Power System Simulation Using Semi-Analytical Methods

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.

• About the Editor xiiiList of Contributors xvPreface xvii1 Power System Simulation: From Numerical to Semi-Analytical 1Kai Sun1.1 Timescales of Simulation 11.2 Power System Models 31.2.1 Overview 31.2.1.1 Simplifying a Power System Model 31.2.1.2 A Practical Power System Model 41.2.2 Generator Models 51.2.2.1 Sixth-Order Model 61.2.2.2 Fourth-Order Model 71.2.2.3 Second-Order Model 91.2.3 Controller Models 91.2.3.1 Governor and Turbine Models 91.2.3.2 Excitation System Model 121.2.3.3 Power System Stabilizer 141.2.4 Load Models 141.2.4.1 Composite Load Model 151.2.4.2 ZIP Load Model 151.2.4.3 Motor Load Model 171.2.5 Network Model 171.2.6 Classical Power System Model 181.3 Numerical Simulation 201.3.1 Explicit Integration Methods 211.3.1.1 Forward Euler Method 221.3.1.2 Modified Euler Method 221.3.1.3 Runge–Kutta Methods 231.3.2 Implicit Integration Methods 241.3.2.1 Stiffness of ODEs 241.3.2.2 Backward Euler Method 261.3.2.3 Trapezoidal-Rule Method 261.3.2.4 Comparison with Explicit Methods 281.3.3 Solving Differential-Algebraic Equations 281.3.3.1 Partitioned Solution Approach 281.3.3.2 Simultaneous Solution Approach 291.4 Semi-Analytical Simulation 301.4.1 Drawbacks with Numerical Simulations 301.4.2 Emerging Methods for Semi-Analytical Power System Simulation 311.4.3 Approaches to Semi-Analytical Solutions 331.4.3.1 Analytical Expansion Approach 331.4.3.2 Analytical Homotopy Approach 351.4.4 Forms of Semi-Analytical Solutions 401.4.4.1 Power Series Form 401.4.4.2 Other Series Forms 401.4.4.3 Fractional Forms 411.4.5 Schemes on Semi-Analytical Power System Simulation 411.5 Parallel Power System Simulation 431.5.1 Parallelization in Space 441.5.1.1 Natural Decoupling 441.5.1.2 Network Partitioning 441.5.2 Parallelization in Time 451.5.3 Parallelization of Semi-Analytical Solutions 481.6 Final Remark 48References 492 Power System Simulation Using Power Series-Based Semi-Analytical Methods 53Bin Wang2.1 Power Series-Based SAS for Simulating Power System ODEs 532.1.1 Power Series-Based SAS for ODEs 532.1.2 SAS-Based Fault-On Trajectory Simulation and Its Application in Direct Methods 562.1.2.1 SAS-Based Simulation of Fault-On Trajectories 562.1.2.2 Application of SAS in Direct Methods 602.2 Power Series-Based SAS for Simulating Power System DAEs 632.2.1 Power Series-Based SAS for Power System DAEs 632.2.2 SAS-Based Simulation of Power System DAEs 662.3 Adaptive Time-Stepping Method for SAS-Based Simulation 692.3.1 Error-Rate Upper Bound 692.3.2 Adaptive Time-Stepping for SAS-Based Simulation 702.4 Numerical Examples 722.4.1 SAS vs. RK4 and BDF 722.4.2 SAS Derivation 742.4.3 Application of SAS-Based Simulation on Polish 2383-Bus Power System 75References 783 Power System Simulation Using Differential Transformation Method 81Yang Liu and Kai Sun3.1 Introduction to Differential Transformation 813.2 Solving the Ordinary Differential Equation Model 853.2.1 Derivation Process 853.2.1.1 Governor Model 853.2.1.2 Turbine Model 863.2.1.3 Power System Stablizer Model 863.2.1.4 Synchronous Machine Model 863.2.1.5 Exciter Model 883.2.2 Solution Algorithm 893.2.3 Case Study 913.2.3.1 Scanning Contingencies 923.2.3.2 Numerical Stability 943.2.3.3 Accuracy and Time Performance 983.3 Solving the Differential-Algebraic Equation Model 1013.3.1 Basic Idea 1023.3.2 Derivation Process 1043.3.2.1 Current Injection of Generators 1043.3.2.2 Current Injection of Loads 1053.3.2.3 Transmission Network Equation 1063.3.3 Solution Algorithm 1063.3.4 Case Study 1073.3.4.1 Accuracy and Time Performance 1083.3.4.2 Robustness 1103.4 Broader Applications 1123.5 Conclusions and Future Directions 113References 1144 Accelerated Power System Simulation Using Analytic Continuation Techniques 117Chengxi Liu4.1 Introduction to Analytic Continuation 1184.1.1 Direct Method (or Matrix Method) 1214.1.2 Continued Fractions (i.e. Viskovatov Method) 1224.2 Finding Semi-Analytical Solutions Using Padé Approximants 1234.2.1 Semi-Analytical Solution Using Padé Approximants 1244.2.1.1 Offline Solving Differential Equations Using Power Series Expansion 1244.2.1.2 Offline Transforming Power Series Expansion to the Padé Approximants 1264.2.1.3 Online Evaluating SAS Within a Time Window 1274.2.2 Padé Approximants of Power System Differential Equations 1284.2.3 Examples 1304.2.3.1 Case A. Test on the IEEE 9-Bus Power System 1304.2.3.2 Case B. Test on the IEEE 39-Bus Power System 1334.3 Fast Power System Simulation Using Continued Fractions 1364.3.1 The Proposed Two-Stage Simulation Scheme 1374.3.1.1 Solving Power System DAEs Using a Partitioned Dynamic Bus Method 1384.3.2 Continued Fractions-Based Semi-Analytical Solutions 1404.3.2.1 Online Evaluation of SAS Over a Time Interval 1404.3.2.2 Transformation from Power Series to Continued Fractions 1414.3.3 Adaptive Time Interval Based on Priori Error Bound of Continued Fractions 1434.3.3.1 Priori Error Bound of Continued Fractions 1434.3.3.2 Adaptive Time Interval for Analytical Solution-Based Dynamic Simulations 1454.3.4 Examples 1464.4 Conclusions 152References 1525 Power System Simulation Using Multistage Adomian Decomposition Methods 155Nan Duan5.1 Introduction to Adomian Decomposition Method 1555.1.1 Solving Deterministic Differential Equations 1555.1.2 Solving Stochastic Differential Equations 1565.2 Adomian Decomposition of Deterministic Power System Models 1575.2.1 Applying Adomian Decomposition Method to Power Systems 1575.2.2 Convergence and Time Window of Accuracy 1615.2.3 Adaptive Time Window 1665.2.4 Simulation Scheme 1675.2.4.1 Offline Stage 1675.2.4.2 Online Stage 1675.2.5 Examples 1695.2.5.1 Fixed Time Window 1705.2.5.2 Adaptive Time Window 1765.2.5.3 Time Performance 1795.2.5.4 Simulation of a Contingency With Multiple Disturbances 1825.3 Adomian Decomposition of Stochastic Power System Models 1825.3.1 Single-Machine Infinite Bus System With a Stochastic Load 1845.3.2 Examples 1885.3.2.1 Stochastic Loads with Low Variances 1885.3.2.2 Stochastic Loads with High Variances 1895.3.2.3 Comparison of Time Performances 1905.3.2.4 Control Informed by Stochastic Simulation 1915.4 Large-Scale Power System Simulations Using Adomian Decomposition Method 192References 1936 Application of Homotopy Methods in Power Systems Simulations 197Gurunath Gurrala and Francis C. Joseph6.1 Introduction 1976.2 The Homotopy Method 1986.2.1 Multi-stage MHAM 2006.2.2 Stability of Homotopy Analysis 2016.2.3 Application to a Linear System 2086.2.4 Application to a Nonlinear System 2096.3 Application of Homotopy Methods to Power Systems 2126.3.1 Generator Model for Transient Stability 2126.3.1.1 Single Machine Infinite Bus with IEEE Model 1.1 2146.4 Multimachine Simulations 2176.4.1 Impact of Number of Terms Considered 2206.4.2 Effect of c 2216.5 Application of Homotopy for Error Estimation 2266.5.1 MHAM-Assisted Adaptive Step Size Adjustment for Modified Euler Method 2276.5.2 Non-iterative Adaptive Step Size Adjustment 2286.5.3 Simulation Results 2306.5.4 Tracking of LTE 2306.5.5 Accuracy with Variation of Desired LTE 2326.5.6 Computational Time and Speedup 2346.6 Summary 236References 2367 Utilizing Semi-Analytical Methods in Parallel-in-Time Power System Simulations 239Byungkwon Park7.1 Introduction to the Parallel-in-Time (Parareal Algorithm) Simulation 2397.1.1 Overview of Parareal Algorithm 2397.1.2 The Derivation of Parareal Algorithm 2427.1.3 Implementation of Parareal Algorithm 2447.1.3.1 Standard Coarse Operator 2447.1.3.2 Fine Operator 2457.2 Examination of Semi-Analytical Solution Methods in the Parareal Algorithm 2457.2.1 Adomian Decomposition Method 2467.2.2 Homotopy Analysis Method 2477.2.3 Summary 2497.3 Numerical Case Study 2527.3.1 Validation of Parareal Algorithm 2537.3.2 Benefits of Semi-Analytical Solution Methods 2557.3.3 Results with the High Performance Computing Platform 2597.3.4 Results with Variable Order Variable Step Adaptive Parareal Algorithm 2607.4 Conclusions 264References 2658 Power System Simulation Using Holomorphic Embedding Methods 267Rui Yao, Kai Sun, and Feng Qiu8.1 Holomorphic Embedding from Steady State to Dynamics 2678.1.1 Holomorphic Embedding Formulations 2698.1.1.1 Classic Formulation from Trivial Germ Solution 2698.1.1.2 Continuation from Practical States 2738.1.1.3 Enabling Dynamic Modeling 2768.1.2 VSA Using Holomorphic Embedding 2778.1.2.1 Extend Effective Range by Using Padé Approximation 2778.1.2.2 Multistage Holomorphic Embedding 2778.1.2.3 Partial-QSS Voltage Stability Analysis Scheme 2788.1.2.4 Full-Dynamic Simulation 2798.1.3 Test Cases 2808.1.3.1 IEEE 14-Bus System 2808.1.3.2 NPCC 140-Bus System 2828.1.3.3 Polish Test System 2878.1.4 Summary of the Section 2888.2 Generic Holomorphic Embedding for Dynamic Security Analysis 2898.2.1 General Holomorphic Embedding 2908.2.1.1 Dynamic Simulation Formulation 2908.2.1.2 Approximation with Holomorphic Embedding 2918.2.1.3 General Computation Flow 2928.2.1.4 Rules for Deriving Holomorphic Embedding Coefficients 2948.2.1.5 Some Properties of Holomorphic Embedding 2958.2.2 Solve State after Instant Switches 2978.2.3 Overall Dynamic Simulation Process 2988.2.4 Test Cases 2998.2.4.1 Modified IEEE 39-Bus System 2998.2.4.2 2383-Bus Polish System 3018.2.5 Summary of Section 3038.3 Extended-Term Hybrid Simulation 3048.3.1 Steady-State and Dynamic Hybrid Simulation 3058.3.1.1 Switching from Dynamic to Quasi-Steady-State (QSS) Models 3058.3.1.2 Switching from Steady-State to Dynamic Models 3068.3.1.3 Efficient Determination of Steady State Using Holomorphic Embedding Coefficients 3068.3.2 Extended-Term Simulation Framework 3098.3.2.1 Event-Driven Simulation Based on Holomorphic Embedding 3098.3.2.2 Overall Work Flow of Extended-Term Simulation 3108.3.3 Experiments 3108.3.3.1 2-Bus Test System 3108.3.3.2 4-Bus Test System 3138.3.3.3 Simulation of Restoration on New England Test System 3168.3.4 Summary of Section 3188.4 Robust Parallel or Distributed Simulation 3188.4.1 Steady-State Contingency Analysis: Problem Formulation and State of the Art 3198.4.1.1 Problem Formulation 3198.4.1.2 Holomorphic Embedding-Based Contingency Analysis 3208.4.2 Partitioned Holomorphic Embedding (PHE) 3218.4.2.1 Interface-Based Partitioning 3218.4.2.2 Comparative Complexity Analysis 3258.4.3 Parallel and Distributed Computation 3268.4.3.1 Parallel Partitioned Holomorphic Embedding (P 2 HE) 3268.4.3.2 Parallelism Among Contingency Analysis Tasks 3278.4.4 Experiment on Large-Scale System 3298.4.5 Summary of Section 329References 330Index 337