Preface and Acknowledgments ................................... xxi
1 INTRODUCTION ................................................. 1
1.1 Overview of Transmission Planning ....................... 1
1.1.1 Basic Tasks in Transmission Planning ............. 1
1.1.2 Traditional Planning Criteria .................... 3
1.2 Necessity of Probabilistic Transmission Planning ........ 6
1.3 Outline of the Book ..................................... 8
2 BASIC CONCEPTS OF PROBABILISTIC PLANNING .................... 11
2.1 Introduction ........................................... 11
2.2 Probabilistic Planning Criteria ........................ 12
2.2.1 Probabilistic Cost Criteria ..................... 12
2.2.2 Specified Reliability Index Target .............. 13
2.2.3 Relative Comparison ............................. 13
2.2.4 Incremental Reliability Index ................... 13
2.3 Procedure of Probabilistic Planning .................... 14
2.3.1 Probabilistic Reliability Evaluation ............ 14
2.3.2 Probabilistic Economic Analysis ................. 17
2.4 Other Aspects in Probabilistic Planning ................ 17
2.5 Conclusions ............................................ 18
3 LOAD MODELING ............................................... 21
3.1 Introduction ........................................... 21
3.2 Load Forecast .......................................... 22
3.2.1 Multivariate Linear Regression .................. 22
3.2.2 Nonlinear Regression ............................. 26
3.2.3 Probabilistic Time Series ....................... 28
3.2.4 Neural Network Forecast ......................... 34
3.3 Load Clustering ........................................ 37
3.3.1 Multistep Load Model ............................ 37
3.3.2 Load Curve Grouping ............................. 40
3.4 Uncertainty and Correlation of Bus Loads ............... 42
3.5 Voltage- and Frequency-Dependent Bus Loads ............. 44
3.5.1 Bus Load Model for Static Analysis .............. 45
3.5.2 Bus Load Model for Dynamic Analysis ............. 46
3.6 Conclusions ............................................ 46
4 SYSTEM ANALYSIS TECHNIQUES .................................. 49
4.1 Introduction ........................................... 49
4.2 Power Flow ............................................. 50
4.2.1 Newton-Raphson Method ........................... 50
4.2.2 Fast Decoupled Method ........................... 51
4.2.3 DC Power Flow ................................... 52
4.3 Probabilistic Power Flow ............................... 53
4.3.1 Point Estimation Method ......................... 54
4.3.2 Monte Carlo Method .............................. 55
4.4 Optimal Power Flow (OPF) ............................... 57
4.4.1 OPF Model ....................................... 58
4.4.2 Interior Point Method (IPM) ..................... 60
4.5 Probabilistic Search Optimization Algorithms ........... 64
4.5.1 Genetic Algorithm (GA) .......................... 64
4.5.2 Particle Swarm Optimization (PSO) ............... 69
4.6 Contingency Analysis and Ranking ....................... 72
4.6.1 Contingency Analysis Methods .................... 72
4.6.2 Contingency Ranking ............................. 75
Index .................................................. 75
4.7 Voltage Stability Evaluation ........................... 76
4.7.1 Continuation Power Flow Technique ............... 76
4.7.2 Reduced Jacobian Matrix Analysis ................ 78
4.8 Transient Stability Solution ........................... 80
4.8.1 Transient Stability Equations ................... 80
4.8.2 Simultaneous Solution Technique ................. 81
4.8.3 Alternate Solution Technique .................... 82
4.9 Conclusions ............................................ 83
5 PROBABILISTIC RELIABILITY EVALUATION ........................ 85
5.1 Introduction ........................................... 85
5.2 Reliability Indices .................................... 86
5.2.1 Adequacy Indices ................................ 86
5.2.2 Reliability Worth Indices ....................... 88
5.2.3 Security Indices ................................ 89
5.3 Reliability Worth Assessment ........................... 90
5.3.1 Methods of Estimating Unit Interruption Cost .... 90
5.3.2 Customer Damage Functions (CDFs) ................ 91
5.3.3 Application of Reliability Worth Assessment ..... 92
5.4 Substation Adequacy Evaluation ......................... 93
5.4.1 Outage Modes of Components ...................... 94
5.4.2 State Enumeration Technique ..................... 95
5.4.3 Labeled Bus Set Approach ........................ 96
5.4.4 Procedure of Adequacy Evaluation ................ 97
5.5 Composite System Adequacy Evaluation ................... 99
5.5.1 Probabilistic Load Models ...................... 100
5.5.2 Component Outage Models ........................ 101
5.5.3 Selection of System Outage States .............. 102
5.5.4 System Analysis ................................ 103
5.5.5 Minimum Load Curtailment Model ................. 104
5.5.6 Procedure of Adequacy Evaluation ............... 105
5.6 Probabilistic Voltage Stability Assessment ............ 107
5.6.1 Optimization Model of Recognizing Power Flow
Insolvability .................................. 108
5.6.2 Method for Identifying Voltage Instability ..... 110
5.6.3 Determination of Contingency System States ..... 111
5.6.4 Assessing Average Voltage Instability Risk ..... 113
5.7 Probabilistic Transient Stability Assessment .......... 114
5.7.1 Selection of Prefault System States ............ 114
5.7.2 Fault Probability Models ....................... 115
5.7.3 Selection of Fault Events ...................... 117
5.7.4 Transient Stability Simulation ................. 117
5.7.5 Assessing Average Transient Instability Risk ... 118
5.8 Conclusions ............................................ 120
6 ECONOMIC ANALYSIS METHODS .................................. 123
6.1 Introduction .......................................... 123
6.2 Cost Components of Projects ........................... 124
6.2.1 Capital Investment Cost ........................ 124
6.2.2 Operation Cost ................................. 124
6.2.3 Unreliability Cost ............................. 125
6.3 Time Value of Money and Present Value Method .......... 125
6.3.1 Discount Rate .................................. 125
6.3.2 Conversion between Present and Future Values ... 126
6.3.3 Cash Flow and Its Present Value ................ 127
6.3.4 Formulas for a Cash Flow with Equal Annual
Values ......................................... 128
6.4 Depreciation .......................................... 131
6.4.1 Concept of Depreciation ........................ 131
6.4.2 Straight-Line Method ........................... 132
6.4.3 Accelerating Methods ........................... 133
6.4.4 Annuity Method ................................. 135
6.4.5 Numerical Example of Depreciation .............. 135
6.5 Economic Assessment of Investment Projects ............ 137
6.5.1 Total Cost Method .............................. 137
6.5.2 Benefit/Cost Analysis .......................... 139
6.5.3 Internal Rate of Return Method ................. 140
6.5.4 Length of Cash Flows ........................... 141
6.6 Economic Assessment of Equipment Replacement .......... 142
6.6.1 Replacement Delay Analysis ..................... 142
6.6.2 Estimating Economic Life ....................... 143
6.7 Uncertainty Analysis in Economic Assessment ........... 144
6.7.1 Sensitivity Analysis ........................... 145
6.7.2 Probabilistic Analysis ......................... 145
6.8 Conclusions ........................................... 147
7 DATA IN PROBABILISTIC TRANSMISSION PLANNING ................ 149
7.1 Introduction .......................................... 149
7.2 Data for Power System Analysis ........................ 150
7.2.1 Equipment Parameters ........................... 150
7.2.2 Equipment Ratings .............................. 155
7.2.3 System Operation Limits ........................ 161
7.2.4 Bus Load Coincidence Factors ................... 161
7.3 Reliability Data in Probabilistic Planning ............ 163
7.3.1 General Concepts of Reliability Data ........... 163
7.3.2 Equipment Outage Indices ....................... 164
7.3.3 Delivery Point Indices ......................... 171
7.4 Other Data ............................................ 176
7.4.1 Data of Generation Sources ..................... 176
7.4.2 Data for Interconnections ...................... 177
7.4.3 Data for Economic Analysis ..................... 177
7.5 Conclusions ........................................... 178
8 FUZZY TECHNIQUES FOR DATA UNCERTAINTY ...................... 181
8.1 Introduction .......................................... 181
8.2 Fuzzy Models of System Component Outages .............. 182
8.2.1 Basic Fuzzy Models ............................. 183
8.2.2 Weather-Related Fuzzy Models ................... 186
8.3 Mixed Fuzzy and Probabilistic Models for Loads ........ 190
8.3.1 Fuzzy Model for Peak Load ...................... 190
8.3.2 Probabilistic Model for Load Curve ............. 190
8.4 Combined Probabilistic and Fuzzy Techniques ........... 192
8.4.1 Probabilistic Representation for Region-
Divided Weather States ......................... 192
8.4.2 Hybrid Reliability Assessment Method ........... 193
8.5 Example 1: Case Study Not Considering Weather
Effects ............................................... 196
8.5.1 Case Description ............................... 196
8.5.2 Membership Functions of Reliability Indices .... 198
8.6 Example 2: Case Study Considering Weather Effects ..... 202
8.6.1 Case Description ............................... 202
8.6.2 Membership Functions of Reliability Indices .... 204
8.6.3 Comparisons between Fuzzy and Traditional
Models ......................................... 211
8.7 Conclusions ........................................... 212
9 NETWORK REINFORCEMENT PLANNING ............................. 215
9.1 Introduction .......................................... 215
9.2 Probabilistic Planning of Bulk Power Supply System .... 216
9.2.1 Description of Problem ......................... 216
9.2.2 Economic Comparison between Two Options ........ 217
9.2.3 Reliability Evaluation Method .................. 217
9.2.4 Reliability Comparison between Two Options ..... 219
9.2.5 Effect of the Existing HVDC Subsystem .......... 221
9.2.6 Summary ........................................ 224
9.3 Probabilistic Planning of Transmission Loop Network ... 225
9.3.1 Description of Problem ......................... 225
9.3.2 Planning Options ............................... 225
9.3.3 Planning Method ................................ 227
9.3.4 Study Results .................................. 229
9.3.5 Summary ........................................ 234
9.4 Conclusions ........................................... 234
10 RETIREMENT PLANNING OF NETWORK COMPONENTS .................. 237
10.1 Introduction .......................................... 237
10.2 Retirement Timing of an Aged AC Cable ................. 238
10.2.1 Description of Problem ......................... 239
10.2.2 Methodology in Retirement Planning ............. 239
10.2.3 Application to Retirement of the Aged AC
Cable .......................................... 244
10.2.4 Summary ........................................ 247
10.3 Replacement Strategy of an HVDC Cable ................. 247
10.3.1 Description of Problem ......................... 247
10.3.2 Methodology in Replacement Strategy ............ 249
10.3.3 Application to Replacement of the Damaged
HVDC Cable ..................................... 251
10.3.4 Summary ........................................ 257
10.4 Conclusions ........................................... 257
11 SUBSTATION PLANNING ........................................ 259
11.1 Introduction .......................................... 259
11.2 Probabilistic Planning of Substation Configuration .... 260
11.2.1 Description of Problem ......................... 260
11.2.2 Planning Method ................................ 261
11.2.3 Comparison between the Two Configurations ...... 266
11.2.4 Summary ........................................ 272
11.3 Transformer Spare Planning ............................ 272
11.3.1 Description of Problem ......................... 272
11.3.2 Method of Probabilistic Spare Planning ......... 273
11.3.3 Actual Example ................................. 276
11.3.4 Summary ........................................ 280
Conclusions ................................................ 280
12 SINGLE-CIRCUIT SUPPLY SYSTEM PLANNING ...................... 283
12.1 Introduction .......................................... 283
12.2 Reliability Performance of Single-Circuit Supply
Systems ............................................... 285
12.2.1 Delivery Point Reliability Indices ............. 285
12.2.2 Contributions of Different Components to
Reliability Indices ............................ 286
12.3 Planning Method of Single-Circuit Supply Systems ...... 288
12.3.1 Basic and Weighted Reliability Indices ......... 288
12.3.2 Unit Incremental Reliability Value Approach .... 293
12.3.3 Benefit/Cost Ratio Approach .................... 294
12.3.4 Procedure of Single-Circuit Supply System
Planning ....................................... 296
12.4 Application to Actual Utility System .................. 298
12.4.1 Short List Based on Weighted Reliability
Index .......................................... 298
12.4.2 Financial Justification of Reinforcement ....... 301
12.4.3 Ranking Priority of Single-Circuit Systems ..... 302
12.5 Conclusions ........................................... 307
APPENDIX A ELEMENTS OF PROBABILITY THEORY AND STATISTICS ...... 309
A.l Probability Operation Rules ........................... 309
A.1.1 Intersection ................................... 309
A.1.2 Union .......................................... 310
A.1.3 Conditional Probability ........................ 310
A.2 Four Important Probability Distributions .............. 310
A.2.1 Binomial Distribution .......................... 310
A.2.2 Exponential Distribution ....................... 311
A.2.3 Normal Distribution ............................ 311
A.2.4 Weibull Distribution ........................... 312
A.3 Measures of Probability Distribution .................. 313
A.3.1 Mathematical Expectation ....................... 313
A.3.2 Variance and Standard Deviation ................ 313
A.3.3 Covariance and Correlation Coefficient ......... 314
A.4 Parameter Estimation .................................. 314
A.4.1 Maximum Likelihood Estimation .................. 314
A.4.2 Mean, Variance, and Covariance of Samples ...... 315
A.4.3 Interval Estimation ............................ 315
A.5 Monte Carlo Simulation ................................ 316
A.5.1 Basic Concept .................................. 316
A.5.2 Random-Number Generator ........................ 317
A.5.3 Inverse Transform Method ....................... 317
A.5.4 Three Important Random Variates ................ 318
A.5.4.1 Exponential Distribution Random
Variate ............................... 318
A.5.4.2 Normal Distribution Random Variate .... 318
A.5.4.3 Weibull Distribution Random Variate ... 319
APPENDIX В ELEMENTS OF FUZZY MATHEMATICS ...................... 321
B.l Fuzzy Sets ............................................ 321
B.1.1 Definition of Fuzzy Set ........................ 321
B.l.2 Operations of Fuzzy Sets ....................... 322
B.2 Fuzzy Numbers ......................................... 323
B.2.1 Definition of Fuzzy Number ..................... 323
B.2.2 Arithmetic Operation Rules of Fuzzy Numbers .... 323
B.2.2.1 Addition ............................... 323
B.2.2.2 Subtraction ............................ 323
B.2.2.3 Multiplication ......................... 323
B.2.2.4 Division ............................... 324
B.2.2.5 Maximum and Minimum Operations ......... 324
B.2.3 Functional Operation of Fuzzy Numbers ........... 324
B.3 Two Typical Fuzzy Numbers in Engineering
Applications .......................................... 325
B.3.1 Triangular Fuzzy Number ........................ 325
B.3.2 Trapezoidal Fuzzy Number ....................... 325
B.4 Fuzzy Relations ....................................... 326
B.4.1 Basic Concepts .................................. 326
B.4.1.1 Reflexivity ........................... 327
B.4.1.2 Symmetry .............................. 327
B.4.1.3 Resemblance ........................... 327
B.4.1.4 Transitivity .......................... 327
B.4.1.5 Equivalence ........................... 327
B.4.2 Operations of Fuzzy Matrices ................... 327
APPENDIX С ELEMENTS OF RELIABILITY EVALUATION ................. 329
C.l Basic Concepts ........................................ 329
С 1.1 Reliability Functions .......................... 329
С 1.2 Model of Repairable Component .................. 330
C.2 Crisp Reliability Evaluation .......................... 331
C.2.1 Series and Parallel Networks ................... 331
C.2.1.1 Series Network ........................ 331
C.2.1.2 Parallel Network ...................... 332
C.2.2 Minimum Cutsets ................................ 333
C.2.3 Markov Equations ............................... 333
C.3 Fuzzy Reliability Evaluation .......................... 335
C.3.1 Series and Parallel Networks Using Fuzzy
Numbers ........................................ 335
C.3.2 Minimum Cutset Approach Using Fuzzy Numbers .... 336
C.3.3 Fuzzy Markov Models ............................ 338
C.3.3.1 Approach Based on Analytical
Expressions ........................... 338
C.3.3.2 Approach Based on Numerical
Computations .......................... 339
References .................................................... 341
Index ......................................................... 349
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