Reliability Evaluation of Engineering Systems:

Transcription

1 Reliability Evaluation of Engineering Systems: Concepts and Techniques Roy Billinton PhD, DSc, FEIC, FRSC, FIEEE, PE c. J. MacKenzie Professor of Electrical Engineering University of Saskatchewan and Ronald N Allan PhD, FSRS, MIEEE, MIEE, CEng Senior Lecturer in Electrical Power Systems University of Manchester Institute of Science and Technology Springer Science+Business Media, LLC

2 Longman Scientific & Technical Longman Group UK Limited, Longman House, Bumt Mill, Harlow, Essex CM20 2JE, England and Associated Companies throughout the world. Co-published in the United States by Plenum Press Springer Science+Business Media New York 1983 Originally published by Plenum Press in 1983 Softcover reprint ofthe hardcover 1 st edition 1983 AII rights reserved; no part of this publicat ion may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publishers. First published in Great Britain by Pitman Publishing Limited 1983 Reprinted 1985 (twice) Reprinted by Longman Scientific & Technical 1987 Library of Congress Cataloging in Publication Data Billinton, Roy. Reliability evaluation of engineering systems. lncludes bibliographical references and index. 1. Reliability (Engineering) I. Alian, Ronald N. (Ronald Norman) II. Title. TA169.B ' ISBN ISBN (ebook) DOI /

3 Contents Preface ix 1 Introduction 1 2 Basic probability theory Probability concepts Permutations and combinations General concepts Permutations Combinations Comparison of permutations and combinations Application in probability evaluation Practical engineering concepts Venn diagrams Rules for combining probabilities Rule I-independent events Rule 2-mutually exclusive events Rule 3-complementary events Rule 4-conditional events Rule 5-simuItaneous occurrence of events Rule 6--occurrence of at least one of two events Rule 7 -application of conditional probability Probability distributions Random variables Density and distribution functions Mathematical expectation Variance and standard deviation Conclusions 34 Problems 35 3 Application of the binomial distribution Binomial distribution concepts Properties of the binomial distribution General characteristics Binomial coefficients 41 iii

4 iv Contents Expected value and standard deviation Engineering applications Restricting the assessment Implication of economics Effect of redundancy Effect of partial output (derated) states Effect of unavailability Effect of one unit in reserve Non-identical capacities Non-identical unavailabilities Conclusions 60 Problems 61 4 Network modelling and evaluation of simple systems Network modelling concepts Series systems Parallel systems Series-parallel systems Partially redundant systems Standby redundant systems Redundancy concepts Perfect switching Imperfect switching Standby redundancy calculations Conclusions 79 Problems 79 5 Network modelling and evaluation of complex systems Modelling and evaluation concepts Conditional probability approach Cut set method Cut set concepts Application of cut sets Approximate evaluation Deducing the minimal cut sets Application and comparison of previous techniques Tie set method Connection matrix techniques Event trees General concepts Continuously operated systems Cut and tie set deduction Standby and sequential logic systems Fault trees 113

5 Contents v 5.9 Multi-failure modes Conclusions 120 Problems Probability distributions in reliability evaluation Distribution concepts Terminology of distributions General reliability functions Evaluation of the reliability functions Shape of reliability functions The Poisson distribution General concepts Derivation of the Poisson distribution Relationship with the binomial distribution The normal distribution General concepts Probability density function Evaluation of probabilities The exponential distribution General concepts Reliability functions A posteriori failure probability Mean value and standard deviation The Weibull distribution The gamma distribution The Rayleigh distribution The lognormal distribution The rectangular (or uniform) distribution Summary of reliability functions Conclusions 167 Problems System reliability evaluation using probability distributions Introduction Series systems Parallel systems Partially redundant systems Mean time to failure Standby systems General concepts Perfect switching Imperfect switching Effect of spare components Non-identical components 188

6 vi Contents Failures in the standby mode Wearout and component reliability Maintenance and component reliability Conclusions 202 Problems Disel'ete Markov chains Introduction General modelling concepts Stochastic transitional probability matrix Time dependent probability evaluation Limiting state probability evaluation Absorbing states Application of discrete Markov techniques Conclusions 222 Problems Continuous Markov processes Introduction General modelling concepts Transition rate concepts Evaluating time dependent probabilities Evaluating limiting state probabilities State space diagrams General concepts Single repairable component Two repairable components Three component system Larger number of components Standby redundant systems Mission-orientated systems Stochastic transitional probability matrix Evaluating limiting state probabilities Single repairable component Two identical repairable components Evaluating time dependent state probabilities Differential equations method Matrix multiplication method Reliability evaluation in repairable systems Mean time to failure Evaluation concepts Stochastic transitional probability matrix method Application of techniques to complex systems Conclusions 250 Problems 251

7 Contents vii 10 Frequency and duration techniques Introduction Frequency and duration concepts Application to multi-state problems Two component repairable system State probabilities Frequency of encountering individual states Mean duration of individual states Cycle time between individual states Frequency of encountering cumulated states Recursive evaluation of cumulative frequency Mean duration of cumulated states Frequency balance approach Two stage repair and installation process General concepts One component system-no spare available One component system-one spare available One component system-two spares available Two component system-one spare available Limiting number of spares Application of the techniques Conclusions 280 Problems Approximate system reliability evaluation Introduction Series systems Parallel systems Two component system Systems with more than two components Network reduction techniques Minimal cut set/failure modes approach Inclusion of scheduled maintenance Common mode failures General concepts Modelling and evaluation techniques Conclusions 301 Problems Systems with non-exponential distnbutions Introduction Method of stages Stages in series Stages in parallel Series stages in series with two parallel stages 307

8 viii Contents 12.6 Time dependent and limiting state probabilities Conclusions 312 Problem Epilogue 314 Appendix I-Rules of Boolean algebra 31S Appendix 2-The normal distribution function 316 Appendix 3-Elementary matrix algebra 317 A3.1 Concepts of matrices 317 A3.2 Square matrix 317 A3.3 Column matrix (or vector) 317 A3.4 Row matrix (or vector) 318 A3.S Transposed matrix 318 A3.6 Diagonal matrix 318 A3.7 Identity (or unit) matrix 318 A3.8 Symmetric matrix 319 A3.9 Determinant of a matrix 319 A3.10 Co-factors 319 A3.11 Evaluation of determinants 319 A3.12 Addition of matrices 320 A3.13 Subtraction of matrices 321 A3.14 Multiplication of matrices 321 A3.1S Multiplication by a constant 322 A3.16 Inverse of a matrix 322 A3.17 Solution of simultaneous equations 323 A3.18 Cramer's rule for solving simultaneous equations Appendix 4-Differential equations and Laplace transforms A4.1 Differential equations 326 A4.2 Laplace transforms 326 A4.3 Solving differential equations using Laplace transforms 328 Appendix S-Confidence levels and limits 331 AS.l Introduction 331 AS.2 Unavailability at selected confidence levels 331 AS.3 Failure rate at selected confidence levels 334 AS.4 Conclusions 338 Problems 338 References 339 Solutions 342 Index 34S

9 Preface This book has evolved from our deep interest and involvement in the development and application of reliability evaluation techniques. Its scope is not limited to anyone engineering discipline as the concepts and basic techniques for reliability evaluation have no disciplinary boundaries and are applicable in most, if not all, engineering applications. We firmly believe that reliability evaluation is an important and integral feature of the planning, design and operation of all engineering systems; from the smallest and most simple to the largest and most complex. Also, we believe that all engineers involved with such systems should be aware of, and appreciate, not only the benefits which can accrue from reliability assessment, but also how such assessments can be made. Our primary objective has been to compile a book which provides practising engineers and engineering graduates who have little or no background in probability theory or statistics, with the concepts and basic techniques for evaluating the reliability of engineering systems. It is hoped that the material presented will enable them to reach quickly a level of self-confidence which will permit them to assimilate, understand and appreciate the more detailed applications and additional material which is available in the journals and publications associated with their own discipline. We have attempted to structure the book in such a way that only one new concept or technique is introduced and applied at a time. We have also made frequent use of numerical examples to consolidate the concepts and techniques. We believe that this structure will permit the reader to become confident with the application and understanding of reliability evaluation and enable the book to be used either as a self-tutorial text or as the text for a formally taught reliability evaluation course at undergraduate and postgraduate level. It would not have been possible for us to have written this book without our involvement and close association with many individuals and organizations: the students who have been on our respective (post) graduate research programmes, our colleagues on IEEE, CEA and lee committees and at our respective universities, and the engineers in the various industries with which we have been involved. Several typists have helped in the preparation of the manuscripts and ix

10 x Preface we would like to express our appreciation to them as a group. Finally, but by no means least, we would like to thank our respective wives, Joyce and Diane, for their perseverance and constant encouragement and for their assistance in reading and checking the manuscript. August 1982 Roy Billinton Ron Allan