Risk assessment of power systems : models, methods, and applications

CONTENTS Preface xvii 1 Introduction 1 1.1 Risk in Power Systems 1 1.2 Basic Concepts of Power System Risk Assessment 3 1.2.1 System Risk Evaluation 3 1.2.2 Data in Risk Evaluation 5 1.2.3 Unit Interruption Cost 6 1.3 Outline of the Book 8 2 Outage Models of System Components 13 2.1 Introduction 13 2.2 Models of Independent Outages 14 2.2.1 Repairable Forced Failure 14 2.2.2 Aging Failure 16 2.2.2.1 Probability of Transition to Aging Failure 16 2.2.2.2 Unavailability Due to Aging Cailure 18 2.2.2.3 Explicit Expression of Equation 2.10 20 2.2.3 Nonrepairable Chance Failure 21 2.2.4 Planned Outage 22 2.2.5 Semiforced Outage 25 2.2.6 Partial Failure Mode 26 2.2.7 Multiple Failure Mode 28 2.3 Models of Dependent Outages 28 2.3.1 Common-Cause Outage 29 2.3.1.1 Composite Model 29 2.3.1.2 Individual Model 30 2.3.1.3 Comparison between Composite and 32 Individual Models vii 2.3.2 Component-Group Outage 34 2.3.3 Station-Originated Outage 36 2.3.4 Cascading Outage 37 2.3.5 Environment-Dependent Failure 38 2.4 Conclusions 40 3 Parameter Estimation in Outage Models 43 3.1 Introduction 43 3.2 Point Estimation of Mean and Variance of Failure Data 44 3.2.1 Sample Mean 44 3.2.2 Sample Variance 46 3.3 Interval Estimation of Mean and Variance of Failure Data 46 3.3.1 General Concept of Confidence Interval 47 3.3.2 Confidence Interval of Mean 48 3.3.3 Confidence Interval of Variance 51 3.4 Estimating Failure Frequency of Individual Components 51 3.4.1 Point Estimation 52 3.4.2 Interval Estimation 52 3.5 Estimating Probability from a Binomial Distribution 54 3.6 Experimental Distribution of Failure Data and Its Test 55 3.6.1 Experimental Distribution of Failure Data 55 3.6.2 Test of Experimental Distribution 56 3.7 Estimating Parameters in Aging Failure Models 58 3.7.1 Mean Life and Its Standard Deviation in the Normal Model 59 3.7.2 Shape and Scale Parameters in the Weibull Model 61 3.7.3 Example 64 3.8 Conclusions 68 4 Elements of Risk Evaluation Methods 69 4.1 Introduction 69 4.2 Methods for Simple Systems 70 4.2.1 Probability Convolution 70 4.2.2 Series and Parallel Networks 71 4.2.2.1 Series Network 72 4.2.2.2 Parallel Network 73 4.2.3 Markov Equations 74 4.2.4 Frequency-Duration Approaches 76 4.2.4.1 Frequency of Encountering a State 77 4.2.4.2 Frequency of Transition between Two States 77 4.2.4.3 Frequency of Encountering a State Set 77 4.2.4.4 Mean Duration of Residing in Each System State 78 4.2.4.5 Mean Duration of Residing in a State Set 78 4.3 Methods for Complex Systems 79 viii CONTENTS 4.3.1 State Enumeration 79 4.3.2 Nonsequential Monte Carlo Simulation 82 4.3.3 Sequential Monte Carlo Simulation 84 4.4. Conclusions 87 5 Risk Evaluation Techniques for Power Systems 89 5.1 Introduction 89 5.2 Techniques Used in Generation-Demand Systems 90 5.2.1 Convolution Technique 90 5.2.1.1 Discrete Generation Probability Distribution 90 5.2.1.2 Discrete Load Probability Distribution 91 5.2.1.3 Index Calculation 92 5.2.2 State Sampling Method 94 5.2.3 State Duration Sampling Method 96 5.3 Techniques Used in Radial Distribution Systems 98 5.3.1 Analytical Technique 99 5.3.2 State Duration Sampling Method 101 5.4 Techniques Used in Substation Configurations 102 5.4.1 Failure Modes and Modeling 103 5.4.2 Connectivity Identification 105 5.4.3 Stratified State Enumeration Method 107 5.4.4 State Duration Sampling Method 111 5.5 Techniques Used in Composite Generation and Transmission 113 Systems 5.5.1 Basic Procedure 114 5.5.2 Component Failure Models 114 5.5.3 Load Curve Models 116 5.5.4 Contingency Analysis 117 5.5.4.1 AC Power-Flow-Based Sensitivity Technique 117 5.5.4.2 DC Power-Flow-Based Contingency Analysis 118 5.5.5 Optimization Models for Load Curtailments 119 5.5.5.1 AC Power-Flow-Based OPF Model 119 5.5.5.2 DC Power-Flow-Based OPF Model 120 5.5.6 State Enumeration Method 121 5.5.7 State Sampling Method 123 5.6 Conclusions 125 6 Application of Risk Evaluation to Transmission 127 Development Planning 6.1 Introduction 127 6.2 Concept of Probabilistic Planning 128 6.2.1 Basic Procedure 128 6.2.2 Cost Analysis 129 6.2.3 Present Value 130 CONTENTS ix 6.3 Risk Evaluation Approach 130 6.3.1 Risk Evaluation Procedure 131 6.3.2 Risk Cost Model 131 6.4 Example 1: Selecting the Lowest-Cost Planning Alternative 133 6.4.1 System Description 133 6.4.2 Planning Alternatives 134 6.4.3 Risk Evaluation 135 6.4.4 Overall Economic Analysis 137 6.4.4.1 Approach 137 6.4.4.2 Data 139 6.4.4.3 Results 140 6.4.5 Summary 141 6.5 Example 2: Applying Different Planning Criteria 141 6.5.1 System and Planning Alternatives 141 6.5.2 Study Conditions and Data 143 6.5.2.1 Study Conditions 143 6.5.2.1 Data 143 6.5.3 Risk and Risk Cost Evaluation 144 6.5.4 Overall Economic Analysis 146 6.5.4.1 Cash Flows in Multiple Stage Investments 147 6.5.4.2 Performance-Based Cost Efficiency Criterion 147 6.5.4.3 Combined Criterion of Single Contingency and 148 Cost Efficiency 6.5.5 Summary 149 6.6 Conclusions 150 7 Application of Risk Evaluation to Transmission Operation Planning 151 7.1 Introduction 151 7.2 Concept of Risk Evaluation in Operation Planning 152 7.3 Risk Evaluation Method 155 7.4 Example 1: Determining the Lowest-Risk Operation Mode 157 7.4.1 System and Study Conditions 157 7.4.2 Assessing Impacts of Load Transfer 158 7.4.3 Comparing Different Reconfigurations 159 7.4.4 Selecting Operation Mode Under the N– 2 Condition 161 7.4.5 Summary 162 7.5 Example 2: A Simple Case by Hand Calculations 163 7.5.1 Basic Concept 163 7.5.2 Case Description 163 7.5.3 Study Conditions and Data 164 7.5.4 Risk Evaluation 166 7.5.4.1 Calculating the Failure State Probability 166 7.5.4.2 Evaluating EENS by Assuming One-Hour 167 Switching Time x CONTENTS 7.5.4.3 Evaluating EENS by Assuming Two-Hour 168 Switching Time 7.5.5 Summary 170 7.6 Conclusions 170 8 Application of Risk Evaluation to Generation Source Planning 173 8.1 Introduction 173 8.2 Procedure for Reliability Planning 174 8.3 Simulation of Generation and Risk Costs 175 8.3.1 Simulation Approach 175 8.3.2 Minimization Cost Model 176 8.3.3 Expected Generation and Risk Costs 177 8.4 Example 1: Selecting Location and Size of Cogenerators 178 8.4.1 Basic Concept 178 8.4.2 System and Cogeneration Candidates 179 8.4.3 Risk Sensitivity Analysis 181 8.4.4 Maximum Benefit Analysis 183 8.4.5 Summary 187 8.5 Example 2: Making a Decision to Retire a Local 187 Generation Plant 8.5.1 Case Description 187 8.5.2 Risk Evaluation 188 8.5.3 Total Cost Analysis 189 8.5.3.1 Investment Cost 190 8.5.3.2 Operation Cost 190 8.5.3.3 Risk Cost 190 8.5.4 Summary 191 8.6 Conclusions 192 9 Selection of Substation Configurations 193 9.1 Introduction 193 9.2 Load Curtailment Model 194 9.3 Risk Evaluation Approach 197 9.3.1 Component Failure Models 197 9.3.2 Procedure of Risk Evaluation 197 9.3.3 Economic Analysis Method 198 9.4 Example 1: Selecting Substation Configuration 198 9.4.1 Two Substation Configurations 198 9.4.2 Risk Evaluation 199 9.4.2.1 Study Condition Data 199 9.4.2.2 Results 200 9.4.3 Economic Analysis 203 9.4.4 Summary 204 CONTENTS xi 9.5 Example 2: Selecting Transmission Line Arrangement 205 Associated with Substations 9.5.1 Description of Two Options 205 9.5.2 Risk Evaluation and Economic Analysis 206 9.5.2.1 Study Conditions and Data 206 9.5.2.2 Results 207 9.5.2.3 Economic Analysis 209 9.5.3 Summary 209 9.6 Conclusions 209 10 Reliability-Centered Maintenance 211 10.1 Introduction 211 10.2 Basic Tasks in RCM 212 10.2.1 Comparison between Maintenance Alternatives 212 10.2.2 Lowest-Risk Maintenance Scheduling 213 10.2.3 Predictive Maintenance Versus Corrective Maintenance 213 10.2.4 Ranking the Importance of Components 214 10.3 Example 1: Transmission Maintenance Scheduling 215 10.3.1 Procedure of Transmission Reliability-Centered 215 Maintenance 10.3.2 Description of the System and Maintenance Outage 217 10.3.3 The Lowest-Risk Schedule of the Cable Replacement 218 10.3.4 Summary 218 10.4 Example 2: Workforce Planning in Maintenance 219 10.4.1 Problem Description 220 10.4.2 Procedure 220 10.4.3 Case Study and Results 221 10.4.4 Summary 222 10.5 Example 3: A Simple Case Performed by Hand Calculations 223 10.5.1 Case Description 223 10.5.2 Study Conditions and Data 224 10.5.3 EENS Evaluation 224 10.5.4 Summary 226 10.6 Conclusions 226 11 Probabilistic Spare-Equipment Analysis 229 11.1 Introduction 229 11.2 Spare-Equipment Analysis Based on Reliability Criteria 230 11.2.1 Unavailability of Components 230 11.2.1.1 Unavailability Due to Repairable Failures 230 11.2.1.2 Unavailability Due to Aging Failures 231 11.2.1.3 Total Unavailability 231 11.2.2 Group Reliability and Spare-Equipment Analysis 231 11.3 Spare-Equipment Analysis Using the Probabilistic Cost Method 233 xii CONTENTS 11.3.1 Failure Cost Model 233 11.3.2 Unit failure Cost Estimation 234 11.3.3 Annual Investment Cost Model 235 11.3.4 Present-Value Approach 235 11.3.5 Procedure for Spare-Equipment Analysis 235 11.4 Example 1: Determining Number and Timing of Spare 236 Transformers 11.4.1 Transformer Group and Data 236 11.4.2 Spare-Transformer Analysis Based on Group 236 Failure Probability 11.4.3 Spare-Transformer Plans Based on the Probabilistic 238 Cost Model 11.4.3.1 Evaluating Failure Cost Deductions Due 238 to Spares 11.4.3.2 Benefit/Cost Analysis 238 11.4.4 Summary 240 11.5 Example 2: Determining Redundancy Level of 500 kV Reactors 240 11.5.1 Problem Description 241 11.5.2 Study Condition and Data 242 11.5.3 Redundancy Analysis 244 11.5.4 Summary 245 11.6 Conclusions 246 12 Reliability-Based Transmission-Service Pricing 249 12.1 Introduction 249 12.2 Basic Concept 250 12.2.1 Incremental Reliability Value 250 12.2.2 Impacts of Customers on System Reliability 252 12.2.3 Reliability Component in Price Design 253 12.3 Calculation Methods 254 12.3.1 Unit Incremental Reliability Value (UIRV) 254 12.3.2 Generation Credit for Reliability Improvement (GCRI) 255 12.3.3 Load Charge for Reliability Degradation (LCRD) 255 12.3.4 Load Charge Rate Due to Generation Credit (LCRGC) 255 12.4 Rate Design 256 12.4.1 Charge Rate for Wheeling Customers 256 12.4.2 Charge Rate for Native Customers 256 12.4.3 Credit to Generation Customers 257 12.5 Application Example 257 12.5.1 Calculation of the UIRV 258 12.5.2 Calculation of the GCRI 258 12.5.3 Calculation of the LCRD 259 12.5.4 Calculation of the LCRGC 260 12.5.5 Calculations of Charge Rates 260 12.6 Conclusions 261 CONTENTS xiii 13 Probabilistic Transient Stability Assessment 263 13.1 Introduction 263 13.2 Probabilistic Modeling and Simulation Methods 264 13.2.1 Selection of Prefault System States 264 13.2.2 Fault Models 264 13.2.2.1 Probability of Fault Occurrence 264 13.2.2.2 Probability of Fault Location 265 13.2.2.3 Probability of Fault Type 265 13.2.2.4 Probability of Unsuccessful Automatic Reclosure 266 13.2.2.5 Probability Model of Fault-Clearing Time 266 13.2.3 Monte Carlo Simulation of Fault Events 267 13.2.4 Transient Stability Simulation 268 13.3 Procedure 268 13.3.1 Procedure for the First Type of Study 268 13.3.2 Procedure for the Second Type of Study 270 13.4Examples 270 13.4.1 System Description and Data 270 13.4.1.1 System Load Data 272 13.4.1.2 Fault Model Data 272 13.4.2 Transfer Limit Calculation in the Columbia River System 273 13.4.3 Generation Rejection Requirement in the Peace 275 River System 13.4.4 Summary 278 13.5 Conclusions 279 Appendix A Basic Probability Concepts 281 A.1 Probability Calculation Rules 281 A.1.1 Intersection 281 A.1.2 Union 281 A.1.3 Full Conditional Probability 282 A.2 Random Variable and its Distribution 282 A.3 Important Distributions in Risk Evaluation 283 A.3.1 Exponential Distribution 283 A.3.2 Normal Distribution 283 A.3.3 Log-Normal Distribution 284 A.3.4 Weibull Distribution 285 A.4 Numerical Characteristics 286 A.4.1 Mathematical Expectation 287 A.4.2 Variance and Standard Deviation 287 A.4.3 Covariance and Correlation Coefficients 287 Appendix B Elements of Monte Carlo Simulation 289 B.1 General Concept 289 xiv CONTENTS B.2 Random Number Generators 290 B.2.1 Multiplicative Congruent Generator 290 B.2.1 Mixed Congruent Generator 291 B.3 Inverse Transform Method of Generating Random Variates 292 B.4 Important Random Variates in Risk Evaluation 292 B.4.1 Exonential Distribution Random Variate 292 B.4.2 Normal Distribution Random Variate 293 B.4.3 Log-Normal Distribution Random Variate 294 B.4.4 Weibull Distribution Random Variate 294 Appendix C Power-Flow Models 297 C.1 AC Power-Flow Models 297 C.1.1 Power-Flow Equations 297 C.1.2 Newton–Raphson Method 298 C.1.3 Fast Decoupled Method 298 C.2 DC Power-Flow Models 299 C.2.1 Basic Equation 299 C.2.2 Line-Flow Equation 300 Appendix D Optimization Algorithms 303 D.1 Simplex Methods for Linear Programming 303 D.1.1 Primal Simplex Method 303 D.1.2 Dual Simplex Method 305 D.2 Interior Point Method for Nonlinear Programming 306 D.2.1 Optimality and Feasibility Conditions 306 D.2.2 Procedure for Algorithms 308 Appendix E Three Probability Distribution Tables 311 Table 1: Relationship between area Q and z under the standard 312 normal distribution Table 2: Relationship between area
and t
(n) under the 313 t-distribution Table 3: Relationship between area
and 2
(n) under the 314 2 distribution References 315 Index 321 About the Author 325