About the Editor xiii
List of Contributors xv
Preface xvii
1 Power System Simulation: From Numerical to Semi-Analytical 1
Kai Sun
1.1 Timescales of Simulation 1
1.2 Power System Models 3
1.2.1 Overview 3
1.2.1.1 Simplifying a Power System Model 3
1.2.1.2 A Practical Power System Model 4
1.2.2 Generator Models 5
1.2.2.1 Sixth-Order Model 6
1.2.2.2 Fourth-Order Model 7
1.2.2.3 Second-Order Model 9
1.2.3 Controller Models 9
1.2.3.1 Governor and Turbine Models 9
1.2.3.2 Excitation System Model 12
1.2.3.3 Power System Stabilizer 14
1.2.4 Load Models 14
1.2.4.1 Composite Load Model 15
1.2.4.2 ZIP Load Model 15
1.2.4.3 Motor Load Model 17
1.2.5 Network Model 17
1.2.6 Classical Power System Model 18
1.3 Numerical Simulation 20
1.3.1 Explicit Integration Methods 21
1.3.1.1 Forward Euler Method 22
1.3.1.2 Modified Euler Method 22
1.3.1.3 Rungeâ"Kutta Methods 23
1.3.2 Implicit Integration Methods 24
1.3.2.1 Stiffness of ODEs 24
1.3.2.2 Backward Euler Method 26
1.3.2.3 Trapezoidal-Rule Method 26
1.3.2.4 Comparison with Explicit Methods 28
1.3.3 Solving Differential-Algebraic Equations 28
1.3.3.1 Partitioned Solution Approach 28
1.3.3.2 Simultaneous Solution Approach 29
1.4 Semi-Analytical Simulation 30
1.4.1 Drawbacks with Numerical Simulations 30
1.4.2 Emerging Methods for Semi-Analytical Power System Simulation 31
1.4.3 Approaches to Semi-Analytical Solutions 33
1.4.3.1 Analytical Expansion Approach 33
1.4.3.2 Analytical Homotopy Approach 35
1.4.4 Forms of Semi-Analytical Solutions 40
1.4.4.1 Power Series Form 40
1.4.4.2 Other Series Forms 40
1.4.4.3 Fractional Forms 41
1.4.5 Schemes on Semi-Analytical Power System Simulation 41
1.5 Parallel Power System Simulation 43
1.5.1 Parallelization in Space 44
1.5.1.1 Natural Decoupling 44
1.5.1.2 Network Partitioning 44
1.5.2 Parallelization in Time 45
1.5.3 Parallelization of Semi-Analytical Solutions 48
1.6 Final Remark 48
References 49
2 Power System Simulation Using Power Series-Based Semi-Analytical Methods 53
Bin Wang
2.1 Power Series-Based SAS for Simulating Power System ODEs 53
2.1.1 Power Series-Based SAS for ODEs 53
2.1.2 SAS-Based Fault-On Trajectory Simulation and Its Application in Direct Methods 56
2.1.2.1 SAS-Based Simulation of Fault-On Trajectories 56
2.1.2.2 Application of SAS in Direct Methods 60
2.2 Power Series-Based SAS for Simulating Power System DAEs 63
2.2.1 Power Series-Based SAS for Power System DAEs 63
2.2.2 SAS-Based Simulation of Power System DAEs 66
2.3 Adaptive Time-Stepping Method for SAS-Based Simulation 69
2.3.1 Error-Rate Upper Bound 69
2.3.2 Adaptive Time-Stepping for SAS-Based Simulation 70
2.4 Numerical Examples 72
2.4.1 SAS vs. RK4 and BDF 72
2.4.2 SAS Derivation 74
2.4.3 Application of SAS-Based Simulation on Polish 2383-Bus Power System 75
References 78
3 Power System Simulation Using Differential Transformation Method 81
Yang Liu and Kai Sun
3.1 Introduction to Differential Transformation 81
3.2 Solving the Ordinary Differential Equation Model 85
3.2.1 Derivation Process 85
3.2.1.1 Governor Model 85
3.2.1.2 Turbine Model 86
3.2.1.3 Power System Stablizer Model 86
3.2.1.4 Synchronous Machine Model 86
3.2.1.5 Exciter Model 88
3.2.2 Solution Algorithm 89
3.2.3 Case Study 91
3.2.3.1 Scanning Contingencies 92
3.2.3.2 Numerical Stability 94
3.2.3.3 Accuracy and Time Performance 98
3.3 Solving the Differential-Algebraic Equation Model 101
3.3.1 Basic Idea 102
3.3.2 Derivation Process 104
3.3.2.1 Current Injection of Generators 104
3.3.2.2 Current Injection of Loads 105
3.3.2.3 Transmission Network Equation 106
3.3.3 Solution Algorithm 106
3.3.4 Case Study 107
3.3.4.1 Accuracy and Time Performance 108
3.3.4.2 Robustness 110
3.4 Broader Applications 112
3.5 Conclusions and Future Directions 113
References 114
4 Accelerated Power System Simulation Using Analytic Continuation Techniques 117
Chengxi Liu
4.1 Introduction to Analytic Continuation 118
4.1.1 Direct Method (or Matrix Method) 121
4.1.2 Continued Fractions (i.e. Viskovatov Method) 122
4.2 Finding Semi-Analytical Solutions Using Pad© Approximants 123
4.2.1 Semi-Analytical Solution Using Pad© Approximants 124
4.2.1.1 Offline Solving Differential Equations Using Power Series Expansion 124
4.2.1.2 Offline Transforming Power Series Expansion to the Pad© Approximants 126
4.2.1.3 Online Evaluating SAS Within a Time Window 127
4.2.2 Pad© Approximants of Power System Differential Equations 128
4.2.3 Examples 130
4.2.3.1 Case A. Test on the IEEE 9-Bus Power System 130
4.2.3.2 Case B. Test on the IEEE 39-Bus Power System 133
4.3 Fast Power System Simulation Using Continued Fractions 136
4.3.1 The Proposed Two-Stage Simulation Scheme 137
4.3.1.1 Solving Power System DAEs Using a Partitioned Dynamic Bus Method 138
4.3.2 Continued Fractions-Based Semi-Analytical Solutions 140
4.3.2.1 Online Evaluation of SAS Over a Time Interval 140
4.3.2.2 Transformation from Power Series to Continued Fractions 141
4.3.3 Adaptive Time Interval Based on Priori Error Bound of Continued Fractions 143
4.3.3.1 Priori Error Bound of Continued Fractions 143
4.3.3.2 Adaptive Time Interval for Analytical Solution-Based Dynamic Simulations 145
4.3.4 Examples 146
4.4 Conclusions 152
References 152
5 Power System Simulation Using Multistage Adomian Decomposition Methods 155
Nan Duan
5.1 Introduction to Adomian Decomposition Method 155
5.1.1 Solving Deterministic Differential Equations 155
5.1.2 Solving Stochastic Differential Equations 156
5.2 Adomian Decomposition of Deterministic Power System Models 157
5.2.1 Applying Adomian Decomposition Method to Power Systems 157
5.2.2 Convergence and Time Window of Accuracy 161
5.2.3 Adaptive Time Window 166
5.2.4 Simulation Scheme 167
5.2.4.1 Offline Stage 167
5.2.4.2 Online Stage 167
5.2.5 Examples 169
5.2.5.1 Fixed Time Window 170
5.2.5.2 Adaptive Time Window 176
5.2.5.3 Time Performance 179
5.2.5.4 Simulation of a Contingency With Multiple Disturbances 182
5.3 Adomian Decomposition of Stochastic Power System Models 182
5.3.1 Single-Machine Infinite Bus System With a Stochastic Load 184
5.3.2 Examples 188
5.3.2.1 Stochastic Loads with Low Variances 188
5.3.2.2 Stochastic Loads with High Variances 189
5.3.2.3 Comparison of Time Performances 190
5.3.2.4 Control Informed by Stochastic Simulation 191
5.4 Large-Scale Power System Simulations Using Adomian Decomposition Method 192
References 193
6 Application of Homotopy Methods in Power Systems Simulations 197
Gurunath Gurrala and Francis C. Joseph
6.1 Introduction 197
6.2 The Homotopy Method 198
6.2.1 Multi-stage MHAM 200
6.2.2 Stability of Homotopy Analysis 201
6.2.3 Application to a Linear System 208
6.2.4 Application to a Nonlinear System 209
6.3 Application of Homotopy Methods to Power Systems 212
6.3.1 Generator Model for Transient Stability 212
6.3.1.1 Single Machine Infinite Bus with IEEE Model 1.1 214
6.4 Multimachine Simulations 217
6.4.1 Impact of Number of Terms Considered 220
6.4.2 Effect of c 221
6.5 Application of Homotopy for Error Estimation 226
6.5.1 MHAM-Assisted Adaptive Step Size Adjustment for Modified Euler Method 227
6.5.2 Non-iterative Adaptive Step Size Adjustment 228
6.5.3 Simulation Results 230
6.5.4 Tracking of LTE 230
6.5.5 Accuracy with Variation of Desired LTE 232
6.5.6 Computational Time and Speedup 234
6.6 Summary 236
References 236
7 Utilizing Semi-Analytical Methods in Parallel-in-Time Power System Simulations 239
Byungkwon Park
7.1 Introduction to the Parallel-in-Time (Parareal Algorithm) Simulation 239
7.1.1 Overview of Parareal Algorithm 239
7.1.2 The Derivation of Parareal Algorithm 242
7.1.3 Implementation of Parareal Algorithm 244
7.1.3.1 Standard Coarse Operator 244
7.1.3.2 Fine Operator 245
7.2 Examination of Semi-Analytical Solution Methods in the Parareal Algorithm 245
7.2.1 Adomian Decomposition Method 246
7.2.2 Homotopy Analysis Method 247
7.2.3 Summary 249
7.3 Numerical Case Study 252
7.3.1 Validation of Parareal Algorithm 253
7.3.2 Benefits of Semi-Analytical Solution Methods 255
7.3.3 Results with the High Performance Computing Platform 259
7.3.4 Results with Variable Order Variable Step Adaptive Parareal Algorithm 260
7.4 Conclusions 264
References 265
8 Power System Simulation Using Holomorphic Embedding Methods 267
Rui Yao, Kai Sun, and Feng Qiu
8.1 Holomorphic Embedding from Steady State to Dynamics 267
8.1.1 Holomorphic Embedding Formulations 269
8.1.1.1 Classic Formulation from Trivial Germ Solution 269
8.1.1.2 Continuation from Practical States 273
8.1.1.3 Enabling Dynamic Modeling 276
8.1.2 VSA Using Holomorphic Embedding 277
8.1.2.1 Extend Effective Range by Using Pad© Approximation 277
8.1.2.2 Multistage Holomorphic Embedding 277
8.1.2.3 Partial-QSS Voltage Stability Analysis Scheme 278
8.1.2.4 Full-Dynamic Simulation 279
8.1.3 Test Cases 280
8.1.3.1 IEEE 14-Bus System 280
8.1.3.2 NPCC 140-Bus System 282
8.1.3.3 Polish Test System 287
8.1.4 Summary of the Section 288
8.2 Generic Holomorphic Embedding for Dynamic Security Analysis 289
8.2.1 General Holomorphic Embedding 290
8.2.1.1 Dynamic Simulation Formulation 290
8.2.1.2 Approximation with Holomorphic Embedding 291
8.2.1.3 General Computation Flow 292
8.2.1.4 Rules for Deriving Holomorphic Embedding Coefficients 294
8.2.1.5 Some Properties of Holomorphic Embedding 295
8.2.2 Solve State after Instant Switches 297
8.2.3 Overall Dynamic Simulation Process 298
8.2.4 Test Cases 299
8.2.4.1 Modified IEEE 39-Bus System 299
8.2.4.2 2383-Bus Polish System 301
8.2.5 Summary of Section 303
8.3 Extended-Term Hybrid Simulation 304
8.3.1 Steady-State and Dynamic Hybrid Simulation 305
8.3.1.1 Switching from Dynamic to Quasi-Steady-State (QSS) Models 305
8.3.1.2 Switching from Steady-State to Dynamic Models 306
8.3.1.3 Efficient Determination of Steady State Using Holomorphic Embedding Coefficients 306
8.3.2 Extended-Term Simulation Framework 309
8.3.2.1 Event-Driven Simulation Based on Holomorphic Embedding 309
8.3.2.2 Overall Work Flow of Extended-Term Simulation 310
8.3.3 Experiments 310
8.3.3.1 2-Bus Test System 310
8.3.3.2 4-Bus Test System 313
8.3.3.3 Simulation of Restoration on New England Test System 316
8.3.4 Summary of Section 318
8.4 Robust Parallel or Distributed Simulation 318
8.4.1 Steady-State Contingency Analysis: Problem Formulation and State of the Art 319
8.4.1.1 Problem Formulation 319
8.4.1.2 Holomorphic Embedding-Based Contingency Analysis 320
8.4.2 Partitioned Holomorphic Embedding (PHE) 321
8.4.2.1 Interface-Based Partitioning 321
8.4.2.2 Comparative Complexity Analysis 325
8.4.3 Parallel and Distributed Computation 326
8.4.3.1 Parallel Partitioned Holomorphic Embedding (P 2 HE) 326
8.4.3.2 Parallelism Among Contingency Analysis Tasks 327
8.4.4 Experiment on Large-Scale System 329
8.4.5 Summary of Section 329
References 330
Index 337
