Modeling and Stability Analysis of Inverter-Based Resources 1032348291, 9781032348292

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Modeling and Stability Analysis of Inverter-Based Resources Renewable energy sources interface with ac grids via inverters and are termed inverter-based resources (IBRs). They are replacing traditional fossil fuel-based synchronous generators at a dazzling speed. In turn, unprecedented dynamic events have occurred, threatening power grid reliability. Modeling and Stability Analysis of Inverter-Based Resources provides a fundamental understanding of IBR dynamics. Developing reliability solutions requires a thorough understanding of challenges, and in this case, of IBR-associated dynamics. Modeling and stability analysis play an indispensable role in revealing a mechanism of dynamics. This book covers the essential techniques of dynamic model building for IBRs, including type-3 wind farms, type-4 wind farms, and solar photovoltaics. Besides modeling, this book offers readers the techniques of stability analysis. The text includes three parts. Part I concentrates on tools, including electromagnetic transient simulation, analysis, and measurement-based modeling. Part II focuses on IBR modeling and analysis details. Part III highlights generalized dynamic circuit representation—a unified modeling framework for dynamic and harmonic analysis. This topic of IBR dynamic modeling and stability analysis is interesting, challenging, and intriguing. The authors have led the effort of publishing the 2020 IEEE Power and Energy Society’s TR-80 taskforce report, “Wind Energy Systems Subsynchronous Oscillations: Modeling and Events,” and the two taskforce papers on investigation of real-world IBR dynamic events. In this book, the authors share with readers many insights into modeling and analysis for real-world IBR dynamic events investigation.

Taylor & Francis Taylor & Francis Group

http://taylorandfrancis.com

Modeling and Stability Analysis of Inverter-Based Resources

Lingling Fan and Zhixin Miao

First edition published 2024 by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2024 Lingling Fan and Zhixin Miao Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-1-032-34829-2 (hbk) ISBN: 978-1-032-34749-3 (pbk) ISBN: 978-1-003-32365-5 (ebk) DOI: 10.1201/9781003323655 Typeset in Nimbus Roman by KnowledgeWorks Global Ltd. Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.

Contents Preface...............................................................................................................................................ix About the authors..............................................................................................................................xi

Part I Chapter 1

Tools Introduction .............................................................................................................3 1.1 1.2 1.3

Chapter 2

Tools: simulation and analysis ................................................................................ 7 2.1 2.2

2.3

2.4

2.5 Chapter 3

Why this book? .............................................................................................. 3 Book structure.................................................................................................4 Features and highlights ...................................................................................5

Electromagnetic transient (EMT) simulation .................................................7 2.1.1 Why to use an EMT package ............................................................. 8 2.1.2 How to master EMT........................................................................... 9 Per unit analysis............................................................................................20 2.2.1 Per unit as scaling ............................................................................ 22 2.2.2 Individual versus aggregated models ............................................... 23 2.2.3 Equations with coefficients ..............................................................24 2.2.4 System strength and short circuit ratio (SCR) .................................25 Analytical dynamic model building..............................................................25 2.3.1 Frame conversion in space ...............................................................25 2.3.2 Frame conversion between the time and frequency domains ..........26 2.3.3 A three-phase phase-locked loop (PLL) .......................................... 26 2.3.4 A single-phase PLL.......................................................................... 27 2.3.5 An RLC circuit................................................................................. 31 Small-signal analysis ....................................................................................34 2.4.1 Modal analysis .................................................................................35 2.4.2 Block diagram and frequency-domain analysis ...............................36 2.4.3 Interactions of two RLC branches in a circuit ................................. 36 Summary.......................................................................................................42

Tools: measurement-based modeling .................................................................... 44 3.1 3.2 3.3 3.4

A tutorial example ........................................................................................ 45 Model development ..................................................................................... 50 3.2.1 Modeling the dc side dynamics of a solar PV.................................. 51 3.2.2 Modeling PLLs with sophisticated structures..................................53 Frequency scan: dq admittance ....................................................................61 3.3.1 Admittance measurement based on frequency scans....................... 62 Frequency scan: sequence-domain admittance............................................. 63 3.4.1 Injection frequency above the fundamental frequency ....................63 3.4.2 Injection frequency at the mirror frequency ................................... 65

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3.4.3

3.5

Part II Chapter 4

Inverter-Based Resources: Detailed Examination Control of IBR power plants ................................................................................. 73 4.1

4.2 4.3 4.4 Chapter 5

Grid-following control..................................................................................73 4.1.1 The actuator: the converter voltage.................................................. 73 4.1.2 More about frame conversion .......................................................... 76 4.1.3 Outer control design.........................................................................79 4.1.4 Inner current control ........................................................................87 4.1.5 PLL design .......................................................................................94 4.1.6 EMT simulation results....................................................................96 Plant-level control......................................................................................... 98 4.2.1 Bandwidth of IBR controls ............................................................ 100 4.2.2 EMT simulation results..................................................................102 Synchronization techniques ........................................................................103 Summary.....................................................................................................105

Analytical modeling of a GFL-IBR..................................................................... 106 5.1 5.2 5.3

5.4 5.5 5.6

5.7

5.8 Chapter 6

Relationship between sequence-domain and dq-domain admittance matrices...................................................................................66 3.4.4 Measurement procedure...................................................................66 3.4.5 An example of admittance measurement ......................................... 66 Summary.......................................................................................................69

A simplified linear model ........................................................................... 106 5.1.1 Application: weak grid voltage instability mechanism.................. 108 Analysis of real-world 0.1-Hz oscillation event ......................................... 113 Current source-based model with PLL included ........................................117 5.3.1 A second-order model.................................................................... 118 5.3.2 PLL and negative impedance ......................................................... 119 5.3.3 A linearized model......................................................................... 121 PLL weak grid stability: inclusion of grid dynamics.................................. 127 Nonlinear analytical models of a GFL-IBR................................................ 134 5.5.1 The testbed .....................................................................................134 5.5.2 State-space representation..............................................................135 Applications of the nonlinear analytical models ........................................138 5.6.1 Parameterization of a simplified model ......................................... 138 5.6.2 Weak grid oscillation analysis ....................................................... 140 5.6.3 Interactions of PLL and the rest of the system .............................. 151 GFL with static-frame current control........................................................156 5.7.1 The study system and the key challenges ...................................... 156 5.7.2 Comparison of the analytical model and the EMT model ............. 159 5.7.3 Applications of the analytical model ............................................. 159 5.7.4 Analysis of a real-world 20-Hz oscillation event........................... 161 Summary.....................................................................................................166

Grid-forming (GFM) control...............................................................................168 6.1 6.2 6.3

Multi-loop GFM: from GFL to GFM .........................................................170 GFM1: advanced GFL ................................................................................170 GFM2: the conventional design.................................................................. 171

Contents

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6.4 6.5 6.6 Chapter 7

Type-3 wind farms............................................................................................... 179 7.1 7.2 7.3

7.4 Chapter 8

8.2 8.3

8.4

Chapter 9

Analysis of induction generator effect (IGE) ............................................. 179 7.1.1 Example 1: subsynchronous resonance due to induction generator effect ....................................................................................... 181 An EMT test case of SSR ........................................................................... 183 Analytical modeling.................................................................................... 187 7.3.1 dq-frame dynamic models ............................................................. 189 7.3.2 Steady-state computation ............................................................... 193 7.3.3 Simscape SPS implementation ...................................................... 198 7.3.4 Applications ................................................................................... 200 Summary..................................................................................................... 202

Power networks with multiple IBRs.................................................................... 203 8.1

Part III

GFM3: minimal edits.................................................................................. 174 Virtual synchronous machines (VSM)........................................................ 176 Summary..................................................................................................... 177

Inter-IBR oscillation mode ......................................................................... 203 8.1.1 Block diagram construction and integration techniques ................ 203 8.1.2 Simulation results .......................................................................... 206 8.1.3 Modal analysis ............................................................................... 206 8.1.4 Two decoupled circuits .................................................................. 209 A three-generator power grid...................................................................... 211 8.2.1 Nonlinear analytical modeling ...................................................... 211 8.2.2 Admittance-based modeling .......................................................... 218 Frequency-domain modal analysis ............................................................. 223 8.3.1 Admittance-based stability analysis............................................... 223 8.3.2 Basic mode shape analysis............................................................. 224 8.3.3 Extended mode shape analysis....................................................... 226 Summary..................................................................................................... 228

Generalized Dynamic Circuits Generalized dynamic circuits .............................................................................. 231 9.1

9.2

9.3

Induction machines..................................................................................... 232 9.1.1 Steady-state circuit representation ................................................. 232 9.1.2 Dynamic circuit in the dq frame .................................................... 234 9.1.3 From steady-state circuits to generalized circuits.......................... 236 9.1.4 Applications: stability analysis ...................................................... 237 Unbalanced topologies................................................................................ 240 9.2.1 Sequence networks......................................................................... 240 9.2.2 Expanding sequence networks to a dynamic circuit ...................... 240 9.2.3 Case study: an induction machine served by an unbalanced network ............................................................................................... 242 Synchronous machines ............................................................................... 246 9.3.1 An induction machine with unbalance in rotor circuits ................. 246 9.3.2 Steady-state circuit of a synchronous machine ............................. 247 9.3.3 From a dynamic model to a generalized circuit............................. 250 9.3.4 Case study: starting a synchronous machine ................................. 252

Contents

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9.4 9.5

9.3.5 Summary ........................................................................................255 IBRs ...........................................................................................................255 9.4.1 From dq admittance to a two-port circuit representation............... 256 9.4.2 Two-port circuit representation by derivation................................ 258 Summary and notes.....................................................................................266

References ..................................................................................................................................... 267 Index..............................................................................................................................................271

Preface Inverter-based resources (IBRs), such as wind and solar photovoltaics (PV), are replacing synchronous generators in power grids at a dazzling speed. In turn, power grids worldwide have experienced dynamic and stability issues due to IBRs. Some have led to large-scale power disruption events. Reliable operation has become the top priority. This requires a thorough understanding of IBR dynamics. This book, Modeling and Stability Analysis of Inverter-Based Resources, aims to offer readers a fundamental understanding of IBR dynamics in power grids. This book presents the essential techniques of dynamic model building for IBRs. The purpose of dynamic model building is for time-domain simulation and stability analysis. Dynamic model building translates physics and power electronics control into mathematical models consisting of blocks with suitable interconnections. Model building procedures will be covered in this book for several IBRs: type-3 wind farms, type-4 wind farms, and solar PVs. Besides modeling, this text offers readers the techniques of stability analysis. The authors will combine engineering insights with mathematic rigor into stability analysis of various dynamic phenomena observed in real-world power grids. The book builds on the research and notes of the authors from the past decade. It covers four sets of techniques in dynamic modeling and analysis: electromagnetic transient (EMT) test bed building and tuning, ordinary differential equations- (ODE-) based model building, stability analysis techniques, and measurement data-based model identification. Two features of the book are worthwhile to be mentioned. First, we prepare case studies associated with real-world IBR dynamics. Remarks drawn from numerical case studies are compared with real-world observations. Those case studies serve the purpose of mechanism analysis of real-world events to understand root causes. Another feature of the book is its tutorial style and availability of example codes and models; both help readers self-learn EMT simulation and dynamic modeling. The only requirements are fundamental circuit and control knowledge. Readers can refer to those codes and models and effectively learn the techniques offered by the book. The authors would like to acknowledge the U.S. Department of Energy (DOE) Solar Energy Technologies Office (SETO), the Wind Energy Technologies Office (WETO), the Electric Power Systems Research, and the National Science Foundation for providing funds that make it possible to conduct this research, which is highly relevant to the practical world. Specifically, the authors would like to acknowledge Drs. Jian Fu, Kemal Celik, and Guohui Yuan for recognizing the importance of their research. The authors would like to acknowledge many colleagues collaborating with them and providing support over the years: Shahil Shah, Deepak Ramasubramanian, Yunzhi Cheng, Pengwei Du, Ryan Quint, Chun Li, Babak Badrzadeh, Nilesh Modi, Jayanth Ranganathan Ramamurthy, Xiaodong Liu, Jay Liu, Songzhe Zhu, Song Wang, Bikash Pal, Julia Matevosyan, Carlos Aldazabal, Ranjit X. Amgai, and others. The authors are thankful to many people who helped them along their academic journeys: Subrayan Yuvarajan, Roger Green, Rajesh Kavasseri, Alexander Domijian, John Wiencek, Don Morel, Salvatore Morgera, Lee Stafanakos, and Chris Ferekides.

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Preface

Several students and former students have helped proofread this book and their work greatly improved the clarity. Thanks to Zhengyu Wang, Li Bao, Rabi Kar, and Rahul H. Ramakrishna for their meticulous work. Last, but not least, the authors would like to acknowledge Ms. Nora Konopka of Taylor & Francis Group for entrusting them with the production of this book. MATLAB is a registered trademark of The MathWorks, Inc. For product information please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA, 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: [email protected] Web: www.mathworks.com

About the authors Lingling Fan is a professor in the Department of Electrical Engineering at the University of South Florida. Before joining the academia, she has worked in the grid operating industry with Midwest ISO, for six years (2001–2007). She received a BS and MS in electrical engineering from Southeast University (Nanjing, China) in 1994 and 1997, respectively. She obtained a PhD in electrical engineering from West Virginia University, Morgantown in 2001. Dr. Fan is active in research in control, computing, and dynamic analysis of power systems, power electronics, and electric machines. Her research has been sponsored by the Department of Energy, Midwest ISO, Duke Energy, National Science Foundation, Electric Power Research Institute, Florida Cyber Security Center, Jabil and more. She has authored/co-authored two books, Modeling and Analysis of Double Fed Induction Generator Wind Energy Systems (Elsevier Press, 2015) and Control and Dynamics in Power Systems and Microgrids (CRC Press, 2017). Dr. Fan has served as consulting editor for IEEE Transactions on Sustainable Energy. Currently, she serves as editor-in-chief of IEEE Electrification Magazine and associate editor of IEEE Transactions on Energy Conversion. She was elevated to IEEE fellow class 2022 for her contributions to stability analysis and control of inverter-based resources. She is the recipient of USF’s Outstanding Research Achievement Award in 2022 and has been featured in IEEE Power and Energy Society social media in 2022 to celebrate World Engineering Day and National Women’s History Month. Zhixin Miao is a professor in the Department of Electrical Engineering at the University of South Florida (USF). He received a BS in electrical engineering from the Huazhong University of Science and Technology in Wuhan, China in 1992, MS in electrical engineering from the Nanjing Automation Research Institute (NARI) in Nanjing, China in 1997, and PhD in electrical engineering from West Virginia University in Morgantown in 2002. He worked as a power system protection engineer. From 1992–1999 in NARI and a transmission planning engineer at Midwest ISO, St. Paul, MN from 2002–2009. His research interests include digital twins, power system computer and hardware simulations, microgrids, and renewable energy integration. Dr. Miao serves as an associate editor for IEEE Transactions on Sustainable Energy.

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Part I Tools

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1.1

1

Introduction

WHY THIS BOOK?

A good book immediately brings us into a field and exposes us to a systematic view. An excellent book offers rigorous analysis and an in-depth view as additions. As researchers, we benefit from excellent books, e.g., Bergen’s book on power system analysis [1], Yazdani & Iravani’s book on basics of voltage source converters [2]. This book is aimed to offer an in-depth understanding and various techniques in modeling and stability analysis of inverter-based resources (IBRs) in power grids. IBRs refer to those energy sources interfacing with the ac grid through power electronics converters. Type-3 and type-4 wind turbines, solar photovoltaic (PV), and battery energy storage systems (BESS) are all IBRs. The power grid has been operating for more than a century with synchronous generators as the main workhorse. This is changing and we are witnessing the increasing penetrations of IBRs in power grids. The Texas power grid, with a peak load size of approximate 98 GW, experienced 69% record high wind penetration (27 GW) on April 10, 2022 and 29.77% solar penetration (11.9 GW) on March 4, 2023. By 2030, the United State has a plan to integrate 30 GW offshore wind into its power grid. Modeling and stability analysis of IBRs is of vital importance to power grid reliable operation. and it is a new research area. We want to share with the readers not only research outcomes but also the process of research, with a special focus on the strategies of mechanism analysis of real-world IBR dynamics. We hope the book is thought-provoking and will benefit the readers on research skills and reasoning. The book should be useful to power grid operators and IBR control designers. Operators can better understand the features of IBR controls. IBR control designers can better understand the influence of grid on IBR performance. Technologies evolve along with applications. In IBR grid integration, even after devices have been manufactured and tested individually, once they are put into a grid, unprecedented dynamics may occur since a complete check of grid conditions is not possible. There are just too many grid conditions. Interested readers may check the talk titled “When IBRs Meet the Grid” [3], delivered by the first author in February 2023 at the U.S. Department of Energy Solar Energy Technologies Office webinar platform, for a list of grid conditions and the possible interactions of IBRs and grids. Regarding unprecedented dynamics, one example is the type-3 wind farm subsynchronous resonance (SSR). Only when type-3 wind farms have been put into the grid in scale, and only when it happened to have contingencies leaving turbines radially connected to series capacitors, SSR would appear. After the 2009 Texas wind farm SSR event, wind turbine manufacturers designed SSR damping controls for the turbines to operate stably even when they are radially connected to series capacitors. This solved the issue for the next eight years. However, when the grid condition changes and when IBR penetrations become higher, SSR may occur again. The Texas 2017 wind farm SSR events are such cases. As we are writing this book, SSR events were reported again in Texas in January and May 2023. Another example is weak grid stability issues for IBR-penetrated power grids. When the penetration becomes higher, the grid strength reduces. A reduced grid strength leads to high sensitivity of voltage towards real and reactive power injection. Hence, voltage stability related issues show up. In Australia, 7-Hz oscillations were observed in solar PV farms in its west Murray zone. Such a low grid strength condition perhaps has never been thought about in the IBR control design stage. Therefore, it can be seen that technologies improve through use and iterations are necessary. Without the advancements of power electronics converters, large-scale deployment of wind and

DOI: 10.1201/9781003323655-1

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Modeling and Stability Analysis of Inverter-Based Resources

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solar would not be possible. On the other hand, with large-scale deployment, new challenges appear and new solutions are proposed. In turn, IBR control gets enhanced to accommodate various grid conditions. Developing solutions requires a thorough understanding of the challenges, in this case, IBR associated dynamics. Modeling and stability analysis is the main tool that we rely on and it plays an indispensable role to understand IBR dynamics. Domain knowledge of power grid modeling and power electronic converter controls, as well as computing and analysis knowledge, is required. This requirement is similar to that for research of Flexible Alternating Current Transmission Systems (FACTS) and High-Voltage Direct Current (HVDC). This category of research is interesting, challenging, and intriguing. Relevant and impactful research outcomes can be developed by focusing on the real-world needs and solving puzzles that both the industry and academic community have an interest. The authors have led the effort of publishing the IEEE Power and Energy Society’s TR-80 taskforce report “Wind Energy Systems Subsynchronous Oscillations: Modeling and Events” in July 2020, and the two taskforce papers on investigation of real-world IBR dynamic events [4, 5]. In this book, the authors will present modeling and analysis techniques employed in real-world IBR dynamic events investigation.

1.2

BOOK STRUCTURE

The book has three parts. Part I focuses on tools, including simulation and analysis tools and measurement-based tools. Part II focuses on IBR modeling and analysis details. Part III is on generalized dynamic circuit representation—a unified modeling framework for dynamic and harmonic analysis. In Part I, the readers will be introduced to various tools used in dynamical system modeling and analysis. Chapter 2 serves as an introduction to electromagnetic transient (EMT) simulation, analytic modeling, and linear analysis. Chapter 3 focuses on measurement-based model extraction. In Part II, more detailed treatment focusing on three-phase grid connected inverters, type-3 wind turbines, and multi-IBR systems, will be offered. Part II also offers numerous application examples. Specifically, Chapter 4 examines a grid-following inverter based IBR’s controls and the design principles. The controls have been implemented in EMT testbeds for demonstration. Readers may find Matlab codes used for design and EMT models for testing. In Chapter 5, analytical models of grid-following IBR with different levels of complexity are examined. Moving forward, Chapter 6 presents several types of grid-forming controls. Each control has been demonstrated using EMT simulation. In Chapter 7, the SSR phenomena in a type-3 wind farm’s are first demonstrated in an EMT testbed. Next, an analytical model is built to have the full details of induction machine electromagnetic dynamics, rotor-side-converter control, grid-sideconverter control, and dc-link capacitor dynamics. This analytical model can be used for timedomain simulation as well as eigenvalue analysis. Chapter 8 covers analytical modeling techniques when there are two or more IBRs in a power grid. Part III presents our new research outcome: generalized dynamic circuits. This part can be viewed a new tool or a new modeling framework capable of facilitating unbalanced fault analysis, harmonic analysis, and dynamic analysis. This is a powerful modeling framework suitable for systems operating under multiple frequencies. A typical example is an induction machine where its stator and rotor are operating in different frequencies. In circuit representation, the stator and rotor circuits are treated as a whole. For IBRs, particularly when they are black boxes, we need a modeling framework that can efficiently integrate models extracted measurements. In Part II, although we have touch based on dynamic analysis by use of IBR’s measured dq admittance, we have not come up with an efficient framework that can deal with multiple operating frequencies, which often appear during unbalance. This framework is presented in Part III. We feel that this is a milestone work capable of a variety of power system analysis.

Introduction

1.3

5

FEATURES AND HIGHLIGHTS

A feature of this book is that we have extensively used frequency-domain modeling and analysis. The entire Chapter 9 is on a new modeling and analysis framework by use of operational calculus or Laplace transform variable s. From the beginning of introducing analysis tools in Chapter 2, readers can find frequency-domain analysis examples. Interested readers may compare our book with several classic books. In the classic book “Power System Stability and Control” by P. Kundur (McGraw-Hill, 1994) [6], while transfer functions have been used in synchronous modeling, the tool for small-signal stability is modal analysis or eigenvalue analysis based on state-space models, regardless of low-frequency electromechanical oscillations or subsynchronous resonances. A. Bergen, in his book “Power System Analysis” [1], used transfer function-based block diagrams and Root Locus diagrams to examine stability. Root-locus diagrams are still in the arena of eigenvalues. Only in “Voltage-Sourced Converters in Power Systems” by Yazdani & Iravani [2], Bode diagrams were extensively used for control design and stability check. While both Kundur and Bergen are viewed as system experts, Yazdani and Iravani are viewed as experts for power electronics and power systems. Indeed, frequency-domain analysis can lead to straightforward design and analysis, and has been favored by many practical converter control designers. L. Hannefors of ABB extensively used this tool in many of his papers on converter control design. See e.g., [7, 8]. Both authors were trained for power systems. With such a background, it took us years to appreciate and master frequency-domain analysis. In this book, we share our experience in utilizing this great tool for control design and stability analysis. Readers may find interesting case studies in Chapter 5 on the 0.1-Hz oscillations caused by plant-level voltage control delay and the 20-Hz/80Hz oscillations caused by slow inner current control. In both cases, frequency-domain analysis has been employed. Of course, eigenvalue-based modal analysis results are also presented. For largescale networks, eigenvalues provide direct check on stability. This book has many example models and codes meticulously prepared. For a learner, it is very important to have examples and codes to know the exact concept. Take the following example of using per unit values. Researchers with power systems background are used to per unit-based analysis while researchers with power electronics background are more used to physical valuebased analysis. This reflects in our book and [2] where we adopt per unit systems while the authors of [2] adopt physical values most of the time. We know that per unit based analysis is great for computing and we have remembered many rules of per unit calculation. With this understanding, can we quickly write down the transfer function between the per unit current and voltage across a choke filter? 1 Assume that its resistor is Rpu = 0.01 pu and its inductance is Lpu = 0.15 pu. Can 0.01+0.15s describe the current and the voltage relationship, or represent the admittance? To answer this question, let’s substitute s by j377 rad/s. The resulting impedance is 0.01 + j377 × 0.15. The reactance is too large. At the nominal frequency, the impedance should be 0.01 + j0.15 p.u, since X pu is also 0.15 pu if Lpu is 0.15 pu. In fact, the admittance should be 1 . 0.01 + 0.15 377 s The reason is that R and sL share the same impedance base. Therefore, if we per unitize the impedance R + sL, the resulting expression is R sL Lpu R + sL = + = Rpu + s = 0.01 + 3.98 × 10−4 s. Zbase Zbase ω0 Lbase ω0 The admittance that relates the per unit current to the per unit value should be 100 of 1+15s . This example shows that numerical examples and codes matter.

100 1+0.0398s , instead

Modeling and Stability Analysis of Inverter-Based Resources

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In Chapter 4 of the book, the sizes of the dc-link capacitors for solar PV farms and wind turbines/farms of different power rate from the MATLAB’s demo systems are given. For a 2-MW type-4 wind turbine, the size is 0.09 F. If we have this typical number in mind, we can estimate that for a 400-kW PV inverter, the size of the dc-link capacitor should be one-fifth of 0.09 F, or 0.018 F. The 400-kW PV demo case shows the dc-link capacitor size as 0.05 F, while the 250-kW PV demo case shows the dc-link capacitor size as 0.0272 F. These numbers give readers an idea on the range of dc-link capacitor sizes. In addition to examples and codes, readers may find many relevant online videos of the authors. Some of those videos are listed in below. Typing the titles of the talks in Google should lead to the videos and/or presentations slides. 1. Mechanism Analysis of Dynamic Phenomena in Power Grids with High Penetrations of InverterBased Resources (IBRs). A webinar for DOE SETO System Integration 2023 webinar series, 2/22/2023 2. Analytical model building for IBRs. A webinar for the Universal Interoperability for Gridforming inverters (UNIFI) Consortium’s Fall 2022 webinar series, 10/31/2022. Youtube video 3. Replication and identification of causes of grid oscillations. A talk at the Energy System Integration Group (ESIG) Fall 2022 Technical Workshop, 10/26/2022. Youtube video. 4. Inverter-based Resources Subsynchronous Oscillations: Events and Mechanism Analysis. A webinar for IEEE PES Live Webinar, July 6, 2022. IEEE PES resource center. 5. Analysis of Inverter-based Resource Dynamic Events. A presentation for the Spring 2022 seminar series of the UC Berkley PES/PELS student chapter, March 10, 2022. Youtube video. We hope readers enjoy this book.

2

Tools: simulation and analysis

In this chapter, three types of tools of simulation and analysis will be discussed: Electromagnetic Transient (EMT) simulation, analytical dynamic model building, and small-signal analysis. Per unit analysis is very important. Therefore, this topic is covered in a separate section. Time-domain dynamic simulation can be carried out in an EMT environment (e.g., MATLAB’s Simscape) or by designing a nonlinear model. For the latter task, MATLAB’s Simulink is an excellent platform, compared to coding. Simulink offers a graphic user interface (GUI) so that once we develop a block diagram, we can do coding in module by module. This platform also allows us to put scopes flexibly for monitoring and debugging. For a tutorial, please refer to the first author’s book Control and Dynamics in Power Systems and Microgrids published by the CRC Press in 2017 [9] (Chapter 2). While the second approach requires mathematic modeling skill, the requirement is much relaxed when modeling is conducted in an EMT environment. In this chapter, we cover EMT simulation in Section 2.1 and cover analytical model building in Section 2.3. In between, we cover per unit analysis in Section 2.2, since analytical models are usually built in per unit system. Just time-domain simulation is not enough. We also need analysis. Small-signal analysis is based on linear systems. These systems can be directly obtained if a nonlinear analytical model is available and Jacobian linearization is possible. Otherwise, we may extract models from measurement data relying on system identification techniques. The topic is covered in Section 2.3 along with analytical model building. In addition, Section 2.4 is devoted to small-signal analysis to cover both eigenvalue-based modal analysis and block diagram-based frequency-domain analysis. An example is presented in this section to explain both approaches. Extracting linear models from measurement data is becoming popular since high-fidelity IBR EMT models provided by OEMs to generator owners and grid operators are usually vendor-specific and site-specific black boxes. With such models, measurement-based characterization is important to understand a model’s behavior. This topic is extensively covered in Chapter 3.

2.1

ELECTROMAGNETIC TRANSIENT (EMT) SIMULATION

Electromagnetic Transient (EMT) simulation is a simulation tool initiated by Hermann Dommel in his Ph.D. dissertation in 1968. In the next several decades, the industry and the U.S. federal funding agencies sponsored the development of EMT simulation tools. EMT simulation leads to accurate dynamics, including switching transients, traveling waves, not to mention electromagnetic transient dynamics in circuits, machines and power electronics devices. Conventional time-domain simulation is based on numerical integration of ordinary differential equations (ODE) or differential algebraic equations (DAE). Dommel’s treatment of any ODE is to first discretize it and then convert it to a Norton circuit consisting of a current source and a resistor. ODEs related to an inductor or a capacitor are all treated this way. With each component treated as Norton circuit, the entire circuit can be easily assembled. Many people/research groups contributed to EMT simulation. To date, there are three major groups: the Manitoba community, which includes the PSCAD and RTDS vendors and the University of Manitoba with Professor Aniruddha Gole as the main educator and developer; the Quebec community, which includes EMTP-RV and OPAL-RT vendors and Polytechnique Montr´ea as the main training center and Professor Jean Mahseredjian as the main developer; and the Europe community which focuses on applications and development of an open-source free package ATP.

DOI: 10.1201/9781003323655-2

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8

Modeling and Stability Analysis of Inverter-Based Resources

EMT simulation has been used in various areas, including protection, HVDC and IBR grid integration. People use its capability to generate accurate simulation results of circuits. The simulation results are close to reality. That is, we directly see three-phase waveforms from simulation, same as from an oscilloscope. In phasor-based power system dynamic simulation software, e.g., PSS/E, PSLF and MATLAB toolboxes such as Power System Toolbox developed by Chow and others [10], or Power System Analysis Toolbox [11] by Federico Milano, the simulation results are phasors, instead. We see flat lines of voltage magnitude or phase angles at steady state. Outputs from these package are not directly related to the physical systems. Interpretation is required. In the modeling and analysis perspective, additional layers of abstraction are required. If unbalance and harmonics have to be considered, the phasors considered are no longer limited to a single ac frequency. Complicated derivation of dynamic phasor-based models are required. This area is an active research area whenever a new device is being developed. For converters, the most sophisticated converter is modular multi-level converter (MMC) which has been used in HVDC. Many researchers are currently working on refining the phasor-based models for MMC. See e.g., Dragan Jovcic’s papers [12, 13]. There is a unique advantage of phasor-based models compared to EMT models. In an EMT testbed, instantaneous current and voltage at steady state are time varying. On the other hand, one unique feature of phasor-based models is that the state variables are constants at steady state. This makes the models suitable for steady-state analysis and small-signal model extraction via direct Jacobian linearization. The latter is an important feature. As long as we have linear models, there are mature technologies of linear algebra and frequency-domain analysis available to us. Even in such cases, to guarantee an accurate derivation of phasor-based models, benchmarking with the corresponding EMT model is the first and a critical step to guarantee success. 2.1.1

WHY TO USE AN EMT PACKAGE

To study IBR dynamics, EMT simulation is an indispensable tool. Part of the reason is that IBR’s dynamics cover a wide frequency range. An EMT package usually has a library of component models well developed and tested, e.g., machines, transmission lines, cables, etc. Electromagnetic dynamics are heavily involved in many IBR dynamics observed in real world. EMT software packages thus provide a fast venue for us to construct a testbed for simulation. To use an EMT package is to have a direct access to research achievements accumulated for decades from many researchers and software developers. Packages such as MATLAB/Simscape and PSCAD all have nice GUI with libraries of components, e.g., machines, converters, transmission lines, sensors, phase-locked-loops (PLL), meters, etc. The environment can be treated as a virtual lab. One may find machines, passive components, and meters from the library. One can then grab those devices and make connections to form a circuit. In MATLAB/Simscape, the two main libraries to use are the Simscape Specialized Power Systems (SPS) library and Simulink’s basic library. There are several unique features of Simscape simulation vs simulation conducted by just Simulink blocks. 1. Simscape is a virtual lab. It is GUI based and almost bug free. One does not need to think about math and differential equations. To construct a testbed, you do not need to code and debug. 2. Same as a physical lab, one has to keep in mind that there are two sets of components: those for physical circuits that can handle large power, and those for signals. For example, a current sensor has three ports. Two ports are for connection in the circuit. One port is for signal and it is to be connected to a scope for monitoring. Similarly, if you’d like to put a switch in a circuit, you need to find a breaker from the Simscape library. A switch from Simulink is for signals only.

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3. Simscape has to have powerGUI block. This block makes sure that the GUI blocks can be translated into math equations for numeral integration or simulation. In general, discrete simulation is adopted and a step size of 50 μs is adequate for 60-Hz ac circuits. The step size set in the powerGUI should match that in the settings for a discrete simulation solver. These are the basics. With those in mind, the rest is to practice and learn from examples. Simscape’s SPS have many useful examples, e.g., starting a synchronous machine, operating an induction machine, a type-3 wind farm system, a solar PV grid integration system, and a microgrid with three parallel inverters. Similarly, other vendors all provide demo cases. This saves researchers a great deal of time to start from scratch up. One nice demo example in Simscape power system is a two-terminal voltage sourced converter (VSC)-HVDC system. Both the rectifier and the inverter are modeled in great details. It is worth to note that real-world converter controls contain logics usually not available in demo systems, e.g., fault-ride through schemes. Another example is the type-3 wind farm average model which includes an induction machine, a rotor-side converter with its control, a grid-side converter with its control, and dc-link dynamics. Such demo systems help expedite learning and research. 2.1.2

HOW TO MASTER EMT

In this book, EMT models are built in MATLAB Simscape SPS environment. To quickly learn EMT simulation, it is vital to work out a simple circuit from scratch up and refer to many demo examples for further learning. In short, we advocate this approach: “learning by doing” and always learn from examples. 2.1.2.1

A simple circuit with an LC mode

EMT modeling task: Set up a test bed of a single-phase current source connected to a grid; and demonstrate subcycle overvoltage upon ramping down of the current source. The objective of this example is to demonstrate that the subcycle overvoltage is caused by the LC resonance mode due to the shunt compensation and grid inductance. A solar PV can be viewed as a current source. In real world operation, momentary cessation, or quickly ramping down the current, occurs when a low voltage is sensed. This can trigger the LC mode. Fig. 2.1 shows the model built in MATALB SPS. Besides the circuit consisting of a current source, a shunt capacitor, and the Th´evenin equivalent representation of the grid as a voltage source behind an RL impedance, there is the PowerGUI block in the upper left corner showing the simulation time step. This block has many other useful functions, e.g., fast Fourier transform (FFT). To have the current and voltage measurements, current and voltage sensors have also been put into the circuit. Finally, a real and reaction power calculation block outputs real and reactive power with the instantaneous capacitor voltage measurement and the current to the grid as the inputs. Fig. 2.2 shows the simulation results in the four scopes. At 0.5 s, the current source ramps down from 1 to 0.5. At 0.51 s, it further ramps down to be 0. This triggers the dynamics in the circuit and the capacitor voltage shows overvoltage right after 0.5 s. It can be seen that upon current ramping down, overvoltage is observed in the capacitor voltage. The arrangement in Fig. 2.1 can be refined to lead to a more concise configuration with better visual effect. Measurements can be obtained through enabling each component’s measurement. This saves all the sensors. Also for any complicated switching sequences, we can use a MATLAB function with time clock as the input and write a few lines of codes to have the output. Finally, all measurements can be collected and put into a single scope with multiple channels. Fig. 2.3 shows the refined configuration of the model and Fig. 2.4 shows the scope data for 1 second. Fig. 2.5 further shows the zoom-in data focusing on the 0.45 s-0.65 s period.

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10

Figure 2.1: Example 1. The system parameters are R = 0.05 Ω, L = 0.5/ω H, C = 0.2/ω F, where ω = 377 rad/s. It can be seen that the capacitor voltage shown in Fig. 2.5 has a fundamental component and another high harmonic component at about 180 Hz from 0.5 s to 0.56 s. The FFT analyzer in the PowerGUI block can help determine the harmonic component. Fig. 2.6 is the screen copy of the FFT analysis results. The FFT analyzer examines the 10 cycles of the capacitor voltage starting from 0.5 s. The high-order harmonic component is shown to have a frequency about 186 Hz and the amplitude is about 15% of the fundamental component. It will be better to follow up with a circuit analysis to examine the dynamics. We are interested in the capacitor voltage vc . This voltage can be expressed by the grid voltage vg and the current source is in the frequency domain easily. Based on the nodal voltage expression, we can see the following:   vg 1 sC + vc = is + R + sL R + sL R + sL 1 =⇒vc = 2 (2.1) is + 2 vg . s LC + sRC + 1 s LC + sRC + 1

 
 



 
  


 


 

 

 


 


Figure 2.2: Simulation results of Example 1.

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Figure 2.3: Example 1 in MATLAB SPS with better visual effect. The system has a characteristic polynomial s2 LC + sRC + 1 = 0 and the eigenvalues of the system are the roots of the polynomial: R 1 =0 s2 + s + L LC ω2 R =0 =⇒s2 + ωs + X XB ω −0.1ω ± j √XB =⇒s ≈ 2

Figure 2.4: Simulation results in the scope in Fig. 2.3.

(2.2) (2.3)

12

Modeling and Stability Analysis of Inverter-Based Resources

Figure 2.5: Zoom-in of Fig. 2.4. √ In this example, X = 0.5 and B = 0.2, hence the oscillation frequency is 10 ω or 190 Hz. Based on (2.1), an analytical model based on transfer functions can also be built. We also compare the capacitor voltage from the EMT testbed with that from the analytical model. Fig. 2.7 shows the screen shot of the two models and Fig. 2.8 shows the data in the scope. It can be seen that the linear model leads to the same output as the EMT testbed.

Figure 2.6: Results of the FFT analyzer.

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Figure 2.7: The screen copy of the MATLAB SPS with two models: EMT and analytical model. We have shown that an analytical model can be used for simulation. On the other hand, the main purpose of an analytical model is for linear analysis. For the system shown in (2.1), the frequency response plot or Bode plot the transfer function relating the current source to the capacitor voltage is shown in Fig. 2.9. It can be seen that when X = 0.5, the frequency response shows a resonance peak at 190 Hz. If the grid is stronger or X is less, the peak is located at higher frequency region. The resonance peak indicates the existence of an oscillation mode at that frequency. The Bode diagrams also show that if the grid is weaker, more severe overvoltage may be observed, since the magnitude at the resonance is higher. 2.1.2.2

A two-level three-phase VSC

EMT model building task: set up a test bed of a two-level three-leg voltage source converter with sinusoidal pulse width modulation (PWM). Fig. 2.10 shows the screen copy of the EMT testbed of a three-phase two-level converter. The dc side of the converter is connected to a constant voltage source at 200 V. Six IGBT blocks

Figure 2.8: Comparison of the simulation results from the EMT testbed and the analytical model.

Modeling and Stability Analysis of Inverter-Based Resources

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Bode Diagram

Magnitude (dB)

40

20

0

-20

-40 90 X=0.5 X=0.25 X=0.1

Phase (deg)

45 0 -45 -90 101

102 Frequency (Hz)

103

Figure 2.9: Bode plots of the transfer function relating the current source to the capacitor voltage Ls+R for three different grid impedance. LCs2 +RCs+1 and gate signals are explicitly modeled. The gate signals are generated by the comparison of the control signals against the triangular carrier waves of 1980 Hz. Those signals are shown in Fig. 2.11. Finally, the converter voltages, the filtered converter voltage, and the voltage after the choke filter or the voltage at the point of common coupling (PCC) are shown in Fig. 2.12. At 0.005 s, the control signals or the modulation signals’ amplitudes change from 1 to 0.8. This leads to the PCC bus voltage amplitude reduce from 100 V to 80 V. Note that the phase voltage measured at the converter’s terminal is discrete polluted with the switching frequency harmonics. A moving average filter (MAF) has bee used to obtain the filtered voltage. This voltage is shown to be the same as the PCC voltage. The MAF has a signal passing an integrator. The output will be subtracted by the output generated by an integrator with a delayed signal. Furthermore, division by the delay time leads to the

Figure 2.10: A three-phase two-level voltage source converter. The choke filter parameters: R = 0.02 Ω, L = 0.003 H, the shunt capacitor C = 30e − 6, and the load RL = 1 Ω.

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Figure 2.11: The control signals, carrier signals, and the PWM pulses. average value of a signal. In this case, the delay time is one cycle of the switching frequency and Tw = 1/1980 s. The system can be expressed as follows: y 1 − e−Tw s = . u Tw s

(2.4)

The frequency response of the MAF is compared with that of a first-order low-pass filter (LPF) 1/(T s + 1), where T = 1/(2π × 500) s and shown in Fig. 2.13. It can be seen that MAF is very effective to filter out harmonics above 1000 Hz.

Figure 2.12: The converter voltage, the filtered converter voltage, and the PCC voltage.

Modeling and Stability Analysis of Inverter-Based Resources

16

Bode Diagram

Magnitude (dB)

10 MAF LPF 0

-10

Phase (deg)

-20 0 -45 -90 -135 -180 101

102

103 Frequency (Hz)

Figure 2.13: Frequency responses of an MAF and a LPF. 2.1.2.3

A grid-integrated VSC

Example 2 shows that a VSC can be viewed as a controllable three-phase voltage source. For grid interconnection dynamics research, we may utilize the average model, which requires the input of three-phase modulation signals: ma , mb , and mc . The inverter’s three-phase output voltages amplify those signals: va = ma

Vdc Vdc Vdc , vb = mb , vc = mc 2 2 2

(2.5)

In this example, we will use the universal bridge’s average model option to represent a threephase dc/ac inverter. We will also set the modulation signals to have the desired real power and reactive power exporting to the grid. The modulation signals will be computed by steady-state analysis. In the first step, we may grab a dc voltage source, a capacitor, and a universal bridge as the inverter. The dc voltage source along with a resistor will be connected with the capacitor and the inverter in parallel. When the modulation signals are left unspecified, the default value of 0 is assumed. Hence, the inverter ac voltage is 0. The ac side of the universal is connected to an RL branch and then to a constant voltage source representing a grid. The next step is to put a few sensors to measure the instantaneous voltage and current. We may put three sensors, one right after the converter and the other right before the grid voltage source. The three-phase sensors can be configured to output either per unit values or physical values. Real power and reactive power are computed based on those current and voltage instantaneous measurements. The power measurement (3ph, instantaneous) has the three-phase voltage and current as two inputs and have real and reactive power as the two outputs. Note that this block assumes that the inputs and outputs are all in physical values. If the input is per unit value, the output should be scaled down by 2/3 to have the correct per unit value. Per unit system computing will be discussed in more details in Section 3.2. Also a scaling factor 1/Pb with Pb = 106 W is used so that the real power and reactive power are in the units of MW and MVAr, respectively. Fig. 2.14 shows the testbed built in Simscape SPS. With the physical circuit set up, the last step is to configure the modulation signals ma , mb , and mc to be fed to the universal bridge’s reference signal port, notated as Uref .

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Figure 2.14: Example 3: a grid-integrated VSC. Parameters: dc-side resistor: 0.05 Ohm, dc-link capacitor: 0.09 F. Ac side: resistor: 0.016 Ohm, inductor: 0.2122 mH. Dc voltage source: 1000 V. Ac voltage source: 400 V. ma , mb , and mc are three-phase balanced sinusoidal signals. In this example, we directly use a dq0/abc block to set the modulation signals. The inputs to the dq0/abc block are the dq0 signals and the rotating angle ωt. The rotating angle is obtained by passing a constant speed 377 rad/s to an integrator. The dq0 signals are set as vd = cos(δ ) (where δ = π6 ), vq = sin(δ ) and v0 = 0. The abc √ √ 2 to achieve nominal voltage amplitudes first, and signals will be amplified by a scaling factor 400 3 then be divided by half of the dc-link voltage. The resulting outs are finally the modulation signals Uref . Fig. 2.15 shows the simulation results captured by the scope of seven channels. The last two subplots are P and Q measured right before the grid voltage source. It can be seen that with this set of modulation signal, the converter exports 0.91 MW and −0.45 MVAr to the grid. The dc-link capacitor voltage is 946.5 V and the dc current is 1070 A. Based on the design of the modulation signals, the converter output voltages are:  √ 400 π 2 √ sin ωt + 6 3   √ 400 π 2π √ sin ωt + − vb (t) = 2 6 3 3   √ 400 π 2π vc (t) = 2 √ sin ωt + + 6 3 3 va (t) =

(2.6)

The grid voltage is also sine based. Hence the two voltage phasors of the converter and the grid π √ e j 6 and Vg = 400 √ . The per-phase line current phasor is are V = 400 3 3 π

ej 6 −1 400 I= √ . 3 0.016 + j0.08

Modeling and Stability Analysis of Inverter-Based Resources

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The complex power measured at the grid side is ∗

S = 3Vg I = (0.91 − j0.45) × 106

The steady-state phasor-impedance based analysis results match the simulation results.







 Figure 2.15: Example 3: simulation results. Frame conversion: The dq0/abc block is to conduct the following computation based on dq0 and the angle θ : π

va = ℜ{(vd + jvq )e j(θ − 2 ) } = ℜ{(vq − jvd )e jθ } vb = ℜ{(vq − jvd )e j(θ − vc = ℜ{(vq − jvd )e

2π ) 3

j(θ + 2π 3 )

} }

(2.7)

Fig. 2.16 shows the two types of dq frames. The first dq frame has the angle θ notating the q-axis relative to the static phase-a axis. The second dq frame has the angle θ notating the d-axis relative to the static phase-a axis.

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Figure 2.16: Two types of dq frames. Equation (2.7) is based on the first dq frame. Suppose there is a vector V in the dq frame. Then this vector can be expressed as V = vd + jvq based on the d-axis. Note that since the q-axis against the static frame is θ , then the d-axis relative to the static frame is θ − π2 . Therefore, the vector V viewed in the static frame is π V e j(θ − 2 ) = (vq − jvd )e jθ .

This is the analytic form of phase a voltage. Taking its real part, we obtain the phase a signal as va = vq cos θ + vd sin θ . If vd = cos δ and vq = sin δ , the resulting phase a signal is va = sin(θ + δ ). When θ = ωt, va = sin(ωt + δ ). Similarly, we may obtain the phase b signal and phase c signal. With the zero-sequence component added, in the vector/matrix format, we get: ⎤⎡ ⎤ ⎡ ⎤ ⎡ sin θ cos θ 1 vd va ⎣vb ⎦ = ⎣sin(θ − 2π ) cos(θ − 2π ) 1⎦ ⎣vq ⎦ (2.8) 3 3 2π vc v sin(θ + 2π ) cos(θ + ) 1 0 3 3 For a three-phase balanced set, when θ = 0, va = vq . Hence, this dq frame has its q-axis aligned with phase a initially, or d-axis lagging phase a by 90 degree. If we set vd = 1 and vq = 0 (v0 = 0 for balanced sets), the resulting phase a signal is a sine signal va = sin θ . The inverse of dq0/abc is abc/dq0 conversion, or Park’s transform. The expression is as follows: ⎡ ⎤⎡ ⎤ ⎡ ⎤ sin θ sin(θ − 2π ) sin(θ + 2π ) va vd 3 3 ⎣vq ⎦ = 2 ⎣cos θ cos(θ − 2π ) cos(θ + 2π )⎦ ⎣vb ⎦ (2.9) 3 3 3 1 1 1 v v0 c 2 2 2

The dq frame may have its d-axis aligned with phase a initially. This frame is also shown in Fig. 2.16 as the second type. Setting d as 1 and q as 0 results in cosine signals in phase a. The dq0/abc conversion and abc/dq0 conversion expressions are as follows: ⎡ ⎤ ⎡ ⎤⎡ ⎤ cos θ − sin θ 1 vd va ⎣vb ⎦ = ⎣cos(θ − 2π ) − sin(θ − 2π ) 1⎦ ⎣vq ⎦ (2.10) 3 3 2π vc v ) − sin(θ + ) 1 cos(θ + 2π 0 3 3

Modeling and Stability Analysis of Inverter-Based Resources

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⎡ ⎤ ⎡ cos θ vd 2 ⎣vq ⎦ = ⎣− sin θ 3 1 v0 2

cos(θ − 2π 3 ) sin(θ − 2π 3 ) 1 2

⎤⎡ ⎤ cos(θ + 2π va 3 ) ⎦ ⎣vb ⎦ sin(θ + 2π ) 3 1 vc 2

(2.11)

It is much easier to come up with conversion by use of the complex vectors. This way we also avoid memorizing the vector/matrix expressions. Suppose that we use the second type of dq frame (with the d-axis aligned with the static phase a-axis initially) to conduct abc to dq conversion. This is essentially to change the view point of the complex vector from the static frame to the dq frame. In the static frame, the three-phase set can be combined to a single space vector: � 2π 2π 2� (2.12) va + e j 3 vb + e− j 3 vc �v = 3

For a balanced three-phase set, the space vector is a rotating vector with a constant speed. This speed matches the frequency. This equation has a physics origin and is associated with the formation of a rotating magnetic field by three-phase stator current. We may verify that for a balanced set shown below, va = vˆ cos(θ ), 2π ), 3 2π vc = vˆ cos(θ + ). 3

vb = vˆ cos(θ −

The resulting space vector has the following format: � � 2π 2π 2π 2 2π �v = vˆ cos(θ ) + e j 3 cos(θ − ) + e− j 3 cos(θ + 3 ) 3 3 � � 2π 2π 2π  2π 2π 2π −θ 2π 1 2π jθ − 3 −θ + 3 ) + e j(− 3 +θ + 3 ) + e j(− 3 ) = vˆ e jθ +  e− + e j( 3 +θ − 3 ) +  e j(  3 3 = vˆ e jθ .

(2.13)

Thus, if θ = ωt, the space vector is rotating with a constant speed ω. Viewing this space vector in the dq frame, we may subtract the angle between the d-axis and the phase-a axis θ . Hence: � 2π 2π 2� V =�ve− jθ = va + e j 3 vb + e− j 3 vc e− jθ 3 ⎡ ⎤ (2.14) � va � 2 − jθ 2π 2π = e− j(θ − 3 ) e− j(θ + 3 ) ⎣vb ⎦ e 3 vc

2.2

Separating the real and imaginary parts, we get again the vector/matrix format: ⎡ ⎤ � � � va � 2π 2 cos θ vd ) cos(θ + ) cos(θ − 2π 3 3 ⎣v b ⎦ = 2π vq 3 − sin θ − sin(θ − 2π 3 ) − sin(θ + 3 ) vc

(2.15)

PER UNIT ANALYSIS

Often time, an analytical model for dynamic simulation is based on the per unit (pu) system. Therefore, we first talk about per unit analysis before addressing the topic on analytical model building.

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Power system steady-state analysis has two unique features different from the simple circuit analysis learned in an undergraduate course: three phase and with transformers. For a three-phase system with balanced sources and loads and a symmetric topology, per-phase analysis yields to analysis of a single-phase system. By treating every generator and load as Y-connected component and identifying a neutral point in every generator and load, all neutral points will have the same voltage potential and they can be directly connected. This fact is based on the assumption that for balanced systems, the sum of three-phase currents is zero. Therefore, even if there exists impedance between two neutral points, the current flowing into the neutral impedance is 0 and the two neutral points have the same voltage potential. Therefore, per-phase analysis takes consideration only the phase a to neutral. Transformers make a system having different ac voltage levels. Direct forming a network admittance matrix becomes difficult. Per unit based analysis can help “get rid of” an ideal transformer by use of voltage bases with their ratios aligned with the transformer voltage ratios. Besides “getting ride of” transformer ratios, per unit analysis has at least three more salient advantages. 1. First, scaling makes variables assume numerical values in a small range. This makes numerical simulation easier. 2. Second, the numbers in per unit are easy for sanity check. For example, it is very easy to remember the typical values of a synchronous generator synchronizing reactance in pu. Thus, one can quickly check whether a parameter of a synchronizing reactance is reasonable or not. One can also quickly check if voltage or power is in the reasonable range or not. 3. Third, per unit analysis leads to more simplified equations (e.g., without coefficients) and straightforward circuit representations for ac machines. We will show the latter using a synchronous machine circuit example. Equations In many cases, equations will be simplified. Coefficients will be gone. For example, the per-phase and three-phase complex power can be expressed as the following: Sφ = V I

∗

S3φ = 3V I

(2.16) ∗

where V and I are based on root mean square (RMS) of per-phase voltage and current. In per unit, the two equations become the same: pu  pu ∗ . Spu = V I

(2.17)

(2.18)

Synchronous machine circuit One may recall that the synchronous generator circuit representation from Bergen’s book vs. that from Krause’ book. Bergen’s book (chapter 6) [14] shows the excitation circuit is related to the d-axis circuit via coupling magnetic field. On the other hand, Krause’ book [15] has the stator d-axis circuit and the excitation circuit connected together with a shunt magnetizing inductance Lm . In Krause’s book, another layer of scaling of the field current has been used. This circuit is much more concise and has been adopted also by MATLAB/Simscape. The paragraph below describes how the field current i f and the d-axis current id are related. The flux linkage linked the d-axis winding is contributed by both id and i f . Same for the flux linkage linked the field winding. Note that the effect of the d-axis component of the stator’s three-phase current (its magnitude is notated as id ) in generating a magnetic field is same as a dc current with the value of 1.5id in the rotor winding. 3 λd = LLs id + Md id + Md i f d 2 3 λ f d = M f d id + (M f d + Ll f )i f d 2

(2.19) (2.20)

Modeling and Stability Analysis of Inverter-Based Resources

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where M f d = Md assuming that the winding turns of the d-axis winding and the field winding are the same. Let Lmd  1.5Md , then 2 λd = (LLs + Lmd )id + Lm i f d 3 2 λ f d = Lmd id + (Lmd + 1.5Ll f ) i f d 3

(2.21) (2.22)

Define i�f d  23 i f d , also define Ll� f = 1.5Ll f , the above relationship becomes λd = LLs id + Lmd (id + i�f d ) λ f d = Lmd (id + i�f d ) + Ll� f i�f d .

(2.23)

With this relationship, it is possible to have a shunt branch Lmd connecting the stator d-axis winding and the field winding. In synchronous machine’s per unit system, the stator windings and the field winding share the same power base Pb and the same voltage base vb if the turn ratio is 1. 3 Pb = vb ib , 2

(2.24)

where vb and ib are the bases for voltage and current amplitudes. They are also the bases of dq variables. For the field winding, since at steady state, voltage and current are dc variables, the real power base can be expressed as Pb = vb i�b

(2.25)

where vb is field voltage base and i�b is the field current’s base. The field voltage base and the stator voltage base are the same. It can be seen that the current bases of the stator windings and the rotor windings are different. If the stator winding’s current base is ib (per phase amplitude, or dq amplitude), then the field winding’s current base i�b = 1.5ib . In turn, the impedance base of the field winding is 23 of that of the stator winding: Zb� = 23 Zb . Therefore, we may directly apply the above per unit system to lead to the same equation (2.23). These advantages of per unit analysis have been appreciated by researchers working in the fields of digital simulation, power system analysis, and electric machines. On the other hand, per unit analysis is less used in the field of power electronics. Part of the reasons may be that hardware experiments are more popular in power electronics and for those testbeds physical values are more straightforward. 2.2.1

PER UNIT AS SCALING

Per unit analysis in nothing but scaling. The system should have only one power base, but several voltage bases, depending on transformers and how it can be separated into zones. For example, a 400 V inverter-based source is connected to a 13.2 kV system through a step-up transformer 400 V/13.2 kV. This system has two zones. Usually, a transformer’s nominal voltages are used as the base voltages. For example, the first step of per unit analysis is to set the power base at 1 MW (note: this is the three-phase power base) and two voltage bases: zone 1 400 V and zone 2 13.2 kV. Note the voltage bases are Line-line RMS values. The ratio of the voltage bases should align with the transformer’s voltage ratio. In case that zone 2’s voltage base is set at 14 kV, then zone 1’s base should be set as 400 13.2 × 14 V.

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In step 2, the other bases are defined or calibrated. Current base – this base is in fact the per-phase RMS current’s base. For example, zone 1’s current base should be Ib1 = √

Sb 106 A =√ 3Vb1 3 × 400

Zone 2’s current base should be Ib2 = √

Sb 106 =√ A. 3Vb2 3 × 13.2 × 103

Impedance base—this base is in fact the per-phase impedance’s base. V2 Vb1 Zb1 = √ = b1 . Sb 3Ib1

(2.26)

Next, the rest bases are defined. Those base include instantaneous element’s base. For example, the base of dq current and abc instantaneous current may use the base of current amplitude. In zone 1, this number should be: Sb √ 106 √ iˆb1 = 2= √ 2 VLL,b1 3400 The base of dq voltage and abc voltage in zone 1 should be: √ √ 2 2400 vˆb1 = √ VLL,b1 = √ . 3 3 Relationship among bases is important to be clarified. Based on the above definition, we can see that Pb = Qb = Sb =

√

3 3VLL,b Ib = 3Vφ ,b Ib = vˆb iˆb . 2

(2.27)

If the converter voltage phasor expressed in per unit is 0.94 + j0.3, we can further find out the modulation indices ma , mb , mc for the PWM of the dc/ac converter. The dc voltage is assumed to be 1000 V. Then V = vd + jvq = 0.94 + j0.3 = 0.987∠17.7◦ , =⇒vd = 0.94, vq = 0.3. if the dq frame is aligned with the grid voltage

(2.28) (2.29)

Hence, 400 √ 1 sin(ωt + 17.7◦ ) ma = 0.987 × √ 2 500 3 400 √ 1 sin(ωt + 17.7◦ − 120◦ ) 2 mb = 0.987 × √ 500 3 400 √ 1 2 sin(ωt + 17.7◦ + 120◦ ) mc = 0.987 × √ 500 3 2.2.2

(2.30) (2.31) (2.32)

INDIVIDUAL VERSUS AGGREGATED MODELS

When per unit values are adopted, converting an individual model (e.g., a single 1.5 MW wind turbine) into an aggregated model (e.g., a 150-MW wind farm) is easy and straightforward.

Modeling and Stability Analysis of Inverter-Based Resources

24

Assume that 100 turbines are connected in parallel and the power base of the aggregated model is 100 times of the power base of the individual wind turbine: Pb� = 100Pb . The voltage base is kept the same and the current base becomes 100 times of the original base: Ib� = 100Ib . Hence, the aggregated model’s impedance base Zb� is 1% of the original base: Zb� = 0.01Zb . Since all turbines are connected in parallel, the aggregated impedance Zaggregated = 0.01Z. The per unit value of an impedance based on Pb� and Vb is pu

Zaggregated =

Zaggregated 0.01Z = = Z pu . � Zb 0.01Zb

(2.33)

Similarly, current of the aggregated model is 100 times of the individual turbine. Its per unit value, however, keeps the same since the based has been scaled up by 100 times. The voltage of the aggregated model is same as that of the individual turbine. Its per unit value also does not change. With the voltage, current, impedance’s per unit values keep the same for the individual turbine and the aggregated model, it can be seen that control gains in per unit also do not change. 2.2.3

EQUATIONS WITH COEFFICIENTS

In a few occasions, even in per unit, those equations have coefficients. Instantaneous power equation is one such example. Instantaneous power can be expressed by abc variables or dq-frame variables. The latter will be covered in detail in future chapters. 3 (2.34) p = va ia + vb ib + vc ic = (vd id + vq iq ). 2 At steady state, the abc-frame voltages and currents are time varying periodic while dq-frame variables are constants. We also know that Pb = 32 vˆb iˆb . 3 (vd id + vq iq ) va ia + vb ib + vc ic p = = 2 3 3 ˆ ˆ Pb i v ˆ 2 b b 2 vˆb ib

(2.35)

This leads to the following equation in per unit: 2 p = (va ia + vb ib + vc ic ) = vd id + vq iq (2.36) 3 We have mentioned previously that the three-phase power block meter has instantaneous voltages (va , vb , and vc ) and currents (ia , ib , and ic ) as input and outputs real and reactive power. The computing equations for the real and reactive power are P = va ia + vb ib + vc ic 1 Q = √ [(vb − vc )ia + (vc − va )ib + (va − vb )ic ] 3

(2.37)

Scaling the equations by use of 1/Pb leads to va ia + vb ib + vc ic 3 ˆ 2 vˆb ib 2 pu pu pu = (vpu ipu + vb ib + vpu c ic ) 3 a a 1 (vb − vc )ia + (vc − va )ib + (va − vb )ic Qpu = √ 3 ˆ 3 2 vˆb ib    1  2 pu pu pu pu pu pu √ (vb − vc ) ia + (vc − va ) ib + (va − vb ) ic = 3 3 Ppu =

(2.38) (2.39) (2.40) (2.41)

Tools: simulation and analysis

25

It can be seen that if the per unit values are fed into the block, a scaling factor of 2/3 should be considered to arrive at per unit values of real and reactive power. 2.2.4

SYSTEM STRENGTH AND SHORT CIRCUIT RATIO (SCR)

Per unit values help a quick understanding of the grid condition. System strength or short circuit capacity can be directly related to per unit grid impedance. At a point of interconnection (POI), the entire grid may be viewed as a Thevenin equivalent with a constant voltage source Vg behind an impedance Zg . At the nominal condition, Vg = 1 p.u. Therefore, the short circuit current and the short circuit capacity of the POI bus is |Z1g | . Short circuit ratio (SCR) is the ratio of the short circuit capacity and the power rate of the IBR connected to the POI. Therefore, if the power base is selected as the power rate of the IBR, SCR is 1 |Zg | . Based on the definition used in HVDC [16], a strong grid has an SCR greater than 5. A weak grid has an SCR in the range of 2 to 5. A very weak grid has a SCR below 2.

2.3

ANALYTICAL DYNAMIC MODEL BUILDING

A nonlinear analytical model has the capability of providing time-domain simulation results and providing linear models for further analysis, e.g., modal analysis. This model building approach has been adopted for machines and power grids. Many details and application examples can be found in Kundur’s book [6]. On the other hand, the practical value of nonlinear analytical model building has not been fully realized in the new era of IBR-dominated power grids. One reason is that this modeling approach is not popular in the power electronics field. Literature from the power electronics community, e.g., [17, 7], show that analysis is usually carried out by block-diagram based frequency-domain analysis, while simulation is carried out by use of hardware experiments. While block diagrams offer great insights to dynamics, they have to be derived manually. When the system becomes complex, manual derivation is not feasible. One salient advantage of analytical models is that based on the analytical models, block diagrams can be easily found. This feature has not been explored sufficiently by the power system community. Majority of the case studies in Kundur’s book adopts modal analysis, e.g., eigenvalues, mode shapes, or participation factors. In this book, extracting block diagrams from analytical models will be demonstrated. In short, with a nonlinear analytical model, we can carry out simulation, modal analysis based on a state-space model, and frequency-domain analysis based on block diagrams. Different from EMT testbed building, building a nonlinear analytical model requires mathematic derivation to arrive at a set of first-order ordinary differential equations (ODE), or state-space models. Once the ODEs are available, numerical integration of a set of ODEs can be invoked by a solver. A nonlinear analytical model is in the form of a continuous nonlinear ODEs with state variables constant at steady state and the derivatives of the state variables 0 at steady state. To arrive at this model, discrete events have to be dealt with. For example, power electronic converter’s pulse width modulation (PWM) leads to a controllable voltage source at fundamental frequency. In an analytical model, PWM will not be included and the end result of PWM – a controllable voltage source – is directly used. The resulting model is called an average model. The average model still has instantaneous currents and voltages as state variables. To have state variables as constants at steady state, frame conversion and/or domain conversion have to be conducted. 2.3.1

FRAME CONVERSION IN SPACE

The well-known Clarke’s transform and Park’s transform are frame conversion techniques. Clarke’s transform converts a three-phase abc variables into αβ variables with β -axis leading the α-axis by

Modeling and Stability Analysis of Inverter-Based Resources

26

Yabc

Yq

abc dq

‫ ݌ܭ‬൅

‫݅ܭ‬ ‫ݏ‬

ω

ͳ ‫ݏ‬

θ3//

Figure 2.17: A three-phase PLL. The d-axis aligns to the phase a axis when θPLL is 0. 90 degree. In essence, the resulting frame from Clarke’s transform is still a static frame. On the other hand, Park’s transform is to view variables in a static frame from a rotating frame, or a dq frame. If the rotating speed of the frame is same as the electric frequency of the variables in a static frame, the variables viewed in the dq frame are dc variables. This frame conversion is not only useful in modeling but also in control implementation. In modeling a synchronous machine, the stator variables are all referred to the rotor frame. Thus, the three-phase stator ac currents can be viewed as an equivalent rotor dc current. From there, the dqframe circuits can be built. This model saves simulation time if the generator is operating at the nominal frequency. We only dealt with state variables that are constants at steady state. In control design, to track an ac reference signal, proportional integral (PI) control is no longer popular. If it is to be used, a high bandwidth is required. The ac reference signal and the measurement can all be converted to dc signals in a dq frame. In the dq frame, PI control can still be used for reference tracking. If frame conversion is not used, proportional resonant (PR) control has to be used for ac reference tracking. 2.3.2

FRAME CONVERSION BETWEEN THE TIME AND FREQUENCY DOMAINS

In order to have dc variables after the abc to dq frame conversion, the abc variables have to be a balanced set. If the set contains a negative sequence component, the conversion leads to a dc component and a second-harmonic component. This is not desired for modeling. To deal with unbalance and harmonic, domain conversion, instead of frame conversion, has to be used to arrive at a phasor model. This conversion essentially extracts the phasors of each harmonic component. In turn, dynamics will be expressed by phasors, instead of instantaneous variables. In the literature, this approach is termed as the dynamic phasor modeling approach. 2.3.3

A THREE-PHASE PHASE-LOCKED LOOP (PLL)

A three-phase PLL has a set of three-phase voltage as the input while outputs the voltage’s angle, frequency, and magnitude. The main control objective of a PLL is to have its output angle track the input angle of the voltage. This angle is a ramp signal in the form of θ = ω0t + Δθ . Hence, for tracking design, double integrators will be used. Fig. 2.17 shows the control block diagram of a PLL. The three-phase abc signals are passed through the Park’s transformation to have dq signals. For the Park’s transformation, the angle input is the PLL’s output angle. The q-axis signal is enforced to 0 through a PI control which outputs the frequency deviation. Finally, the angle is found by integrating the frequency. We can see that the q-axis voltage vq after Park’s transformation is the projection of the voltage space vector on the q-axis of the PLL frame. In Section 2.1, we have shown that a balanced voltage set can be formed as a space vector: �v = ve ˆ jθ , where θ is va (t)’s angle.

(2.42)

Tools: simulation and analysis

27

Park’s transformation is to view this space vector from the PLL frame’s perspective: V = ve ˆ j(θ −θPLL ) . =⇒ vd = vˆ cos(θ − θPLL ) vq = vˆ sin(θ − θPLL ).

(2.43)

If the PLL frame aligns with the voltage space vector, or the angle θPLL is the same as θ , then vq = 0. This is not an analytical model since the input signals at steady state are periodic. To develop an analytical model, we will use frame conversion technique. A dq frame with a constant rotating speed at ω0 is introduced. This dq frame is different from the PLL dq frame and is notated as the grid dq frame. At steady state, the two dq frames align with each other. Hence, the input signals will be the dq-axis voltages: vgd and vgq (where superscript g notates the grid dq frame), or the voltage magnitude vˆ and phase angle Δθ , while the outputs are Δω, ΔθPLL , and voltage magnitude v. ˆ This frame conversion leads to a nonlinear analytical model. The two models are implemented in MATLAB Simscape SPS and shown in Fig. 2.18. Note that for the analytical model, an αβ 0/dq0 block is used. This block is for dq0g /dqPLL conversion. A low-pass filter is added between vq and the PI controller. Dynamic performance is compared for the same phase angle step change and voltage magnitude change (shown in Fig. 2.19). It can be seen that two models lead to the same performance, when DeltaθPLL are compared. 2.3.4

A SINGLE-PHASE PLL

For a single-phase PLL, Fig. 2.20 shows the implementation in MATLAB/Simscape SPS. The input of the PLL has only one signal while the output is an angle. The input signal is multiplied by the

Figure 2.18: A three-phase PLL and its analytical model.

Modeling and Stability Analysis of Inverter-Based Resources

28

Figure 2.19: Simulation results comparison of a three-phase PLL and its analytical model. In the left column, the low-pass filter has a cutoff frequency of 28 Hz. In the right column, the low-pass filter has a cutoff frequency of 50 Hz. cosine of the output angle to generate an error. This error is passed through a PI controller and an integral controller. If the input signal can be expressed as v(t) = vˆ sin(ω0t + Δθ ), and the output signal can be expressed as θPLL = ω0t + ΔθPLL , then the error signal e(t) is expressed as follows. e(t) = vˆ sin(ω0t + Δθ ) cos(ω0t + ΔθPLL ) 1 = (sin(Δθ − ΔθPLL ) + sin(2ω0t + Δθ + ΔθPLL )) 2

(2.44)

For the analytical model, we only consider the first component in e(t), while ignoring the 2nd harmonic component. This treatment is to convert a signal from time domain to frequency domain. In this case, we examine only the dc component. It can be seen that e(t) ≈

1 sin(Δθ − ΔθPLL ). 2

(2.45)

Fig. 2.21 shows comparison of the simulation results of the single-phase PLL and its analytical model. It can be seen that the analytical model captures the dynamics of the dc component. The analytical can further be linearized by treating sin(x) ≈ x. For the linearized model, the open-loop gain is:   1 Ki 1 Loop Gain = . (2.46) Kp + 2 s s If a low-pass filter

1 1+τs

is used, the loop gain becomes the following   1 1 Ki 1 LoopGain = . Kp + 2 1 + τs s s

(2.47)

Tools: simulation and analysis

29

Figure 2.20: A single-phase PLL and its analytical model. It can be seen from the simulation results (shown in Fig. 2.22) that if K p = 180, Ki = 3200, when τ = 0.05 s, the system is stable. On the other hand, when τ = 0.06 s, the system is unstable. Based on Bode diagrams (shown in Fig. 2.23), it can be seen that when τ = 0.06 s, the system has a negative gain margin, while it has a positive gain margin if τ = 0.05 s.

Figure 2.21: Simulation results comparison of a single-phase PLL and its analytical model.

Modeling and Stability Analysis of Inverter-Based Resources

30

Figure 2.22: Simulation results comparison of a single-phase PLL with different low-pass filter parameters. Besides using the open-loop gain to check stability, for a simple system, we may also use the closed-loop system poles to check stability. The poles of the closed-loop system should be all located in the left half plane. Otherwise, the system is unstable. For this case, the closed-loop system transfer function is as follows. Gcl =

K p s + Ki LoopGain ΔθPLL = = 3 2 . Δθ 1 + LoopGain τs + s + 0.5K p + 0.5Ki

(2.48)

The poles of Gcl are the roots of the denominator polynomial. We may use Routh-Hurwitz stability criterion to check the denominator’s roots.

Figure 2.23: Bode diagrams of the open-loop gain of the single-phase PLL with different low-pass filters.

Tools: simulation and analysis

31 s3 s2 s

τ 1 0.5K p − 0.5Ki τ

0.5K p 0.5Ki

Based on the Routh Hurwitz criterion, all elements of the first column should be positive. Therefore 0.5K p − 0.5Ki τ > 0, =⇒ τ
0, the impedance can be obtained by (8.21). Zd,i = −

Vi = Rd,i + jXd,i Ii

where Vi and Ii are voltage and the current injecting to the network from bus i. ⎡ ⎤ Xd,i s −Xd,i Rd,i + ⎢ ⎥ ω0 Z dq d,i = ⎣ Xd,i ⎦ Xd,i Rd,i + s ω0

(8.21)

(8.22)

The s-domain admittance of RL load on bus i is expressed in (8.23). −1 Z dq Y dq d,i = (Z d,i ) .

(8.23)

If the load is a RC load, i.e., Qd < 0, the load admittance is calculated by Yd,i = − Then,

Ii = Gd,i + jBd,i Vi

(8.24)

⎡ ⎤ Bd,i s −Bd,i Gd,i + ⎢ ⎥ ω0 Y dq d,i = ⎣ Bd,i ⎦ Bd,i Gd,i + s ω0

(8.25)

dq Y dq Y dq Y dq net ((2i − 1) : 2i, (2i − 1) : 2i) = Y l,i +Y c,i +Y d,i

(8.26)

Finally, the diagonal element of Y dq net can be calculated by sum of (8.19), (8.20), (8.23) or (8.25).

The s-domain n-bus network admittance modeling process can be included in Algorithm 1. 8.2.2.2

Synchronous generators

Next, the admittance of synchronous generators will be found using Jacobian linearization of the block modeled in Park’s transformation. The simplified synchronous generator can be replaced by a RL circuit admittance. The IBR admittance can also be found via Jacobian linearization of a Simulink block.

Power networks with multiple IBRs

221

Algorithm 1: n−bus network admittance modeling Get Y bus via MATPOWER toolbox; for i = 1 : n, j = 1 : n do if i �= j then Zi j = Y1i j = Ri j + jXi j ;

⎡

Ri j +

⎢ Y dq net ((2i − 1) : 2i, (2 j − 1) : 2 j) = −⎣

Xi j s ω0

Xi j

⎤−1 −Xi j ⎥ Xi j ⎦ ; Ri j + s ω0

else Y dq Y dq , j ∈ Ni ; l,i = ∑ ⎡ j ij ⎤ Bi s −Bi ⎢ ω0 ⎥ 1 Y dq c,i = ⎣ Bi ⎦ , Bi = 2 ∑ j Bi j , j ∈ Ni ; Bi s ω0 if Qd, j > 0 then ⎡ ⎤−1 Xd,i s −Xd,i Rd,i + ⎢ ⎥ ω0 Y dq d,i = ⎣ Xd,i ⎦ , Rd,i + jXd,i = −Vi /Ii ; Xd,i Rd,i + s ω0 else ⎡ ⎤ Bd,i s −Bd,i Gd,i + ⎢ ⎥ ω0 Y dq d,i = ⎣ Bd,i ⎦ , Gd,i + jBd,i = −Ii /Vi ; Bd,i Gd,i + s ω0 end dq Y dq Y dq Y dq net ((2i − 1) : 2i, (2i − 1) : 2i) = Y l,i +Y c,i +Y d,i end end

For the simplified RL circuit synchronous generator, we can get the impedance directly by (8.27). ⎤ ⎡ Xs,i s −Xs,i Rs,i + ⎥ ⎢ ω0 (8.27) Z dq G,i = ⎣ Xs,i ⎦ Xs,i Rs,i + s. ω0 The s-domain admittance of synchronous generator on bus i can be found by (8.28):

8.2.2.3

IBR

� � dq −1 Y dq = Z . G,i G,i

(8.28)

Since the nonlinear state-space IBR dynamic block has been built in Simulink system, the admittance of an IBR can be extracted by using MATLABT function linmod. In Fig. 8.21, the bus 2 IBR system s-domain admittance is abstracted by defining voltage as input and current as the output.

Modeling and Stability Analysis of Inverter-Based Resources

222

Figure 8.21: Extract IBR admittance from a Simulink block. 8.2.2.4

Total admittance

The whole system admittance can be obtained by adding generators admittance and IBR admittance to generation bus on the diagonal elements in Y matrix. (8.29) gives an example of ⎡ dq Y dq Y net(1:2,1:2) +Y G,1 ⎢ ... ⎢ Y total = ⎢ ⎣ ... ... 8.2.2.5

... Y dq Y dq net(3:4,3:4) +Y IBR,2 ... ...

⎤ ... ⎥ . . .⎥ ⎥ . . .⎦ ...

... ... Y dq Y dq net(5:6,5:6) +Y IBR,3 ...

(8.29)

Eigenvalue analysis

The eigenvalue analysis can be accessed from above mentioned two approaches. For the nonlinear state-space model, the integrated system will be linearized via linmod and use command eig(A) to get the eigenvalue from A matrix. For the linear admittance model, the eigenvalue of the whole system can be found by as the zeros of the total admittance matrix Y total (s) or the poles of the impedance matrix Z total (s), where Z total (s) = Y total (s)−1 [78, 79] . Fig. 8.22 shows the eigenvalue analysis conducted from two different approaches: nonlinear state-space model and linear admittance model. In this case, Two IBRs both adopt battery on DC

1000 800

State-space model Admittance model

State-space model Admittance model

10

600

Imaginary Axis (Hz)

Imaginary Axis (Hz)

400 200 0 -200

5

0

-5

-400 -600

-10

-800 -1000 -1400

-1200

-1000

-800

-600

-400

Real Axis (sec -1 )

(a)

-200

0

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Real Axis (sec -1 )

(b)

Figure 8.22: Eigenvalue analysis from two models. (b) is a zoom-in view of low frequency modes in (a).

Power networks with multiple IBRs

223

side dynamic assumption. It can be seen that two models can lead to the same set of eigenvalues for the same modeling assumptions.

8.3

FREQUENCY-DOMAIN MODAL ANALYSIS

In this section, we demonstrated two approaches of building linear models: linearizing a nonlinear analytical model, or assembling linearized blocks and forming a network admittance. Both lead to the same results. The first approach requires a transparent model. This requirement is relaxed in the second approach. Since the linear model, e.g., admittance of an IBR, can be found from measurement. In Chapter 3, we have demonstrated how to find a type-4 wind’s dq admittance through measurement. Therefore, the second approach offers a new small-signal analysis framework suitable for black box IBRs. Currently, due to intellectual property protection, IBR models provided by manufacturers have limited information available. Finding models applicable for analysis via measurement is a feasible way. In [80], this approach is demonstrated for a100% IBR-penetrated 9-bus grid. The three-IBRs are first measured for their dq admittance measurements. Through data fitting, the s-domain admittance are obtained. In the assembling step, the s-domain admittance network is formed. Based on this total admittance, eigenvalue analysis and further modal analysis has been carried out. The frequency-domain modal analysis is very different from the time-domain modal analysis. Hence, this section is devoted to frequency-domain analysis. 8.3.1

ADMITTANCE-BASED STABILITY ANALYSIS

The foundation of admittance-based analysis is briefly described in this subsection. The relationship between a grid’s nodal voltage vector and its current injection vector is as follows. i(s) = Y (s) v(s).

(8.30)

If the current injection is treated as the input, while the nodal voltage is treated as the output, then the closed-loop system transfer function G(s) may be expressed as G(s) =

v(s) = Y (s)−1 . i(s)

(8.31)

The poles of a closed-loop transfer function G(s) are the eigenvalues of the system matrix A, where A, B, C, D are the minimal state-space realization of G(s). x˙ = Ax + Bu, y = Cx + Du

(8.32)

The closed-loop system’s transfer function G(s) has the following relationship with A, B, C, D. G(s) = C(sI − A)−1 B =

1 C adj(sI − A)B + D φ (s)

(8.33)

where φ (s) is the characteristic function of G(s) and n

φ (s)  det(sI − A) = ∏(s − λi ).

(8.34)

i=1

It can be seen that the poles of G(s) are the roots of the characteristic function φ (s). Pole of G(s) are also the eigenvalues of the system matrix A.

224

Modeling and Stability Analysis of Inverter-Based Resources

If we derive G(s)’s expression from its inverse Y (s), then we found that G(s) is associated with the determinant of Y (s) in the following way: G(s) = Y −1 (s) =

1 adj(Y (s)). det (Y (s))

(8.35)

Thus, it can be seen that φ (s) is the numerator of the determinant of Y (s). Therefore, the system eigenvalues are the zeros of det(Y (s)). The same statement can be found in [65] by Semlyen on finding closed-loop system eigenvalues through computing the zeros of the network admittance matrix, though the reasoning is different. Semlyen started from the voltage and current relationship as follows. Y (s)v = 0

(8.36)

where v is the union of the nodal voltage and the total current into a node is zero. To find a non-trivial solution v, Y (s) has to be singular. This means that s will make the determinant of Y (s) zero: det(Y (s)) = 0 and s is an eigenvalue of the system. Besides the aforementioned approach based on eigenvalues, stability can also be analyzed by an open-loop system gain. The entire system or circuit can be viewed as an interconnected circuit with two admittance, then i(s) = (Y1 (s) +Y2 (s)) v(s) = Y2 (s) (Z2 (s)Y1 (s) + I2×2 ) v(s), =⇒ Z2 (s)i(s) = (Z2 (s)Y1 (s) + I2×2 ) v(s).

(8.37) (8.38)

where Z2 (s) = Y2 (s)−1 and I2×2 is the identity matrix. The transfer function between the input Z2 (s)i(s) and the output v(s) is: v(s) = (Z2 (s)Y1 (s) + I2×2 )−1 Z2 (s)i(s)

(8.39)

If Z2 (s) is a stable system, the stability of the overall system can be determined by the loop gain of the open-loop system: Z2 (s)Y1 (s), or the ratio of two impedances (admittances). The loop gain is a multi-input multi-output (MIMO) system. Therefore, generalized Nyquist stability criterion may be applied. the frequency-domain eigenvalues of the loop gain should be used for Nyquist stability check. 8.3.2

BASIC MODE SHAPE ANALYSIS

Z total (s)). The eigenvalues An eigenvalue λi can be found from the impedance matrix using pole(Z of the IEEE 9-bus system with three IBRs are presented in Fig. 8.23. It can be seen that there is a 3-Hz mode very close to the imaginary axis that makes the system have a poorly damped 3-Hz oscillations. Time-domain simulation results in Fig. 8.24 also show that the system has a poorlydamped 3-Hz mode. The bus impedance matrix, Z total (s), will be evaluated for an eigenvalue λi by letting s = λi . Since the eigenvalue is a pole of the bus impedance matrix, it is expected that an eigenvalue of the bus impedance matrix evaluated at the system eigenvalue, Z total (λi ), will approach infinity. Eigenvalue decomposition of Z total (λi ) leads to matrix factorization as follows: Z total (λi ) = Q Γ Q −1 ,

(8.40)

where Q is the right eigenvector matrix of Z total (λi ) and Γ is a diagonal matrix consisting of 2n eigenvalues: γi , i = 1, . . . , 2n.

Power networks with multiple IBRs

Imaginary axis(Hz)

10

225

Reduced 3-node Original 9-node

3 Hz

5

0

-5

-10 -40

-30

-20

-10

0

10

Real axis(sec -1 )

(a)

(b)

Figure 8.23: (a) Poles of Z total (s) in the low-frequency range. (b) Magnitude plot of Γ regarding Z(λ3Hz ) eigenvalue decomposition. Fig. 8.23(b) shows the magnitudes of the 18 eigenvalues of the bus impedance matrix evaluated at the 3-Hz dominant mode Ztotal (λ3Hz ). γ1 is the one with the maximum magnitude. It can be seen from Fig. 8.23(b) that compared to γ1 , the rest eigenvalues have negligible magnitudes. The relationship between the nodal voltage vector and the nodal current injection vector can be written as follows: V (λi ) = Z (λi )II (λi ) � = Q c,1

Q c,2

···

2n

⎡

�⎢ ⎢ Q c,2n ⎢ ⎣

γ1

γ2

..

. γ2n

−1 = ∑ Q c,i γi Q −1 r,i I ≈ Q c,1 γ1 Q r,1 I

⎤ ⎡ −1 Q r,1 ⎥⎢ Q −1 r,2 ⎥⎢ ⎥⎢ .. ⎦⎢ ⎣ . Q −1 r,2n

⎤

⎥ ⎥ ⎥I ⎥ ⎦

i=1

V1m V2m V3m

Vm (pu)

1.08 1.06 1.04 1.02 1 10

10.5

11

11.5

1.5

P (pu)

12 P1 P2 P3

1

0.5 10

10.5

11

11.5

12

Time (s)

Figure 8.24: PSCAD 9-bus system time-domain dynamic responses.

Modeling and Stability Analysis of Inverter-Based Resources

226

−1 where Q c,i is the ith column vector of Q and Q −1 r,i is the ith row vector in Q . γ1 is the eigenvalue with maximal absolute value, or the one reaching infinity. The corresponding 1st column eigenvector is expressed as: T  Q c,1 = Vd1 Vq1 . . . Vdn Vqn . (8.41)

Vdj and Vqj can be used to examine how the eigenvalue λi of the system contributes to Bus j’s voltage in dq-frame. For the 9-bus system, n is 9 and Vd1 and Vq1 represent the contributions of the eigenvalue λi to Bus 1 voltage. The column vector Q c,1 is the mode shape vector of the eigenvalue λi . Each element is a complex number reflecting the observability of the eigenvalue λi in the corresponding measurement. 8.3.3

EXTENDED MODE SHAPE ANALYSIS

The basic mode shape vector is not sufficient enough to evaluate the observability of an eigenvalue on measurements such as nodal voltage magnitudes, angles, or IBRs’ output currents, real and reactive power. To find any mode shape for a general measurement, the relationship of the measurement of interest against the nodal voltage in the dq-frame is explored. 8.3.3.1

Voltage magnitudes

Take Bus 1’s voltage magnitude as an example, it has the following relationship with its dq-frame voltages:  Vm1 =

2 +V 2 . Vd1 q1

(8.42)

Their linearized relationship at a particular operation point can be found as ΔVm1 =

∂Vm1 ∂Vm1 ΔVd1 + ΔVq1 . ∂Vd1 ∂Vq1

(8.43)

Thus the mode shape of the bus voltage magnitude, notated as Vm1 , can be found as: Vm1 =

8.3.3.2

∂Vm1 ∂Vm1 Vd1 + Vq1 ∂Vd1 ∂Vq1

(8.44)

IBR current magnitudes

An IBR’s dq-frame current and its terminal voltage are related with its dq admittance. Using IBR1 as an example, the following relationship exists:     Vd1 Id1 (λ ) = Y dq (8.45) IBR,1 3Hz Iq1 Vq1 Hence, the mode shapes of the dq-frame current, notated as Id1 and Iq1 , can be calculated via (8.46).     Vd1 Id1 (λ ) = Y dq (8.46) IBR,1 3Hz Iq1 Vq1 Similarly, the current magnitude mode shape of Bus 1 can be computed from (8.47). Im1 = where Im1 =



2 + I2 . Id1 q1

∂ Im1 ∂ Im1 Id1 + Iq1 , ∂ Id1 ∂ Iq1

(8.47)

Power networks with multiple IBRs

8.3.3.3

227

Real and reactive power

The power computation equation (8.48) can be linearized. The mode shapes of active power and reactive power of IBR1, notated as P1 and Q1 , can be computed via (8.49).

 8.3.3.4

P1 Q1



=



Vd1 Vq1

P1 = V1d I1d +V1q I1q Q1 = −V1d I1q +V1q I1d    Id1 Id1 Vq1 + −Vd1 Iq1 −Iq1

(8.48a) (8.48b) Iq1 Id1



Vd1 Vq1



(8.49)

Bus angles

The mode shape of Bus 1’s angle, notated as A1 , can be found using (8.50). A1 =

∂ θ1 ∂ θ1 Vd1 + Vq1 , ∂Vd1 ∂Vq1

(8.50)

  V where θ1 = arctan Vq1 . d1 Fig. 8.25(a) shows the voltage magnitude mode shapes for Buses 1, 2, and 3 corresponding to the dominant 3-Hz oscillation mode. The corresponding voltage phase angle mode shapes are presented in Fig. 8.25 (b). Fig. 8.25(c) and (d) show the mode shape plots of P and Q. The mode shape plots all show that the 3-Hz mode is more related to IBR1 and IBR3, while relatively less related to IBR2. IBR1 and IBR3 are the two GFM converters. Furthermore, mode

Figure 8.25: 9-bus system mode shapes: Voltage magnitudes, bus voltage angles, real power of IBRs and reactive power of IBRs.

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shapes of voltage magnitude, phase angle, real and reactive power all show that IBR1 and IBR3 oscillate against each other for this mode.

8.4

SUMMARY

In Chapter 8, analytical model building is applied for a power grid with multiple IBRs. Special care is needed when dealing with the power network dynamics. In addition to the ODE-based state-space modeling, a passive network can be modeled by transfer functions and converted to a state-space model. Furthermore, Chapter 8 demonstrates a different small-signal modeling framework. In this framework, an IBR is treated as dq admittance. A total network admittance is assembled and used for frequency-domain analysis. In this chapter, we have demonstrated modal analysis for both timedomain state-space models and for frequency-domain admittance models.

Part III Generalized Dynamic Circuits

Taylor & Francis Taylor & Francis Group

http://taylorandfrancis.com

9

Generalized dynamic circuits

The current power system analysis methods are built upon many scientists and engineers’ brilliant work, e.g., Heaviside’s operational calculus, Steinmetz’s phasor-based steady-state analysis and induction machine equivalent circuit, Fortescue’s symmetric component theory and unbalance treatment, and Park’s synchronous machine dynamic modeling. Today we take for granted to use phasor-impedance based circuit representations for steadystate power system analysis. A review of history shows that such a simple and straightforward method is not possible if Heaviside had not invented operational calculus for linear differential equation solving and Steinmetz had not invented complex phasor-impedance representation for ac electric circuit analysis. In Heaviside’s operational calculus, a derivative operator is notated as “p”. With this notation, calculus is simplified to algebraic calculation. This operator has been used popularly in the literature of the last century, e.g., Park’s 1929 paper [81]. “p” is equivalent to the Laplace transform variable s. For an RLC circuit, the instantaneous voltage imposed on the circuit and current flowing into the circuit can be easily related by the use of impedance in s-domain or frequency domain: 1 v(s) = R + Ls + , i(s) Cs

(9.1)

with v(s) and i(s) are also in the Laplace domain. If s is evaluated at the operating frequency jω, the resulting relationship is the steady-state impedance of an RLC circuit. 1 v( jω) = R + jωL + . i( jω) jωC

(9.2)

Note that, v( jω) and i( jω) are the phasors of the voltage and current at that frequency. Therefore it can be clearly seen that the steady-state impedance is a special realization of a dynamic impedance evaluated at the operating frequency. The impedance is associated with phasors of the input and output sinusoidal signals of that frequency. In another word, v(s) and i(s) in the impedance equation may be viewed as phasors associated to each frequency point. If we set up this perspective, many steady-state circuits can now be seamlessly converted to dynamic circuits. Recall that in Chapter 7 (Type-3 wind farms) Section 7.1, we have shown that an induction machine’s slip can be expressed by a dynamic expression using Laplace transform variable s. The procedure of the conversion is as follows: slip = 1 −

jωm ωm = 1− ωs jωs

(9.3)

where ωm is the rotating speed in rad/s and ωs is the synchronous speed or the stator’s operating frequency in rad/s. We may replace jωs by s. This leads to an expression of the slip in the s-domain with complex coefficients. slip(s) = 1 −

jωm s

(9.4)

When the stator’s perturbing frequency is below the rotating speed, the slip at that frequency is negative. This is encouraging since there exist many well-developed steady-state circuits. Some include both positive- and negative-sequence voltage and current phasors, e.g., a sequence network used DOI: 10.1201/9781003323655-9

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to deal with unbalanced faults. Some include multiple harmonic components, e.g., an induction machine with rotor circuit unbalance [82]. Depending on the rotating speed ωm , the stator circuits have both the nominal operating frequency ωs and its mirror 2ωm − ωs , while the rotor circuits may have the positive-, negative-, and zero-sequence components at the slip frequency. Converting those circuits into dynamic circuits is to essentially obtain phasor models for a system with multifrequency harmonics. In another word, these are dynamic phasor models. In this chapter, we will expand phasor-impedance circuit representations of various power grid components (IBRs included) to dynamic phasor-impedance representations. This chapter includes four sections, covering induction machines, unbalanced topologies, synchronous generators, and IBRs, respectively. In each section, applications of those circuits for dynamic analysis and harmonic analysis will be demonstrated.

9.1

INDUCTION MACHINES

In this section, we derive s-domain dynamic circuits of an induction machine viewed from different frames. We first present the well-known steady-state and dynamic circuits of an induction machine. We further show from both the steady-state representation and the dynamic representation, a generalized dynamic circuit can be found. Compared to the known dq dynamic circuits coupled by flux-related speed voltages, the generalized dynamic circuit has a concise form. It relates voltage with current using impedance components. This representation is straightforward and can be used for stability analysis. What’s more, the generalized dynamic circuit representation shows that a steady-state circuit is a special realization by evaluating s at an operating frequency jω. 9.1.1

STEADY-STATE CIRCUIT REPRESENTATION

For a three-phase induction machine, Steinmetz designed the steady-state circuit representation around the 1900s [83] and it has since been presented in classic machine books. The circuit is presented in Fig. 9.1. This circuit has the premise that the machine’s stator operating frequency is ωs . The stator’s voltage and current are positive sequence. The four phasors (V s , V r , I s , and I r ) notate the stator and rotor voltages and currents. In a classical machine book, e.g., Fitzgerald’s machine book [84], it usually takes several pages to explain the derivation of this circuit. Here, we show a brief derivation by the use of space vectors. The three-phase stator or rotor current is combined into a space vector is or ir . The stator current space vector is rotating at ωs . The rotor current space vector is rotating at the slip frequency: slip × ωs . If the slip is negative, the rotor current vector is indeed rotating in the opposite direction, if viewed from the rotor frame. is = I s e jωs t ir = I r e jslip×ωs t

Figure 9.1: The steady-state circuit representation of a three-phase induction machine.

(9.5) (9.6)

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Since the rotor has a mechanical motion and the rotating speed is ωm , the rotor current space vector is rotating at a speed of ωs , if viewed from the stator frame. Thus, the two current space vectors have the same rotating speed. The angle between them is usually close to 180 degrees since the sum of these two currents is the magnetizing current, which is usually much smaller compared � m , where to the two currents. The magnetizing current generates the magnetic field ψ � m = Lm (�is +�ir e jωm t ) = Lm (I s + I r )e jωs t , viewed from the stator frame, ψ � mr = Lm (�is e− jωm t +�ir ) = Lm (I s + I r )e jslip×ωs t , viewed from the rotor frame. ψ

(9.7) (9.8)

where Lm is the magnetizing inductance. Based on Faraday’s law: a rotating magnetic field induces electromotive forces (emfs) in both the stator circuits and the rotor circuits. Therefore: �es = jωs Lm (I s + I r ) e jωs t ,   

(9.9)

Es

�er = jslip × ωs Lm (I s + I r ) e jslip×ωs t .   

(9.10)

Er

If we extract the phasors of �es and �er , we see that

E r = slip × E s , =⇒ E s =

Er = ωs Lm (I s + I r ). slip

(9.11)

Based on this fact, the rotor circuit, after scaling its voltage and impedance by a factor of 1/slip, while keeping its current intact, can be combined with the stator circuit. This leads to the Steinmetz’s circuit shown in Fig. 9.1. This concise and elegant representation is a multi-frequency harmonic circuit. The stator circuit’s frequency is ωs while the rotor circuit’s frequency is slip × ωs . The circuit integrates two circuits operating at two frequencies, and the mechanical motion all together. In the end, it represents the relationship between the stator voltage and current phasors at the stator frequency ωs and the rotor voltage and current phasors at the rotor slip frequency slip × ωs . It has to be noted that, when the slip is negative, the rotor circuit flows a set of negativesequence current. The voltage and current phasors in Fig. 9.1 represent the phasors evaluated at a negative frequency, e.g., −1 Hz. These phasors are the conjugates of the negative-sequence phasors. Below is an explanation to show that a negative-sequence set forms a rotating space vector at the counter revolution direction, and the phasor corresponding to the negative frequency component is the conjugate of the negative-sequence phasor. Take the example of a negative-sequence set with the following expressions, va (t) = V cos(ωt + θ ), 2π ), 3 2π ), vc (t) = V cos(ωt + θ − 3

vb (t) = V cos(ωt + θ +

(9.12) (9.13) (9.14)

where phase b voltage leads phase a voltage by 120 degrees while phase c leads phase b by 120 degrees. The phasor of the negative-sequence voltage is notated as V 2 = Ve jθ based on phase a voltage’s time-domain expression. If we use the definition of space vector, this set forms a space

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vector as follows.  2π 2π 2 va (t) + e j 3 vb (t) + e− j 3 vc (t) 3  2π j(ωt+θ 2π 2π 2π V  j(ωt+θ   + 3 e 3 ) + e j 3 e− j(ωt+θ + 3 ) +  ej e  ) + e− j(ωt+θ ) + = 3    2π 2π  j 2π − 2π 3 3 ) + e− j 3 e− j(ωt+θ − 3 ) e− e j(ωt+θ  ∗  ∗ =Ve− jθ e− jωt = V 2 · e− jωt = V 2 · e jωt

�v(t) =

(9.15)

The above derivation shows that the formed space vector has a phasor corresponding to the −ω frequency component and that phasor is the conjugate of the negative-sequence phasor. We may have the following equation: [�v(t)]∗ = V 2 · e jωt .

(9.16)

Eq. 9.16 indicates that the negative-sequence phasor corresponds to ω component in the conjugate of the formed space vector. When the slip is negative, in the Steinmetz’s circuit, the rotor circuit current phasor I r is the phasor of the negative slip frequency. This phasor is the conjugate of the negative-sequence phasor I 2r : I r = (I 2r )∗ . 9.1.2

DYNAMIC CIRCUIT IN THE DQ FRAME

Circuits that are suitable for dynamic simulation and analysis have also been derived and presented in the classic books, e.g., Krause’ book Analysis of electric machinery and drive systems [15]. Fig. 9.2 shows such a circuit representation in a rotating dq frame. This dq frame rotates at a constant speed of ωs . Viewed from this frame, the three-phase stator (rotor) currents and voltages at steady state are all constants. Fig. 9.2 reflects the differential equations of an induction machine. The stator and rotor voltage complex vectors V s and V r can be expressed by the stator and rotor winding flux linkages: dψ s , dt dψ r V r = Rr I s + jωr ψ r + , dt V s = Rs I s + jωs ψ s +

(9.17)

where ωr is the slip frequency and ωr = ωs − ωm . The flux linkages are associated with the stator and rotor currents as follows: ψ s = (Lls + Lm )I s + Lm I r , (9.18) ψ r = (Llr + Lm )I r + Lm I s .

Figure 9.2: The dynamic circuit of an induction machine in the dq frame.

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Figure 9.3: The intermediate dynamic circuit of an induction machine in the dq frame. ωr is the frequency in the rotor circuit. The circuit representation, however, is not straightforward, since it has flux linkages explicitly included into the circuit. For a concise representation, we aim to express voltage by current only. In addition, the generalized circuit representation is expected to easily show as the equivalent of the Steinmetz’s steady-state circuit at steady state. If the two speed voltages associated with the fluxes in the circuit ( jωs ψ s and jωr ψ r ) can be replaced by impedance branches, this circuit reflects only voltage and current relationship and is considered more straightforward. We may do so by taking into the consideration that ψ s = Lls I s + Lm (I s + I r ), ψ r = Llr I r + Lm (I s + I r ).

(9.19)

Therefore, we can create two circuits where the speed voltage effect can be represented by two complex reactances, one in the series branch with the stator (or rotor) current flowing through, and the other in the shunt branch with the sum of the two currents flowing through. The circuits are shown in Fig. 9.3. Fig. 9.3 can be further simplified by scaling the voltage of the rotor circuit by a factor of s+ jωs s+ jωr . This scaling activity makes sure that the two shunt branches in two circuits now share the same voltage. Evaluated at s = j0, this factor is indeed 1/slip. In the dq frame, the slip’s dynamic expression is slip(s) =

s + j(ωs − ωm ) s + jωr = . s + jωs s + jωs

(9.20)

Since we want to keep the current intact, all impedance in the rotor circuit should be scaled by this factor. This leads to a dynamic circuit, as shown in Fig. 9.4. This circuit is associated with the complex phasors formed by the dq-frame variables. If we evaluate s at j0, a steady-state circuit arrives. This circuit is same as the Steinmetz’s circuit.

Figure 9.4: The generalized dynamic circuit of an induction machine in the dq frame.

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Figure 9.5: The generalized dynamic circuit of an induction machine viewed from the static frame. Fig. 9.4 is based on the dq-frame. The dq-frame is rotating against the static frame by a constant speed at ωs . A space vector viewed from the static frame has the following relationship with the complex vector in the dq frame: �v(t) = V e jωs t .

(9.21)

In the Laplace domain, the two have the following relationship: �v(s) = V (s − jωs ), or �v(s + jωs ) = V (s).

(9.22)

Similarly, any impedance’s expressions in the static frame and in the dq frame are related by the frequency shift jωs : Z(s) = Z dq (s − jωs ).

(9.23)

With this knowledge, we can find the generalized dynamic circuit in the static frame, shown in Fig. 9.5. A close examination of the circuit in Fig. 9.5 shows that if we replace s by jωs , we have the Steinmetz’s circuit. 9.1.3

FROM STEADY-STATE CIRCUITS TO GENERALIZED CIRCUITS

While the above derivation involves converting differential equations into their Laplace transforms and relating variables in different frames through frequency shift in Laplace transform expressions, there is another approach to come up with dynamic circuit representations by use of steady-state circuits. The steady-state circuit represents the positive-sequence phasor/impedance relationship at a particular perturbation frequency, i.e., when s = jωs . If ωs is any frequency, the circuit indeed represents the frequency-domain relationship between the stator terminal voltage and current. If we move forward to replace jωs by s, we now obtain a dynamic circuit. Thus, an important technique of dynamic circuit/model derivation is as follows: Dynamic circuits and models can be found by the full use of steady-state phasor-impedance circuits. 9.1.3.1

Example 1: Positive-sequence induction machine dynamic circuit

Examining the steady-state circuit, we may see that the impedances in the circuit that contain ωs are the stator leakage inductor ( jωs Lls ), the magnetizing inductor ( jωs Lm ), the rotor leakage inductor ( jωs Llr ), and finally the equivalent rotor resistance (Rr /slip where slip = 1 − ωm /ωs ). For the passive components such as inductors, their impedances in the s-domain can be quickly found by replacing jωs as s. For slip, some manipulation is necessary to have jωs explicitly presented. slip = 1 −

ωm jωm jωm . = 1− . =⇒ slip(s) = 1 − ωs jωs s

(9.24)

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Therefore, the slip can be further expressed in the s-domain. This derivation technique has been presented by the authors in 2012 [64]. It has to be noted that the derivation is very concise and it takes the advantage of prior knowledge of induction machine steady-state equivalent circuit derivation. This approach has a solid theoretic foundation. 9.1.3.2

Example 2: Induction machine dynamic circuit in negative sequence

It is very easy to conduct reference frame transform utilizing Laplace transform’s characteristics. For example, the space vector viewed from the static frame and the complex vector viewed from a dq-frame with a rotating speed ωs have the following relationship:�v = V e jωs t . In turn, the frequency domain relationship becomes the follows: �v(s) = V (s − jωs ), or �v(s + jωs ) = V (s).

(9.25)

An RL circuit’s impedance in the static frame is R + sL and in the dq frame is R + (s + jωs )L. R + (s + jωs )L can also be viewed as the positive-sequence impedance. Similarly, the impedance in the second-harmonic domain becomes R + (s + j2ωs )L. In short, the Laplace transform makes frame conversion very easy. In the negative-sequence circuit, the equivalent rotor resistor is Rr /slip� . We may further find the negative-sequence circuit’s slip transfer function as follows. slip� = 2 − slip = 2 − (1 −

ωm ) ωs

ωm jωm = 1+ , ωs jωs jωm . =⇒ slip� (s) = 1 + s = 1+

(9.26)

An alternative approach is to come with the above expression using the Laplace transform’s characteristics. If the Laplace transform of an analytic signal x(t) is X(s), then the Laplace transform of x∗ (t)—the conjugate of the analytic signal —is (X(s∗ ))∗ . For example, a space vector e jωt formed by a set of positive-sequence voltage has the Laplace transform as s−1jω . Then its conjugate e− jωt formed by a set of negative-sequence voltage has the Laplace transform as  ∗ 1 1 . = s∗ − jω s + jω The slip in the negative sequence circuit can be directly found from the slip in the positive sequence circuit by the use of the conjugate property:   jωm ∗ jωm � ∗ ∗ . (9.27) = 1+ slip (s) = (slip(s )) = 1 − ∗ s s Fig. 9.6 shows the negative-sequence dynamic circuit of an induction machine. Note that this circuit is associated with the conjugate of the space vector. When evaluating s at jωs , v∗ ( jωs ) refers to the negative-sequence phasor V 2s at ωs , according to (9.16). 9.1.4

APPLICATIONS: STABILITY ANALYSIS

The generalized circuit representation is very useful for dynamic analysis. In the authors’ work [85], the positive-sequence circuit of a type-3 wind turbine presented in Fig. 9.7(a) was used for subsynchronous resonance (SSR) analysis when a type-3 wind farm is radially connected to a transmission line with series compensation. Fig. 9.7(b) presents the negative-sequence circuit derived from the authors’ another paper [86].

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Figure 9.6: The generalized negative-sequence dynamic circuit of an induction machine viewed from the static frame. The entire circuit can be viewed to have two parts: the wind turbine impedance Zsr (the shunt impedance Zg may be ignored due to its large magnitude) and the network impedance Znet . A circuit analysis problem can be converted to a feedback system with the total voltage source as the input and the current as the output: Δi =

1 ΔV 1  ΔV =  Z Zsr + Znet Znet sr 1 + Znet

(9.28)

If the network admittance 1/Znet is a stable system and the input is changed to ΔV /Znet , the system’s stability can be checked by examining the loop gain: the ratio of the two impedances: Loop Gain =

Zsr . Znet

(9.29)

Bode plots and Nyquist diagrams may be used for visual check of stability. See Fig. 9.8a. The Nyquist diagram examines the effect of wind speed on stability. It can be seen that lower wind speed makes SSR stability worse. The Bode diagrams examine the effect of compensation level on SSR stability. It can be seen that the higher compensation level leads to a less stable system. In addition to stability check, the generalized circuit can be used to explain other phenomena. For example, it is found that the dq-frame state-space model of the system has both subsynchronous mode and supersynchronous mode [67]. While the two modes appear to originate both from the LC resonance mode in the static frame, they have different characteristics. Increasing the rotor resistance makes the subsynchronous mode more unstable, while the supersynchronous mode more stable.

Figure 9.7: (a) The positive-sequence dynamic circuit of a type-3 wind farm connected to a grid through an RLC circuit. (b) The negative-sequence dynamic circuit of a type-3 wind farm connected to a grid through an RLC circuit. .

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(a)

(b)

Figure 9.8: (a) Frequency-domain stability analysis using the generalized circuit representation [85]. (b) Total impedance and total admittance. While eigenvalue analysis is a powerful tool, it has difficulty to explain why. On the other hand, frequency-domain analysis based on the generalized circuit can quickly lead to further insights and answer why. For example, Fig. 9.8b presents the entire system’s impedance and admittance over −100 Hz to 100 Hz. It has to be noted that compared to the normal Bode plots, this plot displays not only positive frequency domain, but also negative frequency domain responses. Due to the complex coefficients in the slip transfer function, the negative frequency domain responses are no longer just the mirror of the positive frequency domain responses. Rather, asymmetry is assumed. It can be seen that the impedance at 42 Hz has a negative resistance while the impedance at −42 Hz has a positive resistance. The dq frame is to view the frequency response from a rotating frame. In the frequency domain, this is equivalent to moving the y-axis from 0 Hz to 60 Hz of the static frame. The dqframe’s 0 Hz is the static frame’s 60 Hz. Thus, 42 Hz becomes −18 Hz subsynchronous mode while −42 Hz becomes −102 Hz supersynchronous mode. Obviously, increasing the rotor resistance makes the resistance of −18 Hz more negative while the resistance at −102 Hz more positive. Therefore, it can be seen that the generalized circuit can provide many more insights.

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9.2

UNBALANCED TOPOLOGIES

While we have demonstrated the concept and the powerful applications of a generalized indiction machine circuit, to make this concept useful for power grids with both synchronous generators and IBRs as resources, generalized dynamic circuits for those sources have to be developed. Unlike a three-phase induction machine, those sources do not have symmetric topologies. Thus, we expect to see many challenging issues in circuit derivation. In this section, we investigate how to integrate unbalanced topologies into generalized circuits. This type of generalized circuits can be used to investigate many more challenging scenarios in stability analysis with unbalanced topologies considered. For example, if phase a of the induction motor is connected to the ground, how to analyze such kind of systems? Of course, we may resort to dynamic phasor (DP)-based modeling, as been done by A. Stankovic for asymmetry faults in [87]. On the other hand, tremendous efforts are required for DP-based modeling. In addition, the outcome of DPs is a state-space model in the time domain, which again requires further manipulation to lead to straightforward insights on dynamic phenomena. In the following, we examine how to integrate unbalanced topologies into generalized dynamic circuits. We also use an example case to demonstrate the effect of unbalance on dynamic performance and the quantitative analysis. 9.2.1

SEQUENCE NETWORKS

In steady-state analysis, unbalanced faults in a three-phase symmetric network can be dealt efficiently with by the use of the sequence network interconnection technique. For example, for a single-line-ground (SLG) fault, the three-phase fault current and the fault bus voltage have the following characteristics: V a = 0, I b = I c = 0.

(9.30)

In turn, the sequence-domain voltages and currents have the following relationship: +

−

0

+

−

0

V +V +V = 0, I = I = I =

Ia . 3

(9.31)

Thus, the sequence networks can be interconnected in series. Fig. 9.9 shows the sequence networks interconnected in series, for a single-line to ground fault. The fault current in sequence can be found by circuit analysis of the sequence network interconnection. Can we expand such techniques to form the s-domain sequence circuits? 9.2.2

EXPANDING SEQUENCE NETWORKS TO A DYNAMIC CIRCUIT

In order to check the possibility, we examine the same SLG fault using the time-varying space vectors, instead of the phasors at the nominal frequency. The boundary conditions are expressed in the time domain as follows: va (t) = 0, ib (t) = ic (t) = 0.

(9.32)

The space vector expression has been developed to reflect that three-phase currents can formulate a rotating magnetomotive force (mmf). The space vector aggregates the three-phase variables to form a single variable. In addition, we bring into the picture the conjugate of the space vector and

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Figure 9.9: Sequence networks interconnected in series for an SLG fault. the zero-sequence component in the time domain:  2π 2π �v 1  = va + e j 3 vb + e− j 3 vc , 2 3  2π 2π (�v)∗ 1 = va + e− j 3 vb + e j 3 vc , 2 3 1 0 v = (va + vb + vc ). 3

(9.33) (9.34) (9.35)

The above relationship leads to the expression of abc variables in terms of the space vector, its conjugate and the zero-sequence component: �v (�v)∗ + , 2 2 2π � 2π (� v v)∗ , vb = v0 + e− j 3 + e j 3 2 2 2π � 2π (� v v)∗ . vc = v0 + e j 3 + e− j 3 2 2 va = v0 +

(9.36) (9.37) (9.38)

Thus, the boundary condition of voltage leads to the following va = 0, =⇒

�v (�v)∗ + + v0 = 0, 2 2

(9.39)

according to (9.36) while the boundary condition of currents leads to the followin ib = ic = 0, =⇒

�i (�i)∗ 1 = = i0 = ia . 2 2 3

(9.40)

according to (9.33), (9.34), and (9.35). If the space vectors �v and �i as viewed to be related to the positive sequence circuit, then their conjugates can be viewed as to be related with the negative sequence circuit. With the zero-sequence circuit available, the three circuits can be interconnected at the faulted bus to have a series connection. Therefore, it looks promising that we can even expand steady-state sequence circuit interconnection to dynamic sequence circuit interconnection.

242

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(a)

(b)

Figure 9.10: (a) An induction motor connected to a series compensated network is subject to unbalanced fault. (b) Simulation results. Before t = 1 s, the motor is connected to both the RL and the RLC circuits. At t = 1 s, the parallel RL circuit is tripped leaving the motor radially connected to the RLC circuit. At t = 2, phase a is connected to the ground. We will use a case study to derive the details of the dynamic sequence network. Adaption of the work can also be found in the authors’ paper [88]. 9.2.3

CASE STUDY: AN INDUCTION MACHINE SERVED BY AN UNBALANCED NETWORK

Fig. 9.10a presents a test bed of a 200-hp 460-V induction motor connected to a series compensated network. The motor speed is fixed at 0.73 p.u. At t = 1 s, the parallel RL circuit is tripped leaving the motor radially connected to the RLC circuit. This RLC circuit has 50% compensation level. At t = 2, phase a is connected to the ground. Fig. 9.10b presents the simulation results. It can be clearly seen that once the IM is radially connected to the capacitor, 26-Hz oscillations in the torque and 34-Hz oscillations in the stator currents become undamped. This 26-Hz mode is due to the LC resonance. After phase a of the terminal bus connects to the ground, the torque has only 120-Hz ripples due to unbalance while the stator currents have only 60-Hz fundamental components. It can be clearly seen that the 26-Hz mode has improved damping due to unbalance. Indeed, in the literature of SSR control, unbalance has been pointed out to have positive impact [89] in 1993 by Edris. In a 2011 paper [90], the dynamic phasor modeling approach was adopted

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243

for model derivation and finding eigenvalues under unbalance. Reference [90] pointed out that large levels of phase imbalance can lead to a significant movement in damping of the subsynchronous network modal frequencies. 9.2.3.1

Circuit 1

If the space vector �v and �i are viewed to be related with an impedance Z(s), then their conjugates (�v)∗ and (�i)∗ are related with an impedance (Z(s∗ ))∗ , based on the rule of the Laplace transform. For an RLC circuit, both Z(s) and (Z(s∗ ))∗ are the same: R + sL + 1/(sC). However, if there is a complex coefficient, the two impedances are not the same. For example, the induction machine’s rotor equivalent resistance is expressed as s Rr , s − jωm and its expression for the conjugate should be s Rr . s + jωm With the zero-sequence circuit available, the three circuits can be interconnected at the faulted bus to have a series connection. Fig. 9.11 presents the interconnected circuit model. 9.2.3.2

Circuit 2

The circuit in Fig. 9.11 is viewed based on the space vectors and the conjugates. At unbalanced conditions, the space vector has both positive- and negative-sequence components. To be able to associate with the steady-state interconnected sequence network, we further derive the circuit to be related with sequence components. The boundary conditions are re-examined. A space vector can

Figure 9.11: Circuit 1: the example system subject to an SLG fault.

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be expressed as the sum of the positive- and negative-sequence components: ∗  �v(t) =�v1 (t) + [�v2 (t)]∗ = V 1 (t)e jω t + V 2 (t)e jω t ,   �i(t) =�i1 (t) + [�i2 (t)]∗ = I 1 (t)e jω t + I 2 (t)e jω t ∗ ,

(9.41) (9.42)

where the subscript 1 and 2 notate the positive- and negative-sequence components, respectively. One has to keep mind that �v1 (t) (�i1 (t)) and �v2 (t) (�i1 (t)) are having the same rotating direction and speed. Hence the first boundary condition 0.5�v + 0.5(�v)∗ + v0 = 0 is equivalent to 0.5(�v1 (t) + [�v2 (t)]∗ ) + 0.5([�v1 (t)]∗ +�v2 (t)) + v0 (t) = 0, 0.5(�v1 (t) + [�v2 (t)]∗ ) + 0.5([�v1 (t)]∗ +�v2 (t)) + 0.5(v0 (t) + v∗0 (t)) = 0, =⇒ �v1 +�v2 + v0 = 0.

(9.43)

In the above derivation, we have used the fact that v0 (t) is a signal and is always real. Therefore, its conjugate is itself. Another boundary condition 0.5�i = 0.5(�i)∗ = i0 = 13 ia is equivalent to 0.5�i = 0.5(�i)∗ = i0 =⇒�i1 + [�i2 ]∗ = [�i1 ]∗ +�i2 = 2i0 , =⇒ �i1 =�i2 = i0 .

(9.44)

Based on �i1 + [�i2 ]∗ = [�i1 ]∗ +�i2 , we know that �i1 =�i2 , since �i1 and �i2 are rotating at the same speed, while �i∗1 and �i∗2 are rotating in the opposite directions. The relations in (9.43) and (9.44) hold in both time domain and frequency domain. Based on (9.43) and (9.44), the interconnected network is built and shown in Fig. 9.12. Note the impedance relating�v1 and�i1 is the same as that relating�v with�i. The impedance relating�v2 and�i2 is the same as that relating (�v)∗ and (�i)∗ . The advantage of the circuit in Fig. 9.12 is that it can be directly related to the steady-state sequence network. The balanced source voltage only appears in the positivesequence network.

Figure 9.12: Circuit 2: the example system subject to an SLG fault.

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245

(i) The circuit presented in Fig. 9.12 is a dynamic circuit with unbalanced topology modeled. If we substitute s by jω where ω is the synchronous frequency, the resulting circuit is the same steady-state circuit for SLG faults. (ii) Indeed, for unbalanced systems that can be represented by a steady-state phasor-impedance sequence network, we may directly come up with the corresponding dynamic circuit by replacing jω using the Laplace transform variable s. 9.2.3.3

Stability analysis

Compared to the steady-state sequence network which is mainly used for fault analysis, the dynamic circuit is capable of stability analysis. Below is a demonstration. We ignore the shunt magnetizing branch sLm for simplicity since its impedance magnitude is one order greater than the rotor impedances in the 20-40 Hz range. The total positive-sequence impedance of the induction machine (IM) is ZIM1 = Rs +

s Rr + s(Lls + Llr ). s − jωm

(9.45)

The total negative-sequence impedance is ZIM2 = Rs +

s Rr + s(Lls + Llr ). s + jωm

(9.46)

For the balanced system, the loop gain is Rs + s− sjωm Rr + s(Lls + Llr ) ZIM1 (s) L1 (s) = = 1 ZRLC (s) R + sL + sC

(9.47)

For the SLG case, the loop gain is L2 (s) =

Z1 (s) + Z2 (s) , ZRLC (s)

(9.48)

where Z1 (s) and Z2 (s) are as follows: Z1 (s) =

ZIM1 ZRLC ZIM2 ZRLC , Z2 (s) = . ZIM1 + ZRLC ZIM2 + ZRLC

Fig. 9.13 presents the Bode diagrams of the two loop gains. It can be clearly seen that for the balanced system, at about 34 Hz when the phase shifts from 180◦ to −180◦ , L1 ’s gain is at 0 dB, indicating instability. On the other hand, for the unbalanced system, the loop gain’s phase keeps in the range of 0 to 180 degrees in the 0-80 Hz range. At −26 Hz when L2 ’s gain is 0 dB, the phase margin is about 30 degrees. Hence, the Bode diagram shows no stability issue for the SLG case. The dynamic circuit in Fig. 9.12 also reveals that unbalanced topology mitigates the effect of the equivalent rotor resistor sRr /(s − jωm ), which is negative if the excitation frequency is less than 42 Hz (corresponding to 0.7 pu rotating speed). This leads to the improvement of SSR stability. The second circuit is a generalized circuit for unbalanced systems and the well-known steadystate sequence network is a special realization when it is evaluated at the nominal frequency. This research connects dynamic modeling and unbalanced treatment. It further shows that the steadystate sequence network can be directly converted to an s-domain dynamic circuit suitable for both fault analysis and stability analysis.

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246 50

-26 Hz

dB

0

-50

-100 -80

34 Hz

-60

-40

-20

0

20

40

60

80

0

20

40

60

80

200

Degree

100

Balanced case SLG case

0 -100 -200 -80

-60

-40

-20

Freq (Hz)

Figure 9.13: Loop gains for stability check.

9.3

SYNCHRONOUS MACHINES

While for the symmetric circuits such as a three-phase induction machine, the dynamic circuits can be obtained with ease, for other components, such ease cannot be found. For example, for a synchronous generator with a round rotor, its steady-state circuit is a simple Th´evenin equivalent with a voltage source behind an impedance at the nominal frequency. On the other hand, we do have the knowledge that a synchronous machine behaves like an induction machine during the starting up process if the excitation voltage source in the rotor is taken out. A synchronous machine is much more complicated as it can be viewed as an induction machine with unbalanced three-phase rotor circuits. The well-known Th´evenin equivalent circuit has the underlying assumption that the generator speed can approximated as nominal. Therefore, such a circuit cannot be used to find dynamic circuits by extension. Indeed, for a synchronous generator operating at any asynchronous speed, steady-state circuit representation is missing from the textbooks on power system analysis. 9.3.1

AN INDUCTION MACHINE WITH UNBALANCE IN ROTOR CIRCUITS

Since a synchronous machine may be viewed as an induction machine with unbalanced rotor impedance, we may seek to represent it for steady state first. In this area, fortunately, Garbarino and Gross developed an equivalent circuit for this unbalanced system in 1950 [82] to explain Goerges phenomenon, i.e., unbalanced rotor impedance may lead to an induction motor operating at half of the synchronous speed, which was discovered by Hans Goerges in 1896. In [82], the circuit analysis results have been validated by hardware experiments. The authors have examined [82] and presented a detailed explanation on how to develop the equivalent circuit step by step using the theory of symmetrical components, along with an understanding of rotating machine electromagnetic in [91]. In the following, the results from [91] are briefly described. A testbed of an induction machine with unbalanced rotor impedance and the simulation results of free acceleration using MATLAB/Simscape are shown in Fig. 9.14. It can be seen that when there is an extra rotor resistance in phase a and this resistance is 1 Ω, the induction motor can only rise to approximately half of the nominal speed for free acceleration.

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247

Figure 9.14: The induction machine’s free acceleration testbed and the simulation results. The speed achieves 0.507 p.u. In turn, the slip is 0.493 p.u. While the source voltage is 60 Hz, it has to be mentioned that the machine’s stator circuit has two components: 60 Hz and (1 − 2 × slip) × 60 = 0.84 Hz. This 0.84-Hz component is due to the negative sequence component in the rotor current. The rotor current has a slip frequency of slip × 60 = 28.58 Hz. Due to unbalanced topology, a negative-sequence component is expected. This component leads to a rotating magnetomotive force (mmf) at (−slip × ωs + ωm ) in the airgap, viewed from the stator perspective. This frequency is (1 − 2 × slip) × 60 Hz. This mmf induces an emf at the same frequency. Thus, the stator circuit will have a component at 0.84 Hz. The resulting steady-state circuit is presented in Fig. 9.15. It can be clearly seen that a synchronous machine can assume a similar representation. For example, if RL is assumed to be infinite, this machine is a synchronous machine just without excitation voltage. Furthermore, we may again treat this circuit as the realization for any stator frequency perturbation at ωs . Replacing jωs by the Laplace transform variable s, a dynamic circuit of a synchronous generator can be found! Thus, it is encouraging to see that a synchronous machine can be indeed represented by a two-port circuit, with one port associated with the perturbation frequency at ωs and the other port associated with the mirror frequency at (2ωm − ωs ). 9.3.2

STEADY-STATE CIRCUIT OF A SYNCHRONOUS MACHINE

A round rotor synchronous machine without the dc excitation voltage source can be viewed as an inductor machine with its rotor circuits’ two phases short-circuited, while another left open. The

Figure 9.15: The steady-state circuit of an induction machine with unbalanced rotor resistance.

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Figure 9.16: The positive-sequence circuit of an IM viewed from the rotor terminal. rotor’s terminal voltages and the currents have the following relationship: V r,bn = V r,cn , I rb + I rc = 0, I ra = 0.

(9.49)

Applying the symmetrical component transform leads to the following relationship for the sequence voltage and current components. +

−

+

−

V r = V r , I r + I r = 0.

(9.50)

Thus, the positive- and negative-sequence circuits are interconnected at the rotor terminal as two parallel circuits. This network representation is based on the view point of the rotor. Hence, the perturbing frequency is the slip frequency notated as ωr . To view the circuit from the stator side, we may conduct impedance scaling and voltage scaling. Fig. 9.16 shows the positive-sequence circuit of an IM viewed from the rotor terminal. This circuit is very similar to the Steinmetz’s circuit except the viewpoint is at the rotor terminal. The stator side voltage and impedance are scaled by a factor of slip s to be interconnected to the rotor side circuit. Fig. 9.17 shows the negative-sequence circuit of an IM viewed from the rotor terminal. Note that when the rotor circuit is unbalanced, there presents a negative-sequence component at the slip frequency in the rotor current. This rotor current space vector rotates at a speed of −sωs . In the static frame, this space vector rotates at a speed −sωs + ωm = (1 − 2s)ωs . This current introduces an electromagnetic field with (1 − 2s)ωs rotating speed. Furthermore, in the stator circuit, this field induces an emf and further currents at the mirror frequency of (1 − 2s)ωs or 2ωm − ωs . The phasors of emfs in the stator and rotor windings are related by a ratio of (1 − 2s) : −s, as shown in Fig. 9.17. Through stator side voltage and impedance scaling, an interconnected circuit is formed. Furthermore, conjugate of the circuit is used to represent the relationship between negative-sequence phasors. Finally, the positive- and negative-sequence circuits are connected in parallel. Further scaling the voltages and impedances by a factor of 1/s leads to a circuit viewed from the stator. Fig. 9.18 presents the network. This network representation can also be found in the 1950 paper by Garbarino and Gross [82] (Fig. 3). This steady-state circuit is viewed from the stator’s side when the perturbing frequency is ωs , the synchronous frequency. Thus, after replacing jωs with s in the reactance representation, and replacing the slip with s− sjωm , where ωm is the rotor speed, we may quickly arrive at the dynamic

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249

Figure 9.17: The negative-sequence circuit of an IM viewed from the rotor terminal. circuit representation. If we notate the rotor resistance and inductance as a synchronous machine’s field resistance R f and leakage inductance Ll f , the final representation is shown in Fig. 9.20. This derivation philosophy is to treat the steady-state circuit representation as a special case of the generalized dynamic circuit when the stator side’s perturbation frequency is ωs . In the dynamic circuit, jωs can all be replaced by the Laplace transform variable s. Hence, the stator leakage reactance, the rotor leakage reactance, and the magnetizing reactance, such as jωs Lls , jωs Ll f , and jωs Lm can be expressed as sLls , sLl f and sLm . Furthermore, slip can also be expressed by use of its s-domain expression. The two equivalent resistances are shown as follows: 2Rr s s −Rs = = 2Rr , Rs . slip s − jωm 1 − 2slip s − j2ωm

(9.51)

Thus, a steady-state circuit representation can be converted to a generalized dynamic circuit.

Rs Vs

jXls I+s

Rr slip

jXm

jXlr I+r

Rr slip

jXlr I-r

-Rs 1-2slip

jXls

jXm

I-s

Figure 9.18: The sequence network of an induction machine with its rotor circuits phase b and phase c short circuited while phase a left open.

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9.3.3

FROM A DYNAMIC MODEL TO A GENERALIZED CIRCUIT

We now examine a synchronous generator with an excitation circuit and work towards the goal of generalized dynamic circuit starting from the dq-frame model. The dynamic model in time domain is presented as follows. First, the relationship among the stator voltages, the stator currents, and the stator flux is expressed in (9.52). vd = −id Rs − ωm λq + λ˙d , vq = −iq Rs + ωm λd + λ˙q ,

(9.52)

where the flux linkages are related to the dq stator current and the excitation current i f d as follows: λd = (Lmd + Lls )(−id ) + Lmd i f d , λq = (Lmq + Lls )(−iq ).

(9.53)

Next, for the rotor excitation circuit, the voltage and current relationship is: v f d = i f d R f + λ˙ f d ,

(9.54)

where λ f d = Lmd (−id ) + (Lmd + Ll f )i f d . For arriving at a generalized circuit representation, we aim to present the voltage and current relationship explicitly. The dynamic model ((9.52) and (9.54)) does not meet the requirement since the speed voltage terms are related to the flux linkages. To get rid of the speed voltages, we seek to use complex vector in the Laplace domain. From (9.52), it can be found that V1 (s) = −Rs I1 (s) + (s + jωm )Λ1 (s) V2 (s) = −Rs I2 (s) + (s − jωm )Λ2 (s)

(9.55) (9.56)

where F1 (s) = fd (s) + j fq (s), F2 (s) = fd (s) − j fq (s), with F being the complex vector V , I or Λ, and f being v, i, or λ . 9.3.3.1

Physical meaning of the complex vectors F1 (s) and F2 (s)

Harnefors initiated the use of complex vector in the analysis of dq-frame converter control [92] and this concept has been used extensively in the literature on power electronics control. Among them, asymmetrical-dynamics-induced mirror frequency has been discussed in [93, 94]. We will examine the prime and mirror frequency components. Suppose that the dq-frame currents id (t) and iq (t) viewed from the rotor are sinusoidal signals with a frequency of ωr . The machine has a rotating speed of ωm . The d-axis current id creates two space vectors, one with a rotating speed of ωr + ωm = ωs and the other at the speed of −ωr + ωm = 2ωm − ωs . Similarly, the q-axis current iq also creates two space vectors of the two frequencies. Hence, in the stator side, the stator current should have two harmonic components: the primary component at ωs and the mirror frequency component at 2ωm − ωs . Only if the two signals have the same magnitudes and iq lags id by 90 degrees, the two space vectors at the mirror frequency are canceled by each other with the stator currents having only ωs component. The space vector formed by the three-phase currents (notated as �i(t)) has the following relationship with id (t) and iq (t): �i(t) = (id (t) + jiq (t))e jωm t . Notate the analytic form of id (t) as i˜d (t) and the analytical form of iq (t) as i˜q (t). Then the space vector �i(t) can be written as: ⎛ ⎞ ⎜1 � � 1� �⎟ ⎟ �i(t) = ⎜ ⎜ i˜d (t) + ji˜q (t) + i˜∗d (t) + ji˜∗q (t) ⎟ e jωm t 2 ⎝�2 �� � � �� �⎠ I1 (t)

I2∗ (t)

(9.57)

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251

Note that, i˜d (t), i˜q (t), and I1 (t) are analytic signals with a frequency of ωr , while i˜∗d (t), i˜∗q (t), and I2∗ (t) are analytic signals with a frequency of −ωr . Both I1 (t) and I2 (t) are analytical signals with a frequency of ωr . In the Laplace domain, it can be seen that I1 (s) = id (s) + jiq (s), I2 (s) = id (s) − jiq (s).

(9.58)

Therefore, the complex vectors represent the primary frequency component and the conjugate of the mirror frequency component, viewed in the rotor frame. Furthermore, the primary frequency component, notated as i1 viewed from the stator frame has the following relationship with I1 (s): i1 (s + jωm ) = I1 (s), or i1 (s) = I1 (s − jωm ). The mirror-frequency component’s conjugate, notated as i2 has the following time-domain format I2 (t)e− jωm t . Hence, their Laplace-domain expressions are related as I2 (s) = i2 (s − jωm ), =⇒ I2 (s − jωm ) = i2 (s − j2ωm ). 9.3.3.2

Generalized circuit development

For simplicity, round rotor structure is assumed and the dq-axis inductances are the same: Lmd = Lmq = Lm . The flux linkages and the currents have the following relationship.        L + Lls −I1 L Λ1 0 (9.59) = m + m ifd Λ2 Lm 0 Lm + Lls −I2 It can be seen that the excitation current contributes to both flux linkage complex vectors. On the other hand, (9.54) is still expressed by the d-axis flux linkage λ f d . We will express λ f d by the use of the complex vectors of the current.   −I1 −I2 (9.60) + λ f d = Lm + (Lm + Ll f )i f d , 2 2 =⇒ 2λ f d = Lm (−I1 − I2 ) + 2(Lm + Ll f )i f d .

(9.61)

Thus, the flux linkage linked with the excitation circuit has contributions from the two current complex vectors and the excitation current. Based on the above relationship, we may construct a circuit representing the formation of the three flux linkages first: stator flux linkage’s complex vectors and the rotor excitation field flux linkage λ f d . The circuit is shown in Fig. 9.19(a), where both −I1 and i f d contribute to Λ1 , and both −I2 and i f d contribute to Λ2 . Since the induced voltage from Λ1 is sΛ1 , it has to be amplified to s+ sjωm times to be connected to V1 (s) through the resistor Rs . Similarly, the induced voltage from Λ2 is sΛ2 . This voltage has to be amplified by s− sjωm times to be connected to V2 (s). Two transformer like symbols are used to represent the amplification. These two symbols represent voltage changes from one side to the other. The currents at the two sides keep the same. In step 2, the input voltage is viewed from the stator and notated as v1 (s). Note that v1 (s) = V1 (s − jωm ). We will replace s by s − jωm in Fig. 9.19(a). In turn, the previous circuit components viewed from the rotor frame have their s replaced by s − jωm . For the rightmost part of the circuit, the terminal voltage and currents are v2 (s − j2ωm ) and i2 (s − j2ωm ). In the last step, the two transformer like symbols are taken out. The excitation voltage v f d is assumed to be 0. The middle part of the circuit between the two transformers has every impedance scaled by s− sjωm . This scaling effort makes sure that the voltages at the two sides of the first transformer are now the same while the current is kept intact. For the rightmost part of the circuit, the

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Modeling and Stability Analysis of Inverter-Based Resources

Figure 9.19: Circuit representation of a round rotor synchronous machine. (a) Step 1. (b) Step 2. s scaling factor is s− j2ω . The resulting circuit is Fig. 9.20. It can be seen that while all inductance has m its impedance represented as sL, the rotor resistance and the stator resistance in the mirror-frequency s . domain are scaled by s− sjωm and s− j2ω m If the perturbation frequency is the synchronous frequency ωs , we may replace s by jωs . The 2R equivalent rotor resistance now becomes slipf and the stator resistance in the mirror-frequency doRs . We can see that the resulting steady-state circuit in Fig. 9.20 is the same as that in main is 2slip−1 Fig. 9.18. In short, a synchronous generator can be viewed as a two-port circuit connecting the primary frequency components with the mirror-frequency components.

9.3.4

CASE STUDY: STARTING A SYNCHRONOUS MACHINE

The proposed circuit can be used to build a simulation model of starting a synchronous machine from standstill. Both the excitation voltage source v f d and the mechanical torque Tm are set to 0. Based on the circuit in Fig. 9.20, at any given speed and for a given stator voltage, the stator current phasors and the field current phasors can be found by ignoring the EMT dynamics. Furthermore, the torque can be computed from these current phasors. The average torque is expressed as follows:    (9.62) Tem,avg = Lm Imag i1 (s)[i f d (s − jωm )]∗ + +i f d (s − jωm )[i2 (s − j2ωm )]∗

where s is evaluated at the nominal frequency of 377 rad/s.

Figure 9.20: Circuit representation of a round rotor synchronous machine without excitation voltage source.

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Figure 9.21: The phasor model implemented in MATLAB/Simulink and the EMT testbed of starting a synchronous machine. The machine parameters: stator resistor, leakage, dq mutual inductance and inertia H: 0.2917, 0.0113, 3.0314, 3.0314, 0.1492; rotor resistance and leakage: 0.4667, 0.0490. During starting process, the torque has both a dc component and a harmonic component. Since we are interested in the dynamics of accelerating a machine, we may compute only the useful torque, i.e., the average torque, for electromechanical dynamics simulation. Combined with the circuit analysis and torque computing, the swing equation can be integrated and dynamic simulation results are produced based on this first-order model. For comparison, an EMT testbed with a generator model from the MATLAB/Simscape library is also built. The machine’s d-axis has a field winding and a damping winding and its q-axis has two damping windings. Thus, the generator model contains both the first-order swing dynamics and the sixth-order electromagnetic dynamics. For a fair comparison, the impedances of three damping windings in the EMT model have been increased to large numbers. Fig. 9.21 shows the simulation model implemented in MATLAB/Simulink. The results from the simulation model are compared with those based on a generator model in the MATLAB/Simscape library which contains details of both swing dynamics (one order) and sixth-order electromagnetic dynamics. The machine’s d-axis has a field winding and a damping winding and its q-axis has two damping windings. Since the proposed circuit considers only the field winding, for a fair comparison, the impedances of three damping windings in the EMT model have been increased to large numbers. Three cases are examined for a varying field resistance. When R f = 2.45 p.u., the machine can be accelerated from standstill to full speed. When R f = 0.7 p.u., the machine can still be accelerated from standstill to full speed. However, in between, it staggered at half of full speed for a while.

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Modeling and Stability Analysis of Inverter-Based Resources

Figure 9.22: Comparison of the simulation results of the phasor model (blue lines) and the EMT testbed (orange lines) for three scenarios of different field resistance. (1) Full speed is achieved. (2) The machine achieves full speed after staggering at half speed. (3) The machine achieves only half speed. The machine parameters in p.u.: stator resistor, leakage, dq mutual inductance and inertia: 0.2917, 0.0113, 3.0314, 3.0314, 0.1492; rotor resistance and leakage: 0.4667, 0.0490. When R f = 0.4667 p.u., the machine can only be accelerated to 51% of the full speed. The last case demonstrates Goerges phenomenon [95]. Fig. 9.22 shows the simulation results. The phasor model has the machine speed, the average torque, and the field current’s phasor’s magnitude exported. From the EMT testbed, the measured torque has to be passed to a second-order low-pass filter first to have its dc component, or the average torque is taken out. This filtered torque will be compared with the average torque from the phasor model. The measured field current from the EMT testbed is sinusoidal and with a varying frequency, while the phasor model exports the field current phasor’s magnitude. These two are plotted together. From the comparison of Fig. 9.22, we can see that even we ignored the electromagnetic dynamic in the phasor model, the phasor model with only first-order swing dynamics included can accurately capture the machine electromechanical behavior. In addition, the computed average torque and the field current phasor can also accurately reflect the true values in general. In addition, harmonic analysis can be carried out using the circuit at two conditions: (i) at the beginning of the starting process when the mechanical speed is still 0, and (ii) at the steady-state condition of Goerge’s phenomenon when the machine’s speed keeps at 0.51 p.u. The results from the phasor model have all been verified using the measurements from the EMT testbed. In summary, the circuit gives accurate harmonic component analysis results. Fig. 9.23 shows the simulation results from the EMT testbed and the phasor model. For case 1, the machine speed is 0 and both the stator and the field currents are of 60 Hz. The first subplot shows the 60-Hz field current measurement from the EMT testbed and the field current phasor’s magnitude from the phasor model. The second subplots shows the three-phase stator current from the EMT testbed. It is an unbalanced set with the abc phase magnitudes at 0.3267 p.u., 1.0069 p.u., and 1.3219 p.u., respectively. The phasor model gives the phasors of the positive-sequence and the negative-sequence currents. They are: 0.732 p.u., at 158 degrees and 0.6482 p.u. at 4 degrees. Based on these two phasors, it can be confirmed that abc currents have the magnitudes of 0.3267 p.u., 1.0069 p.u., and 1.3219 p.u. For case 2, the machine speed is 0.51 p.u. and the slip frequency is 29 Hz. The phasor model gives the phasors of the 29-Hz field current, the 60-Hz stator current component (0.6248 p.u. at

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Figure 9.23: Simulation results from the EMT model and the phasor model. (a) speed is 0. (b) speed is 0.51 p.u. 110 degree), and the 2-Hz stator current component (0.109 p.u. at −131 degree). These results also match the fast Fourier transform (FFT) analysis results of the EMT model’s phase stator currents (shown in Fig. 9.24). 9.3.5

SUMMARY

We present a generalized circuit representation for a round synchronous machine suitable for dynamic and harmonic analysis. This representation takes the full advantage of the Laplace transform variable s in dealing with frame conversion and calculus. The resulting circuit representation of a synchronous machine is a two-port circuit connecting the stator’s primary frequency components with the stator’s mirror-frequency components. This circuit can also be viewed as a model representing the relationship of dynamic phasors of the primary frequency components and the mirrorfrequency components.

9.4

IBRS

Fig. 9.15 shows a two-port circuit for a synchronous machine. If the rotating speed ωm is at the nominal speed ω0 , we see a similar phenomenon of IBR: frequency coupling. An IBR interfaces with the power grid through a voltage-sourced inverter. The inverter uses phase-locked-loop for synchronization and its control is usually implemented in the dq frame. If an

Modeling and Stability Analysis of Inverter-Based Resources

256

Figure 9.24: FFT results of the stator phase a current when the speed is 0.51 p.u. The 60-Hz component has a magnitude of 0.6283 p.u. while the 2-Hz component has a magnitude of 0.109 p.u. IBR’s ac side is perturbed by a voltage signal at ω p , the resulting current contains both the primary frequency ω p but also the mirror frequency (2ω0 − ω p ) (where ω0 is the synchronous frequency) [96]. Thus, a dynamic circuit is expected to describe the relationship between the coupling harmonics. 9.4.1

FROM DQ ADMITTANCE TO A TWO-PORT CIRCUIT REPRESENTATION

At an operating condition, the dq admittance of an IBR can be measured. Hence, we assume that the m (s) which represents the dq-frame current dq admittance of an IBR is available and notated as Ydq and voltage relationship. The relationship is equivalent to another relationship when two complex variables are introduced: V1 = vd + jvq and V2 = vd − jvq .      Ydd (s) Ydq (s) vd (s) id (s) =− , (9.63) iq (s) Yqd (s) Yqq (s) vq (s)    m (s) Ydq

       1 1 id j I (s) 1 j 1 j m =⇒ 1 Ydq = (s) = 1 − j iq −j I2 (s) 2 1 −j    Y (s) Y12 (s) V1 (s) = − 11 , ∗ (s) Y ∗ (s) Y12 V2 (s) 11    

m (s) Y12

It can be seen that

Y11 = and Y12 =

 1 Ydd +Yqq + j(Yqd −Ydq ) , 2  1 Ydd −Yqq + j(Ydq +Yqd ) . 2

1 j



(9.64)

Generalized dynamic circuits

257

If Ydd = Yqq and Ydq = −Yqd , the 2-by-2 matrix is diagonal or decoupled and there is no frequency coupling phenomenon. It is necessary to examine the physical meaning of the complex vectors V1 = vd + jvq and V2 = vd − jvq . The time-domain expressions of vd , vq are related to the abc-frame voltage space vector as follows. �v(t) = (vd (t) + jvq (t)) · e jωt

(9.65)

If vd (t) is a sinusoidal signal at a frequency of ω p , vd (s) with s being evaluated at jω p represents the complex Fourier coefficient or the phasor of vd (t). ∗  vd (t) = vd ( jω p )e jω p t + vd ( jω p )e jω p t ∗  vq (t) = vq ( jω p )e jω p t + vq ( jω p )e jω p t (9.66)

Hence

�v(t) = V1 (t)e jωt + [V2 (t)]∗ e jωt       v1 (t)

where

(9.67)

v∗2 (t)

V1 (t) = [vd ( jω p ) + jvq ( jω p )] e jω p t V2 (t) = [vd ( jω p ) − jvq ( jω p )] e jω p t It can be seen that in the time-domain and Laplace domain, the following relationships exist: v1 (t) = V1 (t)e jωt , v2 (t) = V2 (t)e− jωt v1 (s + jω) = V1 (s), v2 (s − jω) = V2 (s).

(9.68)

Both V1 (t) and V2 (t) are the analytic signals viewed in the dq frame. V1 ( jω p ) is the Fourier coefficient of the primary frequency component of the space vector v1 (t) at ω p + ω = ωs and V2 ( jω p )∗ is the Fourier coefficient of the mirror frequency component of the space vector v∗2 (t) at (2ω − ωs ). Based on above relationship, the following relationship can be found.      Y (s − jω) Y12 (s − jω) v1 (s) i1 (s) = − 11 ∗ (s − jω) Y ∗ (s − jω) i2 (s − j2ω) Y12 v2 (s − j2ω) 11 Fig. 9.25 presents a simple IBR grid integrated system. It can be seen that an IBR can be modeled as a two-port circuit.

Figure 9.25: A circuit representation of an IBR as a two-port component. The two-port s-domain circuit has a clear physical meaning with its terminal variables related to the harmonic signals in the physical circuit.

Modeling and Stability Analysis of Inverter-Based Resources

258

Ldc V

Cdc

idc + Vdc -

it Rf

Lf

vt m

R

L

vg

Cf

δ

Figure 9.26: Topology of an IBR grid integration system. 9.4.2

TWO-PORT CIRCUIT REPRESENTATION BY DERIVATION

A test case of an IBR grid integration system is examined for its two-port circuit representation. Fig. 9.26 shows the testbed. The dc-side of the IBR is represented by an ideal constant voltage source behind an inductor (1mH) and a dc-link capacitor of 7500 μF. The measured dc voltage is the voltage across the capacitor, and the measured dc current is the current flowing into the inverter. The ac side has an RLC filter and an RL transmission line connected with an infinite bus. The converter is assumed to have only the modulation order given. The modulation vector, based on the grid frame, is m δ . Therefore, the converter’s voltage and the dc-link voltage have the following relationship: Vt =

Vdc Vdc m δ =m , 2 2

(9.69)

where the modulation index m = m δ = m cos δ + jm sin δ . To convert every variable to per unit variable, we assume, Vdcbase = 2Vacbase , where Vac refers to the amplitude of the per-phase ac voltage. ∗ Since Vdc Idc = 32 V t I , we can form the dc side current base as 3 Idcbase = Iacbase . 4 The ac circuit relationship is first presented. For simplicity, the effect of the shunt capacitor filter is ignored. The total ac side resistance and the inductance are defined as Rtot = Rf + R, Ltot = Lf + L. For the ac circuit , the voltage and current relationship is as follows. �vt −�vg = (Rtot + sLtot )�it

(9.70)

where, �vt = V t e jω0 t , �vg = V g e jω0 t , �it = I t e jω0 t . Hence, the relationship of the phasors is as follows. V t (s) −V g (s) = (Rtot + (s + jω0 )Ltot ) I t (s)

∗ ∗ V t (s) −V g (s)

= (Rtot + (s −

∗ jω0 )Ltot ) I t (s)

(9.71) (9.72)

The dc side contains an ideal dc voltage source, a series inductor, and a dc-link capacitor. From these components, we can express the dc side relationship as, Vdc (s) = Vth (s) − Idc (s)Zdc (s)

(9.73)

where Vth (s) is the Th´evenin equivalent voltage viewed from the dc-bus to the dc side and Zdc is the equivalent impedance of the series inductor and the dc-link capacitor. Zdc (s) = Ldc s||

1 Cdc s

Generalized dynamic circuits

259

In per unit, the dc voltage and the ac voltage are related as follows: �vt =m Vdc e jω0 t , =⇒ V t = m Vdc . �vt∗

− jω0 t

∗

=m Vdc e

,

∗ =⇒ V t

(9.74)

∗

= m Vdc .

(9.75)

In the Laplace domain, the ac voltage space vectors, phasors are associated with the dc voltage as follows: �vt (s + jω0 ) = V t (s) = m Vdc (s) �vt∗ (s −

∗ jω0 ) = V t (s)

(9.76)

∗

= m Vdc (s)

(9.77)

Since Pdc = Pac , based on the relationship in (9.74) and (9.75), we can form the relationship between the currents in dc and ac side as   Vdc Idc = 0.50 �vt · (�it )∗ + (�vt )∗ ·�it   ∗ = 0.5 mVdc e jω0 t · I t e− jω0 t + m∗Vdc e− jω0 t · I t e jω0 t   ∗ = 0.5 mVdc · I + m∗Vdc · I t ∗

=⇒ 2Idc = m I t + m∗ I t .

(9.78)

Based on the dc voltage (current) and ac voltage (current) relationship, we can express the equivalent circuit as a transformer connected between the dc and ac voltage with modulation index, m as the turns ratio. The dc current may be split to mI¯t∗ and m∗ I¯t . The circuit is shown in Fig. 9.27(a). Since (9.78) shows the dc current is doubled, we need to modify the dc impedance to accommodate the change so that we can rewrite (9.73) as, Vdc (s) = Vth (s) − 2Idc (s)

Zdc (s) 2

(9.79)

Fig. 9.27(a) expresses the modulation index as a phasor; it is always preferred to have the transformation ratio as a simple real value. To achieve that, we can designate the converter voltage phase angle as zero while the grid voltage has an angle of −δ . Hence: V t −δ −V g −δ = (Rtot + jω0 Ltot )I t −δ ∗ Vt

δ

∗ −V g

δ = (Rtot −

∗ jω0 Ltot )I t

δ

(9.80) (9.81)

Fig. 9.27(a) can be reformed as Fig. 9.27(b). To further simplify the circuit, we remove the two transformers in Fig. 9.27(b). The resulting circuit is Fig. 9.27(c). It can be seen that Fig. 9.27(c) represents the steady-state equivalent circuit for a balanced system, where Zdc = 0 and frequency of operation for V g is 60 Hz. 9.4.2.1

Negative-sequence circuit

Under an unbalanced condition, a second harmonic component will be created in the instantaneous power in the ac side. Based on energy conservation, the dc side also has the 2nd harmonics. We may express the 2nd harmonic component of the dc-link voltage as ∗

vdc2 (t) = V dc,+2 e j2ω0 t +V dc,+2 e− j2ω0 t

(9.82)

In the ac side, the two components are manifested as a 3rd harmonic component and a negativesequence component: ∗

m e jω0 t vdc2 = mV dc,+2 e j3ω0 t + mV dc,+2 e− jω0 t .

(9.83)

Modeling and Stability Analysis of Inverter-Based Resources

260 jω0L

R

Vt

V*t

It

-jω0L I*t

0.5Zdc

Vg

2Idc

R

Vg

VTH 1 : m -δ

m δ :1 ;ĂͿ R

jω0L

Vt

Vt

It 2Idc

m :1

R

jω0L

R

I*t

0.5Zdc

Vg -δ

Vg δ

VTH 1:m

;ďͿ Vt

Vt

-jω0L

R

I*t

It

Vg -δ

-jω0L

2

0.5m Zdc

2Idc m

mVTH

Vg δ

;ĐͿ

Figure 9.27: Equivalent circuit for the positive-sequence components. The converter voltage can be represented as, �vt = V t,+1 e jω0 t +V t,−1 e− jω0 t +V t,+3 e j3ω0 t

(9.84)

where ∗

V t,−1 = m ·V dc,+2 V t,+3 = m ·V dc,+2

(9.85) (9.86)

In the above equations, (+2) represents the second harmonic, (−1) represents the negative sequence component, and (+3) represents the positive sequence third harmonics components. In turn, the ac current now has a positive-sequence, a negative-sequence, and a 3rd harmonic component. �it = I t e jω0 t + I t e− jω0 t + I t e j3ω0 t ,+1 ,−1 ,+3

(9.87)

Based on the power balance relationship, it can be seen that the 2nd harmonics components in the dc power and the ac power should be the same. Assume that the dc-link voltage ripple is very small compared to its dc component. Hence, the dc-side 2nd harmonic component in the instantaneous power is:   ∗ pdc2 = Vdc idc2 = Vdc I dc,+2 e j2ω0 t + I dc,+2 e− j2ω0 t (9.88)

Generalized dynamic circuits

R

j3ω0L

261

Vt,+3

V*t,-1

Vdc,+2

R

I*t,-1

It,+3

Idc,+2

0.5Zdc(j2ω0)

m :1 R

jω0L

Vt,+3

j3ω0L

V*g,-1

1:m V*t,-1

jω0L

It,+3

R

I*t,-1 Idc,+2 m

0.5m2Zdc(j2ω0)

V*g,-1

Figure 9.28: Negative-sequence equivalent circuit for the hybrid dc/ac circuit. In the ac side, the harmonic components in the ac voltage are assumed to be very small compared to the fundamental component. Hence, the ac-side 2nd harmonic component in the instantaneous power is:   ∗ ∗ pac2 = Real V t,+1 I t,+3 +V t,+1 I t,−1   ∗ (9.89) = Real m∗Vdc I t,+3 + mVdc I t,−1 Hence, the relationship between the dc and ac side currents is: ∗

2I dc,+2 = m I t,−1 + m∗ I t,+3 .

(9.90)

Based on the dc/ac voltage and current relationship, an equivalent circuit representing the 120Hz dc component vs. the ac side 180-Hz component and −60-Hz component has been derived and presented in Fig. 9.28. 9.4.2.2

The generalized circuit

Based on the above analysis, generalized circuits can be found based on either the positive-sequence steady-state circuit, or the negative-sequence steady-state circuit. The generalized circuit is shown in Fig. 9.29. Fig. 9.29(a) views the circuit based on the dc side, while Fig. 9.29(b) views the circuit based on the ac side. When s is substituted by 0, Fig. 9.29(a) is the positive-sequence circuit. When s is substituted by j2ω0 , Fig. 9.29(a) is the negative-sequence circuit. When s is substituted by − jω0 , Fig. 9.29(b) is the positive-sequence circuit. When s is substituted by jω0 , Fig. 9.29(b) is the negative-sequence circuit. If the dc-system has an oscillation component at 10 Hz, the ac system will show +70 Hz and +50 Hz oscillation components. In the generalized circuit representation Fig. 9.29(a), s should be substituted by j2π × 10 rad/s to accurately reflect the frequency coupling. On the other hand, in the generalized circuit representation Fig. 9.29(b), s should be substituted by − j2π × 50 rad/s to accurately reflect the frequency coupling. While the circuit representation appears simple, the physical meaning of the ac components needs careful contemplation.

Modeling and Stability Analysis of Inverter-Based Resources

262

R

(s+jω0)L

vt(s+jω0)

it(s+jω0) vg (s+jω0)

v*t(s-jω0)

(s-jω0)L

R

i*t(s-jω0) 0.5m Zdc(s) 2

Idc(s) m

v*g (s-jω0)

mVTH(s)

;ĂͿ R

it(s+j2ω0) vg (s+j2ω0)

v*t(s)

(s+jj2ω0) (s+j2ω0)Lvt(s+

Idc(s+jω0) m

sL

R

i*t(s) 0.5m Zdc(s+jω0) 2

mVTH(s+jω0)

v*g (s)

;ďͿ

Figure 9.29: Generalized equivalent circuits. Using the dc system view circuit, we can find that the ac impedance viewed from the dc side becomes:  −1 1 2 1 + . (9.91) m2 Zac (s + jω0 ) Zac (s − jω0 ) 9.4.2.3

EMT model and validation

We use the time-domain simulation results from an EMT testbed to validate the circuit model described in the previous section. The EMT model is built in MATLAB SPS and shown in Fig. 9.30. A simple IBR system is built as an open-loop model with the inverter controlled by a constant modulation index, m. (9.30) shows the topology of the EMT model. The converter is operating at the constant modulation mode. The modulation’s magnitude m = 0.95 while the angle is 20◦ .

Figure 9.30: The EMT testbed.

Generalized dynamic circuits

263

Table 9.1 Parameters of the EMT testbed Parameters Power base ac grid voltage base dc voltage source dc side inductor Ldc and capacitor Cdc choke filter RL parameters 400-V/13.2-kV transformer impedance 13.2-kV line impedance

Values 1 MW 400 V, 13.2 kV 1025.62 V 0.001 H, 0.4 F 0.1 Ohm, 46 × 10−6 H 0.02 + j0.0721 pu 0.005 + j0.5 pu

Table 9.1 lists the parameters of the EMT testbed. At t = 2s, a three-phase voltage source is superimposed on the nominal frequency voltage source. The amplitude of the voltage source is 0.2 pu while the frequency of the voltage source is set to 70 Hz or 80 Hz. Fig. 9.31 shows the simulation results of 80-Hz perturbation in the ac voltage. The plots show the dc-link voltage, the dc current to the inverter, the grid current, the converter voltage, and the grid voltage. It can be seen that the dc voltage and dc current have 20-Hz oscillations. Fig. 9.32 shows the dc-link voltage and dc current for 70-Hz and 80-Hz injections. The harmonic components are listed in Table 9.2. Analysis based on the generalized circuit leads to the similar results with negligible errors. The MATLAB codes are shown below. In this code, the 80-Hz injection is examined.

Figure 9.31: The EMT testbed’s simulation results.

Modeling and Stability Analysis of Inverter-Based Resources

264

Figure 9.32: The EMT testbed’s simulation results. omega = 2*pi*60; pp = 0.2; fp = 80; f = fp -60; s = 1i*2*pi*f; % dc side parameters P_base = 1e6; % Vdc_base = 400/sqrt(3)*sqrt(2)*2; Idc_base = P_base/Vdc_base; Zdc_base = Vdc_base/Idc_base; Vb1 = 400; vb1 = Vb1/sqrt(3)*sqrt(2); Vb2 = 13.2e3; vb2 = Vb2/sqrt(3)*sqrt(2); Zb1 = Vb1ˆ2/1e6; Zb2 = Vb2ˆ2/1e6; Iac_base = Vb1*sqrt(2/3)/Zb1; L_dc =1e-3/Zdc_base;

Table 9.2 Harmonic components Experiment dc side harmonic dc voltage harmonic magnitude dc current harmonic magnitude ac current supersynchronous magnitude ac current subsynchronous magnitude converter voltage magnitude

70-Hz 10 Hz 29.86 V 394 A 0.265 pu 0.0015 pu 0.0217 pu

80-Hz injection 20 Hz 91.91 V 268.8 A 0.194 pu 0.0997 pu 0.066 pu

Generalized dynamic circuits

265

C_dc =0.04*Zdc_base; Vdc_source = 1025.6215/Vdc_base; Zdc = 1/(1/(L_dc*s)+(C_dc*s)) m = 0.95; Vg_80Hz = pp; Xgrid_pu = 0.05; Rg = 0.1/Zb1+0.02+Xgrid_pu/5; Lg = 46e-6/Zb1+0.0721/omega+Xgrid_pu/omega; Z_left = Rg + (s+1i*omega)*Lg Z_right = Rg + (s-1i*omega)*Lg Y_total = 1/Z_left+1/Z_right+1/(0.5*mˆ2*Zdc); Z_total = 1/Y_total Vt = 1/Y_total*(Vg_80Hz/Z_left) Vt = 0.0628 + 0.0268i I_right = Vt/Z_right; I_left = (Vg_80Hz-Vt)/Z_left; Vdc = Vt/m; Idc = Vdc/Zdc; [abs(Vdc)*Vdc_base, abs(Idc)*Idc_base] ans = 1x2 46.9391

137.5879

[abs(I_right), abs(I_left), abs(Vt)] ans = 1x3 0.1015

0.1932

0.0683

It has to be noted that for the dc voltage and dc current harmonic component, the values from the analysis are half of those from FFT. This is due to the fact that the FFT gives the magnitude of a real signal, while the analysis gives the magnitude of an analytical signal. For example, FFT shows that the magnitude of a 20-Hz signal vˆ cos(2π × 20t) is v. ˆ On the other hand, in this analysis, our definition of phasors is based on the analytical signals: vˆ cos(2π × 20t) = In the computing codes,

vˆ 2

vˆ j2π×20t vˆ − j2π×20t + ·e . ·e 2 2

will be identified as the magnitude of the 20-Hz harmonic.

Modeling and Stability Analysis of Inverter-Based Resources

266

9.5

SUMMARY AND NOTES

In this chapter, we present our most recent research on generalized dynamic circuit building. From state-space modeling building to generalized dynamic circuit building, it took us a decade. Dynamic model building starts from ordinary differential equations (ODE). While working in the area of type3 wind farm SSR and IBR impedance modeling in frequency domain, it is natural to ask whether we can find the connection between the two. For three-phase power systems, dynamic models are usually built in dq frames. On the other hand, when viewing a circuit, we think about abc instantaneous voltage and current, or phasors in harmonic domains. After reading many papers in the power electronics area, where people have spent time dealing with dq domain and sequence domain impedance, we came to realize that we can do more to bridge the gap between dynamic modeling and circuit analysis. Our contributions are twofold. We establish the generalization procedure and expand the steady-state circuits well known in the power system analysis arena, e.g., the Steinmetz’s circuit, the interconnected sequence networks for fault analysis, into the entire frequency domain. We also establish the powerful applications based on generalized dynamic circuits, including steady-state analysis, harmonic analysis, and dynamic analysis. This step is a big move. We continued the work by Heaviside and Steinmetz, the two brilliant power engineers a century ago, and worked out something insightful. Indeed, operational calculus and the Laplace transform variable s fascinate us.

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Index ac voltage control (AVC), 85 admittance, 37 analytical model of a PLL, 136 analytical model of grid dynamics, 136 analytical models, 25, 106, 187 average models, 16 bandwidth, 80, 90, 100 block diagram, 36 block diagrams, 143 Bode diagrams, 15, 29, 237 Bode stability criterion, 36 bus impedance matrix, 224 Clarke transformation, 76 closed-loop transfer function, 85 cross-coupling compensation, 88 current control implementation, 88 data-driven modeling, 44 dc voltage control (DVC), 82 delay, 98, 113 doubly-fed induction generators, 179 dq admittance, 61 droop control, 101 dynamic phasors, 252 eigenvalue analysis, 142 eigenvalues, 11, 30, 35, 200 eigenvectors, 35 electromagnetic transient (EMT), 7 EMT simulation, 96, 102, 183 Energizing, 104 Faraday’s Law, 233 feedback systems, 37 feedforward, 88 frame conversion, 32, 76 frequency scan, 61, 63 frequency-domain analysis, 28, 36 frequency-domain data fitting, 49 frequency-domain impedance, 231 frequency-domain modal analysis, 223 frequency-domain mode shape, 224, 226 frequency-domain stability analysis, 237 Frequency-power feedback system, 101 gain margin, 29 GFL control blocks, 97

GFL-IBR, 134 grey-box model identification, 47 grid forming (GFM), 168 grid strength, 25 grid-following control (GFL), 73, 106 harmonic analysis, 254 impedance, 37 induction generator effect (IGE), 179 Induction machine dynamic model, 234 induction machine steady-state circuit, 232 induction machines, 232 initialization, 193 inter-area oscillations, 228 inter-IBR oscillations, 203 inverse Park’s transformation, 19, 76 Jacobian linearization, 106 Laplace transform, 231 linear voltage expression, 121 load flow, 194 loop gain, 83, 89, 122, 130 low-pass filter (LPF), 54 meshed networks, 203 modal analysis, 35 mode shape, 35 model reduction, 51 negative impedance, 119 negative sequence, 234 nonlinear analytical models , 134 nonlinear model, 118 Nyquist diagrams, 237 Nyquist stability criterion, 36 operational calculus, 231 oscillations, 113 Pad´e approximation, 98 Park’s transformation, 18 per unit, 5, 20 phase-locked loop (PLL), 26, 27, 53 plant-level control, 98, 113 PLL, 117, 119 PLL closed-loop transfer function, 122 PLL design, 94 271

INDEX

272

PLL weak grid stability, 127 poles, 30 power plant control, 73 proportional resonant (PR), 156 Proportional resonant (PR) control, 91 reactive power control, 80 real power control, 80 Routh-Hurwitz stability criterion, 30 second-order generalized integrator (SOGI)-PLL, 53 sequence networks, 240 sequence-domain admittance, 63 short circuit ratio, 25 simplified model, 252 simplified models, 106 single-phase PLL, 27 slip, 231 small-signal model, 121 space vector, 20, 31, 135, 233 SSR, 179 stability analysis, 28, 145 Start-up transients, 104 state-space modeling, 135 state-space models, 31, 37, 187

stationary frame, 156 steady-state computation, 193 Steady-state impedance, 231 Steinmetz, 232 step excitation, 54 subsynchronous resonance (SSR), 183, 200, 242 symmetrical component theory, 240 synchronization, 100, 103 synchronous machine circuit, 247 synchronous machine startup, 252 synchronous machines, 246 system identification, 44, 145 the root-locus diagram, 41 time-domain data fitting, 54 transfer function, 31 transfer function estimation, 47, 54 two-port circuit, 255 type-3 wind turbines, 179 unbalance, 231, 240, 242, 246 virtual synchronous machines (VSM), 176 voltage stability, 108 voltage synchronization, 103 voltage-reactive power feedback system, 101