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.

Renchang Dai, PhD, is a Consulting Analyst and Project Manager for Puget Sound Energy, Washington, USA. He is a founding member of GE Energy Consluting Smart Grid CoE and an IEEE Senior Member, and has worked and published extensively on graph based power system analysis software.

Guangyi Liu, PhD, is Chief Scientist and Smart Grid CoE at Envision Digital, USA. He is an IEEE Senior member and has extensive experience developing software for graph-based power system analysis across numerous applications.

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¿/ ¿) 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