Multistate Systems Reliability Theory with Applications (Wiley Series in Probability and Statistics) - Hardcover

Natvig, Bent

Hardcover

ISBN 10: 0470697504 ISBN 13: 9780470697504

Publisher: Wiley, 2010

Synopsis

Most books in reliability theory are dealing with a description of component and system states as binary: functioning or failed. However, many systems are composed of multi-state components with different performance levels and several failure modes. There is a great need in a series of applications to have a more refined description of these states, for instance, the amount of power generated by an electrical power generation system or the amount of gas that can be delivered through an offshore gas pipeline network.

This book provides a descriptive account of various types of multistate system, bound-for multistate systems, probabilistic modeling of monitoring and maintenance of multistate systems with components along with examples of applications.

Key Features:

• Looks at modern multistate reliability theory with applications covering a refined description of components and system states.
• Presents new research, such as Bayesian assessment of system availabilities and measures of component importance.
• Complements the methodological description with two substantial case studies. 

Reliability engineers and students involved in the field of reliability, applied mathematics and probability theory will benefit from this book.

About this title

About the Author

Bent Natvig is Professor in Mathematics Statistics, University of Oslo.
Member of Research Education Committee, Faculty of Mathematics and Natural Sciences, 2000-. Member of User Committee, G.H. Sverdrup's Building, 2000-. Guest/Invited speaker at numerous lectures around Europe and America for the past 30 years. Speaking on reliability theory and mathematics and statistics. Research interests include: Reliability theory and risk analysis, Bayesian statistics, Bayesian forecasting and dynamic models and queuing theory. Currently Associate Editor of Methodology and Computing in Applied Probability.

From the Back Cover

Most books in reliability theory are dealing with a description of component and system states as binary: functioning or failed. However, many systems are composed of multi-state components with different performance levels and several failure modes. There is a great need in a series of applications to have a more refined description of these states, for instance, the amount of power generated by an electrical power generation system or the amount of gas that can be delivered through an offshore gas pipeline network.

The book provides a descriptive account of various types of multistate system, bound-for multistate systems, probabilistic modeling of monitoring and maintenance of multistate systems with components along with examples of applications.

Key Features:

• Looks at modern multistate reliability theory with applications covering a refined description of components and system states.
• Presents new research, such as Bayesian assessment of system availabilities and measures of component importance.
• Complements the methodological description with two substantial case studies.

Reliability engineers and students involved in the field of reliability, applied mathematics and probability theory will benefit from this book.

From the Inside Flap

Most books in reliability theory are dealing with a description of component and system states as binary: functioning or failed. However, many systems are composed of multi-state components with different performance levels and several failure modes. There is a great need in a series of applications to have a more refined description of these states, for instance, the amount of power generated by an electrical power generation system or the amount of gas that can be delivered through an offshore gas pipeline network.

The book provides a descriptive account of various types of multistate system, bound-for multistate systems, probabilistic modeling of monitoring and maintenance of multistate systems with components along with examples of applications.

Key Features:

• Looks at modern multistate reliability theory with applications covering a refined description of components and system states.
• Presents new research, such as Bayesian assessment of system availabilities and measures of component importance.
• Complements the methodological description with two substantial case studies.

Reliability engineers and students involved in the field of reliability, applied mathematics and probability theory will benefit from this book.

Excerpt. © Reprinted by permission. All rights reserved.

Multistate Systems Reliability Theory with Applications

By Bent Natvig

John Wiley & Sons

Copyright © 2011 John Wiley & Sons, Ltd
All right reserved.
ISBN: 978-0-470-69750-4

Chapter One

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 bihnary 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 ITLITL = {1, ..., n} be the set of components and S_{i, i} = 1, ..., n the set of states of the ith component. We claim {0, M} [??] S_i< [??] 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 x_i, i = 1, ..., n denote the state or performance level of the ith component at a fixed point of time and x = (x₁, ..., x_n). It is assumed that the state, f, of the system at the fixed point of time is a deterministic function of x, i.e. f = f(x). Here x takes values in S₁ x S₂ x ... x S_n and f takes values in S. The function f is called the structure function of the system. We often denote a multistate system by (ITLITL, f). Consider, for instance, a system of n components in parallel where S_i = {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,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (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 x_i = 0 if neither of the branches work, 1 if one branch works and 3 if two branches work. The states of the system are given in Table 1.1.

Note, for instance, that the state 1 is critical both for each component and the system as a whole in the sense that the failing of a branch leads to the 0 state. In binary theory the functioning state comprises the states {1, 2, 3} and hence only a rough description of the system's performance is possible. It is not hard to see that the structure function is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where I (·) is the indicator function.

The following notation is needed throughout the book.

(i, x) = (x₁, ..., x_i-1, ..., x_i+1, ..., x_n). y < x means y_i = x_i for i = 1, ..., n, and y_i < x_i for some i.

Let A [??] C. Then

x^A = vector with elements x_i, i [element of] A,

A^c = subset of ITLITL complementary to A.

1.2 An offshore electrical power generation system

In Figure 1.2 an outline of an offshore electrical power generation system, considered in Natvig et al. (1986), is given. The purpose of this system is to supply two nearby oilrigs with electrical power. Both oilrigs have their own main generation, represented by equivalent generators A₁ and A₃ each having a capacity of 50 MW. In addition, oilrig 1 has a standby generator A₂ that is switched into the network in case of outage of A₁ or A₃, or may be used in extreme load situations in either of the two oilrigs. The latter situation is, for simplicity, not treated in this book. A₂ is in cold standby, which means that a short startup time is needed before it is switched into the network. This time is neglected in the following model. A₂ also has a capacity of 50 MW. The control unit, U, continuously supervises the supply from each of the generators with automatic control of the switches. If, for instance, the supply from A₃ to oilrig 2 is not sufficient, whereas the supply from A₁ to oilrig 1 is sufficient, U can activate A₂ to supply oilrig 2 with electrical power through the standby subsea cables L.

The components to be considered here are A₁, A₂, A₃, U and L. We let the perfect functioning level M equal 4 and let the set of states of all components be {0, 2, 4}. For A₁, A₂ and A₃ these states are interpreted as

0: The generator cannot supply any power;

2: The generator can supply a maximum of 25 MW;

4: The generator can supply a maximum of 50 MW.

Note that as an approximation we have, for these generators, chosen to describe their supply capacity on a discrete scale of three points. The supply capacity is not a measure of the actual amount of power delivered at a fixed point of time. There is continuous power-frequency control to match generation to actual load, keeping electrical frequency within prescribed limits.

The control unit U has the states

0: U will, by mistake, switch the main generators A₁ and A₃ off without switching A₂ on;

2: U will not switch A₂ on when needed;

4: U is functioning perfectly.

The subsea cables L are actually assumed to be constructed as double cables transferring half of the power through each simple cable. This leads to the following states of L

0: No power can be transferred;

2: 50% of the power can be transferred;

4: 100% of the power can be transferred.

Let us now, for simplicity, assume that the mechanism that distributes the power from A₂ to platform 1 or 2 is working perfectly. Furthermore, as a start, assume that this mechanism is a simple one either transferring no power from A₂ to platform 2, if A₂ is needed at platform 1, or transferring all power from A₂ needed at platform 2. Now let F₁ (A₁, A₂, U) = the amount of power that can be supplied to platform 1, and F₂ (A₁, A₂, A_{3, U, L) = the amount of power that can be supplied to platform 2. F1 will now just take the same states as the generators whereas F2 can also take the following states}

1: The amount of power that can be supplied is a maximum of 12.5 MW;

3: The amount of power that can be supplied is a maximum of 37.5 MW.

Number the components A₁, A₂, A₃, U, L successively 1, 2, 3, 4, 5. Then it is not too hard to be convinced that F₁ and F₂ are given respectively by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

Let us still assume that the mechanism that distributes the power from A₂ to platform 1 or 2 is working perfectly. However, let it now be more advanced, transferring excess power from A₂ to platform 2 if platform 1 is ensured a delivery corresponding to state 4. Of course in a more refined model this mechanism should be treated as a component. The structure functions are now given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

noting that max(x₁ + x₁I (x₄ = 4) - 4, 0) is just the excess power from A₂ which one tries to transfer to platform 2.

1.3 Basic definitions from binary theory

Before going into the specific restrictions that we find natural to claim on the structure function F, it is convenient first to recall some basic definitions from the traditional binary theory. This theory is nicely introduced in Barlow and Proschan (1975a).

Definition 1.1: A system is a binary monotone system (BMS) iff its structure function F satisfies:

(i) F (x) is nondecreasing in each argument

(ii) F (0) = 0 and F (1) = 1 0 = (0, ..., 0), 1 = (1, ..., 1).

The first assumption roughly says that improving one of the components cannot harm the system, whereas the second says that if all components are in the failure state, then the system is in the failure state and, correspondingly, that if all components are in the functioning state, then the system is in the functioning state.

We now impose some further restrictions on the structure function F.

Definition 1.2: A binary coherent system (BCS) is a BMS where each component is relevant, i.e. the structure function F satisfies: [??]i [element of] {1, ..., n}, [??] (·_i, x) such that F(1_i, x) = 1, F(0_i, x) = 0.

A component which is not relevant is said to be irrelevant. We note that an irrelevant component can never directly cause the failure of the system. As an example of such a component consider a condenser in parallel with an electrical device in a large engine. The task of the condenser is to cut off high voltages which might destroy the electrical device. Hence, although irrelevant, the condenser can be very important in increasing the lifetime of the device and hence the lifetime of the whole engine. The limitation of Definition 1.2 claiming relevance of the components, is inherited by the various definitions of a multistate coherent system considered in this book.

Let ITLITL₀ (x) = {i|x_i = 0} and ITLITL₁ (x) = {i|x_i = 1}.

Definition 1.3: Let F be the structure function of a BMS. A vector x is said to be a path vector iff F(x) = 1. The corresponding path set is ITLITL₁ (x). A minimal path vector is a path vector x such that F(y) = 0 for all y < x. The corresponding minimal path set is ITLITL₁ (x).

Definition 1.4: Let F be the structure function of a BMS. A vector x is said to be a cut vector iff F(x) = 0. The corresponding cut set is ITLITL₀ (x). A minimal cut vector is a cut vector x such that F(y) = 1 for all y > x. The corresponding minimal cut set is ITLITL₀ (x).

We also need the following notation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] We then have the following representations for the series and parallel systems respectively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

Consider a BCS with minimal path sets P₁, ..., P_p and minimal cut sets K₁, ..., K_k. Since the system is functioning iff for at least one minimal path set all the components are functioning, or alternatively, iff for all minimal cut sets at least one component is functioning, we have the two following representations for the structure function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

Definition 1.5: The monotone system (A, ?) is a module of the monotone system (ITLITL, F) iff

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where ? is a monotone structure function and A [??] C.

Intuitively, a module is a monotone subsystem that acts as if it were just a supercomponent. Consider again the example where a condenser is in parallel with an electrical device in a large engine. The parallel system of the condenser and the electrical device is a module, which is relevant.

(Continues...)

Excerpted from Multistate Systems Reliability Theory with Applicationsby Bent Natvig Copyright © 2011 by John Wiley & Sons, Ltd. Excerpted by permission of John Wiley & Sons. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Edition details

• Publisher: Wiley
• Publication date: 2010
• Language: English
• ISBN 10: 0470697504
• ISBN 13: 9780470697504
• Binding: Hardcover
• Edition number: 1
• Number of pages: 262