Risk Assessment of Power Systems
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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
