Power System Simulation Using Semi-Analytical Methods E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 462

Veröffentlichungsjahr: 2023

Beschreibung

POWER SYSTEM SIMULATION USING SEMI-ANALYTICAL METHODS Robust coverage of semi-analytical and traditional numerical methods for power system simulation In Power System Simulation Using Semi-Analytical Methods, distinguished researcher Dr. Kai Sun delivers a comprehensive treatment of semi-analytical simulation and current semi-analytical methods for power systems. The book presents semi-analytical solutions on power system dynamics via mathematical tools, and covers parallel contingency analysis and simulations. The book offers an overview of power system simulation and contingency analysis supported by data, tables, illustrations, and case studies on realistic power systems and experiments. Readers will find open-source code in MATLAB along with examples for key algorithms introduced in the book. You'll also find: * A thorough background on power system simulation, including models, numerical solution methods, and semi-analytical solution methods * Comprehensive explorations of semi-analytical power system simulation via a variety of mathematical methods such as the Adomian decomposition, differential transformation, homotopy analysis and holomorphic embedding methods * Practical discussions of semi-analytical simulations for realistic large-scale power grids * Fulsome treatments of parallel power system simulation Perfect for power engineers and applied mathematicians with an interest in high-performance simulation of power systems and other large-scale network systems, Power System Simulation Using Semi-Analytical Methods will also benefit researchers and postgraduate students studying power system engineering.

Leseprobe

Table of Contents

Cover

Table of Contents

Title Page

Copyright

About the Editor

List of Contributors

Preface

1 Power System Simulation: From Numerical to Semi‐Analytical

1.1 Timescales of Simulation

1.2 Power System Models

1.3 Numerical Simulation

1.4 Semi‐Analytical Simulation

1.5 Parallel Power System Simulation

1.6 Final Remark

References

2 Power System Simulation Using Power Series‐Based Semi‐Analytical Methods

2.1 Power Series‐Based SAS for Simulating Power System ODEs

2.2 Power Series‐Based SAS for Simulating Power System DAEs

2.3 Adaptive Time‐Stepping Method for SAS‐Based Simulation

2.4 Numerical Examples

References

3 Power System Simulation Using Differential Transformation Method

3.1 Introduction to Differential Transformation

3.2 Solving the Ordinary Differential Equation Model

3.3 Solving the Differential‐Algebraic Equation Model

3.4 Broader Applications

3.5 Conclusions and Future Directions

References

4 Accelerated Power System Simulation Using Analytic Continuation Techniques

4.1 Introduction to Analytic Continuation

4.2 Finding Semi‐Analytical Solutions Using Padé Approximants

4.3 Fast Power System Simulation Using Continued Fractions

4.4 Conclusions

References

5 Power System Simulation Using Multistage Adomian Decomposition Methods

5.1 Introduction to Adomian Decomposition Method

5.2 Adomian Decomposition of Deterministic Power System Models

5.3 Adomian Decomposition of Stochastic Power System Models

5.4 Large‐Scale Power System Simulations Using Adomian Decomposition Method

References

6 Application of Homotopy Methods in Power Systems Simulations

6.1 Introduction

6.2 The Homotopy Method

6.3 Application of Homotopy Methods to Power Systems

6.4 Multimachine Simulations

6.5 Application of Homotopy for Error Estimation

6.6 Summary

References

7 Utilizing Semi‐Analytical Methods in Parallel‐in‐Time Power System Simulations

7.1 Introduction to the Parallel‐in‐Time (Parareal Algorithm) Simulation

7.2 Examination of Semi‐Analytical Solution Methods in the Parareal Algorithm

7.3 Numerical Case Study

7.4 Conclusions

References

8 Power System Simulation Using Holomorphic Embedding Methods

8.1 Holomorphic Embedding from Steady State to Dynamics

8.2 Generic Holomorphic Embedding for Dynamic Security Analysis

8.3 Extended‐Term Hybrid Simulation

8.4 Robust Parallel or Distributed Simulation

References

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1 Constants and variables of generator models.

Table 1.2 Constants and variables of governor and turbine models.

Table 1.3 Constants and variables of excitation system models.

Chapter 2

Table 2.1 Coefficients of in .

Table 2.2 Stability analysis of 10 faults on the IEEE 9‐bus system.

Table 2.3 Largest error of angle differences (in degree) by SAS.

Table 2.4 Stability analysis of 20 selected faults on the WECC 179‐bus syst...

Table 2.5 Largest absolute SAS errors in time domain.

Table 2.6 Time cost for deriving higher order SASs of ‐bus system.

Table 2.7 Comparison of time cost between SAS and FE on polish ‐bus system...

Chapter 3

Table 3.1 Comparison of numerical stability under three scenarios.

Table 3.2 Comparison of maximum time step length to maintain the numerical ...

Table 3.3 Comparison of time step length under different error tolerances (...

Table 3.4 Comparison of computation time under different error tolerances (...

Table 3.5 Comparison of accuracy and time performance.

Table 3.6 Comparison of number of LU factorization.

Chapter 4

Table 4.1 A portion of Padé table for the exponential function exp(

x

).

Table 4.2 Comparison of simulation between CF‐SAS and forward Euler.

Table 4.3 Total time cost between CF‐SAS and PS‐SAS.

Table 4.4 Time cost using ideal parallel computation (compared with 58.03 s...

Chapter 5

Table 5.1 Parameters of the SMIB system.

Table 5.2

T

max

vs. time constants of the system.

Table 5.3 The number of computing units for the fourth‐order model system....

Table 5.4 The number of computing units for the sixth‐order model system.

Table 5.5 Time performance on the fourth‐order model system.

Table 5.6 Time Performance on the sixth‐order model system.

Table 5.7 Influence of parallel capability on time performance.

Table 5.8 Time performance comparison of stochastic load cases

Chapter 6

Table 6.1 MHAM terms for the mechanical equations of the generator.

Table 6.2 MHAM terms for the field and damper windings.

Table 6.3 MHAM terms for the second damper windings.

Table 6.4 Execution time (s) for all test systems [11].

Table 6.5 Maximum time step, execution time and maximum errors [11].

Table 6.6 Execution time (s) for each network [23].

Chapter 7

Table 7.1 Comparison of the multistage Adomian decomposition method (MADM) ...

Table 7.2 Number of Parareal iterations for the New England system with dif...

Table 7.3 Number of Parareal iterations for the Polish system with differen...

Table 7.4 Number of Parareal iterations for the Polish system with differen...

Table 7.5 The full Eastern Interconnection simulation result (s) with the h...

Table 7.6 The number of iterations and approximate CPU time in the 2383‐bus...

Chapter 8

Table 8.1 Accuracy and efficiency of HE‐I in IEEE 14‐bus system case.

Table 8.2 Performance of holomorphic embedding in static VSA.

Table 8.3 Computational efficiency comparison in NPCC system.

Table 8.4 Computational efficiency comparison in Polish system.

Table 8.5 Holomorphic embedding coefficients for elementary operations and ...

Table 8.6 Computation time of different methods in IEEE 39‐bus system.

Table 8.7 Holomorphic embedding coefficients (power series) of rotor speed ...

Table 8.8 Connections between main system and lower‐level systems.

Table 8.9 Computation time of 100 analysis on 21 447‐bus system.

List of Illustrations

Chapter 1

Figure 1.1 Categories of power system simulation by timescale.

Figure 1.2 Components of a bulk power system model.

Figure 1.3 Governor and turbine models.

Figure 1.4 IEEE Type 1 speed‐governor model.

Figure 1.5 Linear excitation system with a power system stabilizer.

Figure 1.6 IEEE Type 1 exciter.

Figure 1.7 Composite load model.

Figure 1.8 Classical power system model.

Figure 1.9 Comparison of the forward Euler method, modified Euler method, se...

Figure 1.10 Illustration of the backward Euler and Trapezoidal‐rule methods....

Figure 1.11 Illustration of the analytical homotopy approach.

Figure 1.12 Illustration of a semi‐analytical solution approach.

Figure 1.13 Illustration of the parareal algorithm.

Chapter 2

Figure 2.1 One‐line diagram of the IEEE 9‐bus system.

Figure 2.2 Comparison of fault‐on trajectories of 9‐bus system: SAS vs. RK4....

Figure 2.3 Comparison of extended fault‐on trajectories of ‐bus system.

Figure 2.4 One‐line diagram of the WECC 179‐bus system.

Figure 2.5 Fault‐on trajectories of 179‐bus system: SAS vs. RK4.

Figure 2.6 Flow chart of SAS‐based simulation of power system DAEs using the...

Figure 2.7 Typical curve of the error‐rate upper bound.

Figure 2.8 Closed‐form solution compared with nd and th order SASs.

Figure 2.9 Error curves of RK4, BDF, and rd order SAS.

Figure 2.10 Adaptive time steps by BDF and SASs with orders from 3 to 8.

Figure 2.11 Simulation results of SAS‐based approach and the FE method. (a) ...

Figure 2.12 Adaptive time windows determined in the SAS‐based simulation....

Figure 2.13 Simulated rotor speed by FE with 2 ms fixed time window.

Chapter 3

Figure 3.1 39‐bus system under a stable contingency. (a) Rotor angles and (b...

Figure 3.2 39‐bus system under a marginal stable contingency. (a) Rotor angl...

Figure 3.3 39‐bus system under marginally unstable contingency. (a) Rotor an...

Figure 3.4 2383‐bus system under a stable contingency. (a) Rotor angles and ...

Figure 3.5 2383‐bus system under a marginally stable contingency. (a) Rotor ...

Figure 3.6 2383‐bus system under a marginally unstable contingency. (a) Roto...

Figure 3.7 Accuracy of the differential transformation method when faults oc...

Figure 3.8 Accuracy of the differential transformation method when faults oc...

Figure 3.9 Accuracy of the differential transformation method with and witho...

Figure 3.10 Accuracy of the differential transformation method with and with...

Figure 3.11 Error propagation for the three scenarios with different time st...

Figure 3.12 Recursive process to solve power series coefficients.

Figure 3.13 Trajectories of the stable scenario for the 2383‐bus system. (a)...

Figure 3.14 Trajectories of the unstable scenario for the 2383‐bus system. (...

Figure 3.15 Robustness against different disturbances.

Figure 3.16 Robustness against different percentages of constant power load....

Chapter 4

Figure 4.1 Domains of Convergence of

f

1

(

s

),

f

2

(

s

), and

f

3

(

s

).

Figure 4.2 Demonstration of the process of online stage.

Figure 4.3 Comparison of Taylor series and Padé appro...

Figure 4.4 Comparison of time intervals between Taylor...

Figure 4.5 Comparison of curves between 5th‐order Taylor series and 5th‐orde...

Figure 4.6 Map of IEEE 10‐generator, 39‐bus New England system.

Figure 4.7 Curves of SASs based on the 5th‐order Padé and...

Figure 4.8 Curves of SASs based on the 5th‐order Padé & Tay...

Figure 4.9 Time step of IEEE 10‐generator, 39‐bus New England system with th...

Figure 4.10 Flow chart of the proposed CF‐SAS approach.

Figure 4.11 Partitioned method for solving power system Differential‐algebra...

Figure 4.12 Flow chart of determination of length of time interval.

Figure 4.13 Simulation results of CF‐SAS (adaptive time interval) and the re...

Figure 4.14 (a) Time interval and (b) maximum error of voltage using the 3‐,...

Figure 4.15 Time interval using the 4‐order CF‐SAS with different inertia pa...

Chapter 5

Figure 5.1 Comparison of semi‐analytical solutions with numerical results.

Figure 5.2 Different terms of the semi‐analytical solutions and the time win...

Figure 5.3 Relationships between

T

max

,

T

2

,

T

1,

and

H

3

.

Figure 5.4

T

max

′

s with respect to selected

H

3

′

s.

Figure 5.5 Using an initial state with

ω

(0)...

Figure 5.6 Using an initial state with

w

(0) = 1.38...

Figure 5.7 Flowchart of the semi‐analytical scheme for power system simulati...

Figure 5.8 IEEE 10‐generator 39‐bus system.

Figure 5.9 Comparison of the simulation results given by the R‐K 4 and the t...

Figure 5.10 Comparison of the simulation results given by the R‐K 4 and the ...

Figure 5.11 Comparison of the simulation results of rotor speeds given by th...

Figure 5.12 Comparison of the simulations using the sixth‐order generator mo...

Figure 5.13 Estimation of

I

D,max

. (a) Contingency 1, (b) Contingency 2.

Figure 5.14 Comparison of rotor angles given by the R‐K 4 and the three‐term...

Figure 5.15 The comparison of adaptive changing of time window length.

Figure 5.16 Comparison of the simulation results with a topology change at

t

Figure 5.17 Single‐machine infinite bus system with constant impedance load ...

Figure 5.18 Semi‐analytical stochastic simulation results of generator 1 rot...

Figure 5.19 Mean, variance, and skewness comparisons of the generator 1 roto...

Figure 5.20 Semi‐analytical stochastic simulation results of generator 1 rot...

Figure 5.21 Mean, variance, and skewness comparisons of the generator 1 roto...

Figure 5.22 Results of the semi‐analytical stochastic simulation without con...

Figure 5.23 Results of the semi‐analytical stochastic simulation with the EM...

Figure 5.24 ADM maximum rotor angle error w.r.t Trapezoidal Method for IEEE ...

Chapter 6

Figure 6.1 Stability of Homotopy Method with Forward Euler, Modified Euler, ...

Figure 6.2 Error of Homotopy Method with respect to Forward Euler, Modified ...

Figure 6.3 Second order system step response

w

0

= 314...

Figure 6.4 The prefault, fault, and post fault circuits.

Figure 6.5 Swing Curve for a SMIB, 3‐phase midline fault cleared by tripping...

Figure 6.6 The pre‐ and post‐switch on circuits.

Figure 6.7 The pre‐fault, fault, and post‐fault SMIB networks.

Figure 6.8 Generator Response to a Line Switching Operation, SMIB with IEEE ...

Figure 6.9 Generator Response to a self‐cleared Line Fault, SMIB with IEEE 1...

Figure 6.10 3 Term MHAM application to Power System Stability Computation.

Figure 6.11 Impact of Number of Terms, of Gen‐10, 10 Generator 39 Bus Syst...

Figure 6.12 (a) δ and

E

fd

response of Generator‐9 for Self Clearing 3...

Figure 6.13 Maximum Rotor Angle Error using MHAM w.r.t. Midpoint Trapezoidal...

Figure 6.14 Maximum Rotor Angle Error using MHAM w.r.t. Midpoint Trapezoidal...

Figure 6.15 Maximum Rotor Angle Error w.r.t. Midpoint Trapezoidal Method for...

Figure 6.16 Response of Gen‐7 for the Respective Maximum Time Steps, Sel...

Figure 6.17 Flowchart of Adaptive Modified Euler.

Figure 6.18 Flowchart of Adaptive Modified Euler with MHAM.

Figure 6.19 LTE of 10 Generator 39 Bus System, 4 cycle self cleared fault....

Figure 6.20 LTE of Polish System, 4 cycle self cleared fault.

Figure 6.21 LTE of PEGASE9241 System, 4 cycle self cleared fault.

Figure 6.22 LTE of PEGASE13659 System, 4 cycle self cleared fault.

Figure 6.23 of Generator‐1, 10 Generator 39 Bus System, 4 cycles self clea...

Figure 6.24 of Generator‐10, Polish System, 4 cycles self‐cleared fault at...

Figure 6.25 of Generator‐2, PEGASE9241 System, 4 cycles self‐cleared fault...

Figure 6.26 of Generator‐1, PEGASE13659 System, 4 cycles self‐cleared faul...

Figure 6.27 Relative Speedup of MHAM‐assisted Adaptive ME over Adaptive ME....

Chapter 7

Figure 7.1 Graphical illustration of the Parareal algorithm.

Figure 7.2 MADM and MHAM results followed by the three‐phase fault at bus 1 ...

Figure 7.3 MADM and MHAM results followed by the three‐phase fault at bus 1 ...

Figure 7.4 Differential transformation results followed by the 3‐phase fault...

Figure 7.5 High performance computing result for trajectories of state varia...

Figure 7.6 Rotor angle trajectories in the Polish system using the different...

Figure 7.7 The window size and the number of iterations of each window in th...

Figure 7.8 The time step length and order of the differential transformation...

Chapter 8

Figure 8.1 Equivalent circuit of induction motor.

Figure 8.2 Flowchart of partial‐QSS VSA.

Figure 8.3 Imbalance of power flow equations of HE‐II.

Figure 8.4 Illustration of induction motor steady‐state solutions (denoted b...

Figure 8.5 Slip of motor on bus 4 in IEEE 14‐bus system. The horizontal line...

Figure 8.6 Difference of voltage between HE‐III and the modified Euler metho...

Figure 8.7 Imbalance of equations: HE‐III and the modified Euler methods.

Figure 8.8 Percentage by load type on 54 load buses in NPCC system. (a) ZIP ...

Figure 8.9 Motor slips of NPCC system obtained by partial‐QSS with holomorph...

Figure 8.10 Bus voltage of NPCC system (ZIP+Motor load).

Figure 8.11 Motor slip in NPCC system (ZIP+Motor load).

Figure 8.12 Comparison between holomorphic embedding (full‐dynamic) and modi...

Figure 8.13 Comparison of steps and equation mismatch between partial‐QSS an...

Figure 8.14 Comparison of results between holomorphic embedding (full‐dynami...

Figure 8.15 Stages in transient stability analysis.

Figure 8.16 Voltage magnitude curves in IEEE 39‐bus system.

Figure 8.17 Comparison of errors in IEEE 39‐bus system.

Figure 8.18 Computation time and average stage length under different .

Figure 8.19 Comparison of robustness between HE and ME‐NR. The fault is appl...

Figure 8.20 Dynamics of Polish system simulated by holomorphic embedding (un...

Figure 8.21 Dynamics of Polish system simulated by holomorphic embedding (st...

Figure 8.22 Illustration of event‐tracking errors using holomorphic embeddin...

Figure 8.23 Flowchart of extended‐term simulation.

Figure 8.24 2‐bus test system.

Figure 8.25 Event detection time error with holomorphic embedding, modifie...

Figure 8.26 4‐bus test system.

Figure 8.27 Frequency and voltage of 4‐bus system.

Figure 8.28 Difference of frequency and voltage between holomorphic embeddin...

Figure 8.29 Difference of (a) voltage and (b) frequency between holomorphic ...

Figure 8.30 Selected system state trajectory of IEEE 39‐bus system under res...

Figure 8.31 Illustration of main system and lower‐level systems.

Figure 8.32 Separate modeling of main system and lower‐level system(s).

Guide

Cover

Table of Contents

Title Page

Copyright

About the Editor

List of Contributors

Preface

Begin Reading

Index

WILEY END USER LICENSE AGREEMENT

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Amin Moeness

Desineni Subbaram Naidu

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Jim Lyke

Hai Li

Brian Johnson

Jeffrey Reed

Diomidis Spinellis

Adam Drobot

Tom Robertazzi

Ahmet Murat Tekalp

Power System Simulation Using Semi‐Analytical Methods

Edited by Kai Sun University of Tennessee, Knoxville, TN, USA

Copyright © 2024 by The Institute of Electrical and Electronics Engineers, Inc.

All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

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About the Editor

Kai Sun is a professor at the Department of Electrical Engineering and Computer Science in the University of Tennessee, Knoxville. He received his bachelor's degree in Automation in 1999 and his PhD degree in Control Science and Engineering in 2004, both from Tsinghua University in Beijing. Before joining the university in 2012, Dr. Sun was a project manager with the Electric Power Research Institute in Palo Alto, California, from 2007 to 2012 for R&D programs in the areas of grid operations, planning, and renewable integration. Earlier, he worked as a research associate at Arizona State University in Tempe and a postdoctoral fellow at the University of Western Ontario in Canada. Dr. Sun has served in the editorial boards of IEEE Transactions on Power Systems, IEEE Transactions on Smart Grid, IEEE Open Access Journal of Power and Energy, and IEEE Access. He received the EPRI Technology Innovation Excellence Award in 2008, EPRI Chauncey Award in 2009, NSF CAREER Award in 2016, NASPI CRSTT Most Valuable Players Award in 2016, and several Best Conference Paper Awards by IEEE PES General Meetings. His research areas include power system dynamics, stability, and control.

List of Contributors

Nan Duan

Transmission Planning

Midcontinent Independent System Operator, Inc.

Carmel

IN

USA

Gurunath Gurrala

Department of Electrical Engineering

Indian Institute of Science

Bangalore

Karnataka

India

Francis C. Joseph

Department of Electrical Engineering

Indian Institute of Science

Bangalore

Karnataka

India

Chengxi Liu

School of Electrical Engineering and Automation

Wuhan University

Wuhan

China

Yang Liu

Division of Energy Systems and Infrastructure Analysis

Argonne National Laboratory

Lemont

IL

USA

Byungkwon Park

School of Electrical Engineering

Soongsil University

Dongjak‐Gu

Seoul

South Korea

Feng Qiu

Division of Energy Systems

Argonne National Laboratory

Lemont

IL

USA

Kai Sun

Department of Electrical Engineering & Computer Science

University of Tennessee

Knoxville

TN

USA

Bin Wang

Department of Electrical and Computer Engineering

University of Texas at San Antonio

San Antonio

TX

USA

Rui Yao

Division of Energy Systems

Argonne National Laboratory

Lemont

IL

USA

Preface

Power system simulation is an essential tool for investigating electrical power systems, used by researchers, engineers, and other stakeholders. It is based on a mathematical model and uses time‐domain simulation to predict the behavior of a power system under anticipated operating conditions, whether it is currently operational or undergoing a new design. Simulation results provide insights into system stability, reliability, efficiency, and the performance of new equipment and functionalities. However, traditional simulation programs are facing significant challenges in accurately and efficiently simulating increasingly complex and uncertain bulk power systems with the growing penetration of renewable energy resources, particularly when fast, real‐time power system simulation is required.

The purpose of this book is to enhance the speed of time‐domain simulations of bulk power systems by introducing a semi‐analytical methodology that integrates various simulation techniques. This new approach differs from traditional simulations that use low‐order numerical integrators. Instead, it employs a high‐order analytical form that approximates the dynamic response of the system with improved accuracy. By embedding symbolic variables and parameters, the resulting semi‐analytical solution allows for the decomposition of its computation into sequential or concurrent processes, reducing the time required for simulations. Additionally, a significant amount of computation can be performed offline before a simulation run is needed or be parallelized, making the simulations even faster. These semi‐analytical methods can be integrated or interfaced with existing numerical integrators to create more robust power system simulators that can be used with high‐performance parallel computers.

The book is structured to provide comprehensive and systematic coverage of such emerging methods for finding semi‐analytical solutions and accelerating power system simulation, as documented in the literature. Chapters 2–8 describe these methods in detail, including their accuracy and time performance, and demonstrate their applications to realistic bulk power system models. These methods are compared to traditional numerical methods, providing a thorough evaluation of their effectiveness.

In the rest of the book, Chapter 1 provides an overview of power system simulation, including timescales, models, and numerical methods of simulation. It also introduces the basic idea of a semi‐analytical solution for power systems and the fundamental approaches for semi‐analytical simulation. Furthermore, the chapter briefs readers on approaches for parallel power system simulation and discusses where a semi‐analytical method can fit.

Chapter 2 begins by presenting semi‐analytical solutions in a power series form. These solutions are derived from Taylor's series expansion of true solutions for power systems modeled by nonlinear differential equations. The chapter then extends the power series‐form solutions to differential‐algebraic equation models to accelerate simulations of large‐scale power systems.

Chapter 3 presents a systematic approach for deriving and evaluating a high‐order power series‐form semi‐analytical solution for fast power system simulation. The approach applies the Differential Transformation Method to nonlinear differential equations and algebraic equations of power system models, which are transformed into formally linear equations about power series coefficients. Solving these coefficients from low orders to high orders in a recursive manner allows growing a power series‐form semi‐analytical solution to a high order for desired accuracy.

Chapter 4 describes applications of analytic continuation to prolong the time interval of accuracy with a power series‐form semi‐analytical solution. Two methods, Padé Approximation and Continued Fractions, are applied to transform a power series‐form semi‐analytical solution into analytical, fractional forms with improved accuracy and convergence.

Chapter 5 introduces the Adomian Decomposition Method, which decomposes nonlinear functions of power system models into Adomian polynomials. This method uses a truncated sum as the semi‐analytical solution in either a power series form or a non‐power series form. The chapter also discusses applications of a semi‐analytical solution to parallel simulation, taking advantage of intrinsic parallelism in its analytical form, and to efficient stochastic simulation thanks to embedded symbolic variables.

Chapter 6 presents the Homotopy Analysis Method for semi‐analytical power system simulation. This method offers more freedom in form and algorithm for a semi‐analytical solution. By designing the homotopy characterized by an embedded variable, the method deforms a solved, linear model into the nonlinear power system model to derive the semi‐analytical solution. Additionally, the method introduces auxiliary parameters to adjust the convergence characteristics of the solution. The chapter compares this method with numerical integrators and other similar semi‐analytical methods, including the Adomian Decomposition Method.

Chapter 7 describes a Parareal algorithm enhanced by semi‐analytical methods for parallel‐in‐time power system simulation. The chapter first introduces the prediction‐correction mechanism using the coarse and fine operators of the Parareal algorithm. Then, it compares the algorithm, respectively adopting three semi‐analytical methods as the coarse operator, which are the Differential Transformation Method, Adomian Decomposition Method, and Homotopy Analysis Method described in Chapters 3, 5, and 6, respectively.

Finally, Chapter 8describes the extension of the Holomorphic Embedding Method for steady‐state and quasi‐steady‐state analyses to a semi‐analytical method for extended‐term time‐domain simulations. The method requires complex analyticity in the approximation of a power system solution by considering a complex embedding variable. The chapter also discusses the partitioning and parallelization of the Holomorphic Embedding Method for high‐performance simulations.

As emerging tools, the semi‐analytical methods described in this book suggest a new paradigm for accelerating simulations of large‐scale, bulk power systems. It is hoped that the book will inspire researchers, developers, and engineers in the power system field to develop powerful power system simulators utilizing advanced high‐performance computing technology.

I am deeply grateful to the talented contributors who have authored the chapters of this timely book. Their expertise and dedication have made this work possible. Specifically, I would like to thank my former students and postdoctoral researchers, Bin Wang, Nan Duan, Yang Liu, Nan Duan, Chengxi Liu, and Rui Yao, as well as my collaborators, Byungkwon Park and Feng Qiu, Gurunath Gurrala, and Francis C. Joseph.

My research on power system simulation has been supported by the US National Science Foundation under NSF Award Numbers EEC‐1041877 and ECCS‐1610025, as well as the US Department of Energy through Oakridge National Laboratory and Argonne National Laboratory.

This monograph is dedicated to the memory of my PhD advisor, Professor Da‐Zhong Zheng, whose knowledgeable mentoring, thoughtful guidance, and insightful encouragement proved invaluable throughout my academic journey.

Finally, I want to express my deepest gratitude to my loving family – my wife Fang and my sons Yi and Rei – who have always been there to support me, spark my curiosity with insightful inquiries, and encourage me to explore new ideas and perspectives.

Knoxville, TN, USAAugust 2023

Kai Sun

1Power System Simulation: From Numerical to Semi‐Analytical

Kai Sun

Department of Electrical Engineering & Computer Science, University of Tennessee, Knoxville, TN, USA

1.1 Timescales of Simulation

A power system is composed of a vast network of generators, loads, and control devices, each with their own intricate and diverse dynamics. Therefore, power system simulators usually focus on specific timescales or types of dynamic behaviors to simplify the modeling and computation of the simulated system. In general, there are three types of power system simulations based on their timescales, models, and primary objectives, which are electromagnetic transient simulation, transient stability simulation, and quasi‐steady‐state simulation, as illustrated in Figure 1.1.

Electromagnetic transient(EMT) simulation provides a high‐resolution solution to the three‐phase alternating current and voltage of each circuit element of a power system, typically at a microsecond scale. EMT simulations are indispensable for investigating protective actions against short‐circuit faults and electromagnetic dynamics of existing or new equipment and controllers.

Transient stability simulation, on the other hand, is concerned with the slower electromechanical dynamics of generators, motors, and their controllers. It computes the time variation of phasors of voltages and currents, which are approximate representations of periodical quantities based on a common synchronous frequency. Although this phasor approximation can introduce errors in the actual frequency deviations, it is acceptable when the frequencies are close to the synchronous frequency. Under normal or nonextreme abnormal conditions, the frequencies at buses of a power transmission system must not deviate by more than 1 Hz before triggering an under‐ or over‐frequency protection action. In each transient stability simulation, phasors and the system state are computed typically at a time step of milliseconds to cycles, which is sufficient to capture authentic electromechanical dynamics and much larger than the microsecond‐level time step of EMT simulation. Transient stability simulation is a crucial tool in power system studies that support grid operations and planning. It is often an essential functionality of the Energy Management System (EMS) used by electric utilities and regional transmission organizations. It is a critical component of dynamic security assessment (DSA) programs that evaluate the system's angular stability, frequency regulation, and postfault voltage recovery, all of which are essential for ensuring the reliable and secure operation of the power system.

Figure 1.1 Categories of power system simulation by timescale.

Quasi‐steady‐state (QSS) simulation studies how a power system behaves under slowly changing operating conditions, such as load variations and generation redispatches, at a low resolution from tens of seconds to minutes. It is important for analyzing steady‐state generation, load and power‐flow controls, long‐term voltage stability, and the early stages of cascading outages. In such simulations, the algebraic power‐flow equations form the core of the simulation model, and the dynamics of the system are either ignored or significantly simplified with differential equations.

This book focuses on accelerating transient stability simulation of electromechanical dynamics in power systems. While fast EMT dynamics are not covered, the simulation of QSS behaviors in a power system is also studied in Chapter 8, which combines transient stability simulation and QSS simulation into extended‐term simulations. While many methods introduced in later chapters are primarily geared toward transient stability simulation, they can also be applied to EMT and QSS simulations. This is because all three types of simulation use nonlinear differential and algebraic equations for their models, and the semi‐analytical methods introduced in the book are applicable to these equations.

The remainder of this chapter provides an introduction to power system modeling for transient stability simulation, including mathematical models of basic power system components. Power system simulation is formulated as an initial value problem based on the simulation model. Two approaches for resolving the problem are described: the conventional numerical solution approach and the emerging semi‐analytical solution approach. Additionally, the chapter covers parallel power system simulation.

It should be noted that this chapter does not aim to provide a comprehensive introduction to power system modeling for transient stability simulation. Rather, it provides the minimum background information for readers to follow the formulations and methods for power system simulations in the rest of the book. Readers who are interested in more details on power system modeling and simulations may refer to books by Kundur [1], Padiyar [2], and Anderson and Fouad [3].

1.2 Power System Models

1.2.1 Overview

1.2.1.1 Simplifying a Power System Model

If the EMT dynamics, electromechanical dynamics, and QSS behaviors of a power system are to be considered, the resulting mathematical model will comprise a set of highly stiff ordinary differential equations (ODEs).

For example, let us consider a power system model in the form of a two‐timescale nonlinear system:

(1.1)

where x and y are state vectors that represent electromechanical and faster EMT dynamics, respectively. The matrix ε, which is nonsingular, has all eigenvalues close to zero and corresponds to time constants associated with EMT dynamics. The vector p includes parameters whose values specify the operating conditions and can vary slowly in an explicit function of time to simulate QSS behaviors of the system.

If the EMT dynamics are neglected, resulting in a value of ε equal to 0, the mathematical model takes the form of differential‐algebraic equations (DAEs) shown in (1.2).

(1.2)

This model includes ODEs about state vector x for the dynamical devices, such as generators, motors, and associated controllers, and their corresponding f functions. Additionally, it contains algebraic equations for the control laws, such as power‐flow equations, and their corresponding g functions. The state vector y in (1.1), which changes rapidly with EMT dynamics, is now considered a vector of nonstate variables such as bus voltages or line currents. Their changes in the electromechanical timescale are instantaneous and dependent on state variables in x. As a result, Eq. (1.2) is a simplification of the full ODE model (1.1) that reduces EMT dynamics.

While this book primarily focuses on simulating electromechanical dynamics and transient stability of power systems using the DAE model (1.2), it should be noted that most of the semi‐analytical methods to be introduced in the book can be extended to simulate (1.1) as well. This is because there is no fundamental difference in mathematics between the ODEs on x and the ODEs on y in (1.1). In fact, semi‐analytical methods are more easily applicable to a purely ODE model like (1.1) than a DAE model like (1.2).

Furthermore, to simplify the transient stability simulation process over an extended simulation period, a reduced version of the DAE model in can be used, where fast elements in are ignored and assumed to be zero, resulting in a simpler DAE model with fewer state variables. Moreover, assuming  = 0 with all state variables for QSS simulation leads to a purely algebraic equation model, which can be expressed as:

(1.3)

In QSS simulation, a slowly time‐variant p(t) models the desired sequence of conditions of interest, such as load and generation variations. Explicitly assuming p(t) to change with time can reflect changes in conditions over time.

1.2.1.2 A Practical Power System Model

In practice, transient stability simulation typically involves the consideration of a time‐invariant model that operates at a constant condition represented by p = p0. A widely used model for this purpose is the bus injection representation model (1.4).

(1.4)

Here, Ybus is the bus admittance matrix of the network and Ibus, which is a complex vector‐valued function of state vector x and bus voltage phasors in Vbus, determines the current injections into the network. In contrast to the general DAE model in (1.2), model (1.4) equates y to the complex vector Vbus on all bus voltages, and its g function constrains the balance between the current from the source or load at each bus and the current injected to the bus.

In DAE models (1.2) and (1.4), the algebraic equations represent the power network that connects all buses by power lines and transformers, including AC power‐flow equations. The ODEs model the dynamic devices of a power system, such as:

Synchronous machine models for synchronous generators and condensers

Auxiliary component/controller models of generators such as turbines, speed governors, exciters, and

power system stabilizer

s (

PSS

s)

Network controller/compensator models such as

FACTS

(

flexible AC transmission system

) devices

Induction motor models for three‐phase and one‐phase motor loads

Other dynamic or controllable loads

Inverter‐based resources such as renewable generators and battery‐based energy storage systems.

In power grid simulations, power engineers often use a bulk power system model that includes the main power plants and the transmission network, while the distribution system and loads under each transmission substation are often aggregated into an equivalent composite load model. As illustrated in Figure 1.2, a bulk power system model needs to include the following elementary models:

Synchronous generator models

Excitation system models, which optionally include a PSS

Turbine and speed governor models

Transmission network model

Bus load models (including passive shunt devices)

Other energy resources (such as renewable generators and energy storage systems).

1.2.2 Generator Models

This section introduces the sixth‐order, fourth‐order, and second‐order synchronous generator models that are widely used in studies on power system stability and control.

Figure 1.2 Components of a bulk power system model.

1.2.2.1 Sixth‐Order Model

A detailed model for a synchronous machine is provided by equations in (1.5), which can be used to represent conventional synchronous generators or synchronous motors, including synchronous condensers used for reactive power support. This sixth‐order model is based on a synchronous frequency of the system, denoted as f0, which typically equals either 50 or 60 Hz. The model consists of six ODEs (1.5a)–(1.5f) and four algebraic equations (1.5g)–(1.5j) that describe the machine's interface with the power network.

(1.5a)

(1.5b)

(1.5c)

(1.5d)

(1.5e)

(1.5f)

(1.5g)

(1.5h)

(1.5i)

(1.5j)

The first two ODEs, (1.5a) and (1.5b), are known as swing equations, which describe the rotor angle δ (in radians) with respect to a revolving system reference angle and the deviation Δω = ω−ω0 of the electrical speed ω (in radians per second) from synchronous angular frequency ω0 = 2πf0.

This model is also referred to as a “2.2 model” because the remaining ODEs comprise two ODEs in the d (direct)‐axis plus two ODEs in the q (quadrature)‐axis of the rotor, which is 90° ahead of the d‐axis, as defined in ANSI/IEEE standard 100‐1977. ODEs (1.5c) and (1.5d) describe internal transient voltage components e′q and e′d in q‐axis and d‐axis, respectively, with transient time constants T′d0 and T′q0 (in seconds), while ODEs (1.5e) and (1.5f) describe internal subtransient voltage components e″q and e″d in two axes, respectively, with faster time constants T″q0 and T″d0 (in seconds). These four time constants can be estimated by open‐circuit frequency response tests on the machine. Table 1.1 lists the other constants and variables with their units, where “p.u.” stands for “per unit.”

Table 1.1 Constants and variables of generator models.

Notation

Meaning

H

(second)

Inertial time constant, which is equal to its rated kinetic energy divided by the VA base when operated at the synchronous frequency.

D

(p.u.)

Damping coefficient representing integrated damping effects such as the effect of its damper windings, the control by the power system stabilizer, and nearby frequency‐dependent load.

x′

d

,

x′

q

,

x″

d

,

x″

q

(p.u.)

d

‐ and

q

‐axes, transient and subtransient reactances or inductances.

r

a

(p.u.)

Armature resistance.

P

m

(p.u.)

Mechanical power or torque controlled by its governor and turbine.

P

e

(p.u.)

Electrical power injected into the grid and coupled with other sources and loads through the grid.

e

fd

(p.u.)

Field voltage controlled by an automatic voltage regulator (AVR) with the excitor.

v

d

,

v

q

,

i

d

,

i

q

(p.u.)

d

‐ and

q

‐axes components of terminal voltages and currents in the machine‐based local reference frame.

v

x

,

v

y

,

i

x

,

i

y

(p.u.)

Real (

x

‐axis) and imaginary (

y

‐axis) components of terminal voltages and currents in a system‐wide common reference frame that lags the

q

‐axis by rotor angle

δ

.

In (1.5e) through (1.5j), we can eliminate vd, vq, id, and iq in order to represent terminal voltages, and currents by vx, vy, ix, and iy, which are all based on the system‐wide common reference frame. The terminal current phasor I = ix + jiy can then be expressed as a function of variables that include the terminal voltage phasor V = vx + jvy and the state variables in (1.4), and then a complete set of equations in (1.4) for the entire grid can be constructed after integrating the models of all sources, loads, and the network.

It is important to note that while model (1.5) accurately represents a round‐rotor generator, such as a thermal generating unit, it may not be accurate for salient‐pole rotor generators, such as hydraulic generating units. To model a salient‐pole rotor generator, one ODE in the q‐axis, i.e., is often removed. The resulting model is referred to as a “2.1 model,” with two ODEs in the d‐axis and one ODE in the q‐axis.

1.2.2.2 Fourth‐Order Model

In academic studies, a simplified fourth‐order model given by the following equations in (1.6), along with (1.5i) and (1.5j), is often used to represent a synchronous machine. This model is often referred to as a “1.1 model” and is obtained from (1.5) by assuming that T″d0 and T″q0 in (1.5e) and (1.5f) are zero and by eliminating all subtransient voltages and reactances with a double prime.

(1.6a)

(1.6b)

(1.6c)

(1.6d)

(1.6e)

(1.6f)

Similarly, by removing (1.6d), a “1.0 model” can be obtained for a salient‐pole rotor generator.

1.2.2.3 Second‐Order Model

If both (1.6c) and (1.6d) are removed and x′q is set equal to x′d, the simplified “0.0 model” can be derived, which is commonly referred to as the second‐order classical model as given in (1.7).

(1.7a)

(1.7b)

(1.7c)

(1.7d)

This model assumes a constant internal electromotive force E′ = e′d + je′q with its angle equal to the rotor angle δ. This simplified model is frequently used for fast, approximate analysis of transient stability in power systems.

1.2.3 Controller Models

To ensure accurate power system simulation, it is crucial to model the controllers associated with each synchronous generator. These controllers include the turbine and governor for frequency and active power control, and the excitation system for automatic voltage regulation and reactive power control. In addition, for generators that contribute to undesired power system oscillations, PSSs are often installed with a nonvoltage auxiliary input to improve oscillation damping. In this section, we will introduce some commonly used examples of controller models that can be incorporated into the generator model for more accurate power system simulations.

1.2.3.1 Governor and Turbine Models

The block diagram in Figure 1.3 depicts a linear frequency control system for each synchronous generator.

Figure 1.3 Governor and turbine models.

The system consists of a speed governor and a first‐order turbine model, represented by Eqs. (1.8) and (1.9), respectively.

(1.8)

(1.9)

Table 1.2 provides a list of constants and variables used in the model. The speed governor, modeled by Eq. (1.8), regulates the opening of the turbine valve to maintain a constant rotational speed. The turbine model, given by Eq. (1.9), determines the active power output of the generator as a function of the valve position and the generator rotor speed deviation from the synchronous speed. The model assumes that the valve position directly controls the active power output of the generator. The model depicted in Figure 1.3 is a simplified linear approximation of an actual speed‐governing system. It is employed in some chapters of the book for transient stability simulations of large‐scale power systems.

Table 1.2 Constants and variables of governor and turbine models.

Notation

Meaning

T

g

(second)

Governor time constant

T

t

(second)

Turbine time constant

P

ref

(p.u.)

Setting point of the active power output determined by generation dispatch

(p.u.)

Speed deviation in per unit, namely (

ω

 − 

ω

0

)/

ω

0

R

(p.u.)

Speed regulation factor for the droop control

P

g

(p.u.)

Output of the speed governor, i.e. the input to the turbine

A more practical speed‐governing system model is presented in Figure 1.4, which is the IEEE Type‐1 speed‐governor model (IEEEG1) from the reference [4]. This model includes additional factors that are crucial for accurate representation of this type of real‐world speed‐governing system. These factors include:

Upper and lower limits of the speed governor output

P

g

are considered.

Speed deviation Δ

ω

first goes through a deadband of, e.g., 30 mHz to avoid oversensitive speed regulation with small frequency variations.

Nonlinearities, e.g. hysteresis and saturation, can be modeled with the turbine.

The first‐order turbine model can be extended to multiple sequentially or parallelly connected first‐order turbines to model, e.g., low‐pressure to high‐pressure turbines.

Figure 1.4 IEEE Type 1 speed‐governor model.

Source: [4].

1.2.3.2 Excitation System Model

A simplified excitation control system for generators is illustrated in Figure 1.5 and modeled by Eq.s (1.10) to (1.13). It consists of four blocks: the exciter, amplifier, stabilizer, and transducer, and also receives the control signal Vs from a PSS for damping improvement.

(1.10)

(1.11)

(1.12)

(1.13)

Figure 1.5 Linear excitation system with a power system stabilizer.

The control system regulates the field voltage, EFD, in order to automatically regulate the targeted voltage Vt measured through the transducer. The targeted voltage, Vt, can be the same as the voltage V in (1.5i) if the terminal voltage of the generator is being controlled; it can also be a remote bus voltage if that is estimated via a load compensator based on the equivalent impedance from the terminal to the remote bus. The function of the stabilizer inside the excitation control system is responsible for ensuring the stability of the closed‐loop transfer function on the exciter. To improve the damping of power system oscillation, the exciter also receives the output signal Vs from a PSS, which will be introduced later. It is worth noting that EFD is the same as efd of (1.6c), with the exception that it is calculated using an alternative per‐unit system. This amplifies the small per‐unit value of efd to a value close to or bigger than 1 p.u. For more information, refer to Kundur [1].

Table 1.3 summarizes the variables and constants of the model. Time constants are given in seconds, and gains and voltage signals are expressed in per unit.

Table 1.3 Constants and variables of excitation system models.

Notation

Meaning

V

R

, K

A

, T

A

Output voltage signal, gain, and time constant of the amplifier.

E

FD

, K

E

, T

E

Field voltage and related parameters of the exciter.

V

F

, K

F

, T

F

Output voltage signal, gain, and time constant of the excitation stabilizer.

V

c

, K

R

, T

R

Output voltage signal, gain, and time constant of the voltage transducer.

V

ref

,

V

s

Voltage setting point and the control input from PSS.

K

STAB

Gain of PSS.

T

W

,

T

1

,

T

2

Time constants of the washout filter and phase compensator.

Figure 1.6 shows a practical simulation model using the IEEE Type‐1 exciter (IEEET1) from reference [5]. The model includes additional factors compared to the model in Figure 1.5, such as:

Upper and lower limits of the amplifier to ensure

V

RMIN

 ≤ 

V

R

 ≤ 

V

RMAX

.

V

OEL

and

V

UEL

are outputs from the over‐excitation limiter and under‐excitation limiter, respectively. These help prevent the generator from exceeding its reactive power capacity.

Consideration of the saturation effect of the exciter. An exponential correction is added to the constant

K

E

in

(1.11)

and

(1.12)

to account for this effect.

Figure 1.6 IEEE Type 1 exciter.

Source: [5].

1.2.3.3 Power System Stabilizer

The excitation system of a synchronous generator may use a large gain KA in its amplifier to improve the steady‐state values of the controlled voltage, at the terminal bus or a remote bus, and the synchronizing torque of the generator. However, this can reduce the damping torque of the generator, making it more vulnerable to power system oscillation. To address this, a PSS, as shown earlier in Figure 1.5, is used to add damping against oscillation by controlling the exciter using a nonvoltage auxiliary signal, such as the frequency deviation or its estimation.

Figure 1.5 has provided a simplified PSS model that includes three basic components: the gain, washout (high‐pass) filter, and phase compensation, which can be modeled by:

(1.14)

In this model, the gain KSTAB specifies the expected amount of damping to be added by the PSS. TW is used to block low‐frequency signals while keeping signals associated with power system oscillation unchanged. T1 and T2 together provide phase‐lead compensation to the excitor for a range of frequencies centered at the frequency of generator oscillation that needs to be damped. The phase‐lead compensation is necessary to compensate the phase‐lag characteristic with the transfer function from the output signal Vs to the electrical torque Te.

1.2.4 Load Models

In a bulk power system model, generators and the transmission and subtransmission facilities with voltages above a specific voltage level, such as 100 kV, are included, while distribution networks and electric loads in the area around each transmission substation are not modeled in detail. Instead, they are aggregated into an equivalent single‐load device connected to the substation, which is represented as one bus in the bulk power system model. A load model is required to represent the equivalent load device on the bus.

1.2.4.1 Composite Load Model

One method for modeling the load at each bus is to use a composite load model. The devices to be reduced and merged into a bus for a composite load model may include substation step‐down transformers, subtransmission and distribution feeders, distribution transformers, shunt capacitors, voltage regulators, customer wiring, etc. Figure 1.7 illustrates the structure of a composite load model, which is composed of static load components, where the active and reactive loads are dependent on the bus voltage, and dynamic load components, such as industrial motors and air conditioning motors, requiring ODEs to model.

1.2.4.2 ZIP Load Model

Load modeling typically employs static voltage‐dependent loads to represent the majority of bus loads, with or without frequency dependency. Dynamic loads, such as motor loads, are typically only considered on selected buses where a significant industrial load is present or the air conditioning load is high. A widely used static load model in power systems is the ZIP load model, which consists of three parallel load components for both active and reactive powers. These components, respectively, represent constant power, constant current, and constant impedance, as shown in Eq. (1.15):

Figure 1.7 Composite load model.

(1.15)

Here, V is the voltage phasor at the load bus, P0 and Q0 are the nominal active and reactive powers, and V0 is the nominal voltage magnitude at the bus. The parameters pZ, pI, and pP define the percentages of the constant impedance, current, and power components in active power, while the parameters qZ, qI, and qP do the same for reactive power. The parameters Kpf and Kqf represent the sensitivities of the active and reactive loads to deviations of the actual frequency f from the nominal, synchronous frequency f0.

The static ZIP load model in Eq. (1.15) is also known as a polynomial load model since it maps the load characteristics onto a second‐order polynomial function of the bus voltage magnitude. Alternatively, the model can be approximated by exponential functions in Eq. (1.16), where a and b are, respectively, the sensitivities of active and reactive power loads to changes in the bus voltage magnitude. When a (or b) is close to 0, 1, or 2, the active (reactive) load behaves similarly to a constant power, current, or impedance load, respectively.

(1.16)

In the absence of detailed load characteristics, the following assumptions are commonly made in industrial practice for transient stability simulation of bulk power systems:

When |

V|

is around

V

0

,

a

 = 1 and

b

 = 2 are assumed, treating

P

and

Q

as a constant current load and a constant impedance load, respectively.

When |

V|

becomes low, such as |

V|

 < 0.7 

V

0

, the entire bus load is converted to a constant impedance load with

a

 = 

b

 = 2 to mitigate numerical issues in computations.

1.2.4.3 Motor Load Model

To provide an example of a dynamical load model, the third‐order single‐cage induction motor model is introduced below, which is often used to model a small motor of the composite load model shown in Figure 1.7