Exploring solution methods for fault trees constrained by location

Fault Tree Analysis (FTA) is used extensively to evaluate the logical dependency of a system on its constituent components. Fault trees (FTs) can be used to identify and correct weaknesses in a design before a system goes to production. Effective methods have been developed over the course of several decades for finding minimal cut sets (MCS). Cut sets identify combinations of component failures that cause the system to fail. Other methods focus on probability risk assessment, in which component failure probabilities are evaluated to determine which failure events are most probable under normal operating conditions. However, traditional FTs do not contain information about the physical location of the components that make up the system. Thus, they cannot identify vulnerabilities induced by the proximity relationships of those components. Components that are sufficiently close to each other could be defeated by a single event with a large enough radius of effect. Events such as the Deepwater Horizon explosion and subsequent oil spill demonstrate the potentially devastating risk posed by such vulnerabilities. Adding positional information to the logical information contained in the FT can capture proximity relationships that constitute vulnerabilities in the overall system but are not contained in the logical structure alone. Thus, existing FTA methods cannot address these concerns. Making use of the positional information would require extensions to existing solution methods or possibly new methods altogether. In practice, fault trees can grow very large, exceeding one thousand components for a large system, which causes a combinatorial explosion in the number of possible solutions. Traditional methods cope with this problem by limiting the number of solutions; generally this is an acceptable limitation since those methods will find the most likely events capable of defeating the fault tree. However, adding more information to the tree and searching for different criteria (such as conditional probabilities) can render that trade invalid and motivates the search for alternate means to find vulnerabilities in the system. Candidate methods for this type of problem should be able to find “hot spots” in the physical space of very large real world systems where a destructive event would damage multiple components and cause the overall system to fail. In the present research, a test set of medium to large fault tree systems was generated using Lindenmayer systems. These systems vary in size from tens of components to over a thousand and vary in terms of complexity as measured by the proportion of operator types and size of minimal cut sets. Two solution approaches were explored in this research that use graph clustering to integrate positional information with FT solutions as an initial attempt to solve spatially constrained fault trees. These methods were applied to the set of test fault trees to evaluate their performance in finding solutions to this type of problem. The first method uses xfta, a freely available FT solver from OpenPSA, to find minimal cut sets, then performs k-means clustering on the resulting cut sets to determine whether a spatial vulnerability exists. This method works well for smaller fault trees for which all minimal cut sets can be determined. However, for large, complex fault trees, there remains the possibility that crucial vulnerabilities are not identified since the overall proportion of MCS that can be evaluated in practical time can be less than one in a million. The second method performs a modified k-means cluster on the entire set of components to find groups of spatially related components, then feeds the groups into a fault tree evaluator. This method also works, though not very effectively, for smaller fault trees or when the radius of effect is large relative to the physical space. Neither method provides a deterministic means to solve large complex fault trees, leaving open the question of whether better methods exist to solve this type of problem. The combinatorial effect combined with the addition of positional information increases the difficulty of finding solutions in the search space. This research is presented in the hope of stimulating interest in the research community to find better methods of finding and correcting vulnerabilities using fault trees with location information.