System Reliability Assessment and Optimization Methods and Applications

This book offers a comprehensive overview of recently developed methods for assessing and optimizing system reliability. It consists of two main parts, for treating assessment methods and optimization methods, respectively.

The first part covers methods of multi-state system reliability modelling and evaluation, Markov processes, Monte Carlo simulation and uncertainty analysis. The methods considered range from piecewise-deterministic Markov processes to belief function analysis. The second part covers optimization methods of mathematical programming and evolutionary algorithms, and problems of multi-objective optimization and optimization under uncertainty. The methods of this part range from non-dominated sorting genetic algorithm to robust optimization.

The book also includes the application of the assessment and optimization methods considered on real case studies, particularly with respect to the reliability assessment and optimization of renewable energy systems, and bridges the gap between theoretical method development and engineering practice.

Yan-Fu Li is Full Professor at the Department of Industrial Engineering and Deputy-Director of the Institute for Quality & Reliability at Tsinghua University, China. He received his Ph.D in Industrial Engineering from National University of Singapore in 2010.

Enrico Zio is Full Professor at the Center for Research on Risks and Crises (CRC) of Ecole de Mines, ParisTech, PSL University, France, and at the Energy Department of Politecnico di Milano, Italy. He received his Ph.D in nuclear engineering from Politecnico di Milano and in Probabilistic Risk Assessment from MIT in 1996 and 1998, respectively.

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

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