# Reliability Analysis for Asset Management of Electric Power Grids

# Reliability Analysis for Asset Management of Electric Power Grids

ISBN: 978-1-119-12519-8

Dec 2018

520 pages

$104.99

Product not available for purchase

## 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.

Table of contents page

**1 Introduction 17**

1.1 Electric power grids 17

1.2 Asset Management of Electric Power Grids 18

1.3 Maintenance styles 19

1.3.1 Corrective Maintenance 21

1.3.1.1 CM inspections 21

1.3.1.2 CM servicing 21

1.3.1.3 CM replacement 21

1.3.1.4 Evaluation of CM 22

1.3.2 Period Based Maintenance 22

1.3.2.1 PBM inspections 23

1.3.2.2 PBM servicing 23

1.3.2.3 PBM replacement 23

1.3.2.4 Evaluation of PBM 24

1.3.3 Condition Based Maintenance 24

1.3.3.1 CBM inspections 25

1.3.3.2 CBM servicing 26

1.3.3.3 CBM replacement 26

1.3.3.4 Introduction to the Health Index 26

1.3.3.5 Evaluation of CBM 27

1.3.4 Risk Based Maintenance 28

1.3.4.1 Corporate Business Values and the Risk Matrix 29

1.3.4.2 RBM inspections 31

1.3.4.3 RBM servicing 32

1.3.4.4 RBM replacement 32

1.3.4.5 Evaluation of RBM 32

1.3.5 Comparison of maintenance styles 32

1.4 Incident management 34

1.5 Summary asset management 35

1.6 Questions for Chapter 1 36

**2 Basics of statistics and Probability 38**

2.1 Outcomes, sample space and events 38

2.2 Probability of events 42

2.3 Probability versus statistical distributions 43

2.4 Fundamental statistical functions 45

2.4.1 Failure distribution F 46

2.4.2 Reliability R 47

2.4.3 Probability or distribution density f 47

2.4.4 Probability or distribution mass f 48

2.4.5 Hazard rate h and the bath tub model 48

2.4.6 Cumulative hazard function H 50

2.5 Mixed distributions 51

2.5.1 Competing processes 51

2.5.2 Inhomogeneous populations 53

2.5.2.1 Theorem of Bayes 55

2.5.2.2 Failure distribution of an inhomogeneous population 57

2.5.3 Early failures interpreted as infant mortality 60

2.6 Multivariate distributions and Power Law 62

2.6.1 Ageing dose and Power law 62

2.6.2 Accelerated ageing 65

2.6.3 Multi-stress ageing 68

2.6.4 Cumulative distribution, ageing dose, and CBM 69

2.7 Summary 71

2.8 Questions for Chapter 2 73

**3 Measures in statistics 74**

3.1 Expected values and moments 74

3.1.1 Operations and means 76

3.1.2 Bayesian mean 77

3.1.3 The moments of a distribution 78

3.1.4 Moment generating function* 78

3.1.5 Characteristic function* 79

3.1.6 Central moments of a distribution 81

3.1.7 The first four central and normalized moments 81

3.1.8 Mean, standard deviation and variance of a sample 83

3.2 Median and other quantiles 85

3.3 Mode 86

3.4 Merits of mean, median and modal value 87

3.5 Measures for comparing distributions 88

3.5.1 Covariance 88

3.5.2 Correlation 91

3.5.3 Cross-correlation and autocorrelation 92

3.6 Similarity of distributions and compliance 93

3.6.1 Similarity of counting in discrete sets 94

3.6.2 Similarity of two discrete distributions 96

3.6.3 Similarity of two continuous distributions 98

3.6.4 Significance of similarity 102

3.6.5 Singularity issues and alternative similarity indices 105

3.7 Compliance 106

3.8 Summary 107

3.9 Questions for Chapter 3 108

**4 Specific distributions 110**

4.1 Fractions and ranking 110

4.1.1 Uniform distribution 111

4.1.1.1 Continuous uniform distribution characteristics 111

4.1.1.2 Discrete uniform distribution characteristics 113

4.1.1.3 Moment generating function and characteristic function* 114

4.1.2 Beta distribution or Rank distribution 116

4.1.2.1 Beta distribution characteristics 116

4.1.2.2 Moment generating function and characteristic function* 120

4.2 Extreme value statistics 121

4.2.1 Weibull distribution 122

4.2.1.1 Weibull-2 distribution 122

4.2.1.2 Weibull-2 distribution moments and mean 124

4.2.1.3 Weibull-2 distribution characteristics 126

4.2.1.4 Moment generating function* 126

4.2.2 Weibull-3 distribution 127

4.2.3 Weibull-1 distribution 128

4.2.4 Exponential distribution 128

4.2.4.1 Exponential distribution and average hazard rate 128

4.2.4.2 Exponential distribution characteristics 133

4.3 Mean and variance statistics 133

4.3.1 Normal distribution 133

4.3.1.1 Characteristics of the Normal distribution 135

4.3.1.2 Moments, moment generating function and characteristic function* 137

4.3.1.3 Central limit theorem* 138

4.3.2 Lognormal distribution 141

4.3.2.1 Characteristics of the Lognormal distribution 143

4.3.2.2 Moment generating function and characteristic function* 144

4.3.2.3 Lognormal versus Weibull 144

4.4 Frequency and hit statistics 145

4.4.1 Binomial distribution 146

4.4.1.1 Mean and variance 148

4.4.1.2 Characteristics of the Binomial distribution 149

4.4.1.3 Moment generating function* 150

4.4.2 Poisson distribution 151

4.4.2.1 Characteristics of the Poisson distribution 151

4.4.2.2 Derivation of the Poisson distribution 152

4.4.2.3 Homogeneous Poisson Process 152

4.4.2.4 Non-Homogeneous Poisson Process 153

4.4.2.5 Poisson versus Binomial distribution 154

4.4.3 Hypergeometric Distribution 155

4.4.3.1 Mean and variance of the Hypergeometric distribution 156

4.4.3.2 Characteristics of the Hypergeometric distribution 158

4.4.4 Normal distribution approximation of the Binomial distribution 159

4.4.5 Multinomial distribution 160

4.4.5.1 Mean, variances and moment generating function 162

4.4.6 Multivariate Hypergeometric distribution 162

4.5 Summary 163

4.6 Questions for Chapter 4 164

**5 Graphical Data-analysis 167**

5.1 Data quality 167

5.2 Parameter-free graphical analysis 168

5.2.1 Basic graph of a population sample 168

5.2.2 Censored data 170

5.2.3 Kaplan-Meier plot 174

5.2.4 Confidence intervals around a known distribution 177

5.2.5 Confidence intervals with data 181

5.2.6 Alternative confidence intervals 182

5.3 Model based or parametric graphs 184

5.4 Weibull plot 186

5.4.1 Weibull plot with expected plotting position 187

5.4.2 Weibull plot with median plotting position 190

5.4.3 Weibull plot with expected probability plotting position 191

5.4.4 Weibull plot with censored data 192

5.4.5 Confidence intervals in Weibull plots 194

5.5 Exponential plot 196

5.5.1 Exponential plot with expected plotting position 196

5.5.2 Exponential plot with median plotting position 197

5.5.3 Exponential plot with censored data 198

5.5.4 Exponential plot with confidence intervals 198

5.6 Normal distribution 201

5.6.1 Normal plot with expected plotting position 202

5.6.2 Normal probability plot with confidence intervals 205

5.6.3 Normal plot and Lognormal data 205

5.7 Power law reliability growth 205

5.7.1 Duane and Crow AMSAA plots and models 205

5.7.2 NHPP model in Duane and Crow-AMSAA plots 209

5.8 Summary 211

5.9 Questions 212

**6 Parameter estimation 215**

6.1 General aspects with parameter estimation 215

6.1.1 Fundamental properties of estimators 216

6.1.1.1 Bias 216

6.1.1.2 Efficiency 217

6.1.1.3 Consistency 218

6.1.2 Why working with small data sets? 219

6.1.3 Asymptotic behaviour of estimators 220

6.2 Maximum Likelihood estimators 220

6.2.1 ML with uncensored data 220

6.2.2 ML for sets including censored data 221

6.2.3 ML for the Weibull distribution 222

6.2.3.1 ML estimators for Weibull-2 uncensored data 222

6.2.3.2 ML estimators for Weibull-2 censored data 223

6.2.3.3 Expected ML estimators for the Weibull-2 distribution 224

6.2.3.4 Formulas for bias and scatter 226

6.2.3.5 Effect of the ML estimation bias in case of Weibull-2 228

6.2.4 ML for the Exponential distribution 229

6.2.5 ML for the Normal distribution 230

6.3 Linear Regression 231

6.3.1 The LR method 231

6.3.1.1 LR by unweighted least squares 232

6.3.1.2 LR by weighted least squares 236

6.3.1.3 LR with censored data 239

6.3.1.4 LR with fixed origin 240

6.3.1.5 Which is the (co)variable? 241

6.3.2 LR for the Weibull distribution 242

6.3.2.1 LR by unweighted LS for the Weibull distribution 243

6.3.2.2 LR by weighted LS for the Weibull distribution 245

6.3.2.3 Processing censored data with the Adjusted Rank Method 246

6.3.2.4 Processing censored data with the Adjusted Plotting Position Method* 248

6.3.2.5 Expected LS and WLS estimators for the Weibull-2 distribution 251

6.3.2.6 Formulas for bias and scatter for LS and WLS 252

6.3.2.7 Comparison of bias and scatter in LS, WLS and ML 255

6.3.3 LR for the Exponential distribution 258

6.3.3.1 LR by unweighted LS for the Exponential distribution 260

6.3.3.2 LR by weighted LS for the Exponential distribution 261

6.3.3.3 Processing censored data with the Adjusted Rank Method 262

6.3.3.4 Processing censored data with the Adjusted Plotting Position Method* 264

6.3.3.5 Expected LS and WLS estimator for the Exponential distribution 265

6.3.4 LR for the Normal distribution 266

6.3.4.1 LR by unweighted LS for the Normal distribution 267

6.3.4.2 Processing censored data with the Adjusted Rank Method 269

6.3.4.3 Processing censored data with the Adjusted Plotting Position Method* 271

6.3.4.4 Expected LS estimators for the Normal distribution 271

6.3.5 LR applied to Power law reliability growth 272

6.4 Summary 272

6.5 Questions for Chapter 6 273

**7 System and component Reliability 275**

7.1 The basics of system reliability 275

7.2 Block diagrams 276

7.3 Series systems 277

7.4 Parallel systems and redundancy 280

7.5 Combined series and parallel systems, common cause 281

7.6 Reliability and expected life of k-out-of-n systems* 283

7.7 Analysis of complex systems 285

7.7.1 Conditional method 286

7.7.2 Up-table method 288

7.7.3 Minimal paths and minimum blockades* 291

7.8 Summary 293

7.9 Questions for Chapter 7 294

**8 System states, reliability and availability 298**

8.1 States of components and systems 298

8.2 States and transition rates of 1 component systems 299

8.2.1 One component system with mere failure behaviour 299

8.2.2 One component system with failure and repair behaviour 301

8.3 System state probabilities via Markov chains 304

8.3.1 Component and system states 305

8.3.2 System states and transition rates for failure and repair 306

8.3.3 Differential equations based on the state diagram 308

8.3.4 Differential equations based on the transition matrix 309

8.4 Markov-Laplace method for reliability and availability 310

8.5 Lifetime with absorbing states and spare parts 312

8.6 Mean lifetimes MTTFF and MTBF 315

8.7 Availability and steady state situations 318

8.8 Summary of Chapter 8 320

8.9 Questions 320

**9 Application to Asset and Incident management 323**

9.1 Maintenance styles 323

9.1.1 Period Based Maintenance optimization for lowest costs 323

9.1.1.1 Case description 323

9.1.1.2 References to introductory material 323

9.1.1.3 PBM cost optimization analysis 324

9.1.1.4 Remarks 327

9.1.2 Corrective versus Period Based Replacement and Redundancy 328

9.1.2.1 Case description 328

9.1.2.2 References to introductory material 329

9.1.2.3 Analysis of Corrective versus Period Based Replacement and redundancy 329

9.1.2.4 Remarks 332

9.1.3 Condition based maintenance 333

9.1.3.1 References to introductory material 333

9.1.3.2 Analysis of Condition versus Period Based Replacement 333

9.1.3.3 Remarks 335

9.1.4 Risk Based Maintenance 336

9.1.4.1 References to introductory material 336

9.1.4.2 Analysis of Risk versus Condition Based Maintenance 336

9.1.4.3 Remarks 339

9.2 Health Index 340

9.2.1 General considerations of Health Index 340

9.2.1.1 References to introductory material on HI concept considerations 340

9.2.1.2 Analysis of the Health Index concept 340

9.2.1.3 Remarks 342

9.2.2 Combined Health Index 342

9.2.2.1 References to introductory material 342

9.2.2.2 Analysis of the Combined Health Index concept 343

9.3 Testing and quality assurance 343

9.3.1 Accelerated ageing to reduce Child mortality 343

9.3.2 Tests with limited test object numbers and sizes 344

9.4 Incident Management (Determining end of trouble) 347

9.4.1 Component failure data and confidence intervals 348

9.4.1.1 References to introductory material on 'component breakdown data' 348

9.4.1.2 Analysis of the case 'Component breakdown data' 348

9.4.1.3 Remarks 351

9.4.2 Failures in a cable with multiple problems and stress levels 352

9.4.2.1 References to introductory material on the case 354

9.4.2.2 Analysis of the case 354

9.4.2.3 Remarks 357

9.4.3 Case of cable joints with five early failures 357

9.4.3.1 References to introductory material on the case 357

9.4.3.2 Analysis of the case 358

9.4.3.3 Prognosis using a Weibull plot and confidence intervals 358

9.4.3.4 Estimation of sample size using the Similarity index 360

9.4.3.5 Redundancy and urgency 363

9.4.3.6 Remarks 364

9.4.4 Joint failure data with five early failures and large scatter 365

9.4.4.1 References to introductory material on the case 365

9.4.4.2 Analysis of the case 365

9.4.4.3 Prognosis using a Weibull plot and confidence intervals 366

9.4.4.4 Estimation of sample size using the similarity index 368

9.4.4.5 Remarks 370

**10 Miscellaneous subjects 371**

10.1 Basics of Combinatorics 371

10.1.1 Permutations and combinations 371

10.1.2 The Gamma function 372

10.2 Power functions and asymptotic behaviour 373

10.2.1 Taylor and Maclaurin series 373

10.2.2 Polynomial fitting 375

10.2.2.1 Polynomial interpolation 375

10.2.2.2 Polynomial and linear regression 378

10.2.3 Power function fitting 381

10.3 Regression analysis 384

10.4 Sampling from a population and simulations 390

10.4.1 Systematic sampling 391

10.4.2 Numerical integration and expected values 395

10.4.3 Ranked samples with size n and confidence limits 400

10.4.3.1 Behaviour of population fractions 401

10.4.3.2 Confidence limits for population fractions 402

10.4.4 Monte Carlo experiments and random number generators 406

10.4.5 Alternative sampling and fractals 409

10.5 Hypothesis testing 410

10.6 Approximations for the Normal distribution 411

**11 References 416**

Appendix A Weibull Plot 420

Appendix B Laplace Transforms 421

Appendix C Taylor Series 422

Appendix D SI Prefixes 423

Appendix E Greek Characters 424

Appendix F Standard Weibull and Exponential Distribution 425

Appendix G Standardized Normal Distribution 430

Appendix H Standardized Lognormal Distribution 435

Appendix I Gamma Function 440

Appendix J Plotting Positions 444