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On the Exact Analysis of Non-Coherent Fault Trees: The ASTRA Package (PSAM-0285)

## Excerpt

Fault Tree Analysis (FTA) is widely used for safety and more recently, for security studies. Depending on the type of variables fault trees can be Coherent / Non-Coherent and the corresponding structure functions are monotonic / non-monotonic. Apart from Event Trees (ET) analyzed using the fault tree linking approach, non coherent fault trees have never been particularly popular among practitioners, in spite of their usefulness in system modeling. This was due to the lack of powerful analysis methods, which has led to the use of procedures applied to the approximated coherent tree. However, these procedures work well only if components' failure probability are “sufficiently” low, i.e. when the success probability can reasonably be approximated to unity.

A very efficient approach to FTA is based on Binary Decision Diagrams (BDD) Bryant [1], Brace et al. [2]. A BDD is a compact graph representation of Boolean functions. The Binary Decision Diagrams (BDD) approach applied to FTA offers several advantages e.g. it works equally well on coherent and non-coherent trees, it offers the possibility to perform the exact probabilistic analysis without the need to determine the system failure modes (Minimal Cut Sets, MCS, or Prime Implicants PI) Rauzy [3], Rauzy and Dutuit [4].

Different algorithms have been proposed in the past for the probabilistic quantification of non-coherent fault trees. These algorithms can be classified in two groups. The first encompasses algorithms that determine the system unconditional failure and repair frequencies, directly from the set of Prime Implicants, whereas algorithms of the second group determine them on the basis of the probability of system's critical states. For example, Inagaki and Henley [5] and Liu and Pan [6] proposed methods belonging to the first group, whereas the methods proposed by Becker and Camarinopoulos [7] and Beeson and Andrews [8] belong to the second group. Moreover the problem of determining the importance of components in non-coherent trees has been tackled among the other, by Jackson [9], Zhang and Mei [10] and Beeson and Andrews [8]. All these methods, when applied to the set of Prime Implicants, represented in the form of SOP (Sum of Products) or in the form of DSOP (Disjunctive SOP) of Prime Implicants, are very time consuming, hence bound approximations and/or cut off techniques are necessarily to be applied.

The aim of this paper is to summarise the new procedure implemented in the JRC proprietary ASTRA 3.0 package for performing the logical and probabilistic analysis of coherent and non-coherent fault trees entirely based on the BDD approach Contini [11]. The fault tree is modularized and, to each module, the algorithm for the construction of the BDD in which the variables, associated to the nodes of the graph, are dynamically labeled with the information about their type (Single Positive, Single Negated and Double Form) is applied. The BDD with labeled variables will be referred to as Labeled BDD (LBDD). The information about the variable type is useful for driving the application of the algorithms for the logical and probabilistic analysis. In fact algorithms of different complexity are used for different variables and the dynamic labeling allows for an efficient choice of the needed ones. Concerning the probabilistic analysis the Becker—Camarinopoulos method [7] for the determination of the unconditional failure and repair frequencies has been adapted to the LBDD data structure [11]. Moreover, components importance measures are also calculated for any type of variables extending the work of Zang and Mei [10]. The main aspects of the computational methods implemented in ASTRA 3.0 for the analysis of coherent and non-coherent fault trees, and its performances will be also briefly described in the paper. The software is currently under the testing and validation phase.

• Summary
• Introduction
• Analysis Procedure with ASTRA 3.0
• Experimental Results
• Conclusions and on Going Developments
• Acknowledgments
• References
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