Series Editor’s Foreword by Dr. Andre V. Kleyner xv Preface xvii Acknowledgments xix List of Abbreviations xx Notations xxii Part I The Fundamentals 1 1 Reliability Assessment 3 1.1 Definitions of Reliability 3 1.1.1 Probability of Survival 3 1.2 Component Reliability Modeling 6 1.2.1 Discrete Probability Distributions 6 1.2.2 Continuous Probability Distributions 8 1.2.3 Physics-of-Failure Equations 13 1.3 System Reliability Modeling 15 1.3.1 Series System 15 1.3.2 Parallel System 16 1.3.3 Series-parallel System 16 1.3.4 K-out-of-n System 17 1.3.5 Network System 18 1.4 System Reliability Assessment Methods 18 1.4.1 Path-set and Cut-set Method 18 1.4.2 Decomposition and Factorization 19 1.4.3 Binary Decision Diagram 19 1.5 Exercises 20 References 22 2 Optimization 23 2.1 Optimization Problems 23 2.1.1 Component Reliability Enhancement 23 2.1.2 Redundancy Allocation 24 2.1.3 Component Assignment 25 2.1.4 Maintenance and Testing 26 2.2 Optimization Methods 30 2.2.1 Mathematical Programming 30 2.2.2 Meta-heuristics 34 2.3 Exercises 36 References 37 Part II Reliability Techniques 41 3 Multi-State Systems (MSSs) 43 3.1 Classical Multi-state Models 43 3.2 Generalized Multi-state Models 45 3.3 Time-dependent Multi-State Models 46 3.4 Methods to Evaluate Multi-state System Reliability 48 3.4.1 Methods Based on MPVs or MCVs 48 3.4.2 Methods Derived from Binary State Reliability Assessment 48 3.4.3 Universal Generating Function Approach 49 3.4.4 Monte Carlo Simulation 50 3.5 Exercises 51 References 51 4 Markov Processes 55 4.1 Continuous Time Markov Chain (CMTC) 55 4.2 In homogeneous Continuous Time Markov Chain 61 4.3 Semi-Markov Process (SMP) 66 4.4 Piecewise Deterministic Markov Process (PDMP) 74 4.5 Exercises 82 References 84 5 Monte Carlo Simulation (MCS) for Reliability and Availability Assessment 87 5.1 Introduction 87 5.2 Random Variable Generation 87 5.2.1 Random Number Generation 87 5.2.2 Random Variable Generation 89 5.3 Random Process Generation 93 5.3.1 Markov Chains 93 5.3.2 Markov Jump Processes 94 5.4 Markov Chain Monte Carlo (MCMC) 97 5.4.1 Metropolis-Hastings (M-H) Algorithm 97 5.4.2 Gibbs Sampler 98 5.4.3 Multiple-try Metropolis-Hastings (M-H) Method 99 5.5 Rare-Event Simulation 101 5.5.1 Importance Sampling 101 5.5.2 Repetitive Simulation Trials after Reaching Thresholds (RESTART) 102 5.6 Exercises 103 Appendix 104 References 115 6 Uncertainty Treatment under Imprecise or Incomplete Knowledge 117 6.1 Interval Number and Interval of Confidence 117 6.1.1 Definition and Basic Arithmetic Operations 117 6.1.2 Algebraic Properties 118 6.1.3 Order Relations 119 6.1.4 Interval Functions 120 6.1.5 Interval of Confidence 121 6.2 Fuzzy Number 121 6.3 Possibility Theory 123 6.3.1 Possibility Propagation 124 6.4 Evidence Theory 125 6.4.1 Data Fusion 128 6.5 Random-fuzzy Numbers (RFNs) 128 6.5.1 Universal Generating Function (UGF) Representation of Random-fuzzy Numbers 129 6.5.2 Hybrid UGF (HUGF) Composition Operator 130 6.6 Exercises 132 References 133 7 Applications 135 7.1 Distributed Power Generation System Reliability Assessment 135 7.1.1 Reliability of Power Distributed Generation (DG) System 135 7.1.2 Energy Source Models and Uncertainties 136 7.1.3 Algorithm for the Joint Propagation of Probabilistic and Possibilistic Uncertainties 138 7.1.4 Case Study 140 7.2 Nuclear Power Plant Components Degradation 140 7.2.1 Dissimilar Metal Weld Degradation 140 7.2.2 MCS Method 145 7.2.3 Numerical Results 147 References 149 Part III Optimization Methods and Applications 151 8 Mathematical Programming 153 8.1 Linear Programming (LP) 153 8.1.1 Standard Form and Duality 155 8.2 Integer Programming (IP) 159 8.3 Exercises 164 References 165 9 Evolutionary Algorithms (EAs) 167 9.1 Evolutionary Search 168 9.2 Genetic Algorithm (GA) 170 9.2.1 Encoding and Initialization 171 9.2.2 Evaluation 172 9.2.3 Selection 173 9.2.4 Mutation 174 9.2.5 Crossover 175 9.2.6 Elitism 178 9.2.7 Termination Condition and Convergence 178 9.3 Other Popular EAs 179 9.4 Exercises 181 References 182 10 Multi-Objective Optimization (MOO) 185 10.1 Multi-objective Problem Formulation 185 10.2 MOO-to-SOO Problem Conversion Methods 187 10.2.1 Weighted-sum Approach 188 10.2.2 ε-constraint Approach 189 10.3 Multi-objective Evolutionary Algorithms 190 10.3.1 Fast Non-dominated Sorting Genetic Algorithm (NSGA-II) 190 10.3.2 Improved Strength Pareto Evolutionary Algorithm (SPEA 2) 193 10.4 Performance Measures 197 10.5 Selection of Preferred Solutions 200 10.5.1 “Min-Max” Method 200 10.5.2 Compromise Programming Approach 201 10.6 Guidelines for Solving RAMS+C Optimization Problems 201 10.7 Exercises 203 References 204 11 Optimization under Uncertainty 207 11.1 Stochastic Programming (SP) 207 11.1.1 Two-stage Stochastic Linear Programs with Fixed Recourse 209 11.1.2 Multi-stage Stochastic Programs with Recourse 217 11.2 Chance-Constrained Programming 218 11.2.1 Model and Properties 219 11.2.2 Example 221 11.3 Robust Optimization (RO) 222 11.3.1 Uncertain Linear Optimization (LO) and its Robust Counterparts 223 11.3.2 Tractability of Robust Counterparts 224 11.3.3 Robust Optimization (RO) with Cardinality Constrained Uncertainty Set 225 11.3.4 Example 226 11.4 Exercises 228 References 229 12 Applications 231 12.1 Multi-objective Optimization (MOO) Framework for the Integration of Distributed Renewable Generation and Storage 231 12.1.1 Description of Distributed Generation (DG) System 232 12.1.2 Optimal Power Flow (OPF) 234 12.1.3 Performance Indicators 235 12.1.4 MOO Problem Formulation 237 12.1.5 Solution Approach and Case Study Results 238 12.2 Redundancy Allocation for Binary-State Series-Parallel Systems (BSSPSs) under Epistemic Uncertainty 240 12.2.1 Problem Description 240 12.2.2 Robust Model 241 12.2.3 Experiment 243 References 244 Index 245

RELIABILITY ANALYSIS, SAFETY ASSESSMENT AND OPTIMIZATION METHODS AND APPLICATIONS IN ENERGY SYSTEMS AND OTHER APPLICATIONS This book is a comprehensive overview of the recently developed methods for assessing and optimizing system reliability and safety. It consists of two main parts, for assessment and optimization methods, respectively. The former covers multi-state system modelling and reliability evaluation, Markov processes, Monte Carlo simulation and uncertainty treatments under poor knowledge. The reviewed methods range from piecewise-deterministic Markov process to belief functions. The latter covers mathematical programs, evolutionary algorithms, multi-objective optimization and optimization under uncertainty. The reviewed methods range from non-dominated sorting genetic algorithm to robust optimization. This book also includes the applications of the assessment and optimization method on real world cases, particularly for the reliability and safety of renewable energy systems. From this point of view, the book bridges the gap between theoretical development and engineering practice.