Fault Tree Analysis Using Binary Decision Diagrams Lab Seminar May 4th, 2006 Seung Ki, Shin

Contents Introduction Classification of Fault Trees Shortcoming of Conventional Fault Tree Analysis Binary Decision Diagrams Approach Example of Non-Coherent Fault Tree Summary & Further Study References

Introduction The fault tree diagram itself is an excellent way of deriving the failure logic for a system. Conventional fault tree analysis techniques have several disadvantages when analyzing some kinds of fault trees. It is possible to overcome these disadvantages using Binary Decision Diagram (BDD) method.

Classification of Fault Trees Static Coherent Non-coherent Dynamic Static fault tree : Composed of Boolean gates Dynamic fault tree : Adding sequential notion to static fault tree

Classification of Fault Trees Coherent fault tree Logical gates are restricted to AND and OR gates. Top event is described in terms of Minimal Cut Sets. Minimal Cut Set : Combination of component failure events which are necessary and sufficient to cause the top event. ex) Non-coherent fault tree Inverse gates besides AND and OR gates. (NOT, NAND, NOR, and XOR gates) Top event is described with Prime Implicants from Boolean algebra. Prime Implicant : Combination of basic events (success or failure) which is both necessary and sufficient to cause the top event.

Shortcoming of Conventional Fault Tree Analysis Inclusion-Exclusion Expansion (IEE) ( are minimal cut sets/prime implicants ) For complex systems an analysis may produce hundreds of thousands of minimal cut sets. Then it is impossible to calculate the exact probability using IEE. Truncation of the expansion is used to simplify the calculation. It is justified for coherent fault trees. For non-coherent fault trees, this approximation is not valid and creates considerable inaccuracies in evaluating top event probability.

Shortcoming of Conventional Fault Tree Analysis The prime implicants are frequently reduced to their coherent approximations by assuming any working states for the components in the expression are set to TRUE. This approximation may induce considerable inaccuracies.

Binary Decision Diagrams Approach The binary decision diagram (BDD) method was utilized by Bryant and later developed by Rauzy. BDD provides an alternative logic form to the fault tree structure to express the system failure causes. Exact system failure probability can be deduced without the need to resort to any approximations. The BDD structure has the additional advantage that its quantification does not require the minimal cut sets/prime implicants.

Binary Decision Diagrams Approach 1 Terminal Node <Fault Tree> <Binary Decision Diagram>

Binary Decision Diagrams Approach Conventional Method Minimal Cut Sets : Probability of Top Event (Inclusion-Exclusion Expansion) Binary Decision Diagrams Approach Disjoint Path : Probability of Top Event * Due to the binary branching each path in the BDD is mutually exclusive and so the probability of system failure is obtained by simply summing the probability of each disjoint path leading to a terminal one node.

Example of Non-Coherent Fault Tree - Each component failure probability : 0.1

Example of Non-Coherent Fault Tree Exact Calculation Using IEE Method Min cut set : Probability of top event : 0.094851 Two Conventional Approximations Truncation (after one term) Coherent approximation Min cut set becomes Very inaccurate !

Example of Non-Coherent Fault Tree Binary Decision Diagram Approach 1 Disjoint Path Probability 0.00729 0.006561 0.081 Total 0.094851 Simple & Exact !

Summary & Further Study When analyzing non-coherent systems, it is shown that analysis methods based on traditional fault tree analysis are both inaccurate and inefficient. It has been shown that analysis procedures based on binary decision diagrams to represent the system failure logic can produce all minimal cut sets for problems which defeat conventional approaches. The size of the resulting BDD is determined by the ordering that has to be given to the basic events in the fault tree before the BDD is constructed. To improve the efficiency of the BDD analysis, it is important to seek a BDD of minimal size when a certain fault tree is given.

References J.D. Andrews, S.J. Dunnett, “Event Tree Analysis Using Binary Decision Diagrams”, IEEE, 2000. A. Rauzy, “New algorithms for fault trees analysis”, Reliability Engineering and System Safety, 1993. R.M. Sinnamon, J.D. Andrews, “Improved Efficiency in Qualitative Fault Tree Analysis”, Quality and Reliability Engineering International, 1997.