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Reliability Analysis for Asset Management of Electric Power Grids

Reliability Analysis for Asset Management of Electric Power Grids

Robert Ross

ISBN: 978-1-119-12519-8 December 2018 Wiley-IEEE Press 520 Pages

 E-Book

$104.99

Description

A practical guide to facilitate statistically well-founded decisions in the management of assets of an electricity grid

Effective and economic electric grid asset management and incident management involve many complex decisions on inspection, maintenance, repair and replacement. This timely reference provides statistically well-founded, tried and tested analysis methodologies for improved decision making and asset management strategy for optimum grid reliability and availability.

The techniques described are also sufficiently robust to apply to small data sets enabling asset managers to deal with early failures or testing with limited sample sets. The book describes the background, concepts and statistical techniques to evaluate failure distributions, probabilities, remaining lifetime, similarity and compliancy of observed data with specifications, asymptotic behavior of parameter estimators, effectiveness of network configurations and stocks of spare parts. It also shows how the graphical representation and parameter estimation from analysis of data can be made consistent, as well as explaining modern upcoming methodologies such as the Health Index and Risk Index.

Key features:

  • Offers hands-on tools and techniques for data analysis, similarity index, failure forecasting, health and risk indices and the resulting maintenance strategies.
  • End-of-chapter problems and solutions to facilitate self-study via a book companion website.

The book is essential reading for advanced undergraduate and graduate students in electrical engineering, quality engineers, utilities and industry strategists, transmission and distribution system planners, asset managers and risk managers.

Preface xvii

Acknowledgements xxi

List of Symbols and Abbreviations xxiii

About the Companion website xxix

1 Introduction 1

1.1 Electric Power Grids 1

1.2 Asset Management of Electric Power Grids 2

1.3 Maintenance Styles 4

1.3.1 Corrective Maintenance 5

1.3.1.1 CM Inspections 6

1.3.1.2 CM Servicing 6

1.3.1.3 CM Replacement 6

1.3.1.4 Evaluation of CM 6

1.3.2 Period-Based Maintenance 7

1.3.2.1 PBM Inspections 8

1.3.2.2 PBM Servicing 8

1.3.2.3 PBM Replacement 9

1.3.2.4 Evaluation of PBM 9

1.3.3 Condition-Based Maintenance 9

1.3.3.1 CBM Inspections 11

1.3.3.2 CBM Servicing 11

1.3.3.3 CBM Replacement 11

1.3.3.4 Introduction to the Health Index 11

1.3.3.5 Evaluation of CBM 13

1.3.4 Risk-Based Maintenance 13

1.3.4.1 Corporate Business Values and the Risk Matrix 14

1.3.4.2 RBM Inspections 17

1.3.4.3 RBM Servicing 17

1.3.4.4 RBM Replacement 17

1.3.4.5 Evaluation of RBM 18

1.3.5 Comparison of Maintenance Styles 18

1.4 Incident Management 20

1.5 Summary 21

2 Basics of Statistics and Probability 25

2.1 Outcomes, Sample Space and Events 26

2.2 Probability of Events 29

2.3 Probability versus Statistical Distributions 30

2.4 Fundamental Statistical Functions 33

2.4.1 Failure Distribution F 33

2.4.2 Reliability R 34

2.4.3 Probability or Distribution Density f 35

2.4.4 Probability or Distribution Mass f 35

2.4.5 Hazard Rate h and the Bath Tub Model 36

2.4.6 Cumulative Hazard Function H 38

2.5 Mixed Distributions 38

2.5.1 Competing Processes 39

2.5.2 Inhomogeneous Populations 41

2.5.2.1 Bayes’ Theorem 42

2.5.2.2 Failure Distribution of an Inhomogeneous Population 45

2.5.3 Early Failures Interpreted as Child Mortality 47

2.6 Multivariate Distributions and Power Law 49

2.6.1 Ageing Dose and Power Law 49

2.6.2 Accelerated Ageing 52

2.6.3 Multi-Stress Ageing 54

2.6.4 Cumulative Distribution, Ageing Dose and CBM 56

2.7 Summary 59

3 Measures in Statistics 63

3.1 Expected Values and Moments 63

3.1.1 Operations and Means 65

3.1.2 Bayesian Mean 66

3.1.3 The Moments of a Distribution 67

3.1.4 Extra: Moment Generating Function 67

3.1.5 Extra: Characteristic Function 68

3.1.6 Central Moments of a Distribution 69

3.1.7 The First Four Central and Normalized Moments 70

3.1.8 Mean, Standard Deviation, and Variance of a Sample 71

3.2 Median and Other Quantiles 73

3.3 Mode 75

3.4 Merits of Mean, Median and Modal Value 75

3.5 Measures for Comparing Distributions 77

3.5.1 Covariance 77

3.5.2 Correlation 80

3.5.3 Cross-Correlation and Autocorrelation 81

3.6 Similarity of Distributions 82

3.6.1 Similarity of Counting in Discrete Sets 82

3.6.2 Similarity of Two Discrete Distributions 85

3.6.3 Similarity of Two Continuous Distributions 87

3.6.4 Significance of Similarity 90

3.6.5 Singularity Issues and Alternative Similarity Indices 95

3.7 Compliance 96

3.8 Summary 97

4 Specific Distributions 101

4.1 Fractions and Ranking 101

4.1.1 Uniform Distribution 102

4.1.1.1 Continuous Uniform Distribution Characteristics 103

4.1.1.2 Discrete Uniform Distribution Characteristics 104

4.1.1.3 Extra: Moment Generating Function and Characteristic Function 106

4.1.2 Beta Distribution or Rank Distribution 106

4.1.2.1 Beta Distribution Characteristics 107

4.1.2.2 Extra: Moment Generating Function and Characteristic Function 111

4.2 Extreme Value Statistics 112

4.2.1 Weibull Distribution 113

4.2.1.1 Weibull-2 Distribution 113

4.2.1.2 Weibull-2 Distribution Moments and Mean 115

4.2.1.3 Weibull-2 Distribution Characteristics 117

4.2.1.4 Extra: Moment Generating Function 117

4.2.2 Weibull-3 Distribution 117

4.2.3 Weibull-1 Distribution 118

4.2.4 Exponential Distribution 119

4.2.4.1 Exponential Distribution and Average Hazard Rate 119

4.2.4.2 Exponential Distribution Characteristics 123

4.3 Mean and Variance Statistics 124

4.3.1 Normal Distribution 124

4.3.1.1 Characteristics of the Normal Distribution 125

4.3.1.2 Extra: Moments, Moment Generating Function and Characteristic Function 127

4.3.1.3 Extra: Central LimitTheorem 128

4.3.2 Lognormal Distribution 131

4.3.2.1 Characteristics of the Lognormal Distribution 133

4.3.2.2 extra: Moment Generating Function and Characteristic Function 134

4.3.2.3 Lognormal versus Weibull 134

4.4 Frequency and Hit Statistics 134

4.4.1 Binomial Distribution 135

4.4.1.1 Mean and Variance 138

4.4.1.2 Characteristics of the Binomial Distribution 138

4.4.1.3 : Moment Generating Function 139

4.4.2 Poisson Distribution 140

4.4.2.1 Characteristics of the Poisson Distribution 140

4.4.2.2 Derivation of the Poisson Distribution 141

4.4.2.3 Homogeneous Poisson Process 141

4.4.2.4 Non-Homogeneous Poisson Process 142

4.4.2.5 Poisson versus Binomial Distribution 143

4.4.3 Hypergeometric Distribution 144

4.4.3.1 Mean and Variance of the Hypergeometric Distribution 145

4.4.3.2 Characteristics of the Hypergeometric Distribution 146

4.4.4 Normal Distribution Approximation of the Binomial Distribution 147

4.4.5 Multinomial Distribution 149

4.4.5.1 Mean, Variances and Moment Generating Function 150

4.4.6 Multivariate Hypergeometric Distribution 151

4.5 Summary 152

5 Graphical Data Analysis 157

5.1 Data Quality 158

5.2 Parameter-Free Graphical Analysis 158

5.2.1 Basic Graph of a Population Sample 159

5.2.2 Censored Data 160

5.2.3 Kaplan–Meier Plot 165

5.2.4 Confidence Intervals Around a Known Distribution 168

5.2.5 Confidence Intervals with Data 172

5.2.6 Alternative Confidence Intervals 174

5.3 Model-Based or Parametric Graphs 176

5.4 Weibull Plot 178

5.4.1 Weibull Plot with Expected Plotting Position 179

5.4.2 Weibull Plot with Median Plotting Position 182

5.4.3 Weibull Plot with Expected Probability Plotting Position 182

5.4.4 Weibull Plot with Censored Data 183

5.4.5 Confidence Intervals in Weibull Plots 185

5.5 Exponential Plot 188

5.5.1 Exponential Plot with Expected Plotting Position 188

5.5.2 Exponential Plot with Median Plotting Position 189

5.5.3 Exponential Plot with Censored Data 190

5.5.4 Exponential Plot with Confidence Intervals 190

5.6 Normal Distribution 193                   

5.6.1 Normal Plot with Expected Plotting Position 194

5.6.2 Normal Probability Plot with Confidence Intervals 196

5.6.3 Normal Plot and Lognormal Data 196

5.7 Power Law Reliability Growth 197

5.7.1 Duane and Crow AMSAA Plots and Models 197

5.7.2 NHPP Model in Duane and Crow AMSAA Plots 200

5.8 Summary 202

6 Parameter Estimation 207

6.1 General Aspects with Parameter Estimation 207

6.1.1 Fundamental Properties of Estimators 209

6.1.1.1 Bias 209

6.1.1.2 Efficiency 209

6.1.1.3 Consistency 210

6.1.2 Why Work with Small Data Sets? 211

6.1.3 Asymptotic Behaviour of Estimators 212

6.2 Maximum Likelihood Estimators 212

6.2.1 ML with Uncensored Data 213

6.2.2 ML for Sets Including Censored Data 214

6.2.3 ML for the Weibull Distribution 214

6.2.3.1 ML Estimators for Weibull-2 Uncensored Data 215

6.2.3.2 ML Estimators for Weibull-2 Censored Data 215

6.2.3.3 Expected ML Estimators for theWeibull-2 Distribution 216

6.2.3.4 Formulas for Bias and Scatter 218

6.2.3.5 Effect of the ML Estimation Bias in Case ofWeibull-2 220

6.2.4 ML for the Exponential Distribution 221

6.2.5 ML for the Normal Distribution 222

6.3 Linear Regression 223

6.3.1 The LR Method 223

6.3.1.1 LR by Un weighted Least Squares 224

6.3.1.2 LR by Weighted Least Squares 228

6.3.1.3 LR with Censored Data 231

6.3.1.4 LR with Fixed Origin 232

6.3.1.5 Which is the (Co)variable? 233

6.3.2 LR for the Weibull Distribution 234

6.3.2.1 LR by Un weighted LS for the Weibull Distribution 235

6.3.2.2 LR by Weighted LS for the Weibull Distribution 236

6.3.2.3 Processing Censored Data with the Adjusted Rank Method 238

6.3.2.4 Extra: Processing Censored Data with the Adjusted Plotting Position Method 240

6.3.2.5 Expected LS and WLS Estimators for theWeibull-2 Distribution 242

6.3.2.6 Formulas for Bias and Scatter for LS and WLS 243

6.3.2.7 Comparison of Bias and Scatter in LS,WLS and ML 246

6.3.3 LR for the Exponential Distribution 249

6.3.3.1 LR by Un weighted LS for the Exponential Distribution 251

6.3.3.2 LR by Weighted LS for the Exponential Distribution 252

6.3.3.3 Processing Censored Data with the Adjusted Rank Method 253

6.3.3.4 EXTRA: Processing Censored Data with the Adjusted Plotting Position Method 255

6.3.3.5 Expected LS and WLS Estimator for the Exponential Distribution 256

6.3.4 LR for the Normal Distribution 257

6.3.4.1 LR by Un weighted LS for the Normal Distribution 258

6.3.4.2 Processing Censored Data with the Adjusted Rank Method 260

6.3.4.3 EXTRA: Processing Censored Data with the Adjusted Plotting Position Method 261

6.3.4.4 Expected LS Estimators for the Normal Distribution 262

6.3.5 LR Applied to Power Law Reliability Growth 263

6.4 Summary 263

7 System and Component Reliability 267

7.1 The Basics of System Reliability 267

7.2 Block Diagrams 268

7.3 Series Systems 269

7.4 Parallel Systems and Redundancy 272

7.5 Combined Series and Parallel Systems, Common Cause 273

7.6 EXTRA: Reliability and Expected Life of k-out-of-n Systems 276

7.7 Analysis of Complex Systems 277

7.7.1 Conditional Method 278

7.7.2 Up-table Method 280

7.7.3 EXTRA: Minimal Paths and Minimum Blockades 283

7.8 Summary 285

8 System States, Reliability and Availability 291

8.1 States of Components and Systems 291

8.2 States and Transition Rates of One-Component Systems 292

8.2.1 One-Component System with Mere Failure Behaviour 293

8.2.2 One-Component System with Failure and Repair Behaviour 294

8.3 System State Probabilities via Markov Chains 297

8.3.1 Component and System States 298

8.3.2 System States and Transition Rates for Failure and Repair 300

8.3.3 Differential Equations Based on the State Diagram 301

8.3.4 Differential Equations Based on the Transition Matrix 302

8.4 Markov–Laplace Method for Reliability and Availability 303

8.5 Lifetime with Absorbing States and Spare Parts 306

8.6 Mean Lifetimes MTTFF and MTBF 310

8.7 Availability and Steady-State Situations 312

8.8 Summary 314

9 Application to Asset and Incident Management 317

9.1 Maintenance Styles 317

9.1.1 Period-Based Maintenance Optimization for Lowest Costs 317

9.1.1.1 Case Description 317

9.1.1.2 References to Introductory Material 318

9.1.1.3 PBM Cost Optimization Analysis 318

9.1.1.4 Remarks 320

9.1.2 Corrective versus Period-Based Replacement and Redundancy 321

9.1.2.1 Case Description 322

9.1.2.2 References to Introductory Material 322

9.1.2.3 Analysis of Corrective versus Period-Based Replacement and Redundancy 323

9.1.2.4 Remarks 326

9.1.3 Condition-Based Maintenance 326

9.1.3.1 References to Introductory Material 327

9.1.3.2 Analysis of Condition versus Period-Based Replacement 327

9.1.3.3 Remarks 329

9.1.4 Risk-Based Maintenance 330

9.1.4.1 References to Introductory Material 330

9.1.4.2 Analysis of Risk versus Condition-Based Maintenance 331

9.1.4.3 Remarks 333

9.2 Health Index 334

9.2.1 General Considerations of Health Index 334

9.2.1.1 References to Introductory Material on Health Index Concept Considerations 335

9.2.1.2 Analysis of the Health Index Concept 335

9.2.1.3 Remarks 336

9.2.2 Combined Health Index 337

9.2.2.1 References to Introductory Material 337

9.2.2.2 Analysis of the Combined Health Index Concept 337

9.3 Testing and Quality Assurance 338

9.3.1 Accelerated Ageing to Reduce Child Mortality 338

9.3.2 Tests with Limited Test Object Numbers and Sizes 339

9.4 Incident Management (Determining End of Trouble) 342

9.4.1 Component Failure Data and Confidence Intervals 342

9.4.1.1 References to Introductory Material 343

9.4.1.2 Analysis of the Case 343

9.4.1.3 Remarks 345

9.4.2 Failures in a Cable with Multiple Problems and Stress Levels 347

9.4.2.1 References to Introductory Material 348

9.4.2.2 Analysis of the Case 348

9.4.2.3 Remarks 352

9.4.3 Case of Cable Joints with Five Early Failures 352

9.4.3.1 References to Introductory Material 353

9.4.3.2 Analysis of the Case 353

9.4.3.3 Prognosis Using a Weibull Plot and Confidence Intervals 354

9.4.3.4 Estimation of Sample Size Using the Similarity Index 356

9.4.3.5 Redundancy and Urgency 359

9.4.3.6 Remarks 359

9.4.4 Joint Failure Data with Five Early Failures and Large Scatter 360

9.4.4.1 References to Introductory Material 360

9.4.4.2 Analysis of the Case 361

9.4.4.3 Prognosis Using a Weibull Plot and Confidence Intervals 361

9.4.4.4 Estimation of Sample Size Using the Similarity Index 364

9.4.4.5 Remarks 364

10 Miscellaneous Subjects 367

10.1 Basics of Combinatorics 367

10.1.1 Permutations and Combinations 367

10.1.2 The Gamma Function 368

10.2 Power Functions and Asymptotic Behaviour 369

10.2.1 Taylor and Maclaurin Series 370

10.2.2 Polynomial Fitting 371

10.2.2.1 Polynomial Interpolation 371

10.2.2.2 Polynomial and Linear Regression 374

10.2.3 Power Function Fitting 378

10.3 Regression Analysis 380

10.4 Sampling from a Population and Simulations 386

10.4.1 Systematic Sampling 387

10.4.2 Numerical Integration and Expected Values 389

10.4.3 Ranked Samples with Size n and Confidence Limits 396

10.4.3.1 Behaviour of Population Fractions 396

10.4.3.2 Confidence Limits for Population Fractions 399

10.4.4 Monte Carlo Experiments and Random Number Generators 401

10.4.5 Alternative Sampling and Fractals 405

10.5 Hypothesis Testing 407

10.6 Approximations for the Normal Distribution 408

10.6.1 Power Series 409

10.6.2 Power Series Times Density f (y) 409

10.6.3 Inequalities for Boxing R(y) and h(y) for Large y 410

10.6.4 Polynomial Expression for F(y) 410

10.6.5 Power Function for the Reliability Function R(y) 410

10.6.6 Wrap-up of Approximations 412

Appendix A Weibull Plot 413

Appendix B Laplace Transforms 415

Appendix C Taylor Series 417

Appendix D SI Prefixes 419

Appendix E Greek Characters 421

Appendix F Standard Weibull and Exponential Distribution 423

Appendix G Standardized Normal Distribution 429

Appendix H Standardized Lognormal Distribution 435

Appendix I Gamma Function 441

Appendix J Plotting Positions 447

J.1 Expected Ranked Probability <Fi,n> for n = 1,…, 30 448

J.2 Expected Ranked Probability <Fi,n> for n = 31,…, 45 449

J.3 Expected Ranked Probability <Fi,n> for n = 45,…, 60 450

J.4 Median Ranked Probability FM,i,n for n =1,…, 30 452

J.5 Median Ranked Probability FM,i,n for n = 31,…, 45 453

J.6 Median Ranked Probability FM,i,n for n =46,…, 60 454

J.7 Probability of Expected Ranked Weibull Plotting Position F(<Zi,n>) for n = 1,…, 30 456

J.8 Probability of Expected Ranked Weibull Plotting Position F(<Zi,n>) for n = 31,…, 45 457

J.9 Probability of Expected Ranked Weibull Plotting Position F(<Zi,n>) for n = 46,…, 60 458

J.10 Expected Ranked Weibull Plotting Position <Zi,n> for n = 1,…, 30 460

J.11 Expected Ranked Weibull Plotting Position <Zi,n> for n = 31,…, 45 461

J.12 Expected Ranked Weibull Plotting Position <Zi,n> for n = 46,…, 60 462

J.13 Weights for Linear Regression of Weibull-2 Data for n = 1,…, 30 464

J.14 Weights for Linear Regression of Weibull-2 Data for n = 31,…, 45 465

J.15 Weights for Linear Regression of Weibull-2 Data for n = 46,…, 60 466

References 469

Index 473

vii
Contents
Preface xvii
Acknowledgements xxi
List of Symbols and Abbreviations xxiii
About the Companion website xxix
1 Introduction 1
1.1 Electric Power Grids 1
1.2 Asset Management of Electric Power Grids 2
1.3 Maintenance Styles 4
1.3.1 Corrective Maintenance 5
1.3.1.1 CM Inspections 6
1.3.1.2 CM Servicing 6
1.3.1.3 CM Replacement 6
1.3.1.4 Evaluation of CM 6
1.3.2 Period-Based Maintenance 7
1.3.2.1 PBM Inspections 8
1.3.2.2 PBM Servicing 8
1.3.2.3 PBM Replacement 9
1.3.2.4 Evaluation of PBM 9
1.3.3 Condition-Based Maintenance 9
1.3.3.1 CBM Inspections 11
1.3.3.2 CBM Servicing 11
1.3.3.3 CBM Replacement 11
1.3.3.4 Introduction to the Health Index 11
1.3.3.5 Evaluation of CBM 13
1.3.4 Risk-Based Maintenance 13
1.3.4.1 Corporate Business Values and the Risk Matrix 14
1.3.4.2 RBM Inspections 17
1.3.4.3 RBM Servicing 17
1.3.4.4 RBM Replacement 17
1.3.4.5 Evaluation of RBM 18
1.3.5 Comparison of Maintenance Styles 18
1.4 Incident Management 20
1.5 Summary 21
COPYRIGHTED MATERIAL
viii Contents
2 Basics of Statistics and Probability 25
2.1 Outcomes, Sample Space and Events 26
2.2 Probability of Events 29
2.3 Probability versus Statistical Distributions 30
2.4 Fundamental Statistical Functions 33
2.4.1 Failure DistributionF 33
2.4.2 ReliabilityR 34
2.4.3 Probability or Distribution Densityf 35
2.4.4 Probability or Distribution Massf 35
2.4.5 Hazard Rate h and the Bath Tub Model 36
2.4.6 Cumulative Hazard FunctionH 38
2.5 Mixed Distributions 38
2.5.1 Competing Processes 39
2.5.2 Inhomogeneous Populations 41
2.5.2.1 Bayes’Theorem 42
2.5.2.2 Failure Distribution of an Inhomogeneous Population 45
2.5.3 Early Failures Interpreted as Child Mortality 47
2.6 Multivariate Distributions and Power Law 49
2.6.1 Ageing Dose and Power Law 49
2.6.2 Accelerated Ageing 52
2.6.3 Multi-Stress Ageing 54
2.6.4 Cumulative Distribution, Ageing Dose and CBM 56
2.7 Summary 59
3 Measures in Statistics 63
3.1 Expected Values and Moments 63
3.1.1 Operations and Means 65
3.1.2 Bayesian Mean 66
3.1.3 The Moments of a Distribution 67
3.1.4 EXTRA: Moment Generating Function 67
3.1.5 EXTRA: Characteristic Function 68
3.1.6 Central Moments of a Distribution 69
3.1.7 The First Four Central and Normalized Moments 70
3.1.8 Mean, Standard Deviation, and Variance of a Sample 71
3.2 Median and Other Quantiles 73
3.3 Mode 75
3.4 Merits of Mean, Median and Modal Value 75
3.5 Measures for Comparing Distributions 77
3.5.1 Covariance 77
3.5.2 Correlation 80
3.5.3 Cross-Correlation and Autocorrelation 81
3.6 Similarity of Distributions 82
3.6.1 Similarity of Counting in Discrete Sets 82
3.6.2 Similarity of Two Discrete Distributions 85
3.6.3 Similarity of Two Continuous Distributions 87
3.6.4 Significance of Similarity 90
3.6.5 Singularity Issues and Alternative Similarity Indices 95
Contents ix
3.7 Compliance 96
3.8 Summary 97
4 Specific Distributions 101
4.1 Fractions and Ranking 101
4.1.1 Uniform Distribution 102
4.1.1.1 Continuous Uniform Distribution Characteristics 103
4.1.1.2 Discrete Uniform Distribution Characteristics 104
4.1.1.3 EXTRA: Moment Generating Function and Characteristic Function 106
4.1.2 Beta Distribution or Rank Distribution 106
4.1.2.1 Beta Distribution Characteristics 107
4.1.2.2 EXTRA: Moment Generating Function and Characteristic Function 111
4.2 Extreme Value Statistics 112
4.2.1 Weibull Distribution 113
4.2.1.1 Weibull-2 Distribution 113
4.2.1.2 Weibull-2 DistributionMoments and Mean 115
4.2.1.3 Weibull-2 Distribution Characteristics 117
4.2.1.4 EXTRA: Moment Generating Function 117
4.2.2 Weibull-3 Distribution 117
4.2.3 Weibull-1 Distribution 118
4.2.4 Exponential Distribution 119
4.2.4.1 Exponential Distribution and Average Hazard Rate 119
4.2.4.2 Exponential Distribution Characteristics 123
4.3 Mean and Variance Statistics 124
4.3.1 Normal Distribution 124
4.3.1.1 Characteristics of the Normal Distribution 125
4.3.1.2 EXTRA: Moments, Moment Generating Function and Characteristic
Function 127
4.3.1.3 EXTRA: Central LimitTheorem 128
4.3.2 Lognormal Distribution 131
4.3.2.1 Characteristics of the Lognormal Distribution 133
4.3.2.2 EXTRA: Moment Generating Function and Characteristic Function 134
4.3.2.3 Lognormal versus Weibull 134
4.4 Frequency and Hit Statistics 134
4.4.1 Binomial Distribution 135
4.4.1.1 Mean and Variance 138
4.4.1.2 Characteristics of the Binomial Distribution 138
4.4.1.3 EXTRA: Moment Generating Function 139
4.4.2 Poisson Distribution 140
4.4.2.1 Characteristics of the Poisson Distribution 140
4.4.2.2 Derivation of the Poisson Distribution 141
4.4.2.3 Homogeneous Poisson Process 141
4.4.2.4 Non-Homogeneous Poisson Process 142
4.4.2.5 Poisson versus Binomial Distribution 143
4.4.3 Hypergeometric Distribution 144
4.4.3.1 Mean and Variance of the Hypergeometric Distribution 145
4.4.3.2 Characteristics of the Hypergeometric Distribution 146
x Contents
4.4.4 Normal Distribution Approximation of the Binomial Distribution 147
4.4.5 Multinomial Distribution 149
4.4.5.1 Mean, Variances and Moment Generating Function 150
4.4.6 Multivariate Hypergeometric Distribution 151
4.5 Summary 152
5 Graphical Data Analysis 157
5.1 Data Quality 158
5.2 Parameter-Free Graphical Analysis 158
5.2.1 Basic Graph of a Population Sample 159
5.2.2 Censored Data 160
5.2.3 Kaplan–Meier Plot 165
5.2.4 Confidence Intervals Around a Known Distribution 168
5.2.5 Confidence Intervals with Data 172
5.2.6 Alternative Confidence Intervals 174
5.3 Model-Based or Parametric Graphs 176
5.4 Weibull Plot 178
5.4.1 Weibull Plot with Expected Plotting Position 179
5.4.2 Weibull Plot with Median Plotting Position 182
5.4.3 Weibull Plot with Expected Probability Plotting Position 182
5.4.4 Weibull Plot with Censored Data 183
5.4.5 Confidence Intervals inWeibull Plots 185
5.5 Exponential Plot 188
5.5.1 Exponential Plot with Expected Plotting Position 188
5.5.2 Exponential Plot with Median Plotting Position 189
5.5.3 Exponential Plot with Censored Data 190
5.5.4 Exponential Plot with Confidence Intervals 190
5.6 Normal Distribution 193
5.6.1 Normal Plot with Expected Plotting Position 194
5.6.2 Normal Probability Plot with Confidence Intervals 196
5.6.3 Normal Plot and Lognormal Data 196
5.7 Power Law Reliability Growth 197
5.7.1 Duane and Crow AMSAA Plots and Models 197
5.7.2 NHPP Model in Duane and Crow AMSAA Plots 200
5.8 Summary 202
6 Parameter Estimation 207
6.1 General Aspects with Parameter Estimation 207
6.1.1 Fundamental Properties of Estimators 209
6.1.1.1 Bias 209
6.1.1.2 Efficiency 209
6.1.1.3 Consistency 210
6.1.2 WhyWork with Small Data Sets? 211
6.1.3 Asymptotic Behaviour of Estimators 212
6.2 Maximum Likelihood Estimators 212
6.2.1 ML with Uncensored Data 213
6.2.2 ML for Sets Including Censored Data 214
Contents xi
6.2.3 ML for theWeibull Distribution 214
6.2.3.1 ML Estimators for Weibull-2 Uncensored Data 215
6.2.3.2 ML Estimators for Weibull-2 Censored Data 215
6.2.3.3 Expected ML Estimators for theWeibull-2 Distribution 216
6.2.3.4 Formulas for Bias and Scatter 218
6.2.3.5 Effect of the ML Estimation Bias in Case ofWeibull-2 220
6.2.4 ML for the Exponential Distribution 221
6.2.5 ML for the Normal Distribution 222
6.3 Linear Regression 223
6.3.1 The LR Method 223
6.3.1.1 LR by Unweighted Least Squares 224
6.3.1.2 LR byWeighted Least Squares 228
6.3.1.3 LR with Censored Data 231
6.3.1.4 LR with Fixed Origin 232
6.3.1.5 Which is the (Co)variable? 233
6.3.2 LR for theWeibull Distribution 234
6.3.2.1 LR by Unweighted LS for theWeibull Distribution 235
6.3.2.2 LR byWeighted LS for theWeibull Distribution 236
6.3.2.3 Processing Censored Data with the Adjusted Rank Method 238
6.3.2.4 EXTRA: Processing Censored Data with the Adjusted Plotting Position
Method 240
6.3.2.5 Expected LS andWLS Estimators for theWeibull-2 Distribution 242
6.3.2.6 Formulas for Bias and Scatter for LS andWLS 243
6.3.2.7 Comparison of Bias and Scatter in LS,WLS and ML 246
6.3.3 LR for the Exponential Distribution 249
6.3.3.1 LR by Unweighted LS for the Exponential Distribution 251
6.3.3.2 LR byWeighted LS for the Exponential Distribution 252
6.3.3.3 Processing Censored Data with the Adjusted Rank Method 253
6.3.3.4 EXTRA: Processing Censored Data with the Adjusted Plotting Position
Method 255
6.3.3.5 Expected LS andWLS Estimator for the Exponential Distribution 256
6.3.4 LR for the Normal Distribution 257
6.3.4.1 LR by Unweighted LS for the Normal Distribution 258
6.3.4.2 Processing Censored Data with the Adjusted Rank Method 260
6.3.4.3 EXTRA: Processing Censored Data with the Adjusted Plotting Position
Method 261
6.3.4.4 Expected LS Estimators for the Normal Distribution 262
6.3.5 LR Applied to Power Law Reliability Growth 263
6.4 Summary 263
7 Systemand Component Reliability 267
7.1 The Basics of System Reliability 267
7.2 Block Diagrams 268
7.3 Series Systems 269
7.4 Parallel Systems and Redundancy 272
7.5 Combined Series and Parallel Systems, Common Cause 273
7.6 EXTRA: Reliability and Expected Life of k-out-of-n Systems 276
xii Contents
7.7 Analysis of Complex Systems 277
7.7.1 ConditionalMethod 278
7.7.2 Up-tableMethod 280
7.7.3 EXTRA: Minimal Paths and Minimum Blockades 283
7.8 Summary 285
8 SystemStates, Reliability and Availability 291
8.1 States of Components and Systems 291
8.2 States and Transition Rates of One-Component Systems 292
8.2.1 One-Component System with Mere Failure Behaviour 293
8.2.2 One-Component System with Failure and Repair Behaviour 294
8.3 System State Probabilities via Markov Chains 297
8.3.1 Component and System States 298
8.3.2 System States and Transition Rates for Failure and Repair 300
8.3.3 Differential Equations Based on the State Diagram 301
8.3.4 Differential Equations Based on the Transition Matrix 302
8.4 Markov–Laplace Method for Reliability and Availability 303
8.5 Lifetime with Absorbing States and Spare Parts 306
8.6 Mean Lifetimes MTTFF and MTBF 310
8.7 Availability and Steady-State Situations 312
8.8 Summary 314
9 Application to Asset and IncidentManagement 317
9.1 Maintenance Styles 317
9.1.1 Period-Based Maintenance Optimization for Lowest Costs 317
9.1.1.1 Case Description 317
9.1.1.2 References to Introductory Material 318
9.1.1.3 PBM Cost Optimization Analysis 318
9.1.1.4 Remarks 320
9.1.2 Corrective versus Period-Based Replacement and Redundancy 321
9.1.2.1 Case Description 322
9.1.2.2 References to Introductory Material 322
9.1.2.3 Analysis of Corrective versus Period-Based Replacement and
Redundancy 323
9.1.2.4 Remarks 326
9.1.3 Condition-Based Maintenance 326
9.1.3.1 References to Introductory Material 327
9.1.3.2 Analysis of Condition versus Period-Based Replacement 327
9.1.3.3 Remarks 329
9.1.4 Risk-Based Maintenance 330
9.1.4.1 References to Introductory Material 330
9.1.4.2 Analysis of Risk versus Condition-Based Maintenance 331
9.1.4.3 Remarks 333
9.2 Health Index 334
9.2.1 General Considerations of Health Index 334
9.2.1.1 References to Introductory Material on Health Index Concept
Considerations 335
Contents xiii
9.2.1.2 Analysis of the Health Index Concept 335
9.2.1.3 Remarks 336
9.2.2 Combined Health Index 337
9.2.2.1 References to Introductory Material 337
9.2.2.2 Analysis of the Combined Health Index Concept 337
9.3 Testing and Quality Assurance 338
9.3.1 Accelerated Ageing to Reduce Child Mortality 338
9.3.2 Tests with Limited Test Object Numbers and Sizes 339
9.4 Incident Management (Determining End of Trouble) 342
9.4.1 Component Failure Data and Confidence Intervals 342
9.4.1.1 References to Introductory Material 343
9.4.1.2 Analysis of the Case 343
9.4.1.3 Remarks 345
9.4.2 Failures in a Cable with Multiple Problems and Stress Levels 347
9.4.2.1 References to Introductory Material 348
9.4.2.2 Analysis of the Case 348
9.4.2.3 Remarks 352
9.4.3 Case of Cable Joints with Five Early Failures 352
9.4.3.1 References to Introductory Material 353
9.4.3.2 Analysis of the Case 353
9.4.3.3 Prognosis Using aWeibull Plot and Confidence Intervals 354
9.4.3.4 Estimation of Sample Size Using the Similarity Index 356
9.4.3.5 Redundancy and Urgency 359
9.4.3.6 Remarks 359
9.4.4 Joint Failure Data with Five Early Failures and Large Scatter 360
9.4.4.1 References to Introductory Material 360
9.4.4.2 Analysis of the Case 361
9.4.4.3 Prognosis Using aWeibull Plot and Confidence Intervals 361
9.4.4.4 Estimation of Sample Size Using the Similarity Index 364
9.4.4.5 Remarks 364
10 Miscellaneous Subjects 367
10.1 Basics of Combinatorics 367
10.1.1 Permutations and Combinations 367
10.1.2 The Gamma Function 368
10.2 Power Functions and Asymptotic Behaviour 369
10.2.1 Taylor and Maclaurin Series 370
10.2.2 Polynomial Fitting 371
10.2.2.1 Polynomial Interpolation 371
10.2.2.2 Polynomial and Linear Regression 374
10.2.3 Power Function Fitting 378
10.3 Regression Analysis 380
10.4 Sampling from a Population and Simulations 386
10.4.1 Systematic Sampling 387
10.4.2 Numerical Integration and Expected Values 389
10.4.3 Ranked Samples with Size n and Confidence Limits 396
10.4.3.1 Behaviour of Population Fractions 396
xiv Contents
10.4.3.2 Confidence Limits for Population Fractions 399
10.4.4 Monte Carlo Experiments and Random Number Generators 401
10.4.5 Alternative Sampling and Fractals 405
10.5 Hypothesis Testing 407
10.6 Approximations for the Normal Distribution 408
10.6.1 Power Series 409
10.6.2 Power Series Times Density f (y) 409
10.6.3 Inequalities for Boxing R(y) and h(y) for Large y 410
10.6.4 Polynomial Expression for F(y) 410
10.6.5 Power Function for the Reliability Function R(y) 410
10.6.6 Wrap-up of Approximations 412
Appendix A Weibull Plot 413
Appendix B Laplace Transforms 415
Appendix C Taylor Series 417
Appendix D SI Prefixes 419
Appendix E Greek Characters 421
Appendix F StandardWeibull and Exponential Distribution 423
Appendix G Standardized Normal Distribution 429
Appendix H Standardized Lognormal Distribution 435
Appendix I Gamma Function 441
Appendix J Plotting Positions 447
J.1 Expected Ranked Probability <Fi,n> for n = 1,…, 30 448
J.2 Expected Ranked Probability <Fi,n> for n = 31,…, 45 449
J.3 Expected Ranked Probability <Fi,n> for n = 45,…, 60 450
J.4 Median Ranked Probability FM,i,n for n =1,…, 30 452
J.5 Median Ranked Probability FM,i,n for n = 31,…, 45 453
J.6 Median Ranked Probability FM,i,n for n =46,…, 60 454
J.7 Probability of Expected Ranked Weibull Plotting Position F(<Zi,n>) for
n = 1,…, 30 456
J.8 Probability of Expected Ranked Weibull Plotting Position F(<Zi,n>) for
n = 31,…, 45 457
J.9 Probability of Expected Ranked Weibull Plotting Position F(<Zi,n>) for
n = 46,…, 60 458
J.10 Expected Ranked Weibull Plotting Position <Zi,n> for n = 1,…, 30 460
J.11 Expected Ranked Weibull Plotting Position <Zi,n> for n = 31,…, 45 461
J.12 Expected Ranked Weibull Plotting Position <Zi,n> for n = 46,…, 60 462
J.13 Weights for Linear Regression of Weibull-2 Data for n = 1,…, 30 464
Contents xv
J.14 Weights for Linear Regression of Weibull-2 Data for n = 31,…, 45 465
J.15 Weights for Linear Regression of Weibull-2 Data for n = 46,…, 60 466
References 469
Index 473