About the Authors xiii
Preface xv
Acknowledgments xvii
Part I Theory and Approaches 1
1 Introduction 3
1.1 Power System Analysis 6
1.1.1 Power Flow Calculation 6
1.1.2 State Estimation 6
1.1.3 Contingency Analysis 7
1.1.4 Security-Constrained Automatic Generation Control 7
1.1.5 Security-Constrained ED 8
1.1.6 Electromechanical Transient Simulation 9
1.1.7 Photovoltaic Power Generation Forecast 10
1.2 Mathematical Model 10
1.2.1 Direct Methods of Solving Large-Scale Linear Equations 10
1.2.2 Iterative Methods of Solving Large-Scale Linear Equations 11
1.2.3 High-Dimensional Differential Equations 11
1.2.4 Mixed Integer-Programming Problems 11
1.3 Graph Computing 12
1.3.1 Graph Modeling Basics 13
1.3.2 Graph Parallel Computing 14
References 14
2 Graph Database 17
2.1 Database Management Systems History 17
2.2 Graph Database Theory and Method 18
2.2.1 Graph Database Principle and Concept 18
2.2.1.1 Defining a Graph Schema 19
2.2.1.2 Creating a Loading Job 20
2.2.1.3 Graph Query Language 21
2.2.2 System Architecture 25
2.2.3 Graph Computing Platform 25
2.3 Graph Database Operations and Performance 26
2.3.1 Graph Database Management System 26
2.3.1.1 Parallel Processing by MapReduce 27
2.3.1.2 Graph Partition 29
2.3.2 Graph Database Performance 35
References 38
3 Graph Parallel Computing 41
3.1 Graph Parallel Computing Mechanism 41
3.2 Graph Nodal Parallel Computing 44
3.3 Graph Hierarchical Parallel Computing 46
3.3.1 Symbolic Factorization 47
3.3.2 Elimination Tree 51
3.3.3 Node Partition 56
3.3.4 Numerical Factorization 57
3.3.5 Forward and Backward Substitution 58
References 59
4 Large-Scale Algebraic Equations 61
4.1 Iterative Methods of Solving Nonlinear Equations 61
4.1.1 Gauss–Seidel Method 61
4.1.2 PageRank Algorithm 62
4.1.2.1 PageRank Algorithm Mechanism 63
4.1.2.2 Iterative Method 66
4.1.2.3 Algebraic Method 67
4.1.2.4 Convergence Analysis 69
4.1.3 Newton–Raphson Method 72
4.2 Direct Methods of Solving Linear Equations 75
4.2.1 Introduction 75
4.2.2 Basic Concepts 76
4.2.2.1 Data Structures of Sparse Matrix 76
4.2.2.2 Matrices and Graphs 78
4.2.3 Historical Development 80
4.2.4 Direct Methods 81
4.2.4.1 Solving Triangular Systems 81
4.2.4.2 Symbolic Factorization 82
4.2.4.3 Fill-Reducing Ordering 82
4.3 Indirect Methods of Solving Linear Equations 83
4.3.1 Stationary Methods 83
4.3.1.1 Jacobi Method 83
4.3.1.2 Gauss–Seidel Method 85
4.3.1.3 SOR Method 86
4.3.1.4 SSOR Method 86
4.3.2 Nonstationary Methods 88
4.3.2.1 CG Method 88
4.3.2.2 Gmres 89
4.3.2.3 BCG (bi-CG) 90
References 91
5 High-Dimensional Differential Equations 95
5.1 Integration Methods 95
5.1.1 An Overview of Integration Methods and their Accuracy 95
5.1.1.1 One-Step Methods 96
5.1.1.2 Linear Multistep Methods 99
5.1.2 Integration Methods for Power System Transient Simulations 100
5.1.3 Transient Analysis Accuracy 100
5.1.4 Transient Analysis Stability 101
5.1.4.1 Absolute Stability 101
5.1.4.2 Stiff Stability 102
5.2 Time Step Control 103
5.2.1 Adaptive Time Step 104
5.2.1.1 Change by Iteration Number 105
5.2.1.2 Change by Estimated Truncation Error 105
5.2.1.3 Change by State Variable Derivative 106
5.2.2 Multiple Time Step 106
5.2.3 Break Points 109
5.3 Initial Operation Condition 110
5.4 Graph-Based Transient Parallel Simulation 115
5.5 Numerical Case Study 117
5.6 Summary 123
References 124
6 Optimization Problems 125
6.1 Optimization Theory 125
6.2 Linear Programming 125
6.2.1 The Simplex Method 127
6.2.1.1 Basic Feasible Solution 127
6.2.1.2 The Simplex Iteration 128
6.2.2 Interior-Point Methods 132
6.3 Nonlinear Programming 138
6.3.1 Unconstrained Optimization Approaches 139
6.3.1.1 Line Search 140
6.3.1.2 Trust Region Optimization 141
6.3.1.3 Quasi-Newton Method 141
6.3.1.4 Double Dogleg Optimization 142
6.3.1.5 Conjugate Gradient Optimization 143
6.3.2 Constrained Optimization Approaches 145
6.3.2.1 Karush–Kuhn–Tucker Conditions 145
6.3.2.2 Linear Approximations of Nonlinear Programming with Linear Constraints 145
6.3.2.3 Linear Approximations of Nonlinear Programming with Nonlinear Constraints 147
6.4 Mixed Integer Optimization Approach 147
6.4.1 Branch-and-Bound Approach 148
6.4.2 Machine Learning for Branching 150
6.5 Optimization Problems Solution by Graph Parallel Computing 151
6.5.1 Simplex Method Based on Graph Parallel Computing 151
6.5.2 Interior-Point Method Based on Graph Parallel Computing 154
References 156
7 Graph-Based Machine Learning 159
7.1 State of Art on PV Generation Forecasting 159
7.2 Graph Machine Learning Model 160
7.3 Convolutional Graph Auto-Encoder 162
7.3.1 Auto-Encoder 162
7.3.2 Auto-Encoder on Graphs 163
7.3.3 Probability Distribution Function Approximation 164
7.3.4 Convolutional Graph Auto-Encoder 167
7.3.5 Graph Feature Extraction Artificial Neural Network (R(G)) 169
7.3.6 Encoder (Q) and Decoder (P) 170
7.3.7 Estimation of P(V∗/ π) 171
References 171
Part II Implementations and Applications 175
8 Power Systems Modeling 177
8.1 Power System Graph Modeling 177
8.2 Physical Graph Model and Computing Graph Model 178
8.3 Node-Breaker Model and Graph Representation 180
8.4 Bus-Branch Model and Graph Representation 189
8.5 Graph-Based Topology Analysis 190
8.5.1 Substation-Level Topology Analysis 190
8.5.2 System-Level Network Topology Analysis 196
References 198
9 State Estimation Graph Computing 199
9.1 Power System State Estimation 199
9.2 Graph Computing-Based State Estimation 201
9.2.1 State Estimation Graph Computing Algorithm 201
9.2.1.1 Build Node-Based State Estimation 201
9.2.1.2 Graph-Based State Estimation Parallel Algorithm 203
9.2.2 Numerical Example 209
9.2.3 Graph-Based State Estimation Implementation 215
9.2.3.1 Graph-Based State Estimation Graph Schema 215
9.2.3.2 Nodal Gain Matrix Formation 216
9.2.3.3 Build RHS 219
9.2.4 Graph-Based State Estimation Computation Efficiency 220
9.3 Bad Data Detection and Identification 223
9.3.1 Chi-Squares Test 224
9.3.2 Advanced Bad Data Detection 224
9.3.3 Bad Data Identification 228
9.3.3.1 Normalized Residual 228
9.3.3.2 Largest Normalized Residual for Bad Data Identification 229
9.4 Graph-Based Bad Data Detection Implementation 229
References 231
10 Power Flow Graph Computing 233
10.1 Power Flow Mathematical Model 233
10.2 Gauss–Seidel Method 234
10.3 Newton–Raphson Method 242
10.3.1 Build Jacobian Graph 245
10.3.2 Graph-Based Symbolic Factorization 247
10.3.3 Graph-Based Elimination Tree Creation and Node Partition 249
10.3.4 Graph Numerical Factorization 251
10.3.5 Build Right-Hand Side 253
10.3.6 Graph Forward and Backward Substitution 254
10.3.7 Graph-Based Newton–Raphson Power Flow Calculation 255
10.4 Fast Decoupled Power Flow Calculation 257
10.4.1 Build B_P and B_PP Graphs 259
10.5 Ill-Conditioned Power Flow Problem Solution 261
10.5.1 Introduction 261
10.5.2 Determine the Feasibility of the Power Flow 262
10.5.3 Problem Formulation for Determining the Feasibility of Power Flow 263
10.5.4 Power Flow Feasibility Verification 264
10.5.5 Find a Feasible Solution for the Power Flow Problem 266
References 271
11 Contingency Analysis Graph Computing 273
11.1 dc Power Flow 273
11.2 Bridge Search 276
11.3 Conjugate Gradient for Postcontingency Power Flow Calculation 282
11.4 Contingency Analysis Using Convolutional Neural Networks 294
11.4.1 Convolutional Neural Network 295
11.4.2 Convolutional Neural Network Components 297
11.4.2.1 Convolutional Neural Network Input 297
11.4.2.2 Convolutional Neural Network Output 297
11.4.2.3 Convolutional Neural Network Convolutional Layer 297
11.4.2.4 CNN Pooling Layer 298
11.4.2.5 CNN Fully Connected Layer 299
11.4.3 Evaluation Metrics 299
11.4.3.1 Accuracy 299
11.4.3.2 Precision 300
11.4.3.3 Recall 300
11.4.4 Implementation of Convolutional Neural Network 300
11.5 Contingency Analysis Graph Computing Implementation 302
References 306
12 Economic Dispatch and Unit Commitment 309
12.1 Classic Economic Dispatch 309
12.1.1 Thermal Unit Economic Dispatch 309
12.1.2 Hydrothermal Power Generation System Economic Dispatch 315
12.2 Security-Constrained Economic Dispatch 320
12.2.1 Generation Shift Factor Matrix 323
12.2.2 Graph-Based SCED Modeling 325
12.2.3 Graph-Based SCED 327
12.2.3.1 Buildup Simplex Graph 328
12.2.3.2 Graph-Based Simplex Method 331
12.2.3.3 Update Power Flow 331
12.2.3.4 Graph-Based SCED Implementation 333
12.3 Security-Constrained Unit Commitment 334
12.3.1 SCUC Model 334
12.3.2 Graph-Based SCUC 335
12.4 Numerical Case Study 336
12.4.1 Graph-Based SCED Modeling 336
12.4.2 Basic Feasible Solution 340
12.4.3 Economic Dispatch Optimal Solution 342
References 342
13 Automatic Generation Control 345
13.1 Classic Automatic Generation Control 345
13.1.1 Speed Governor Control 345
13.1.2 Speed Droop Function 347
13.1.3 Frequency Supplementary Control 353
13.1.4 Fundamentals of Automatic Generation Control 355
13.2 Network Security-Constrained Automatic Generation Control 358
13.3 Security-Constrained AGC Graph Computing 361
References 364
14 Small-Signal Stability 365
14.1 Small-Signal Stability of a Dynamic System 365
14.2 System Linearization 366
14.3 Small-Signal Stability Mode 367
14.4 Single-Machine Infinite Bus System 367
14.4.1 Classical Generator Model 367
14.4.2 Third-Order Generator Model 369
14.4.3 Numerical Case Study 373
14.4.3.1 Stable Case 373
14.4.3.2 Instable Case 376
14.5 Small-Signal Oscillation Stabilization 378
14.6 Eigenvalue Calculation 379
14.6.1 Graph-Based Small-Signal Stability Analysis 382
14.6.2 Buildup Small-Signal Stability Graph 383
14.6.3 Numerical Example 383
References 388
15 Transient Stability 391
15.1 Transient Stability Theory 391
15.1.1 Stability Region and Boundary 391
15.1.2 Energy Function Method 391
15.1.2.1 Controlling UEP Method 392
15.1.2.2 Stability-Region-Based Controlling UEP Method 393
15.2 Transient Simulation Model 393
15.2.1 Generator Rotor Model 393
15.2.2 Generator Electro-Magnetic Model 394
15.2.3 Excitation System Model 394
15.2.4 Governor Model 396
15.2.5 PSS Model 397
15.3 Transient Simulation Approach 397
15.3.1 Transient Simulation Algorithm 398
15.3.2 Steady-State Equilibrium Condition 398
15.3.3 Generator Injection Current 400
15.4 Transient Simulation by Graph Parallel Computing 401
15.4.1 Transient Simulation Graph 401
15.4.2 Loading Data into Graph 403
15.4.3 Graph-Based Transient Simulation Implementation 406
15.5 Numerical Example 406
15.5.1 Power Flow Data 406
15.5.2 Dynamic Data 406
15.5.3 Power Flow Results 409
15.5.4 Steady-State Equilibrium Point 410
15.5.5 Generator Injection Current Calculation 415
15.5.6 Calculate Bus Voltage 416
15.5.7 Simulation Results 416
References 421
16 Graph-Based Deep Reinforcement Learning on Overload Control 425
16.1 Introduction 425
16.2 DDPG Algorithm 426
16.2.1 Terminology 426
16.2.2 Q Function 427
16.2.3 Q Value Approximation 427
16.2.4 Policy Gradient 428
16.3 Branch Overload Control 429
16.3.1 States 429
16.3.2 Actions 430
16.3.3 Rewards 430
16.4 Graph-Based Deep Reinforcement Learning Implementation 430
References 433
17 Conclusions 435
Appendix 437
Index 481
