Series and Parallel Systems

In Section 20.2, equations for tlie reliability of series and parallel systems are established. Various reliability relations are developed in Section 20.3. Sections 20.4 and 20.5 introduce several probability distribution models lliat are extensively used in reliability calculations in hazard and risk analysis. Section 20.6 deals witli tlie Monte Carlo teclinique of mimicking observations on a random variable. Sections 20.7 and 20.8 are devoted to fault tree and event tree analyses, respectively. 

Many systems consisting of several components can be classified as series, parallel, or a combination of both. However, llie majority of industrial and process plants (units and systems) have series of parallel configurations. 

A series system is one in which Uie entire system fails to operate if any one of its components fails to operate. If such a system consists of n components tliat function independently, llien tlie reliability of the system is lire product of tlie reliabilities of tlie individual components. If Rs denotes lire reliability of a series system and R, denotes tlie reliability of the i component i = 1,. .., n, llien 

A parallel system is one lliat fails to operate only if all its components fail to operate. If R, is tlie reliability of the i component, llien (1-Ri) is tlie probability tliat tlie i component fails i = 1,. .., n. Assuming lliat all n components function independently, tlie probability tliat all n components fail is (1-Ri)(l-R2)...(l-Rn). Suiitracting lliis product from unity yields the following formula for Rp, tlie reliability of a parallel system.  

Tlie reliability formulas for series and parallel systems can be used to obtain tlie reliability of a system lliat combines features of a series and a parallel system. Consider, for example, tlie system diagrammed in Fig. 20.2.1. Components A, B, C, and D have for llieir respective reliabilities 0.90, 0.80, 0.80, and 0.90. The system fails to operate if A fails, if B and C both fail, or if D fails. Component B and C constitute a parallel subsystem connected in series to components A and D. The reliability of the parallel subsystem is obtained by applying Eq. (20.2.2), which yields 

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The modular design of the HyPM fuel cells allows scaling for higher power requirements using a variety of configurations, such as series and parallel systems. Potential applications for the technology include vehicle propulsion, auxiliary power units (APU), stationary applications including backup and standby power units, combined heat and power units and portable power applications for the construction industry and the military. 

In Section 20.2, equations for the reliability of series and parallel systems are established. Various reliability relations are developed in Section... 

The reliability formulas for series and parallel systems can be used to obtain the reliability of a system that combines features of series and parallel systems. 

Equipment arrangement involves the concepts of series and parallel systems as well as standby systems. Redundancy can just be duplication of the procedure, but it also can be having some information confirm the remainder—an error-checking code. For example, giving a meeting date as Thursday, February 25, lets the Thursday confirm the 25. A ZIP code confirms the names of city and state on a postal address. 

As a starting point for coming up with a rule for which dependencies to include, we will take a look at the error that is introduced when falsely including independence for simple series and parallel systems. 

For cold-standby redundancy a detection and switching mechanism is required to sense the presence of a failed component and to activate a standby component. The reliability of a series and parallel system with... 

The generalized reliability indices of series and parallel systems of events is presented as ... 

The analysis of reliability indices of series and parallel systems with 6 equireliable elements, the reliability index of which is j8= 0-3 (Pandey 1998 a, b) are shown in Figure 3 and 4, respectively. The correlation coefficients are given in product... 

More generally, we are interested in problems that can be defined by one or more limit-state functions. In series systems, global failure occurs when at least one limit-state functions is violated, whereas in parallel systems, it occurs when all limit-state functions are violated. So, the failure events for series and parallel systems with k limit-state functions, are, respectively ... 

For the formulation of a reliability assessment problem as a PC problem space, we distinguish between series and parallel systems. 

The problems in the following example are found in (Sorensen 2004) to illustrate series and parallel systems. 

The results obtained with the classical approach for series and parallel systems analysis, where simple (SB) and Ditlevsen (DB) bounds are considered (see (Sorensen 2004, Notes 6 and 7) for details) and with the PCTM algorithm are shown in Table 4. 

In this paper we propose to use the Probabilistic Continuous Constraints framework to deal with reliability assessment problems. Given its grounding on continuous constraint solving, this framework computes safe bounds for the reliability of series and parallel systems, contrary to classical approaches. The various kinds of approximations used by these approaches may turn the computed reliability value of little practical use, since they do not provide any bounds to the errors incurred. This is particularly significant in systems modeled by means of nonlinear constraints. 

A structural system is an assemblage of components, whose performance is described by individual component limit state function. Therefore, in a structural system reliability problem, the failure domain may be described as the union and/or intersection of several limit state surfaces. In particular, a series system (also known as the weakest-link) reliability problem is defined as the union of the failure domains (Fig. lb), and a parallel system reliability problem is defined as the intersection of the failure domains (Fig. Ic). fii analytic terms, the probability of failure for series and parallel systems with n components is defined, respectively, as... 

A cut-set is made as a combination of series and parallel systems generating a generalized system reliability problem. A cut-set (Fig. 2) is any set of components whose joint failure represents a failure of the system (Ditlevsen and Madsen 1996 Au and Beck 2003b Jalayer et al. 2007a). Analytically, the probability of failure for a cut-set system is evaluated as... 

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