Systematic evaluation of fault trees using real-time model checker UPPAAL

Fault tree analysis, the most widely used safety analysis technique in industry, is often applied manually. Although techniques such as cutset analysis or probabilistic analysis can be applied on the fault tree to derive further insights, they are inadequate in locating flaws when failure modes in fault tree nodes are incorrectly identified or when causal relationships among failure modes are inaccurately specified. In this paper, we demonstrate that model checking technique is a powerful tool that can formally validate the accuracy of fault trees. We used a real-time model checker UPPAAL because the system we used as the case study, nuclear power emergency shutdown software named Wolsong SDS2, has real-time requirements. By translating functional requirements written in SCR-style tabular notation into timed automata, two types of properties were verified: (1) if failure mode described in a fault tree node is consistent with the system's behavioral model; and (2) whether or not a fault tree node has been accurately decomposed. A group of domain engineers with detailed technical knowledge of Wolsong SDS2 and safety analysis techniques developed fault tree used in the case study. However, model checking technique detected subtle ambiguities present in the fault tree.     

Development of an analytical method to break logical loops at the system level

Logical loops or circular logics are interpreted as circular supporting relations among systems or their fault trees. The logical loops are located in the merged fault tree that is created by combining the system fault trees. The inconsistent manual breaking of the logical loops could be one of the major sources of the uncertainty in the fault tree analysis. This paper presents an analytical method to break the logical loops at the system level. The analytical solution at the system level is obtained in a mathematical way without an actual manipulation of the fault tree. Then, the actual manipulation of the fault tree in the analytical solution is performed and the resultant broken fault tree is solved by the fault tree quantifier.The analytical solution is consistent regardless of the knowledge, experiences, skills, and practices of the reliability specialist. The analytical solution of this method is easy to understand or trace how to break the logical loops. Reliability analysts and concerned people can communicate how to break the logical loops with the help of the mathematical expression of the analytical solution.     

On structured fault tree construction by modularizing control loops

The objectives of this article are: (1) to develop the fault trees for control loops in a way that they all appear in a proper form and as modules in the fault tree for the whole system; and also (2) to modularize each control loop in a system properly to establish its unit model. These two methods are essentially equivalent. Either of them can be applied to facilitate constructing fault trees for the whole system. To accomplish such equivalent or parallel objectives, we will first take the feedback and feedforward control loops in the heat exchanger system as examples and then: (1) construct the fault trees of deviations in controlled variable for the two control loops in the way that all of their non-basic terminal events should be deviations in variables on those streams that connect to the loop from their outside in the system; and (2) establish the unit model for each of such two loops. One of the purposes of (2), which will not be addressed here, is to regard each control loop as a dummy unit and hence reduce the system to facilitate system input and fault tree construction eventually for complete automation purpose.     

A fault tree analysis strategy using binary decision diagrams

The use of binary decision diagrams (BDDs) in fault tree analysis provides both an accurate and efficient means of analysing a system. There is a problem, however, with the conversion process of the fault tree to the BDD. The variable ordering scheme chosen for the construction of the BDD has a crucial effect on its resulting size and previous research has failed to identify any scheme that is capable of producing BDDs for all fault trees. This paper proposes an analysis strategy aimed at increasing the likelihood of obtaining a BDD for any given fault tree, by ensuring the associated calculations are as efficient as possible. The method implements simplification techniques, which are applied to the fault tree to obtain a set of ‘minimal’ subtrees, equivalent to the original fault tree structure. BDDs are constructed for each, using ordering schemes most suited to their particular characteristics. Quantitative analysis is performed simultaneously on the set of BDDs to obtain the top event probability, the system unconditional failure intensity and the criticality of the basic events.     

An enhanced component connection method for conversion of fault trees to binary decision diagrams

Fault tree analysis (FTA) is widely applied to assess the failure probability of industrial systems. Many computer packages are available, which are based on conventional kinetic tree theory methods. When dealing with large (possibly non-coherent) fault trees, the limitations of the technique in terms of accuracy of the solutions and the efficiency of the processing time become apparent. Over recent years, the binary decision diagram (BDD) method has been developed that solves fault trees and overcomes the disadvantages of the conventional FTA approach. First of all, a fault tree for a particular system failure mode is constructed and then converted to a BDD for analysis. This paper analyses alternative methods for the fault tree to BDD conversion process.For most fault tree to BDD conversion approaches, the basic events of the fault tree are placed in an ordering. This can dramatically affect the size of the final BDD and the success of qualitative and quantitative analyses of the system. A set of rules is then applied to each gate in the fault tree to generate the BDD. An alternative approach can also be used, where BDD constructs for each of the gate types are first built and then merged to represent a parent gate. A powerful and efficient property, sub-node sharing, is also incorporated in the enhanced method proposed in this paper. Finally, a combined approach is developed taking the best features of the alternative methods. The efficiency of the techniques is analysed and discussed.     

Development of measures to estimate truncation error in fault tree analysis

The fault tree quantification uncertainty from the truncation error has been of great concern for the reliability evaluation of large fault trees in the probabilistic safety analysis (PSA) of nuclear plants. The truncation limit is used to truncate cut sets of the gates when quantifying the fault trees. This paper presents measures to estimate the probability of the truncated cut sets, that is, the amount of truncation error. The functions to calculate the measures are programmed into the new fault tree quantifier FTREX (Fault Tree Reliability Evaluation eXpert) and a Benchmark test was performed to demonstrate the efficiency of the measures.The measures presented in this study are calculated by a single quantification of the fault tree with the assigned truncation limit. As demonstrated in the Benchmark test, lower bound of truncated probability (LBTP) and approximate truncation probability (ATP) are efficient estimators of the truncated probability. The truncation limit could be determined or validated by suppressing the measures to be less than the assigned upper limit. The truncation limit should be lowered until the truncation error is less than the assigned upper limit. Thus, the measures could be used as an acceptability of the fault tree quantification results. Furthermore, the developed measures are easily implemented into the existing fault tree solvers by adding a few subroutines to the source code.     

An analytic solution for a fault tree with circular logics in which the systems are linearly interrelated

A circular logic or a logical loop is defined as the infinite circulation of supporting relations due to their mutual dependencies among the systems in the fault tree analysis. While many methods to break the circular logic have been developed and used in the fault tree quantification codes, the general solution for a circular logic is not generally known as yet. This paper presents an analytic solution for circular logics in which the systems are linearly interrelated with each other. To formulate the analytic solution, the relations among systems in the fault tree structure are described by the Boolean equations. The solution is, then, obtained from the successive substitutions of the Boolean equations, which is equivalent to the attaching processes of interrelated system's fault tree to a given fault tree. The solution for three interrelated systems and their independent fault tree structures are given as an example.     

Approximate estimation of system reliability via fault trees

In this article, we show how fault tree analysis, carried out by means of binary decision diagrams (BDD), is able to approximate reliability of systems made of independent repairable components with a good accuracy and a good efficiency. We consider four algorithms: the Murchland lower bound, the Barlow-Proschan lower bound, the Vesely full approximation and the Vesely asymptotic approximation. For each of these algorithms, we consider an implementation based on the classical minimal cut sets/rare events approach and another one relying on the BDD technology. We present numerical results obtained with both approaches on various examples.     

A dynamic fault tree

The fault tree analysis is a widely used method for evaluation of systems reliability and nuclear power plants safety. This paper presents a new method, which represents extension of the classic fault tree with the time requirements. The dynamic fault tree offers a range of risk informed applications. The results show that application of dynamic fault tree may reduce the system unavailability, e.g. by the proper arrangement of outages of safety equipment. The findings suggest that dynamic fault tree is a useful tool to expand and upgrade the existing models and knowledge obtained from probabilistic safety assessment with additional and time dependent information to further reduce the plant risk.     

Posbist fault tree analysis of coherent systems

When the failure probability of a system is extremely small or necessary statistical data from the system is scarce, it is very difficult or impossible to evaluate its reliability and safety with conventional fault tree analysis (FTA) techniques. New techniques are needed to predict and diagnose such a system's failures and evaluate its reliability and safety. In this paper, we first provide a concise overview of FTA. Then, based on the posbist reliability theory, event failure behavior is characterized in the context of possibility measures and the structure function of the posbist fault tree of a coherent system is defined. In addition, we define the AND operator and the OR operator based on the minimal cut of a posbist fault tree. Finally, a model of posbist fault tree analysis (posbist FTA) of coherent systems is presented. The use of the model for quantitative analysis is demonstrated with a real-life safety system.     

Fault tree construction of hybrid system requirements using qualitative formal method

When specifying requirements for software controlling hybrid systems and conducting safety analysis, engineers experience that requirements are often known only in qualitative terms and that existing fault tree analysis techniques provide little guidance on formulating and evaluating potential failure modes. In this paper, we propose Causal Requirements Safety Analysis (CRSA) as a technique to qualitatively evaluate causal relationship between software faults and physical hazards. This technique, extending qualitative formal method process and utilizing information captured in the state trajectory, provides specific guidelines on how to identify failure modes and relationship among them. Using a simplified electrical power system as an example, we describe step-by-step procedures of conducting CRSA. Our experience of applying CRSA to perform fault tree analysis on requirements for the Wolsong nuclear power plant shutdown system indicates that CRSA is an effective technique in assisting safety engineers.     

Two fault classification methods for large systems when available data are limited

In this paper, we consider the problem of fault diagnosis for a system with many possible fault types. Two approaches are presented that are useful for initial diagnosis of system-wide faults, assuming that no data are available before commissioning the system but the possibility of the occurrence of each symptom is known for each fault. The first method uses a fault tree approach to reduce the solution space before applying the geometric classification method, the assumption being that no unwanted symptoms are possible. This method is nonparametric and thus does not require any data to estimate the underlying distribution of faults and symptoms. The second method is based on the Bayes classification approach to utilize the subjective information and the limited data that may be available. The two methods are generic and applicable to a variety of industrial processes.     

Integrating cyber attacks within fault trees

In this paper, a new method for quantitative security risk assessment of complex systems is presented, combining fault-tree analysis, traditionally used in reliability analysis, with the recently introduced Attack-tree analysis, proposed for the study of malicious attack patterns. The combined use of fault trees and attack trees helps the analyst to effectively face the security challenges posed by the introduction of modern ICT technologies in the control systems of critical infrastructures. The proposed approach allows considering the interaction of malicious deliberate acts with random failures. Formal definitions of fault tree and attack tree are provided and a mathematical model for the calculation of system fault probabilities is presented.     

Identification of independent modules in fault trees which contain dependent basic events

The reliability performance of a system is frequently a function of component failures of which some are independent whilst others are interdependent. It is possible to represent the system failure logic in a fault tree diagram, however only the sections containing independent events can be assessed using the conventional fault tree analysis methodology. The analysis of the dependent sections will require a Markov analysis. Since the efficiency of the Markov analysis largely depends on the size of the established Markov model, the key is to extract from the fault tree the smallest sections which contain dependencies. This paper proposes a method aimed at establishing the smallest Markov model for the dependencies contained within the fault tree.     

Enhancing software safety by fault trees: experiences from an application to flight critical software

The fault tree analysis is a well-established method in system safety and reliability assessment. We transferred the principles of this technique to an assembler code analysis, regarding any incorrect output of the software as the undesired top-level event. Starting from the instructions providing the outputs and tracking back to all instructions contributing to these outputs a hierarchical system of references is generated that may graphically be represented as a fault tree. To cope with the large number of relations in the code, a tool suite has been developed, which automatically creates these references and checks for unfulfilled preconditions of instructions. The tool was applied to the operational software of an inertial measurement unit, which provides safety critical signals for artificial stabilization of an aircraft. The method and its implementation as a software tool is presented and the benefits, surprising results, and limitations we have experienced were discussed.     

Analysis of large fault trees based on functional decomposition

With the advent of the Binary Decision Diagrams (BDD) approach in fault tree analysis, a significant enhancement has been achieved with respect to previous approaches, both in terms of efficiency and accuracy of the overall outcome of the analysis. However, the exponential increase of the number of nodes with the complexity of the fault tree may prevent the construction of the BDD. In these cases, the only way to complete the analysis is to reduce the complexity of the BDD by applying the truncation technique, which nevertheless implies the problem of estimating the truncation error or upper and lower bounds of the top-event unavailability.This paper describes a new method to analyze large coherent fault trees which can be advantageously applied when the working memory is not sufficient to construct the BDD. It is based on the decomposition of the fault tree into simpler disjoint fault trees containing a lower number of variables. The analysis of each simple fault tree is performed by using all the computational resources. The results from the analysis of all simpler fault trees are re-combined to obtain the results for the original fault tree.Two decomposition methods are herewith described: the first aims at determining the minimal cut sets (MCS) and the upper and lower bounds of the top-event unavailability; the second can be applied to determine the exact value of the top-event unavailability. Potentialities, limitations and possible variations of these methods will be discussed with reference to the results of their application to some complex fault trees.     

Gastric esophageal surgery risk analysis with a fault tree and Markov integrated model

Reliability methods have been widely used in risk analysis of medical surgeries. In this study, the authors combine a fault tree with Markov models to assess time independent- and dependent factors together. Dynamics are integrated in the traditional fault tree, and meanwhile the processes of solving Markov are simplified with the modular approach. Continuous time Markov chains are adopted in evaluating the failure probability of a gastric esophageal surgery after categorizing basic events in the fault tree, and a certain time dependent variables, such as failure rate of medical equipment, surgery frequency, and rescue timeliness are involved into risk analysis. A case is studied with data collected from a general hospital, to illustrate the operational process of the proposed method. Results based on the inputs show that taking rescue actions into consideration can reduce the gap between the result of fault tree analysis and the reality. Sensitivity analysis for measuring the impacts of the above time relevant variables is conducted, as well as limitations of the Markov model are discussed.     

SMV model-based safety analysis of software requirements

Fault tree analysis (FTA) is one of the most frequently applied safety analysis techniques when developing safety-critical industrial systems such as software-based emergency shutdown systems of nuclear power plants and has been used for safety analysis of software requirements in the nuclear industry. However, the conventional method for safety analysis of software requirements has several problems in terms of correctness and efficiency; the fault tree generated from natural language specifications may contain flaws or errors while the manual work of safety verification is very labor-intensive and time-consuming. In this paper, we propose a new approach to resolve problems of the conventional method; we generate a fault tree from a symbolic model verifier (SMV) model, not from natural language specifications, and verify safety properties automatically, not manually, by a model checker SMV. To demonstrate the feasibility of this approach, we applied it to shutdown system 2 (SDS2) of Wolsong nuclear power plant (NPP). In spite of subtle ambiguities present in the approach, the results of this case study demonstrate its overall feasibility and effectiveness.     

An ordering heuristic to develop the binary decision diagram based on structural importance

Fault tree analysis is often used to assess risks within industrial systems. The technique is commonly used although there are associated limitations in terms of accuracy and efficiency when dealing with large fault tree structures. The most recent approach to aid the analysis of the fault tree diagram is the Binary Decision Diagram (BDD) methodology. To utilise the technique the fault tree structure needs to be converted into the BDD format. Converting the fault tree requires the basic events of the tree to be placed in an ordering. The ordering of the basic events is critical to the resulting size of the BDD, and ultimately affects the performance and benefits of this technique. A number of heuristic approaches have been developed to produce an optimal ordering permutation for a specific tree. These heuristic approaches do not always yield a minimal BDD structure for all trees. This paper looks at a heuristic that is based on the structural importance measure of each basic event. Comparing the resulting size of the BDD with the smallest generated from a set of six alternative ordering heuristics, this new structural heuristic produced a BDD of smaller or equal dimension on 77% of trials.     

Component-based modeling of systems for automated fault tree generation

One of the challenges in the field of automated fault tree construction is to find an efficient modeling approach that can support modeling of different types of systems without ignoring any necessary details. In this paper, we are going to represent a new system of modeling approach for computer-aided fault tree generation. In this method, every system model is composed of some components and different types of flows propagating through them. Each component has a function table that describes its input-output relations. For the components having different operational states, there is also a state transition table. Each component can communicate with other components in the system only through its inputs and outputs. A trace-back algorithm is proposed that can be applied to the system model to generate the required fault trees. The system modeling approach and the fault tree construction algorithm are applied to a fire sprinkler system and the results are presented.