Power System Simulation Using Semi-Analytical Methods
Description
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.
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Kai Sun, PhD, is a Professor with the Department of Electrical Engineering and Computer Science at the University of Tennessee in Knoxville. He is the author of Power System Control under Cascading Failures: Understanding, Mitigation and System Restoration and has co-authored more than ten IEEE journal papers on semi-analytical methods for power system simulation.
Content
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 Runge-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 Padé Approximants 123
4.2.1 Semi-Analytical Solution Using Padé 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 Padé Approximants 126
4.2.1.3 Online Evaluating SAS Within a Time Window 127
4.2.2 Padé 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 Padé 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
1
Power 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 e, 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 e 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,...
