜕ma 𝜕ΔPvR ∕𝜕fvR ⎥ ⎢ ⎥ ⎢ 0 𝜕ΔQvR ∕𝜕𝜑 ⎥ ⎢ ΔQvR ⎥ ⎢ 𝜕ΔQvR ∕𝜕evR 𝜕ΔQvR ∕𝜕ma 𝜕ΔQvR ∕𝜕fvR ⎥ ⎢ ⎥ ⎢ 0 𝜕ΔUvR ∕𝜕fvR 0 0 ⎥ ⎢ ΔUvR ⎥ = − ⎢ 𝜕ΔUvR ∕𝜕evR ⎥ ⎢ ⎥ ⎢ 𝜕ΔQ ∕𝜕e 𝜕ΔQ ∕𝜕m 𝜕ΔQ ∕𝜕f 𝜕ΔQ ∕𝜕B 𝜕ΔQ ∕𝜕𝜑 ΔQ ⎥ ⎢ 0−vR ⎥ ⎢ 0−vR vR 0−vR a 0−vR vR 0−vR eq 0−vR ⎥ ⎢ ⎥ ⎢ 0 𝜕ΔP0−vR ∕𝜕𝜑 ⎦ ⎣ ΔP0−vR ⎦ ⎣ 𝜕ΔP0−vR ∕𝜕evR 𝜕ΔP0−vR ∕𝜕ma 𝜕ΔP0−vR ∕𝜕fvR ⎡ ΔevR ⎤ ⎥ ⎢ ⎢Δma ⎥ ⎥ ⎢ × ⎢ ΔfvR ⎥ ⎥ ⎢ ⎢ΔBeq ⎥ ⎥ ⎢ ⎣ Δ𝜑 ⎦

(3.43)

Subsequent evaluations of the nodal power equations are carried out using the (1,2…) improved set of values being furnished by the iterative process: (e(1,2…) , fvR , B(1,2…) , eq vR (1,2…) (1,2…) ma ,𝜑 ). The entries making up the Jacobian matrix in (3.43) are the following: 𝜕PvR 𝜕CRvR 𝜕CI vR = CRvR + evR + fvR 𝜕evR 𝜕evR 𝜕evR

𝜕QvR 𝜕CRvR 𝜕CI vR = −CI vR + fvR − evR 𝜕evR 𝜕evR 𝜕evR

𝜕PvR 𝜕CRvR 𝜕CI vR = CI vR + evR + fvR 𝜕fvR 𝜕fvR 𝜕fvR

𝜕QvR 𝜕CRvR 𝜕CI vR = CRvR + fvR − evR 𝜕fvR 𝜕fvR 𝜕fvR

𝜕PvR 𝜕CRvR 𝜕CI vR = evR + fvR 𝜕𝜑 𝜕𝜑 𝜕𝜑

𝜕QvR 𝜕CRvR 𝜕CI vR = fvR − evR 𝜕𝜑 𝜕𝜑 𝜕𝜑

𝜕PvR 𝜕CRvR 𝜕CI vR = evR + fvR 𝜕ma 𝜕ma 𝜕ma

𝜕QvR 𝜕CRvR 𝜕CI vR = fvR − evR 𝜕ma 𝜕ma 𝜕ma

𝜕PvR 𝜕CRvR 𝜕CI vR = evR + fvR 𝜕EDC 𝜕EDC 𝜕EDC

𝜕QvR 𝜕CRvR 𝜕CI vR = fvR − evR 𝜕EDC 𝜕EDC 𝜕EDC

𝜕P0−vR 𝜕CR0 = EDC 𝜕evR 𝜕evR

𝜕Q0−vR 𝜕CI 0 = −EDC 𝜕evR 𝜕evR

𝜕P0−vR 𝜕CR0 = EDC 𝜕fvR 𝜕fvR

𝜕Q0−vR 𝜕CI 0 = −EDC 𝜕fvR 𝜕fvR

𝜕P0−vR 𝜕CR0 = EDC 𝜕𝜑 𝜕𝜑

𝜕Q0−vR 𝜕CI 0 = −EDC 𝜕𝜑 𝜕𝜑

𝜕P0−vR 𝜕CR0 = EDC 𝜕Beq 𝜕Beq

𝜕Q0−vR 𝜕CI 0 = −EDC 𝜕Beq 𝜕Beq

𝜕P0−vR 𝜕 = EDC CR 𝜕ma 𝜕ma 0

𝜕Q0−vR 𝜕CI 0 = −EDC 𝜕ma 𝜕ma

𝜕P0−vR 𝜕CR0 = CR0 + EDC 𝜕EDC 𝜕EDC

𝜕Q0−vR 𝜕CI 0 = −CI 0 − EDC 𝜕EDC 𝜕EDC

3.6 The Voltage Source Converter Model

The various current derivatives are given explicitly in Appendix 3.A at the end of this chapter. Similar to the power mismatch terms (3.18), (3.19) and (3.26), relevant mismatch terms for the VSC model, which has been assumed to be connected between buses vR and 0, are given below: ΔPvR = PvR,net − PvR,cal = (PvR,gen − PvR,load ) − PvR,cal ΔQvR = QvR,net − QvR,cal = (QvR,gen − QvR,load ) − QvR,cal ΔP0 = P0,net − P0,cal = (P0,gen − P0,load ) − P0,cal ΔQ0 = Q0,net − Q0,cal = (Q0,gen − Q0,load ) − Q0,cal sch ΔP0−vR = −P0−vR + ΔP0

ΔQ0−vR = 0 + ΔQ0

(3.44)

sch where P0−vR is the active power which is scheduled to arrive at node 0 and the negative sign is to signify that it ﬂows from node vR towards node 0. This value is normally set to zero but it could be the value of any suitable DC load. On the other hand, the scheduled value for the reactive power ﬂow arriving at node 0 is always set to zero, which correctly implies that no reactive power exists at the DC bus. Similar to Eqs. (3.21) and (3.22), the state variable increments calculated at iteration (r) are the diﬀerence between the state variable value at that iteration and their values at the previous iteration:

Δe(r) = e(r) − e(r−1) vR vR vR (r) (r) (r−1) ΔfvR = fvR − fvR (r) (r−1) ΔB(r) eq = Beq − Beq (r) (r−1) Δm(r) a = ma − ma

Δ𝜑(r) = 𝜑(r) − 𝜑(r−1)

3.6.5

(3.45)

Non-Regulated Power Flow Solutions

If no voltage regulation at node vR applies because either the amplitude modulation index limit has been reached or one explicitly wishes to do so, ma is removed from the list of state variables and the mismatch voltage ΔV vR becomes deactivated in the linearized expression (3.43). Notice that no new Jacobian terms are required: ⎡ ΔPvR ⎤ ⎡ 𝜕ΔPvR ∕𝜕evR ⎢ ΔQvR ⎥ ⎢ 𝜕ΔQvR ∕𝜕evR ⎢ΔQ ⎥ = − ⎢𝜕ΔQ 0−vR ∕𝜕evR ⎢ 0−vR ⎥ ⎢ ⎣ ΔP0−vR ⎦ ⎣ 𝜕ΔP0−vR ∕𝜕evR

𝜕ΔPvR ∕𝜕fvR 0 𝜕ΔQvR ∕𝜕fvR 0 𝜕ΔQ0−vR ∕𝜕fvR 𝜕ΔQ0−vR ∕𝜕Beq 𝜕ΔP0−vR ∕𝜕fvR 0

𝜕ΔPvR ∕𝜕𝜑 ⎤ ⎡ ΔevR ⎤ 𝜕ΔQvR ∕𝜕𝜑 ⎥ ⎢ ΔfvR ⎥ 𝜕ΔQ0−vR ∕𝜕𝜑⎥ ⎢ΔBeq ⎥ ⎥⎢ ⎥ 𝜕ΔP0−vR ∕𝜕𝜑 ⎦ ⎣ Δ𝜑 ⎦ (3.46)

There are instances outside the scope of the STATCOM application in which it may be desirable to relax the VSC’s DC voltage, such as in some HVDC applications. In such

119

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3 Power Flows

cases, a rearrangement of state variables in the linearized expression (3.43) also takes place: ⎡ ΔP ⎤ ⎡ 𝜕ΔP ∕𝜕e 𝜕ΔPvR ∕𝜕ma 𝜕ΔPvR ∕𝜕fvR 0 𝜕ΔPvR ∕𝜕𝜑 𝜕ΔPvR ∕𝜕EDC ⎤ vR vR vR ⎢ ⎥ ⎥ ⎢ ⎢ ΔQ ⎥ ⎢ 𝜕ΔQ ∕𝜕e 𝜕ΔQvR ∕𝜕ma 𝜕ΔQvR ∕𝜕fvR 0 𝜕ΔQvR ∕𝜕𝜑 𝜕ΔQvR ∕𝜕EDC ⎥ vR vR vR ⎥ ⎢ ⎥ ⎢ ⎢ ΔU ⎥ ⎥ ⎢ 𝜕ΔU ∕𝜕e 0 𝜕ΔUvR ∕𝜕fvR 0 0 0 vR vR vR ⎥ ⎢ ⎥ ⎢ =− ⎢ΔQ ⎥ ⎢𝜕ΔQ ∕𝜕evR 𝜕ΔQ0−vR ∕𝜕ma 𝜕ΔQ0−vR ∕𝜕fvR 𝜕ΔQ0−vR ∕𝜕Beq 𝜕QΔ0−vR ∕𝜕𝜑 𝜕ΔQ0−vR ∕𝜕EDC ⎥ 0−vR 0−vR ⎢ ⎥ ⎥ ⎢ ⎢ ΔP ⎥ ⎢ 𝜕ΔP 0 𝜕ΔP0−vR ∕𝜕𝜑 𝜕ΔP0−vR ∕𝜕EDC ⎥ 0−vR ∕𝜕evR 𝜕ΔP0−vR ∕𝜕ma 𝜕ΔP0−vR ∕𝜕fvR ⎢ 0−vR ⎥ ⎥ ⎢ ⎢ ΔP ⎥ ⎢ 𝜕ΔP ∕𝜕e 𝜕ΔP0 ∕𝜕ma 𝜕ΔP0 ∕𝜕fvR 0 𝜕ΔP0 ∕𝜕𝜑 𝜕ΔP0 ∕𝜕EDC ⎥⎦ ⎣ ⎣ 0 ⎦ 0 vR ⎡ Δe ⎤ ⎢ vR ⎥ ⎢ Δm ⎥ a⎥ ⎢ ⎢ Δf ⎥ ⎢ vR ⎥ ⎢ ΔB ⎥ ⎢ eq ⎥ ⎢ Δ𝜑 ⎥ ⎥ ⎢ ⎢ΔE ⎥ ⎣ DC ⎦

3.6.6 3.6.6.1

(3.47)

Practical Implementations Control Strategy

As illustrated in Figure 3.5b, the VSC is assumed to be connected between a sending bus vR and a receiving bus 0, with the former being the VSC’s AC bus and the latter being the VSC’s DC bus. The voltage EDC is kept constant by the action of a small DC capacitor bank with rated capacitance C DC , which in steady-state draws no current. In the Newton-Raphson power ﬂow solution, the voltage magnitude V vR is regulated within system-dependent maximum and minimum values, aﬀorded by the following basic relationship: (√ ) | | | | 2 2 VvR = k1 ma EDC − R1 + X1 + |Zreac | |I 1 | (3.48) | | | | Note that in the VSC’s linear range of modulation, the index ma takes values within the bounds: 0 < ma < 1

(3.49)

However, in power systems reactive power control applications, it is unlikely that values of ma lower than 0.5 will be used. The reason is that the voltage magnitude at the VSC’s AC bus must be kept within practical limits because too high a voltage may induce insulation coordination failure at the point of connection with the power grid and too low a voltage may induce a condition of voltage collapse. To illustrate this point and using realistic values of R1 = 0.0002 p.u., X 1 = 0.002 p.u., Rreac = 0 p.u., X reac = 0.02 p.u. and EDC = 2 p.u., and considering low-current operation, say 0.1 p.u., V vR will take a value of 0.590 36 p.u. with ma = 0.5. 3.6.6.2

Initial Parameters and Limits

Three VSC parameters require initialization. The amplitude modulation ratio ma and its phase angle 𝜑 are normally set at 1 and 0, respectively. The VSC is assumed to operate

3.6 The Voltage Source Converter Model

within the linear region, whereas the phase angle 𝜑 is assumed to have no limits. The third parameter is the equivalent shunt susceptance Beq , which is given an initial value that lies within the range Beq+ and Beq− . 3.6.7

VSC Numerical Examples

A power ﬂow computer program is an essential tool to carry out the steady-state solution of even simple electrical power networks. The computer program VSC_Power_Flows, written in protected Matlab code, is suitable to carry out the power ﬂow solution of small and medium-size power systems; it resides in the following repository: www.wiley.com/go/acha_vsc_facts The program is general as far as the topology of the network is concerned; it caters for any number of PV and PQ buses. Any bus in the network can be designated to be a slack bus – contrary to convention, there can be more than one of these buses. Models of the following conventional network elements are included: overhead transmission lines, underground cables, LTC transformers, constant-power loads and ﬁxed shunt and series compensation. The model of the VSC is also included in this code. However, no checking of reactive power limits violations is carried out for generators or for VSCs. For illustration purposes, the code of the main program, VSC_Power_Flows, is given below: Program 3.1 A program written in Matlab to calculate positive sequence power ﬂows using the Newton-Raphson method. %***- - - Main Program %Read system data DataPowerFlowsR_1VSC; %% Set up the nodal admittance matrix [Ybus] = YBus(nbb,ntl,tlsend,tlrec,tlresis,tlreac,... tlcond,tlsuscep,nsh,shbus,shresis,shreac); %% Determine Vreal and Vimag by iteration, together with the VSC %% and LTC parameters [Vreal,Vimag,mVSC,phiVSC,BVSC,tftap]= VSC_NewtonRaphsonR℩ (tol,itmax,ngn,nld,nbb,bustype,genbus,loadbus,Pgen,Qgen,Pload,… Qload,Ybus,Vreal,Vimag,nvsc,VSCsend,VSCrec,RVSC1,XVSC1,GVSC0,… RVSCR,XVSCR,RVSCF,XVSCF,BVSC0,mVSC,phiVSC,VSCVCtrl,VSCVM,… VSCPL,CVSC,ntf,tfsend,tfrec,tfresis,tfreac,tftap,TFVMT,TFVCtrl); %% Calculate the power flows and power losses in transmission %% lines [PQtlsend,PQtlrec,PQtlloss,PQbus,PQgen,PQshunt,PQtotalloss]=… PQflowsR(nbb,ngn,ntl,ntf,nsh,nld,genbus,tlsend,tlrec,… tlresis,tlreac,tlcond,tlsuscep,tfsend,tfrec,Tfresis,Tfreac,… tftap,shbus,shresis,shreac,loadbus,Pload,Qload,Vreal,Vimag); %% Calculate the power flows and power losses in LTC %% transformers if ntf > 0 ; [PQbus,PQTFsend,PQTFrec,PQTFLoss]… = TF_PQflowsR… (Vreal,Vimag,PQbus,ntf,tfsend,tfrec,tfresis,tfreac,tftap); end %% Calculate the power flows in SVC if nvsc > 0 ; [PQbus,PQVSCsend,PQVSCrec,PQVSC,PQVSCLoss]=… VSC_PQflowsR(Vreal,Vimag,PQbus,nvsc,VSCsend,VSCrec,RVSC1,XVSC1,… GVSC0,BVSC,RVSCR,XVSCR,RVSCF,XVSCF,mVSC,phiVSC,CVSC); end %% End of Main Program

121

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3 Power Flows

1∠0° 0.2586

2

1.05∠ –1.70° 0.0052

0.2552

0.8632∠ –1.78° 0

1 0.5213

0.5551

0.7551 2

0.7603

0.0041

0.25 + j0.20 Figure 3.7 VSC providing voltage support at Node 2.

To illustrate the application of the theory so far covered and the usefulness of the computer program provided, the VSC model is applied in a rather contrived test case where the voltage source converter (VSC) is connected at the receiving end of a loaded transmission line. In this particular example, and in order to simplify matters, a connecting transformer is neglected and so are the impedances of the smoothing reactor and the shunt ﬁlter. Three cases are considered: (i) the VSC is used to provide reactive power; (ii) the VSC is used to draw reactive power; and (iii) the VSC is used to supply a DC power load. Test Case 1. The three-node system shown in Figure 3.7 comprises one generator, one transmission line and one AC-DC converter (VSC), which is represented by the elements shown within the broken-line rectangle. The generator node is taken to be the slack bus where the voltage magnitude is kept at 1 p.u. and its phase angle provides a reference for all other phase angles in the network, excepting bus 0 since this is a DC bus and the voltage is always a real quantity. The VSC consumes 0.0052 p.u. of active power from the system to account for its internal losses while supplying 0.7551 p.u. of reactive power to the system. The equivalent susceptance (in capacitive mode) produces 0.7603 p.u. of reactive power and its capacitive susceptance stands at Beq = 0.6803 p.u. The VSC switching losses are 0.41%, corresponding to an actual conductance G0 ≈ 0.1%. The DC bus voltage is controlled at 2 p.u. and the voltage magnitude at bus 2 is kept at 1.05 with ma = 0.8632. The phase shifter angle takes a value of −1.78∘ . The line current drawn by the VSC is 0.7192∠+87.91∘ . The following parameters are used by the computer program in the function DataPowerFlowsR_1VSC to solve this system: %% Bus data - notice that power generation and load are treated %% as attributes of the node % bustype = type of bus % VM = nodal voltage magnitude in per-unit % VA = nodal voltage phase angle in degrees % Pgen and Qgen = nodal active and reactive power generation in % per-unit % Pload and Qload = nodal active and reactive power load in % per-unit bustype(1)=1; VM(1)=1; VA(1)=0; Pgen(1)=0; Qgen(1)=0; Pload(1)=0; Qload(1)=0; bustype(2)=3; VM(2)=1.05; VA(2)=0; Pgen(2)=0; Qgen(2)=0;

3.6 The Voltage Source Converter Model Pload(2)=0.25; Qload(2)=0.20; bustype(3)=2; VM(3)=2; VA(3)=0; Pgen(3)=0; Qgen(3)=0; Pload(3)=0; Qload(3)=0; %% Transmission line data % tlsend = sending end of transmission line % tlrec = receiving end of transmission line % tlresis = series resistance of transmission line in per-unit % tlreac = series reactance of transmission line in per-unit % tlcond = shunt conductance of transmission line in per-unit % tlsuscep = shunt susceptance of transmission line in per-unit tlsend(1)=1; tlrec(1)=2; tlresis(1)=0.01; tlreac(1)=0.10; tlcond(1)=0; tlsuscep(1)=0; %LTC transformer data % tfsend = sending end of transformer - if tfsend(1)=0 then the % number of transformers in the network are zero % tfrec = receiving end of transformer % tfresis = series resistance of transformer in per-unit % tfreac = series reactance of transformer in per-unit % tftap = transformer’s tap % TFVM = target voltage magnitude in per-unit % TFVCtrl = voltage control with the transformer’s tap. 1 and 2 % is for voltage control at the sending and receiving nodes, % respectively, 0 is for no control. tfsend(1)=0; tfrec(1)=0; Tfresis(1)=0; Tfreac(1)=0; tftap(1)=0; TFVMT(1)=1.05; TFVCtrl(1)=1; %% Shunt data % shbus = shunt element bus number - if shbus(1)=0 then the % number of shunt elements in the network are zero % shresis = resistance of shunt element in per-unit % shreac = reactance of shunt element in per-unit % +ve for inductive reactance and -ve for capacitive reactance shbus(1)=0; shresis(1)=0; shreac(1)=0; %% VSC data % Topology connection to the VSC’s AC and DC buses- if % VSCsend(1)=0 then the number of VSC in the network are zero VSCsend(1)=2; VSCrec(1)=3; % RVSC1 = series resistance in-per unit % XVSC1 = series reactance in per-unit % GVSC0 = shunt conductance in per-unit corresponding to % switching losses at nominal current % BVSC0 = shunt susceptance in per-unit corresponding to the % VSC’s nominal MVA % RVSCR = resistance of the smoothing reactor in-per unit % XVSCR = reactance of the smoothing reactor in-per unit % RVSCF = resistance of the shunt filter in-per unit % XVSCF = reactance of the shunt filter in-per unit % VSCMVA = nominal power in per-unit RVSC1(1)=0.002; XVSC1(1)=0.010; GVSC0(1)=0.002; BVSC0(1)=0.50; RVSCR(1)=0; XVSCR(1)=0; RVSCF(1)=0; XVSCF(1)=0; VSCMVA(1)=1; % VSC Initial values % mVSC = initial value of amplitude modulation ratio (00

(4.11)

It should be noted that the KKT conditions above guarantee a local optimum for general non-convex functions only. However, if the problem is convex, the KKT conditions will guarantee a global optimum. From an engineering perspective and speciﬁcally referring to the OPF problem, the set of KKT conditions is used to test whether or not a technically feasible solution is also a local optimum for the problem in (4.8). This means that for an actual power system, the optimum operating point, i.e. the set of complex nodal voltages and control variables set-points, satisﬁes the set of equality constraints, i.e. nodal active and reactive power balance equations, and all equipment operates within design limits while maintaining a given objective function at its optimum value.

4.5 AC OPF Formulation In its most general form, the OPF is inherently a nonlinear and non-convex formulation resulting from the nonlinear nature of the power ﬂow equations given in (4.1). The full AC OPF formulation in polar coordinates is given in Eq. (4.12), which is, in turn, a derivation of the general optimization form given in (4.8). For simplicity, only the pre-contingency operating state, i.e. k = 0, is shown. It is assumed here that no controls, other than the ones associated with the conventional synchronous generators, are active. min 𝜃,V

s.t.

F(𝛉, V, Pgen ) ⎧h(𝛉, V) = 0 ⎪ ⎪g(𝛉, V) ≤ gmax ⎨ min max ⎪𝛉 ≤ 𝛉 ≤ 𝛉 ⎪Vmin ≤ V ≤ Vmax ⎩

(4.12)

In Eq. (4.12), the vector function h(𝜽, V) is deﬁned explicitly as a function of the system nodal power balance equations: ( h(𝛉, V) =

) ⎛ gen load Pg,l − Pd,l − Plcalc ⎞ hP (𝛉, V) ⎟=0 = ⎜ gen hQ (𝛉, V) ⎜Q − Qload − Qcalc ⎟ g,l d,l l ⎠ ⎝

∀l ∈ L, g ∈ G, d ∈ D (4.13)

In Eq. (4.13), sets L, G, and D are deﬁned as the sets of all the network buses, generators and loads, respectively. Equation (4.13) arises from an application of Kirchhoﬀ ’s current law, which holds true for a feasible solution. These equations ﬁnd applicability in conventional AC power systems as well as in hybrid AC-DC systems. There may be other control equality constraints associated with power ﬂow controllers or FACTS equipment, which may be suitably added to the set of equality constraints in Eq. (4.12).

4.5 AC OPF Formulation

Similarly, the function g(𝜽, V) in Eq. (4.14) is deﬁned as a complex-valued function over the set of state variables deﬁning the inequality constraints. For an actual system, the inequality constraints generally refer to the transmission line thermal limits, but depending on the nature of the problem, e.g. when a DC approximation is used, they could include active power limits and/or current limits. In general, the set of inequality constraints include the following: ⎧ ‖ ‖2 ⎪ ‖Sl ‖ ≤ Slmax ∀l ∈ L ⎪ gen,min‖ ‖ gen gen,max (4.14) g(𝛉, V) = ⎨ Pg ≤ Pg ≤ Pg ∀g ∈ G ⎪ gen,min gen gen,max ≤ Qg ≤ Qg ∀g ∈ G ⎪ Qg ⎩ Normally, the angle constraints apply to the slack bus only. On the other hand, all buses must satisfy the voltage magnitude constraints. The inequality constraint violations are enforced through an appropriate constraint-handling method, which is explained in Sections 4.5.5 and 4.5.6. 4.5.1

Objective Function

More often than not, the OPF objective function F(𝜽, V, Pgen ) is the fuel cost–output power function of the thermal synchronous generators normally employed in economic dispatch assessments [3], which may be assumed to have the following general quadratic form [7, 8]: ∑ gen gen2 F(𝛉, V, Pgen ) = (ag + bg Pg + cg Pg ) (4.15) g∈G

In Eq. (4.15) the coeﬃcients a, b and c deﬁne the generator’s quadratic cost curve. The generator’s active power dispatch may be initialized ﬁrst by solving a lossless economic dispatch problem, ignoring the transmission constraints in Eq. (4.12) and assuming an equal marginal cost for the generators’ outputs. This criterion of equal marginal costs provides a suitable starting point for the OPF algorithm [3]. 𝜕 F(𝛉, V, Pgen ) = 𝜆ini ∀g ∈ G (4.16) 𝜕Pg However, the OPF solution determines an optimum feasible dispatch problem taking into account all the transmission constraints and optimality conditions. Alternative objective functions may be used which largely depend on the nature of the problem being solved. For operational planning, the aim of the TSO is normally to obtain the most economic power dispatch for all the generators and in a free market environment this will depend on the market design and market rules. The objective could also be the minimization of active power transmission losses, reactive power losses and even equipment placing. 4.5.2

Linearized System of Equations

The Lagrangian function for the AC OPF problem given in (4.12) may be expressed as follows: ∑ ∑ L(𝛉, V, 𝛌, 𝛍) = F(𝛉, V, Pgen ) + 𝜆i hi (𝛉, V) + 𝜇j gj (𝛉, V) (4.17) i∈E

j∈A⊂I

165

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4 Optimal Power Flows

To solve the function in (4.17) towards the optimum, the Lagrangian may be linearized using Newton’s method, resulting in a system of linearized equations, given by Eq. (4.18). ( ) Δz 2 = −∇L(z, 𝛚) (4.18) ∇ L(z, 𝛚) × Δ𝛚 To simplify the formulation we may assume that z = (𝜽, V, Pgen )t and 𝝎 = (𝝀, 𝝁)t are vectors of system variables and multipliers, respectively. If the set of active inequality constraints is empty, i.e. A = 0, then Eq. (4.18) does not include the multipliers μ and may be written down as follows: [ ][ ] [ ] ∇2zz L ∇2z𝛌 L Δz 𝜕L∕𝜕z =− (4.19) ∇2𝛌z L ∇2𝛌𝛌 L Δ𝛌 𝜕L∕𝜕𝛌 In Eq. (4.19), the Hessian and Jacobian terms of the Lagrangian Eq. (4.17), i.e. matrices of second- and ﬁrst-order partial diﬀerentials, respectively, as well as the gradient terms, are deﬁned as follows: ⎧∇2 L = ∇2 F + 𝛌t ∇2 h(z) zz zz ⎪ zz ⎪∇2z𝛌 L = ∇2𝛌z L = ∇z h(z) ⎪ 2 ⎨∇𝛌𝛌 L = 0 ⎪ ⎪𝜕L∕𝜕z = ∇z L(z) ⎪𝜕L∕𝜕𝛌 = h(z) ⎩

(4.20)

A full iterative solution of Eq. (4.19) requires the convergence of an inner loop to a speciﬁed tolerance. In principle at least, convergence of this inner loop may be achieved in a true quadratic convergence fashion. Accurate values of state variables and multipliers are obtained upon convergence of the inner loop. However, it is likely that one or more variables may have violated their operational limits. Hence, all inequality constraints are checked at this point for limits violations and the active set is updated by enforcing the violated inequalities using their corresponding multipliers, i.e. 𝝁. In other words, if the active set is no longer empty then their corresponding multipliers are non-zero, i.e. 𝝁 ≠ 0. This means that the Lagrangian function now takes the form of Eq. (4.17), including all the equality constraints as well as the inequalities constraints which have been violated, becoming part of the active set. This would complete one global iteration. Note that the updating of the active set takes place following a Gauss-type procedure. The iterative process is ready to start the next global iteration, which calls for the solution of the linearized Lagrangian expression in Eq. (4.18). It involves a recalculation of the state variables and multipliers using the inner iterative loop governed by the Newton-Raphson method. Upon convergence, all inequality constraints are checked again for any change in their status concerning limits violations, i.e. the active set is updated. If one or more inequality constraints have changed their status, then a new global iteration is started; conversely, if no further change in the active set is detected, then the overall iterative process, including global and internal iterations, comes to an end. It should be remarked that the active set is updated outside the inner loop, a procedure that impairs the overall convergence of the OPF solution using Newton’s method. Furthermore, and although in theory the inner loop may be executed in a true

4.5 AC OPF Formulation

convergent fashion, a decelerating factor (0 < 𝛽 < 1) is used at the point of updating the state variables and Lagrange multipliers at the end of each local iteration. This is particularly the case during the local iterations of the ﬁrst two global iterations [9]. Use of a deceleration factors impairs the quadratic convergence characteristics of the Newton-Raphson procedure, i.e. the number of local iterations will increase. However, experience has shown that this is a very powerful resource owing to the highly nonlinear nature of the problem at hand. 4.5.3

Augmented Lagrangian Function

An alternative to the original Lagrangian function given in (4.17) is to augment it by a penalty function for active inequality constraints [3]. The augmented Lagrangian function is then formed as shown in Eq. (4.21): ∑ ∑ 𝜆i hi (𝛉, V) + 𝜓j (𝜇j , gj (𝛉, V)) (4.21) L(𝛉, V, 𝛌, 𝛍) = F(𝛉, V, Pgen ) + i∈E

j∈A⊂I

The penalty function for the active set in (4.21), in more explicit form, is given in Eq. (4.22): { 𝛼 𝜇j (gj (𝛉, V) − gjmax ) + ‖gj (𝛉, V) − gjmax ‖2 ∀j ∈ A ⊂ I 2 (4.22) 𝜓j = zero otherwise This equation is, essentially, an augmentation of the term 𝝁t g in the Lagrangian in Eq. (4.17) by the penalty term: 𝛼2 ‖g − g max ‖2 . This is the reason why the augmented Lagrangian term is used. The penalty function 𝜓 combines both the method of multipliers and penalty functions for enforcing the active inequality constraints. The non-zero penalty term 𝛼 is started with a rather large value. However, some care needs to be exercised in its assignation since too large a value may turn the numerical problem into an ill-conditioned one [3]. Nevertheless, the eﬀect of the penalty function 𝜓 is that it discourages the solution search from moving in a direction that increases the violation; hence, upon convergence, all the active inequality constraints are enforced to their respective limits – this means that at this point the third KKT constraint should hold, i.e. 𝜇j (gj (𝛉, V) − gjmax ) = 0. It should be noted that for the sake of simplicity, only the upper limits are shown explicitly. However, the above explanation applies with no change when inequality constraints violate their lower limits, the only thing required is to substitute the upper limit by the lower limit in expression (4.22). In summary, the iterative algorithm comprises an inner Newton-Raphson loop and an outer global loop where the algorithm checks for any active inequality constraint violations. The multipliers associated with the active inequalities in the i − th iteration are updated in the outer loop and take the following general form: { (i−1) 𝜇j + 𝛼 (i−1) ‖gj (𝛉, V) − gjmax ‖2 if 𝜇j(i−1) ≥ 𝛼 (i−1) (gj (𝛉, V) − gjmax ) ∀j ∈ A ⊂ I 𝜇j(i) = zero otherwise (4.23) It should be remarked that only violations for the upper limits are shown explicitly. This would conclude the ﬁrst global (outer) iteration. The second loop now includes both equality and inequality constraints shown in Eq. (4.17) [3].

167

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4 Optimal Power Flows

4.5.4

Selecting the OPF Solution Algorithm

It should be said that there is a wide range of algorithms for solving a nonlinear optimization problem such as the OPF. However, the method of the augmented Lagrangian function is a popular choice in the electrical power industry and it has been chosen here as the solution method. The steps for solving the OPF are given below for two key processes: (i) for forming the augmented Lagrangian function; (ii) for linearizing the constraints and solving towards the optimum solution using Newton’s method. Nonetheless, the reader is encouraged to study alternative methods for solving nonlinear problems, such as interior point methods (IPMs) [6] and gradient methods [10]. These alternative solutions are beyond the subject matter of this book. Step One. Initialize the OPF procedure using a lossless economic dispatch. Step Two. Set global iterations to 1. Step Three. Form the Lagrangian function system L(z, 𝝎). Step Four. Form the linearized Lagrangian function and solve for Δz and Δ𝝎. Step Five. If Δz is within the tolerance limits, stop and go to Step Six. Otherwise, repeat Steps Three and Four, successively, until the tolerance criterion fulﬁls. If the maximum number of inner iterations is exceeded, terminate the iterative process. Step Six. Check for binding inequalities. If the active set is empty, the OPF solution has been reached and the overall process is stopped. Otherwise, go to Step Seven. Step Seven. Update the active set and go to Step Three. If the maximum of global iterations permitted has been exceeded, terminate the process.

4.5.5

Control Enforcement in the OPF Algorithm

In OPF solutions using Newton’s method, the equality constraint set is expanded to include control constraints. The control constraints are any constraints explicitly used by the control equipment in the power grid. A good example of this is the automatic voltage regulator (AVR) controllers used to regulate nodal voltages at buses where synchronous generators are connected, as long as they operate within their reactive power limits [3]. Another example of this is the nodal voltage regulation that transformers with tap-changing capabilities may exert when operating within their maximum/minimum control range. Other controllers may include power electronic mechanisms, such as FACTS and HVDC converters, and energy storage systems. In any case, suitable control variables are deﬁned and the OPF solution ensures that such control variables attain values that establish an optimal operation of the power grid subject to all the economic, operational and security constraints that have been deﬁned a priori. The control constraints may be deﬁned, in general form, by the following function: C(x, u) = 0

(4.24)

The Lagrangian for the function (4.24) is given in (4.25). This is used to enforce the control actions deﬁned in (4.24) throughout the solution process: LC = 𝛌C C(x, u)

(4.25)

4.5 AC OPF Formulation

Of particular interest in this book is the voltage source converter’s operational design limits for both active and reactive powers, which are enforced by means of a set of equality constraints of the type (4.25). These are expanded to include any voltage control and power ﬂow control in STATCOMs and VSC-HVDC systems.

4.5.6

Handling Limits of State Variables

Pure penalty functions may be used to enforce violated limits on state variables, such as nodal voltage magnitudes for voltage-controlled buses using AVRs, load tap changers (LTCs) or any other kind of voltage regulator. A suitable penalty function may be a quadratic equation of the form given in (4.26) [9]: Φ(x1 , … , xn ) =

1 W (S )(Δx)2 2

(4.26)

The term SW is a penalty factor which is a large non-zero integer values. The vector Δx = (Δx1 , …, Δxn )t is deﬁned as the vector of all violated state variables, which must be enforced to their respective limits. The penalty function in (4.26) has the Jacobian and Hessian terms given in Eqs. (4.27) and (4.28), respectively. ( )t 1 W 𝜕 𝜕 2 2 ∇x Φ(x1 , … , xn ) = S Δx , … , Δx (4.27) 2 𝜕Δx1 1 𝜕Δxn n ( 2 )t 1 𝜕2 𝜕 2 2 Δx , … , Δx (4.28) ∇2xx Φ(x1 , … , xn ) = SW n 2 𝜕Δx21 1 𝜕Δx2n It is observed that by choosing a penalty function with a quadratic form, it yields a Hessian matrix which is positive deﬁnite (i.e. ∇2xx Φ > 0). This enables application of convexiﬁcation around the violated limit, Δx, which in turn helps to improve the numerical eﬃciency of the solution algorithm. The reader should be aware that selecting a good initial value for the penalty term is a highly empirical exercise, which is rooted in experience and trial and error [3]. Choosing too large a value may lead to ill-conditioning of the solution whereas small values may lead to a poor rate of convergence and possible stagnation. Experience shows that a value of 1010 is a good penalty factor in most cases.

4.5.7

Handling Limits of Functions

Sometimes it becomes necessary to enforce a control action throughout the OPF solution process to represent the action of control equipment in the power system. One classic example is the synchronous generator’s reactive power control. Under normal circumstances, the generator keeps constant nodal voltage at the high-voltage bus of its connecting transformer by way of its AVR system, using ﬁeld excitation control [8]. However, synchronous generators operating under stressful operating conditions may hit their reactive power limit and the AVR is no longer able to maintain nodal voltage at the speciﬁed magnitude. To model such an operating condition in the OPF solution, it becomes necessary to enforce the reactive power limit for the violated generator in question by deﬁning an explicit Lagrangian function in the form of Eq. (4.24) [3].

169

170

4 Optimal Power Flows

1

2

3

4

P1gen + jQ1gen

P2gen + jQ2gen jQ1shunt

P1load + jQ1load

Figure 4.2 A contrived power system diagram.

4.5.8

A Simple Network Model

In this section, the process of formulating the full AC OPF problem given in Eq. (4.12) is described in step-by-step fashion, applied to the contrived four-node power system given in Figure 4.2. This example is designed to show how the various components in the power system are modelled and formulated as an optimization problem. The process involves (i) identifying the state variables, (ii) identifying necessary equality and inequality constraints, and (iii) choosing an appropriate solution method – for this particular example, the augmented Lagrangian method has been chosen. In the following steps the sets of buses, generators and shunt compensators are deﬁned as L = {1, .., 4}, G = {1, 2}, S = {1} respectively. All the appropriate state variables are indexed over their corresponding sets. For example, all nodal voltages are indexed over the set of nodes, L, all generators over the set G, and the shunt compensator over the set S. 4.5.8.1

Step One – Identifying State and Control Variables

The state variables applied to the ﬁctitious four-node power system in Figure 4.2 are given in Eq. (4.29): gen

gen

z = (𝜃2 , .., 𝜃4 , V1 , … , V4 , Bshunt , P1 , P2 )t 1

(4.29)

Appropriate upper and lower limits apply to all variables given in (4.28). As stated earlier, the angle on the slack bus (system reference bus) is ﬁxed to zero. All other nodal voltage magnitudes as well as angles are included in the system state variables vector. The vector of control variables in this case includes the power output of both generators as well as the reactive shunt susceptance of the shunt compensator. 4.5.8.2

Step Two – Identifying Constraints

The equality constraints include the nodal power balance equations at each node in the general form given in (4.13). For this reason, it is necessary to ﬁrst calculate the nodal power injections at each bus using Eq. (4.30). Sl =

4 ∑ m=1

Vl .Ylm .Vn ej(𝜃l −𝜃m −𝛾lm )

∀l ∈ L = {1, .., 4}

(4.30)

4.5 AC OPF Formulation

In (4.30) the term Y lm is the corresponding admittance element in the system admittance matrix for the branch connecting nodes l and m, where m goes from 1 to 4. It should be noted that this also includes the transformer’s reactance element connecting nodes 2 and 3. The reactive power constraint for the shunt compensator in node 3 is given in (4.31). = −jVl2 Bshunt ∀l = {3}, s = {1} Qshunt s s,l

(4.31)

The nodal power balance equations in the form of equality constraints may be formed as shown in Eq. (4.32), which follows the convention set given in Eq. (4.13). Moreover, the set of inequality constraints is formed following the convention set in Eq. (4.14). The ensuing inequality constraint set is shown in Eq. (4.33). ⎧ ⎪ ⎪h1 ⎪ ⎪ ⎪ ⎪ ⎪h2 ⎨ ⎪ ⎪h3 ⎪ ⎪ ⎪ ⎪h4 ⎪ ⎩

=

=

(

)

(

)

hP1 hQ1 hP2 hQ2

⎛ P − Pgen ⎞ 1 1,1 ⎟ =⎜ =0 ⎜Q − Qgen ⎟ ⎝ 1 1,1 ⎠ ⎛ P + Pload ⎞ 2 1,2 ⎟ =⎜ =0 ⎜Q + Qload ⎟ ⎝ 1 1,2 ⎠

(4.32)

= hQ3 = Q3 − Qshunt =0 1,3 ( ) ⎛ P − Pgen ⎞ hP4 4 2,4 ⎟ =⎜ =0 = ⎜Q − Qgen ⎟ hQ4 ⎝ 4 2,4 ⎠ gen,min

+ jQg

gen,max

+ jQg

g(𝛉, V) = ‖(Pg ≤ ‖(Pg

gen,min

gen

gen

)‖2 ≤ ‖(Pg + jQg )‖2

gen,max

)‖2

∀g ∈ G = {1, 2}

(4.33)

In this example, the thermal limits of transmission lines have been neglected but in actual systems these limits are also enforced as part of the overall OPF formulation. 4.5.8.3

Step Three – Forming the Lagrangian Function

Equation (4.34) shows the Lagrangian function at the start of the solution process where the set of active inequality constraints is empty. L(𝛉, V, 𝛌) = F(𝛉, V, Pgen ) +

4 ∑

𝜆i hi (𝛉, V)

(4.34)

i=1

Notice that in (4.34), the objective function is deﬁned as the summation of the individual generators’ quadratic cost functions as shown in (4.35): F(𝛉, V, Pgen ) =

2 ∑

gen

gen2

(ag + bg Pg + cg Pg

)

(4.35)

g=1

It should be noted that the initial dispatch of the generators to enable the calculation of the starting point of the objective function is carried out through a lossless economic dispatch, as stated in Section 4.4.1.

171

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4 Optimal Power Flows

4.5.8.4

Step Four – Linearized System of Equations

The linearized system of equations for this test network is shown in (4.36). It contains the Hessian and the Jacobian terms of the Lagrangian (4.35). ⎛∇L ⎞ ⎞ ⎛ Δ𝛉 ⎞ ⎛ ∇22 L ∇2 L 0 ∇𝛉 h 0 𝛉V ⎜ 𝛉 ⎟ ⎟ ⎟⎜ ⎜ 𝛉 2 2 2 ⎜ ∇V L ⎟ ⎟ ⎜ ΔV ⎟ ⎜∇V𝛉 L ∇V2 L ∇VB L ∇V h 0 ⎜ ⎟ ⎟ ⎟⎜ ⎜ 2 2 (4.36) ∇B2 L ∇B h 0 ⎟ ⎜ ΔBs ⎟ = − ⎜ ∇B L ⎟ ⎜ 0 ∇BV L ⎜ ⎟ ⎟ ⎟⎜ ⎜ ∇B h 0 ∇Pgen h ⎟ ⎜ Δ𝛌 ⎟ ⎜ Δh ⎟ ⎜ ∇𝛉 h ∇V h ⎜∇ gen L ⎟ ⎜ 0 0 0 ∇Pgen h ∇2Pgen2 F ⎟⎠ ⎜⎝ΔPgen ⎟⎠ ⎝ P ⎠ ⎝ It should be noted that each element in the matrix of coeﬃcients in (4.36) is itself a submatrix, containing Hessian (second-order) terms and Jacobian (ﬁrst-order) terms of the Lagrangian, as deﬁned by Eq. (4.21). 4.5.8.5

Step Five – Implementation of the Augmented Lagrangian

If and when the augmented Lagrangian function is required in order to enforce the active set, throughout the course of the solution the active constraints are enforced to their respective limits using the penalty functions in the form given in (4.22). This yields the following augmented Lagrangian function: L(𝛉, V, 𝛌, 𝛍) = F(𝛉, V, Pgen ) +

4 ∑

𝜆i hi (𝛉, V) +

i=1

∑

𝜓j (𝜇j , gj )

(4.37)

j∈A⊂I

In (4.37), the general complex function g is deﬁned by (4.33), corresponding to the set of functions for active inequality constraints. For example, take the case when the active set includes a voltage magnitude violation in node 2 (i.e. V2(2) > V2max ) at the end of the ﬁrst global iteration. This involves the formation of the following augmented Lagrangian function for the next iteration: L(2) (𝛉, V, 𝛌, 𝛍) = F(𝛉, V, Pgen ) +

4 ∑

𝜆i hi (𝛉, V) + 𝜇1 (V2 − V2max ) +

i=1

𝛼 (2) (V − V2max )2 2 2 (4.38)

Hence, the linearized system of equations in (4.36) is augmented by the respective Hessian and Jacobian terms corresponding to the new augmented Lagrangian function. In the solution process, the Hessian term becomes a rather large positive value (i.e. ∇2V 2 L >> 0), which precludes the search direction from increasing the violation: 2

∑ 𝜕2 𝜕 2 (2) L (𝛉, V, 𝛌, 𝛍) = 𝜆i 2 hi (𝛉, V) + 𝛼 2 𝜕V2 𝜕V2 i=1 4

(4.39)

The Jacobian term, on the other hand, penalizes the gradient vector and prevents it from deviating any further from the limit boundaries of the solution space: ∑ 𝜕2 𝜕 (2) L (𝛉, V, 𝛌, 𝛍) = 𝜆i 2 hi (𝛉, V) + 𝜇2 + 𝛼(V2(2) − V2max ) 𝜕V2 𝜕V2 i=1 4

(4.40)

This will also ensure the third KKT condition is satisﬁed at the solution. It should be noted that a pure quadratic penalty function in the form of Eq. (4.26) may be used for enforcing the control in, for example, voltage-regulated buses.

4.5 AC OPF Formulation

4.5.9

Recent Extensions in the OPF Problem

Over the past decade, advances in computation technology, coupled with rapid developments in power systems theory and practice, have enabled both system operators and the wider power systems research community to extend considerably the basic OPF formulation given in (4.1). Such extensions include approximation methods, such as convex relaxation, which approximates the non-convex OPF problem to a convex problem, reducing the associated computational burden when solving large systems. The OPF can also be approximated to be a linear mathematical program by setting all voltage magnitudes to constant, essentially solving a DC-like problem instead of a full AC power ﬂow problem [11]. It should be noted that this is only in the sense that the formulation uses only real numbers as opposed to complex numbers. There are other forms of formulating the OPF problem. For instance, the well-known IPM has been used to formulate the inequality-constrained OPF problem by imposing logarithmic barrier functions on the objective function to create equality-constrained problems, which could then be solved using the method of Lagrange multipliers [2, 6, 12]. The IPM is essentially a Lagrangian method which creates a dual objective function to be solved using the method of Lagrange multipliers described in Section 4.4. Moreover, the deterministic OPF may be extended to be a non-deterministic formulation to account for uncertainties in input data such as renewable power generation outputs, ﬂexible load patterns (e.g. electric vehicles and demand-side management integration) and power systems contingencies [4]. These extensions are beyond the scope of this book and the reader is encouraged to refer to the relevant literature for more in-depth discussions on these topics. 4.5.10

Test Case: IEEE 30-Bus System

In this section, the IEEE 30-bus system1 is used as test system. The OPF solution for this system is formulated and solved for a range of operating conditions using general-purpose, nonlinear mathematical programming solvers. The process of formulating and solving the OPF problem follows the approach described in Sections 4.4.1–4.5.7. 4.5.10.1

Test System

The IEEE 30-bus test system comprises 6 generators, 34 transmission lines, 4 transformers, 30 load points and 2 shunt compensators, which are placed at nodes 10 and 24. The system voltage levels are 132/33 kV and the base power is taken to be 100 MVA. 4.5.10.2

Problem Formulation

Following the general expression in (4.12), the benchmark test system is formulated as a full AC OPF problem. To begin with, no controls other than the generators’ power outputs apply. It follows that the vector of state/control variables is deﬁned as z = (𝜽, V, Pgen )t and the problem is bound to the equality and inequality constraints given in Eqs. (4.13) and (4.14), respectively. The objective of the OPF solution is to ﬁnd a feasible minimum-cost solution for the synchronous generators’ power dispatch subject to constraints. The fuel costs–output powers characteristics of the synchronous generators are modelled as polynomial 1 System data may be found from the following link: http://labs.ece.uw.edu/pstca/pf30/pg_tca30bus.htm.

173

174

4 Optimal Power Flows

Table 4.1 Generator cost function coeﬃcients. Generator number

Bus number

a ($ MW−2 )

1

1

0.0384

20

0

2

2

0.25

20

0

3

5

0.01

40

0

4

8

0.01

40

0

5

11

0.01

40

0

6

13

0.01

40

0

b ($ MW−1 )

c ($)

Table 4.2 Transformer data (load tap changers). Branch number

Sending bus

Receiving bus

Transformer tap

Upper limit

Lower limit

11

6

9

0.978

1.3

0.7

12

6

10

0.969

1.3

0.7

15

4

12

0.932

1.3

0.7

36

28

27

0.968

1.3

0.7

Table 4.3 Shunt devices. Bus number

Initial susceptance (MVAR)

10

19

24

4.3

quadratic cost functions, as in Eq. (4.15). The cost functions are calculated in units of US $ h−1 . The synchronous generators data is given in Table 4.1. Transformers and shunt compensators data are given in Tables 4.2 and 4.3, respectively. Table 4.4 shows the synchronous generators limits. 4.5.10.3

OPF Test Cases

Further to the base case (benchmark OPF solution), four other test cases are solved to investigate various aspects of the OPF formulation, using the IEEE 30-bus system. The benchmark test case solution veriﬁes the model objects created in the advanced interactive multidimensional modelling system (AIMMS) environment2 and compares it against the solution furnished by a well-known Matlab-based power system simulation package, i.e. MATPOWER.3 The model object created in AIMMS is a fully customizable 2 AIMMS is a multi-purpose optimization model-based language. A free academic license may be obtained from: https://aimms.com. 3 MATPOWER is a Matlab-based open source software available to download from: http://www.pserc .cornell.edu/matpower.

4.5 AC OPF Formulation

Table 4.4 Generator limits. Generator number

Active power (upper limit) (MW)

Reactive power (upper limit) (MVAR)

Reactive power (lower limit) (MVAR)

1

360.2

10

0

2

140

50

−40

3

100

40

−40

4

100

40

−40

5

100

24

−24

6

100

24

−24

model of the IEEE 30-bus system, which follows the problem formulation given in Eq. (4.12). This model can be modiﬁed with ease to accommodate further linear and nonlinear constraints for exerting control actions at various points of the power system. Three test cases with voltage regulation are carried out using the four tap-changing transformers available in the IEEE 30-bus system. The transformers’ taps are modelled as additional constraints using the model given in Chapter 3, with the taps treated as continuous variables. If required, additional control constraints in the form of Eq. (4.26) may be added to exert nodal voltage control. The three OPF solutions with voltage regulation are solved using the augmented Lagrangian method embedded within the MINOS solver. Furthermore, the OPF solutions have been carried out with an interior-point solver (IPOPT) and with an additional solver called CONOPT, which uses a generalized gradient method and has been developed for solving highly nonlinear problems. In all cases solved as part of this exercise, the OPF solutions converge to exactly the same results and within the same bounds deﬁned by the constraints. 4.5.10.4

Benchmark Test Case (With No Voltage Control)

There is no nodal voltage regulation in the base test case and the transformers’ taps are taken to be ﬁxed parameters. A summary of results is presented in Tables 4.5 and 4.6 as given by the solvers AIMMS and MATPOWER, respectively. It should be noted Table 4.5 Optimal power ﬂow solution: real powers (benchmark test). Generator number

Final solution (AIMMS) (MW)

Final solution (MATPOWER) (MW)

1

212.23

212.23

2

36.23

36.23

3

29.35

29.35

4

12.93

12.93

5

4.4

4.4

6

0

0

175

176

4 Optimal Power Flows

Table 4.6 Optimal power ﬂow solution: reactive powers (benchmark test). Generator number

Final solution (AIMMS) (MVAR)

Final solution (MATPOWER) (MVAR)

1

0.00

0.00

2

27.17

27.17

3

29.94

29.94

4

40.00

40.00

5

9.10

9.10

6

7.72

7.72

that their respective solutions converge to exactly the same optimal values. Given the nonlinearity of the considered constraints – nodal power balance equations – the KKT conditions only guarantee a local optimum for the primal problem, which is deﬁned by Eq. (4.12). 4.5.10.5

Test Case with Voltage Control Using Variable Transformers Taps (Case I)

In this test case, the vector of state/control variables is extended to include the variable tap T for the on-load tap changer transformers. Hence, the vector takes the following form: zcont = (𝜽, V, Pgen , T)t . No further constraints are added to the constraints set used in the benchmark test case. However, during the course of the OPF solution, the tap is now treated as a variable as opposed to a constant parameter. 4.5.10.6

Test Case with Nodal Voltage Regulation (Case II)

In this test case, an additional constraint is added to regulate the nodal voltages of the transformer buses to keep the values obtained in the benchmark test case. The nodal voltages are kept ﬁxed by adding a constraint to the constraints set (4.26). The target nodal voltage magnitudes and the associated transformers’ taps values, which are now part of the state variables vector, are given in Table 4.7. It should be noticed from this table that the OPF solution has converged to the same tap ratios as in the benchmark test case, which in that particular solution were ﬁxed values. This result further validates the accuracy response of the OPF model using the AIMMS solver. Table 4.7 Voltage regulation at transformer buses – controlled test II. Transformer number

Regulated bus

Target voltage

Transformer ﬁnal tap value

1

9

1.0422

0.978

2

10

1.0385

0.969

3

12

1.0498

0.932

4

27

1.0215

0.968

4.5 AC OPF Formulation

Table 4.8 Voltage regulation at transformer buses – controlled test III. Transformer number

Regulated bus

Target voltage

Transformer ﬁnal tap value

1

9

1.05

0.901

2

10

1.01

1.145

3

12

1.00

1.041

4

27

1.00

0.977

As expected, there are no changes in the results compared to the nodal voltage magnitudes in the benchmark test case – both test cases converge to the same solution. 4.5.10.7

Test Case with Nodal Voltage Regulation (Case III)

The OPF model now considers a case of variable operating conditions. The test considers that the nodal voltage magnitudes at the transformers’ receiving end buses are controlled to predetermined values. The same process applies as in the previous test case, i.e. nodal voltage regulation I, where the transformers’ taps are taken to be state variables. Table 4.8 shows the transformers’ taps values at the optimum solution, corresponding to the target voltage values shown in Table 4.8. Constraint Handling

1. There is a voltage constraint violation at bus 11, which is handled with the use of the augmented Lagrangian function method. The voltage is enforced to its corresponding upper limit of 1.06 p.u. 2. Likewise, the synchronous generators connected at nodes 8 and 13 are bound to their respective upper limits of 40 and 24 MVAR, respectively. This contrasts with the reactive power behaviour of the slack generator, which is kept to its lower limit, i.e. zero. 4.5.10.8

A Summary of Results

AIMMS uses the nonlinear solver CONOPT whereas MATPOWER uses Matlab’s interior point solver MIPS. A summary of the OPF results for both benchmark and voltage-controlled tests is given in Table 4.9. It can be observed that the best optimal solution is obtained when the transformers’ taps are free to vary within their limits and no additional constraints are imposed on the system. In this case the total value of the objective function is $8902.69 h−1 for the six generators. Conversely, the worst solution among the ﬁve tests carried out corresponds to the ﬁnal case where nodal voltage magnitudes are controlled at predetermined values, signifying that this is a more constrained operating point. In this case the total value of the objective function gives a cost of $8909.3 h−1 , which increases the cost by $6.7 h−1 . The nodal voltage proﬁles for the ﬁve test cases are shown in Figure 4.3 for the 30 nodes of the test system, where the changes in nodal voltage magnitudes are shown when comparing the tests with voltage control and the benchmark tests. In all instances, the OPF algorithms converge to local optimal solutions, with all the constraints satisﬁed at their limits.

177

178

4 Optimal Power Flows

Table 4.9 OPF summary results – IEEE 30-bus system.

Test

Convergence time

Optimality of primal

Final objective solution

MATPOWER MIPS – Interior point

Benchmark

3.9 s

Locally optimal

$8906.14 h−1

AIMMS

MINOS – Augmented Lagrangian

Benchmark

0.05 s – 11 Locally optimal Global iterations

$8906.14 h−1

AIMMS

MINOS – Augmented Lagrangian

Controlled I – Variable tap changer

0.05 sec – 11 Locally optimal Global iterations

$8902.69 h−1

AIMMS

MINOS – Augmented Lagrangian

Controlled II – Voltage regulation I

0.05 sec – 7 Locally pptimal Global iterations

$8906.18 h−1

AIMMS

MINOS – Augmented Lagrangian

Controlled III – Voltage regulation II

0.03 sec – 6 Locally optimal Global iterations

$8909.30 h−1

Model object

OPF solver

Nodal voltage magnitudes – IEEE 30–bus system tests

Nodal voltage magnitude (p.u.)

1.06

MATPOWER AIMMS (Benchmark) Controlled I Controlled II Controlled III

1.04

1.02

1

0.98

0.96 0

5

10

15 Bus number

20

25

30

Figure 4.3 IEEE 30-bus test results: nodal voltage magnitudes.

The synchronous generators’ power dispatches for both active and reactive powers are shown in Figure 4.4 for the benchmark and the third test case, i.e. nodal voltage regulation to a priori set values. Notice the changes in the reactive power dispatch brought about by changes in the nodal voltage magnitude settings. The change in the optimum active power dispatch is minimum.

4.6 Generalization of the OPF Formulation for AC-DC Networks

0.4 Active power dispatch – benchmark Active power dispatch – controlled III

Reactive power (p.u.) – 100 MVA base

Real power (p.u.) – 100 MVA base

2.5

2

1.5

1

0.5

0

Reactive power dispatch – benchmark Reactive power dispatch – controlled III

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

1

2

3 4 5 Generator number

6

1

2

3 4 5 Generator number

6

Figure 4.4 Generators’ power outputs at optimum.

4.6 Generalization of the OPF Formulation for AC-DC Networks The original OPF formulation shown in (4.12) is extended in this section to incorporate explicitly all the control constraints necessary to model combined AC-DC network. If vector z = (𝜽, V, Pgen )t contains the original vector of state variables for the AC network, then this is expanded to encompass the state variables corresponding to the DC network and DC equipment, as described in Eq. (4.41). min

z,xdc ,udc

s.t.

F(z, xdc , udc ) ⎧hacdc (z, xdc , udc ) = 0 ⎪ dc ⎪C (xdc , udc ) = 0 ⎪gacdc (z, xdc , udc ) ≤ gacdcmax ⎪ dc dcmax ⎨g (xdc , udc ) ≤ g ⎪zmin ≤ z ≤ zmax ⎪ max min ≤ xdc ≤ xdc ⎪xdc ⎪ min max ⎩udc ≤ udc ≤ udc

(4.41)

The objective function is taken to be the generators’ cost functions and for the sake of simplicity in the formulation, any post-contingency operating states are neglected to begin with. These will be incorporated as the iterative solution progresses and there are constraints violations. In Eq. (4.41) the vector zdc = (xdc , udc )t is a vector containing DC state and control variables, to be deﬁned at a later stage. Moreover, the explicit equality constraints associated with DC equipment are contained in the complex function set Cdc (xdc ,udc ), which

179

180

4 Optimal Power Flows

possesses a set of Lagrangian functions. Each piece of DC equipment has an associated inequality constraints set deﬁned by gdc , as well as limits on the DC state and control variables. The OPF formulation in (4.41) represents a more generalized formulation than the one given in (4.12). It is now possible to solve the power ﬂow equations of a combined AC-DC system using a uniﬁed reference frame. However, the assumption of the existence of a uniﬁed frame-of-reference for solving power ﬂow equations hinges on deﬁning the appropriate set of nodal power balance equations – hacdc , one that makes no notional diﬀerence between the DC and AC networks. In other words, and from the perspective of a mathematical programming solver, there should be no distinctions between the AC and DC sides of the network. This assumption is illustrated in Figure 4.5. The Lagrangian function of the equation set in (4.41) is given by Eq. (4.42), assuming that the active set is empty at this stage. ∑ ∑ dc Lacdc (z, zdc ) = F(z, zdc ) + 𝜆i hacdc (z, zdc ) + 𝜆dc (4.42) i j Cj (zdc ) i∈E

j∈Edc

dc t In (4.42) the vector 𝛌dc = (𝜆dc 1 , … , 𝜆N ) is the vector containing Lagrange multipliers pertaining to DC equipment or DC systems. The set Edc is the set of all DC control constraints deﬁned by the complex function C(z) = 0. The linearized system of equations given in (4.18) and (4.19) for the AC OPF problem may be expanded to include the compound AC-DC problem, as follows:

⎡ ∇2zz Lacdc ⎢∇2 Lacdc ⎢ zdc z ⎢ ∇z (hacdc ) ⎢ ⎣ ∇z (Cdc )

∇2zzdc Lacdc ∇2zdc zdc Lacdc ∇zdc (hacdc ) ∇zdc (Cdc )

∇z (hacdc ) ∇zdc (hacdc ) 0 0

∇z (Cdc ) ⎤ ⎡ Δz ∇zdc (Cdc )⎥⎥ ⎢⎢ Δzdc ⎥ ⎢ Δ𝛌 0 ⎥⎢ 0 ⎦ ⎣Δ𝛌dc

⎤ ⎡ ∇z Lacdc ⎥ ⎢ acdc ⎥ = − ⎢∇zdc L acdc ⎥ ⎢ Δh ⎥ ⎢ ⎦ ⎣ ΔCdc

⎤ ⎥ ⎥ (4.43) ⎥ ⎥ ⎦

This system of linear equations may be solved by iteration towards a candidate optimum, using Newton’s method. The necessary optimality conditions introduced in Section 4.3 guarantee that a candidate solution for the system of equations in (4.43) is a local optimum. It should be remarked that the key to integrating AC-DC power ﬂows in a single frame of reference rests on the ability to deﬁne a suitable nodal power balance equation set – hacdc .

AC network

DC network (notional AC network)

DC constraints: Cdc = 0

Coupled constraints set: hac – dc = 0 Figure 4.5 Network model for AC-DC uniﬁed formulations.

4.7 Inclusion of the VSC Model in OPF

4.7 Inclusion of the VSC Model in OPF The VSC equivalent circuit, for which the full model was developed in Chapter 3 in rectangular coordinates, is shown schematically in Figure 4.6. It should be brought to the reader’s attention that there are alternative VSC models for both power ﬂows and OPFs, which can be used instead of the VSC model shown in Figure 4.6. For example, [13] presents a converter model where the VSC is described by a controllable voltage source. However, such a model is more contrived than the one presented here. For instance, it does not explicitly contain a DC bus. Nonetheless, the lack of a DC bus in the voltage source model does not prevent it from being used in the STATCOM and VSC-HVDC power ﬂow applications. but it is amenable only to iterative, numerical solutions of the sequential kind. That is, the AC and DC subsystems are solved separately, in sequences, exchanging partial results between the AC and DC networks at each iteration until there are no discernible changes in results in both networks. This contrasts with the numerical solutions presented in Chapters 3–6 of this book, where iterative uniﬁed solutions of the AC and DC subsystems are carried out. This is possible owing to the use of the model shown in Figure 4.6, which is essentially the model of a STATCOM and represents the VSC stations of the many forms of VSC-HVDC systems. 4.7.1

VSC Power Balance Equations

Following the generalized OPF formulation introduced in Section 4.6, the VSC nodal power balance equations may be written down, in compact form, as follows: hvsc c (𝛉vR , VvR , Edc , Beq , ma , Φ) = 0

(4.44)

∀c ∈ Cv

The function in (4.44) represents the coupled AC-DC nodal power ﬂows at the sending node vR and at the receiving node 0 of the converter station. The set Cv is deﬁned to be the set comprising all the VSC converter stations in the system. Figure 4.6 VSC converter station with its uniﬁed equivalent circuit.

+ EDC

ʹ

VvR

VvR

CDC Yreac

Yfilter

– maeJφ 1: k2mae jφ + EDC –

ʹ

V vR Y1

G0

jBeq

181

182

4 Optimal Power Flows

Building on the model presented in Chapter 3, it is noted that the AC and DC sides of the converter are coupled together by way of the AC voltage equation given in (4.45), which is valid for any VSC converter station – for simplicity, the converter index c is being dropped: V 1 = k2 ma Edc = k2 ma Edc ej𝜙

(4.45)

It should be noted that this equation is in fact equation (3.32) used in the previous chapter and it is reproduced here to enable a smooth development of the VSC-OPF formulation. The factor k 2 allows for application of Eq. (4.45) to any kind of AC-DC converter, including modular multi-level converters. For a two-level, three-phase VSC √

the constant is: k2 = 38 . For the VSC station given in Figure 4.6, the nodal power equations can be calculated using Eq. (4.46): vsc

vsc

vsc ∗

Sc = diag(V c ).I c

vsc

vsc

= diag(V c ).{Yvscc .V c }∗

∀c ∈ Cv

(4.46)

Expanding Eq. (4.46) results in an equation similar to the one derived in the previous chapter for the full VSC model, but all the variables are expressed using polar coordinates: ) ( SvRc = S 0c )( ( ) {( )} ∗ ∗ VvRc ej𝜃vR 0 Y vRvRc −Y vR0c e−j𝜑c VvRc e−j𝜃vRc . ∗ ∗ 0 Edcc Edcc −Y vR0c ej𝜑c (Gsw − jk22 m2a Beq ) + Y 0c (4.47)

∀c ∈ Cv where Y vRvR = Y ﬁlter + ΔY = YvRvR ej𝛾vRvR Y vR0 = k2 ma ΔY = YvR0 ej𝛾vR0 Y 00 = k22 m2a ΔY = Y00 ej𝛾00 ΔY =

Y 1 Y reac (Y 1 + Y reac )

(4.48) (4.49)

Note that the index c has been dropped. Expressions for the nodal power equations at the sending and receiving ends of the VSC may be derived by developing the complex voltage–current relationship in (4.47). The ensuing equations are given in (4.50). { 2 SvR = (PvR + jQvR ) = VvR YvRvR e−j𝛾vRvR − VvR Edc YvR0 ej(𝜃vR −𝛾vR0 −𝜙) 2 S0 = (P0 + jQ0 ) = Edc [(Gsw − jk22 m2a Beq ) + Y00 e−j𝛾00 ] − VvR Edc YvR0 ej(𝜙−𝜃vR −𝛾vR0 ) (4.50) When integrating this model into the OPF formulation, one reactive power constraint needs to be included in the form of a control constraint on the DC side of the VSC to enforce reactive power ﬂow to zero at the DC side. It should be noted that there is no

4.7 Inclusion of the VSC Model in OPF

reactive power in DC circuits. To force zero reactive power ﬂow on the DC circuit, a notional variable susceptance, Beq , is employed in the AC side of the converter. The zero reactive power constraint is: Q0 = 0

(4.51)

The VSC formulation in OPF requires the equality constraints given in Eqs. (4.50) and (4.51), which are added to the system Lagrangian function. The vector of state and control variables for the VSC is given in Eq. (4.52): zvsc = (𝜃vR , VvR , Edc , Beq , ma , 𝜑)t

(4.52)

Moreover, the vector of Lagrange multipliers for the equality constraints in Eqs. (4.50) and (4.51) is deﬁned as follows: 𝛌vsc = (𝛌vR , 𝜆P0 , 𝜆Q0 )t

(4.53)

where the term 𝝀vR = (𝜆vRP , 𝜆vRQ )t is the vector of Lagrange multipliers associated with the active and reactive power balance equations at the VSC’s AC terminal, respectively. Once the vector of variables/multipliers is identiﬁed for the VSC station, the power balance equations for the VSC station, in the form of equality constraints, given in Eq. (4.44), may be written down as follows: Lvsc (zvsc , 𝛌vsc ) = 𝛌tvR hvR + 𝛌tP CPdc + 𝛌tQ CQdc 0

0

(4.54)

Note that in Eq. (4.54), the following nodal power balance equations are deﬁned as equality constraints for both AC and DC sides of the converter station: ⎧ gen ⎛ vR ⎞ ⎛ load ⎞ ⎪ vR ⎜hPc ⎟ ⎜ PvRc − Pg,c + Pd,c ⎟ = = 0 ∀c ∈ Cv, g ∈ G, d ∈ D = h ⎪ c ⎜hvR ⎟ ⎜Q − Qgen + Qload ⎟ ⎪ g,c ⎝ Qc ⎠ ⎝ vRc d,c ⎠ ⎨ dc sch ∀c ∈ Cv ⎪CP = P0c − P0c = 0 ⎪ ⎪CQdc = Q0 − Qsch ∀c ∈ Cv 0c = 0 c ⎩

(4.55)

In Eq. (4.55), two control constraints are deﬁned explicitly for the DC side of the VSC station to enable: (i) control/schedule of DC power ﬂow to a predetermined value, Psch , and (ii) zero reactive power output at the DC side of the converter station to be maintained. The latter point means that within the OPF formulation, the reactive power at the DC side of the converter station is always set to zero, Qsch 0 = 0. 4.7.2

VSC Control Considerations

The VSC control strategies, described in Chapter 3 for power ﬂows, apply with little change to the case of OPF. The VSC can be used to control rather eﬀectively the nodal voltage magnitude at the VSC’s AC side, i.e. node vR, while keeping the DC side voltage, Edc , constant. To exert voltage control within the OPF solution, we use penalty functions similar to the one described in Section 4.5.3. In the simulation scenarios given in this chapter, the controls are automatically handled by the solver. The VSC variables are initialized in a similar manner as for the conventional power ﬂow solution. All the VSC stations in the

183

184

4 Optimal Power Flows

simulations presented in this chapter operate within their linear regions of ma operation, assuming purely sinusoidal voltage and current waveforms. The limits on the equivalent susceptance, Beq , deﬁne the boundaries of VSC station operation in terms of reactive power operation. Similarly, limits on the active power operation of the VSC may also be enforced by the introduction of suitable constraints. 4.7.3

VSC Linearized System of Equations

The linearized system of equations, given in this section in compact form, possesses the same structure as the linearized system of equations given in Eq. (4.43) but for the VSC with a Lagrangian function given in Eq. (4.54). )( ) ( ) ( Δzvsc 𝜕Lvsc ∕𝜕zvsc Hvsc Jvsc =− (4.56) Jtvsc 0 Δ𝛌vsc 𝜕Lvsc ∕𝜕𝛌vsc In Eq. (4.56) the following Hessian and Jacobian terms are deﬁned as follows: ⎧ t t dc t dc 2 2 2 vR ⎪Hvsc = 𝛌vR ∇z2vsc h + 𝛌P0 ∇z2vsc CP + 𝛌Q0 ∇z2vsc CQ ⎨ ⎪Jvsc = ∇zvsc hvR + ∇zvsc CPdc + ∇zvsc CQdc ⎩ Similarly, the gradient terms are deﬁned as follows: ⎧ vsc 𝜕CQdc vR 𝜕CPdc t ⎪ 𝜕L = 𝛌t 𝜕h + 𝛌t + 𝛌 vR P0 Q0 ⎪ 𝜕zvsc 𝜕zvsc 𝜕zvsc 𝜕zvsc ⎨ ⎪ 𝜕Lvsc vR dc dc ⎪ 𝜕𝛌 = Δh + ΔCP + ΔCQ ⎩ vsc

(4.57)

(4.58)

4.8 The Point-to-Point and Back-to-Back VSC-HVDC Links Models in OPF The VSC OPF formulation is extended to model the point-to-point VSC-HVDC link and, as a particular case, the back-to-back VSC-HVDC link. In this respect, a general frame-of-reference to solve OPF problems aﬀorded by Eq. (4.41) becomes a rather useful device for modelling these kinds of VSC-HVDC links, with the variables/multipliers explicitly deﬁned for the two VSC stations together with their corresponding Lagrangian functions. The power balances that exist in the point-to-point and back-to-back VSC-HVDC links, useful for the OPF solution, are shown in Figure 4.7a,b, respectively. Each converter station is connected in shunt with its corresponding AC system using a step-up transformer, which may be taken to be an LTC transformer for the sake of modelling convenience, although in practice they may well be conventional transformers. Nonetheless, the VSC interfacing transformer, whether a conventional transformer or an LTC one, is handled as separate components within the OPF formulation. The VSC and the transformer each will have their own Lagrangian function and equality constraints, i.e. nodal power balance equations at the VSC and transformer interfacing nodes.

4.8 The Point-to-Point and Back-to-Back VSC-HVDC Links Models in OPF

S0R

GDC

PDCR

VvR

PDCI

VvI

EDCI

EDCR

SvR

S0I

SvI maR

maI Inverter

Rectifier (a) S0I

S0R

VvR

VvI EDC

SvR

SvI

maR

maI

Rectifier

Inverter (b)

Figure 4.7 Power balances in VSC-HVDC links: (a) point-to-point, (b) back-to-back (interfacing transformers not shown).

4.8.1

VSC-HVDC Link Power Balance Formulation

From the modelling viewpoint, there will be two sets of nodal power balance equations for each VSC station (i.e. rectiﬁer and inverter), similarly to Eq. (4.50), with the pro2 viso that the DC power loss, i.e. RDC IDC , should be taken into account in point-to-point VSC-HVDC topologies. Naturally, this power loss is nil in back-to-back VSC-HVDC topologies. In order to identify the power balance equations for the DC link, the circuit in Figure 4.8 proves rather useful. 0 0 The terms S0R and S0I correspond to the VSC nodal power equations at the DC sides of the inverter and the rectiﬁer converters, as given by Eq. (4.50). To derive an expression for the nodal power injections at both DC nodes, the active power losses in the cable 0R

0

GDC

S0R

0I

0

S0I

S0R

S0I

PDCR 0

Re S0R = Re S0R + PDCR

PDCI 0

Re S0I = Re S0I + PDCI

Figure 4.8 The VSC-HVDC link for a point-to-point conﬁguration with the DC link nodal power balance equations.

185

186

4 Optimal Power Flows

require consideration. Since there is no reactive power ﬂow in a DC cable, the reactive power loss is nil and the following equation set represents the nodal power balance of the DC link in Figure 4.8: ⎧Re{S } = Re{S0 } + P 0R DCR 0R ⎪ 0 ⎪Im{S0R } = Im{S0R } + 0 (4.59) 0 ⎨ ⎪Re{S0I } = Re{S0I } + PDCI 0 ⎪ ⎩Im{S0I } = Im{S0I } + 0 ) ( 0 0 t where S0R S0I is the vector of nodal power injections at the DC side, which were derived using Eqs. (4.46) and (4.47) and given by Eq. (4.50). The expression (4.60) encapsulates the DC active power losses in the cable, which is derived following a similar procedure as in (4.46) and (4.47). ) ( ) {( )( )} ( PDCR EDCR GDC −GDC EDCR = EDCI −GDC GDC EDCI PDCI ) ( 2 GDC − GDC EDCI EDCR EDCR (4.60) = 2 EDCI GDC − GDC EDCI EDCR Substituting Eqs. (4.50) and (4.60) into (4.59) yields the full set of nodal power injections for the rectiﬁers and the inverter stations: 2 SvR = (PvR + jQvR ) = VvR YvRvR e−j𝛾vRvR − VvR EDCR YvR0 ej(𝜃vR −𝛾vR0 −𝜙R ) 2 S0R = (P0R + jQ0R ) = EDCR [(GswR − jk22 m2aR BeqR ) + Y00R e−j𝛾00R ]

− VvR EDCR YvR0 ej(𝜙R −𝜃vR −𝛾vR0 ) + PDCR SvI = (PvI + jQvI ) = VvI2 YvIvI e−j𝛾vIvI − VvI EDCI YvI0 ej(𝜃vI −𝛾vI0 −𝜙I ) 2 S0I = (P0I + jQ0I ) = EDCI [(GswI − jk22 m2aI BeqI ) + Y00I e−j𝛾00I ]

− VvI EDCI YvI0 ej(𝜙I −𝜃vI −𝛾vI0 ) + PDCI

(4.61)

It should be noted that the terms Y vR0 and Y vI0 are functions of maR and maI – refer to the VSC equivalent admittance deﬁnition given in Eq. (4.48). From Eqs. (4.59) and (4.60), and taking into account the equality constraints given in Eq. (4.55), the complete set of equality constraints for the point-to-point VSC-HVDC link is given in Eq. (4.62). This includes any binding equality constraint for enforcing active power control on its DC side. ⎧ gen ⎛ vi ⎞ ⎛ load ⎞ ⎪hvi = ⎜ hPc ⎟ = ⎜ Pvic − Pg,c + Pd,c ⎟ = 0 ∀c ∈ Cv, g ∈ G, d ∈ D ⎪ c ⎜hvi ⎟ ⎜Q − Qgen + Qload ⎟ g,c ⎪ ⎝ Qc ⎠ ⎝ vic d,c ⎠ ⎪ 0 0i and i = {R, I} ⎨hc = Re{S0ic − S0ic } − PDCic = 0 ∀c ∈ Cv ⎪ dc sch ⎪CPi = P0ic − P0i =0 ∀c ∈ Cv c ⎪ ⎪C dc = Q0i − Qsch = 0 ∀c ∈ Cv 0ic c ⎩ Qi (4.62)

4.8 The Point-to-Point and Back-to-Back VSC-HVDC Links Models in OPF

It would be appropriate at this point to bring to the reader’s attention that for the back-to-back VSC-HVDC link, the DC power loss is set to zero in the formulation. Hence, Eq. (4.59) simpliﬁes to the following: 0 ⎧ ⎪Re{S0R } = Re{S0R } + 0 0 ⎪ ⎪Im{S0R } = Im{S0R } + 0 ⎨ 0 ⎪Re{S0I } = Re{S0I } + 0 ⎪ ⎪Im{S } = Im{S0 } + 0 0I 0I ⎩

4.8.2

(4.63)

VSC-HVDC Link Control

There are several control options available for the VSC rectiﬁer and inverter stations. It was stated in Chapter 3 that in a point-to-point VSC-HVDC conﬁguration, the rectiﬁer station is normally used to regulate power at its DC side whereas the inverter station is used to regulate its DC voltage. These control laws may be added to the general OPF formulation by simply deﬁning additional control constraints in the form of Eq. (4.24) for power ﬂow control at either side of the converter stations. Alternatively, pure penalty functions may be used for enforcing nodal voltage magnitudes at either side of the converter, i.e. AC or DC. The DC power ﬂow control is already included as an equality power constraint on the DC side, given by the set of constraints in Eq. (4.62). More speciﬁcally, when the rectiﬁer station exerts DC power ﬂow control at its DC, we have: sch CPdc = P0R − P0R =0

(4.64)

R

For the case of the point-to-point conﬁguration shown in Figure 4.7a, and assuming that the rectiﬁer regulates DC power ﬂow and that the inverter regulates DC nodal voltage, the Lagrangian function may be written down as follows: Ldc link (zvscR , zvscI , 𝛌R , 𝛌I ) = 𝜆tvR hvR + 𝜆tvI hvI + 𝜆0R h0R + 𝜆0I h0I + + 𝜆P0R CPdc + 𝜆Q0R CQdc + 𝜆Q0I CQdc R

R

I

(4.65)

In Eq. (4.65), the following vectors of state/controls variables and Lagrange multipliers are deﬁned for the DC link: { zvscR = (zvR , z0R )t (4.66) zvscI = (zvI , z0I )t { 𝛌R = (𝜆vR , 𝜆0R , 𝜆PQR )t (4.67) 𝛌I = (𝜆vI , 𝜆0I , 𝜆PQI )t Furthermore, in Eqs. (4.66) and (4.67) the following subvectors are deﬁned: { zvR = (𝜃vR , VvR , BeqR , maR , 𝜙R )t zvI = (𝜃vI , VvI , BeqI , maI , 𝜙I )t

(4.68)

187

188

4 Optimal Power Flows

{ z0R = (EdcR )t z0I = (EdcI )t { 𝛌PQR = (𝜆P0R , 𝜆Q0R )t 𝛌PQI = (𝜆P0I , 𝜆Q0I )t 4.8.3

(4.69)

(4.70)

VSC-HVDC Full Set of Equality Constraints

The schematic diagram of a point-to-point VSC-HVDC link and its associated constraints is shown in Figure 4.9. However, note that not all the constraints are active at the same time and that their inclusion in the active set depends on which converter control modes have been selected. The only exceptions to the above are the zero reactive power constraints, i.e. CQdc = 0 and CQdc = 0, which are always binding given that they R I correspond to physical constraints. The vector of state/control variables for the AC-DC system is given in Eqs. (4.71)–(4.74), including subvectors for the AC and DC networks and the DC link. ZACDC = (ZAC , zvscR , zvscI )t

(4.71)

ZAC = (Pgen , 𝛉AC , VAC , T)t

(4.72)

zvscR = (zvR , z0R )

(4.73)

zvscI = (zvI , z0I )

t

t

(4.74)

The following vectors are deﬁned for the Lagrange multipliers associated to the nodal power balances (and other constraints) for both the AC system and the DC link. 𝚲ACDC = (𝚲AC , 𝛌R , 𝛌I )t

(4.75)

𝚲AC = (𝛌AC )t

(4.76)

where 𝛌R = (𝜆vR , 𝜆0R , 𝜆PQR ) 𝛌I = (𝜆vI , 𝜆0I , 𝜆PQI )

t

(4.77)

t

(4.78)

As shown in Figure 4.9, three sets of constraints are required for the entire AC-DC system: the set of AC constraints, the set of DC constraints and the set of coupled AC-DC GDC

hacR = 0

h2vI = 0

h1vR = 0 h10R = 0

h20I = 0

dc = 0 CPR

dc = 0 CPI

dc = 0 CQR

dc = 0 CQI

hacI = 0

Figure 4.9 Hybrid AC-DC network model with the full set of coupled AC-DC constraints.

4.8 The Point-to-Point and Back-to-Back VSC-HVDC Links Models in OPF

constraints. They are deﬁned as follows: ( ) ⎧ hP ac ⎪h = =0 hQ ⎪ ⎪ gen ⎪ ⎛ vi ⎞ ⎛ load ⎞ ⎪hvi = ⎜ hPc ⎟ = ⎜ Pvic − Pg,c + Pd,c ⎟ = 0 ∀c ∈ Cv, g ∈ G, d ∈ D c ⎪ ⎜hvi ⎟ ⎜Q − Qgen + Qload ⎟ g,c and i = {R, I} d,c ⎠ ⎝ Qc ⎠ ⎝ vic ⎨ ⎪ 0 ⎪h0i c = Re{S0ic − S0ic } − PDCic = 0 ∀c ∈ Cv ⎪ dc sch ⎪CPi = P0ic − P0i =0 ∀c ∈ Cv c ⎪ ⎪C dc = Q − Qsch = 0 ∀c ∈ Cv 0ic 0i ⎩ Qi c

(4.79) The Lagrangian function for the overall AC-DC system is: Lacdc (ZACDC , 𝚲ACDC ) = F(ZAC ) + Lac (ZAC , 𝚲AC ) + Ldc (zvscR , zvscI , 𝛌R , 𝛌I )

(4.80)

Note that no active inequality constraints are assumed to exist at this point of the solution; when the inequality limits become violated, as the solution proceeds they will be enforced using either penalty functions or the augmented Lagrangian method. The Lagrangian function of the AC system takes the following form: Lac (ZAC , 𝚲AC ) = 𝚲tAC hac (ZAC )

(4.81)

The function hac (ZAC ) refers to the vector of nodal power balance equations at the AC side of the system, excluding the AC nodes of the DC link. Similarly, the Lagrangian term pertaining to the DC link is deﬁned in Eq. (4.65), with the constraint set deﬁned in Eq. (4.62). The Lagrangian function for the DC link is: Ldc link (zvscR , zvscI , 𝛌R , 𝛌I ) = 𝛌t vR hvR + 𝛌t vI hvI + 𝜆0R h0R + 𝜆0I h0I + + 𝜆P0R CPdc + 𝜆Q0R CQdc + 𝜆Q0I CQdc R

4.8.4

R

(4.82)

I

Linearized System of Equations

With reference to Eqs. (4.71)–(4.82), the linearized system of equations for the entire AC-DC system may be written as follows: ⎛H HACR HACI JAC JACR JACI ⎞ ⎛ ZAC ⎜ AC ⎟⎜ ⎜ Ht ⎟ ⎜zvscR H H J J J R RI RAC VRR VRI ⎜ ACR ⎟⎜ HI JIAC JVIR JVII ⎟ ⎜ zvscI ⎜ HtACI HtRI ⎜ t ⎟⎜ JtACR JtACI 0 0 0 ⎟ ⎜ 𝚲AC ⎜ JAC ⎜ Jt t t 0 0 0 ⎟⎟ ⎜⎜ 𝛌R ⎜ RAC JVRR JVRI t t ⎜ Jt 0 0 0 ⎟⎠ ⎜⎝ 𝛌I ⎝ IAC JVIR JVII

⎞ ⎛g ⎟ ⎜ AC ⎟ ⎜ gR ⎟ ⎜ ⎟ ⎜ gI ⎟ = − ⎜ ac ⎟ ⎜ Δh ⎟ ⎜Δhvsc ⎟ ⎜ R ⎟ ⎜Δhvsc ⎠ ⎝ I

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(4.83)

189

190

4 Optimal Power Flows

In Eq. (4.83), the Hessian terms corresponding to the original AC Hessian, deﬁned in Eqs. (4.19) and (4.20), are: HAC = ∇2Z2 Lacdc (ZAC ) = ∇2Z2 F(ZAC ) + 𝚲tAC ∇2Z2 hac (ZAC ) AC

AC

(4.84)

AC

The additional terms in Eq. (4.83), arising from the inclusion of the DC link, in the Hessian submatrix are: HACR = ∇2ZAC zvscR Ldc link = 𝛌t vR ∇2ZAC zvscR hvR

(4.85)

HACI = ∇2ZAC zvscI Ldc link = 𝛌t vI ∇2ZAC zvscI hvI

(4.86)

HR = ∇2z2 Ldc link = 𝛌t vR ∇2z2 hvR + 𝜆0R ∇2z2 h0R + 𝜆P0R ∇2z2 P0R + 𝜆Q0R ∇2z2 Q0R vscR

vscR

vscR

vscR

vscR

(4.87) HI = ∇2z2 Ldc link = 𝛌t vI ∇2z2 hvI + 𝜆0I ∇2z2 h0I + 𝜆Q0I ∇2z2 Q0I

(4.88)

HRI = ∇2zvscR zvscI Ldc link = 0

(4.89)

vscI

vscI

vscI

vscI

The Jacobian term associated with the AC network corresponds to Eq. (4.20) and it is reproduced here for the sake of completeness: JAC = ∇2Z

AC 𝚲AC

Lacdc = ∇ZAC hac (ZAC )

(4.90)

The additional Jacobian terms in Eq. (4.83) are the following: JACR = ∇2Z

Lacdc = ∇ZAC hvR

(4.91)

JACI = ∇2Z

Lacdc = ∇ZAC hvI

(4.92)

AC 𝛌R

AC 𝛌I

JRAC = ∇2z

vscR 𝚲AC

JIAC = ∇2z

vscR 𝛌R

JVII = ∇2z

vscI 𝛌I

(4.93)

Lacdc = ∇zvscI hac

(4.94)

L = ∇zvscR (hvR + h0R + P0R + Q0R )

(4.95)

vscI 𝚲AC

JVRR = ∇2z

Lacdc = ∇zvscR hac

L = ∇zvscI (hvI + h0I + Q0I )

(4.96)

JVRI = ∇2z

L=0

(4.97)

JVIR = ∇2z

L=0

(4.98)

vscR 𝛌I

vscI 𝛌R

The gradient terms are: 𝜕 Lacdc 𝜕ZAC 𝜕 gR = Lacdc 𝜕zvscR 𝜕 Lacdc gI = 𝜕zvscR

gAC =

(4.99) (4.100) (4.101)

4.9 Multi-Terminal VSC-HVDC Systems in OPF

4.9 Multi-Terminal VSC-HVDC Systems in OPF The point-to-point VSC-HVDC link model presented in Section 4.8 is expanded in this section to model any number of VSC converters making up a multi-terminal VSC-HVDC system of an arbitrary conﬁguration. To illustrate this point, refer to the schematic circuit shown in Figure 4.10, comprising n VSC stations, i.e. n terminals, where n = nR + nI , with nR and nI being the number of stations working as rectiﬁers and inverters, respectively. The nodal power constraints at each DC bus of the multi-terminal VSC-HVDC system are shown on the ﬁgure. These constraints are an extension of the point-to-point VSC-HVDC formulation, representing the nodal power balance at the DC node, and active and reactive power ﬂow control constraints. It should be remarked that the reactive power ﬂow constraint is always set to zero to maintain zero reactive power ﬂow in the DC link – refer to Eq. (4.51). In the OPF multi-terminal VSC-HVDC solution, one VSC is selected to maintain constant DC voltage at its DC bus, acting essentially as a slack node and becoming a voltage reference for the other nodal voltages in the DC grid. Notice that this concept is not diﬀerent from that employed in Chapter 3 for the conventional power ﬂow solution of multi-terminal VSC-HVDC systems. However, in the OPF solution the implementation is carried out quite simply by enforcing a DC nodal voltage constraint, as shown in Section 4.5.6. It will become apparent in Section 4.9.2 – in the test case – that the nature of the uniﬁed formulation makes it possible to treat this as simply another constraint on the state variables and the enforcement is carried out by the chosen method in the underlying optimization solver. 0I

hn

h0R =0 1

R +1

=0

CPdc = 0

CPdc = 0

CQdc R1

CQdc = 0

I1

R1

=0

I1

DC power grid

0R

0I

hn = 0

hn = 0

R

CPdc

RnR dc CQ RnR

=0 =0

Converters: n = nR + nI Rectifiers: 1 to nR Inverters: nR + 1 to n

Figure 4.10 Multi-terminal VSC-HVDC system with n-VSC stations.

CPdc = 0 RnI

CQdc = 0 RnI

191

192

4 Optimal Power Flows

4.9.1

The Expanded, General Formulation

The number of nodal power balance equations used in (4.79) for the case of the point-to-point VSC-HVDC link is expanded to encompass the n VSC terminal making up the multi-terminal VSC-HVDC system – one for each VSC station. In the generic Figure 4.10 is assumed that nR stations are operating as rectiﬁers and that nI stations are operating as inverters. Hence, the following vectors of state/control variables are set up: ZAC−DC = (ZAC , ZvscR , ZvscI )t

(4.102)

ZAC = (P , 𝛉AC , VAC , T)

(4.103)

gen

ZvscR = (ZvR , Z0R ) ZvscI = (ZvI , Z0I )

t

t

(4.104)

t

(4.105)

In Eqs. (4.102)–(4.105) the vector ZAC is for the AC power network, the vector ZvscR = (zvR1 , zvR2 , . … zvRnR , z0R1 , z0R2 , . … z0RnR )t is for the nR -terminals in the system acting as rectiﬁers and ZvscI = (zvI1 , zvI2 , . … zvInI , z0I1 , z0I2 , . … z0InI )t is for the nI -terminals in the system acting as inverters. Similarly, the vector of Lagrange multipliers for the multi-terminal system can be deﬁned as follows: 𝚲AC−DC = (𝚲AC , 𝚲R , 𝚲I )t

(4.106)

𝚲AC = (𝛌AC )t

(4.107)

𝚲R = (𝛌vR , 𝛌0R , 𝛌PQR )

t

(4.108)

𝚲I = (𝛌vI , 𝛌0I , 𝛌PQI )t

(4.109)

In Eqs. (4.106)–(4.109) the vector ΛAC is for the AC power network, the vector 𝛌vR = (𝜆vR1 , 𝜆vR2 , . … 𝜆vRnR )t and 𝛌0R = (𝜆0R1 , 𝜆0R2 , . … 𝜆0RnR )t is for the nR -terminals in the system acting as rectiﬁers and 𝛌vI = (𝜆vI1 , 𝜆vI2 , . … 𝜆vInR )t and 𝛌0I = (𝜆0I1 , 𝜆0I2 , . … 𝜆0InI )t is for the nI -terminals in the system acting as inverters. The terms 𝝀PQR and 𝝀PQI are Lagrange multipliers, which are used to enforce the active and reactive power constraints at the DC bus of the corresponding receiving and sending end converters. The linearized system of Eq. (4.83) may be expanded to accommodate the linearized representation of the AC system and that of the multi-terminal VSC-HVDC system, as shown below: ⎛ [H] [H]ACR [H]ACI [J]AC [J]ACR [J]ACI ⎞ ⎛ [Z]AC AC ⎜ ⎟⎜ ⎜[H]t ⎟ ⎜[Z]vscR [H] [H] [J] [J] [J] R RI RAC VR VRI ⎜ ACR ⎟⎜ [H]I [J]IAC [J]VIR [J]VI ⎟ ⎜ [Z]vscI ⎜ [H]tACI [H]tRI ⎜ t ⎟⎜ [J]tACR [J]tACI [𝟎] [𝟎] [𝟎] ⎟ ⎜ [𝚲]AC ⎜ [J]AC ⎜ [J]t [J]tVR [J]tVRI [𝟎] [𝟎] [𝟎] ⎟⎟ ⎜⎜ [𝚲]R ⎜ RAC t ⎜ [J]t [J]tVI [𝟎] [𝟎] [𝟎] ⎟⎠ ⎜⎝ [𝚲]I ⎝ IAC [J]VIR

⎞ ⎛ [g] AC ⎟ ⎜ ⎟ ⎜ [g]R ⎟ ⎜ ⎟ ⎜ [g]I ⎟ = −⎜ ac ⎟ ⎜ [Δh] ⎟ ⎜[Δh]vsc R ⎟ ⎜ ⎟ ⎜[Δh]vsc I ⎠ ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (4.110)

4.9 Multi-Terminal VSC-HVDC Systems in OPF

The Hessian and Jacobian entries in (4.110) have the same patterns as those given in Eqs. (4.84)–(4.98) for the case of the point-to-point VSC-HVDC link, but for the multi-terminal system there will be n terms as opposed to two terms. This higher dimensionality of the multi-terminal VSC-HVDC system is reﬂected in (4.110) using brackets. 4.9.2

Multi-Terminal VSC-HVDC Test Case

In this section, the ﬁve-bus AC test system in [3] is modiﬁed to embed in it a multi-terminal VSC-HVDC transmission system, as shown in Figure 4.11. This is now a hybrid AC-DC test system comprising two synchronous generators and three VSC terminals, eﬀectively forming an interconnected AC-DC transmission system. The data for the modiﬁed hybrid test system is given in Tables 4.9–4.14. This example serves the purpose of illustrating the applicability of the uniﬁed OPF formulation presented in this chapter for solving AC-DC systems and to assess the impact that converter losses have on the overall system performance. The system has been modelled in the AIMMS solution environment. The optimization solution uses the augmented Lagrangian algorithm already available in AIMMS. 4.9.2.1

DC Network

The data for the three converter stations, including interfacing transformers and ﬁlters, is given in Table 4.10. The data for the three DC lines is given in Table 4.11. The DC voltage information for the three buses is given in Table 4.12. The relevant VSC constraints are given in Table 4.13.

45+j15 VAC1

40+j5 VAC4

VAC3

VSC2 VDC2 VDC1

VSC1 VAC2

VDC3

VSC3 VAC5

20+j10

60+j10

Figure 4.11 Modiﬁed ﬁve-bus power system [3] to include a tree-terminal VSC-HVDC system.

193

194

4 Optimal Power Flows

Table 4.10 VSC data. Base DC voltage (kV)

Transformer resistance (p.u.)

Transformer inductance (p.u.)

Shunt ﬁlter susceptance (p.u.)

Switching losses (nominal current)

VSC No

AC bus

DC bus

Rated power (MVA)

1

VAC2

VDC1

100

345

0.0015

0.1121

0

1%

2

VAC3

VDC2

100

345

0.0015

0.1121

0

1%

3

VAC5

VDC3

100

345

0.0015

0.1121

0

1%

Table 4.11 DC lines data.

Line

Line resistance (p.u.)

DC voltage (kV)

DC power (MW)

1

0.0260

345

100

2

0.0365

345

100

3

0.0260

345

100

Table 4.12 DC bus data.

DC bus

Max voltage (p.u.)

Min voltage (p.u.)

VDC 1

1.10

0.9

VDC 2

1.01

1.01

VDC 3

1.10

0.9

Table 4.13 VSC constraints.

Power constraints

AC active power (MW)

AC reactive power (MW)

AC output voltage (p.u.)

VSC

Max

Max

Max

Min

Min

Min

1

500

−500

500

−500

1.20

0.8

2

500

−500

500

−500

1.20

0.8

3

500

−500

500

−500

1.20

0.8

4.9.2.2

AC Network

The AC voltage magnitude constraints at the nodes are given in Table 4.14 and the generator constraints are given in Table 4.15. 4.9.2.3

Objective Function

The objective is to produce a minimum-cost solution by reducing the generators’ fuel costs. A simple cost function is chosen for the two generators, given by the following

4.9 Multi-Terminal VSC-HVDC Systems in OPF

Table 4.14 AC voltage magnitude constraints. Max voltage (p.u.)

AC node

Min voltage (p.u.)

AC 1 (slack)

1.02

1.0

VAC 2

1.02

1.0

VAC 3

1.10

0.9

VAC 4

1.10

0.9

VAC 5

1.10

0.9

Table 4.15 Generator constraints. Max P (MW)

Generator

Min P (MW)

Max Q (MVAR)

Min Q (MVAR)

Voltage reference

1

250

10

500

−500

1.02

2

40

0

40

−40

1.02

linear function. gen

gen

f (Pg ) = cPg 4.9.2.4

∀g ∈ G

(4.111)

Summary of OPF Results

Two OPF solutions are carried out: (i) with no converter’s internal losses, and (ii) with converter internal losses included. In the latter case, the power loss is taken to be a function of the actual VSC’s output current, such as described by the multi-terminal VSC-HVDC model in Chapter 3. In both cases the DC voltage at converter VSC2 is kept at a reference value of 1.01 p.u. The test system has been implemented in the solution environment aﬀorded by AIMMS. For the case when the converters are taken to incur no switching power loss, two diﬀerent solutions tools were used to ﬁnd out the optimum operating point. Both application tools furnished the same solution. One OPF algorithm was coded using Matlab scripting and the other uses AIMMS. The case when the VSC’s switching power losses are taken into account was only carried out using AIMMS. Initially, values of 1% are assigned to the three VSC converters, as seen in Table 4.10. A summary of the results is given in Table 4.16. It is observed from these results that the converters contribute a small portion of the overall system power loss. It is in fact about 5.5% of the total power loss, with the rest corresponding to power loss in the seven AC transmission lines and three DC transmission lines. The converter results for both solution cases are given in Tables 4.17 and 4.20, respectively. In both solution cases the nodal voltage magnitudes on the AC and DC sides of the network have been kept within their limits and no constraint violations have been observed during the solution process. Note that since the problem was formulated as a nonlinear program, the results are guaranteed only to be locally optimal, owing to its nonconvex

195

196

4 Optimal Power Flows

Table 4.16 A summary of the hybrid test system results.

Model object

Final objective solution

Test

AIMMS MINOS – Augmented Lagrangian

Lossless converters

0.02 s – 7 Global Locally iterations optimal

0 MW

AIMMS MINOS – Augmented Lagrangian

Lossy converters

0.05 s – 11 Locally Global iterations optimal

0.23 MW 4.14 MW $8119 h−1

OPF solver

Optimality of primal

Total Total converter system losses losses

Convergence time

3.79 MW $8102 h−1

nature. In both solutions of the test case, the converters are free to set their AC voltages within the allowable limits, except VSC2, which sets the DC link’s reference voltage – it acts as slack bus for the DC network. A selection of the most relevant VSC converter, DC network and AC network operational parameters is shown below. For the sake of clarity and comprehensiveness, the results are separated into DC and AC network results (Tables 4.17–4.22).

DC Network 4.9.2.5

Converter Outputs – No Converter Losses

Table 4.17 Converters outputs summary.

Converter No.

AC bus voltage (phase angle)

AC bus voltage (magnitude)

DC bus voltage

Output active power (MW)

Output reactive power (MVAR)

DC bus power (MW)

Converter losses (MW)

0

VSC1

1.004

−3.201

1.017

54.78

8.22

−54.73

VSC2

0.996

−4.625

1.010

−20.49

13.66

20.50

0

VSC3

0.994

−5.023

1.008

−33.74

10.08

33.76

0

Table 4.18 DC power lines.

DC line

Sending end power (MW)

Receiving end power (MW)

VDC1 – VDC2

28.64

−28.43

VDC1 – VDC3

26.09

−25.85

VDC2 – VDC3

7.93

−7.91

DC Network

Table 4.19 Converters PWM performance. Converter No.

Amplitude modulation ratio

Phase angle (∘ )

VSC1

0.8143

−11.74

VSC2

0.8338

−1.475

VSC3

0.8243

−0.261

4.9.2.6

Converter Outputs – With Converter Losses

Table 4.20 Converters outputs summary. Output reactive power (MVAR)

DC bus power (MW)

37.55

3.85

−37.52

0.14

−12.34

9.68

12.35

0.02

7.98

24.72

0.07

Output active power (MW)

Converter No.

AC bus voltage (magnitude)

AC bus voltage (phase angle)

DC bus voltage

VSC1

1.006

−3.147

1.015

VSC2

0.992

−4.922

1.010

VSC3

0.991

−5.486

1.008

−24.71

Table 4.21 DC lines.

DC line

Sending end power (MW)

Receiving end power (MW)

VDC1 – VDC2

19.07

−18.98

VDC1 – VDC3

18.31

−18.20

VDC2 – VDC3

6.661

−6.660

Table 4.22 Converters PWM performance. Converter No.

Amplitude modulation ratio

Phase angle (∘ )

VSC1

0.8127

−9.010

VSC2

0.8232

−2.996

VSC3

0.8183

−1.580

Converter losses (MW)

197

4 Optimal Power Flows

DC bus voltages

1.02

No converter loss Converter losses

DC voltage (pu)

198

1

1

2 DC bus

3

Figure 4.12 DC network voltage proﬁle.

It is apparent from these results that in the ﬁrst test case, power ﬂow is entering into the DC meshed system through VSC1: importing 54.78 MW. In turn, 20.50 MW and 33.75 MW are put back into the AC system through converters VSC2 and VSC3, respectively. In the second test case, the power import into the DC mesh system is also through VSC1 but with a reduced value of 37.55 MW and power is put back into the AC network through converters VSC2 and VSC3, at values of 12.34 MW and 24.72 MW, respectively. The diﬀerent imports of power ﬂow through VSC1 are due to the diﬀerent DC voltages that exist at the bus DC1 for both test cases. The DC voltages are shown in Figure 4.12 for the three DC buses. It is seen from the ﬁgure that VSC1 keeps a higher voltage in the case when power losses are not included. This diﬀerence in voltage magnitude at VSC1 makes clear the importance of having a realistic converter loss model included in the OPF formulation. Note that in both cases, VSC2 provides the reference voltage for the whole of the DC network, acting as slack bus for the DC network. It should also be noted that apart from the voltage regulation in bus DC2 by VSC2, there is no other active control. The PWM’s amplitude modulation ratios and phase angles of the VSC converters are given in Tables 4.19 and 4.22 for both test cases. The results show that the three converters operate within their linear regions, i.e. below 1. It is also seen from the VSC’s phase angles that converter VSC1 carries the largest load in both cases, which reﬂects in their largest phase angles, with values of −11.74∘ and −9.01∘ , respectively.

AC Network

AC Network Table 4.23 shows the generator outputs for both test cases. Table 4.23 Generators outputs. Generator Bus

No converter loss

With converter loss

Active power (MW)

Reactive power (MVAR)

VAC1

128.79

−7.72

VAC2

40

4.9.2.7

9.50

Active power (MW)

129.14

−8.38

40

15

Power Flows in AC Transmission Lines – With No Converter Losses

Table 4.24 Active power ﬂows in the AC network.

AC Lines

Sending end active power (MW)

Receiving end active power (MW)

VAC1 – VAC2

94.590 8

−92.869 9

VAC1 – VAC3

34.198 7

−33.293 3

VAC2 – VAC3

13.884 6

−13.769 9

VAC2 – VAC4

17.676 3

−17.490 4

VAC2 – VAC5

26.529 7

−26.250 6

VAC3 – VAC4

22.565 3

−22.514 0

VAC4 – VAC5

0.004 41

−0.004 39

Table 4.25 Reactive power ﬂow in the AC network.

AC Lines

Sending end reactive power (MVAR)

Receiving end reactive power (MVAR)

VAC1 – VAC2

−5.3197

4.3353

VAC1 – VAC3

−2.4048

0.0220

VAC2 – VAC3

−1.8131

−1.8440

VAC2 – VAC4

−1.7356

−1.6990

VAC2 – VAC5

−1.2903

−0.8673

VAC3 – VAC4

−1.0922

−0.7332

VAC4 – VAC5

−2.5678

−2.3709

Reactive power (MVAR)

199

200

4 Optimal Power Flows

4.9.2.8

Power Flows in AC Transmission Lines – With Converter Losses

Table 4.26 Active power ﬂows in the AC network.

AC lines

Sending end active power (MW)

Receiving end active power (MW)

VAC1 – VAC2

92.591 4

−90.940 6

VAC1 – VAC3

36.552 4

−35.524 1

VAC2 – VAC3

17.740 1

−17.551 6

VAC2 – VAC4

21.193 8

−20.925 7

VAC2 – VAC5

34.455 8

−20.925 8

VAC3 – VAC4

20.416 8

−33.985 0

VAC4 – VAC5

1.300 11

−1.298 61

Table 4.27 Reactive power ﬂows in the AC network.

AC lines

Sending end reactive power (MVAR)

VAC1 – VAC2

−6.874 13

Receiving end reactive power (MVAR)

5.671 7

VAC1 – VAC3

−1.504 30

−0.474 0

VAC2 – VAC3

−0.318 76

−3.107 9

VAC2 – VAC4

−0.449 91

−2.731 8

VAC2 – VAC5

0.190 78

−1.767 2

VAC3 – VAC4

−2.394 57

0.555 5

VAC4 – VAC5

−2.823 80

−2.079 4

The active and reactive power ﬂows in the AC transmission lines are shown in Tables 4.24–4.27. They correspond to the cases with converter losses and no converter losses and complement the power ﬂows results in the DC transmission lines given in Tables 4.18 and 4.21, respectively.

4.10 Conclusion The basic principles of the OPF problem as a nonlinear optimization problem have been explained in this chapter. TSOs use the OPF application tool for the operational planning and near real-time dispatch of utility-size power grids. The theoretical requirements for developing a uniﬁed OPF formulation of hybrid AC-DC systems have been laid out and applied to develop a comprehensive frame-ofreference suitable for the optimal solution of modern power systems. This involved developing the OPF descriptor of the AC-DC kernel, namely the VSC. This was followed by expanding the kernel model to build up models of VSC-HVDC links and multi-terminal VSC-HVDC systems of an arbitrary conﬁguration.

References

It has been shown that in order to achieve the more complex VSC-HVDC solutions, the use of a set of appropriately deﬁned constraints is mandatory in order to model the DC grid. This yields a uniﬁed formulation for achieving OPF solutions of combined AC-DC systems using Newton’s method, exhibiting true quadratic convergence for the inner loop. The equipment and operational constraints are enforced in the outer loop and quadratic convergence is not possible here. Nevertheless, the uniﬁed formulation makes it simple for power system modellers to quickly formulate and solve combined AC-DC networks with no additional modiﬁcations to the solver structure, as demonstrated in this chapter using AIMMS. A simple test system has been solved using this solution environment. The example illustrates one possible role that a multi-terminal DC system may play in future power systems, namely that a multi-terminal DC system may be used to re-route power, thus alleviating congestion, all while adhering to security and economic constraints. It should be remarked that the OPF formulation presented in this chapter is capable of solving the full network, AC and DC networks, in a uniﬁed manner, with its ensuing strong convergence characteristics.

References 1 B. Stott and O. Alsaç, (2012) “Optimal power ﬂow – basic requirements for real-life

2 3 4 5 6 7 8 9

10 11

12 13

problems and their solutions”, White paper, Arizona, USA, 1 June 2012. https://pdfs .semanticscholar.org/673e/416fa38c04611e90b12cbdb1a74ca08367db.pdf Vaahedi, E. (2014.). Practical Power System Operation. IEEE Press. Acha, E., Fuerte-Esquivel, C.R., Ambriz-Perez, H., and Angeles-Camacho, C. (2004). FACTS Modelling and Simulation in Power Networks. John Wiley & Sons. Capitanescu, F. (2018). Critical review of recent advanced and further developments needed in AC optimal power ﬂow. Electric Power Systems Research vol. 136: 57–68. Bertsekas, D.P. (1982). Constrained Optimization and Lagrange Multiplier Methods. Academic Press. Boyd, S. and Vandenberghe, L. (2009). Convex Optimization. Cambridge University Press. Saccomano, F. (2003). Electric Power Systems Analysis and Control. IEEE Press. Stevenson, W.D. and Grainger, J. (1994). Power System Analysis. McGraw-Hill. Kazemtabrizi, B. and Acha, E. (2014). An advanced STATCOM model for optimal power ﬂows using Newton’s method. IEEE Transactions on Power Systems vol. 29 (2): 514–525. Dommel, H.W. and Tinney, W.F. (1968). Optimal power ﬂow solutions. IEEE Transactions on Power Apparatus & Systems PAS-97: 37–47. Zimmerman, R.D., Murillo-Sanchez, C.E., and Thomas, R.J. (2011). MATPOWER: steady-state operations, planning and analysis tools for power systems research and education. IEEE Transactions on Power Systems 26 (1): 12–19. Pardalos, P. and Resende, M.G.C. (2002). Handbook of Applied Optimization. Oxford University Press. B. Kazemtabrizi and E. Acha, (2012), “A Comparison Study Between Mathematical Models of Static VAR Compensators Aimed at Optimal Power Flow Solutions”, 16th IEEE Mediterranean Electrotechnical Conference, pp. 1125–1128.

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5 State Estimation 5.1 Introduction The wide-area blackout that aﬀected the northeast region of the United States of America in 1965 prompted the electrical utilities to embark on new procedures that ensured a higher level of service reliability. From that point onwards, the electricity supply industry started making use of concepts such as safety analysis, safety indexes, stability analysis and optimization, all of which nowadays are standard features in energy management system (EMS) technology. The power network’s measurement and control systems were very basic at the time when the wide-area blackout occurred; they could only monitor, in real time, the status of switches, the system frequency and the voltage magnitudes and powers measured by current transformers and voltage transformers at the substation level. A major drawback was that the readings arriving at the control centre were contaminated with the noise inherent to the measurement chain and calibration errors. The early eﬀorts to develop a more reliable measurement system were aimed at acquiring, at few seconds’ intervals, the network’s topology (switches’ status) and analogue (measurement equipment) information in order to conﬁgure a real-time database. This process of data acquisition, detection and signalling of the power network, together with the use of graphical displays and storage of all the events, comprise the supervisory control and data acquisition (SCADA) system. Once the database is conﬁgured, all the applications that reside in the EMS take as data input the operating state of the power network corresponding to that point in time. Such a snapshot of the system is used as input information to a large number of power systems applications; hence, the ﬁdelity and accuracy of this information are reﬂected in the ﬁnal, overall results provided by the EMS and in the ensuing analysis and decisions taken. There are three fundamental issues in real-time power systems applications, which are not satisfactorily addressed by the type of power ﬂow algorithms described in Chapters 3 and 4 of this book: (i) The number of measurements available is usually much larger than the minimum amount of information required to determine the system state (i.e. there is redundancy).

VSC-FACTS-HVDC: Analysis, Modelling and Simulation in Power Grids, First Edition. Enrique Acha, Pedro Roncero-Sánchez, Antonio de la Villa Jaén, Luis M. Castro and Behzad Kazemtabrizi. © 2019 John Wiley & Sons Ltd. Published 2019 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/acha_vsc_facts

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5 State Estimation

(ii) Owing to inherent errors in the instrumentation, there is always a degree of inconsistency between the values of the available measurements. (iii) The loss of certain measurements, for periods of time, is something to be expected; hence, the number and the distribution of measurements will change from one instant to the next. Owing to the basic assumptions under which the classic power ﬂow formulation is developed, its use in real-time applications becomes impractical. A paradigm change in which the power systems equations are solved, which fully took advantage of the rich source of real-time data that is available in a control centre, is credited to F. Schweppe and his team at the Massachusetts Institute of Technology (MIT), who in 1970 published a method to carry out state estimation in electrical power networks [1]. The state estimation process allows the determination of the state variables of a power system using statistical criteria to the entire set of available measurements, in real time. Thus, the estimated state computed at a given point in time yields the statistically optimal state that can be obtained from the available information and models that describe the power system’s behaviour. The state estimators have evolved to become crucial elements in the control of today’s electrical power networks [2, 3]. Over the years, a number of HVDC and FACTS equipment models have been developed and incorporated into state estimation formulations. The early publications [4–7] addressed the combined solution of AC and DC power networks interconnected by thyristor-based HVDC links. The work reported in [8] describes a sequential solution method with improved rate of convergence by using a decoupled method. In [9], simpliﬁed state estimation models of the thyristor-controlled series compensator (TCSC) and the uniﬁed power ﬂow controller (UPFC) were put forward and in [10] a more elaborated model for the UPFC is reported where the controller’s operating constraints are taken into account. New models for the TCSC, UPFC and static var compensator (SVC) are proposed in [11, 12], where additional control variables are incorporated as state variables, with their initial conditions speciﬁed to enhance the algorithm’s rate of convergence. Additional models of FACTS equipment are presented in [13, 14]. In reference [15], a basic model of the voltage source converter is developed using ﬁrst principles and then extensions are made to include the representation of the STATCOM and VSC-HVDC links. This model incorporates an explicit representation of the VSC’s DC circuit and takes into account the converter power losses. In [16] the model of a multi-terminal VSC-HVDC system is introduced. The outputs of phasor measurement units (PMUs) were identiﬁed, from the outset, to be measurements capable of improving the estimation process of a power network [17, 18].

5.2 State Estimation of Electrical Networks The measurements that are used to form the real-time database are carried out at the substations using current and potential transformers (i.e. transducers). The amplitudes of the instantaneous voltage and current waveforms measured by the transducers are used to calculate the RMS values of voltage and current together with the corresponding values of active power and reactive power. This information is concentrated in remote terminal units (RTUs) together with information of the contact positions of the switchgear equipment and passed onto the SCADA systems located in the control centres.

5.2 State Estimation of Electrical Networks

In modern substations where the standard IEC 61850 is employed, the measurements and the status of the switchgear equipment are directly processed by intelligent electronic devices (IEDs). The captured information is shared with other IEDs through a local area network (LAN) and from there onto the SCADA systems. As with any kind of measurement carried out in the electrical power system, there is a degree of built-in noise. The noise results from errors introduced during the various stages of the measurement chain, comprising the transducers, the SCADA system and the communication system. For instance, if several voltage measurements were available at a given bus, it is very unlikely that the measured values would be exactly the same. Likewise, the power balance at a given bus may diﬀer from zero. In the control centre, after receiving information from the various substations, the priority is to obtain a reliable assessment of the operational state of the electrical power system, corresponding to the point in time when the measurements were captured. The power system state estimator carries out this function in various steps, as indicated in

Circuit breakers status

Analogue measurements

Connectivity

TOPOLOGY PROCESSOR

OBSERVABILITY

STATE ESTIMATION

BAD DATA

System state Figure 5.1 State estimation process.

Parameters

205

206

5 State Estimation

Figure 5.1. It uses the dynamic information received from the substations and the static information corresponding to the electrical parameters of the equipment that make up the power network and the network’s connectivity. The various steps involved in the state estimation process are succinctly described below: • Topology processor. All the information sent from the substations to the EMS is collected in this module; a model of the power network is built and the available measurements are assigned to the newly established model. • Observability. In this module, an analysis of the distribution of the network measurements is carried out. It checks whether or not the available information is suﬃcient to calculate the estimated system state and determines the observable areas of the network. • Estimation. The state estimation algorithm resides in this module; it furnishes an estimate of the system state. • Bad data identiﬁcation. The process of detection and identiﬁcation of bad data (measurements that are grossly in error) is carried out in this module. In the absence of bad data, the estimated state is given a clean bill of health. However, if bad data is detected, this is deleted from the data set and the estimation process is repeated.

5.3 Network Model and Measurement System 5.3.1

Topological Processing

It is quite normal for a power system network to change its topology as a result of control actions, maintenance operations and power network contingencies. The state estimator is a power systems application tool that runs in real time and uses the topology that the power network exhibits at the point in time when the state is estimated. The topology processor module carries out such a task. It uses a static database that contains the connection status of the various pieces of equipment making up the network: transmission lines, transformers, switches, measuring devices, etc. Moreover, a dynamic database is used – it comprises the real-time information received at the control centre from the substations. This information contains the measurements received from the measuring equipment installed at the various substations, such as voltage, current, power, discrete information corresponding to the operational status (open or closed) of the switchgear equipment (e.g. circuit breakers and disconnectors). This information is used to build a network model known as the bus-branch model. The branches are associated to the transmission lines and transformers that are energized. The branches are joined at their respective ends through electrical nodes. Hence, the number of nodes that are part of the energized power network at a given point in time is determined in this module. 5.3.2

Network Model

In general, the various models used to determine the estimation of the network’s state assume that the power system operates under steady-state, balanced conditions – this is similar to the simpliﬁed assumptions adopted in the power ﬂow formulation presented in Chapters 3, 4 and 6. Hence, an AC power network with n nodes requires a state vector x with 2n variables to fully describe its operational state.

5.3 Network Model and Measurement System

Figure 5.2 Nodal power balances at a generic node i.

i P i Qi Pij

j Psi Qsi

One available option is to use the polar representation, i.e. voltage magnitude and phase angle: x = [V1 , … , Vn , 𝜃1 , … , 𝜃n ]T

(5.1)

where V i and 𝜃 i are the voltage magnitude and phase associated with a generic node i, where i = 1…n. This basic state vector may be extended to incorporate additional variables aiming at achieving a more convenient description of a given piece of equipment and to estimate the value of the new state variable based on available measurements. A load tap changer (LTC) transformer is a good example of this, where the position of the transformer tap may be estimated. To this end, the oﬀ-nominal turns ratio of the transformer tap position, which is a non-voltage variable, is incorporated as an additional variable in the state vector. Likewise, HVDC and FACTS equipment may also contribute non-voltage state variables to the state vector – this issue will be discussed later in the chapter. A generic node i is depicted in Figure 5.2; it emphasizes the relationship between the branch and the node elements, amenable to the construction of the branch-node model used in power systems state estimation. The bus-branch model is amenable to a straightforward balance of active and reactive powers at each node of the power network, say at the generic node i: ∑ Pij (5.2) Pi = Psi + j⊂Ω

Qi = Qsi +

∑

Qij

(5.3)

j⊂Ω

where Pi and Qi are the net active and reactive powers injected at node i; Pij and Qij are power ﬂows of active and reactive power through the branch i − j; Ω is the set of all nodes connecting to node i; and the variables Psi and Qsi represent the active and reactive powers of the shunt element connected to node i. Both branches and shunt components will be described with reference to the particular model employed to represent a given piece of equipment. In this application, the equipment model should be formulated in terms of only electrical parameters (i.e. voltage, current and power) as a function of the state variables and their own electrical parameters. Having determined the values of the state variables and using readily available information of the parameters of the equipment making up the network, it follows that the power ﬂows throughout the branches and shunt elements of the power network are determined with ease. In turn, using the power balance equations (5.2) and (5.3), the power injections at each node of the network are determined.

207

208

5 State Estimation

In a conventional state estimator, the following premise is taken at the outset: given an electrical magnitude z, it is always possible to determine a function h(•), such that the magnitude z may be expressed as a function of the state vector x, i.e. (5.4)

z = h(x)

The models of equipment used in conventional state estimation are similar to those used in the conventional power ﬂow problem presented in Chapter 3. However, in this chapter the nodal voltages used as state variables are expressed in polar form. The nodal power expressions of the conventional power system equipment are given below: • Transmission lines are described by Pij = Vi2 (Gsi + Gij ) − Vi Vj (Gij cos 𝜃ij + Bij sin 𝜃ij ) Qij =

−Vi2 (Bsi

+ Bij ) − Vi Vj (Gij sin 𝜃ij − Bij cos 𝜃ij )

(5.5) (5.6)

where Gij and Bij are the series conductance and susceptance of the branch, Gsi and Bsi are the shunt conductance and susceptance at node i and 𝜃 ij = 𝜃 i − 𝜃 j . • LTC transformers are described by Pij = Vi2 Gij − TV i Vj (Gij cos 𝜃ij + Bij sin 𝜃ij ); Pji = T 2 Vj2 Gij − TV i Vj (Gij cos 𝜃ji + Bij sin 𝜃ji )

(5.7)

Qij = −Vi2 Bij − TV i Vj (Gij sin 𝜃ij − Bij cos 𝜃ij ); Qji = −T 2 Vj2 Bij − TV i Vj (Gij sin 𝜃ji − Bij cos 𝜃ji )

(5.8)

where T is the oﬀ-nominal tap ratio of the LTC transformer. These equations are the starting point to develop power equations with which to calculate the active and reactive powers at both ends of the branches. In nodes with shunt-connected devices, the nodal powers may be expressed in terms of their conductance and susceptance, say at node i: Psi = Vi2 Gshi and Qsi = −Vi2 Bshi . Other measurements of electrical quantities such as the nodal voltage magnitudes, phase angles and LTC transformer taps may be directly expressed since these are variables already included in the state vector. The measurements associated to the injections of active and reactive powers at a given node may be formulated using the power balance equations given by (5.2) and (5.3) and the power equations (5.5)–(5.8). 5.3.3

The Measurements System Model

This section addresses the incorporation of the available measurements in the power system into the estimation process. As already pointed out, the measurements distribution in a power grid is not always the same. In fact, the number, distribution and type of measurements available in a power network are closely related to the voltage level at which the measurements are carried out. The high-voltage transmission system often has the largest number of measurements relating to voltage, active power and reactive power. In these networks the number of measurements is usually high, exhibiting high redundancy. In low-voltage distribution networks the number of measurements tends to be low, with current and voltage magnitude being the most popular types of measurements.

5.3 Network Model and Measurement System

The raw measurement set used in the estimation process may change from one estimation process to the next, this being a function of the information that is available in the SCADA system at the point in time when the estimation is carried out. This feature represents a major diﬀerence between the state estimation formulation and the power ﬂow formulation. In the latter, an n-node network will have 2n speciﬁed nodal quantities and 2n state variables. From the outset, the type of node is intrinsically linked, in a semi-ﬁxed format, to the type of speciﬁed variables, e.g. voltage magnitude, phase angle voltage, active power and reactive power. In general, in the state estimation problem, the existing redundancy in the measurement vector with m measurements will be much greater than the number of 2n state variables. This redundancy in the measurements enables the state estimation process to obtain an operational ﬁltered status of the power system and to develop functions for detection and identiﬁcation of bad data. In the topology processor, once the bus-branch model has been determined, the available measurements are assigned to the network model. These measurements may be classiﬁed according to the element associated with the measurement: a) Nodal measurements such as nodal voltage magnitude, injections of net active and reactive powers at a node or active and reactive powers drawn by a shunt element. b) Measurements in branches such as the active and reactive power ﬂows in transmission lines and transformers. c) Measurements of parameters associated with the model representation of a piece of equipment. A case in point is the tap position of an LTC transformer singled out for estimation, with the transformer’s tap position being an available measurement. The raw value of the measurements may be expressed as a function of the system’s state variables using (5.9). Moreover, the measurements model in the state estimation adds an error term to the exact value of the measurement. ⎡ z̃ 1 ⎤ ⎡ h1 (x1 , x2 , … , x2n ) ⎤ ⎡ e1 ⎤ ⎢ z̃ ⎥ ⎢ h (x , x , … , x2n ) ⎥ ⎢ e2 ⎥ (5.9) z̃ = ⎢ 2 ⎥ = ⎢ 2 1 2 ⎥+⎢⋮⎥ ⋮ ⋮ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣z̃ m ⎦ ⎣hm (x1 , x2 , … , x2n )⎦ ⎣em ⎦ where z̃ is a column vector containing the measured values and ei (with i = 1, …, m) is an error term associated to measurement i. The error term ei associated to a given measurement z̃ i is a random variable with a Gaussian distribution of zero mean and standard deviation 𝜎 i . Hence, the higher the quality of a measuring instrument, the smaller its standard deviation will be. By way of example, the measured voltage magnitude at a given node i is carried out directly on a state variable and its formulation may be expressed as follows: (5.10) Ṽ i = Vi + eVi The active and reactive power ﬂows at one end of a transmission line are formulated using Eqs. (5.5) and (5.6), as follows: (5.11) P̃ ij = V 2 (Gsi + Gij ) − Vi Vj (Gij cos 𝜃ij + Bij sin 𝜃ij ) + epij i

̃ ij = −V 2 (Bsi + Bij ) − Vi Vj (Gij sin 𝜃ij − Bij cos 𝜃ij ) + eqij Q i

(5.12)

The measurements errors are taken to be random variables independent from each other. Hence, the covariance matrix of errors is a diagonal matrix of size m × m, where m

209

210

5 State Estimation

is the total number of measurements available. The entries in this matrix are the square of the standard deviations: ⎤ ⎡⋱ Rz = ⎢ 𝜎i2 ⎥ ⎥ ⎢ ⋱⎦ ⎣

(5.13)

The matrix of weighted measurements is calculated as the inverse of the covariance matrix: 𝟏 W = R− z

⎡⋱ =⎢ ⎢ ⎣

1 𝜎i2

⎤ ⎥ ⎥ ⋱⎦

(5.14)

The diagonal entries may be interpreted as the relative weight of a measurement bearing in mind the quality of the measuring equipment used to acquire it – the higher its quality, the higher the weight of its associated measurement [19]. A special case in the state estimation formulation is the transit node, where the balance of active and reactive power injected at the node is zero. To implement this feature in the state estimation model it is necessary to incorporate a virtual measurement comprising injections of active and reactive power of zero values at the node. Since the power balances (5.2) and (5.3) in the transit node are exact, the error terms are always zero, i.e. not random variables. One way to combine the virtual and the conventional measurements is to give the former an error term with a standard deviation of a very low value. It follows that the virtual measurements possess a higher relative weight than the conventional measurements and the estimated values of the virtual injections is almost zero. This approach simpliﬁes the use of such measurements but it may introduce ill-conditioning problems in the state estimation formulation. To circumvent the problem, the virtual measurements are introduced as equality constraints during the optimization process of the state estimation solution. This method is presented in Section 5.4.2. As a corollary to this discussion, most FACTS devices models in state estimation incorporate power balances which must be fulﬁlled in an exact manner. In such cases the power balances are included as virtual measurements in the state estimation process. This issue is addressed in Section 5.6.

5.4 Calculation of the Estimated State 5.4.1

Solution by the Normal Equations

In the previous section, the network model with the measurements in terms of the state variables was formulated. In compact form, Eq. (5.9) may be expressed as follows: z̃ = h(x) + e

(5.15)

It should be emphasized that the main objective of the state estimation process is to determine, using the measurements available at a given point in time, the value of the state variables that best describe the operating state of the power grid. This requires establishing a methodology to solve the system of equations (5.15). This system

5.4 Calculation of the Estimated State

comprises m non-linear equations, each associated with a measurement, and 2n unknowns associated with 2n state variables. Under certain conditions, which are discussed in the observability section, the system of equations (5.15) is usually over-determined. The most popular solution method in power systems state estimation is the weighted least-squares estimation (WLS). In the WLS method, the estimated value of the state variables is obtained by solving an optimization problem, which consists of minimizing the following objective function: J(x) =

m ∑

wii ri 2

(5.16)

i=1

where the variable ri , termed residual of a measurement, is obtained as follows: ri = z̃ i − hi (x)

(5.17)

This quantiﬁes the diﬀerence between the measured value and the estimated value of the measurement. The variable wii is the relative weight associated to the measurement z̃ i and corresponding to the diagonal element in the weights matrix deﬁned in (5.14). The objective function (5.16) may then be formulated in matrix form as follows: J(x) = [̃z − h(x)]T W[̃z − h(x)]

(5.18)

Hence, the estimated value of the state vector x̂ by means of the WLS estimator corresponds to the solution of the following optimization problem: min [̃z − h(x)]T W[̃z − h(x)]

(5.19)

The solution process involves the checking of the ﬁrst-order optimality conditions: 𝜕J(x) g(x) = =0 (5.20) 𝜕x which yields −HT (x)W[̃z − h(x)] = 0

(5.21)

where 𝜕h(x) (5.22) 𝜕x is the Jacobian matrix. Expanding the function g(x) into its Taylor series around the state vector xk , an approximated function may be established by neglecting the higher-order terms: H(x) =

g(x) ≈ g(xk ) + G(xk )Δx

(5.23)

where 𝜕g(xk ) 𝜕x k G(x ) is called the gain matrix and can be computed by G(xk ) =

(5.24)

G(xk ) = HT (xk )WH(xk )

(5.25)

The solution of the non-linear equation (5.25) may be obtained by iteration using the system of equations known as the normal equation: G(xk )Δxk+1 = HT (xk )W[̃z − h(xk )] k+1

where Δx

=x

k+1

k

−x .

(5.26)

211

212

5 State Estimation

The steps to determine the estimated values of the state vector using the WLS algorithm are: 1) Set the iteration counter k = 0 and voltage magnitudes and transformer turns ratio at unity values, and voltage phase angles at zero values, i.e. ﬂat start. 2) Determine the Jacobian matrix HT (xk ) and compute the estimated value of the measurements h(xk ). 3) Set up and solve the system of normal equations deﬁned in (5.26) to obtain Δxk + 1 . 4) Test for convergence: max(Δxk + 1 ) < 𝛿, where 𝛿 is the convergence threshold speciﬁed a priori. If the convergence criterion has been satisﬁed, stop. 5) Update the state vector xk + 1 = xk + Δxk + 1 . Go to step 2. The implementation of this algorithm should be carried out in an eﬃcient manner if the aim is the solution of utility-size power networks. These networks may comprise thousands of nodes and several thousands of available measurements. The various matrices involved in the problem are huge and their solution requires the use of sparse matrix techniques. On the other hand, in solving the normal equations (5.26), the inverse of the gain matrix is calculated not directly but using factorization techniques such as Cholesky’s factorization or orthogonal factorization [20]. Once the network state x̂ has been estimated, it is then possible to obtain the values of the estimated measurements using Eq. (5.27): ̂ ẑ = h(x)

(5.27)

where ẑ is a column vector containing the estimated values of the measurements. In turn, the covariance matrix of the estimated measurements Rẑ can be calculated using the following expression: Rẑ = HG−𝟏 HT

(5.28)

The diagonal elements of the covariance matrix are the variances associated to each one of the estimated measurements. It may be argued that, in general, thanks to the redundancy presented by the measurement system and the estimation process itself, the estimated measurements have greater accuracy than the values of the raw measurements obtained at the substation level. It follows that the standard deviations of the estimated measurements have comparable or lower standard deviations than those associated with the errors of the raw measurements. Hence, the estimated status would be, on the whole, more accurate. 5.4.2

Equality-Constrained WLS

As discussed, the transit nodes must incorporate virtual measurements with zero active and reactive power injections and these measurements could be incorporated as equality constraints into the process of calculating the estimated state. The optimization problem to be solved is formulated as follows: min

J = [̃z − h(x)]T W[̃z − h(x)]

subject to

c(x) = 0

where c(x) contains the virtual measurements.

(5.29)

5.4 Calculation of the Estimated State

The estimated state may be calculated using the following Lagrangian function: L = [̃z − h(x)]T W[̃z − h(x)] − 𝜆T c(x)

(5.30)

Applying optimality conditions principles to Eq. (5.30) yields the following nonlinear system of equations, which can be solved by iteration using a Gauss-Newton’s method: ][ ] [ T [ T ] H WΔz H WH CT Δx = (5.31) C 𝟎 −𝛌 − c(x) It should be noted that the calculation of the system state using this approach bears a high degree of resemblance to the power system optimization method presented in Chapter 4. 5.4.3

Observability Analysis and Reference Phase

When parts of the network lack suﬃcient measurements, it is not possible to develop the estimation process. To proceed with the state estimation process, it becomes necessary to include a minimum number of pseudo-measurements. These pseudo-measurements are incorporated into the estimation process by resorting to historical databases. This process is developed in the observability module. Another key aspect which is taken care of in the observability module is the allocation of the phase angle reference. Each observable island present in the electrical system is assigned a phase angle reference. However, it should be noted that the node with the assigned phase angle has no inﬂuence on the estimated values of voltage magnitudes or powers throughout the network. This point is further elaborated by means of the example network shown in Figure 5.3, where all branches have a series impedance value of 0.05 + j0.10 p.u. Notice that the operational status is the same in both representations of the network, with the only diﬀerence being the selection of reference node and the assigned value of phase angle. In Case (a), node 1 is assigned to be the reference and is given a zero value whereas in Case (b), node 2 is assigned to be the reference and is given a value of 30∘ . It may be seen that the power ﬂows and nodal power injections are the same in both cases. The only discernible diﬀerence is in the phase angle voltages. In Case (b), all the phases are rotated by 33∘ with respect to the phases in Case (a), with 3∘ resulting from the change in the selection of reference node and 30∘ coming from the adopted reference value. Nonetheless, the phase angle diﬀerences across each one of the branches are identical in both cases. Since the power ﬂows in a given branch depend on the voltage magnitudes and phase angle diﬀerence across the branch, both network states yield identical power ﬂows. It follows that the operating state of the power network is independent of the value assigned to the reference phase angle; although it is usually taken to be zero, it can in fact take any other value. The value assigned to the reference phase angle does not change during the estimation process; the estimated and the assigned values coincide. Hence, it is normal practice to assign a zero value to the reference phase angle and to remove this variable from the set of state variables. This reduces the number of state variables to 2n−1. Alternatively, all the phase angles are incorporated in the state vector. This enables the straightforward assignment of any reference phase angle to the measurements

213

V1 = 1.05 p.u. θ1 = 0°

P12 = 0.66 p.u. Q12 = 0.21 p.u.

P31 = –1.06 p.u. Q31 = –0.38 p.u.

V2 = 1.0 p.u. θ2 = –3°

P23 = 0.47 p.u. Q23 = 0.27 p.u.

(a)

Figure 5.3 Phase reference selection.

V1 = 1.05 p.u. θ1 = 33°

P12 = 0.66 p.u. Q12 = 0.21 p.u.

V3 = 0.95 p.u. θ2 = –5°

P31 = –1.06 p.u. Q31 = –0.38 p.u.

V2 = 1.0 p.u. θ2 = 30°

P23 = 0.47 p.u. Q 23 = 0.27 p.u.

(b)

V3 = 0.95 p.u. θ2 = 28°

5.4 Calculation of the Estimated State

vector; it involves adding a phase angle pseudo-measurement (which is usually zero) corresponding to the node which has been selected to be the reference phase angle. This is the approach implemented in the program weighted least squares state estimator (WLS-SE), which will be introduced in the section below. When the synchrophasor measurements provided by the PMUs are incorporated into the state estimation process, all the phase angles may be given explicit representation in the state vector. As discussed in Chapter 1, synchrophasors already have a common reference time, which is why it is not necessary to include a phase reference pseudo-measurement; this issue is addressed in Section 5.7. 5.4.4

Weighted Least Squares State Estimator (WLS-SE) Using Matlab Code

A computer program with which to calculate the estimated state of power systems of small size has been developed using the Matlab environment. The algorithm employed is the equality-constrained WLS described in the previous sections. The main program comprises four steps: (i) (ii) (iii) (iv)

All the necessary data with which to carry out the estimation process is read. The estimated state is calculated. An analysis of bad data is carried out. Results are displayed.

Appendix 5.A gives additional information concerning the characteristics of the state estimation application and the input data format required by the WLS-SE computer program. To illustrate the application of the state estimation process, a small test network is used [21]. As shown in Figure 5.4, this power network comprises ﬁve buses and seven transmission lines. The branch parameters are given in Table 5.1. Two test cases are included in this section. The measurements sets included in the WLS-SE program have been carefully selected to ensure that some of the key concepts are highlighted. However, it should be noted that the measurements used in all test cases are editable (see Appendix 5.A). In the ﬁrst test case, a state estimation process is developed with all the measurements taken to have exact values, i.e. with no noise. The measurements set comprises the voltage magnitudes at all nodes; the phase angle Node 1

Node 3

Node 2 Figure 5.4 Tutorial network for WLS-SE.

Node 4

Node 5

215

216

5 State Estimation

Table 5.1 WLS-SE tutorial network parameters.

Branch

Resistance (p.u)

Reactance (p.u)

Total susceptance (p.u)

1

2

0.02

0.06

0.06

1

3

0.08

0.24

0.05

2

3

0.06

0.18

0.04

2

4

0.06

0.18

0.04

2

5

0.04

0.12

0.03

3

4

0.01

0.03

0.02

4

5

0.08

0.24

0.05

reference assigned to node 1; the injections of active and reactive powers at the nodes 1, 2 and 5; the active and reactive power ﬂows at both ends of the transmission lines linking nodes 1–2 and 1–5; and active and reactive power ﬂows at only one end of the remaining transmission lines. The measurements of voltage magnitude, power injection and power ﬂow are given typical values of standard deviations: 0.004, 0.01 and 0.008 p.u., respectively. The ﬁrst test case uses the following data ﬁles: Network ﬁle:

Network_tutorial

Measurements ﬁle:

Measure_tutorial_network

SE conﬁguration ﬁle:

Conﬁg_1

Running the main program WLS_SE.p yields the estimated state shown in Table 5.2. The estimation process takes four iterations to converge to a speciﬁed tolerance of 1e-5 for all the variables involved. As expected, since exact measurements were used, the ﬁnal value of the objective function is zero. Also, notice that the furnished estimated state should coincide with a power ﬂow solution that uses the same input data as in this example, bearing in mind that the input data for the power ﬂow application is more restricted than for the state estimation application. Table 5.2 WLS-SE results for the tutorial network with exact measurements. Node

Nodal voltages Magnitude

Phase (deg)

1

1.0600

0

2

1.0450

−4.9790

3

1.0100

−12.7197

4

1.0188

−10.3247

5

1.0204

−8.7834

5.5 Bad Data Identiﬁcation

A second test case assesses the situation when the exact measurements contain noise. This uses a new estimator conﬁguration (ﬁle_conﬁguration = ‘Conﬁg_2’) which adds random noise to each one of the exact measurements. Network ﬁle:

Network_tutorial

Measurements ﬁle:

Measure_tutorial_network

SE conﬁguration ﬁle:

Conﬁg_2

It can be observed that due to the presence of noise in the measurements, the estimate diﬀers from the exact state obtained previously. It can also be seen that in this case the value of the objective function may not be considered to be zero. It is interesting to see that the standard deviations of the estimated measurements have values that are, in general, lower than the corresponding values of the raw measurements. For instance, at Node 1 the standard deviation associated with the estimated voltage magnitude takes a value of 0.0018 p.u., which contrasts with the standard deviation of the measurement which stands at 0.0040 p.u. This represents a 55% reduction in the value of standard deviation. This result illustrates one of the beneﬁts of having redundancy in the measurements, with the estimation process signiﬁcantly increasing the accuracy of the estimated operating status of the network. It should be mentioned that owing to the random nature of the noise incorporated in the measurements, each run will yield diﬀerent results. This may be further corroborated by carrying out new simulations including new measurements to the estimation process.

5.5 Bad Data Identiﬁcation 5.5.1

Bad Data

In addition to improving the accuracy of the estimated state of the power network, a measurement set with in-built redundancy may be used to detect and identify bad data. A measurement is considered bad data when the diﬀerence between the measured value and the estimated value is larger than what can be considered to be noise introduced by the measurement chain. The presence of bad data is undesirable because it distorts the estimated state. Prior to the estimation process, the measurements are ﬁltered and it is possible to identify at this stage some bad data directly from the measurements values. Consider, for example, the case of measurements of voltage magnitude whose values are much lower than the nominal value, some of them even being zero. These values are discarded and not included in the vector of raw measurements. However, pre-ﬁltering may not remove all the erroneous measurements and the actual estimation process incorporates a module to identify and eliminate the remaining bad data. This is carried out at the end of the estimation process when a new and more comprehensive test is done to eliminate the remaining erroneous measurements; it takes advantage of the statistic nature of the estimated state. This test, which is described below, searches for the largest normalized residual.

217

218

5 State Estimation

In realistic environments it is not possible to know the exact value of the measurements, hence statistical criteria may be applied to ﬁnd out the wrong measurements with a certain level of conﬁdence. The reference value of the measurement is taken to be an estimated state. In order to develop this method it becomes necessary to have a previously estimated state. Any measurement identiﬁed as bad data must be removed from the available set of measurements. The process comes to an end when a new estimation of the state is carried out and no more bad data is found. 5.5.2

The Largest Normalized Residual Test

The most popular test to detect and identify bad data in the state estimation process is the largest normalized residual. This test is based on the statistical properties of the estimated residuals r̂i associated to the measurements, which are calculated after having obtained the estimated state using: ̂ = z̃ i − ẑ i r̂i = z̃ i − hi (x)

(5.32)

The covariance matrix associated to the estimated residuals is given as: Rr̂ = Rz − HG− 𝟏 HT

(5.33)

where the diagonal elements of this matrix are the variances of the residuals. The normalized residuals are: r̂ (5.34) riN = √ i Rr̂ (i, i) √ where Rr̂ (i, i) is the standard deviation of the residual estimate r̂i . The normalized residual of a measurement, calculated in this manner, follows a normal distribution of zero mean and unity standard deviation. This enables veriﬁcation of whether or not a measurement is bad data by simply comparing the normalized residual value and a pre-speciﬁed threshold value. A popular threshold value in power systems state estimation is 3; a measurement with a normalized residual higher than 3 will have a high probability of being bad data. Bad data and any other measurement associated to it will exhibit normalized residuals which may also surpass the threshold value of 3. It may be shown that in the case of a single bad data, the normalized residual with the highest value corresponds to the erroneous measurement. The test for detection and identiﬁcation of bad data may be implemented using the following algorithm: Obtain an estimated state x̂ . Calculate the diagonal elements of the covariance matrix of the residuals diag(Rr̂ ). Calculate the normalized residuals of the measurements riN (i = 1, … , m). If the largest normalized residual is smaller than 3, then stop. Delete from the estimation process the measurement with the highest normalized residual. 6) Go to step 1.

1) 2) 3) 4) 5)

Notice that only one bad data is eliminated per cycle. Upon completion of this process the estimated state of the power grid becomes known together with the

5.5 Bad Data Identiﬁcation

subset of measurements classiﬁed as bad data. This subset does not participate in the estimation process. A special type of measurement is the critical measurement. If one or more of them are eliminated from the set of measurements, the entire system becomes unobservable. The estimator has no redundancy associated with these critical measurements, so their estimated values coincide with their measured values. Furthermore, their residual values are zero. It should be noted that in this situation a zero value of the residual bears no relationship whatsoever to a higher quality of the estimated value. The estimated variance associated to the residual of a critical measurement is zero and this prevents the critical measurements from being incorporated into the bad data identiﬁcation test, since they would present singularity problems at the point of calculating the normalized residuals. It may be concluded that the errors in critical measurements can be neither detected nor identiﬁed. In cases of reduced redundancy, the test of the largest normalized residual enables the presence of an error to be detected but not its identiﬁcation. It shows as a number of measurements with the same values of normalized residuals. Hence, the error is detected but the measurement that causes the error cannot be identiﬁed. 5.5.3

Bad Data Identiﬁcation Using WLS-SE

The estimator WLS-SE enables an analysis of bad data using the largest normalized residual test. Consider the second test case presented in Section 5.4.4, which includes the addition of Gaussian noise to the exact measurements. It may be seen that, in general, the value of the normalized residuals associated to the noisy measurements is lower than the threshold value of 3. Owing to the random nature of the incorporated noise, each execution of the WLS-SE program will yield diﬀerent values of measurements and associated normalized residuals. It is to be expected that in some cases the random error assigned to a measurement exhibits a high value and that the normalized residual can exceed the threshold value of 3. To run the third test case, the WLS-SE program uses the ﬁles: Network ﬁle:

Network_tutorial

Measurements ﬁle:

Measure_tutorial_network_bad_data

SE conﬁguration ﬁle:

Conﬁg_1

It can be seen how in the estimation process of the largest residual, a value as large as 16.3 is present. It corresponds to the measurement of active power ﬂowing between nodes 1 and 2. It is quite clear that such a large value of normalized residual indicates the presence of bad data. To corroborate this, it is suﬃcient to eliminate the measurement from the ﬁle containing the raw measurements. Running the WLS-SE program again, it is veriﬁed that all the normalized residuals have zero values, i.e. the rest of the measurements have their exact values. It should be remarked how a single measurement with a wrong value may yield several measurements with non-zero normalized residuals, some of them going above the threshold value of 3. This is due to the adverse eﬀect that the bad data inﬂicts on the outcome of the estimate process, with its impact being more severe in the vicinity of the wrong measurement.

219

220

5 State Estimation

New simulations can be carried out by simply changing the erroneous measurement, or by modifying the number of available measurements in the vicinity of the erroneous measurement and with it the local redundancy.

5.6 FACTS Device State Estimation Modelling in Electrical Power Grids 5.6.1

Incorporation of New Models in State Estimation

The inclusion of models of new equipment into the state estimation processor should enable a more detailed estimation of the operating state of the electrical power grid. This would be the case because of the incorporation of new measurements, which increases redundancy, thus improving the accuracy and identiﬁcation of bad data. The incorporation of new equipment representation in the network model requires: a) enlargement of the state vector to incorporate all the new basic variables b) addition of the set of constraints characterizing the new devices. Such constraints usually involve new parameters pertaining to the devices being modelled c) augmentation of the measurements set so that the new devices can be monitored. In general, the incorporation of a FACTS device representation into the power system model would increase the node count. A VSC-based FACTS device comprises an AC circuit and a DC circuit; normally, the AC circuit will connect to a point of the AC power grid through a coupling transformer. Hence, the AC point of connection between the two devices introduces a new node, as illustrated in Figure 5.5. The VSC’s coupling transformer is suitably represented by a conventional branch, which enables the incorporation of measurements at each end of the branch. In addition to the newly created AC nodes, the inclusion of VSCs in the circuit introduces DC nodes. These nodes may be isolated, such as in the case of STATCOMs, or may be interconnected, as is the case when two or more VSCs are used to form HVDC links, forming meshed networks in the case of multi-terminal VSC-HVDC systems. It will be shown below, for each of the devices, that each DC node available would enable the incorporation of suitable DC measurements. Figure 5.5 Schematic representation of the VSC and its coupling transformer.

Connecting node Coupling transformer

VSC

5.6 FACTS Device State Estimation Modelling in Electrical Power Grids

5.6.2

Voltage Source Converters

A VSC on its own may not be considered to be a FACTS device, let alone a VSC-HVDC link. However, the VSC constitutes the kernel used by a number of compound devices, some of which are addressed in the remainder of this chapter. Hence, a suitable state estimation model of the VSC is developed below which will then be used to form the state estimation models of the STATCOM, UPFC, VSC-HVDC and multi-terminal VSC-HVDC. As elucidated previously, the VSC model introduces two new nodes into the overall state estimation model, one associated with its AC side and the other with its DC side, as shown in Figure 5.6. Node v on the AC side connects to the low-voltage side of the coupling transformer, which may be represented in a conventional manner. The state variables associated with this node are voltage magnitude, V v , and phase angle, 𝜃 v . The DC node enables the connection of the VSC to a connecting DC system, which may be as simple as a DC bus, with a capacitor connected to it, or as complex as a meshed DC power grid. In order to add generality to the model description, the state variables making up the DC part of the state vector are not explicitly stated. Instead, the DC state variables are given, speciﬁcally, for each of the VSC-based devices. In this application, it is desirable to take account of the VSC power losses and to include them in the active power balance set. Moreover, these power losses may be separated into two terms [15], one for switching losses and the other for conduction losses: c s Ploss = Ploss + Ploss

(5.35)

c , also termed ohmic losses, are As indicated in Chapter 3, the conduction losses, Ploss well represented by means of a series resistor, R1 , connected to the AC node, whereas s the switching losses, Ploss , may be represented by means of a shunt conductance, Gsw , connected to the DC node. Hence, the VSC power losses are deﬁned by the

Figure 5.6 Representation of the VSC model in state estimation.

Vi θi

i

Vv θv

v I AC

AC side

VSC

IDC

DC side + EDC –

221

222

5 State Estimation

following relationship: 2 2 + Gsw EDC Ploss = R1 IAC

(5.36)

where I AC is the AC current delivered by the VSC and EDC is the DC voltage of the VSC station. The active power balance in the VSC may be expressed as follows: PAC + PDC + Ploss = 0

(5.37)

where PAC represents the active power delivered by the convertor at its AC node whereas PDC is the power delivered by the converter at its DC node. The power PDC may be calculated using available information of voltage and current: PDC = EDC IDC

(5.38)

where I DC is the DC current of the VSC station. Substituting (5.38) into (5.37) and combining with the VSC’s power loss equation (5.36), the active power balance is reformulated as follows: 2 2 + Gsw EDC =0 PAC + EDC IDC + R1 IAC

(5.39)

In order to include this equation into the VSC model, it becomes necessary to express all the variables involved in terms of state variables. In general, the DC voltage, EDC , is taken to be a state variable. Concerning the DC current I DC , it may also be taken directly to be a state variable or, alternatively, it may be expressed to be a function of other DC state variables. The variables PAC , and the square of the AC current I AC , delivered by the VSC at its AC node, are elaborated below. The powers contributed by the VSC at its AC node may be derived by resorting to power balances, active and reactive, at node v: PAC = Pvi − Pv

(5.40)

QAC = Qvi − Qv

(5.41)

where Pvi and Qvi are the active and reactive power ﬂows through the coupling transformer; Pv and Qv are the nodal active and reactive power injections at node v. In turn, the square of the current through the VSC, I AC , may be expressed in terms of Eqs. (5.40) and (5.41), as follows: 2 IAC =

2 PAC + Q2AC

Vv2

(5.42)

The power ﬂows, Pvi and Qvi , may be expressed in terms of the AC state variables (nodal voltages), using the conventional model of the coupling transformer (5.7) and (5.8). In cases when the power injections Pv and Qv are diﬀerent from zero, owing to an element connected to node v, the associated power equations are expressed as a function of state variables. By way of example, let us assume that the element connected to node v is a shunt element. The associated injected power may be calculated using the shunt admittance and the nodal voltage magnitude at node v. Notice that when it is not possible, or not desirable, to include in the formulation the associated model of the power injections Pv and Qv , then the power variables must be incorporated into the state vector as new state variables.

5.6 FACTS Device State Estimation Modelling in Electrical Power Grids

It is often the case that the VSC is directly connected to the coupling transformer, leading to zero power injections, Pv and Qv , at this node. In this case, the VSC powers at the AC node are calculated using the power equations (5.7) and (5.8): PAC = Pvi = Gsc Vv2 − Vv Vi (Gsc cos 𝛾 + Bsc sin 𝛾) QAC = Qvi =

−Bsc Vv2

− Vv Vi (Gsc sin 𝛾 − Bsc cos 𝛾)

(5.43) (5.44)

where Gsc and Bsc are the real and imaginary parts of the coupling transformer’s series admittance, the tap of the coupling transformer is assumed T = 1 and 𝛾 = 𝜃 v − 𝜃 i . On the other hand, the square of the AC current leaving the transformer can be obtained by substituting the Eqs. (5.43) and (5.44) into (5.42), giving the following result: 2 2 = Ivi2 = (Gsc + B2sc )(Vv2 + Vi2 − 2Vv Vi cos 𝛾) IAC

(5.45)

In this chapter we shall assume that the VSC is directly connected to the coupling transformer, so that expressions (5.43)–(5.45) are applicable. It should be noted that other transformer models, such as tap-changing or phaseshifting transformers, may be included instead, but slightly diﬀerent equations will be required – refer to Sections 3.2.3 and 3.2.4. The VSC model, described above, may incorporate new measurements relating to the actual control of the VSC, such as the modulation index ma of the PWM control, and the phase-angle control 𝛾 = 𝜃 v − 𝜃 i . Notice that this phase-angle control signal 𝛾, representing the phase-angle diﬀerence between the terminal buses of the coupling transformer, is directly formulated by means of the associated state variables at the two ends of the coupling transformer. The modulation index ma relates the line-to-line RMS voltage magnitude in the AC side of the converter to the DC-side voltage [22]. As explained in Section 3.6.1, this constitutes a coupling between the AC and DC sides of the converter: √ 3 VvLL = (5.46) m E 8 a DC where VvLL is the RMS value of the fundamental-frequency, line-to-line voltage and EDC is the DC-side voltage. The above equation, expressed in per unit, is integrated into the VSC model, considLL and EB, DC for the AC side and DC side, respectively, ering as base voltages VB,AC (5.47)

Vv = k1 ma EDC where k1 =

EB,DC LL VB,AC

√ •

3 8

(5.48)

V v and EDC are the AC-side and DC-side voltages expressed in per unit, the former being a line-line voltage. In the examples included in this chapter it is considered that the base voltage of the AC and DC side have the same value. In this way, the constant k 1 takes the value 0.6124. In the remainder of this chapter, the VSC model is used to develop the state estimation models of VSC-FACTS and VSC-HVDC equipment.

223

224

5 State Estimation

5.6.3

STATCOM

In this section, we apply the generic model of the VSC developed above to obtain the state estimation model of a STATCOM. As explained in Section 1.2.2, the combination of a VSC and its connecting transformer is termed, in the parlance of power systems engineers, a STATCOM. As illustrated in Figure 5.6, it uses only one node on its DC side, with a capacitor connected to it. The STATCOM model incorporated into the state estimator uses as state variables the AC voltage magnitude and its phase angle at node v, namely V v and 𝜃 v , and the DC voltage, EDC : XSTATCOM = [Vv , 𝜃v , EDC ]

(5.49)

It should be noted that this model assumes that the power delivered by the VSC at its DC node is zero; hence, the DC current is also zero. This would correspond to the case when the DC capacitor is fully charged, a fact that happens during steady-state operation. The model is completed by incorporating the active power balance (5.39) as either virtual measurement or equality constraint. It follows that the only power exchanged between the DC and AC nodes, through the coupling transformer, is the power corresponding to the converter losses, i.e. conduction and switching power losses. In this way, incorporating the converter power losses (5.36) in the active power restriction equation yields: 2 Pvi + R1 Ivi2 + Gsw EDC =0

(5.50)

This restriction may be formulated as a function of the state variables on the AC node employing the expressions (5.43)–(5.45), in order to formulate the active power ﬂow, Pvi , and the square of the current ﬂow, I vi , through the coupling transformer. Table 5.3 summarizes the measurements set associated with this STATCOM model. A special case of the STATCOM operation is when it is possible to have power exchanges between the AC and DC sides of the device, something that is achievable when there is an energy storage device connected to the DC node, such as a battery pack. This type of FACTS equipment, introduced in Section 1.2.4, is known as BESS. From the vantage of the state estimation application, this VSC model is extended to Table 5.3 STATCOM measurements. Conventional measurements

AC voltage AC power ﬂows

Ṽ i = Vi + 𝜀Vi Ṽ v = Vv + 𝜀Vj P̃ vi = Gsc Vv2 − Vv Vi (Gsc cos 𝛾 + Bsc sin 𝛾) + 𝜀pvi ̃ = −B V 2 − V V (G sin 𝛾 − B cos 𝛾) + 𝜀 Q vi

DC voltage

Control measurements

sc

v

v

i

sc

sc

qvi

The reverse power ﬂows are realized by simply exchanging the subscripts v and i and changing the sign of the phase-angle 𝛾. Ẽ = E + 𝜀 DC

DC

EDC

Vv k1 EDC

Modulation index

̃a = m

Phase-angle control signal

𝛾̃ = 𝛾 + 𝜀𝛾

+ 𝜀ma

5.6 FACTS Device State Estimation Modelling in Electrical Power Grids

Table 5.4 Extended STATCOM measurements. Measurements

DC current DC power

ĨDC = IDC + 𝜀IDC P̃ DC = EDC IDC + 𝜀PDC

incorporate the output DC current, I DC , into the state vector. Hence, the new model of the STATCOM has the following state vector: amp

XSTATCOM = [Vv , 𝜃v , EDC , IDC ]

(5.51)

Furthermore, the active power balance is extended to incorporate a new term in order to take account of the power exchanged with the DC side of the STATCOM. The restriction term (5.50) becomes: 2 EDC IDC + Pvi + R1 Ivi2 + Gsw EDC =0

(5.52)

With this extended model, it is now possible to incorporate new measurements on the DC side of the STATCOM – shown in Table 5.4. It should be noted that this extended model is suitable to use in cases when it is not physically possible to exchange power between the AC and DC sides of the STATCOM owing to, say, the lack of an energy storage source. In such a case, it suﬃces to set the DC current restriction to zero, i.e. I DC = 0. 5.6.4

STATCOM Model in WLS-SE

The software WLS-SE includes three test cases, which use the extended STATCOM model, developed in the previous section. The basic network of the tutorial section is expanded to include one STATCOM model. The STATCOM’s coupling transformer is connected to Node 3 of the tutorial network and a new node is created, Node 6, to connect the STATCOM. The transformer impedance is 0.01 + j 0.1 p.u. The STATCOM series resistance is 0.02 p.u. and the shunt conductance is 0.002 p.u. The study case 4, using WLS-SE, employs the following data ﬁles: Network ﬁle:

Network_tutorial_STATCOM

Measurements ﬁle:

Measure_STATCOM_case_a

SE conﬁguration ﬁle:

Conﬁg_1

In this example, the STATCOM is set to inject reactive power at Node 3, as shown in Figure 5.7. The active power taken from the network at Node 3 is exactly the power required to supply the power losses of the VSC and its coupling transformer, since this VSC is unable to contribute active power at all. This test case was solved using the extended model of the STATCOM; hence, it has become necessary to use a zero DC current restriction. This example uses exact measurements, including the measurements associated to the VSC control and DC voltage.

225

226

5 State Estimation

Node 6

Node 1

Node 3

Node 2

Node 4

Node 5

Figure 5.7 Tutorial network including one STATCOM.

It may be of some instructive value to assess the performance of the STATCOM model when the VSC’s power losses are assumed to be negligibly small. This requires that both the series resistance and shunt conductance of the VSC be set to zero in the relevant data ﬁle. This corresponds to the test case 5 and in WLS-SE. These changes have been implemented in the following data ﬁles: Network ﬁle:

Network_tutorial_STATCOM_VSC_no_losses

Measurements ﬁle:

Measure_STATCOM_case_a

SE conﬁguration ﬁle:

Conﬁg_1

The exact measurements are the same as those used in the previous example; however, the VSC power losses have not been included in the model. The results furnished by WLS-SE show that the measurements of active power ﬂows though the coupling transformer 3-6 has a normalized residue value of 1.0, which incorrectly indicates a noisy value of the measurements given that in all fairness, it is the correct value. This example highlights how incomplete/inaccurate models may distort the estimated state, having a negative impact on the normalized residual values of some of the measurements associated with the device. In test case 6, which also uses the STATCOM connected to Node 3, WLS-SE uses the following data ﬁles: Network ﬁle:

Network_tutorial_STATCOM

Measurements ﬁle:

Measure_STATCOM_case_b

SE conﬁguration ﬁle:

Conﬁg_2

In this example, the STATCOM possesses the ability to inject power at the DC node, which ﬂows towards the AC network. It employs the full STATCOM model – without the restriction that nulliﬁes the DC current on the VSC’s DC node. Instead, it incorporates this current as a measurement. Notice that this case also includes the

5.6 FACTS Device State Estimation Modelling in Electrical Power Grids

power measurement on the DC side of the VSC. In this example, the state estimator conﬁguration incorporates noise in all the measurements. 5.6.5

Uniﬁed Power Flow Controller

As introduced in Section 1.2.4, the UPFC comprises two VSCs connected in a back-to-back conﬁguration on their DC sides. At their AC sides, one VSC is connected to the secondary winding of a transformer whose primary winding is connected in shunt to the AC power grid. At the opposite end, a second VSC is connected to the secondary winding of a transformer whose primary winding is connected in series with the AC power grid, as shown in Figure 5.8. The state estimation model developed in this section includes the power losses associated with both converters. Hence, the generic VSC model introduced in Section 5.6.2, is suitable to represent each one of the VSCs making up the UPFC. In addition to the nodal voltages at the AC side of both VSCs, namely nodes s and r, the model incorporates the DC voltage and the DC current as state variables. Accordingly, the state variable set associated to the UPFC model is: XUPFC = [Vs , 𝜃s , Vr , 𝜃r , EDC , IDC ]

(5.53)

The complete UPFC model requires the active power balance equations for each VSC. The balance in the shunt-connected VSC is the following: 2 EDC IDC + Psk + Rssh Isk2 + Gpsh EDC =0

(5.54)

where Rssh and Gpsh are the parameters associated to the power losses in this converter. The restriction (5.54) is expressed as a function of the AC state variables given in (5.43)–(5.45), to formulate the active power ﬂow Psk and the square de la current, I sk , ﬂowing through the shunt-connected, coupling transformer, s − k. The following expressions have been used in the formulation of the measurements of power ﬂow through the series-connected transformer. ∗ Skm = Pkm + jQkm = Vk Ikm

Smk = Pmk + jQmk = k

(5.55)

∗ Vm Imk

(5.56)

Vm θm

Vk θk Boost

Shunt

IDC s Vs θ s

+ EDC –

Figure 5.8 State estimation UPFC model.

r Vr θr

m

227

228

5 State Estimation

where the currents can be expressed as: Ikm = (Vk − Vr − Vm )Yb

(5.57)

Imk = −Ikm

(5.58)

with the parameter Y b being the series admittance of the shunt transformer. The power ﬂows can be formulated using the state variables as: Skm = Vk (Vk∗ − Vm∗ )Yb∗ − Vk Vr∗ Yb∗ Smk =

Vm (Vm∗

−

Vk∗ )Yb∗

+

Vm Vr∗ Yb∗

(5.59) (5.60)

In the series-connected VSC, the active power balance equation is: 2 =0 −EDC IDC + Pr + Rssr Ir2 + Gpsr EDC

(5.61)

where Rssr and Gpsr are the parameters associated to the power losses of this converter. The variables Pr and I r correspond to the active power and current delivered by the series-connected converter at node r. In order to express these constraints as a function of the AC state variables, we use the active and reactive power expressions in the series-connected converter: Sr = Pr + jQr = Vr Ir∗

(5.62)

Considering that I r = I mk , the power ﬂows are computed as: Sr = Vr (Vr∗ − Vk∗ )Yb∗ + Vr Vm∗ Yb∗

(5.63)

The square of the current, I r , in the converter may be expressed as follows: Ir2 =

Pr2 + Q2r

(5.64)

Vr2

The measurements set of the UPFC is summarized in Table 5.5. 5.6.6

The UPFC Model in WLS-SE

In this section we analyze the results of a case included in WLS-SE, where a UPFC is added to the tutorial’s test system. The UPFC is connected to Node 3 in series with the transmission line 8-4 (previously labelled as 3-4). The shunt-connected VSC is added to the grid through the coupling transformer 6-3, as shown in Figure 5.9. The series impedance of both transformers is: 0.01 + j 0.1 p.u. The ﬁles corresponding to this case are: Network ﬁle:

Network_tutorial_UPFC

Measurements ﬁle:

Measure_UPFC

SE conﬁguration ﬁle:

Conﬁg_1

Notice that there is no nodal power injection at node 8, i.e. it is the connection point of the UPFC and line 8-4 (transit node). To represent such condition in WLS-SE, we require equality power constraints with null injected powers at node 8. Execution of this example shows that the measurement of active power ﬂow from node 8 to node 4 is ﬂagged as bad data, with the rest of the measurements being exact measurements.

5.6 FACTS Device State Estimation Modelling in Electrical Power Grids

229

Table 5.5 UPFC measurements. Conventional measurements

AC voltage

Control measurements

Ṽ r = Vr + 𝜀Vr Ṽ k = Vk + 𝜀Vk Ṽ s = Vs + 𝜀Vs Ṽ m = Vm + 𝜀Vm

Modulation index

̃ as = m

Vs + 𝜀mas k1 EDC

Vr + 𝜀mar k1 EDC 𝜃̃sk = 𝜃s − 𝜃k + 𝜀𝜃sk 𝜃̃rm = 𝜃r − 𝜃m + 𝜀𝜃rm 𝜃̃rk = 𝜃r − 𝜃k + 𝜀𝜃rk 𝜃̃km = 𝜃k − 𝜃m + 𝜀𝜃km ̃ ar = m

AC power ﬂows

P̃ sk = GTs Vs2 − Vs Vk (GTs cos 𝜃sk + BTs sin 𝜃sk ) + 𝜀psk ̃ = −B V 2 − V V (G sin 𝜃 − B cos 𝜃 ) + 𝜀 Q sk Ts s s k Ts sk Ts sk qsk The opposite ﬂows can be obtained by interchanging the indexes s and k. P̃ km = GTb Vk2 − Vk Vm (GTb cos 𝜃km + BTb sin 𝜃km )−

Phase-angle control signal

−Vk Vr (GTb cos 𝜃kr + BTb sin 𝜃kr ) + 𝜀pkm ̃ Qkm = −BTb Vk2 − Vk Vm (GTb sin 𝜃km − BTb cos 𝜃km )− −Vk Vr (GTb sin 𝜃kr − BTb cos 𝜃kr ) + 𝜀qkm The opposite ﬂows can be obtained by interchanging the indexes k and m. P̃ r = GTb Vr2 − Vr Vk (GTb cos 𝜃rk + BTb sin 𝜃rk )+ +Vr Vm (GTb cos 𝜃rm + BTb sin 𝜃rm ) + 𝜀pr ̃ = −B V 2 − V V (G sin 𝜃 − B cos 𝜃 )+ Q r Tb r r k Tb rk Tb rk DC voltage DC current DC power

Node 1

+Vr Vm (GTb sin 𝜃rm − BTb cos 𝜃rm ) + 𝜀qr ̃E = E + 𝜀 DC DC EDC ĨDC = IDC + 𝜀IDC P̃ = E I + 𝜀 DC

DC DC

PDC

Node 3

Node 4

Node 6

Node 2

Node 7

Node 5

Figure 5.9 Tutorial network including a UPFC.

It is interesting to assess the case that results from removing the restrictions of null power injection at node 8. This situation would present itself if we were to have power injections at the node but no available measurements. The results show normalized residues larger than 4.5. In this case, two of the measurements have large values: the active power ﬂow 8-4 and the active power injection at node 4. The test of the largest

230

5 State Estimation

normalized residue detects the presence of an error in the measurements set, but it is not possible to identify the measurement that causes the error. This anomalous situation is due to the lack of suﬃcient measurement redundancy around the bad data, i.e. there is insuﬃcient information to identify the bad data. 5.6.7

High Voltage Direct Current Based on Voltage Source Converters

The VSC generic model is now applied to develop the VSC-HVDC model for state estimation. It incorporates two AC nodes, AC1 and AC2, which use the following conventional state variables (nodal complex voltages) for their representation: V AC1 , 𝜃 AC1 , V AC2 , 𝜃 AC2 . Moreover, the VSC-HVDC introduces two DC nodes, which are linked through a DC cable of resistance, RDC , as shown in Figure 5.10. The model described below represents well the converters when operating either as a rectiﬁer or as an inverter. It should be recalled that the DC resistance, RDC , is taken to have a zero value in cases of a back-to-back conﬁguration. The new electrical quantities associated to the DC circuit are the DC voltages and currents at the VSC-1 and VSC-2 nodes, together with the modulation indexes of the two VSCs: [EDC1 , EDC2 , IDC1 , IDC2 , ma1 , ma2 ]

(5.65)

where the subscripts 1 and 2 refer to the DC nodes of the VSC-1 and VSC-2, respectively. In turn, using these variables it is possible to calculate the injected powers at the respective DC nodes. Four out of the six variables associated to the DC circuit may be derived from two basic variables, using the available relationships of modulation indexes (5.47): ma1 =

VAC1 k1 EDC1

ma2 =

VAC2 k1 EDC2

(5.66)

and two other relationships which occur naturally in the DC circuit: (5.67)

IDC1 = −IDC2 and IDC1 =

EDC1 − EDC2 RDC

(5.68)

It is said that the variables associated to the DC circuit possess two-degree freedom. It should be noted that the relationship (5.68) is undeﬁned when applied to the back-to-back conﬁguration. In order to overcome this singularity in the model, we select an alternative set of state variables, independent of the DC cable resistance. DC link Vk θk

VAC1 θAC1

VSC1 Figure 5.10 VSC-HVDC model.

IDC1

IDC2

EDC1

EDC2

VAC2 θAC2

VSC2

Vm θ m

5.6 FACTS Device State Estimation Modelling in Electrical Power Grids

These are the voltage and current at one of the DC buses. Hence, the alternative state vector contains the following set of variables: XVSC−HVDC = [VAC1 , 𝜃AC1 , VAC2 , 𝜃AC2 , EDC1 , IDC1 ]

(5.69)

Using this state variable set, the DC electrical quantities at the VSC-2 are deﬁned by the following relationships: IDC2 = −IDC1

(5.70)

EDC2 = EDC1 − IDC1 RDC

(5.71)

Notice that these relationships are well suited to the null value of the DC resistance, RDC , of the back-to-back HVDC link, where EDC2 = EDC1 . In order to complete the model, it is necessary to incorporate a power balance restriction equation (5.39) at each VSC. At the VSC-1, where we have taken the state variables, the active power balance is: 2 2 + Gsw1 EDC1 =0 PAC1 k + EDC1 IDC1 + R11 IAC1 k

(5.72)

where R11 and Gsw1 are the parameters associated with the VSC-1’s power losses. This restriction is expressed in terms of the AC state variables used in (5.43)–(5.45) in order to formulate the active power ﬂow, PAC1k , and the square of the current ﬂow, I AC1k , through the coupling transformer AC1 – k. For the VSC-2, the active power balance expression is formulated as follows: 2 2 PAC2 m + EDC2 IDC2 + R12 IAC2 m + Gsw2 EDC2 = 0

(5.73)

where R12 and Gsw2 are the parameters responsible for the power losses in VSC-2. Employing the relationships (5.70) and (5.71), the power balance equation (5.73) may be expressed as a function of the state variables in the DC circuit: 2 PAC2 m − (1 + 2RDC Gsw2 )EDC1 IDC1 + (RDC + R2DC Gsw2 )IDC1 2 2 + Gsw2 EDC1 + R12 IAC2m =0

(5.74)

The active power, PAC2 m , and the square of the current, I AC2 m , in the coupling transformer are formulated as a function of the state variables applying the expressions (5.43)–(5.45) of the coupling transformer AC2 – m. The measurements of the VSC-HVDC state estimation model developed above are summarized in Table 5.6. It should be remarked that the VSC-HVDC model just developed for the pointto-point case is also valid for the back-to-back case, where it is only necessary to set the resistance of the DC cable, RDC , to zero. 5.6.8

VSC-HVDC Model in WLS-SE

In this section, we present two examples relating to the VSC-HVDC, which are included in WLS-SE, using the VSC-HVDC models presented in the section above. In test case 7, the transmission line connected between nodes 3 and 4 in the tutorial network is replaced by a point-to-point VSC-HVDC link, as shown in Figure 5.11. Coupling to the AC grid is carried out using transformers 3-6 and 4-7, each having a series impedance of 0.01 + j 0.1 p.u. The rectiﬁer and inverter VSCs are connected

231

232

5 State Estimation

Table 5.6 VSC-HVDC measurements. Conventional measurements

AC voltage

AC power ﬂows

Control measurements

Ṽ AC1 = VAC1 + 𝜀VAC1 Ṽ k = Vk + 𝜀Vk Ṽ AC2 = VAC2 + 𝜀VAC2 Ṽ m = Vm + 𝜀Vm

Modulation index

̃ a2 m

2 P̃ AC1k = GAC1k VAC1 − VAC1

Phase-angle control signal

Vk (GAC1k cos 𝛾1 + BAC1k sin 𝛾1 ) + 𝜀pAC1k

2 ̃ Q AC1k = −BAC1k VAC1 − VAC1

VAC1 + 𝜀ma1 k1 EDC1 VAC2 = + 𝜀ma2 k1 EDC2

̃ a1 = m

𝛾̃1 = 𝜃AC1 − 𝜃k + 𝜀𝛾1 𝛾̃2 = 𝜃AC2 − 𝜃m + 𝜀𝛾2

Vk (GAC1k sin 𝛾1 − BAC1k cos 𝛾1 ) + 𝜀qAC1k

DC voltage DC current DC power

The opposite ﬂows can be obtained by interchanging the indexes AC1 and k, and changing the sign of the phase-angle 𝛾 1 . The coupling transformer AC2-m has a similar formulation. Ẽ Ẽ =E +𝜀 =E −R I +𝜀 DC1

DC1

EDC1

DC2

DC DC1

DC2

DC1

DC DC1

ĨDC1 = IDC1 + 𝜀IDC1 ĨDC2 = −IDC1 + 𝜀IDC2 P̃ DC1 = EDC1 IDC1 + 𝜀PDC1 P̃ = R I2 − E I +𝜀

Node 1

DC1 DC1

Node 3

EDC2

PDC2

Node 6

Node 2

Node 7

Node 4

Node 5

Figure 5.11 Tutorial network including VSC-HVDC.

to nodes 6 and 7, each having a series resistance and a shunt conductance of 0.02 p.u. and 0.002 p.u. It should be borne in mind that these parameters impact directly on the conduction and switching losses, respectively. The DC cable which links both VSCs has a resistance value of 0.02 p.u. In this test case, the estate variables have been taken to correspond to the VSC at node 6. The data ﬁles corresponding to this example are: Network ﬁle:

Network_tutorial_VSC_HVDC_ptp

Measurements ﬁle:

Measure_VSC_HVDC_ptp

SE conﬁguration ﬁle:

Conﬁg_1

5.6 FACTS Device State Estimation Modelling in Electrical Power Grids

Execution of the program reveals the existence of bad data. The error is identiﬁed to be a wrong measurement in the DC circuit; more speciﬁcally, it is the DC voltage of the VSC at node 7. In order to verify that this is the case, the wrong measurement is eliminated and the program is run again, showing that all the normalized residuals have a null value, i.e. it yields exact measurements. For test case 8, WLS-SE uses the following ﬁles: Network ﬁle:

Network_tutorial_VSC_HVDC_btb

Measurements ﬁle:

Measure_VSC_HVDC_btb

SE conﬁguration ﬁle:

Conﬁg_2

In this example we replace the transmission line 3-4 of the tutorial network with a back-to-back VSC-HVDC. Except for the DC resistance, which is taken to be zero, all other parameters of the VSC-HVDC are kept the same as in the previous test case. When analyzing the results, keep in mind that noise has been added to the measurements to make them more realistic. 5.6.9

Multi-terminal HVDC

So far, all the devices addressed in this chapter have comprised one and two DC nodes. In contrast, a multi-terminal VSC-HVDC system with n VSCs will have one or many DC nodes, up to n nodes. This point is discussed in more detail in Section 1.2.6. In any case, the n converters are linked on their DC sides through a network of DC cables or, indeed, a single DC bus, such as in the case of a back-to-back MT-VSC-HVDC. In a multi-terminal system, each VSC sets the border between an AC system and the DC system. The coupling of a VSC with the AC grid is carried out through a coupling transformer, as shown in Figure 5.12. The ﬁrst step towards establishing the model of the multi-terminal VSC-HVDC is to select its state variables. As with the other VSC-based devices presented in this chapter, the state vector incorporates the AC nodal voltages of each VSC. Concerning the state variables of the DC system, the selection is the voltages at the DC nodes of each VSC. Hence, the state vector of the MT-VSC-HVDC is the following extended state vector: XMulti = [VAC1 , VAC2 … VACn , 𝜃AC1 , 𝜃AC2 … 𝜃ACn , EDC1 , EDC2 … EDCn ]

(5.75)

We now look at the available measurements for the multi-terminal system and their formulations. Concerning the measurements of the AC sides of the VSCs, the formulation is the same as the one employed by the model described in Section 5.6.2. Concerning the DC grid, the injected currents at the DC nodes may be derived from the nodal DC conductance matrix, GDC , and the DC nodal voltages: IDC = GDC EDC

(5.76)

where EDC and IDC are vectors of voltages and injected currents, respectively, at all the DC nodes. Accordingly, the current I DCi injected by the VSC connected to the DC node i may be expressed as a function of the state variables using: IDCi = gDCi EDC

(5.77)

233

234

5 State Estimation

Va θa

VAC1 θAC1

EDC1

Vb θb

VAC2 θAC2

EDC2

.

AC network

.

DC network

. Vk θk

VACn θACn

EDCn

Figure 5.12 Multi-terminal network.

where gDCi is a row vector extracted from the nodal conductance matrix corresponding to node i. On the other hand, an inner measurement of current in the DC grid, between nodes i and j, would have the following form: IDC ij =

EDCi − EDCj RDCij

(5.78)

where RDCij is the resistance of the DC link between nodes i and j. The injected power by this converter to node i of the DC grid is: PDCi = EDCi IDCi = EDCi gDCi EDC

(5.79)

To complete the model, it is only necessary to incorporate an active power balance equation for each VSC in the system. Let us consider the case of a VSC which connects to nodes i and r on its DC and AC sides, respectively. Moreover, the VSC connects to the AC grid through a coupling transformer, using nodes r and j, so that the power balance equation is formulated as follows: 2 EDCi gDCi EDC + Prj + R1i Irj2 + Gswi EDCi =0

(5.80)

where Gswi and R1i are the parameters associated to the power losses of this converter. Table 5.7 summarizes the measurements of the MT-VSC-HVDC model developed above.

5.6 FACTS Device State Estimation Modelling in Electrical Power Grids

235

Table 5.7 Multi-terminal measurements. Conventional measures

Control measures

AC voltage

Ṽ r = Vr + 𝜀Vr Ṽ j = Vj + 𝜀Vj

Modulation index

AC power ﬂows

P̃ rj = Grj Vr2 − Vr Vj (Grj cos 𝛾rj + Brj sin 𝛾rj ) + 𝜀prj ̃ = −B V 2 − V V (G sin 𝛾 − B cos 𝛾 ) + 𝜀 Q rj rj r r j rj rj rj rj qrj

Phase-angle control signal

DC voltage

The opposite ﬂows can be obtained by exchanging the indexes r and j. Ẽ =E +𝜀

DC current

ĨDCi = gDCi EDC + 𝜀IDCi (Injected)

DCi

ĨDCij = DC power

Vr k1 Edci

+ 𝜀m

𝛾̃rj = 𝜃r − 𝜃j + 𝜀𝜃rj

EDC

EDCi − EDCj RDCij

+ 𝜀IDCij (In DC link i-j)

P̃ DCi = EDCi gDCi EDC + 𝜀PDC (Injected) P̃ DCij =

5.6.10

DCi

̃ ri = m

2 EDCi − EDCi EDCj

RDCij

+ 𝜀IDCij (In DC link i-j)

MT-VSC-HVDC Model in WLS-SE

In this section, the state estimation of an MT-VSC-HVDC system incorporated into WLS-SE is analyzed. The system has been formed by incorporating into the tutorial grid a three-node DC grid. The DC grid connects to the AC grid in nodes 2, 3 and 4, as shown in Figure 5.13. The impedances of the coupling transformers 2-8, 3-6 and 4-7 are 0.02 + j 0.1, 0.01 + j 0.7 and 0.01 + j 0.7, respectively. The resistances of the DC lines 1-2, 2-3 and 1-3 are 0.2, 0.1 and 0.15, respectively. The ﬁles with the relevant information for this case, in WLS-SE, are: Network ﬁle:

Network_tutorial_Multiterminal

Measurements ﬁle:

Measure_Multiterminal

SE conﬁguration ﬁle:

Conﬁg_1

The results furnished by the estimator show some wrong measurements. The largest normalized residue is 11.6, which is associated to the measurement of the AC voltage magnitude at node 4. By eliminating this measurement and executing once more the estimation process, it is found that one more bad data is present. In this case the largest normalized residue is 3.5, which corresponds to the measurement of power in the DC link 1-3. By eliminating this measurement and executing the estimation process again, it shows that all measurements are exact, since all the normalized residues carry null values. This example illustrates how the method of the largest normalized residue is used to identify one or more bad data. However, a word of caution is needed; this characteristic of the largest normalized residue cannot be generalized. In this example, the correlation

236

5 State Estimation

AC node 3

AC node 4 AC node 6

DC node 1

AC node 7

DC node 2

DC node 3

AC node 8 AC node 2 Figure 5.13 Multi-terminal DC network and coupling transformers.

between the two wrong measurements is weak and this is the reason it has been possible to identify them in two stages. When the wrong measurements are strongly correlated, the method of the largest normalized residue may fail. Furthermore, in the presence of unfortunate combinations of errors, some valid measurements may be ﬂagged as bad data by the method of the largest normalized residue.

5.7 Incorporation of Measurements Furnished by PMUs 5.7.1

Incorporation of Synchrophasors in State Estimation

As stated in Section 1.4 of Chapter 1, PMU devices are used to acquire synchronized measurements of voltage and current phasors in power grids, termed synchrophasors. In power systems state estimation, the synchrophasors ﬁnd quite a natural place in the state estimation processes, enabling a larger amount of information over the operational state of the power grid. In general, a state estimation containing synchrophasor measurements would exhibit higher precision, owing to the higher quality of these types of measurements [23]. The implementation of a conventional state estimator in an electrical power grid enables the operational status of the power grid at a given point in time to be determined. The power grid is taken to remain static throughout one estimation cycle; this involves the time that passes between the measurements’ capture at the substations and their transmission to the control centre. Throughout this interval, the measurements are taken to remain stationary. It is unavoidable that some small diﬀerences will exist between the various measurements at their time of capture. Nevertheless, if

5.7 Incorporation of Measurements Furnished by PMUs

the operating conditions do not change signiﬁcantly, this does not seem to aﬀect the estimation process unduly. However, the measured synchrophasors do not allow for any slack at the point in time at which they are taken, given that any delay impacts adversely the measured value. Hence, all the synchrophasor measurements aimed at state estimation applications should be synchronized to correspond to the same sampling time. One way to incorporate the synchrophasors into the estate estimation process is to carry out the implementation in two stages [24]. In the ﬁrst stage, we carry out a conventional state estimation process, without using the synchrophasor measurements. In the second stage, we use the synchrophasor measurements together with the results from the ﬁrst stage to start a new, more reﬁned, state estimated process. It should be noted that the second stage can be formulated in such a way that the estimation process be treated as a linear process and, accordingly, non-iterative. The advantage of this approach is that the results from the ﬁrst stage can be obtained with a commercial, conventional state estimator. The disadvantage of using this approach is that it does not exploit fully the great potential that synchrophasors have in improving the grid’s state estimation. If the synchrophasor measurements are not incorporated during the ﬁrst stage, neither the grid’s observability nor the treatment of bad data will be improved. Furthermore, the calculated state estimation will not be optimal since all the available measurements, conventional and non-conventional, were not considered simultaneously. An alternative approach is to incorporate the synchrophasors in the state estimation process directly, as a new type of measurement. In this way, the calculated state estimation will be optimum and the synchrophasors will improve both the grid’s observability and the treatment of bad data. The section below is used to describe this approach in more detail. 5.7.2

Synchrophasors Formulations

The PMU measurements, which are involved in the state estimation process, are: – synchrophasor of nodal voltages – synchrophasor of branch currents – synchrophasor of nodal current injections. In state estimation PMU applications, the voltage and current waveforms are assumed to be purely sinusoidal which, in generic form, may be expressed by the following time expression: x(t) = Xm cos(𝜔t + 𝜙)

(5.81)

where X m is the signal amplitude, 𝜙 is the signal phase shift with respect to an arbitrary reference and 𝜔 = 2𝜋f , with f being the frequency. This time expression has an associated synchrophasor, which may be written down in two diﬀerent, commutable ways: the polar and rectangular representations. In the former case, the synchrophasor X is expressed in terms of the RMS value of X m and its argument 𝜙, i.e. X = Xej𝜙

√ where X = Xm ∕ 2 is the RMS value of X m .

(5.82)

237

238

5 State Estimation

In case of the rectangular representation, the synchrophasor is expressed in terms of its real and imaginary parts, X Re and X Im , respectively, X = XRe + j XIm

(5.83)

where X Re = X cos 𝜙 and X Im = X sin 𝜙 are the real and imaginary parts of X, respectively. The standard IEEE C37.118 2005 ‘Standard for Synchrophasors for Power Systems’ allows both output formats for a PMU synchrophasor, the polar and the rectangular representations. Each representation has distinctive advantages when applied to either voltage or current measurements. It is customary for state estimators to represent the nodal voltages in polar form. Hence, this provides an incentive for representing the synchrophasors of voltage in polar form, i.e. using magnitude and phase. These will not require a diﬀerent treatment from those employed for the conventional measurements of voltage magnitude: Ṽ i = Vi + 𝜀Vi

(5.84)

Concerning the phase voltages, these show similar characteristics to the magnitudes, in the sense that they are also direct synchrophasor measurements of a state variable. Even though conventional state estimators do not normally incorporate phase voltage measurements, their inclusion in the state estimator presents no diﬃculty at all, simply being expressed as: 𝜃̃i = 𝜃i + 𝜀𝜃i

(5.85)

On the other hand, the current synchrophasors possess a non-linear representation whether they are expressed in polar form or rectangular form. In the former case, it becomes necessary to use the current magnitude as measurement. This type of measurement is widely used in distribution systems. However, its formulation introduces ill-conditioning terms in the Jacobian matrix, for reduced values of the current magnitude. This anomalous condition shows up at the ﬁrst iteration when the iterative solution is started from a ﬂat voltage proﬁle, with currents taking a null value, leading to some Jacobian terms having undetermined values. An alternative to overcome this problem is to use the square of the current as the measurement, leading to well-conditioned Jacobian terms. However, the quality of the measurement worsens as the associated error increases. Hence, for current synchrophasors it is recommended to use the rectangular form. This way, ill-conditioning problems are avoided and the original precision of the measurement is preserved. Using the rectangular representation, the current ﬂow, in phasor form, from node i to node j, is: Iij = IRe,ij + j IIm,ij

(5.86)

Their real and imaginary parts may be expressed as a function of the state variables through the following non-linear equations: IRe,ij = (Gij + Gsi ) Vi cos 𝜃i − (Bij + Bsi ) Vi sin 𝜃i − Gij Vj cos 𝜃j + Bij Vj sin 𝜃j (5.87) IIm,ij = (Gij + Gsi ) Vi sin 𝜃i + (Bij + Bsi ) Vi cos 𝜃i − Bij Vj cos 𝜃j − Gij Vj sin 𝜃j (5.88) where Gij + j Bij is the series admittance of branch i − j and Gsi + j Bsi is the shunt admittance at node i.

5.7 Incorporation of Measurements Furnished by PMUs

The synchrophasor of the total current injected into node i may be expressed as the sum of the real and the imaginary parts, respectively, of all the current ﬂowing through the branches which connect to that node as follows: ∑ IRe,i = Ir,ij (5.89) j∈Ω

IIm,i =

∑

(5.90)

Ii,ij

j∈Ω

where Ω is the set of nodes connecting to node i. 5.7.3

Phase Reference

As pointed out in Section 5.4, in a conventional state estimation process, i.e. one where no synchrophasor measurements are employed, it is necessary to assign a phase reference to each observable island in the power grid. However, when synchrophasor measurements are incorporated into the estimation process, there is no need to adopt a phase reference. In this case, the adopted phase reference is determined by the synchronizing signal transmitted across the GPS. Only one synchrophasor measurement is suﬃcient (either voltage or current) for the estimation process to have a phase reference. Accordingly, when synchrophasor measurements are available, no pseudo-measurements of phase reference should be included in the state estimation process. For this reason, as in the conventional estimator, when synchrophasor measurements are included in the state estimation process, all the nodal phases of the power grid are incorporated as state variables in the state vector. One potential problem that arises from not adopting the customary zero value of phase reference is that the nodal phase angles furnished by PMUs may be very far away from zero. The reason is that at nodes where no synchrophasor measurements are available, the initial values of nodal phase angles state variables are normally taken to be zero, a situation which will impinge on the convergence of the state estimation process. An alternative to ameliorate this convergence problem is to adopt as initial value of nodal phase state variables the measured value, and at nodes where no synchrophasor measurements are available, the average value of all the synchrophasor phase measurements available. 5.7.4

PMU Outputs in WLS-SE

In this section, we show the results of one test case in WLS-SE where synchrophasor measurements of voltage and currents are included. We use the tutorial power grid, which includes a PMU at node 2. The available outputs enable the incorporation of voltage synchrophasors at node 2 and current synchrophasors in the transmission lines that connect at node 2. The ﬁles for this example are: Network ﬁle:

Network_tutorial

Measurements ﬁle:

Measure_PMU

SE conﬁguration ﬁle:

Conﬁg_2

239

240

5 State Estimation

It should be noted that two voltage magnitude measurements appear at node 2: one corresponding to the conventional measurements system and the other to the output of the PMU device. Since the measurements incorporate noise, both measurement values diﬀer; however, the quality of the synchrophasor measurement is higher. It should be remarked that since we have included synchrophasor measurements, no pseudo-measurement of phase reference is required in this case.

5.A Appendix 5.A.1 Input Data and Output Results in WLS-SE 5.A.1.1

Input Data

The input information for the program WLS_SE.p has been grouped in three main blocks. Each block contains an editable m-ﬁle. The ﬁrst block contains the network data, the second block incorporates the measurements and the third block contains the state estimator conﬁguration. Table 5.A.1 gives the ﬁle names used in the examples presented in Chapter 5. All the information can be modiﬁed using the Matlab editor. The variables information is given below.

5.A.1.2

Network Data

The node information is included in the variable ‘ac_nodes’ and the network branches in the variable ‘ac_branch’. The variable ‘ac_nodes’ is a matrix which contains one row per Table 5.A.1 Data ﬁles. Case

Network_data

Measurement_data

Conﬁg_data

1

Network_tutorial

Measure_tutorial_network

Conﬁg_1

2

Network_tutorial

Measure_tutorial_network

Conﬁg_2

3

Network_tutorial

Measure_tutorial_network_ bad_data

Conﬁg_1

4

Network_tutorial_STATCOM

Measure_STATCOM_case_a

Conﬁg_1

5

Network_tutorial_STATCOM_ no_losses

Measure_STATCOM_case_a

Conﬁg_1

6

Network_tutorial_STATCOM

Measure_STATCOM_case_b

Conﬁg_2

7

Network_tutorial_VSC_HVDC_ptp

Measure_VSC_HVDC_ptp

Conﬁg_1

8

Network_tutorial_VSC_HVDC_btb

Measure_VSC_HVDC_btb

Conﬁg_2

9

Network_tutorial_UPFC

Measure_UPFC

Conﬁg_1

10

Network_tutorial_Multiterminal

Measure_Multiterminal

Conﬁg_1

11

Network_tutorial

Measure_PMU

Conﬁg_1

5.A.1 Input Data and Output Results in WLS-SE

Table 5.A.2 Variable ‘ac_node’ format. ac_nodes

1

Number of node

2

Not used

3

Not used

4

VSC type: (0) Without VSC converter (3) VSC STATCOM (4) HVDC sending VSC (5) HVDC receiving VSC (6) UPFC shunt VSC (7) UPFC series VSC (8) Multi-terminal VSC-HVDC

5

VSC series resistance related to switching losses

6

VSC shunt conductance related to conduction losses

7

In type 5 (VSC-HVDC) dc link resistance In type 8 (multiterminal HVDC) related dc-side node

8

In type 4 (VSC-HVDC) receiving node In type 5 (VSC-HVDC) sending node

Table 5.A.3 Variable ‘ac_branch’ format. ac_branch

1

‘From’ node

2

‘To’ node

3

Series resistance

4

Series reactance

5

Total shunt susceptance

6

Not used

7

4 in boosting transformer (UPFC) 0 in other case

node. The nodes are assigned the row number that they occupy in the matrix. The information associated to each node is entered column-wise, according to the format indicated in Table 5.A.2 The variable ‘ac_branch’ contains one row per branch. The branches are assigned the row number that they occupy in the matrix. The information associated to each branch is entered column-wise, according to the format indicated in Table 5.A.3. In the case of a multi-terminal HVDC grid, the additional variables ‘Adc_inc’ and ‘dc_link’ are required. The variable ‘Adc_inc’ is the node-branch incidence matrix for the dc grid. The variable ‘dc_link’ is a column vector containing in each row the resistance of the corresponding dc branch.

241

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5 State Estimation

5.A.1.3

Measurements Data

There are two variables in this block, ‘z’ and ‘c’. The variable ‘z’ is a matrix containing one row per available measurement. In turn, the information associated with each measurement is entered, column-wise, in the matrix, following the format given in Table 5.A.4. The variable ‘c’ is a matrix containing one row per equality restriction included, with the information associated with each restriction entered, column-wise, in the matrix, following the format given in Table 5.A.5. Table 5.A.4 Variable ‘z’ format. z

1

Type of measurement: 1 Voltage magnitude (Vi) 2 Voltage phase angle (Oi) 3 Active power ﬂow (Pij) 4 Reactive power ﬂow (Qij) 7 Active power injection (Pi) 8 Reactive power injection (Qi) 12 Real part of the current synchrophasor through a branch (Re(Iij)) 13 Imaginary part of the current synchrophasor through a branch (Im(Iij)) 14 Real part of the current injection synchrophasor (Re(Ii)) 15 Imaginary part of the current injection synchrophasor (Im(Ii)) 25 dc voltage in STATCOM, HVDC VSC-sending and UPFC dc-link (VSC s Edc) 26 dc current in STATCOM, HVDC VSC-sending and UPFC dc-link (VSC s Idc) 27 dc power in STATCOM, HVDC VSC-sending and UPFC dc-link (VSC s Pdc) 28 Modulation index STATCOM, HVDC VSC-sending and UPFC VSC-shunt (VSC m) 29 dc voltage in HVDC VSC-receiving (VSC r Edc) 30 dc current in HVDC VSC-receiving (VSC r Idc) 31 HVDC VSC-receiving dc power (VSC r Pdc) 32 Modulation index in HVDC VSC-receiving and UPFC VSC-series (VSC r m) 33 Modulation index in Multiterminal HVDC VSC (m VSC Mul) 34 Nodal phase diﬀerence (Oi-Oj) 40 UPFC active power ﬂow at the ac-side node in series-VSC (UPFC psVSC) 41 UPFC reactive power ﬂow at ac-side node in series-VSC (UPFC qsVSC) 45 Multiterminal HVDC dc voltage (Vdc Mul) 46 Multiterminal HVDC dc current injection at VSC (Idc_i Mul) 48 Multiterminal HVDC dc current ﬂow in dc-link (Idc_ij Mul) 49 Multiterminal HVDC dc power ﬂow in dc-link (Pdc_i Mul)

2

For a node measurement number. For a branch ﬂow measurement number. This value is positive if the measurement is taken in the node ‘from’ of the corresponding branch in variable ‘ac_branch’. If the measurement is taken in the opposite end of the branch, i.e. node ‘to’, the branch number takes a negative value

3

Measured value

4

Quality of measurement (standard deviation)

5.A.2 Output Results

Table 5.A.5 : Variable ‘c’ format. c

1

Type of constraint: 7 Active power injection (Pi) 8 Reactive power injection (Qi) 26 Current at VSC dc side in STATCOM (VSC s Idc) 50 STATCOM, HVDC VSC-sending, UPFC VSC-shunt active power balance (VSC s bal) 51 HVDC VSC-receiving active power balance (VSC r bal) 52 UPFC VSC-receiving active power balance (UPFC r bal) 53 Multiterminal HVDC VSC active power balance (Multi. bal)

2

Related number of the ac node.

3

Not used. This value must be 0

The elimination of a measurement from the estimation process does not require the corresponding row to be deleted. It is suﬃcient to comment out the relevant row using the character ‘%’. In general, for all the cases presented in this chapter, all the possible measurements have been included in the relevant matrices and the measurements that have not been included in the estimation process have simply been commented out. This facilitates the incorporation of new measurements in new study cases. It has been stated above that it is possible to eliminate measurements; however, some caution needs to be exercised since the WLS-SE program has not been ﬁtted with an observability analysis. It should be remarked that to carry out a successful state estimation calculation, it is necessary to count with a number and distribution of measurements that warrant it. 5.A.1.4

State Estimator Conﬁguration

This block uses three variables: ‘noise’, ‘converg’ and ‘iter_ﬁn’. The variable ‘noise’ speciﬁes whether a Gaussian noise has been added to the measurement prior to the state estimation process. If the variable is given a value of 1, then random noise is incorporated to the measurement value. The value of the variable ‘converg’ is the convergence tolerance used in the iterative process. The variable ‘iter_ﬁn’ is the maximum number of allowed iterations.

5.A.2 Output Results Once the estimation process has run, the program shows the results on the screen. The information is presented in three blocks. The ﬁrst block contains information relating to the general characteristics of the estimation process, including the data ﬁles used and the conﬁguration parameters of the state estimator. The second block presents

243

244

5 State Estimation

the parameters associated to the power grid just estimated. The third block includes information relating to the estimation process and the estimated values. In the next section the estimated values of the state variables are presented and then the estimated values of the measurements. In order to identify bad data, the normalized residual value of each measurement is speciﬁed. For completeness, all the restrictions that have been incorporated into the estimation process are listed.

References 1 Schweppe, F.C., Wildes, J., and Rom, D.B. (1970). Power system static-state estima-

2 3 4

5

6

7

8

9 10 11

12 13

14

tion, part I: exact model, part II: approximate model, part III: implementation. IEEE Transactions on Power Apparatus and Systems PAS-89 (1): 120–135. Monticelli, A. (1999). State Estimation in Electric Power System. A Generalized Approach. Kluwer Academic Publishers. Abur, A. and Gomez Exposito, A. (2004). Power System State Estimation: Theory and Implementation. Marcel Dekker. Sirisena, H.R. and Brown, E.P.M. (1981). Inclusion of HVDC links in AC power system state estimation. Proceedings of the Institution of Electrical Engineers 128 (3): 147–154. Leite Da Silva, A.M., Prada, R.B., and Falcao, D.M. (1985). State estimation for integrated multi-terminal DC/AC systems. IEEE Transactions on Power Apparatus and Systems PAS-104: 2349–2355. Jegatheesan, R. and Duraiswamy, K. (1987). AC/multi-terminal DC power system state estimation—a sequential approach. Electric Machines and Power Systems 12: 27–42. Sinha, A.K., Roy, L., and Srivastava, H.N. (1994). A decoupled second order state estimator for AC-DC power systems. IEEE Transactions on Power Systems 9 (3): 1485–1491. Ding, Q., Chung, T.S., and Zhang, B. (2001). An improved sequential method for AC/MTDC power system state estimation. IEEE Transactions on Power Systems 16 (3): 506–512. A. Abur, B. Gou and E. Acha, “State estimation of networks containing power ﬂow control devices”, in Proc. 13th PSCC, Trondheim, June–July 1999, pp. 427–433. Xu, B. and Abur, A. (2004). State estimation of systems with UPFCs using the interior point method. IEEE Transactions on Power Systems 19 (3): 1635–1641. E.A. Zamora and C.R. Fuerte-Esquivel, “Static state estimation of power containing series and shunt FACTS controllers”, in Proc. 15th PSCC, Liege, Belgium, 22–26 August 2005. Zamora, E.A. and Fuerte-Esquivel, C.R. (2011). State estimation of power systems containing facts controllers. Electric Power Systems Research, vol. 81,: 995–1002. Qifeng, D., Boming, Z., and Chung, T.S. (2000). State estimation for power systems embedded with FACTS devices and MTDC systems by a sequential solution approach. Electric Power Systems Research, 55,: 147–156. Rakpenthai, C., Premrudeepreechacharn, S., and Uatrongjit, S. (2009). Power system with multi-type FACTS devices state estimation based on predictor-corrector interior point algorithm. Electrical Power and Energy Systems, 31,: 160–166.

References

15 de la Villa Jaén, A., Acha, E., and Gómez Expósito, A. (2008). Voltage source con-

16

17

18

19

20 21 22 23 24

verter modeling for power system state estimation: STATCOM and VSC-HVDC. IEEE Transactions on Power Systems, 23 (4): 1552–1559. J. Cao, W. Du and H.F. Wang, “The incorporation of generalized VSC MTDC model in AC/DC power system state estimation”, International Conference on Sustainable Power Generation and Supply (SUPERGEN 2012), 8–9 September 2012. Thorp, J.S., Phadke, A.G., and Karimi, K.J. (1985). Real-time voltage phasor measurements for static state estimation. IEEE Transactions on Power Apparatus and Systems 104 (11): 3098–3107. A. Gómez-Expósito, A. Abur, P. Rousseaux, A. de la Villa Jaén and C. Gómez-Quiles, “On the Use of PMUs in Power System State Estimation”, 17th Power System Computation Conference, Stockholm, Sweden, August 22–26, 2011. de la Villa Jaén, A., Beloso Martínez, J., Gómez-Expósito, A., and González Vázquez, F. (2018.). Tuning of measurement weights in state estimation: theoretical analysis and case study. IEEE Transactions on Power Systems, 33 (4): 4583–4592,. Acha, E. (2015). Nodal Analysis & Sparsity. ADP-London, Kindle Publishing. Stagg, G.W. and El-Abiad, H. (1968). Computer Methods in Power Systems Analysis. McGraw-Hill. Mohan, N., Undeland, T.M., and Robbins, W.P. (1995). Power Electronics. Converters, Applications, and Design. Wiley. Phadke, A.G. and Thorp, J.S. (2008). Synchronized Phasor Measurements and Their Applications. Springer. Zhou, M., Centeno, V.A., Thorp, J.S., and Phadke, A.G. (2006). An alternative for including phasor measurements in state estimation. IEEE Transactions on Power Systems 21 (4): 1930–1937.

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6 Dynamic Simulations of Power Systems 6.1 Introduction As elucidated in Chapter 1, today’s electric power systems are among the most complex dynamic systems ever built, requiring the intervention of skilful engineers, round the clock. Power systems’ engineers ought to be able to assess in advance the various issues that emerge when the power network undergoes drastic changes, such as when new equipment and, particularly, new technologies are added to it, for instance FACTS and VSC-HVDC technologies. This is even more so when one considers the natural dynamic behaviour exhibited by such large systems, which always depend on the whims of societal behaviour and are exposed to the forces of nature. A great deal of study and research eﬀort is required to master the topic of power system dynamics. At present, comprehensive models of VSC equipment are still required – reliable models with which to evaluate the dynamic impact of these devices in the network, prior to their installation. At the design stage, simulation studies of credible disturbances will reveal how the new equipment will perform and what impact it will have on the power network. Careful consideration needs to be given to the control systems of the VSC to assess the eﬀectiveness, or otherwise, of the voltage and power controls to counteract voltage, power and frequency oscillations in the network – the assessments must be beyond doubt. Interest in HVDC transmission systems has long been associated with the need to transfer electrical energy over very long distances. However, other applications have emerged in more recent times where the HVDC links are gaining ground, for instance in the tapping of oﬀshore wind energy resources. There is general agreement that the operation of today’s power networks is becoming increasingly complex owing to the use of HVDC systems and similar equipment. Moreover, the intermittence of the wind may pose a major risk to the integrity of the power system. In this new scenario, it becomes important that greater attention is given to the dynamics of the power grid than has been the case so far; it is recommended that the HVDC controllers be suitably tuned and that advanced control functions be implemented. The installation of FACTS-type power ﬂow controlling devices seems to oﬀer a realistic solution to ameliorate potential dynamic problems that may arise, as well as to enable the transmission of larger amounts of electrical energy between areas without endangering the stability of the interconnected networks. As discussed in Chapter 1, VSC-HVDC systems are rapidly becoming the most versatile and cost-eﬀective power transmission option in modern power systems networks. VSC-FACTS-HVDC: Analysis, Modelling and Simulation in Power Grids, First Edition. Enrique Acha, Pedro Roncero-Sánchez, Antonio de la Villa Jaén, Luis M. Castro and Behzad Kazemtabrizi. © 2019 John Wiley & Sons Ltd. Published 2019 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/acha_vsc_facts

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6.2 Modelling of Conventional Power System Components The conventional, large synchronous generator continues to be the dominant source of electrical energy in most power systems around the globe. Owing to the wide range of complex dynamic phenomena, to which these synchronous machines are subjected, various models with diﬀerent ranges of applicability can be found in the open literature. The mechanical equations of a synchronous generator are well established [1–3] and they are only outlined here. Three basic assumptions are made in deriving its equations: (i) the machine rotor speed does not depart greatly from the synchronous speed; (ii) the machine rotational losses due to windage and friction are neglected; and (iii) the mechanical power shaft is smooth, meaning that the power coming from the shaft is constant except for the control action exerted by the speed governor. Therefore, the accelerating power of the generator Pa is the diﬀerence between the mechanical input power Pm [p.u.] supplied by the prime mover and the electrical output power Pe [p.u.]. The machine acceleration is expressed as follows: P P − Pe d2 𝛿 = a = m 2 Mg Mg dt

(6.1)

where the second-time derivative of the rotor angle 𝛿 [rad] represents the acceleration of the machine and Mg is the angular momentum. The angular momentum may be further deﬁned in terms of the system base frequency f 0 [Hz] and the inertia constant H [MWs/MVA], which does not change much regardless of the size of the machine, Mg =

H 𝜋f0

(6.2)

Eddy currents induced in the rotor iron or in the damping windings produce torques which oppose the motion of the rotor relative to the synchronous speed. To account for these eﬀects, a deceleration power is introduced into the mechanical equation. Additionally, decomposing (6.1) into two ﬁrst-order diﬀerential equations leads to the following expressions: d𝜔 𝜋f0 (6.3) = [P − Pe − D(𝜔 − 𝜔0 )] dt H m d𝛿 (6.4) = 𝜔 − 𝜔0 dt where 𝜔 [rad s−1 ] stands for the mechanical angular speed and D [s/rad] is the damping coeﬃcient. 6.2.1

Modelling of Synchronous Generators

In early power system stability studies, the generators were represented by the classical model, which assumes constant ﬂux linkages in both axes and neglects transient saliency [4]. In this simple model, an internal voltage behind the d-axis transient reactance Xd′ is determined and its magnitude is assumed constant. To enable a more realistic representation of the power system behaviour, the transient saliency and the varying ﬂux linkages that exist in both axes of the synchronous generator ought to be considered. Figure 6.1 depicts the phasor diagram of the synchronous generator during the transient

6.2 Modelling of Conventional Power System Components

Ei Ed

E′q d axis

IqXq Ef

Eq

Iq

q axis

E′ Iq X′q

V

E′d

Id Xd

IRa

I

Id

Id X′d

(a) Ei E″q d axis

Ed Eq E′

Iq

Iq Xq

q axis

Ef

E″ E″d

Iq X″q

V I

Id

Id X″d I d Xd

IRa (b)

Figure 6.1 Phasor diagram of the synchronous generator: (a) transient state, (b) sub-transient state.

and sub-transient states [5, 6]. Based on these phasor diagrams, the mathematical model of the synchronous generator can be developed. Taking the case of a synchronous generator undergoing a transient condition, the voltage behind the synchronous reactance is obtained with the internal transient ﬂux voltages, Ed′ and Eq′ . Then it is necessary to ﬁnd the voltages which will represent the ﬂux linkages of the rotor windings. The equations that describe the time variation of the voltages representing the rotor ﬂux linkages in the dq reference frame are: ′ Td0 ′ Tq0

dEq′ dt dEd′

= Ef − Eq′ + (Xd − Xd′ )Id

= −Ed′ − (Xq − Xq′ )Iq dt Ed′ = Ra Id + Xq′ Iq + Vd Eq′

= Ra Iq −

Xd′ Id

+ Vq

(6.5) (6.6) (6.7) (6.8)

′ ′ where Td0 and Tq0 are the open-circuit transient time constants; Ef is the excitation voltage; V d and V q are the generator’s terminal voltages; I d and I q are the generator’s terminal currents; Xd′ and Xq′ are the transient reactances; X d and X q are the synchronous reactances; and Ra is the armature resistance. Notice that except for Ef , all other quantities are given on a per-axis basis: d-axis and q-axis, respectively.

249

250

6 Dynamic Simulations of Power Systems

If a more detailed synchronous machine model representation is required, then the sub-transient synchronous generator model must be used. The equations that govern the sub-transient behaviour of the synchronous generator are: ′′ Td0 ′′ Tq0

dEq′′

= Eq′ − Eq′′ + (Xd′ − Xd′′ )Id

(6.9)

= Ed′ − Ed′′ − (Xq′ − Xq′′ )Iq dt Ed′′ = Ra Id + Xq′′ Iq + Vd

(6.10)

Eq′′

(6.12)

dt dEd′′

= Ra Iq −

Xd′′ Id

+ Vq

(6.11)

′′ ′′ where Td0 and Tq0 are the open-circuit sub-transient time constants, Ed′′ and Eq′′ are the internal sub-transient ﬂux voltages, and Xd′′ and Xq′′ are the sub-transient reactances. The active and reactive power at the synchronous generator terminal in the dq reference frame can be expressed as:

6.2.2

Pg = Vd Id + Vq Iq

(6.13)

Qg = Vq Id − Vd Iq

(6.14)

Synchronous Generator Controllers

Conventional power plants are made up of several components to enable the power conversion from mechanical to electrical in an eﬃcient manner. The main elements that ﬁnd a representation in most dynamic analyses are the synchronous generators, voltage regulators, speed governors, turbines, boilers and protection equipment. The most relevant models are outlined below. 6.2.2.1

Speed Governors

The speed governor is a frequency controller which plays a crucial role in power system operation. Its main functions are to: • keep the generator speed as close as possible to its nominal value • guarantee a fast and automatic participation of the generator following any change in generation or load, aiming at maintaining a balanced generation-load pattern in the system. To adjust the amount of steam or water input into the turbine, the speed governor senses the diﬀerence between the actual rotor speed and the angular frequency and adjusts the turbine valve accordingly. Figures 6.2a,b show the block diagrams used to represent the speed governing system of a steam turbine and hydro turbine, respectively. These are simpliﬁed representations of the models in [7]. From Figure 6.2a the following mathematical expression is derived: dPGV P − KΔ𝜔 − PGV = set dt TGV

(6.15)

where Δ𝜔 = 𝜔 − 𝜔0 , K is the governor gain in per-unit, Pset is the power set point in per-unit, T GV is the governor time constant in seconds and PGV is the output power from

6.2 Modelling of Conventional Power System Components

PGV(max) Pset

+ –

1 1 + sTGV

ω

Turbine and generator PGV(min) Δω

K

+–

ω0

(a) Pset ω0

+ –

Δω

PGV(max)

ΔP + K – 1 + sTGV PGV(min)

Turbine and generator

v

(b) Figure 6.2 Simpliﬁed IEEE speed governor model for a (a) steam turbine, (b) hydro turbine. Source: ©IET, 1987.

the control valve in per-unit. For the case of a hydro turbine, the diﬀerential equation representing its speed-governing system is given by (6.16), where PGV = Pset − ΔP. dΔP KΔ𝜔 − ΔP = dt TGV 6.2.2.2

(6.16)

Steam Turbine and Hydro Turbine

A turbine is a ﬂuid machine through which ﬂuid passes continuously, transferring a portion of its energy to an impeller with vanes or blades. The turbine transforms the energy of steam ﬂow or water ﬂow into mechanical energy through a momentum exchange between the working ﬂuid and the impeller, which carries out the energy exchange. Conventional power-generating plants utilize predominantly hydro turbines or steam turbines which drive a synchronous generator to produce electrical energy. The design of each type of turbine diﬀers greatly from the others owing to their vastly diﬀerent applications. All compound steam-turbine systems utilize governor-controlled valves at the inlet to the high-pressure or very high-pressure chambers to control the steam ﬂow. The steam chest and inlet piping to the ﬁrst turbine cylinder and reheaters and crossover piping downstream all introduce delays between valves movements and changes in steam ﬂow. The main objective in modelling the steam system for stability studies is to take account of these delays. For such a purpose, ﬂows in and out of the steam vessels may be related by simple time constants. For steam turbines, a comprehensive model put forward by an IEEE committee report is selected [7] and shown in Figure 6.3. This block diagram represents a cross-compound, three-stage, single-reheat steam-turbine system. The equations corresponding to this model are: P − PHP dPHP = GV dt TCH dPIP PHP − PIP = dt TRH

(6.17) (6.18)

251

252

6 Dynamic Simulations of Power Systems

FHP

PGV

1 1 + sTCH

PHP

++

++

FIP

FLP

1 1 + sTRH

PIP

1 1 + sTCO

Pm

PLP

Figure 6.3 Simpliﬁed IEEE steam-turbine model. Source: ©IET, 1987.

dPLP P − PLP = IP dt TCO

(6.19)

Pm = PHP FHP + PIP FIP + PLP FLP

(6.20)

where T CH is the steam chest time constant in seconds (including the high-pressure HP stage of the turbine time constant), T RH is the reheater time constant (including the intermediate-pressure IP stage of the turbine time constant), T CO is the steam storage or cross-over time constant (including the low-pressure LP stage of the turbine time constant), F HP is the HP turbine power fraction, F IP is the IP turbine power fraction, F LP is the LP turbine power fraction and Pm is the equivalent generator input mechanical power. In a hydro turbine, the motion energy of the water is 1 – sTW PGV Pm transformed into rotational motion of a shaft. Typically, 1 + 0.5sTW this turbine is equipped with an automatic speed regulaFigure 6.4 IEEE hydro-turbine tor, which is essentially a device sensitive to variations in turbine speed. The impeller receives torque from the water model. Source: ©IET, 1987. pressure, which allows the shaft to obtain a mechanical power. The speed controller is responsible for keeping the turbine at a constant speed in the steady-state regime. Figure 6.4 shows the hydro-turbine model which is selected from the IEEE committee report [7]. The equation arising from the Laplace-based block representing the hydro-turbine model is: P − TW Ṗ GV − Pm dPm (6.21) = GV dt 0.5TW where T W is the water time constant in seconds and Pm is the equivalent generator input mechanical power in per-unit. 6.2.2.3

Automatic Voltage Regulator

The basic function of an automatic voltage regulator (AVR) is to provide the synchronous generator’s ﬁeld winding with a variable direct current, enabling an eﬀective control of the generator terminal voltage and, hence, the enhancement of system stability. In addition, the excitation system can provide a broad number of control and protection functions necessary for the satisfactory performance of the generator and the power system. The protective functions ensure that the capacity limits of both the synchronous machine and the excitation control system will not be exceeded. Several

6.2 Modelling of Conventional Power System Components

Vref V

–+ –

VR(max) KA 1 + sTA Vx

VR(min)

VR

1 1 + sTE

Ef

KF s 1 + sTF

Figure 6.5 Simpliﬁed IEEE-type I AVR model.

AVR models for transient stability studies exist in the open literature. The model used in this book is shown in Figure 6.5. It is a simpliﬁed version of the STA1 IEEE model [8] with neither saturation function nor transductor function. The latter represents the inherent delay in measuring the voltage at the generator’s terminal. From Figure 6.5 the following expressions are derived for the AVR model: KA (Vref − Vx − V ) − VR dV R = dt TA ̇ KF Ef − Vx dV x = dt TF dEf VR − Ef = dt TE

(6.22) (6.23) (6.24)

where V R(max) is the maximum regulator voltage, V R(min) is the minimum regulator voltage, K A is the ampliﬁer gain, T A is the ampliﬁer time constant, sec, K F is the stabilizer gain, T F is the stabilizer time constant, sec, and T E is the ﬁeld circuit time constant. 6.2.2.4

Transmission Line Model

In fundamental frequency power system studies, such as power ﬂows, state estimation and dynamic studies, it is common to represent a transmission line by its nominal 𝜋-circuit, as discussed in (Section 3.2.1). However, during a network disturbance the system frequency may deviate from its nominal value and therefore the series impedance and shunt susceptance of the line may change signiﬁcantly. Hence, for dynamic studies, both parameters may incorporate a certain degree of frequency dependency, as [9]: ( ) f +f (6.25) Zs = Rs + jXs0 k m 2f ( 0 ) 2f0 Ysh = Gsh + jB0sh (6.26) fk + fm where f k and f m are the frequency at buses k and m, respectively, Xs0 and B0sh are the series reactance and shunt susceptance obtained at the rated frequency f 0 . 6.2.2.5

Load Model

The impact of the load models on the power system dynamics is a matter of great importance. Generally, the load varies with respect not only to voltage magnitude but also to frequency. The case of load dependency with respect to voltage, which is relevant to power ﬂows and state estimation, was discussed in (Section 3.2.7). For a

253

254

6 Dynamic Simulations of Power Systems

load connected to bus k, the active power PLk and reactive power QLk drawn by the loads may be modelled by a polynomial representation multiplied by a linear factor to account for their voltage and frequency dependence, respectively [10]: [ ( ) ( )] Vk Vk 2 [1 + Kpf ( fk − f0 )] (6.27) + cp PLk = P0k ap + bp V0k V0k [ ( ) ( )] Vk Vk 2 [1 + Kqf ( fk − f0 )] + cq (6.28) QLk = Q0k ap + bq V0k V0k where P0k and Q0k are the nominal active and reactive powers at rated frequency f 0 and voltage V 0k , the parameters ap , bp , cp , aq , bq , cq represent the degree of dependency of √ the active and reactive powers with respect to voltage magnitude V k = (ek 2 + jf k 2 ), and K pf and K qf are the degree of dependency of the active and reactive powers with respect to the frequency f k of the bus to which the load is connected.

6.3 Time Domain Solution Philosophy The power ﬂows in a network are calculated very eﬀectively by modelling the power system by a set of nonlinear algebraic equations corresponding to active and reactive power injections at the nodes, valid for a predeﬁned set of system generation and load pattern. Conversely, for dynamic power system assessments, the mathematical model which describes the dynamic behaviour of generators and their controls can be represented by a set of diﬀerential equations. By suitable combination of both sets of algebraic and diﬀerential equations, the complete power system model aimed at time domain simulations of large-scale power networks may be expressed as an algebraic-diﬀerential problem of the form: 0 = f (x, y)

(6.29)

ẏ = g(x, y, t)

(6.30)

where f and g are nonlinear vector functions and x and y are vectors of variables that are computed at discrete points in time t. Eq. (6.29) corresponds to the network’s power balance equations whereas (6.30) corresponds to the diﬀerential equations of all the synchronous generators and controlling devices. Since each generator is coupled only to the electrical system, expression (6.30) is a collection of separate subsets of independent block-like equations. Nevertheless, to eﬃciently solve Eqs. (6.29) and (6.30) as a function of time, a uniﬁed framework is used ([11, 12]. The Newton-Raphson method is selected in this chapter to carry out such a task since it yields reliable numerical solutions and retains the quadratic convergence of the conventional Newton-Raphson power ﬂow, as discussed in Section 3.5.2. 6.3.1

Numerical Solution Technique

On way to solve the algebraic-diﬀerential set (6.29) and (6.30) is to transform the differential equations into algebraic equations using the implicit trapezoidal method, a robust technique known for giving reasonably accurate results even when relatively large

6.3 Time Domain Solution Philosophy

integration time steps are selected [13, 14]. The implicit trapezoidal method assumes linearity during the integration time step. The transformed algebraic equation of (6.30) for the time interval Δt is: Δt ̇ Y(t) = Y(t−Δt) + + Ẏ (t) ) (6.31) (Y 2 (t−Δt) Rearranging (6.31) as a mismatch equation yields: (6.32) FY = Y(t−Δt) + 0.5Δt Ẏ (t−Δt) − (Y(t) − 0.5Δt Ẏ (t) ) = 0 The ﬁrst step in the application of the trapezoidal method is to express each diﬀerential equation in the form of (6.32). Notice that they are expressed in the form of a mismatch equation, in the same fashion as that of the network’s active and reactive power mismatch equations – refer to Section 3.5. This enables a suitable combination of both kinds of equations in this uniﬁed frame of reference. The overall dynamic models presented in this chapter are developed in an allencompassing frame-of-reference where the nonlinear algebraic equations of the transmission network, synchronous generators and their controls are linearized around a base operating point and combined with the discretized diﬀerential equations arising from the control devices and synchronous generators, for uniﬁed iterative solutions using the Newton-Raphson method [9]. One such iterative solution is valid for a given point in time and its rate of convergence remains quadratic. To this end, the set of diﬀerential equations transformed into algebraic equations is appended to the existing set of algebraic equations representing the Jacobian matrix of the conventional power ﬂow method, as shown in (6.33): ⎡ 𝜕ΔP ⎤ ⎥ ⎢ 𝜕Y ⎥ ⎡ Δe ⎤ ⎡ ΔP ⎤ ⎢ Jacobian 𝜕ΔQ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (6.33) ⎢ΔQ ⎥ = ⎢ 𝜕Y ⎥ ⎢Δf ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ FY ⎦ ⎢ 𝜕FY 𝜕FY 𝜕FY ⎥ ⎣ΔY ⎦ ⎥ ⎢ 𝜕Y ⎦ ⎣ 𝜕e 𝜕f where ΔP and ΔQ are the active and the reactive power mismatch vectors, respectively; F Y is a vector that contains the discretized diﬀerential equations of each generator or controlling device; Δe, Δf and ΔY represent the vectors of incremental changes in the real and imaginary parts of the nodal voltages, as well as the state variables arising from each diﬀerential equation. In this uniﬁed solution, all the state variables are adjusted simultaneously to compute the new equilibrium point of the power system at every time step. The simultaneous solution of both sets of equations plays a key role in achieving a reliable assessment of the dynamic stability of a power network. To exemplify the procedure of aggregating the equations arising from any dynamic component into the conventional Jacobian matrix, the solution method is applied to the dynamic equations brought about by the synchronous generator transient model represented by Eqs. (6.3)–(6.6): F𝜔 = 𝜔(t−Δt) + 0.5Δt 𝜔̇ (t−Δt) − (𝜔(t) − 0.5Δt 𝜔̇ (t) )

(6.34)

F𝛿 = 𝛿(t−Δt) + 0.5Δt 𝛿̇ (t−Δt) − (𝛿(t) − 0.5Δt 𝛿̇ (t) )

(6.35)

′ ′ FE′ q = E′ q(t−Δt) + 0.5Δt Ė q(t−Δt) − (E′ q(t) − 0.5Δt Ė q(t) )

(6.36)

255

256

6 Dynamic Simulations of Power Systems

FE′ d = E′ d(t−Δt) + 0.5Δt Ė ′ d(t−Δt) − (E′ d(t) − 0.5Δt Ė ′ d(t) )

(6.37)

where, 𝜋f0 − Pe(t) − D(𝜔(t) − 𝜔0 )] [P H m(t) 𝜋f 𝜔̇ (t−Δt) = 0 [Pm(t−Δt) − Pe(t−Δt) − D(𝜔(t−Δt) − 𝜔0 )] H 𝛿̇ (t) = 𝜔(t) − 𝜔0 𝜔̇ (t) =

(6.38) (6.39) (6.40)

𝛿̇ (t−Δt) = 𝜔(t−Δt) − 𝜔0

(6.41)

Ė ′ q(t) = T ′ d0 −1 [Ef (t) − E′ q(t) + (Xd − X ′ d )Id(t) ]

(6.42)

Ė ′ q(t−Δt) = T ′ d0 −1 [Ef (t−Δt) − E′ q(t−Δt) + (Xd − X ′ d )Id(t−Δt) ]

(6.43)

Ė ′ d(t) = T ′ d0 −1 [−E′ d(t) − (Xq − X ′ q )Iq(t) ]

(6.44)

Ė ′ d(t−Δt) = T ′ d0 −1 [−E′ d(t−Δt) − (Xq − X ′ q )Iq(t−Δt) ]

(6.45)

The set of Eqs. (6.34)–(6.45) must be solved together with the equations of the whole network when including a synchronous generator dynamic model. Hence, the following linearized equation provides the computing engine with which the time-domain solution is carried out: ⎡ ΔP ⎤ ⎡ Δe ⎤ ⎢ΔQ ⎥ ⎢ Δf ⎥ ⎢ ⎥ ⎥ ]⎢ [ F J11 J12 ⎢ Δ𝜔 ⎥ ⎢ 𝜔 ⎥ (6.46) ⎢F ⎥=− ⎥ ⎢ J21 J22 ⎢ Δ𝛿 ⎥ ⎢ 𝛿 ⎥ ⎢ FE ′ q ⎥ ⎢ΔE′ q ⎥ ⎢ ⎥ ⎢ ′ ⎥ ⎣FE′ d ⎦ ⎣ΔE d ⎦ where J11

⎡ 𝜕ΔP 𝜕ΔP ⎤ ⎢ 𝜕e 𝜕f ⎥ =⎢ , 𝜕ΔQ 𝜕ΔQ ⎥ ⎢ ⎥ ⎣ 𝜕e 𝜕f ⎦

J21

⎡ 𝜕F𝜔 𝜕F𝜔 ⎤ ⎢ 𝜕e 𝜕f ⎥ ⎢ ⎥ ⎢ 𝜕F𝛿 𝜕F𝛿 ⎥ ⎢ 𝜕e 𝜕f ⎥ = ⎢ 𝜕F ′ 𝜕F ′ ⎥ , Eq⎥ ⎢ Eq ⎢ 𝜕e ⎥ 𝜕f ⎢ ⎥ ⎢ 𝜕FE′ d 𝜕FE′ d ⎥ ⎢ ⎥ 𝜕f ⎦ ⎣ 𝜕e

⎡ 𝜕ΔP ⎢ 𝜕𝜔 J12 = ⎢ 𝜕ΔQ ⎢ ⎣ 𝜕𝜔 ⎡ 𝜕F𝜔 ⎢ 𝜕𝜔 ⎢ ⎢ 𝜕F𝛿 ⎢ 𝜕𝜔 J22 = ⎢ 𝜕F ′ ⎢ Eq ⎢ 𝜕𝜔 ⎢ ⎢ 𝜕FE′ d ⎢ ⎣ 𝜕𝜔

𝜕ΔP 𝜕𝛿 𝜕ΔQ 𝜕𝛿 𝜕F𝜔 𝜕𝛿 𝜕F𝛿 𝜕𝛿 𝜕FE′ q

𝜕ΔP 𝜕E′ q 𝜕ΔQ 𝜕E′ q 𝜕F𝜔 𝜕E′ q 𝜕F𝛿 𝜕E′ q 𝜕FE′ q

𝜕𝛿 𝜕E′ q 𝜕FE′ d 𝜕FE′ d 𝜕𝛿

𝜕E′ q

𝜕ΔP ⎤ 𝜕E′ d ⎥ , 𝜕ΔQ ⎥ ⎥ 𝜕E′ d ⎦ 𝜕F𝜔 ⎤ 𝜕E′ d ⎥ ⎥ 𝜕F𝛿 ⎥ 𝜕E′ d ⎥ ⎥ 𝜕FE′ q ⎥ . 𝜕E′ d ⎥⎥ 𝜕FE′ d ⎥ ⎥ 𝜕E′ d ⎦

6.3 Time Domain Solution Philosophy

The submatrix J 11 is the conventional Jacobian matrix which comprises the ﬁrst-order partial derivatives of the nodal active and reactive power mismatches with respect to the real and imaginary parts of the nodal voltages. Likewise, J 12 contains the partial derivatives arising from the nodal active and reactive powers with respect to the corresponding variables of the synchronous generator. The matrix J 21 consists of partial derivatives of the synchronous generator’s discretized diﬀerential equations with respect to the real and imaginary parts of the nodal voltages. Lastly, J 22 is a matrix that accommodates the ﬁrst-order partial derivatives of the synchronous generator’s discretized diﬀerential equations with respect to its own state variables. 6.3.2

Benchmark Numerical Example

A typical power network is used to illustrate the time domain solution using the Newton-Raphson-based dynamic power ﬂow program. The New England test system [15] shown in Figure 6.6 is a 39-bus network containing 10 synchronous generators, Gen 8

Gen 1

37

30

26

28

25

29

27

2 18

38

17

1

Gen 9

16 3 Gen 10

21

24

15

22

23

35

39 4

14

36 Gen 6

19 5 6

20 12

7 31 8

11

34

13

Gen 4

10 9

33

Gen 2

Gen 5 32

Figure 6.6 New England test system.

Gen 3

Gen 7

257

258

6 Dynamic Simulations of Power Systems

34 transmission lines, 12 transformers and 19 loads. For the steady-state solution, the generator number 10 which is connected at node 39 is taken to be the slack generator, while the rest are assumed to be PV generators. The data and parameters for the power system are given in per-unit values on a 100 MVA base in Tables 6.1–6.4. The generators Table 6.1 Synchronous machine parameters. Bus

Pnom (p.u)

Ra (p.u)

X d (p.u)

X q (p.u)

X′ d (p.u)

X′ q (p.u)

T′ d0 (s)

T′ q0 (s)

H (s)

D (s/rad)

30

2.500

0.00

0.1000

0.069

0.0310

0.0080

10.2

0.01

42.0

0.5

31

5.732

0.00

0.2950

0.282

0.0697

0.1700

6.56

1.50

30.3

0.5

32

6.500

0.00

0.2495

0.237

0.0531

0.0876

5.70

1.50

35.80

0.5

33

6.320

0.00

0.2620

0.258

0.0430

0.1660

5.69

1.50

28.60

0.5

34

5.080

0.00

0.6700

0.620

0.1320

0.1660

5.40

0.44

26.0

0.5

35

6.500

0.00

0.2540

0.241

0.0500

0.0814

7.30

0.40

34.80

0.5

36

5.600

0.00

0.2950

0.292

0.0490

0.1860

5.66

1.50

26.40

0.5

37

5.400

0.00

0.2900

0.280

0.0570

0.0911

6.70

0.41

24.30

0.5

38

8.300

0.00

0.2106

0.205

0.0570

0.0587

4.79

1.96

34.50

0.5

39

10.1927

0.00

0.0200

0.019

0.0060

0.0080

7.00

0.70

500

0.5

Table 6.2 Transmission line parameters. Buses

R (p.u)

X (p.u)

B/2 (p.u)

Buses

R (p.u)

X (p.u)

B/2 (p.u)

1

2

0.0350

0.0410

0.345

13

14

0.0009

0.0101

0.08625

1

39

0.0100

0.0250

0.375

14

15

0.0018

0.0217

0.183

2

3

0.0130

0.0130

0.125

15

16

0.0009

0.0094

0.0855

2

25

0.0070

0.0086

0.070

16

17

0.0007

0.0089

0.0671

3

4

0.0013

0.0213

0.1107

16

19

0.0016

0.0195

0.152

3

18

0.0011

0.0133

0.1069

16

21

0.0008

0.0135

0.1274

4

5

0.0008

0.0128

0.0671

16

24

0.0003

0.0059

0.034

4

14

0.0008

0.0129

0.0691

17

18

0.0007

0.0082

0.06595

5

6

0.0002

0.0026

0.217

17

27

0.0013

0.0173

0.1608

5

8

0.0008

0.0112

0.0738

21

22

0.0008

0.0140

0.12825

6

7

0.0006

0.0092

0.0565

22

23

0.0006

0.0096

0.09225

6

11

0.0007

0.0082

0.06945

23

24

0.0022

0.0350

0.1805

7

8

0.0004

0.0046

0.039

25

26

0.0032

0.0323

0.2565

8

9

0.0023

0.0363

0.1902

26

27

0.0014

0.0147

0.1198

9

39

0.0010

0.0250

0.600

26

28

0.0043

0.0474

0.3901

10

11

0.0004

0.0043

0.03645

26

29

0.0057

0.0625

0.5145

10

13

0.0004

0.0043

0.03645

28

29

0.0014

0.0151

0.1245

6.3 Time Domain Solution Philosophy

Table 6.3 Transformer parameters. Buses

Rs (p.u)

X s (p.u)

tap

Buses

Rs (p.u)

X s (p.u)

tap

2

30

0.000

0.0181

1.025

19

33

0.0007

0.0142

1.070

6

31

0.000

0.0250

1.070

20

34

0.0009

0.0180

1.009

10

32

0.000

0.0200

1.070

22

35

0.000

0.0143

1.025

12

11

0.0016

0.0435

1.006

23

36

0.0005

0.0272

1.000

12

13

0.0016

0.0435

1.006

25

37

0.0006

0.0232

1.025

19

20

0.0007

0.0138

1.060

29

38

0.0008

0.0156

1.025

Table 6.4 Load parameters. Bus

PL (p.u)

QL (p.u)

Bus

PL (p.u)

QL (p.u)

Bus

PL (p.u)

QL (p.u)

3

3.220

0.024

18

1.580

0.300

26

1.390

0.170

4

5.000

1.840

20

6.800

1.030

27

2.810

0.755

7

2.330

0.840

21

2.740

1.150

28

2.060

0.276

8

5.220

1.760

23

2.475

0.846

29

2.835

0.269

12

0.085

0.880

24

3.086

−0.922

31

0.0912

0.046

15

3.200

1.530

25

2.240

0.472

39

16

3.294

0.323

11.040

2.500

1–5 are driven by a hydro-turbine whilst generators 6–10 are driven by steam-turbines. All synchronous generators are represented by their transient model. Each generator has an AVR with the following parameters: T A = 0.10 s, K A = 75.0, T F = 1.0 s, K F = 0.03 and T E = 0.80. The parameters for the generators employing hydro-turbine-governing systems are K = 0.06, T GV = 0.30 s and T W = 0.5 s, whereas for those using steam-turbine-governing systems are K = 0.06, T GV = 0.30 s, F HP = 0.3, F IP = 0.4, F LP = 0.3, T CH = 0.3, T RH = 7.0 and T CO = 0.2. The network is subjected to a disconnection of the transmission lines connected between buses 2–25, 2–3 and 3–4, at t = 0.1 s (which in steady-state are approximately transmitting 230 MW, 380 MW and 75 MW, respectively). The simulation runs for 5 s with a time step of 1 ms. Because of such a drastic change in the network topology, a rearrangement of power ﬂows takes place in several transmission lines of the system together with changes in the powers drawn by the loads (owing to their voltage dependency). All this brings about variations in power transfers which are accompanied by changes in the angular speed of the synchronous generators as well as in the active and reactive power ﬂows in some of the transmission lines neighbouring the tripped lines, as seen from Figures 6.7–6.9, respectively. Also, such power ﬂuctuations cause signiﬁcant voltage oscillations in several nodes of the power system, as shown in Figure 6.10. These sample results are representative of what the rest of the nodes in the power network experience following the perturbation.

259

6 Dynamic Simulations of Power Systems

Angular speed [rad/s]

377.8

ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9 ω10

377.6 377.4 377.2 377 376.8 376.6

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.7 Synchronous generators’ angular speed.

Active power flow [p.u.]

6 P14 – 4 P13 – 14 P14 – 15 P16 – 15 P16 – 17 P27 – 17 P26 – 27

5 4 3 2 1 0 –1

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.8 Active power ﬂow behaviour in selected transmission lines.

2 Reactive power flow [p.u.]

260

Q14 – 4 Q13 – 14 Q14 – 15 Q16 – 15 Q16 – 17 Q27 – 17 Q26 – 27

1.5 1 0.5 0 –0.5 –1

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

Figure 6.9 Reactive power ﬂow behaviour in selected transmission lines.

4

4.5

5

6.4 Modelling of the STATCOM for Dynamic Simulations

1.06

V4 V5 V14 V15 V16 V17 V18 V26 V27

Voltage [p.u.]

1.04 1.02 1 0.98 0.96

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.10 Voltage performance at diﬀerent nodes of the network.

6.4 Modelling of the STATCOM for Dynamic Simulations The power electronics equipment that emerged from the FACTS initiative [16] and its ramiﬁcations has a common purpose: to alleviate one or more operational problems at key locations of the power grid. A case in point are the VSC-based devices such as the STATCOM and VSC-HVDC. The STATCOM was designed to regulate reactive power, at its point of connection with the grid, in response to either fast or slow network voltage variations. Its superior performance, compared with that of the SVC, arises from the fast action of the PWM-driven insulated gate bipolar transistor (IGBT) valves, which enable the VSC to maintain a smooth voltage proﬁle at its connecting node, even in the face of rather severe disturbances in its vicinity. It was pointed out in Section 1.2.2 of Chapter 1 that the concept of a controllable voltage source behind a coupling impedance has been a popular modelling resource to represent the steady-state, fundamental frequency operation of the VSCs of the STATCOM and VSC-HVDC links [17]. More recently, it has been argued that this simple concept explains well the operation of these devices from the standpoint of the AC network but its usefulness reduces when the requirement involves the assessment of variables relating to their DC buses [18]. The situation is very much the same when looking at the dynamic regime where the standard approach has also been the use of a controllable voltage source. Aiming at alleviating such a key shortcoming, this chapter presents enhanced STATCOM and VSC-HVDC models for assessing the dynamic operating regime of large-scale power grids. Similar to the VSC model developed in Section 3.6 for the purpose of steady-state studies, the VSC model presented below uses a complex tap-changing transformer at its kernel, one where its primary and secondary sides yield, notionally speaking, the AC and DC sides of the VSC, respectively. The AC terminal of the VSC combines easily with the model of the interfacing transformer to make up both the STATCOM and HVDC models, which are elegant and yield great physical insight when employed to carry out time-domain simulations. Building on the basic concepts and structural characteristics of the STATCOM presented in Chapters 1 and 2, we now turn our attention to develop a comprehensive model for dynamic studies. As shown in Figure 6.11, the VSC station uses a DC capacitor, which

261

262

6 Dynamic Simulations of Power Systems

IDC

+ EDC –

Figure 6.11 A STATCOM and its control variables.

Control system γ ma IDCR

VvR

Vk

plays a key role in the converter dynamics. It should be recalled that in steady-state the capacitor is assumed to be fully charged, hence its voltage is kept constant at the nominal value EDCnom and the current in the capacitor is zero, ic = 0. However, disturbances on the AC network aﬀect the DC link voltage and current, pushing the capacitor into a charging/discharging state. The dynamic behaviour of the VSC is adequately captured provided the dynamics of the DC capacitor is suitably represented. For instance, to compute the voltage oscillations at the DC bus, Eqs. (6.47) and (6.48) are used. These yield information of the current I DCR , which is the actual current injected at the converter’s DC bus, and the capacitor’s current ic = −I DCR − I DC . ic = CDC IDCR =

dEDC dt

P0 EDC

(6.47) (6.48)

where the power P0 is the nodal active power injected at the DC bus, which has been previously derived in Section 3.6.3. Therefore, the following expression serves the purpose of calculating the STATCOM’s DC voltage dynamics: dEDC −I − IDC = DCR dt CDC

(6.49)

where I DC represents the DC current control variable acting upon the DC voltage. This expression enables the calculation of the voltage dynamics in the DC bus. Its capacitor value is estimated as a function of the energy stored in the capacitor: 2 Wc = 1∕2CDC EDC . At least from the conceptual point of view, the capacitor’s energy storage bears a resemblance to the kinetic energy of the rotating electrical machinery, where this energy can be further related to the inertia constant that impacts the machine’s motion equation. Taking this reasoning further, the electrostatic energy stored in the DC capacitor can be associated with an equivalent inertia constant H c by means of the following relationship: W c = H c Snom , where Snom corresponds to the rated apparent power of the VSC. By drawing a parallel between the VSC and a bank of thyristor-switched capacitors of the same rating, this time constant may be taken to be: H c ≈ 𝜔s −1 . However, H c ≈ 5 ms has been found to be a more suitable value, to avoid numerical instabilities [19].

6.4 Modelling of the STATCOM for Dynamic Simulations

Figure 6.12 DC voltage controller.

Kpedc EDC

+ +

+ – Kiedc s

IDC

IDCaux

EDCnom Figure 6.13 DC power controller for the VSC’s DC side.

Kppdc P0

+ +

+ + Kipdc s

γ γaux

0

Hence, the per-unit value of the capacitor is: CDC =

2Snom Hc 2 EDC

(6.50)

The VSC controls the voltage at the DC bus, an action carried out by controlling the DC current entering or leaving the converter, I DC . The implementation of the DC voltage controller is shown in Figure 6.12, where the error between the actual voltage EDC and EDCnom is processed through a PI controller to obtain new values of the DC current I DC at every time step. Hence, the DC voltage dynamics is largely determined by the gains K pedc and K iedc . The equations arising from the block diagram corresponding to the DC voltage controller are: dI DCaux (6.51) = Kiedc (EDC − EDCnom ) dt IDC = Kpedc (EDC − EDCnom ) + IDCaux

(6.52)

The converter keeps the capacitor charged at the required voltage level by making its output voltage lag the AC system voltage by a small angle [16]. This angular diﬀerence is computed as 𝛾 = 𝜃 vR − 𝜑, where 𝜑 is the angle of the phase-shifting transformer and 𝜃 vR = tan−1 (f vR /evR ) represents the angle of the VSC’s terminal voltage. During the dynamic regime, the power balance on the DC side of the converter, P0 = 0, is achieved by suitable control of the angle 𝛾 [20]. The PI controller responsible for this is shown in Figure 6.13. The diﬀerential equation that represents the dynamic behaviour of the power balance controller is: d𝛾aux (6.53) = Kipdc P0 dt 𝛾 = Kppdc P0 + 𝛾aux

(6.54)

The modulation index is responsible for keeping the voltage magnitude V vR =(evR 2 + f vR 2 )1/2 at the desired value; to such an end, the AC-bus voltage controller

263

264

6 Dynamic Simulations of Power Systems

Figure 6.14 AC-bus voltage controller.

nom

I1 VvR

– +

VvR0

Kma 1 + sTma

dma

+ +

ma

ma(0)

depicted in Figure 6.14 is employed. This is a ﬁrst-order controller which yields small changes in the modulation index dma by comparing the actual voltage V vR and the scheduled AC voltage V vR0 = (evR0 2 + f vR0 2 )1/2 . The modulation index increases or decreases according to the operating requirements [20]. The diﬀerential equation for the AC-bus controller is given by (6.55): d(dma ) Kma (VvR0 − VvR ) − dma (6.55) = dt Tma To guarantee the VSC operation within limits, a limit checking of the terminal current must take place, that is, I1 ≤ I1nom . Hence, the overall dynamics of the STATCOM is well captured by means of Eqs. (6.49)–(6.55). 6.4.1

Discretization and Linearization of the STATCOM Diﬀerential Equations

The STATCOM diﬀerential equations representing the DC voltage dynamics, the DC current controller and the AC voltage controller are discretized as follows: FEDC = EDC(t−Δt) + 0.5Δt Ė DC(t−Δt) − (EDC(t) − 0.5Δt Ė DC(t) ) (6.56) FIDCaux = IDCaux(t−Δt) + 0.5Δt İ DCaux(t−Δt) − (IDCaux(t) − 0.5Δt İ DCaux(t) )

(6.57)

F𝛾aux = 𝛾aux(t−Δt) + 0.5Δt 𝛾̇ aux(t−Δt) − (𝛾aux(t) − 0.5Δt 𝛾̇ aux(t) )

(6.58)

Fdma = dma(t−Δt) + 0.5Δt dṁ a(t−Δt) − (dma(t) − 0.5Δt dṁ a(t) )

(6.59)

The discretized diﬀerential equations of the VSC are appended to those of the active and reactive power balances of the network, at the AC terminal of the VSC, that is, at node vR. These equations were derived in (Section 3.6.3) but for the sake of ﬂuidity of the text, they are reproduced below: ΔPvR = −PvR − PvR,load − PvR,cal

(6.60)

ΔQvR = −QvR − QvR,load − QvR,cal

(6.61)

where 2 ) − EDC PvR = GvRvR (e2vR + fvR ×(evR [GvR0 cos 𝜙 − BvR0 sin 𝜙] + fvR [GvR0 sin 𝜙 + BvR0 cos 𝜙]) 2 ) − EDC QvR = −BvRvR (e2vR + fvR ×(−evR [GvR0 sin 𝜙 + BvR0 cos 𝜙] + fvR [GvR0 cos 𝜙 − BvR0 sin 𝜙])

(6.62) (6.63)

Eqs. (6.56)–(6.61) make up the necessary set of equations that must be solved together with the equations of the whole network, including those of the synchronous generators and their controls, to carry out dynamic power ﬂow solutions of a power system containing STATCOM equipment. To solve the nonlinear set of equations,

6.4 Modelling of the STATCOM for Dynamic Simulations

the Newton-Raphson method is employed for reliable dynamic simulations. Hence, the linearized matrix Eq. (6.64) provides the computing framework, with which the time-domain solutions are performed: ⎡ ΔPvR ⎤ ⎡ ΔevR ⎤ ⎢ΔQvR ⎥ ⎥ ⎢ ] ⎢ ΔfvR ⎥ [ ⎢F ⎥ ⎢ EDC ⎥ = − J11 J12 ⎢ ΔEDC ⎥ J21 J22 ⎢ΔIDCaux ⎥ ⎢FIDCaux ⎥ ⎢ F𝛾 ⎥ ⎢ Δ𝛾aux ⎥ ⎢ aux ⎥ ⎥ ⎢ ⎣ Fdma ⎦ ⎣ Δdma ⎦

(6.64)

where J 11 comprises the ﬁrst-order partial derivatives of the nodal active and reactive power mismatches with respect to the real and imaginary parts of the AC-side voltage. Likewise, J 12 contains the partial derivatives arising from the nodal active and reactive powers with respect to the VSC state variables. The matrix J 21 consists of partial derivatives of the VSC’s discretized diﬀerential equations with respect to AC voltages. Lastly, J 22 is a matrix that accommodates the ﬁrst-order partial derivatives of the VSC discretized diﬀerential equations with respect to its own control variables. [ 𝜕ΔPvR 𝜕ΔPvR ] [ 𝜕ΔPvR 𝜕ΔP ] 0 0 𝜕dmvR 𝜕evR 𝜕fvR 𝜕EDC a J12 = 𝜕ΔQvR J11 = 𝜕ΔQvR 𝜕ΔQvR 𝜕ΔQ 0 0 𝜕dmvR 𝜕evR 𝜕fvR 𝜕EDC a 𝜕FEDC ⎤ 𝜕FEDC ⎡ 𝜕F 𝜕F ⎡ EDC 0 0 EDC ⎤ ⎢ 𝜕E 𝜕dma ⎥ ⎥ ⎢ ⎥ ⎢ 𝜕F DC 𝜕F ⎢ 𝜕evR 𝜕fvR ⎥ IDCaux ⎢ IDCaux 0 ⎥ 0 0 ⎥ ⎢ 0 ⎥ ⎢ 𝜕E 𝜕IDCaux J21 = ⎢ 𝜕F𝛾aux 𝜕F𝛾aux ⎥ , (6.65) J22 = ⎢ 𝜕F DC 𝜕F𝛾aux 𝜕F𝛾aux ⎥ ⎥ ⎢ 𝛾aux ⎥ ⎢ 0 ⎢ 𝜕evR 𝜕fvR ⎥ ⎢ 𝜕EDC 𝜕𝛾aux 𝜕dma ⎥ ⎢ 𝜕Fdma 𝜕Fdma ⎥ ⎢ 𝜕Fdma ⎥ ⎥ ⎢ ⎥ ⎢ 0 0 0 ⎣ 𝜕evR 𝜕fvR ⎦ ⎣ 𝜕dma ⎦ where the derivatives of the mismatch powers at terminal vR with respect to the terminal voltage are presented in (Section 3.6.4). The new derivative terms corresponding to VSC’s discretized diﬀerential equations are given below in explicit form: 𝜕FEDC 𝜕evR 𝜕FEDC 𝜕fvR 𝜕FEDC 𝜕EDC

𝜕F𝛾aux

𝜕P0 𝜕evR 𝜕P −1 −1 = 0.5Δt CDC EDC 0 𝜕fvR −1 −1 = 0.5Δt CDC EDC

𝜕fvR 𝜕F𝛾aux 𝜕EDC 𝜕F𝛾aux

= [

𝜕P0 −2 − EDC P0 𝜕EDC 𝜕FEDC 𝜕P −1 −1 = 0.5Δt CDC EDC 0 𝜕dma 𝜕ma 𝜕FIDCaux = −0.5Δt Kiedc 𝜕EDC 𝜕FIDCaux = 1.0 𝜕IDCaux 𝜕F𝛾aux 𝜕P = −0.5ΔtK ipdc 0 𝜕evR 𝜕evR

−1 −1 1.0 + 0.5Δt CDC EDC

]

𝜕𝛾aux 𝜕F𝛾aux 𝜕dma 𝜕Fdma 𝜕evR 𝜕Fdma 𝜕fvR 𝜕Fdma 𝜕dma

𝜕P0 𝜕fvR 𝜕P0 = −0.5ΔtK ipdc 𝜕EDC = −0.5ΔtK ipdc

= 1.0 = −0.5ΔtK ipdc

𝜕P0 𝜕ma

−1 2 −1∕2 = 0.5Δt Kma Tma evR (e2vR + fvR ) −1 2 −1∕2 = 0.5Δt Kma Tma fvR (e2vR + fvR )

= 1.0 + 0.5ΔtT −1 ma

265

266

6 Dynamic Simulations of Power Systems

It is worth emphasizing that the dynamic simulation is started from a steady-state equilibrium point, which is obtained a priori using the conventional Newton-Raphson power ﬂow algorithm, detailed in Chapter 3, to ensure a smooth dynamic solution. 6.4.2

Numerical Example with STATCOMs

The New England test system, with suitable modiﬁcations, is shown in Figure 6.15. It incorporates STATCOMs at nodes 5 and 27 where large voltage ﬂuctuations are expected to occur following a disturbance at any of the main corridors of the electrical network. The rest of the parameters are the same as those used in the benchmark example of Section 6.3.2. The taps of the load tap changer (LTC) transformers are kept constant during the dynamic simulation. It should be noted that the time response of the servomotor that drives the tap changer is much slower than the time response of the controller that drives the modulation ratio of the VSC. Table 6.5 gives the parameters Gen 8

Gen 1

37

30

26

28

25

29

27

2 18

38

17

1

STATCOM 2

16 3 Gen 10

21

24

15

Gen 9 22

23

35

39 4

14

36 Gen 6

19 5 6 7

STATCOM 1

20

31 8

34 11

13

Gen 4

10 9

33

12

Gen 2

Gen 5 32

Gen 3

Figure 6.15 Test system used to incorporate two STATCOMs.

Gen 7

6.4 Modelling of the STATCOM for Dynamic Simulations

Table 6.5 STATCOM parameters. Bus Snom (MVA) E DC (p.u) G0 (p.u) R1 (p.u) X 1 (p.u) K pedc K iedc K ppdc

K ipdc K ma T ma

X ltc (p.u)

5

100.0

2.00

2e-3

2e-3

0.01

0.20 7.50 0.001 0.15 7.5 0.02

0.05

27

100.0

2.00

2e-3

2e-3

0.01

0.20 7.50 0.001 0.15 7.5 0.02

0.05

Table 6.6 Computed STATCOMs variables by the power ﬂow solution. Qgen (MVAR)

ma

𝝋 (∘ )

Beq (p.u)

LTC’s tap

Ploss (MW)

STATCOM 1

12.258

0.8311

1.6759

0.1100

1.0060

0.0125

STATCOM 2

35.646

0.8685

2.2568

0.3177

1.0165

0.1142

Table 6.7 Initial values of the STATCOM variables for the dynamic simulation. E DC (p.u)

IDC,aux (p.u)

𝜸 aux (rad)

dma

STATCOM 1

2.00

0.0

0.2286e-3

0.0

STATCOM 2

2.00

0.0

0.7078e-3

0.0

used in the simulation, on the STATCOM’s base, of Snom = 100 MVA. Table 6.6 presents a summary of the steady-state power ﬂow results; note that both STATCOMs inject reactive power to the grid in order to uphold their target voltages at 1.01 p.u and 1.04 p.u, respectively. During steady-state operation, the total power losses incurred by the STATCOMs stand at 0.0125 MW and 0.1142 MW, respectively. The reason for such a diﬀerence in these values lies in the quite diﬀerent currents ﬂowing through the VSCs, which stand at 0.1120 p.u and 0.3380 p.u, respectively. It is also worth recalling that the switching losses are functions of the actual operating conditions, i.e. these losses are scaled by the quadratic ratio of the actual current magnitude to the nominal current of the equipment. In this example, the rated current is 1 p.u for both VSCs. Once the steady-state power ﬂow solution is determined, the initial values of the control variables with which the dynamic simulation will be carried out are determined. These parameters are shown in Table 6.7. Notice that the following relationships hold for the steady-state regime: I DC = I DC,aux , 𝛾 = 𝛾 aux and ma = ma (0) . The power network undergoes a disturbance and the dynamic response of the STATCOMs is assessed. The transmission lines connecting buses 25–2, 2–3 and 3–4 are tripped at t = 0.1 s, with the result that several nodes in the surrounding area experience important voltage drops, as seen from the results shown in Figure 6.10, when no reactive power compensation is available. In contrast, when reactive power support is available in the form of STATCOMs, owing to their very rapid response, the voltage drops are not as severe, as shown by the results presented in Figure 6.16, where the

267

6 Dynamic Simulations of Power Systems

1 STATCOM 1 STATCOM 2

Qgen [p.u.]

0.8 0.6 0.4 0.2 0

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.16 Reactive power provided by the STATCOMs.

STATCOM 1 STATCOM 2

1.04 Voltage [p.u.]

268

1.03 1.02 1.01 1 0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.17 Voltage performance at the VSCs’ AC network nodes.

reactive power peaks reach values of about 46 MVAR and 98 MVAR, respectively. The nodal voltage at the AC terminal of the converters (low-voltage side of the LTC) changes little owing to the very fast and eﬀective control, as can be seen in Figure 6.17. Undoubtedly, the STATCOMs operation assists not only AC voltage of the converters, but also their neighbouring nodes, throughout the transient period; in the sense that the network voltages are stabilized much faster, as shown in Figure 6.18. This feature makes the STATCOM a very attractive choice to install in weak nodes of the power network. Following the disturbance, abrupt changes occur in the VSCs’ currents, disrupting the power balance on the DC side of the VSCs. This induces variations on the DC voltage, forcing the DC voltage controller to respond by modulating the DC current. This is shown in the results presented in Figures 6.19 and 6.20. It is seen that both control variables stabilize after just a few seconds of the occurrence of the perturbation. It should be noted that the DC current returns to zero once the VSCs and the whole network reach a new equilibrium point. This is an expected behaviour and fully validates the idea that the DC capacitor’s current must be zero during steady-state operation.

6.4 Modelling of the STATCOM for Dynamic Simulations

1.06

V4 V5 V14 V15 V16 V17 V18 V26 V27

Voltage [p.u.]

1.04 1.02 1 0.98 0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.18 Voltage performance at several nodes of the network.

STATCOM 1 STATCOM 2

2 EDC [p.u.]

1.995 1.99 1.985 1.98 1.975 1.97

0

0.5

1

1.5

2 2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.19 STATCOM’s DC-bus voltages.

4

× 10–3 STATCOM 1 STATCOM 2

2

IDC [p.u.]

0 –2 –4 –6 –8 –10 0

0.5

1

1.5

Figure 6.20 STATCOM’s DC current.

2

2.5 Time [s]

3

3.5

4

4.5

5

269

6 Dynamic Simulations of Power Systems

2

× 10–3 STATCOM 1 STATCOM 2

γ [rad]

1.5

1

0.5 0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.21 STATCOM’s angle 𝛾.

1 STATCOM 1 STATCOM 2

0.8 Ploss [%]

270

0.6 0.4 0.2 0

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.22 Total active power losses incurred by the STATCOMs.

The performance of the angle 𝛾 of the DC power controller is shown in Figure 6.21. The reason for the increase in 𝛾 is explained by recalling that this angle represents the angular aperture between the phase-shifting angle 𝜙 and the terminal voltage angle 𝜃 vR . As expected, the more current passes through the electronic switches, the more power losses are incurred by the VSCs. The result in Figure 6.22 proves this point rather well. The DC power controller adjusts 𝛾 during the transient period aiming at re-establishing the power balance on the VSC’s DC side. Notice that the power losses of STATCOM 2 reach almost 1% during the transient period and those of the STATCOM 1 reach approximately 0.2%. Likewise, the controls of the modulation ratios of the two STATCOMs start exerting voltage regulation just after the disturbance has occurred, as shown in Figure 6.23, whose behaviour is governed by (6.55). From this equation, it is inferred that when the terminal voltage VvR is smaller than VvR0 , the derivative of the modulation index with respect to time is positive, this being the reason for the increases in the modulation ratio

6.4 Modelling of the STATCOM for Dynamic Simulations

STATCOM 1 STATCOM 2

0.9

ma

0.88 0.86 0.84 0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.23 Dynamic behaviour of the modulation index of the STATCOMs.

STATCOM 1 (i) (ii) (iii)

2

EDC [p.u.]

EDC [p.u.]

STATCOM 2

2.01

2.005

1.995 1.99

(i) (ii) (iii)

2 1.99 1.98

1.985 0

0.5

1 Time [s]

1.5

2

1.97

0

0.5

1 Time [s]

1.5

2

Figure 6.24 DC-bus voltages for diﬀerent ratings of the capacitors.

of both STATCOMs. Notice that for a very short time, the modulation ratio of STATCOM 2 undergoes a rapid increase, reaching a value of 0.943. Following the transient event, it quickly settles down to a new steady-state value of 0.884. The internal variables of STATCOM 2 are more sensitive to the disturbance since this device is located closer to the transmission line that connects buses 17 and 18, which becomes the only path available to supply the large loads connected at nodes 3 and 18. A parametric analysis is carried out to examine more deeply the impact of the size of the capacitors on the dynamics of both the STATCOMs’ internal variables along with their reactive power injection into the power grid. It is interesting to reproduce the system’s conditions during the disturbance but altering the inertia of the capacitors. The following values are selected: (i) 5 ms, (ii) 10 ms and (iii) 20 ms. The dynamic performance of the DC voltage and modulation ratio is shown in Figures 6.24 and 6.25, respectively. Signiﬁcant diﬀerences are shown with increases in the electrostatic energy stored in the capacitors. It can be seen that the DC voltage dip decreases with increases in the equivalent inertia of the capacitor and that this is accompanied by a DC voltage overshot, which rapidly damps out. The reactive power generated by the STATCOM, for diﬀerent values of the capacitor’s inertia H c , is presented in Figure 6.26. For this parametric analysis, the gains of the STATCOM control loops have not been altered.

271

6 Dynamic Simulations of Power Systems

STATCOM 1

STATCOM 2

0.845

0.89

0.84

0.88

0.835 0.83

(i) (ii) (iii)

ma

ma

0.9

(i) (ii) (iii)

0.85

0.87 0

1

2 3 Time [s]

4

5

0

1

2 3 Time [s]

4

5

Figure 6.25 Modulation ratios for diﬀerent ratings of the capacitors.

STATCOM 1 (i) (ii) (iii)

(i) (ii) (iii)

0.8 Qgen [p.u.]

0.3 0.2 0.1

STATCOM 2

1

0.4 Qgen [p.u.]

272

0.6 0.4 0.2

0

1

2 3 Time [s]

4

5

0

0

1

2 3 Time [s]

4

5

Figure 6.26 Reactive power provided by the VSCs for diﬀerent ratings of the capacitors.

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations If two VSC converters are interconnected on their corresponding DC buses, using a cable with a resistance RDC , a VSC-HVDC system forms; this is termed point-to-point conﬁguration. If the cable does not exist, this may be assumed to be a cable of zero resistance and the VSC-HVDC conﬁguration is termed back-to-back. Electric power is taken from one point of the AC network, converted to DC power in the rectiﬁer station, transmitted through the DC cable and then converted back to AC power in the inverter station and injected into the receiving AC network. In addition to transferring power in DC form, the VSC-HVDC system is capable of supplying/absorbing reactive power at both AC networks, thus providing independent dynamic voltage control. Steady-state and dynamic operating regimes are both ﬁelds that need to be covered to the satisfaction of the network’s planners and operators, therefore a wide range of power systems-oriented VSC-HVDC models is required, bearing in mind the main operational features of the VSC-HVDC equipment. In large-scale power system studies, the tendency has been to keep the HVDC models as simple as possible in order to keep simulation times and memory requirements at manageable levels. Hence, representing the VSC stations of the VSC-HVDC link as controllable voltage sources has been a popular option. This reduces modelling complexity but sacriﬁces the acquisition of important VSC parameters since some internal variables may not be readily available.

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

Control system maR Vk

γR

IDCI maI IDCR

VvR CDC

Rectifier

LDC IRx

RDC Rx

LDC Ix

IIx

DC link

IDCI VvI

Vm

CDC

Inverter

Figure 6.27 Dynamic model of the VSC-HVDC link with control variables.

It has been shown recently that more realistic behaviours of VSC-HVDC systems may be attained by using more advanced models, which are also useful for large-scale power networks. The new models of VSC-HVDC links are made up of basic electric circuit elements, which are suitably combined to represent the main operational features of the VSC-HVDC links during steady-state and dynamic regimes [21]. The most relevant operational parameters of the VSCs become explicitly available and the dynamic performance of the VSC-HVDC model encapsulates very well the behaviour of the AC and DC circuits of the actual equipment. Figure 6.27 depicts a schematic representation of the VSC-HVDC link together with its controls. This model synthesizes rather well the four degrees of freedom found in actual VSC-HVDC installations: simultaneous voltage support on its two AC terminals, DC voltage control at the inverter and DC power regulation at the rectiﬁer end. The capacitors’ and inductors’ dynamics play a crucial role in the operation of the HVDC link during voltage and power variations in the external AC networks. In this application, the DC voltage control is exerted to preserve stable operation of the DC link. The converter acting as inverter is usually tasked with this responsibility. The following relationships hold at the capacitors’ nodes: icR = −I DCR − I Rx and icI = −I DCI − I Ix , where icR and icI are the capacitor’s current, I DCR and I DCI are the currents leaving the rectiﬁer’s DC bus and the inverter’s DC bus, respectively. Combining these current relationships with those of the capacitor’s current ic = C DC dE/dt and inductor’s voltage Ei = LDC dI/dt, the ensuing diﬀerential equations yields the voltage and current dynamics in the DC link: − IRx dI Rx − EDCRx −I E dEDCR , = DCR = DCR dt CDC dt LDC −IDCI − IIx dI Ix EDCI − EDCIx dEDCI , = = dt CDC dt LDC

(6.66) (6.67)

−1 where IDCR = P0R EDCR and the power P0R has been derived in (Section 3.8.1). The per-unit value of the capacitance is estimated using (6.50) and an equivalent inertia constant H c representing the electrostatic energy stored in the capacitor. A similar expression may be derived for the per-unit value of the inductor through an inertia constant that accounts for the electromagnetic energy stored in the inductor:

LDC =

2Snom Hi 2 IDC

(6.68)

273

274

6 Dynamic Simulations of Power Systems

Figure 6.28 VSC-HVDC dynamic controller for the DC voltage. Source: ©Elsevier, 2015.

Kpedc EDCI

+ +

+ – EDCInom

Kiedc s

IDCI IDCIaux

where H i and I DC stand for the ﬁctitious inertia constant and nominal current of the inductor. It may be noticed that the current balance of the back-to-back converter given by (6.66)–(6.67) is akin to the power balance inside the HVDC link for steady-state operation – when the time derivative is zero. When the current/power balance becomes disrupted, voltage variations appear in the DC link and this would need to be brought under quick control. The dynamic control of the DC voltage may be carried out by using the DC current entering the inverter, I DCI , as the control variable, as shown in Figure 6.28. The error between the actual voltage EDCI and the reference EDCInom is passed through a PI controller, with gains K pedc and K iedc , to obtain new values of the DC current I DCI . The diﬀerential and algebraic equations arising from the DC voltage dynamic controller are: dI DCIaux = Kiedc (EDCI − EDCInom ) dt

(6.69)

IDCI = Kpedc (EDCI − EDCInom ) + IDCIaux

(6.70)

At the same time, the rectiﬁer must ensure that its output active power is kept at the scheduled value Psch . From Figure 6.27, it can be inferred that the power entering the inverter station is the scheduled power Psch minus the power loss incurred in the DC cable RDC . The DC power ﬂowing from the node Rx towards the node Ix may be expressed in terms of DC voltages, as follows: 2 PDCR = (EDCRx − EDCRx EDCIx )R−1 DC

(6.71)

Likewise, the power ﬂowing from the inverter towards the rectiﬁer through the DC cable may be expressed as: 2 PDCI = (EDCIx − EDCIx EDCRx )R−1 DC

(6.72)

Notice that during steady-state, it is true that EDCR = EDCRx and EDCI = EDCIx , since the currents taken by both capacitors is zero. The angular aperture 𝛾 R between the phase-shifting angle of the rectiﬁer 𝜑R and the voltage angle 𝜃 vR = tan−1 ( f vR /evR ) is related to the power exchange taking place between the network and the rectiﬁer’s DC bus at any point in time. Hence, the angular diﬀerence 𝛾 R = 𝜃 vR − 𝜑R is the parameter that requires regulation with the aim of achieving the scheduled active power transfer Psch in the DC link. The pursued power balance on the DC side is P0R + Psch = 0, as shown in Figure 6.29 [21].

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

Figure 6.29 DC power transfer controller of the VSC-HVDC link model. Source: ©Elsevier, 2015.

Kppdc Psch

γR

+ +

+ + Kipdc s

γRaux

P0R

The equations enabling the calculation of the dynamic behaviour of the DC power transmission are: d𝛾Raux = Kipdc (Psch + P0R ) dt 𝛾R = Kppdc (Psch + P0R ) + 𝛾Raux

(6.73) (6.74)

On the other hand, the AC voltage dynamic control of the VSC-HVDC link requires two control loops. The modulation indices maI and maR are responsible for controlling the AC voltage magnitude at the receiving and sending ends of the VSC-HVDC at the scheduled values, V vI0 and V vR0 . The ﬁrst-order control blocks shown in Figure 6.30 are employed to provide reactive power support to the network. Each control is designed in such a way that the modulation indices maI and maR are readjusted at every time step according to the diﬀerence between the scheduled voltage magnitudes and the actual voltage at the nodes where the converters are connected, V vI = (evI 2 + f vI 2 )1/2 and V vR = (evR 2 + f vR 2 )1/2 . The diﬀerential equations for the AC-bus voltage controllers are: d(dmaR ) KmaR (VvR0 − VvR ) − dmaR = dt TmaR

(6.75)

d(dmaI ) KmaI (VvI0 − VvI ) − dmaI = dt TmaI

(6.76)

Figure 6.30 AC bus voltage controllers. Source: ©Elsevier, 2015. (a) Rectiﬁer station and (b) inverter station.

nom

I1R VvR

– +

VvR0

KmaR 1 + sTmaR

dmaR

+ +

maR

(0)

maR

(a) nom

I1I VvI

– +

VvI0

KmaI 1 + sTmaI

dmaI

+ +

(0)

(b)

maI

maI

275

276

6 Dynamic Simulations of Power Systems

6.5.1 Discretization and Linearization of the Diﬀerential Equations of the VSC-HVDC The diﬀerential equations arising from the VSC-HVDC controllers are discretized and arranged in a suitable form to facilitate their accommodation into the expanded Jacobian matrix [21]. They are: FE = EDCR(t−Δt) + 0.5Δt Ė DCR(t−Δt) − (EDCR(t) − 0.5Δt Ė DCR(t) ) (6.77) DCR

FEDCI = EDCI(t−Δt) + 0.5Δt Ė DCI(t−Δt) − (EDCI(t) − 0.5Δt Ė DCI(t) ) FI = IRx(t−Δt) + 0.5Δt İ Rx(t−Δt) − (IRx(t) − 0.5Δt İ Rx(t) ) Rx

FIIx = IIx(t−Δt) + 0.5Δt İ Ix(t−Δt) − (IIx(t) − 0.5Δt İ Ix(t) ) FIDCIaux = IDCIaux(t−Δt) + 0.5Δt İ DCIaux(t−Δt) − (IDCIaux(t) − 0.5Δt İ DCIaux(t) )

(6.78) (6.79) (6.80) (6.81)

F𝛾Raux = 𝛾Raux(t−Δt) + 0.5Δt 𝛾̇ Raux(t−Δt) − (𝛾Raux(t) − 0.5Δt 𝛾̇ Raux(t) )

(6.82)

FdmaR = dmaR(t−Δt) + 0.5Δt dṁ aR(t−Δt) − (dmaR(t) − 0.5Δt dṁ aR(t) )

(6.83)

FdmaI = dmaI(t−Δt) + 0.5Δt dṁ aI(t−Δt) − (dmaI(t) − 0.5Δt dṁ aI(t) )

(6.84)

The expressions (6.77)–(6.84) establish the dynamic behaviour of the VSC-HVDC model. The ﬁrst ﬁve diﬀerential equations capture the DC voltage and current performance of the DC link when the energy balance is disrupted. Likewise, the equation involving the angular aperture 𝛾 R (6.82) deals with the power unbalance present in the DC link. On the other hand, the expressions given in (6.83)–(6.84) enable the computation of the new values of the modulation indices with which the AC voltage set points are achieved. In order to link the control variables of the VSC-HVDC with the state variables of the network, at nodes vR and vI, the algebraic power mismatch Eqs. (6.85)–(6.88) are used. The power balance equations at the internal nodes Rx, Ix and 0I, given by the algebraic relationships (6.89)–(6.91), complete the dynamic model: ΔPvR = −PvR − PvR,load − PvR,cal

(6.85)

ΔQvR = −QvR − QvR,load − QvR,cal

(6.86)

ΔPvI = −PvI − PvI,load − PvI,cal

(6.87)

ΔQvI = −QvI − QvI,load − QvI,cal

(6.88)

ΔPRx = −EDCRx IRx + PDCR

(6.89)

ΔPIx = −EDCIx IIx + PDCI

(6.90)

ΔP0I = EDCI IDCI − P0I

(6.91)

The active and reactive power injections, derived in (Section 3.8.1), are restated below: 2 PvR = GvRvR (e2vR + fvR )

− EDCR (evR [GvR0 cos 𝜑R + BvR0 sin 𝜑R ] − fvR [GvR0 sin 𝜑R − BvR0 cos 𝜑R ]) (6.92) 2 ) QvR = −BvRvR (e2vR + fvR

− EDCR (evR [GvR0 sin 𝜑R − BvR0 cos 𝜑R ] + fvR [GvR0 cos 𝜑R + BvR0 sin 𝜑R ]) (6.93) 2 P0R = (GswR + G00R )EDCR

− EDCR (evR [GvR0 cos 𝜑R + BvR0 sin 𝜑R ] + fvR [GvR0 sin 𝜑R − BvR0 cos 𝜑R ]) (6.94)

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

PvI = GvIvI (e2vI + fvI2 ) − EDCI (evI [GvI0 cos 𝜑I − BvI0 sin 𝜑I ] + fvI [GvI0 sin 𝜑I + BvI0 cos 𝜑I ]) QvI =

−BvIvI (e2vI

+

(6.95)

fvI2 )

− EDCI (−evI [GvI0 sin 𝜑I + BvI0 cos 𝜑I ] + fvI [GvI0 cos 𝜑I − BvI0 sin 𝜑I ]) (6.96) 2 P0I = (GswI + G00I )EDCI

− EDCI (evI [GvI0 cos 𝜑I + BvI0 sin 𝜑I ] + fvI [GvI0 sin 𝜑I − BvI0 cos 𝜑I ])

(6.97)

Eqs. (6.77)–(6.91) constitute the set of mismatch equations that must be assembled together with the equations of the whole network, synchronous generators and their corresponding controllers. The linearized form of the VSC-HVDC dynamic model is given by: ] [ J J (6.98) ΔF = − 11 12 Δz J21 J22 ]T [ ΔPvR ΔQvR ΔPvI ΔQvI ΔPRx ΔPIx ΔP0I · · · ΔF = · · · FEDCR FEDCI FIRx FIIx FIDCIaux F𝛾Raux FdmaR FdmaI [ ]T ΔevR ΔfvR ΔevI ΔfvI ΔEDCRx ΔEDCIx Δ𝜑I ··· Δz = · · · ΔEDCR ΔEDCI ΔIRx ΔIIx ΔIDCIaux Δ𝛾Raux ΔdmaR ΔdmaI where J 11 comprises the ﬁrst-order partial derivatives of the network power mismatch equations and VSC-HVDC mismatch equations with respect to the network’s state variables and the state variables of the VSC-HVDC. Likewise, J 12 contains the ﬁrst-order partial derivatives arising from the algebraic mismatch equations with respect to the control variables of the VSC-HVDC link. The matrix J 21 consists of partial derivatives of the VSC-HVDC’s discretized diﬀerential equations with respect to the AC voltages, the phase-shifting angle 𝜑I and the DC voltage EDCR . Lastly, J 22 is a matrix that accommodates the ﬁrst-order partial derivatives of the VSC-HVDC’s discretized diﬀerential equations with respect to their own control variables. ⎡ 𝜕ΔPvR 𝜕ΔPvR 0 0 0 0 0 ⎤ ⎢ 𝜕evR ⎥ 𝜕fvR ⎢ ⎥ ⎢ 𝜕ΔQR 𝜕ΔQvR ⎥ 0 0 0 0 0 ⎥ ⎢ 𝜕e 𝜕fvR ⎢ vR ⎥ ⎢ 𝜕ΔPvI 𝜕ΔPvI 𝜕ΔPvI ⎥ 0 0 0 ⎢ 0 ⎥ 𝜕evI 𝜕fvI 𝜕𝜑I ⎥ ⎢ ⎢ 𝜕ΔQvI 𝜕ΔQvI 𝜕ΔQvI ⎥ ⎥, 0 0 0 J11 = ⎢ 0 ⎢ 𝜕evI 𝜕fvI 𝜕𝜑I ⎥ ⎢ ⎥ 𝜕ΔPRx 𝜕ΔPRx ⎢ 0 ⎥ 0 0 0 0 ⎢ ⎥ 𝜕EDCRx 𝜕EDCIx ⎢ ⎥ 𝜕ΔPIx 𝜕ΔPIx ⎢ ⎥ 0 ⎥ 0 0 0 ⎢ 0 𝜕E 𝜕E DCRx DCIx ⎢ ⎥ ⎢ 𝜕ΔP0I 𝜕ΔP0I 𝜕ΔP0I ⎥ ⎢ 0 ⎥ 0 0 0 ⎣ 𝜕evI 𝜕fvI 𝜕𝜑I ⎦

277

278

6 Dynamic Simulations of Power Systems

J12

J21

⎡ 𝜕ΔPvR ⎢ 𝜕EDCR ⎢ ⎢ 𝜕ΔQvR ⎢ 𝜕E ⎢ DCR ⎢ ⎢ 0 ⎢ ⎢ =⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣

0

0

0

0 0

𝜕ΔPvR 𝜕dmaR

0

0

0

0 0

𝜕ΔQR 𝜕dmaR

𝜕ΔPvI 𝜕EDCI

0

0

0 0

0

𝜕ΔQvI 𝜕EDCI

0

0

0 0

0

0

𝜕ΔPRx 𝜕IRx

0

0 0

0

0

0

𝜕ΔPIx 0 0 𝜕IIx

0

𝜕ΔP0I 𝜕EDCI

0

0

0 0

0

⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ 𝜕ΔPvI ⎥ ⎥ 𝜕dmaI ⎥ 𝜕ΔQvI ⎥ ⎥ 𝜕dmaI ⎥ ⎥ 0 ⎥⎥ ⎥ ⎥ 0 ⎥ ⎥ 𝜕ΔP0I ⎥ ⎥ 𝜕dmaI ⎦ 0

⎡ 𝜕FEDCR 𝜕FEDCR 0 0 0 0 0 ⎤⎥ ⎢ 𝜕evR 𝜕fvR ⎢ ⎥ 𝜕FEDCI 𝜕FEDCI 𝜕FEDCI ⎥ ⎢ 0 0 0 ⎢ 0 ⎥ 𝜕evI 𝜕fvI 𝜕𝜑I ⎥ ⎢ ⎢ ⎥ 𝜕FIRx ⎢ 0 0 0 0 0 0 ⎥ ⎢ ⎥ 𝜕EDCRx ⎢ ⎥ 𝜕FIIx ⎢ ⎥ 0 ⎥ 0 0 0 0 ⎢ 0 𝜕E =⎢ DCIx ⎥, ⎢ 0 ⎥ 0 0 0 0 0 0 ⎢ ⎥ ⎢ 𝜕F ⎥ 𝛾Raux 𝜕F𝛾Raux ⎢ 0 0 0 0 0 ⎥ ⎢ 𝜕evR ⎥ 𝜕fvR ⎢ ⎥ ⎢ 𝜕FdmaR 𝜕FdmaR ⎥ 0 0 0 0 0 ⎥ ⎢ 𝜕e 𝜕f vR ⎢ vR ⎥ ⎢ ⎥ 𝜕FdmaI 𝜕FdmaI ⎢ 0 ⎥ 0 0 0 0 ⎣ ⎦ 𝜕evI 𝜕fvI

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

J22

⎡ 𝜕FEDCR ⎢ 𝜕E ⎢ DCR ⎢ ⎢ 0 ⎢ ⎢ ⎢ 𝜕FIRx ⎢ 𝜕E ⎢ DCR ⎢ ⎢ 0 ⎢ =⎢ ⎢ ⎢ 0 ⎢ ⎢ 𝜕F𝛾Raux ⎢ ⎢ 𝜕EDCR ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎣

0 𝜕FEDCI 𝜕EDCI 0 𝜕FIIx 𝜕EDCI 𝜕FIDCIaux

𝜕FEDCR 𝜕IRx 0 𝜕FIRx 𝜕IRx 0

0

𝜕FEDCR

0

0

𝜕dmaR

𝜕IIx

0

0

0

0

0

0

0

0

0

0

𝜕FEDCI

𝜕FIIx 𝜕IIx

𝜕FIDCIaux

𝜕EDCI

0

0

𝜕IDCIaux

0

0

0

0

0

0

0

0

0

0

0

𝜕dmaR

0

0

0

0

0

0

𝜕F𝛾Raux 𝜕F𝛾Raux 𝜕𝛾Raux 𝜕dmaR 𝜕FdmaR

⎤ ⎥ ⎥ 𝜕FEDCI ⎥ ⎥ 𝜕dmaI ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥⎥ ⎥ 𝜕FdmaI ⎥ ⎥ 𝜕dmaI ⎦ 0

(6.99)

It can be seen that the derivative terms corresponding to the two converters working in a coordinated fashion as part of the HVDC link are not diﬀerent from those corresponding to the VSC operating as a STATCOM. Hence, they can be found by simply using subscripts vR or vI in the equation already available for the STATCOM. However, two new algebraic equations have been added to link the DC voltages and currents between the two converters, one diﬀerential equation to capture the inverter’s modulation index dynamics and one more relating to the controller responsible for regulating the angle 𝛾 R to suitably guarantee the correct transfer of power. The additional derivative terms arising from both converters acting as a single VSC-HVDC system are given below: 𝜕ΔPRx = −IRx + (2EDCRx − EDCIx )R−1 DC 𝜕EDCRx

𝜕FIIx

𝜕FIIx

𝜕ΔPRx = −EDCRx 𝜕IRx

= −L−1 DC EDCIx 𝜕EDCI 𝜕EDCIx 𝜕FIRx 𝜕FIIx = = 1.0 𝜕IIx 𝜕IRx

𝜕ΔPRx = −EDCRx R−1 DC 𝜕EDCIx

𝜕ΔP0I 𝜕P = − 0I 𝜕evI 𝜕evI

=−

279

280

6 Dynamic Simulations of Power Systems

𝜕ΔPIx = −IIx + (2EDCIx − EDCRx )R−1 DC 𝜕EDCIx

𝜕ΔP0I 𝜕P = − 0I 𝜕fvI 𝜕fvI

𝜕ΔPIx = −EDCIx 𝜕IIx

𝜕ΔP0I 𝜕P = − 0I 𝜕𝜑I 𝜕𝜑I

𝜕ΔPIx = −EDCIx R−1 DC 𝜕EDCRx 𝜕FIRx 𝜕FIRx =− = −L−1 DC EDCRx 𝜕EDCR 𝜕EDCRx

𝜕ΔP0I 𝜕P0I = IDCI − 𝜕EDCI 𝜕EDCI 𝜕ΔP0I 𝜕P = − 0I 𝜕dmaI 𝜕maI

The steady-state condition needed to start the dynamic simulation is calculated using the conventional Newton-Raphson power ﬂow algorithm including the VSC-HVDC link model, derived in Chapter 3. Such a solution provides accurate initial conditions to ensure reliable dynamic simulations, with a smooth start. 6.5.2

Validation of the VSC-HVDC Link Model

The range of validity of the response of the VSC-HVDC link model is established in this section by carrying out a comparison against the widely used electromagnetic transient (EMT)-type simulation software Simulink. Both types of simulation tools enable dynamic assessments of electrical power networks but they take a fundamentally diﬀerent solution approach. Simulink represents every component of the power grid by means of RLC circuits and their corresponding diﬀerential equations require discretization at rather small time steps, in the order of microseconds, to ensure a stable numerical solution. Conversely, the solution of the RMS-type model requires only ‘one phase’ of the network, i.e. positive sequence network, and uses fundamental frequency phasors of voltages and currents as opposed to the three-phase representation of the network and instantaneous waveforms of voltages and currents used in EMT-type solution approaches such as Simulink. The VSC-HVDC model comparison is carried out using a simple power system comprising two independent AC networks (2000 MVA, 230 kV, 50 Hz), which are interconnected through a VSC-HVDC link (200 MVA, ±100 kV DC) with a DC cable length of 75 km, as shown in Figure 6.31. Both converter stations comprise a step-down transformer, AC ﬁlter, converter reactor, DC capacitors, DC ﬁlters and ﬁxed oﬀ-nominal transformers’ tap positions. The model of the power system including the VSC-HVDC link together with its parameters can be found in the section of ‘demos’ in Simulink under the title: VSC-Based HVDC Transmission System (Detailed Model). The parameters of the VSC-HVDC model described in this chapter are shown in Table 6.8. To ensure a reliable numerical solution, the EMT-type simulation package discretizes the power system and the control system 2000 MVA 230 kV, 50 Hz

Vk

LTC

VvR

Source 200 MVA impedance 230:100 kV

Ldc

Phase reactor AC filter 40 MVAr

Rdc

Cdc Rectifier

Ldc

Phase reactor

Cdc DC link

Inverter

AC filter 40 MVAr

VvI

LTC

Vm

2000 MVA 230 kV, 50 Hz

Source 200 MVA 100:230 kV impedance

Figure 6.31 Test system used to validate the VSC-HVDC model. Source: ©Elsevier, 2015.

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

Table 6.8 VSC-HVDC parameters. Snom (p.u)

Psch (p.u)

RDC (p.u)

EDCI (p.u)

G0I , G0R (p.u)

R1I , R1R (p.u)

2.0

2.0

0.02135

2.00

4e-3

1e-3 X ltc (p.u)

X 1I , X 1R (p.u)

RfI , RfR (p.u)

X fI , X fR (p.u)

BfI , BfR (p.u)

Rltc (p.u)

0.5e-3

7.5e-4

0.075

0.40

2.5e-3

0.075

H c , H i (s)

K pedc

K iedc

K ppdc , K ipdc

K maR , K maI

T maR , T maI

0.014

0.60

35.0

0.0, 5.0

25.0

0.02

Simulink model DC voltage [p.u.]

2.2

EdcRx EdcIx EdcRx EdcIx

Step change in DC power

2.1 2 1.9 1.8

RMS model

1.7 0

0.5

Step change in DC voltage 1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.32 DC voltage performance for the RMS-type and Simulink models. Source: ©Elsevier, 2015.

equations with a sample time of 7.406 μs and 74.06 μs, respectively. This contrasts with the RMS-type model, where an integration step of 1 ms is used. Initially the rectiﬁer station is set to control the active power transmission at Psch = 200 MW (2 p.u.). The inverter is responsible for controlling the DC voltage at EdcInom = 200 kV (2 p.u.). The rectiﬁer and inverter stations are set to comply with a ﬁxed reactive power command of 0 p.u. and −0.1 p.u., respectively. In order to reach the steady-state equilibrium point in Simulink, the simulation is run up to t = 1 s. At this point, the active power transmission is reduced from 200 MW to 100 MW, that is, a −50% step is applied to the reference DC power. Furthermore, at t = 3 s, a step change of −5% is applied to the reference DC voltage of the inverter, i.e. the DC voltage is decreased from 2 p.u to 1.9 p.u. The DC voltages corresponding to cases where step changes in the reference DC power and DC voltage are applied are shown in Figure 6.32. As expected, some diﬀerences may be seen in these results, owing to the two very diﬀerent solution techniques. Nevertheless, the dynamic performance of the DC voltages of the RMS-type model follows well the dynamic pattern obtained by the switching-based HVDC model in Simulink. The diﬀerence between the two solution approaches at the start of the simulation (0.5 s of the simulation) is explained by the very diﬀerent manner in which the two simulations are initialized: the RMS model of the VSC-HVDC system uses accurate starting conditions furnished by a power ﬂow solution whereas the Simulink model starts from

281

6 Dynamic Simulations of Power Systems

2.5 Simulink model RMS model

Step change in DC power DC power [p.u.]

282

2 Step change in DC voltage

1.5 1 0.5

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.33 DC power performance for the RMS-type and Simulink models. Source: ©Elsevier, 2015.

its customary zero initial condition, i.e. the currents and voltages of the inductors and capacitors, respectively, are set to zero at t = 0 s. From the physical vantage, this is akin to assuming that the system was in a de-energized state and that it is energized at a time 0+ . Conversely, the RMS solution assumes that the system and the VSC-HVDC are operating under normal steady-state. Similar conclusions can be drawn when analyzing the dynamic response of the DC power following the application of the step changes in DC power and DC voltage, as shown in Figure 6.33. As for the change in the DC power reference, it can be seen that the power stabilizes in no more than 0.5 seconds; this shows the rather quick response and robustness aﬀorded by the dynamic controls of the VSC-HVDC link even in the event of a drastic change in the transmitted DC power. On the other hand, the step change in the DC voltage reference causes momentary power ﬂow oscillations in the DC link which are also damped out quite rapidly. The dynamic behaviour of the modulation indices of the converters is depicted in Figure 6.34. The step change in the DC power reference yields a very abrupt variation in the modulation indices; the dynamic performance of the modulation indices as calculated by both Simulink and the RMS-type VSC-HVDC model follows the same trend although an exact match was not expected. After the ﬁrst disturbance, a steady-state error of 0.87% and 1.67% is obtained for the modulation indices of the rectiﬁer and the inverter, respectively. Similarly, once the oscillations due to the step change in the reference DC voltage have been damped out, the diﬀerences in the modulation indices stand at 0.07% and 1.16%, respectively. These variations in the modulation indices may be explained by the very diﬀerent modelling and solution approaches employed by the two simulation techniques. For the sake of comparison, Table 6.9 shows the VSC-HVDC results as obtained by the RMS-type model and the Simulink model at diﬀerent points in time of simulation. Table 6.9 also shows the computing times required to simulate the test system using both the RMS-type VSC-HVDC model and the EMT-type simulation tool Simulink, with the former model being approximately nine times faster than the EMT simulation. The very signiﬁcant saving in computing time makes the RMS-type VSC-HVDC link

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

Modulation index

1.1 1

maR maI maR mI

Step change in DC power

RMS model

0.9 0.8 Step change in DC voltage

Simulink model 0.7

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.34 Modulation indices performance for the RMS-type and Simulink models. Source: ©Elsevier, 2015. Table 6.9 VSC-HVDC variables for the RMS-type model and Simulink model. Time

Developed RMS-type model

Simulink model

E DCR

E DCI

maR

maI

− PDCI

E DCR

E DCI

maR

maI

− PDCI

t = 1− s

2.0211

2.000

0.8553

0.8296

1.9791

2.0207

1.9985

0.8499

0.8301

1.9795

t = 3− s

2.0106

2.000

0.8389

0.8172

0.9947

2.0103

1.9998

0.8476

0.8005

0.9954

t = 5s

1.9112

1.900

0.9329

0.8711

0.9942

1.9113

1.8990

0.9336

0.8827

0.9909

Computing time: 19.76 seconds

Computing time: 180.39 seconds

model a recommendable option in cases of large-scale power system simulations, particularly in studies that require long simulation times such as those involving synchronous generators’ frequency variations and long-term voltage stability issues. 6.5.3

Numerical Example with an Embedded VSC-HVDC Link

The New England test system is modiﬁed, as shown in Figure 6.35, to illustrate the performance of the VSC-HVDC model, with the parameters shown in Table 6.10, when the network is subjected to a disturbance. The rest of the parameters correspond to those of the original case. The transmission line connecting nodes 4 and 14 is replaced by a VSC-HVDC link. The DC cable resistance is assumed to be of 0.08% (100 MVA base), which is the same as that of the replaced transmission line. The rectiﬁer and inverter stations exert voltage control at their respective AC terminals at V vR = 1.01 p.u and V vI = 1.03 p.u, respectively. For the steady-state condition, the voltage magnitudes of the high-voltage sides of the LTC transformers, which correspond to nodes 4 and 14, are held ﬁxed at the same voltage levels as those of the converters’ terminals, V vR and V vI . Under these conditions, the LTCs’ taps are computed using the steady-state power

283

284

6 Dynamic Simulations of Power Systems

Gen 8

Gen 1

37

30

26

29

28

25 27

2 18

38

17

1

Gen 9

16 3 Gen 10 Psch 39 4

22

23

Pk

Pm

Qk

Qm Rectifier

21

24

15

Inverter

14

35

36 Gen 6

19

5 6

20

7

33

Gen 7

12 31

34

8

13

11

Gen 4

10 9

Gen 2

Gen 5 32

Gen 3

Figure 6.35 Modiﬁed New England test system to incorporate a VSC-HVDC link.

Table 6.10 VSC-HVDC parameters. Snom (p.u)

Psch (p.u)

RDC (p.u)

EDCI (p.u)

G0I , G0R (p.u)

R1I , R1R (p.u)

3.0

1.0

0.0008

2.00

6e-3

6.66e-4

X 1I , X 1R (p.u)

RﬁltI , RﬁltR (p.u)

X ﬁltI , XﬁltR (p.u)

BﬁltI , BﬁltR (p.u)

Rltc (p.u)

X ltc (p.u)

3.33e-3

0.0

0.02

0.30

0.0

0.01667

H c , H i (s)

K pedc

K iedc

K ppdc , K ipdc

K maI , K maR

T maI , T maR

15e-3

0.05

1.00

0.002, 0.075

25.0

0.02

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

ﬂow algorithm, described in Chapter 3, and their values are kept constant during the dynamic solution. In addition to providing reactive power control, the HVDC link performs power regulation at the DC bus of the rectiﬁer station, at Psch = 100 MW. This implies that active power is drawn from node 4 and injected into node 14. During steady-state operation, the rectiﬁer station delivers 153.291 MVAR to the network to uphold its target voltage magnitude with a modulation index of 0.8484, whereas the inverter station operates with a modulation index of 0.8412, injecting 27.3 MVAR into the grid. In the case of the power ﬂowing through the HVDC link, the active power entering the rectiﬁer station stands at 100.869 MW and the active power leaving the inverter station takes a value of 99.667 MW. The diﬀerence between these two powers is the total power loss in the HVDC system, including the power loss in the DC cable. Taking as a reference the nominal apparent power of each converter Snom , the total power losses stand at 0.4%, with 0.29% corresponding to the rectiﬁer station and 0.104% to the inverter station, while the power loss produced by Joule’s eﬀect in the DC cable stands at 0.006%. It should be recalled that its magnitude is dependent on the length of the DC transmission line. Table 6.11 shows the main VSC-HVDC results, as given by the steady-state power ﬂow solution and used to start the dynamic simulation. Using the information in Table 6.11, it is a straightforward matter to proceed to the calculation of the initial values of the control variables taking part in the dynamics of the VSC-HVDC link, as shown in Table 6.12. These parameter values are employed to initialize the dynamic simulation where the modiﬁed test network is assumed to undergo the same disturbance as the original network, that is, the transmission lines connecting nodes 25-2, 2-3 and 3-4 are tripped at t = 0.1 s. Figure 6.36 shows the voltage magnitudes at key nodes of the network following the disturbance. During the transient period, the target voltage set point is achieved very quickly by the action of the AC-bus voltage controllers that regulate the converters’ modulation indices maR and maI , as shown in Figure 6.37. The prompt action of both controllers leads to a rapid reactive power injection at both converters’ AC terminal, as can be seen in Figure 6.38. This results in a very eﬀective damping of the voltage oscillations, enabling a smooth voltage recovery throughout the grid. The disturbance in the AC system disrupts the power balance in the DC link and the voltage dips that take place at both converters’ AC terminals reduce the power being Table 6.11 VSC-HVDC results given by the power ﬂow solution. Converter Pk , Pm (MW) Qk , Qm (MVAR) E DC (p.u.) ma

𝝋 (∘ )

Rectiﬁer

−100.869

153.291

2.0004

0.8484 −3.7380

Inverter

99.667

27.300

2.0000

0.8412

Beq (p.u.) LTC’s tap Ploss (MW)

1.2433

1.0252

0.8697

7.9373 −0.0062

1.0044

0.3124

Table 6.12 Initial VSC-HVDC’s control variables for the dynamic simulation. E DCI (p.u)

2.000

IDCIaux (p.u)

IDCI (p.u)

𝜸 Raux (rad)

𝜸 R (rad)

dmaR

dmaI

0.4999

0.4999

0.0232

0.0232

0.0

0.0

285

6 Dynamic Simulations of Power Systems

1.06

V4 V5 V14 V15 V16 V17 V18 V26 V27

Voltage [p.u.]

1.04 1.02 1 0.98 0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.36 Voltage performance at diﬀerent nodes of the network.

0.865 Modulation indices

maR 0.86

maI

0.855 0.85 0.845 0.84

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.37 Dynamic behaviour of the converters’ modulation index.

Qgk

2 Reactive power [p.u.]

286

Qgm 1.5 1 0.5 0

0.5

1

1.5

2

2.5 Time [s]

Figure 6.38 Reactive power generated by the HVDC link.

3

3.5

4

4.5

5

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

0.505

DC current [p.u.]

–IDCR IDCI

0.5 0.495 0.49 0.485 0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.39 DC current behaviour of the rectiﬁer and the inverter.

2.004 EDCR DC voltage [p.u.]

2.002

EDCRx EDCI

2

EDCIx

1.998 1.996 1.994

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.40 Voltage behaviour in the DC link.

transferred through the DC link. Accordingly, the rectiﬁer’s current I DCR drops abruptly from 0.4999 p.u. to 0.4849 p.u., as illustrated in Figure 6.39. A momentary mismatch between the DC currents of both converters takes place because this cannot be instantly re-established due to the time constants of the current controller in the inverter station, which have ﬁnite values. This leads to DC voltage oscillations, as shown in Figure 6.40. Once the controller starts responding, the current I DCI hunts the rectiﬁer current I DCR , aiming at overcoming the voltage drop, as shown in Figure 6.39. This enables a speedy recovery of the DC link voltages. In this case, the cable resistance is relatively small and so is the voltage drop in the DC transmission line. There are minor diﬀerences between the voltage plots EDCR and EDCI and the voltages EDCRx and EDCIx , respectively. The oscillations in the voltages EDCRx and EDCIx are smoothened to some extent by the damping eﬀect of the DC inductors. The simulation results for the active powers and the DC power transfer through the HVDC link, following the disconnection of the transmission lines, are illustrated in Figure 6.41. The power Pgk is the active power entering the high-voltage side of the LTC

287

6 Dynamic Simulations of Power Systems

AC powers & DC power [p.u.]

1.02 –Pgk Pgm

1.01

PdcR 1 0.99 0.98

0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

Figure 6.41 VSC-HVDC’s AC active powers and DC power transfer behaviour.

0.0236

0.8 ϕR

0.0235

0.6

ϕI

0.0234

0.4

θ4 θ14

0.0233

0.2

0.0232 0.0231

ϕR, ϕI, θ4,θ14 [rad]

γR

γR [rad]

288

0 0

0.5

1

1.5

2

2.5 Time [s]

3

3.5

4

4.5

5

–0.2

Figure 6.42 Dynamic performance of various angles involved in the VSC-HVDC link.

transformer coupling the rectiﬁer station whereas Pgm stands for the active power at the high-voltage side of the inverter’s LTC. The power diﬀerence represents the total power losses in the VSC-HVDC link, including the ohmic losses in the DC cable. The power transfer PdcR is also shown in this graph. As shown in Figures 6.39 and 6.40, the voltage and current controls operate quite eﬀectively, leading to a fast power recovery in spite of the rather severe network disturbance. Since the power ﬂow from rectiﬁer to inverter has been brought back to its initial target, Psch = 1.0 p.u, the power angle 𝛾 R exhibits only a marginal increase, as observed in Figure 6.42. This small increase is to comply with the new steady-state operating conditions where diﬀerent currents and, therefore, diﬀerent power losses are incurred. Figure 6.42 also shows the behaviour of the phase-shifting angles of the rectiﬁer and inverter converters, 𝜑R and 𝜑I . It can be seen that these angles follow the same pattern as the AC voltage angles, 𝜃 = tan−1 (f /e), of the nodes where the VSC-HVDC system is connected.

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

6.5.4 Dynamic Model of the VSC-HVDC Link with Frequency Regulation Capabilities VSC-HVDC systems ﬁnd applicability in a wide range of power systems applications, such as interconnecting AC networks of the same or diﬀerent frequencies, supplying electrical energy to remote islands and oﬀshore oil and gas platforms, in-feeding of high load points in city centres and the evacuation of the electrical energy from oﬀshore wind parks. Two independent AC networks interconnected by means of an HVDC link exhibit a natural decoupling in terms of both voltage and frequency and it is said that the two AC power systems are interconnected in an asynchronous manner. These kinds of interconnections are primarily aimed at preventing the excursions of oscillations between AC systems, for instance between a strong and a weak electric network. Nevertheless, there are applications where it is desirable to exert the inﬂuence of the strong network upon the weak network, through the DC link. Such a situation would arise when power imbalances occur in an AC power network with little or no inertia and fed by a VSC-HVDC link. In a power network with no inertia, even very small power imbalances would induce rather large frequency rises or drops, depending on the nature of the power unbalance. Furthermore, all AC networks contain a degree of frequency-sensitive loads and VSC converters do not possess the ability to strengthen on their own the inertial response or to aid the primary frequency control of AC networks. This issue is timely because electrical networks with poor inertia are likely to become more common [22]. There has been research progress in resolving this problem, using a range of equipment models and control schemes embedded in software with which simulation studies have been carried out [23–25]. Microgrid structures and control techniques have also been a matter of great research activity, with particular attention being paid to grid-forming, grid-feeding and grid-supporting topologies [26]. These types of grids are likely to become cost-eﬀective solutions for the interconnection of distributed generators in power grids. Hence, power control performance in AC microgrids, with their strong impact on frequency, is receiving research attention [27–29]. Basically, frequency deviations in a network arise from a mismatch between the mechanical power and the electrical power supplied by the synchronous generators. These frequency deviations are larger in the smaller synchronous generators owing to their lighter rotating masses. The generator’s inertia is the parameter that determines the size of the frequency deviation following a power imbalance. Besides the inertial response exhibited by synchronous generators, most of them are ﬁtted with speed-governing controls to bring a degree of power controllability into the generating system, which is said to exert primary-frequency control. In principle at least, a parallel may be drawn between the kinetic energy stored in the rotating mass of a synchronous generator, reﬂected in its inertia constant, and the electrostatic and electromagnetic energies stored in the converters’ DC capacitors and DC inductors of a VSC-HVDC link, which will have an inertia constant H = H c + H i . Of course, the latter is numerically rather small, but this analogy is useful in the developments presented below. Furthermore, the diﬀerence between the DC power entering the inverter EDCI I DCI and the power P0I will give rise to a frequency deviation in the

289

290

6 Dynamic Simulations of Power Systems

network connected to the inverter. Hence, the angular speed 𝜔I and the angle 𝜑I of the inverter station are expressed as follows [30]: d𝜔I (6.100) = K𝜔 (EDCI IDCI − P0I ) dt d𝜑I (6.101) = 𝜔I − 2𝜋fInom dt where K 𝜔 = 𝜋 f Inom /H and f Inom represents the nominal electrical frequency [Hz] of the low-inertia AC grid. Its value may be the same as or diﬀerent from that of the main utility grid connected to the rectiﬁer terminals. It should be remarked that the angle 𝜑I , as well as the angular speed 𝜔I , might be taken to be reference signals for the low-inertia grid. Given that the total ﬁctitious inertia of the inverter H is very small (let us say 10 ms), in the event of relatively small power disturbances in the low-inertia AC grid, large sags/swells in the angular speed will take place. By way of example, take the case of a low-inertia grid operating at 60 Hz, where the value of K 𝜔 would be around 18.85*103 . If the power imbalance is in the order of ΔP = −1*10−3 p.u. MW and it takes one second to exert power control, Δt = 1 s, then the speed deviation would be Δ𝜔 ≈ K 𝜔 ⋅ΔP⋅Δt ≈ −18.85 rad s−1 , which in terms of frequency is f I ≈ 57 Hz, a rather large frequency value which is not operational. This is a critical issue in networks with frequency-sensitive loads. In cases where the AC grid does not possess frequency-control supporting equipment, e.g. synchronous generators with speed governors or battery-energy storage systems, there is a need to ﬁnd a solution to mitigate any possible frequency excursion. In this sense, a solution may be to import power from a large, near utility grid through an HVDC link. Notice that this requirement departs from the traditional idea of transmitting a ﬁxed amount of power between two networks interconnected by an HVDC link. To this end, an auxiliary control loop, such as the one shown in Figure 6.43, is used to enable the HVDC link to import from the strong system the power required to overcome the power mismatches in the low-inertia AC system (passive network). Note that in this application, the inverter station acts as a power electronic source with active and reactive power control capabilities [30]. The control loop in Figure 6.43 measures the actual angular speed of the low-inertia AC network 𝜔I and compares it with its nominal speed, 𝜔Inom = 2𝜋f Inom . The error is processed by a PI controller to adjust the angular aperture between the AC system’s voltage phase angle 𝜃 vR and the phase-shifting angle of the rectiﬁer 𝜑R . It should be remarked that the objective of the controller in Figure 6.29 is to maintain a ﬁxed amount of power transfer in the DC link whereas the objective of the VSC-HVDC controller depicted in Figure 6.43 is to maintain the electrical frequency of the network connected to the inverter within safe limits by varying the power drawn from the grid Figure 6.43 Frequency controller of the VSC-HVDC. Source: ©IEEE, 2015.

Kppdc ωInom

+ +

+ – ωI

Kipdc s

γRaux

γR

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

connected to the rectiﬁer. It follows that the value of the DC power transfer should be suﬃcient and variable to bring the frequency of the low-inertia AC grid back to its nominal steady-state value. The equations that enable the assessment of the dynamic performance of the DC power when the inverter station of the HVDC system acts as power electronic source with frequency regulation capabilities are: d𝛾Raux = Kipdc (𝜔Inom − 𝜔I ) dt

(6.102)

𝛾R = Kppdc (𝜔Inom − 𝜔I ) + 𝛾Raux

(6.103)

6.5.4.1 Linearization of the Equations of the VSC-HVDC Model with Frequency Regulation Capabilities

The set of power ﬂow equations for the two interconnected AC networks together with the discretized diﬀerential equations of the synchronous generators and their controllers, as well as the algebraic and discretized diﬀerential equations of the VSC-HVDC link, are solved simultaneously using the trapezoidal method and the Newton-Raphson algorithm. The mismatch expressions that must be solved together with the mismatch equations of the interconnected networks, at every integration step Δt, are [30]: ΔPvR = −PvR − PvR,load − PvR,cal

(6.104)

ΔQvR = −QvR − QvR,load − QvR,cal

(6.105)

ΔPvI = −PvI − PvI,load − PvI,cal

(6.106)

ΔQvI = −QvI − QvI,load − QvI,cal

(6.107)

ΔPRx = −EDCRx IRx + PDCR

(6.108)

ΔPIx = −EDCIx IIx + PDCI FEDCR = EDCR(t−Δt) + 0.5Δt Ė DCR(t−Δt) − (EDCR(t) − 0.5Δt Ė DCR(t) ) FE = EDCI(t−Δt) + 0.5Δt Ė DCI(t−Δt) − (EDCI(t) − 0.5Δt Ė DCI(t) )

(6.109)

DCI

FIRx = IRx(t−Δt) + 0.5Δt İ Rx(t−Δt) − (IRx(t) − 0.5Δt İ Rx(t) ) FI = IIx(t−Δt) + 0.5Δt İ Ix(t−Δt) − (IIx(t) − 0.5Δt İ Ix(t) ) Ix

FIDCIaux = IDCIaux(t−Δt) + 0.5Δt İ DCIaux(t−Δt) − (IDCIaux(t) − 0.5Δt İ DCIaux(t) )

(6.110) (6.111) (6.112) (6.113) (6.114)

F𝛾Raux = 𝛾Raux(t−Δt) + 0.5Δt 𝛾̇ Raux(t−Δt) − (𝛾Raux(t) − 0.5Δt 𝛾̇ Raux(t) )

(6.115)

FdmaR = dmaR(t−Δt) + 0.5Δt dṁ aR(t−Δt) − (dmaR(t) − 0.5Δt dṁ aR(t) )

(6.116)

FdmaI = dmaI(t−Δt) + 0.5Δt dṁ aI(t−Δt) − (dmaI(t) − 0.5Δt dṁ aI(t) )

(6.117)

F𝜑I = 𝜑I(t−Δt) + 0.5Δt 𝜑̇ I(t−Δt) − (𝜑I(t) − 0.5Δt 𝜑̇ I(t) )

(6.118)

F𝜔I = 𝜔I(t−Δt) + 0.5Δt 𝜔̇ I(t−Δt) − (𝜔I(t) − 0.5Δt 𝜔̇ I(t) )

(6.119)

The set of equations representing the VSC-HVDC dynamic model is reformulated to endow the HVDC link with frequency regulation capabilities. The model incorporates the two diﬀerential Eqs. (6.118)–(6.119), which enable the computation of the reference angle and the angular speed of the otherwise independent AC network connected to the inverter. Furthermore, contrary to the customary objective of the HVDC to maintain

291

292

6 Dynamic Simulations of Power Systems

a ﬁxed amount of power in the DC link, Eq. (6.115) represents the DC power controller that acts upon changes in the frequency of the grid connected to the inverter. This set of equations yields the following linearized form of the VSC-HVDC model: (6.120)

ΔF = −JHVDC Δz where

[ ΔF = [

ΔPvR ΔQvR ΔPvI ΔQvI ΔPRx ΔPIx FEDCR FEDCI … … FIRX FIIx FIDCIaux F𝛾Raux Fdmar FdmaI F𝜑I F𝜔I

]T

ΔevR ΔfvR ΔeVI ΔfvI ΔEDCRx ΔEDCIx ΔEDCR ΔEDCI · · · Δz = · · · ΔIRx ΔIIx ΔIDCIaux Δ𝛾Raux ΔdmaR ΔdmaI Δ𝜑I Δ𝜔I

(6.121) ]T (6.122)

The Jacobian matrix J HVDC accommodates the ﬁrst-order partial derivatives of the algebraic equations and discretized diﬀerential equations of the HVDC link model. It should be stressed that Eqs. (6.108)–(6.109) and (6.115) provide the link between the set of equations corresponding to both AC networks. Eq. (6.115) relates the angular aperture of the AC voltage of the rectiﬁer to the actual frequency at the inverter station, for power regulation purposes. Eqs. (6.108)–(6.109) relate the voltages in the DC link through the transmission line resistance. The VSC-HVDC is used to interconnect two otherwise independent networks and the overall Jacobian matrix J bears the structure shown in (6.123): 1 ⎡ 2 ⎢ JACN 1 ⎢ ⋱ ⋮ ⎢ k ⎢ J = vR ⎢ ⎢ ⎢ ⋰ ⎢ ⎢ 0 ⎢ ⎣

0 ⋰ JHVDC ⋱ JACN 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ vI ⎥ ⎥m ⎥⋮ ⎥ ⎥1 ⎦

(6.123)

where JACN1 and JACN2 stand for the Jacobian matrices of the main AC utility grid and the weak AC network upon which frequency support is being exerted, respectively. It should be emphasized that the structure of the Jacobian (6.123) has a similar structure to the nodal admittance matrix of the overall power system. It is also worth stressing that the way the equations of the converters have been assembled enables the uniﬁed dynamic solution of both interconnected AC networks together with the VSC-HVDC system, following a straightforward numerical process. 6.5.4.2

Validation of the VSC-HVDC Link Model Providing Frequency Support

The dynamic response of the VSC-HVDC link model with frequency regulation capabilities is compared against a model assembled in the Simulink environment, which uses an EMT-type solution. The comparison is carried out using a rather simple electrical network comprising two independent 50 Hz AC networks interconnected by a VSC-HVDC link. The test system is shown in Figure 6.31. The parameters of the HVDC link are given in Table 6.13. A similar frequency control loop to the one shown in Figure 6.43 is implemented in the Simulink model to endow it with frequency regulation capabilities.

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

Table 6.13 Parameters of the VSC-HVDC link with frequency regulation capabilities. Snom (p.u)

Psch (p.u)

RDC (p.u)

EDCI (p.u)

G0I , G0R (p.u)

R1I , R1R (p.u)

2.0

0.2

0.02135

2.00

4e-3

0.0

X 1I , X 1R (p.u)

RﬁltI , RﬁltR (p.u)

X ﬁltI , XﬁltR (p.u)

BﬁltI , BﬁltR (p.u)

Rltc (p.u)

X ltc (p.u)

0.0

7.5e-4

0.075

0.40

2.5e-3

0.075

H c , H i (s)

K pedc

K iedc

K ppdc , K ipdc

K maI , K maR

T maI , T maR

0.014

2.50

10.0

0.5e-3, 38e-3

25.0

0.02

Frequency at the inverter's AC terminal 0.6 (a) (b)

49.8 49.6 49.4 49.2 0.5

(a) (b)

0.5 PdcI [p.u.]

fI [Hz]

50

VSC-HVDC DC power

0.4 0.3 0.2

1

1.5 2 Time [s]

2.5

3

0.1 0.5

1

1.5 2 Time [s]

2.5

3

Figure 6.44 Dynamic behaviour of the frequency at the inverter’s AC terminal and DC power. Source: ©IEEE, 2015. (a) Simulink model; (b) RMS model.

The initial DC power setpoint of the rectiﬁer station is 20 MW (0.2 p.u.) while the inverter is set to control the DC voltage at EdcInom = 2 p.u. The simulation in Simulink is run up to t = 1.0 s in order to reach the steady-state. To assess the dynamic response of the HVDC link in the face of rather critical frequency deviations, a sudden frequency drop is imposed by the ideal voltage source connected at the AC terminal of the inverter, as shown in Figure 6.44, something expected to occur in low-inertia networks. Figure 6.44 also depicts the dynamic performance of the DC power transfer for both solution approaches. Some diﬀerences exist but it is clear that the dynamic response of the RMS-type HVDC model follows quite well the pattern furnished by the switching-based HVDC model in Simulink. The abrupt frequency drop seen by the inverter station is rapidly taken care of by the frequency control loop of the HVDC link. This regulator quickly increases the amount of power drawn from the utility grid connected at the rectiﬁer’s terminal. The momentary excess of energy in the DC link gives rise to voltage swells in both EDCRx and EDCIx , as seen in Figure 6.45. However the DC voltage controller operates very eﬀectively to damp out the oscillations after only a few milliseconds. The dynamic performance of the modulation indices, as obtained with both HVDC link models with frequency regulation capabilities (i.e. Simulink and RMS-type), is shown in Figure 6.45. Both responses follow the same trend, although an exact match was not expected, owing to the very diﬀerent modelling approaches used by the two models. The computing times required to simulate the test system using the RMS-type VSC-HVDC model and the EMT-type model are 16.51 s and 160.42 s, respectively.

293

6 Dynamic Simulations of Power Systems

VSC-HVDC DC voltages

VSC-HVDC modulation indices 0.9

(a) EDCRx (a) EDCIx (b) EDCRx (b) EDCIx

2.06 2.04 2.02

0.85 0.8

2 1.98 0.5

(a) maR (a) maI (b) maR (b) maI

ma

2.08

EDC [p.u.]

294

0.75 1

1.5 2 Time [s]

2.5

3

0.5

1

1.5 2 Time [s]

2.5

3

Figure 6.45 Dynamic performance of the DC voltages and modulation indices. Source: ©IEEE, 2015. (a) Simulink model; (b) RMS model.

Both solutions agree quite well with each other but the RMS-type model introduced in this chapter outperforms the switching-based model in Simulink by 10 times, in terms of computing speed. This is mainly due to two reasons: (i) the RMS-type model uses an integration time step of 1 ms whereas a sample time of 7.406 μs is needed for the EMT-type model in order to ensure a reliable numerical solution, and (ii) the RMS-type model requires only the positive sequence representation of the network whereas a three-phase representation is used in the EMT-type model. 6.5.4.3 Numerical Example with a VSC-HVDC Link Model Providing Frequency Support HVDC Link Feeding into a Low-Inertia AC Network Figure 6.46 shows the schematic repre-

sentation of a 10-MVA VSC-HVDC link, which interconnects two independent AC networks with very diﬀerent characteristics. The main utility grid (strong AC network) is represented by the Thevenin equivalent of a large power system with a total demand of 60 GW operating with a lagging power factor of 0.95, which is coupled to a reactive tie-line of value X s = 0.06 p.u. The low-inertia AC grid (weak network) consists of a 1 MW hydro-generator, a 2 MW wind turbine and a 7 MVA constant load with a lagging power factor of 0.95. The synchronous machine of the hydro-generator is represented by a two-axis model with an inertia constant of 1 ms, resulting in a network with very low inertia. The parameters of the VSC-HVDC and the low-inertia AC network are: Low-inertia AC Network

Main utility grid

4 Vk

IDCR LDC Pgk

VvR

CDC

Qgk Rectifier

IRx

RDC Rx

LDC IDCI Ix

DC link

IIx

VvI

CDC Inverter

Vm Pgm

Z3

Qgm

Z2

1

WG 3 (PL, QL)

Z1 2

Figure 6.46 VSC-HVDC link feeding into a low-inertia AC network. Source: ©IEEE, 2015.

HG

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

VSC stations: Snom = 0.1 p.u, EDCInom = 2.0 p.u, RDC = 5 p.u, G0R = G0I = 2e-4 p.u, R1R = R1I = 0.02 p.u, X 1R = X 1I = 0.7 p.u., H c = 7e-4 s, H i = 7e-3 s, K pedc = 0.05, K iedc = 1.0, K ppdc = 15e-4, K ipdc = 30e-4, K maR = 2.5, T maR = 0.02, K maI = 2.5, T maI = 0.02. Shunt AC ﬁlters at nodes vR and vI: Bﬁlt = 0.1 p.u. LTC transformers: ZLTC = j0.5 p.u. AC distribution line parameters: ZL1 = ZL2 = Z L3 = 0.1 + j0.15 p.u. The rectiﬁer and inverter stations are set to exert voltage control at their respective AC nodes at 1 p.u. and 1.025 p.u., respectively. From the results provided by the power ﬂow solution, it is noticed that to meet the power equilibrium at the weak network, an import of 3.9175 MW from the main utility grid is required at the rectiﬁer station of the DC link. The power injected by the inverter station, which stands at 3.7030 MW, takes account of the power losses incurred in the low-inertia AC grid: PvI + Pg − Pd = 52.95 kW. The inverter plays the role of a slack generator from the power ﬂow solution standpoint, where the hydro-generator is treated as a PV node with the reference angle provided by the inverter which stands at zero, as shown in Table 6.14. During steady-state operation, the power losses incurred by the rectiﬁer and inverter converters are 16.5 kW and 24.4 kW, respectively, whereas the DC link power loss stands at 173.7 kW. To show that the VSC-HVDC link can provide dynamic frequency regulation to low-inertia AC networks, the load is increased by 5% at t = 0.5 s. Two cases are considered below, when the HVDC link is set to provide frequency support and when it is not. Because of the momentary power imbalances, a rearrangement of power ﬂows takes place, causing voltage oscillations and frequency deviations in the AC grid, as shown in Figures 6.47 and 6.48, respectively. The simulated event leads to temporary frequency deviations of approximately 1 Hz when the VSC-HVDC does not provide frequency support, given the low inertia of the Table 6.14 VSC-HVDC results given by the power ﬂow solution. ma

𝝋 (∘ )

2.0932

0.7803

−2.8295

1.0007

2.0000

0.8570

0

1.0165

Pk , Pm (MW)

Qk , Qm (MVAR)

E DC (p.u.)

Rectiﬁer

3.9175

0.1020

Inverter

3.7030

3.4375

VSC-HVDC without frequency control 1.03 V1 1.025 V2 V3 1.02 V4 1.015 1.01

0

1

2 3 Time [s]

4

5

1.03 AC voltage [p.u.]

AC voltage [p.u.]

Converter

LTC’s tap

VSC-HVDC with frequency control V1 V2 V3 V4

1.025 1.02 1.015 1.01

0

1

2 3 Time [s]

Figure 6.47 Voltage behaviour in the low-inertia AC network. Source: ©IEEE, 2015.

4

5

295

6 Dynamic Simulations of Power Systems

VSC-HVDC with frequency control

VSC-HVDC without frequency control 60

Inverter HG

59.8

Frequency [Hz]

Frequency [Hz]

60 59.6 59.4 59.2 59 0

1

2 3 Time [s]

4

59.95 59.9 59.85 59.8

5

Inverter HG

0

1

2 3 Time [s]

4

5

Figure 6.48 Frequency behaviour in the low-inertia AC network. Source: ©IEEE, 2015.

VSC-HVDC without frequency control 4 PDCR –Pgk 3.9 Pgm 3.8 3.7

0

1

2 3 Time [s]

4

5

VSC-HVDC with frequency control DC & Active powers [p.u.]

DC & Active powers [p.u.]

296

4.4 PDCR –Pgk Pgm

4.2 4 3.8 0

1

2 3 Time [s]

4

5

Figure 6.49 Performance of VSC-HVDC link’s DC and AC powers. Source: ©IEEE, 2015.

network, as seen in Figure 6.48. However, when frequency control is exerted through the power converters, the frequency drops only to about 59.8 Hz. The power injected to the low-inertia grid, through the inverter, Pgm , helps to mitigate the temporary power imbalances brought about by the load increase. Figure 6.49 shows the dynamic performances of the powers involved in the VSC-HVDC link. Notice that when the converters do not support the frequency, the powers Pgk and PDCR are practically constant, with a short duration power peak in Pgm . This is the expected behaviour. As shown in Figure 6.48, the frequency is successfully controlled just after 3 s of the occurrence of the perturbation. This is due to the quick response of both the rectiﬁer and the inverter, which regulate the angular aperture 𝛾 R and the DC current I DCI , respectively, as shown in Figure 6.50. It should be stressed that although 𝛾 R controls the power being drawn from the main utility grid to provide frequency regulation to the low-inertia network, the current I DCI , which is responsible for controlling the DC voltage, impacts directly on the performance of the DC power PDCR . By examining the performance of the DC voltages shown in Figure 6.51, it is observed that the controller takes less time to bring EDCI back to its nominal value than the time required to lead the frequency in the low-inertia grid back to its rated value, as shown in Figure 6.48.

6.5 Modelling of VSC-HVDC Links for Dynamic Simulations

VSC-HVDC with frequency control

0.031

0.0205 DC current [p.u.]

γR

γR [rad]

0.03 0.029 0.028 0.027

0

1

2 3 Time [s]

4

IDCI 0.02 0.0195 0.019 0.0185

5

VSC-HVDC with frequency control

0

1

2 3 Time [s]

4

5

Figure 6.50 Dynamic performances of 𝛾 R and IDCI . Source: ©IEEE, 2015.

VSC-HVDC with frequency control EDCR EDCRx

2.1 2.02

EDCI EDCIx

2.01 2 0

1

2 3 Time [s]

4

VSC-HVDC with frequency control Modulation indices

DC voltages [p.u.]

2.12

5

0.78

maR

0.775 0.858 maI

0.856 0.854 0

1

2 3 Time [s]

4

5

Figure 6.51 Performance of the DC voltages and modulation indices. Source: ©IEEE, 2015.

Small voltage oscillations appear in the low-inertia network because of the power imbalance, as shown in Figure 6.47. During the transient period, the voltage set point is achieved very quickly by the action of the modulation index controllers, as seen from Figure 6.51. Parametric Analysis of the VSC-HVDC Link Feeding into a Low-Inertia AC Network The

VSC-HVDC model provides frequency regulation to networks with near-zero inertia, as is the case of a system ﬁtted with only one small hydro-generator, one wind generator and a ﬁxed load. However, the value of the gains corresponding to the HVDC frequency controller plays a key role in determining the frequency behaviour of the low-inertia grid. To explore this point, a load increase of 5% occurring at t = 0.5 s is assessed for diﬀerent values of gains in the frequency controller, as given in Table 6.15. Figure 6.52 shows the dynamic performance of the frequency for diﬀerent gain values in the frequency control loop, namely k ppdc and k ipdc . In cases (i) to (iii), the proportional gain k ppdc is increased while the integral gain k ipdc is kept constant; it stands out that the frequency f I improves in terms of what is called the inertial response of the network, with the frequency deviation, just after the disturbance, narrowing from 0.56 Hz to 0.12 Hz.

297

6 Dynamic Simulations of Power Systems

Table 6.15 Diﬀerent gains for the frequency controller. Gains

Scenarios (i)

(ii)

(iii)

(iv)

(v)

(vi)

k ppdc

1e-4

15e-4

30e-4

15e-4

15e-4

15e-4

k ipdc

25e-4

25e-4

25e-4

10e-4

30e-4

50e-4

Frequency in the low-inertia network

DC voltage of the VSC-HVDC 2.025

60 (i) (ii) (iii) (iv) (v) (vi)

59.8 59.6 59.4

0

(i) (ii) (iii) (iv) (v) (vi)

2.02

1

2 3 Time [s]

4

EDCI [p.u.]

fI [Hz]

298

2.015 2.01 2.005

5

2 0.4

0.6

0.8 Time [s]

1

Figure 6.52 Frequency in the low-inertia network and DC voltage of the inverter for diﬀerent gains of the frequency control loop. Source: ©IEEE, 2015.

For cases (iv) to (vi), k ppdc is kept constant while k ipdc increases gradually. This results in improving what is termed the primary frequency response given that a faster frequency recovery is achieved when increasing k ipdc . Notice that for the cases being simulated, the smaller the frequency deviations, the bigger the DC voltage oﬀsets, as appreciated in Figure 6.52. The reason is that the power needed to bring the frequency back to its nominal value is injected faster into the network, leading to larger over-voltages in the DC link.

6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations The construction and operation of a multi-terminal HVDC system represent a much larger challenge than for a point-to-point HVDC link. At this point, it becomes necessary to develop more versatile and accurate models of multi-terminal HVDC systems to aid power system engineers to confront the challenges relating to the planning, operation and control of such AC-DC systems. In a multi-terminal VSC-HVDC arrangement, the key objective is to balance the powers of the diﬀerent VSC stations to ensure a reliable operation of both the whole HVDC system and the various AC networks which connect to it. Power reallocation and control of the voltage proﬁle on the meshed DC network are primary functions pursued in a multi-terminal scheme. The control system of each VSC station enables a rapid response to recover quickly after faults occurring either in the AC system or in the DC

6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations

system. It is expected that all the beneﬁts gained with the use of two-terminal HVDC systems would be passed onto the multi-terminal schemes. It is also expected that the great resilience exhibited by meshed AC networks, where the power supply to users is maintained even at times when the AC power grid is operating under stress, will be inherited by the multi-terminal VSC-HVDC systems. At the planning stage, a holistic assessment of the multi-terminal VSC-HVDC system together with the interconnected AC networks will require the availability of suitable VSC models. System-wide dynamic studies of practical networks would beneﬁt from adopting a modular approach to enable eﬃcient simulations with suﬃcient accuracy of results. Moreover, the availability of essential state and control variables pertaining to the converter stations would be most useful and welcome. It was shown in Chapter 3 that the fundamental VSC station model possesses a high degree of modelling ﬂexibility, where the aggregation of any number of VSC stations is carried out with relative ease to give rise to a generic multi-terminal arrangement [31]. The same modular philosophy is followed for the dynamic operating regime in this chapter. 6.6.1

Three-terminal VSC-HVDC Dynamic Model

Three types of converters were deﬁned in Section 3.9 to conform to speciﬁc control strategies and pairing of AC networks: the slack converter controlling its DC voltage (VSCSlack ), the converter controlling its DC power (VSCPsch ) and the converter feeding into a passive network (VSCPass ). Such a converter classiﬁcation aimed at steady-state analysis is also suitable for the dynamic modelling of multi-terminal systems addressed in this chapter. To illustrate this point, the basic three-terminal VSC-HVDC system, depicted in Figure 6.53, which uses the three types of converters, is used. Then the three-terminal system is expanded quite naturally to build the model of a generic multi-terminal VSC-HVDC system. The ensuing framework makes provisions for any number of VSC units, which is commensurate with the number of terminals in the HVDC system, suitably accommodated in a uniﬁed frame-of-reference suitable for dynamic simulations of power networks using only its positive-sequence representation [32]. The AC-DC transmission system shown in Figure 6.53 comprises three VSC stations. For simplicity of representation, the associated phase reactor and AC ﬁlter capacitor of the converters have been omitted from the ﬁgure, but they are taken into account. On the DC side of each VSC, the DC bus connects to a network forming a meshed DC power grid with cables of resistance RDC4 , RDC5 and RDC6 . Notice that in a multi-terminal scheme, any converter can play the role of a rectiﬁer or inverter to satisfy the power exchanges between the converters and the DC grid, as well as to satisfy the requirements of the AC network connected at their corresponding AC terminal. Therefore, the subscripts R and I, formerly standing for rectiﬁer and inverter, respectively, have been changed to numbers. In connection with Figure 6.53, the voltage and current of the capacitors and inductors, respectively, are computed by Eqs. (6.124)–(6.125): − I1x dEDC2 − I2x dEDC3 − I3x −I −I −I dEDC1 , , (6.124) = DC1 = DC2 = DC3 dt CDC1 dt CDC2 dt CDC3 dI 1x E E E − EDC4 dI 2x − EDC5 dI 3x − EDC6 , , (6.125) = DC1 = DC2 = DC3 dt LDC1 dt LDC2 dt LDC3

299

300

6 Dynamic Simulations of Power Systems

ma1

IDC1

VSCSlack Vk

Vv1

Control system IDC3 ma3

IDC1 1 LDC1 RDC1 CDC1 I1x

4

RDC4

RDC2 LDC2 5

8

7

RDC6

LTC1

γ2

I 2 DC2

ma2

VSCPsch

I2x C DC2

RDC5 RDC3 LDC3 9

6

I3x

Vm

Vv2 LTC2

3 IDC3

Vn

Vv3

CDC3

LTC3 VSCPass Figure 6.53 Representation of a three-terminal VSC-HVDC link with its control variables. Source: ©IEEE, 2016. Figure 6.54 Dynamic controller for the DC voltage of the slack converter VSCSlack . Source: ©IEEE, 2016.

Kpe EDC1

+ +

+– EDCnom

Kie s

IDC1 IDC1aux

Figure 6.54 shows the DC voltage controller for the slack converter VSCSlack . The PI controller processes the error between the actual voltage of the converter’s DC bus and its nominal DC voltage, EDC1 and EDCnom , respectively. This results in a new value of the DC current I DC1 . Simultaneously, the other two VSC stations are responsible for controlling the power at its corresponding DC bus, but their power control objectives are diﬀerent from each other. The aim of the converter of type VSCPsch is to achieve a ﬁxed, scheduled DC power transfer. In contrast, the converter of type VSCPass , whose main goal is to provide frequency support to the network connected at its AC terminal, regulates the power injection into the passive network as a function of the power deviations in the AC grid. Figure 6.55 shows the control loops for these two converter stations. Notice that the converter of type VSCPsch aims at a power balance at its DC bus by regulating the angular aperture between its phase-shifting angle and its corresponding AC voltage angle. On the other hand, the converter of type VSCPass acts upon variations on the frequency measured at its AC terminal, increasing or decreasing its DC current depending on the actual operating conditions [32]. In two-terminal HVDC systems, the modulation index of each VSC station is responsible for controlling the voltage magnitudes at their corresponding AC terminal. In the case of the three-terminal VSC-based transmission system, there are three control loops aimed at controlling the voltage magnitudes at each AC terminal. Each control loop

6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations

Kpω

Kpp P0v2

++ Psch

γ2

+ + Kip s

ω3nom

+ +

+–

γ2aux

Kiω s

ω3

IDC3 IDC3aux

(b)

(a)

Figure 6.55 Source: ©IEEE, 2016. (a) DC power controller of the station VSCPsch . (b) Frequency controller of the station VSCPass . nom

nom

Iv1 Vv1

–+

Iv2

dma1 + Kma1 + 1 + sTma1

spec

Vv2

ma1

Kma2 1 + sTma2

spec

(0)

ma1

Vv1

–+

dma2 + +

ma2

(0)

ma2

Vv2 nom

Iv3 Vv3

–+

dma3 + Kma3 + 1 + sTma3

spec

ma3

(0)

Vv3

ma3

Figure 6.56 Modulation index controllers of the three-terminal HVDC system.

comprises a ﬁrst-order control block whose objective is to readjust the corresponding modulation index according to the diﬀerence between the scheduled voltage magnitudes and the actual voltage at the nodes where the converters are connected, as shown in Figure 6.56. Notice that each control loop acts autonomously, needing no feedback from the other two modulation index controllers – this implies that no communication is needed between converters in the multi-terminal system, as far as AC voltage control is concerned. Unlike point-to-point HVDC connections where the resistor of the DC smoothing inductor can be simply added to the DC cable resistor, in a multi-terminal connection, the DC resistance of the DC smoothing inductor produces an additional node in the DC system. However, its inclusion may be carried out with no particular diﬃculty. The calculated power injections at each node of the DC system are computed as shown in (6.126). For practical reasons, this expression only includes the computation of the power injections at the DC buses of a purely resistive network, which is formed from bus 4 to bus 9. It should be clear that during steady-state conditions, the voltages at the DC buses of the three converters equal the voltages at buses 4, 5 and 6, respectively. 2 PDCj,cal = EDCj GDCjj + EDCj

9 ∑ j=4 m∈j

EDCm GDCjm

(6.126)

301

302

6 Dynamic Simulations of Power Systems

The three-terminal HVDC arrangement, shown in Figure 6.53, includes the most representative AC-DC components that signiﬁcantly aﬀect its dynamics. Reliable dynamic solutions call for the use of an approach similar to the one employed in point-to-point HVDC conﬁgurations. The diﬀerential equations, shown in (6.127)–(6.140), are derived from the voltage and current dynamics of the capacitors and inductors, as well as those of the controllers of the three-terminal VSC-HVDC model [32]. (6.127) FE = EDC1(t−Δt) + 0.5Δt Ė DC1(t−Δt) − (EDC1(t) − 0.5Δt Ė DC1(t) ) DC1

FEDC2 = EDC2(t−Δt) + 0.5Δt Ė DC2(t−Δt) − (EDC2(t) − 0.5Δt Ė DC2(t) ) FE = EDC3(t−Δt) + 0.5Δt Ė DC3(t−Δt) − (EDC3(t) − 0.5Δt Ė DC3(t) ) DC3

(6.128) (6.129)

= I1x(t−Δt) + 0.5Δt İ 1x(t−Δt) − (I1x(t) − 0.5Δt İ 1x(t) ) = I2x(t−Δt) + 0.5Δt İ 2x(t−Δt) − (I2x(t) − 0.5Δt İ 2x(t) )

(6.130)

= I3x(t−Δt) + 0.5Δt İ 3x(t−Δt) − (I3x(t) − 0.5Δt İ 3x(t) ) = IDC1aux(t−Δt) + 0.5Δt İ DC1aux(t−Δt) − (IDC1aux(t) − 0.5Δt İ DC1aux(t) )

(6.132)

F𝛾2aux = 𝛾2aux(t−Δt) + 0.5Δt 𝛾̇ 2aux(t−Δt) − (𝛾2aux(t) − 0.5Δt 𝛾̇ 2aux(t) ) FI = IDC3aux(t−Δt) + 0.5Δt İ DC3aux(t−Δt) − (IDC3aux(t) − 0.5Δt İ DC3aux(t) )

(6.134)

FI1x FI2x FI3x FIdc1aux

dc3aux

(6.131) (6.133) (6.135)

F𝜑3 = 𝜑3(t−Δt) + 0.5Δt 𝜑̇ 3(t−Δt) − (𝜑3(t) − 0.5Δt 𝜑̇ 3(t) )

(6.136)

F𝜔3 = 𝜔3(t−Δt) + 0.5Δt 𝜔̇ 3(t−Δt) − (𝜔3(t) − 0.5Δt 𝜔̇ 3(t) )

(6.137)

Fdma1 = dma1(t−Δt) + 0.5Δt dṁ a1(t−Δt) − (dma1(t) − 0.5Δt dṁ a1(t) )

(6.138)

Fdma2 = dma2(t−Δt) + 0.5Δt dṁ a2(t−Δt) − (dma2(t) − 0.5Δt dṁ a2(t) )

(6.139)

Fdma3 = dma3(t−Δt) + 0.5Δt dṁ a3(t−Δt) − (dma3(t) − 0.5Δt dṁ a3(t) )

(6.140)

The above set of expressions determines the whole dynamics in the three-terminal HVDC system. However, the formulation requires the AC and DC power balance equations to be completed. The active and reactive power mismatch equations at the AC terminal of each VSC unit are given by: ΔPv1 = −Pv1 − Pv1,load − Pv1,cal

(6.141)

ΔQv1 = −Qv1 − Qv1,load − Qv1,cal

(6.142)

ΔPv2 = −Pv2 − Pv2,load − Pv2,cal

(6.143)

ΔQv2 = −Qv2 − Qv2,load − Qv2,cal

(6.144)

ΔPv3 = −Pv3 − Pv3,load − Pv3,cal

(6.145)

ΔQv3 = −Qv3 − Qv3,load − Qv3,cal

(6.146)

With reference to Figure 6.53, the purely resistive network, formed by nodes 7, 8 and 9, produces the mismatch Eqs. (6.150)–(6.152). Furthermore, the coupling between this resistive network and the DC side of the power converters is given by Eqs. (6.147)–(6.149), corresponding to DC nodes 4, 5 and 6. ΔPDC4 = EDC4 I1x − PDC4,cal

(6.147)

ΔPDC5 = EDC5 I2x − PDC5,cal

(6.148)

ΔPDC6 = EDC6 I3x − PDC6,cal

(6.149)

ΔPDC7 = −Pd7 − PDC7,cal

(6.150)

6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations

ΔPDC8 = −Pd8 − PDC8,cal

(6.151)

ΔPDC9 = −Pd9 − PDC9,cal

(6.152)

ΔPDC1 = EDC1 IDC1 − P0v1

(6.153)

where the powers Pd7 , Pd8 and Pd9 correspond to the powers drawn by the DC loads at the nodes that make up the delta circuit of DC cables. The expressions PDC4,cal to PDC9,cal represent the calculated powers at the corresponding nodes, which are computed using (6.126). In addition to the above expressions, the Eq. (6.153) enables the computation of the power exchange between the AC and DC sides of the slack converter, VSCSlack . The last equation suggests that the internal power equilibrium of the slack converter is attained ‘instantaneously’, something that does not occur in converters of type VSCPsch and VSCPass . In these two types of converters, whose aim is to provide power regulation and frequency support, the internal power equilibrium is reached with a time delay, imposed by the speed of response of their corresponding dynamic controllers, depicted in Figure 6.55. Such a time delay causes a momentary energy mismatch, charging and discharging of the capacitors and inductors of the DC grid. Therefore it can be inferred that the time response of these two types of controllers will directly impact the magnitude of the DC voltage deviations when the AC-DC system undergoes a disturbance. The overall three-terminal HVDC model for dynamic simulations is deﬁned by Eqs. (6.127)–(6.153), with the linearized form shown in (6.154). The uniﬁed frame-of-reference suitably combines the whole set of discretized diﬀerential equations and algebraic equations arising from the AC-DC system formed by the three VSC stations [32]: i

i

i

0 JSdc ⎤ ⎡𝚫𝚽Slack ⎤ ⎡JSlack 0 ⎡FSlack ⎤ ⎢ 0 JPsch 0 JPsdc ⎥ ⎢ 𝚫𝚽Psch ⎥ ⎢ FPsch ⎥ ⎥ = −⎢ 0 ⎢F 0 JPass JPadc ⎥ ⎢ 𝚫𝚽Pass ⎥ ⎥⎢ ⎥ ⎢ ⎢ Pass ⎥ ⎣ JdcS JdcPs JdcPa Jdc ⎦ ⎣ 𝚫𝚽DC ⎦ ⎣ FDC ⎦

(6.154)

where 0 is a zero-padded matrix of suitable order and the other entries of the Jacobian matrix are: [ ′ [ ′ ] [ ′ ] ] J J J JSlack = ′′Slack , JPsch = ′′Psch , JPass = ′′Pass , J Pass J Slack J Psch [ [ [ [ ] ] ] ]T JSdc = 𝟎 J′Sdc 𝟎 , JPsdc = 𝟎 J′Psdc 𝟎 , JPadc = 𝟎 J′Padc 𝟎 , JdcS = 𝟎 J′dcS 𝟎 , [ [ ]T ]T (6.155) JdcPs = 𝟎 J′dcPs 𝟎 , JdcPa = 𝟎 J′dcPa 𝟎 Matrices J’Slack , J’Psch and J’Pass accommodate the derivative terms of the AC power mismatches of the three types of converters with respect to their corresponding AC state variables and control variables. Notice that J’Pass also makes provision for the derivatives with respect to the electrical reference variables of the passive network, 𝜑3 and 𝜔3 . On the other hand, the derivative terms of the discretized diﬀerential equations arising from the dynamic controllers of the VSC stations with respect to their corresponding AC state variables and control variables are suitably accommodated in matrices J′′ Slack , J′′ Psch and J′′ Pass .

303

304

6 Dynamic Simulations of Power Systems

J′Slack

J′′ Slack

J′Psch

J′′ Psch

⎡ 𝜕ΔPv1 𝜕ΔPv1 𝜕ΔPv1 0 𝜕ΔPv1 ⎤ ⎢ 𝜕ev1 𝜕fv1 𝜕𝜙1 𝜕dma1 ⎥ ⎥ ⎢ 𝜕ΔQv1 ⎥ ⎢ 𝜕ΔQv1 𝜕ΔQv1 𝜕ΔQv1 0 =⎢ , 𝜕fv1 𝜕𝜙1 𝜕dma1 ⎥⎥ ⎢ 𝜕ev1 ⎢ 𝜕ΔP 𝜕ΔPdc1 ⎥ dc1 𝜕ΔPdc1 𝜕ΔPdc1 ⎥ ⎢ 0 ⎣ 𝜕ev1 𝜕fv1 𝜕𝜙1 𝜕dma1 ⎦ 𝜕FIDC1aux ⎤ ⎡ 0 ⎥ 0 0 ⎢ 0 𝜕IDC1aux ⎥ =⎢ ⎥ ⎢ 𝜕Fdm 𝜕Fdm 𝜕F dma1 a1 a1 ⎥ ⎢ 0 0 ⎣ 𝜕ev1 𝜕fv1 𝜕dma1 ⎦

(6.156)

𝜕ΔPv2 ⎤ ⎡ 𝜕ΔPv2 𝜕ΔPv2 0 ⎥ ⎢ 𝜕e 𝜕f 𝜕dm v2 v2 a2 ⎥, =⎢ ⎢ 𝜕ΔQv2 𝜕ΔQv2 𝜕ΔQv2 ⎥ 0 ⎥ ⎢ 𝜕fv2 𝜕dma2 ⎦ ⎣ 𝜕ev2 ⎡ 𝜕F𝛾2aux 𝜕F𝛾2aux 𝜕F𝛾2aux ⎢ 𝜕e 𝜕fv2 𝜕𝛾2aux v2 =⎢ ⎢ 𝜕Fdm 𝜕Fdm a2 a2 ⎢ 0 ⎣ 𝜕ev2 𝜕fv2

𝜕F𝛾2aux ⎤ 𝜕dma2 ⎥ ⎥ 𝜕Fdma2 ⎥ ⎥ 𝜕dma2 ⎦

J′Pass

𝜕ΔPv3 𝜕ΔPv3 ⎤ ⎡ 𝜕ΔPv3 𝜕ΔPv3 0 0 ⎢ 𝜕e 𝜕fv3 𝜕dma3 𝜕𝜑3 ⎥ v3 ⎥ =⎢ ⎢ 𝜕ΔQv3 𝜕ΔQv3 𝜕ΔQv3 𝜕ΔQv3 ⎥ 0 0 ⎥ ⎢ 𝜕fv3 𝜕dma3 𝜕𝜑3 ⎦ ⎣ 𝜕ev3

J′′Pass

𝜕FIDC3aux ⎡ 0 0 0 0 ⎢ 𝜕IDC3aux ⎢ ⎢ 𝜕Fdma3 𝜕Fdma3 𝜕Fdma3 ⎢ 0 0 𝜕fv3 𝜕dma3 ⎢ 𝜕ev3 =⎢ 𝜕F𝜔3 𝜕F𝜔3 ⎢ 𝜕F𝜔3 𝜕F𝜔3 0 ⎢ 𝜕e 𝜕fv3 𝜕dma3 𝜕𝜔3 v3 ⎢ ⎢ 𝜕F𝜑3 ⎢ 0 0 0 0 ⎣ 𝜕𝜔3

(6.157)

0 ⎤⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 𝜕F𝜔3 ⎥ 𝜕𝜑3 ⎥ ⎥ 𝜕F𝜑3 ⎥ ⎥ 𝜕𝜑3 ⎦

(6.158)

The terms arising from deriving both the AC power mismatches and the discretized diﬀerential equations of the dynamic controllers of the three VSC stations with respect to the capacitor’s DC voltage are located in matrices J’Sdc , J’Psdc and J’Padc . Vectors J’dcS , J’dcPs and J’dcPa contain the partial derivatives of the discretized diﬀerential equations of the converters’ DC voltage dynamics (voltage equations of the capacitors) with respect to their corresponding AC state variables and control variables.

6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations T ⎡ 𝜕ΔPv1 ⎤ ⎡ 𝜕ΔPv3 ⎤ ⎡ 𝜕FEDC1 ⎤ ⎢ 𝜕E ⎢ 𝜕EDC3 ⎥ ⎢ 𝜕e ⎥ ⎥ ⎡ 𝜕ΔPv2 ⎤ v1 ⎥ DC1 ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ 𝜕E ⎥ 𝜕ΔQv3 ⎥ 𝜕FEDC1 ⎥ ⎢ 𝜕ΔQv1 ⎥ ⎢ ⎢ DC2 ⎢ ⎥ ⎥ ⎢ 𝜕E ⎢ 𝜕E ⎢ ⎥ ⎥ ⎢ 𝜕ΔQv2 ⎥ DC1 ⎥ ⎢ ⎢ DC3 ⎥ ⎢ 𝜕fv1 ⎥ ⎢ ⎥ = ⎢ 𝜕ΔPDC1 ⎥ , J′Psdc = ⎢ 𝜕EDC2 ⎥ , J′Padc = ⎢ 0 ⎥ , J′dcS = ⎢ 𝜕FEDC1 ⎥ , ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎢ 𝜕F𝛾2aux ⎥ ⎢ 𝜕EDC1 ⎥ ⎢ 0 ⎥ ⎢ 𝜕𝜑1 ⎥ ⎢ ⎥ ⎢ 𝜕FIDC1aux ⎥ ⎢ 𝜕F ⎥ ⎢ 0 ⎥ ⎢ 𝜕EDC2 ⎥ 𝜔3 ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎢ 0 ⎥ ⎢ 𝜕EDC1 ⎥ ⎢ 𝜕EDC3 ⎥ ⎢ 𝜕FEDC1 ⎥ ⎣ ⎦ ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎥ ⎣ ⎣ 0 ⎦ ⎣ 𝜕dma1 ⎦ ⎦

J′Sdc

J′dcPs

T ⎡ 𝜕FEDC3 ⎤ T 𝜕F ⎡ EDC2 ⎤ ⎢ 𝜕ev3 ⎥ ⎢ ⎢ 𝜕ev2 ⎥ ⎥ ⎥ ⎢ 𝜕FEDC3 ⎥ ⎢ 𝜕F ⎢ 𝜕f ⎥ ⎢ EDC2 ⎥ ⎢ v3 ⎥ ⎢ 𝜕f ⎥ = ⎢ v2 ⎥ , J′dcPa = ⎢ 0 ⎥ ⎢ ⎢ 𝜕FEDC2 ⎥ ⎥ ⎥ ⎢ 𝜕FEDC3 ⎥ ⎢ ⎢ 𝜕dm ⎥ ⎢ 𝜕𝛾2aux ⎥ a3 ⎥ ⎢ ⎢ 𝜕FE ⎥ DC2 ⎥ ⎢ 0 ⎥ ⎢ ⎣ 𝜕dma2 ⎦ ⎢ ⎥ ⎣ 0 ⎦

(6.159)

Matrix Jdc , shown in (6.160), is further subdivided into several matrix blocks to illustrate further the variables involved. This matrix represents the coupling between the DC circuit and the three VSC stations. It contains the derivatives of the discretized diﬀerential equations of inductors currents and capacitors voltages together with the nodal power mismatches of the internal DC nodes, 4 to 9, with respect to suitable DC state and control variables. ⎡Jdc1 Jdc2 Jdc5 ⎤ Jdc = ⎢Jdc3 Jdc4 0 ⎥ (6.160) ⎥ ⎢ 0 J J ⎣ dc6 dc7 ⎦ 𝜕F ⎤ ⎤ ⎡ I1x ⎡ 𝜕FI1x 0 0 0 0 ⎥ ⎥ ⎢ 𝜕I ⎢ 𝜕E ⎥ ⎥ ⎢ 1x ⎢ dc1 𝜕FI2x 𝜕FI2x ⎥ ⎥ ⎢ ⎢ Jdc1 = ⎢ 0 0 ⎥ , Jdc2 = ⎢ 0 0 ⎥, 𝜕I2x 𝜕Edc2 ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ 𝜕FI3x ⎥ 𝜕FI3x ⎥ ⎥ ⎥ ⎢ 0 ⎢ 0 0 0 ⎣ ⎣ 𝜕Edc3 ⎦ 𝜕I3x ⎦

Jdc3

⎤ ⎡ 𝜕FEDC1 0 0 ⎥ ⎢ 𝜕I ⎥ ⎢ 1x 𝜕FEDC2 ⎥ ⎢ =⎢ 0 0 ⎥ 𝜕I2x ⎥ ⎢ ⎢ 𝜕FEDC3 ⎥ ⎥ ⎢ 0 0 ⎣ 𝜕I3x ⎦

(6.161)

305

306

6 Dynamic Simulations of Power Systems

Jdc4

⎤ ⎤ ⎡ 𝜕FEDC1 ⎡ 𝜕FI1x 0 0 ⎥ 0 0 ⎥ ⎢ 𝜕E ⎢ 𝜕E ⎥ ⎥ ⎢ DC1 ⎢ DC4 𝜕FI2x 𝜕FEDC2 ⎥ ′ ⎥ ⎢ ⎢ =⎢ 0 0 ⎥ , Jdc5 = ⎢ 0 0 ⎥, 𝜕EDC2 𝜕EDC5 ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ 𝜕FEDC3 ⎥ 𝜕FI3x ⎥ ⎥ ⎥ ⎢ 0 ⎢ 0 0 0 ⎣ ⎣ 𝜕EDC3 ⎦ 𝜕EDC6 ⎦

J′dc6

⎤ ⎡ 𝜕ΔPDC4 0 0 ⎥ ⎢ 𝜕I 1x ⎥ ⎢ 𝜕ΔPDC5 ⎥ ⎢ 0 ⎥ =⎢ 0 𝜕I 2x ⎥ ⎢ 𝜕ΔPDC6 ⎥ ⎢ 0 ⎥ ⎢ 0 𝜕I3x ⎦ ⎣

(6.162)

𝜕ΔPDC4 ⎡ 𝜕ΔPDC4 ⎤ 0 0 0 0 ⎥ ⎢ 𝜕E 𝜕EDC7 DC4 ⎢ ⎥ 𝜕ΔPDC5 𝜕ΔPDC5 ⎢ ⎥ 0 0 0 ⎥ ⎢ 0 𝜕EDC5 𝜕EDC8 ⎢ ⎥ 𝜕ΔPDC6 ⎥ 𝜕ΔPDC6 ⎢ 0 0 0 ⎢ 0 𝜕EDC6 𝜕EDC9 ⎥⎥ Jdc7 = ⎢ (6.163) 𝜕ΔPDC7 𝜕ΔPDC7 𝜕ΔPDC7 ⎥ ⎢ 𝜕ΔPDC7 0 0 ⎢ ⎥ 𝜕EDC7 𝜕EDC8 𝜕EDC9 ⎥ ⎢ 𝜕EDC4 ⎢ 𝜕ΔPDC8 𝜕ΔPDC8 𝜕ΔPDC8 𝜕ΔPDC8 ⎥ 0 ⎢ 0 ⎥ 𝜕E 𝜕EDC7 𝜕EDC8 𝜕EDC9 ⎥ DC5 ⎢ ⎢ 𝜕ΔPDC9 𝜕ΔPDC9 𝜕ΔPDC9 𝜕ΔPDC9 ⎥ 0 ⎢ 0 ⎥ 𝜕EDC6 𝜕EDC7 𝜕EDC8 𝜕EDC9 ⎦ ⎣ [′ ] [ ] 𝟎 J where Jdc5 = J′dc5 𝟎 and Jdc6 = dc6 . 𝟎 𝟎 Notice that the diagonal matrices shown in (6.161) and (6.162), whose orders correspond to the number of VSC stations, may be expanded with ease to conform to any kind of multi-terminal arrangement of converters. The mismatch vectors and increments of the state and control variables for the three VSC units and the DC circuit are: [ ]T FSlack = ΔPv1 ΔQv1 ΔPDC1 FIDC1aux Fdma1 [ ]T FPsch = ΔPv2 ΔQv2 F𝛾2aux Fdma2 [ ]T FPass = ΔPv3 ΔQv3 FIDC3aux Fdma3 F𝜔3 F𝜙3 [ ]T FI2x FI3x FEDC1 FEDC2 FEDC3 ··· F FDC = I1x (6.164) · · · ΔPDC4 ΔPDC5 ΔPDC6 ΔPDC7 ΔPDC8 ΔPDC9

6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations

[ 𝚫𝚽Slack = Δev1 [ 𝚫𝚽Psch = Δev2 [ 𝚫𝚽Pass = Δev3 [ ΔI1x 𝚫𝚽DC = …

Δfv1 Δ𝜑1 ΔIDC1aux Δdma1 ]T Δfv2 Δ𝛾2aux Δdma2

]T

Δfv3 ΔIDC3aux Δdma3 Δ𝜔3 Δ𝜑3

]T

ΔI2x ΔI3x ΔEDC1 ΔEDC2 ΔEDC3 … ΔEDC4 ΔEDC5 ΔEDC6 ΔEDC7 ΔEDC8 ΔEDC9

]T (6.165)

Notice that vector 𝚫𝚽DC contains the DC voltages as state variables as well as the DC currents of the inductors. The increments of the state variables of the converters and DC circuit, calculated at iteration i, are used to update the state variables as follows: i+1 𝚽Slack = 𝚽iSlack + 𝚫𝚽iSlack i+1 𝚽Psch = 𝚽iPsch + 𝚫𝚽iPsch i+1 𝚽Pass = 𝚽iPass + 𝚫𝚽iPass i+1 𝚽DC = 𝚽iDC + 𝚫𝚽iDC

6.6.2

(6.166)

Validation of the Three-Terminal VSC-HVDC Dynamic Model

The three-terminal VSC-HVDC system, shown in Figure 6.57, is used to carry out a comparison of the RMS-type model against an EMT-type model, which was assembled in the simulation environment aﬀorded by the package SymPowerSystems of Simulink. With no loss of generality, the AC sub networks connected to VSCSlack and VSCPsch are represented by equivalent networks (2000 MVA, 230 kV, 50 Hz), whereas the passive network fed by VSCPass is represented by a 50 MW load. Each VSC in this multi-terminal HVDC system is rated at 200 MVA, ±100 kV DC. The resistances of the DC cables, RDC4 to RDC6 , have value of 1.39 × 10−2 Ω/km, with the cables lengths being 75 km, 100 km and 150 km, respectively. The rest of the parameters, given on a 100 MVA base, are given in Table 6.16. Initially the slack converter VSCSlack controls its DC voltage at EDCnom = 2 p.u., the converter VSCPsch controls its DC power at Psch = 150 MW and the converter VSCPass feeds the 50 MW load. The three converters controlling the reactive power ﬂow to a null ref value, i.e. Qv = 0. The simulation in Simulink is run up to t = 1.5 s to reach steady-state conditions. The following step changes in parameters are applied: (i) a 50 MW reduction of the scheduled power of the converter VSCPsch is applied at t = 1.5 s, and (ii) the load fed by the converter VSCPass is increased by 40% (20 MW) at t = 3 s. Both disturbances cause voltage variations in the DC network, as shown in Figure 6.58. The slack converter controls the voltage at its corresponding DC bus and impacts positively on the DC voltages of the converters VSCPsch and VSCPass . It can be seen that the pattern of the dynamic response of the RMS-type multi-terminal VSC-HVDC model follows relatively well the pattern of the EMT-type model solution.

307

2000 MVA 230 kV, 50 Hz

Source impedance

VSCSlack Vk

LTC

Vv1

200 MVA 230:100 kV

Phase reactor

AC filter 40 MVAr

IDC1 LDC1 RDC1 1

4

CDC1

I1x

RDC2 LDC2 IDC2

RDC4

2

7

8

RDC6

RDC5

5

VSCPsch

I2x CDC2

RDC3 LDC3 3 IDC3 9

6

AC filter 40 MVAr

LTC

Vm

I3x

AC filter 40 MVAr VSCPass

2000 MVA 230 kV, 50 Hz

Source 200 MVA 230:100 kV impedance

Vn Phase reactor Vv3 LTC

CDC3

Figure 6.57 Three-terminal VSC-HVDC link used to carry out the validation test.

Phase reactor Vv2

200 MVA 100:230 kV

Passive network 50 MW

6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations

Table 6.16 Parameters of the three-terminal VSC-HVDC link. Parameters of the three VSC

Snom ( p.u)

G0 (p.u)

R (p.u)

X (p.u)

Rf (p.u)

2.0

4e-3

0.0

0.0

7.5e-4

X f (p.u)

Bf (p.u)

Rltc (p.u)

X ltc (p.u)

H c , H i (s)

0.40

2.5e-3

0.075

14e-3, 14e-3

K pe , K ie

K pp , K ip

K p𝜔 , K i𝜔

K ma

T ma

VSCSlack

1.5, 15

—

—

50.0

0.20

VSCPsch

—

0.0, 5.0

—

50.0

0.20

VSCPass

—

—

0.025, 0.25

50.0

0.20

DC voltage [p.u.]

DC voltage [p.u.]

DC voltage [p.u.]

0.075 Gains

(a) EMT-type model

VSCSlack

2.1

(b) RMS-type model

2 1.9

1

1.5

2

2.5

3.5

4

(a) EMT-type model

VSCPsch

2.1

3 Time [s]

4.5

5

(b) RMS-type model

2 1.9

1

1.5

2

2.5

3.5

4

(a) EMT-type model

VSCPass

2.1

3 Time [s]

4.5

5

(b) RMS-type model

2 1.9

1

1.5

2

2.5

3 Time [s]

3.5

4

4.5

5

Figure 6.58 DC voltages of the VSC units comprising the three-terminal VSC-HVDC link.

As expected, the power reduction of 50 MW in Psch of the converter VSCPsch produces larger over-voltages than those experienced by the power increase of 20 MW in the load fed by the converter VSCPass . These DC voltage deviations aﬀect the DC power performance of the three converters, as seen in Figure 6.59. It can be seen that some transient power peaks computed by the Simulink model during the applied step changes are not captured by the RMS-type model. The reason for this is the very diﬀerent solution approaches taken by models. However, the rest of the DC power responses follow the same trend reasonably well.

309

6 Dynamic Simulations of Power Systems

DC power [p.u.]

2.5

(a) VSCSlack

(a) VSCPsch

(a) VSCPass

(b)VSCSlack

(b) VSCPsch

(b) VSCPass

2 Change in the scheduled power of VSCPsch 1.5 Response of VSCSlack to balance the powers in the DC network

1

Change in the load fed by VSCPass

0.5 1

1.5

2

2.5

3 Time [s]

3.5

4

4.5

5

Figure 6.59 DC power behaviour at the DC bus of the three VSC stations. Source: ©IEEE, 2016. (a) EMT-type model. (b) Developed RMS-type model.

(a) VSCSlack

0.95 Modulation ratio

310

(a) VSCPsch

(a) VSCPass

(b)VSCSlack

Change in the scheduled power of VSCPsch

0.9

(b) VSCPsch

(b) VSCPass

Change in the load fed by VSCPass

0.85 0.8 0.75

1

1.5

2

2.5

3 Time [s]

3.5

4

4.5

5

Figure 6.60 Modulation ratio of the three VSC stations. Source: ©IEEE, 2016. (a) EMT-type model. (b) RMS-type model.

The dynamic responses of the modulation indices are shown in Figure 6.60. The most noticeable diﬀerences between the two types of responses appear when there is a change in the load fed by the station VSCPass , i.e. the oscillations of the modulation ratio of VSCPass are larger than those calculated by the RMS-type model. In terms of execution times, the RMS-type multi-terminal VSC-HVDC model outperforms by more than eight times the EMT-type model. The former takes 1.04 min to solve the three-terminal VSC-HVDC system whereas the latter takes 8.5 min. 6.6.3

Multi-Terminal VSC-HVDC Dynamic Model

The HVDC dynamic model of the three-terminal system, synthesized by Eq. (6.154), shows the way in which the three types of VSC stations are combined in a uniﬁed frame-of-reference, suitable for an iterative solution using the Newton-Raphson method at each time step.

6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations

Expanding this representation, a generalized multi-terminal arrangement comprising m converter stations of type VSCSlack , n converter stations of type VSCPsch and r converter stations of type VSCPass is shown below [32]: i

i

⎡ FSlack1 ⎤ ⎡JSlack1 · · · 𝟎 ⎤ ⎢ ⋮ ⎥ ⎢ ⋮ ⋱ ⋮ 𝟎 𝟎 [JSdc ] ⎥ ⎢F ⎢ 𝟎 ··· J ⎥ ⎥ Slackm ⎢ Slackm ⎥ ⎢ ⎥ JPsch1 · · · 𝟎 ⎢ FPsch1 ⎥ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ ⋮ ⋱ ⋮ 𝟎 [JPsdc ]⎥ 𝟎 ⎢F ⎥ = −⎢ ⎥ 𝟎 · · · JPschn ⎢ Pschn ⎥ ⎢ ⎥ JPass1 · · · 𝟎 ⎢ FPass1 ⎥ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ ⋮ ⋱ ⋮ [JPadc ]⎥ 𝟎 𝟎 ⎢F ⎢ ⎥ ⎥ 𝟎 · · · JPassr ⎢ Passr ⎥ ⎢ ⎥ [JdcS ] [JdcPs ] [JdcPa ] [Jdc ] ⎦ ⎣ [FDC ] ⎦ ⎣ i

⎡ 𝚫Slack1 ⎤ ⎢ ⋮ ⎥ ⎢𝚫 ⎥ ⎢ Slackm ⎥ ⎢ 𝚫Psch1 ⎥ ⎢ ⋮ ⎥ ×⎢ ⎥ 𝚫 ⎢ Pschn ⎥ ⎢ 𝚫Pass1 ⎥ ⎢ ⋮ ⎥ ⎢𝚫 ⎥ ⎢ Passr ⎥ ⎣ [𝚫DC ] ⎦

(6.167)

In expression (6.167), the mismatch terms F and the vectors of state variables increments 𝚫𝚽 take the forms given by (6.164)–(6.165) depending on the type of converter, i.e. VSCSlack , VSCPsch or VSCPass . Likewise, matrices J with subscripts Slack, Psch and Pass take the form given by (6.155). Some caution must be exercised because the non-zero entries of vectors JSdc , JPsdc , JPadc , JdcS , JdcPs and JdcPa need to be suitably staggered, as exempliﬁed in the three-terminal case. Accordingly, the vectors and matrices entries, which have a location corresponding to any of the DC network variables, 𝚫𝚽DC , have the following structure: ⎡𝟎 … 𝟎 J′Sdc1 … 𝟎 𝟎 … 𝟎 ⎤ [JSdc ] = ⎢⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⎥ , ⎥ ⎢ ′ ⎣𝟎 … 𝟎 𝟎 … JSdcm 𝟎 … 𝟎 ⎦ ⎡𝟎 … 𝟎 J′Psdc1 … 𝟎 𝟎 … 𝟎 ⎤ [JPsdc ] = ⎢⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⎥ , ⎥ ⎢ ′ ⎣𝟎 … 𝟎 𝟎 … JPsdcn 𝟎 … 𝟎 ⎦ ⎡𝟎 … 𝟎 J′Padc1 … 𝟎 𝟎 … 𝟎 ⎤ [JPadc ] = ⎢⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⎥ ⎥ ⎢ ′ ⎣𝟎 · · · 𝟎 𝟎 · · · JPadcr 𝟎 · · · 𝟎 ⎦

311

312

6 Dynamic Simulations of Power Systems T

⎡𝟎 … 𝟎 J′dcS1 … 𝟎 𝟎 … 𝟎 ⎤ [JdcS ] = ⎢⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⎥ , ⎥ ⎢ ′ ⎣𝟎 · · · 𝟎 𝟎 · · · JdcSm 𝟎 · · · 𝟎 ⎦ T

⎡𝟎 … 𝟎 J′dcPs1 … 𝟎 𝟎 … 𝟎 ⎤ [JdcPs ] = ⎢⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⎥ , ⎥ ⎢ ′ ⎣𝟎 · · · 𝟎 𝟎 · · · JdcPsn 𝟎 · · · 𝟎 ⎦ T

⎡𝟎 … 𝟎 J′dcPa1 … 𝟎 𝟎 … 𝟎 ⎤ [JdcPa ] = ⎢⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⎥ ⎥ ⎢ ′ ⎣𝟎 · · · 𝟎 𝟎 · · · JdcPar 𝟎 · · · 𝟎 ⎦

(6.168)

The entries in (6.168) are vectors with the generic j − th terms, depending on the number and type of converter, with j = 1, …, m, j = 1, …, n and j = 1, …, r for converters VSCSlack , VSCPsch and VSCPass , respectively. They are given below:

J′Sdc

⎡ 𝜕ΔPvj ⎤ ⎡ 𝜕ΔPvj ⎤ ⎢ ⎥ ⎢ 𝜕E ⎥ ⎡ 𝜕ΔPvj ⎤ ⎢ 𝜕EDCj ⎥ DCj ⎢ ⎥ ⎢ 𝜕ΔQ ⎥ ⎢ 𝜕E ⎥ ⎢ 𝜕ΔQvj ⎥ vj ⎥ ⎢ ⎢ DCj ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ 𝜕E 𝜕ΔQvj DCj ⎢ 𝜕EDCj ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ = ⎢ 𝜕ΔPDCj ⎥ , J′Psdc = ⎢ 𝜕EDCj ⎥ , J′Padc = ⎢ 0 ⎥ , ⎢ ⎢ ⎥ ⎥ ⎢ 𝜕EDCj ⎥ ⎢ 0 ⎥ ⎢ 𝜕F𝛾jaux ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ 𝜕FIDCjaux ⎥ 𝜕F𝜔j ⎥ 𝜕EDCj ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ 𝜕E ⎥ ⎢ ⎥ ⎢ 𝜕EDCj ⎥ ⎣ 0 ⎦ ⎢ DCj ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎣ 0 ⎦ ⎣ ⎦

J′dcS

T T ⎡ 𝜕FEDCj ⎤ ⎡ 𝜕FEDCj ⎤ T ⎡ 𝜕FEDCj ⎤ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ 𝜕evj ⎥ ⎢ 𝜕evj ⎥ ⎢ 𝜕evj ⎥ ⎢ 𝜕F ⎥ ⎢ 𝜕F ⎥ ⎢ 𝜕F ⎥ ⎢ EDCj ⎥ ⎢ EDCj ⎥ E DCj ⎥ ⎢ ⎢ 𝜕f ⎥ ⎢ 𝜕f ⎥ ⎢ 𝜕fvj ⎥ ⎢ vj ⎥ ⎢ vj ⎥ ′ ′ ⎥ ,J = ⎢ 𝜕FEDCj ⎥ , JdcPs = ⎢ =⎢ 0 ⎥ ⎢ 𝜕FEDCj ⎥ dcPa ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ 𝜕𝜑j ⎥ ⎢ 𝜕FEDCj ⎥ ⎢ 𝜕𝛾jaux ⎥ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ 0 ⎥ ⎢ 𝜕dmaj ⎥ ⎢ 𝜕FEDCj ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ 𝜕dm ⎥ ⎢ 𝜕FEDCj ⎥ ⎢ 0 ⎥ ⎣ aj ⎦ ⎢ 𝜕dm ⎥ ⎢ ⎥ aj ⎦ ⎣ ⎣ 0 ⎦

(6.169)

The Jacobian matrix of the DC network takes the general form of Eq. (6.170): ⎡Jdc1 Jdc2 [Jdc5 ] ⎤ [Jdc ] = ⎢Jdc3 Jdc4 𝟎 ⎥ ⎥ ⎢ ⎣ [Jdc6 ] 𝟎 Jdc7 ⎦ where Jdc5

[′ ] [′ ] Jdc6 𝟎 = Jdc5 𝟎 and Jdc6 = . 𝟎 𝟎

(6.170)

6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations

The matrices from Jdc1 to Jdc4 , J′ dc5 and J′ dc6 are diagonal matrices with dimensions corresponding to the total number of converters making up the multi-terminal system, i.e. nvsc = m + n + r, with j = 1, …, nvsc and k = nvsc + 1, …, 2nvsc . The order of matrix Jdc7 corresponds to the number of DC nodes ndc minus the number of converters nvsc , with ⟨j, k⟩ = nvsc + 1, …, ndc . These matrices are given in explicit form as:

Jdc1

⎡ 𝜕FIjx ⎡ 𝜕FIjx ⎡ 𝜕FEDCj ⎤ ⎤ ⎤ 0 0 0 0 0 0 ⎥ ⎢ 𝜕I ⎢ 𝜕E ⎢ 𝜕I ⎥ ⎥ ⎢ jx ⎢ DCj ⎢ jx ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ = ⎢ 0 ⋱ 0 ⎥ , Jdc2 = ⎢ 0 ⋱ 0 ⎥ , Jdc3 = ⎢ 0 ⋱ 0 ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ 𝜕FIjx ⎥ 𝜕FIjx ⎥ 𝜕FEDCj ⎥ ⎢ ⎢ ⎢ ⎢ 0 0 ⎢ 0 ⎢ 0 ⎥ ⎥ ⎥ 0 0 ⎣ ⎣ ⎣ 𝜕Ijx ⎦ 𝜕EDCj ⎦ 𝜕Ijx ⎦ (6.171)

Jdc4

⎤ ⎡ 𝜕FEDCj ⎤ ⎡ 𝜕FIjx 0 0 ⎥ ⎢ 0 0 ⎥ ⎢ 𝜕E ⎥ ⎢ 𝜕EDCj ⎥ ⎢ DCk ⎥ ⎢ ⋱ 0 ⎥, = ⎢ 0 ⋱ 0 ⎥ , J′dc5 = ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎢ 𝜕FEDCj ⎥ 𝜕FIjx ⎥ ⎢ ⎥ ⎢ 0 0 0 ⎢ 0 ⎣ 𝜕EDCj ⎥⎦ 𝜕EDCk ⎦ ⎣

J′dc6

⎡ 𝜕ΔPDCk 0 ⎡ 𝜕ΔPDCj 0 ⎤⎥ ··· ⎢ 𝜕Ijx ⎢ 𝜕E ⎥ ⎢ ⎢ DCk ⋱ 0 ⎥ , Jdc7 = ⎢ ⋮ ⋱ =⎢ 0 ⎥ ⎢ ⎢ ⎢ ⎢ 𝜕ΔPDCj 𝜕ΔPDCk ⎥ 0 ⎥ ⎢ 0 ⎢ ··· 𝜕Ijx ⎦ ⎣ ⎣ 𝜕EDCk

𝜕ΔPDCj ⎤ 𝜕EDCk ⎥ ⎥ ⋮ ⎥ ⎥ 𝜕ΔPDCj ⎥ ⎥ 𝜕EDCk ⎦

(6.172)

The remaining terms, corresponding to the DC side of the multi-terminal HVDC system, with j = 1, …, nvsc and k = nvsc + 1, …, ndc , are: [ ]T [FDC ] = FIjx · · · FIjx FEDCj · · · FEDCj ΔPDCk · · · ΔPDCk (6.173) [ ]T [𝚫𝚽DC ] = ΔIjx · · · ΔIjx ΔEDCj · · · ΔEDCj ΔEDCk · · · ΔEDCk

(6.174)

This modelling approach is modular and developing new converter models with diﬀerent control strategies would not represent major diﬃculties in terms of their aggregation into the generalized multi-terminal HVDC link model. During the derivation of the multi-terminal HVDC model, synthetized by (6.167), it has been assumed that each VSC unit connects to a capacitor and a smoothing inductor coupled to a resistor on its DC side. However, any modiﬁcations to this arrangement should pose no diﬃculty. Unlike the steady-state model of the multi-terminal HVDC system, the DC voltage of the slack converter is part of the variables to be computed through the time-domain solution, where at least one slack converter must exist in the multi-terminal arrangement, to ensure the voltage stability in the DC network.

313

314

6 Dynamic Simulations of Power Systems

The calculated power at each node j of the DC network is computed using expression (6.175). The active and reactive power equations that link the multi-terminal HVDC system with the AC networks are given by (6.176)–(6.177), respectively. ndc ∑

2 GDCjj + EDCj PDCj,cal = EDCj

Pk,cal = (e2k + fk2 )Gkk +

∑

EDCm GDCjm

(6.175)

j=nvsc +1 m∈j

[ek (Gkm ek − Bkm fk ) + fk (Bkm ek + Gkm fk )]

∑

(6.176)

m∈k

Qk,cal = −(e2k + fk2 )Bkk +

[ fk (Gkm ek − Bkm fk ) − ek (Bkm ek + Gkm fk )]

(6.177)

m∈k

where k represents a generic node. 6.6.4 Ring

Numerical Example with a Six-Terminal VSC-HVDC Link Forming a DC

The six-terminal VSC-HVDC network, shown in Figure 3.19, which was solved in Chapter 3 for the purpose of a steady-state assessment, is also used here to show the applicability of the multi-terminal VSC-HVDC dynamic model. It is useful to recall that AC1 , AC3 and AC5 stand for the UK Grid, the NORDEL Grid and the UCTE Grid, whose total power demands are 60 GW, 30 GW and 450 GW, respectively. For the sake of simplicity, these AC power networks were represented by their Thevenin equivalents, determined as Rth + jX th = [P + jPtan(𝜃)]−1 , 𝜃 = cos−1 (pf ), pf = 0.95, coupled to reactive tie lines. The network AC2 , which represents the in-feed point of the Valhall oil platform, has a power demand of 78 MW. The equivalent power injections of networks AC4 and AC6 , representing the collector points of the wind parks at the German Bight and Dogger Bank, are rated at 400 MW each. For the purpose of the dynamic simulation, all the AC networks are taken to operate at 50 Hz. 6.6.4.1

Disconnection of a DC Transmission Line

The DC transmission line that connects the nodes d and e, which carries 126 MW during steady-state, is tripped at t = 0.5 s. This disturbance causes the DC ring to open, becoming a radial DC grid. In order to redistribute the power ﬂows in the longitudinal DC network, the DC voltages are adjusted, as seen in Figure 6.61. The DC voltage of the slack converter is ﬁttingly controlled and so are the rest of the DC voltages. Accordingly, the power imbalances throughout the DC grid are mitigated promptly, in only a few milliseconds. The dynamic behaviour of the modulation ratio of the converters is shown in Figure 6.62, where it is observed that only a small readjustment took place following the disconnection of the DC transmission line. The frequencies of the passive networks fed by the converters VSCb , VSCd and VSCf are given in Figure 6.62. It becomes clear from this simulation study that the disturbance in the DC network does not impact adversely the operation of the island AC networks. 6.6.4.2

Three-Phase Fault Applied to AC3

The behaviour of the six-terminal VSC-HVDC system to AC faults is assessed by simulating a three-phase-to-ground short-circuit fault at node 5. This is applied at

6.6 Modelling of Multi-terminal VSC-HVDC Systems for Dynamic Simulations

DC voltage [p.u.]

2.004

VSCa VSCb VSCc VSCd VSCe VSCf

2.002 2 1.998 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

DC power [p.u.]

Time [s] Pab Pbc Pcd Pde Pef Pfa

2 0 –2 –4 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

Figure 6.61 DC voltages and power ﬂows in the DC grid. VSCa VSCc VSCe

0.86 0.859 0.858 0.857 0.4

0.6 0.8 Time [s]

1

0.805 Modulation ratio

Modulation ratio

0.861

VSCb VSCd VSCf

0.8 0.795 0.79

0.4

0.6 0.8 Time [s]

1

Frequency [Hz]

50.01

VSCb VSCd VSCf

50.005 50 49.995 0.3

0.4

0.5

0.6

0.7 Time [s]

0.8

0.9

1

1.1

Figure 6.62 Modulation ratio of the VSCs and frequency of the passive grids fed by VSCb , VSCd , VSCf .

t = 0.5 s and lasts 120 ms (six cycles of the 50 Hz power system). The point in fault is the high-voltage side of the LTC transformer corresponding to the converter station VSCc . The disturbance causes a severe voltage drop at the AC terminal of the converter, as shown in Figure 6.63. As expected, the AC power injected at the fault point, Pvc , follows a similar trend to that of the nodal voltage, dropping from 250 MW to 0 MW for the duration of the fault. Since the 250 MW can no longer be delivered to AC3 , this power ﬂows through the DC ring reaching the converter station VSCa , i.e. the slack converter, as conﬁrmed by the dynamic performance of Pva in Figure 6.63.

315

AC voltage [p.u.]

6 Dynamic Simulations of Power Systems

V4 V5 Vvc

1

0.5

0

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

Active power [p.u.]

5

2

Pva Pvc Pve

4 3 2 1 0

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

2

Figure 6.63 AC voltages and AC powers during the three-phase fault.

DC voltage [p.u.]

2.02

VSCa VSCb VSCc VSCd VSCe VSCf

2.01 2 1.99 1.98

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

4 DC power [p.u.]

316

Pab Pbc Pcd Pde Pef Pfa

2 0 –2 –4

2

0

0.2

0.4

0.6

0.8

1 Time [s]

1.2

1.4

1.6

1.8

2

Figure 6.64 Voltages and powers in the DC network during the three-phase fault.

The short-circuit fault also causes transient voltage rises and power ﬂuctuations throughout the DC network, as seen in Figure 6.64. While the voltage magnitude at node 5 drops to zero for the duration of the three-phase fault, the reactive power injection, Qvc , of the converter station VSCc suddenly increases, as shown in Figure 6.65. Hence, the modulation index controller, ref which is exerting a ﬁxed reactive power setpoint, Qv = 0, forces the modulation ratio of the converter to decrease rapidly so as to comply with its command.

Modulation ratio

6.7 Conclusion

VSCa VSCb VSCc VSCd VSCe VSCf

0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Reactive power [p.u.]

Time [s] 8

Qva Qvc Qve

6 4 2 0 –2 –4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time [s]

Figure 6.65 Modulation ratio of the converters and reactive power injected by VSCc .

6.7 Conclusion This chapter has covered the dynamic models of synchronous generators and their controls: the AVR, hydro and steam turbines, and the speed-governing system. More signiﬁcantly, it has introduced RMS-type dynamic models of the most signiﬁcant VSC-based equipment: the STATCOM, back-to-back, point-to-point and multi-terminal VSC-HVDC systems. All the key aspects of their modelling implementation within the context of a dynamic power ﬂow algorithm have been elucidated. A salient feature of this modelling approach is that the VSC model does not rely on the customary equivalent voltage source but rather on the use of a complex phase-shifting transformer, a kernel in the modelling of the VSC making up the STATCOM and VSC-HVDC systems of various kinds. The ensuing models yield a reasonably accurate representation of their AC and DC voltage dynamics, where the incorporation of the diﬀerential equation that governs both the behaviour of the DC capacitor and DC inductor has turned out to be of paramount importance. Indeed, the AC bus voltage control and the regulation of active power between the AC networks interconnected through a VSC-HVDC link are promising features of this technology. As for the voltage control, each VSC model is ﬁtted with a ﬁrst-order regulator that, upon variations in the network’s voltage, adjusts the modulation ratio in a dynamic fashion to bring the voltage magnitude back to the setpoint value. The control system responsible for keeping the power balance in the converters, achieved by regulating the angular aperture between the AC terminal voltage angle and the ideal transformers’ phase-shifting angle, is a new controller with many potential applications yet to be fully developed. In the case of a VSC-HVDC model with frequency support capabilities, the steady-state VSC-HVDC model computes the power that the HVDC link must transfer

317

318

6 Dynamic Simulations of Power Systems

to satisfy the load connected at the inverter’s end from the inertial response standpoint. In cases where the low-inertia network contains no synchronous generation of its own, that is, a passive network, the inverter provides the angular reference for this network; it is the angle of the inverter’s phase-shifting transformer 𝜑I . Thus, the AC node of the inverter would act as a slack bus for the low-inertia system. The power ﬂowing through the DC link is not dynamically pegged to a speciﬁc constant value; instead, the DC voltages are adjusted to enable a DC power transfer that meets the power requirements of the low-inertia grid fed through the inverter, to mitigate the frequency deviations. The theory presented in this chapter has shown to be useful for assessing the dynamic model of HVDC multi-terminal systems, with an arbitrary number of VSC units and an arbitrary DC network topology. The classiﬁcation of VSC stations in three diﬀerent types of converters – the slack converter, the power-scheduled converter and the passive converter – is key to this development. Their main dynamic features are DC voltage control, DC power and frequency regulation, respectively. In summary, the converters’ controllers act independently to cope with speciﬁc tasks involving voltage and power transfer control. In the modelling approach presented in this chapter, the nonlinear equations describing the models of the various VSC-based devices are linearized together with their associated discretized diﬀerential equations. Furthermore, the state variables corresponding to the VSC units are combined with the state variables of the synchronous generators and their controls as well as with the network’s state variables, all this expressed in a single frame-of-reference for uniﬁed iterative solutions. The implicit trapezoidal integration method embedded within a Newton-Raphson iterative technique has been selected to determine, in time domain, the dynamic response of the whole AC-DC network. The sanity of the RMS results, in time domain, has been checked by comparing selected results with comparable time-domain results obtained with Simulink. The overall satisfaction with the closeness of the results is good, considering the very diﬀerent modelling and solution approaches that the two methods take to solve the same problem. Provided a suitable initialization of the Simulink solution is employed, from the outset, the responses furnished by Simulink are taken to be more accurate than the method put forward in this chapter, since it uses a more detailed representation of the actual equipment; however, this is achieved at the expense of longer computing times. It should be borne in mind that the execution time is an issue that becomes more acute as the size of the AC-DC power system increases.

References 1 Concordia, C. (1951). Synchronous Machines, Theory and performance. Wiley. 2 Kimbark, E.W. (1948). Power System Stability, vol. 3, Synchronous Machines. IEEE

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31

32

interfaced distributed generators in a microgrid system. IEEE Transactions on Industrial Applications 46: 1078–1088. Eghtedarpour, N. and Farjah, E. (2014). Power control and management in a hybrid AC/DC microgrid. IEEE Transactions on Smart Grid 5: 1494–1505. Sao, C.K. and Lehn, P.W. (2008). Control and power management of converter fed microgrids. IEEE Transactions on Power Systems 23: 1088–1098. Castro, L.M. and Acha, E. (2015). On the provision of frequency regulation in low inertia AC grids using HVDC systems. IEEE Transactions on Smart Grid 7 (6): 2680–2690. Acha, E. and Castro, L.M. (2016). A generalized frame of reference for the incorporation of multi-terminal VSC-HVDC systems in power ﬂow solutions. Electric Power Systems Research 136: 415–424. Castro, L.M. and Acha, E. (2016). A uniﬁed modeling approach of multi-terminal VSC-HVDC links for dynamic simulations of large-scale power systems. IEEE Transactions on Power Systems 31 (6): 5051–5060.

321

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment 7.1 Introduction Electromagnetic transient studies are very useful in the analysis and testing of systems of an electrical nature. They constitute the best solution by which to obtain the system behaviour in steady-state and with regard to its transient response, thus providing valuable information about the dynamic performance of diﬀerent systems under any operation condition. Power systems are described by a set of diﬀerential equations which can be either linear or nonlinear. These systems are under the inﬂuence of certain phenomena such as switching actions, short-circuits and disturbances, among others, which can produce abnormal waveforms, fast changes in the voltage and current or electromechanical transients. Furthermore, modern power systems involve a high degree of complexity owing to, among other things, the increasing use of FACTS and HVDC systems which are based on power electronic converters. In this scenario, a mathematical tool which is able to obtain numerical solutions that accurately describe the behaviour of any power system is therefore necessary. Various commercial electromagnetic transient simulators exist which not only provide models of basic electrical elements but also power electronic devices and advanced control functions, and are able to achieve an accurate time response for a great variety of power systems with diﬀerent conﬁgurations. Some of the commercial software packages used for electromagnetic transient analysis are: • ATP-EMTP – Alternative Transients Program-Electromagnetic Transients Program • PSCAD/EMTDC – Power System Computer Aided Design/Electromagnetic Transient Direct Current • SPICE • Matlab Power System Toolbox. This chapter deals with the simulation of the transient response of diﬀerent voltage source-converter based devices. Although any of the above-mentioned packages could be used, the authors have chosen PSCAD/EMTDC owing to the fact that it has been designed to simulate power systems in addition to the implementation of diﬀerent control functions and power-electronic devices. PSCAD/EMTDC is capable of simulating transient responses in both electric and control systems. Since its ﬁrst development in VSC-FACTS-HVDC: Analysis, Modelling and Simulation in Power Grids, First Edition. Enrique Acha, Pedro Roncero-Sánchez, Antonio de la Villa Jaén, Luis M. Castro and Behzad Kazemtabrizi. © 2019 John Wiley & Sons Ltd. Published 2019 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/acha_vsc_facts

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7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

1975, PSCAD/EMTDC has evolved up to the current version, PSCAD X4 (release of version 4.6.2 in June 2017). Detailed information on the features of PSCAD/EMTDC can be found in [1]. The electromagnetic transient responses of four diﬀerent FACTS and HVDC test systems are studied throughout this chapter. Section 7.2 analyzes the behaviour of a STATCOM based on a conventional two-level voltage source converter connected to a distribution grid. An extension of the STATCOM using a three-level ﬂying capacitor converter is developed in Section 7.3. The case of a point-to-point HVDC system of two terminals based on use of multilevel voltage source converters is studied in Section 7.4. Finally, a multi-terminal HVDC system, built with multilevel voltage source converters, is simulated in Section 7.5. The design of the various schemes used to control the diﬀerent power systems is comprehensively explained in the chapter by means of the root-locus technique, and all four examples have been simulated using the educational licence of the latest available version of PSCAD X4.

7.2 The STATCOM Case Reactive-power compensation was traditionally carried out by means of SVCs which have the ability to inject either inductive or capacitive current into the electrical grid. However, these devices may cause resonance problems with other elements of the grid. The modern alternative is the STATCOM, which has the same ability as the SVC but with added advantages: its transient response is faster than that achieved with the SVC, and its enhanced controllability is greater since it is based on the VSC principle, which makes it easier to regulate voltage magnitude and reactive power support. The STATCOM may be used both in transmission systems as a reactive power compensator to enable voltage regulation and in distribution systems to improve power quality at the point of common coupling (PCC) – in the latter case it is termed distribution STATCOM or DSTATCOM [2]. The most basic structure of a three-phase STATCOM comprises a two-level VSC with a DC energy storage device which is connected to the AC mains by means of a coupling transformer. The VSC can be operated by a control system. Figure 7.1 shows the schematic diagram of the DSTATCOM in which a capacitor is used as the DC energy storage device and two diﬀerent loads are connected at the PCC. Figure 7.2 depicts the one-line diagram of the STATCOM connected to the distribution system at the PCC, where the VSC is represented by an ideal voltage source u, vs is the grid voltage, v is the voltage at the PCC and i is the current injected into the grid by the STATCOM. The parameters R and L are the resistance and inductance of the coupling transformer, respectively. The state-variable model of the system shown in Figure 7.2 can be expressed using Park’s transformation in the synchronous reference frame (SRF) together with the PCC voltage. Moreover, the three-phase system may be assumed to be balanced and the following equation is then obtained [3]: ] [ ] [ ][ [ ] ⎡− R 𝜔 ⎤ id − v 1 0 u 1 d id d d ⎥ + (7.1) =⎢ L ⎢ −𝜔 − R ⎥ i dt iq L 0 1 u q q ⎣ L⎦

7.2 The STATCOM Case

Figure 7.1 Basic scheme of a STATCOM connected to a distribution system. Source: ©MDPI, 2014.

Brk PCC Load 2

AC mains

Load 1

STATCOM Coupling transformer Control system

VSC Figure 7.2 One-line equivalent circuit of the STATCOM connected at PCC. Source: ©MDPI, 2014.

R

L

i

PCC v zs

u

vs

where the state variables are the d and q components of the current, namely id and iq , the variables ud and uq are the components of the VSC (control inputs), and 𝜔 is the angular speed of the SRF, i.e. the angular frequency of the grid voltage. This implies that the PCC voltage has only the d component, vd , which is a signal that can be measured. In these conditions, all the sinusoidal variables of frequency 𝜔 become DC-like magnitudes and the instantaneous active power, p, and the instantaneous reactive power, q, injected into the grid by the converter are [4]: p = vd id

(7.2)

q = −vd iq

(7.3)

p and q are therefore DC quantities that can be controlled by the id and iq components, respectively. Unfortunately, model (7.1) is coupled, and changes in the current id will therefore produce variations in iq and vice versa. This issue may be circumvented by

323

324

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

obtaining an equivalent decoupled model of the following form: ][ ] [ ] [ [ ] ⎡− R 0 ⎤ id zd 1 0 1 d id ⎥ (7.4) =⎢ L R⎥ i + L 0 1 ⎢ 0 dt iq zq − q ⎣ L⎦ where zd and zq are the yet-to-be-deﬁned control inputs. Once variables zd and zq have been calculated using a speciﬁc control law, the variables ud and uq will be obtained as: [ ][ ] [ ] [ ] [ ] z ud 0 −𝜔 id v (7.5) =L + d + d uq iq zq 0 𝜔 0 Furthermore, the power pc extracted from the capacitor of the DC energy storage by the VSC can be written as: dvc 1 1 p =− p (7.6) = dt Cvc c Cvc vsc where vc is the voltage of the DC capacitor C and pvsc is the power of the lossless VSC: (7.7)

pvsc = ud id + uq iq

The average voltage of the DC capacitor must be controlled to reduce the energy extracted from the DC capacitor and to guarantee proper control of the reactive power injected into the grid. Moreover, a fast controller will make it possible to reduce the capacitor value. Equation (7.6) is nonlinear. In order to obtain a simpler capacitor-voltage equation, (7.6) can be written as: d(v2c ) dvc 2 (7.8) = = − pvsc dt dt C where v2c is the new state variable. Since the VSC is assumed to be lossless, the power extracted from the capacitor equals the power injected into the grid plus the losses in the coupling transformer: 2vc

(7.9)

pvsc = p + pT = ud id + uq iq The substitution of (7.1) into (7.9) yields: pT

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ di2q

di2d

L L pvsc = Ri2d + Ri2q + + 2 dt 2 dt ⏟⏞⏟⏞⏟ ⏟⏞⏞⏞⏞⏞⏞ ⏟⏞⏞⏞⏞⏞⏞⏟ pR

p

⏞⏞⏞ + vd id

(7.10)

pL

where the term pR represents the losses in the transformer resistance, which are normally quite small (R ≈ 0). Although the term pL (losses in the inductance) will always be zero in steady-state, it will not be zero during transients owing to changes in the current. A current controller should be designed for maximum speed of response in order to reach the steady-state as quickly as possible. A simple approach aimed at fulﬁlling this premise is to assume that pvsc ≈ p [5]. The block diagram of the control scheme for a DSTATCOM is depicted in Figure 7.3. It shows two inner control loops with two identical PI regulators for the d and q components of the current injected into the grid (i.e. the active and reactive powers).

pmax vc_ref

+

Π +

vc

+

Σ

−

PI pref control pmin

+

Π +

zmax id_ref +

Σ

−

id

Vrms

PI control zmin

imax Vrms_ref + vA vB vC

Voltage meas. and filtering

ABC DQ

Σ

PI control

Vector Vrms magnitude

θ

PLL

Figure 7.3 Control system scheme for the DSTATCOM case. Source: ©MDPI, 2014.

id vd iq

zmax

iq_ref

imin

vd vq

−

zd

+

Σ

−

iq

PI control zmin

zq

θ

Decoupling equations

ud uq

DQ ABC

uA uB uC To PWM generators

326

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

The outputs of these controllers are the compound variables zd and zq . The voltage of the DC capacitor is controlled by means of an outer PI controller that is slower than the current controllers and which includes an anti-windup action. It should be noted that this PI regulator controls the state variable v2c . The output of this controller is the reference of the active power that is necessary to maintain the capacitor voltage at the desired value. This reference is divided by the value of the PCC voltage in order to obtain the reference of the d axis current. Another outer PI regulator is also tailored in order to maintain the voltage at the PCC: the output of this controller is the reference of the q component of the current (i.e. the DSTATCOM injects or absorbs reactive power) and, as in the case of the DC voltage regulator, it is slower than the current PI controllers. Since the overall control system is designed in an SRF, a phase locked loop (PLL) is used to obtain the angle required to carry out such a transformation. The inputs of the PLL are the ﬁltered measurements of the grid voltages. This transformation is a power invariant dq0 transformation [6] and will be used throughout the chapter. The chosen transformation has the additional advantage that for balanced sinusoidal conditions, the RMS value of the line-to-neutral voltages and the line currents of the three-phase system equal the magnitude of the resulting vectors in the SRF. Equation (7.11) shows this transformation: 1 1 ⎤ ⎡ √1 √ √ ⎥ ⎢ 2 2 2 ⎡x0 ⎤ √ ⎢ ⎥ ⎡ xA ⎤ ) ( ) ( 2⎢ 4𝜋 ⎥ ⎢ x ⎥ 2𝜋 ⎢xd ⎥ = (7.11) B cos 𝜃 − cos 𝜃 − ⎢ ⎥ 3 ⎢ cos 𝜃 3 3 ⎥ ⎢⎣xC ⎥⎦ ⎣ xq ⎦ ⎥ ⎢ ) ( ) ( 4𝜋 ⎥ ⎢− sin 𝜃 − sin 𝜃 − 2𝜋 − sin 𝜃 − ⎣ 3 3 ⎦ where xA , xB and xC are the electrical magnitudes of the three-phase system, and x0 , xd and xq are the transformed variables in the SRF, which rotates at an angular speed 𝜔 such as 𝜃 = ∫ 𝜔dt. The decoupled Eq. (7.5) is used to obtain the outputs ud and uq , which are transformed into variables of a three-phase system by means of the inverse matrix of the dq0 transformation. These control variables are used to drive the PWM process and to generate the necessary ﬁring signals of the VSC using a sinusoidal PWM scheme [7]. A conventional two-level, three-phase VSC is used to illustrate the control laws/theory described above. It should be noted that for proper operation of the DSTATCOM, the inner control loop (i.e. the PI controllers for the d and q components of the current) should respond much faster than the outer control loop. Two diﬀerent alternatives can be used to implement the proportional-integral control law: zx = kix

∫

(ix

ref

− ix )dt − kpx ix

zx = kix

∫

(ix

ref

− ix )dt + kpx (ix

(7.12) ref

− ix )

(7.13)

where k ix and k px are the controller gains and the subscript x represents either the d axis or the q axis. For equal gain values, the associated closed-loop system will have the same poles for both equations. However, Eq. (7.13) yields zeros in the transfer function of the closed-loop system, which results in a more adverse transient response than that

7.2 The STATCOM Case

obtained with control law (7.12). Hence, Eq. (7.12) is normally used in the design of current regulators. The PI controller for the voltage of the DC capacitor is designed according to the following equation: pref = kic

∫

(v2c ref − v2c )dt + kpc (v2c ref − v2c )

(7.14)

where k ic and k pc are the controller gains. This PI regulator also includes an anti-windup action with its respective upper and lower limits, which resets the integral term of the controller when the limits are reached. This action avoids the integral windup eﬀect [6]. Finally, the voltage regulator at the PCC can be designed only by integrating the error between the reference and the measured voltage: iq

ref

= kiv

∫

(Vrms

ref

− Vrms )dt

(7.15)

where k iv is the integral gain, Vrms_ref is the PCC voltage reference and Vrms is the ﬁltered value of the measured voltage. Since the Thevenin equivalent is seen from PCC changes with load variations, these parameters should be avoided in the voltage control design. Moreover, the dynamics of the ﬁlter used in the measurement of the voltage must be taken into account. The parameter k iv must therefore be chosen with care. A three-phase distribution system such as the one described in Figure 7.1 has been simulated using PSCAD/EMTDC. The RMS line-to-line voltage of the grid is 13.8 kV, with rated frequency equal to 50 Hz. Two diﬀerent loads are connected to the PCC with the following features: • Load 1: Active power 20 MW and reactive power 15 MVAR. • Load 2: Active power 60 MW and reactive power 50 MVAR. Load 1 is permanently connected and a circuit breaker is used to control the connection of Load 2 to the system. A three-phase transformer with a winding ratio 20 kV/62.5 kV is used to connect the VSC to the system. The transformer’s primary and secondary windings are star and delta connected, respectively. The DSTATCOM is connected to the secondary side of the transformer. The respective resistance and leakage inductance of the transformer referred to the 20 kV, primary side are 5 mΩ and 4 mH. A 660 μF capacitor is used in the DC side of the STATCOM as the DC energy storage system. The parameters of the diﬀerent controllers have been obtained using the root-locus technique, which has been successfully applied in [8]: the desired poles for the inner control loop have been placed at s1 = s2 = −1000 rad s−1 , as shown in Figure 7.4, and the corresponding control gains of Eq. (7.13) are k ix = 4000 and k px = 7.995. The time response for a step input in the reference is plotted in Figure 7.5, which shows that no overshoot takes place and the steady-state is reached in approximately 10 ms. The PI regulator for the DC capacitor voltage has been designed to obtain a time response that is much slower than that obtained with the current controllers. In this situation, the dynamics of the inner control loop can be neglected in comparison with the dynamics of the DC voltage control loop. In this example, the chosen poles have been placed at s1 = −20 rad/s and s2 = −100 rad/s and the resulting parameters for control law (7.14) are k ic = −0.66 and k pc = −0.0396. Figure 7.6 shows the location of the closed-loop system poles, whereas Figure 7.7 exhibits the time response for a step

327

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

40

Root locus for current control

30 20

Image axis

10 0 –10 –20 –30 –40 –2000 –1800 –1600 –1400 –1200 –1000 –800 Real axis

–600

–400

–200

0

Figure 7.4 Pole location for the inner control loop. Source: ©MDPI, 2014.

Step response − current control 1 0.9 0.8 0.7 Amplitude

328

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Time (s)

Figure 7.5 Time response of the inner loop obtained for a step reference. Source: ©MDPI, 2014.

7.2 The STATCOM Case

Root locus for DC voltage control

20 15 10

Image axis

5 0 –5 –10 –15 –20 –100

–90

–80

–70

–60

–50 –40 Real axis

–30

–20

–10

0

Figure 7.6 Location of the poles for DC voltage control loop. Source: ©MDPI, 2014.

Step response − DC Voltage control 1.4

1.2

Amplitude

1

0.8

0.6

0.4

0.2

0

0

0.05

0.1

0.15 0.2 Time (s)

0.25

0.3

0.35

Figure 7.7 Time response of the DC voltage control loop for a step reference. Source: ©MDPI, 2014.

329

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment BRK P = −9.981 Q = 0.8112 V = 0.97

12.12e–3 [H]

3.174 [ohm]

#2

#1

Ist

9.522 [ohm]

40.413e–3 [H]

A V

0.5e–3 [ohm]

330

BRK P

Timed Breaker Logic [email protected]

660 [uF]

A

P B C

N N

VSC 2 Levels Gs Gs

Vdc Vdc_st Uabc

Sinusoidal PWM

Uabc

Figure 7.8 Circuit of a VSC-based STATCOM implemented in PSCAD/EMTDC.

input in the reference: no signiﬁcant overshoot is present (less than 10%) and the time response settles in approximately 0.15 s. With regard to the PCC voltage control, the measurement ﬁlter of the voltage has the following ﬁrst-order diﬀerential equation: dvf

(7.16) + vf = vm dt where vm is the measured voltage, vf is the ﬁltered voltage and T f is the smoothing time constant. For this example, T f = 10 ms, and the parameter k iv has been chosen to be equal to 100 after an adjusting process involving diﬀerent simulations. The implementation of the STATCOM in the distribution system of Figure 7.1 is shown in Figure 7.8, while the two-level, three-phase VSC implemented in PSCAD/EMTDC is plotted in Figure 7.9. Note that the VSC was previously modelled as an ideal voltage source for solely control design purposes, but the implementation of the VSC in PSCAD/EMTDC is carried out using elements available in the master library, such as IGBTs and diodes. Furthermore, the three diﬀerent control schemes are Tf

7.2 The STATCOM Case

P

I1

I3

D1

G1

I5

D3

G3

G5

A

I2

B

I4

D2

G2

D5

C

I6

D4

G4

D6

G6

N

Gs 1

2 G1

3 G2

4 G3

5 G4

Gs

6 G5

G6

Figure 7.9 Implementation of the two-level VSC using IGBTs and diodes from the PSCAD/EMTDC master library.

depicted in Figures 7.10–7.12. Finally, Figure 7.13 shows the sinusoidal PWM scheme implemented in PSCAD/EMTDC with a switching frequency of 3150 Hz. The experiment is carried out as follows: Load 1 and the STATCOM are initially connected to the grid; the DC capacitor is discharged and, following a transient period, the DC voltage regulator controls the level of charge of the capacitor until it reaches 100 kV. At t = 0.3 s, Load 2 is connected, producing a balanced voltage sag of approximately 15%. In order to compensate this voltage sag, the PCC voltage control system is connected at t = 0.5 s and the STATCOM injects reactive power to increase the voltage at the PCC. The simulation ends at t = 1 s.

331

iref

1 iref_d

* 2

Pi

2 iref_q

* 50

idq

1

*

Cosine

vdq

2 id

1

iq

ua ub uc

2 vd

Uabc

vq

[DQ_Transformation] Cosine [DQ_Transformation] Sine

Sine

* 4e–3 vd

iq B iref_d

B +

−

1 sT

D + −

+

+

−

ud

−30.0

Cos

Cos a

* 7.9950

id

Cosine Sine

* 2

Pi

* 50

*

ud

* 4e–3 vq

uq

Sin

ds

ds

Sin

ds

d

qs

qs

d

qs

q

Cos

b c

B iref_q

iq

D + −

B

+ +

−

+ +

DsQs->ABC DQ->DsQs

q

id 1 sT

Sin

uq

* 7.9950

Figure 7.10 Inner control loop (current controllers) for the STATCOM implemented in PSCAD/EMTDC.

DQ->DsQs

ua ub uc

7.2 The STATCOM Case

Vdc_ref

*

*

D

+

1 sT

–

+

D

N

* −1

+

iref_d

iref_d

Clear EN Vdc_m

*

clc * 0.0396

*

vd

EN

A

TIME

B Comparator

0.1

A

2.0

EN

B Comparator

|X|

iref_d

clc

Figure 7.11 Control system of the voltage of the DC capacitor implemented in PSCAD/EMTDC.

V_ref

D

+

1 sT

*

−

* −1

iref_q

EN

Vpcc

A

TIME

EN B

0.5

Comparator

Figure 7.12 Control system of the PCC voltage implemented in PSCAD/EMTDC.

Vdc_link

* 0.5

Vdc_med A Vm_A

*

Vtri

Vtri

G1

ComparB ator

Vm_A

A B

Vtri

Comparator

G2

Vdc_med A Vm_B

Vmod_A

* 1.0

Vmod_B

* 1.0

Vm_B

Vmod_C

* 1.0

Vm_C

Vm_A Vtri

G3

ComparB ator

Vm_B

B Vtri

A

Gs

1

2

Vm_C

3

G4

A B

Vtri

Comparator

Comparator

G6

Vmod_C

Uabc

Vmod_B

1 2

G1 G2 G3 G4 G5 G6

Vtri

G5

B Comparator

Vmod_A

Vm_C

A

Figure 7.13 Sinusoidal PWM modulation scheme for two-level VSC of the STATCOM implemented in PSCAD/EMTDC.

333

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

1.1 1.05 1

vrms (p.u.)

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.2

0.25

0.3

0.35

0.4

0.45 0.5 Time (s)

0.55

0.6

0.65

0.7

Figure 7.14 RMS voltage at the PCC in p.u.

10

5

vpcc (kV)

334

0

–5

–10 0.2

0.25

0.3

0.35

0.4

0.45 0.5 Time (s)

Figure 7.15 Line-to-neutral voltage at the PCC in kV.

0.55

0.6

0.65

0.7

7.2 The STATCOM Case

Figure 7.14 shows the RMS voltage at the PCC produced by PSCAD/EMTDC. When the STATCOM starts operating as a voltage regulator, the voltage sag is cancelled out and the voltage at the PCC reaches its target value of 1 p.u. The transient response of the line-to-neutral voltage at the PCC is plotted in Figure 7.15 for phase A: it will be observed that the voltage amplitude is kept constant when the STATCOM compensates the voltage sag and the voltage waveform has a low distortion. The d- and q-axis currents are shown in Figure 7.16a, while the active and reactive powers that the STATCOM exchanges with the grid are plotted in Figure 7.16b. It will be observed that the active power is proportional to id , the reactive power is proportional to −iq and the transient responses are decoupled. The STATCOM provides the necessary reactive power to overcome the voltage sag, while the active power absorbed from the grid remains close to zero. Figure 7.17 shows the DC capacitor voltage, which is kept constant and equal to 100 kV during steady-state operation. Notice that there is a small transient disturbance in the 1 0 id

id, iq (kA)

–1 –2 –3 –4

iq

–5 –6 0.2

0.25

0.3

0.35

0.4

0.45 0.5 Time (s) (a)

0.55

0.6

0.65

0.7

0.65

0.7

70 P (MW), Q (MVAR)

60

Q

50 40 30 20 10

P

0 –10 0.2

0.25

0.3

0.35

0.4

0.45 0.5 Time (s) (b)

0.55

0.6

Figure 7.16 STATCOM waveforms: (a) d and q components of the current injected into the grid; (b) active and reactive powers injected into the grid.

335

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

102

100

98 vc (kV)

336

96

94

92

90 0.1

0.2

0.3

0.4 Time (s)

0.5

0.6

0.7

Figure 7.17 DC capacitor voltage in kV.

response when Load 2 is connected and when the STATCOM begins to operate in voltage control mode. These disturbances are quickly corrected by the control system. Since no switching losses are included and the transformer losses are very low in this study case, the active power that the STATCOM needs to absorb from the grid in order to maintain the capacitor voltage level at 100 kV is very small, as shown in Figure 7.16b. It should be noted that the regulator is designed to attain control of v2c , and this action implies that the control maintains the DC capacitor voltage at its reference value, as shown in Figure 7.17. It is quite clear from these results that the injection of reactive power into the grid can be used to regulate voltage at the PCC and to improve on the poor voltage quality caused by the voltage sags. Moreover, the only exchange of active power between the grid and the STATCOM is owing to the ohmic losses in the coupling transformer.

7.3 STATCOM Based on Multilevel VSC The previous example illustrates how a simple STATCOM can be used to compensate the necessary reactive power in an electrical grid in order to maintain the voltage at the PCC and how it is implemented in PSCAD/EMTDC. Two-level converters are normally used for low-voltage and low-power applications. Nonetheless, as the rating of STATCOMs continues to increase in the realm of reactive-power compensation, the power electronic converters are beginning to be higher-voltage points of the grid, and the use of two-level VSCs is more diﬃcult to justify owing to the high voltages that the switches

7.3 STATCOM Based on Multilevel VSC

must block. Multilevel converter topologies have thus been advanced as a means to reduce the voltage stress on the switching devices [9]. A number of multilevel converter topologies have been put forward, although the most popular are neutral-point-clamped converters (NPC), ﬂying capacitor converters (FC) and H-bridge converters [10]. All of them have beneﬁts and drawbacks, and various PWM techniques can be used to draw on the best control characteristics of these converters [11]. This case study addresses the control capabilities of a multilevel STATCOM based on the FC topology, which has been shown to be more ﬂexible than NPC topology when it comes to increasing the number of levels [12]. Figure 7.18 shows a three-phase, three-level FC converter where the capacitor C 1 is the ﬂying capacitor, one per leg, which is charged at 2vc∕2 = vc . Hence, each switch device must block only half of the voltage in comparison to the voltage that the switch of a conventional two-level converter would withstand, thus enabling the use of devices of a lower rating, albeit with a higher number of switches. The study case developed in Section 7.2 is again used, but this time the two-level VSC is replaced with a three-level FC converter, as shown in Figure 7.18. No parameter change takes place but the phase-shifted pulse width modulation method is used to control the output voltage of the FC converter: for a converter with n levels, n-1 triangular carriers with frequency f sw must be compared with a common sinusoidal modulating waveform per phase with frequency f m , and f sw ≫ f m . The switching instants are obtained at the intersection between the modulating signal and the various carrier signals. A shifting phase of 2𝜋∕n−1 is introduced at each carrier signal, thus obtaining an eﬀective switching frequency of (n − 1)f sw and resulting in a signiﬁcant improvement in the total harmonic distortion of the output voltage [13]. A control scheme with which to balance the FC voltages is also needed to cancel out voltage imbalances in the ﬂying capacitors. These voltage imbalances may be caused by Ta1

Tb1 Da1

Tc1 Db1

Dc1

+ vc

C

−

Ta2 + vC1 −

Tb2

Da2 + A

C1 Ta3

vC1 −

Tc2

Db2

Dc2

+ B

C1 Tb3

Da3

vC1

C

C1

−

Tc3

Db3

Dc3

+ vc −

C Ta4

Tc4

Tb4 Da4

Db4

Figure 7.18 Topology of a three-phase, three-level ﬂying capacitor converter.

Dc4

337

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

asymmetrical conditions in the circuit parameters in the bridge, or by diﬀerences in the switching of the valves. A closed-loop control system modiﬁes the modulating signal by adding a square waveform to increase or reduce the voltage in the ﬂying capacitors [14]. The control method put forward in this publication for the case of n number of converter levels and m switches connected to the positive terminals of the various ﬂying capacitors, where m = 1, …, n − 1, makes use of a square waveform vsq which is added to the modulating signal: vsq = A ⋅ D ⋅ sign(i)

(7.17)

where A is the amplitude of the waveform vsq ; D is a function that indicates that the duty cycle decreases (D = − 1), increases (D = 1) or remains unchanged (D = 0); and variable i stands for the phase current while ‘sign’ is the sign function. The FC voltage is compared with the reference voltage and the result of this comparison is used to calculate the value of D by means of a logic function. Figure 7.19 shows the test system implemented in PSCAD/EMTDC with the three-level FC converter and the control blocks necessary for its correct operation, BRK P = −0.02396 Q = 69.92 V = 0.9983

1 Vc1A Vc1A Vc1B

DA

DA

Vc1C

DB

DB

Iabc

DC

DC

Vc1B Vc1C

BRK

Timed breaker logic [email protected]

Ist VD Vdc_st

Voltage balance DA DA

VD Vdc_st Uabc Uabc

DB

DC

DB

Sinusoidal PWM 3-Level VSC

Vcon Vcon

DC GAs GAs

A

GBs GBs

B

GCs

GCs

C

Vdp P

1320 [uF]

Vdn N

1320 [uF]

Figure 7.19 STATCOM based on the three-level FC converter implemented in PSCAD/EMTDC.

12.12e–3 [H]

3.174 [ohm]

40.413e–3 [H]

#1

EN

9.522 [ohm]

Ist

A V

#2

338

7.3 STATCOM Based on Multilevel VSC

namely the PWM control process and the method used to balance the ﬂying capacitor voltages. The parameters of the two loads and the grid remain the same as in the example shown in Section 7.2. The DC capacitor is made of two identical capacitors of 1320 μF each, connected in series and with a midpoint grounded. Experience has shown that by splitting the overall value of capacitor in this manner, the simulation process speeds up. The PSCAD/EMTDC scheme of the FC converter is shown in Figure 7.20. For this example, the value of the ﬂying capacitors has been set at 110 μF in order to maintain a trade-oﬀ between capacitor size and voltage ripple. The balancing control scheme of the ﬂying capacitor voltages is depicted in Figure 7.21. The amplitude A of the square waveform is initially set at 20 kV in order to quickly charge the ﬂying capacitors to the reference voltage (i.e. half of the voltage of the DC capacitor). This amplitude value is quite high and yields very signiﬁcant harmonic components in the output voltage of the FC converter. In order to reduce this problem to manageable levels, the amplitude value is reduced to 1 kV after 200 ms, which represents a good compromise to maintain the balance of the ﬂying capacitor voltages and to reduce the harmonic content in the output voltage. Figure 7.22 shows the implementation of the phase-shifted PWM method to control the FC converter: the two diﬀerent carrier signals are displaced from one to another by 𝜋 radians and compared with the modulating signal of each phase in order to obtain the switching instants. The modulating signals are the result of the waveforms obtained by Vdp

IA1

IB1

DA1

GA1

DB1

GB1

C1AP

C1BP IA2

IA3

GB2 B IB3

DA3

GA3

DB2

IA4

C

IB4

DB4

GB4

GC4

Vdn

1 2 3 4 GB1 GB2 GB3 GB4 1 2 3 4 GC1 GC2 GC3 GC4

Vc1B

C1CP

Vc1C

C1BN

C1CN

3

C1AN

C1BP

Vc1A

Vc1A

Vc1B Vc1C

GCs

C1AP

2

GBs

1 2 3 4 GA1 GA2 GA3 GA4

1

GAs

DC2

IC3

DC3

IC4

DC4

GC3 C1CN

DA4

GA4

IC2 GC2

DB3

GB3 C1BN

C1AN

110 [uF]

110 [uF]

110 [uF]

A

DC1

C1CP IB2

DA2

GA2

IC1 GC1

Vcon

Figure 7.20 Three-level ﬂying capacitor converter implemented in PSCAD/EMTDC.

339

1

2 ia

EN

* 1.0

−1.0

Vc1ref

B

3

Vc1ref D

ib

ic

+

+ + D + − F

F

D

+

F

−

B + +

+

* F

F

ena

P1B Vc1ref D

ena

Vc1B

+

D − F

D F

B + +

+

* F

P1C

ens

Vc1C

Vc1A

SiaD

ib

SibD

ic

SicD

P1A

A

A

Ctrl = 1

*

*

SiaD

Dvolt

*

*

SiaD

Dvolt

D1B D2B

1.0 Dvolt 20.0 Dvolt

D1A D2A

* −1.0

0.2

Vc1ref

P1A

D

ena

ia

TIME

*

D

D1A

*

P1B

SibD

* −1.0

D2A

B Ctrl

Dvolt

DA

DB

* SibD

D1C D2C

Iabc

−1.0

–1.0

* 0.5

VD

DC

B Comparator

Figure 7.21 Voltage-balance method for the ﬂying capacitors implemented in PSCAD/EMTDC.

*

D1B

P1C

Dvolt

* Dvolt

D2B

* −1.0

*

*

SicD

Dvolt

*

*

SicD

Dvolt

D1C

D2C

A GA1 Comparator

B Vtri_1

Vtri_1

VD_med

Vmod_BD1

Vtri_2

GB1 B

Vtri_1

A

Vmod_CD1

Comparator

GA2 Comparator

B Vtri_2

GB2 B

Comparator

A

Vmod_BD2 Vtri_2

GC1 B

Vtri_1

A

Vmod_AD2 *

A

A

Vmod_AD1 *

Vmod_CD2

Comparator

GC2 B

Vtri_2

Comparator

VD_med * 0.5

GB2

GB3

GC2

GC3

GA4

GB1

GB4

GC1

GC4

GA4 4

D1A

D

GB4

D2A

GBs

+ +

Vmod_AD1

D1B

D

D

+ +

GC4

Vmod_BD1

+

D1C

D

Vmod_AD2

D2B

D

+

Vmod_BD2

+

+ +

Vmod_CD1

F Vmod_C

D2C

D

F Vmod_B

+ +

Vmod_CD2

F Vmod_C

GCs

Figure 7.22 Phase-shifted PWM modulation scheme for the three-level FC converter implemented in PSCAD/EMTDC.

1

2 D2C

DC

2

D1C

1

D2B

DB

2

D1B

1

D2A

DA

D1A

3

Vmod_C

2

Vmod_B

1

+ F Vmod_B

F Vmod_A

Vmod_A

GA3

GA3

GA1

F Vmod_A

4

GA2

GA2 VD_med

GAs

4

1

2 3 GB3

GB2

GB1

1

2 3 GC3

GC2

GC1

1 Uabc

2 3

GA1

VD

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

1.1 1.05 1

vrms (p.u.)

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.2

0.25

0.3

0.35

0.4

0.45 0.5 Time (s)

0.55

0.6

0.65

0.7

Figure 7.23 RMS voltage at the PCC obtained with the three-level FC converter.

10

5

vpcc (kV)

342

0

–5

–10 0.2

0.25

0.3

0.35

0.4

0.45 0.5 Time (s)

0.55

Figure 7.24 Instantaneous line-to-neutral voltage at the PCC.

0.6

0.65

0.7

7.3 STATCOM Based on Multilevel VSC

the current controller plus the square signals generated to maintain the voltage balance of the ﬂying capacitors. The frequency of the carrier signals (i.e. the switching frequency) is set at 1950 Hz. Figure 7.23 shows the RMS voltage at the PCC: the voltage sag caused by the connection of Load 2 is compensated when the multilevel STATCOM injects reactive power at time t = 0.5 s. The corresponding instantaneous line-to-neutral voltage at PCC is shown in Figure 7.24: it should be noted that the waveform is very similar to that obtained with the two-level VSC. A thorough analysis of the harmonic content is shown in Figure 7.25: the harmonic spectrum of the voltage obtained with the two-level VSC is higher than that produced with the three-level FC converter. The voltage total harmonic distortions (THDs) are 2.5% and 1.4%, respectively. Furthermore, the harmonics caused by the switching frequency are located at around 3150 Hz for the two-level case and at around 0.5

Amplitude (kV)

0.4 0.3 0.2 0.1 0

0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz) (a)

0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz) (b)

0.5

Amplitude (kV)

0.4 0.3 0.2 0.1 0

Figure 7.25 Detail of the spectrum of the line-to-neutral voltage at the PCC with: (a) conventional two-level VSC, and (b) three-level FC converter.

343

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

150

VSC voltage (kV)

100 50 0 –50

–100 –150 0.6

0.62

0.64

0.66

0.68

0.7 0.72 Time (s) (a)

0.74

0.76

0.78

0.8

0.62

0.64

0.66

0.68

0.7 0.72 Time (s) (b)

0.74

0.76

0.78

0.8

150 100 VSC voltage (kV)

344

50 0 –50

–100 –150 0.6

Figure 7.26 Line-to-line output voltage of the VSC obtained with: (a) conventional two-level VSC, and (b) three-level FC converter.

3900 Hz for the three-level case – it should be recalled that the switching frequency was set at 1950 Hz for the FC converter. A detailed view of the converter-output voltage for both types of converters is plotted in Figure 7.26, in which the three levels in the output voltage of the FC converter (Figure 7.26b) can be clearly appreciated. This represents a clear advantage over the waveform of the two-level converter (Figure 7.26a). It could be argued that the former follows a sinusoidal waveform more closely than the latter. Figure 7.27 shows the d and q current components and the active and reactive powers exchanged with the grid. The time responses of the two current components are again decoupled; the active power exchanged with the grid is proportional to id and is very close to zero: the small value is necessary to compensate the losses of the transformer and the switching internal losses of the valves. In this test case, no switching losses are considered. The reactive power is proportional to −iq , and the injected amount of

7.3 STATCOM Based on Multilevel VSC

1 0 id

id, iq (kA)

–1 –2 –3 –4

iq

–5 –6 0.2

0.25

0.3

0.35

0.4

0.45 0.5 Time (s) (a)

0.55

0.6

0.65

0.7

0.65

0.7

70 P (MW), Q (MVAR)

60 Q

50 40 30 20 10

P

0 –10 0.2

0.25

0.3

0.35

0.4

0.45 0.5 Time (s) (b)

0.55

0.6

Figure 7.27 Waveforms of the STATCOM obtained with the three-level FC converter: (a) d and q components of the current injected into the grid; and (b) active and reactive powers injected into the grid.

reactive power increases to compensate the voltage sag and to maintain the RMS voltage at around 1 p.u., as shown in Figure 7.23. Figure 7.28 shows the time response of the DC-capacitor voltage: the steady-state value is kept constant at around 100 kV. There are three disturbances at diﬀerent time instants that the control system of the DC-capacitor voltage quickly corrects, as can be appreciated in the ﬁgure. The voltages of the ﬂying capacitors are plotted in Figure 7.29. The amplitude of the square waveform was initially set at 20 kV in order to charge the capacitors to the reference voltage of 50 kV and, after 200 ms, the amplitude decreases to 1 kV. Figure 7.29 shows that in the time interval t = 0.2 s and t = 0.5 s, the control system of the voltage balance is not able to maintain the voltages at their reference value, showing small deviations. This is owing to the fact that the value of the current during this interval is almost

345

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

102

100

vc (kV)

98

96

94

92

90 0.1

0.2

0.3

0.4 Time (s)

0.5

0.6

0.7

Figure 7.28 Voltage of the DC capacitor in kV.

50 40 vfcA (kV)

346

30 20 10 0

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (s) (a) Figure 7.29 Voltages of the three ﬂying capacitors: (a) phase A, (b) phase B, and (c) phase C.

0.7

7.4 Example of HVDC based on Multilevel FC Converter

50

vfcB (kV)

40 30 20 10 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.4

0.5

0.6

0.7

Time (s) (b) 50

vfcC (kV)

40 30 20 10 0

0

0.1

0.2

0.3 Time (s) (c)

Figure 7.29 (Continued)

zero, as shown in Figure 7.27a. This issue is resolved at t = 0.5 s when the q component of the current changes drastically and the voltages of the ﬂying capacitors are eﬀectively balanced.

7.4 Example of HVDC based on Multilevel FC Converter The following example illustrates the implementation in PSCAD/EMTDC of a monopolar, point-to-point HVDC system with two terminals which employs two three-level FC converters. Figure 7.30 shows the schematic diagram of the HVDC link: this connects two diﬀerent electrical grids, one to the AC side of the rectiﬁer VSC 1 and the other to the VSC 2. In each case, the connection is carried out by using coupling transformers. Each converter has a capacitor on its DC side as an energy-storage device, and both converters are connected through a DC line. The DC line is modelled as an inductor in series

347

348

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

Grid 1

Coupling transformer 1

VSC 1

VSC 2

Coupling transformer 2

Grid 2

Figure 7.30 Basic scheme of an HVDC system.

with a resistance; these parameters reduce with the length of the line and become zero in the back-to-back HVDC schemes. In these cases, the resistance and the inductance are not taken into account and both converters share one equivalent capacitor. In the scheme shown in Figure 7.30, the active power ﬂow is bidirectional since the two converters can be operated either in inverter mode or in rectiﬁer mode. Moreover, both converters are able to provide ancillary services such as VAR support, since each VSC controls its own reactive power. One of the converters controls the DC voltage in order to maintain the power balance, e.g. VSC 1, and the other converter controls the active power setpoint, e.g. VSC 2, [11]. The DC voltage control system should be designed to counteract disturbances, such as power imbalances and losses, and even possible modelling errors caused by inaccuracies in the data set and random variations. In Figure 7.30, the state-space equation of the capacitor C 1 associated with terminal 1 is: 2vc1

d(v2c1 ) dvc1 2 (p − ploss − p1 ) = = dt dt C1 2

(7.18)

where vc1 is the voltage of the VSC 1 capacitor, p2 is the power of terminal 2, ploss is the term owing to losses and p1 is the power of terminal 1. From the discussion above, it follows that the control system of terminal 1 coincides with that of the STATCOM case and this is plotted in Figure 7.31 for completeness. The control scheme of terminal 2, meanwhile, is shown in Figure 7.32, where it will be observed that the converter VSC 2 does not control the DC voltage but rather follows an active-power reference. It should be noted that the value of this reference can be either positive or negative since the active-power ﬂow is bidirectional. pmax

vc1

zmax

id_ref1 PI pref1 ÷ Σ Σ control + – + – id1 pmin Vrms1

vc_ref 1 + Π + +

Π +

PI control zmin

imax Vrms_ref1 + vA1 Voltage vB1 meas. and vC1 filtering

ABC DQ

Σ

vd

–

PI control imin

iq_ref1 +

Σ

– iq1

PI control zmin

zd1 id1 Decoupling ud1 vd1 iq1 uq1 zmax equations zq1

Vector Vrms1 vq magnitude

θ1 PLL

Figure 7.31 Control system for terminal 1 of the HVDC. Source: ©MDPI, 2014.

θ1 DQ ABC

uA1 uB1 uC1 To PWM generators

7.4 Example of HVDC based on Multilevel FC Converter zmax pref2

i ÷ d_ref2 Σ + – id2 Vrms2

PI control zmin

imax Vrms_ref 2 + vA2 Voltage vB2 meas. and vC2 filtering

ABC DQ

vd

Σ

–

PI control

imin

iq_ref 2 +

Σ

– iq2

PI control

zd2 id2 Decoupling ud2 vd2 iq2 uq2 zmax equations zq2

θ2 DQ ABC

uA2 uB2 uC2 To PWM generators

zmin

Vector V rms2

vq magnitude

θ2 PLL

Figure 7.32 Control system for terminal 2 of the HVDC.

The method used to design the various PI regulators is the same as that explained in Section 7.2. The two-terminal HVDC system implementation in PSCAD/EMTDC is shown in Figure 7.33. The two electrical grids with diﬀerent network parameters and various kinds of loads connected to their respective points of common coupling have the following characteristics: the line-to-line voltage and frequency of the grid connected to terminal 1 are 13.8 kV and 50 Hz, while the rated voltage and frequency of the grid connected to terminal 2 are 13.8 kV and 60 Hz. The loads connected to both power systems are used to illustrate the ancillary service capabilities that the HVDC system should be able to provide to the two electrical grids, e.g. voltage support. The load characteristics are as follows: • Grid 1: – Load 1. Constant active power 30 MW. – Load 2. Active power 4.76 MW and inductive reactive power 12.12 MVAR. Load 2 is connected by means of a circuit breaker at time instant t = 0.25 s. • Grid 2: – Load 1. Constant active power 80 MW and constant inductive reactive power 15 MVAR. – Load 2. Constant capacitive reactive power 40 MVAR. Load 2 is connected by means of a circuit breaker at time instant t = 1.2 s. The coupling transformers have winding ratios of 20 kV/62.5 kV with a star/delta connection and the same parameters as those used in the transformers in the example shown in Section 7.2. Each terminal uses a 660 μF capacitor as DC energy storage: these capacitors are made up of two capacitors of 1320 μF in series with their midpoints grounded in order to speed up the simulation process, as explained in the example in Section 7.3. The DC line is modelled as a resistance of 2.5 Ω in series with an inductance of 2 mH. The three-level FC converters use 47 μF ﬂying capacitors. The reference for the DC voltage is 100 kV and it is controlled by terminal 1. The switching frequencies of the converters are 3150 Hz and 3780 Hz for terminal 1 and terminal 2, respectively. In both cases, the frequency modulation index is mf = 63. The parameters of the various controllers have been obtained by using the root-locus technique and they have the same values as those obtained in Section 7.2, since

349

P=0 Q=0

BRK2

80 [MW] 15 [MVAR]

30 [MW]

Vcon_2

Vcon Vdp P

B

#2

C

Vdn N

DA1 DB1 DC1 DA

DB

2e-3 [H]

2.5 [ohm]

1320 [uF] 1320 [uF]

A

#1

I2

Vcon_1

Terminal 1 Timed breaker logic [email protected]

Pr Vdp

DA2 DB2 DC2

GCs GBs GAs

DC

DA

DB

Uabc1

Uabc

Sinusoidal PWM 3-Level VSC

GAs

Figure 7.33 Two-terminal HVDC system implemented in PSCAD/EMTDC.

Vdc_2 Uabc Uabc2

A

Sinusoidal PWM 3-Level VSC

C GAs GBs GCs

DC GAs

VD GBs

Terminal 2

Vcon

B Vdn

GCs VD Vdc_1

A V

50e–3 [H]

40 [ohm]

I1

A V

BRK

P=0 Q=0 V=0

40 [MVAR]

1320 [uF] 1320[uF]

A V

BRK P=0 Q=0 V=0

GBs GCs

BRK2

#2

#1

Timed breaker logic [email protected]

7.4 Example of HVDC based on Multilevel FC Converter

the parameters of the coupling transformers and the DC capacitors have remained unchanged. The control scheme used to balance the ﬂying capacitor voltages in Section 7.3 has been used again. The amplitude of the square waveform is initially set at 12 kV in order to charge the ﬂying capacitors very quickly, and this amplitude is then reduced to 600 V at time instant t = 0.2 s to maintain the voltages of the ﬂying capacitors at around their reference values. The event sequence is the following: the current controllers, the DC voltage regulator at terminal 1 and the controllers used to balance the ﬂying capacitor voltages in both terminals are, initially, the only control systems enabled. At this time interval, the reference of active power, i.e. the reference of id , in terminal 2 is set at zero. After 250 ms, the DC voltage reaches its reference value of 100 kV and the control systems of the PCC voltages are enabled at both terminals. This action releases the full STATCOM capabilities of the two converters to their respective electrical grids. At time instant t = 0.5 s, the active-power reference in terminal 2 is set at 40 MW, i.e. terminal 2 injects active power into grid 2, and at t = 0.9 s this reference value changes to −10 MW, i.e. terminal 2 absorbs active power from grid 2. The simulation ends at t = 1.5 s. Figure 7.34 shows the most relevant parameters of grid 1, while those of grid 2 are plotted in Figure 7.35. The HVDC system is able to compensate in steady-state the variations of voltage at their respective points of common coupling, supplying or absorbing the necessary reactive power to maintain these voltage values at 1 p.u., as shown in Figures 7.34a and 7.35a, respectively. A salient point of this compensation is when Load 2 of grid 2 is connected at t = 1.2 s: the converter at terminal 2 of the HVDC link injects reactive power into the grid and, at this instant, Load 2 provides a very signiﬁcant amount of additional reactive power which is quickly absorbed by the converter at terminal 2 in order to maintain the PCC voltage equal to 1 p.u., as can be seen in Figure 7.35a and, more importantly, in Figure 7.35b. The HVDC transmits power from one terminal to another: when the active-power reference at terminal 2 is set at 40 MW, the converter at terminal 1 absorbs active power from grid 1 and transmits it in DC form through the cable to reach the converter at terminal 2 and into grid 2, excepting the losses. When the power reference changes from 40 to −10 MW, the converter at terminal 2 absorbs active power from grid 2 and this energy, excepting the losses, is injected into the converter at terminal 1 and into grid 1. These results are illustrated in Figures 7.34b and 7.35b. As in the previous examples, the power is proportional to the d component of the current, whereas the reactive power is proportional to the q component of the current, as can be seen by comparing Figures 7.34b,c for the case of grid 1 and Figures 7.35b,c for the case of grid 2. Figure 7.36 shows a detailed view of the voltages at the points of common coupling for both grids. The time interval is 0.6 s < t < 0.8 s, where the converter at terminal 1 absorbs 40.7 MW from grid 1 and injects, approximately, 19 MVAR into grid 1. The converter at terminal 2 injects 40 MW and 18 MVAR into grid 2. The multilevel-converter conﬁguration achieves its operation with low-distortion output voltages, signifying that the harmonic content of the PCC voltages is very much reduced. The voltages of the DC capacitors are plotted in Figure 7.37 for both converters. A detailed view of these voltages is shown in Figure 7.37c for the capacitor voltage of the converter at terminal 1 and in Figure 7.37d for the capacitor voltage of the converter at terminal 2. Note that the capacitor voltage of the converter at terminal 1 has a value of 100 kV in steady-state and that the diﬀerence between the two capacitor voltages

351

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

1.1

vrms 1 (p.u.)

1.05 1 0.95 0.9 0.2

0.3

0.4

0.5

0.6

0.7

0.8 0.9 Time (s)

1

1.1

1.2

1.3

1.4

1.5

1

1.1

1.2

1.3

1.4

1.5

1.3

1.4

1.5

(a)

P1 (MW), Q1 (MVAR)

40 20

Q1

0 –20 P1

–40 –60 0.2

0.3

0.4

0.5

0.6

0.7

0.8 0.9 Time (s) (b)

2 id 1

1 id1, iq1 (kA)

352

0 –1

iq1

–2 –3 –4 0.2

0.3

0.4

0.5

0.6

0.7

0.8 0.9 Time (s) (c)

1

1.1

1.2

Figure 7.34 Main electrical magnitudes of grid 1: (a) RMS voltage at PCC, (b) active and reactive powers, and (c) d and q components of the current injected into the grid by the VSC 1.

7.4 Example of HVDC based on Multilevel FC Converter

1.1

vrms2 (p.u.)

1.05 1 0.95 0.9 0.2

0.3

0.4

0.5

0.6

0.7

0.8 0.9 Time (s) (a)

1

1.1

1.2

1.3

1.4

1.5

1.2

1.3

1.4

1.5

P2 (MW), Q2 (MVAR)

60 P2 40

Q2

20 0 –20 0.2

0.3

0.4

0.5

0.6

0.7

0.8 0.9 Time (s)

1

1.1

(b) 4

id 2

id2, iq2 (kA)

2

iq2

0 –2 –4 0.2

0.3

0.4

0.5

0.6

0.7

0.8 0.9 Time (s) (c)

1

1.1

1.2

1.3

1.4

1.5

Figure 7.35 Main electrical magnitudes of grid 2: (a) RMS voltage at PCC, (b) active and reactive powers, and (c) d and q components of the current injected into the grid by the VSC 2.

353

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

12 8

vpcc1 (kV)

4 0 –4 –8 –12 0.6

0.62

0.64

0.66

0.68

0.7 0.72 Time (s)

0.74

0.76

0.78

0.8

0.74

0.76

0.78

0.8

(a) 12 8 4 vpcc2 (kV)

354

0 –4 –8 –12 0.6

0.62

0.64

0.66

0.68

0.7 0.72 Time (s) (b)

Figure 7.36 Detail of the PCC voltages for the time interval 0.6 s < t < 0.8 s: (a) voltage at PCC 1, and (b) voltage at PCC 2.

is owing to the losses in the DC line. When the converter at terminal 2 injects active power into grid 2, its capacitor voltage is lower than the capacitor voltage of the converter at terminal 1, i.e. 100 kV in steady-state. This can be observed in the time interval 0.5 s < t < 0.9 s. Conversely, when the converter at terminal 2 absorbs active power from grid 2, i.e. t > 0.9 s, the capacitor voltage of the converter at terminal 2 is higher than the capacitor voltage of the converter at terminal 1. Figures 7.38 and 7.39 show the ﬂying capacitor voltages of the three-level converters associated with terminal 1 and terminal 2, respectively. The transient response quickly

7.4 Example of HVDC based on Multilevel FC Converter

120

vc1 (kV)

100 80 60 40 20 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) (a)

1

1.1 1.2 1.3 1.4 1.5

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s)

1

1.1 1.2 1.3 1.4 1.5

1

1.1 1.2 1.3 1.4 1.5

1

1.1 1.2 1.3 1.4 1.5

120

vc2 (kV)

100 80 60 40 20 0

(b)

vc1 (kV)

105

100

95

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) (c)

vc2 (kV)

105

100

95

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) (d)

Figure 7.37 Detail of the PCC voltages for the time interval 0.6 s < t < 0.8 s: (a) voltage at PCC 1, and (b) voltage at PCC 2.

355

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

60

vfcA1 (kV)

50 40 30 20 10 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s)

1

1.1 1.2 1.3 1.4 1.5

1

1.1 1.2 1.3 1.4 1.5

1

1.1 1.2 1.3 1.4 1.5

(a) 60

vfcB1 (kV)

50 40 30 20 10 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) (b)

60 50 vfcC1 (kV)

356

40 30 20 10 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) (c)

Figure 7.38 Voltages of the three ﬂying capacitors of terminal 1: (a) phase A, (b) phase B, and (c) phase C.

7.4 Example of HVDC based on Multilevel FC Converter

60

vfcA2 (kV)

50 40 30 20 10 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s)

1

1.1 1.2 1.3 1.4 1.5

1

1.1 1.2 1.3 1.4 1.5

1

1.1 1.2 1.3 1.4 1.5

(a) 60

vfcB2 (kV)

50 40 30 20 10 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) (b)

60

vfcC2 (kV)

50 40 30 20 10 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) (c)

Figure 7.39 Voltages of the three ﬂying capacitors of terminal 2: (a) phase A, (b) phase B, and (c) phase C.

357

358

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

reaches 50 kV owing to the high amplitude of the square waveform (i.e. 12 kV) used at the initial time interval. The amplitude of the square waveform is then reduced to 600 V, which is a suﬃciently high value to balance the voltages of the three capacitors in both converters throughout the rest of the simulation experiment.

7.5 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters There is a great deal of interest in exploring the feasibility of multi-terminal HVDC systems as technical solutions that may yield more competitive solutions than a conventional AC transmission system from the technical and economic vantage points. For instance, the perceived wisdom is that HVDC systems require less investment and smaller footprints for equal power-transmission capabilities. Furthermore, multi-terminal HVDC systems exhibit additional advantages such as the ability to connect a number of power grids in an asynchronous manner, unlike point-to-point HVDC systems which can only exchange active power between two terminals [15]. A number of multi-terminal HVDC system topologies have been put forward in the literature, such as those reported in [15] and in [16] relating to the oﬀshore wind farm installations in the North Sea under current consideration. A generic multi-terminal HVDC system is shown in Figure 7.40 in which four terminals are connected to a common DC bus. This conﬁguration allows the connection of four diﬀerent electrical grids, and it has been suggested that this would establish a typical conﬁguration of DC distribution systems for city centres [17], although many other topologies are possible.

Grid 1

Coupling transformer 1

VSC 1

VSC 2

Coupling Grid 2 transformer 2

DC BUS

Grid 3

Coupling transformer 3

VSC 3

Figure 7.40 Scheme of a multi-terminal HVDC system.

VSC 4

Coupling Grid 4 transformer 4

7.5 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters

In multi-terminal HVDC systems, one of the terminals is dedicated to the control of the DC bus voltage and in many ways acts as the voltage reference, whereas the other terminals operate in active-power control mode. All the terminals are initially disconnected from the DC bus and they work in DC voltage control mode. This feature allows the converter capacitors at each terminal to be independently energized from the AC side in order to attain the same voltage value. At this time, all the converters are connected to the DC bus and one and only one of the converters takes charge of controlling the DC bus voltage while the others can be used to control the exchange of active power with their respective AC sides. The block diagram for the terminal which controls the voltage at the DC bus is depicted in Figure 7.3, which coincides with that of DSTATCOM. Figure 7.41 shows the general control scheme for the remaining terminals of the HVDC system: the subscript n applies to any of the non-voltage control type converters; instead, they have a setpoint for active power. The terminal VSCs initially charge their capacitors at the desired voltage levels. When this voltage level is reached, the terminal converters are connected to their respective DC buses and their control systems switch to the active-power control mode. The rest of the control functions used in the multi-terminal HVDC system are identical to those described for the terminal that controls the DC bus voltage. The PSCAD/EMTDC implementation of a four-terminal HVDC system is shown in Figure 7.42. Each terminal comprises a three-level FC converter with a DC capacitor of 330 μF which plays the role of DC energy storage system. Terminal 1 controls the DC bus voltage while the other three terminals operate in their active-power control mode. A total equivalent capacitor of 1320 μF is obtained when the four terminals are connected to the DC bus. The main parameters of the coupling transformers associated with each terminal are summarized in Table 7.1, while the parameters of the diﬀerent AC grids are shown in Table 7.2.

p pmax ref n 2 vc_ref n + Π + vc n + Π +

+

Σ

–

zmax id_ref n Σ + – Vrms n idn

PI pref n 1 control pmin

÷ sw

PI control zmin

imax Vrms_ref n

Σ

+ – vAn vBn Voltage meas. and vCn filtering

ABC DQ

vd vq

PI control imin

iq_ref n Σ + – iqn

PI control

zdn idn Decoupling udn vdn iqn uqn zmax equations zqn

zmin

Vector V rms n magnitude

θn PLL

Figure 7.41 Control scheme of a terminal with active-power reference.

θn DQ ABC

uAn uBn uCn To PWM generators

359

BRK_G1

BRK_G2

P=0 Q=0 V=0

P=0 Q=0 V=0

40 [MVAR]

A V

30 [MW]

A V

80 [MW] 15 [MVAR] 12 [MVAR]

I1

I2

5 [MW]

Vcon_1

Terminal 1

Vcon Vdp P

A #2

Vdn

C

Timed breaker logic [email protected]

BRK_T2 1e–3 [H]

2 [ohm]

Vcon_2

Terminal 2

Vcon Pr2 Vdp 330 [uF]

#1

B

660[uF]660[uF]

BRK_G1

BRK_T2

Timed breaker logic [email protected]

A

BRK_G2

B Vdn

#2

Timed breaker logic [email protected]

#1

C

N

DA

DB

GCs GBs GAs

GAs GBs GCs 0.2[ohm]

DC GCs

DA

DB1

Vdc_1 Uabc Uabc1

DC

Sinusoidal GBs PWM 3-Level VSC GAs

0.1e–3 [H]

VD

DA1 DB

DC1 BRK_T3

Terminal 3

25 [MW] 20 [MVAR]

VD

B

#2

C

Vdn

DA2 DB DB2 DC DC2

Voltage balance

P=0 Q=0 V=0

A B

#2

#1

Eabc Vdn

DA

20 [MVAR] 30 [MW]

Vcon Pr4 Vdp

330 [uF]

#1

I3

Terminal 4

Vcon_4

Vdp Pr3

330 [uF]

A V

Sinusoidal Vdc_2 PWM Uabc 3-Level VSC Uabc2

GBs

Vcon_3

A

Vc1A Vc1A2 Vc1B Vc1B2 Vc1C Vc1C2 Iabc I2 VD Vdc_2

DA

GCs

GAs

Vcon

P=0 Q=0 V=0

DB

BRK_T4

Voltage balance

EN EN

DC2 DB2 DA2 DC

0.08e–3 [H]

Vc1A Vc1A1 Vc1B Vc1B1 Vc1C Vc1C1 Iabc I1 VD Vdc_1

DA1 DB1 DC1

0.2 [ohm]

EN EN

A V

I4

C

N

0.1[ohm]

GCs GBs GAs DA3 DB3 DC3 DA

DB

DC GCs

VD Vdc_3 Uabc Uabc3

EN EN

BRK_T3

Timed breaker logic [email protected]

Sinusoidal GBs PWM 3-Level VSC GAs

Timed breaker logic [email protected]

GCs

DC4 DB4 DA4 DC

DB

DA

GCs VD

Sinusoidal Vdc_4 PWM Uabc Uabc4 GAs 3-Level VSC GBs

EN EN

Vc1A4 Vc1B

DA DA3 DB DB3

Iabc

DC

I3

DC3 VD

BRK_T4

GBs

Vc1A

Vc1 A Vc1A3 Vc1 B Vc1B3 Vc1 C Vc1C3

Vdc_3

GAs

Voltage balance

Figure 7.42 Four-terminal HVDC system implemented in PSCAD/EMTDC.

Vc1B4 Vc1C Vc1C4 Iabc I4 VD Vdc_4

DA DA4 DB DB4 DC DC4

Voltage balance

7.5 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters

Table 7.1 Main parameters of the coupling transformers. Coupling transformers

Winding voltages

Leakage inductance (mH)

Copper losses (m𝛀)

Operation frequency (Hz)

Transformer 1

20 kV/62.5 kV (Y/Δ)

4

5

50

Transformer 2

20 kV/62.5 kV (Y/Δ)

4

5

60

Transformer 3

15 kV/62.5 kV (Y/Δ)

3

5

50

Transformer 4

24.5 kV/62.5 kV (Y/Δ)

5

8

50

Table 7.2 Electric parameters of the diﬀerent grids.

Grids

Rated voltage (kV)

Grid inductance (mH)

Grid resistance

Operation frequency (Hz)

Grid 1

13.8

1.5

–

50

Grid 2

13.8

1.8

–

60

Grid 3

11

1.3

100 mΩ

50

Grid 4

20

1

–

50

The various loads connected at the PCC of each grid have the following characteristics: • Grid 1: – Load 1. Constant active power of 30 MW. The load is permanently connected. – Load 2. Constant active power of 5 MW, constant inductive reactive power of 12 MVAR. This load is connected by means of a circuit breaker at the time instant t = 0.25 s. • Grid 2: – Load 1. Constant active power of 80 MW, constant inductive reactive power of 15 MVAR. The load is permanently connected. – Load 2. Constant capacitive reactive power of 40 MVAR. The load is connected by means of a circuit breaker at the time instant t = 1.1 s. • Grid 3: – Load. Constant active power of 25 MW, constant inductive reactive power of 20 MVAR. The load is permanently connected. • Grid 4: – Load. Constant active power of 30 MW, constant inductive reactive power of 20 MVAR. The load is permanently connected. The state-variable model of each of the terminals is the same as that described in Section 7.2 and the locations of the closed-loop poles of the inner control loop are therefore chosen to be the same as those used in the STATCOM test case, i.e. s1 = s2 = − 1000 rad s−1 . However, the pole locations of the closed-loop system for the

361

362

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

Table 7.3 Parameters of the inner control loop for each terminal. Terminal

kix Gain

kpx Gain

Terminal 1

k ix = 4000

k px = 7.995

Terminal 2

k ix = 4000

k px = 7.995

Terminal 3

k ix = 3000

k px = 5.995

Terminal 4

k ix = 5000

k px = 9.995

Table 7.4 Parameters of the DC voltage control loop for each terminal. Terminal

kic Gain

kpc Gain

Terminal 1

k ic = 0.66

k pc = 0.0462

Terminal 2

k ic = 0.33

k pc = 0.0198

Terminal 3

k ic = 0.33

k pc = 0.0198

Terminal 4

k ic = 0.33

k pc = 0.0198

DC voltage control loop have been placed at s1 = −20 rad s−1 and s2 = −50 rad s−1 for terminal 1, which is responsible for controlling the voltage at the DC bus, and s1 = −20 rad s−1 and s2 = −100 rad s−1 for the remaining three terminals. It should be noted that the last three converters control the voltage of their respective DC capacitors only until they have reached the reference voltage and they are then switched to the active-power control mode. The parameters of the inner control loop and the DC voltage control loop are summarized in Table 7.3 and in Table 7.4, respectively. The parameter of the PCC voltage control loop has the same setting as that used in the example in Section 7.2, i.e. k iv = 100. The three-level FC converters use the control scheme described in Section 7.3 to balance the ﬂying capacitor voltages of the converters at the four terminals: the initial amplitude of the square waveform has been set at 12 kV and is reduced to 600 V when the ﬂying capacitor voltages have been charged to their reference voltages, i.e. at instant t = 0.2 s. The modulation process of all the FC converters uses a frequency modulation index mf = 63, and the switching frequency for terminals 1, 3 and 4 is therefore 3150 Hz. The switching frequency for terminal 2 is 3780 Hz. The reference value for the voltage at the DC bus is 100 kV. The sequence of events from this experiment is as follows. All the terminals are initially disconnected from the DC bus and work in DC voltage control mode. The loads

7.5 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters

of the diﬀerent grids are connected to the corresponding point of common coupling by following the load characteristics previously stated for each grid. All the terminals provide VAR support at t = 0.25 s, thus compensating possible voltage sags/swells caused by the connection/disconnection of loads. The DC side of the converter at terminal 3 is connected to the DC side of terminal 1 at t = 0.4 s, working in active-power control mode with a constant reference value of −20 MW, i.e. the order is that terminal 3 absorbs active power from grid 3. At the time instant t = 0.5 s, the DC side of the converter at terminal 2 is connected to the DC bus with an active-power reference value of 40 MW, which is changed from 40 to −10 MW at t = 0.9 s. Finally, the DC side of the converter at terminal 4 is connected to the DC bus at t = 1.3 s with a constant setpoint for active power equal to 40 MW. The total simulation time is 1.6 s. Figure 7.43 shows the active powers injected by the four terminals in their respective grids – if the measured active power is negative, the terminal absorbs active power from the grid. It should be noted that all measured values equal their respective references in steady-state, i.e. zero tracking error. Furthermore, it can be seen that the four terminals maintain the power balance in steady-state owing to the control of the voltage of the DC bus. Figure 7.44 plots the voltages of the DC capacitors at each converter, showing that the converter at terminal 1 maintains the voltage at 100 kV in steady-state and the differences between this value and those measured on the DC sides of the rest of terminals are caused solely by the losses in the various DC lines. The results shown in Figures 7.43 and 7.44 demonstrate that the multi-terminal HVDC system can correctly control the active-power ﬂow between the four diﬀerent grids. The various electrical magnitudes of the four grids are plotted in Figures 7.45–7.48. In particular, the RMS voltage at PCC, the reactive powers injected by the FC converters and a detailed view of the time responses of the line-to-neutral voltage at PCC are shown. In all cases, the RMS voltages at PCC are kept equal to 1 p.u. in steady-state owing to the reactive power compensation carried out by the FC converters. A point of interest is the situation shown in Figures 7.46a,b where it can be appreciated that the converter at terminal 2 injects/absorbs reactive power in order to cancel out the voltage sags/swells produced by the connection of diﬀerent loads while maintaining the RMS value of the PCC voltage at 1 p.u. The details of the voltages at PCC have also been plotted to show how close to a sinusoidal waveform these voltages really are. The currents injected by the four FC converters into their respective grids are shown in Figure 7.49. Similarly to the previous examples, the active power is proportional to the d-axis current and the reactive power is proportional to the q-axis current, a fact that can be appreciated by a careful comparison of Figure 7.43 with Figure 7.49, and Figures 7.45–7.48 with Figure 7.49. The three ﬂying capacitor voltages of all four FC converters are plotted in Figures 7.50–7.53. As shown in these ﬁgures, once the ﬂying capacitor voltages reach their reference values, i.e. 50 kV, the control scheme of the FC voltages is able to compensate the voltage variations and therefore guarantees the voltage balance of the ﬂying capacitors.

363

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

P1 (MW)

40 20 0 –20 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1

1.2

1.4

1.6

1

1.2

1.4

1.6

1

1.2

1.4

1.6

Time (s) (a)

P2 (MW)

40 20 0 –20 0.2

0.4

0.6

0.8 Time (s) (b)

10 P3 (MW)

0 –10 –20 –30 0.2

0.4

0.6

0.8 Time (s) (c)

40 P4 (MW)

364

20 0 0.2

0.4

0.6

0.8 Time (s) (d)

Figure 7.43 Measured active powers of the four terminals: (a) active power of terminal 1, (b) active power of terminal 2, (c) active power of terminal 3, and (d) active power of terminal 4.

7.5 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters

vc1 (kV)

105

100

95 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1

1.2

1.4

1.6

1

1.2

1.4

1.6

1

1.2

1.4

1.6

Time (s) (a)

vc2 (kV)

105

100

95 0.2

0.4

0.6

0.8 Time (s) (b)

vc3 (kV)

105

100

95 0.2

0.4

0.6

0.8 Time (s) (c)

vc4 (kV)

105

100

95 0.2

0.4

0.6

0.8 Time (s) (d)

Figure 7.44 Voltage at the DC side of each terminal: (a) DC voltage of terminal 1, (b) DC voltage of terminal 2, (c) DC voltage of terminal 3, and (d) DC voltage of terminal 4.

365

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

1.04

vrms 1 (p.u.)

1.02 1 0.98 0.96 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1

1.2

1.4

1.6

Time (s) (a) 20

Q 1 (MVAR)

15 10 5 0 0.2

0.4

0.6

0.8 Time (s) (b)

12 8 vpcc 1 (kV)

366

4 0 –4 –8 –12 0.6

0.62

0.64

0.66

0.68

0.7 0.72 Time (s)

0.74

0.76

0.78

0.8

(c) Figure 7.45 Electrical magnitudes of grid 1: (a) RMS voltage at PCC, (b) reactive power, and (c) detail of the line-to-neutral voltage at PCC.

7.5 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters

1.1

vrms 2 (p.u.)

1.05 1 0.95 0.9 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1

1.2

1.4

1.6

Time (s) (a) 40

Q 2 (MVAR)

30 20 10 0 –10 –20 0.2

0.4

0.6

0.8 Time (s) (b)

12

vpcc 2 (kV)

8 4 0 –4 –8 –12 0.6

0.62

0.64

0.66

0.68

0.7 0.72 Time (s)

0.74

0.76

0.78

0.8

(c) Figure 7.46 Electrical magnitudes of grid 2: (a) RMS voltage at PCC, (b) reactive power, and (c) detail of the line-to-neutral voltage at PCC.

367

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

vrms 3 (p.u.)

1

0.95

0.9 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1

1.2

1.4

1.6

Time (s) (a) 40

Q3 (MVAR)

30 20 10 0 0.2

0.4

0.6

0.8 Time (s) (b)

10 5 vpcc 3 (kV)

368

0 –5 –10 0.6

0.62

0.64

0.66

0.68

0.7 0.72 Time (s)

0.74

0.76

0.78

0.8

(c) Figure 7.47 Electrical magnitudes of grid 3: (a) RMS voltage at PCC, (b) reactive power, and (c) detail of the line-to-neutral voltage at PCC.

7.5 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters

1.02

vrms 4 (p.u.)

1.01

1

0.99

0.98 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1

1.2

1.4

1.6

Time (s) (a) 25

Q 4 (MVAR)

20 15 10 5 0 0.2

0.4

0.6

0.8 Time (s) (b)

18

vpcc 4 (kV)

12 6 0 –6 –12 –18 0.6

0.62

0.64

0.66

0.68

0.7 0.72 Time (s)

0.74

0.76

0.78

0.8

(c) Figure 7.48 Electrical magnitudes of grid 4: (a) RMS voltage at PCC, (b) reactive power, and (c) detail of the line-to-neutral voltage at PCC.

369

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

3 id1, iq1 (kA)

2

id1

1 0

iq1

–1 –2 0.2

0.4

0.6

0.8 (a)

1

1.2

1.4

1.6

1

1.2

1.4

1.6

1

1.2

1.4

1.6

id2, iq2 (kA)

4 2

id2

0

iq2

–2 –4 0.2

0.4

0.6

0.8 (b)

1 id3, iq3 (kA)

0 –1

id3

–2 iq3

–3 –4 0.2

0.4

0.6

0.8 (c)

3 2 id4, iq4 (kA)

370

id4

1 0

iq4

–1 –2 0.2

0.4

0.6

0.8 (d)

1

1.2

1.4

Figure 7.49 d and q components of the currents injected by: (a) terminal 1, (b) terminal 2, (c) terminal 3, and (d) terminal 4.

1.6

7.5 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters

60

vfcA1 (kV)

50 40 30 20 10 0

0

0.2

0.4

0.6

0.8 Time (s) (a)

1

1.2

1.4

1.6

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1

1.2

1.4

1.6

1

1.2

1.4

1.6

60

vfcB1 (kV)

50 40 30 20 10 0

(b) 60

vfcC1 (kV)

50 40 30 20 10 0

0

0.2

0.4

0.6

0.8 Time (s) (c)

Figure 7.50 Voltages of the three ﬂying capacitors of terminal 1: (a) phase A, (b) phase B, and (c) phase C.

371

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

60

vfcA2 (kV)

50 40 30 20 10 0

0

0.2

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1

1.2

1.4

1.6

(a) 60

vfcB2 (kV)

50 40 30 20 10 0

0

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1

1.2

1.4

1.6

0

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1

1.2

1.4

1.6

60 50 vfcC2 (kV)

372

40 30 20 10 0

(c) Figure 7.51 Voltages of the three ﬂying capacitors of terminal 2: (a) phase A, (b) phase B, and (c) phase C.

7.5 Example of a Multi-Terminal HVDC System Using Multilevel FC Converters

60

vfcA3 (kV)

50 40 30 20 10 0

0

0.2

0.4

0.6

0.8 Time (s) (a)

1

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1.6

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1

1.2

1.4

1.6

60

vfcB3 (kV)

50 40 30 20 10 0

60

vfcC3 (kV)

50 40 30 20 10 0

(c) Figure 7.52 Voltages of the three ﬂying capacitors of terminal 3: (a) phase A, (b) phase B, and (c) phase C.

373

7 Electromagnetic Transient Studies and Simulation of FACTS-HVDC-VSC Equipment

60

vfcA4 (kV)

50 40 30 20 10 0

0

0.2

0.4

0.6

0.8 Time (s)

1

1.2

1.4

1.6

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1.2

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(a) 60

vfcB4 (kV)

50 40 30 20 10 0

0

0.2

0.4

0.6

0.8 Time (s) (b)

60 50 vfcC4 (kV)

374

40 30 20 10 0

0

0.2

0.4

0.6

0.8 Time (s) (c)

Figure 7.53 Voltages of the three ﬂying capacitors of terminal 4: (a) phase A, (b) phase B, and (c) phase C.

References

7.6 Conclusions This chapter has presented time-domain simulations of several examples of FACTS devices, including HVDC systems. The simulation software PSCAD/EMTDC has been chosen owing to its easy use and control implementation. The reader can of course implement these examples with other simulation software packages. The control systems designed for the diﬀerent cases are very simple and easy to implement, but they are useful to show how to obtain decoupled control of the active and reactive powers exchanged between FACTS and the grid. These control schemes have demonstrated an eﬃcient behaviour with which to provide the necessary reactive power and therefore maintain the PCC voltage at a desired value, and they have allowed the active-power ﬂow between several electrical grids. Moreover, these controllers are independent of the topology of the power-electronic converter, up to certain limits, and they can either be used with a conventional two-level VSC or implemented in three-level FC converters. Multilevel voltage source converters, operated by an eﬃcient control strategy, have proved to be very useful in reducing the switching frequency and the voltage stress of the switching devices in HVDC system applications. Finally, the reader can add more complexity to these examples by increasing the number of levels of the multilevel FC converters, using more advanced controllers or with diﬀerent grid topologies.

References 1 PSCAD (2016). User’s Guide. Manitoba: HVDC Research Centre. 2 Mishra, M.K., Ghosh, A., and Joshi, A. (2003). Operation of a DSTATCOM in

voltage control mode. IEEE Transactions on Power Delivery 18 (1): 258–264. 3 Krause, P.C. (1986). Analysis of Electric Machinery. McGraw-Hill Inc. 4 Akagi, H., Kanazawa, Y., and Nabae, A. (1984). Instantaneous reactive power com-

5 6 7 8

9

10

pensators comprising switching devices without energy storage components. IEEE Transactions on Industry Applications IA−20 (3): 625–630. García-González, P. and García-Cerrada, A. (2000). Control system for a PWM-based STATCOM. IEEE Transaction on Power Delivery 15 (4): 1252–1257. Kundur, P. (1994). Power System Stability and Control. McGraw-Hill, Inc. Mohan, N., Undeland, T.M., and Robbins, W.P. (2002). Power Electronics: Converters, Applications, and Design, 3e. Wiley. Roncero-Sánchez, P. and Acha, E. (2014). Design of a control scheme for distribution static synchronous compensators with power-quality improvement capability. Energies 7 (4): 2476–2497. Gupta, R., Ghosh, A., and Joshi, A. (2008). Switching characterization of cascaded multilevel-inverter-controlled systems. IEEE Transactions on Industrial Electronics 55 (3): 1047–1058. Rodríguez, J., Lai, J.-S., and Peng, F.Z. (2002). Multilevel inverters: a survey of topologies, controls, and applications. IEEE Transactions on Industrial Electronics 49 (4): 724–738.

375

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11 Arrillaga, J., Liu, Y.H., and Watson, N.R. (2007). Flexible Power Transmission. The

HVDC Options. Chichester: Wiley. 12 Feng, C., Liang, J., and Agelidis, V.G. (2007). Modiﬁed phase-shifted pwm control

13

14

15

16 17

for ﬂying capacitor multilevel converters. IEEE Transactions on Power Electronics 22 (1): 178–185. Liang, Y. and Nwankpa, C.O. (1999). A new type of STATCOM based on cascading voltage-source inverters with phase-shifted unipolar SPWM. IEEE Transactions on Industry Applications 35 (5): 1118–1123. Xu, L. and Agelidis, V.G. (2004). Active capacitor voltage control of ﬂying capacitor multilevel converters. IEE Proceedings on Electric Power Applications 151 (3): 313–320. Häusler M.; Schaltanlagen C.E., “Multiterminal HVDC for high power transmission in Europe,” in Proceedings of the Central European Power Exhibition and Conference (CEPEX99), Poznan, Poland, 1999. Haileselassie, T.M. (2008). Control of Multi-terminal VSC-HVDC, Master of Science in Energy and Environment. Norwegian University of Science and Technology. Sood, V.K. (2004). HVDC and FACTS Controllers. Applications of Static Converters in Power Systems. Boston: Kluwer Academic Publishers.

377

Index a AC-bus voltage controllers 264 VSC-HVDC link modelling 275 AC conductors 17 AC-DC distribution systems 37–40 AC-DC networks 179–180 AC-DC smart grids 41 AC-DC transmission system 300, 301 AC distribution feeders 37 AC ﬁlters 17 AC network 194, 199–200 AC OPF formulation 164–165 augmented Lagrangian function 167 control enforcement in OPF Algorithm 168–169 functions, handling limits of 169–170 IEEE 30-bus system 174–179 linearized system of equations 165–167 objective function 165 OPF solution algorithm 168 recent extensions in OPF problem 173–174 simple network model 170–173 state variables, handling limits of 169 AC power grids 31 AC power systems 40 AC power transmission reinforcement 18 active inequality constraints 163 active power ﬂows, in AC network 199, 200 AC transmission line 139 AC transmission system 17, 19 AC voltage controller 26 AC voltage magnitude constraints 195

advanced interactive multidimensional modelling system (AIMMS) 175, 178 advanced power electronic converters 2 AIMMS see advanced interactive multidimensional modelling system (AIMMS) All-HVDC transmission system 19 alternative multilevel converter topology 85–88 arduous algebra 117, 126 assessment 99 augmented Lagrangian function 167 implementation of 172–173 automatic voltage regulator (AVR) 104, 168, 252–253

b back-to-back HVDC link 13 back-to-back VSC-HVDC link 30, 135, 137 bad data identiﬁcation 206, 217–218 largest normalized residual test 218–219 using WLS-SE 219–220 balancing control scheme, of ﬂying capacitor voltages 339, 340 battery energy storage systems (BESS) 2, 31 voltage and power controls 26 benchmark test case 176 BESS see battery energy storage systems (BESS) bipolar junction transistor (BJT) 57–58

VSC-FACTS-HVDC: Analysis, Modelling and Simulation in Power Grids, First Edition. Enrique Acha, Pedro Roncero-Sánchez, Antonio de la Villa Jaén, Luis M. Castro and Behzad Kazemtabrizi. © 2019 John Wiley & Sons Ltd. Published 2019 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/acha_vsc_facts

378

Index

block-diagonal matrix of nodal voltages 108 bus-branch model 209

c cascaded H-Bridge VSC 81–84 classical HVDC systems 89 classical HVDC transmission 12 closed-loop control system 338 CO2 emissions reduction 41 commercial electromagnetic transient simulators 321 common DC bus model, multi-terminal VSC-HVDC system model with 147–148 complex power model 117 complex-valued vector function 165 compound steam-turbine systems 251 compound symbols 55 compound transformers modelling 102 compound VSC equipment for AC applications 10–12 CONOPT 176 constraint handling 178 contrived four-node power system 170 contrived power system diagram 170 control constraints 168 control enforcement in OPF Algorithm 168–169 controllable-shunt compensation 21 controlled ﬂux 30 controlled injections 37 control strategy 120 conventional AC power ﬂow studies 103 conventional electrical energy systems, ﬂexible electrical energy systems vs., 15–16 distribution (see distribution) generation (see generation) transmission (see transmission) conventional Jacobian matrix 255–257 conventional power ﬂow method formulations 100, 160 Jacobian matrix of 255 conventional power ﬂows 159 conventional power plants 250

conventional power system components modelling 248 synchronous generators controllers 250–254 modelling 248–250 conventional power transmission corridor 17 conventional state estimation process 239 in electrical power grid 236 conventional three-phase AC distribution feeder 37 conventional transformers modelling 100–101 conventional turbine-governor control of synchronous generator 26 conventional two-level VSC line-to-line output voltage of VSC 344 line-to-neutral voltage at PCC 343 convergence characteristics of Newton-Raphson method 108–109 converters outputs 196–197 converters PWM performance 197 convex relaxation 173 CSC-HVDC installations 94 CSC-HVDC links 12–13 CSC-HVDC technology 17 CSC technology 91 current controllers 351 customers’ voltage quality 40 custom power electronic technology 1, 34 Custom Power equipment 34

d Darlington conﬁgurations 58 DC bus data 194 DC capacitors 8 voltage 335, 336 time response of 345, 346 DC-DC boost converter 20, 30 based on VSCs 41 DC distributed energy resources 37 DC distribution system 40 DC lines data 194, 197 DC link 37 DC microgrid 42 DC network 14, 193–194, 196–198

Index

DC network voltage proﬁle 198 DC nodal voltage constraint 191 DC power controller, for VSC’s DC side 263, 270 DC power grids with multiple voltage levels 40 DC power lines 196 DC power systems 40 DC power transfer controller, of VSC-HVDC link modelling 274–275 DC transmission systems 19 DC voltage controller 263 for slack converter VSCSlack 300 DFIG see doubly-fed induction generator (DFIG) DFIG-based wind farm 31 DFT see Discrete Fourier Transform (DFT) DG see distributed generation (DG) diﬀrential equations, discretization and linearization of STATCOM 264–266 VSC-HVDC link modelling 276–280 digital measuring system 44 diode 55–56 Discrete Fourier Transform (DFT) 44 distributed generation (DG) 36 distribution 33–34, 48 AC-DC distribution systems 37–40 DC Power Grids with Multiple Voltage Levels 40 dynamic voltage support 35–36 ﬂexible reconﬁgurations 36–37 load compensation 35 smart grids 40–43 distribution feeders 38 distribution systems 36 doubly-fed induction generator (DFIG) 30 DSTATCOM 34, 35, 322, 323 control system scheme for 324–326 operation of 326 dynamic SVC 21 dynamic voltage support 35–36

e electrical power-consuming equipment 100

electrical power grids conventional state estimator in 236 FACTS device state estimation modelling (see FACTS device state estimation modelling, electrical power grids) electrical power network 100 electrical power transmission 100 electrical symbol of diode 55, 56 electromagnetic transient studies 321 electromechanically based switching centre 37 electronic symbols of semiconductor switches 54 embedded VSC-HVDC Link 283–288 EMS technology see energy management system (EMS) technology EMT-type simulation tool Simulink 280–283 energy management system (EMS) technology 203 equality-constrained WLS 212–213 equality constraints 170 excitation system 104

f FACTS device state estimation modelling, electrical power grids multi-terminal HVDC 233–235 new models, state estimation incorporation 220 STATCOM 223–225 uniﬁed power ﬂow controller 227–228 voltage source converters 221–223 high voltage direct current based on 230–231 WLS-SE MT-VSC-HVDC model in 235–236 STATCOM model in 225–227 UPFC model in 228–230 VSC-HVDC model 231–233 FACTS-HVDC-VSC equipment 321–322, 375 HVDC based on multilevel FC converter 347–358 multi-terminal HVDC system using multilevel FC converters 358–374

379

380

Index

FACTS-HVDC-VSC equipment (contd.) STATCOM based on multilevel VSC 336–347 STATCOM case 322–336 FACTSs see ﬂexible alternating current transmission systems (FACTSs) FACTS-type power ﬂow controlling devices, installation of 247 ﬁctitious four-node power system 170 Finnish distribution system 40 ﬁrst-order optimality conditions 163 ﬁve-level FC conﬁguration 82 ﬁve-level NPC conﬁguration 79 ﬂexible alternating current transmission systems (FACTSs) 1–2, 6, 16, 100 ﬂexible electrical energy systems 1–4 vs. conventional electrical energy systems (see conventional electrical energy systems, ﬂexible electrical energy systems vs.) ﬂexible transmission system equipment (see ﬂexible transmission system equipment) phasor measurement units (PMUs) 43–46 ﬂexible reconﬁgurations 36–37 ﬂexible transmission system equipment 5–6 compound VSC equipment for AC applications 10–12 CSC-HVDC links 12–13 with renewable energy sources 4 SSC 9–10 STATCOM 7–9 SVC 6–7 VSC-HVDC 13–15 ﬂying capacitors (FC) 337 voltage-balance method for 339, 340 voltages 345–347 balancing control scheme of 339, 340 high voltage direct current 354, 356–358 Fourier series 44 four-terminal HVDC system, implementation in PSCAD/EMTDC 359, 360

four-terminal VSC-HVDC system 15 frequency compensation 24–27 frequency controller, of VSC-HVDC 290 frequency response of BESS 27 FSIG-based wind farm 31 full VSC station model 115–116 fully controlled semiconductor valves 55 functional block diagram of PMUs 45 functions, handling limits of 169–170 fundamental-harmonic amplitude 74

g galvanic isolation 57 gate turn-oﬀ (GTO) thyristor 55, 59–60 Gauss-Seidel technique 99 Gauss-type procedure 166 generalized gradient method 176 generation 27–28, 46–47 solar power generation 30–33 wind power generation 28–30 generator constraints 195 generator cost function coeﬃcients 174 generator limits 175 Geostationary Operational Environmental Satellite (GOES) 44 Global Positioning System (GPS) satellites 43 GOES see Geostationary Operational Environmental Satellite (GOES) GPS satellites see Global Positioning System (GPS) satellites GTO thyristor see gate turn-oﬀ (GTO) thyristor

h handling limits of functions 169–170 of state variables 169 H-bridge converters 337 Hessian matrix 169 Hessian submatrix 190 highly complex systems 159 high switching frequency 85 high temperature superconductivity (HTSC) 48 high-voltage AC transmission line (HVAC) 16

Index

high voltage direct current (HVDC) systems 61 based on multilevel FC converter 347–358 based on VSC 88–94 control system for 348–349 DC capacitors voltages 351, 354, 355 parameters of 351–353 PCC voltages for time interval 351, 354, 355 three ﬂying capacitors voltages 354, 356–358 transmission lines 12, 16 transmission systems 247 Hingorani, N.G. 1 HTSC see high temperature superconductivity (HTSC) HVAC see high-voltage AC transmission line (HVAC) HVDC see high voltage direct current (HVDC) systems HVDC power transmission for increased power throughputs, HVAC vs., 16–19 hybrid AC-DC network model 188 hybrid AC-DC transmission circuit 17 hybrid HVDC transmission system 19 hybrid test system results 196 hybrid VSC 88 hydro turbine 251–252

i ICTs see information and communication technologies (ICTs) IEDs see intelligent electronic devices (IEDs) IEEE 30-bus system 171, 174–179 IEEE hydro-turbine model 252 IEEE speed governor model 251 IEEE steam-turbine model 252 IEEE-type I AVR model 253 IGBT see insulated-gate bipolar transistor (IGBT) induction generators 30 information and communication technologies (ICTs) 3

insulated-gate bipolar transistor (IGBT) 59, 60 insulated-gate bipolar transistor (IGBT)-based converters 6 intelligent electronic devices (IEDs) 205 intermittent generation 24 inverter’s AC system 14 IPOPT 176 iterative process 166

j Jacobian matrix 161 of conventional power ﬂow method 255 of DC network 147 of DC network, three-terminal VSC-HVDC dynamic model 312 of Newton-Raphson power ﬂow method 108 Jacobian term 190 Jinping-Sunan link 89

k Karush-Kuhn-Tucker (KKT) conditions 163–164 KCL see Kirchhoﬀ’s Current Law (KCL) key drivers 3 Kirchhoﬀ’s current law (KCL) 105, 164 KKT conditions see Karush-Kuhn-Tucker (KKT) conditions

l Lagrange multipliers, for multi-terminal system 192 Lagrangian function method 162–164, 172, 178, 187, 189 largest normalized residual test, bad data identiﬁcation 218–219 light-triggered thyristors 57 linearized system of equations 107–108, 143–146, 161, 165–167, 172, 189–190 line-frequency diodes 56 load compensation 35 load modelling 102, 253–254 load tap changers (LTCs) 115, 169, 175, 207, 266 transformers modelling 101

381

382

Index

lossless economic dispatch problem 165 low-inertia AC network, VSC-HVDC link modelling 294–297 parametric analysis of 297–298

m Matlab code, WLS-SE using 215–217 MaxSine HVDC 86 MCT see MOS-controlled thyristor (MCT) measurements system model 208–210 meshed AC transmission systems 14 metal-oxide-semiconductor ﬁeld-eﬀect transistor (MOSFET) 59 microprocessor 45 midpoint shunt compensation 23 modern BESS 24, 25 modern compensation equipment 100 modern control switching techniques 95 modiﬁed ﬁve-bus power system 193 modulation index controllers, of three-terminal HVDC system 301 monopolar VSC-HVDC 130 MOS-controlled thyristor (MCT) 60–61 MOSFET see metal-oxide-semiconductor ﬁeld-eﬀect transistor (MOSFET) MT-VSC-HVDC see multi-terminal VSC-HVDC systems (MT-VSC-HVDC) multilevel converter topologies 337 multilevel FC converters HVDC based on 347–358 multi-terminal HVDC system using 358–374 multilevel FC VSC 80–81 multilevel NPC VSC 76–80 multilevel voltage source converters 375 STATCOM based on 336–347 multilevel VSC topologies 74 multi-terminal DC system 201 multi-terminal HVDC system 233–235 active powers of terminals 363, 364 control scheme of terminal with active-power reference 359 coupling transformers parameters 359, 361 d and q components of currents 363, 370

DC voltage of terminal 363, 365 electrical magnitudes 363, 366–369 electric parameters 359, 361 ﬂying capacitors voltages 363, 371–374 four-terminal HVDC system implementation 359, 360 inner control loop and DC voltage control loop parameters 362 PCC loads 361 scheme of 358 using multilevel FC converters 358–374 multi-terminal HVDC technology 47 multi-terminal VSC-HVDC systems (MT-VSC-HVDC) 3, 41, 146–147, 298–299, 310–314 with common DC bus model 147–148 model in WLS-SE 235–236 in OPF 191–200 six-terminal VSC-HVDC link 314–317 test case 150–153 three-terminal VSC-HVDC dynamic model (see three-terminal VSC-HVDC dynamic model) uniﬁed solutions of AC-DC networks 148 uniﬁed vs. quasi-uniﬁed power ﬂow solutions 148–149

n NavstarGPS satellite transmission 44 N-channel MOSFET 59 necessary optimality conditions (Karush-Kuhn-Tucker conditions) 163–164 network model 206–208 for AC-DC uniﬁed formulations 180 network nodal admittance 102–103 neutral-point-clamped converters (NPC) 337 New England test system 257–261 STATCOM modelling 266–272 VSC-HVDC link modelling 283–288 Newton-Raphson algorithm 116 Newton-Raphson method 99, 121, 153, 160, 161, 166, 255, 265 in rectangular coordinates 106–107

Index

convergence characteristics of 108–109 incorporation of PMU information in 111–112 initialization of 109–111 linearized equations 107–108 Newton-Raphson power ﬂow algorithm 105 in rectangular coordinates 109 Newton’s method 163, 168, 201 nodal admittance matrix 105, 131 nodal complex power equations 106, 132 nodal power constraints 191 nodal power expressions 139 nodal power ﬂow equations 105–106 nodal transfer admittance matrix 100 nodal voltage magnitudes 170 nodal voltage proﬁles 178 nodal voltage regulation, test case with 177 nonlinear mathematical programs 162 non-regulated power ﬂow solutions 119–120 normalized harmonic components 72 NPC see neutral-point-clamped converters (NPC) numerical solution technique 254–257

o objective functions 159, 165, 194–195 1 kV AC distribution feeders 39 operational limits 162 optimal power ﬂows (OPF) 159–160 AC-DC networks 179–180 AC OPF formulation (see AC OPF Formulation) formulation 161–162 Lagrangian methods 162–164 multi-terminal VSC-HVDC systems in 191–200 point-to-point and back-to-back VSC-HVDC links models in 184–190 in polar coordinates 160–161 VSC model in (see voltage source converter (VSC) model)

p passive AC power grids 142 passive distribution systems 37 PCC see point of common coupling (PCC) PDC see phasor data concentrators (PDC) peculiarities of power ﬂows formulation 103–105 penalty functions 169 permanent magnet synchronous generator (PMSG) 30 phase angle voltages 110 phase-control thyristors 57 phase locked loop (PLL) 326 phase-locked oscillator (PLO) 45 phase-shift control operation 69–72 phase-shift control scheme 80 phase-shifted PWM modulation scheme, three-level FC converter 339, 341 phase-shifting transformers modelling 101 phasor data concentrators (PDC) 46 phasor measurement units (PMUs) 43–46 measurements incorporation outputs in WLS-SE 239–240 phase reference 239 synchrophasors formulations 237–239 synchrophasors incorporation 236–237 physical’ slack node 111 PI regulators 324, 326, 327, 349 PLL see phase locked loop (PLL) PLO see phase-locked oscillator (PLO) PMSG see permanent magnet synchronous generator (PMSG) PMUs see phasor measurement units (PMUs) point of common coupling (PCC) control system of 331, 333 instantaneous line-to-neutral voltage 342, 343 line-to-neutral voltage at 334, 335 RMS voltage at 334, 335 with three-level FC converter, RMS voltage at 342, 343 point-to-point VSC-HVDC linking two equivalent AC subsystems 136

383

384

Index

point-to-point VSC-HVDC system 140 polar coordinates, power ﬂows in 160–161 post-contingency control function 162 power-angle characteristic 21 power-angle curves 22 power-constraining equations 142 power converters 17 power distribution 93 power electronic-based compensators 29 power electronics technology 16, 30 controllers 34 converters 74 equipment 2 for VSC-based bridges 53 HVDC systems based on VSC 88–94 power semiconductor switches (see power semiconductor switches) voltage source converters (see voltage source converters (VSCs)) power engineers 109 power ﬂows 99 in AC transmission lines 199, 200 formulation, peculiarities of 103–105 multi-terminal VSC-HVDC system model 146–153 Newton-Raphson method in rectangular coordinates 106–112 nodal power ﬂow equations 105–106 power network modelling 100–103 STATCOM model 125–129 three-terminal VSC-HVDC system model 139–146 voltage solution 137–139 voltage source converter model 112–125 VSC-HVDC systems modelling 129–139 powerful PV-based system 31 power generators 100 power network modelling 100–103 power quality 48 power responses 33 power semiconductor switches 53–55 bipolar junction transistor (BJT) 57–58 diode 55–56

gate turn-oﬀ thyristor (GTO) 59–60 insulated-gate bipolar transistor (IGBT) 59 metal-oxide-semiconductor ﬁeld-eﬀect transistor (MOSFET) 59 MOS-controlled thyristor (MCT) 60–61 switch selection process, considerations for 61 thyristor 56–57 Power System Computer Aided Design/Electromagnetic Transient Direct Current (PSCAD/EMTDC) 321–322 control system of voltage, DC capacitor 333 four-terminal HVDC system implementation 359, 360 inner control loop 332 PCC voltage control system 333 phase-shifted PWM modulation scheme for three-level FC converter implementation 339, 341 sinusoidal PWM modulation scheme 333 STATCOM based three-level FC converter 338 three-level ﬂying capacitor converter implementation 339 two-level VSC using IGBTs and diodes implementation 330–331 two-terminal HVDC system implementation 349, 350 voltage-balance method for ﬂying capacitors implementation 339, 340 VSC-based STATCOM circuit implemented in 330 power systems application 5 power systems, dynamic simulations 247, 317–318 conventional power system components modelling 248 synchronous generator controllers 250–254 synchronous generators modelling 248–250

Index

multi-terminal VSC-HVDC system modelling 298–299, 310–314 six-terminal VSC-HVDC link 314–317 three-terminal VSC-HVDC dynamic model (see three-terminal VSC-HVDC dynamic model) STATCOM modelling 261–264 New England test system 266–272 STATCOM diﬀrential equations, discretization and linearization 264–266 time domain solution philosophy 254–261 VSC-HVDC link modelling AC-bus voltage controllers 275 capacitors’ and inductors’ dynamics 273 DC power transfer controller of 274–275 diﬀrential equations, discretization and linearization of 276–280 dynamic controller, for DC voltage 274 dynamic model, with control variables 273 dynamic model, with frequency regulation capabilities 289–298 New England test system 283–288 transferring power in DC form 272 validation of 280–283 power transformers 40, 101 PSCAD/EMTDC see Power System Computer Aided Design/Electromagnetic Transient Direct Current (PSCAD/EMTDC) pseudo-measurements 213 pulse width modulated-output schemes and half-bridge VSC 62–66 pulse width-modulated VSCs 62 pulse width modulation (PWM) control 112 with bipolar switching 67–69 techniques for multilevel VSCs 85 with unipolar switching 69 PV generator 30, 32

PWM control see pulse width modulation (PWM) control

q quadratic convergence characteristics

149

r reactive-power compensation 322 reactive power ﬂows in AC network 199, 200 reactive power production process 8 real-time power systems application, fundamental issues in 203–204 rectiﬁer’s AC system 14 reducing global emissions 28 renewable energy sources, ﬂexible transmission system with 4 reverse-recovery time 56 RMS-type VSC-HVDC model 282, 283, 293–294 root-locus technique 327

s satellites systems 44 SCADA system see supervisory control and data acquisition (SCADA) system Schottky diodes 56 SCIG see squirrel-cage induction generator (SCIG) SCR see silicon-controlled rectiﬁer (SCR) semiconductor valves 54 semi-controlled device 56 semi-controlled semiconductor valves 54–55 series and shunt compensation modelling 102 seven-level cascaded H-bridge VSC 83, 84 shunt devices 175 shunt dynamic compensation 19 silicon-controlled rectiﬁer (SCR) 56–57 simple network model 170–173 simple test system 201 Simulink 280–283, 307 single-phase full-bridge VSC 66–67 phase-shift control operation 69–72 PWM with bipolar switching 67–69 square-wave mode 69

385

386

Index

single-phase half-bridge VSC 63 single-phase VSCs 62 sinusoidal PWM modulation scheme 333 sinusoidal waveform 46, 88 six-terminal VSC-HVDC link DC transmission line disconnection 314, 315 three-phase-to-ground short-circuit fault 314–317 six-terminal VSC-HVDC network 150 smart grids 3, 40–43 SMES systems see superconductor magnetic energy storage (SMES) systems smoothing line reactor 116 solar power generation 30–33 speed-governing system 25 speed governors 250–251 square-wave mode 69 square-wave operation of three-phase VSC 75 of VSC 67 square-wave scheme 77 square-wave VSCs 62 squirrel-cage induction generator (SCIG) 28 SRF see synchronous reference frame (SRF) SSSC 9–10 standard AC transmission line 17 STATCOM see static compensator (STATCOM) STATCOM based three-level FC converter 338 STATCOM case 322–336 based on multilevel VSC 336–347 STATCOM-type compensation 129 state estimation 203–204 bad data identiﬁcation 217–220 calculation of equality-constrained WLS 212–213 observability analysis and reference phase 213–215 solution by normal equations 210–212 WLS-SE using Matlab code 215–217 of electrical networks 204–206 FACTS device state estimation modelling, electrical power grids

(see FACTS device state estimation modelling, electrical power grids) measurements incorporation, PMUs outputs in WLS-SE 239–240 phase reference 239 synchrophasors formulations 237–239 synchrophasors incorporation 236–237 network model and measurement system 206–210 process of 206 WLS-SE input data 240–243 measurements data 242–243 network data 240–241 output results 243–244 state estimator conﬁguration 243 state estimation UPFC model 227 state-variable model 322, 323 state variables, handling limits of 169 static compensator (STATCOM) 2, 6–9, 20–22, 112, 125–129, 223–225, 261–264 diﬀrential equations, discretization and linearization of 264–266 New England test system 266–272 STATCOM diﬀrential equations, discretization and linearization 264–266 waveforms 335 in WLS-SE 225–227 static VAR compensator (SVC) 6–7, 20, 122 static V-I characteristics of VSC 9 steady-state analysis 104 steady-state model, of multi-terminal HVDC system 313 steady-state performance of shunt VAR compensation 20 steam turbine 251–252 Steinmetz, Charles P. 43 superconductor magnetic energy storage (SMES) systems 31 supervisory control and data acquisition (SCADA) measurements 104

Index

supervisory control and data acquisition (SCADA) system 203–205, 209 SVC see static var compensator (SVC) switching centres 37 switching loss model 115 switch-on and switch-oﬀ processes 56 switch selection process, considerations for 61 symmetrical transmission system 20 synchronous generators 25 controllers automatic voltage regulator 252–253 load model 253–254 speed governors 250–251 steam turbine and hydro turbine 251–252 transmission line model 253 modelling 248–250 phasor diagram of 248, 249 synchronous reference frame (SRF) 322, 323 control system 326 synchrophasor 44, 45

t tap-changing transformer 101 target nodal voltage magnitudes 177 TCRs see thyristor-controlled reactors (TCRs) TCSC see thyristor-controlled series compensator (TCSC) test system 35 three-level FC converter line-to-line output voltage of VSC 344 line-to-neutral voltage at PCC 343 phase-shifted PWM modulation scheme for 339, 341 RMS voltage at PCC with 342, 343 STATCOM waveforms obtained with 344–345 three-level ﬂying capacitor converter 337, 339 three-level ﬂying capacitor VSC 81 three-level NPC VSC 77–78 three-level three-phase FC VSC 80 three-level three-phase NPC VSC 77 three-node system 122

three-phase AC transmission lines 16 three-phase distribution system 327 three-phase hybrid VSC 90 three-phase MMC 89 three-phase multilevel VSC 74–76 alternative multilevel converter topology 85–88 cascaded H-Bridge VSC 81–84 multilevel FC VSC 80–81 multilevel NPC VSC 76–80 PWM techniques for multilevel VSCs 85 three-phase STATCOM 322 three-phase two-level VSC 72 three-phase VSC 72–74 three-terminal HVDC system for dynamic simulations 303 modulation index controllers of 301 three-terminal VSC-HVDC dynamic model 298–307 validation of 307–310 three-terminal VSC-HVDC system model 139–142 linearized system of equations 143–146 power mismatches 142–143 VSC types 142 thyristor 56–57 thyristor-controlled reactors (TCRs) 6 thyristor-controlled series compensator (TCSC) 204 time-domain simulations 375 time domain solution philosophy benchmark numerical example 257–261 numerical solution technique 254–257 topological processing 206 traditional low-voltage system 40 transformer data 175 transient performance of shunt VAR compensation 20–24 transmission 16, 47–48 frequency compensation 24–27 HVAC vs. HVDC power transmission for increased power throughputs 16–19 VAR compensation 19–24

387

388

Index

transmission lines modelling 100, 253 transmission system operators (TSOs) 159 transmission system planning framework 160 trapezoidal method 254–255 triangular signals 85 ‘true’ steady-state condition 43 TSOs see transmission system operators (TSOs) two-level converters 336 two-terminal HVDC system, PSCAD/EMTDC implementation 349, 350 two-terminal HVDC systems 300–301

u uncontrolled semiconductor valves 54 uniﬁed OPF formulation of hybrid AC-DC systems 200 uniﬁed power ﬂow controller (UPFC) 10, 111, 204, 227–228 model in WLS-SE 228–230 uniﬁed solutions of AC-DC networks 148 uniﬁed vs. quasi-uniﬁed power ﬂow solutions 148–149 Universal Time Coordinated (UTC) 45 UPFC see uniﬁed power ﬂow controller (UPFC) upgraded network 127 upstream PQ disturbances and actions 34 UTC see Universal Time Coordinated (UTC)

v VAR see volt-ampere reactives (VAR) voltage-balance method, for ﬂying capacitors implementation 339, 340 voltage control 3 test case with 177 voltage-controlled buses 108 voltage-controlled node 104 voltage sags/swells 35

voltage source converter (VSC) model 53, 61–62, 112–113 constraints 194 control considerations 183–184 control strategy 120 data 194 full station model 115–116 initial parameters and limits 120–121 linearized system of equations 117–119, 184 nodal admittance matrix representation 113–115 nodal power equations 117 non-regulated power ﬂow solutions 119–120 numerical examples 121–125 in OPF 181–184 power ﬂow model 130 pulse width modulated-output schemes and half-bridge VSC 62–66 single-phase full-bridge VSC 66–72 technology 37 three-phase multilevel VSC 74–88 three-phase VSC 72–74 voltage source converters (VSCs) discretized diﬀerential equations of 264 dynamic behaviour of 262 FACTS device state estimation modelling, electrical power grids 221–223 high voltage direct current based on 230–231 volt-ampere reactives (VAR) 7 compensation 19–24 equipment 100 systems 95 VSC see voltage source converters (VSCs) VSC-based STATCOM circuit, PSCAD/EMTDC implemented 330 VSC-DC cable 132 VSC-HVDC-infeed of micro-grids 105 VSC-HVDC link modelling AC-bus voltage controllers 275 capacitors’ and inductors’ dynamics 273 DC power transfer controller of 274–275

Index

diﬀrential equations, discretization and linearization of 276–280 dynamic controller, for DC voltage 274 dynamic model, with control variables 273 dynamic model, with frequency regulation capabilities 289–291 linearization of equations 291–292 low-inertia AC network 294–297 validation of 292–294 New England test system 283–288 transferring power in DC form 272 validation of 280–283 VSC-HVDC systems modelling 93–94, 129–131 back-to-back 135 linearized equations 133–135 nodal power equations 131–133 numerical examples 135–139 parameter data for 136 WLS-SE 231–233 VSC-HVDC technology 13–15, 92, 95 full set of equality constraints 188–189 installations 92, 93 link 11, 46, 91 link control 187–188 link power balance formulation 185–187

w weighted least-squares (WLS) algorithm 212 weighted least squares state estimator (WLS-SE) bad data identiﬁcation using 219–220 FACTS device state estimation modelling, electrical power grids MT-VSC-HVDC model in 235–236 STATCOM model in 225–227 UPFC model in 228–230 VSC-HVDC model 231–233 PMUs outputs in 239–240 state estimation input data 240–243 measurements data 242–243 network data 240–241 output results 243–244 state estimator conﬁguration 243 using Matlab code 215–217 wind power generation 28–30 wind power plants 46 WLS-SE see weighted least squares state estimator (WLS-SE)

z zero-distance, 135

389

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