Graph Database and Graph Computing for Power System Analysis Understand a new way to model power systems with this comprehensive and practical guide Graph databases have become one of the essential tools for managing large data systems. Their structure improves over traditional table-based relational databases in that it reconciles more closely to the inherent physics of a power system, enabling it to model the components and the network of a power system in an organic way. The authors’ pioneering research has demonstrated the effectiveness and the potential of graph data management and graph computing to transform power system analysis. Graph Database and Graph Computing for Power System Analysis presents a comprehensive and accessible introduction to this research and its emerging applications. Programs and applications conventionally modeled for traditional relational databases are reconceived here to incorporate graph computing. The result is a detailed guide which demonstrates the utility and flexibility of this cutting-edge technology. The book’s readers will also find: Design configurations for a graph-based program to solve linear equations, differential equations, optimization problems, and more Detailed demonstrations of graph-based topology analysis, state estimation, power flow analysis, security-constrained economic dispatch, automatic generation control, small-signal stability, transient stability, and other concepts, analysis, and applications An authorial team with decades of experience in software design and power systems analysis Graph Database and Graph Computing for Power System Analysis is essential for researchers and academics in power systems analysis and energy-related fields, as well as for advanced graduate students looking to understand this particular set of technologies.

Table of Contents:
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