Transient Characteristics, Modelling and Stability Analysis of Microgird 9811584028, 9789811584022

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Zhikang Shuai

Transient Characteristics, Modelling and Stability Analysis of Microgird

Transient Characteristics, Modelling and Stability Analysis of Microgird

Zhikang Shuai

Transient Characteristics, Modelling and Stability Analysis of Microgird

Zhikang Shuai College of Electrical and Information Engineering Hunan University Changsha, Hunan, China

ISBN 978-981-15-8402-2 ISBN 978-981-15-8403-9 https://doi.org/10.1007/978-981-15-8403-9

(eBook)

© Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

Microgrid provides possible solutions to get distributed generators and loads together and becomes controllable units of the traditional power system. Several concerns are raised to promote the application of microgrid. In some specific scenarios, like in remote island, microgrid plays an important role in providing reliable power. However, extreme operating conditions and unpredictable weather conditions are expected in remote island, such as high humidity, lack of sustainable primary resources, risk of a weapon attack, etc. These specific environmental conditions would put forward high reliability and strong anti-disturbance ability of microgrid. In other operating scenarios, there would be several extreme operating conditions as well, such as grounded faults, off-grid switching and so on. These extreme conditions need to be dealt with carefully since stability and safety of the microgrid are challenged in these conditions. This book investigates the transient characteristics and transient stability of the microgrid. The whole book can be divided into three parts. The first part is about transient characteristics analysis among converters with different control schemes and corresponding fault ride through control strategy is proposed. It is pointed out that voltage-controlled converters are facing large inrush current, which can help the design of control system and protective devices in the microgrid. The second part and the third part both talk about the transient stability problem of the microgrid. In the second part, time-domain simulation method for transient stability analysis is discussed based on a reduced mathematical model. In the third part, transient stability is investigated through a theoretical method, such as Lyapunov function and linear matrix inequality. The key technical issues are discussed and possible solutions are given in the book. These works are helpful and inspiring to understand the problems faced by the transient operation of the microgrid. The main results of this book are original from the authors who perform the related research activities long-term and sustainably. I have no doubt that this book

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Foreword

can provide wide use for researchers, engineers and graduate students who carry on the related topics. College of Electrical and Information Engineering, Hunan University, Changsha, Hunan, China

An Luo

Preface

With more and more distributed resources are connected to the power system, the microgrid concept has been proposed to regulate distributed resources and different loads as a controllable unit. This makes microgrid to become a typical local power electronic system with a high proportion of power electronic devices. In the past few years, microgrid technology has developed rapidly in both industrial application (such as high reliable power supply, wide range of frequency oscillation elimination, special power supply, circulation suppression, etc.) and theoretical researches (such as mathematical modelling, topology transformation, control optimization and small-signal stability analysis, etc.). These researches deal with several problems encountered by the promotion of microgrid technology. Since power electronic devices have limited thermal capacity and weak antidisturbance ability, microgrid is subjected to serious transient stability and poor reliability problem. In recent years, it is reported that power electronic devices are exposed to burn out and the microgrid is prone to transient instability during the transient period in the existing microgrid test bed. To promote the industrial application of microgrid, it is important to carry out theoretical insights into the transient operation of microgrid, especially under extreme conditions. This is the core topic discussed in the book. There are two main concerns raised by the reliable operation of the microgrid during the transient period: (1) the safety of power electronic devices; (2) the stability of the microgrid under disturbances. This book intends to bring state-of-the-art researches on the transient operation of microgrid in recent years. The book has collected new research ideas and achievements such as fault characteristics comparison among current-controlled IIDGs and voltage-controlled IIDGs, fault ride through control strategies, mathematical modelling and model reduction method in spatial scale and multi-time scale and transient angle stability analysis of virtual synchronous generators. The motivation of this book is to give the authors’ perspectives on this topic and provide basic discussions on several problems encountered by microgrid during transient period. The authors hope that this book would help to attract more attention on this topic and can inspire researches in the near future. vii

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Preface

The main research results of this book are original from authors who carried out the related research together for almost 5 years, which is a comprehensive summary for the authors’ latest research results. This book is likely to be of interest to university researchers, R&D engineers and graduate students in electrical engineering who wish to learn the core principles, methods and applications of microgrid during the transient period. Changsha, China 2020

Zhikang Shuai

Acknowledgements

This book is supported by the Nature Science Foundation of China (NSFC) under Grant 51622702 and 51977066. The author would like to thank Prof. Z. John Shen from Illinois Institute of Technology (IIT) for his great support and valuable comments on this book. Special thanks go to the postgraduate students Ms. Xia Shen, Ms. Lili He, Ms. Huijie Cheng, Mr. Feng Zhao, Mr. Yelun Peng and Mr. Chao Shen for their contributions and proofreading. Finally, the author would like to thank the long-term support and encouragement from the family.

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Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Trends and Development of Power Electronic System . . . . . . . 1.2 Features of Microgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Transient Characteristics of Microgrid . . . . . . . . . . . . . . . . . . 1.4 Transient Stability Problem of Microgrid . . . . . . . . . . . . . . . . 1.5 Challenges of Transient Characteristics and Stability Analysis of Microgird . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

1 1 3 4 7

. . .

10 11 12

Transient Characteristics of Current Controlled IIDGs During Grid Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Principles and Control Structures . . . . . . . . . . . . . . . . . . 2.1.1 Topology of Three-Phase Inverter . . . . . . . . . . . . . . . . 2.1.2 Principle of Constant Current Control . . . . . . . . . . . . . 2.1.3 Principle of PQ Control . . . . . . . . . . . . . . . . . . . . . . . 2.2 Transient Characteristics of Constant Current Controlled IIDGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Fault Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Fault Current Calculation . . . . . . . . . . . . . . . . . . . . . . 2.3 Transient Characteristics of PQ Controlled IIDGs . . . . . . . . . . . 2.3.1 Transient Characteristics of PQ Controlled IIDGs During Symmetrical Fault . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Transient Characteristics of PQ Controlled IIDGs During Asymmetrical Fault . . . . . . . . . . . . . . . . . . . . . 2.3.3 Influence of Current Limiter . . . . . . . . . . . . . . . . . . . . 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 17 18 19 19 20 20 21 22 41 42 43

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3

Contents

Transient Characteristics of Voltage Controlled IIDGs During Grid Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Principle and Control Structures . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Principle of V/f Control . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Principle of Droop Control . . . . . . . . . . . . . . . . . . . . . 3.1.3 Principle of Virtual Synchronous Control . . . . . . . . . . . 3.2 Transient Characteristics of V/f Controlled IIDG During Grid Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Fault Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Fault Current Calculation . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Influencing Factors of Fault Current Characteristics . . . 3.3 Transient Characteristics of Droop Controlled IIDG During Grid Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Fault Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Fault Current Calculation . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Influencing Factors of Fault Current Characteristics . . . 3.4 Transient Characteristics of VSG Controlled IIDG During Grid Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Fault Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Fault Current Calculation . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Influencing Factors of Fault Current Characteristics . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46 47 49 50 51 51 58 62 63 64 66 71 71 72 76 77 78

4

Fault Ride Through Control Methods of VSG Controlled IIDGs . . 81 4.1 Typical Topology of VSG Controlled IIDGs . . . . . . . . . . . . . . . 81 4.2 Problem Description of VSG Controlled IIDGs During Fault . . . 82 4.2.1 Instantaneous Inrush Current of VSG Controlled IIDGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2.2 Analysis for Maximum Withstanding Time of VSG Controlled IIDGs During Fault . . . . . . . . . . . . . . . . . . 83 4.2.3 Difficulties in Restraining Instantaneous Inrush Current of VSG Controlled IIDGs . . . . . . . . . . . . . . . . . . . . . . 86 4.3 Fault Ride Through Control Methods of VSG Controlled IIDGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3.1 Current Limiting Control Method Based on Virtual Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.2 Fast Inrush Current Restraining Method Based on Control Mode Switching . . . . . . . . . . . . . . . . . . . . 91 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5

Full-Order Modeling and Dynamic Stability Analysis of Microgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1 Full-Order Modeling of Microgrid . . . . . . . . . . . . . . . . . . . . . . 101 5.1.1 Coordination Transformation for DERs . . . . . . . . . . . . 102

Contents

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5.1.2

6

7

Modeling of Inverters with Different Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Modeling of Network . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Modeling of Different Kinds of Loads . . . . . . . . . . . . 5.1.5 Full-Order Modeling of Microgrid . . . . . . . . . . . . . . . 5.2 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Parameter Stability-Region of Microgrid . . . . . . . . . . . . . . . . . 5.3.1 Bifurcation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Parameter Stability-Region Analysis . . . . . . . . . . . . . 5.3.3 Verification of Bifurcation Instability . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

103 109 110 113 114 118 119 120 126 130 131

Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Multi-Time Scale Property of Microgrid . . . . . . . . . . . . . . . . . 6.2 Wide Frequency Range Stability Problem Classification . . . . . 6.3 Time-Scale Model Reduction of Microgrid . . . . . . . . . . . . . . . 6.3.1 Singular Perturbation Theory . . . . . . . . . . . . . . . . . . 6.3.2 Singular Perturbation Reduction of Microgrid . . . . . . 6.3.3 Verification of Reduced Order Model . . . . . . . . . . . . 6.4 Comparative Study of Different Reduced Models . . . . . . . . . . 6.4.1 Eigenvalue Comparative Analysis . . . . . . . . . . . . . . . 6.4.2 Numerical Comparative Simulation . . . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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133 133 135 136 136 138 146 148 150 153 158 159

Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Concept of Dynamic Equivalent Modeling for Multi-Microgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Dynamic Equivalent Model of External Microgrid . . . . . . . . . . 7.2.1 The Division of External Microgrid . . . . . . . . . . . . . . . 7.2.2 Simplification of Network . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Aggregation of Buses . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Aggregation of DERs . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Verification of the Dynamic Equivalent Model . . . . . . . . . . . . . 7.3.1 Evaluation for the Studied System . . . . . . . . . . . . . . . . 7.3.2 Evaluation of Testing Multi-Microgrid with 15 Buses . . . 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 161 163 164 165 167 168 171 172 178 183 184

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9

10

Contents

Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic Phasor Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Concept of Dynamic Phasor Method . . . . . . . . . . . . . . . . . . . . 8.2 Dynamic Phasor Modeling of Asymmetrical Microgrid . . . . . . . 8.2.1 Dynamic Phasor Model of VSG . . . . . . . . . . . . . . . . . 8.2.2 Dynamic Phasor Model of Single-Phase PV . . . . . . . . . 8.2.3 Aggregation of DG Model . . . . . . . . . . . . . . . . . . . . . 8.2.4 Dynamic Phasor Model of Load and Network . . . . . . . 8.2.5 Dynamic Phasor Model of Asymmetrical Microgrid . . . 8.3 Eigenvalue Analysis of Asymmetrical Microgrid . . . . . . . . . . . . 8.3.1 Case Study 1: Load Disturbance Test . . . . . . . . . . . . . 8.3.2 Case Study 2: Asymmetrical Short-Circuit Fault Test . . . 8.4 Improved Voltage Unbalance Compensation Strategies for Asymmetrical Microgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Small-Signal Analysis of the Voltage Unbalance Compensation Control . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Compensation Method to Improve the Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transient Angle Stability of Grid-Connected VSG . . . . . . . . . . . . . 9.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Full-Order Model of a VSG . . . . . . . . . . . . . . . . . . . . 9.1.2 Model Reduction of a VSG . . . . . . . . . . . . . . . . . . . . . 9.2 Transient Angle Stability Mechanism . . . . . . . . . . . . . . . . . . . . 9.2.1 Transient Angle Stability of VSG . . . . . . . . . . . . . . . . 9.2.2 Deteriorative Effect of Q-V Droop on Transient Angle Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Simulation and Experiment Results . . . . . . . . . . . . . . . 9.3 Stability Region Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Derivative of Lyapunov Function . . . . . . . . . . . . . . . . 9.3.2 Proposed Lyapunov Method Considering the Influence of Reactive Power Loop . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Influence of Different Parameters . . . . . . . . . . . . . . . . 9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Model of Paralleled VSGs . . . . . . . . . . . . . . . . . . . . 10.1.2 Model of Paralleled SGs and VSGs . . . . . . . . . . . . . . 10.2 Transient Angle Stability Mechanism . . . . . . . . . . . . . . . . . . . 10.2.1 Transient Angle Stability of Paralleled VSGs . . . . . . .

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185 185 188 188 191 192 193 194 199 201 201 204 205 210 214 216 217 217 218 220 222 223 224 226 228 228 233 235 242 242 245 245 245 249 252 252

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10.2.2

Transient Angle Stability of Paralleled SGs and VSGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Differences Between Paralleled VSGs and Paralleled SGs and VSGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Stability Improvement of Paralleled SGs and VSGs . . 10.2.5 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Stability Region Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Lyapunov Function of Paralleled SGs and VSGs . . . . 10.3.2 Influence of Different Parameters . . . . . . . . . . . . . . . 10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Re-synchronization Phenomenon of Microgrid . . . . . . . . . . . . . . . 11.1 Re-synchronization Phenomenon of VSG . . . . . . . . . . . . . . . . 11.1.1 Mechanism of Re-synchronization . . . . . . . . . . . . . . . 11.1.2 Influence of Different Parameters on Re-synchronization . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Re-synchronization Phenomenon of Paralleled Systems . . . . . . 11.2.1 Re-synchronization of Paralleled VSGs . . . . . . . . . . . 11.2.2 Re-synchronization of Paralleled SGs and VSGs . . . . 11.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 253 . . . . . . . .

254 256 257 260 261 266 268 270

. 271 . 271 . 272 . . . . . . .

275 278 283 283 285 287 290

Abbreviations

APOC CHIL DE DEM DERs DFIGs DGs DP DSP EAPO EARPO EEAC EMF EMF EP ERPO ES ESS FCL FRT HB HBD IIDGs IM IMG KVL LCL LMI LPF

Active Power oscillation control Control hardware-in-loop Dynamic equivalent Dynamic equivalent model Distributed renewable energy resources Doubly fed induction generators Distributed generators Dynamic phasor Digital signal processor Elimination of active power oscillation Elimination of active and reactive power oscillation Extended equal area criterion Electromotive force Transient electromotive force Equilibrium point Elimination of reactive power oscillation Energy storage Energy storage system Fault current limiter Fault ride through Hopf bifurcation Hysteresis band Inverter-interfaced distributed generators Induction motor Isolated microgrid Kirchhoff’s voltage law Inductor–capacitor–inductor filter Linear matrix inequalities Low pass filter

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LVRT MMG MPPT NSCC NUEP PCL PI PLL PSS PSTs PWM P-δ RCG RESs RPOC SNB SOFC SOGI-PLL SPSC SVIB TS UEP VPA VSG VUC VUF ZIP ZSCC

Abbreviations

Low voltage ride through Multi-microgrid Maximum power point track Negative sequence current control The next unstable equilibrium point Peak current limitation Proportional-integral Phase locked loop Power system stabilizer Phase-shifting transformers Pulse width modulation Power-angle Reference current generation Renewable energy resources Reactive power oscillation control Saddle node bifurcation A tubular solid oxide fuel cell Second order generalized integrator phase locked loop Symmetrical positive sequence current A single VSG connected to infinite bus Takagi-Sugeno Unstable equilibrium point Virtual power angle Virtual synchronous generator Voltage unbalanced compensation Voltage unbalance factor Constant impedance, constant current, constant power Zero sequence current control

Chapter 1

Introduction

In this chapter, trends and development of power electronic system will be introduced and microgrid is taken as the research target particularly. Then, the features of microgrid different from traditional power system are summarized. These features determine that the transient operation characteristics and transient stability of microgrid need to be paid attention to particularly. Afterwards, brief introductions and state-of-art researches on transient characteristics and transient stability of microgrid are provided. Finally, based on these introductions, the challenges of microgrid during transient period will also be discussed.

1.1

Trends and Development of Power Electronic System

Along with the development of human society, large amount of energy sources is needed to support the development. However, traditional energy sources, like oil, coil, etc., are limited and harmful to the environment. Thus, renewable energy sources, like solar, wind, etc., are widely investigated to solve energy crisis and environment problems. Since the output of renewable energy sources needs to be transferred into viable electricity, power electronic devices like inverters are employed as the interfaces to improve the controllability of renewable energy sources and ensure the power quality. Due to the advantages of power electronic devices in power conversion, the usage of power electronic devices dramatically increased in recent years. The increasing penetration of power electronic devices has significantly challenged the traditional power system in terms of intermittent power generation, bidirectional power flow, limited overcurrent capacity, etc. In order to effectively manage renewable energy sources, several concepts of the power electronic system are proposed. 1. Large-scale power electronic system. The output power of renewable energy resources (RESs), like solar, wind, etc., highly depends on the places, weather © Springer Nature Singapore Pte Ltd. 2021 Z. Shuai, Transient Characteristics, Modelling and Stability Analysis of Microgird, https://doi.org/10.1007/978-981-15-8403-9_1

1

2

1 Introduction

and other uncontrollable elements. For example, the solar energy is adequate in western-north china while hydropower resources are concentrated in the southern china. Thus, to make the best of RESs in different places, large-scale renewable energy system is built according to the features of the local places, like wind farm, Photovoltaic plant, etc. Large-scale renewable energy system can aggregate large amount of the DERs to construct a high penetration of power electronic system. Since the output power of RESs is unpredictable, it probably results in the voltage drop and other stability problems due to the power fluctuation. Large-scale renewable energy system can be connected to the traditional power system through specific nodes, thus the stability problem can be solved conveniently. Moreover, such a system can provide large amount of energy with less loss through HVDC, etc. 2. Distributed power electronic system. Distributed renewable energy sources (DERs) are widely adopted for its convenience and efficiency, including energy storage system (ESS), PV, wind turbine and diesel generator, etc. DERs can provide flexibility and convenience of the connection of renewable energies and attracts great attention. However, the output of DERs is unpredictable which affects the regulation of system energy. In this condition, the concept of microgrid was proposed which aggregates DERs, loads and energy storage (ES) units. The microgrid can be used to enhance power reliability and reduce operation cost with proper control and design. However, many challenges have been raised simultaneously. The feature of such a system is that the power supply radius is small and the grounded faults spread in a faster speed. One of severe challenges is that fault current in a microgrid probably damages power electronic devices because of the limited thermal overcurrent capacity of semiconductor devices. Moreover, inverter-interfaced distributed generators (IIDGs) response quite differently from synchronous generators (SGs) during fault period so that fault current calculation methods of conventional power systems cannot be fully applicable. This probably makes protection systems invalid during transient period. Thus, the transient response of such a system needs to be studied comprehensively. Along with the development of power electronic technique, power electronic devices are widely adopted in the power system to transfer renewable power to available electricity. Compared with the large renewable energy system, distributed renewable energy system is more applicable in the distribution system. Moreover, distributed renewable energy system is more appealing in the future development due to its limited environmental requirements. As an important concept of distributed power electronic system, microgrid has attracted great attention for its various advantages and are widely adopted in the island power supply system, remote area, etc. Thus, in this book, the research is focused on the distributed power electronic system and microgrid is taken as the research target.

1.2 Features of Microgrid

1.2

3

Features of Microgrid

With more and more power electronic devices (e.g. PV, wind farm, storage system, etc.) are connected to the traditional power system, the system operation characteristics undergoes a significant change, like bidirectional power flow in distribution network, flexible control strategy, etc.. The response of power electronic devices is quite different from that of the traditional synchronous generators. As a result, the operation of microgrid, which is aggregated with the power electronic devices, is different from the traditional power system, especially during transient period. To take insight into the transient stable operation of microgrid, features of the system need to be concluded. It can help to give guidance of the system design and planning. Figure 1.1 shows typical topology of microgrid. Compared with traditional power system, features of microgrid are drawn as follows. 1. Limited thermal capacity. Power electronic devices, like IGBT, MOSFET, JEFT, etc., are fully-controlled devices so that they can realize power conversion flexibly. When choosing devices’ parameters, economy and reliability should be considered at the same time. Generally speaking, these devices are usually chosen as the two times of the rated value. However, large inrush current would occur during transient period which probably threat the safety of the devices. Although SiC, GaN and other material devices with better overcurrent capacity are emerging, it still remains the problem on safe operation of power electronic devices during transient period. Distribution PCC Network Breaker

BUS Line1

Domestic consumer

GND Line2 Wind turbine AC-DC

Induction motor

DC-AC

Line3 PV generator

DC-DC

DC-AC

Breaker

Industrial load

Line4

Energy storage

DC-DC

DC-AC

Fig. 1.1 Topology of a microgrid

Microgrid

Diesel generator

4

1 Introduction

2. Strong nonlinearity. Power electronic devices can realize power conversion by controlling the power electronic switches on and off. Along with the switches state changing, the system topology varies at the same time. Thus, the power electronic system is time-varying nonlinear system which brings the difficulties in the system modelling and analysis. Moreover, over-modulation and limiting units make the system operation condition far more complicated. Limiting units like current limiter would probably turn the closed-loop system into the open-loop system. In addition, when the inverter operates at the over-modulation mode, the system would turn into a positive-feedback system instead of a negative-feedback system which probably drives the system into an unstable state. 3. Multi-time-scale dynamic response. Since microgrid can realize power conversion fast and flexibly, it presents dynamic response characteristics in a wide frequency range compared with the traditional power system. At the power electronic devices level, power electronic switches present us level and other distributed energy sources (PV, Wind turbine, ES, etc.) response at a range from us to min level. At the control system level, voltage and current control loops are at the ms level and power control loop response at the 100 ms to s level, while the optimization control of the system s is always in a range of min level. 4. Small inertia and weak anti-disturbance ability. Different from the traditional SGs, it is known that there is no physical rotating unit in power electronic devices. Thus, the power electronic devices can only provide limited inertia to the system through equipping storage system, like flywheel storage, super capacitor, etc., and adding proper control strategy, like virtual synchronous control, droop control, etc. When the power electronic system is exposed to system disturbances, like load fluctuation, grounded fault, etc., it probably leads to the power oscillation and instability of the system due to the small inertia and damping coefficiency. 5. Intermitted power supply. Due to the environment pollution and energy crisis, RESs, like solar energy, wind, etc., are widely investigated to deal with these problems. Therefore, PV panel, wind turbine and other energy transformation units are connected through power electronic devices to realize energy conversion. However, it is known that RESs rely on the weather and other uncontrollable conditions, like the temperature, earth environment, etc. Thus, the output power of power electronic devices is intermitted in a range of long time-scale. Though storage systems can balance the output power oscillation, the expense of the system would increase incredibly.

1.3

Transient Characteristics of Microgrid

It is known that power electronic devices have limited thermal capacity. Thus, it is necessary to design fault ride through control (FRT) strategy to protect the devices during transient period. Before considering the design of the FRT strategy, the

1.3 Transient Characteristics of Microgrid

5

transient characteristics of the system should be studied thoroughly to guide the control design. Many researches discussed the response of different kinds of IIDGs when subjecting to large disturbances [1–24]. It is known that transient characteristics of IIDGs are highly depends on their control strategies [1]. The diversification of control schemes adopted by IIDGs in a microgrid leads to the complexity on fault current estimation. From the perspective of the control strategy, IIDGs can be divided into voltage-controlled IIDGs and current-controlled IIDGs. Different control strategies lead to different response characteristics during transient period. To extend conventional fault analysis methods to IIDGs, current-controlled IIDGs are modeled as ideal current sources and voltage-controlled IIDGs are modeled as ideal voltage sources during fault period in [2, 3]. However, the accuracy of fault models is not enough during the sub-transient (the first cycle) and transient (2–6 cycles) periods. To better reflect transient response of IIDGs, dynamic characteristics of control systems needs to be considered. In the following part, the discussion on transient characteristics of IIDGs would be divided into current-controlled IIDGs and voltage-controlled IIDGs. For current-controlled IIDGs, they can regulate output current directly so that fault current can be described by analyzing dynamics of control loops [4]. The basic method to estimate fault current of current-controlled IIDGs is to establish their mathematical model. It is pointed in [1] that constant current-controlled IIDGs deliver constant current under both symmetrical and asymmetrical faults. When another widely used current control strategy, called PQ control, is adopted, fault current is relevant to the voltage dips. It means the equivalent fault model of PQ-controlled IIDGs consists of a controlled current source paralleled with admittance. Since current-controlled IIDGs can be well controlled by controllers, their fault current can be described by mathematical model precisely. Moreover, for current-controlled IIDGs, primary energy, such as wind turbine or PV system, also decides dynamic response of dc side voltage. Thus, it is also necessary to discuss the primary energy types on fault current of IIDGs and several researches are carried out, which can be found in [5–12]. For voltage-controlled IIDGs, it is much more complicated than currentcontrolled IIDGs because they cannot regulate output current directly [13]. A feasible idea to study fault current of voltage-controlled IIDGs is to find out the dominate factors in transient response by analyzing the time constants of different control loops. And then fault analysis of IIDGs can be simplified into mathematical descriptions on the interaction between two voltage sources. V/f control, droop control and virtual synchronous control are three voltage control strategies which are widely used and these three control strategies have different control targets and features. V/f control is used to stabilize the voltage and frequency to normal value in islanded microgrid. But this control strategy is just suitable for IIDGs with sufficient energy storage and extra load is undertaken by V/f-controlled IIDGs. To realize distributed power sharing, droop control is proposed. This control strategy can help paralleled IIDGs to realize power sharing without communication. However, this control strategy performs not well enough during dynamic period since it lacks inertia and

6

1 Introduction

damping. Thus, virtual synchronous control is proposed. The general idea of virtual synchronous control is to mimic dynamic response of synchronous generators. It can provide virtual inertia and virtual damping for IIDGs which can benefits dynamic response of IIDGs. Based on the above basic idea, inrush fault current of V/f control, droop control and virtual synchronous control is identified in [1, 14], which consists of a gradually attenuated periodic component and a dc component. To consider the effect of primary energy on fault current, fuel cell and tubular solid oxide fuel cell (SOFC) are chosen as the typical voltage-controlled IIDGs. The dynamic models of these two microsources were discussed in [15–21]. It shows that these two voltagecontrolled IIDGs would supply larger short-circuit current than current-controlled IIDGs. Thus, special attention needs to be paid to limit short-circuit current of voltage-controlled IIDGs. The comparisons of fault current among currentcontrolled IIDGs and voltage-controlled IIDGs are given in [1]. Fault current of voltage-controlled IIDGs has larger peak value and shorter peak time than currentcontrolled IIDGs. These fault characteristics require that the control targets of fault current limiting strategies of current- and voltage-controlled IIDGs are different. Several researches are dedicated to limit fault current of IIDGs. In general, there are two kinds of fault current limit strategies: active current limiting method [14] and passive current limiting method [1]. Active current limiting method represents that fault current is limited by means of control while passive current limiting method means limiters are designed to limit fault current. For current-controlled IIDGs, fault capacity is limited due to fault characteristics of primary energy. The control targets of current-controlled IIDGs during fault period can be designed to limit power oscillation (active power and reactive power) and current amplitude [22]. Different control targets can be realized by controlling output current of IIDGs directly. Compared with current-controlled IIDGs, fault current of voltage-controlled IIDGs is larger and serious. Thus, it is urgent to carry out fault current limiting method of voltage-controlled IIDGs. It is discovered that inrush current would occur in voltagecontrolled IIDGs and it probably damages semiconductor devices [14]. This fault characteristic requires fault current limiting strategies react in short time. Existing fault current limiting methods include switching to current control mode [14], virtual impedance control [27], etc. The former control strategy can design and limit output current directly while has limited ability in supporting frequency and voltage in microgrid during fault period. The latter control strategy can maintain voltage control mode while it has limited performances in limiting inrush current especially in short time scale. When it comes to passive current limiting method, different limiters in different control loops have great impacts on fault response of IIDGs, which are widely adopted in both current-controlled IIDGs and voltage-controlled IIDGs [1]. Limiters are usually used in control systems to achieve the rational operation of a microgrid. Due to limited thermal overcurrent capacities of semiconductor devices, current limiters are necessary for the safe operation of inverters [1]. Meanwhile, modulating wave limiters are applied to avoid being driven into over-modulation range [23]. Specific to droop-controlled IIDGs, droop limiters are needed for system stability and high-power quality [24]. During transient period, system electrical parameters

1.4 Transient Stability Problem of Microgrid

7

undergo significant change so that control systems can be easily driven into the states that limiters function. These nonlinear limiters would change operation states of IIDGs and make fault current totally different from normal operation conditions. For the reliability of protection devices, the impacts of different limiters also need serious attention.

1.4

Transient Stability Problem of Microgrid

In this section, a summary of the definition, analysis and classification of the microgrid stability is presented. The power system stability has been recognized as a significant issue for the secure operation since 1920s. The nature of microgrid stability issue is considerably different from that of a traditional power system, which is reflected in the factor leading to instability, instability phenomena, and disturbance types. Compared with bulk power system, microgrid is relatively small and a limited number of generators are integrated. A higher uncertainty of disturbance including generator shut down, load change and faults, results in the relatively large and fast deviation of frequency and load. As a result, microgrids are more prone to instability. Recently, the cases of microgrid instability have been reported in China, USA and Europe. The important features of microgrid relevant to system stability are: small size, uncertainty of renewable energy generation, diversity of power generation, low system inertia, limited current capacity, asymmetrical three-phase loading, and high R/X ratio. These intrinsic differences between microgrid and conventional power system make the stability definitions and classification for microgrids has its characteristics. The classification of microgrid stability can be carried out according to the physical cause, the disturbance size, the involved physical components, and the time-span. The relative disturbance size is the useful classification that can be divided into small-signal analysis and large-signal analysis. Small-signal analysis means that a disturbance in microgrid is considered small and a linearized model can adequately describe the system behavior. Small-signal stability analysis determines in which range zeros the system parameters can maintain stability. The most common phenomenon of small-signal instability is a wide frequency range of oscillations caused by the poor damped dominant modes after a small disturbance [25, 26]. Eigenvalue analysis for the state-space model and impedance analysis method based on frequency model are useful analysis methods to analyze the small-signal stability of microgrid on system level [25, 26]. Great effort has been made on the accurate modeling, participation analysis of microgrid with various connected components. It is found that the time delay from the hierarchical control structure, the poor parameter design of power sharing and inner control loops, and the interaction between generator and load are the major root causes of small-signal instability problems in microgrid. However, there is no explicit knowledge about what degree of disturbance is small. Moreover, as the microgrid has low inertia and shows high nonlinear dynamics, even a relatively

8

1 Introduction

small disturbance could cause a large deviation of system state. The small-signal analysis from the linearized model might not describe the microgrid stability accurately. The small signal analysis needs to be extended to large-signal stability analysis. Large-signal analysis, which is also called transient stability analysis, focuses on the disturbance that is considered relatively large, including grounded faults, abrupt power loss, off grid switching, etc. These disturbances frequently occur and may result in serious impact on the system stable operation, even leading to the system break down, thus it needs more attention. Moreover, according to the characteristics of microgrid as discussed in the previous paragraph, the microgrid stability is confronted with great challenges compared with the conventional power system due to its small size. Some cases of the microgrid instability have been reported in the actual operation microgrid and demonstration projects, like in CERTs [27, 28]. Therefore, transient stability analysis is really urgent to carry out to speed up the application of microgrid technique. Different from small-signal stability, transient stability analysis determines in which range of zones the system state variables can maintain stability. And once state variables are out of the range, the system is judged to undergo irreversible instability. These research results can help to design and operate the microgrid in stable state even the system is confronted with large disturbances [29]. However, relatively researches about transient stability analysis are rare at present. Several reasons account for this condition. The first one is that the mathematical model order of inverter-interfaced microgrid is high due to the complexity of its control system and physical system [30, 31]. And the mathematical model presents highly nonlinear characteristics, which means that the small-signal stability analysis method, such as impedance-based criterion, root-locus method and bifurcation theory, etc., cannot be adopted directly into the analysis [29]. The second reason is that the microgrid presents wide time-scale characteristics, which gives difficulties in the insight into the system instable mechanism. The microgrid wide time-scale can be divided into current control time-scale, voltage control time-scale and power control time-scale. It should be pointed out if the hierarchical control is considered, the response characteristics of the microgrid are more complex. Furthermore, different timescales are coupled which result in the difficulty in the transient stability analysis [32]. The third reason is that there is no systematic method that can be adopted to deal with the transient stability analysis. The most commonly used method is timedomain simulation method which is based on the differential-algebraic equation (DAE). Several calculation soft wares are developed such as PSCAD/EMTDC, Matlab/Simulink, etc. However, this method is time-consuming and cannot give guidance to the system design since only one case can be simulated in one condition [33]. Another method is called Lyapunov’s direct method, which determines the attraction region of the system in state variables domain [34, 35]. It is realized by constructing a candidate Lyapunov function based on the system reduced model. This method can give guidance to the system design and determine the system stable region quantifiably. However, there is no systematic method to construct a candidate Lyapunov function for the system model higher than three orders. As discussed

1.4 Transient Stability Problem of Microgrid

9

Microgrid Stability

Control System Stability

Electric Machine Stability

Power Supply and Balance Stability

Voltage Stability

Converter Stability

System Voltage Stability

Frequency Stability

DC-Link Voltage Stability

Small Disturbance

Large Disturbance

Small Disturbance

Large Disturbance

Short Term

Long Term

Short Term

Long Term

Fig. 1.2 Classification of stability in microgrid Ref. [38]

above, the mathematical model of microgrid system is usually higher than threeorder, even model-reduction method is adopted. Since there are two common methods can be used to study transient stability and several problems related to these two methods are needed to be solved, this book would carry out transient stability analysis using mathematical simulation and Lyapunov’s direct method. Corresponding contents are shown in Chaps. 5–11. In traditional power system, there are systematic definition and classification on transient stability [36, 37]. When it comes to the study on transient stability of microgrid, classifications of different transient stability problems are given in [38], as shown in Fig. 1.2. Recent researches show that traditional transient angle stability may occur in virtual synchronous generators (VSGs) or droop-controlled inverters due to their dynamic response emulation of synchronous generators [35, 39, 40]. It was pointed out in [36, 37] that transient instability results from sustained imbalance between the electromagnetic and mechanical torque in the SGs. Different from SGs, VSGs can regulate output active power as well as reactive power through active and reactive power control loops. During transient period, like grounded fault, there would be a sudden increase in the output reactive power of an inverter, which would lead to an abrupt decrease in the internal voltage amplitude of the inverter through reactive power control loop. This will reduce transient angle stability margin of VSGs. The effect of reactive power control loop on transient stability has been identified in [35] and paralleled operation between VSGs and SGs are given in [39]. Moreover, the effect of the current limiter on the transient instability mechanism of droop-controlled inverters was analyzed in [41, 42]. Furthermore, for current-controlled inverters based on phase lock loop (PLL), transient instability phenomenon is also discovered and instability mechanism is identified [43, 44]. The

10

1 Introduction

inherent similarity and differences among current-controlled inverters, voltagecontrolled inverters and SGs on transient instability mechanism are studied. This book will focus on the transient angle stability of voltage-controlled inverters and the latest researches will be presented in the following contents.

1.5

Challenges of Transient Characteristics and Stability Analysis of Microgird

Although many theoretical researches on transient characteristics, modelling and stability analysis have been carried out, there are still many works need to be performed. In order to move forward the practical application of microgrid, challenges can be summarized as follows: 1. Problem of transient characteristics study and fault ride through control. Power electronic devices are widely adopted to realize power conversion. It is known that power electronic devices have limited thermal capacity. The devices are usually chosen as the two times of the rated value to balance the reliability and cost. However, large inrush current would occur during transient period and probably burn the devices down. Thus, it is quite important to take insight into the transient characteristics of power electronic devices to provide basis for devices protection. Furthermore, fault ride through control strategy is also introduced to protect power electronic devices due to the flexibility and controllability of power electronic devices. But the burn down of power electronic devices ranges in few milliseconds. Therefore, fast monitoring system is the key of the system protection. Up to now, there have been many reports on how to protect power electronic devices, but when it comes to semi-conductor devices monitoring during transient period, the related researches are urgently needed. 2. Problem of complexity of mathematical model. Simulation based on the mathematical model is an important method to investigate on the transient response of the microgrid. Compared with the traditional power system, the mathematical model of the microgrid is far more complicated due to the diversification of the control system and system structure. Thus, the efficiency of the microgrid simulation is low and it needs a lot of time especially in a system with large amount of power electronic devices since the conventional simulation calculation is usually based on derivative-algebraic equation. Hence, there is a compromise between high efficiency and accuracy of the simulation system. To accomplish this purpose, reduced model with high accuracy under different conditions needs to be established. 3. Problem of complicated transient stability analysis. The microgrid is usually confronted with transient operation condition, such as heavy load switching, grounded fault, off-grid and grid-connected mode switching etc. Different from traditional power system, transient stability of the microgrid has its unique difficulties and characteristics. The microgrid is nonlinear system which results

1.6 Structure of the Book

11

in the research methods in small-disturbance analysis are not applicable to transient stability analysis. In general, systematic theory to study transient stability problem is absent. Though there is related research theory such as Lyapunov’s theory, it’s just applicable to simple systems. However, nonlinear units, such as amplitude limiting unit and nonlinear modulation characteristics, and timevarying control strategies lead to the difficulties in transient stability analysis. Moreover, the microgrid has complicated dynamic response characteristics due to the multi-time-scale control and circuit system. Thus, appropriate theory and detailed stability problem identification need to be carried out.

1.6

Structure of the Book

Transient Characteristics, Modelling and Stability Analysis of Microgrid

According to the preliminary study, this book would bring latest research result on the transient characteristics, modelling and stability analysis of the microgrid. The overall structure of the book is shown in Fig. 1.3. Specific contents within each chapter will be mentioned below. In part I, transient characteristics of inverter-interfaced distributed generators are given. Both current-controlled and voltage-controlled inverters are taken as the Part I: Transient characteristics of inverterinterfaced distributed generators

Part II: Systematic modelling and model reduction methods for Microgrids

Part III: Transient angle stability and resynchronization stability analysis in Microgrid

Fig. 1.3 Structure of the book

Transient Characteristics of Current Controlled IIDGs during Grid Fault Transient Characteristics of Voltage Controlled IIDGs during Grid Fault Fault Ride Through Control Methods of Virtual Synchronous Generator Controlled IIDGs

Full-order Modeling and Dynamic Stability Analysis of Microgrid Time-scale Model Reduction of Microgrid based on Singular Perturbation Theory Spatial-scale Model Reduction of Multi-Microgrid based on Dynamic Equivalent Theory Modeling and Stability Analysis of Asymmetrical Microgrid based on Dynamic Phasor Theory Transient Angle Stability of Grid-connected Virtual Synchronous Generator Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs Re-synchronization Phenomenon Microgrid

12

1 Introduction

research targets. Moreover, influence of common factors including line impedance, fault occurring moment, fault type, and nonlinear limit on fault current characteristics are elaborated. In addition, some typical fault fault-ride through control strategies at present are also introduced based on the theoretical analysis of fault characteristics to eliminate large fault current. In part II, transient stability analysis of microgrid is explored through timedomain simulation method. First, the detailed model of IIDGs, including a number of states from inner control loops and filters, is established. Conceivably, the complete model of microgrid with tens or hundreds of inverters becomes complexity and high-order. The numerical transient analysis of such microgrid model becomes complex and time-consuming. Then, this part introduces two kinds of reduction methods: time-scales reduction and spatial model reduction methods. Furthermore, considering the natural three-phase asymmetrical structure of microgrid, dynamic phasor modeling method is presented to deal with the mathematical modeling of microgrid under asymmetrical conditions. This work can help to simplify the transient analysis and reduces the computational burden for numerical simulation. In part III, transient stability analysis of microgrid is explored through theoretical method, such as Lyapunov’s direct method and Linear Matrix Inequality (LMI). First, transient angle stability (synchronization stability) is identified in voltagecontrolled IIDGs and virtual synchronous generator (VSG) is taken as the research target to carry out the research. Both single VSG and paralleled system are explored for transient angle stability. Dominate parameters that affects transient angle stability are pointed out quantitatively throughout the theoretical method. Furthermore, re-synchronization stability is discovered in a VSG because of the strong damping characteristics of microgrid. These theoretical analysis give design guide to improve the transient operation ability of the microgrid.

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14

1 Introduction

26. X. Wang, F. Blaabjerg, Harmonic stability in power electronic-based power systems: Concept, modeling, and analysis. IEEE Trans. Smart Grid 10(3), 2858–2870 (2019) 27. A.D. Paquette, D.M. Divan, Virtual impedance current limiting for inverters in microgrids with synchronous generators. IEEE Trans. Ind. Appl. 51(2), 1630–1638 (2015) 28. M.C. Pulcherio et al., Evaluation of control methods to prevent collapse of a mixed-source microgrid. IEEE Trans. Ind. Appl. 52(6), 4566–4576 (2016) 29. M. Kabalan, P. Singh, D. Niebur, Nonlinear Lyapunov stability analysis of seven models of a DC/AC droop controlled inverter connected to an infinite bus. IEEE Trans. Smart Grid 10(1), 772–781 (2019) 30. Y. Peng, Z. Shuai, J. Shen, J. Wang, C. Tu, Y. Cheng, Reduced order modeling method of inverter-based microgrid for stability analysis, in 2017 IEEE Applied Power Electronics Conference and Exposition (APEC), Tampa, FL, (IEEE, Piscataway, 2017), pp. 3470–3474 31. Z. Shuai, Y. Peng, X. Liu, Z. Li, J.M. Guerrero, Z.J. Shen, Dynamic equivalent modeling for multi-microgrid based on structure preservation method. IEEE Trans. Smart Grid 10(4), 3929–3942 (2019). https://doi.org/10.1109/TSG.2018.2844107 32. Z. Shuai, Y. Peng, X. Liu, Z. Li, J.M. Guerrero, J. Shen, Parameter stability region analysis of islanded microgrid based on bifurcation theory. IEEE Trans. on Smart Grid 10(6), 6580–6591 (2019). https://doi.org/10.1109/TSG.2019.2907600 33. Z. Shuai, Y. Peng, J.M. Guerrero, Y. Li, Z.J. Shen, Transient response analysis of inverterbased microgrids under asymmetrical conditions using a dynamic Phasor model. IEEE Trans. Indus. Electron. 66(4), 2868–2879 (2019) 34. D. Marx, P. Magne, B. Nahid-Mobarakeh, S. Pierfederici, B. Davat, Large signal stability analysis tools in DC power systems with constant power loads and variable power loads—A review. IEEE Trans. Power Electron. 27(4), 1773–1787 (2012) 35. Z. Shuai, C. Shen, X. Liu, Z. Li, Z.J. Shen, Transient angle stability of virtual synchronous generators using Lyapunov’s direct method. IEEE Trans. Smart Grid 10(4), 4648–4661 (2019) 36. P. Kundur, Power System Stability and Control (McGraw-Hill Educ, New York, 1994) 37. P. Kundur et al., Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions. IEEE Trans. Power Syst. 19(3), 1387–1401 (2004) 38. M. Farrokhabadi et al., Microgrid stability definitions, analysis, and examples. IEEE Trans. Power Syst. 35(1), 13–29 (2020) 39. H. Cheng, Z. Shuai, C. Shen, X. Liu, Z. Li, Z.J. Shen, Transient angle stability of paralleled synchronous and virtual synchronous generators in islanded microgrids. IEEE Trans. Power Electron. 35(8), 8751–8765 (2020). https://doi.org/10.1109/TPEL.2020.2965152 40. L. Xiong et al., Static synchronous generator model: a new perspective to investigate dynamic characteristics and stability issues of grid-tied PWM inverter. IEEE Trans. Power Electron. 31 (9), 6264–6280 (2016) 41. H. Xin, L. Huang, L. Zhang, Z. Wang, J. Hu, Synchronous instability mechanism of P-f droopcontrolled voltage source converter caused by current saturation. IEEE Transactions on Power System 31(6), 5206–5207 (2016) 42. L. Huang, H. Xin, Z. Wang, L. Zhang, K. Wu, J. Hu, Transient stability analysis and control design of droop-controlled voltage source converters considering current limitation. IEEE Trans. Smart Grid 10(1), 578–591 (2019) 43. Q. Hu, L. Fu, F. Ma, F. Ji, Large signal synchronizing instability of PLL-based VSC connected to weak AC grid. IEEE Trans. Power Syst. 34(4), 3220–3229 (2019) 44. H. Wu, X. Wang, Design-oriented transient stability analysis of PLL-synchronized voltagesource converters. IEEE Trans. Power Electron. 35(4), 3573–3589 (2020)

Chapter 2

Transient Characteristics of Current Controlled IIDGs During Grid Fault

A microgrid can be used to enhance power reliability and reduce operation cost with proper control and design. However, many challenges have been raised simultaneously. One of severe challenges is that the fault current in a microgrid probably damages power electronic devices because of the limited thermal overcurrent capacity of semiconductor devices. Moreover, IIDGs response quite differently from SGs during fault period so that fault current calculation methods of conventional power systems cannot be fully applicable. This probably makes protection systems invalid during transient period. Thus, to enhance the reliability of protection systems as well as to ensure the safety of power electronic devices, fault response of IIDGs needs to be restudied comprehensively. Current control and voltage control are two kinds of widely used control modes for converters. To extend conventional fault analysis methods to IIDGs, this chapter investigates the symmetrical and asymmetrical fault response of current controlled IIDGs.

2.1

Basic Principles and Control Structures

The current controlled IIDGs are designed to deliver power to the grid, which cannot be operated in islanded mode if there is no voltage controlled IIDGs or local synchronous generators supporting the voltage amplitude and frequency of the microgrid. Generally, the current controlled inverters are used as an interface to integrate most of the stochastic RESs, such as PV and wind turbines, into the grid. The current control strategies are mainly divided into constant current (CC) control and PQ control. In this section, the basic principles and control structures of CC control and PQ control will be discussed.

© Springer Nature Singapore Pte Ltd. 2021 Z. Shuai, Transient Characteristics, Modelling and Stability Analysis of Microgird, https://doi.org/10.1007/978-981-15-8403-9_2

15

16

2.1.1

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

Topology of Three-Phase Inverter

Commonly, the three-phase inverters consist of the three-phase three-wire or threephase four-wire inverters depending on whether there is a neutral connection. There are two simple ways of providing a neutral connection for three-phase four-wire inverter systems, which are the split DC-link topology and four-leg topology. In split DC-link topology, the neutral point is provided by connecting the neutral path to the midpoint of the DC-link split capacitors. In four-leg topology, the neutral point is provided by connecting the neutral path to the midpoint of the additional fourth leg. Since an additional leg with switching semiconductors is needed in the three-phase four-leg inverter, the structure and control are more complicated and the cost increases. While, the three-phase four-wire inverter with split capacitor does not have these obvious shortcomings, which is one of the more common ways of providing a neutral point for three-phase four-wire systems. Therefore, the threephase four-wire inverter with split capacitor will be introduced in this section. Figure 2.1a presents the topology of three-phase three-wire inverter connected to the grid. In power circuit, Cdc is the DC side capacitor. Vdc is the DC side voltage. The AC side of inverter is connected to the grid via an LC filter consisting of Lf and Cf. iic and ioc are the inductance current of inverter and the grid side current, respectively. Figure 2.1b shows the topology of three-phase four-wire inverter connected to the grid, which includes the three leg and two DC-link split capacitors to provide a zero sequence current channel. In power circuit, Cdc1 and Cdc2 are the two DC-link split capacitor. Vdc1 and Vdc2 are the DC voltage of two DC-link split capacitors, respectively. The variables of AC side are the same as that of three-phase three-wire inverter. Moreover, since there is no zero sequence current channel for three-phase threewire inverters, their asymmetrical output current only contains positive and negative Fig. 2.1 The diagram of three-phase inverter connected to the grid, (a) three-phase three-wire inverter, (b) three-phase four-wire inverter

Inverter

(a)

PCC iic Lf

Lg

ioc

vgc N

Cdc

Vdc

vgb vga

Cf iiabc

(b)

voab c

Inverter

Vdc1

Cdc1

PCC iic Lf

ioc

vgc N

0

Vdc2

Lg vgb

Cdc2

vga

Cf iiabc

voab c

2.1 Basic Principles and Control Structures

17

sequence components. Therefore, under asymmetrical faults, the three-phase threewire inverter only has four controllable freedom degrees. Instead, the three-phase four-wire inverter has six controllable freedom degrees for its zero-sequence current channel. Hence, it has better control performance than the three-phase three-wire inverter under the asymmetrical faults in theory. In particular, the fault analysis methods and fault characteristics of the three-phase four-wire inverter are also suitable to the three-phase three-wire inverter. Therefore, this chapter focus on the fault characteristics of the three-phase four-wire inverter.

2.1.2

Principle of Constant Current Control

Constant current controlled inverters are usually used as an interface to integrate the PV and wind turbine systems into the grid, which only operated in grid-connected mode [1–4]. CC control aims at delivering constant current to the system and guaranteeing high current quality. The control system is implemented in the abc frame (natural reference frame), as shown in Fig. 2.2. i*iabc and iiabc are the reference value and actual value of the output inductance current of inverter, respectively. voabc is the output capacitor voltage of inverter. Kp and Ki are the proportional gain and integral gain, respectively. Kpwm equals to 1. R and L represent the inverter output impedance. According to Fig. 2.2, the input-output transfer function of CC control can be derived as follows iiabc ¼ Gi ðsÞiiabc

ð2:1Þ

where Gi ðsÞ ¼

Kps þ Ki Ls2 þ ðK p þ RÞs þ K i

ð2:2Þ

It can be observed that the output inductance current of inverter is always controlled to track the reference signals. The magnitude, frequency and initial angle of reference current is given and will not change even the system is subjected to faults. Therefore, the output inductance current of inverter always remains constant.

Fig. 2.2 General control block diagram of CC control

voab c ∗ iiabc

+–

K p s + Ki s

++

Kpwm

+–

1 R + sL

iiabc

18

2.1.3

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

Principle of PQ Control

PQ control aims at delivering the reference power to the grid, which are widely used in PV and wind turbine generation systems [5–8]. To track the reference reactive and active power, the reference current of a PQ controller can be produced by Eq. (2.3) according to the power relationship between output voltage and current and output power. iid* and iiq* are the d-axis and q-axis reference current, respectively. vod and voq represent the d-axis and q-axis output capacitor voltage, respectively. "

iid iiq

# ¼

 vod 1 v2od þ v2oq voq

voq vod



P

 ð2:3Þ

Q

The control block diagram of a PQ controller is shown in Fig. 2.3. iid and iiq are the d-axis and q-axis output current of inverter, respectively. Kip and Kii are the proportional gain and integral gain, respectively. Kpwm equals to 1. The whole system is implemented in a dq reference frame where synchronous angle is aligned to q axis. The outer power loop is to generate and limit the value of the reference current so that it can be tracked by a current controller. The proportional-integral (PI) controller can realize zero steady-state error between the reference current and output current. The difference between PQ controller and CC controller is that the reference current in CC controller will not change whether the grid voltage sags or not, while the reference current in a PQ controller is relevant to PCC voltage and reference active and reactive power.

Fig. 2.3 General control block diagram of PQ control

Voq ∗ iiq

+–

s

++ +

Kpwm

iiq

ωL

ωL

ωL +–

1 Ls + R

– +–

ωL

iiq

iid iid∗

K ip s + K ii

K ip s + K ii s

+–+

Vod

Kpwm

+––

1 Ls + R

iid

2.2 Transient Characteristics of Constant Current Controlled IIDGs

2.2

19

Transient Characteristics of Constant Current Controlled IIDGs

For current controlled IIDGs, its fault response is mostly dependent on the control loops rather than its physical parameters. To investigate the fault characteristics of CC controlled IIDG, the fault models and fault current calculation method are analyzed, respectively.

2.2.1

Fault Models

It can be seen from Eq. (2.1) that the output current of CC controlled IIDGs is only relevant to the reference current. Since reference current remains constant, CC controlled IIDGs would deliver constant current to the grid whether the grid voltage sags or not. In other words, CC controlled IIDGs can be regarded as a constant current source during symmetrical or asymmetrical faults. Therefore, the positive, negative and zero sequence fault models of CC controlled IIDGs can be described by  0 Fig. 2.4. Where, Cf is the filter capacitor of inverter. Z þ g , Z g , Z g are the positive, negative, and zero sequence components of the grid equivalent impedance, respec 0 tively. θþ g , θg , θ g are the impedance angle of the positive, negative, and zero  0 sequence grid equivalent impedance, respectively. V þ g , V g , V g are the positive, negative, and zero sequence components of the grid voltage, respectively. The variables meaning of the following icons is consistent with this section. It can be concluded that the CC controlled IIDG can be equivalent to the constant current source in parallel with the filter capacitor in the positive sequence network, while in the negative and zero sequence networks, it can only be equivalent to the filter capacitor. Note that current controlled IIDGs only regulate the inductance current, so the filter capacitor C does appear in sequence networks [9].

(a) *

Gi iiabc

(b) + g

Z ∠θ Cf

+ g

Vg+ Cf

(c)

Z g– ∠θ g–

Vg− Cf

Z g0 ∠θg0

Vg0

Fig. 2.4 Fault models of CC controlled IIDGs, (a) positive sequence, (b) negative sequence, (c) zero sequence

20

2.2.2

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

Fault Current Calculation

For CC controlled IIDG, the invariance in reference current lead to its constant output current whether symmetrical or asymmetrical faults occur, as described by Eq. (2.1). Therefore, only the asymmetrical faults are considered. When the grid-connected system is subjected to asymmetrical faults, like two-phase-to-ground fault, symmetrical component method can be used to illustrate fault characteristics of IIDGs. As analyzed in [10], CC controlled IIDG can still regulate three-phase output current independently because the control system is implemented in abc reference frame. Since the inverter output current is always constant, the inverter internal voltage is adjusted to realize balanced output current. According to Eq. (2.4), we get that inverter internal voltage viabc are asymmetrical [11]. L

diiabc þ Riiabc ¼ viabc  vgabc dt

ð2:4Þ

The fault response of CC controlled IIDGs is shown in Fig. 2.5. It can be observed that when two-phase-to-ground faults occur at tf, the three-phase output current remains balanced and inverter internal voltage viabc are asymmetrical. And, the inverter output current remains constant during pre-fault and pro-fault periods.

2.3

Transient Characteristics of PQ Controlled IIDGs

As discussed in Sect. 2.2, CC controlled IIDGs can be viewed as constant current sources during fault period. However, for PQ controlled IIDGs, it is much more complicated, which has many different optimal control targets during symmetrical and asymmetrical faults [9, 11]. Therefore, in this section, the fault models and fault current calculation methods of PQ controlled IIDGs will be discussed under symmetrical and asymmetrical faults, respectively. Moreover, under asymmetrical faults, the PQ controlled IIDGs with different control targets are considered, respectively.

viabc [p.u.]

(a)

1 0.5 0 –0.5 –1

(b)

v ia

v ib

0

iiabc [p.u.]

Fig. 2.5 Fault response of CC controlled IIDGs when subjected to two-phase-toground fault, (a) inverter internal voltage, (b) inverter output current

2 1 0 –1 –2

tf iia

0

tf

v ic

t iic

iib

t

2.3 Transient Characteristics of PQ Controlled IIDGs

21

2.3.1

Transient Characteristics of PQ Controlled IIDGs During Symmetrical Fault

2.3.1.1

Fault Models

According to Eq. (2.3), when a symmetrical fault occurs, the sudden drop in the grid voltage magnitude will lead to the increase in the reference current. The transfer function can be derived from the control block diagram of PQ controlled IIDGs, shown in Eq. (2.5). iidq ¼

k s þ ki p  i Ls þ R þ k p s þ ki idq

ð2:5Þ

2

It can be seen from Eq. (2.5) that the inverter output current is only relevant to the reference current. Since the reference current is related to the grid voltage, therefore, PQ controlled IIDG can be equivalent to a controlled current source in parallel with filter capacitor when symmetrical faults occur, as shown in Fig. 2.6. Where, Cf is the filter capacitor of inverter. Zg and θg are the grid equivalent impedance and impedance angle, respectively. Vg represents the grid voltage. It can be concluded that the PQ controlled IIDG can be equivalent to a controlled current source in parallel with the filter capacitor [9].

2.3.1.2

Fault Current Calculation

According to Eq. (2.5), the symmetrical fault current of a PQ controlled IIDG is similar to the transient response of a second-order system. By applying the inverse Laplace transformation to Eq. (2.5), analytical solutions of fault current can be derived as follows: i  h iidq ¼ iidq þ iidq0  iidq Aer1 ðtt f Þ þ Ber2 ðtt f Þ A¼

kp r1 þ ki Lðr 1  r 2 Þ

B¼

ð2:6Þ

kp r2 þ ki Lðr 1  r 2 Þ

ð2:7Þ

where r1 and r2 are the roots of characteristics equations shown in Eq. (2.5). iidq0 are the initial value of output current in the dq reference frame. When subjected to symmetrical faults, the reference current in dq-axis iidq* only consists of the dc component so that they can be traced by a PI controller. Current in the abc frame can Fig. 2.6 Fault model of PQ controlled IIDGs under symmetrical fault

Gi i *id q

Cf

Z g ∠θ g

Vg

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

Fig. 2.7 Fault response of PQ controlled IIDGs when subjected to three-phase symmetrical fault, (a) threephase PCC voltage, (b) inverter output current

(a) vpabc [p.u.]

22

vpa

1.95

iiabc [p.u.]

(b)

vpb

vpc

1 0.5 0 –0.5 –1 2

iia 3 calculation results 2 1 0 –1 –2 –3 1.95 2

2.1

2.05

Time [s]

2.05

iib

iic

2.1

be derived by applying the inverse Park transform to Eq. (2.6). Based on the above analysis, the fault current of PQ controlled IIDGs is well regulated by Eq. (2.6). Hence, they can be viewed as a current source whose magnitude undergoes a step response when symmetrical fault occurs. The symmetrical fault response of PQ controlled IIDGs is shown in Fig. 2.7. It can be observed that when the symmetrical fault occurs at t ¼ 2 s, the inverter output current undergoes a transient response process and then gradually stabilizes to a steady-state value, which is mainly related to the drop depth of PCC voltage and control parameters of PQ controller. It should be noted that the fault current of current-controlled IIDGs may exceed the limited current of the semiconductor and the devices can be damaged. Thus, it is necessary to limit fault current during transient period. There are two options for current limitation. One is inherent current limitation through current limiters and the other one is to limit current by control system. The influence of current limiter would be elaborated in following part. When fault ride through control strategy is adopted, the fault current depends on the adopted control strategy. Symmetrical fault is one of specific fault types of asymmetrical faults. The fault current under asymmetrical faults considering fault ride through control strategy is given in following part, which is also suitable for symmetrical faults.

2.3.2

Transient Characteristics of PQ Controlled IIDGs During Asymmetrical Fault

In order to analyze the fault characteristics of the PQ controlled IIDGs during asymmetrical faults, the grid voltage needs to be firstly described in this section. The asymmetrical three-phase voltage is the sum of the positive, negative, and zero sequence components. To simplify the analysis, only the fundamental frequency component is considered. Therefore, the grid voltage can be expressed as

2.3 Transient Characteristics of PQ Controlled IIDGs

2

V ga

3

2

3

sin ðωt þ ϕþ v Þ

23

2

3

sin ðωt þ ϕ v Þ

6 6 7 6 ∘ 7 ∘ 7  þ6 6 þ 7 7 7 vg ¼ 6 4 V gb 5 ¼ V m 4 sin ðωt þ ϕv  120 Þ 5 þ V m 4 sin ðωt þ ϕv þ 120 Þ 5 ∘

∘

sin ðωt þ ϕ v  120 Þ

sin ðωt þ ϕþ v þ 120 Þ 3 2 0 sin ðωt þ ϕv Þ 6 ∘ 7 0 7 þ V 0m 6 4 sin ðωt þ ϕv þ 120 Þ 5 ∘ sin ðωt þ ϕ0v þ 120 Þ

V gc

ð2:8Þ ‐ 0 where V þ m , V m and V m represent the positive, negative, and zero sequence voltage  0 amplitude, respectively. ϕþ v , ϕv , and ϕv represent the initial phase of the positive, negative, and zero sequence voltage, respectively. Similarly, the output current can be also expressed as:

2

I oa

3

2

3

sin ðωt þ ϕþ i Þ

2

3

sin ðωt þ ϕ i Þ

6 6 7 6 ∘ 7 ∘ 7  þ þ 6  6 7 7 7 io ¼ 6 4 I ob 5 ¼ I om 4 sin ðωt þ ϕi  120 Þ 5 þ I om 4 sin ðωt þ ϕi þ 120 Þ 5 I oc

∘

∘

2

sin ðωt þ ϕþ i þ 120 Þ sin ðωt þ ϕ0i Þ

3

sin ðωt þ ϕ i  120 Þ

6 ∘ 7 0 7 þ I 0om 6 4 sin ðωt þ ϕi þ 120 Þ 5 ∘

sin ðωt þ ϕ0i þ 120 Þ ð2:9Þ  0 where I þ om , I om and I om represent positive, negative and zero sequence current  0 amplitudes, respectively. ϕþ i , ϕi and ϕi represent the initial phase of the positive, negative, and zero sequence current, respectively. According to the instantaneous power theory, when the asymmetrical faults occur in the grid, the instantaneous three-phase active power p, reactive power q and the zero sequence active power p0 can be expressed as in the stationary reference frame.

3 3 2 3 2 p vα i α þ vβ i β P þ Pc2 cos ð2ωtÞ þ Ps2 sin ð2ωtÞ 7 6 7 6 7 6 4 q 5 ¼ 4 vα iβ  vβ iα 5 ¼ 4 Q þ Qc2 cos ð2ωtÞ þ Qs2 sin ð2ωtÞ 5 p0 v0 i 0 P0 þ P0c2 cos ð2ωtÞ þ P0s2 sin ð2ωtÞ 2

ð2:10Þ

Then, the instantaneous three-phase active power P3ϕ and reactive power Q3ϕ of the inverter can be expressed as.

24

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

"

P3ϕ

#

Q3ϕ

" ¼

#

 Pc2 þ P0c2 cos ð2ωtÞ Qc2 Q   Ps2 þ P0s2 sin ð2ωtÞ þ Qs2 P þ P0



þ

ð2:11Þ

where, P, P0 and Q are the average components of the active power, zero sequence active power and reactive power, respectively. Pc2 and Ps2 are the cosine and sine components of the double frequency components of active power, respectively. Qc2 and Qs2 are the cosine and sine components of the double frequency components of reactive power, respectively. P0c2, P0s2 are the cosine and sine components of the double frequency component of zero sequence active power, respectively. The power components in Eq. (2.11) can be formulated as: 2

P þ P0

3

2

6 6P þ P 7 6 0c2 7 6 c2 6 7 6 6 6 Ps2 þ P0s2 7 3 6 7¼ 6 6 7 26 6 Q 7 6 6 7 6 6 5 4 6 Qc2 4 Qs2

vþ od

vþ oq

v od

v oq

v0Re

v od

v oq

vþ od

vþ oq

v0Re

v oq

v od

vþ oq

vþ od

v0Im

vþ oq

vþ od

v oq

v od

0

v oq

v od

vþ oq

vþ od

0

v od

v oq

vþ od

vþ oq

0

v0Im

32

iþ od

3

76 7 v0Im 7 iþ oq 7 76 7 6 0 76  7 vRe 76 iod 7 76 7 i 0 7 76 oq 7 7 76 6 0 7 7 0 54 iRe 5 i0Im 0

ð2:12Þ

where, vod+, voq+, vod, voq, iod+, ioq+, iod, and ioq are the positive and negative sequence components of output voltage and current in the dq reference frames, respectively. vRe0, vIm0, iRe0, and iIm0 are the real and imaginary components of the voltage and current in the zero sequence reference frame, respectively. It can be seen from Eq. (2.12) that Pc2, Ps2, Qc2, Qs2, P0c2 and P0s2 are six power fluctuation components, therefore, a three-phase four-wire inverter with a zero sequence current channel can establish six control equations. Six power fluctuation components can be combined into active power fluctuation phenomenon (Pc2, Ps2, P0c2 and P0s2) and reactive power fluctuation phenomenon (Qc2and Qs2), respectively. After the three-phase grid voltage is determined, there are six control freedom degrees (iod+, ioq+, iod, ioq, iRe0 and iIm0) in the inverter to regulate the output current. Therefore, by controlling the corresponding output current, the corresponding control target can be achieved. Similarly, six current control freedom degrees can be also combined into the positive sequence current components (iod+ and ioq+), negative sequence current components (iod and ioq), and zero sequence current components (iRe0 and iIm0), respectively. By combining different control targets, such as active and reactive power fluctuation suppression, the output performance and FRT capacity of PQ controlled IIDGs can be optimized during asymmetrical faults. The optimal control targets of PQ controlled IIDGs mainly include the active power fluctuation suppression, reactive power fluctuation suppression, zero

2.3 Transient Characteristics of PQ Controlled IIDGs

25

sequence current or negative sequence current suppression. Although these control objectives can improve the operating performance of the grid-connected system under asymmetrical faults, other operating performances are expensed and the inverter output current will be more than the rated current. For example, when the active and reactive power fluctuations are suppressed, the maximum phase output current of the inverter reaches up to five times rated current during two-phase-toground faults, which results in a serious overcurrent phenomenon. Therefore, it is necessary to analyze the asymmetrical fault current characteristics of PQ controlled IIDGs with different optimal control targets. In this section, the control objectives of PQ controlled IIDGs are classified into four categories according to different fault current characteristics, including the Symmetrical Positive Sequence Current (SPSC) Control, the Active Power Oscillation (APOC) Control, Reactive Power Oscillation (RPOC) Control, and the Active and Reactive Power Oscillation (ARPOC) Control. Under these four kinds of control objectives, the fault models and fault current calculation of the PQ controlled IIDGs are discussed to analyze the different fault characteristics, respectively.

2.3.2.1

Case 1: IIDGs with Symmetrical Positive Sequence Current (SPSC) Control

Based on Eq. (2.12) and the six control freedom degrees to regulate the output current. In this case, the PQ controlled IIDG aims at outputing the symmetrical positive sequence current only, both the negative and zero sequence currents are controlled to zero.

Fault Model According to the control objective in this section, the constraint equations of the negative and zero sequence currents can be obtained as (

 i od ¼ ioq ¼ 0

i0Re ¼ i0Im ¼ 0

ð2:13Þ

Moreover, the average components of active and reactive power of PQ-controlled IIDGs with different control targets are controlled equal to the active and reactive reference value, which can be expressed as follows. (

P3ϕ ¼ P ¼ Pref Q3ϕ ¼ Q ¼ Qref

ð2:14Þ

26

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault (a)

(b) +

*

Gi i odq

Cf

(c) Z ∠θ g − g

+

Z g ∠θ g

Vg+ Cf

−

Vg− Cf

Z g0 ∠θg0

Vg0

Fig. 2.8 Fault models of PQ controlled IIDGs with APSC Control, (a) positive sequence, (b) negative sequence, (c) zero sequence

To simplify the analysis, it is assumed that the d-axis (positive and negative sequence) or real-axis (zero sequence) components in the dq reference frame are in phase with the positive, negative, and zero sequence components of the A phase voltage in this section. The q-axis (positive sequence and negative sequence) or imaginary-axis (zero sequence) components of the output voltage are approximately equal to zero. By substituting Eqs. (2.13) and (2.14) into (2.12), the controllable current components under this control target can be expressed as Eq. (2.15). Therefore, the fault models of PQ controlled IIDG with SPSC Control can be established, as shown in Fig. 2.8. 8 > < iþ  2 Pref , iþ   2 Qref oq od 3 Vþ 3 Vþ od od > : i ¼ i ¼ i0 ¼ i0 ¼ 0 od

oq

Re

ð2:15Þ

Im

It can be concluded that the PQ controlled IIDG with SPSC Control can be equivalent to a controlled current source in parallel with the filter capacitor in the positive sequence network, while in the negative and zero sequence networks, it can only be equivalent to the filter capacitor. The inverter output current is mainly related to the reference power and the positive sequence component of the inverter output voltage.

Fault Current Calculation Equation (2.9) can be rewritten as Eqs. (2.16)–(2.18). Equations (2.16), (2.17) and (2.18) present the three-phase fault current, the initial phase of fault current and the fault current amplitude of inverter, respectively. Based on this, the fault current characteristics of PQ controlled IIDGs with different current components injections can be analyzed. 8 > < ioa ¼ I oaðpeakÞ cos ðωt þ δa Þ iob ¼ I obðpeakÞ cos ðωt þ δb Þ > : ioc ¼ I ocðpeakÞ cos ðωt þ δc Þ

ð2:16Þ

2.3 Transient Characteristics of PQ Controlled IIDGs

27

8 þ þ  0  0 > 1 I om sin φi þ I om sin φi þ I om sin φi > δ ¼ tan > a þ  0 0  > > Iþ > om cos φi þ I om cos φi þ I om cos φi >    >   <  0  0 I þ sin φþ i  120 þ I om sin φi þ 120 þ I om sin φi ð2:17Þ δa ¼ tan 1 omþ   þ  0 0  > I om cos ðφi  120 Þ þ I om ðφi þ 120 Þ þ I om cos φi > >  þ   >   >  0 0 > Iþ > om sin φi þ 120 þ I om sin φi  120 þ I m sin φi 1 > : δa ¼ tan   þ  0 0  Iþ om cos ðφi þ 120 Þ þ I om ðφi  120 Þ þ I om cos φi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8  þ  u  þ 2   2  0 2   > u I om þ I om þ I om þ 2I þ > om I om cos φi  φi > t > I oaðpeakÞ ¼ >     > > I 0om cos φ  φ0i þ 2I 0om I þ cos φ0i  φþ þ2I  > i i om om > > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >   > u  þ 2   2  0 2 < u I om þ I om þ I om þ 2I þ I cos φþ  φ þ 120 i i om om I obðpeakÞ ¼ t    0   > þ 0 0 þ 0 > þ2I  > om I om cos φi  φi þ 120 þ 2I om I om cos φi  φi þ 120 > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >  >  u  þ 2   2  0 2 > > u I om þ I om þ I om þ 2I þ I cos φþ  φ  120 > i i om om > t > >    0 : I oaðpeakÞ ¼   þ 0 0 þ 0 þ2I  om I om cos φi  φi  120 þ 2I om I om cos φi  φi  120 ð2:18Þ When the negative and zero sequence currents are controlled to zero, and only the positive sequence current are injected into the grid from PQ controlled IIDG, Eq. (2.19) can be obtained by substituting Eqs. (2.15) into (2.18).

I oaðpeakÞ ¼ I obðpeakÞ ¼ I ocðpeakÞ

2 ¼ 3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2ref þ Q2ref vþ od

ð2:19Þ

According to the symmetrical component method, the magnitude of vþ od is related to the fault type and fault degree. Based on the previous assumption, the post-fault voltage is recorded as k times the pre-fault voltage (k ¼ 0.4). When single-phase-to2þk  1k ground fault occurs, vþ od ¼ 3 V om , vod ¼ 3 V om . When two-phase-to-ground fault þ 1þ2k k1  occurs, vod ¼ 3 V om , vod ¼ 3 V om . Therefore, according to the expression of rated current, Eq. (2.19) can be expressed as per-unit value, respectively. 3 Single-phase-to-ground fault: I aðpeakÞ ¼ I bðpeakÞ ¼ I cðpeakÞ ¼ 2þk .    3 Two-phase-to-ground fault: I aðpeakÞ ¼ I bðpeakÞ ¼ I cðpeakÞ ¼ 2kþ1 . Where, k is recorded as the ratio of the post-fault voltage and the pre-fault voltage, and its range is 0–1. Under this control target, the inverter output current is three-phase balanced during asymmetrical faults. The simulation verifications are carried out as below. Here, k is set as 0.4, and asymmetrical fault types are set as single-phase-to-ground fault and two-phase-toground fault, respectively. It should be note that only the drop of voltage amplitude is taken into consideration, and the abrupt change in phase and frequency are not considered after asymmetrical faults. According to the symmetrical component

28

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

0.9

Vgb

Vgc

Calculation result

Ioa

Iob

Ioc

Simulation result

Pout Qout

0.95

1 t [s]

1.05

1.1

Voltage [p.u.]

1 0.8 0.6 0.4 0.2

Vga

1 0.5 0 –0.5 –1

Current [p.u.]

4 2 0 –2 –4

Voltage sag

Voltage balance

4 2 0 –2 –4

Power [p.u.]

Voltage [p.u.] Current [p.u.]

(b)

1 0.5 0 –0.5 –1

Power [p.u.]

(a)

1 0.8 0.6 0.4 0.2 0.9

Voltage sag

Voltage balance

Vga Vgb

Vgc

Iob

Ioc

Calculation result

Simulation result

Ioa Pout Qout

0.95

1

1.05

1.1

t [s]

Fig. 2.9 Output current and power waveforms of the grid-connected inverter when asymmetrical faults occur in the grid and the inverter control target is to eliminate both negative and zero sequence currents, (a) single-phase-to-ground fault, (b) two-phase-to-ground fault

method, the positive, negative and zero sequence voltage components can be obtained, respectively. Therefore, the reference current under this control target can be calculated according to Eq. (2.15). The specific simulation results are as shown in Fig. 2.9: It can be seen from Fig. 2.9 that, before asymmetrical faults, the inverter output current is balanced and its value is equal to the rated current. The output power is constant. After asymmetrical faults, since the negative and zero sequence current components are controlled to zero, the inverter output current is balanced and its amplitude exceeds the rated current. The output active and reactive powers contain double frequency fluctuations. Moreover, the calculated amplitudes of three-phase fault current are consistent with the simulation results, therefore, the aforementioned fault current calculation can be accurate to evaluate the fault current amplitude. Additionally, in the case of the same voltage drop depth and power capacity of inverter, the output current and power fluctuation range of the two-phase-to-ground fault are larger than those of the single-phase-to-ground fault.

2.3.2.2

Case 2: IIDGs with Active Power Oscillation (APOC) Control

It can be seen from Eq. (2.12) that the symmetrical current control, active and reactive power fluctuations cannot be suppressed at the same time due to the constraint of the control freedom degrees. Therefore, the fault models of PQ controlled IIDGs with active power fluctuation suppression will be discussed in this section. Moreover, according to different positive/negative/zero sequence current

2.3 Transient Characteristics of PQ Controlled IIDGs

29

injections to the grid-connected system, this case can also be divided into two conditions respectively.

Fault Analysis When Positive and Negative Sequence Currents Are Injected into the Grid-Connected System Fault Model Since the active power fluctuations are controlled to zero, the constraint equations of the active power fluctuations can be obtained as:

P0c2 ¼ P0s2 ¼ 0 Pc2 ¼ Ps2 ¼ 0

ð2:20Þ

In addition, only the positive and negative sequence currents are injected into the grid, and the zero sequence current is controlled to zero. Thus, the constraint equation of the zero sequence current can be expressed as: i0Re ¼ i0Im ¼ 0

ð2:21Þ

In such case, the controllable current components to eliminate active power fluctuations can be obtained by substituting Eqs. (2.14), (2.20) and (2.21) into (2.12), as shown in Eq. (2.22). Therefore, the fault models of PQ controlled IIDG with active power fluctuation suppression under the positive and negative sequence currents injection can be established, as shown in Fig. 2.10. 8 þ 2 Pref :V þ 2 Qref :V od > od þ þ > i ¼ ¼ , i > q od > 3 M 3 N > > >   < P :V 2 ref od  2 Qref :V od  iod ¼  , iq ¼  3 M 3 N > > > 0 0 > i ¼ i > Re Im > >  2   2  2   2 : M ¼ Vþ  V od , N ¼ V þ þ V od od od

ð2:22Þ

It can be concluded that the PQ controlled IIDG under the positive and negative sequence currents injection can be equivalent to a controlled current source in (a)

(b) +

+ Gi iodq

Cf

(c)

+

Z g ∠θ g

– Vg+ Gi iodq

Cf

− g

−

Z ∠θ g

Vg−

Cf

Z g0 ∠θg0

Vg0

Fig. 2.10 Fault models of PQ controlled IIDGs with active power fluctuation suppression under only positive and negative sequence currents injection, (a) positive sequence, (b) negative sequence, (c) zero sequence

30

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

parallel with the filter capacitor in the positive and negative sequence networks, while in the zero sequence network, it can only be equivalent to the filter capacitor. The inverter output current is mainly related to the reference power, the positive and negative sequence components of the inverter output voltage.

Fault Current Calculation Since the zero sequence current and active power fluctuation are controlled to zero, only the positive and negative sequence currents are injected into the grid from PQ controlled IIDG. By substituting Eqs. (2.22) into (2.18), (2.23) can be obtained. 8 ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2   þ 2   2   2  þ 2   2  > > u > vod þ vod Qref vod þ vod > u Pref > > u þ > 2 > > M N2 2u > u > I ¼ ! > oa ð peak Þ 2  2  > 3u > u   > Qref Pref > þ  t  > þ2vod vod þ cos φþ > i  φi 2 2 > > M N > > > > ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > u > 2   þ 2   2   2  þ 2   2  > u > vod þ vod Qref vod þ vod > u Pref > > > u þ < 2 M N2 2u ð2:23Þ I obðpeakÞ ¼ u ! u 2  2  > 3u >  þ Qref Pref >  > t þ   > þ2vod vod þ cos φi  φi þ 120 > > M2 N2 > > > > ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > u 2   þ 2   2   2   þ 2   2  > > u > vod þ vod Qref vod þ vod > u Pref > > u > þ > 2 u > M N2 > 2 > > I ocðpeakÞ ¼ u ! u   >   2 2 > 3u >  >  t þ2vþ v Pref þ Qref >  > cos φþ > i  φi  120 od od 2 2 : M N  Similarly, the magnitudes of vþ od and vod are related to the fault type and fault degree. By simplifying Eq. (2.23), (2.23) can be expressed as per-unit value according to the expression of rated current. The case of signal-phase-to-ground fault

8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u > > u ð3λÞ2 ð 9Þ 2 1  λ 2 >  > t > I ¼ þ  2 2 > oaðpeakÞ > ð1 þ 2kÞ2 < 2k þ 2k þ 5 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      u > > u3ð λ Þ 2 k 2 þ k þ 1 > 33 k 2 þ k þ 1 1  λ 2 >   t > I ¼ I ¼ þ >  2 2 ocðpeak Þ > : obðpeakÞ ð1 þ 2k Þ2 2k þ 2k þ 5 The case of two-phase-to-ground fault

ð2:24Þ

2.3 Transient Characteristics of PQ Controlled IIDGs

(b)

1 0.8 0.6 0.4 0.2 0.9

Vgc Vgb

Voltage [p.u.]

Voltage sag

Vga

Calculation result

Simulation result Ioa

Iob

Ioc

Pout

Qout

0.95

1 t [s]

1.05

1.1

1 0.5 0 –0.5 –1

Voltage balance

Voltage sag

Vgc Vgb

Vga

Calculation result

Current [p.u.]

4 2 0 –2 –4

Voltage balance

4 2 0 –2 –4

Power [p.u.]

Current [p.u.]

1 0.5 0 –0.5 –1

Power [p.u.]

Voltage [p.u.]

(a)

31

1 0.8 0.6 0.4 0.2 0 0.9

Simulation result Ioa

Iob

Ioc

Pout

Qout

0.95

1 t [s]

1.05

1.1

Fig. 2.11 Output current and power waveforms of the grid-connected inverter when asymmetrical faults occur in the grid and the inverter control target is to eliminate both zero sequence current and active power fluctuation, (a) single-phase-to-ground fault, (b) two-phase-to-ground fault

8 ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u > 2 2 2 > u k 9 1  λ 9 ð λ Þ > > I oaðpeakÞ ¼ t þ > 2 > > ð2 þ k Þ2 < 5k2 þ 2k þ 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2   u 2 2 > 2 > uðλÞ 7k þ k þ 1 > 9 7k þ k þ 1 1  λ >   > I ¼ I ocðpeakÞ ¼ t þ >  2 2 > : obðpeakÞ k 2 ð1 þ 2kÞ2 5k þ 2k þ 2

ð2:25Þ

where, λ is the power factor. k is recorded as the ratio of the post-fault voltage and the pre-fault voltage, and its range is 0–1. The simulation verifications are carried out as below. Here, k is set as 0.4, and the asymmetrical fault types are set as single-phase-to-ground fault and two-phase ground fault, respectively. According to the symmetrical component method, the positive, negative, and zero sequence voltage components can be obtained, respectively. Thus, the reference current under this control target can be calculated according to Eq. (2.22). The specific simulation results are shown in Fig. 2.11. It can be seen from Fig. 2.11 that, after asymmetrical faults, the inverter output current is asymmetrical, and the output active power of inverter is constant, while the reactive power has double frequency fluctuation. Moreover, the calculated amplitudes of three-phase fault current are consistent with the simulation results, therefore, the aforementioned fault current calculation can be accurate to evaluate the fault current amplitude. During single-phase-to-ground faults, the healthy phase current is approximately equal, and larger than the fault phase current. During two-phase-toground faults, the fault phase current is larger than the healthy phase current, and the maximum phase fault current exceeds the rated current. Additionally, in the case of

32

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

same voltage drop depth and power capacity of inverter, the output current and power fluctuation range of the two-phase-to-ground fault is larger than that of the single-phase-to-ground fault.

Fault Analysis When Positive and Zero Sequence Currents Are Injected into the Grid-Connected System Fault Model From the aforementioned analysis, it can be seen that the zero sequence current is only related to the average component and the double frequency component of active power. When the negative sequence current is controlled to zero, the reactive power fluctuation cannot be suppressed. Therefore, this could be the secondary condition of the case that only the active power fluctuations are suppressed. Since the active power fluctuations are eliminated, the constraint equations of active power fluctuations are expressed as follows.

P3ϕc2 ¼ Pc2 ¼ 0 P3ϕs2 ¼ Ps2 ¼ 0

ð2:26Þ

In this condition that the zero sequence current are considered, Eq. (2.14) should be rewritten as (

P3ϕ ¼ P þ P0 ¼ Pref Q3ϕ ¼ Q ¼ Qref

ð2:27Þ

Moreover, since only positive and zero sequence currents are injected into the grid from PQ controlled IIDGs, the constraint equation of negative sequence current can be obtained as  i d ¼ iq ¼ 0

ð2:28Þ

By substituting Eqs. (2.26)–(2.28) into (2.12), all controllable current components under this control target can be expressed as Eq. (2.29). Hence, the fault models of PQ controlled under positive and zero sequence currents injection can be obtained, as shown in Fig. 2.12. 8 2 Pref 2 Qref > þ > iþ > od  3 þ  , iq   3 þ > V od þ V od V od > <  i ¼ i ¼ 0 od oq > >  > V V > 0 d þ 0 > : iRe  0d iþ od , iIm  0 ioq V Re V Re

ð2:29Þ

2.3 Transient Characteristics of PQ Controlled IIDGs (a)

(b) +

+ Gi iodq

33

C

(c)

+

Z g ∠θ g

Vg+

Z ∠θ g − g

C

−

Vg−

Gi i

0

C

Z g0 ∠θg0 Vg0

Fig. 2.12 Fault models of PQ controlled IIDGs under only positive and zero sequence currents injection, (a) positive sequence, (b) negative sequence, (c) zero sequence

It can be concluded that the PQ controlled IIDG under the positive and zero sequence currents injection can be equivalent to a controlled current source in parallel with the filter capacitor in the positive and zero sequence networks, while only the filter capacitor appears in the negative sequence network. The inverter output current is mainly related to the reference power, the positive, negative and zero sequence components of the inverter output voltage.

Fault Current Calculation Since the negative sequence current and active power fluctuations are controlled to zero, only the positive and zero sequence current components are injected into the grid. By substituting Eqs. (2.29) into (2.18), (2.30) can be obtained. 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 !  2 u    > 2 > u Qref Pref > v od > u >  þ 2 þ  þ 2 þ > u v0Re þ 1 > > u vod þ v vod > od 2 > u > I ¼ ! > oa ð peak Þ u     > 2 2 3 u  > >   Qref Pref > t 2 vod > þ  φ0i cos φþ >  þ    i > 0 2 2 > vRe > vod þ v vþ > od od > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >  2 !   u >

 2 > u  v  2 > Q P ref > ref od > u >  þ 2 þ  þ 2 þ > u v0Re þ 1 < u vod þ v vod od 2u I obðpeakÞ ¼ u !     > 2 2 3u  >  > Qref Pref  > t 2 vod 0 > þ cos φþ >     i  φi þ 120 0 2 2 > þ þ  > vRe vod þ vod vod > > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > >  2 !  2 u    > > 2 u  > Q P ref v ref > od u > >  þ  þ  þ 2 þ u v0Re þ 1 >  2 > u v þ v vod > od od > 2 > > I ocðpeakÞ ¼ u ! u >     > 2 2 3u  >  Qref Pref >  t 2 vod > 0 > þ cos φþ     > i  φi  120 v0 2 : þ þ 2  v þv v Re od

od

od

ð2:30Þ Similarly, Eq. (2.30) can be expressed as per-unit value according to the expression of rated current, respectively. The case of signal-phase-to-ground fault

34

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

8  I ¼0 > > < oaðpeakÞ   > > : I obðpeakÞ ¼ I ocðpeakÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi  2   3 ¼ 3 λ2 þ 2þk 1  λ2 :

ð2:31Þ

The case of two-phase-to-ground fault 8  I oaðpeakÞ ¼ 0 > > > < > I ¼ I ocðpeakÞ > > : obðpeakÞ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 2

2 ! 2 u 9 1λ λ ð Þ Þ ¼ t3 þ 2kþ1 k2

ð2:32Þ

where, λ is the power factor. k is recorded as the ratio of the post-fault voltage and the pre-fault voltage, and its range is 0–1. The simulation verifications are carried out as follows. Here, k is set as 0.4, and the asymmetrical fault types are set as single-phase-to-ground fault and two-phase grounded fault, respectively. According to the symmetrical component method, the positive, negative, and zero sequence voltage components can be obtained respectively, and the reference current under this control target can be calculated according to Eq. (2.29). The specific simulation results are shown in Fig. 2.13. It can be seen from Fig. 2.13 that, after asymmetrical faults, the inverter output current is asymmetrical, and the output active power of inverter is constant, while the reactive power has double frequency fluctuation. Moreover, the calculated amplitudes of three-phase fault current are consistent with the simulation results, therefore,

0.9

Voltage [p.u.]

1 0.8 0.6 0.4 0.2

Ioc

Current [p.u.]

4 2 0 –2 –4

Vgc

4 Calculation result 2 0 –2 –4 Simulation result 1 0.8 0.6 0.4 0.2

Voltage sag

Voltage balance

Vga

Vgb

Calculation result

Simulation result

Ioa

Iob Pout

Qout

0.95

1 t [s]

1.05

1.1

Voltage sag

Voltage balance

Power [p.u.]

Voltage [p.u.]

1 0.5 0 –0.5 –1

Current [p.u.]

(b)

Power [p.u.]

(a)

1 0.5 0 –0.5 –1

0.9

Vga

Ioa

Vgb

Iob

Vgc

Ioc

Pout

Qout

0.95

1 t [s]

1.05

1.1

Fig. 2.13 Output current and power waveforms of the grid-connected inverter when asymmetrical faults occur in the grid and the inverter control target is to eliminate both negative sequence current and active power fluctuation, (a) single-phase-to-ground fault, (b) two-phase-to-ground fault

2.3 Transient Characteristics of PQ Controlled IIDGs

35

the aforementioned fault current calculation can be accurate to evaluate the fault current amplitude. During the single-phase-to-ground faults, the healthy phase currents are the largest, and the fault phase current is equal to zero. On the contrary, during the two-phase-to-ground faults, the fault phase currents are the largest, and the healthy phase current is equal to zero. The maximum phase fault current also exceeds the rated current. Additionally, the output current and power fluctuation range of the two-phase-to-ground faults is larger than that of the single-phase-toground faults.

2.3.2.3

Case 3: IIDGs with Reactive Power Oscillation (RPOC) Control

In addition to the IIDGs with symmetric current control and active oscillation suppression control mentioned in the previous sections, the reactive power oscillation suppression can also be performed with different control freedom combination as introduced as follows.

Fault Model Since the reactive power fluctuation is controlled to zero, the constraint equation of the reactive power fluctuation can be obtained as: Qc2 ¼ Qs2 ¼ 0

ð2:33Þ

Similarly, the constraint equation of the zero sequence current can be written as: i0Re ¼ i0Im ¼ 0

ð2:34Þ

In such case, by substituting Eqs. (2.14), (2.33) and (2.34) into (2.12), the controllable current components to eliminate reactive power fluctuations can be obtained, as shown in Eq. (2.35). Therefore, the fault models of PQ controlled IIDG with reactive power fluctuation suppression under the positive and negative sequence currents injection can be obtained, as shown in Fig. 2.14.

(a)

(b) +

+ Gi iodq

C

(c)

+

Z g ∠θ g

– Vg+ Gi iodq

C

− g

−

Z ∠θ g

Vg−

C

Z g0 ∠θg0

Vg0

Fig. 2.14 Fault models of PQ controlled IIDGs with reactive power fluctuation suppression under only positive and negative sequence currents injection, (a) positive sequence, (b) negative sequence, (c) zero sequence

36

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

8 þ 2 Pref :V þ 2 Qref :V od > þ od þ > , i i ¼ ¼ > oq > od 3 N 3 M > > >   < P :V 2 2 Qref :V od ref od  i ¼  ¼  , i od oq 3 N 3 M > > > 0 0 > iRe ¼ iIm ¼ 0 > > >  2   2  2   2 : M ¼ Vþ  V od , N ¼ V þ þ V od od od

ð2:35Þ

It can be concluded that the PQ controlled IIDG under the positive and negative sequence currents injection can be equivalent to a controlled current source in parallel with the filter capacitor in the positive and negative sequence networks, while only the filter capacitor appears in the zero sequence network. The inverter output current is mainly related to the reference power, the positive and negative sequence components of the inverter output voltage.

Fault Current Calculation When reactive power fluctuation and zero sequence current are controlled to zero, only the positive and negative sequence current components are injected into the grid. Equation (2.18) can be simplified into Eq. (2.36). 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     u   2  > þ 2 2 > u P2 vþ 2 þ v 2 > Q v þ vod > ref od u ref od od > > u þ > > > N2 M2 2u > > I oaðpeakÞ ¼ u ! > u > 3u 2 2 > > > t þ2vþ v Pref þ Qref cos φþ  φ  > > i od i od > > N2 M2 > > > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >     u >   2    2  > þ 2 þ 2 2 u P2 > > v þ v Q v þ vod ref od u ref > od od > > u þ < N2 M2 2u I obðpeakÞ ¼ u ! u > 3u 2 2 > > > t þ2vþ v Pref þ Qref cos φþ  φ þ 120  > > i od i od 2 > N M2 > > > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >     > u >   2  > þ 2 2 u P2 vþ 2 þ v 2 > Q v þ vod > ref od u ref od od > > u > þ > 2 2 > N M > 2u > > I ocðpeakÞ ¼ u ! u > > 3 2 2 u > > t þ2vþ v Pref þ Qref cos φþ  φ  120  > > > i i od od : N2 M2

ð2:36Þ

Similarly, Eq. (2.36) can be expressed as per-unit value according to the expression of rated current, respectively. The case of signal-phase-to-ground fault:

2.3 Transient Characteristics of PQ Controlled IIDGs

8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  u 2 2 > > u   9 ð λ Þ 4k þ 4k þ 1 > 2 > > I oaðpeakÞ ¼ t  2 þ 1  λ > 2 > < 2k þ 2k þ 5 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2  ffi u > 2 2 2 > u > 9 ð λ Þ k þ k þ 7 k þ k þ 7 1  λ > > I ¼ I ocðpeakÞ ¼ t  > 2 þ > : obðpeakÞ ð1 þ 2kÞ2 2k 2 þ 2k þ 5

37

ð2:37Þ

The case of two-phase-to-ground fault: 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u > > u 9ðλÞ2 ð2 þ kÞ2 1  λ2 >  > t I oaðpeakÞ ¼  > 2 þ > > k2 < 5k2 þ 2k þ 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     2  u > > u9ðλÞ2 7k 2 þ k þ 1 > 7k þ k þ 1 1  λ2 > I  t > ¼ I ocðpeakÞ ¼ >  2 2 þ > : obðpeakÞ k2 ð2 þ k Þ2 5k þ 2k þ 2

ð2:38Þ

where, λ is the power factor. k is recorded as the ratio of the post-fault voltage and the pre-fault voltage, and its range is 0–1. The simulation verifications are carried out as follow. Here, k is set as 0.4, and the asymmetrical fault types are set as single-phase grounded faults and two-phase grounded faults, respectively. According to the symmetrical component method, the positive, negative and zero sequence voltage components can be obtained respectively, and the reference current under this control target can be calculated according to Eq. (2.35). The specific simulation results are shown in Fig. 2.15. It can be seen from Fig. 2.15 that, after asymmetrical faults, the inverter output current is asymmetrical, and the output reactive power of inverter is constant, while the active power has double frequency fluctuation. Moreover, the calculated amplitudes of three-phase fault current are consistent with the simulation results, therefore, the aforementioned fault current calculation can be accurate to evaluate the fault current amplitude. During single-phase-to-ground faults, the fault phase current is approximately equal, and less than the fault phase current. During two-phase-toground faults, the healthy phase current is larger than the fault phase current, and the maximum phase fault current exceeds the rated current. Additionally, the output current and power fluctuation range of the two-phase-to-ground faults is larger than that of the single-phase-to-ground faults.

2.3.2.4

Case 4: IIDGs with Active and Reactive Power Oscillation (ARPOC) Control

It can be seen from Eq. (2.12) that Pc2, Ps2, Qc2, Qs2, P0c2, and P0s2 are the power fluctuation components, and six control constraint equations can be established for the three-phase four-wire inverter with zero sequence current channel. When the grid

38

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

Vga Vgb Vgc Calculation result

Simulation result

0.9

Ioa

Iob

Ioc

Pout

Qout

0.95

1 t [s]

1.05

1.1

Voltage [p.u.]

1 0.8 0.6 0.4 0.2

Voltage sag

1 0.5 0 –0.5 –1

Current [p.u.]

4 2 0 –2 –4

Voltage balance

4 2 0 –2 –4

Power [p.u.]

Voltage [p.u.] Current [p.u.]

(b) 1 0.5 0 –0.5 –1

Power [p.u.]

(a)

1 0.8 0.6 0.4 0.2

Voltage balance

Voltage sag

Vga

Vgb

Vgc

Calculation result

Simulation result

0.9

Ioa

Iob

Ioc

Pout

Qout

0.95

1 t [s]

1.05

1.1

Fig. 2.15 Output current and power waveforms of the grid-connected inverter when asymmetrical faults occur in the grid and the inverter control target is to eliminate both zero sequence current and reactive power fluctuation, (a) single-phase-to-ground fault, (b) two-phase-to-ground fault

voltage is determined, there are six control freedom degrees (iod+, ioq+, iod, ioq, iRe0, and iIm0) to regulate inverter output current, therefore, the active and reactive power fluctuations can be suppressed simultaneously. In this case, the symmetrical current injection will have a trade off with the ARPOC realization.

Fault Model The constraint equations of active and reactive power fluctuations can be obtained as: 8 > < P3ϕc2 ¼ Pc2 þ P0c2 ¼ 0 P3ϕs2 ¼ Ps2 þ P0s2 ¼ 0 > : Qc2 ¼ Qs2 ¼ 0

ð2:39Þ

By substituting Eqs. (2.33), (2.39) into (2.12), all controllable current components with zero-sequence current loop can be expressed as Eq. (2.40). Thus, the fault model of PQ controlled IIDG under positive, negative and zero sequence currents injection can be established, as shown in Fig. 2.16.

2.3 Transient Characteristics of PQ Controlled IIDGs (a)

(b) +

+ Gi iodq

39

C

(c)

+

Z g ∠θ g

– Vg+ Gi iodq

C

Z g− ∠θ g−

Vg−

Gi i

0

C

Z g0 ∠θg0 Vg0

Fig. 2.16 Fault models of PQ controlled IIDGs under positive, negative and zero sequence current injection, (a) positive sequence, (b) negative sequence, (c) zero sequence

8 Qref :V þ > 2 Pref :V þ 2 od þ od > > iþ  , i     > od > 3 V þ þ V  2 oq 3  V þ 2 þ V  2 > > od od od od > < V V od þ  od þ  iod  þ iod , ioq   þ ioq > V od V od > > > >   > 2V 2V > þ 0 þ > : i0Re  0od iod , iIm  0od ioq V Re V Re

ð2:40Þ

It can be concluded that the PQ controlled IIDG under the positive and zero sequence currents injection can be equivalent to a controlled current source in parallel with the filter capacitor in the positive, positive and zero sequence networks. The inverter output current is mainly related to the reference power, the positive, negative and zero sequence components of the inverter output voltage.

Fault Current Calculation Since the active and reactive power fluctuations are controlled to zero, and the positive, negative, and zero sequence current components are injected into the grid. By substituting Eqs. (2.40) into (2.18), (2.41) can be obtained. 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  2 u > > u 1 þ ðk Þ2 þ 2kn þ 2k cos φþ  φ þ > n n > u i i k0 > > > þ u > I u ¼ I oa ð cal Þ > om t >   4kn  0  > ð2kn Þ2 > 0 þ > cos φ > i  φi þ k cos φi  φi > k > 0 0 > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >  2 u > > > u 1 þ ðk Þ2 þ 2kn þ 2k cos φþ  φ þ 120∘  > n n < u i i k0 u u I obðcalÞ ¼ I þ om 2 > t ð2kn Þ   4k   > > > þ cos φ  φ0i þ 120∘ þ n cos φ0i  φþ þ 120∘ > i i > k0 k0 > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > >  2 u >  > u 1 þ ðk Þ2 þ 2kn þ 2k cos φþ  φ  120∘  > > n n u i i > k0 > > þ u > I u ¼ I > oc ð cal Þ om > t ð2k n Þ2 >   4k   > > þ cos φ  φ0i  120∘ þ n cos φ0i  φþ  120∘ : i i k0 k0 ð2:41Þ

40

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

Similarly, Eq. (2.41) can be expressed as per-unit value according to the expression of rated current, respectively. The case of signal-phase-to-ground fault 8 > > > > > < > > > >   > : I obðcalÞ ¼ I ocðcalÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s

2 λ 1  λ2  I oaðcalÞ ¼ 3 þ 3 ð1 þ 2kÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s

2 λ 1  λ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼3 þ k2 þ 3k þ 3 3 ð1 þ 2k Þ2

ð2:42Þ

The case of two-phase-to-ground fault 8 > > > > > < > > > >   > : I obðcalÞ ¼ I ocðcalÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s

2 λ 1  λ2  I oaðcalÞ ¼ 3 þ 3k ð2 þ k Þ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s

2 ffi 3 λ 1  λ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3k2 þ 3k þ 1 ¼ þ 2 k 3k ð2 þ k Þ

ð2:43Þ

where, λ is the power factor. k is recorded as the ratio of the post-fault voltage and the pre-fault voltage, and its range is 0–1. The simulation verifications are carried out below. Here, k is set as 0.4, and the asymmetrical fault types are set as single-phase-to-ground fault and two-phase ground fault, respectively. According to the symmetrical component method, the positive, negative and zero sequence voltage components can be obtained respectively, and the output reference current value under this optimized control target can be calculated according to Eq. (2.40). The specific simulation results are shown in Fig. 2.17. It can be seen from Fig. 2.17 that, in the case that both active and reactive power fluctuations are suppressed, the inverter output current is asymmetrical after the asymmetrical faults, and the output active and reactive powers of inverter are constant. Moreover, the calculated amplitudes of three-phase fault current are consistent with the simulation results, therefore, the aforementioned fault current calculation can be accurate to evaluate the fault current amplitude. During the singlephase-to-ground faults, the healthy phase currents are equal and larger than the fault phase current. While, during the two-phase-to-ground faults, the fault phase currents are equal and larger than the healthy phase current. In both cases, the maximum phase current exceeds the rated current, and the overcurrent phenomenon of two-phase-to-ground faults is more serious than that of single-phase-to-ground faults. Additionally, the output current and power fluctuation range of the two-phase-to-ground faults is larger than that of the single-phase-to-ground faults.

2.3 Transient Characteristics of PQ Controlled IIDGs

(b)

4 2 0 –2 –4

Voltage [p.u.]

Vga Vgb

Vgc

Calculation result

Ioa

Simulation result

1 0.8 0.6 0.4 0.2 0.9

Voltage sag

Current [p.u.]

1 0.5 0 –0.5 –1

Voltage balance

Iob Ioc

Pout

Power [p.u.]

Power [p.u.]

Current [p.u.]

Voltage [p.u.]

(a)

41

Qout

0.95

1 t [s]

1.05

1.1

Voltage balance

1 0.5 0 –0.5 –1 4 0 –4

Voltage sag

Vga

Vgb

Vgc

Calculation result

Ioa

Simulation result

1 0.8 0.6 0.4 0.2 0.9

Iob

Ioc

Pout

Qout

0.95

1 t [s]

1.05

1.1

Fig. 2.17 Output current and power waveforms of grid-connected inverters when asymmetrical faults occur in the grid and the inverter control target is to eliminate both active and reactive power fluctuations, (a) single-phase-to-ground fault, (b) two-phase-to-ground fault

2.3.3

Influence of Current Limiter

Due to limited overcurrent capacities of semiconductor devices, current limiters are applied in PQ control loops. According to mathematical Eq. (2.6), power control loops might produce a large reference current value that exceeds the threshold ith (i.e. two times greater than the rated value) when faults occur. There are two options for current limitation. One is to use fault ride through current strategy, as aforementioned. The benefits of this method are that the power flow of IIDGs during fault period is controllable. However, it also puts foreward requirements for the control system and makes the controller complicated. Different from the control system, inherent current limitation through current limiters is simple but power flow during fault period is uncontrollable. This part would discuss the influence of current limiters on fault current. In this case, the reference current becomes irrelevant to PCC voltages and will be locked at ith. Assuming that current loops can realize trace in a short time, the amplitude of fault current will be determined and Eq. (2.6) can be simplified into Eq. (2.44) 8  > < iLq ¼ iLq lim rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 > : iLd ¼ i2  i th

ð2:44Þ

Lq lim

Figure 2.18 shows how PQ-controlled IIDGs react when current limiters are applied. It is observed that the amplitude of fault current cannot reach the amplitude

2 Transient Characteristics of Current Controlled IIDGs During Grid Fault

iabc [pu]

42

2

i*

ia

ith

ic

ib

0 –2 0

tf

t

Fig. 2.18 Fault current of PQ-controlled IIDGs with current limiters when subjected to two-phase grounded fault (PCC voltage dip to 33% of the rated value)

of the reference current i* and are determined by ith. The assumption is made that phase lock loop (PLL) can track phase angle of the faulted system precisely, even asymmetrical faults occur. In this condition, conclusions can be made that PQ-controlled IIDGs with current limiters can be seen as a constant current source and the amplitude of the current source is determined by the limiter’s threshold value.

2.4

Summary

In this chapter, the symmetrical component method, the topology and control principle of the three-phase four-wire inverter with split capacitor on the DC side are firstly described. Then, the transient fault characteristics of CC controlled and PQ controlled IIDGs with different control targets have been investigated comprehensively. Fault models and fault current calculation methods are introduced. Specific conclusions are as follows: 1. For CC controlled IIDG, the fault current is described by a general mathematical expression during symmetrical and asymmetrical faults. The output current is constant whether symmetrical or asymmetrical grid faults occur, which is not influenced by the power circuit. 2. For PQ controlled IIDGs, according to whether there are positive, negative or zero sequence components in the inverter output current, the inverter can have several control targets including the symmetrical current control, active power fluctuation suppression, and reactive power fluctuation suppression. The fault current mathematical expressions of PQ controlled inverter with different combined control targets can be obtained, respectively. 3. The amplitude calculations of the fault current are verified by simulation results. It is concluded that the control targets of PQ controlled IIDGs can be realized under different reference currents. Meanwhile, their other operating performances are sacrificed. Generally, the studied fault current characteristics of CC controlled and PQ controlled IIDGs can provide the foundations for the design of FRT control methods of current controlled IIDGs.

References

43

References 1. M.P. Kazmierkowski, L. Malesani, Current control techniques for three-phase voltage-source PWM converters: a survey. IEEE Trans. Ind. Electron. 45(5), 691–703 (1998) 2. M.P. Kazmierkowski, M.A. Dzieniakowski, Review of current regulation methods for VS-PWM inverters, in Conf. Rec. IEEE IECON’94, (IEEE, Piscataway, 1994), pp. 567–575 3. P. Enjeti, P.D. Ziogas, J.F. Lindsay, M.H. Rashid, A novel current controlled PWM inverter for variable speed AC drives, in Conf. Rec. IEEE-IAS Annu. Meeting, (IEEE, Piscataway, 1986), pp. 235–243 4. A.B. Plunkett, A current controlled PWM transistor inverted drive, in Conf. Rec. IEEE-IAS Annu. Meeting, (IEEE, Piscataway, 1979), pp. 785–792 5. F. Blaabjerg, R. Teodorescu, M. Liserre, A.V. Timbus, Overview of control and grid synchronization for distributed power generation systems. IEEE Trans. Ind. Electron. 53(5), 1398–1409 (2006) 6. K. Ma, W. Chen, M. Liserre, et al., Power controllability of a three-phase converter with an asymmetrical AC source. IEEE Trans. Power Electron. 30(3), 1591–1604 (2014) 7. G. Azevedo, G. Vazquez, A. Luna, et al., Photovoltaic inverters with fault ride-through capability, in Proceedings 2009 IEEE International Symposium on Industrial Electronics, (IEEE, Piscataway, 2009), pp. 549–553 8. A. Camacho, M. Castilla, J. Miret, et al., Active and reactive power strategies with peak current limitation for distributed generation inverters during asymmetrical grid faults. IEEE Trans. Indus. Electron. 62(3), 1515–1525 (2015) 9. Z. Shuai, C. Shen, X. Yin, X. Liu, Z.J. Shen, Fault analysis of inverter-interfaced distributed generators with different control schemes. IEEE Trans. Power Delivery 33(3), 1223–1235 (2018) 10. N. Nimpitiwan, G.T. Heydt, R. Ayyanar, S. Suryanarayanan, Fault current contribution from synchronous machine and inverter based distributed generators. IEEE Trans. Power Delivery 22 (1), 634–641 (2007) 11. Z. Shuai, M. Xiao, J. Ge, Z. J. Shen, Overcurrent and its restraining method of PQ-controlled three-phase four-wire converter under asymmetrical grid Fault. IEEE J. Emer. Sel. Top. Power Electron. 7(3), 2057–2069 (2019)

Chapter 3

Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

The fault current of voltage-controlled IIDGs is more serious than that of currentcontrolled inverters, which may affect the safety of power electronic devices seriously. Therefore, voltage-controlled IIDGs should be equipped with fault ride through (FRT) capability and have a reliable protection system. The studies on fault current and fault transient characteristics can help the design of optimal FRT strategies and protection systems. However, IIDGs response quite differently from SGs during fault period so that conventional fault current calculation methods cannot be fully applicable. This probably makes protection systems invalid and leads the failure of FRT strategies. Therefore, it is important to investigate inverter fault characteristics of voltage-controlled IIDGs. This chapter investigates the transient fault characteristics of three voltage controlled IIDGs, including V/f controlled IIDGs, droop controlled IIDGs and virtual synchronous generators (VSG). Firstly, the operation principles and control structures of the V/f control, droop control and virtual synchronous control are introduced. Then, the fault models and fault current mathematical expressions for three voltage-controlled IIDGs are introduced in detail. Finally, the main influencing factors of fault current are analyzed, covering control loops, fault type, line impedances, fault occurring moment and nonlinear limiters.

3.1

Principle and Control Structures

In previous studies on transient characteristics of voltage-controlled IIDGs, they are modeled as ideal voltage sources. However, the fault model is not accurate enough. To better reflect transient response of IIDGs, dynamic characteristics of control systems needs to be considered. Thus, the basic principles and control structures of voltage controlled IIDGs should be studied firstly. The voltage control strategies are mainly V/f control, droop control and virtual synchronous control. V/f controlled IIDGs aim to generate the constant ac voltage as a reference to maintain voltage and frequency stability in microgrid. The objective of droop controlled IIDGs and VSGs © Springer Nature Singapore Pte Ltd. 2021 Z. Shuai, Transient Characteristics, Modelling and Stability Analysis of Microgird, https://doi.org/10.1007/978-981-15-8403-9_3

45

46

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

is to participate in the regulation of the ac grid voltage amplitude and frequency by controlling output active and reactive power.

3.1.1

Principle of V/f Control

V/f control is widely applied to maintain voltage and frequency stability in a microgrid operating in islanded mode [1]. Figure 3.1 shows the typical topology and control block diagram of V/f controlled IIDG in a microgrid. The power circuit consists of the DC bus capacitance, three-phase three-leg inverter and an LC filter. ii represents the inductance current. vo and io are the output voltage and current of the inverter, respectively. In V/f control system, v
oabc is the reference voltage of control block. v
od and v
oq is the d-axis and q-axis value of setted reference voltage in synchronous reference frame, respectively. i
id and i
iq is the d-axis and q-axis value of reference current, respectively. v
id and v
iq is the d-axis and q-axis value of modulation voltage, respectively. The controller of a V/f controlled IIDG is composed of two parts. First part is the voltage controller, which is designed to control the capacitor voltage of inverter to match the reference voltagev
oabc . Second part is the current controller, which regulates the current supplied by the inverter. Therefore, the controlled current flowing through the inductor Lf charges the capacitor Cf to keep the output voltage close to the reference provided to the voltage control loop. Figure 3.2 shows the control block diagram of voltage controller and current controller including all feed-back and feed-forward terms. In the voltage outer loop, the error between the reference vodq* and the measured voltage vodq is the input to a PI controller whose output is added with measured current iodq and decoupling component, this sum establishes the current reference iidq*. Note that, decoupling components are needed to negate any transient errors caused by coupling among any of the phases. Similar, the inner current loop regulates the inductance current, tracking the reference current provided by the outer voltage loop by a PI controller. The outputs of current controller then delivered to pulse width modulation (PWM) block to drive semiconductor switches, in turns to generate output potential of the Fig. 3.1 Typical topology of V/f control

Lf

Zl1

io

PCC

Edc

PWM modulation

vo

Cf

ii

* voabc

abc/dq

abc/dq *

vid v iq*

iid

iiq

Current controller

vod* *

iid * iiq

voq*

Voltage controller

vod voq

3.1 Principle and Control Structures Vq

ioq Vq∗

+–

+ ++

ωC

Vq Vd Vd∗

K vp s + Kvi s

Kvp s + Kvi s

∗ iLq

+–

iLq

ωC +–

47

++ +

Vqref

ωL

iLd ∗ iLd +–+

Kip s+K ii s

ioq iLq

+––

ωL

∗ Vabc abc dq/ Vabc K pwm abc /dq

K ip s + Kii s

+–+

Vdref

+––

Vq

ωC 1 Ls + R

Vd

iod

1 Cs ωC

ωL

ωL +–

1 Ls + R

+––

iLd

– +–

1 Cs

Vd

iod

Fig. 3.2 General control block diagram of V/f control

inverter. According to Fig. 3.2, the transfer function can be described by (3.1) (Detailed description for the simplification of voltage and current control loops can be found in [2]) V odq ¼ Gv ðsÞV
odq  Z ðsÞiodq

ð3:1Þ

in which Gv ðsÞ ¼

kvp s þ kvi τi Cs3 þ Cs2 þ kvp s þ kvi

τ i s2 Z ðsÞ ¼ 3 τi Cs þ Cs2 þ k vp s þ kvi

ð3:2Þ

where τi represents time constant of the inner current control loop.

3.1.2

Principle of Droop Control

Droop control is one of the most widely used control strategies for distributed generators due to its advantages of load sharing and plug and play characteristics. The droop controlled IIDGs can participate in regulating the amplitude and frequency of grid voltage in both grid-connecting mode and islanded mode. Figure 3.3 shows the typical topology and controller system of a droop controlled IIDG [3]. The power processing section is the same with V/f controlled IIDG. In droop control system, V* and ω* are the references of amplitude and angular frequency of capacitor voltage respectively. P* and Q* are the no-load active and reactive powers, respectively. Vo and ωo are the amplitude and angular frequency of capacitor voltage, respectively. Other variables are the same with the control system of V/f control. The control system of droop control includes power controller, voltage controller and current controller. Similar to V/f control, the voltage and current control loops aim at regulating output voltage and current of inverter. Power controller loops distinguish droop control from V/f control, the power controller is

48

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

Fig. 3.3 Typical topology of droop control

Lf

Cf PWM modulation

Z l1

io

V dc vo

V *ω * P * Q*

ωo

ii

Vo*

abc/dq vid*

Fig. 3.4 Internal structure of power controller

vod voq iod ioq

viq*

PCC

Power controller

iid

iiq

iid*

Current controller

vod iod + voq ioq vod ioq – voq iod

Power calculation module

p

ωc

s + ωc q

ωc

* iq

i

p q

vo io

ω o*

abc/dq vod* voq* vod voq

Voltage controller

ω* – mp( p –P* )

ωo

1

θ

s

s + ωc

V *– mq( q –Q* )

Filter

Droop regulation module

Vo

implemented to regulate the exchange of active and reactive powers with the grid, in order to keep the grid voltage frequency and amplitude under control. The power controller comprises power calculation module, filter module and droop regulation module. The internal structure of the power control part is shown in Fig. 3.4. p and q are the instantaneous power components, which are calculated from the measured output voltage and current in power calculation module, as shown in (3.3). p ¼ vod iod þ voq ioq , q ¼ vod ioq  voq iod

ð3:3Þ

The instantaneous power components are passed through filter modules, shown in (3.4), to obtain the real and reactive powers p and q corresponding to the fundamental component. ωc represents the cut-off angular frequency of low-pass filters. p¼

ωc p, s þ ωc

q¼

ωc q s þ ωc

ð3:4Þ

The basic idea of droop regulation module is to mimic the governor of a synchronous generator. In a conventional power system, SGs will share any increase in the load by decreasing the frequency according to their governor droop characteristics. This principle is implemented in droop regulation by decreasing the reference angular frequency ω* when there is an increase in the load. Similarly, reactive power is shared by introducing the droop characteristics in reference of voltage amplitude V*. The droop characteristics is shown in (3.5). mp and mq are the

3.1 Principle and Control Structures

49

droop gains for P-f and Q-V curve. The voltage amplitude and angular frequency are delivered to transmission module to generate reference voltages of inner controllers.


ωo ¼ ω
 mp ðp  P
Þ

ð3:5Þ

V o ¼ V
 mq ðq  Q
Þ

3.1.3

Principle of Virtual Synchronous Control

VSG control is designed to provide inertia and damping for microgrid, it mimics the transient characteristics of SG by emulating its fundamental swing equation [4]. Compared with V/f control and droop control, VSGs have considerable inertia that enables them to resist small disturbances. Therefore, VSGs have remarkable advantages from the viewpoint of dynamic stability. The typical topology of a VSG is presented in Fig. 3.5. Lf, Rf and Cf are the filter inductor, resistance and capacitor, respectively. Rl1 and Ll1 are the line impedance between the VSG and point of common coupling (PCC). Rl2 and Ll2 are the line impedance from PCC to the load. Rs and Ls are the line impedance from PCC to the grid. io and v represent the output current and output fundamental potential of VSG respectively. V and θ are the amplitude and phase of v respectively. vs is the grid
voltage. V* and θ_ are the references of the output potential and angular frequency respectively. P* and Q* are the references of the active and reactive powers, respectively. VSGs comprise a power part and an electric control part. The power part includes the DC bus capacitance, inverter, and filter, which are used to convert the DC power to industrial frequency AC power. the core of the electric control part is the virtual synchronized control shown in Fig. 3.6. J is the virtual inertia. Dp is the frequencydrooping coefficient. K is the voltage integral coefficient. and Dq is the voltageThree-phase inverter

Power part v Rf

io

Lf

PCC Rl1

Ll1

Rs

Ls

vs

+ Udc Cf

Drive signal

Rl2 Grid

PWM

Electronic part

Ll2

io

Synchronverter control unit V*

·

θ*

Load Q*

Fig. 3.5 Typical topology of a VSG

P*

50

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault ΔT

Dp

P*

1 ω*

Tm

1 Js

+

+

Fig. 3.6 Control block and equivalent circuit of VSG

-

ω

ω* 1 s

θ

Te Eq.(3.6) e

Q 1 Ks

+ Q* +

io

Mfif + Dq

V* -

V

drooping coefficient. Tm and Te are the mechanical torque and electromagnetic toque, respectively. Mfif is the field excitation. Q is the output reactive power of VSG. The algorithm mainly comprises the active and reactive power control loops. The active and reactive power control loops realize power droop and inertia characteristics, respectively. The VSG’s basic control principles are given by (3.6) and (3.7). Equation (3.6) gives the power calculation and how the reference voltage is formed. Equations (3.7) and (3.8) give the frequent inertia loop and voltage inertia loop of the VSG, respectively. 8 > < T e ¼ M f i f hio , sin θi _ f i f hio , sin θi Q ¼ θM > : _ f i f sin θ v ¼ θM  
 1 T m  T e þ Dp θ_  θ_ θ_ ¼ Js  1 
M fif ¼ Q  Q þ D q ðV
 V Þ Ks

ð3:6Þ

ð3:7Þ ð3:8Þ

The inertia of the VSG is advantageous to some extent, because it can offset the responding lag of the system regulation and restrain the active and reactive power’s high-frequency ripple wave, which plays the same role that the synchronized generator does in a grid.

3.2

Transient Characteristics of V/f Controlled IIDG During Grid Fault

In this section, the transient fault characteristics of V/f controlled IIDG are studied. Firstly, the fault model using the instantaneous symmetric component method is introduced. Then, the fault current is analyzed and the mathematical model of fault

3.2 Transient Characteristics of V/f Controlled IIDG During Grid Fault

51

current is derived. It is found that the main influencing factors of fault current characteristics of V/f controlled IIDG are control parameters, fault types, line impedances, fault occurring moment and nonlinear limiters. Finally, the effects of these influencing factors are analyzed carefully.

3.2.1

Fault Models

By converting the transfer function in dq reference frame, as shown in (3.1), to the abc reference frame, the transfer function can be described in (3.9), where Gv is the inverter voltage gain, Zo is the output impedance. voabc ¼ Gv ðsÞv
oabc  Z o ðsÞioabc

ð3:9Þ

where Gv(s) and Z0(s) can be found in Eq. (3.2). Thus, V/f controlled IIDG with LC filter can be equivalent to a voltage source Gv(s)voabc* in series with Zo. voabc* is the given reference instantaneous voltage. It is assumed that dynamics of Gv(s) can be ignored due to its fast response characteristics so that IIDGs can be characterized by quasi steady-state equivalent fault model [5]. Therefore, V/f controlled IIDG can be equivalent to a constant voltage source voabc* in series with an output impedance Zo, as shown in Fig. 3.7, where V and θ are the amplitude and phase of reference voltage voabc*, respectively.

3.2.2

Fault Current Calculation

3.2.2.1

Mathematical Model of Fault Current

Figure 3.8 shows topology of the studied system. Rl1 and Ll1 are the line impedance between the inverter and PCC. Rl2 and Ll2 are the line impedance from PCC to the load. Rs and Ls are the line impedance from PCC to the grid. The DG is equipped with a V/f controlled inverter. Here, SL represents the load impedance. Whether a symmetrical fault or an asymmetrical fault occurs, the equivalent voltage source of V/f controlled IIDG remains constant and is still a symmetric positive sequence voltage, according to the above analysis. If asymmetrical fault occurs, the grid side contains positive, negative and zero-sequence components. Fig. 3.7 Fault model of V/f controlled IIDGs

Zo

+ V∠ θ

–

52

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

Fig. 3.8 Topology of the studied system

vs

Rs

Ls

PCC

Rl2

Ll2

vf SL

Grid Rl1

Ll1

DG

However, for symmetrical faults, only positive sequence will appear in gird side. Obviously, symmetrical fault is a special case in asymmetrical faults, the fault analysis method drawn by asymmetrical fault will be suitable for symmetrical fault. Therefore, only the asymmetrical fault will be analyzed in this chapter. When an asymmetrical fault occurs, the output impedance of the V/f controlled IIDG is always constant, which is different from the output impedance of SG who experiences the sub transient, transient, and steady states after asymmetric faults. Besides, since the output impedance are all three-phase symmetric, the positive, negative, and zero-sequence output impedances of IIDG are independent and equal. Therefore, during asymmetric faults, the V/f controlled IIDG is equivalent to a controlled symmetrical voltage source in series with an equivalent output impedance [6]. Moreover, the asymmetry at the fault node can be replaced with connecting a set of asymmetric voltage sources, which is represented as vf in Fig. 3.8. vf can be obtained according to the pre-fault voltage at the fault node and fault boundary conditions. Moreover, using the instantaneous symmetric component method, vf can be separated into the positive, negative, and zero-sequence voltages, as shown in (3.10) and (3.11). Where, a ¼ e j2π/3and a2 ¼ ej2π/3. vfa, vfb, and vfc are the three 0 phase instantaneous voltages. vþ fa , vfa , and vfa are the positive, negative, and zerosequence instantaneous voltages of phase A at the fault node, respectively. It should be noted that the instantaneous symmetric method adopted in this chapter is defined in the time domain and can obtain the instantaneous positive, negative and zerosequence components by performing symmetric transformation on the instantaneous voltage or current. Compared to the traditional symmetric component method, it is more suitable for transient analysis of asymmetric fault. 2

vþ fa 6 v 4 fa v0fa

2

3 pffiffiffi   3 j90  1 vfb  vfc 7 6 vfa  2 vfb þ vfc  2 e 7 pffiffiffi 7 16 6  5¼ 6   7 3 4 vfa  1 vfb þ vfc þ 3 ej90 vfb  vfc 7 5 2 2 vfa þ vfb þ vfc 3

ð3:10Þ

3.2 Transient Characteristics of V/f Controlled IIDG During Grid Fault

(a)

(b) Zl1

·

+

53

I·o+

Zl2

I·f+

+ Zl1

Zs · Vf+ · Vs

Zo

Zs

–

–

–

Zl2

I·s-

I·f-

+ · Vf-

–

(c) Zl2

0 I·f

+ ·0 Vf

Zs

– Fig. 3.9 The sequence networks of the studied system during asymmetric faults: (a) Positive sequence network, (b) Negative sequence network, (c) Zero-sequence network

3 2 2 þ 32 þ 3 2 þ 3 avfa vfc vþ a vfa fb 6 v 7 6 av 76 v 7 6 a2 v 7 4 fb 5 ¼ 4 fa 54 fc 5 ¼ 4 fa 5 2

v0fb

v0fa

v0fc

ð3:11Þ

v0fa

As shown in Fig. 3.8, the grid and load are neutral-grounded systems with positive, negative, and zero-sequence current loops. The V/f controlled inverter is a three-phase three-wire inverter with only positive and negative sequence current loops. During asymmetric faults, only the positive and negative sequence currents are injected into the fault node by the inverter. In addition, given that the line parameters are three-phase symmetric after replacing the fault node with a set of voltage sources, the positive and negative sequence impedances of studied system are independent and equal. Therefore, to analyze the asymmetric fault current, positive, negative, and zerosequence networks can be established separately, as shown in Fig. 3.9, where V_ represents the equivalent controlled voltage source of the inverter. V_ s is grid voltage. þ  0 V_ f , V_ f and V_ f are the positive, negative, and zero-sequence voltages at the fault þ  node. Moreover, I_ o and I_ o are the positive and negative sequence currents of IIDG, þ  0 respectively. and I_ f , I_ f , and I_ f are the positive, negative, and zero-sequence currents at the fault node, respectively. According to the Thevenin’s theorem, the portion in the dashed box in Fig. 3.9a is equivalent to a positive sequence voltage source in series with an equivalent line impedance, and the portion in the dashed box of Fig. 3.9 is equivalent to a negative sequence voltage source in series with an equivalent line impedance, as shown in þ  Fig. 3.10a and b, respectively. Where, V_ eq and V_ eq represent the equivalent positive and negative sequence voltage sources respectively, as shown in (3.12)–(3.13). Z 0eq represents the equivalent impedances between the equivalent grid voltage and PCC

54

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

(a)

+

(b) Zeq'

Zol1 I·o+

· V

I·o-

+ V· eq

-

Zeq'

Zol1

+ –

+ V· eq-

–

Fig. 3.10 The simplified sequence networks of studied system when analyzing fault current of V/f controlled IIDG: (a) Positive sequence network, (b) Negative sequence network

after asymmetric faults, as shown in (3.14). Zol1 consists of the equivalent output impedance Zo of V/f controlled IIDG and line impedances Rl1 and Ll1 between the IIDG and PCC, as shown in (3.15). þ V_ eq ¼

þ Z l2 Zs V_ þ V_ Z s þ Z l2 s Z s þ Z l2 f

ð3:12Þ

 Zs V_ Z s þ Z l2 f

ð3:13Þ

Z s Z l2 Z s þ Z l2

ð3:14Þ

Z ol1 ¼ Z o þ Z l1

ð3:15Þ

 V_ eq ¼

Z 0eq ¼

The sequence component of Vf+ and Vf expressions during different fault occurs are shown in (3.16). Vf_ab-c-g+ and Vf_ab-c-g, Vf_a-g+ and Vf_a-g, Vf_b-c-g+ and Vf_b-c+   g , Vf_b-c and Vf_b-c are sequence component when ABC-G, A-G, BC-G and B-C fault occur respectively. Vpre is pre-fault voltage of fault point. The equivalent positive and negative sequence networks are both symmetric. Thus, the positive and negative sequence fault currents can be calculated using the symmetric fault current analysis method. In the normal operation, the equivalent voltage source of the inverter, the equivalent grid voltages, and the total line impedances of studied system are presented in (3.17). V þf V þf V þf

¼ 0, V f

abcg ¼ 0  ag ¼ 2=3  V pre , V f ag ¼ 1=3  V pre  bcg ¼ 1=3  V pre , V f bcg ¼ 1=3  V pre  bc ¼ 1=2  V pre , V f bc ¼ 1=2  V pre abcg

V þf 8 > < v ¼ V sin ðωt þ θÞ Z t ¼ Z ol1 þ Z eq ¼ Rt þ jωLt > : veq ¼ V eq sin ðωt þ α1 Þ

ð3:16Þ

ð3:17Þ

where, v, V, and θ are the instantaneous value, amplitude, and phase angle of the inverter equivalent voltage, respectively. Zt represents the total line impedance

3.2 Transient Characteristics of V/f Controlled IIDG During Grid Fault

55

between the inverter and the grid equivalent source. and veq, Veq, and α1 are the instantaneous value, amplitude, and phase angle of the equivalent grid voltage, respectively. (

Z 0t ¼ Z ol1 þ Z 0eq ¼ R0t þ jωL0t    v0eq ¼ V þ eq sin ωt þ θ p þ V eq sin ðωt þ θ n Þ

ð3:18Þ

When a asymmetric fault occurs, Veq changes into the positive voltage amplitude þ  V_ eq and negative voltage amplitude V_ eq . the phase angle α1 changes to the phase angles of the positive and negative sequences θ+ and θ. and the total line impedance Zt changes to Z 0t , as shown in (3.18). According to KVL, the mathematical differential equations of phase A during fault period are obtained, as shown in (3.19) and (3.20). diþ þ o ¼ V sin ðωt þ θÞ  V þ eq sin ðωt þ θ Þ dt  0 dio   R0t i þ L o t dt ¼ V eq sin ðωt þ θ Þ

0 R0t iþ o þ Lt

ð3:19Þ ð3:20Þ

Since the equivalent voltage source of the inverter is a constant voltage source after fault occurs, the solution of (3.19) and (3.20) contains a general solution and a particular solution. Assuming fault occurs at t ¼ 0, the positive and negative sequence fault currents can be calculated.   1 0 þ þ 0 ð Þ  V sin ð ωt þ θ  φ Þ þ V sin ωt þ θ  φ eq j Z 0t j  1  V sin ðθ  φÞ  V eq sin ðα1  φÞ et=T a  j Zt j   1 þ 0 t=T a V sin ðθ  φ0 Þ  V þ 0 eq sin ðθ  φ Þ e jZtj   1  1  0   0 i ¼ sin ð ωt þ θ  φ Þ  sin ð θ  φ Þ et=T a V V o j Z 0 t j eq j Z 0t j eq   0 L0t ωLt ωLt 0 T a ¼ 0 , φ ¼ arctan , φ ¼ arctan Rt Rt R0t iþ o ¼

ð3:21Þ

ð3:22Þ

where, Ta is attenuation time constant. φ and φ0 are total line impedance angles before and after faults. Thus, given that asymmetric fault current is sum of positive and negative sequence current, the asymmetric fault current can be obtained.

56

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

iop ¼



 1 0 V sin ðωt þ θ  φ0 Þ  V þ 0 eq sin ωt þ θ p  φ j Zt j 

ioap ¼

 io ¼ iþ o þ io ¼ iop þ ioap

V eq sin ðωt þ θn  φ0 ÞÞ j Z 0t j



ð3:23Þ

ð3:24Þ

1   V sin ðθ  φÞ  V eq sin ðα1  φÞ j Zt j     1 0  0 t=T a  0 V sin ðθ  φ0 Þ  V þ eq sin θ p  φ  V eq sin ðθ n  φ Þ e j Zt j ð3:25Þ

where iop and ioap are the periodic component and DC component of the fault current, respectively. Equation (3.23) is the mathematical expression of the phase A asymmetric fault current. By substituting θ  2π/3, θ+  2π/3, θ  2π/3, θ  4π/ 3, θ+  4π/3, and θ  4π/3 for α, θ+, and θ in (3.23) and (3.24), the asymmetric fault current of phases B and C can be obtained, respectively.

3.2.2.2

Fault Current Estimation

Noted that V/f controlled IIDG can be regarded as a constant voltage source before and after faults, fault current can be calculated through established model. Detailed steps of the algorithm are as follows: Step 1: Obtain equivalent voltage source V and output impedance Zo according to (3.1). Obtain impedance Zl1 and Zl2 and load ZL to get pre-fault and post-fault total line impedance Zt and Zt0. Finally, calculate the attenuation time constant Ta. Step 2: Obtain the voltage of the fault point Vf. Then, Calculate post-fault þ  equivalent voltage source V_ f andV_ f . Step 3: Substituting the voltage source V into (3.24) and (3.25), the periodic component and dc component of the asymmetric fault current can be obtained. Then, the full asymmetrical fault current can be calculated by (3.23). The calculation steps can be illustrated through Fig. 3.11. In order to validate the correctness of introduced algorithm, the system in Fig. 3.7 is simulated in PSCAD/ ETMDC. Specific parameters are shown in Table 3.1. Figure 3.12 shows simulation results of the fault current of V/f controlled inverter when the grid is subjected to single-phase-to-ground (A-G), two-phase-to-ground fault (BC-G) and phase-to-phase fault (B-C), respectively. To give intuitive comparasion, calculation results obtained from algorithm written in Matlab are superimposed on the simulation results in Fig. 3.12. The theoretical calculation results by Fig. 3.11 and the simulation results are consistent, which validate the practical of introduced fault current calculation method.

3.2 Transient Characteristics of V/f Controlled IIDG During Grid Fault Fig. 3.11 Flow chart of fault current estimation

57

START Obtain V, Z0, Zt, Zt’ Calculate Ta Obtain Vf, Vf+, VfSubstitute V to (3.24) and (3.25)

END

Fig. 3.12 Validation of the correctness of the introduced algorithm using simulation results: (a) Single-phase-to-ground fault (A-G), (b) Two-phaseto-ground fault (BC-G), (c) Phase-to-phase fault (B-C)

(a) [A] i 200 oabc

Calculation results

Simulation results

100 0 -100 -200

(b) 200

tf oabc

Calculation results

Simulation results

100 0 -100 -200

(c) [A] i 200 oabc

tf

Calculation results

Simulation results

100 0 -100 -200

tf

Table 3.1 System parameters Symbol Nominal voltage Vn Nominal frequency ω* ZLoad Lf

Value 321.5 V 314 rad/s 7.26 Ω 3 mH

Symbol Cf Zc Zg ZL

Value 20 μF 0.3 Ω + 2.3 mH 0.2 Ω + 0.3 mH 0.3 Ω + 0.3 mH

58

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

3.2.3

Influencing Factors of Fault Current Characteristics

3.2.3.1

Influence of Line Impedance and Fault Occurring Moment

According to (3.23), the asymmetric fault current of V/f controlled inverter can be expressed as follows. þ   io ¼ I þ pm sin ðωt þ φ Þ þ I pm sin ðωt þ φ Þ   þ   þ I m sin φ1  I þ sin φ  I sin φ et=T a pm pm

ð3:26Þ

0 þ  j V_  V_ eq j  j V_ eq j j V_  V_ eq j þ Im ¼ , I pm ¼ , I pm ¼ j Zt j j Z 0t j j Z 0t j

ð3:27Þ

where, Im and φ1 are the amplitude and phase of the output current of the inverter +   before faults, respectively. I þ pm , I pm , φ , andφ are the amplitudes and phases of the positive and negative sequence fault currents, respectively. From (3.26), the maximum inrush current phase of the asymmetric fault current is simplified as.

 0 0 io ¼ I pm sin ðωt þ φ2 Þ þ I m sin φ1  I pm sin φ2 eRt t=Lt rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ 0 _ þ 2 _  2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V  V eq þ V eq 2  2 I pm ¼ I þ pm þ I pm ¼ j Z 0t j

ð3:28Þ

ð3:29Þ

where Ipm and φ2 are the periodic component amplitude and phase of the asymmetric fault current, respectively. From (3.28) and (3.29), it can be seen that the amplitude of the asymmetric fault current is related to the line impedance. The larger the line impedance is, the smaller the fault current amplitude is. The closer the fault node to the inverter is, the larger the fault current amplitude is. Moreover, the initial value of the DC component is: I ap ¼ I m sin φ1  I pm sin φ2

ð3:30Þ

Given that φ1 and φ2 are determined by the fault occurring moment, the initial value of the DC component is also related to the fault occurring moment. In addition, the attenuation time constant of the DC component is only determined by the line impedances.

3.2 Transient Characteristics of V/f Controlled IIDG During Grid Fault

3.2.3.2

59

Influence of Fault Type

According to the KVL equations and fault boundary conditions of the positive, negative, and zero-sequence networks in Fig. 3.9, the composite sequence networks at the fault node under different types of asymmetric faults can be obtained respectively, as shown in Fig. 3.13. Moreover, the fault node voltages under different asymmetric faults can be given by their composite sequence networks. Equations (3.31)–(3.33) are the positive, negative, and zero-sequence components of the fault node voltages under singlephase-to-ground faults, two-phase-to-ground faults, and phase-to-phase faults, respectively.   8  0 _ > V sg Z þ Z > f f þ > > V_ f ¼  þ > >  > Z f þ Z f þ Z 0f > < Z f V_ sg > _ f ¼ V > > > Z þf þ Z f þ Z 0f > > > > 0 þ  : V_ f ¼ V_ f  V_ f

ð3:31Þ

Z f Z 0f V_ sg

þ  0 V_ f ¼ V_ f ¼ V_ f ¼

ð3:32Þ

Z þf Z f þ Z þf Z 0f þ Z f Z 0f Z f V_ sg 1 ¼ V_ sg 2 Z þf þ Z f

þ  V_ f ¼ V_ f ¼

ð3:33Þ



UVF ¼

Fig. 3.13 Composite sequence networks at the fault node under different types of asymmetric faults: (a) Single-phase-to-ground fault, (b) Two-phase-toground fault, (c) Phase-tophase fault

j V_ f j

· Vf+

– + (b)

I·f-

–

+

+

+

·+ Vf

–

–

· Vf-

–

I·fZf-

0 I·f

+

·0 Vf

–

(c)

0 I·f

I·f+

Zf0

I·f+

+ · Vsg

· Vsg

Zf-

Zf+

(a)

Zf+

ð3:34Þ

þ

j V_ f j

I·f+ Zf0

Zf+ · Vsg

+ –

+ · Vf+

–

I·fZf-

60

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

_ ðZ s þZ fl1 ÞZ l2 Z VþZ V_ where, V_ sg ¼ sZ s þZflfl s , Z þf ¼ Z f ¼ Z fl1 þZ s þZ l2 , Z 0f ¼ Z s þ Z l2 . V_ sg represents pre-fault voltage at the fault node. Z þf , Z f and Z 0f are the equivalent input impedances of the fault node in positive, negative and zero-sequence networks, respectively. Since there is no zero-sequence current from VSG injecting into the fault node, Z þf , Z f and Z 0f meet the equation Z þf ¼ Z f 6¼ Z 0f during short-circuit fault period. It can be seen from (3.31)–(3.34) that the magnitude, phase, and voltage unbalance factor (VUF) of the fault node voltages with different asymmetric faults are very different. Thus, the fault type directly determines the drop depth and VUF of the equivalent grid voltage, which affects the periodic and DC components of the asymmetric fault current.

3.2.3.3

Influence of Nonlinear Limiter

A. Influence of Current Limiter Due to limited overcurrent capacities of semiconductor devices, current limiters are applied in control loops. When fault occurs, since the capacitor voltage remains constant, control loops might produce a large reference current value that exceeds the threshold imax (In general, imax is 1.2 times greater than the rated value). In this case, the reference current becomes irrelevant to PCC voltages and will be locked at imax. Assuming that current loops can realize trace in a short time, the amplitude of fault current will be determined as shown in (3.35). 8 < iLdref ¼ iLdref
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi : iLqref ¼ imax 2  iLdref
2

ð3:35Þ

Note that after current limiter acts, control loops only regulate the current flowing through inductance, so the filter capacitors Cf do appear in sequence networks. Fault models of V/f controlled inverter after current limiter acts can be described by Fig. 3.14. In positive sequence networks, inverters are equivalent to current sources in parallel with filtering capacitors. However, only filtering capacitors appear in negative sequence networks.

(a)

(b) Zl1

+ IL ∠ θ

Cf

–

Zeq'

Zl1

+

·+ Veq

Cf

–

Zeq'

+

· Veq-

–

Fig. 3.14 Fault models of V/f controlled inverter under the current limiter function: (a) Positive sequence fault model, (b) Negative sequence fault model

3.2 Transient Characteristics of V/f Controlled IIDG During Grid Fault 100

Ioabc [A]

50

I max

Simulation results

61

Calculation results

0 -50 -100 0.42

0.46

0.5

0.54

0.58

0.62

t [s]

Fig. 3.15 The comparison of the theoretical calculation results and the simulation results of the fault current when current limiter is considered

(a)

+

(b) ZLfLc Vmax∠ θ

–

Zl1

Zeq'

+

·+ Veq

ZLfLc

Zl1

Zeq'

+

· Veq-

–

–

Fig. 3.16 Fault models of V/f controlled inverter under the function of modulation wave limiter: (a) Positive sequence fault model, (b) Negative sequence fault model

Fault response of the inverter is shown in Fig. 3.15. Also, the calculation results obtained by the algorithm established in Matlab are superimposed on the simulation results. It is clear that two results are almost identical. So, conclusions are made that V/f controlled inverters with current limiters can be seen as a constant current source and the amplitude of the current source is determined by the limiter’s threshold value.

B. Influence of Modulation Wave Limiter When modulation wave limiters are adopted, it is possible that modulation signals are limited at a preset value. The amplitude and phase of the modulation signals are decided by the threshold value of the limiter. Vset,d and Vset,q represents the magnitude of modulation wave in dq axis respectively and Vmax represents RMS value of inverter internal voltage. In this condition, fault current of a V/f controlled inverter is similar to that of an ideal three-phase voltage source. The setting principle of Vset,d and Vset,q is as shown in (3.36) and the corresponding equivalent circuits are shown in Fig. 3.16. In normal operation conditions, positive sequence output impedance of V/f controlled IIDGs is shaped by control loops. However, when modulation wave limiters are considered, positive sequence impedance is decided by LC filters (shown in Fig. 3.16) because voltage loops and current loops lose the control ability and the closed loop system turns into an open loop system.

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

* [V] Viabc

* [A] I idq

62

600 400 200 0 -200 -400

Current limiter acts

Fault occurs

I id*

I iq*

Modulation wave limiter acts

Vmax

1000

Via*

Vib*

Vic*

0

-1000 1000

Ioabc [A]

I max

Simulation results

Calculation results

0

-1000 0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

t [s]

Fig. 3.17 Fault characteristics when the modulation wave limiter functions

V set,d ¼ V max cos θ, V set,q ¼ V max sin θ

ð3:36Þ

In this part, we take the example that the inverter is operated in rated state before the fault, its rated current is 335A and the BC-G fault occurs at t ¼ 0.5 s to verify the theoretical analysis. From Fig. 3.17, the fault current reaches 1.2 times of I* at t ¼ 0.5083 s, then current limiter acts and the inverter outputs three-phase symmetrical current. Then, at t ¼ 0.5113 s, the modulation wave reaches 0.5 times of Vdc, which will trigger the modulation wave limiter. It can be seen from the Fig. 3.16 that the calculation results of fault current are almost identical to the simulation results. Therefore, when modulation wave limiter works, the inverter is equivalent to a constant voltage source, and fault current is the result of the interaction between the positive sequence constant voltage source and the grid voltage source. In this condition, the closed loop system turns to an open loop system. To ensure the normal and reliable operation of a microgrid, such extreme situation should be avoided.

3.3

Transient Characteristics of Droop Controlled IIDG During Grid Fault

In this section, the transient fault characteristics of droop-controlled IIDG are presented. Firstly, fault models are constructed. Then, the transient process of the fault current is described by an iterative calculation. Based on that, the main

3.3 Transient Characteristics of Droop Controlled IIDG During Grid Fault

63

influencing factors of the fault current are analyzed, including the control system and nonlinear limiters.

3.3.1

Fault Models

Similar to V/f controlled IIDG, droop-controlled IIDG can be equivalent to a positive voltage source in series with output impedance, according to the same block diagram of voltage and current controllers. However, due to the existence of power controller, when a fault occurs, the output power of inverters would increase which leads to a decrease of the reference voltage, as indicated in Eq. (3.5). And the decrease of the reference voltage amplitude and angular frequency will further change the output power. The relationship between output power and voltage is called coupling in this chapter and this coupling distinguishes fault model of droopcontrolled IIDGs from V/f controlled IIDGs. Therefore, a droop-controlled IIDG should be equivalent to a controlled positive voltage source in series with output impedance, as shown in Fig. 3.18. The amplitude and phase of equivalent voltage source is controlled by the outputs of power controller, which can be obtained by (3.37) and (3.38). 

ωc
V ¼ V  mq qQ s þ ωc   Z ωc

pP θ ¼ 2π ω  mp dt s þ ωc


ð3:37Þ ð3:38Þ

Obviously, it is different from V/f controlled inverter that the mathematical model of droop control includes the coupling characteristics (the coupling between output power and output voltage) of the power loop and the dynamics of low-pass filters. This makes this model more accurate and complex compared with the existing models and brings difficulties in calculating the fault current.

vod voq iod ioq

vod iod + voq ioq vod ioq – voq iod

p

ωc

s + ωc q

ωc

s + ωc

p q

ωo* – mp( p – P* )

ωo

1

s

θ ·

V Vo*– mq( q –Q* )

Fig. 3.18 Control block and equivalent circuit of droop-controlled IIDG

Vo

+ –

Zo

64

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

3.3.2

Fault Current Calculation

3.3.2.1

Mathematical Model of Fault Current

There is a similarity between droop-controlled and V/f controlled IIDGs that they are both equivalent to voltage sources in series with output impedance. As discussed in Sect. 3.2.2, fault current of an ideal voltage source includes a periodic steady-state component and an exponential damping component. It is the same for droopcontrolled IIDGs. The exponential damping component is irrelevant to the control scheme adopted by inverters and can be calculated by (3.25). While the periodic component shown in (3.24) needs to be determined according to the control scheme adopted by inverters. Specific to droop control, according to the droop characteristics, the amplitude and phase of the equivalent voltage source would decrease to a new steady-state value, which would lead to a decrease in the amplitude and phase of the periodic fault current. While the equivalent voltage source V is expressed as a constant in (3.24) and (3.25), the transient process of the asymmetric fault current could not be described. Thus, (3.37) and (3.38) should be combined to calculate the asymmetric fault current theoretically. Owing to the droop characteristics, the coupling relationship makes it difficult to describe the transient process of the asymmetric fault current. To accurately estimate asymmetrical fault current, the equivalent voltage source should be iteratively calculated to reflect the change in the equivalent voltage magnitude and phase, which will be discussed carefully in the following part.

3.3.2.2

Fault Current Estimation

To solve the coupling problem between the output power and the output voltage, iterative method is a feasible way, which is first proposed in [7] and used in many literatures in estimating fault current [8, 9]. In this section, an iterative mathematical method is introduced to obtain the amplitude V and phase θ of inverter equivalent voltage source, then the fault current can be estimated accurately. Assuming the fault duration is T, the iterative process is decomposed into n intervals. The time step ts should be selected based on (3.39), where tf is the time constant of low-pass filter.   min t v , t f ts < 10

ð3:39Þ

For convenience, the fault occurring moment is set as t ¼ 0. ts is set as 100 μs. The detailed steps of the iterative calculation are summarized below. Detailed steps of the algorithm are as follows: Step 1: Calculate the pre-fault output current and the voltage of the fault point Vfabc. Obtain the post-fault grid equivalent voltage Veq+, Veq and impedance Zt, and then calculate the attenuation time constant Ta.

3.3 Transient Characteristics of Droop Controlled IIDG During Grid Fault

65

Step 2: Calculate the instantaneous output active power Pk and reactive power Qk of the inverter by substituting the voltage and current at previous moment into instantaneous power formula. pffiffiffi  k1  k1   k1   3  k1 k1 k1 k1 k1 ioa þ voc iob voa  vk1 Q ¼  voa ob ioc þ vob  voc 3 k

ð3:40Þ

Step 3: Using (3.41) to calculate output real power and reactive power filtered by the first-order low-pass filter. þ Pkf ¼ Pk1 f

  ts  k ts  k P  Pk1 , Qkf ¼ Qk1 þ Q  Qk1 f ts þ t f ts þ t f

ð3:41Þ

Step 4: According to the droop characteristics, the phase increment Δθk and voltage increment ΔVk with respect to the pre-fault voltage V0 in the kth interval can be derived by (3.42) and (3.43). Thereby, Vk and θk can be obtained.   Δθk ¼ mp Pk  P
 2πt s   ΔV k ¼ mq Qk  Q
θk ¼ θk1 þ Δθk V k ¼ V 0 þ ΔV k

ð3:42Þ ð3:43Þ

Step 5: The voltage source vk is determined by calculated voltage source amplitude Vk and phase angle θk. Substituting the voltage source into (3.24) and (3.25), the periodic component and dc component of the asymmetric fault current is obtained. Step 6: Substituting (3.24) and (3.25) into (3.23), the asymmetric fault current iok in kth interval is obtained. Then, return to Step 2 until the fault is cleared. In this way, the post-fault current at any time can be calculated. The detailed flow chart for fault current estimation is shown in Fig. 3.19. In order to validate the correctness of proposed calculation algorithm, the system with droopcontrolled IIDG in Fig. 3.20 is simulated in PSCAD/ETMDC. Specific parameters are shown in Table 3.2. Figure 3.20 shows simulation results and calculation results of the fault current of droop-controlled IIDG when grid is subjected to single-phase-to-ground (A-G), two-phase-to-ground fault (BC-G) and phase-to-phase fault (B-C), respectively. The calculation results with the proposed algorithm and the simulation results are consistent, which validate feasible of proposed fault current calculation method.

66

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

Fig. 3.19 Flow chart of fault current estimation of droop-controlled IIDGs

START

k=0, K=K0 Obtain Veq+, Zt, VeqCalculate Ta Calculate P k and Q k using (3.40) Calculate Pfk and Qfk using (3.41) Calculate V k and θ k using (3.42) and Substitute V k and θ k into (3.24) and (3.25) k=k+1

k ωmin ω0 < ωmin

ð3:44Þ

V
 m q ðQ  Q
Þ V min

V 0 > V min V 0 < V min

ð3:45Þ

However, nonlinear limiters in power loop would not affect the operation of voltage and current control loops. Thus, the only difference between the equivalent fault models with or without power loop limiter is the amplitude and angle of the equivalent voltage source. Its equivalent fault models are shown in Fig. 3.25, which are the same as the situation without limiter. Figure 3.26 shows the fault response of droop-controlled IIDG when droop limiter is considered. In this condition, it can be observed that the amplitude of fault current is larger at steady state because limiters prevent inverter internal voltage

3.4 Transient Characteristics of VSG Controlled IIDG During Grid Fault

71

decreasing to a value lower than the threshold. The two results are almost identical. Therefore, when droop limiter is considered, the inverter can be equivalent to a controlled positive sequence voltage source in series with output impedance. However, as to the short-circuit fault, usually, the droop limiter will become invalid because the current limiter will work firstly for its fastest response performance. Only for a system with large droop coefficients, the droop limiter may work in a case where the fault current is not large enough to trigger the current limiter, but this situation is not common in practice. It should be noted that the output signal of power control loop is delivered to the inner voltage and current control loops. If modulation wave limiters and current limiters in inner loops are triggered, the fault characteristics of droop-controlled IIDGs are the same with that of V/f-controlled IIDGs.

3.4

Transient Characteristics of VSG Controlled IIDG During Grid Fault

In this section, the transient fault characteristics of VSG controlled IIDG are studied. Firstly, fault model of VSG controlled IIDG is derived. Then, the transient process of the fault current is described by iterative calculation. The main influencing factors of the fault current mainly are control loops of the inverter, fault type, line impedances, and fault occurring moment.

3.4.1

Fault Models

From the analysis in Sect. 3.3, droop control regulates the grid voltage and frequency by shaping the internal potential of the inverter. Therefore, the same as droopcontrolled IIDG, VSG controlled IIDG can be viewed as a controlled voltage source in series with LC filter. The phase and amplitude of voltage source are determined by VSG controller, as shown in (3.46) and (3.47). τv and τf are the time constants of the voltage-drooping and frequency-drooping loops, respectively.  1 Tm  Te _
J þ θ ,τf ¼ J D D s þ 1 P P DP 
1 Q Q K K _ fif ¼  þ V
, τv ¼ V ¼ θM
K _ D s þ 1 q Dq θ Dq θ_ _ θ_ ¼

ð3:46Þ ð3:47Þ

Dq θ

Equation (3.48) represents the output potential of VSG. where, va, vb, and vc are the instantaneous values of the output potential. Obviously, the output potential is three-phase symmetric, which is similar to the induced excitation electromotive force due to the rotor winding in the SG. In addition, given that the current flowing

72

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

Fig. 3.27 Control block and equivalent circuit of VSG

· +θ *

Dp – *

P

1 Tm · θ* +

+ – Te

1 Js

·

1 θ s

θ

io

Eq.(3.6)

Q* +

Q –

Z

v 1 Ks

·

Mf if

V

θ

V θ

– Equivalent circuit

+ VSG control block

+

–

Dq + V*

into the filter capacitor from the inductor is significantly less than the inductor current, the filter capacitor current can be ignored in the fundamental frequency calculations. va ¼ V sin θ, vb ¼ V sin ðθ  2π=3Þ, vc ¼ V sin ðθ þ 2π=3Þ Z o ¼ R f þ jωL f

ð3:48Þ ð3:49Þ

Thus, to simplify the analysis in this study, the filter capacitor Cf is not considered. Based on that, the output resistance Rf and filter inductor Lf are regarded as the equivalent output impedance Zo of VSG, as shown in (3.14). Therefore, the DGs with the virtual synchronous control are equivalent to a symmetric controlled voltage source in series with an equivalent output impedance, as shown in the right part of Fig. 3.27. It can be seen that the equivalent model of droop-controlled IIDG and VSG controlled IIDG is consistent, but the values of the two equivalent models are different. For droop control, the equivalent impedance is usually quite small and is depended on the parameters of inner voltage and current control loops. Thus, the equivalent voltage source approximately equals to capacitor voltage, which is determined by the outputs of droop controller. However, for VSG controlled IIDG, the equivalent voltage source is actually the inverter internal voltage. And the equivalent impedance equals to the impedance of LC filter.

3.4.2

Fault Current Calculation

3.4.2.1

Mathematical Model of Fault Current

According to the analysis in Sect. 3.4.1, VSG controlled IIDG is similar to droopcontrolled IIDGs that they are both equivalent to a controlled positive voltage

3.4 Transient Characteristics of VSG Controlled IIDG During Grid Fault

73

sources in series with output impedance. As discussed in Sect. 3.3.2, fault current of a positive voltage source controlled by droop characteristics includes a periodic steady-state component and an exponential damping component. The exponential damping component is irrelevant to the control scheme adopted by inverters and can be calculated by (3.25). While the estimation of periodic component shown in (3.24) needs to be calculated iteratively since the amplitude and phase of the equivalent voltage source would decrease to a new steady-state value. Same with droop-controlled IIDG, to accurately estimate asymmetrical fault current, the equivalent voltage source of VSG controlled IIDG should be iteratively calculated to reflect the change in the equivalent voltage magnitude and phase, which will be discussed carefully in the following part.

3.4.2.2

Fault Current Estimation

Given that the angular frequency gradually stabilize to the grid frequency and the angular frequency deviation is small enough to be ignored during transient fault period. Step 1: Calculating the output active power, electromagnetic toque, and reactive powers at the present moment by the voltage source and current of VSG at the previous moment, as shown in (3.50). pffiffiffi  k1  k1   k1   3  k1 k1 k1 Q ¼ via  vk1 ioc þ vib  vick1 ik1 iob ib oa þ vic  via 3 k

ð3:50Þ

Step 2: Obtaining the attenuation rate of the field excitation and angular frek k1 quency, (dMfif/dt)k and €θ by substituting Qk, Vk1, T ke and θ_ into (3.7) and (3.8), respectively. k Step 3: Integrating (dMfif/dt)k and €θ , the variations on the field excitation and k angular frequency ΔMfifk, Δθ_ can be obtained, as shown in (3.51). And, the variation of the phase angle can be also described by (3.52). Then, the field excitation, angular frequency and phase angle in kth interval can be expressed as (3.53)  k ΔM f i f k ¼ dM f i f =dt t s Δθ_ ¼ €θ t s k

k

Δθk ¼ Δθ_
t s k

ð3:51Þ ð3:52Þ

74

3 Transient Characteristics of Voltage Controlled IIDGs During Grid Fault

Fig. 3.28 Fault current estimation flow chart of VSG

START

k=0, K=K0 Calculate P k and Q k using (3.50) Substitute V k, Q k, Te k into (3.7) and (3.8) Substitute into (3.51) (3.52 )and (3.53) Calculate V k using (3.54) Substitute V k into (3.24) and (3.25) Substitute (3.24) and (3.25) into (3.23) k=k+1

k 2Irated Max(ioa,iob,ioc) > 2Irated

FRT control methods Virtual impedance Control mode switching

current value of VSG controlled IIDGs and voltage sag depth, the appropriate FRT control method can be selected to ride through the faults. It is known that semiconductor devices are chosen to output two times current of the rated value at most. When the voltage dips is not deep, the inrush current of VSG is not serious so that virtual impedance control can limit output current effectively in this condition. However, when the PCC voltage drops to a low value, the inrush current is large and VSGs are needed to be switched to current control mode to avoid the damage of VSGs. From engineering experiences, the preset value to judge the degree of voltage drop is 0.7 times of the rated value. The corresponding relationships are shown in Table 4.1. In case of shallow voltage sag and small fault current of inverter, the current limiting method based on virtual impedance is adopted to maintain the voltage control mode of VSG controlled IIDGs. For the case of deep voltage sag and large fault current of inverter, the fast inrush current restraining method is adopted to protect the inverter and output additional current to support the grid. In this section, the control principles and specific realization processes of the current limiting method based on virtual impedance and fast inrush current restraining method based on control mode switching are introduced, respectively.

4.3.1

Current Limiting Control Method Based on Virtual Impedance

4.3.1.1

Control Principle

The current limiting control method based on virtual impedance reduces the inner electric potential of inverter to limit the fault current by adding a virtual impedance in the control loop. The implementation algorithm of the virtual impedance current limiting control is shown in Fig. 4.5. In normal operation, the inverter operates in VSG control mode. When the grid fault occurs, S1 switches to 2, and the virtual impedance is added to limit the fault current. The criteria for fault detection are shown in (4.13). During the fault period, the output current of inverter changes sharply, which brings wide-band signals, while the virtual inductance has a certain amplification effect on the high-frequency harmonics. Therefore, adding low-pass filter (LPF) to filter high-frequency harmonics and enhance the ability to resist highfrequency disturbances will also bring about the delay of current limiting protection. After a lot of simulation, the cut-off frequency of LPF is 3000 Hz, which can

88

4 Fault Ride Through Control Methods of VSG Controlled IIDGs +

Dp Tm

1 θ&

P*

+

n

+ -

θ&*

-

1

1 θ&

Js Te

θ

s

Eqn.(3.6)

Q

1

Driver Signals

S1 PWM

Q

+

Virtual impedance

-

+

1

1

Ks

Rv+ sLv

io

2

v +-

*

LPF

Synthesize Dp +

Amplitude detection

V

V*

Inrush-current detection io

Amplitude detection vp

Fig. 4.5 The control block diagram of current limiting method based on virtual impedance

improve the power quality without affecting the control effect of the current limiting based on virtual impedance.

max ðioa , iob , ioc Þ  ith V m  V th

ð4:13Þ

From the analysis in Chap. 3, it can be seen that the fault current of VSG controlled IIDGs mainly includes a decaying DC component and a periodic component. The amplitude of the DC component is closely related to the fault occurring moment and the line impedance, attenuated by the time constant Ta ¼ L0/R0, and the peak value of fault current is mainly related to the DC component. Therefore, the current limiting control based on virtual impedance should not only limit the periodic component of fault current, but also limit the decaying DC component. By increasing the resistive virtual impedance, the decay time constant of the DC component is changed to (4.14). T a ¼ ðL0 þ Lv Þ=ðR0 þ Rv Þ

ð4:14Þ

And the periodic component of the fault current can be described as: ip ¼

pffiffiffi j Ε  V0g j 2 0 sin ðωt þ φ00 Þ Z eq þ Z v

ð4:15Þ

Therefore, by reasonably increasing the virtual impedance, the decay rate of the DC component can be accelerated, meanwhile, the periodic component amplitude of the fault current can be limited. Additionally, the VSG control is less sensitive to line

4.3 Fault Ride Through Control Methods of VSG Controlled IIDGs

89

impedance characteristics, and virtual impedance with increased virtual resistance can still maintain system stability. Therefore, the current limiting based on virtual impedance does not need to consider power coupling of VSG control. 8 V m  0:9V  > < 0,   Lv ¼ K L ðV   V m Þ 1  et=T , 0:5V  < V m < 0:9V  >   : V m  0:5V  0:5V  K L 1  et=T , 8 V m  0:9V  > < 0,   Rv ¼ K R ðV   V m Þ 1 þ et=T , 0:5V  < V m < 0:9V  >   : V m  0:5V  0:5V  K R 1  et=T ,

ð4:16Þ

ð4:17Þ

Moreover, since the grid-connected system often suffers from different degrees of short-circuit faults, the virtual impedance needs to be adaptively selected according to the voltage drop depth, as shown in (4.16) and (4.17). When the output voltage of inverter is greater than 0.9 V*, the inverter operates normally. When the output voltage drops below 0.5 V*, the virtual impedance is maintained at the maximum value in order to maintain the system stability and maintain the power transfer capability of the inverter. When the voltage drops below 0.9 V* and is greater than 0.5 V*, the virtual impedance is adaptively selected, according to the output voltage drop depth of inverter. KL and KR represent virtual reactance and virtual resistance coefficient, respectively. In order to reduce the decay time constant of the DC component of fault current, KL and KR meet (4.18). K R ¼ 2K L

ð4:18Þ

In addition, in order to further accelerate the decay of the DC component, the initial value of the virtual resistance is twice the steady-state value and gradually decreases with the time constant T. And, the virtual reactance increases gradually from zero to the steady-state value with the time constant T, as shown in (4.16) and (4.17). When the time constant is very small, it is equivalent that the high line resistance can be changed to the low line resistance instantaneously, which may have a great impact on the microgrid. Therefore, the time constant T cannot be too small, meanwhile, it cannot affect the operation of the inverter during steady state fault. Here, T is twice the line frequency period. The current limiting control based on virtual impedance limits the fault current by adding virtual impedance. In order to ensure the microgrid stability and power transmission capability, the value of virtual impedance has an upper limitation. Therefore, in the case of deep voltage drop and large inrush current, the current limiting method based on virtual impedance is difficult to restrain the inrush current.

90

4.3.1.2

4 Fault Ride Through Control Methods of VSG Controlled IIDGs

Experiment Results

To verify the aforementioned current limiting method based on virtual impedance, the grid-connecting system shown in Fig. 4.6 is built in the control hardware-in-loop (CHIL) experiments. The parameters of the VSG controller and the network system are shown in Tables 4.2 and 4.3, respectively.

A. Hardware Implementation of Fault Detection The fault detection of the current limit methods can be implemented by increasing few extra hardware circuits like resistors, RC filter and comparators with only two subroutines for the control program in the DSP controller. Therefore, the cost of the introduced method is very low and easy to be accepted by the industrial application. Figure 4.7 shows the fault detection circuit.

Fig. 4.6 Typical distributed network structure

V& g

PCC SL

Grid

Lg

Rg

Lf

Rf

Ll

Rl

Load

E& s

VSG

Table 4.2 Parameters of the experimental system Parameters Nominal frequency V* Lf Rf Lload

Values 50 Hz 310 V 4 mH 0.1 ohm 46.219 mH

Parameters DC-bus voltage Vg Lg Rg Rload

Values 800 V 310 V 0.6 mH 0.1 ohm 14.52 ohm

Table 4.3 Parameters of the controller Parameters DP fc (carrier frequency) P*(rated active power)

Values 12.665 6.4 kHz 5 kW

Parameters Dq KL/KR Q*(rated reactive power)

Values 160.7061 0.026/0.0065 5 kvar

4.3 Fault Ride Through Control Methods of VSG Controlled IIDGs

91

Fig. 4.7 Fault detection circuit

Vcc Iprotect From Current Hall sensor R1 Rsample

R2

DSP

LM393 C1 -Iprotect LM393

B. Experiment Verification of the Current Limiting Method Based on Virtual Impedance Before the VSG is connected to the grid, a phase-locked-loop (PLL) and a virtual load were needed for the preliminary synchronization operation [3]. It is switched to the grid-connected mode at t ¼ 3.00 s, and the PLL and virtual load mentioned previously are removed. When t ¼ 4.00 s, a three-phase symmetry short-circuit fault occurs on the load line. At t ¼ 4.6 s, the fault is cleared. The comparison of the conventional VSG and the VSG with the current limiting method based on virtual impedance is shown in Fig. 4.8. Figure 4.8(a) shows the experimental result of three-phase current when the conventional VSG is adopted. The peak current is 60 A, which is about three times the rated current. When the fault is cleared, the inverter and grid maybe damaged by the overshoot current. Figure 4.8(b) shows the experimental result of the three-phase current of the VSG with the current limiting method based on virtual impedance. The peak current is 40 A. That is just what we set in the control unit. In addition, the speed of detecting and restraining is very fast. About only 10 μs after the current went beyond the limit of 40 A, which is about two times the rated current, and the inrush current is restrained. Figure 4.9 shows the dynamic results of the output current during restraining process and after fault clears. We can see that during the restraining process, a small over current is generated. After the fault clears, there is no inrush current, and the oscillation is very small.

4.3.2

Fast Inrush Current Restraining Method Based on Control Mode Switching

The large inrush current caused by a grid fault will threaten the safety of the inverter. Moreover, in Chap. 3, it is concluded that the VSG has large instantaneous inrush current and fast response characteristics. The fault current can inrush to the maximum allowable current of power electronic devices within a few microseconds. Therefore, it requires a method that can restrain VSG instantaneous inrush current

4 Fault Ride Through Control Methods of VSG Controlled IIDGs

Fig. 4.8 Experimental results of (a) fault current with current limiting method based on virtual impedance, (b) fault current without current limiting method based on virtual impedance

(a) Fault clears

20A/div

92

Fault occurs 125ms/div (b)

20A/div

Fault clears

Fault occurs

125ms/div

very fast. On the other hand, as shown in Sect. 4.3.1.2, limiting the inrush current in controller is an effective and efficient method, but it is difficult to be implemented by the indirect current control. Based on the above reasons, a very fast inrush current restraining method based on control mode switching is introduced to restraining the instantaneous inrush current [4].

4.3.2.1

Control Principle

The hysteresis loop control can be used to restrain the rapidly rising fault current during fault period for its quick response to disturbances and incomparable advantages for restraining the inrush current. The control block diagram of the FRT control method is shown in Fig. 4.10. Related operation modes in chronological order are shown in Table 4.4. The operation strategy is as follows: 1. When the grid is in the normal state, the VSG controlled IIDGs operate in the VSG control mode (PD–QD–mode).

4.3 Fault Ride Through Control Methods of VSG Controlled IIDGs

93

(a)

10A/div

Fault occurs

25ms/div

(b)

10A/div

Fault clears

25ms/div

Fig. 4.9 Dynamic results of (a) fault current during restraining process, (b) output current after fault clears

P* 1 Pset

2 S2

Tm

1 θ&

+

n

Q* 1 Qset 2

1 Js

Te

e

-

S4 +

1 s

θ&

Eqn.(3.6)

Q

θ

PWM generation

1

+

S1

2 Synthesize

iset

1 Ks

+

θ&*

-

+ -

+

Dp

Sp

-

Inrush-current detection

hysteresis S3

Amplitude detection vp

1

io 1

Sq

-

Dp +

Amplitude detection

2 V

V*

Fig. 4.10 FRT control method for VSG controlled IIDGs

is

Rs + sLs

- vp

v +

94

4 Fault Ride Through Control Methods of VSG Controlled IIDGs

Table 4.4 Operation modes of the introduced FRT control method State (1) (2) (3) (4)

S1, S3 1 2 2 1

S2, S4 1 2 2 1

Sp, Sq ON OFF OFF ON

Grid fault NO YES NO NO

Mode PD–QD–mode H–PQ–tracking–mode PQ–tracking–mode PD–QD–mode

2. When a grid fault occurs, the VSG controlled IIDGs switch to the hysteresis control. At the same time, a virtual impedance is introduced to connect the VSG controlled IIDGs with the grid for tracking the hysteresis current. (H–PQ– tracking–mode). 3. When the grid fault is cleared, the VSG controlled IIDGs should wait for a short time until the virtual current is drives very close to io. (PQ–tracking–mode). 4. When the VSG controlled IIDGs re-switches to the PD–QD–mode, the output current adjusts to the normal state without impact.

4.3.2.2

Instantaneous Inrush Current Restraining

In Fig. 4.10, Pset, Qset are the references values of active and reactive power in PQ tracking mode. Sp, Sq, S1, S2, S3, and S4 are mode change switch. iset is the reference of the hysteresis current. When the grid is in the normal state, Sp and Sq are switched on. S1, S2, S3, and S4 are all connected to position 1. Therefore, the VSG controlled IIDGs operate in the PD–QD–mode. When a serious grid fault occurs, in order to detect the faults and trigger the mode switch to H–PQ–tracking–mode fast, three phase instantaneous current should be detected. When any phase of the inrush current output by the VSG controlled IIDGs is larger than the protection reference value, Sp and Sq are switched off. S1, S2, S3, and S4 are all switched to position 2. The VSG controlled IIDGs operate in the H– PQ–tracking–mode. The design method of the reference current phase A for hysteresis control is shown in (4.19). The reference of Phase B and C can be obtained via rotating the phase A by 120 degrees and 120 degrees, respectively.   isetA ¼ I set  sin θg þ δ

ð4:19Þ

Here, the magnitude of the reference current Iset is 2.5 times of the inverter rated current. θg is the angle of phase A of PCC voltage. The phase of the reference current depends on both the voltage phase of phase A and the reactive power compensation requirement δ. δ is designed by the reactive power insert rate rules in the IEEE std. 1547(TM). The instantaneous inrush current of VSG controlled IIDGs is restrained via a hysteresis controller to track the reference current. The principle of the hysteresis

4.3 Fault Ride Through Control Methods of VSG Controlled IIDGs Fig. 4.11 Principle of hysteresis control: (a) Hysteresis band, actual and reference currents, (b) Gate signal

95

Hysteresis band

(a)

Upper band

Actual curent

Lower band

Refrence curent 0

π /4

π /2

3 π /4

π

ωt [rad]

(b)

0

π /4

π /2

3 π /4

π

ωt [rad]

Fig. 4.12 Design method of Pset and Qset

θ

i oa i ob i oc

abc /dq

i od i oq

1.5 θ& ⋅ Mf i f 1.5 θ& ⋅ Mf i f

P set Q set

control is shown in Fig. 4.11. When the actual current is below the lower boundary of a hysteresis band (HBD) around the reference current, the upper switch is turned on and the lower switch is turned off, which causes the actual current to increase. When the actual current exceeds the upper boundary of the HBD, the upper switch is turned off and the lower switch is turned on, which causes the current to decrease. As a result, the PWM signal for the upper switch is generated as follows 8 1 > < ON if i < isetA  HBD 2 upper switch ¼ > : OFF if i < isetA þ 1 HBD 2

ð4:20Þ

The PWM signal for lower switch is complementary and the PWM signals for upper switch and the power electronics switches of the phase B and C can be determined accordingly.

4.3.2.3

Smooth re-Switching Control

In Fig. 4.12, Rs’ + sLs’ is a virtual per-phase impedance connect the VSG with grid whose value is equal to the equivalent impedance from the inverter to the PCC, and the virtual current is is

96

4 Fault Ride Through Control Methods of VSG Controlled IIDGs

is ¼

  1 0 v  vp Rs þ sLs 0

ð4:21Þ

Pset and Qset are adjusted as (4.22) to drive the is to i. (

  _ f i f isd Pset ¼ 1:5 ud isd þ uq isq  1:5ud isd ¼ 1:5  θM   _ f i f isq Qset ¼ 1:5 ud isq  uq isd  1:5ud isq ¼ 1:5  θM

ð4:22Þ

When Pset and Qset forming according to Fig. 4.12 are set as the references of VSG, and switch it to the H–PQ–tracking–mode or PQ–tracking–mode. By (4.22) and Fig. 4.12, if it is stable, isd  iod, isq  ioq. Thus, the current in the virtual impedance can be consistent with the output current of the VSG, which will be very helpful for the re-switching process. isd and isq are d-axis current and q-axis current of is, respectively. iod and ioq are d-axis current and q-axis current of io, respectively. It can be obtained by PCC voltage amplitude detection when the grid fault is cleared. Then, the VSG will switch back to the PD–QD–mode. The operation process is as follows: At first, the VSG operates in PQ–tracking–mode and should wait for a short time until is is driving very close to io, then S1, S2, S3, and S4 are switched to position 1, and Sp, Sq are switched on. For is is very close to io, the impact during the re-switching process can be restricted. A nonimpact re-switching process is realized. In fact, it will affect the re-switch performance when there are differences between Rs’, Ls’ and Rs, Ls, respectively. However, in simulation and experiment results, it is found that the influence is small.

4.3.2.4

Experiment Results

To demonstrate the effective and excellent performance of the FRT control method, the CHIL experimental results during grid fault are carried out. The system parameters are the same as shown in Tables 4.2 and 4.3. The fault detection of the FRT method is also implemented by increasing few extra hardware circuits, as shown in Fig. 4.7. Figure 4.13 shows a comparison of the conventional VSG and the VSG with the introduced control strategy. The three-phase symmetry short-circuit fault occurs on the load line and this result in the voltage of PCC dropping to 80%. 0.6 s later, the fault is cleared. Figure 4.13(a) shows the experimental result three-phase current of when the conventional VSG was adopted. The peak current was 113 A. When the fault is cleared, the inverter and grid are shocked by the overshoot current. Figure 4.13(b) shows the experimental result of the three-phase current of the VSG with the introduced control strategy. The peak current was 40 A. That is just what we set in the control unit. In addition, the speed of detecting and restraining is very fast. About only 10 μs after the current went beyond the limit of 40 A, the mode switched, and the current was restrained.

4.3 Fault Ride Through Control Methods of VSG Controlled IIDGs

(a)

Fault clears

40A/div

Fig. 4.13 Experimental results of the inrush current with restraining strategy and without restraining strategy: (a) inrush current without restraining strategy, (b) inrush current with restraining strategy

97

Fault occurs

250ms/div

(b) (2)

(3)

(4)

40A/div

(1)

Re -switch Fault clears Fault occurs

250ms/div

By comparing Fig. 4.13(a), (b), at the moment of fault occurs, it can be found that the peak of inrush current of conventional VSG reached 7.27 times the rated current, it is 113A. However, when the introduced control strategy is enabled, the peak of inrush current is stable at 2.5 times the rated current, it is 40 A, only about 35.40% of the conventional VSG, the effect is very significant. Figure 4.14 shows the dynamic results of the restraining and re-switching process. We can see that during the restraining and switching process, a small over current is generated. At a moment of re-switching, the output current is only a small amount of asymmetry, impact and oscillation is very small, it achieves a non-impact of the re-switch. Figure 4.15 shows the PCC voltage amplitude waveform, where the purple curve shows the PCC voltage amplitude of conventional VSG meeting with grid faults. Green curve shows the PCC voltage amplitude of the method introduced in this

98

(a)

ia

ic

ib

40A/div

Fig. 4.14 Dynamic results of (a) inrush current during restraining process, (b) inrush current during the re-switching process

4 Fault Ride Through Control Methods of VSG Controlled IIDGs

Fault occurs

10ms/div (b)

40A/div

Re-switch

25ms/div

section has been utilized. By the waveform, for the traditional VSG, since it is removed by protection device protect device for the instantaneous overcurrent, the amplitude of PCC voltage drops to 245.8 V. With the introduced method of this section, the instantaneous inrush current of the VSG is limited in the safe range, thus the amplitude of PCC voltage drops to 252.6 V. It is about 6.8 V higher than the conventional VSG, with about 2.186% of the rated voltage of the PCC. As a result, the effectiveness of the introduced method has been proved. In this case, only a VSG is considered to support the grid voltage. Actually, the number of VSGs will be more than one, thus its support for the fault grid will be more obvious.

4.4 Summary

99

Fig. 4.15 Comparison of amplitude of PCC voltage during restraining

The proposed strategy is adopted

50V/div

Conventional synchronverter

100ms/div

4.4

Summary

In this chapter, the fault inrush current, the maximum withstanding time of VSG controlled IIDGs during grid fault and the difficulties in restraining its instantaneous inrush current are introduced. Based on that, two FRT control methods are introduced. Also, the experiment results are presented to demonstrate their effective and excellent performance. The conclusions are as follows: 1. The VSG is implemented without an inner current control loop. Thus, the output potential cannot change abruptly, and there is a large fault inrush current after short-circuit faults. 2. The peak time tp of fault current is approximately 5 ms, and the time required for the fault current to reach two times of the rated current tm is approximately within the range of one hundred of microseconds to 5 ms, which is the maximum withstanding time of the inverter after faults. 3. In case of small voltage sag and small fault current of inverter, the current limiting method based on virtual impedance can be adopted to limit fault current and maintain the frequency stability of VSG controlled IIDGs at the same time, especially in islanded microgrid. And the experiment results shows that fault current can be effectively limited. 4. For the case of deep voltage sag and large fault current of inverter, the FRT control method based on control mode switching can be adopted to limit the output current quickly and supporting the grid voltage simultaneously. It is realized by switching the VSG control to the hysteresis current control. 5. The operation of two introduced FRT control methods is simple because it only requires a few extra hardware circuits. Hence, the introduced FRT control methods can be economic and widely applicable. In particular, the FRT capability

100

4 Fault Ride Through Control Methods of VSG Controlled IIDGs

of IIDGs has positive significance to guide the security and planning of distribution grid simultaneously.

References 1. Q.-C. Zhong, T. Hornik, Control of Power Inverters in Renewable Energy and Smart Grid Integration (Wiley-IEEE Press, 2013). Translated into Chinese by Q.-C. Zhong et al, published by China Machine Press in August 2016 2. Q.-C. Zhong, G. Weiss, VSG contolled IIDGss: inverters that mimic synchronous generators. IEEE Trans. Ind. Electron. 58(4), 1259–1267 (2011) 3. Q.-C. Zhong et al., Self-synchronized VSG contolled IIDGss: inverters without a dedicated synchronization unit. IEEE Trans. Power Electron. 29(2), 617–630 (2014) 4. Z. Shuai, W. Huang, C. Shen, J. Ge, Z. John Shen, Characteristics and restraining method of fast transient inrush fault currents in VSG controlled IIDGs. IEEE Trans. Ind. Electron. 64(9), 7487–7497 (2017) 5. Z. Shuai, J. Ge, W. Huang, Y. Feng, J. Tang, Fast inrush voltage and current restraining method for droop controlled inverter during grid fault clearance in distribution network. IET Gener. Transm. Distrib. 12(20), 4597–4604 (2018) 6. L. He, Z. Shuai, X. Zhang, X. Liu, Z. Li, Z. J. Shen, Transient characteristics of synchronverters subjected to asymmetric faults. IEEE Trans. Power Del. 34(3), 1171–1183 (2019)

Chapter 5

Full-Order Modeling and Dynamic Stability Analysis of Microgrid

With more and more distributed renewable energy sources (DERs) and ES devices connected to the microgrid via DC-AC inverters, the simulation based on switching model is complicated and time-consuming. It is difficult to use traditional switching model to study transient response of microgrid. Detailed mathematical model are important method to carry out transient stability anlaysis and can help to reduce simulation time. In this chapter, the full-order modeling method of the inverter-based microgrid is discussed, including elements of the inverter, line cables, and load etc. Afterwards, numerical simulations are performed to verify the model’s accuracy through a typical microgrid example. Finally, the numerical bifurcation theory is also presented to study the parameter stability region of the inverter-based microgrid, followed by the hardware-in-loop verification.

5.1

Full-Order Modeling of Microgrid

In the commercial electromagnetic transient simulation software represented by Matlab/Simulink and PSCAD/EMTDC, mathematical models described by a switching function (i.e., a switch model) are usually employed. The switching model can achieve an accurate description of the inverter switching process and is suitable for electromagnetic transient simulation of microgrid. However, since the switching model contains high frequency components of the switching process, it is not suitable for designing the microgrid. Meanwhile, for ensuring the accuracy, the calculation step size should be much smaller than the switching period, which greatly increases the calculation duration of time domain simulation. For the dynamic stability analysis of the microgrid, the dynamics of different elements in the system should be considered carefully. The full-order model of the microgrid for dynamic stability analysis requires comprehensive dynamic variables at wide time scales [1–4]. However, conventional models used for power system stability analysis usually only include rotor motion of the synchronous machine on © Springer Nature Singapore Pte Ltd. 2021 Z. Shuai, Transient Characteristics, Modelling and Stability Analysis of Microgird, https://doi.org/10.1007/978-981-15-8403-9_5

101

102

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

the electromechanical time scale and excitation control on the electromagnetic time scale. Hence, the time scale range considered by the conventional power system model is no longer applicable to the microgrid [4–6]. Therefore, it is necessary to investigate the mathematical expressions of multiple types of DERs, loads and line network etc. in a microgrid and to establish a full-order model of microgrid for dynamic stability analysis.

5.1.1

Coordination Transformation for DERs

As shown in Fig. 5.1, the three-phase three-wire circuit is typical topology for DC-AC inverters, where ii, io, and uo are the filter current, output current and output voltage of inverter, respectively. Rf, Lf, Rg, and Lg are the filter impedance and grid side impedance, respectively. Dynamics of the intermittent distributed renewable energy can also be decoupled with the microgrid via inverter. Hence, the study of DERs that includes the primary renewable energy and ES are often simplified into a grid-side inverter in an AC microgrid. Since the switching frequency of inverter is usually much higher than the microgrid’s fundamental frequency, the high-frequency components are often ignored. Thus, the switching process can be described via duty cycle, namely the average model can be obtained. The average model has the advantages of clear physical meaning and intuitive expression. However, the physical quantities in an AC microgrid are AC timevarying variables. They cannot be directly used for stability analysis and parameter design. To address this issue, the microgrid model in the three-phase stationary coordinate system needs to be converted into a dq coordinate system. Through the 3s/2r transformation, the fundamental sinusoidal variables can be transformed into the DC variables. Thereby, a stable equilibrium point (EP) for stability analysis in the steady state can be obtained. Meanwhile, it is also more convenient for parameters design. The 3s/2r transformation with constant amplitude can be expressed as

Fig. 5.1 Topology of a three-phase inverter

vi

Rf

vo

Lf

Rg

Lg

DERs or ESS AC bus

n PWM

ii ,vo

g

Control System of Inverter

io

5.1 Full-Order Modeling of Microgrid

103

Fig. 5.2 3s/2r coordinates transformation

b

qi

Q

ωi

β

θ

di D

ωcom a

c

2


3 2 cos θ þ π 3 7
7 7 2 sin θ þ π 7 7 3 5 1 2


 2 cos θ  π 3
 2  sin θ  π 3 1 2

cos θ

6 6 26 T ¼ 6  sin θ 36 4 1 2

ð5:1Þ

where θ is the angle between the d-axis of the rotating coordinate system and the a-axis of the stationary coordinate system. Assuming that the rotational angular velocity of the system’s common dq coordinate system is ωcom, the variables on the rotating coordinate system (d-q)i can be expressed as (Fig. 5.2): 

xDi xQi



 ¼ Ti

xdi xqi



 ¼

cos β sin β

‐ sin β cos β



xdi xqi

 ð5:2Þ

Therefore, the coordinate transformation can turn the complex AC components into DC components, and lay the foundation for the modeling of various DERs.

5.1.2

Modeling of Inverters with Different Control Strategies

5.1.2.1

Modeling of Droop Controlled Inverter

The droop control inverter which mimics the primary drooping characteristics of a conventional SG is a typical voltage-controlled inverter [6, 7]. The basic principles of droop control have been introduced in the previous Chap. 3 and full-order mathematical model would be further elaborated to give insight to droop-controlled inverters in this chapter. The simplified control principle is shown in Fig. 5.3. As can be observed from Fig. 5.3, droop control can be realized by controlling internal voltage according to its output power. The whole control system can be divided into three parts: droop power control loop, voltage control loop and current control loop. The current ii,abc through the filter, the output voltage vo,abc of the filter

104

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid * ii,dq

* vo,dq

Voltage controller

Droop Controller

vi,dq Current controller

dq

vi,abc

ii,abc

Inverter

Rf

Lf

ii,dq

vo,dq

Bus

io,abc Lg

Cf

Rg

vo,abc abc

io,dq

Fig. 5.3 Block diagram of droop-controlled inverter

vod voq iod ioq

1.5(vodi iodi + voqi ioqi )

ω ωc P * ω - m p ( P - Pn ) ωc + s

1 s

1.5(voqi iodi – vodi ioqi )

ωc Q ωc + s

* vod

V * – mq Q

θ

Fig. 5.4 Block diagram of the power control loop

uod

iod u

* od

+ -

uod uoq * uoq

PI

++ –

* fd

i

ω nC f ω nC f +-

PI

++ +

Voltage control loop

ioq

i*fq

* fd

i i fd i fq

+–

i*fq

+–

PI

++ –

uid

ω nL f ω nL f PI

Current control loop

++ +

uoq

uiq

Fig. 5.5 Block diagram of the voltage and current control loops

capacitor and the output current io,abc to the grid side are measured and then delivered to the control system. The d-axis of the dq coordinate system coincides with the output voltage vector on the filter capacitor. By using coordination transformation, three-phase signals can be transformered into constant DC values. Figure 5.4 shows the power control block diagram of a typical droop controller. It can achieve power sharing of active and reactive power by P-f and Q-V drooping features. The droop equations can be described as 

ω ¼ ω  m p ð P  P  Þ vod ¼ V   mq Q

ð5:3Þ

where ω*, V* and P* are the rated angular frequency, voltage and active power, respectively. mp and mq are the active and reactive droop coefficients, respectively. P and Q are the instantaneous active and reactive, respectively. Figure 5.5 shows the block diagrams of the current and voltage control loops. As can be observed, the droop controller compares the filtered active power P and reactive power Q to the command values, and outputs the frequency command ωo* and voltage reference vo*. The inner-control loop is designed under the dq synchronous reference

5.1 Full-Order Modeling of Microgrid

105

frame for voltage tracking with zero steady-state error. The PI controller for voltage control compares the vo* and 0 to the capacitor voltage vod and voq, respectively, and outputs the current reference iid* and iiq*. The obtained current references are sent to the current controller where the measured current iid and iiq are compared to iid* and iiq*, respectively. Command voltage vid, viq are given via the current controller and transformed into abc reference frame for PWM generation. The first order low pass filter is typically used to filter high frequency components in the instantaneous power and to provide some control inertia for the droopcontrolled inverter. The state equations of first-order low-pass filter can be described as 8   dP > < ¼ ωc 1:5vod iod þ 1:5voq ioq  P dt   > : dQ ¼ ωc 1:5voq iod  1:5vod ioq  Q dt

ð5:4Þ

The state equation of inner control loop can be expressed as 8 dx1 > > > > dt > > > > dx > < 2 dt dx > > 3 > > > dt > > > > : dx4 dt

  ¼ K vi vod  vod   ¼ K vi 0  voq   ¼ K ci iid  iid
 ¼ K ci iiq  iiq

ð5:5Þ

where Kvi is the integral coefficient of the voltage PI controller, and Kci is the integral coefficient of the current PI controller. The filter inductor current reference output by the voltage control loop can be given as: (

  iid ¼ iod  ωC f voq þ K vp vod  vod þ x1   iiq ¼ ioq þ ωC f vod þ K vp 0  voq þ x2

ð5:6Þ

The voltage modulation value vi* output by the current control loop can be given: 8   < vid ¼ ωL f iiq þ K cp iid  iid þ x3
 : viq ¼ ωL f iid þ K cp iiq  iiq þ x4

ð5:7Þ

LCL filters are often used in high-power generation systems. Compared to L filters, LCL filters have smaller filter inductance. Meanwhile, the inverter can adopt a lower switching frequency to reduce power loss. Ignoring the high-frequency harmonic components of the PWM modulation and assuming that the inverter directly outputs the terminal voltage reference, the state equation of LCL filter can be given as

106

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

8  diid  > > ¼ R f iid þ vid  vod þ ωL f iiq =L f > > dt > > >  diiq  > > > ¼ R f iiq þ viq  uoq  ωL f iid =L f > > dt > > >  > dvod  > < ¼ iid  iod þ ωvoq C f =C f dt   dv > > > oq ¼ iiq  ioq  ωvod C f =C f > > dt > > >  diod  > > ¼ Rg iod þ vod  vbd þ ωLg ioq =Lg > > > dt > > > di > : oq ¼ Rg ioq þ voq  vbq  ωLg iod =Lg dt

ð5:8Þ

The full-order model of a single droop control inverter can be up to 12 order, which can be described by differential and algebraic equations, where (5.4, 5.5) and (5.9) are the differential equations of the model, (5.3), (5.6) and (5.7) are the algebraic equations.

5.1.2.2

Modeling of VSG

Inverters that are controlled as VSGs can compensate for the loss of inertia and the spinning reserve capacity that result from the reduction of SGs. The basic control principle has been introduced carefully in part II, Chap. 3. Here, only the electronic part of the VSG is given in Fig. 5.6. In order to combine the VSGs’ model to that of the microgrid, it is necessary to convert the three-phase AC variables into the DC variables in the dq coordinate system. Assuming the output voltage vector of the VSG is locked on the d-axis. Accordingly, (5.4), (5.5) and (5.6) are transformed to (5.9), (5.10), and (5.11) with the 3s/2r transform.

ΔT

Dp

P*

1 ω*

Tm

1 Js

+

+

Fig. 5.6 Block diagram of VSG

-

ω

ω* 1 s

θ

Te Eq.(3.6) e

Q 1 Ks

+ *

Q

+

io

Mfif + Dq

V* -

V

5.1 Full-Order Modeling of Microgrid

107

vid ¼ M f i f  ω

ð5:9Þ

3 T e ¼ M f i f  iid 2 3 Q ¼  ω  M f i f  iiq 2

ð5:10Þ ð5:11Þ

Assuming the output voltage vik is equal to reference vik*, the state equations of the LC filter are thus: diidk ¼ vidk  vodk  iidk Rsk þ iiqk ωk Lsk dt dilqk Lsk ¼ viqk  voqk  iiqk Rsk  iidk ωk Lsk dt dv C k odk ¼ iidk  iodk þ voqk ωk C k dt dvoqk Ck ¼ iiqk  ioqk  vodk ωk C k dt Lsk

ð5:12Þ ð5:13Þ ð5:14Þ ð5:15Þ

Adding three state equations of the control part, (5.16) to (5.18), the integrated nonlinear model of the VSG is built and presented by Eqs. (5.9) to (5.18). Jk

dωk Pk  ¼   T ek  Dpk ðωk   ωk Þ dt ωk

ð5:16Þ

dθk ¼ ωk dt

ð5:17Þ

dM f ifk 1  Qk þ Dqk ðV k   V k Þ  Qk ¼ Kk dt

ð5:18Þ

It should be noted that the subscript d or q represents the value of the amount of alternating current on the d-axis or q-axis. The magnitude of the output voltage can be expressed as V¼

5.1.2.3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vod 2 þ voq 2

ð5:19Þ

Modeling of Current Controlled Inverter

The current controlled inverters are usually adopted when renewable energy sources such as wind power or photovoltaic are connected to the microgrid. The control principle of the commonly used current-controlled inverter is given in Fig. 5.7, where ub is the PCC voltage, ui is the output voltage of the current control loop,

108 Fig. 5.7 Control block diagram of current controlled inverter

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

Pref Qref

Power control loop

ref iodq θ

Current control loop

ui,dq dq

ui,abc

Threephase inverter

uo,abc

Lg

Rg

PCC

io,abc

io,dq

ub,dq

ub,abc

PLL

θ abc

Fig. 5.8 Control block diagram of phaselocked loop

u ba u bb u bc

abc

u bd

– dq u bq +

K pllp +

K plli s

ω com

1

θ

s

0

superscript ref represents the rated value, subscript abc, and dq represent the threephase value and transformation value in dq coordinate respectively. As shown in Fig. 5.7, the d-axis is in phase with the grid voltage vector. PQ control can be achieved by tuning the active and reactive power injected into the microgrid through the d-axis and q-axis current components control. Moreover, PLL is a necessity for the directional vector control of the current-controlled inverter to achieve grid-connected operation. The dynamic characteristics of PLL units directly determine the dynamic performance of the current-controlled inverter. The synchronous reference frame PLL is often used as the frequency measurement when the grid voltage is balanced. The control block diagram is shown in Fig. 5.8. Where Kpllp and Kplli are the proportional and integral coefficients of the PI controller, respectively. When the grid vector is in phase with the d-axis, the q-axis variable ubq of the grid voltage will be 0. Therefore, ubq behaves as a DC component and will tend to zero when it is input to a PI controller with non-steady-state error. Afterwards, the voltage phase locking to the grid can be finished. At this time, the phase angle output by the PLL is the grid voltage’s A phase. Control system of the PLL in a dq coordinate system can be represented by a second-order dynamic equation system: 8 dθ > < ¼ K pllp ubq þ K plli xpll dt dx > : pll ¼ ubq dt

ð5:20Þ

where xpll is the state variable introduced by the PI controller’s integration link.

5.1 Full-Order Modeling of Microgrid Fig. 5.9 Control block diagram of phaselocked loop

109

Pn

×

ubd

÷

Qn

×

ubd

÷

ubd

i*od

+–

2/3

PI

iod

ω n Lf

ioq

ω n Lf

+–

2/3

PI

i*oq



8 2 Pn > _ > δ1 ¼ K ci  iod > > 3 ubd > > >

> >

  > > > uid ¼ ωLg ioq þ K cp iod  iod þ δ1 > > >
 > > : uiq ¼ ωLg iod þ K cp i  ioq þ δ2 oq

++ –

uiq

++ +

uiq

ubq

ð5:21Þ

Figure 5.9 is the block diagram of the inner current control loop of the inverter. When the phase-locking has been achieved by PLLs, the inner loop control can be represented by a four-order state differential equation system (5.21).

5.1.3

Modeling of Network

Since the dynamic response of DERs is much faster than that of SGs in the traditional power system, the dynamic response of the line current directly affects the dynamic behavior of DERs. Hence, it is necessary to consider the dynamics of the network in the microgrid model. Thereafter, the traditional node admittance matrices will no longer be suitable. At the same time, since the scale of microgrid is relatively small compared with conventional power grid, the state equation can be directly established to describe the dynamics of the line current. Considering length of the line cable is short, the dynamics of the line current between the network node i and j can be expressed by the state equation as (5.22). 8   di > < lD ¼ Rline ilD þ ubiD  ubjD þ ωcom Lline ilQ =Lline dt > : dilQ ¼ Rline ilQ þ ubiQ  ubjQ  ωcom Lline ilD =Lline dt

ð5:22Þ

where Rline and Lline are the equivalent resistance and inductance of the line, respectively. When introducing a state equation to describe the network, the node voltage needs to be defined separately. Supposing the system nodes are connected to

110

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

ground through a large resistor Rvir, the node voltages are described as (5.23) according to the KCL equation. ubDi ¼ Rvir ubQi ¼ Rvir


X
X

ioD  ioQ 

X X

iloadD þ iloadQ þ

X X

 ilD  ilQ

ð5:23Þ ð5:24Þ

where io, iload, and il represent the output current of DER, load current, and line current, respectively.

5.1.4

Modeling of Different Kinds of Loads

In a microgrid, many different types of loads are connected to the system. Loads are often treated as equivalent resistances in the grid model. However, the load dynamics directly affect the stable operation of microgrid. This section will discuss the model of static load, and ZIP loads in the microgrid framework, followed by the induction motor load model.

5.1.4.1

Modeling of ZIP Load

The ZIP load model is often used to describe the static load in the power system. The ZIP load model can be developed as the active and reactive power consumed by static load with change of voltage 8

U 2b Ub > > > < PZIP ¼ P0 a1 U 2 þ a2 U 0 þ a3 0

2 > U Ub > > þ b3 : QZIP ¼ Q0 b1 b2 þ b2 U0 U0

ð5:26Þ

where P0 and Q0 are the rated active and reactive power of the ZIP load, respectively. U0 represents the rated voltage. a1 and b1, a2 and b2, a3 and b3 represent the percentage coefficients of the rated active and reactive power of the constant impedance load, constant current load, and constant power load, respectively. According to the previous section on the line model, the line currents are used to describe the network dynamics in a microgrid, thus, (5.26) cannot be directly incorporated into the microgrid model. Since current is the input signal form of the network model, the ZIP load is equivalent here to a controlled parallel admittance. The equivalent conductance GZIP and susceptance BZIP can be expressed as

5.1 Full-Order Modeling of Microgrid

111

8

a1 a2 a3 > > > < GZIP ¼ P0 U 2 þ U 0 U b þ U 2 b 0

> b b b > 1 2 3 > þ þ : BZIP ¼ Q0 U 20 U 0 U b U 2b

ð5:27Þ

Thereafter, the active current component of ZIP load can be expressed as 8 di > < PD ¼ ðGZIP ubD  iPD Þ=τ dt > : diPQ ¼ ðGZIP ubQ  iPQ Þ=τ dt

ð5:28Þ

The reactive current component of the ZIP load can be expressed as 8 di > < QD ¼ ðBZIP ubQ  iQD Þ=τ dt > : diQQ ¼ ðB u  i Þ=τ ZIP bQ QQ dt

ð5:29Þ

where τ is the time constant of ZIP.

5.1.4.2

Modeling of Induction Motor Load

Dynamic induction motor (IM) loads are the important load type in power system. In order to reflect the electromagnetic transient of IM, the model containing the fourorder flux equations, and the one-order rotor motion equation are established. The stator flux linkage can be expressed in the dq coordinate system as 8 ψ sD > > > < ψ sQ > ψ rQ > > : ψ rQ

¼ Ls isD þ Lm irD ¼ Ls isQ þ Lm irQ ¼ Lr irD þ Lm isD ¼ Lr irQ þ Lm isQ

ð5:30Þ

where Lm is the linkage inductance, Ls and Lr are the stator inductance and rotor inductance, respectively. The voltage equations of IM are written as: 8 usD ¼ Rs isD þ ψ_ sD  ωcom ψ sQ > > > < usQ ¼ Rs isQ þ ψ_ þ ωcom ψ sQ sD > _ 0 ¼ R i þ ψ  ð ω  ωr Þψ rQ r rD com rD > > : 0 ¼ Rr irQ þ ψ_ rQ þ ðωcom  ωr Þψ rD

ð5:31Þ

112

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

Fig. 5.10 Bock diagram of paralleled IM loads and their equivalent model

IM load 1 ub TL

IM load 2 IM load k

is1 is2

Σ

ise

isk

ub kTL Dynamic equivalent method

IM equivalent model

ise

where Rs and Rr are the stator resistance and rotor resistance, respectively. Substituting (5.30) into (5.31) and the cooperating rotor motion equation, the dynamic model of the IM can be written as: 8 2  np dωr np Lm  > > isQ ψ rD  isD ψ rQ  T L ¼ > > dt JL J > r > > > > dψ 1 L rD > > ¼  ψ rD þ ðωcom  ωÞψ rQ þ m isD > > T dt Tr r > > > > dψ 1 L > > rD ¼  ψ rD þ ðωcom  ωÞψ rQ þ m isD > < dt Tr Tr dψ rQ 1 L > > ¼  ψ rQ  ðωcom  ωÞψ rD þ m isQ > > T dt Tr r > > > > > Rs L2r þ Rr L2m disD Lm Lm u > > ¼ ψ þ ω ψ  isD þ ωcom isQ þ bD com > rD rQ 2 > dt σL L T σL L σLs σL L s r r s r > s r > > > > > disQ Rs L2r þ Rr L2m ubQ Lm L > : ψ rQ  m ωcom ψ rD  isQ  ωcom isD þ ¼ dt σLs Lr T r σLs Lr σLs σLs L2r ð5:32aÞ where J is the rotor inertia coefficient. σ ¼ 1Lm2/LsLr denotes the leakage factor and Tr ¼ Lr/Rr Tr ¼ LrRr represents the rotor electromagnetic time constant. The load torque TL can be described by the following nominal as: T L ¼ T 0 þ T 1 ω þ T 2 ω2

ð5:32bÞ

In general, the incremental IM load at a given bus can be obtained by placing identical IMs in parallel. However, to track the increment of IM loads, the number of integrated IMs increases at the given bus, which will change the structure and increase the size of MG model. On the other hand, the paralleled IMs can be presented as a single-unit equivalent IM based on the dynamic equivalent method. By doing so, the continuous load increment can be presented as the parameters change of the equivalent IM. Figure 5.10 presents the signal flow of an IM model, the input signals of IM loads are bus voltage ub and load torque TL, stator current is is the output signal. To provide the same dynamic behavior with the original paralleled IMs, the dynamic equivalent model of IMs must meet the following requirements:

5.1 Full-Order Modeling of Microgrid

113

1. The voltage and current on the connected bus do not change. 2. The active and reactive power assumption of equivalent IMs must be equal to the sum of all the power assumption at the connected bus of paralleled IMs. 3. The electromagnetic time constant of the equivalent IM equal to that of the paralleled IMs. According to the first condition, the stator current and bus voltage of the equivalent IM should satisfy isDe ¼

X

isD ¼ KisD , isQe ¼

X

isQ ¼ KisQ

ð5:33Þ

As the stator voltage keeps unchanged, the stator and rotor flux of the equivalent IM are obtained as: ψ rDe ¼ ψ rD , ψ rQe ¼ ψ rQ , ψ sDe ¼ ψ sD , ψ sQe ¼ ψ sQ

ð5:34Þ

According to the second condition, the load torque and rotor angular velocity are: ωre ¼ ωr , T Le ¼

X

T L ¼ KT L

ð5:35Þ

From the third condition, the electrical parameters of the equivalent IM should meet the relationship below: T se ¼ T s ¼ Ls =Rs , T re ¼ T r ¼ Lr =Rr

ð5:36Þ

Substituting (5.33, 5.34, 5.35 and 5.36) into the flux and voltage Eqs. (5.32a and 5.32b), the parameters of the equivalent motor load rated at PIM ¼ KPnW can be described by an IM load factor K Lme ¼ Lm =K, Lse ¼ Ls =K, Rse ¼ Rs =K, Lre ¼ Lr =K Rre ¼ Rr =K, J e ¼ KJ, T Le ¼ KT L

ð5:37Þ

Therefore, based on a typical motor load rated at PnW, the parameters of the equivalent motor load rated at PIM ¼ KPnW can be described by (5.37). Thus, the continuous increment of IM load is described by increasing load coefficient K.

5.1.5

Full-Order Modeling of Microgrid

In order to combine sub-modules of various system elements that were originally built in their respective coordinate system into a global model, coordinate transformation is necessary [8–10]. The voltage-controlled DERs can directly control the amplitude and frequency of the output voltage. Hence it can be used as the main

114

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

Fig. 5.11 Signal transmission diagram of the overall model of microgrid

Voltagecontrolled inverter,i ω

Voltagecontrolled inverter,j

i o,i u b,i

com

Currentcontrolled inverter,i

i o,j u b,j

Load model,z

i o,k u b,k

i o,z u b,z

Model of line network in a AC microgrid

power source in a microgrid. On the contrary, the current-controlled inverters need to synchronize the voltage with PLLs and then realizes the output power control. Therefore, in the full-order model of microgrid, the output voltage’s frequency of one voltage-controlled inverter is usually taken as the system reference frequency ωcom, and the synchronous rotating coordinate system of this inverter is used as the common coordinate system as well. The current-controlled inverters, loads, and other elements that cannot directly control the frequency are directly modeled on the system common coordinate. The signal transmission of the microgrid model is shown in Fig. 5.11. The input signals of the inverter model are the bus voltage and the output signals are the current. Therefore, in order to integrate the voltage-controlled inverter model into the microgrid model, their output current signals need to be converted from the local coordinate to the common coordinate. The transformation equation is 

iDk iQk



 ¼ Tk

idk iqk



 ¼

cos θk sin θk

 sin θk cos θk



idk iqk

 ð5:38Þ

The bus voltage needs to be converted from the common coordinate to the local coordinate. The transformation equation is 

udk uqk

 ¼

T 1 k



udk uqk





cos θk ¼  sin θk

sin θk cos θk



uDk uQk

 ð5:39Þ

Through combining the state equations of each inverter, load and line network in the microgrid, the differential algebraic equations can be achieved to fully describe the dynamic behaviors of a microgrid.

5.2

Model Verification

In order to verify correctness of the dynamic modeling method mentioned in the previous section, a 9-node inverter-dominated microgrid model is established in this section. The mathematical model is compared with the detailed switching model built in the Matlab/SimPowerSystem environment to verify its accuracy. Topology

5.2 Model Verification Fig. 5.12 Topology of microgrid

115 645

633

632

634

IM

Droop 684

671

692

Droop

Syn

675

PQ

680 ZIP

Table 5.1 Droop-controlled DG parameters Parameter P1*, P2* V1*, V2* mp1, mp2 mq1, mq2 ωc Kvp1, Kvp2 Kvi1, Kvi2

Value 40 kW, 20 kW 330 V, 330 V 0.75e4 rad/W, 1.5e4 rad/W 5e4 V/Var, 7e4 V/Var 31.4 rad/s 1.2, 1 380, 275

Table 5.2 VSG parameters

Parameter P* V* Dp J Dq

Table 5.3 Current-controlled DG parameters

Parameter Pref Qref Kcp, Kci

Parameter Lf1, Lf2 Rf1, Rf2 Cf1, Cf2 Lg1, Lg2 Rg1, Rg2 Kcp1, Kcp2 Kci1, Kci2

Value 10 kW 330 V 10.61 0.1061 kg/m2 500

Value 10 kW 0 kVar 1, 200

Value 1.2 mH, 1.2 mH 0.1 Ω, 0.1 Ω 0.55 mF, 0.45 mF 0.6 mH, 0.4 mH 0.3 Ω, 0.2 Ω 4, 3.2 550, 480

Parameter Lf Rf Cf Lg Rg

Parameter Lf Rf Kpllp, Kplli

value 3.5 mH 0.1 Ω 25 μF 2.8 mH 0.25 Ω

value 3.5 mH 0.1 Ω 7, 100

of the microgrid is shown in Fig. 5.12, where four DERs, three types of loads are included. The main parameters of the DERs, loads and line cables are shown in Tables 5.1, 5.2, 5.3, 5.4 and 5.5. The equivalent series resistance and inductance of line cable are 0.17 Ω/km and 0.3 mH/km, respectively. Firstly, accuracy of the mathematical model to describe the slow dynamic process in the case of small disturbances is verified. When t ¼ 2 s, a 50 Ω grounding

116

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

Table 5.4 Line cable parameters

Length (m)a 150 50 80 95

Node A-B 632–645 632–633 632–671 671–680

Node A-B 633–634 671–684 671–692 692–675

Length (m) 135 90 180 120

Table 5.5 Load parameters ZIP and resistance load b1, b2, b3 c1, c2, c3 P0 Q0 R632, R633, R671

(b)

20 15

P[kW]

IM load parameter Lm Lr, Rr Ls, Rs J T0, T1, T2

P2

10

P3

5 0

1

P1 P2

10

P3

5

2 t[s]

0

3

Value 0.043927 mH 0.0449 mH, 0.228 Ω 0.0449 mH, 0.087 Ω 1.662 kg/m2 65, 0, 0

20 15

P1

P[kW]

(a)

Value 0.2, 0.3, 0.5 0.2, 0.3, 0.5 10 kW 5 kVar 17 Ω, 20 Ω, 50 Ω

1

2 t[s]

3

Fig. 5.13 Output active power of voltage-controlled inverter with step change of resistive load (a) Mathematical model; (b) Switching model

(a)

(b) 155.3 ωr [rads/s]

ωr [rads/s]

155.3 155.2 155.1 155

1

2 t[s]

3

155.2 155.1 155

1

2 t[s]

3

Fig. 5.14 Angular velocity of IM with step change of resistive load (a) Mathematical model (b) Switching model

resistance load is connected in parallel at node 671. The output power of the droop controlled and virtual synchronous controlled inverters after first-order filtering is shown in Fig. 5.13. The rotor speed of the IM is shown in Fig. 5.14. The output current of the current-controlled inverter is shown in Fig. 5.15. As can be observed in Figs. 5.13, 5.14 and 5.15, when the resistance loads increase, the output power of the voltage controlled DERs increase. Besides, due to the node voltage drop in the microgrid, the rotor speed of IM decreases. On the

5.2 Model Verification

(a)

117

(b)

30

25

10

io4 [A]

io4 [A]

20 0 -10

20

i o4 = iod2 4 + ioq2 4

15

-20 -30 1.8

1.9

2 t[s]

2.1

10

2.2

1

1.5

2 t[s]

2.5

3

Fig. 5.15 Output current of the current-controlled inverter with step change of resistive load (a) Mathematical model (b) Switching model

(a)

(b)

400

250 200

io1 [A]

io1 [A]

200 0

-200 -400 1.5

150 100

i o1 = iod2 1 + ioq2 1

50 2 t[s]

2.5

0 1.5

2 t[s]

2.5

Fig. 5.16 Output voltage response of the droop-controlled DER1 when a three-phase short-circuit fault occurs (a) Mathematical model (b) Switching model

contrary, the current-controlled DERs are not affected by load fluctuations, and the output current remains unchanged. Compared with the detailed switching model established in the Matlab/SimPowerSystem, it can be known that the mathematical model established in this section can accurately describe the slow dynamics of the microgrid. In order to verify the accuracy of the mathematical model to describe the fast dynamics in the microgrid under large disturbance, the three-phase grounding shortcircuit fault is triggered at node 633 at t ¼ 2 s. The short-circuit resistance is 0.5 Ω, and the fault is cleared after 0.05 s. When using the mathematical model and the detailed switching, the output currents of droop-controlled inverters model are compared in Figs. 5.16 and 5.17, while the output currents of VSG are compared in Fig. 5.18. As shown in Figs. 5.16, 5.17 and 5.18, the dynamic behavior of the microgrid under large disturbances can also be accurately described. Meanwhile, since the mathematical models use the DC variables to describe the AC signals in the detailed switching model, a larger calculation step size can be employed to achieve relatively accurate numerical simulation, which greatly reduces the computation burden.

118

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

(a)

(b)

200

200

io2 [A]

250

io2 [A]

400

0

150 100

-200 -400 1.5

i o 2 = iod2 2 + ioq2 2

50 2 t[s]

0 1.5

2.5

2 t[s]

2.5

Fig. 5.17 Output voltage response of the droop-controlled DER2 when a three-phase short-circuit occurs (a) Mathematical model (b) Switching model

(a)

(b)

100

60

io3 [A]

io3 [A]

50 0 -50 -100 1.5

80

40

i o 3 = iod2 3 + ioq2 3

20 2 t[s]

2.5

0 1.5

2 t[s]

2.3

2.5

Fig. 5.18 Output voltage response of the Syn when a three-phase short-circuit occurs (a) Mathematical model (b) Switching model

5.3

Parameter Stability-Region of Microgrid

It is known that simulations rely on the mathematical calculation of the switching model. It is time-consuming and can not give insight into the effects of different parameters on system stability. Mathematical model can be used to realize parameters optimization through theoretical analysis. As can been seen in the previous section, the full-order model of the microgrid has been established. Based on the established mathematical model, parameter-domain design would be carried out in this section. In practical case, multiple types of load are considered in microgrid system, such as resistance load, constant power load and IM load. To achieve stable operation, the parameter stability region of microgrid with multitype loads need to be predicted. Impedance analysis method and eigenvalue analysis method have been widely used to investigate the stability of the linearized system under a set of specific parameter configuration. However, when system parameters change, the equilibrium is accordingly changed and deviates from the predefined linearization point. Thereby, these two typical small-signal analysis methods are no longer applicable. This section extends the small-signal analysis of bifurcation theory to analyze the parameter stability region of the MG. Firstly, the concept and analytical methods of bifurcation theory are introduced. Secondly, the numerical bifurcation method is used to investigate the stability region of the MG with static ZIP load and dynamic IM load. Thirdly, the influence of control parameters is studied as well.

5.3 Parameter Stability-Region of Microgrid

5.3.1

119

Bifurcation Theory

Bifurcation theory is an important tool for studying the parameter stability of dynamic nonlinear systems. Compared with the small-signal analysis that analyzes the perturbation of a fixed equilibrium, the bifurcation analysis can perform parametric stability analysis that traces equilibrium solutions as the parameters change. Given a dynamic nonlinear system described by the state equation: x_ ¼ f ðx, μÞ

ð5:40Þ

where x is the state variables and μ is a set of bifurcation parameter vectors consisting of system parameters. The bifurcation phenomenon at μ0 is defined as: when μ changes continuously and passes through a critical value μ0, the system will undergo structural mutation. When the bifurcation parameter vector is μ0, the EP is called the bifurcation point, and all the bifurcation points will form the bifurcation boundary. In practical engineering applications, only partial bifurcation phenomenon at the EP is often studied. The partial bifurcation phenomenon will lead to the direct instability of the power system. Hence, the partial boundary will constitute the parameter stability boundary of the system. The numerical bifurcation analysis in this section focuses on the partial bifurcation of the microgrid and the system parameter stability domain formed by the partial bifurcation boundary. The common bifurcation phenomena in power system can be divided into static bifurcation and dynamic bifurcation. Generally, the Saddle Node Bifurcation (SNB) represents the static bifurcation phenomenon in which the system’s real eigenvalue at the EP changes from positive to negative during the continuous change of the bifurcation parameter. At the SNB point, the system has a zero eigenvalue and the system’s Jacobian matrix is singular. When SNB phenomenon occurs, the number of the system EPs changes, the stable EP disappears, and the system will directly collapse. The Hopf Bifurcation (HB) represents the dynamic bifurcation phenomenon in which the conjugate eigenvalue at the EP will cross the imaginary axis during the continuous change of the bifurcation parameter. At the HB point, the system has a pair of conjugate eigenvalues with a real zero. When the HB phenomenon occurs, the stable system EP disappears, the limit cycle appears, and the system begins to oscillate. The HB can also be classified into the supercritical HB and the subcritical HB according to the stability of the limit cycle. When the supercritical HB occurs, the system limit cycle is asymptotically stable and exhibits continuous oscillation. When the subcritical HB occurs, the system limit cycle is unstable. The unstable condition manifests itself as the oscillation with increasing amplitude or even unstable. The detection of the bifurcation function can be performed synchronously with the numerical iterative operation of the EP as the parameter changes. Assuming the detection function is j ¼ j(x). When the detection function has different symbols in

120

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

the two iterations, it is considered that the bifurcation phenomenon is detected. The detection functions of SNB and HB can be written as:   ∂f xsys , μ φSNB xsys , μ ¼ det ∂xsys  

  ∂f xsys , μ M I φH xsys , μ ¼ det 2 ∂xsys 

where,

5.3.2

L



ð5:41Þ ð5:42Þ

represents the Bialternate product, and I is the unit matrix of n-th order.

Parameter Stability-Region Analysis

A typical microgrid is taken as an example in this section to carry out parameter stability domain analysis with numerical bifurcation method. As shown in Fig. 5.19, the analyzed microgrid contains two droop-controlled inverters, static ZIP loads, and dynamic IM loads. The parameters of inverters and loads are shown in Tables 5.1 and 5.2 respectively. The line cable’s impedance is 0.164 + 0.082 Ω/km, and the line length is 0.5 km. This section uses MATLAB’s software package MATCONT to realize the numerical calculation of the bifurcation boundary of the nonlinear system.

5.3.2.1

Case 1: Numerical Bifurcation Analysis on mp-mq Plane

In the droop-controlled-inverter-dominated microgrid, the power control loop of the inverter directly affects the dominant dynamic behavior of the microgrid. Hence, the parameter stability domain of the power droop coefficients is investigated in this section. The influence of the control parameters and the circuit parameters on the system stability domain are also studied. In order to ensure the power sharing between the two DERs, the ratio of the active droop coefficients is kept at 1:2, and the ratio of the reactive droop coefficient is kept at 5:7. The initial point of the bifurcation analysis is the EP when the parameters in Tables 5.1 and 5.2 are applied. Figure 5.20 plots the bifurcation diagram of mp and mq, where the red curves denote the HB boundaries of test system, respectively. As shown, when the active Fig. 5.19 The microgrid with two DERs

Bus1 Droop DG1 Coupling impedance1 Droop DG2 Coupling impedance2

Bus2 Line impedance

M IM load ZIP load

5.3 Parameter Stability-Region of Microgrid Fig. 5.20 The parameter stability region on mpmq plane

121

20e-4 (28e-4)

mq1 (mq2)

15e-4 (21e-4) 10e-4 (14e-4)

Instable Region II Subcritical HB II

5e-4 (7e-4) 0

Stable Region I

1.5e-4 (3e-4)

3e-4 (6e-4)

Instable Region I 4.5e-4 (9e-4)

6e-4 (12e-4) mp1 (mp2)

7.5e-4 (15e-4)

9e-4 (18e-4)

10.5e-4 (21e-4)

60

Q1[kVar]

Fig. 5.21 Phase plane of the microgrid before and after subcritical HB. Red curve: stable operation; Blue curve: subcritical HB

Subcritical HB I

50 40 30 Start point 20

0

20

40

60

80

P 1 [kW ]

droop coefficient mp or the reactive droop coefficient mq increases to a certain value, the micro-grid will undergo subcritical HB phenomenon. Subcritical HB1 and HB2 are caused by excessive values of mp or mq, respectively. It can be seen that the two HB boundaries bound the parameter stability region on mpmq plane. The proposed Microgrid tends to lose its stability with the increase of mp or mq. The Lyapunov coefficients on the two HB curves are smaller than 0. Thus, increasing mp or mq will lead to the subcritical HB1 and HB2, respectively. Besides, increasing reactivepower droop gain mq extends the upper limit value of active power gains mp. While mp has little effect on the upper limit of mq and raise the lower limit of mq. The feasible region of mq narrows when mp increases. Thus, when a large mp is designed to improve transient response, a large mq should be selected to make sure the stability of microgrid. The phase plane of active power P1 and reactive power Q1 before and after subcritical HB1 are plotted in Fig. 5.21. The red curve denotes the trajectory of operation point before the subcritical HB1 when mp1 ¼ 4.5e4, mq1 ¼ 5e4 (mp2 ¼ 9e4, mq2 ¼ 7e4). The blue curve denotes the trajectory after the subcritical HB when mp1 ¼ 7.5e4, mq1 ¼ 5e4 (mp2 ¼ 15e4, mq2 ¼ 7e4). The subcritical HB1 gives rise to an unstable limit circle around the stable equilibrium. As illustrated, the trajectory after the HB diverges from the start point, and the system becomes oscillatory. Then, the impacts of the cut-off frequency ωc on the stability region of mpmq plane are studied. The cut-off frequency of two DGs are both changed. It can be seen

122

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

Fig. 5.22 HB boundaries for microgrid with different cut-off frequency mp1[rad/W] (mp2[V/Var])

30e-4 (42e-4) HB, ωc=15.7rads/s 20e-4 (28e-4) HB, ωc=31.4rads/s 10e-4 (14e-4) HB, ωc=47.1rads/s

Stable Region 0

Fig. 5.23 HB boundaries for microgrid with different coupling inductance

0

7.5e-4 (15e-4)

15e-4 22.5e-4 (30e-4) (45e-4) mp1[rad/W] (mp2[V/Var])

25e-4 (35e-4)

30e-4 (60e-4)

37.5e-4 (75e-4)

HB, Lg1=1.2mH Lg2=0.8mH

20e-4 (28e-4) mp1[rad/W] (mp2[V/Var])

Instable Region

HB,Lg1=0.6mH Lg2=0.4mH

15e-4 (21e-4) 10e-4 (14e-4)

HB,Lg1=0.3mH Lg2=0.2mH

5e-4 (7e-4) 0 0

7.5e-4 (15e-4)

15e-4 (30e-4)

22.5e-4 (45e-4)

30e-4 (60e-4)

37.5e-4 (75e-4)

mp1[rad/W] (mp2[V/Var])

from Fig. 5.22, decreasing the inverter cut-off frequency ωc broadens the stability region. Besides, the extension part of the stability region is the area with a larger reactive power gain mq. The extended range of the active power droop gain is only valid when the reactive droop gain increases correspondingly. Figure 5.23 plots the stability region on mpmq plane when different coupling inductors are selected. As presented in Fig. 5.23, the increase of coupling inductance tends to increase the overall stability region on mpmq plane, especially in terms of active power gain. That means a weaker coupling between inverters will improve the system stability. It can be noticed that not only a greater inductor but also a substantial virtual impedance can be utilized to broaden the parameter stability region. Figure 5.24 plots the parameter stability region on mpmq plane when the active power of ZIP load increases from 10 to 30 kW. The stable region of microgrid shrinks with the increase of ZIP load. The increasing active power load has minor effect on mq but narrows the stability region of mp.

5.3 Parameter Stability-Region of Microgrid Fig. 5.24 HB boundaries for microgrid with different ZIP load mp1[rad/W] (mp2[V/Var])

20e-4 (28e-4)

P0 =30kW P0 =20kW P0 =10kW

15e-4 (21e-4)

Instable Region

10e-4 (14e-4) 5e-4 (7e-4) 0

5.3.2.2

123

Stable Region

1.5e-4 (3e-4)

3e-4 (6e-4)

Instable Region 4.5e-4 6e-4 (9e-4) (12e-4) mp1[rad/W] (mp2[V/Var])

7.5e-4 (15e-4)

9e-4 (18e-4)

10.5e-4 (21e-4)

Case 2: Numerical Bifurcation Analysis on P-V Plane

For the microgrid in islanded mode, the power assumption of a local load can only be provided by the IIDGs. Power unbalance is the major physical mechanism of instability, the maximum power supply of microgrid should be calculated to make sure its stability. In this part, the bifurcation analyses are performed on P-V plane to figure out the load fluctuations on the system stability. The theoretical maximum load carrying capability is calculated. As presented previously that the parameter stability region on mpmq plane is highly related to the coupling inductor of inverters. Thus, a large inductor is recommended for improving the stability of microgrid with parallel inverters. However, the installation of bulk inductor is costly. Thus, virtual impedance control has been proposed in many literatures for stability improvement, harmonic mitigation and also for fault ride through. In general, the voltage droop from the virtual inductor that reacts to output current is added on the voltage reference. The modified voltage references can be presented by: (

vodk ¼ V   mqk Qk þ ωLvk ioqk  Rvk iodk voqk ¼ ωLvk iodk  Rvk ioqk

ð5:43Þ

where Lvk and Rvk are the inductance and resistance values of the virtual impedance of the inverter k, respectively. This section discusses the parameter stability region of microgrid when virtual inductor control is implemented. The physical coupling impedance of inverter 1 and 2 are designed as 0.3 + j0.24 πΩ and 0.2 + j0.16 πΩ, respectively. The virtual inductor of inverters Lvir is initially set to 4 mH. Figure 5.25 plots the equilibrium solution manifold for different types of load on the active-power versus bus voltages (P-V) plane. The nominal reactive-power Q0 is set to zero. The constant-impedance load, constant-current load, and constant-power load are obtained by setting percentages of ZIP load as a1, b1 ¼ 100%; a2, b2 ¼ 100%, and a3, b3 ¼ 100%, respectively. It can be seen from Fig. 5.25 that

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

Fig. 5.25 Equilibrium solution manifold on the P0Ubus2 plane for load increasing. SNB is saddlenode bifurcation detected in the curve of IM load and constant-power load

Constant impedance load 300 Ubus2[V]

124

Constant current load

250 IM load

200

SNB

Constant power load

150 100 0

Fig. 5.26 SNB boundaries for microgrid with different lengths of connection line

10

20

30

60

P0[kW]

SNB

40 Pn[kW]

50

Instable region

60

70

80

IM load Constant Power load

50 SNB 40 Stable region 30 0.5

1

1.5

2

Lline[km]

the voltage at bus 2 decreases when the load increases. Then, an SNB point is found in the equilibrium solution manifold of the constant-power load when P0 ¼ 58.17 kW. An SNB point can also be found in the curve for IM load when the IM load PIM ¼ 42.54 kW. The PCC voltage decreases sharply when the load close to the SNB boundary and their bifurcation point of PCC voltage are around 200 V, which indicates the voltage instability phenomenon. The stable equilibrium will disappear after the SNB, and the nonlinear system loses its stability when the SNB occurs. For the bifurcation curves of constant-impedance and constant-current loads, the bifurcation phenomena do not exist. Therefore, in a ZIP load model, the constant-power component has a significant effect on the stability of microgrid. In addition, the stability margin of IM load is smaller than that of ZIP load. Figure 5.26 presents the influence of the connection line with fixed X/R ratio. The increasing the length of connection line tends to shrink the stability region. Besides, the parameter stability region of IM loads is smaller than that of constant power load. As shown in Fig. 5.26, the increase of virtual coupling inductor moves the SNB boundary to a lower level. Thus, the application of virtual inductance control tends to broaden the parameter stability region on mpmq plane, but shrinks the stability region on P-V plane. That means, both SNB and Hopf boundaries should be considered for virtual inductance design to keep the safe operation of the microgrid. Since, the power assumption is related to the voltage regulation. The influence of reactive-power droop control on the SNB is investigated. The Fig. 5.27 plots stability region of mq when virtual inductor is added. In this case, the SNB occurs before the HB for a relatively large load consumption. Moreover, the mq limit decrease sharply with load increase. A small value of reactive power gain should

5.3 Parameter Stability-Region of Microgrid 70

Instable region

60

P0 [kW]

Fig. 5.27 SNB boundaries for microgrid when different virtual inductances are selected

125

IM load Constant Power load

50 40 30

Stable region

20 3

4

5

6

Lvir [mH]

65

IM load Constant Power load

Instable region 60 P0 [kW]

Fig. 5.28 SNB boundaries of P0 for microgrid when different reactive-power droop gain mq are selected

55 50 45 Stable region 40 -4 2.5e (3.5e-4)

mq1 (mq2)

7.5e-4 (10.5e-4)

10e -4 (14e-4)

60 55

P0[kW]

Fig. 5.29 SNB boundaries of P0 for microgrid when different nominal voltage of droop controller V* are selected

5e-4 (7e-4)

Instable region IM load Constant Power load

50 45 40 Stable region 35 300

310

320

330

Un[V]

be designed for improving the stability when system under heavy load condition. For the mq design, there is a trade-off between the accurate reactive-power sharing and maximum power transmission. Figure 5.28 plots the SNB when nominal voltage change. It can be seen that decreasing nominal voltage shrink the stable range of load assumption. Since, the nominal voltage may be scheduled on secondary control level, its stable range should be fully evaluated. From the bifurcation analysis above, several conclusions are obtained as follows: 1. For the microgrid with ZIP loads and IM loads, increasing the active power gain mp or reactive power gain mq leads to the subcritical HB phenomena. Microgrid becomes oscillating after the subcritical HB whose boundaries constitute the

126

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

parameter stability region on the mpmq plane. Decreasing the value of cut-off frequency ωc or increasing the value of coupling inductance broadens the parameter stability region on mpmq plane. The connected loads have minor effect on the HB boundaries. 2. The increase of an IM load or constant power component of a ZIP load results in the SNB phenomena. The bus voltages of the microgrid collapse after the SNBs. SNB boundaries constitute the parameter stability region for load increasing. Decreasing the value of reactive power gain mq or impedance of connection line will broaden the stability region of loads. Increasing the value of nominal voltage V* or virtual inductance shrink the stability region of load.

5.3.3

Verification of Bifurcation Instability

In order to verify the bifurcation phenomena and the parameter stability region analysis in the former section, a detailed switching model of the studied microgrid is built in the MATLAB/SymPowerSystem environment. The parameters from Tables 5.1 and 5.2 are used. To validate the parameter stability region composed of HB boundaries, the simulation results when active power gains mp1 and mp2 switch from mp1 ¼ 1.5e4 and mp2 ¼ 3e4 to mp1 ¼ 7.5e4 and mp2 ¼ 15e4 coordinately at 3 s are shown in Fig. 5.30. The waveform starts to oscillate at 3 s, after which the magnitude of the oscillation gradually become larger. This kind of nonlinear phenomenon belongs to the subcritical HB, and coincides with the analysis in Sect. 5.3. The parameter stability region of the ZIP load shown in Fig. 5.21 is validated by the simulation results, as shown in Fig. 5.31. The active power in the ZIP load continuously increases by connecting the input reference value port of the load module in MATLAB/SymPowerSystem to a ramp signal source. When the ZIP load is composed of a 100% constant-power load component, the voltage decreases and collapses at 3.2 s. Figure 5.32 shows the voltage at bus 2 with a 37.5 kW IM load in large time scale. The parameters of this IM load are designed according to Table 6.2. At 2 s, a 15 kW IM load with an IM load factor K ¼ 15/22.5 is connected to bus 2. The bus voltage decreases continuously after the step change of IM load, and the angular velocity will eventually decrease to 0 rads/s. Figure 5.33 is plotted to validate the influence of virtual inductance control and droop control on the SNB boundaries. The constant power load with 50 kW nominal active power is connected to bus 2. In Fig. 5.33(a) the virtual inductance is switched from 4 mH to 6 mH at 2 s, the voltage collapse after 2 s. In Fig. 5.33(b), the nominal voltage in power controller is switched from 330 V to 300 V at 2 s, the bus voltage continuously decreases, and eventually collapse after 3.3 s. In order to further verify the aforementioned bifurcation phenomenon, the CHIL method is also proposed for the experiment verification in this section. Figure 5.34

5.3 Parameter Stability-Region of Microgrid (a) 100 P1/P2 [kW]

Fig. 5.30 Simulation results of DG1 when the active gains are switched from mp1 ¼ 1.5e4 and mp2 ¼ 3e4 to mp1 ¼ 7.5e4 and mp2 ¼ 15e4 at 3 s. (a) Active power. (b) Output current, (c) Output voltage

127

50

P1

0

P2

-50 2.5

(b)

3 t [s]

3.5

3 t[s]

3.5

3 t [s]

3.5

400

ioabc1 [A]

200 0

-200 -400 2.5

(c)

600

uoabc1 [V]

400 200 0 -200 -400 -600 2.5

500

ub1 [V]

Fig. 5.31 The phase voltage at bus 2 with the increase of the rated active power at a ZIP load that consists of 100% constant active power load. The SNB appears and then system collapses

0

-500

0

0.5

1

1.5

2

2.5

3

3.5

t [s]

shows the physical map of the HIL simulation platform. All the electrical components including power electronic inverters, network, ZIP load and IM load are simulated at real time in RT-LAB [3]. The two droop-controlled inverters are controlled via two real DSP controllers, respectively. The parameters of droop controllers and electrical circuit are the same as shown in Tables 5.1 and 5.2.

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

Fig. 5.32 The phase voltage at bus 2 with the increase of the IM load. (a) The votage at the connection point of IM. (b) Angular velocity of IM rotor

(a) 300

ub2 [V]

128

0

-300

0

2

4

6

8

10

6

8

10

t [s]

ω [rads/s]

(b) 100 0

-100

0

2

4 t [s]

(a) 500

ub2 [V]

Fig. 5.33 The phase voltage at bus 2 with 50 kW ZIP load. (a) The virtual inductance of is switched from 4 to 6 mH at 2 s. (b) The nominal voltage of droop controller is switched from 330 V to 300 V at 2 s

0

-500 1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

t [s]

ub2 [V]

(b) 500

0

-500 1.5

2

2.5 t [s]

3

3.5

Figure 5.35 shows the waveforms of the DG1 after the subcritical HB. The active power gains of two DG mp1 and mp2 are switched from mp1 ¼ 1.5e4, mp2 ¼ 3e4 to mp1 ¼ 7.5e4, mp2 ¼ 15e4 coordinately. Since the output ranges of the analog out ports are limited, the maximum permissible magnitude of output current signals is 100 A. The output voltages and currents of the DGs fluctuate after the subcritical bifurcation, which coincides with the bifurcation analysis and the simulation results.

5.3 Parameter Stability-Region of Microgrid

RT-LAB Real-time Simulator

Monitoring jacks

Digital input

129

Analogy output

TMS320F28335

Monitoring panel Controller

Fig. 5.34 RT-LAB platform

(a) 100A/div

200ms/div

(b) 200V/div

200ms/div

Fig. 5.35 Experiment results of the subcritical HB. (a) Output currents. (b). Output voltages

Figure 5.36 shows the experiment result of the SNB as load increases. The voltage of Bus 2 decreases with the continuous increment of the constant power load, and eventually, the voltage collapses. This result is in accordance with the simulated result given in Fig. 5.28(a), (b). The SNB phenomenon caused by the increment of the IM load is verified. A 37.5 kW IM load is connected to the Bus 2 at first. Then, a 15 kW IM load is integrated. The voltage of Bus 2 droops irreversibly, which coincides with the simulation result as shown in Fig. 5.29.

130

5 Full-Order Modeling and Dynamic Stability Analysis of Microgrid

Fig. 5.36 Experiment results of the SNBs. (a) The IM load steps from 37.5 to 52.5 kW. (b) Constant active-power load increases

(a) 100V/div

500ms/div

(b) 250V/div

500ms/div

5.4

Summary

This chapter firstly discusses the full-order modeling method of the microgrid. In symmetric condition, the procedures of the combination of different DERs, various types of loads, network into a microgrid are elaborated. Secondly, the parameter stability region of a droop controlled-inverter-dominated microgrid including ZIP load and IM load are analyzed with bifurcation theory. According to the research in this chapter, the following conclusions can be drawn: 1. Full-order mathematical model can imitate transient characteristics of the switching model. Furthermore, the mathematical model can reduce calculation burden and simulation time than the switching model. 2. The dynamic model built in the dq rotating coordinate can fully describe the dynamic behavior of symmetric microgrid. By establishing the equivalent impedance model of ZIP loads and IM load, the system stability with continuous growth of multi-type load characteristics can be analyzed. 3. The SNB and HB boundaries constitute the parameter stability region of MG with ZIP and IM load. The HB of power droop gain is related to dynamics from the power controller and network of DGs. The SNB of load are related to the coupling inductance, connection line and voltage control.

References

131

References 1. Z. Shuai, Y. Peng, X. Liu, Z. Li, J.M. Guerrero, J. Shen, Parameter stability region analysis of islanded microgrid based on bifurcation theory. IEEE Trans. Smart Grid. https://doi.org/10. 1109/TSG.2019.2907600 2. Z. Shuai, Y. Peng, J.M. Guerrero, Y. Li, Z.J. Shen, Transient response analysis of inverterbased microgrids under asymmetrical conditions using a dynamic Phasor model. IEEE Trans. Ind. Electron. 66(4), 2868–2879 (2019) 3. Z. Shuai, Y. Peng, X. Liu, Z. Li, J.M. Guerrero, Z.J. Shen, Dynamic equivalent modeling for multi-microgrid based on structure preservation method. IEEE Trans. Smart Grid. https://doi. org/10.1109/TSG.2018.2844107 4. Z. Shuai, Y. Hu, Y. Peng, C. Tu, J. Shen, Dynamic stability analysis of synchronverter -dominated microgrid based on bifurcation theory. IEEE Trans. Ind. Electron. 64(9), 7467–7477 (2017) 5. M. Hamzeh, S. Emamian, H. Karimi, J. Mahseredjian, Robust control of an islanded microgrid under asymmetrical and nonlinear load conditions. IEEE J. Emerg. Select. Topics Power Electr. 4(2), 512–520 (2016) 6. N. Pogaku, M. Prodanovic, T.C. Green, Modeling, analysis and testing of autonomous operation of an inverter-based microgrid. IEEE Trans. Power Electron. 22(2), 613–625 (2007) 7. J.C. Vasquez, J.M. Guerrero, M. Savaghebi, J. Eloy-Garcia, R. Teodorescu, Modeling, analysis, and design of stationary-reference-frame droop-controlled parallel three-phase voltage source inverters. IEEE Trans. Ind. Electron. 60(4), 1271–1280 (2013) 8. S. Leitner, M. Yazdanian, A. Mehrizi-Sani, A. Muetze, Small-signal stability analysis of an inverter-based microgrid with internal model–based controllers. IEEE Trans. Smart Grid 9(5), 5393–5402 (2018) 9. X. Tang, W. Deng, Z. Qi, Investigation of the dynamic stability of microgrid. IEEE Trans. Power Syst. 29(2), 698–706 (2014) 10. J. Alipoor, Y. Miura, T. Ise, Stability assessment and optimization methods for microgrid with multiple VSG units. IEEE Trans. Smart Grid 9(2), 1462–1471 (2018)

Chapter 6

Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

Since the simulations based on the switching model are time consuming, full-order mathematical model has been established in the previous chapter. However, the fullorder mathematical model is still complicated due to the multiple time-scale characteristics of microgrid. The complicated full-order model often has a high order. These characteristics will make the numerical simulation time consuming and inconvenient for stability analysis. Thus, model reduction is needed to solve this problem. This chapter first analyzes the multi-time scale characteristics of dynamics of microgrid. Secondly, the singular perturbation reduction method of microgrid model is proposed. The obtained microgrid reduced model accurately preserves the dominant dynamics of microgrid, which is suitable for system-level stability analysis. Finally, through numerical simulation and eigenvalue analysis, accuracy of the proposed reduced-order model and other reduced-model in the existing literature are compared to obtain the advantages of different models.

6.1

Multi-Time Scale Property of Microgrid

The DERs’ dynamics have a decisive influence on the overall dynamics of the microgrid. Further, the inverter-interfaced DERs is the main power supply in a microgrid, whose dynamics mainly depend on the overall control strategy of the grid-connected inverter [1–4]. Moreover, the inverters with diverse control strategies often have very fast dynamics, which is quite different from that of the traditional SG. Specifically, hierarchical control that has been widely applied in microgrids to achieve upper level power management has dynamics around minute scale. On the other hand, the power control loop of inverter’s main control system has dynamics around second scales. While the voltage control loop and inner current control loop often has dynamics around millisecond scale. © Springer Nature Singapore Pte Ltd. 2021 Z. Shuai, Transient Characteristics, Modelling and Stability Analysis of Microgird, https://doi.org/10.1007/978-981-15-8403-9_6

133

134

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

Inner current control

Response of control system

Upper level optimize control Power control

Line current Dynamics of circuit

100μs

DC side voltage of Inverter

Capacitor voltage and inductor current of the filter circuit

ms

10ms

Motor rotor mechanical speed

100ms

s

10s

Fig. 6.1 Time scale distribution of different elements in a microgrid

In addition, different physical elements in a microgrid also have significant differences in response or motion speed due to their physical properties. For example, the rotating components in the microgrid has the dynamics around second scale, the inverter’s DC voltage are around hundred-millisecond scale, the filter circuit of the inverter and the line cables are both around millisecond scale, and the power electronic switching devices are around microsecond scale. Generally, there are great differences on the dynamic time scales among various elements in a microgrid. Due to different elements’ interactions on different time scales, the multi-time scale dynamics of the microgrid are cross-coupled [5–7]. Firstly, the coupling often behaves as the coupling between the controller action and the circuit component response at the equipment level. For example, the performance of voltage and current control loops directly affect the dynamics of the capacitor voltage and the inductor current. Secondly, the coupling is also reflected that the dynamics on the short-term scale affects that on the long-term scale. For example, the dynamics of the line current with the millisecond-level time scale directly affects the dynamic stability of the power control loop with second-level time scale. Thirdly, at the system level, the coupling represents the mutual interactions among various DERs. This coupling appears through the connection of line cables due to the short electrical distance between DERs: the dynamic performance of a single DER is directly affected by others in the microgrid. As shown in Fig. 6.1, the dynamic behavior of a typical microgrid includes the control behavior of the inverter and the other devices, the rotor response of the motor, and the response of the circuit. It can be seen that the power level and the control behavior of the voltage and current levels are distributed on the time scale of s~ms. The voltage and current response behavior of circuit components are distributed on the time scale of ms~μs. The microgrid is a typical nonlinear system. Different dynamics interact on multiple time scales. The excitation on short time scales may significantly affect the dynamic response over long time scales. Since the change speed of different dynamics behave differently, the numerical analysis of high-order stiff systems usually requires the a very small calculation step to solve the fast system dynamic process, which easily leads to problems of large calculation

6.2 Wide Frequency Range Stability Problem Classification

135

burden, etc. Therefore, it is necessary to study the reduced order simplification method of multi-time scale model of microgrid. Making sure that the coupling of dynamics at different time scales can be reduced to a certain extent under the premise of accurately preserving the dominant system dynamics. Finally, the efficiency of numerical calculation is improved while reducing the complexity of stability analysis.

6.2

Wide Frequency Range Stability Problem Classification

As shown in Fig. 6.1, dynamic response of microgrid exhibits wide frequency range characteristics. When microgrid is subjected to small or large disturbances, a wide frequency range stability problem would occur. According to the different time scale, stability problem can be divided into harmonic stability [8], voltage stability, synchronization stability and frequency stability, which is shown in Fig. 6.2. It can be seen that the time-scale of four different stability problems is illustrated and dominant components affecting four different stability problems are given below. Harmonic instability, which is also named as resonance instability, is usually caused by the interactions between fast controller dynamic (such as current controller) and passive circuit components (like inductances or capacitors). Since microgrid is equipped with more and more converters, harmonic instability in microgrid becomes hot issues. The time scale of harmonic instability lies around 1 ms. And passive damping by adding resistors in the circuit or active damping by design control system is widely used to deal with harmonic instability problem. Voltage instability in microgrid can be divided into DC voltage instability or AC voltage instability, even in AC microgrid. It is because converters in AC microgrid usually transfer DC power to AC side. Voltage instability in DC side is relevant to the control parameters of DC voltage controller. In AC side, voltage stability is decided by the power balance between the provided power delivered from the transmission line and the power assumption of the loads. Constant power loads (CPLs) and induction motors (IMs) have great influences on voltage stability. Synchronization stability represents for the ability of converters to maintain synchronization with the grid. Power control loop in voltage-controlled converters and phase lock loop (PLL) in current-controlled converters are the most commonly used synchronization units. Similar to traditional synchronous generators, Fig. 6.2 Different stability problem in wide range time scale and its dominant factors

Synchronization Stability

1ms

Frequency Stability

Voltage Stability

Harmonic Stability

10ms

100ms

1s

min

t

-Current Control -Voltage Control -Power Control -Secondary Control -PLL Loop -Power Management -Circuit Impedance -Load Type -Current Limiter -DC Voltage -Time Delay

136

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

synchronization stability is also discovered in microgrid. Furthermore, synchronization stability in microgrid is also somewhat different from that in traditional power system. Current limiters and reactive power control loop have unneglected effect on the synchronization stability in microgrid. Frequency stability represents for the ability of the system to maintain nominal frequency when subjected to disturbances, which is decided by the power balance of the whole system. In microgrid, secondary controller and power management controller are designed to regulate voltage and frequency to nominal value in a large time scale. Frequency stability is relevant to these controller parameters. Moreover, less inertia provided by converters would make frequency of microgrid more prone to instability area in short term. It should be known that systematic classification on microgrid stability is limited at present and more researches are needed to give insights into microgrid stability in both instability mechanism and improvement control. First of all, mathematical model in different time scales are needed to be established for further investigation. In this chapter, preliminary works of model reduction are done to simply divide the system state variables into slow variables and fast variables. This work can give dynamic characteristics of microgrid state variables intuitively.

6.3 6.3.1

Time-Scale Model Reduction of Microgrid Singular Perturbation Theory

This section studies the time-scale model reduction method of the inverterdominated microgrid based on the singular perturbation theory. The model construction and reduction process are proposed, and a reduced model of a typical microgrid is developed by the proposed method. Generally, the singular perturbation method is for approximate model solutions of dynamic mathematical models with small parameters. If it is directly assumed that the value of the small parameter in the system is zero, the state matrix of the mathematical model will be singular. The singular perturbation method decomposes the numerical calculation of the high-order system model into the solving problems of subsystems on different time scales [7–12]. By ignoring the dynamics of the state variables with fast changing speed in the system, the approximate simplified solution of the original system is obtained. The reduced model only preserves the state variables with slower change process, and the model order will be greatly reduced. Moreover, the state equations related to the fast-dynamic process is described by the boundary layer system. Thus, the quasi-steady state expressions can be preserved. By decoupling the fast state and slow state in the system, the stiff of the original system model is eliminated, and the time required for numerical calculation is greatly reduced. For a detailed microgrid’s dynamic model composed of (n + m) state equations, the system state variables can be decomposed into n slow variables and m fast

6.3 Time-Scale Model Reduction of Microgrid

137

variables. In the singular perturbation model, the fast variable is multiplied by a small perturbation parameter ε, and the original model can be expressed as a singular perturbation form x_ = f ðx, z, t, εÞ

ð6:1Þ

ε_z = gðx, z, t, εÞ

ð6:2Þ

Where x represents the slow state variables describing the slow transient in the system, z represents the fast state variable describing the fast change process, t is the time variable. The functions f 2 Rn and g 2 Rm are continuously differentiable functions. ε ¼ diag{ε1, ε2, . . . ., εm} denotes a diagonal matrix whose elements are the ratios of physical parameters that reflect the “true smallness” [7]. Subsequently, the state equations of the fast state variables z are transformed into the boundary layer system by setting ε to be 0. Equations (6.1) and (6.2) constitute the singular perturbation model of the system. Afterwards, a new time variable τ is introduced τ¼

t  t0 ε

ð6:3Þ

For a particular time t0, when ε is small enough, the time interval t  t0 will stretch to obtain a time interval τ. When ε approaches 0, τ will tend to infinity. Assuming that the fast variables can return to a steady state at the initial point t0, then they can always reach this steady state within the τ interval. When there is perturbation parameter ε ¼ 0 in (6.1) and (6.2), the model order is reduced from n + m to n. Substituting (6.3) into (6.2), a boundary layer system that describes fast variables can be obtained dz ¼ gðx, z, t, 0Þ dτ

ð6:4Þ

The change speed of the fast variable z in the model can be z_ = g=ε. In the time interval τ, the slow variables can be regarded as a fixed parameter. In a reduced model of describing slow variables, (6.2) can be approximated as an algebraic equation 0 ¼ gðx, z, t, εÞ

ð6:5Þ

When (6.5) has a solution, it can be expressed as the boundary layer system of the system: z ¼ hðx, t Þ

ð6:6Þ

When ε is very small, the fast variables can quickly converge to the equilibrium point of (6.6). This equilibrium point is described only by the system slow variables.

138

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

Thus, transient process of the fast variables in the original system can be filtered out. This steady-state solution is also the equilibrium solution to (6.2). Substituting (6.6) into (6.1), the reduced order model of the original system can be obtained. x_ = f ðx, hðx, t Þ, t, 0Þ

ð6:7Þ

In order to ensure the accuracy of the reduced-order model, the EPs of the reduced-order model and the original full-order model should be the same. The fast variables in the original model can quickly converge from the initial point to the equilibrium point of the boundary layer system. Assuming that the initial point of the fast variables is z0 ¼ z(t0). Since the initial state of the steady-state solution of the boundary layer system depends on the initial point of the slow variables, the initial value z ¼ h(x(t0), t(t0)) may deviate from the initial point z0 of the fast variables in the original model. Therefore, the reduced model may have an error in describing the dynamic process of the fast variables in the short time scale after disturbances. However, when applying a reduced model for time-domain numerical simulation, the system is usually assumed to be in a steady state before the disturbance occurs. At this time, the initial value of the steady-state solution of the boundary layer system is equal to the initial point of its fast variables, and the prediction error of the reduced system will be very small.

6.3.2

Singular Perturbation Reduction of Microgrid

In order to obtain the singular perturbation model of the original system, it is important to properly determine the fast variables and select the perturbation parameters of the state equation corresponding to the fast variables. Singular perturbation reduction in conventional power system usually relies on engineering experience to determine the fast variables, such as ignoring the inductor current dynamics and capacitor voltage dynamics in line network. Due to the small inertia and fast response speed of the inverter-interfaced DERs, the dynamics of the fast variables may interact with that of the slow variables, which will inevitably affect the system’s dominant dynamics. Therefore, the selection of fast variables in a microgrid needs to be evaluated. In [1–3], the singular perturbation reductions of the microgrid model are carried out. In [1], the variables such as line current and inner loop controller are defined as system fast variables. Literature [2] defines the output current of inverter and capacitor voltage. However, it didn’t verify rationality for selecting these ignored fast variables. Ref. [3] assumes that there is large difference between the time constants of the differential equation of the line current and load current, and defines the current variable with a smaller time constant as the fast variable to reduce the order. However, the model itself is overly simplified. Generally, none of the above literatures analyzes the rationality of selecting those fast variables in a typical microgrid reduced model.

6.3 Time-Scale Model Reduction of Microgrid Fig. 6.3 Topology of microgrid

Droop 1

L g1

139

R g1

I o1

Droop 2

L g2

Droop 3

L g3

I o2

PCC 1 Zline1 load 1

R g2 PCC 2 Zline2

PCC 4 Zline4

Il AC

Zline3

Table 6.1 DER Control parameters

Symbol P* V* mp mq ωc

Table 6.2 Circuit parameters

Symbol Lf1, Lf2, Lf3 Lg1, Lg2, Lg3 Cf1, Cf2, Cf3 Lline1 Lline2 Lline3 Lline4 Lload1 Lload3

R g3

I o3

PCC 3 load 3

Value 10 kW 320 V 0.0003 0.0006 31.4 rads/s

value 1 mH 0.8 mH 0.55mF 0.4 mH 0.2 mH 0.6 mH 1 mH 3 mH 5 mH

Symbol Kvp Kvi Kcp Kci

Value 2 300 2 400

Symbol Rf1, Rf2, Rf3 Rg1, Rg2, Rg3 Rvir Rline1 Rline2 Rline3 Rline4 Rload1 Rload3

value 0.2 Ω 0.4 Ω 10,000 Ω 0.2 Ω 0.1 Ω 0.3 Ω 0.2 Ω 25 Ω 20 Ω

This section will take the microgrid with three paralleled classic droop controlledinverter as an example to analyze the interaction of different state variables. Moreover, this section will elaborate the process of how to determine the fast variables, construct the singularly perturbed form and establish the reduced order model. The microgrid topology is shown in Fig. 6.3, which includes three droopcontrolled inverters, the node 1 and node 3 are connected to the impedance loads, and the node 4 is connected to the conventional power grid through line cables. The inverter and line parameters are shown in Tables 6.1 and 6.2, respectively. The fullorder model of the system can be established by using modeling method described in the previous chapter. It is found that there are 51 state variables in the full-order model. The state vectors for describing the inverter and the line in the full-order model of the system are as follows:

140

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory


T xg1 ¼ θ1 , P1 , Q1 , x11 , x21 , x31 , x41 , iid1 , iiq1 , vod1 , voq1 , iod1 , ioq1
T xg2 ¼ θ2 , P2 , Q2 , x12 , x22 , x32 , x42 , iid2 , iiq2 , vod2 , voq2 , iod2 , ioq2
T xg3 ¼ θ3 , P3 , Q3, x13 , x23 , x33 , x43 , iid3 , iiq3 , vod3 , voq3 , iod3 , ioq3 xl ¼ ½ilD1 , ilQ1 , ilD2 , ilQ2 , ilD3 , ilQ3 , ilD4 , ilQ4 , iloadD1 , iloadQ1 , iloadD3 , iloadQ3 T

ð6:8Þ ð6:9Þ ð6:10Þ ð6:11Þ

Then the state variables in the system full-order model can be expressed as: h iT xsys ¼ xTg1 , xTg2 , xTg3 , xTl

ð6:12Þ

The system state equations can be expressed as:   x_ sys ¼ f sys xsys

ð6:13Þ

The full-order model preserves the inverter’s control process such as power sharing and voltage closed-loop control in a microgrid. Meanwhile, it describes the dynamics of electrical variables such as filter circuit and line network, which can accurately predict the dynamics of the microgrid under actual disturbances. After the mathematical dynamic model of the system is built in the MATLAB/ Simulink environment, the equilibrium point (EP) can be calculated by using the trim function. Then the linmod function is called to obtain the state matrix of the system full-order mathematical model. After combining all state variables of the model in linearized mode, the dynamic response of each state variable can be described. On the other hand, by applying mathematically transform on the state matrix, it is possible to introduce the mutually decoupled system modalities to describe the response motion process of state variables in the original model. According to the theory of small signal analysis, the eigenvalues of state matrix characterize the system dynamics. Assuming that there are conjugate eigenvalues in the system: λ ¼ σ  jω

ð6:14Þ

Then each pair of conjugate eigenvalues corresponds to a system oscillation mode: eσt sin ðωt þ θÞ

ð6:15Þ

Therefore, the real part σ describes the damping of the system oscillation modal, and the imaginary part jω corresponds to the angular velocity of the oscillation. A negative real part indicates that the corresponding oscillation will decay with time, while a positive real part indicates an increasing-amplitude oscillation. The oscillation damping is defined as

6.3 Time-Scale Model Reduction of Microgrid

141

Table 6.3 System eigenvalue Lable 1,2 3,4 5,6 7,8 9,10 11,12 13,14 15,16 17,18 19,20 21,22 23,24 25,26 27,28

Eigenvalue 143029411.48  j314.15 66861627.97  j314.15 48157399.50  j314.16 18687954.0  j314.15 35839.099  j314.15 19499.689  j314.14 1030.32  j2952.69 1030.78  j2907.03 1018.33  j2729.20 915.46  j2576.497 912.1  j2538.32 887.18  j2395.35 336.33  j264.02 311.38  j258.21

σ ξ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 σ þ ω2

Lable 29,30 31,32 32,33 34,35 36 37,38 39,40 41,42 43 44 45 46,47 48,49 50,51

Eigenvalue –243.68  j294.61 200.0 200.0 200.0 200.0 164.12  j10.20 163.10  j 7.83 157.9  j3.06 49.71 47.61 38.36 14.15  j21.13 10.33  j33.57 9.87  j39.40

ð6:16Þ

This damping evaluates the attenuation of the modal oscillation. The real eigenvalues in the state matrix characterize the non-oscillation modes in the system. Assuming there is a real eigenvalue λ ¼ σ, and the corresponding non-oscillation mode to be eσt. Then the positive real eigenvalue represents a non-periodic unstable mode, while the negative real eigenvalue represents a progressively stable mode. The system eigenvalues are shown in Table 6.3. Observing the real part of eigenvalues, it can be found that the real part of a pair of conjugate eigenvalues farthest from the imaginary axis is about 108, and the nearest pair of eigenvalues is about 101. This indicates that the rate of change of different modes in the system is significantly different, resulting in the system behave with stiff features. The mode closest to the imaginary axis has the slowest change rate, which can be defined as the dominant mode to constrain the system stability margin. The system dynamics mainly depend on the dominant mode, while the non-dominant mode can quickly converge to the equilibrium state. Therefore, the reduced model should preserve the dominant mode in the original system as much as possible and eliminate the mode with fast change rate. This section uses the participation factor analysis to guide the selection of fast variables in the singular perturbation model. State variables that are less correlated with the dominant mode are selected as fast variables and their dynamic processes are ignored. The participation factor is a measure of the relative participation of the kth state variable in the ith mode, and vice versa. The participation factor of the ith mode and jth state is given by:

142

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

Fig. 6.4 System participation factor distributions

Participation factor

0.5 0.3 0.1 Y axis X axis Mode away from the imaginary axis Modal near the imaginary axis

pki ¼ uki vik

©

ilined , ilineq iod , ioq ild, ilq uod, uoq x3, x4 x1, x2, PˈQ

ð6:17Þ

where uij and vji denotes the left eigenvectors and right eigenvectors, respectively. The participation factor measures the relationship between the ith mode and the kth state variable xk. The participation of a subsystem on the mode i is defined by summing up all participation factors of dynamic states describing this subsystem, the response of the state variable xk over time can be expressed as: Δxk ðt Þ ¼

n X

pki eλi t

ð6:18Þ

i¼1

where Δxk is the deviation of the state variable relative to its equilibrium. This equation shows that the i-th mode triggered by the initial value Δxk(0) ¼ 1 will participate in the time domain response of Δxk with the coefficient pki. For all mode or state variables in the system, there are: n X i¼1

pki ¼

n X

pki ¼ 1

ð6:19Þ

k¼1

The participation factor analysis can be used to determine the dominant mode and associated state variables in the system. By setting the state variables with high correlation with the dominant mode as the slow variables, the accuracy of the dominant mode can be preserved to the greatest extent. Figure 6.4 shows the distribution of the system participating factors. On the x-axis, the eigenvalues are arranged in order according to the distance from the imaginary axis, and the y-axis includes the state variables in the microgrid. It can be seen that the dynamic variables related to power control loop and inner control loop in the microgrid are mainly related to the mode near the imaginary axis, and the variables such as inverter voltage and current, and line current are associated with the mode away from the imaginary axis. Therefore, the dynamics of the microgrid state variables is mainly separated on two-time scales.

6.3 Time-Scale Model Reduction of Microgrid

143

Table 6.4 Dominant eigenvalues and their participating variables Dominant eigenvalue 9.63  j39.40 10.58  j33.57 14.14  j21.13 38.36 47.61 49.71

Main participation variable (p > 0.1) θ3, P3 θ2, θ1, P2, P1 θ1, P1, θ2, P2 Q1, Q2, Q3 Q3, Q1 Q2, Q1, P2,

Non-negligible participation variables (0.1 > p > 0.01) θ2, P2, Q3, Q2, iod3, ioq3, θ1, P1, ilined3, ilineq3, iod2, ioq2 Q1, x13, ilined2 Q1, Q2, θ3 P3, iod1, ioq1, iod2, ioq2, ilineq1, Q3, ilined1, ilined2, ilineq2 θ3, P3, Q1, ilined4 Q2, ilineq4, Q3, iod1, ioq1, iod2, ioq2, ilineq1 P1, P2, P3, θ1, θ2, θ3, ilined4, ilineq4 P3, P1 θ3, θ1, Q1, ioq3, ioq1, iod1, x13 iod3 Q3, P1, θ2, P3, θ1, ioq2, θ3, iod1, x12, ioq1

Table 6.4 shows the dominant modes in the system and the main variables associated. It can be seen that the conjugate dominant eigenvalues are mainly related to the dynamic variables describing the dynamic process of the power control loop. It is worth noting that the dynamic process of output current and line current also have a non-negligible effect on the low frequency dominant mode. This is because the low-frequency dominant mode reflects the interaction of different inverters when participating in power sharing. Different inverters need to be electrically interconnected through coupled inductors and line networks. Therefore, the output current on the coupled inductor and the line current directly affect the low frequency dominant mode. In order to establish a reduced-order model that accurately preserves the dominant mode in the system, the phase angle θ, the active power P, the reactive power Q, the output current io, and the line current iline are set as the slow variables that retain the dynamics. While the other state variables can be set as the fast variables. The overall flow chart of the singular perturbation reduction procedure is shown in Fig. 6.5. After determining the slow and fast variables of the microgrid studied in this chapter, the full-order model of the microgrid can be converted into a singular perturbation form, where the slow variable vectors are expressed as:
x ¼ θ1 , P1 , Q1 , iod1 , ioq1 , θ2 , P2 , Q2 , iod2 , ioq2 , θ3 , P3 , Q3 , iod3 , ioq3 , ilD1 , ilQ1 , ilD2 , ilQ2 , ilD3 , ilQ3 T

ð6:20Þ

The fast variables can be expressed as:
z ¼ x11 , x21 , x31 , x41 , iid1 , iiq1 , vod1 , voq1 , x12 , x22 , x32 , x42 , iid2 , iiq2 , vod2 , voq2 , x13 , x23 , x33 , x43 , iid3 , iiq3 , vod3 , voq3 , iloadD1 , iloadQ1 , iloadD3 , iloadQ3 T ð6:20Þ After selecting the system fast variables, the singular perturbation form of their corresponding differential equations needs to be determined, and the perturbation

144

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

Fig. 6.5 The procedure of model order reduction

Full-order model Classify the dynamic states into the fast and slow subsystems The dynamic states from the fast subsystem belongs to the fast states

The dynamic states from the slow subsystem belongs to the slow states

Transform the full-order model into the singular perturbation form

Set the singular perturbation ratios ε=0, and develop the boundary layer system.

The reduced model for bifurcation analysis

parameters in the equation are extracted. Since it is impossible to quantitatively determine whether the perturbation parameter is sufficiently small, there is currently no general method for extracting the perturbation parameter in the equation. In engineering applications, perturbation parameters are usually determined based on empirical knowledge of physical processes and controls, with the following typical selection methods. 1. When the state equation which describes the physical process can be extracted with the time constant τ, then τ is usually employed as the singular perturbation parameter. This selection type is the most commonly used method. 2. When there is no parameter like time constant in the differential equations, the parasitic small parameters such as mass, capacitance, etc. that may increase the model order will be extracted as the perturbation parameter. The singular perturbation method also provides a reasonable basis for such a simplified model to ignore the dynamics of parasitic parameters. 3. The reciprocal of the feedback gain in the control system can be selected as the perturbation parameters. This is because the high gain of the feedback system usually allows the corresponding feedback variables to quickly recover to their reference values. The boundary layer subsystem in this microgrid example can be described by the following equations. For the state equation of the voltage, current control loop, the integral parameter of the PI controller is selected as the perturbation parameter. Then the state equations of the inverter with voltage closed loop control can be expressed as a singular perturbation form

6.3 Time-Scale Model Reduction of Microgrid

145

1 dx1k ¼ V   mqk Qk  vodk K vk dt

ð6:22Þ

1 dx2k ¼ 0  voqk K vk dt

ð6:23Þ

1 dx3k ¼ iidk   iidk K ck dt

ð6:24Þ

1 dx4k ¼ iiqk   iiqk K ck dt

ð6:25Þ

The state equation of the inverter filter inductor current and output current can be extracted with the time constant L/R as the perturbation parameters. The state equation of the filter capacitor voltage uses the capacitance value as the perturbation parameter. The singular perturbation model of the inverter circuit system can be expressed as follows  Lfk diidk 1  vidk  vodk þ ωLfk iiqk ¼ iidk þ Rfk Rfk dt

ð6:26Þ

 Lfk diiqk 1  v  voqk þ ωLfk iidk ¼ iiqk þ Rfk iqk Rfk dt

ð6:27Þ

dvodk ¼ iidk  iodk þ ωvoqk Cfk dt dvoqk C fk ¼ iiqk  ioqk  ωvodk Cfk dt

C fk

ð6:28Þ ð6:29Þ

The time constant of the state equation of load current are selected as the perturbation parameter as discussed in the first type of the typical selection methods. Then, the singular perturbation form of the state equation of load current can be expressed as Lloadk _ 1 ¼ iloaDk þ ðu þ ωcom Lloadk iloadQk Þ i Rloadk k Rloadk loadDk

ð6:30Þ

Lloadk _ 1 ¼ iloaQk þ ðu  ωcom Llk iloadDk Þ i Rloadk k Rloadk loadQk

ð6:31Þ

At this time, the singular perturbation parameter matrix of the boundary layer system can be expressed as  ε ¼ diag

 1 1 1 1 Lfk Lfk L L , , , , , , Cfk , C fk , loadk , loadk : K vk K vk K ck K ck Rfk Rfk Rloadk Rloadk

ð6:32Þ

When the perturbation parameter ε ¼ 0, (6.22)–(6.32) can be reduced to a quasisteady-state form. Based on the Gaussian elimination method, the quasi-steady

146

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

solution of the boundary layer system is obtained as z = h(x, t). Among them, the quasi-steady state solution of the electrical variable in the boundary layer system are given as

iiqk

vodk ¼ V   mqk Qk

ð6:33Þ

voqk ¼ 0

ð6:34Þ

iidk ¼ iodk   ¼ ioqk þ ωC f V   mqk Qk

ð6:35Þ ð6:36Þ

iloadDk ¼

Rloadk ubDk þ ωcom Lloadk ubQk R2loadk þ ω2com L2loadk

ð6:37Þ

iloadQk ¼

Rloadk ubQk  ωcom Lloadk ubDk R2loadk þ ω2com L2loadk

ð6:38Þ

Substituting (6.33)–(6.38) into the fast variables in the reduced order system, a reduced order system model that accurately describes the slow dynamic can be obtained. After the singular perturbation reduction is completed, the mathematical model order of the microgrid is reduced from 51 to 21, which is 58.8% lower than the full-order model.

6.3.3

Verification of Reduced Order Model

In order to demonstrate accuracy of the reduced-order model, a reduced-order model and a full-order model of one microgrid are built in MATLAB/Simulink. Figure 6.6 shows the distribution of eigenvalues calculated by the reduced model and the 3000 Full-order model Reduced model

Imaginary part[rads/s]

2000

50

1000 0

0

-1000 -50

-2000 -3000 -107

-106

-105

-104 -103 Real part[1/s]

-102

-101

-50 -40 -30 -20 -10

-1

Fig. 6.6 Comparison of eigenvalue spectrum of full-order models and reduced-order models

Q3[kW]

Q2[kW]

Fig. 6.8 Reactive power response of the inverter with the three reduced model when the grid voltage drops at 1 s

Q1[kVar]

P3[kW]

P2[kW]

Fig. 6.7 Active power response of the inverter with the three reduced model when the grid voltage drops at 1 s

P1[kW]

6.3 Time-Scale Model Reduction of Microgrid 13 12 11 10 9 14 13 12 11 10 9 13 12 11 10 9

147

P1 P1_reduced

P2 P2_reduced

P3 P3_reduced

t[s]

8 6 4 2 0 -2 10

Q1 Q1_reduced

Q2 Q2_reduced

5 0 -5 8 6 4 2 0 -2 0.5

Q3 Q3_reduced

1 t[s]

1.5

full-order model. It can be seen that the dominant eigenvalues of the reduced-order system are highly coincidence with the original full-order model. The dominant eigenvalues in the reduced-order model are 9.62  37.70i, 10.56  33.33i, 14.13  21.05i, 38.37, 47.50, 50.29. The high-frequency mode and part of the non-oscillation mode around milliseconds in the original system will be omitted in the reduced model since some dynamics of the double closed-loop control has been ignored. Meanwhile, there is a certain deviation among the modes related to the output current. After that, the accuracy of the reduced-order model is verified by numerical simulation. At 1 s, the amplitude of PCC voltage drops from 310 V to 250 V. The output active power and reactive power of the three inverters in the microgrid are shown in Figs. 6.7 and 6.8, respectively. Since the inverters operate in the gridconnected mode, the output power of the three inverters in steady state is equal to their rated power. After the disturbance, the output active power will return to the rated value. The output reactive power will increase after the voltage droop, thus to provide voltage compensation. It can be seen that the reduced order model based on

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

Fig. 6.9 Output voltage response of the inverter with the three reduced model when the grid voltage drops at 1 s

322

vod1[V]

148

vod1 vod1_reduced

320 318 316 314

vod2[V]

322

vod2 vod2_reduced

320 318 316 314

vod3[V]

322

vod3 vod3_reduced

320 318 316 314 0.5

1 t[s]

1.5

singular perturbation method can accurately describe the transient response process of the filtered active and reactive power after the large disturbance. Generally, the reduced model can accurately predict the original model, and there is almost no deviation. Figure 6.9 shows the time-domain response waveform of the fast variable of inverter’s output voltage. It can be seen that the fast variable’s transient process described by the boundary layer system after the disturbance has a certain deviation from the original model. The voltage time domain response described by the fullorder model has a transient voltage impact after the disturbance. However, the boundary layer model cannot reflect this transient process. The dynamics of the fast variable will quickly converge to the quasi-steady state solution described by the boundary layer model, and the reduced order model can describe the trend of the output voltage after the disturbance. Therefore, the reduced model preserves the dominant mode of the whole system, but there is a certain error in describing the transient mutation of the fast variable. Due to the significant reduction in the order, the simulation speed of the mathematical model is greatly reduced. If the numerical simulation operation is performed on a personal computer with intel i5 processor (main frequency 2.5 GHz), the calculation time required for the full-order model of this example is 11.7 s, while the reducedorder model only needs 5.4 s. Overall, the simulation time is nearly reduced by 53.8%.

6.4

Comparative Study of Different Reduced Models

According to the singular perturbation reduction theory, the singular perturbation form is obtained by mathematically transforming the state equation related to the state variables. By setting the singular perturbation parameter ε ¼ 0, the

6.4 Comparative Study of Different Reduced Models

149

Table 6.5 Reduced-order model comparison Reduced order model R1 R2 R3

Slow variable in reduced order model P, Q, θ, io, ii, P, Q, θ, ii, vo, x1, x2, x3, x4 P, Q, θ

model order 21 27 9

corresponding reduced-order model can be obtained. Therefore, most of the variables in the microgrid model can be set as fast variables. By ignoring their dynamic processes, the corresponding reduced model can be obtained. In the previous literature, the fast variables are usually selected by engineering experience or by observing the transient response time constant of the system variables. Hence, with these methods, various forms of reduced-order models can be obtained. This section compares the reduced-order model proposed in this chapter with two common reduced-order forms in other literatures. The eigenvalue spectrums are analyzed and the transient response under different severity of disturbances is compared, followed by conclusions of the characteristics and applicable scenarios of different reduced-order models. As shown in Fig. 6.3, the eigenvalues away from the imaginary axis are mainly related to the line current and the output current of inverter. Besides, the dynamic process of line current is also often ignored based on the engineering experience of conventional power system. Therefore, in [3], a reduced model that ignores the dynamics of line current and inverter output current is established. The node admittance equation is constructed to describe the relationship of inverters in the network. In order to further simplify the stability analysis, some literatures only preserve the dynamics of filtered active power P, reactive power Q and phase angle θ to obtain the third-order model of inverter. This section intends to specifically compare the two reduced models with the proposed one in the former section. The slow variables and model orders of the three types of reduced order models are shown in Table 6.5. The quasi-steady-state expressions of the fast variables such as the capacitor voltage, the filter current, and the inner control loop in the reduced models R2 and R3 can be derived according to the derivation in Sect. 6.2. The quasi-steady-state expressions of the inverter’s output current and the line current are as follows. ioD ¼

Rg ðvoD  ubD Þ þ ωcom Lg ðvoQ  ubQ Þ R2g þ ω2com L2g

ð6:39Þ

ioQ ¼

Rg ðvoQ  ubQ Þ  ωcom Lg ðvoD  ubD Þ R2g þ ω2com L2g

ð6:40Þ

ilineD ¼

Rline ðub1D  ub2D Þ þ ωcom Lline ðub1Q  ub2Q Þ R2line þ ω2com L2line

ð6:41Þ

150

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

ilineQ ¼

Rline ðub1Q  ub2Q Þ  ωcom Lline ðub1D  ub2D Þ R2line þ ω2com L2line

ð6:42Þ

In the reduced models R2 and R3, when the line current is not the observation object of numerical simulation, the relationship between inverters can be described by using the node admittance matrix of the microgrid. The admittance matrix calculates the voltage of each node according to the output current of the inverters. At this time, each node no longer needs to define the node voltage by using a virtual resistor, and the magnitude of the line current can be calculated according to (6.41) and (6.42).

6.4.1

Eigenvalue Comparative Analysis

First, the different reduced-order models are compared by eigenvalue analysis. The parameters of different reduced-order models are shown in Tables 6.1 and 6.2. Comparisons of the eigenvalue distributions of the three types of reduced order models and the full-order model are shown in Fig. 6.10. According to the participation factor analysis, the eigenvalues of the full-order model can be divided into four clusters. a-Cluster: these eigenvalues are the low-frequency dominant modes in the system, which are mainly related to the state variables of the power controller. b-Cluster: these eigenvalues represent the intermediate-frequency modes with dynamics around millisecond timescale, which are primarily related to voltage and current controllers. c-Cluster: these eigenvalues represent the high frequency mode, which are mainly related to the filter inductor current and the inverters’ output voltage.

Imaginary part[rads/s]

4000

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

(c)

2000

0

(d)

(a) (b)

-2000

-4000 8 -10

-107

-106

-105

-104

-103

-102

Real part [1/s]

Fig. 6.10 Comparison of eigenvalue spectrums of different reduced models

-101

-100

6.4 Comparative Study of Different Reduced Models (a)

Imaginary part[rads/s]

Fig. 6.11 Comparison of dominant eigenvalues of reduced models with different active power droop coefficients. (a) when Active droop coefficient mp ¼ 0.0003. (b) when Active droop coefficient mp ¼ 0.0009

50 0.34

Imaginary part[rads/s]

0.26

0.14

0.21

0.48 0.75

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

0 0.75 0.48 0.34

-50 -15

(b)

151

80

0.26

0.21

0.17

0.14

0.118

-10 Real part[1/s]

0.16

0.11

0.08

-5

0.052

0.034 0.016

0.24

40

0.45

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

0 -40

0.45 0.24

-80 -15

0.16

0.11

0.08

-10

0.052 0.034 0.016

-5

0

Real part[1/s]

d-Cluster: these eigenvalues are far from the imaginary axis of the complex plane, which represent the mode with dynamics around microsecond timescale and are sensitive to the line current and the inverters’ output current. The modes in the b-cluster and the c-cluster constitute the description of the dynamics of the inverter’s power control and double closed voltage/current control loop. As can be observed in Fig. 6.10, the three types of reduced-order models all preserve the low-frequency dominant in the full-order model. Since the reduced models R1 and R3 both ignore the inner control loop, their eigenvalues of the b cluster and the c cluster are missing. Therefore, there will be relatively large deviations in describing the dynamics around millisecond timescale. Besides, only the reduced-order model R1 retains the eigenvalues of d-Cluster, modes related to the line current and inverters’ output current in the reduced models R1 and R3 have been all omitted. Comparisons of the low-frequency dominant eigenvalues between the reduced models when there are different active power droop coefficients are shown in Fig. 6.11. It can be seen that when the droop coefficient mp is small, the reduced models R1 and R2 both can describe the dominant mode accurately, and the obtained eigenvalues in R1 are more accurate. However, the dominant eigenvalues calculated by R3 shows a large deviation. When the droop coefficient is increased to 0.0009, the dominant eigenvalues move towards the virtual axis of the complex plane, and the system stability margin decreases. Under this condition, the dominant modes deviation predicted by the reduced model R2 become larger, and the prediction result

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

Fig. 6.12 Comparison of dominant eigenvalues of reduced order models with different reactive power droop coefficients. (a) When Active droop coefficient mq ¼ 0.0012. (b) when Active droop coefficient mq ¼ 0.0018

(a) 50 Imaginary part[rads/s]

152

0.56 0.74 0.9

0.32

0.22 0.14 0.07

0 0.9 0.74 0.56 -30

-50 -40

(b) 60 Imaginary part[rads/s]

0.42

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

0.76

40

0.88

20

0.97

0.42 0.32 -20 Real part[1/s] 0.62

40 0.97

-40

0.88

-60 -80

0.48 0.36 0.24 0.12

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

0 -20

0.22 0.14 0.07 -10 0

0.76 -60

30

20

10

0.62 0.48 0.36 0.24 0.12 -40 -20 0 Real part[1/s]

tends to be optimistic. On the contrary, the dominant modes and system stability margin predicted by model R1 remain very accurate. Comparisons of the low-frequency dominant eigenvalues between the reduced models when there are different reactive power droop coefficients are described in Fig. 6.12. Due to the line cables of the microgrid are resistive, the P-f and Q-V control are easily coupled with each other. Therefore, as the reactive power droop coefficient mq increases, the damping ratio of the dominant low frequency eigenvalues decrease and start to move to the right half of the complex plane. Under the different values of reactive power droop coefficient, the dominant eigenvalues calculated by the reduced model R1 always coincides with that of the full-order model. The dominant low-frequency eigenvalues predicted by the reduced-order models R2 and R3 are farther away from the right half-plane than the full-order model, and the results tend to be optimistic. At the same time, when the reactive power droop coefficient is small, the three reduced-order models can accurately predict the eigenvalues related to reactive power control on the real axis. When the reactive power droop coefficient increases, the real eigenvalues predicted by the reduced models R2 and R3 show large deviations, while the reduced-order model R1 can always accurately predict the real eigenvalues related to reactive power control. By referring literature [4], it can be known that the voltage and current control loops in the DQ synchronous rotating coordinate system can affect the highfrequency mode in the microgrid, which may cause the undamped oscillation at harmonic frequency.

6.4 Comparative Study of Different Reduced Models (a)

Imaginary part [rads/s]

Fig. 6.13 Eigenvalue distribution when integral gain Kvk of voltage control loop vary, 300  Kvk  4300. (a) Eigenvalue trajectory of full-order model. (b) Eigenvalue trajectory of reduction model R2

5000

0

-5000 -1200 -1000

Imaginary part [rads/s]

(b)

153

-800

-600 -400 -200 Real part[1/s]

0

200

5000

0

-5000 -3500 -3000 -2500 -2000 -1500 -1000 -500 Real part [1/s]

0

500

In these reduced models, only the model R2 retains the inverters’ double closed control loops. Hence, it is attempted to verify the eigenvalue distributions of the medium-high frequency mode with the change of the voltage control loop parameters. Figure 6.13 plots trajectory of the system eigenvalues as the integral gain Kvi of the voltage controller increases. Figure 6.13(a) depicts the trajectory of the full-order model. It can be seen that when Kvi is increased to 4300, the eigenvalues of the medium-high frequency mode will pass through the imaginary axis, and the system will suffer from the unstable harmonics. Figure 6.13(b) shows the results of the reduced-order model R2. It can be seen that its description of the medium-high frequency mode shows a large deviation. This is because the medium-high frequency oscillation mode in the system reflects the interaction of the inverters’ voltage control loops, while the interactions among different inverters come from the filter inductor current and the line current. Therefore, ignorance of the dynamics of the filter current and the line current is likely to cause a prediction error of the medium frequency mode associated with the inner control loops.

6.4.2

Numerical Comparative Simulation

The accuracy of the three reduced-order models will also be verified by numerical simulations in two cases. The accuracy of each model to describe fast and slow variables under small disturbances are compared in case 1. At t ¼ 1 s, the magnitude of the PCC voltage drops from 310 V to 280 V. Figure 6.14 shows the active power

154

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory (a) 18

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

17

P1 [kW]

16

16

14

15

12

14 1

1.005

1.01

1.2

1.3

10 8 0.8

0.9

1

1.1

1.4

1.5

t [s] (b) 40

iod1 [A]

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

34 32 30

35 30 25

1

20

1.005

1.01

1.2

1.3

15 10 0.8

0.9

1

1.1

1.4

1.5

t [s] (c) 322

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

322

vod1 [V]

320 319

318

1

1.02

316 314 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

t[s]

Fig. 6.14 Transient response comparisons of different reduced models during voltage dips. (a) Active power. (b) Output current. (c) Output voltage

P1, the output current iod1 and the output voltage vod1 of inverter 1. As seen that all the three reduced-order models can accurately describe the inverters’ output active power under small disturbances, and all the dynamics of active power can be remained. Moreover, the reduced-order model R1 accurately preserves the dynamics of the output current with higher accuracy. Although models R2 and R3 ignore the dynamics of the output current, the state variables of the output current can quickly converge to the quasi-steady state solution of the boundary layer system. Model R2 retains the inner voltage control loop, hence it can describe the abrupt change of the output voltage under disturbance, and neither model R1 nor R3 can accurately predict this transient voltage response. In Case 2, the accuracy of the three models to describe the fast and slow variables is compared under large transient disturbances. At t ¼ 1 s, there is a three-phase ground short-circuit fault occurs at the network node 1. The fault is cleared after

6.4 Comparative Study of Different Reduced Models

80

60

70

40

60

20

50

P1 [kW]

(a) 80

155

Full-order model Reduced model R1 Reduced model R2 Reduced model R3 1.01

1

1.02

0 -20 -40 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

t[s] (b) 200

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

-20

iod1 [A]

150 -40

100

-60 1.02

50

1.06

1.1

0 -50 -100 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

t[s] (c) 380

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

vod1 [V]

360 340 320

308

300

306 304

280 260 0.8

302 1.06

0.9

1

1.1

1.2

1.08

1.3

1.1

1.4

1.5

t[s]

Fig. 6.15 Comparison of numerical simulation waveforms of different reduced-order models in short-circuit faults (a) Active power, (b) Output current, (c) Output voltage

0.04 s, and the fault resistance is 1 Ω. Figure 6.15 shows the transient response of the inverter after the fault. It can be seen that all the models can accurately reflect the dynamic response of the inverter when large transient disturbance occurs. However, there are certain deviations of the power and current waveforms described by the reduced-order models R2 and R3 after the fault is cleared. On the contrary, the reduced model R1 can accurately describe the system dynamics after the disturbance is cleared. However, when the fault is cleared, the microgrid has experienced severe voltage spikes. At this time, only the reduced-order model R2 can reflect the transient process of the change of the capacitor voltage, and has the ability to describe the voltage spike. In Case 3, the line connected to the conventional power grid in the microgrid is disconnected, and the microgrid operates in island mode. The three-phase

156

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

Table 6.6 Time-consumption comparison of numerical simulation calculations with different models

Case1 Case2 Case3

Full-order model 20.2 s 48.4 s 58.2 s

Reduced-order model R1 11.1 s 25.2 s 19.56 s

Reduced-order model R2 15.3 s 30.8 s 21.14 s

Reduced-order model R3 7.0 s 12.6 s 8.58 s

short-circuit fault scenario in Case 2 is repeated. Figure 6.15 shows the inverters’ transient response predicted by different reduced-order models after the fault occurs. It can be observed that there is no rated voltage support provided by the power grid in the island mode, the inverters can achieve autonomously active power sharing according to the load condition in the microgrid. It can be seen that the reduced-order model R2 can accurately describe the power and current surge of the inverter after the fault occurs since it retains the inverters’ inner control loops. On the contrary, the reduced-order models R1 and R3 have certain errors. After the fault is cleared, the power of the inverter fluctuates even more, and the reduced-order model R1 can more accurately describe the transient response trend of the inverters’ variables than models R2 and R3. From the above three simulation cases, it can be concluded that when studying the dynamics of the microgrid under small disturbances, the reduced-order model R3 can predict the time-domain response waveform of the main variables more accurately, and can be less time-consuming. However, when studying the transient characteristics of the microgrid under large disturbances, the reduced-order model R1 can be more accurate to describe the dynamics of the dominant mode in the microgrid. However, since the voltage double closed loop in R1 is neglected, it is hard to reflect the transient mutation of the output voltage at the time of fault occurrence and clear. On the other hand, the reduced-order model R2 retains the double-closed-control loops, so it can be more accurate to describe the sudden change of voltage under transient faults. But there is a certain deviation when describing the dominant dynamics of the system after the disturbance ends. Comparing the time consumption of numerical simulation with the three models under the above three cases: The computer’s CPU clock frequency is 2700 MHZ, and the variable step size calculation is performed with the ode 15 s algorithm in Matlab. The simulation durations of the examples are both 1.5 s, and the initial values of the slow variables in all models are consistent with the full-order model and close to the steady-state EP of the system. Table 6.6 shows the time-consuming results of different models. It can be seen that all the three types of reduced-order models can effectively reduce the simulation time. Among them, model R3 takes the shortest time, and the simulation time of the models R1 and R2 are close with each other, but the calculation efficiency of the model R1 is slightly higher than that of R2. On the other hand, at the same simulation time setting, the required calculation time for Case 2 is much higher than Case 1. This is because the transient change rate

6.4 Comparative Study of Different Reduced Models (a) 80

80

157

1.5 Full-order model Reduced model R1 Reduced model R2 Reduced model R3

0

60

P1 [kW]

-10

40 20 50

1

-20 1.05

1.008

1.1

0 -20 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

t[s] (b) 200

iod1 [A]

150 100

140

- 20

120 100

50

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

0

160

1

- 40

1.01

1.05

1.1

0 -50 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

t[s] (c)

380

Full-order model Reduced model R1 Reduced model R2 Reduced model R3

310

vod1 [V]

360

309 308

340

307 1.05

320

1.1

300 280 260 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

t[s]

Fig. 6.16 Comparison of numerical simulation waveforms of different reduced-order models in short-circuit faults: (a) Active power, (b) Output current, (c) Output voltage

of the system state variables is faster under large disturbances. When using the variable step size simulation mode, a smaller simulation step size is needed to ensure the accuracy of the model numerical solution, so the calculation time of the numerical simulation will be longer (Fig. 6.16). According to the performance of different reduced-order models in small-signal analysis and transient numerical simulation, the characteristics of different reducedorder models can be summarized in Table 6.7. The reduced-order model R1 retains the dynamics of the output current and the line current, and can accurately calculate the eigenvalues of the dominant system mode. It is suitable for predicting the small

158

6 Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

Table 6.7 Comparison of different models Model type Fullorder model R1 R2 R3

Orders High

Accuracy of stability analysis High

Simulation velocity in time domain Slow

Accuracy of dominant modes High

Accuracy of dynamics of voltage Yes

Medium Medium Low

High Medium Low

Medium Medium Fast

High Medium Medium

No Yes No

signal stability margin of the system. When applied in numerical simulation, it can effectively reduce the simulation time, and can accurately predict the time domain dynamics of the state variables related to the system dominant mode after the disturbance is removed. Therefore, it is suitable for transient stability analysis under large disturbances. The reduced-order model R2 takes into account the dynamic process of double closed control loops. According to the eigenvalue analysis, the medium-high frequency mode and the low-frequency dominant mode in the system are completely preserved, but there is a certain deviation when describing eigenvalues of the dominant mode. In the numerical simulation, the reduced order model R2 can predict the dynamics of system variables under small disturbances more accurately. However, it has a certain deviation in describing the slow dynamics of the system after the large disturbance removed. Therefore, the reduced-order model R2 is suitable for the small signal analysis of the mode distribution and dynamics of microgrid control system under small disturbances. The reduced model R3 retains only dynamics of three state variables of active power, reactive power, and output voltage’s phase angle. According to the eigenvalue analysis, the low frequency dominant mode is retained, but its error is large. In the numerical simulation, the simulation speed is fast, but only the time domain change of the variable related to the dominant mode under small disturbances can be predicted. Therefore, the reduced-order model R3 is suitable for analytical analysis and observation of dynamics of slow variables such as system frequency and active power under small disturbances.

6.5

Summary

To speed up the simulation efficiency, reduced method of microgrid model based on singular perturbation theory is introduced in this chapter. The singular perturbation reduction model suitable for stability analysis is introduced as well. Finally, the characteristics and application area of several common reduced models are compared by modal analysis and time-domain numerical simulation. The following conclusions can be drawn:

References

159

1. The dynamics of the inverter-based microgrid exhibits multiple time-scale characteristics. Among them, the dynamics of inverter’s power control is around second timescale. The dynamics of voltage and current control loops is around millisecond timescale. The dynamics of inverter’s output current and the line current are around microsecond timescale. 2. According to the results of the participation factor analysis, the low-frequency dominant mode is mainly affected by the state variables related to power control. Although the time constant of the output current and the line current are small, their influence on the dominant mode cannot be ignored. 3. The singular perturbation reduction method reduces the system model order by ignoring part of the dynamics. The reduced-order model proposed in this chapter preserves dynamics of related variables such as power control and line current. Moreover, the double closed-loop control part of the inverter is equivalent as the boundary layer system. Hence, it accurately preserves the dominant mode of the system. 4. According to the comparison of the three reduced models, the model R1 introduced in this book can accurately predict the system dominant mode, and significantly reduce the simulation time. Hence, it is suitable for the prediction of system small signal stability margin and large disturbance transient stability. The model R2 preserves the dynamics of power control and double-closed control loops and ignores the dynamics of output current and line current. It is suitable for predicting dynamic response of microgrid under small disturbance. The model R3 only retains the state variables related to the power control. Its model order is low, and can be less time-consuming in the numerical simulations. But its accuracy of microgrid’s dynamic response is poor, which is suitable for analyzing the dynamics of state variables such as system frequency under small disturbance.

References 1. L. Luo, S.V. Dhople, Spatiotemporal model reduction of inverter-based islanded microgrids. IEEE Trans. Energy Conver. 29(4), 823–832 (2014) 2. M. Rasheduzzaman, J.A. Mueller, J.W. Kimball, Reduced-order small-signal model of microgrid systems. IEEE Trans. Sustain. Energy 6(4), 1292–1305 (2015) 3. V. Mariani, F. Vasca, J.C. Vasquez, et al., Model order reductions for stability analysis of islanded microgrids with droop control. IEEE Trans. Ind. Electron. 62(7), 4344–4354 (2015) 4. P. Vorobev, P.H. Huang, M. Al Hosani, J.L. Kirtley, K. Turitsyn, High-Fidelity model order reduction for microgrids stability assessment. IEEE Trans. Power Syst. 33(1), 874–887 (2018) 5. I.P. Nikolakakos, H.H. Zeineldin, M.S. El Moursi, J.L. Kirtley, Reduced-order model for interinverter oscillations in islanded droop-controlled microgrids. IEEE Trans. Smart Grid 9(5), 4953–4963 (2018) 6. I.P. Nikolakakos, H.H. Zeineldin, M.S. El-Moursi, N.D. Hatziargyriou, Stability evaluation of interconnected multi-inverter microgrids through critical clusters. IEEE Trans. Power Syst. 31 (4), 3060–3072 (2016)

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7. Y. Peng, Z. Shuai, J. Shen, J. Wang, C. Tu, Y. Cheng, Reduced order modeling method of inverter-based microgrid for stability analysis. 2017 IEEE Applied Power Electronics Conference and Exposition (APEC), Tampa, FL, 2017, pp. 3470–3474 8. X. Wang, F. Blaabjerg, Harmonic stability in power electronic-based power systems: concept, modeling, and analysis. IEEE Trans. Smart Grid 10(3), 2858–2870 (2019) 9. A. Dhooge, W. Govaerts, Y. A. Kuznetsov, Matcont: A Matlab package for numerical bifurcation analysis of ODEs, 2003. [Online]. Available: http://sourceforge.net/projects/matcont 10. J. He, Y.W. Li, Analysis, design, and implementation of virtual impedance for power electronics interfaced distributed generation. IEEE Trans. Ind. Appl. 47(6), 2525–2538 (2011) 11. M. Huang, Y. Peng, C.K. Tse, Y. Liu, J. Sun, X. Zha, Bifurcation and large-signal stability analysis of three-phase voltage source converter under grid voltage dips. IEEE Trans. Power Electron. 32(11), 8868–8879 (2017) 12. Z. Shuai, Y. Peng, X. Liu, Z. Li, J.M. Guerrero, et al., Parameter stability region analysis of islanded microgrid based on bifurcation theory. IEEE Trans. Smart Grid 10(6), 6580–6591 (2019). https://doi.org/10.1109/tsg.2019.2907600

Chapter 7

Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent Theory

The microgrid group consisting of multiple microgrids (MMG) can make full use of renewable energy and enhance the reliability and stability of power supply. Compared with a traditional power system, a microgrid group of the same capacity will contain a larger number of DERs. This makes the model of MMG have high orders and complicated structures and the simulation is time-consuming. In Chap. 6, the time-scale model reduction method of a single microgrid is introduced. However, when only the time-scale model reduction method is applied to a microgrid group, the order of the system model is still very high, and the analysis process will be still very complicated. Thus, this chapter studies the spatial-scale reduction method for MMG with dynamic equivalent theory. Firstly, the equivalent modeling method is introduced. Subsequently, the developed 3-MG system is partitioned into a study MG and two external MGs. The study MG of the MMG retains its accurate form, while the external MGs will be replaced by the dynamic equivalent models whose model orders are much lower than that of the original one. The introduced procedure is based on the structure preservation method without the time-consuming iterative calculation. Finally, the validation range of the introduced method is verified with different models using the numerical simulation.

7.1

The Concept of Dynamic Equivalent Modeling for Multi-Microgrid

Dynamic equivalent modeling (DEM) is a method to simplify the system model in space. By dividing the system into research area and external area, the external area is replaced by dynamic equivalent model. The equivalent model in the external region has a lower order than the original model, and has similar external characteristics. Therefore, the complexity of the whole model is reduced and the efficiency of system transient analysis is improved. As shown in Fig. 7.1, microgrid cluster is a © Springer Nature Singapore Pte Ltd. 2021 Z. Shuai, Transient Characteristics, Modelling and Stability Analysis of Microgird, https://doi.org/10.1007/978-981-15-8403-9_7

161

162

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

DGi RES

node voltage i

DGj

Study Of the internal microgrid DGj

Medium and high frequency Study Of the internal microgrid modal interactions 100 HZ ~ 1000 HZ Low frequency dominant modal interaction microgridl 1HZ ~ 10HZ DGj

dynamic equivalence microgridm DGj

SimplifIed network equivalent model l

Simplified network equivalent model m

Fig. 7.1 Dynamic equivalent model of microgrids

complex system composed of multiple interconnected microgrids. As an independent and controllable subsystem, microgrid is connected to the power grid or a microgrid group through transformers, transmission lines and other equipment. The concept of microgrid group originates from the development of microgrid technology. Through the optimization control of upper level, a power dispatching between microgrids can be achieved, making full use of clean renewable energy units, and improving the reliability and stability of microgrid. A single microgrid usually adopts a hierarchical control structure, which coordinates inverters through the upper level controller to control the system frequency and node voltage. From the point of access to the transmission network, a microgrid can be equivalent to an independent and controllable device, while the dynamic equivalent model will retain the external characteristics of the microgrid as an independent device. The microgrid has large number of DERs and complex network structure. The time scale of the dynamic behaviors contained in a microgrid is large. Besides, there are interactions of dynamic behaviors between different time scales. Microgrid group deteriorates the complexity of this interaction from the spatial range. When considering dynamic of a single microgrid, the frequency of the microgrid will be clamped on rated frequency (50 Hz) and can’t reflect the active power distribution if the model of external area is established by using Thevenin’s equivalent method. Thus, it is necessary to study the equivalent method of modeling external area. The method of dynamic equivalent modeling can be divided into two steps. 1. Division of external regions. After determining the research area of concern, the external area is divided into several independent subsystems. 2. Simplifying and modeling for the external regional subsystems. By using the method of aggregating parameters, the model of the equivalent physical structure and control framework can be obtained.

7.2 Dynamic Equivalent Model of External Microgrid

7.2

163

Dynamic Equivalent Model of External Microgrid

In this section, the detailed procedure to construct the dynamic equivalent model (DEM) will be introduced. The systematic procedure for the DEM starts with the network simplification, followed by the aggregation of the terminal buses of the DER. Finally, the DERs with the same structure are aggregated into an equivalent DER. The DEM of the microgrid consists of the simplified network, phase shifting transformers, and equivalent DERs with equivalent terminal buses. The structure of the considered MMGs is shown in Fig. 7.2. Microgrid a, microgrid b and microgrid c can work both in isolated (i.e., Isolated microgrid, IMG) and interconnected modes (i.e. Multi-microgrid, MMG). Bus B and Bus C are defined as the boundary buses which are connected to the study microgrid a through switches S12 and S13, respectively. In this case, without loss of generality, the dynamic performance of microgrid a is under analysis and treated as the study microgrid. Microgrid b and microgrid c are treated as external microgrids and will be replaced by their DEMs. As shown in Fig. 7.2, the MMG system consists of the DERs, networks and loads. The DERs are connected to the microgrid by inverters with droop control or PQ control. Droop control is adopted to achieve power sharing and power balance when the MMG operates in islanded mode. Meanwhile, renewable energies such as photovoltaic and wind turbines are integrated through PQ-controlled inverters to output its maximum power. In transient stability analysis, we assume that the effects of the maximum power point track (MPPT) of the renewable-energy DER can be omitted. Therefore, the references of the active power P and reactive power Q of the PQ controlled DER are treated as constant under a small-time scale in this section. The multistage aggregation procedure for MG c is introduced in Fig. 7.5 as an example.

Grid

MG a (Study MG)

PCC

Droop1

Droop2

S1 S11

A1

A2

BusA

MG b (External MG1)

Line AB

Droop1 S12

B1

PQ1

PQ2

PQ3

B2

B3

B4

BusB

MG c (External MG2)

Line BC

Droop1 Droop2 S13

C1

C2

Droop3 C3

C4

BusC

Fig. 7.2 The structure of the MMG system under study

PQ2

PQ1 C5

C6

C7

C8

164

7.2.1

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

The Division of External Microgrid

The first step for dynamic equivalent modeling method of MMG is to divide external regions. The conditions for division are to determine coherent cluster of the external microgrid. As mentioned in the previous chapter, external microgrid would be divided into different subsystem. And IIDGs with different control strategies in the same subsystem would be used to build equivalent model respectively. For PQ-controlled IIDGs, it is assumed that the dynamic response of the phase lock loop (PLL) is fast enough and only the dynamics of the inner control loops are considered. Based on this assumption, PQ-controlled IIDGs in the same subsystem can be equivalented to single dynamic mathematical model. For droop-controlled IIDGs, they are controlled to mimic droop characteristics of synchronous generators and participate in the dominate low-frequency modals of the system which is corresponding to their power control loops. Thus, droop-controlled IIDGs can be divided into single or several equivalent models according to the dynamic response of droop-controlled IIDGs subjected to disturbances. In traditional power system, synchronous generators would be defined as coherent cluster if their swing equations are of the same or similar under system disturbances. Then, coherent clusters can be replaced by equivalent model. For a single machine to an infinite bus system, the swing equation of single synchronous generator can be written as 8 dδ > > < dt ¼ ω  ω0
 dω 1 EU > > ¼ P  sin δ  Dω : T dt TJ X II

ð7:1Þ

where ω0 and E represent for the frequency and amplitude of ideal grid voltage, respectively. Angular speed ω and angle δ represent for the speed of synchronous generators and angle relative to the power grid. PT is the mechanical power and TJ is the inertia time constant. TJ is usually around 1 ~ 8 s. XII is the equivalent impedance between the droop-controlled IIDGs to the grid and D is the damping coefficient. Combining two first-order equations in Eq. (7.1) into two-order derivative Eq. (7.2) TJ

dδ2 EU dδ ¼ PT  sin δ  D X II dt dt 2

ð7:2Þ

In traditional power system, define curves of power angle versus time δ-t after disturbances as swing curves. If the angle differences between two IIDGs are smaller than certain values, it is considered that these two machines are coherent. The conditions for coherent cluster is that the angle differences are smaller than   5 ~ 10 in 1 ~ 3 s of the postfault system. To verify the coherent characteristics of the IIDGs in microgrid, δ-t curve is observed when subjected to large disturbances such as grounded faults. The

7.2 Dynamic Equivalent Model of External Microgrid (a) 0.5

θ [rads]

Fig. 7.3 Angle differences among different IIDGs when subjected to large disturbances. (a) Angle differences of droopcontrolled IIDGs, (b) Angle differences of PQ-controlled IIDGs

165

IIDG C1 IIDG C2 IIDG C4 IIDG B1

0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0

0.5

1

1.5

2 t [s]

2.5

3

(b) 0.04

4

IIDG C6 IIDG C8 IIDG B2 IIDG B3 IIDG B4

0.035

θ [rads]

3.5

0.03 0.025 0.02 0.015 0.01 1.5

2

t [s]

2.5

3

disturbance is set at node A1, where grounded faults are occurred. The fault condition is that the grounded resistance is around 1 Ω and the fault lasts for around 0.1 s. The simulation results after disturbances are shown in Fig. 7.3. It can be observed from Fig. 7.3a that the changes of power angle of three droop-controlled IIDGs in Microgrid c while they are different from that in Microgrid b. Therefore, in this studied cases, droop-controlled IIDGs would be considered as coherent clusters. To achieve more accurate descriptions on equivalent models, IIDGs of C2 and C4 can be divided into the same coherent cluster, and IIDGs C1 would be separated as a single one. In this condition, equivalent models with better accuracy have been achieved while model becomes more complicated as well. δ-t curves of PQ-controlled IIDGs are illustrated in Fig. 7.3b. It is known that current control strategy is adopted in PQ-controlled IIDGs and they are controlled to follow the angle of PCC voltage. Thus, the angle differences among different PO-controlled IIDGs are small and consistent in general so that PQ-controlled IIDGs in the same subsystem can be considered as coherent clusters.

7.2.2

Simplification of Network

The dynamic equivalent modeling starts with the simplification of the network. Kron reduction is a reliable method to simplify the model of network and load [1–3]. In [2], the coupling inductors are included in the network for the Kron reduction with respect to the nodes of the output voltage. However, the eigenvalue analysis shows a discrepancy in the dominant modes between the original model and its Kron reduced

166

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

Fig. 7.4 Illustration of the Kron reduction of an electrical network with 3 boundary nodes, 5 interior nodes, Kron reduction of the interior nodes results in a reduced network among the boundary nodes with the Kron-reduced matrix

Boundary nodes Interior nodes Kron reduction

Impedance Equivalent Impedance

model when the damping ratio of the modes is smaller than 0.2. The Kron reduction in [2] can reconstruct the configuration of network including coupling inductors. The coupling inductors of DERs are eliminated after the Kron reduction. However, the dynamics on the coupling inductors (output current of DERs) manipulates the output power of DER and has major effect on the dominant nodes. In this section, the Kron reduction is applied to network with respect to the nodes of PCC. The coupling inductors of the DERs are excluded from the network and retain its original form after the Kron reduction. The illustration of the Kron reduction is shown in Fig. 7.4. The PCC nodes connected to the inverter or the external area of the microgrid are defined as boundary nodes, and the rest of the nodes are defined as interior nodes. The boundary nodes are preserved after the Kron reduction, while the interior nodes are eliminated. The admittance matrix Ynet 2 Rn  n of the network is used to obtain the Kronreduced network [3]. The static relationship of the output currents io and voltages of the PCC nodes ub can be described by the admittance matrix. "

I_ o 0

#

 ¼ Y net

ubb ubi



 ¼

Y1 Y3

Y2 Y4



ubbo ubin

 ð7:1Þ

where vectors I_ o and ubbo denote the inject currents and voltage of the boundary nodes respectively, and ubin is the voltage of the interior nodes. Thus, the Kronreduced matrix can be obtained as: Yeq = Y12Y2Y421Y3. The equivalent impedance of the connected line between nodes i and j can be obtained by Rij + ωLijj ¼ Yeq(i, j)1 ¼ Yeq( j, i)1. The equivalent admittance of the load connected at node i can be Pj¼n Yeq ði, jÞ. obtained by the summation of the elements of column i as Yloadi ¼ i¼1 If, the imaginary part of Yloadi ¼ Im(Yloadi) < 0, the equivalent load can be described by the series RL load as Rloadi + ωLloadij ¼ Yloadi 1. If the imaginary part of the Yloadi ¼ Im (Yloadi) > 0, the equivalent load can be described by the parallel RC load as 1/Rloadi + ωCloadij ¼ Yloadi. The Kron-reduced network is then constructed using the parameters obtained from the Kron-reduced matrix. MG c after the network

7.2 Dynamic Equivalent Model of External Microgrid Fig. 7.5 Aggregation Procedure of the microgrid c (External microgrid 2) to obtain the DEM: (a) External microgrid 2 (MGc) in the MMG system under study, (b) The first step: simplifying the network equation using Kron reduction, (c) The second step: aggregation of the terminal buses, (d) The third step: Aggregation of the DERs

167

(a) Droop1

PQ2

PQ1

Droop3

Droop 2

(b) Droop1

(c)

Droop3

Group1

Droop1

a&1

(d)

Droop2

PQ1

Group2

Droop2

a& 2

Droop3

a& 3

Group1

PQ1

PQ2

a&4

a&5

Group2

Droop_eq

a&1

PQ2

a& 2

PQ_eq

a& 3

a&4

a&5

simplification is shown in Fig. 7.5b, where the redefined impedances are depicted as hollow rectangles.

7.2.3

Aggregation of Buses

After the simplification of the network, the DERs with the same control structure constitute a critical cluster and will be replaced by an equivalent DER. The terminal buses of the DERs within the critical cluster are reconnected to an equivalent bus. The concept of critical cluster comes from the transient stability analysis of conventional power systems. However, there is a difference between this concept in the introduced method and in conventional methods. In this work, DERs with similar dynamic performance in a microgrid are treated as a critical cluster. The dynamic behavior of a DER depends on its control structure and input signals [4]. The inputs of a DER are terminal voltages that are coherent in a microgrid. The DERs in a microgrid is electrically interconnected via short-impedance cables with small impedance. Thus, the PQ DERs and droop DERs within an external microgrid are treated as a critical cluster respectively.

168

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

The aggregation of the equivalent buses should ensure that the current and voltage of the tie buses will not be changed. In order to satisfy the above constraint, the terminal buses of the clustered DERs are reconnected to the equivalent terminal buses by phase-shifting transformers (PSTs), whose transform ratios vector are defined as a = ubn =ube

ð7:2Þ

where the voltages of the equivalent buses are denoted as ue. The injected current ie in the equivalent bus should satisfy ioe ¼ aT ioi

ð7:3Þ

where the equivalent bus voltage is obtained by averaging the voltage of terminal buses in each group. The equivalent buses and PSTs are included in the critical cluster. To ensure the inject currents from the PST to the PCC node are well defined, small virtual resistors are connected in series in each branch of the PST. The impedance of virtual inductor is chosen sufficiently small such that its introduction would have minimum influence on the power flow of the system. MG c after the aggregation of terminal buses is given in Fig. 7.5c. The terminal buses of the droop-controlled DERs and PQ-controlled DERs are reconnected to an equivalent bus respectively, where a_ j presents the transform ratio of the transformer connecting to the equivalent bus.

7.2.4

Aggregation of DERs

In this step, the PQ-controlled DERs and droop-controlled DERs in the external microgrids are aggregated into an equivalent model of a PQ-controlled DER and an equivalent model of droop DER respectively. The physical structures of the equivalent DER are the same as those of the original ones. The parameters of the equivalent DER are calculated by the aggregation method without iterative procedures. By doing so, less computation effort is required compared with the measurement-based modeling method [5]. The following condition should be satisfied in the aggregation procedure. The output power of the equivalent DERs should be the same as the sum of the total output of DERs in a critical cluster. The aggregation of DERs need to calculate the equivalent parameters of the coupling inductance, power controller, inner control loop and the LC filter.

7.2 Dynamic Equivalent Model of External Microgrid

7.2.4.1

169

Aggregation of Droop-Controlled DER

1. Equivalent of the Coupling Inductor. The output current of the equivalent DER in the common reference frame is equal to the sum of the DERs in a group according to the mentioned condition. The quasi-static solution of the output currents can be presented as isoDi ¼

Rgi uoDi  Rgi ubDi þ ωi Lgi uoQi  ωi Lgi ubQi R2gi þ ω2i L2gi

ð7:4Þ

isoQi ¼

Rgi uoQi  Rgi ubQi  ωi Lgi uoDi þ ωi Lgi ubDi R2gi þ ω2i L2gi

ð7:5Þ

Equations (7.4) and (7.5) can be also be written as "

" #" # Rgi ωi Lgi uoDi 1 ¼ 2  Rgi þ ω2i L2gi ωi Lgi Rgi isoQi uoQi " #" # Rgi ωi Lgi ubDi 1 R2gi þ ω2i L2gi ωi Lgi Rgi ubQi

ð7:6Þ

or isoi = M i uoi ‐M i ubi

ð7:7Þ

isoDi

#

the corresponding quasi-static equations of the equivalent DER should be isoe = M e uoe ‐M e ube

ð7:8Þ

The output current isoe can be written by summing the Eq. (7.8) in each group as isoe =

X

ioi =

X

M i uoi ‐

X

M i ai ube

ð7:9Þ

where ai is the magnitude of the transformation ratio. The equivalent DER matrices Me can be individually obtained as 

M e1 Me ¼ M e2

M e2 M e1

 ¼

X

M i ai

ð7:10Þ

The output voltage of the equivalent inverter in static state can be obtained in the common reference frame as uoe = M ‐1 e

X

M i uoi

Le and Re can then be calculated from Me in Eq. (7.11) as

ð7:11Þ

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

170

P

M Rge ¼  2 e2 2  ¼ "
M e1 þ M e 2 P

R

ai R2 þωgi2 L2 gi i gi 2
2 # P Rgi ωi Lgi þ ai R2 þω2 L2 ai R2 þω2 L2 gi

i

gi

gi

P

M ¼ Lge ¼  2 e1 ωi M e1 þ M e 22

ωi

"
P

i

gi

L

ω gi ai R2 þω 2 L2 gi i gi 2
2 # P Rgi ωi Lgi þ ai R2 þω2 L2 ai R2 þω2 L2 gi

i

ð7:12Þ

gi

gi

i

ð7:13Þ

gi

2. Equivalent of the Power Controller. Considering that all the droop-controlled DERs in the same group have the same angular frequency ωi, the output active and reactive powers of the equivalent can be obtained by summing the active and reactive powers equation of m DERs in this group X

PNi 

X

Pi ¼

X X 1  ref X 1  ωi  ωn , U n  uref Qi ¼ odi mpi mqi

ð7:14Þ

After that, the parameters of the droop equations of the equivalent DERs can be presented as PNe ¼

X

PNi , ωne ¼ ωn , mpe ¼ P

1 1  , mqe ¼ P  1=mpi 1=mqi

ð7:15Þ

and the nominal voltage Une can be calculated by U ne ¼ U oe þ mqe 

X

Qi

ð7:16Þ

where Uoe is the magnitude of the output voltage. 3. Equivalent of the Inner Control Loop and LC Filter. The objective of the inner control loop is to regulate the output voltage of the DER according to the command of the power controller, which is designed to reject disturbances in a high bandwidth [6]. Considering that the bandwidth of the voltage and current controller is much higher than that of the power controller, the dynamics of the inner control are usually omitted in small-signal stability analysis [2, 7]. However, such simplification may impact the accuracy of the transition stability assessment. Therefore, the inner control loops are preserved, and the weighted mean method is applied for the aggregation. The weight coefficient Li is applied to calculate the equivalent parameters of the turbine and governor models, which is designed to present the relative nominal output power of each inverter as

7.3 Verification of the Dynamic Equivalent Model

Li ¼ PNi =

171

i¼n X

ð7:17Þ

PNi

i¼1

Next, the equivalent parameters can be presented as Lfe ¼

j¼n X

L j Lfj , Rfe ¼

j¼1

K cie ¼

j¼n X

L j Rfj , K cpe ¼

j¼1

K cij L j , C fe ¼

j¼1

7.2.4.2

j¼n X

j¼n X

K cpj L j ,

j¼1 j¼n X

ð7:18Þ

L j Cfj

j¼1

Aggregation of PQ-Controlled DER

The commanded active power and reactive power of the power controller of the equivalent DERs are obtained as Pref e ¼

X

ref Pref i , Qe ¼

X

Qref i

ð7:19Þ

The weighted mean method is also applied to calculate the parameters of the inner control loop. These equivalent parameters can be given by K ae ¼

j¼n X j¼1

L j K ai , K be ¼

j¼n X j¼1

L j K bi , Lge ¼

j¼n X j¼1

L j Lgi , Rge ¼

j¼n X

L j Rgi

ð7:20Þ

j¼1

Finally, the formation of the equivalent microgrid c consists of the simplified network, equivalent droop-controlled DER, equivalent PQ-controlled DER, and PSTs that are given in Fig. 7.5d. The dynamic equivalent method is also applied to microgrid b as another external microgrid. The final formation of the reduced MMG system consists of three parts, the study microgrid (microgrid a), the DEM of microgrid b and the DEM of microgrid c.

7.3

Verification of the Dynamic Equivalent Model

In this section, the numerical evaluation of the DEM is carried out using the detailed model in previous section as a benchmark. At first, the reduced model of the introduced MMG in Fig. 7.2 is tested by the time-domain simulation. The test MMG system in [8] is used to further test the validity range of the introduced method.

172

7.3.1

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

Evaluation for the Studied System

The reduced MMG system is validated by numerical simulation at first. The network and load parameters of the MMG are given in Table 7.1. The length of the interconnection line AB and line BC are 2.5 km and 3 km, respectively, with the impedance parameters Rtie-line ¼ 0.164 Ω/km, Ltie-line ¼ 0.26 mH/km. The order of the DEM for microgrid b is decreased from 33 to 25, and that for the microgrid c is decreased from 73 to 31. The parameters of the droop-controlled DERs are listed in Table 7.2. The parameters of the PQ-controlled DERs are given in Table 7.3. To verify the accuracy of the DEM of the external microgrids, the reduced model of the MMG system is compared against the detailed mathematical model and switching model of the MMG. The switching model of MMG that tracks the switching transient of IGBT is developed in the MATLAB/Simulink environment using the SimPowerSystems toolbox. In steady state, the output active power PB and reactive power QB of the original microgrid b at Bus B are 11.15 kW and 3.52 kvar respectively. The active power and reactive power of the equivalent model of microgrid b are 11.15 kW and 3.54kvar respectively. For microgrid c, the output active power PC and reactive power QC of the original model at bus C are 11.25 kW and 6.61 kvar respectively. The output active power and reactive power of the corresponding equivalent model are 11.25 kW and 6.63 kvar respectively. The negligible deviations of the output reactive power are due to the assumption that the equivalent bus voltages are obtained by averaging the terminal bus voltages in each group. It is acceptable due to the small difference between the terminal buses in a group. These differences are ever small due to the low admittance values of the tie-lines between the DER in a critical cluster. In the first scenario, the load disturbance at bus A1 is arranged to verify the dominant low-frequency modes. The load connected at bus A1 steps up sharply from 1.55 kW to 16.46 kW at t ¼ 2 s, and then drops to 9.15 kW at 3 s. Figure 7.6a–b show the response of the output active power and reactive power of the microgrid b when the load changes. The results of the corresponding switching model are presented in Fig. 7.6c–d. The subscripts eq and or denote the equivalent and original models of the microgrid, respectively. Figure 7.7a–b show the response of the output Table 7.1 Parameters of network and loads Load Bus A1 A2 B1 B2 B3 B4 C1

R [Ω] 20 – 40 – 40 – 10

L [mH] – – – – – – 20

Line cable 0.164 Ω/km 0.26 mH/km 0.2 0.3 0.1 0.4 0.2 0.5 0.4

Loads Bus C2 C3 C4 C5 C6 C7 C8

R [Ω] 35 30 35 25 20 20 30

L [mH] – 1 – 9 – 0.7 0.4

Line cable 0.164 Ω/km 0.26 mH/km 0.8 0.7 1 0.7 0.1 0.7 0.5

7.3 Verification of the Dynamic Equivalent Model

173

Table 7.2 Parameters of droop-controlled DERs System parameter Lf [mH] Rf [Ω] Cf [mF] Lg [mH] Rg [Ω] Pn [w] mp [rad/W] mq [V/Var] Kvp Kvi Kcp Kci

DERA1 1 0.2 0.5 0.5 0.1 5000 0.9e4 0.0005 1.4 300 7 200

DERA2 0.8 0.3 0.55 0.8 0.2 10,000 0.45e4 0.0003 0.9 200 9 400

DERB1 0.8 0.15 0.5 1 0.08 6000 0.75e4 0.0006 1.2 150 10 270

DERC1 1.2 0.2 0.7 1.4 0.08 30,000 0.15e4 0.0004 1.2 200 10 500

DERC2 0.9 0.1 0.5 1.2 0.08 10,000 0.45e4 0.0006 1 300 7 400

DERC3 0.5 0.09 0.4 0.7 0.1 6000 0.75e4 0.0006 0.8 200 5 300

Table 7.3 Parameters of the PQ-controlled DERs System parameters Lf1 [mH] Rf1 [Ω] Pn [W] Qn [Var] Ki Kp

DERB2 1.4 0.1 5000 0 80 1.1

DERB3 1.5 0.1 4000 1000 80 0.8

DERB4 1.8 0.2 6000 500 100 1.2

DERC4 2 0.2 5000 0 95 1

DERC5 2.2 0.1 7000 0 110 1.3

active power and reactive power of the microgrid c. The response of its switching model is shown in Fig. 7.7c–d. As the total capacity of droop-controlled DERs in microgrid b are much smaller than those of droop-controlled DERs in microgrid c, the output oscillation of the microgrid b is much large than that of the microgrid c. The Fig. 7.6 shows the response of DERA1 and DERA2 in the study microgrid (microgrid a). The DERA1 is the nearest to the change load, which takes the larger transient than DERA2. The waveforms of the output active and reactive power match with those of the original model and switching model. The simulations are performed on a personal computer with an Intel Core i5 CPU at 2.7 GHz. The simulation times for the original model and reduced model using MATLAB ode15s solver are 18.1 s and 6.8 s, respectively (Fig. 7.8). The second scenario is designed to test the medium and fast frequency modes under severe disturbance. In scenario two, a three-phase short-circuit fault with 0.5 Ω fault resistance occurs at bus A1 and clears after 5 cycles. The magnitude of the output voltage of DERA1 and DERA2 in the reduced model and original model are shown on Fig. 7.9a, b respectively. The corresponding waveform of the three-phase voltage of the switching model is presented in Fig. 7.9c, d, respectively. The clearance of the fault causes a spike in the output voltage due to the large output reactive power of the DERs during the short-circuit fault period. After the fault is cleared, the output voltage rises at first, and then falls due to the voltage control.

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

Fig. 7.6 The transient response of microgrid b with the load step changes at Bus A1: (a) Active power of the mathematical model, (b) Reactive power of the mathematical model, (c) Active power of the switching model, (d) Reactive power of the switching model

(a) 14 PB[kW]

174

PB_or PB_eq

12

10

2

3

t[s]

4

QB[kVar]

(b) -3 -3.5 -4 -4.5

QB_or QB_eq 2

3

2

3

2

3

t[s]

4

PB[kW]

(c) 14

12

10

t[s]

4

QB[kVar]

(d) -3 -3.5 -4

-4.5

4 t[s]

Figure 7.10 shows the output current response of DERA1 and DERA2. It is to be observed from Figs. 7.9 and 7.10, the DEM has the capability to predict the spike voltage and overload current during a large perturbation. Slight differences in the output variables are observed during the transient period. This is because the equivalent parameters of the inner control loop are calculated by weight averaging the DER parameters. Although the dynamic equivalent method reduces the order of the external microgrids, DERA1 and DERA2 connected to the

7.3 Verification of the Dynamic Equivalent Model

Pc[kW]

(a) -5

P C_or P C_eq

-10

-15

2

3

(b) 12 Qc[kvar]

Fig. 7.7 The transient response of microgrid c with the load step changes at Bus A1. (a) Active power of the mathematical model, (b) Reactive power of the mathematical model, (c) Active power of the switching model, (d) Reactive power of the switching model

175

t[s] 4 Q C_or Q C_eq

6

0

2

3

2

3

t[s] 4

2

3

t[s] 4

t[s]

4

Pc[kW ]

(c) -5

-10

-15

Qc[kvar ]

(d) 12 6

0

dynamic equivalents show a good approximation to those of DGA1 and DGA2 in the original system. In scenario 3, the effect of variation in line impedance on the system response of the reduced model is discussed. The length of the connection line between microgrid a and microgrid b is changed from 0.1 km to 10 km. The differences of the DERs response between the detailed model and reduced model are presented. Figure 7.11 shows the difference of transient response during the load change as carried out in scenario 1. The difference of voltage responses of DERA1 is presented in Fig. 7.11a. Figure 7.11b shows the difference of the power angle between DERA1 and DERA2. The differences of transients between the detailed model and reduced model are very small according to Fig. 7.11. During the transient process, the reduced model with relatively short connection line shows larger errors but these

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

Fig. 7.8 The transient response of the DERs in microgrid a with the load step changes at Bus A2. (a) Active power of the mathematical model, (b) Reactive power of the mathematical model, (c) Active power of the switching model, (d) Reactive power of the switching model

(a) 14 PA1/PA2[kW]

176

PA1_or PA1_eq PA2_or PA2_eq

10 6 2 -2

2

3

t[s]

4

QA1/QA2[kVAR]

(b) 6 2 -2

QA1_or QA1_eq QA2_or QA2_eq

-6 2

3

t[s]

4

PA1/PA2[kW]

(c) 14 10

PA2

6 2 -2

PA1 2

3

t[s] 4

QA1/QA2[kVAR]

(d) 6 Q A1 2 Q A2

-2 -6 2

3

t[s]

4

errors reduce quickly after the disturbances. This is because the coupling of the fast dynamics of the microgrids weakens with the increasing distance between the study microgrid and external microgrids. Therefore, during the transient process, the deviation of the fast dynamics predicted by the dynamic equivalent model with a longer distance to the study microgrid has minor effect on the system response of the study microgrid.

7.3 Verification of the Dynamic Equivalent Model

(a) 330 uA1 [V]

Fig. 7.9 Output voltage response of the DERs when a three-phase short-circuit occurs at Bus A and clears after 5 cycles. (a) Mathematical model of DERA1, (b) Mathematical model of DERA2, (c) Switching model of DERA1, (d) Switching model of DERA2

177

310

320 300 0.999

290

1.005

0.5

330 310 290 1.1

1

(b) 350

1.105 1.5

uA1_or uA1_eq t[s]

2

350

320

uA2 [V]

330 330 310 0.999

1.1

1.005

1.105 uA2_or uA2_eq

310 0.5

1

1.5

t[s]

2

uA1 [V]

(c) 400

0

-400 0.5

1

1.5

1

1.5

t[s]

2

uA2 [V]

(d) 400

0

-400 0.5

t[s]

2

Figure 7.12 shows the difference of the system response during a short-circuit fault as carried out in scenario 2. The differences of system response during the severe disturbance are larger than those during the load change in scenario 1. However, these errors are all in the acceptable range. During the transient process, the maximum voltage error is smaller than 0.4%, and the maximum angel error is smaller than 0.006 rad.

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

Fig. 7.10 Output current response of the DERs when three-phase short-circuit occurs at Bus A and cleared after 5 cycles. (a) Mathematical model of DERA1, (b) Mathematical model of DERA2, (c) Switching model of DERA1, (d) Switching model of DERA2

(a) 300 iA1 [A]

178

iA1_or iA1_eq

200 100 0

0.5

1

1.5

(b)

2

iA2_or iA2_eq

300

iA2 [A]

t[s]

200 100 0

0.5

1

1.5

1

1.5

1

1.5

t[s] 2

iA1 [A]

(c) 300

0

-300 0.5

t[s]

2

iA2 [A]

(d) 300 0

-300 0.5

7.3.2

t[s]

2

Evaluation of Testing Multi-Microgrid with 15 Buses

In this section, an introduced MMG system from [8] is used to perform a systematic comparison for different reduced methods and to further validate the introduced method. The validated range of the dynamic equivalent reduction is tested by comparing the simulation and eigenvalue results of different reduced methods. The structure of the MMG is shown in Fig. 7.13. The parameters of the network and DER are as shown in Tables 7.1 and 7.2. In this section, microgrid b and microgrid c are interconnected through Bus A1 and Bus B2. Microgrid b and microgrid c are connected through Bus B2 and Bus C6. The length of the cable line connecting A1 and B2 is 1.1 km, and that of the cable connecting B2 and C6 is 1.6 km.

7.3 Verification of the Dynamic Equivalent Model 10-4

(a)

Line AB =0.1km Line AB =1km Line AB =10km

4

difference _uA1 [pu]

Fig. 7.11 The response difference of the DERA1 between the original model and reduced model in scenario 2. The lengthen of connection line between microgrid a and microgrid b is changed from 0.1 km to 10 km. (a) The difference of voltage response, (b) The difference of power angel

179

2 0 -2

1

1.5

2

2.5

3

A2 [rad]

-4 (b) 4 10

difference _θ

t [s]

4

Line AB=0.1km Line AB=1km Line AB=10km

2

0

-2

1

difference_u A1 [pu]

2

2.5

3

3.5

t [s]

4

Line AB=0.1km Line AB=1km Line AB=10km

4 0 -4 -8

(b)

1.5

10-3

(a)

difference_­A2 [rad]

Fig. 7.12 The response difference of the DERA1 between the original model and reduced model in scenario 2. The length of connection line between microgrid a and microgrid b is changed from 0.1 km to 10 km. (a) The difference of voltage response, (b) The difference of power angel

3.5

6

1

1.2

1.4

1.6

1.8

2 t [s]

10-3 Line AB=0.1km Line AB=1km Line AB=10km

4 2 0 -2 1

1.2

1.4

1.6

1.8

2 t [s]

180

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

Grid PCC

A1

A2 Droop

A3 Droop

Droop

S1 S11 MGa B1

B2

B3

Droop

B4 Droop

S12 MGb C1

C2

C3 Droop

C5

C4

C6

C7

C8

Droop

S13 MGc

Fig. 7.13 Structure of the MMG test system

There are two major kinds of the reduced model considered in this section. The first relies on the two-time-scale classification [1, 9, 10]. Neglect the dynamics of the output voltage and line current that belong to the fast dynamics. Ref. [11] also excluded these state variables to obtain a reduced model for harmonic stability analysis. The second type is based on a fact that some fast dynamics such as output current and line current have considerable influence on the low-frequency dominant modes, whereas the inner control loop has a minor effect on the dominant modes. The models in [2, 7, 8, 12] are based on a conventional reduced model that include the dynamics of currents and neglect the effect of the inner control loops. Thus, in this section, these two major types of reduced model are compared with the reduced model introduced in section III [13, 14]. The detailed model in Section II incorporates all the system states are considered as a benchmark. The details of these models are outlined in Table 7.4. Model M1 is the detailed model that includes all the system states, which can be used as a benchmark. M2 is the reduced model considering the slow dynamics related to the power controller and inner control loop. The output voltage and current are described by algebraic equations. M3 is a conventional reduced model eliminating the inner control loops. In this section, microgrid b in the tested MMG is selected as study microgrid to analyze its dynamic behavior. Microgrid a and microgrid c are the external microgrids, which are replaced by DEMs. First, the time-domain simulation was conducted to validate the introduced method for the MMG test system with different network parameters. The effect of line impedance in a single microgrid is investigated. The structure of a MMG in Fig. 7.13 is not changed. The line parameters of microgrid c as shown in Table 7.1 are multiplied by a coefficient Kline, respectively. A short-circuit fault with 0.2 Ω fault resistance is designed at bus B3 and clears after 5 cycles. The time-domain simulation is conducted to compare dynamic behaviors of the equivalent model with different coefficients Kline. The subscripts B1 and B3 denote the DERs connected at B1 and B3, respectively. Figure 7.14 plots the response difference of DGb2 in the

7.3 Verification of the Dynamic Equivalent Model

181

Table 7.4 Details of reduced-order models in comparison Model M1 M2 M3 M4

System statesa All system states θ, P, Q, x1, x2, x3, x4 θ, P, Q, iod, ioq, ilineD, ilineQ States in study microgrid and states in DEMs

References Section II and [15] [1, 10]c [8] Section III

Orders 140 48 84 66

Time b 27.5 s 49.1 s 5.3 s 4.7 s

a

The symbols above are defined in Section II Ode15s is selected as the numerical solver to compare the simulation efficiency c There are some differences on the final formation of the reduced models in [1, 10]. However, both of them were based on the time-scale classification and eliminate the line dynamics. M2 is a common reduced model to present this idea b

10 -3

(a)

Kline=0.5 Kline=1 Kline=2 Kline=5

fdiff [pu]

1

0

-1 0

1

2 T[s]

3

(b) 0.02

Kline=0.5 Kline=1 Kline=2 Kline=5

udiff [pu]

0.01 0

10 -3 10.6 10.2 9.8 9.4

-0.01 -0.02 0.9

4

1.1

1

1.104

1.1

1.2 T[s]

1.3

1.4

1.5

Fig. 7.14 Response difference of DGB3 between the reduced model and original model. The Kline is changed from 0.5 to 5. (a) The frequency difference of DGb3, (b) The voltage difference of DGb3

study area between the detail model and reduced model. The response difference becomes larger when the line impedances get larger. The frequency response is more sensitive to line impedance changes. However, such difference is very small. The maximum errors for frequency response and voltage response is smaller than 0.15% and 1.5%, respectively when microgrid under the sever disturbance. Thus, when the line impedances of microgrid are in the rational region, the effect of different impedances separating different buses in a single microgrid can be negligible. The solver ode15s is applied to alleviate the “stiff” issue caused by the introduction of the virtual resistors in M1, M3 and M4. Because of the Kron reduction, a smaller number of virtual resistors are used in M4. The simulation time of different models is presented in Table 7.4. Although M2 has the minimum dimensionality, the

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

(a) 450 uB1 [V]

Fig. 7.15 Transient behavior of the DERs in microgrid b under the threephases short-circuit fault, (a) Output voltage of DER connected to Bus B1. (a) Output voltage of DER connected to Bus B1, (b) Output current of DER connected to Bus B1, (c) Output voltage of DER connected to Bus B3, (d) Output current of DER connected to Bus B3

uB1_M1 uB1_M2 uB1_M3 uB1_M4

400 350 300 0.9

1

1.1

1.2

1.3

(b)

1.4

1.5 t[s]

iB1_M1 iB1_M2 iB1_M3 iB1_M4

300

iB1 [A]

182

200 100 0 0.9

1

1.1

1.2

1.3

1.4

1.5 t[s]

(c) 450

uB3 [V]

400 350 300 250 200 0.9

1

1.1

1.2

1.3

(d)

iB3_M1 iB3_M2 iB3_M3 iB3_M4

500

iB3 [A]

uB3_M1 uB3_M2 uB3_M3 uB3_M4 1.4 1.5 t[s]

300 100 0.9

1

1.1

1.2

1.3

1.4

1.5 t[s]

algebraic loop issue for numerical simulation influences the computational efficiency. The linearization and the matrix transformation as presented in [10] should be used to solve this issue, but this will make this model cannot be used for large signal analysis. Both M3 and M4 improve the computational efficiency, but only M4 has the capability to predict the voltage oscillation and inrush current of the DERs during large perturbations. Then numerical simulation is used to compare dynamic behavior of different reduced models. Figure 7.15 shows the transient responses of the DERs in microgrid b, when three phases short-circuit fault with 0.2 Ω fault resistance occurs at bus B3

7.4 Summary

183

and clears after 5 cycles. Figure 7.15a, c show the output voltage waveforms of DERB1 and DERB3 respectively. Figure 7.15b, d are the output currents waveform of the DERB1 and DERB3 respectively. Because of the elimination of the dynamics related to the output current, and voltage, the transient output voltage and current predicted by the M2 have a large error. The adjusting processes of the inner control loop create the oscillations and overshoot of the output variables, which cannot be obtained from M3. It can be seen from this figure that the reduced model M4 has the best accuracy to describe the transient behavior of the DERs in the study microgrid under a large perturbation.

7.4

Summary

In this chapter, a dynamic equivalent modeling method is introduced to reduce the order of inverter-based MMG model. First, the detailed model of the MMG system including the droop-controlled DERs, PQ-controlled DERs and network is built. The built MMG model is then divided into the study microgrid and external microgrids. The external microgrids are replaced by the DEMs using the structure preservation method. The detailed model of the study microgrid and the DEMs of external microgrid constitute the reduced model. The accuracy of the simplified model is verified by time-domain numerical simulation. According to the work in this chapter, the following conclusions can be drawn: 1. MMG can be divided into different subsystems by observing the coherent characteristics of the IIDGs in microgrids. If angle differences of IIDGs are smaller than certain value when subjected to disturbances, they can be divided into the same subsystem. 2. The dynamic equivalent model introduced in this chapter simplifies the system structure, reduces the model parameters, and completely retains the dynamics in the study microgrid. It can accurately describe the dynamic response characteristics of the fast and slow variables of the DERs in the study microgrid, and the calculation time of time domain simulation is greatly shortened. 3. Numerical simulations validate that the DEMs of the external microgrids have the same effect on the study microgrid as the detailed models. The comparison of different reduced models indicates that the reduced model developed by the DEMs has an advantage in describing the dynamic behavior of DERs within the study microgrid, incorporating accuracy and simplicity. 4. The introduced dynamic equivalent method can be expected to be used for the online-transient analysis and the parameter design of the DERs within a largescale MMG system.

184

7 Spatial-Scale Model Reduction of Multi-Microgrid Based on Dynamic Equivalent. . .

References 1. L. Luo, S.V. Dhople, Spatiotemporal model reduction of inverter-based islanded microgrids. IEEE Trans. Energy Convers. 29(4), 823–832 (2014) 2. I.P. Nikolakakos, H.H. Zeineldin, M.S. El Moursi, J.L. Kirtley, Reduced-order model for interinverter oscillations in islanded droop-controlled microgrids. IEEE Trans. Smart Grid 9(5), 4953–4963 (2017). https://doi.org/10.1109/tsg.2017.2676014 3. F. Dörfler, F. Bullo, Kron reduction of graphs with applications to electrical networks. IEEE Trans. Circuits Syst. I Regul. Pap. 60(1), 150–163 (2013) 4. D.E. Kim, M.A. El-Sharkawi, Dynamic equivalent model of wind power plant using an aggregation technique. IEEE Trans. Energy Convers. 30(4), 1639–1649 (2015) 5. J. Hua, Q. Ai, Y. Yao, Dynamic equivalent of microgrid considering flexible components. IET Gener. Transm. Distrib. 9(13), 1688–1696 (2015) 6. Y. Wang, X. Wang, F. Blaabjerg, Z. Chen, Small-signal stability analysis of inverter-fed power systems using component connection method. IEEE Trans. Smart Grid 9(5), 5301–5310 (2017). https://doi.org/10.1109/tsg.2017.2686841 7. P. Vorobev, P.H. Huang, M. Al Hosani, J.L. Kirtley, K. Turitsyn, High-fidelity model order reduction for microgrids stability assessment. IEEE Trans. Power Syst. 33(1), 874–887 (2018) 8. P. Nikolakakos, H.H. Zeineldin, M.S. El-Moursi, N.D. Hatziargyriou, Stability evaluation of interconnected multi-inverter microgrids through critical clusters. IEEE Trans. Power Syst. 31 (4), 3060–3072 (2016) 9. V. Mariani, F. Vasca, J.C. Vásquez, J.M. Guerrero, Model order reductions for stability analysis of islanded microgrids with droop control. IEEE Trans. Ind. Electron. 62(7), 4344–4354 (2015) 10. M. Rasheduzzaman, J.A. Mueller, J.W. Kimball, Reduced-order small-signal model of microgrid systems. IEEE Trans. Sustain. Energy 6(4), 1292–1305 (2015) 11. Y. Wang, X. Wang, F. Blaabjerg, Z. Chen, Harmonic instability assessment using state-space modeling and participation analysis in inverter-fed power systems. IEEE Trans. Ind. Electron. 64(1), 806–816 (2017) 12. X. Guo, Z. Lu, B. Wang, X. Sun, L. Wang, J.M. Guerrero, Dynamic phasors-based modeling and stability analysis of droop-controlled inverters for microgrid applications. IEEE Trans. Smart Grid 5(6), 2980–2987 (2014) 13. Z. Shuai, Y. Hu, Y. Peng, C. Tu, J. Shen, Dynamic stability analysis of synchronverter -dominated microgrid based on bifurcation theory. IEEE Trans. Ind. Electron. 64(9), 7467–7477 (2017) 14. Z. Shuai, Y. Peng, X. Liu, Z. Li, J.M. Guerrero, Z.J. Shen, Dynamic equivalent modeling for multi-microgrid based on structure preservation method. IEEE Trans. Smart Grid 10(4), 3929–3942 (2019) 15. N. Pogaku, M. Prodanovic, T.C. Green, Modeling, analysis and testing of autonomous operation of an inverter-based microgrid. IEEE Trans. Power Electron. 22(2), 613–625 (2007)

Chapter 8

Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic Phasor Theory

The previous chapters mainly focus on the symmetrical microgrid. The model of microgrid is modeled based on dq coordinates. However, microgrid is often asymmetrical due to integration of single-phase generators, asymmetrical loads and asymmetrical faults. The mathematical model in the rotating synchronous coordinate system cannot describe the asymmetric structure of the microgrid completely, so it is difficult to analyze the stability of the asymmetric microgrid. This chapter proposes the mathematical modeling method of asymmetric microgrid based on dynamic phasor (DP) method and studies the transient response characteristics, and small signal stability of asymmetric microgrid based on the established dynamic phasor model. Through small-signal modeling, it is found that the poor parameter design of the voltage asymmetrical compensation (VUC) control of the inverter can lead to instability of the microgrid. Thus, the parameter tuning method of VUC control is investigated and experimental results are used to validate the effectiveness of introduced method.

8.1

Concept of Dynamic Phasor Method

The DP concept is a generalized averaging method to describe the time-domain quasi-periodic waveform. The DP based on time-varying fundamental frequency is introduced in this section. For a time-domain waveform x(τ) [1], the Fourier expansion of this waveform in the moving window θ2(θ2π, θ] can be presented by the summation of its Fourier series

© Springer Nature Singapore Pte Ltd. 2021 Z. Shuai, Transient Characteristics, Modelling and Stability Analysis of Microgird, https://doi.org/10.1007/978-981-15-8403-9_8

185

186

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

(a)

(b)

x (τ )

7 W 7

π

W

τ

θ (t1-T ) θ (t1)

θ (τ)

Fig. 8.1 (a) Time-domain waveform and defined moving window at t1, (b) Equivalent periodic waveform for phasor transformation at t1

Z

1 hxik ¼ X k ðt 1 Þ ¼ T

t1

t 1 T

xðτÞejkωτ dτ

ð8:1Þ

where ω is the variable system frequency and θ is the phase angle defined Z

t

θ ðt Þ ¼

ϖ ðτÞdτ

ð8:2Þ

0

A time-moving window τ 2 (t1T, t1) can be defined as shown in Fig. 8.1a. Figure 8.1b presents the equivalent waveform for phasor transformation, and its moving window. The mathematical properties of the DP can be found in [1]. Xk(t) is the Fourier coefficient in complex form, which can be defined as a kth DP. It is defined as follows 1 h xi k ¼ X k ð t Þ ¼ 2π

Z

θ1

θ1 2π

xðθÞejkθ dθ

ð8:3Þ

Xk(t) as the kth DP describes the kth harmonics of x(τ) in complex form. The width of window keeps constant with the change of the frequency (θ ¼ 2π), which makes the Eq. (8.1) be integrable. Therefore, this improved DP introduced here can be utilized for the electrical system with time-variable frequency. Since the DPs of a quasiperiodic waveform are constant at steady state. The DP model can be linearized at steady state for small-signal analysis. The main mathematical characters can be described 8 hxðτÞ  yðτÞik ¼ hxik  hyik ¼ X k ðt Þ  Y k ðt Þ > > > > > haxik ¼ ahxik ¼ aX k ðt Þ > > < l¼1 l¼1 P P xy ¼ x  y ¼ X kl ðt Þ  Y l ðt Þ h i h i h i k kl l > > l¼1 l¼1 > > > > dx > dhxik dX k ðt Þ : þ jkωX k ðt Þ dt k ¼ dt þ jkωhxik ¼ dt

ð8:4Þ

8.1 Concept of Dynamic Phasor Method

187

As the fundamental frequency ω is time-varying. Its mathematical description is essential and should be included in a complete DP model. For a real time-domain waveform x(t), its DPs also have the property X k ðtÞ ¼ X k ðtÞ

ð8:5Þ

where X k ðt Þ is the complex conjugate of Xk(t). Substituting Eq. (8.5) into Eq. (8.1), the real time-domain waveform can be written xðt Þ ¼ X 0 ðt Þ þ

1  X

  Re 2X k ðt Þejkωt

ð8:6Þ

k¼1

It can be seen from the Eq. (8.6) that the real waveform can be presented by the DPs whose order k≧0. In DP modeling, the numbers of DPs for a time-domain waveform are decided according to the accuracy requirement. For the balanced electrical system [2], the inverter model commonly contains fundamental component of DP for the variables in ac side and dc components of DP for the variables in dc side. According to Eq. (8.6), conventional DP modeling requires base frequency be constant to get stable integral. At the same time, the DPs which have same frequency can be calculated. However, in a microgrid, the system frequency is dominated by DGs. When the system is disturbed or rearranged power, the frequency of the system will be changed. Although the fluctuation of frequency is small, it influences power angle and power distribution between DGs. The fluctuation of frequency in the system can’t be ignored. The conventional DP method, which assumes that the fundamental frequency is constant, is not suitable for constructing the mathematical model of microgrid. Therefore, it is necessary to improve the definition formula of DP. Firstly, the integral of fundamental frequency with time is defined as phase angle. Z θ1 ¼

t1

ωs ðτÞdτ

ð8:7Þ

0

The kth dynamic phasor is a kth time-varying complex Fourier coefficient that can be defined hxik ¼ X k ðt 1 Þ ¼

1 2π

Z

θ1 θ1 2π

xðθÞejkθ dθ

ð8:8Þ

At this time, the integral window is fixed as the phase angle of 2π rad that the fundamental frequency component of the time domain waveform passes through in a period. Therefore, the formula can be used to calculate the DP of quasi-periodic waveform with the variation of standard frequency. At this point, the waveform in time domain can be expressed

188

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

xð τ Þ ¼

þ1 X

Xke

k¼1

jθ1

¼

þ1 X

Xk e

jωs τ

ð8:9Þ

k¼1

Compared it with Eq. (8.2), the difference between them is that the system frequency is a time domain variable. The mathematical properties of Eqs. (8.4) to (8.6) are also applicable to DPs based on time-varying frequencies. It is worth emphasizing that the operation rules between DPs are valid only when the base frequency is always the same.

8.2

Dynamic Phasor Modeling of Asymmetrical Microgrid

In this section, a 220 V, 50 Hz test microgrid is built to validate the DP model result. As shown in Fig. 8.2, the test microgrid consists of two VSG-based DGs and a single-PV. Three asymmetrical loads are connected to Bus 1 and Bus 3 respectively. The DP model of inverter-based microgrid is divided into the inverter-based DGs, line network and load. Detailed modeling procedures are shown as follows.

8.2.1

Dynamic Phasor Model of VSG

In this point, the DP model of the VSG-based DG is developed. The output current and voltage on the ac side contain 1st fundamental frequency component, and the variables on dc sides consider the dc and 2nd harmonic component. Because the harmonics of the measured signals in the controller can be filtered using low-pass filter, only dc components (hωi0 and hMfifi0, hTei0 respectively) are considered. The -kth DPs are presented as the complex conjugate of kth DP using Eq. (8.5).

Fig. 8.2 Test system of the asymmetrical microgrid system

Bus 1

Zline1

Bus 3 Z load3

VSG 1 Rs=20Ω Zline3

Zload1 Bus 2

PV

Zline2

VSG 2 Zload2

Utility grid

8.2 Dynamic Phasor Modeling of Asymmetrical Microgrid

8.2.1.1

189

DC Side of VSG

The DP model of the dc side can be written as follow dhidc i0 ¼ udc  hudc i0  Rdc hidc i0 dt X h        i dhudc i0 Cdc d j þ1 iij 1 þ d j 1 iij þ1 ¼ hidc i0  dt j¼a, b, c Ldc

dhidc i2 ¼ hudc i2  Rdc hidc i2  j2Ldc hωi0 hidc i2 dt X     dhudc i2 C dc d j þ1 iij þ1  j2Cdc hωi0 hudc i2 ¼ hidc i2  dt j¼a, b, c Ldc

ð8:10Þ ð8:11Þ ð8:12Þ ð8:13Þ

where Ldc and Rdc denote the inductor and resistor respectively. Vdc* is the voltage of ideal voltage source and Vdc is the voltage of input capacitor. idc and is are the output current from the ideal voltage source and input current of inverter, respectively.

8.2.1.2

Control Part of VSG

The dynamic equations of power controller can be presented by using the DP equations  dhωi0 P ¼   h T e i 0  D p ω   h ωi 0 dt ω    d M fif 0 K ¼ Q þ Dq V   hV i0  hQi0 dt J

ð8:14Þ ð8:15Þ

Where J is moment of inertia and Dp denotes active damping coefficient. Te is electromagnetic toque and ω is output angular frequency of synchronverter. V* denotes the reference terminal voltage amplitude, ω* is the reference frequency, and P* and Q* denote the reference active and reactive power, respectively. Dq is the voltage-drooping coefficient, K is inertia coefficient related to Dq. Mf and if denote the virtual mutual inductance and rotor excitation current respectively, and Mfif is treated as a dynamic state for the voltage control. The DPs of the electromagnetic torque 0 and the reactive power 0 are presented

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

190

X 

hT e i0 ¼

M f i f  iLj  sin θ j

j¼a, b, c

¼

X 

M fif

 0

            i sin θ þ M i i sin θ Lj j f f Lj j 1 1 0 1 0 1

ð8:16Þ

j¼a, b, c

X 

hQi0 ¼

j¼a, b, c

X h

¼

ωM f i f  iLj  cos θ j

 0

            i hωi0 M f i f 0 iLj 1 cos θ j 1 þ hωi0 M f i f 0 iLj 1 cos θ j 1

j¼a, b, c

ð8:17Þ The first DPs of the reference output voltage uij of phase j ( j ¼ a, b, c) can be written         ui,j 1 ¼ ω  M f i f  sin θ j 1 ¼ hωi0 M f i f 0 sin θ j 1

ð8:18Þ

The DP of the output voltage to neutral node uig,j can be presented     1 X   u uig,j 1 ¼ d j 1 hudc i0  0:5hudc i0  3 j¼a, b, c o,j 1

ð8:19Þ

The DP model of each three-phase inverter is modeled at its local frequency at first. The first DPs of the sinθj and cosθj in Eqs. (8.17)–(8.18) at the fundamental angle θ ¼ ωt can be calculated as follow 8 Z 1 θ ejθ  ejθ j1θ 1 > > ¼ dθ ¼ 0  j sin θ e h i > a 1 < 2π θ2π 2 2j > h sin θb i1 ¼ h sin θa i1 e3πj > > : 2 h sin θc i1 ¼ h sin θa i1 eþ3πj 2

8.2.1.3

ð8:20Þ

LC Filter and Coupling Inductor

The output LC filter and the coupling inductance DP model can be represented as follow Lf

dii,j dt

1

        ¼ uig,j 1  ii,j 1 R f  vo,j 1  jhωi0 L f ii,j 1

ð8:21Þ

8.2 Dynamic Phasor Modeling of Asymmetrical Microgrid

Cs Lg

dio,j dt

1

dvo,j dt

1

191

      ¼ ii,j 1  io,j 1  jhωi0 C s vo,j 1

ð8:22Þ

      1 X   ¼ vo,j 1 þ u  io,j 1 Rg  ub,j 1 3 j¼a, b, c b,j 1    jhωio Lg io,j 1

ð8:23Þ

j denotes the phase ( j ¼ a, b, c), vo,j is the output voltage of LC filter. For the threephase three-leg inverter, there is no zero sequence current channel for the filter current ii,j, the summation of the filter current ii are equal to zero.

8.2.2

Dynamic Phasor Model of Single-Phase PV

The basic configuration of a single-phase PV is illustrated in Fig. 8.3. Single stage DC/AC inverter is used for energy conversion. The main elements of the single-stage PV are the PV array, input capacitor C, DC/AC inverter and L filter. The control system of the PV is presented in Fig. 8.4. The control system consists of the MPPT, PLL, the current control loop with PR controller and PWM module [3]. The amplitude pffiffiffi of the reference output current of PV Imag is calculated by equation: Imag ¼ 2Ppv =U b,j , where Ppv is the PV array output power, Ub,j is the RMS value of grid voltage. When the inverter is working under the unit power factor mode, the angle of the output current is provided by the PLL that measures the angel of bus voltage. Here, the effects of the MPPT and the dynamics of PLL are not taken into consideration [4]. Since the DPs of the reference output current i* is in phase with DPs of the grid side voltage ug, the DP of the i* can be written

Ra,j PV Array

La,j

DC/AC Inverter

C

ub,j

Fig. 8.3 Single-phase PV system

PPV

udc idc

MPPT

Imag

2/ U b, j

sinθ PLL

Fig. 8.4 Basic control of the PV system

i*

PR controller i

PWM uout

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

192

    hi i1 ¼ I mag  ug,j 1 =2U g,j 

ð8:24Þ

The PR controller is used to track the ac signal i*. Defining the intermediate states x1 and x2 in the PR controller, the dynamic equations of the PR controller can be presented according to [3]. 8   dh x i > < 1 1 ¼ 0:5 iref 1  hii1  2jhωi0 hx1 i1 dt > : dhx2 i1 ¼ 0:5i   hii ref 1 1 dt

ð8:25Þ

The first component of the DP for the output voltage uout can be written    huout i1 ¼ K p iref 1  hii1 þ K r hx1 i1 þ hx2 i1

ð8:26Þ

Considering the first DP of the dynamic in the L filter, the DP equation of output current can be written Ls

diout,j dt

1

        ¼ uout,j 1  iout,j 1 R f  ub,j 1  jhωi0 Ls iout,j 1

ð8:27Þ

Substitute Eqs. (8.26) into (8.27), the DP model of the single-phase PV consists of the Eqs. (8.25) and (8.27).

8.2.3

Aggregation of DG Model

The angular frequency of the output voltage varies during the transient process. As the DP model of each DGs is defined on its local fundamental frequency. To connect DGs into a complete microgrid model, the output of each DG should be transformed into a common fundamental frequency. The relationship of the first DP of variable with different frequency ω is carried out 

xp,1

 p,1

¼e

jθqp

hxiq,1

ð8:28Þ

R where θqp ¼ (ωq  ωp)dt, hxip, 1 is the first DP of x with frequency ωp, and hxiq, 1 is the first DP of x with frequency ωq. One of the DG is selected as the master DG whose frequency is specified as the common fundamental frequency ωcom, and the rest of the DGs are the slave DGs. The master DG provides common fundamental frequency to all the subsystem of microgrid. As the fundamental frequency of PV is the frequency of the bus voltage measured by the PLL. Thus, PV should be taken as slave DG due to its incapability

8.2 Dynamic Phasor Modeling of Asymmetrical Microgrid

193

of frequency manipulation. The DPs of the output current of slave DGs are redefined on the common fundamental frequency.   io,j m,1 ¼ e

jθsm



vo,j

 s,1

ð j ¼ a, b, cÞ

ð8:29Þ

R where θsm ¼ (ωs  ωm)dt, subscripts m denotes the common fundamental frequency ωcom, s denotes salve DGs. The bus voltage should be transformed into the local frequency as the input of each DG, which can be written 

ig,j

 s,1

¼e

jθms



ug,j

 m,1

ð j ¼ a, b, cÞ

ð8:30Þ

R where θms ¼ (ωm  ωs)dt. When the microgrid is in grid-connected mode, the utility grid can be equivalent as the ideal voltage, whose voltage and frequency are constant.

8.2.4

Dynamic Phasor Model of Load and Network

The load connected to microgrid is equivalent to the series connection of the resistors and inductance (RL load). The dynamic equations of the RL load connected at node i are Lloadi,j

diloadi,j ¼ ubi,j  Rloadi,j iloadi,j , ðj ¼ a, b, cÞ dt

ð8:31Þ

The DP model of RL loads are defined on the common fundamental frequency ωcom, which can be written Lloadi,j

diloadi,j dt

1

      ¼ ubi,j 1  iloadi,j 1 Rloadi,j  jωcom Lloadi,j iloadi,j 1

ð8:32Þ

The DP model of network is developed using the algebraic equations in matrix form for a concise presentation. The network model is defined on the common frequency ωcom. It should be noticed that in three-phase framework, each phase of nodes should be defined individually. The series admittance between two nodes ( p, q) is denoted by the 3  3 complex matrix Ypq 2 6 Ydq ¼ 6 4

Rpq,a þ jωcom Lpq,a 0 0

1 

0 Rpq,b þ jωcom Lpq,b 0

3

0

1 

0 Rpq,c þ jωcom Lpq,c

1

7 7 5

194

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

where Rpq,j, Lpq,j and ωcom denote the line resistance, inductance, and the common fundamental frequency respectively. For a network with l Buses, the network matrix can be presented by network matrix Ynet2 Rl  l. Where the elements in this matrix Ynet are the n  n matrix (n ≦ 3) denoted as follow  Ynet ðp, qÞ ¼

Xi¼l i¼1

Ypi ,

‐Ypq , O,

if p ¼ q if p 6¼ q \ ðp, qÞ 2 λ else

where O denotes zero matrix. The set λ ¼ {(i, j)} denotes that there is a connection between the buses i and bus j through a distribution line. If a phase of line does not exist, the corresponding column and row should be zero quantity. To avoid the singularity of network matrix, these rows and columns should be deleted. After delete the zero columns and rows, the final form of network matrix Ynet’ is developed. Thus, the network interactions can be presented by the admittance matrix Ynet based on Ohm’s and Kirchoff’s laws io 2 iload = Ynet 0 ub

ð8:33Þ

where io, iload and ub denote the inject current vector, output load current vector and node voltage vector in complex form respectively as follow Io ¼ ½io1,a , io1,b , io1,c , io2,a , . . . , iol,c T Iload ¼ ½iload1,a , iload1,b , iload1,c , iload2,a , . . . , iload3,c T Ub ¼ ½ub1,a , ub1,b , ub1,c , ub2,a , . . . , ubl,c T The superscript T denotes the transposition of matrix. For the phases of a node that do not exist, the corresponding element in these vectors are deleted. If there is no DG connected to the phase a of node j, ioj,a equal to zero, so does the iloada,j. The node voltage of network can be calculated from Eq. (8.33) 21 ðio 2 iload Þ ub = Ynet

ð8:34Þ

The node voltages of network are treated as the input for each subsystem. Finally, the complete DP model of microgrid can be obtained by combing the DP model of three-phase DGs, single-phase DGs, loads and network.

8.2.5

Dynamic Phasor Model of Asymmetrical Microgrid

To make the developed state-space model suitable for numerical calculation, first dynamic phasor can be presented using a two-dimensional vector, whose two

8.2 Dynamic Phasor Modeling of Asymmetrical Microgrid

195

elements in this vector denotes the value on real axis and imaginary axis of complex plane, respectively. For example, the currents and voltages of LCL filter and distribution line can be presented as follow: h iT h iT    þ þ þ   uþ ) u , u ¼ u , u ) u , u ¼ uo2 , o 1 o ,re o,im o o 1 o ,re o,im  þ  T    T þ  ¼ iþ io 1 ) i ¼ io2 , io 1 ) iþ o,re , io,im o, o,re , io,im D E h iT D E h iT iþf ) iþf ,re , iþf ,im ¼ iþ if ) if ,re , if ,im ¼ if2 , f , 1 1    T iline lm,j 1 ) iline lm,j,re , iline lm,j,im ¼ iline lm,j    T unode,j 1 ) inode,j,re , inode,j,im ¼ unode,j 

ð8:35Þ

where the subscript re and im denote the value on real or imaginary axis, respectively. Therefore, one complex state equation is divided into two state equations. To develop a concise system presentation for the state-space matrix, following vectors are defined:       T T 2T T 2T T i f ¼ iþT , uo ¼ uþT , io ¼ iþT f , if o , uo o , io  T  T δ = if T , uo T , io T , α = hθi0 , hPi0 , hQi0 :

ð8:36Þ ð8:37Þ

For the signal transformations, the transformation from dq to stationary coordidiag nate T1 can be presented using transform matrix T1 as: T1 ¼ (0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1), the phase angle transformation T2 can be transformed in matrix form as  T2e ¼ e ¼ jθ

cos θ

 sin θ

sin θ

cos θ

 ð8:38Þ

The model of power controller in state-space form is α_ ¼ Ap Δα þ BP T1 Δδ þ Bpω ½Δωcom ,     Cpω Δω ¼ Δα , Δuo Cpv 2

0 mp 6 Ap ¼ 4 0 ωc

where

0 

0 0  mq ½033

0

ð8:39Þ ð8:40Þ

3 2 3 2 3T 0 1 0 7 6 7 6 7 0 5, Bpω ¼ 4 0 5, Cpw ¼ 4 mp 5 , Cpv ¼ ωc 0 0

T and

Bp ¼ 32 

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

196

2

½014

6 6 6 6 ½ 0 4 24

½012

D E ωc i þ oq 0 D E þ ωc ioq

  ωc iþ od 0   ωc iþ od 0

½012

½012 D E  þ ωc uod 0 ωc uþ oq 0 D E  þ þ ωc uod 0 ωc uoq

½022

0

½012

3

7 7 7. ½022 7 5

0

The voltage controller can be written in state-space form as:   ½Δx_ 13  ¼ Bv1 Δuo þ Bv2 T1 ½Δδ, h i   Δ_if = Cv ½Δx13  þ Dv1 Δuo þ Dv2 T1 ½Δδ: where x13 ¼ [x1d, x1q, x3d, x3q]T,BV1 ¼ diag (1, 1, 1, 1), Cv ¼ Kic diag(1,1,1,1), Bv2 ¼ ½ ½044 diagð1, 1, 1, 1Þ diag(1,1,1,1), 2 6 6 6 Dv2 ¼ 6 ½044 6 4

K pv

ωn C f

0

0

ωn C f

K pv

0

0

0

0

K pv

ωn C f

0

0

ωn C f

K pv

ð8:41Þ ð8:42Þ

½044  , Dv1 ¼ Kpv 3

7 7 7 diagð1, 1, 1, 1Þ 7: ð8:43Þ 7 5

The current controller is written in state-space form as:   ½Δx_ 24  ¼ Bc1 Δif þ Bc2 T1 ½Δδ,

ð8:44Þ

½Δui  ¼ Cc ½Δx24  þ Dc1 Δif þ Dc2 T1 ½Δδ,

ð8:45Þ

where x24 ¼ [hx2di0, hx2qi0, hx4di0, hx4qi0]T, Bc1 ¼ diag(1,1,1,1), Bc2 ¼ ½ diag2ð1, 1, 1, 1Þ ½048  , Cc ¼ Kicdiag (1,1,1,1), Dc1 ¼ Kpc 3diag (1,1,1,1), K pc ωn L f 0 0 7 6 7 6 ωn L f K pc 0 0 7 6 diagð1, 1, 1, 1Þ ½044 7: Dc2 ¼ 6 7 6 0 0 K pc ωn L f 5 4 0 0 ‐ωn L K pc The LCL filter are written in state-space form as:   Δδ_ ¼ ALCL Δδ þ BLCL1 ½Δui  þ BLCL2 ½Δunode  þ BLCL3 Δω, 2 1

where ALCL

 diag Z f , Z f

6 Lf 6 6 1 ¼6 6 C f diagð1, 11, 1Þ 6 4 ½044

1 diagð1, 1, 1, 1Þ Lf diagðSω , Sω Þ 1 diagð1, 1, 1, 1Þ Lg

½044

ð8:46Þ 3

7 7 7 1 diagð1, 1, 1, 1Þ 7 7 Cf 7 5  1 diag Zg , Zg Lg

8.2 Dynamic Phasor Modeling of Asymmetrical Microgrid

 Sω ¼

0 ω0

BLCL2 ¼

 ω0 , BLCL1 ¼ L1f ½ diagð1, 1, 1, 1Þ 0 

½048 T ,

Rf diagð1, 1, 1, 1Þ  , Zf ¼ ω0 L f

1 ½0 48 Lg ½

197

T

 ω0 L f , Rf

h iT þ   þ þ þ þ BLCL3 ¼ iþf ,im ,iþf ,re ,if ,im ,if ,re ,uþ o,im ,uo,re ,uo,im ,uo,re ,io,im ,io,re ,io,im ,io,re : To transform the inverter model from local coordinate to common coordinate, the dynamic phasors of output currents are transformed from PNZ reference frame at local frequency to abc reference frame at common frequency, and node voltages are from abc reference frame at common frequency to PNZ reference frame at local frequency, which can be written as ½Δio,abc  ¼ T3 T2 ½Δio  þ TC Δθ

ð8:47Þ

½Δunode  ¼ T2r T3r ½Δunode,abc  þ TV Δθ

ð8:48Þ

where T2 = diag(T2e, T2e), T2r = diag(T2e21, T2e21), and T3 denotes the phaseangle transformation and the reference-frame transformation, respectively. TC ¼

∂T3 T2 ½io  ∂T2r T3r ½unode,abc  , TV ¼ : ∂θ ∂θ

ð8:49Þ

A complete inverter model can be obtained by combing the state-space model of power controller, inner-loop controller and LCL filter. There are totally 23 states for a complete DG model, which are ½Δx_ inv  ¼ ½ hθi0

hPi0

hQi0

X13

X24

if

io T :

uo

ð8:50Þ

A complete state-space model for an inverter i can be written as. " ½Δx_ inv,i  ¼ Ainv,i ½Δxinv,i  þ Binv,i ½Δunode,abc  þ ,   BVUC,i Δu‐ o,i þ Bcom,i Δωcom

Δω Δio,abc

#

" ¼

Cinv,ω Cinv,c

# ½Δxinv ,

ð8:51Þ where 2

AP,i

6 BV1,i Cpv,i 6 6 6 BC1,i Dv1,i Cpv,i 6 6B 4 LCL2,i ½TV , 0, 0þ BLCL3,i Cpw,i

½034

Ainv,i ½034

½044 BC1,i Cv,i

½044 ½044

BLCL,i Dc1,i Cv,i

BLCL1,i Cc,i

3

¼

7 Bv2,i T1 7 7 BC1,i Dv2,i T1 þ BC2,i T1 7 7 7 ALCL,i þ BLCL1,i 5

,

Bpi T1

ðDc1,i Dv2,i T1 þ Dc2,i Þ

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

198

2 

Binv,i 

½0116 ¼ BLCL2 T 2r T 3r

½022



 ,



Bcom,i ¼ 

½ Cpω



Bpω

,

½0201

BVUC,i

½034 BV1,i

3

6 6 ¼6 6 BC1,i Dv1,i 4

7 7 7 7 5

½0124 if DGi is the master DG if DGi is the slave DG

½0120 

, Cinvω,i ¼ ½0123 diagð1, 1Þ ,Cinvc,i ¼ ½ Tc ½0618 T3 T2 . Define the master DG as the first DG, a combined model of n inverters can be written as:   ½Δx_ INV  ¼ AINV ½ΔxINV  þ BINV ½Δunode,abc  þ BVUC ΔUþ O

ð8:52Þ

½ΔiO  ¼ CINV ½ΔXINV 

ð8:53Þ

where  T  T ½ΔxINV  ¼ ΔxTinv,1 , . . . , ΔxTinv,n ,½Δunode  ¼ ⋯, ΔuTnode,1 , ⋯, ΔuTnode,n , ⋯ iT   h þ T T , ΔUþ ,[AINV] ¼ diag (Ainv,1 + Bcom,1Cinvω,1, . . ., Ainv, ¼ Δuo,1 , ⋯, Δuþ o,n O  T T T , [CINV] ¼ diag (Cinvc,1, . . ., n + Bcom,nCinvω,n),½BVUC  ¼ BVUC,1 . . . BVUC,n Cinvc,n). [BINV] are defined based on the inverter connecting point to the network nodes as:  ½BINV  ¼ ⋯

⋯

BTinv,n



ð8:54Þ

23n6m

For the microgrid with k lines (including single-phase, double-phase and threephase line), the network can be written as: 

 Δ_iline ¼ Anet ½Δiline  þ Bnet Δunode,abc

ð8:55Þ

where [Δiline] ¼ [Δiline _ 1, a, re, Δiline _ 1, a, im, ⋯Δiline _ k, j, re, Δiline _ k, j, im]T, 02

Anet

Rline1,a B6 Lline1,a 6 ¼ diagB @4 ω

2 h

31 ‐ω 7C 7C, Rline1,a 5A Lline1,a

3 2 R ‐ω 7 6 line1,a 7, ⋯6 Lline1,a Rline1,a 5 4 ω Lline1,a

Bnet ¼ BTnet1,a , BTnet1,b ⋯BTnetk,j

iT

6 , Bnetk,j ¼ 6 4

The node voltage can be written as:

⋯ ⋯

1 Llinek,j 0

0 1 Llinek,j

⋯ ⋯

1 Llinek,j

0

⋯

0

1 Llinek,j

⋯

3 7 7. 5

8.3 Eigenvalue Analysis of Asymmetrical Microgrid

199

½Δunode  ¼ An ðMinv ½Δio,abc  þ Mline ½Δiline Þ

ð8:56Þ

where An ¼ diag (Anode, 1⋯Anode, m), Minv and Mline are the mapping matrix that denote the inverter connection point and connecting line onto network nodes, respectively. The elements in mapping matrix should be put either +1 or 1 if the given current is entering or leaving the node. Now the small-signal state-space model of a complete unbalanced microgrid and hence system state matrix can be obtained and presented as: 

Δx_ INV Δ_iline



 ¼ AMG

   ΔxINV BVUC    þ ΔUO Δiline ½ 0

ð8:57Þ

  AINV þ BINV An Minv Cinv BINV An Mline . where AMG ¼ Bnet An Minv Cinv Anet þ Bnet An Mline  þ  ΔUO as an input signals of microgrid model are calculated according to corresponding VUC control strategy.

8.3

Eigenvalue Analysis of Asymmetrical Microgrid

The parameters of DGs are shown in Table 8.1, the parameters of network and load are shown in Table 8.2. In the test system, two VSG-based DGs are equally rated. The parameters of two DGs are the same so that they share the power equally during transient process. The measured electromagnetic torque Te, reactive power Q and magnitude of output voltage E pass through second-order Butterworth low-pass filter to attenuate the effect of harmonics. The high-order filters have little effect on the dynamics of VSG due to the relatively large time constant of VSG controller. Table 8.1 DG parameters

Parameters P* Q* V* Dp Dq τf τV Lf Rf Cf Lg

Value 10 kW 5 kVar 320 V 20.28 W/rad2 200 Var/V 0.15 s 0.15 s 3 mH 0.2 Ω 35 μF 1.8 mH

Parameters Rg Cdc Ldc Rdc u*dc Ppv Kp Kr Ls Rs

Value 0.3 Ω 1 mF 1 mH 0.2 Ω 900 V 2 kW 3 500 0.8 mH 0.2 Ω

200

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

Table 8.2 Network and load parameters Parameter Zline1 Zline2 Zline3 Zload1 Zload2 Zload3

Value (Phase a) 0.6 + 0.002ωjΩ 0.75 + 0.0025ωjΩ – 25 Ω 30 Ω 30 + 0.05ωjΩ

Fig. 8.5 Eigenvalue spectrum of the asymmetrical microgrid

6

Value (Phase b) 0.6 + 0.002ωjΩ 0.75 + 0.0025ωjΩ – 40 Ω 35 Ω 10 + 0.05ωjΩ

Value (Phase c) 0.6 + 0.002ωjΩ 0.75 + 0.0025ωjΩ 0.35 + 0.0013ωjΩ 40 Ω 30 Ω 10 + 0.05ωjΩ

103

Imaginary [rad/s]

4 2 0 -2 -4 -6 5 -10

-104

-103

-102

-101

-100

Real [1/s], log scale

The dynamic stability of VSG-dominated microgrid and chosen values of droop coefficient have been discussed in [5]. The purpose of this section is to validate the capability of DP model for eigenvalue analysis. A fixed equilibrium of asymmetrical microgrid can be obtained from the DP model. Thus, the linearized state matrix and eigenvalues of the microgrid can be derived without the balanced assumption. The DP model of the test system is developed in MATLAB/Simulink environment. This DP model is linearized around the operating point using the MATLAB function “linmod,” and eigenvalues are calculated by the function “eig”. Finally, the eigenvalue spectrum of asymmetrical microgrid can be obtained. As shown in Fig. 8.5, these eigenvalues can be divided into three clusters. The eigenvalues in cluster 3 are far from the right-half plane, while those in cluster 2 are widely distributed in the frequency region. The dominant eigenvalues in cluster 1 are close to the imaginary axis, and the participation analysis is applied to measure the coupling between the state variables and eigenvalues. From the participation analysis, the eigenvalues in cluster “3” relate to the output current in the coupling inductance of DGs. The eigenvalues in cluster “2” are largely sensitive to the state variables of LC filter, load and dc sides of variables. The dominant modes as shown in cluster “1” largely relate to the state variables of the power controller in the VSG and inner control loop of PV. The dominant low-frequency eigenvalues in cluster 1 and their related states are presented in Table 8.3.

8.3 Eigenvalue Analysis of Asymmetrical Microgrid

201

Table 8.3 Sensitive of dominant eigenvalues Index λ1–2 λ3 λ4 λ5 λ6–7 λ8–9

8.3.1

Eigen values 2.95  7.75j 6.23 7.30 11.52 14.91  3.37j 15.17  629.23j

Related states hθ12i0, hω1i0 hMfif1i0, hMfif1i0, hω1i0, hω2i0 hMfif2i0, hMfif1i0, hω1i0, hω2i0 hMfif2i0, hMfif1i0 hx2i1 hx1i1

Participation 0.49, 0.22 0.26, 0.25, 0.24, 0.23 0.26, 0.25, 0.23, 0.23 0.46, 0.43 0.84 0.85

Case Study 1: Load Disturbance Test

In the first test, a disturbance in load of bus 3 was arranged. This requires the addition of a resistance load Rs in parallel to bus 3, as shown in Fig. 8.2. This disturbance was chosen to be 6.5 kW (Rs ¼ 20 Ω). Figure 8.6a, b show the active and reactive power response of the VSG 1, respectively. Due to the asymmetrical condition of microgrid, the output power of the DGs contains second harmonics. As can be seen in Eq. (8.6), the combination of the DPs hPii0 + 2|hPii2 and hPii0  2|hPii2 corresponds to the upper and lower envelop of the active power in the switching model. hQii0 + 2|hQii2 and hQii0  2|hQii2 corresponds to the upper and lower envelop of the reactive power. The transient responses of the DP model match well with that of the switching model. Figure 8.6c depicts the frequency response of the test system. With the increase of the load, the frequency of the output voltage of VSG-based DG decreases.

8.3.2

Case Study 2: Asymmetrical Short-Circuit Fault Test

In the second test, two phase grounded fault with 1 Ω fault resistance is conducted in phase a and b of the bus 2 and is cleared after 5 cycles. The voltage of bus 2 and the fault response of DGs are presented in Fig. 8.7. As presented in Fig. 8.7a, the bus voltage at phase a and b dip to 47% of the value at steady state. The reference voltages of VSG-based DGs rise after this fault, which leads to the increase of the bus voltage at phase c. Figure 8.7b, c depict the output current of VSG-based DG 1 and DG2, respectively. In Fig. 8.7c, the output current of DG 2 is much larger than that of DG 1, due to that DG is closest to the fault location. As shown in Fig. 8.7d, the output current of the single-phase PV increases abruptly and then decreases to the reference value due to the inner control. The capacitor voltage of the dc sides of VSG 2 is shown in Fig. 8.7e. The DP model predicts the oscillation of dc capacitor voltage under asymmetrical fault. As the second DPs describe the magnitude of the oscillation, the spike voltage predicted by DP model may not exist in switching model. But

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

Fig. 8.6 System responses of the asymmetrical microgrid with a 6.5 kW load step at bus 3. (a) Active response of the DG1, (b) Reactive response of the DG1, (c) Angular frequency response of the system

(a)

15

P1 [kW]

202

P1 0 +2| P1 2 |

P1

0

10

P1 0 – 2| P1 2 | 5 1.3

(b) Q1 [kvar]

6

1.4

Q1 0 + 2| Q1 2 |

1.5

Q1

1.6

0

4 2 0 1.3

Q1 0 – 2| Q1 2 | 1.4

1.5

1.6

(c) 315 ωcom [rad/s]

1.7 t [s]

1.7 t [s] ω com ωcom

314.6

0

314.2 1.3

1.4

1.5

1.6

1.7 t [s]

the DP model predicts the worst scenario, which may destroy the capacitor under asymmetrical fault. Figure 8.7f shows the midpoint to neutral voltage. The waveforms of switching model are filtered by the low-pass filter to extract the fundamental component. A large oscillation with fundamental frequency appears during the asymmetrical faults, which imposes on the output voltage and deteriorates the voltage balance. The three-phase four-leg inverter or isolating transformer can mitigate node to ground voltage, but increase the cost and power loss of microgrid. The simulation time of different scenarios is as presented in Table 8.4. The timedomain simulation of DP model runs much faster than that of the detailed model in MATLAB/SimPowersystem. Although the simulation time of the model relates to computing capability of computer, the simulation time from Table 8.4 reflects the small computation burden of DP model. This is because the DPs describe the magnitude of the ac signals, the states in DP model vary slowly even when instantaneous quantities change abruptly. Therefore, large step time can be chosen for numerical simulation.

8.3 Eigenvalue Analysis of Asymmetrical Microgrid

ubus2 [V]

(a) 500

ubus2a ubus2b ubus2c

0

-500 1.4

1.5

1.6

1.7

1.8 t [s]

(b) 100

io1a io1b io1c

50 0 -50 -100 1.4

1.5

1.6

1.7

1.8 t [s]

(c) 150

io2a io2b io2c

io2 [A]

100 50 0 -50

-100 -150 1.4

1.5

1.6

1.7

(d)

2 i pv

ipv [A]

20

1.8 t [s]

1

i pv 0 -20 1.4

1.5

1.6

1.7

1.8 t [s]

1.7

1.8 t [s]

(e) 1000 udc

udc [V]

950

0

+ 2 udc

udc

900 850 800

1.4

1.5

1.6

(f) 150 100

ung [V]

Fig. 8.7 System responses of the asymmetrical microgrid when a singlephase short circuit occur at phase a of the bus 2. (a) Bus voltage of bus 2, (b) Output current of the DG1, (c) Output current of the DG2, (d) Output current of singlephase PV, (e) DC voltage of DG2, (f) DC midpoint to neutral voltage

203

2 ung

50

1

0 -50 -100 -150 1.4

ung 1.5

1.6

1.7

1.8

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

204

Table 8.4 Simulation time of model Scenarios 1 2

8.4

Time to be simulated 3s 5s

Switching model in SimPowerSystem 2 min 48 s 4 min 23 s

DP model 3s 5s

Improved Voltage Unbalance Compensation Strategies for Asymmetrical Microgrid

A low-voltage microgrid is naturally three-phase asymmetrical due to the connection of asymmetrical loads. VUC strategies of inverters have been introduced to compensate the voltage unbalance on the PCC. However, the small-signal analysis of these VUC strategies have not been investigated in a microgrid with multiple inverters. This lack of discussion may result in a poor performance of the microgrid when the parameters of VUCs are not well designed [4]. Last section develops a detailed state-space model for an asymmetrical microgrid using DP method. The accuracy of the model is verified by numerical simulation. To investigate the dynamic behavior of the asymmetrical microgrid with VUC strategies, small-signal analysis is carried out to investigate the effect of the VUC controller on the dynamic behavior of the microgrid by using DP modeling method in this section. Then, an improved compensation method with better compensation performance is introduced. Finally, Experimental results are used to validate the effectiveness of asymmetrical method. As shown in Fig. 8.8, the control system of inverter is built on dq synchronous coordinate whose rotating frequency is generated by local power controller. Since all the states variables of the control system are dc variables, the inverter controller can be presented by the dc components of DPs. In the introduced control strategy, the VUC control provides the reference of the negative-sequence voltage. The positivesequence and negative-sequence component of the measured three-phase quantities are extracted using the Park transformation with notch filter. Fig. 8.8 Block diagram of DG inverter

PCC if Driving signal PWM ui,abc ωt

uo

Decomposition of positive-sequence and negative-sequence components i +f,dq

i −f,dq

++

+ dq ui ,dq * Voltage and abc current − dq ui ,dq * control loops abc

−ω t

ω

io

+ uo,dq + uo,dq *

ω − * uo,dq

+ io,dq

− uo,dq

− io,dq

Power controller VUC controller

8.4 Improved Voltage Unbalance Compensation Strategies for Asymmetrical Microgrid Fig. 8.9 Testbed details

205

dSPACE 1006 PWM

PWM

DC Supply inverter1

Bearker1 Voltage measure

8.4.1

inverter2 DC voltage: 650V Rated frequency: 50HZ PWM switching frequency: 10kHZ

Bearker2 PCC

Symmetrical resistive load

Danfoss Inverter: 2.2kW

Asymmetrical resistive load

Small-Signal Analysis of the Voltage Unbalance Compensation Control

In this section, the introduced modeling framework is utilized for the small-signal stability analysis of an asymmetrical microgrid with VUC control. An eigenvalue analysis is performed to investigate the dynamic behavior of the typical autonomous VUC controls [6, 7]. Figure 8.9 plots the configuration details of the test system. The testbed includes two Danfoss 2.2 kW inverters. The paralleled inverters are connected to a PCC with symmetrical and asymmetrical resistive load. For the three-phase three-line systems, the VUC control is designed to regulates the negative-sequence components of the output voltage of the inverters, by which the inverters can share the asymmetrical currents and compensates the voltage unbalance at PCC. The autonomous VUC methods are to create a voltage droop for the negative-sequence components according to the local measurements. The severity of asymmetrical can be evaluated using an index such as negative-sequence power Q [6, 7]. The negative-sequence power Q can be defined by using local measurement, which can be found in previous literature. In [7], the Q is defined rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

  2 D E 2ffi Q ¼V þ i i od 0 oq 



0

ð8:58Þ

where urms is the rms value of rated voltage. For autonomous VUC, the concept of droop control has been introduced to create the voltage droops. Figure 8.10 plots a typical autonomous VUC strategy. The filtered Q is multiplied by a negativesequence voltage droop gain Kn, and by the instantaneous negative-sequence of output voltage v o that is also filtered to prevent the sudden change of the control reference. Based on the DP modelling framework, the VUC strategy is added to the test microgrid. The parameters of control system and resistive load are presented in Table 8.5. The DP model is developed in MATLAB environment, and is linearized around an operating point using the MATLAB function “linmod”. Its eigenvalues

206

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

Fig. 8.10 Block diagram of a typical VUC controller

−

u o ,d Q

−

ωc , n ωc , n + s −

u o ,q

Table 8.5 Parameters of the test microgrid

Parameters mp mq Pnom ωnom Enom ωc Kvp,p, Kvp,n Kvi,p, Kvi,n Kcp,p, Kcp,n Kci,p, Kci,n Kn

Value 0.0001 0.0005 2.2 kW 50  2π 200 V 20 rads/s 2 200 5 400 0.015

ωc , n ωc , n + s

*−

u o ,d

 Kn ωc , n ωc , n + s

Parameters ωc,n Lf Rf Cf Lg Rg RAN RBN RCN RAB

*−

u o ,q

Value 100 rads/s 1.8 mH 0.1 Ω 9 μF 1.8 mH 0.12 Ω 115 Ω 115 Ω 115 Ω 57 Ω

are then calculated. Participation analysis can be carried out to evaluate the relationship between the state variables and dynamic modes. Figure 8.11 plots the eigenvalue spectrum of asymmetrical microgrid with the typical autonomous VUC control. The application of the VUC control introduces six modes in cluster λ1–6. Moreover, the VUC controller participates in the mediumfrequency mode cluster λ7–14 that are relate to the negative-sequence components dynamics of the LC filter. Figure 8.12 plots eigenvalue locus with the change of the parameters in VUC controller. With the increase of Kn, the modes in cluster λ1–6 move away from the imaginary axis, which indicate an improved transient performance. However, the modes in cluster λ7–14 move to the imaginary axis. When Kn is equal to 0.06, there are two pairs of the modes on the right half plane and the system becomes unstable. According to the conclusion from [7], the compensation effort will increase with the increase of Kn. Thus, there is a trad-off between the compensation effect and the dynamic performance. When the cut-off frequency ωc,n decreases, the modes in λ7–14 move away from the imaginary axis. However, the modes in λ1–6 move closer to the imaginary axis, which indicates the slow transient responses of the VUC controller. Figure 8.13 plots the eigenvalue locus with the change of the parameters in the negative-sequence voltage controller. It can be observed that the increase of the integral coefficient Kvi,n can improve the transient behavior of the voltage tracking, but deteriorates the system stability. On the other hand, increasing Kvpn only decreases the damping of the modes in cluster λ7–14. Besides, the modes on the

8.4 Improved Voltage Unbalance Compensation Strategies for Asymmetrical Microgrid

207

6000

Imaginary [rads/s]

4000

6000

VUC controller

2000

λ

0.2 0.105

7-14

0.7

λ

0

0

1-6

0.7

-2000 -4000

0.2 0.105 -100

-6000 -2000

-6000 -106

-105

-104

-103 Real [1/s]

-102

-101

-100

Fig. 8.11 Eigenvalue spectrum of microgrid with typical VUC control. The modes in cluster λ1–6 are relate to the dynamics of the VUC controller. The modes in λ7–14 are relate to the negativesequence component of the dynamics of the LC filter

Imaginary [rads/s]

(a) 6000

0.44 0.62

0.32

4000

0.23

0.16

2000

0.84

Kn increases from to 0.08

0.1

0.05

Kn = 0.06

0 -2000

0.84

0.62 -6000 0.44 -3000 -2500 -4000

0.32 -2000

0.23 0.16 -1500 -1000

0.1 0.05 -500 0

500

Real [1/s]

Imaginary [rads/s]

(b) 6000 4000

0.4

2000

0.7

0.28

0.19

0.135

0.095 0.06 0.03

ωcn decreases from 100rads/s to 20rads/s

0 -2000 0.7 -4000 0.4 -6000 -2000

0.28 -1500

0.19 0.135 -1000 real [1/s]

0.095 0.06 0.03 0 -500

Fig. 8.12 Eigenvalue locus as a function of VUC control parameters. (a) 0.01  Kn  0.08, (b) 20rads/s  ωc,n  100rads/s

real axis that are related to the voltage controller move to the imaginary axis when Kvp,n increases. According to the eigenvalue analysis above, the application of VUC control and the negative-sequence control loop influences the dynamics of LC filter. For the VUC controller, a small value of the negative-sequence droop Kn and the cut-off frequency of low-pass filter ωc,n can improve the stability. However, the decrease of

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

208

Imaginary [rads/s]

(a) 6000 4000

0.4 0.56

2000

0.8

0.28

0.2

0.14

0.09

0.04 4e+03

Kvin increases from 200 to 3600

0 -2000

0.8

-4000

0.56 0.4

-6000 -2500

-2000

0.28 -1500

0.2 0.14 -1000

0.09 -500

4e+03 0.04 0

500

Real [1/s]

Imaginary [rads/s]

(b)

1.5

104 0.085

0.058

1

0.042 15

0.028 0.018 0.008

0.5 0

0 Kvpn increases from 2 to 20

-0.5 -1 -1.5 -1500

0.085

0.058 -1000

-15 -100 0.042 -500

0 0.028 0.018 0.008 0

Real[1/s]

Fig. 8.13 Eigenvalue locus as a function of parameters in negative-sequence voltage control. (a) 200  Kvp,n  3600, (b) 2  Kvi,n  20

Kn weakens the compensation effects and the decrease of ωc,n slows down the transient response. For the negative-sequence voltage controller, a small value of integral coefficient Kvi,n improves the stability, but results in a slow transient response for voltage tracking. To validate the small-signal analysis results, the DGs in the test microgrid are equipped with the conventional VUC. In the initial state, only the balanced resistive load is connected to the PCC. The negative-sequence power and output current of the DG1 are observed when an asymmetrical load (RAB ¼ 115 Ω) is activated at 0.6 s. Figure 8.14 plots the dynamic response when the value of the negative-sequence voltage droop Kn is 0.02, and the cut-off frequency ωc,n is set to 100rads/s. The value of negative-sequence power and the magnitude of output current becomes large after the load disturbance, and the microgrid keep stable during this period. Figure 8.15 presents the dynamic response when the value of the Kn and ωc,n are set 0.07 and 100rads/s, respectively. The output current starts to oscillate and microgrid eventually lose its stability. Figure 8.16 presents dynamic response when the value of the Kn and ωc,n are set 0.07 and 20rads/s, respectively. The small value of ωc,n slows the transient responses of VUC controller down, but the unstable oscillations are damped. Figure 8.17 plots the output voltage when the integral coefficient of negativesequence voltage controller is set to 3000. At first, the VUC controller is blocked and

8.4 Improved Voltage Unbalance Compensation Strategies for Asymmetrical Microgrid

Q-[kVar]

0.2 0.1 0 5

io1[A]

Fig. 8.14 Transient response of DGs with Kn ¼ 0.02, ωcn ¼ 100rads/s, when the asymmetrical load is connected at 0.6 s. Upper: negative-sequence power Q. Bottom: output current io1

0 -5 0.2

0.6 t [s]

0.8

1

Q-[kVar]

2 1.5 1 0.5 0 100 50 0 -50 -100 0.2

0.4

io1[A]

Fig. 8.15 Transient response of DGs with Kn ¼ 0.07, ωc,n ¼ 100rads/s, when the asymmetrical load is connected at 0.6 s. Upper: negative-sequence power Q. Bottom: output current io1

209

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t [s]

Q-[kVar]

0.2 0.1 0 5

io1[A]

Fig. 8.16 Transient response of DGs with Kn ¼ 0.07, ωc,n ¼ 20rads/s, when the asymmetrical load is connected at 0.6 s. Upper: negative-sequence power Q. Bottom: output current io1

0 -5 0.2

0.6 t [s]

0.8

1

2000 1000

uo1[V]

Fig. 8.17 The output voltage of DG 1 when the integral coefficient Kvi,n is set to 3000. The VUC controller is activated at 0.2 s

0.4

0 -1000 -2000 0.1

0.2

0.3

t [s]

the reference of negative-sequence voltage controller is set to 0. An oscillation of output voltage can be observed. The VUC controller is activated at 0.2 s, and the complete system lose is stability.

210

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

8.4.2

Compensation Method to Improve the Dynamic Behavior

In this section, an improved VUC control is designed according to the above smallsignal analysis to improve the transient performance and stability margin. Figure 8.18a depicts the theory of the introduced VUC, in order to have a good compensation effect by using a small value of Kn, the compensation reference is designed to act in the opposite phase of the negative-sequence voltage of the PCC. The superscript “’” denotes the variables after the compensation. When the positivesequence voltage is much large than the negative-sequence voltage, the negativesequence current is determined by the positive-sequence voltage and asymmetrical load, and is almost unaffected by the change of the negative-sequence component of output voltage [8]. The negative-sequence component of PCC voltage can be estimated   U pcc ¼ V o  Z g I o

ð8:59Þ

Figure 8.18b depicts the block diagram of the improved VUC controller. The negative-sequence voltage of PCC is estimated using the local measurement. Besides, to improve the current sharing of the negative-sequence components, a current sharing module can be included on secondary control level. The current sharing module includes a current control loop with proportional (P) controller, where the reference is the average value of the negative-sequence output currents of all the DGs. For the parameter design, a relatively small cut-off frequency ωc,n is applied to the low-pass filter. The compensation gain is designed to make sure that the dampingratio of medium-frequency modes is greater than 0.15. A small value of Kn is selected to improve the damping of the oscillatory modes. The introduced VUC control with parameters shown in Table 8.6 is compared with the traditional VUC with parameters in Table 8.5. The DP model of the introduced control are written ! 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

  2 D E 2ffi >   >    > iod 0 þ ioq  hQ i0 hQ i0 ¼ ωcn V > > 0 > < D E

  D E D E       _ u ¼ ω v  R i  ω L i  u > cn g n g od 0 pccd od 0 oq > > D pccd E0 0 >

D E D E D E0  >   > : u_      ¼ω v R i þω L i  u pccq

0

cn

oq

0

g

oq

0

n g

od 0

pccq

ð8:60Þ

0

where Q represents negative-sequence reactive power, upcc represents phase voltage of PCC. vod and iod represent output voltage and output current of inverter. Subscripts ‘dq’ represents variables in dq synchronous coordinate. Superscripts ‘*’ and ‘’ represent reference value of variable and variables in negative sequence respectively. The reference values of negative-sequence voltage are

8.4 Improved Voltage Unbalance Compensation Strategies for Asymmetrical Microgrid

(a)

Im

211

− U pcc

Compensation Reference

− Z g I o−

− U pcc ’

U o− Uo-’

− Z g I o−’ ≈ − Z g I o−

Re

(b) Current sharing module

− io−,avr abc iod ,avr + i i i dq − i od ∑ io− / n + io− − ioq ,avr − ioq Communication Links − uod ωcn − + iod Rg -- − ω ubd * cn + s

DG1

− o1

DG2

− o2

DGn

− on

ωL g

Q−

ωL g − ioq

Rg

-+ +

ωcn ωcn + s

K pn K pn

-K n

− ubd * ωcn

− uoq

*− uod

++

*− uoq

++

ωcn + s

VUC controller

Fig. 8.18 Introduced VUC control. (a) Vector Diagram, (b) Block Diagram Table 8.6 Control parameters of VUC controls

Parameters Kn Kcs ωc,n

Value 0.01 1 50 rads/s

Parameters Kvp,n Kvi,n

8 

 D E       > < v ¼ K i  i Q  u  K h i cs n od 0 pccd 0 od,avr 0 od 0 0 D E

D E D E  D E >     : voq ¼ K cs ioq,avr  ioq  K n hQ i0  u pccq 0

0

0

0

Value 1 200

ð8:61Þ

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

212

6000

Imaginary [rads/s]

4000

0.28

0.19

0.135

0.06 0.03

Traditional VUC

0.4

Proposed VUC

2000 0.7 7-14

0

20

0.8

0.972 -80

0.8 -40

0

-2000 0.7 -4000

0.972

7-14

0.4

-6000 -2000

-20 0.28

0.19

-1600

0.135

-1200 -800 Real [1/s]

0.06 0.03 -400

-120

0

0

Fig. 8.19 Eigenvalue spectrum of the two VUC strategies

Figure 8.19 plots the eigenvalue spectrum of the two VUC control strategies. Under the introduced control structure, the damping of the oscillatory modes related to negative-sequence component is similar with that of the oscillatory modes relating to positive-sequence components. The transient response of the introduced VUC is designed to be slower than that of the voltage control loop. To validate the introduced VUC strategy, the typical VUC control and improved VUC control are compared using the simulation results. The voltage unbalance factor (VUF) of the microgrid with different VUC controls is overserved. The VUF of the PCC can be calculated as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2 u pcc,d þ upcc,q VUF ¼ 100  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2 uþ pcc,d þ upcc,q

ð8:62Þ

In the simulations, the VUC controllers are activated at 0.4 s, then another resistive load (RAB ¼ 115 Ω) is activated at 0.5 s. Figure 8.20 plots simulation results of the typical VUC control. Although a larger Kn have been used, voltage unbalance is slightly decreased. After the step-change of asymmetrical load, the VUF becomes higher. Figure 8.21 plots simulation results of the introduced VUC control. In this case, the voltage unbalance is significantly alleviated. The output current is almost no change after the activation of VUC, which validates that negative-sequence voltage of DGs uo has little effect on the negative-sequence current io. Then the experiment validation is carried out based on the dSPACE. Figure 8.22 presents the experimental results of the testbed with the traditional VUC control. The output voltage of the DGs are as shown in Fig. 8.22a, b, c, d present the three-phase output current, and Fig. 8.22e shows the negative-sequence components of the output current on the dq coordinate. The amplitude and negative-sequence component of three-phase currents of two inverters are different, which indicates the unequal sharing of the asymmetrical load between the two DGs. As depicted in

8.4 Improved Voltage Unbalance Compensation Strategies for Asymmetrical Microgrid 0.4

VUF[pu]

Fig. 8.20 Simulation result of the typical VUC controller. Upper: VUF of PCC, Bottom: output current of DG1

213

0.3 0.2 0.1

io1[A]

0 5

0

-5 0.3

t [s]

0.5

0.6

0.5

0.6

0.4

VUF[pu]

Fig. 8.21 Simulation result of the introduced VUC controller. Upper: VUF of PCC, Bottom: output current of DG1

0.4

0.3 0.2 0.1

io1[A]

0 5

0

-5 0.3

0.4

t [s]

Fig. 8.22f, the PCC voltage is three-phase asymmetrical due to the integration of the single-phase load. Figure 8.23 plots the experimental results for the testbed with the introduced VUC control. In Fig. 8.23c, d, the current difference between two DGs is reduced. As depicted in Fig. 8.23e, negative-sequence components of output currents are equally shared between DG1 and DG2. As shown in Fig. 8.23f, the voltage unbalance is well compensated and the magnitude differences between phase voltages decrease. Figure 8.24 presents the compensation results of voltage compensation of VUC controls. Before the application of VUC control, the negative-sequence of the PCC voltage is relatively high, as a result, the VUF of the PCC is higher than 1. Conventional VUC can only slightly decrease the voltage unbalance. On the other hand, the introduced VUC provides a good compensation results with a relatively small negative-sequence voltage droop Kn.

214

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . . (a) 300

uob1

uoa1

uoc1

(b) 300

0

-150 -300

(c)

-150 0s

6

iob1

ioa1

-300

0.1s

0.05s

(d)

ioc1

0s

6

0.1s

0.05s

ioa2

iob2

ioc2

3

io2 [A]

io1 [A]

uoc2

0

3 0 -3 -6

0 -3

0s

-6

0.1s

0.05s

(e)

(f) od1

i

0s

0.1s

0.05s

400

od2

i

200

0

ioq1

upc c [V]

3

io- [A]

uob2

150 uo2 [V]

uo1 [V]

150

uoa2

ioq- 2

0

-200 -3

0s

0.05s

0.1s

-400

upcc,a 0s

upcc,b

upcc,c

0.05s

0.1s

Fig. 8.22 Experiment results of the asymmetrical testbed with the traditional VUC

8.5

Summary

This chapter develops DP modeling procedure for inverter-based microgrid under asymmetrical condition. Firstly, the DP modeling method for the inverter-based microgrid with time-vary frequency is introduced. According to the characteristics of microgrid and control system, the method of transformation for DP under different frequencies and different reference coordinates is given. A DP model of asymmetrical microgrid has been established according to introduced DP modeling method. Then, the model accuracy is validated using a simulation study. To investigate the influence of the typical VUC control on the stability of microgrids, a smallsignal analysis is performed based on the developed model. The analysis results are used to guide the design of an improved VUC controller. The experimental results

8.5 Summary (b) 300

150

150

uo2 [V]

300

uo1 [V]

(a)

215

0

0 -150

-150 -300

(c)

-300 0s

0.05s

0.1s

(d)

6

0.1s

0s

0.05s

0.1s

0s

0.05s

0.1s

3

io2 [A]

io1 [A]

0.05s

6

3 0

0 -3

-3

-6

-6 0s

(e)

0s

0.05s

0.1s

(f) 400

3

upc c [V]

io- [A]

200

0

0 -200

-3

0s

0.05s

0.1s

-400

Fig. 8.23 Experiment results of the testbed with the introduced VUC

validate the compensation effect of the introduced VUC control. Several conclusions can be obtained: 1. The DP modeling method introduced in this chapter can completely and accurately describe the transient characteristics of asymmetric microgrid. The DP uses the dc variable to describe the dynamic process of ac waveform, which can be accurately described by a large simulation step, and the time of numerical simulation calculation is greatly reduced. 2. The DP model completely describes the asymmetrical microgrid which has stable operating points under the steady-state condition. It is found that there are interactions between inverters equipped with asymmetrical voltage compensation control, and the instability of asymmetrical microgrid may be caused when the parameters of asymmetrical compensation control or negative sequence voltage control are not well designed. 3. An improved asymmetrical voltage compensation control is designed based on the results of small signal analysis in this chapter. The experimental results show

8 Modeling and Stability Analysis of Asymmetrical Microgrid Based on Dynamic. . .

Fig. 8.24 Compensation results of VUF when apply: (a) conventional VUC control, (b) the introduced VUC control

(a) VUF [pu]

216

1.5 1 0.5 0

VUF [pu]

(b)

0s

1s 0.5s 1.5s Traditional VUC control

2s

0s

1s 0.5s 1.5s The proposed VUC control

2s

1.5 1 0.5 0

that this control strategy can effectively suppress the voltage unbalance at the PCC of microgrid, and improve the current sharing of negative-sequence component among different inverters.

References 1. T. Yang, S. Bozhko, J.M. Le-Peuvedic, G. Asher, C.I. Hill, Dynamic phasor modeling of multigenerator variable frequency electrical power systems. IEEE Trans. Power Syst. 31(1), 563–571 (2016) 2. X. Guo, Z. Lu, B. Wang, X. Sun, L. Wang, J.M. Guerrero, Dynamic phasors-based modeling and stability analysis of droop-controlled inverters for microgrid applications. IEEE Trans. Smart Grid 5(6), 2980–2987 (2014) 3. Z. Miao, L. Piyasinghe, J. Khazaei, L. Fan, Dynamic Phasor-based modeling of asymmetrical radial distribution systems. IEEE Trans. Power Syst. 30(6), 3102–3109 (2015) 4. Z. Shuai, Y. Peng, J.M. Guerrero, Y. Li, Z. John Shen, Transient response analysis of inverterbased microgrids under asymmetrical conditions using dynamic phasor model. IEEE Trans. Indus. Electron. 66(4), 2868–2879 (2019). https://doi.org/10.1109/tie.2018.2844828 5. Z. Shuai, Y. Peng, X. Liu, Z. Li, J.M. Guerrero, et al., Parameter stability region analysis of islanded microgrid based on bifurcation theory. IEEE Trans. Smart Grid 10(6), 6580–6591 (2019). https://doi.org/10.1109/tsg.2019.2907600 6. M. Savaghebi, A. Jalilian, J.C. Vasquez, J.M. Guerrero, Autonomous voltage unbalance compensation in an islanded droop-controlled microgrid. IEEE Trans. Ind. Electron. 60(4), 1390–1402 (2013) 7. Y. Han, P. Shen, X. Zhao, J.M. Guerrero, An enhanced power sharing scheme for voltage unbalance and harmonics compensation in an islanded AC microgrid. IEEE Trans. Energy Convers. 31(3), 1037–1050 (2016) 8. L. Meng, X. Zhao, F. Tang, M. Savaghebi, T. Dragicevic, J.C. Vasquez, et al., Distributed voltage unbalance compensation in islanded microgrids by using a dynamic consensus algorithm. IEEE Trans. Power Electron. 31(1), 827–838 (2016)

Chapter 9

Transient Angle Stability of Grid-Connected VSG

Due to the limited thermal capacity and small inertia of inverter-interfaced distributed generators in microgrid, they are more prone to experience transient instability during fault period. Thus, it is important to study transient stability through theoretical method. Transient angle stability refers to the withstanding and synchronization ability of synchronous generators (SGs) to the system when subjected to large disturbance like grid voltage drop [1–3]. It is found that transient angle stability, which is usually discussed in traditional power system with SGs, may occur in microgrid as well. Virtual synchronous generators (VSGs) are proposed to mimic dynamic response of SGs. As a typical control strategy of power converters, it is pointed out in [4–7] that transient angle instability results from the sustained unbalance between the output power and virtual reference power of VSGs. Similar to SGs, VSGs will experience transient angle instability under certain conditions, which probably threatens the system security. In this chapter, the transient angle stability of VSGs is investigated. The mathematical models of VSGs are established for transient stability analysis. Then, the transient instability mechanism is explored. Furthermore, a Lyapunov method is introduced to quantify the stability regions of VSGs.

9.1

Mathematical Model

Although VSGs emulate inertia and damping characteristics of SGs, their differences lie in that VSGs can regulate output active power as well as reactive power through active and reactive power control loops. It will lead to the differences of transient angle stability between SGs and VSGs. The theoretical analysis and simulation results show that VSG is more prone to instability because the reactive power control loop reduces its internal voltage amplitude through Q-V curve during severe faults. To give insight into instability mechanism and to evaluate stability region of VSGs, full-order model and reduced model are elaborated first. © Springer Nature Singapore Pte Ltd. 2021 Z. Shuai, Transient Characteristics, Modelling and Stability Analysis of Microgird, https://doi.org/10.1007/978-981-15-8403-9_9

217

218

9.1.1

9

Transient Angle Stability of Grid-Connected VSG

Full-Order Model of a VSG

VSG is proposed to mimic the dynamic characteristics of SG. The topology of a grid-connected VSG is shown in Fig. 9.1. It includes three inverter legs operated using PWM to transfer power from DC side to AC side and LCL filters to reduce the voltage and current ripple. LT is the transformer equivalent inductor and Ll represents for the equivalent line inductor (also includes Rl). VSG control can be divided into two parts: active power control loop (swing equation) and reactive power control loop (Q-V droop). The active power control loop is responsible to mimic mechanical rotor motion of SGs, as shown in Fig. 9.2a. The reactive power control loop is used to regulate output reactive power of a VSG, which is shown in Fig. 9.2b. Active power control loop and reactive power control loop produce phase angle and voltage magnitude respectively to generate reference sine waves, which are then delivered to the PWM module to produce trigger signals. A virtual speed governor is added into active power control loop to adjust the reference active power according to the system frequency [8]. Figure 9.3 shows the reference frame of VSG and grid, where D-Q frame is the common frame of the gird which rotating at the angular speed of ωg. The frame of

Fig. 9.1 The main topology and control block of a gridconnected VSG

Lt Lf

Grid

Rl

Ll

C Cf

ioabc

uoabc

signal +

E PWM modulation

Fig. 9.2 Control block of a VSG. (a) Active power control loop, (b) Reactive power control loop

LT

Lg

θ

1 Ks

+

E*

-

+

Q*

Q0

Q-V droop E*

Q

P*

Swing Equation

(a)

Governor ω*

Swing Equation Dp

-

+

ω*

Speed Governor ω

+

Kp

ω*

-

+

1

P*

T*

ω0

+

P0

(b)

1 Js

+ -

1 θ s

ω

Te

Q-V Droop E

+

Dq

E*

-

Q0 +

Q*

+

1 Ks

Q

+

E +

E*

9.1 Mathematical Model

219

Fig. 9.3 Reference frame of VSG and grid

Q q

ω δ

d

ωg D

VSG is denoted as dq frame whose speed is ω. The power angle δ is defined as the transformation angle of the VSG with respect to the common reference frame, which can be express as dδ ¼ ω  ωg dt

ð9:1Þ

The equation of reactive power loop is exhibited as follow Q0 ¼ Q  Dq ðE  E Þ

ð9:2Þ

where Dq is the droop coefficient. E* and Q* represents the reference voltage and reactive power. Similar to reactive power control loop, active power loop can be derived according to Fig. 9.2a with slightly more complication P 0 ¼ P   K p ðω  ω  Þ 

T droop ¼ Dp ðω  ωÞ

ð9:3Þ ð9:4Þ

where P* is the rated active power, P0 is reference active power which derived from speed governor. ω* is the reference angular speed and Te is the output torque, which can be calculated by Te ¼ P/ω*. Since the swing equation of VSG can be expressed by J

d2 θ ¼ T   T em  T droop dt 2

ð9:5Þ

where J denotes the virtual inertia. Combining Eqs. (9.3) and (9.5)–(9.7), the complete active power expression of VSG is J

d2 δ ¼ T   T e  D ð ω  ω Þ dt 2

ð9:6Þ

where T* ¼ P*/ω*, and D ¼ Dp + Kp/ω*, D is the equivalent damping. The mathematical equations of reactive power control loop is

220

9

K

Transient Angle Stability of Grid-Connected VSG

dE ¼ Q  Q  Dq ðE  E  Þ dt

ð9:7Þ

where K is the voltage integral coefficient and Dq represents the Q-V coefficient. E0 and Q0 are the reference voltage and reference reactive power respectively. The state equations of the output LCL filter and line impedance are presented did ¼ ed  vd þ ωL f iq dt diq Lf ¼ eq  vq  ωL f id dt dv C f d ¼ id  iod  ωL f vd dt dvq Cf ¼ iq  ioq þ ωL f vq dt di Lt od ¼ vd  vgd  Rl iod þ ωLg ioq dt dioq Lt ¼ vq  vgq  Rl ioq  ωLg iod dt

ð9:8Þ

Lf

9.1.2

ð9:9Þ ð9:10Þ ð9:11Þ ð9:12Þ ð9:13Þ

Model Reduction of a VSG

Although the model is ninth order, it is still complicated for the analysis of transient angle stability under large disturbance. The practical significance of singular perturbation theory is in revealing underlying structural properties and multi-time-scale structures. The detailed theory and reduction process have been discussed in Part II of this book and can be further referred in literature [9] if readers are interested. Parameters of a detailed model will be given as an example in Table 9.1. The

Table 9.1 Parameters of the model

Parameters ω* E* P* Q* J Dp K Dq Vdc

Values 314.159 rad/s 311 V 20 kW 5 kVar 5.0224 10 16 166.7 800 V

Parameters fs Edc Lf Cf Lg LT Ll Rl Vg

Values 5000 Hz 1000 V 4 mH 40 uF 1 mH 2 mH 0.5 mH 0.2 Ω 311 V

9.1 Mathematical Model

221

Fig. 9.4 Eigenvalue spectrum of the system state matrix

8

103

6

2

Imaginary Part

4 2

λ5

λ4

1

λ3

-2

λ3

-4

λ1

λ2

0

λ1

λ4

λ5

-6 -8 -120

-100

-80

-60

-40

-20

0

Real Part

Table 9.2 Participation factors of eigenvalues

Cluster 1

Eigen value 1.86  j6.90

2

12.98 101.96  j4641.52 102.11  j4012.70 104.70  j312.90

State δ ω E iodq vdq iodq vdq idq iodq

Participation 0.49 0.49 0.98 0.16 0.25 0.16 0.25 0.33 0.16

eigenvalue spectrum of the system state matrix is illustrated in Fig. 9.4 and the participation factors of eigenvalues are listed in Table 9.2. It can be seen that different frequency modes can be divided into two clusters, which are denoted as clusters 1 and 2. In order to identify the fast and slow state variables of the system, further information needs to be obtained by observing the participation factor of different state variables in a particular mode. This can be achieved by the participation factor analysis conducted on the system state matrix which is already shown in Table 9.2. State variables whose participation factors for different modes greater than 0.1 are listed. It can be observed that the dominant modes in cluster 1 are sensitive to the state variables of the controller of a VSG and modes in cluster 2 are sensitive to the state variables of the LCL filter block and line currents. Since modes λ1 and λ2 in cluster 1 are the dominant poles, detailed analysis of the dynamics of state variables δ, ω and E are presented at Table 9.1. The regulation of the P-f droop loop of a VSG is shown in Fig. 9.2a. The time constant of the P-f droop loop is τ f ¼ J=Dp

ð9:14Þ

222

9

Transient Angle Stability of Grid-Connected VSG

where a larger inertia J means more energy is stored mechanically. Dp represents the ability of a VSG for preventing the frequency deviation. For a DER, τf is around hundreds of milliseconds. The control topology of Q-V droop loop is shown in Fig. 9.2b. Similar to the P-f droop loop, the time constant of the Q-V droop loop can be calculated by τv ¼ K=Dq

ð9:15Þ

It is known that in SGs, the process of establishing field flux is much faster than the motion of the mechanical rotor. For a VSG, the time constant τv of a Q-V droop loop ranges from a few milliseconds to tens of milliseconds. For a SG, the studied time scale of a transient angle stability problem is from hundreds of milliseconds to a few seconds. It is of the same order for a VSG since a smaller inertia in a VSG incorporates a less damping ability compared to a SG. Thus, it is known that a Q-V droop loop would reach the steady state as soon as possible during the transient period. From this point of view, the derivative equation of inverter’s internal voltage E in Eq. (9.7) can be turned into a quasi-steady equation, which means that E can be treated as a parameter variable rather than a state variable. Based this assumption, the derivative equations of a VSG introduced above turn into quasi-steady equations and the full order model can be simplified into a two-order derivative equation plus a quasi- steady equation, as shown below J

E 2 G  EV g B sin δ  EV g G cos δ d2 δ ¼ T   Dðω  ω Þ 2 ω dt 0 ¼ Q þ E2 B  EV g B cos δ þ EV g G sin δ  Dq ðE  E Þ

ð9:16Þ ð9:17Þ

where X is the total inductance between a VSG and a three- phase voltage source and can be defined as X ¼ ω(Lf + Lt), B ¼ R/(X2 + R2), G ¼ X/(X2 + R2). It is noted that the resistor Rl and ω2LfCf are small so that they can be ignored.

9.2

Transient Angle Stability Mechanism

In Sect. 9.2, transient instability mechanisms of SGs and VSGs are investigated based on the mathematical models and an approximate Lyapunov’s direct method is introduced to estimate the transient stability of VSGs, which takes the effect of reactive power control loop into account. Besides, simulation results are presented to validate the theoretical analysis.

9.2 Transient Angle Stability Mechanism

9.2.1

223

Transient Angle Stability of VSG

The equivalent equation of a VSG connected to an infinite bus is derived above. It is known that a voltage-controlled inverter can be represented by a voltage source in order to study the transient characteristics. Based on this assumption, the equivalent circuit can be established in Fig. 9.5b, where Vg is the equivalent RMS voltage of the grid, and Z represents the equivalent line impedance. Take the grid voltage angle as a reference, the angle difference between a VSG and the grid is the power angle δ. To better illustrate the effect of the reactive power control loop on transient angle stability, the equivalent model of a VSG with constant inverter’s internal voltage (without considering reactive power control loop) is established in Fig. 9.5a as a comparison. Only power angle δ is controllable through P-f droop loop in Fig. 9.5a. Basically, there are two types of transient angle instability for a grid connected VSG, depending on whether the system has equilibrium points (EPs) during disturbance [7]. Thus, the extended equal area criterion (EEAC) method and P-δ curve will be applied in this section to illustrate transient angle stability. To analyze system transient response of a VSG in P-δ curve, Eq. (9.16) can be turned into J

EV g d2 δ ¼ T0  sin δt  Dðω  ω Þ jZ jω dt 2

ð9:18Þ

where T’ ¼ T*E2G/ω* and δt ¼ δφ, Z and φ are the line impedance and the impedance angle, respectively. In transmission grid, resistors are usually ignored because the inductance X is much larger than the resistor R. Under this condition, the transferred active power of a VSG can be represented as P  EVsinδ/X, which is commonly used in the conventional power system. However, the resistor cannot be ignored in the distribution network, which would make the active power is not an odd function of the power angle. Hence, the translation of the coordinate is adopted to change the Eq. (9.18) into an odd function.

(a)

(b) Eq.(9.18) Eq.(9.19)

Eq.(9.18)

VSG

E∠δ

P+jQ

Grid Z=R+jω L

VSG

E∠δ Vg ∠0

P+jQ

Grid Z=R+jω L

Vg ∠0

Fig. 9.5 Equivalent circuit of a VSG connected to power grid: (a) without considering reactive power control, (b) considering reactive power control

224

9.2.1.1

9

Transient Angle Stability of Grid-Connected VSG

Case 1 Existence of Equilibrium Points

Figure 9.6a shows the typical power angle curve of single VSG connected to an infinite bus when a sudden increase of reactance is considered (e.g., a sudden disconnection of one transmission line in a system). According to the swing equation of VSG, the rotor of VSG will accelerate. The operation point moves from a along the P-δ cure to reach a new stable state. If there is equilibrium point (EP) shown by curve II in Fig. 9.6a, it is possible for a VSG to maintain transient stability during fault period, even fault is not cleared. The system won’t be stable unless rotor speed returns to synchronous speed before point d, which is known as unstable equilibrium point (UEP). Otherwise, the power angle will go beyond the UEP and keep accelerating due to Pe > Pm. The system would loss the synchronization.

9.2.1.2

Case 2 None Existence of Equilibrium Points

For the situation that without EPs, for example, three-phase to ground fault, the rotor speed will continuously increase during disturbance, as shown by curve III in Fig. 9.6a. Due to the absence of EPs, load power is always less than electromagnetic power. Hence, the rotor keeps accelerating during the fault. In this situation, the fault clearing time is vital for stable operation. If the protective relay acts in time, the rotor of the SG decelerates after the fault is cleared and it is possible for the rotor speed to recover to the synchronous speed before the UEP.

9.2.2

Deteriorative Effect of Q-V Droop on Transient Angle Stability

As shown in Fig. 9.6b, Curve I describes the relationship between P and δ of the pre-fault and post-fault systems while curve II is the P-δ curve of the system under a fault. There are two EPs denoted by δs and δu where power balance can be reached. δs represents the stable EP and δu is the UEP. Extended Equal Area Criterion (EEAC) will be used to explain the criterion for transient angle stability intuitively. In pre-fault steady state, the system operates at point δs. Then, a grounded fault occurs and the operating point moves from curve I to curve II. Under this condition, power unbalance is introduced and δ increases. When the fault is cleared at δc, the operating point would move from curve II to curve I. The output active power is larger than the reference active power and δ decreases to reach for a stable EP. δmax represents the maximum power angle the system would reach after the fault is cleared. Without considering the energy loss caused by the resistor and damping windings, δmax can be calculated by the condition that the acceleration area S1 equals to the deceleration area S2. The criterion for transient angle stability is that δmax < δu.

9.2 Transient Angle Stability Mechanism Fig. 9.6 P-δ curve during transient period: (a) With or without equilibrium point, (b) Constant inverter’s internal voltage, (c) Considering the effect of reactive power control loop

(a)

225

P

Before fault With EP

P*

Without EP b a

I

II III

Deceleration Pem>P* Equilibrium Pem=P*

d

c

Acceleration PemP0

I

P0

S1

Equilibrium Pem=P0

Pmax2 II

0

(c)

δ

s

δ

δ

c

Acceleration Pem < 1 ¼ x2 dt > : J dx2 ¼ T  EV g B sin ðx1 þ δs Þ  EV g G cos ðx1 þ δs Þ  Dx2 dt ω ω

ð9:20Þ

where T ¼ T’D(ωgω*). First integral method can be used to derive a candidate Lyapunov function for a two-order system [11], which satisfies n X ∂fi ¼0 ∂xi i¼1

ð9:21Þ

By neglecting the damping term Dx2, the system can be viewed as a conservative system, and the detailed process to construct a Lyapunov candidate function can be found in [11]. The damping term Dx2 can be incorporated into the derived function. So the Lyapunov function V(x1, x2) of the system can be expressed in the form as EV g B 1 ½ cos ðδs þ x1 Þ  cos δs  Vðx1 , x2 Þ ¼ Jx22  Tx1 þ 2 ω EV g G D2 2  ½ sin ðδs þ x1 Þ  sin δs  þ Dλx1 x2 þ λx  2J 1 ω

ð9:22Þ

where λ is a constant (0 < λ < 1), Jx22/2 represents the virtual rotor kinetic energy and Tx1 represents the rotor potential energy relative to the reference frame. The third term and the fourth term evaluate the magnetic stored energy and the dissipated energy of the line impedance. The fifth and sixth terms are the approximate dissipated energy of virtual damping. But now, here comes a problem that the positive definite of the introduced Lyapunov function and the semi-negative definite of dE/dt should be proved.

9.3.1.1

Positive Definite of Derived Lyapunov Function

The derived Lyapunov function in Eq. (9.22) can be classified into Vðx1 , x2 Þ ¼ V KE þ V PE þ V DE

ð9:23Þ

It is clear that V(0,0) ¼ 0 and VKE ¼ Jx22/2  0. Thus, we need to prove that VPE and VDE are positive.

230

9

Transient Angle Stability of Grid-Connected VSG

EV g B ½ cos ðδs þ x1 Þ  cos δs  ω EV g G s s   ½ sin ðδ þ x1 Þ  sin δ  ω Z x1 EV g ¼ ð sin ðδs  φ þ uÞ  sin ðδs  φÞÞdu jZ jω 0

VPE ¼ Tx1 þ

ð9:24Þ

Define δts ¼ δsφ. Then Eq. (9.24) can be changed into VPE ¼

Z

EV g jZ jω

x1

ð sin ðδts þ uÞ  sin δts Þdu

ð9:25Þ

0

When π2δts  x1  0, sin(δts + x1)  sinδts. sin(δts + u)sinδts and du are all positive, so VPE  0. When 0  x1  π2δts, sin(δts + x1)  sinδts. sin(δts + u) sinδts and du are all negative, so VPE  0. Thus, we can prove that when π2δts  x1  π2δts, VPE  0. To prove that VDE is positive definite, we can obtain first integral of motion by first multiplying both sides of the second equation of Eq. (9.22) by x1 and integrating with respect to time t Z

t

Z

t

J 2 x_ 2 x1 dt þ

0

0

EV g Dx2 x1 dt ¼ jZ jω

Z

t

½ sin ðδts þ x1 Þ  sin δts x1 dt

ð9:26Þ

0

The left side of Eq. (9.26) can be rewritten as follows: Z 0

t

Z J x_ 2 x1 dt ¼ J x_ 1 x1 jt0 

t

J x_ 1 2 dt Z t ¼ Jx2 x1  J x22 dt

Z 0

0

ð9:27Þ

0

t

1 Dx2 x1 dt ¼ Dx21 2

ð9:28Þ

By substituting equations above into Eq. (9.26), Eq. (9.26) can be rewritten as Z

t

Jx2 x1  J 0

EV g 1 x22 dt þ Dx21 ¼ 2 jZ jω

Z

t

½ sin ðδts þ x1 Þ  sin δts x1 dt

ð9:29Þ

0

From the above equation, we can obtain the following relationship by multiplying D in both sides

9.3 Stability Region Estimation

Z D 0

t

x22 dt

231

D2 2 D EV g ¼ Dx2 x1 þ x  2J 1 J jZ jω

Z

t

½ sin ðδts þ x1 Þ  sin δts x1 dt

ð9:30Þ

0

The last term of Eq. (9.30) can be included into the previous two terms by a constant λ, which is defined as damping coefficient. Z t D2 2 D x22 dt ¼ Dλx1 x2 þ λx 2J 1 0

ð9:31Þ

It is known that the left side of the Eq. (9.31) is positive, so we get that V DE ¼ Dλx1 x2 þ

D2 2 λx  0 2J 1

ð9:32Þ

Thus, we can prove that V(x1, x2) is positive definite.

9.3.1.2

Semi-Negative Definite of dV/dt

The negativeness of time derivative of the derived Lyapunov function can be proved by using chain rule for partial derivatives, which can be evaluated as follows dV ∂V dx1 ∂V dx2 ¼ þ dt ∂x1 dt ∂x2 dt

ð9:33Þ

The results of partial derivative are EV g B EV g G ∂V D2 λx ¼ T  sin ðδs þ x1 Þ  cos ðδs þ x1 Þ þ Dλx2 þ   J 1 ω ω ∂x1 ð9:34Þ EV g D2 ts ts λx ¼ ð sin δ  sin ð δ þ x Þ Þ þ Dλx þ 1 2 1 J jZ jω ∂V ¼ Jx2 þ Dλx1 ∂x2

ð9:35Þ

By substituting Eqs. (9.34) and (9.35) into (9.33), there is EV g x_ 1 dV ¼ ð sin δts  sin ðδts þ x1 ÞÞ þ Dλx2 x_ 1 dt jZ jω þ

D2 λx x_ þ Jx2 x_ 2 þ Dλx1 x_ 2 J 1 1

The second equation of Eq. (9.20) can be rewritten as

ð9:36Þ

232

9

x_ 2 ¼

Transient Angle Stability of Grid-Connected VSG

EV g D ð sin δts  sin ðδts þ x1 ÞÞ  x2 J jZ jω J

ð9:37Þ

By substituting Eqs. (9.35) and (9.37) into (9.20), there is dV Dλ EV g ð sin δts  sin ðδts þ x1 ÞÞx1 ¼ ð1  λÞDx22 þ dt J jZ jω

ð9:38Þ

Since 0 < λ < 1, the first part of Eq. (9.38) is negative and ð sin δts  sin ðδts þ x1 ÞÞx1 < 0

ð9:39Þ

for π2δts  x1  π2δts as proved in the previous part. Equation (9.38) says that Lyapunov V(x1,x2) meets the semi- negativeness of its time derivative for all λ2(0,1) unless the system goes out of the stable region π2δts  x1  π2δts. As a suitable Lyapunov function has been constructed, iteration process of critical fault clearing time based on Lyapunov’s direct method is presented as Fig. 9.10. The corresponding procedure for determining the critical fault clearance time based on Lyapunov direct method is presented as follows: Step1: Power-flow solution of the pre-fault system under steady state. Step2: Obtain the SP and UEP (s.e.p. and u.e.p.) of the post-fault system, which are denoted by (δs, ωg) and (δu, ωg) respectively. Step3: Calculating the critical energy value Vc by substituting (δu, ωg) into Lyapunov function. Step4: Execute forward numerical integration of the faulted system with the pre-fault steady-state operating point as initial conditions. At each time step, calculate V(δ, ω) to compare with Vc. The time the value of V(δ, ω) equals to Vc is defined the critical clearing time tc. The UEP of the studied system can be calculated by δu ¼ πδs. Then, the critical value Vc can be computed from δu and δs   Vc ¼ V δs , δu , ωg

ð9:40Þ

The Lyapunov function without considering the effect of reactive power control loop has been derived above. It should be noted here that V as well as Vc are not relevant to the parameter E in Eqs. (9.22) and (9.40) since the voltage variation is not included in the derived Lyapunov function. However, it is extremely difficult to construct the Lyapunov function when voltage dynamic is considered. The object of this problem is not to tackle a hard problem of constructing an effective Lyapunov function for highly nonlinear equations. Instead, this difficulty will be overcome by translating changes of E from a state variation to a parameter variation. This will be covered in the next section.

9.3 Stability Region Estimation Fig. 9.10 Iteration process of critical fault clearing time based on Lyapunov’s direct method

233

Model of Single -VSG-Infinite-Grid

Power Flow Calculation

Power Flow Solution Power Flow Solution before Fault after Fault

Stable Equilibrium Point (δs, ω*)

Numerical integration

Acquire Post-fault Equilibrium Point

Unstable Equilibrium Point (δu, ω*)

Proposed Lyapunov Function

Calculate Function Energy Value V(δ, ω)

Calculate Critical Energy Value Vc

Critical Clearing Time when V c= V(δ, ω)

9.3.2

Proposed Lyapunov Method Considering the Influence of Reactive Power Loop

Mathematically speaking, transient stability analysis by Lyapunov’s direct method is to investigate whether the state- variables of the system belong to the attraction domain at the time of the fault clearance. It means that the post-fault system state must be known ahead of time. Thus, the numerical integration of the faulted system is inherently needed to obtain the initial value of the post-fault system. In the conventional power system, system state variables, like frequency and voltage phase angle, are monitored to evaluate the operation condition of the system online [1]. For the system consisted with VSGs, state variables, like voltage amplitude,

234

9

Transient Angle Stability of Grid-Connected VSG

voltage phase angle and frequency, can be obtained from the controller directly. Once these necessary state variables are achieved, the operation condition of the system during the transient period can be evaluated based on the introduced method. Since the mathematical model of a VSG connected to an infinite bus has been reduced into a two-order derivative equation plus a quasi-steady equation in Eqs. (9.16)–(9.17), the candidate Lyapunov function Eq. (9.22) can still be used to evaluate the transient stability of a VSG. It means for each determined E, Eq. (9.22) can guarantee the asymptotical stability for state variables within the attraction domain. However, the s.e.p., u.e.p., and the critical energy Vc would not be a constant value for a given fault condition. Instead, δs, δu and Vc need to be updated with a new E according to Eq. (9.17) at each time step. Based on the reduced model in Eqs. (9.16)–(9.17), iteration process of critical fault clearing time considering reactive power loop are given in Fig. 9.11. The corresponding algorithm of the introduced approach is summarized as follows: Step1: Power-flow solution of the pre-fault system under steady state. Step2: Calculate the initial value of inverter’s internal voltage E of the faulted system according to Eq. (9.17). Step3: Obtain s.e.p. and u.e.p. of the post-fault system, which are denoted by (δs, ωg) and (δu, ωg) respectively. Step4: Calculating the critical energy value Vc by substituting (δu, ωg) into the Lyapunov function with E calculated by step 2. Step5: Execute forward numerical integration of the faulted system by Eqs. (9.16)–(9.17) with the pre-fault steady-state operating point as initial conditions. At each time step, we calculate E by Eq. (9.17) and then recalculate δs ¼ δs(E), δu ¼ δu(E) to obtain Vc(δs, δu, 0, E). The time V(δ, ω, E) equals to Vc is called the critical clearing time. Similar to the iteration method that without reactive power control loop, an intuitive figure that illustrates process above is shown in Fig. 9.11. It is observed from the introduced method that additional computation burden is needed to recalculate δs, δu and Vc for each inverter’s internal voltage E during transient period. It means that the introduced method depends on the voltage disturbed trajectory which determines the critical energy value Vc. However, as discussed above, state variables are inherently needed to determine the stable condition of the system [1]. As shown in step 5, the introduced method only needs another variable, inverter’s internal voltage E, to recalculate the critical energy. Moreover, the introduced method avoids the difficulty in finding the UEP, which is important in the system considering voltage dynamics. Because stable EP (δs, ωg) can be easily obtained by assigning the right side of Eq. (9.20) to 0 for each determined E, the UEP δu can be calculated by δu ¼ πδs. Equations to solve for δs, δu and Vc are all algebraic equations, which do not need much storage space.

9.3 Stability Region Estimation

235

Fig. 9.11 Iteration process of critical fault clearing time considering reactive power loop

Model of Single -VSG-Infinite-Grid

Power Flow Calculation Internal Voltage Calculation of Post-fault Converter Power Flow Solution Power Flow Solution before Fault after Fault

Acquire Post-fault Equilibrium Point Numerical Integration

Stable Equilibrium Point (δs, ω*)

Unstable Equilibrium Point (δu, ω*)

Internal Voltage Calculation on Every Time Step during Fault

Proposed Lyapunov Function

Calculate Function Energy Value V(δ, ω)

Calculate Critical Energy Value Vc

Critical Clearing Time when V c= V(δ, ω)

9.3.3

Influence of Different Parameters

An approximate Lyapunov’s direct method has been introduced by Sect. 9.3.2 in this section and the effects of parameters from different part of VSG on the attraction region of the system would be analyzed.

236

9.3.3.1

9

Transient Angle Stability of Grid-Connected VSG

Influence of Reactive Power Control Loop

As analyzed above, the existence of reactive power control loop will lead to the decrease of the inverter’s internal voltage during the transient period, which would deteriorate the transient angle stability of VSGs. The system shown in Fig. 9.2 is taken as the research object, and the attraction region is evaluated. The operation condition is that a three-phase grounded fault occurs and the grid voltage Vg is de creased to 30% of the rated value at 1.5 s. And after the fault is cleared, Vg recovers to the rated value. The detailed parameters have been shown in Table 9.1. The changes of the estimated attraction domain at different time are shown in Fig. 9.12. Vcr represents the attraction domain of the studied system without considering the effect of reactive power control loop. Vcr1 and Vcr2 represent the attraction domain of the studied system at 2 s and 2.12 s respectively. It can be observed that the attraction region without considering the effect of reactive power control is much bigger than the actual attraction domain of the system. This result is consistent with the theoretical analysis. In this situation, a system judged to be stable can probably go unstable if the effect of reactive power control is not considered. This will lead to serious error in transient stability assessment. As time changes, the attraction domain of the studied system would be reduced due to the decrease of the inverter’s internal voltage, which is indicated by Vcr1 and Vcr2 in Fig. 9.12. Figure 9.13 shows Lyapunov’s direct method in determining transient stability. Vp, Vk and Vt are the system potential energy, kinetic energy and total energy respectively. Vcrf and Vcrp represent the critical energies of the system with and without considering the effect of reactive power control loop respectively. It can be observed from both Fig. 9.13a, b that Vcrp is much bigger than Vcrf, which is in accordance with the curves Vcr and Vcr1 in Fig. 9.12. In both conditions, the total energy Vt does not exceed the critical energy Vcrp. Thus, the system could be judged to be stable without the consideration of the effect of reactive power control loop. However, when the introduced Lyapunov’s direct method is adopted, the estimated result is different: the critical energy Vcrf is 62 at 2 s, as shown in Fig. 9.12 Estimated attraction domain of the studied system

10 8

Vcr

6

Vcr1 Vcr2

x2[rad/s]

4 2 0 -2 -4 -6 -8 -10 -2

-1.5 -1

-0.5

0 0.5 x1[rad]

1

1.5

2

2.5

9.3 Stability Region Estimation

237

(a)

(b)

300

200

70

250 80

Vcrp

120

Vt

40

20 1.8 1.9 2 2.1 2.2

Vcrf

0

1

2

3

4 T[s]

150 Vcrp Vk

100 Vcrf

Vp Vk

40

Vt

40 1.9 2 2.1 2.22.3 2.4

200

Vcrf

60

80

0

Vcrf

60 50

Energy[W]

Energy[W]

160

Vp

50 5

6

7

8

0

0

1

2

3

4 T[s]

5

6

7

8

Fig. 9.13 Lyapunov’s direct method to estimate system stability: (a) Stable condition when fault is cleared at 2 s, (b) unstable condition when fault is cleared at 2.12 s

Fig. 9.13a, while its value is 51 at 2.12 s, as shown in Fig. 9.13b, which is identical to Vcr1 and Vcr2 in Fig. 9.12. It can be observed that when the fault is cleared at 2 s, the total energy Vt does not reach the critical energy Vcrf, which means the system is stable. When the fault is cleared at 2.12 s, the total energy Vt exceeds the critical energy, which means the system is unstable in this condition.

9.3.3.2

Influence of Reference Active Power P*

As aforementioned, transient angle instability of a VSG is inherently caused by the unbalance between the reference active power and the output active power. So, the line impedance, damping coefficient and reference active power, which have effects on the response of the swing equation, would be taken as variables. Moreover, Q-V droop coefficient, which determines the dynamic of the inverter’s internal voltage, would also be considered in this part. The reference active power P* varies from 10 kW to 30 kW to identify its effect on the attraction region, and the result is shown in Figs. 9.14 and 9.15. It should be noted that the system parameters are the same with that shown in Tables 9.1 and 9.2. It is observed that the attraction region decreases with the increase of the reference active power, which can be explained by P-δ curve in Fig. 9.9: a large reference active power would lead to a large acceleration area during fault period and a small deceleration area during the post-fault period. This would deteriorate the system transient stability, thus, it is easier for a VSG with large capacity to lose transient stability. Simulation results are shown in Fig. 9.15. It can be observed that when P0 is set as 10 kW, the system can maintain stability after fault clearance. Along with the increase of reference active power, the system is driven into instability area. The simulation results are the same with theoretical analysis using Lyapunov’s direct method.

238

9

Fig. 9.14 Estimated attraction domain with the variation of reference active power

Transient Angle Stability of Grid-Connected VSG 10 P0=20kW P0=10kW

8 6 4

P0=30kW

x2[rad/s]

2 0 -2 -4 -6 -8 -10 -2

Fig. 9.15 Time-domain simulation results with different reference active power

-1.5 -1

-0.5

0 0.5 x1[rad]

1

1.5

2

2.5

326 324 P0=30kW

ω [rad/s]

322

P0=20kW

320 P0=10kW 318 316 314 312 310 -100 -50

0

50

100

150 200

250 300

δ [rad]

9.3.3.3

Influence of Damping Coefficient D

Figure 9.16 gives the estimated attraction domain with the variation of the damping coefficient D (It should be noted that D equals the sum of Dp and Kp/ω*). From the theoretical analysis in Sect. II, a large damping coefficient would consume more energy during transient period and would slow down the increase of the power angle δ, as shown by Eq. (9.20). So a large damping coefficient is beneficial for the system transient angle stability. It can be observed from Fig. 9.16 that the estimated attraction domain increases as the damping coefficient increases, which agree with the theoretical analysis. Simulation results are shown in Fig. 9.17. It can be observed that along with the increase of damping coefficient Dp, the system has stronger ability to return to stable operating point. However, a large damping coefficient may also lead to the performance degradation in terms of frequency control of a VSG. Thus, there is a tradeoff

9.3 Stability Region Estimation

239

Fig. 9.16 Estimated attraction domain with the variation of damping coefficient

10 D=64

8 6

D=32

4

D=8

x2[rad/s]

2 0 -2 -4 -6 -8 -10 -2

Fig. 9.17 Time-domain simulation results with different damping coefficient Dp

-1.5 -1

-0.5

0 0.5 x1[rad]

1

1.5

2

2.5

326 324

ω [rad/s]

322 Dp=8

320 318

Dp=64

316 314

Dp=32

312 310 -100 -50

0

50

100

150 200

250 300

δ [rad]

between the capability to maintain the stability during transient periods and the frequency deviation in steady-state.

9.3.3.4

Influence of Line Impedance Z

Four different parameters, Z1 ¼ 0.01 + j0.15pu, Z2 ¼ 0.02 + j0.3pu, Z3 ¼ 0.03 + j0.45pu and Z4 ¼ 0.1 + j0.28pu, are chosen to identify the effect of the line impedance on the attraction region of the system. Z1, Z2 and Z3 have the same impedance angle, but different amplitudes. Z2 and Z4 have the same amplitude but different impedance angles. Figure 9.18 shows the estimated attraction domain with the variation of line impedance. As indicated by Eq. (9.19), the line transmission capacity is in inverse proportion to the line impedance, which means large impedance may drive the system into the instability situation during transient periods easily. Illustrated intuitively from curves of Z1, Z2 and Z3 in Fig. 9.18, it can be concluded that the larger the line impedance

240

9

Fig. 9.18 Estimated attraction domain with the variation of line impedance

Transient Angle Stability of Grid-Connected VSG 10 8

Z1

6

Z2

4

Z3

x2[rad/s]

2 0 Z4

-2 -4 -6 -8 -10 -2

Fig. 9.19 Time-domain simulation results with different line impedance

-1.5 -1

-0.5

0 0.5 x1[rad]

1

1.5

2

2.5

326 324

ω [rad/s]

322 320

Z1=0.01+j0.15pu Z2=0.02+j0.3pu

318 316 Z3=0.03+j0.45pu

314 312 310 -100 -50

0

50

100

150 200

250 300

δ [rad]

amplitude is, the smaller the attraction region of the system is. The corresponding simulation results are shown in Fig. 9.19. When Z1 is considered, the system can return to stable point after fault clearance. However, when the impedance increases to Z2 or Z3, the system diverges to infinite and cannot recovery to stable point. And the comparison between Z2 and Z4 implies that a smaller impedance angle can increase the attraction region of the system. This is because the resistor can increase the dissipated energy of the system as shown in Eq. (9.22), which would help to maintain the system stability. It should be noted that when Z4 is adopted, the system is stable which is not shown in the figure. The simulation results are consistent with that of theoretical analysis.

9.3 Stability Region Estimation

9.3.3.5

241

Influence of Q-V Droop Coefficient Dq

The effect of Q-V droop coefficient Dq on the attraction region of the system is shown in Fig. 9.20 and corresponding simulation results are illustrated in Fig. 9.21. During the transient period, inverter’s internal voltage would decrease according to the Q-V droop curve. And the larger the Q-V droop coefficient Dq is, the less inverter’s internal voltage drops. A large inverter’s internal voltage is beneficial to the transient stability of a VSG, as shown in Fig. 9.12. It can also be observed from Fig. 9.20 that the attraction domain of the system increases with the increase of the Dq. The result validates the correctness of the theoretical analysis. Simulation results are given in Fig. 9.21. When Dq equals to 333, the system can return to stable operation condition. However, when Dq decreases to 166 and 83, the system loses stability. Thus, large Q-V droop coefficient is beneficial to the transient stability of the system.

Fig. 9.20 Estimated attraction domain with the variation of Q-V droop coefficient

10 8

Dq=333

6

Dq=166

4

Dq=83

x2[rad/s]

2 0 -2 -4 -6 -8 -10 -2

Fig. 9.21 Time-domain simulation results with different Q-V droop coefficient Dp

-1.5 -1

-0.5

0 0.5 x1[rad]

1

1.5

2

2.5

326 324

ω [rad/s]

322 320

Dq=166

Dq=83

318 316 314 Dq=333

312 310 -100 -50

0

50

100

δ [rad]

150 200

250 300

242

9.4

9

Transient Angle Stability of Grid-Connected VSG

Summary

This chapter discussed the transient angle stability of VSGs by taking the reactive power control loop into account. The transient instability mechanism is revealed and the instability phenomenon is illustrated. Moreover, an approximate Lyapunov’s direct method is adopted to evaluate the transient stability of the system. Based on the mechanism analysis, a new control strategy is introduced to improve transient stability of the system. The main conclusions can be drawn as follows: 1. The reactive power control loop will reduce inverter’s internal voltage through Q-V droop curve during transient period, and this may drive the system into instability situation easily. 2. Inverter’s internal voltage can be treated as a parameter variable rather than a state variable in transient angle stability analysis. The introduced approximate Lyapunov’s direct method can predict the transient stability of the system. 3. A VSG with large capacity intends to lose transient angle stability during fault period. The attraction region of the system increases with the increase of the damping coefficient and the Q-V droop coefficient. Large line impedance would reduce the stability margin of the system, and the resistor is beneficial for the transient stability for VSGs. 4. Transient angle instability of a VSG is inherently caused by the unbalance between the reference active power and the output power. By decreasing the reference active power during the fault period, transient stability margin can be improved.

References 1. P. Kundur, Power System Stability and Control (McGraw-Hill Education, New York, 1994) 2. P. Kundur et al., Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions. IEEE Trans. Power Syst. 19(3), 1387–1401 (2004) 3. Union for the Coordination of Electricity Transmission (UCTE). Interim Report of the Investigation Committee on the 28 September 2003 Blackout in Italy. http://www.pserc.wisc.edu/, 2003-10-23 4. C.D. Vournas, P.W. Sauer, M.A. Pai, Relationships between voltage and angle stability of power systems. Int. J. Electr. Power Energy Syst. 18(8), 493–500 (1996) 5. C.W.T.N.J. Balu, D. Maratukulam, Power System Voltage Stability (MCGraw-Hill, New York, 1994) 6. K. Prabha, Power System Stability and Control (McGraw-Hill, New York, 2004) 7. H. Wu, X. Wang, Design-oriented transient stability analysis of grid-connected converters with power synchronization control. IEEE Trans. Ind. Electron. 66(8), 6473–6482 (2019) 8. J. Liu, Y. Miura, T. Ise, Comparison of dynamic characteristics between virtual synchronous generator and droop control in inverter-based distributed generators. IEEE Trans. Power Electron. 31(5), 3600–3611 (2016)

References

243

9. Q. Sun, Y. Zhang, H. He, D. Ma, H. Zhang, A novel energy function-based stability evaluation and nonlinear control approach for energy internet. IEEE Trans. Smart Grid 8(3), 1195–1210 (2017) 10. Z. Shuai, C. Shen, X. Liu, Z. Li, and Z. Shen, Transient angle stability of virtual synchronous generators using Lyapunov’s direct method. IEEE Trans. Smart Grid 10(4), 4648–4661 (2019) 11. N.G. Bretas, L.F.C. Alberto, Lyapunov function for power systems with transfer conductances: extension of the invariance principle. IEEE Trans. Power Syst. 18(2), 769–777 (2003)

Chapter 10

Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs

With the development of VSG techniques, more and more VSGs are connected to the microgrid. However, VSGs has limited thermal capacity and weak grid supporting ability during fault period, especially in islanded microgrid, while SGs have these advantages. Thus, parallel operations of SGs and VSGs become increasingly common in an islanded microgrid. The transient instability mechanism of paralleled systems is more complicated, compared to that of a grid-connected inverter. It will lead to the difficulty of system planning and designing. In this chapter, the transient angle stability of islanded microgrid with paralleled SGs and VSGs is investigated. The mathematical models of paralleled systems are established for transient stability analysis. Then, the transient instability mechanism is explored. Furthermore, a Lyapunov method is employed to establish the nonlinear model of islanded microgrid, by which the stability regions of the paralleled systems are quantified.

10.1

Mathematical Model

Along with the development of power electronic technology, paralleled operation of VSGs and SGs are becoming more and more common nowadays [1–4]. To analyze the stability of paralled system and give comparison between paralleled VSGs and Parallel VSGs and SGs, mathematical model of the paralleled system needs to be established first.

10.1.1 Model of Paralleled VSGs As depicted in Fig. 10.1a, the control parts of a SG consist of an automatic voltage regulator (AVR) and governor. The AVR and speed governor adopt a PI controller © Springer Nature Singapore Pte Ltd. 2021 Z. Shuai, Transient Characteristics, Modelling and Stability Analysis of Microgird, https://doi.org/10.1007/978-981-15-8403-9_10

245

246

10

Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs

Fig. 10.1 Control blocks. (a) SG, (b) VSG

(a) LT iabc Vfield

vabc

Automatic Voltage Regulator Ts*

ωs P

+-

Speed Governor

1 τi s + 1

1 Js s + D s

+

ω*

PCC

Rl

Ll

SG

Engine

SG

Tes

(b) LT

Lf

PCC

Rl

Ll

C iabc

+ -

-

+

+

Tm

+

Js

-

ω*

Speed Governor ω

Te

ω 1

T

Ks

*

Dp

E

-

PWM modulation

1 s

+

signal

θ

vabc

ω* Q*

Q-V control E*

to regulate the terminal voltage and mechanical speed, respectively [5]. The SG is connected to the bus through transformer and transmission lines. LT represents the equivalent transformer inductance. Rl and Ll are equivalent resistance and inductance of transmission line l, respectively. Figure 10.1b shows the control block of a VSG which is comprised of active- and reactive power control loops. To be specific, active power control loop is to emulate rotor motion equations while regulate active power, and reactive power control loop is to control terminal voltage while regulate reactive power [6]. Lf is the filter inductance of VSG. The swing equation of SG is [7]. Js

dωs ¼ T ms  T es  Ds ωs dt

ð10:1Þ

where Js and Ds represent the inertia constant and damping coefficient, respectively. ωs is the rotor angular frequency. Tes is electromagnetic output torque and Tms is generated by the speed governor, which can be expressed as  dT ms 1
 ¼ T s  K p ðωs  ω Þ  T ms τi dt

ð10:2Þ

where Kp is the control parameter of speed governor. τi and ω* are the time-delay constant and reference angular frequency, respectively.

10.1

Mathematical Model

247

Fig. 10.2 Equivalent circuit of a paralleled VSGs system

E1∠δ1

Z filter1

Z line1

Z line 2

Z filter 2

VSG1

Z load = Z load 1 / / Z load 2

E2 ∠δ 2 VSG2

Z load 1

Z load 2

Combining Eqs. (10.1)–(10.2), there is Js

dωs dT ¼ T s  K p ðωs  ω Þ  τi ms  T es  Ds ωs dt dt

ð10:3Þ

The swing equation of VSG is J

dω ¼ T m  T e  Dp ðω  ω Þ dt

ð10:4Þ

where J and Dp denote the virtual inertia and virtual damping coefficient, respectively. ω is the virtual rotor angular frequency. Te is the output torque and Tm is generated by the governor, which can be expressed as T m ¼ T   K s ð ω  ω Þ

ð10:5Þ

According to Eqs. (10.5) and (10.6), we have J

dω ¼ T   K s ð ω  ω Þ  T e  D p ð ω  ω  Þ dt

ð10:6Þ

According to Eqs. (10.3) and (10.6), it is found that speed governors and damping links are the main differences between SGs and VSGs. Figure 10.2 shows the equivalent circuit of a paralleled VSGs system. A VSG can be regarded as a voltage source in series with an impedance [6]. Ek and δk are the amplitude and phase of internal voltage of VSG, respectively (k ¼ 1, 2). Zfliterk and Zload represent the filter- and load impedance, respectively. Zlinek is the total impedance between the export of SG/VSG and the bus, which can be calculated by Z linek ¼ Rl þ jω ðLl þ LT Þ

ð10:7Þ

The voltage-oriented reference frames of VSG1 and VSG2 are shown in Fig. 10.3, where the axis1 (d1-q1) and axis2 (d2-q2) are their reference frames rotating at a frequency of ω1 and ω2, respectively. Assuming that the angular acceleration of VSG1 is greater than that of VSG2 during a fault, the axis (d2-q2) is selected as the common reference frame in paralleled VSGs system. Hence, δ12 is the power angle between VSG1 and VSG2, which leads to

248

10

Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs

Fig. 10.3 Reference frame of VSG1 and VSG2

q2 q1

d1 ω1 δ12 ω2

d2

VSG1 VSG2

δ_ 12 ¼ ω1  ω2

ð10:8Þ

The active power of VSG can be calculated with help of the internal voltage and phase of VSG, which can be represented as P1 ¼ E 21 G11 þ E1 E2 jY 12 j cos ðδ12 þ ϕ12 Þ

ð10:9Þ

P2 ¼ E 22 G22 þ E1 E2 jY 12 j cos ðδ12  ϕ12 Þ

ð10:10Þ

where G11 and B11 are the conductance and susceptance of Y11. G22 and B22 are the conductance and susceptance of Y22. |Y12| and ϕ12 are the amplitude and phase of Y12. 8 > < Y 11 ¼ Y 1 ðY 2 þ Y 3 Þ=ðY 1 þ Y 2 þ Y 3 Þ Y 12 ¼ Y 1 Y 2 =ðY 1 þ Y 2 þ Y 3 Þ > : Y 22 ¼ Y 2 ðY 1 þ Y 3 Þ=ðY 1 þ Y 2 þ Y 3 Þ

ð10:11Þ

8 > < Y 1 ¼ 1=Z t1 Y 2 ¼ 1=Z t2 > : Y 3 ¼ 1=Z load

ð10:12Þ

where

where Ztk is equal to the sum of Zfilterk and Zlinek. From Eqs. (10.3), (10.9) and (10.10), the swing equations of VSG1 and VSG2 can be derived as J1

  E E jY j dω1 ¼ T 01  1 2 12 sin δt1  Dp1 ω1  ω1 dt ω10

ð10:13Þ

10.1

Mathematical Model

J2

  E E jY j dω2 ¼ T 02  1 2 12 sin δt2  Dp2 ω2  ω2 dt ω20

249

ð10:14Þ

where ω10 and ω20 are equal, which can be represented by ω0. 8   E 2 G11 > > > T 01 ¼ T 1  1   K s1 ω1  ω1 > > ω > > <   E 2 G22 T 02 ¼ T 2  2   K s2 ω2  ω2 ω > >  > > > δt1 ¼ δ12 þ ϕ12  90 > >  : t δ2 ¼ δ12  ϕ12  90

ð10:15Þ

Combining Eqs. (10.13) and (10.14), the swing equation of paralleled VSGs system can be deduced as J eq

  Dp1 Dp2 d2 δ012 0   ¼ T  T sin δ  J ð ω  ω Þ  ð ω  ω Þ M em eq 1 2 12 J1 J2 dt 2

ð10:16Þ

d2 δ0

where dt212 is the relative angular acceleration between VSG1 and VSG2. Jeq is the equivalent inertia of paralleled VSGs system. TM and T em sin δ012 are the equivalent input- and output torque of paralleled VSGs system, respectively. 8 J J > J eq ¼ 1 2 > > J þ J2 > 1 > > 0 > > δ ¼ δ γ 12 > 12 >  > > >  J1  J2 > > cot ϕ γ ¼  arccot > 12  90 > J þ J > 1 2 > > < J 2 T 1  J 1 T 2 J 2 E 21 G11  J 1 E 22 G22 TM ¼  > J1 þ J2 ðJ 1 þ J 2 Þω > > >   > > > K K s1 s2 >   > J eq ð ω1  ω Þ  ð ω2  ω Þ > > J1 J2 > > > q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > > 2 2 > J E E Y þ J  2J J cos 2ϕ j j 1 2 12 1 2 > 12 1 2 > : T em ¼ ðJ 1 þ J 2 Þω

ð10:17Þ

10.1.2 Model of Paralleled SGs and VSGs Figure 10.4 shows the equivalent circuit of a paralleled SG-VSG system. A SG can be regarded as a voltage source in series with an impedance [5]. Z 0d is equal to jx0d and

250

10

Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs

Fig. 10.4 Equivalent circuit of a paralleled SG-VSG system

Es ∠δ s

Z d′

Z line1

Z line 2

Z filter 2

DG

E2 ∠δ 2 VSG2

Z load = Z load 1 / / Z load 2

Z load 1

Z load 2

Fig. 10.5 Reference frame of SG and VSG2

q2 qs

ds ωs δs2 ω2

d2

DG VSG2

x0d is transient reactance. The reference frames of SG and VSG2 are shown in Fig. 10.5, where the axis (ds-qs) and axis (d2-q2) are their reference frames rotating at a frequency of ωs and ω2, respectively. The axis (d2-q2) is chosen as the common reference frame for paralleled SG-VSG system. δs2 is the power angle between SG and VSG, which can be expressed as δ_ s2 ¼ ωs  ω2

ð10:18Þ

According to reference [6], the dynamics of inductors can be ignored because those are in the faster time scale compared with the dynamics of mechanical rotor. Hence, the active power of VSG and SG can be calculated by Ps ¼ E2s Gss þ E s E2 jY s2 j cos ðδs2 þ ϕs2 Þ

ð10:19Þ

P2 ¼ E22 G22 þ E s E 2 jY s2 j cos ðδs2  ϕs2 Þ

ð10:20Þ

where Gss and Bss are the conductance and susceptance of Yss. G22 and B22 are the conductance and susceptance of Y22. |Ys2| and ϕs2 are the amplitude and phase of Ys2. 8 > < Y ss ¼ Y s ðY 2 þ Y 3 Þ=ðY s þ Y 2 þ Y 3 Þ Y s2 ¼ Y s Y 2 =ðY s þ Y 2 þ Y 3 Þ > : Y 22 ¼ Y 2 ðY s þ Y 3 Þ=ðY s þ Y 2 þ Y 3 Þ where

ð10:21Þ

10.1

Mathematical Model

251

8 > < Y s ¼ 1=Z ts Y 2 ¼ 1=Z t2 > : Y 3 ¼ 1=Z load

ð10:22Þ

where Z ts ¼ Z 0d þ Z line1 Ztk. From Eqs. (10.4), (10.7), (10.20) and (10.21), the swing equations of SG and VSG can be turned into E E jY j dωs ¼ T 0ms  s 2  s2 sin δts  Ds ωs dt ωs

ð10:23Þ

  E E jY j dω2 ¼ T 02  s 2  s2 sin δts  De2 ω2  ω2 dt ω2

ð10:24Þ

Js J2

where ωs* and ω2* are equal, which can be represented by ω*. 8 E2 Gss > dT > > T 0ms ¼ T s  s   K p ðωs  ω Þ  τi ms > > ω dt > > < 2 E G 22 T 02 ¼ T 2  2   K s2 ðω2  ω Þ ω > >  > t > > ¼ δ þ ϕ δ s2 s2  90 s > >  : t δ2 ¼ δs2  ϕs2  90

ð10:25Þ

Combining Eqs. (10.23) and (10.24), the swing equation of paralleled VSGs system can be written as J eqs d2 δ0

  Dp2 d2 δ0s2 Ds 0  ¼ T  T sin δ  J ω  ð ω  ω Þ Ms ems eqs 2 s2 Js s J2 dt 2

ð10:26Þ

where dt2s2 is the relative angular acceleration between SG and VSG. Jeqs is the equivalent inertia constant of paralleled SG-VSG system. TMs and T ems sin δ0s2 are the equivalent input and output torque of paralleled SG-VSG system, respectively.

252

10

Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs

8 JJ > J eqs ¼ s 2 > > Js þ J2 > > > > > δ0s2 ¼ δs2  γ s > >  > > >  Js  J2 > > γ ¼  arccot cot ϕ > s s2  90 > J þ J > s 2 > < J 2 T s  J s T 2 J 2 E2s Gss  J s E 22 G22 T ¼  > Ms > Js þ J2 ðJ s þ J 2 Þω > > > > > dT > > T ms ¼ T s  K p ðωs  ω Þ  τi ms > > dt > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > E s E 2 jY s2 j J 2s þ J 22  2J s J 2 cos 2ϕs2 > > : T ems ¼ ðJ s þ J 2 Þω

ð10:27Þ

By comparing model of paralleled VSGs system and that of paralleled SG-VSG system, the main differences between two paralleled systems are speed governors and damping links. It is known that transient angle instability of the paralleled system is intrinsically caused by the imbalance between input power and output power. And the differences can influence the input power of paralleled systems. Thus, it is crucial to discuss the influence of their differences on the transient angle stability mechanism of paralleled systems.

10.2

Transient Angle Stability Mechanism

In this Section, P-δ curves and EEAC will be introduced to explore the mechanisms of transient angle instability of both paralleled systems [8]. Moreover, the effects of differences between SGs and VSGs on the transient angle stability of paralleled systems will be discussed. Finally, the hardware-in-loop experiment is performed to validate the theoretical analysis.

10.2.1 Transient Angle Stability of Paralleled VSGs Assuming that Dp1/J1 is equal to Dp2/J2 and Ks1/J1 is as large as Ks2/J2, Eq. (10.16) can be changed into d2 δ0 J eq 212 ¼ dt

 Dp2 T M  J eq ω  T em sin δ012 J 2 12

ð10:28Þ

The damping link belongs to a part of the equivalent input torque, since it owns a similar function with a speed governor based on Eq. (10.28). The equivalent input torque of both paralleled VSGs system can be rewritten as

10.2

Transient Angle Stability Mechanism

Fig. 10.6 The P-δ curve of paralleled VSGs system

253

P Paralleled VSGs

I S2 III Pm

a

Pe>Pm

d f

e

Equilibrium Pe=Pm

c b 0

T 0M ¼

s ¥¥ 0

II

S1

PePm

III Pm

d

a e

f

Equilibrium Pe=Pm

c

0

II

Acceleration PePm

S2 d

III Pm

a

f

e

Equilibrium Pe=Pm

c

0

II

Acceleration PePm

S2 III Pm

δ

d

a e

f

Equilibrium Pe=Pm

c b 0

δ0 δ s

II

S1

δ max δ u

δc

(c)

Acceleration PePm

S2 d

III Pm

f

a e

Equilibrium Pe=Pm

c b 0

II

S1

δ0 δ s

ΔT ¼ K pω ðω  ωs Þ

δc

δ max δ u

Acceleration Pe 0 ATi  M þ M  Ai < 0, 8i 2 f1, 2, . . . , r g:

ð10:38Þ

The existence of M depends on two conditions: the P first one is that any matrix Ai is Hurwitz and the second condition specifies that A ¼ ri¼1 Ai is Hurwitz. Thus, it is necessary (but not sufficient) that all local models are stable. Now, the asymptotic stability of the system is proved if the linear matrix inequation (LMI) Eq. (10.38) is feasible. In this case, a matrix M verifying Eq. (10.38) can be determined and the following Lyapunov function is completely known V ð xÞ ¼ xT  M  x

ð10:39Þ

It is important to note that no knowledge on F ij , wi, and hi is required for the determination of V (x). In fact, we need only local state matrices Ai that can be directly obtained from the nonlinear model by fixing each of q nonlinearities to a constant limit value (min or max value). These limit values determine a domain in which the stability analysis is valid. In other words, the estimation of the domain of attraction of the operating point is deduced from the domain delimited by the aforementioned limit values. Therefore, for estimating the stability region, the following algorithm can be proposed: Step 1: Set all nonlinearities to their operating point values: fjmin ¼ fjmax ¼ fj(0), j j ¼ xmax ¼ 0). j ¼ 1, 2, . . ., q (xmin Step 2: If the LMI problem (11.39) is not feasible, then go to Step 4. j j Step 3: Else, decrease fjmin and increase fjmax by modifying xmin and xmax , j ¼ 1, 2, . . ., q. Then go to Step 2. j j Step 4: The estimated domain of attraction is bordered by xmin and xmax , j ¼ 1, 2, . . ., q.

10.3

Stability Region Estimation

10.3.1.2

263

Lyapunov Function of Paralleled SGs and VSGs Without Improvement Measure

To quantify stability regions of paralleled SG-VSG system without introduced methods, its model is established firstly. Then, the Lyapunov function is derived. The paralleled SG-VSG system can be modeled as [9, 11]. 8 dx1 > > > dt > > > > > dx2 > > > > dt > > > > > > dx3 > > > > dt > > > > > > dx4 > > > > dt > > > > >
> > >  > > > > dx5 ¼ Rload ð sin δ2s0 f ðx9 Þ  cos δ2s0 f ðx8 ÞÞx1  Lt1 f ðx4 Þx2 > > dt Lt1 Rload > > > >   > > > 1 L R t1 t1 > > þ x  ω x  1þ x  sin δ2s0 x8  cos δ2s0 x9 > > Rload 3 Rload 10 4 Rload 5 > > > > > dx > 1 6 > > ¼ ðDe2 x6  1:5f ðx6 Þf ðx9 Þx7  1:5E 20 f ðx6 Þx9 Þ > > J dt > 2 > > > >   1
 > dx7 : Dq2  1:5f ðx8 Þ x7  1:5E20 x8 ¼ K2 dt 8 dx R 8 > ¼ load ½ð sin δ2s0 f ðx4 Þ  cos δ2s0 f ðx5 ÞÞx1  cos δ210 x4 > > dt Lt2 > > >   > > > Lt2 Rt2 Lt2 > >  sin δ x þ f ð x Þx  1 þ þ ω x x > 2s0 5 > Rload 9 6 Rload 8 Rload 20 9 > > > > > > dx9 Rload > > < dt ¼ L ½ð cos δ2s0 f ðx4 Þ þ sin δ2s0 f ðx5 ÞÞx1 þ sin δ2s0 x4 t2   > > Lt2 1 Lt2 Rt2 > >  cos δ2s0 x5  f ðx Þx þ x  ω x  1þ x > > Rload 8 6 Rload 7 Rload 20 8 Rload 9 > > > > > > dx10 > > ¼ K i x2 > > dt > > > > > : dx11 ¼ 1 K x þ x  x  p 2 10 11 τi dt ð10:40Þ with

¼

264

10

Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs

8 x1 ¼ δ2s  δ2s0 , x2 ¼ ωs  ωs0 , x3 ¼ E0q  E 0q0 > > >

x ¼ E  E , x ¼ I  I , x ¼ I  I 7 2 20 8 2d 2d0 9 2q 2q0 > > : x10 ¼ x  x0 , x11 ¼ T ms  T ms0

ð10:41Þ

1 , f ðx4 Þ ¼ x4 þ I 1d0 , f ðx5 Þ ¼ x5 þ I 1q0 x2 þ ωs0 1 > : f ð x6 Þ ¼ , f ðx8 Þ ¼ x8 þ I 2d0 , f ðx9 Þ ¼ x9 þ I 2q0 x6 þ ω20

ð10:42Þ

8 > < f ðx2 Þ ¼

where Des equals to the sum of Kp and Ds. xd and T 0d0 are d-axis synchronous reactance and d-axis open-circuit transient time constant. E0q is the internal voltage of SG. De2 is equivalent to the sum of Dp2 and Ks2. K2 and Dq2 are voltage integral coefficient and Q-V coefficient of VSG2, respectively. Ikd and Ikq represent the d-axis and q-axis components of line current. The subscript “0” refers to the steady-state operating point. Based on the above model, the system is expressed as x_ ¼ AðxÞx with six nonlinear quantities. Then, the Lyapunov function V(x) ¼ xT  M  x can be constructed by following the steps in Sect. 10.3.2.1.

10.3.1.3

Lyapunov Function of Paralleled SGs and VSGs with Improvement Measure

Firstly, the model of paralleled SG-VSG system with introduced method is established firstly. Then, the Lyapunov function is derived to quantify stability regions of paralleled SG-VSG system with introduced method. The paralleled SG-VSG system with the introduced method can be modeled as [12]

10.3

Stability Region Estimation

265

8 dx1 > > ¼ x6  x2 > > dt > > > > > dx2 1 > 0 > ¼ ð D x  1:5f ð x Þf ð x Þx 1:5E f ð x Þx þ x es 2 2 5 3 2 5 10 > q0 > J1 dt > > > > >    dx3 1
> > ¼ 0 x3  xd  x0d x4 > > dt > T > d0 > > > > dx R L > 4 > ¼ load ½ð sin δ2s0 f ðx8 Þ þ cos δ2s0 f ðx9 ÞÞx1 þ t1 f ðx5 Þx2 > > dt L R > t1 load > >  > > > R > >  1 þ t1 x4  cos δ2s0 x8 þ sin δ2s0 x9  > > R > load > > > > > dx5 Rload L 1 > > ¼ ½ð sin δ2s0 f ðx9 Þ  cos δ2s0 f ðx8 ÞÞx1  t1 f ðx4 Þx2 þ x3 > > R dt L R > t1 load load > >   > > > Lt1 Rt1 > >  ω x  1þ x  sin δ2s0 x8  cos δ2s0 x9 > < Rload 10 4 Rload 5   > dx6 1 > > K pω x2  De2 þ K pω x6 1:5f ðx6 Þf ðx9 Þx7  1:5E 20 f ðx6 Þx9 Þ ¼ > > J dt > 2 > > > >   dx7 1
 > > Dq2  1:5f ðx8 Þ x7  1:5E20 x8 ¼ > > K dt > 2 > > > >
  > dx R 8 load > > ¼ sin δ2g0 f ðx4 Þ  cos δ2s0 f ðx5 Þ x1  cos δ210 x4  sin δ2s0 x5 > > dt L t2 > > >   > > > Lt2 Rt2 Lt2 > > þ f ðx Þx  1 þ ω x x þ > > Rload 9 6 Rload 8 Rload 20 9 > > > > > > dx9 Rload > > ½ð cos δ2s0 f ðx4 Þ þ sin δ2s0 f ðx5 ÞÞx1 þ sin δ2s0 x4  cos δ2s0 x5 ¼ > > dt Lt2 > > >   > > > Lt2 1 Lt2 Rt2 > >  f ð x Þx þ x  ω x  1 þ x > > Rload 8 6 Rload 7 Rload 20 8 Rload 9 > > > > >  > dx10 1  > : ¼ K p x2  x10 τi dt ð10:43Þ with 8 x1 ¼ δ2s  δ2s0 , x2 ¼ ωs  ωs0 , x3 ¼ E0q  E 0q0 > > > < x4 ¼ I 1d  I 1d0 , x5 ¼ I 1q  I 1q0 , x6 ¼ ω2  ω20 > x7 ¼ E 2  E 20 , x8 ¼ I 2d  I 2d0 , x9 ¼ I 2q  I 2q0 > > : x10 ¼ T ms  T ms0

ð10:44Þ

10

Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs

Fig. 10.14 Estimated stability regions of paralleled SG-VSG systems with or without the introduced method

40 30 20

x2[rad/s]

266

Without the introduced control

10 0 -10 -20

With the introduced control

-30 -40 -10 -8 -6 -4 -2

0

2

4

6

8

10

x1[rad]

8 > < f ð x2 Þ ¼

1 , f ðx4 Þ ¼ x4 þ I 1d0 , f ðx5 Þ ¼ x5 þ I 1q0 x2 þ ωs0 1 > : f ðx6 Þ ¼ , f ðx8 Þ ¼ x8 þ I 2d0 , f ðx9 Þ ¼ x9 þ I 2q0 x6 þ ω20

ð10:42Þ ð10:45Þ

Based on the above model, the system is expressed as x_ ¼ AðxÞx with six nonlinear quantities. Then, the Lyapunov function can be constructed by following the steps in Sect. 10.3.2.1. The stability regions of paralleled SG-VSG system with or without the introduced method can be quantified based on the Lyapunov functions, as depicted in Fig. 10.14. Kpω is equal to 5. The red curve stands for the stability region of paralleled SG-VSG system without the introduced method and the blue curve represents the stability region of paralleled SG-VSG system with the introduced method. As shown in Fig. 10.14, the estimated attraction domain of paralleled SG-VSG system with the introduced method is larger than that of paralleled SG-VSG system without the introduced method. Namely, the transient angle stability of paralleled SG-VSG system can be improved by adding the stability-enhanced control into VSG control, which is consistent with the theoretical results. Thus, the effectiveness of stability-enhanced control can be quantitatively validated by obtaining the estimated stability regions of paralleled SG-VSG systems with or without the introduced method.

10.3.2 Influence of Different Parameters According to Lyapunov function based on TS fuzzy model, the effects of different parameters on the stability regions of paralleled SG-VSG system are investigated in this part [12].

10.3

Stability Region Estimation

267

x2[rad/s]

Fig. 10.15 Estimated stability regions of the paralleled SGs and VSGs with variations of τi

10 8 6 4 2 0 -2 -4 -6 -8 -10 -8

τi= 0.3s τi= 0.6s τi=1.2s

-6

-4

-2

2

0

4

6

8

x1[rad]

ω21[rad/s]

Fig. 10.16 Time-domain simulation results of the paralleled SGs and VSGs with variations of τi

10.3.2.1

8 7 6 5 4 3 2 1 0 -1 -2 -2

τi=1.2s τi=0.6s τi=0.1s 0

2

4 6 δ21[rad]

8

10

Influence of Time-Delay Constant τ i

The effect of time-delay constant on stability regions is analyzed with τi varied from 0.3 s to 1.2 s, as depicted in Fig. 10.15. It is shown that the attraction domain decreases with the increase of time-delay constant. The result validates the correctness of the theoretical analysis. Simulation results are given in Fig. 10.16. When τi equals to 0.1, the system can return to stable operation condition. However, when τi increases to 0.6 and 1.2, the system loses stability. A larger time-delay constant leads to slower change in the input power during faults, which leads to the rapid increases of power angle and thus deteriorates the transient angle stability of the system.

10.3.2.2

Influence of Proportional Controller Parameter Kp and Ks2

Figure 10.17 shows the estimated attraction domains with different Kp and Ks2. The results show that a larger Kp or a smaller Ks2 is more beneficial for the transient stability of paralleled SGs and VSGs. As indicated in Fig. 10.7c, a larger Kp or a smaller Ks2 would lead to a smaller acceleration area during a fault and a larger

10

Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs

Fig. 10.17 Estimated stability regions of the paralleled SGs and VSGs. (a) variations of Kp, (b) variations of Ks2

(a) 10

x2[rad/s]

268

8 6 4 2 0 -2 -4 -6 -8 -10 -8

Kp=8 Kp=16 Kp=32

-6

-4

-2

4

2

0

6

8

x1[rad]

(b) 20 15

x2[rad/s]

10

Ks2=8 Ks2=16 Ks2=32

5 0 -5 -10 -15 -20 -10 -8 -6 -4 -2

0

2

4

6

8

10

x1[rad]

deceleration area during a post-fault. Namely, it would help the system to return back to stability after the clearance of a fault. Simulation results are shown in Fig. 10.18. It can be observed that along with the increase of proportional controller parameter Kp, the system has stronger ability to return to stable operating point. However, a larger Ks2 leads to the system instability.

10.4

Summary

In this chapter, the transient stability of two islanded systems is investigated in the context of paralleled VSGs and paralleled SG-VSG. The differences of two systems are discussed and the instability mechanisms of paralleled SG-VSG system are elaborated through P-δ curves. Based on the theoretical analysis, a novel control method is introduced to improve the transient stability of paralleled SG-VSG system. Furthermore, a Lyapunov function is introduced based on TS fuzzy model to quantify the transient stability of paralleled system.

Summary

Fig. 10.18 Time-domain simulation results of the paralleled SGs and VSGs. (a) variations of Kp, (b) variations of Ks2

269

(a) 8 6

ω21[rad/s]

10.4

4

Kp=8

2

Kp=16

0

Kp=32

-2 -4 -2

0

2

4 6 δ21[rad]

8

10

8

10

(b) 8

ω21[rad/s]

6 4

Ks2=16

2

Ks2=24

0 -2 -4 -2

Ks2=8 0

2

4 6 δ21[rad]

1. The transient angle instability of paralleled SG-VSG system is worse aggravated by the time delay of SG’s speed governor under large disturbances, compared to that of paralleled VSGs system. 2. The stable operation points can be impacted by the differences of damping links of both systems. Besides, the effects of differences of damping links on the system transient stability are far less than those of speed governors, which can be ignored under transient process. 3. The transient stability of paralleled SG-VSG system can be enhanced by decreasing the input active power, since the transient angle instability results from the unbalance between the input power and output power. 4. The SG with a larger time-delay constant is easier to drive a paralleled SG-VSG system into the unstable region during a fault. The estimated attraction domain of paralleled SG-VSG system is reduced under the condition of SG with a smaller proportional controller parameter. However, the VSG with a smaller proportional controller parameter is more beneficial for transient stability of the system. The Lyapunov method introduced in this chapter allows engineers to estimate stability regions of other systems and provide the guidance for designing the system parameters. The influences of different parameters on the system stability can be investigated based on the Lyapunov function. The novel method introduced in this

270

10

Transient Angle Stability of Islanded Microgrid with Paralleled SGs and VSGs

chapter provides a new perspective for engineers to better improve the system stability.

References 1. Z. Shuai, W. Huang, C. Shen, J. Ge, J. Shen, Characteristics and restraining method of fast transient inrush fault currents in synchronverters. IEEE Trans. Ind. Electron. 64(9), 7487–7497 (2017) 2. S. Krishnamurthy, T. Jahns, R.H. Lasseter, The Operation of Diesel Genset in a CERTS Microgrid, in Proc. IEEE Power Eng. Soc. Conf, (IEEE, Piscataway, 2008), pp. 1–8 3. R.H. Lasseter, R. Lasseter, Distributed generation interface to the CERTS microgrid. IEEE Trans. Power Delivery 24(3), 1598–1608 (2009) 4. D. Lee, L. Wang, Small-signal stability analysis of an autonomous hybrid renewable energy power generation/energy storage system part i: time-domain simulations. IEEE Trans. Energy Convers. 23(1), 311–320 (2008) 5. A. Paquette, M. Reno, R. Harley, D. Divan, Sharing transient loads: causes of unequal transient load sharing in islanded microgrid operation. IEEE Indus Appl Magazine 20(2), 23–34 (2013) 6. Q. Zhong, P. Nguyen, Z. Ma, W. Sheng, Self-synchronized synchronverters: inverters without a dedicated synchronization unit. IEEE Trans. Power Electron. 29(2), 617–630 (2014) 7. P. Kundur, Power System Stability and Control (McGraw-Hill Education, New York, 1994) 8. Y. Xue, T. Van Custem, M. Ribbens-Pavella, Extended equal area criterion justifications, generalizations, applications. IEEE Trans. Power Syst. 4(1), 44–52 (1989) 9. R. Genesio, M. Tartaglia, A. Vicino, On the estimation of asymptotic stability regions: state of the art and new proposals. IEEE Trans. Autom. Control 30(8), 747–755 (1985) 10. D. Marx, P. Magne, B. Nahid-Mobarakeh, S. Pierfederici, B. Davat, Large signal stability analysis tools in DC power systems with constant power loads and variable power loads—a review. IEEE Trans. Power Electron. 27(4), 1773–1787 (2012) 11. Z. Shuai, Y. Hu, Y. Peng, C. Tu, J. Shen, Dynamic stability analysis of synchronverterdominated microgrid based on bifurcation theory. IEEE Trans. Ind. Electron. 64(9), 1223–1235 (2017) 12. H. Cheng, Z. Shuai, C. Shen, X. Liu, Z. Li, Z.J. Shen, Transient angle stability of paralleled synchronous and virtual synchronous generators in islanded microgrids. IEEE Trans. Power Electron. 35(8), 8751–8765 (2020). https://doi.org/10.1109/tpel.2020.2965152

Chapter 11

Re-synchronization Phenomenon of Microgrid

The mechanism of first swing instability in microgrids is investigated. However, there are two possible results after the system experiences the first swing instability: the system is unstable or returns to another stable state. The latter is named as the re-synchronization phenomenon. In this chapter, the re-synchronization phenomenon both in grid-connected and islanded microgrids is found and the physical mechanism is illustrated through power-angle (P-δ) characteristics curve. First, the re-synchronization phenomenon of SG is presented, which intends to summarize the basics of the re-synchronization study. Then, the dynamic model of a single gridconnected inverter for the re-synchronization phenomenon analysis is developed, and the P-δ curve is employed to characterize the transient behavior of the single grid-connected inverter. It reveals that it is more likely for the system to return to stable state after fault clearance even the virtual power angle (VPA) goes beyond unstable equilibrium point (UEP). Furthermore, re-synchronization analyses in paralleled systems are discussed. These findings are validated by simulations in PSCAD/EMTDC.

11.1

Re-synchronization Phenomenon of VSG

In this section, the re-synchronization phenomenon of VSGs is studied. Different from SGs, VSGs own characteristics of inverters and are more flexible to implement power sharing through power control loops. Owing to those differences between SGs and VSGs, the mechanism of VSGs is different from that of SGs and VSGs are more prone to re-synchronization after serious faults are cleared. To better understand it, the re-synchronization mechanism of VSGs is analyzed and then the influences of different parameters are discussed. Finally, simulation results verify the theoretical validity.

© Springer Nature Singapore Pte Ltd. 2021 Z. Shuai, Transient Characteristics, Modelling and Stability Analysis of Microgird, https://doi.org/10.1007/978-981-15-8403-9_11

271

272

11

Re-synchronization Phenomenon of Microgrid

11.1.1 Mechanism of Re-synchronization Since VSGs are introduced to mimic the response characteristics of SGs, there is a transient angle stability problem for VSGs as well. It is pointed out and discussed in Chap. 9 that VSGs are more likely to be driven into the instability area due to the deteriorative effect caused by reactive power control loop. It is also revealed that the existence of current limiter, which introduced to restrict the output current of inverters, would lead to transient angle instability [1]. In Chap. 9, the mechanism of first-swing transient angle instability has been discussed. However, instability is not the one and only state that VSGs or SGs would experience after the power angle goes beyond the UEP. For the rotary SGs in traditional power system, when the SG goes beyond the UEP during fault period, there are two possible results after fault clearance: returns to another stable state, which named re-synchronization here, or exhibits chaotic characteristics [2]. Compared with the traditional SGs, transient response of VSGs exhibits different characteristics, as discussed in [3]. It is found that VSGs are more likely to return to the stable state since the line impedance is small and the damping coefficient is large in power electronic system. But how and why would VSGs return to a new state after the fault is cleared. To answer the question proposed above, consider a single VSG connected to infinite bus (SVIB) directly, which is similar to the model proposed in Chap. 9. It should be noted that there is no transformer in this model as is shown in Fig. 11.1. A VSG consists of a three-leg inverter and a LC filter to filter out high-frequency harmonic component. The current and voltage signals of the main circuit are delivered to the control system to realize VSG emulation. The detailed control scheme is shown in Fig. 11.2. The VSG control can be divided into three parts: power control loop, virtual impedance control loop and voltage and current control loops. Active power control in power control loop aims at regulating output active power and emulating dynamic response of SGs. Reactive power control is designed to realize power sharing and output voltage regulation. In power electronic system, the line impedance presents resistive characteristics and virtual impedance control loop is adopted to guarantee the power decoupling in power control loop. Voltage and current control loops are designed to accelerate the voltage and current tracking speed and improve the output power limit Fig. 11.1 Main topology of a single VSG connected to an infinite bus (SVIB)

θ * Ld

i

* Lq

i

* d

V

* q

V

P* ω * E * Q *

θ

11.1

Re-synchronization Phenomenon of VSG

273

ω*

Vq 1 ω0

ω*

1

1 s

Vq*

θ

s * Vabc

Te

1 Ks

* iLq

Vq

Vd Vd

E*

* iLd

Vd

Fig. 11.2 Control block diagram of a VSG controller

of a VSG. The generated signals are delivered to the PWM modulation unit, driving the switching devices to realize power conversion. Since the mathematical model has been detailed derived in Chap. 9, here are some critical formula listed only for the convenience of readers. The active power control loop can be expressed as follow: J

d2 δ ¼ T   T e  D ð ω  ω Þ dt 2

ð11:1Þ

where T* ¼ P*/ω*, Te ¼ P/ω*, and D ¼ Dp + Kp/ω*, D is the equivalent damping. Similarly, the reactive power control loop is K

dE ¼ Q0  Q  Dq ðE  E 0 Þ dt

ð11:2Þ

where K is the voltage integral coefficient and Dq represents the Q-V coefficient. E* and Q* are the reference voltage and reference reactive power respectively. The output power of the VSG is (

P ¼ E 2 G  EV g B sin δ  EV g G cos δ Q ¼ E 2 B þ EV g B cos δ  EV g G sin δ

ð11:3Þ

The simulation models of Fig. 11.1 are implemented in PSCAD/EMTDC and the system parameters are given in Table 11.1. It is reported in [3] that there is a transient angle stability problem with a VSG. The operation condition is conducted that a grounded fault occurs and the equivalent grid voltage drops to 15% of the nominal value at 1 s. Then, the fault is cleared and the grid voltage recovers to the nominal value at 1.7 s and 1.8 s, respectively. The numerical simulation results are shown in Fig. 11.3. It can be seen that when the fault is cleared at 1.7 s, VPA does not go beyond the UEP and the system will return to the original EP, which can be called first-swing stability, as shown by the red curve I in Fig. 11.3. When the fault is cleared at 1.8 s, VPA of a VSG will go beyond the UEP. From the perspective of transient stability criterion in the power system, it means that the

274

11

Re-synchronization Phenomenon of Microgrid

Table 11.1 Parameters of the grid-connected system Parameters P* Q* ω* E* D Dq J Vdc K C Rl Lf/Ll Vdc

Descriptions Rated active power Rated reactive power Reference angular speed Reference voltage magnitude Equivalent damping coefficient Q-V droop coefficient Inertia value DC link rated voltage Integral constant Capacitor of the filter Line resistance Filter/line reactance DC link rated voltage

Fig. 11.3 Phase portrait of the transient process of SVIB

Values 20 kW 5 kVar 314.159 rad/s 311 V 8 166.7 5.0224 800 V 12 40 uF 0.1 pu 0.1 pu/0.03 pu 800 V

330 fault clear at 1.8s

325

ω (rad/s)

320

II

315 I

310 305

fault clear at 1.7s UEP

EP

NUEP

first swing process re-synchronization process

300 -100

0

100

200 300 δ (degree)

400

500

600

system will step into irreversible instability area and cannot recover to the synchronous state with the grid [3]. However, from the simulation results, it is found that the system can reach another EP, as shown by the blue curve II in Fig. 11.3. This phenomenon means that a VSG in power electronic system possesses strong synchronism capability even manifesting first-swing instability. In other words, there are two possible results when the system loses first-swing stability. One result is that VPA continues to go beyond the next unstable equilibrium point (NUEP), which is called transient angle instability. The other result is that VPA slip to another EP and the system can maintain synchronism with the grid, which can be called re-synchronization stability. Thus, the comprehensive flow chart of the transient stability criterion is given in Fig. 11.4. To better understand the re-synchronization phenomenon, further researches need to be carried out to give insights into the physical mechanism.

11.1

Re-synchronization Phenomenon of VSG

275

Fig. 11.4 Flow chart of the transient stability criterion

Start

VPA>UEP?

N

Y VPA>NUEP?

N

First-swing Stability

Y Transient Angle Instability

Re-synchronization Stability

11.1.2 Influence of Different Parameters on Re-synchronization It is concluded in Chap. 9 that the equivalent model of the system in Fig. 11.1 can be modelled as a voltage source connected to the grid. By combining Eqs. (11.1) and (11.3), it can be derived that J

EV g d2 δ ¼ T0  sin δt  Dðω  ω Þ 2 jZ jω dt

ð11:4Þ

where T’ ¼ T*E2G/ω* and δt ¼ δφ, Z and φ are the equivalent line impedance and residual angle of the impedance angle, respectively. Then based on the coordinate translation, the corresponding relationship between active power P and power angle δ of the model in Fig. 11.1 is shown in Fig. 11.5. As depicted in Fig. 11.5, point a and point e are EP and UEP respectively. And point g represents NUEP. Curve I describes the relationship between P and δ of the pre-fault and post-fault systems while curve II is the P-δ curve of the system under a fault. There are two EPs denoted by δs and δu where power balance can be reached. δs represents the stable EP and δu is the UEP. δc represents VPA of a VSG at the fault clearing time. According to extended EEAC, first-swing stability can be judged by comparing S1 and S2 [2]. If S1 > S2, VPA will go beyond point e and the system is judged to be unstable. Otherwise, the system can maintain first-swing stability. It should be pointed out that the damping loss is seldom taken into account in EEAC. However, it is non negligible for a VSG in power electronic system since the transmission line presents resistive characteristics and droop coefficient plays an important role in the transient process (the system cannot maintain synchronism with the grid without considering the damping loss). In the following researches, only the condition that VPA goes beyond the UEP is considered. Generally, it can be divided into two cases.

276

11

Fig. 11.5 Resynchronization mechanism of a VSG. (a) Case I, (b) Case II

Re-synchronization Phenomenon of Microgrid

(a) Pmax 1

Pe>P0 Pmax 2

PeP0 Pmax 2

Pe δu

When δc > δu at the fault clearing time, the P-δ curve is shown in Fig. 11.5b. It means that the operation trajectory of the system is from point a to point f. S1 and S2 are the acceleration area but S3 is the deceleration area. Likewise, the criterion for re-synchronization stability in this case can be derived as S1 þ S2  S3 þ E loss

ð11:8Þ

where Eloss is the same with Eq. (11.6). The effect of parameters of VSG should also be paid attention. It is pointed out in [3] that the derivative equation of reactive power control loop in Eq. (11.2) can be simplified into an algebraic equation. Combining Eqs. (11.2)–(11.4), it derives J

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 Vg d2 δ 0 V ¼ T  cos δ  A þ B þ V cos δ  A sin δt g g 2jZ jω dt 2 Dp ðω  ω Þ

ð11:9Þ

where A ¼ DqX and B ¼ 4(Q0 + DqE0)X. Thus, P-δ curve of a VSG is no longer an ideal sinusoidal wave when reactive power control loop is considered. The P-δ curve along with the variation of Dq is shown in Fig. 11.6. The value of Dq for curve I, II and III is 40, 400 and 4000 respectively. It can be observed that UEP δu decreases along with the decrease of the Dq. So it is more easily for a VSG to be driven into first-swing instability area and the system experiences complicated re-synchronization process. Moreover, the system operation trajectory is shorter as Dq decreases, which would make the system energy loss Eloss smaller according to Eq. (11.6). Furthermore, S3 in case I will not change along with the variation of Dq while S3 in case II will decrease [4]. The effect of the line impedance Z is also significant. According to Eq. (11.9), P-δ curve along with the variation of |Z| can be plotted, as shown in Fig. 11.7. It can be seen that the transmission power capacity is limited along with the increase of the line impedance dramatically. Furthermore, the UEP would decrease at the same time, which means it is easier for VPA to go beyond UEP. In this

278 Fig. 11.6 P-δ curve along with the variation of Dq

11

Re-synchronization Phenomenon of Microgrid

(a) P III

Dq

II I P’ 0

δ

s

(b)

u1

δ u2δ

δ

u3

δ

t

2

x10 1.5

Pem[kW]

1 0.5 0 -0.5 -1 -1.5 800 15

600 Dq

400 200 0 0

10 5 d] a r [ δ

condition, the system is more likely to be driven into transient angle instability area. It can be seen from Eq. (11.5) that the damping loss energy Eloss is relevant to the damping coefficient D directly. A large D would increase the damping loss of the system and make the system easier to maintain stability during transient period.

11.1.3 Simulation Results To verify the correctness of the theoretical analysis, simulation models in Fig. 11.1 are established in PSCAD/EMTDC and the parameters are shown in Table 11.1. The corresponding simulation results of current and voltage are presented in Figs. 11.9 and 11.10, respectively. The simulation results of case I are shown in Fig. 11.9. The simulation results of case II are shown in Fig. 11.10. The operation condition is that the grid is subjected to a grounded fault at 1 s and the voltage dips to 15% of the rated value. When the fault is cleared at 1.8 s, VPA does not go beyond UEP and the

11.1

Re-synchronization Phenomenon of VSG

Fig. 11.7 P-δ curve along with the variation of Dq

(a)

279

P III II

|Z|

I P’ 0

δ 3δ 2δ 1

(b)

δ u1δ u2δ u3

δ

t

x102

Pem[kW]

2 1 0 -1 -2 3 2 |Z| [oh

15 m]

1

5 0 0

δ

10 [rad]

corresponding simulation results are shown in Fig. 11.8a. It is observed that curve I (blue) exhibits re-synchronization process, which can be illustrated by the operation mechanism in Fig. 11.5a. The corresponding voltage and current simulation results are shown in Fig. 11.9a. Curve II (red) shows the condition that the Q-V droop coefficient Dq increases from 166.7 to 400. It is observed that the system maintains first-swing stability and does not go beyond the UEP. It is because the UEP increases along with the increase of Dq, which is consistent with the analysis in Fig. 11.6. The corresponding time domain simulation results are shown in Fig. 11.8a. Curve III and VI represent the condition that damping coefficient D decreases from 8 to 4 and the line impedance increases from 5 mH to 8 mH, respectively. Due to the decrease of the damping loss and the decrease area, the system is more likely to experience transient angle instability as shown in Fig. 11.8b, which is consistent with the theoretical analysis. The simulation results are shown in Fig. 11.9c, d, respectively. When the fault is cleared at 2.3 s, the simulation results

280 Fig. 11.8 Phase portrait of the transient process. (a) Case I, (b) Case II

11

(a)

Re-synchronization Phenomenon of Microgrid

330 III

325

ω (rad/s)

320 315

fault clear at 1.8s

310

II first swing process re-synchronization process

305 300 -100

(b)

VI

I

0

100

200 300 δ (degree)

400

600

330 325

III

320

ω (rad/s)

500

VI II

I

315

fault clear at 2.3s

310 305 300 -100

first swing process re-synchronization process

0

100

200 300 δ (degree)

400

500

600

are shown in Fig. 11.8b. The re-synchronization mechanism of this case is shown in Fig. 11.5b. The corresponding time-domain simulation results are shown in Fig. 11.10a. It can be seen that when the Q-V droop coefficient Dq increases from 166.7 to 400, the operation trajectory still exhibits re-synchronization process, as shown in Curve II (red) (Shown in Fig. 11.10b). Curve III represents the damping coefficient D decreases from 8 to 4 while curve VI describes the line impedance increases from 5 mH to 8 mH. The corresponding time-domain simulation results are shown in Fig. 11.10c, d respectively. It can be concluded that the less damping coefficient and long line impedance are more likely to drive the system into transient instability area. The simulation results are identical with the theoretical analysis.

Re-synchronization Phenomenon of VSG

281

(a) 400 Vabc[V]

200 0 -200 -400

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 T[s]

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 T[s]

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 T[s]

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 T[s]

6

7

8

9

10

400 iabc[A]

200 0 -200 -400

Vabc[V]

(b)

400 200 0 -200 -400

iabc[A]

400 200 0 -200 -400

Vabc[V]

(c) 400 200 0 -200 -400 400 Iabc[A]

200 0 -200 -400

(d) 400 200

Vabc[V]

Fig. 11.9 Case I: Voltage and current simulation results of SVIB. (a) Nominal value (I), (b) Dq increase from 166.7 to 400 (II), (c) D decrease from 8 to 4 (III), (d) line impedance increase from 5 mH to 8 mH (IV)

0 -200 -400 400

Iabc[A]

11.1

200 0 -200 -400

282

Vabc [V]

(a)

Re-synchronization Phenomenon of Microgrid

400 200 0 -200 -400

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 T[s]

6

7

8

9

10

400

iabc [A]

200 0 -200 -400

Vabc [V]

(b)

400 200 0 -200 -400

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 T[s]

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 T[s]

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 T[s]

6

7

8

9

10

Iabc [A]

400 200 0 -200 -400

Vabc [V]

(c)

400 200 0 -200 -400

Iabc [A]

400 200 0 -200 -400

Vabc [V]

(d)

400 200 0 -200 -400 400 200

Iabc [A]

Fig. 11.10 Case II: Voltage and current simulation results of SVIB. (a) Nominal value (I), (b) Dq increase from 166.7 to 400 (II), (c) D decrease from 8 to 4 (III), (d) line impedance increase from 5 mH to 8 mH (IV)

11

0 -200 -400

11.2

Re-synchronization Phenomenon of Paralleled Systems

11.2

283

Re-synchronization Phenomenon of Paralleled Systems

As discussed above, it is found that SGs and VSGs may exhibit resynchronization phenomenon after VPA goes beyond UEP, which can guarantee their stable operation and provide additional power to support the grid after the fault clearance. In this section, we broach a brief case-study on paralleled VSGs system and paralleled SG-VSG system for the analysis of re-synchronization phenomenon.

11.2.1 Re-synchronization of Paralleled VSGs The topology of paralleled VSGs system is shown in Fig. 11.11. The control strategy is adopted in Fig. 10.1. As is deduced in Chap. 10, the swing equation of paralleled VSGs system can be written as J eq

  Dp1 Dp2 d2 δ012 0   ¼ T  T sin δ  J ð ω  ω Þ  ð ω  ω Þ M e eq 1 2 12 J1 J2 dt 2

ð11:10Þ

According to Eq. (11.10), the swing equation of paralleled VSGs system can be equivalent to that of single machine infinite bus system. The main difference is the power angle δ. The power angle of a grid-connected VSG is the difference between a VSG and power grid while the power angle of paralleled VSGs system is the difference between two VSGs. That is to say, the power angle of paralleled VSGs system is the relative power angle. The parameters of paralleled VSGs system are given in Table 11.2. The numerical simulation results are shown in Fig. 11.12. The red curves I shows the first-swing stability. As shown by the blue curve II and the green curve III, VPA will go beyond the UEP but and the system will return to another EP, when the fault is cleared at 3.5 s and 4.2 s. From the perspective of transient stability criterion in the power system, it means that the system will step into irreversible instability area and cannot recover to the synchronous state with the grid. However, from the simulation results, it is found Fig. 11.11 The topology of paralleled VSGs system

C

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Table 11.2 Parameters of the paralleled VSGs system Parameters P01 P02 ω01 ω02 DP1 DP2 J1 J2 Ks1 Ks2 Vdc1 Vdc2

Descriptions Rated active power of VSG1 Rated active power of VSG2 Reference angular speed of VSG1 Reference angular speed of VSG2 Damping coefficient of VSG1 Damping coefficient of VSG2 Inertia value of VSG1 Inertia value of VSG2 Proportional coefficient of VSG1 Proportional coefficient of VSG2 DC link rated voltage of VSG1 DC link rated voltage of VSG2

ω12[rad/s]

Fig. 11.12 Phase portrait of the transient process

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fault clear

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that the system can reach another EP, as shown by the blue curve II in Fig. 11.12. This phenomenon means that paralleled VSGs system owns strong synchronism capability even manifesting first-swing instability. Namely, there are two possible results when the system loses first-swing stability. One result is that VPA continues to go beyond the next NUEP, which is called transient angle instability. The other result is that VPA slip to another EP and the system can maintain synchronism with the grid, which can be called re-synchronization stability. The phenomenon is similar with that of a grid-connected VSG. Thus, the comprehensive flow chart of the transient stability criterion can be followed as Fig. 11.4. Curves II and III represent the phenomena of cases I and II, respectively. The corresponding voltage and current simulation results are shown in Figs. 11.13, 11.14 and 11.15. When the fault duration is 1.4 s, VPA does not go beyond UEP and the system will return to the original EP, which is shown in Fig. 11.13. Curve II (blue) exhibits re-synchronization process, which can be illustrated by the operation mechanism in Fig. 11.5a. The corresponding voltage and current simulation results are shown in Fig. 11.14.

Re-synchronization Phenomenon of Paralleled Systems (a) 400 200 0 -200 -400

iabc [A]

Fig. 11.13 Voltage and current simulation results of paralleled VSGs system. (a) VSG1, (b) VSG2

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When the fault duration is 2.2 s, curve III (green) shows re-synchronization process and the simulation results are shown in Fig. 11.15. The re-synchronization mechanism of this case is shown in Fig. 11.5b. The simulation results are identical with the theoretical analysis.

11.2.2 Re-synchronization of Paralleled SGs and VSGs The topology of paralleled VSGs system is shown in Fig. 11.16. The control strategy is adopted in Fig. 10.1. As is deduced in Chap. 11, the swing equation of paralleled SG-VSG system can be written as J eqs

  Dp2 d2 δ0s2 Ds 0  ¼ T  T sin δ  J ω  ð ω  ω Þ Ms es eqs 2 s2 Js s J2 dt 2

ð11:11Þ

According to Eq. (11.11), the swing equation of paralleled SG-VSG system can be equivalent to that of single machine infinite bus system. The numerical simulation results are shown in Fig. 11.17. The parameters of paralleled SG-VSG system are given in Table 11.3. As shown by the red curve I in Fig. 11.17, VPA does not go beyond the UEP and the system will return to the original EP when the fault is cleared at 2.6 s. The phenomenon is first-swing stability.

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vabc [V]

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Fig. 11.14 Case I: Voltage and current simulation results of paralleled VSGs system. (a) VSG1, (b) VSG2

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When the fault is cleared at 2.7 s and 3.5 s, VPA will go beyond the UEP. From the perspective of transient stability criterion in the power system, it means that the system will step into irreversible instability area and cannot recover to the synchronous state with the grid. However, from the simulation results, it is found that the system can reach another EP, as shown by the blue curve II and green curve III in Fig. 11.17. This phenomenon means that paralleled SG-VSG system has strong synchronism capability even manifesting first-swing instability. Curves II and III represent the phenomena of cases I and II which is shown in Fig. 11.5, respectively. The corresponding voltage and current simulation results are shown in Figs. 11.18, 11.19 and 11.20. When the fault duration is 0.6 s, the VPA does not go beyond the UEP and the system will return to the original EP, which is shown in Fig. 11.18. Curve II (blue) exhibits re-synchronization process, which can be illustrated by the operation mechanism in Fig. 11.5a. The corresponding voltage and current simulation results are shown in Fig. 11.19. The green curve III illustrates re-synchronization process and the simulation results are shown in Fig. 11.20, when the fault duration is 1.5 s. Clearly, the system can reach another EP and can maintain synchronism. Figure 11.5b shows the re-synchronization mechanism of this case. It can be found that the simulation results are identical with the theoretical analysis.

Summary (a) 400 200 0 -200 -400

iabc [A]

Fig. 11.15 Case II: Voltage and current simulation results of paralleled VSGs system. (a) VSG1, (b) VSG2

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Fig. 11.16 The topology of paralleled SG-VSG system

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To sum up, the paralleled SG-VSG system can still maintain synchronism although the VPA goes beyond the UEP. That is, the phenomenon is the re-synchronization phenomenon.

11.3

Summary

In this chapter, re-synchronization phenomenon for a microgrid which works in gridconnected mode or islanded mode is investigated. The mechanism of re-synchronization phenomenon of SG is introduced firstly. Then, according to

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ωg2[rad/s]

Fig. 11.17 Phase portrait of the transient process

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Table 11.3 Parameters of the paralleled SG-VSG system Parameters P0s P02 ω0s ω02 Ds DP2 Js J2 Ki Ks2 τi

Descriptions Rated active power of SG Rated active power of VSG Reference angular speed of SG Reference angular speed of VSG Damping coefficient of SG Damping coefficient of VSG Inertia value of SG Inertia value of VSG Proportional coefficient of SG Proportional coefficient of VSG2 Time-delay constant of SG

Values 40 kW 20 kW 314.159 rad/s 314.159 rad/s 0.01 0.01 10.04 5.02 8 4 0.6 s

characteristics of VSG and mechanism of re-synchronization phenomenon of SG, re-synchronization phenomenon of SG which is connected to infinite bus is discovered and discussed. It is found that VSGs may exhibit re-synchronization phenomenon after VPA goes beyond UEP. Thirdly, re-synchronization phenomenon of paralleled systems is studied as it is closer to reality than SG-infinite bus system. Several conclusions can be obtained. 1. The existence of Q-V droop loop is more likely to drive the system into re-synchronization area. In the one hand, it decreases inverters’ angle stability. While in other hand, it drives the system into re-synchronization area faster than the inverter without Q-V droop loop. 2. The large damping coefficient and short transmission line in power electronic system can prevent the system from irreversible instability phenomenon. Since re-synchronization process can guarantee the stable operation of the VSGs and provide additional power to support the grid after fault clearance. Designing suitable damping coefficient according to transmission line can help system to recover stability. 3. Re-synchronization analyses in paralleled synchronous and virtual synchronous generators are discussed, which is similar with that of single grid-connected VSG.

Summary

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Fig. 11.18 Voltage and current simulation results of paralleled VSGs system

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Fig. 11.19 Case I: Voltage and current simulation results of paralleled VSGs system

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Fig. 11.20 Case II: Voltage and current simulation results of paralleled VSGs system

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Obtained conclusions in this chapter can help protection devices to judge whether re-synchronization process would happen before cutting off the VSGs in a microgrid. The re-synchronization phenomenon in microgrid investigated in this chapter provides a new perspective for engineers that the system may return to stable state after fault clearance even the VPA goes beyond UEP. Namely, the VSGs needn’t to be cut off after fault. The transient stability criterion introduced in this chapter allows engineers to judge whether the system can return to stable state after fault clearance.

References 1. H. Xin, L. Huang, L. Zhang, Z. Wang, J. Hu, Synchronous instability mechanism of P-f droopcontrolled voltage source converter caused by current saturation. IEEE Trans. Power Syst. 31, 5206–5207 (2016) 2. P. Kundur, Power System Stability and Control (McGraw-Hill Education, New York, 1994) 3. Z. Shuai, C. Shen, X. Liu, Z. Li, Z.J. Shen, Transient angle stability of virtual synchronous generators using Lyapunov’s direct method. IEEE Trans. Smart Grid 10(4), 4648–4661 (2019) 4. C. Shen et al., Re-synchronization capability analysis of virtual synchronous generators in microgrids. 2019 IEEE Energy Conversion Congress and Exposition (ECCE), Baltimore, MD, USA, 2896–2901 (2019)