Multi-scale reliability analysis and updating of complex systems by use of linear programming

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Abstract

Complex systems are characterized by large numbers of components, cut sets or link sets, or by statistical dependence between the component states. These measures of complexity render the computation of system reliability a challenging task. In this paper, a decomposition approach is described, which, together with a linear programming formulation, allows determination of bounds on the reliability of complex systems with manageable computational effort. The approach also facilitates multi-scale modeling and analysis of a system, whereby varying degrees of detail can be considered in the decomposed system. The paper also describes a method for computing bounds on conditional probabilities by use of linear programming, which can be used to update the system reliability for any given event. Applications to a power network demonstrate the methodology.

Introduction

Critical infrastructures, such as water-, sewage-, gas- and power-distribution systems and highway transportation networks are usually complex systems consisting of numerous structural components. In order to guarantee the reliability of such systems against deterioration or natural and man-made hazards, it is essential to have an efficient and accurate method for estimating the failure probability relative to specified system performance criteria and load hazard. Furthermore, for the purpose of developing emergency or recovery plans, it is often of interest to determine the updated reliability of the system or its components for given scenario events or events that have actually occurred. This paper aims at developing methodologies for such analyses, which are well suited for application to complex systems.

System performance criteria usually are defined either in terms of connectivity between input and output nodes, or in terms of availability of specified levels of “flow” (e.g., water flux or pressure, power voltage) at selected nodes. In the case of a connectivity criterion, each component has only two possible states: connected (functioning) or not connected (failed). In the case of a flow criterion, each component as well as the system can have multiple states, e.g., different levels of flow. Mathematically, the two problems are similar, though a multi-state system usually poses more computational challenges. While the methods developed in this paper are applicable to multi-state systems and brief outlines are given, the main focus of the application is on two-state systems. Applications to multi-state systems are currently under development.

Recently, the authors developed a linear programming (LP) method for computing bounds on the reliability of general, two-state systems in terms of marginal or low-order joint component failure probabilities [1]. For a system with n components, the size of the LP problem to be solved is N=2n. This number can be enormously large, e.g., for a system with 100 components N=1.27×1030. Obviously, a direct solution of the LP problem for such a system is not possible. To overcome this problem, in this paper, we propose a multi-scale approach, whereby the system is decomposed into a number of subsystems and a hierarchy of analyses is performed by considering each subsystem or sets of subsystems separately. In addition to computational advantage, this approach allows consideration of details at the subsystem level, which may not be possible to include in the overall system model.

The LP bounding method is next extended to the computation of conditional probabilities for the purpose of system reliability updating. An iterative solution algorithm with a parameterized LP formulation is proposed for this purpose. Example applications to connectivity problems of an electric power substation and a network demonstrate the methodologies developed in this paper.

Section snippets

Review of LP bounds method

We first briefly present the LP formulation for a two-state system and then describe its extension to a multi-state system. Consider a system with n two-state components and let the Boolean variable si∈(0,1), i=1,…,n, denote the state of component i with si=0 denoting its failure state and si=1 denoting its functioning state. There are N=2n distinct realizations of the system, each defined by a distinct realization of the vector s=[s1s2sn]T. Let pj, j=1,…,N, denote the probabilities associated

Multi-scale system reliability analysis

A “divide and conquer” approach to apply the LP bounding method to large systems can be devised as follows: Consider a subset of the components of the system together as a single “super-component” [5] denoted by the bracketed number [1]. If super-component [1] has k components, then the system composed of super-component [1] and the remaining components has nk+1 components. We have thus reduced the size of the LP problem by a factor of 2k−1 We can proceed with this approach introducing

Example: seismic reliability of four-substation power network

Consider the four-substation power network in Fig. 2, which is designed based on the “hypothetical” substation example in Ostrom [7]. In the figure, circuit breakers, switches and transformers are represented by circles, hash-marks and squares, respectively. The equipment items are numbered from 1 to 69. Three power sources (inputs I, II and III) supply the network with electricity. Transformers placed in each substation supply power to the local facilities around the respective substation. We

System reliability updating

In the analysis of system reliability, it is often of interest to compute the conditional probability of a system or subsystem event, given that another system or subsystem event is known or presumed to have occurred. This is known as system reliability updating. Such conditional probabilities are useful in identifying critical components or subsystems within a system, or for post-event planning and decision-making.

Suppose an event A is known to have occurred and we wish to determine the

Example: reliability updating of an electrical substation

Consider Substation 4 of the example system in Fig. 2. Suppose we are interested in determining the conditional failure probabilities of the components of this system, given that the system failure event E②+③→L (lack of connectivity between the connecting input lines ② and ③ and the output line L), or its complement E+L¯, has occurred after an earthquake event of random intensity. Specifically, we wish to compute the conditional probabilities P(Ei|E+L) and P(Ei|E+L¯), i=56, …, 69 Note that

Summary and conclusions

A system decomposition method is developed for the reliability analysis of complex systems characterized by a large number of component states. The decomposition facilitates solution of the system reliability by the LP bounding method, where the large LP problem for the entire system is replaced by several LP problems of much smaller size. This is accomplished through the introduction of super-components consisting of subsets of the system. Probability bounds on the super-components are first

Acknowledgments

The research reported in this paper was conducted while the first author was visiting the Technical University of Denmark with support from the COWI Foundation. An earlier version of the paper was presented as a keynote lecture in a two-part workshop at the DTU on the Reliability Analysis of Complex Systems and on Fatigue and Fracture in Metallic Structures, August 23–25, 2004. The support from COWI Foundation as well as the many insightful discussions that the first author had with his DTU

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