DR. WENYUAN LI, PhD, is recognized as one of the leading authorities on risk assessment of power systems and has been active in power system risk and reliability evaluation for more than twenty-five years. He is a full professor with Chongqing University, China, and a principal engineer at BC Hydro, Canada. He is a fellow of the Canadian Academy of Engineering, the Engineering Institute of Canada, and the IEEE, and received ten international awards due to his significant contributions in the power system risk assessment field.

Preface xix

Preface to the First Edition xxi

1 Introduction 1

1.1 Risk in Power Systems 1

1.2 Basic Concepts of Power System Risk Assessment 4

1.2.1 System Risk Evaluation 4

1.2.2 Data in Risk Evaluation 6

1.2.3 Unit Interruption Cost 7

1.3 Outline of the Book 9

2 Outage Models of System Components 15

2.1 Introduction 15

2.2 Models of Independent Outages 16

2.2.1 Repairable Forced Failure 17

2.2.2 Aging Failure 18

2.2.3 Nonrepairable Chance Failure 24

2.2.4 Planned Outage 24

2.2.5 Semiforced Outage 27

2.2.6 Partial Failure Mode 28

2.2.7 Multiple Failure Mode 30

2.3 Models of Dependent Outages 31

2.3.1 Common-Cause Outage 31

2.3.2 Component-Group Outage 36

2.3.3 Station-Originated Outage 37

2.3.4 Cascading Outage 39

2.3.5 Environment-Dependent Failure 40

2.4 Conclusions 42

3 Parameter Estimation in Outage Models 45

3.1 Introduction 45

3.2 Point Estimation on Mean and Variance of Failure Data 46

3.2.1 Sample Mean 46

3.2.2 Sample Variance 48

3.3 Interval Estimation on Mean and Variance of Failure Data 49

3.3.1 General Concept of Confidence Interval 49

3.3.2 Confidence Interval of Mean 50

3.3.3 Confidence Interval of Variance 53

3.4 Estimating Failure Frequency of Individual Components 54

3.4.1 Point Estimation 54

3.4.2 Interval Estimation 55

3.5 Estimating Probability from a Binomial Distribution 56

3.6 Experimental Distribution of Failure Data and its Test 57

3.6.1 Experimental Distribution of Failure Data 58

3.6.2 Test of Experimental Distribution 59

3.7 Estimating Parameters in Aging Failure Models 60

3.7.1 Mean Life and its Standard Deviation in the Normal Model 61

3.7.2 Shape and Scale Parameters in the Weibull Model 63

3.7.3 Example 66

3.8 Conclusions 70

4 Elements of Risk Evaluation Methods 73

4.1 Introduction 73

4.2 Methods for Simple Systems 74

4.2.1 Probability Convolution 74

4.2.2 Series and Parallel Networks 75

4.2.3 Minimum Cutsets 78

4.2.4 Markov Equations 79

4.2.5 Frequency-Duration Approaches 81

4.3 Methods for Complex Systems 84

4.3.1 State Enumeration 84

4.3.2 Nonsequential Monte Carlo Simulation 87

4.3.3 Sequential Monte Carlo Simulation 89

4.4 Correlation Models in Risk Evaluation 91

4.4.1 Correlation Measures 92

4.4.2 Correlation Matrix Methods 93

4.4.3 Copula Functions 95

4.5 Conclusions 102

5 Risk Evaluation Techniques for Power Systems 105

5.1 Introduction 105

5.2 Techniques Used in Generation-Demand Systems 106

5.2.1 Convolution Technique 106

5.2.2 State Sampling Method 110

5.2.3 State Duration Sampling Method 112

5.3 Techniques Used in Radial Distribution Systems 114

5.3.1 Analytical Technique 114

5.3.2 State Duration Sampling Method 117

5.4 Techniques Used in Substation Configurations 118

5.4.1 Failure Modes and Modeling 119

5.4.2 Connectivity Identification 121

5.4.3 Stratified State Enumeration Method 123

5.4.4 State Duration Sampling Method 127

5.5 Techniques Used in Composite Generation and Transmission Systems 129

5.5.1 Basic Procedure 130

5.5.2 Component Failure Models 131

5.5.3 Load Curve Models 131

5.5.4 Contingency Analysis 133

5.5.5 Optimization Models for Load Curtailments 135

5.5.6 State Enumeration Method 138

5.5.7 State Sampling Method 139

5.6 Conclusions 141

6 Application of Risk Evaluation to Transmission Development Planning 143

6.1 Introduction 143

6.2 Concept of Probabilistic Planning 144

6.2.1 Basic Procedure 144

6.2.2 Cost Analysis 145

6.2.3 Present Value 146

6.3 Risk Evaluation Approach 146

6.3.1 Risk Evaluation Procedure 147

6.3.2 Risk Cost Model 147

6.4 Example 1: Selecting the Lowest-Cost Planning Alternative 149

6.4.1 System Description 149

6.4.2 Planning Alternatives 151

6.4.3 Risk Evaluation 152

6.4.4 Overall Economic Analysis 155

6.4.5 Summary 157

6.5 Example 2: Applying Different Planning Criteria 158

6.5.1 System and Planning Alternatives 158

6.5.2 Study Conditions and Data 159

6.5.3 Risk and Risk Cost Evaluation 161

6.5.4 Overall Economic Analysis 163

6.5.5 Summary 166

6.6 Conclusions 167

7 Application of Risk Evaluation to Transmission Operation Planning 169

7.1 Introduction 169

7.2 Concept of Risk Evaluation in Operation Planning 170

7.3 Risk Evaluation Method 173

7.4 Example 1: Determining the Lowest-Risk Operation Mode 175

7.4.1 System and Study Conditions 175

7.4.2 Assessing Impacts of Load Transfer 177

7.4.3 Comparing Different Reconfigurations 177

7.4.4 Selecting Operation Mode under the N−2 Condition 179

7.4.5 Summary 181

7.5 Example 2: A Simple Case by Hand Calculation 181

7.5.1 Basic Concept 181

7.5.2 Case Description 182

7.5.3 Study Conditions and Data 183

7.5.4 Risk Evaluation 185

7.5.5 Summary 188

7.6 Conclusions 188

8 Application of Risk Evaluation to Generation Source Planning 191

8.1 Introduction 191

8.2 Procedure of Reliability Planning 192

8.3 Simulation of Generation and Risk Costs 193

8.3.1 Simulation Approach 193

8.3.2 Minimization Cost Model 194

8.3.3 Expected Generation and Risk Costs 195

8.4 Example 1: Selecting Location and Size of Cogenerators 196

8.4.1 Basic Concept 196

8.4.2 System and Cogeneration Candidates 197

8.4.3 Risk Sensitivity Analysis 199

8.4.4 Maximum Benefit Analysis 201

8.4.5 Summary 205

8.5 Example 2: Making a Decision to Retire a Local Generation Plant 205

8.5.1 Case Description 206

8.5.2 Risk Evaluation 206

8.5.3 Total Cost Analysis 208

8.5.4 Summary 210

8.6 Conclusions 210

9 Application of Risk Evaluation to Selecting Substation Configurations 211

9.1 Introduction 211

9.2 Load Curtailment Model 212

9.3 Risk Evaluation Approach 215

9.3.1 Component Failure Models 215

9.3.2 Procedure of Risk Evaluation 215

9.3.3 Economic Analysis Method 216

9.4 Example 1: Selecting Substation Configuration 217

9.4.1 Two Substation Configurations 217

9.4.2 Risk Evaluation 218

9.4.3 Economic Analysis 222

9.4.4 Summary 223

9.5 Example 2: Evaluating Effects of Substation Configuration Changes 223

9.5.1 Simplified Model for Evaluating Substation Configurations 223

9.5.2 Problem Description 224

9.5.3 Risk Evaluation 227

9.5.4 Summary 228

9.6 Example 3: Selecting Transmission Line Arrangement Associated with Substations 229

9.6.1 Description of Two Options 229

9.6.2 Risk Evaluation and Economic Analysis 230

9.6.3 Summary 233

9.7 Conclusions 233

10 Application of Risk Evaluation to Renewable Energy Systems 235

10.1 Introduction 235

10.2 Risk Evaluation of Wind Turbine Power Converter System (WTPCS) 237

10.2.1 Basic Concepts 237

10.2.2 Power Losses and Temperatures of WTPCS Components 238

10.2.3 Risk Evaluation of WTPCS 240

10.2.4 Case Study 245

10.2.5 Summary 251

10.3 Risk Evaluation of Photovoltaic Power Systems 251

10.3.1 Two Basic Structures of Photovoltaic Power Systems 251

10.3.2 Risk Parameters of Photovoltaic Inverters 254

10.3.3 Risk Evaluation of Photovoltaic Power System 258

10.3.4 Case Study 263

10.3.5 Summary 270

10.4 Conclusions 272

11 Application of Risk Evaluation to Composite Systems with Renewable Sources 275

11.1 Introduction 275

11.2 Risk Assessment of a Composite System with Wind Farms and Solar Power Stations 276

11.2.1 Probability Models of Renewable Sources and Bus Load Curves 276

11.2.2 Multiple Correlations among Renewable Sources and Bus/Regional Loads 279

11.2.3 Risk Assessment Considering Multiple Correlations 282

11.2.4 Case Study 283

11.2.5 Summary 295

11.3 Determination of Transfer Capability Required by Wind Generation 296

11.3.1 System, Conditions, and Method 296

11.3.2 Wind Generation Model 298

11.3.3 Equivalence of Wind Power in Generation Systems 299

11.3.4 Transfer Capability Required by Wind Generation 303

11.3.5 Summary 309

11.4 Conclusions 310

12 Risk Evaluation of Wide Area Measurement and Control System 313

12.1 Introduction 313

12.2 Hierarchical Structure and Failure Analysis of WAMCS 314

12.2.1 Hierarchical Structure of WAMCS 314

12.2.2 Failure Analysis Technique for WAMCS 315

12.3 Risk Evaluation of Phasor Measurement Units 317

12.3.1 Markov State Models of PMU Modules 317

12.3.2 Equivalent Two-State Model of PMU 324

12.4 Risk Evaluation of Regional Communication Networks in WAMCS 325

12.4.1 Classification of Regional Communication Networks 325

12.4.2 Survival Mechanisms of Regional Networks 328

12.4.3 Risk Evaluation in Two Survival Mechanisms 329

12.4.4 Equivalent Two-State Model of a Regional Communication Network 334

12.5 Risk Evaluation of Backbone Network in WAMCS 335

12.5.1 Equivalent Risk Model of Backbone Communication Network 336

12.5.2 Risk Evaluation of Optic Fiber System 337

12.6 Numerical Results 343

12.6.1 Risk Indices of PMU 343

12.6.2 Risk Indices of Regional Communication Networks 345

12.6.3 Risk Indices of the Backbone Communication Network 347

12.6.4 Risk Indices of Overall WAMCS 348

12.7 Conclusions 349

13 Reliability-Centered Maintenance 351

13.1 Introduction 351

13.2 Basic Tasks in RCM 352

13.2.1 Comparison between Maintenance Alternatives 352

13.2.2 Lowest-Risk Maintenance Scheduling 353

13.2.3 Predictive Maintenance versus Corrective Maintenance 353

13.2.4 Ranking Importance of Components 354

13.3 Example 1: Transmission Maintenance Scheduling 355

13.3.1 Procedure of Transmission Maintenance Planning 355

13.3.2 Description of the System and Maintenance Outage 357

13.3.3 The Lowest-Risk Schedule of the Cable Replacement 358

13.3.4 Summary 359

13.4 Example 2: Workforce Planning in Maintenance 360

13.4.1 Problem Description 360

13.4.2 Procedure 361

13.4.3 Case Study and Results 362

13.4.4 Summary 363

13.5 Example 3: A Simple Case Performed by Hand Calculations 363

13.5.1 Case Description 363

13.5.2 Study Conditions and Data 365

13.5.3 EENS Evaluation 365

13.5.4 Summary 367

13.6 Conclusions 367

14 Probabilistic Spare-Equipment Analysis 369

14.1 Introduction 369

14.2 Spare-Equipment Analysis Based on Reliability Criteria 370

14.2.1 Unavailability of Components 370

14.2.2 Group Reliability and Spare-Equipment Analysis 372

14.3 Spare-Equipment Analysis Using the Probabilistic Cost Method 373

14.3.1 Failure Cost Model 373

14.3.2 Unit Failure Cost Estimation 374

14.3.3 Annual Investment Cost Model 375

14.3.4 Present Value Approach 375

14.3.5 Procedure of Spare-Equipment Analysis 376

14.4 Example 1: Determining Number and Timing of Spare Transformers 376

14.4.1 Transformer Group and Data 376

14.4.2 Spare-Transformer Analysis Based on Group Failure Probability 377

14.4.3 Spare-Transformer Plans Based on the Probabilistic Cost Model 378

14.4.4 Summary 381

14.5 Example 2: Determining Redundancy Level of 500 kV Reactors 381

14.5.1 Problem Description 381

14.5.2 Study Conditions and Data 383

14.5.3 Redundancy Analysis 385

14.5.4 Summary 387

14.6 Conclusions 387

15 Asset Management Based on Condition Monitoring and Risk Evaluation 389

15.1 Introduction 389

15.2 Maintenance Strategy of Overhead Lines 390

15.2.1 Risk Evaluation Using Condition Monitoring Data 391

15.2.2 Overhead Line Maintenance Strategy 397

15.2.3 Case Study 399

15.2.4 Summary 401

15.3 Replacement Strategy for Aged Transformers 402

15.3.1 Transformer Aging Failure Unavailability Using Condition Monitoring Data 403

15.3.2 Transformer Replacement Strategy 407

15.3.3 Case Study 410

15.3.4 Summary 413

15.4 Conclusions 414

16 Reliability-Based Transmission-Service Pricing 417

16.1 Introduction 417

16.2 Basic Concept 418

16.2.1 Incremental Reliability Value 419

16.2.2 Impacts of Customers on System Reliability 420

16.2.3 Reliability Component in Price Design 421

16.3 Calculation Methods 422

16.3.1 Unit Incremental Reliability Value 422

16.3.2 Generation Credit for Reliability Improvement 423

16.3.3 Load Charge for Reliability Degradation 423

16.3.4 Load Charge Rate Due to Generation Credit 424

16.4 Rate Design 424

16.4.1 Charge Rate for Wheeling Customers 424

16.4.2 Charge Rate for Native Customers 425

16.4.3 Credit to Generation Customers 425

16.5 Application Example 425

16.5.1 Calculation of the UIRV 427

16.5.2 Calculation of the GCRI 427

16.5.3 Calculation of the LCRD 427

16.5.4 Calculation of the LCRGC 428

16.5.5 Calculations of Charge Rates 428

16.6 Conclusions 430

17 Voltage Instability Risk Assessment and its Application to System Planning 431

17.1 Introduction 431

17.2 Method of Assessing Voltage Instability Risk 432

17.2.1 Maximum Loadability Model for System States 432

17.2.2 Models for Identifying Weak Branches and Buses 436

17.2.3 Determination of Contingency System States 443

17.2.4 Procedure of Calculating Voltage Instability Risk Indices 444

17.3 Tracing and Locating Voltage Instability Risk for Planning Alternatives 447

17.4 Case Studies 448

17.4.1 Results of the IEEE 14-Bus System 448

17.4.2 Results of the 171-Bus Utility System 453

17.5 Conclusions 456

18 Probabilistic Transient Stability Assessment 459

18.1 Introduction 459

18.2 Probabilistic Modeling and Simulation Methods 460

18.2.1 Selection of Pre-Fault System States 460

18.2.2 Fault Models 461

18.2.3 Monte Carlo Simulation of Fault Events 463

18.2.4 Transient Stability Simulation 464

18.3 Procedure 464

18.3.1 Procedure for the First Type of Study 465

18.3.2 Procedure for the Second Type of Study 465

18.4 Examples 465

18.4.1 System Description and Data 465

18.4.2 Transfer Limit Calculation in the Columbia River System 470

18.4.3 Generation Rejection Requirement in the Peace River System 472

18.4.4 Summary 475

18.5 Conclusions 475

Appendix A Basic Probability Concepts 477

A.1 Probability Calculation Rules 477

A.1.1 Intersection 477

A.1.2 Union 477

A.1.3 Full Conditional Probability 478

A.2 Random Variable and its Distribution 478

A.3 Important Distributions in Risk Evaluation 479

A.3.1 Exponential Distribution 479

A.3.2 Normal Distribution 479

A.3.3 Log-Normal Distribution 481

A.3.4 Weibull Distribution 481

A.3.5 Gamma Distribution 482

A.3.6 Beta Distribution 483

A.4 Numerical Characteristics 483

A.4.1 Mathematical Expectation 483

A.4.2 Variance and Standard Deviation 484

A.4.3 Covariance and Correlation Coefficients 484

A.5 Nonparametric Kernel Density Estimator 485

A.5.1 Basic Concept 485

A.5.2 Determination of the Bandwidth 486

Appendix B Elements of Monte Carlo Simulation 489

B.1 General Concept 489

B.2 Random Number Generators 490

B.2.1 Multiplicative Congruent Generator 490

B.2.2 Mixed Congruent Generator 491

B.3 Inverse Transform Method of Generating Random Variates 491

B.4 Important Random Variates in Risk Evaluation 492

B.4.1 Exponential Distribution Random Variate 492

B.4.2 Normal Distribution Random Variate 493

B.4.3 Log-Normal Distribution Random Variate 494

B.4.4 Weibull Distribution Random Variate 494

B.4.5 Gamma Distribution Random Variate 495

B.4.6 Beta Distribution Random Variate 495

Appendix C Power Flow Models 497

C.1 AC Power Flow Models 497

C.1.1 Power Flow Equations 497

C.1.2 Newton–Raphson Method 497

C.1.3 Fast Decoupled Method 498

C.2 DC Power Flow Models 499

C.2.1 Basic Equation 499

C.2.2 Line Flow Equation 500

Appendix D Optimization Algorithms 503

D.1 Simplex Methods for Linear Programming 503

D.1.1 Primal Simplex Method 503

D.1.2 Dual Simplex Method 505

D.2 Interior Point Method for Nonlinear Programming 506

D.2.1 Optimality and Feasibility Conditions 506

D.2.2 Procedure of the Algorithm 508

Appendix E Three Probability Distribution Tables 511

References 515

Further Reading 523

Index 525

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