Advanced combinatorial method for solving complex fault trees

Since their introduction in form of a canonical representation of logical functions, the Binary Decision Diagrams (BDDs) gained a wide acceptance in numerous industrial applications. This paper summarizes the properties of BDD representation of Minimal Cut Sets (MCS) of Fault Tree (FT) models most typically encountered in nuclear energetics. Cut sets from MCS are defined as paths from the top BDD node to terminal nodes in the BDD, on which a quantitative and qualitative FT analysis (FTA) is performed. The core of the FTA on the BDDs is performed with help of two fundamental algorithms, one for conditional probability evaluation and another for the selection of cut sets. The accuracy of conditional probability evaluation represents an essential feature for an unbiased quantitative analysis, such as the top event probability or the determination of event importance measures. The cut set selection algorithm is shown in a generic version introducing logical predicates for its selection criteria. As it is known, the efficiency of depicted algorithms depends only on the number of BDD nodes used for the FT representation. In order to appraise the compactness of the BDD representation of FT models, their characteristics have herein been evaluated on several real-life models from the Nuclear Power Plant Krško. The extraordinariness of the compactness of the BDD representation reflects in its ability to implement advanced dynamic analysis (i.e. what-if) of FT models. The efficiency of such an approach is recognized by commercial vendors upgrading their FT Tools to new versions by implementing BDD based algorithms.

International Maritime Organization (IMO) implements the principles of risk management through a systematic process called Formal Safety Assessment (FSA). FSA is a structured and systematic methodology, aimed at enhancing maritime safety, including protection of life, health, the marine environment and property, by using risk analysis and cost benefit assessment. One of the most widely used hazard identification and risk analysis techniques in FSA is Fault Tree Analysis (FTA) method. However, only basic conventional methodology of the FTA technique is used in FSA. When dealing with large fault trees, the limitations of FTA in terms of accuracy and the efficiency become apparent. It is necessary for IMO to improve FTA method in FSA. In recent years, the Binary Decision Diagram (BDD) method has been developed that solves fault trees and overcomes the disadvantages of the conventional FTA approach. In this paper, the BDD method is introduced for the application in FSA, as it has never been used in FSA studies. A commonly used construction method and some basic event ordering methods of BDD are described. By applying BDD to FTA, two main issues concerned in FSA, minimal cut set and top event probability, can be obtained efficiently and accurately comparing to the analysis using a conventional FTA method.

The fault tree analysis tool Metaprime allows reliability engineers to make, in an interactive way, exact qualitative and quantitative analysis of very complex fault trees that could not be completely treated by previously available analysis technologies. The qualitative aspect of the analysis of a fault tree consists in computing all or part of the set of prime implicants (i.e., minimal cut sets) of this tree. Particularly interesting parts of this set are prime covers which are subsets of prime implicants that are sufficient to represent all the failure combinations of the system under analysis. When the tree is noncoherent, or/and when it is a multiple-fault tree, it is often possible, but very costly, to build covers with fewer prime implicants, which allows concise representation of the set of failure combinations. This paper proposes original algorithms, now integrated in Metaprime, to generate efficiently prime covers, irredundant prime covers, and minimal prime covers of noncoherent fault trees. Experiments show that these algorithms are robust and provide reliability engineers with efficient strategies for the analysis of complex noncoherent multiple-fault trees. >

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

Minimal cut sets (or prime implicants: minimal combinations of basic event conditions leading to system failure) are important information for reliability/safety analysis and design. To obtain minimal cut sets for general non-coherent fault trees, including negative basic events or multi-valued basic events, a special procedure such as the consensus rule must be applied to the results obtained by logical operations for coherent fault trees, which will require more steps and time. This paper proposes a simple method for a non-coherent fault tree, whose top event is represented as an AND combination of monotonic sub-trees. A “monotonic” sub-tree means that it does not have both positive and negative representations for each basic event. It is proven that minimal cut sets can be obtained by a conventional method for coherent fault trees. An illustrative example of a simple event tree analysis shows the detail and characteristics of the proposed method.

Fussell-Vesely Importance (FVI) is one of the most commonly used importance measures. It is based on the concept of Minimal Cut Sets (MCSs), and it quantifies the contribution of a system component to system failure. In paper (Kvassay et al. 2015), the algorithm for identification of all system MCSs based on Direct Partial Logic Derivatives (DPLDs) has been considered. These MCSs are then used to compute the FVI. However, the FVI of a system component depends only on MCSs that contain the considered component, i.e. it is not necessary to identify all MCSs of the system. This implies that it is useful to develop algorithms that will identify not all system MCSs but only those in which a specific component occurs. Such algorithms can be used to compute the FVI without a priori knowledge of MCSs. In this paper, a new algorithm for this task is proposed. The algorithm is based on a relation between DPLDs and MCSs that is also considered in this paper. of algorithms for identification or generation of MCSs of investigated system. These algorithms are developed mainly on the basis of applied fault trees (Sinnamon & Andrews 1997, Vatn 1992) or networks (Emadi & Afrakhte 2014, Rebaiaia & Ait-Kadi 2013). Another algorithm for identification of system MCSs has been considered in paper (Kvassay et al. 2015). This algorithm is based on tools of logical differential calculus, and it can be used for systems of any type (not only for systems presented as networks or for systems described by fault trees). Logical differential calculus has been developed for analysis of dynamic properties of logic functions. Direct Partial Logic Derivatives (DPLDs) are essential part of this tool (Tapia et al. 1991). Their use in reliability analysis has been considered in several papers (Zaitseva 2003, Zaitseva 2012, Zaitseva et al. 2015). These papers have showed that DPLDs are very appropriate for identification of situations in which failure/repair of a system component results in system failure/repair. This fact has been used to develop algorithms for defining system boundary states and for calculation of some measures that can be used to evaluate system reliability. In this paper, we develop ideas presented in paper (Kvassay et al. 2015) to propose an algorithm that identifies not all system MCSs but only MCSs in which a specified component is presented. Such MCSs are useful in computation of FVI of the considered component. As a result new method for calculation of this IM is developed. This method is

A new method is developed for tracing the unreliability contributions of system components and recognizing the system weak parts using the fault tree analysis (FTA). The method is based on the minimum cut set (MCS) algorithm for evaluating the system reliability using the FTA and the proportional sharing principle (PSP). The fault tree can be expressed as MCSs and the system unreliability can be mathematically expressed by a certain terms of probability of occurrence of MCSs. A unreliability tracing (UT) principle is proposed for allocating the probability of each term to the basic events fairly and reasonably, and then the direct contribution relationship between the system unreliability and basic events can be established. The system UT sharing factors (UTSFs) are derived to easily identify the weakness parts in a system. The applicability of the proposed methods is illustrated by case studies of a simple system and a multiple-output power distribution system.

Fault tree analysis (FTA) is a well-established technique to analyze the safety risks of a system. Two specific prominent FTA methods, largely applied in the aerospace field, are the so-called minimal cut sets (MCS), which uses an approximate evaluation of the problem, and the direct evaluation (DE) of the fault tree, which uses a top-down recursive algorithm. The first approach is only valid for small values of basic event probabilities and has historically yielded faster results than exact solutions for complex fault trees. The second one means exact solutions at a higher computational cost. This article presents several improvements applied to both approaches in order to upgrade the computing performance. First, improvements to the MCS approach have been performed, where the main idea has been to optimize the number of required permutations and to take advantage of the available information from previous solved subsets. Second, improvements to the DE approach have been applied, which deal with a reduction of the number of recursive calls through a deep search for independent events in the fault tree. This could dramatically reduce the computation time for industrial fault trees with a high number of repeated events. Additional implementation improvements have been also applied regarding hash tables, and memory access and usage, but also implementing the so-called “virtual gates”, which enable limitless children on each gate. The results presented hereafter are promising, not only because they show that both approaches have been highly optimized compared to the literature, but also because a DE solution has been achieved, which can compete in time resources (and obviously in precision) with the MCS approach. These improvements are relevant when considering the industrial, and more specifically the aeronautical, implementation and application of both techniques.

α-Cut method based importance measure for criticality analysis in fuzzy probability – Based fault tree analysis

Nuclear power plants play a significant role in the contribution of electricity generation on a global scale. Various reactor designs have advantages over others in different aspects. APR-1400 is a pressurized water reactor that is deemed safe due to the redundancy and independence of the multiple safety systems. Probabilistic safety assessment (PSA) is well known for its effectiveness in the representation of risk and safety analysis of the systems in a nuclear power plant. It provides different scenarios of system failure and accident progression via fault tree analysis. A loss of feedwater (LOFW) accident may occur due to numerous reasons such as spurious closure of valves, component failure of heaters, pumps, tanks, or a loss of offsite power (LOOP) event. In the present research, a methodology has been developed that aims to investigate different factors contributing to the loss of feedwater. This research also aims to analyze LOFW accidents by developing fault tree models for the main feedwater system of the APR-1400 to identify the basic events, which may lead to a loss of feedwater accidents. The results of the top event probabilities, risk decrease factor (RDF), risk increase factor (RIF), minimal cut sets (MCS), basic event probabilities, and sensitivity analysis were compared with the WASH-1400 database. It has been found that the control valve (V04) and main feedwater isolation valve (V05) have more contribution to the LOFW accident. The common cause failure (CCF) analysis has been carried out, and it was found that the flow toward the check valve and steam generator are most critical for CCF.

The subject of research is to determine the functional safety indicators of a fault-tolerant safety-critical system, namely, the minimal cut sets’ probability for a given duration of the system’s operation, using the state transition diagram (STD). The aim is to create a new method for analyzing the functional safety of a fault-tolerant safety-critical system. This method is based on the methodology of developing models of operational reliability behavior in the form of STD. This methodology provides a detailed representation of inoperable states and their relation with pre-failure (inoperable critical) states. The task is to propose a new classification for inoperable states of the STD to obtain all possible emergencies in the same space of inoperable states. This approach allows consideration the correlations between the failures, that it is impossible to use the fault trees. Since the space of inoperable states can reach hundreds and thousands of states, a method is proposed for their automated determination according to the classification. The state space method was used to conduct the validation of the method of functional safety analysis. The following results were obtained: the system of Chapman-Kolmogorov differential equations is formed in accordance with the STD and it provides the dependence of the functional safety indicator – the minimal cut sets’ probability as a function of the operational duration of the fault-tolerant safety-critical system. This dependence is called the emergency function. The method for determining the emergency function is based on the usage of the emergency mask. Note that the proposed model of operational reliability behavior in the form of STD provides the possibility to conduct both the functional safety and the reliability indicators. The value of the minimal cut sets’ probability for a given duration of operation is determined using the fault tree for the validation of the proposed method of functional safety analysis. The fault tree was built by Reliasoft BlockSim software. The obtained value coincides with the value of the minimal cut sets’ probability, which was defined by the emergency function for the same operational duration. Thus, the designer can comprehensively analyze the feasibility of introducing redundancy (structural, temporal, functional). Conclusions: the scientific novelty of the obtained results is the following: the new method for determining safe, critical and catastrophic states in the set of inoperable states is used in the methodology of the STD developing to obtain the stochastic model of operational reliability behavior of fault-tolerant safety-critical system. This technique ensures an automated defining of emergency function by using an improved structural-automatic model.

Dynamic fault tree (DFT) is a powerful modeling approach for reliability analysis of complex system with dynamic failure behaviors. In reality, the tree structure may be highly coupled either by shared basic events or by the high-level dynamic gates. Currently, the application of sequential binary decision diagram (SBDD)-based method for quantitative analysis of such highly coupled DFTs is mainly limited to DFTs whose dynamic gates locate in the bottom of the tree. Moreover, there is no efficient way dealing with the dependencies among different nodes of a SBDD 1-path. This paper makes an improvement to the SBDD-based approach. A generation procedure is proposed to directly construct the binary decision diagram (BDD) model for a DFT with arbitrary tree structure. During the construction, the sequential-dependent information of the tree is derived as several BDD nodes, each indicates a binary-sequential event representing the sequence of two occurred basic events. A topological sorting is applied on each 1-path of the resultant BDD to obtain its contained disjoint cut sequences. Based on this, both qualitative and quantitative analysis can be performed on the DFT with no limitations on tree structure, and its minimal cut sequence set (CSS) is obtained as disjoint. Examples are provided for verification and comparison, and the results illustrate the merits of the proposed approach.

Human and organizational factors (HOFs) play an important role in electric misoperation accidents (EMAs), but research into the reliability of human factors is still in its infancy in the field of EMAs, and further investment in research is urgently required. To analyze the HOFs in EMAs, a hybrid method including the Human Factors Analysis and Classification System (HFACS) and fuzzy fault tree analysis (FFTA) was applied to EMAs for the first time in the paper. HFACS is used to identify and classify the HOFs with 135 accidents, reorganized as basic events (BEs), intermediate events (IEs), and top event (TE), and develop the architecture of fault tree (FT). Fuzzy aggregation is employed to address experts’ expressions and obtain the failure probabilities of the BEs and the minimal cut sets (MCSs) of the FT. The approach generates BEs failure probabilities without reliance on quantitative historical failure statistics of EMAs via qualitative records processing. The FFTA–HFACS model is applied for quantitative analysis of the probability of failure of electrical mishaps and the interaction between accident risk factors. It can assist professionals in deciding whether and where to take preventive or corrective actions and assist in knowledgeable decision-making around the electric operation and maintenance process. Finally, applying this hybrid method to EMAs, the results show that the probability of an EMAs is 1.0410 × 10−2, which is a risk level that is likely to occur and must be controlled. Two of the most important risk factors are habitual violations and supervisory violation; a combination of risk factors of inadequate work preparation and paralysis, and irresponsibility on the part of employees are also frequent errors.

Introducing Temporal Behaviour into Binary Decision Diagrams

Based on the structure and operation mechanism of the dual-stage lithographic system, fault tree analysis for the key subsystem, wafer stage, is performed. Since the system is very complexity, the BDD (binary decision diagram) and DFTA (dynamic fault tree analysis) technique are introduced to make up the disadvantages of the traditional static FTA (fault tree analysis) method. Event “The kinematic accuracy of wafer stage can't meet the goal” is installed as the top event of the fault tree. Central factors affecting the kinematic accuracy of the wafer stage is identified through the study.