Multistate Systems Reliability Theory with Applications

WILEY SERIES IN PROBABILITY AND STATISTICS

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Library of Congress Cataloging-in-Publication Data

Natvig, Bent, 1946-

Multistate systems reliability theory with applications / Bent Natvig.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-69750-4 (cloth)

1. Reliability (Engineering) 2. Stochastic systems. I. Title.

TA169.N38 2011

620′.00452—dc22

2010034245

A catalogue record for this book is available from the British Library.

Print ISBN: 978-0-470-69750-4

ePDF ISBN: 978-0-470-97707-1

oBook ISBN: 978-0-470-97708-8

ePub ISBN: 978-0-470-97713-2

To my Mother, Kirsten, for giving me jigsaws instead of cars, and to Helga for letting me (at least for the last 40 years) run in the sun and work when normal people relax.

Preface

In the magazine Nature (1986) there was an article on a near catastrophe on the night of 14 April 1984 in a French pressurized water reactor (PWR) at Le Bugey on the Rhône river, not far from Geneva. This incident provides ample motivation for multistate reliability theory.

‘The event began with the failure of the rectifier supplying electricity to one of the two separate 48 V direct-current control circuits of the 900 MW reactor which was on full power at the time. Instantly, a battery pack switched on to maintain the 48 V supply and a warning light began to flash at the operators in the control room. Unfortunately, the operators ignored the light (if they had not they could simply have switched on an auxiliary rectifier).

What then happened was something that had been completely ignored in the engineering risk analysis for the PWR. The emergency battery now operating the control system began to run down. Instead of falling precipitously to zero, as assumed in the all or nothing risk analysis, the voltage in the control circuit steadily slipped down from its nominal 48 V to 30 V over a period of three hours. In response a number of circuit breakers began to trip out in an unpredictable fashion until finally the system, with the reactor still at full power, disconnected itself from the grid.

The reactor was now at full power with no external energy being drawn from the system to cool it. An automatic scram system then correctly threw in the control rods, which absorbed neutrons and shut off the nuclear reaction. However, a reactor in this condition is still producing a great deal of heat −300 MW in this case. An emergency system is then supposed to switch in a diesel generator to provide emergency core cooling (otherwise the primary coolant would boil and vent within a few hours). But the first generator failed to switch on because of the loss of the first control circuit. Luckily the only back-up generator in the system then switched on, averting a serious accident.’

Furthermore, Nature writes: ‘The Le Bugey incident shows that a whole new class of possible events had been ignored—those where electrical systems fail gradually. It shows that risk analysis must not only take into account a yes or no, working or not working, for each item in the reactor, but the possibility of working with a slightly degraded system.’

This book is partly a textbook and partly a research monograph, also covering research on the way to being published in international journals. The first two chapters, giving an introduction to the area and the basics, are accompanied by exercises. In a course, these chapters should at least be combined with the two first and the three last sections of Chapter 3. This will cover basic bounds in a time interval for system availabilities and unavailabilities given the corresponding component availabilities and unavailabilities, and how the latter can be arrived at. In addition come applications to a simple network system and an offshore electrical power generation system. This should be followed up by Chapter 4, giving a more in-depth application to an offshore gas pipeline network.

The rest of Chapter 3 first gives improved bounds both in a time interval and at a fixed point of time for system availabilities and unavailabilities, using modular decompositions, leaving some proofs to Appendix 1. Some of the results here are new. Strict and exactly correct bounds are also covered in the rest of this chapter.

In Chapter 5 component availabilities and unavailabilities in a time interval are considered uncertain and a new theory for Bayesian assessment of system availabilities and unavailabilities in a time interval is developed including a simulation approach. This is applied to the simple network system. Chapter 6 gives a new theory for measures of importance of system components covering generalizations of the Birnbaum, Barlow–Proschan and Natvig measures from the binary to the multistate case both for unrepairable and repairable systems. In Chapter 7 a corresponding numerical study is given based on an advanced discrete event simulation program. The numerical study covers two three-component systems, a bridge system and an offshore oil and gas production system.

Chapter 8 is concerned with probabilistic modeling of monitoring and maintenance based on a marked point process approach taking the dynamics into account. The theory is illustrated using the offshore electrical power generation system. We also describe how a standard simulation procedure, the data augmentation method, can be implemented and used to obtain a sample from the posterior distribution for the parameter vector without calculating the likelihood explicitly.

Which of the material in the rest of Chapter 3 and in Chapters 5–8 should be part of a course is a matter of interest and taste.

A reader of this book will soon discover that it is very much inspired by the Barlow and Proschan (1975a) book, which gives a wonderful tour into the world of binary reliability theory. I am very much indebted to Professor Richard Barlow for, in every way, being the perfect host during a series of visits, also by some of my students, to the University of California at Berkeley during the 1980s. I am also very thankful to Professor Arnljot Høyland, at what was then named The Norwegian Institute of Technology, for letting me, in the autumn of 1976, give the first course in moderny reliability theory in Norway based on the Barlow and Proschan (1975a) book.

Although I am the single author of the book, parts of it are based on important contributions from colleagues and students to whom I am very grateful. Government grant holder Jørund Gåsemyr is, for instance, mainly responsible for the new results in Chapter 3, for the simulation approach in Chapter 5 and is the first author of the paper on which Chapter 8 is based. Applying the data augmentation approach was his idea. Associate Professor Arne Bang Huseby, my colleague since the mid 1980s, has developed the advanced discrete event simulation program necessary for the numerical study in Chapter 7. The extensive calculations have been carried through by my master student Mads Opstad Reistadbakk. Furthermore, the challenging calculations in Chapter 5 were done by post. doc Trond Reitan. The offshore electrical power generation system case study in Chapter 3 is based on a suggestion by Professor Arne T. Holen at The Norwegian Institute of Technology and developed as part of master theses by my students Skule Sørmo and Gutorm Høgåsen. Finally, Chapter 4 is based on the master thesis of my student Hans Wilhelm Mørch, heavily leaning on the computer package MUSTAFA (MUltiSTAte Fault-tree Analysis) developed by Gutorm Høgåsen as part of his PhD thesis.

Finally I am thankful to the staff at John Wiley & Sons, especially Commissioning Editor, Statistics and Mathematics Dr Ilaria Meliconi for pushing me to write this book. I must admit that I have enjoyed it all the way.

Bent Natvig

Oslo, Norway

Acknowledgements

We acknowledge Applied Probability for giving permission to include Tables 2.1, 2.2, 2.4 and 2.15 from Natvig (1982a), Figure 1.1 and Tables 1.1 and 3.1 from Funnemark and Natvig (1985) and Figure 1.2 and Tables 2.7–2.10, 3.6 and 3.7 from Natvig et al. (1986). In addition we acknowledge World Scientific Publishing for giving permission to include Figure 4.1 and Tables 4.1–4.15 from Natvig and Mørch (2003).

List of Abbreviations

1

Introduction

In reliability theory a key problem is to find out how the reliability of a complex system can be determined from knowledge of the reliabilities of its components. One inherent weakness of traditional binary reliability theory is that the system and the components are always described just as functioning or failed. This approach represents an oversimplification in many real-life situations where the system and their components are capable of assuming a whole range of levels of performance, varying from perfect functioning to complete failure. The first attempts to replace this by a theory for multistate systems of multistate components were done in the late 1970s in Barlow and Wu (1978), El-Neweihi et al. (1978) and Ross (1979). This was followed up by independent work in Griffith (1980), Natvig (1982a), Block and Savits (1982) and Butler (1982) leading to proper definitions of a multistate monotone system and of multistate coherent systems and also of minimal path and cut vectors. Furthermore, in Funnemark and Natvig (1985) upper and lower bounds for the availabilities and unavailabilities, to any level, in a fixed time interval were arrived at for multistate monotone systems based on corresponding information on the multistate components. These were assumed to be maintained and interdependent. Such bounds are of great interest when trying to predict the performance process of the system, noting that exactly correct expressions are obtainable just for trivial systems. Hence, by the mid 1980s the basic multistate reliability theory was established. A review of the early development in this area is given in Natvig (1985a). Rather recently, probabilistic modeling of partial monitoring of components with applications to preventive system maintenance has been extended by Gåsemyr and Natvig (2005) to multistate monotone systems of multistate components. A newer review of the area is given in Natvig (2007).

The theory was applied in Natvig et al. (1986) to an offshore electrical power generation system for two nearby oilrigs, where the amounts of power that may possibly be supplied to the two oilrigs are considered as system states. This application is also used to illustrate the theory in Gåsemyr and Natvig (2005). In Natvig and Mørch (2003) the theory was applied to the Norwegian offshore gas pipeline network in the North Sea, as of the end of the 1980s, transporting gas to Emden in Germany. The system state depends on the amount of gas actually delivered, but also to some extent on the amount of gas compressed, mainly by the compressor component closest to Emden. Rather recently the first book (Lisnianski and Levitin, 2003) on multistate system reliability analysis and optimization appeared. The book also contains many examples of the application of reliability assessment and optimization methods to real engineering problems. This has been followed up by Lisnianski et al. (2010).

Working on the present book a series of new results have been developed. Some generalizations of bounds for the availabilities and unavailabilities, to any level, in a fixed time interval given in Funnemark and Natvig (1985) have been established. Furthermore, the theory for Bayesian assessment of system reliability, as presented in Natvig and Eide (1987) for binary systems, has been extended to multistate systems. Finally, a theory for measures of component importance in nonrepairable and repairable multistate strongly coherent systems has been developed, and published in Natvig (2011), with accompanying advanced discrete simulation methods and an application to a West African production site for oil and gas.

1.1 Basic Notation and Two Simple Examples

Let S = {0, 1, … ,M} be the set of states of the system; the M + 1 states representing successive levels of performance ranging from the perfect functioning level M down to the complete failure level 0. Furthermore, let C = {1, … ,n} be the set of components and Si, i = 1, … ,n the set of states of the ith component. We claim {0, M} ⊆ Si ⊆ S. Hence, the states 0 and M are chosen to represent the endpoints of a performance scale that might be used for both the system and its components. Note that in most applications there is no need for the same detailed description of the components as for the system.

Let xi, i = 1, … ,n denote the state or performance level of the ith component at a fixed point of time and x = (x1, …, xn). It is assumed that the state, ϕ, of the system at the fixed point of time is a deterministic function of x, i.e. ϕ = ϕ(x). Here x takes values in S1 × S2 × … × Sn and ϕ takes values in S. The function ϕ is called the structure function of the system. We often denote a multistate system by (C, ϕ). Consider, for instance, a system of n components in parallel where Si = {0, M}, i = 1, … ,n. Hence, we have a binary description of component states. In binary theory, i.e. when M = 1, the system state is 1 iff at least one component is functioning. In multistate theory we may let the state of the system be the number of components functioning, which is far more informative. In this case, for M = n,

1.1 1.1

As another simple example consider the network depicted in Figure 1.1. Here component 1 is the parallel module of the branches a1 and b1 and component 2 the parallel module of the branches a2 and b2. For i = 1, 2 let xi = 0 if neither of the branches work, 1 if one branch works