Quantitative analysis of a fault tree with priority AND gates

A method for calculating the exact top event probability of a fault tree with priority AND gates and repeated basic events is proposed when the minimal cut sets are given. A priority AND gate is an AND gate where the input events must occur in a prescribed order for the occurrence of the output event. It is known that the top event probability of such a dynamic fault tree is obtained by converting the tree into an equivalent Markov model. However, this method is not realistic for a complex system model because the number of states which should be considered in the Markov analysis increases explosively as the number of basic events increases. To overcome the shortcomings of the Markov model, we propose an alternative method to obtain the top event probability in this paper. We assume that the basic events occur independently, exponentially distributed, and the component whose failure corresponds to the occurrence of the basic event is non-repairable. First, we obtain the probability of occurrence of the output event of a single priority AND gate by Markov analysis. Then, the top event probability is given by a cut set approach and the inclusion–exclusion formula. An efficient procedure to obtain the probabilities corresponding to logical products in the inclusion–exclusion formula is proposed. The logical product which is composed of two or more priority AND gates having at least one common basic event as their inputs is transformed into the sum of disjoint events which are equivalent to a priority AND gate in the procedure. Numerical examples show that our method works well for complex systems.     

A practical method for accurate quantification of large fault trees

This paper describes a practical method to accurately quantify top event probability and importance measures from incomplete minimal cut sets (MCS) of a large fault tree. The MCS-based fault tree method is extensively used in probabilistic safety assessments. Several sources of uncertainties exist in MCS-based fault tree analysis. The paper is focused on quantification of the following two sources of uncertainties: (1) the truncation neglecting low-probability cut sets and (2) the approximation in quantifying MCSs. The method proposed in this paper is based on a Monte Carlo simulation technique to estimate probability of the discarded MCSs and the sum of disjoint products (SDP) approach complemented by the correction factor approach (CFA). The method provides capability to accurately quantify the two uncertainties and estimate the top event probability and importance measures of large coherent fault trees. The proposed fault tree quantification method has been implemented in the CUTREE code package and is tested on the two example fault trees.     

Improved accuracy in quantitative fault tree analysis

The fault tree diagram defines the causes of the system failure mode or ‘top event’ in terms of the component failures and human errors, represented by basic events. By providing information which enables the basic event probability to be calculated, the fault tree can then be quantified to yield reliability parameters for the system. Fault tree quantification enables the probability of the top event to be calculated and in addition its failure rate and expected number of occurrences. Importance measures which signify the contribution each basic event makes to system failure can also be determined. Owing to the large number of failure combinations (minimal cut sets) which generally result from a fault tree study, it is not possible using conventional techniques to calculate these parameters exactly and approximations are required. The approximations usually rely on the basic events having a small likelihood of occurrence. When this condition is not met, it can result in large inaccuracies. These problems can be overcome by employing the binary decision diagram (BDD) approach. This method converts the fault tree diagram into a format which encodes Shannon's decomposition and allows the exact failure probability to be determined in a very efficient calculation procedure. This paper describes how the BDD method can be employed in fault tree quantification. © 1997 John Wiley & Sons, Ltd.     

基于模糊故障树的军用气象物资包装可靠性分析

应用模糊故障树分析方法对军用气象物资包装可靠性进行了系统分析,简要介绍了模糊故障树分析方法的基本理论,利用专家判断和模糊集理论相结合的方法,评估了故障树底事件发生的模糊失效概率。并以"TFS-1通风干湿表包装"为例,建立了包装系统的故障树,采用下行法求解了引起顶事件发生的最小割集,定量分析计算,得出模糊失效率为0.0705,同时计算了各底事件的重要度。模糊故障树分析方法对于提高军用气象物资包装防护能力,确保物资装备质量,具有非常重要的意义。     

Fault Diagnosis Improvement Using Dynamic Fault Model in Optimal Sensor Placement: A Case Study of Steam Turbine

Health data are collected dominantly through sensors mounted on different locations in the system. Optimization of sensor network has a significant influence on the reliability of system health prognostics process. In this research, the effect of sensors reliability is studied on their placement optimization. Sensors are considered in this study as components in system failure model. This study is aimed to use ‘Priority AND’ gate for evaluating the effect of time dependencies of sensors as well as components failure on the optimal sensor placement. Because of that, PAND gate is added to the fault tree between all sensors and their corresponding components to develop the failure model of each sensor placement scenario. For calculating the probability of top event, a Monte Carlo‐based algebraic approach is proposed. In algebraic approach, temporal operator ‘BEFORE’ is proposed for calculating the probability of ‘PAND’ gate. The result of using ‘BEFORE’ operator is an analytical solution for probability of each cut sequence. Because of the complexity of analytical solution in practical problems, a Monte Carlo simulation is utilized on the solution in this research. Then the probability of each cut sequence is calculated. Consequently, the probability of top event for each scenario is obtained. Finally, all scenarios are ranked based on top event probabilities. As a case study, optimization of sensor placement has been demonstrated on steam turbine and results are discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

The application of Petri nets to failure analysis

Unlike the technique of fault tree analysis that has been widely applied to system failure analysis in reliability engineering, this study presents a Petri net approach to failure analysis. It is essentially a graphical method for describing relations between conditions and events. The use of Petri nets in failure analysis enables to replace logic gate functions in fault trees, efficiently obtain minimal cut sets, and absorb models. It is demonstrated that for failure analysis Petri nets are more efficient than fault trees. In addition, this study devises an alternative; namely, a trapezoidal graph method in order to account for failure scenarios. Examples validate this novel method in dealing with failure analysis.     

Establishment and Analysis of the Fault Tree for the Oil Transfer Pump System in CNPC Work Zone

The reliability of the equipment is very important for the large petrochemical industry, especially for oil pump as the core component of driving equipment. In order to reduce the loss of the enterprise brought by equipment failure, it is need to find those reasons which may lead to equipment failure and take some preventive measures as early as possible. This article analyzes the failure of the oil transfer pump system in CNPC work zone systematically, qualitatively and quantitatively, using the fault tree analysis method. Then 105 groups of minimal cut sets are found, and the probability of system failure after a certain time operation is calculated by using Weibull distribution. Combined with specific requirements of reliability, the work zone may make a scientific decision of plant maintenance cycle according to the conclusion.     

Design of a composite beam using the failure probability‐safety factor method

The paper shows the practical importance of the failure probability‐safety factor method for designing engineering works. The method provides an automatic design tool by optimizing an objective function subject to the standard geometric and code constraints, and two more sets of constraints, that guarantee some given safety factors and failure probability bounds, associated with a given set of failure modes. Since a direct solution of the optimization problem is not possible, the method proceeds as a sequence of three steps: (a) an optimal classical design, based on given safety factors, is done, (b) failure probabilities or bounds of all failure modes are calculated, and (c) safety factors bounds are adjusted. This implies a double safety check that leads to safer structures and designs less prone to wrong or unrealistic probability assumptions, and to excessively small (unsafe) or large (costly) safety factors. Finally, the actual global or combined probabilities of the different failure modes and their correlation are calculated using a Monte Carlo simulation. In addition, a sensitivity analysis is performed. To this end, the optimization problems are transformed into another equivalent ones, in which the data parameters are converted into artificial variables. In this way, some variables of the dual associated problems become the desired sensitivities. The method is illustrated by its application to the design of a composite beam. Copyright 2004 © John Wiley & Sons, Ltd.     

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.     

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.     

Fuzzy Fault Diagnosis of a Diesel Engine Non-start

The diesel locomotive plays an important role in the field of transport,and the engine maintenance work is the prerequisite and guarantee for the locomotive normal working.In this paper,we first establish the fault tree model of locomotive engine 16V240ZJ on the basis of engine non-start as the top event.Then we combines the fuzzy mathematics theory and fault tree analysis method for failure diagnosis of 16V240ZJ engine's abnormal start-up.We obtained the fuzzy probability curve and top events probability c...     

A method for analysing the reliability of a transmission grid

The paper describes a probabilistic method for transmission grid security evaluation. Power system security is the ability of the power system to withstand sudden disturbances such as short circuits. The method presented here uses event and fault trees and combines them with power system dynamic simulations. Event trees model the substation protection and trip operations after line faults. Different event tree end states (fault duration, circuit breaker trips) are simulated with power system dynamic analysis program. The dynamic analysis results (power system post-fault states) are then classified into secure, alert, emergency and system breakdown. The probabilities, minimal cut sets and grid level importance measures (Fussell-Vesely, risk increase and decrease factors) are calculated for the total and partial system breakdown. In this way, the relative importance of the substation devices regarding to the system breakdown can be reached. Also the more and less likely contributing factors to system breakdown are received. With this method, an existing 400 kV transmission grid with its line fault and device failure statistics is analysed.     

A knowledge-based approach to the evaluation of fault trees

A list of critical components is useful for determining the potential problems of a complex system. However, to find this list through evaluating the fault trees is expensive and time consuming. This paper intends to propose an integrated software program which consists of a fault tree constructor, a knowledge base, and an efficient algorithm for evaluating minimal cut sets of a large fault tree. The proposed algorithm uses the approaches of top-down heuristic searching and the probability-based truncation. That makes the evaluation of fault trees obviously efficient and provides critical components for solving the potential problems in complex systems. Finally, some practical fault trees are included to illustrate the results.     

药品智能包装的模糊灰色故障树分析

目的对某款药品智能包装从系统层面进行可靠性分析。方法首先,通过失效分析建立药品智能包装的故障树;其次,利用专家判断知识和模糊数学理论,得到故障树底事件和顶事件的三角形式模糊失效率,以及采用模糊中值法进行底事件模糊重要度计算;最后,运用灰色关联分析求出表征各最小割集与顶事件相关行为关系的灰色关联度。结果定量分析计算显示,药品智能包装模糊失效率为(0.146 59,0.179 69,0.206 91),故障相关性最大的为包装材料本身的功能失效,其次为外界的振动和冲击以及包装尺寸误差等。结论模糊灰色关联故障树同时考虑包装系统的模糊性和灰色性,它可为预防药品智能包装失效、提高智能包装可靠性和安全性提供理论依据。     

Finding minimal cut sets in a fault tree

This paper presents a new method for identification of minimal cut sets in a fault tree. The (non-minimal) cut sets are found by a modification of the well-known MOCUS algorithm. These cut sets are stored in a virtual tree structure which requires far less core space than the MOCUS cut set matrix. The minimal cut sets are found by traversing this virtual tree a number of times. In the first cycle, all cut sets of order one are identified. In the next cycle, all cut sets of order two are identified and compared with the cut sets of order one to exclude non-minimal stes. This procedure is continued until all minimal cut sets are identified. The procedure is very fast. Compared to the standard MOCUS program the computer time is reduced by at least a factor of ten.     

Quantification of Highly Coupled Dynamic Fault Tree Using IRVPM and SBDD

Dynamic fault tree (DFT) is a commonly used method to model systems having sequence‐dependent and function‐dependent failure behaviors. The failure structure function of a DFT can be expressed by logic OR of all minimal cut sequences, that is, minimal cut sequence set (MCSS). The occurrence probability to the top event of a DFT can be calculated using inclusion–exclusion (IE) principle based on enumerating the MCSS. However, the IE‐based approach would have exponential evaluation complexity. Then, a sequential binary decision diagram (SBDD)‐based method is proposed and successfully applied to analyze simple dynamic systems. This method is more efficient than IE‐based method in asymptotic analysis. But this method cannot handle complex systems modeled by different highly coupled dynamic gates. In this paper, we put forward using Independent Random Variable Probabilistic Model‐based plus SBDD‐based methods to quantify an MCSS to obtain the failure probability of a complex DFT. The results obtained by the proposed method are exactly matched with those obtained by the existing methods. In addition, this method enhances the analyzing ability of the original SBDD and retains the advantage of high computational efficiency. The application and advantage of our proposed method is demonstrated by a case study. Copyright © 2014 John Wiley & Sons, Ltd.     

Reliability modelling of repairable systems using Petri nets and fuzzy Lambda–Tau methodology

A methodology is developed which uses Petri nets instead of the fault tree methodology and solves for reliability indices utilising fuzzy Lambda–Tau method. Fuzzy set theory is used for representing the failure rate and repair time instead of the classical (crisp) set theory because fuzzy numbers allow expert opinions, linguistic variables, operating conditions, uncertainty and imprecision in reliability information to be incorporated into the system model. Petri nets are used because unlike the fault tree methodology, the use of Petri nets allows efficient simultaneous generation of minimal cut and path sets.     

The Minimum Cut Sets of System Fault Based on Oriented Matrix Analysis

1　ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｍｍｏｎｌｙ ,ａｇｉｇａｎｔｉｃｃｏｍｐｌｅｘｓｙｓｔｅｍｕｔｉｌｉｚｅｓＦａｕｌｔＴｒｅｅＡｎａｌｙｓｉｓ (ＦＴＡ)ｔｏａｎａｌｙｚｅｉｔｓｒｅｌｉａｂｉｌｉｔｙａｎｄｕｔｉｌｉｚｅｓａｎｕｐｌｉｎｋｏｒｄｏｗｎｌｉｎｋｍｅｔｈｏｄｔｏｏｂｔａｉｎｔｈｅｍｉｎｉｍｕｍｃｕｔｓｅｔｓ.ＹｅｔｉｔｉｓｃｏｍｐｌｅｘｔｏｕｓｅＦＴＡｔｏｂｕｉｌｄａｔｒｅｅａｎｄｉｔｄｅｍａｎｄｓｔｈａｔｔｈｅｂｕｉｌｄｅｒｉｎｔｅｎｓｉｖｅｌｙｒｅａｌｉｚｅａｌｌｃｏｍｐｏｎｅｎｔｓｏ…