تجزیه و تحلیل قابلیت اطمینان در چند مقیاس و به روز رسانی سیستم های پیچیده با استفاده از برنامه ریزی خطی

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.

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.

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 obtained by the solution of several small LP problems. These in turn are used to compute bounds on the failure probability of the entire system by LP. For larger systems, sets of super-components may be combined into super-super-component, and so on. This cascading decomposition of the system allows solution of very large systems. Furthermore, the approach allows multi-scale analysis, where varying degrees of detail in modeling and analysis can be considered for different segments of the system. This facility, however, comes at a cost: the system bounds computed for the decomposed system can be wider than the bounds computed for the intact system with the same level of probability information. A set of guidelines for effective selection of super-components and super-super-components, etc., is formulated. The paper also describes a method for computing bounds on conditional probabilities by LP. The formulation involves the iterative solution of a parameterized LP problem. Conditional probabilities are used for system reliability updating and for post-event planning and decision-making. An application to a power network consisting of 4 substations and 69 components demonstrates the LP decomposition method for computing bounds on the system failure probability. Through this approach, an original LP problem of size 5.76×1017 is reduced to 35 LP problems of size 32,768 or smaller. Numerical results indicate relatively minor widening of the bounds due to the decomposition. The system reliability updating method is demonstrated by computing the conditional failure probabilities of the components in a substation given the failure or survival events of the system. These applications demonstrate the power and facility of LP as a means for system reliability analysis.