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.     

硝铵炸药真空干燥工艺燃爆事故致灾因素的研究

真空干燥是硝铵炸药生产过程中一道容易发生燃爆事故的重要工序。为研究硝铵炸药真空干燥过程中发生燃爆事故的原因及机理，通过事故案例分析和现场调研，确定了导致燃爆事故的各个基本事件及其逻辑关系，并由此构建以燃爆事故作为顶事件的事故树。采用布尔代数化简事故树，得到87个最小割集和9个最小径集，结果显示每个最小径集包含的基本事件都较多，说明真空干燥工艺安全性较低。通过计算各基本事件的结构重要度并排序，得到结构重要度较大的基本事件，由此推断出导致燃爆事故的主要基本事件，并有针对性地提出相应的改进措施与建议，为企业的安全生产提供参考。     

A new importance measure for risk-informed decision making

In this paper, we introduce a new importance measure, the differential importance measure (DIM), for probabilistic safety assessment (PSA). DIM responds to the need of the analyst/decision maker to get information about the importance of proposed changes that affect component properties and multiple basic events. DIM is directly applicable to both the basic events and the parameters of the PSA model. Unlike the Fussell–Vesely (FV), risk achievement worth (RAW), Birnbaum, and criticality importance measures, DIM is additive, i.e. the DIM of groups of basic events or parameters is the sum of the individual DIMs. We discuss the difference between DIM and other local sensitivity measures that are based on normalized partial derivatives. An example is used to demonstrate the evaluation of DIM at both the basic event and the parameter level. To compare the results obtained with DIM at the parameter level, an extension of the definitions of FV and RAW is necessary. We discuss possible extensions and compare the results of the three measures for a more realistic example.     

A study on importance measures and a quantification algorithm in a fire PRA model

In the current quantification of fire probabilistic risk assessment (PRA), when components are damaged by a fire, the basic event values of the components become ‘true’ or one (1), which removes the basic events related to the components from the minimal cut sets, and which makes it difficult to calculate accurate component importance measures. A new method to accurately calculate an importance measure such as Fussell-Vesely in fire PRA is introduced in this paper. Also, a new quantification algorithm in the fire PRA model is proposed to support the new calculation method of the importance measures. The effectiveness of the new method in finding the importance measures is illustrated with an example of evaluating cables’ importance.     

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。     

Differential, criticality and Birnbaum importance measures: An application to basic event, groups and SSCs in event trees and binary decision diagrams

Recent works [Epstein S, Rauzy A. Can we trust PRA? Reliab Eng Syst Safety 2005; 88:195–205] have questioned the validity of traditional fault tree/event tree (FTET) representation of probabilistic risk assessment problems. In spite of whether the risk model is solved through FTET or binary decision diagrams (BDDs), importance measures need to be calculated to provide risk managers with information on the risk/safety significance of system structures and components (SSCs). In this work, we discuss the computation of the Fussel–Vesely (FV), criticality, Birnbaum, risk achievement worth (RAW) and differential importance measure (DIM) for individual basic events, basic event groups and components. For individual basic events, we show that these importance measures are linked by simple relations and that this enables to compute basic event DIMs both for FTET and BDD codes without additional model runs. We then investigate whether/how importance measures can be extended to basic event groups and components. Findings show that the estimation of a group Birnbaum or criticality importance is not possible. On the other hand, we show that the DIM of a group or of a component is exactly equal to the sum of the DIMs of the corresponding basic events and can therefore be found with no additional model runs. The above findings hold for both the FTET and the BDD methods.     

A decision-oriented measure of uncertainty importance for use in PSA

For the interpretation of the results of probabilistic risk assessments it is important to have measures which identify the basic events that contribute most to the frequency of the top event but also to identify basic events that are the main contributors to the uncertainty in this frequency. Both types of measures, often called Importance Measure and Measure of Uncertainty Importance, respectively, have been the subject of interest for many researchers in the reliability field. The most frequent mode of uncertainty analysis in connection with probabilistic risk assessment has been to propagate the uncertainty of all model parameters up to an uncertainty distribution for the top event frequency. Various uncertainty importance measures have been proposed in order to point out the parameters that in some sense are the main contributors to the top event distribution. The new measure of uncertainty importance suggested here goes a step further in that it has been developed within a decision theory framework, thereby providing an indication of on what basic event it would be most valuable, from the decision-making point of view, to procure more information.     

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.     

Fault tree truncation error bounds

The solution of large fault trees can only performed if cut sets of a high order or low probability are neglected. This procedure is non-pessimistic since possible contributions to the top gate failure probability are ignored. Since the number of cut sets neglected is generally large, it is possible that their total contribution is significant compared to the total probability of those cut sets included in the evaluation of the top gate failure probability.In this paper a practical method for the estimation of upper bounds on the total failure probability of cuts sets of a given order is presented. This allows bounds on the contribution of all the cut sets neglected through cut-off procedures in fault tree solutions to be calculated, and so validate the top event failure probability. The method given here is much superior to that suggested by Modarres and Dezfuli in that it produces much lower and hence more useful bounds. The method of Modarres and Dezfuli can be refined, but for realistic fault tree examples the method given here always gives the least pessimistic and most practical bounds.     

基于FTA模型的荔枝冷藏运输环节风险分析

目的解决荔枝冷藏运输环节中安全风险识别及定性分析的难题。方法基于因素空间及故障树分析模型(FTA),分析荔枝冷链运输环节的安全事件集、空间结构(工位)集和简约因素集,建立荔枝冷藏运输环节风险因素关系矩阵,通过矩阵运算获取不同空间结构下荔枝运输安全事故发生的基本事件。结果根据运算求解结果,构建荔枝运输环节的故障树模型,获取了运输环节故障树的最小割集。荔枝冷藏运输事故最小割集数为13个,并分析了各个基本事件的结构重要度。结论通过研究最小割集及事件的结构重要度,进行荔枝冷链运输环节的安全分析,并提出了促进现场安全管理的对策及建议。     

Application of the integrated reliability analysis system (IRAS)

This paper introduces a UNIX-based computer aided reliability assessment system, IRAS, which was developed in the Brite/Euram project BE-4250. It utilises fault propagation models for automatic generation of Fault Trees, Cause–Consequence Diagrams and FMECA. Therefore, it has the following features: a Model Builder which allows the creation of the fault propagation models in a hierarchical manner; a Fault Tree Analysis module that is able to generate Fault Trees on demand and to extract minimal cut sets; an FMECA module that is able to search for and group effects of basic events according to their criticality, severity and probability; a Real Time Fault Location (RTFL) module that enables the fast detection of the most probable cause(s) of system malfunction based on information available from sensors and/or operator. This paper describes the underlying ideas and procedures of IRAS and shows an example application to a Hot Strip Steel Mill.     

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.     

Effects of manufacturing defects on the mechanical properties of carbon fibre reinforced polyethersulphone laminates

The effects of defects on the mechanical properties of carbon fibre reinforced polyethersulphone laminates have been measured. The defects studied were cut fibre plies, omission of polymer films and local delamination produced by the inclusion of foreign matter. Of these it was found that only cut plies had a significant detrimental effect on the strength of a laminate. For specimens with two cut plies, the failure stress, tensile, flexural and compressive in the remaining continuous plies was the same as in the defect-free material, provided that the cut plies were widely separated. However the failure stresses were 15–18% lower in the continuous plies in the specimens containing two cut plies which were more closely spaced and in specimens containing four cut plies.     

On the use of the Balancing Method for calculating component RAW involving CCFs in SSC categorization

In this paper, it is revived and suggested that the method called ‘Balancing Method’ could be used in deriving a component risk achievement worth (RAW) from the basic event RAWs including common cause failure RAW, and the Balancing Method is compared with the other methods. Since the component RAW value generated by the Balancing Method very closely resembles the real component RAW value, the Balancing Method could be widely used in the calculation of the component RAW which plays a key role in the risk informed regulation and applications.     

A practical engineering method for fuzzy reliability analysis of mechanical structures

The fuzzy sets theory in reliability analyses is studied. The structure stress is related to several other variables, such as structure sizes, material properties, external loads; in most cases, it is difficult to be expressed in a mathematical formula, and the related variables are not random variables, but fuzzy variables or other uncertain variables which have not only randomness but also fuzziness. In this paper, a novel approach is presented to use the finite element analysis as a “numerical experiment” tool, and to find directly, by fuzzy linear regression method, the statistical property of the structure stress. Based on the fuzzy stress–random strength interference model proposed in this paper, the fuzzy reliability of the mechanical structure can be evaluated. The compressor blade of a given turbocharger is then introduced as a realistic example to illustrate the approach.     

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.     

Comparison of global sensitivity analysis techniques and importance measures in PSA

This paper discusses application and results of global sensitivity analysis techniques to probabilistic safety assessment (PSA) models, and their comparison to importance measures. This comparison allows one to understand whether PSA elements that are important to the risk, as revealed by importance measures, are also important contributors to the model uncertainty, as revealed by global sensitivity analysis. We show that, due to epistemic dependence, uncertainty and global sensitivity analysis of PSA models must be performed at the parameter level. A difficulty arises, since standard codes produce the calculations at the basic event level. We discuss both the indirect comparison through importance measures computed for basic events, and the direct comparison performed using the differential importance measure and the Fussell–Vesely importance at the parameter level. Results are discussed for the large LLOCA sequence of the advanced test reactor PSA.     

The effect of Ag and Y-24W1 addition on the microstructure and superconducting properties of single grain Y–Ba–Cu–O

We have reported recently the fabrication of high performance, nano-composite Y–Ba–Cu–O (YBCO) bulk superconductors containing nano-sized Y₂Ba₄CuMO (Y-2411, where M is typically a transition metal element) phase inclusions that contribute effectively to flux pinning. However, these studies have revealed that the field trapping capability of YBCO bulk superconductors containing the Y-2411 phase is limited due to the presence of micro- and macro-cracks in the bulk sample microstructure. Here, we report the fabrication of nano-composite YBCO bulk superconductors containing Ag and W-based Y-2411 (i.e. Y₂Ba₄CuWO_y) secondary phase inclusions (SPI's) in an attempt to investigate the influence of Ag content on the formation and propagation of a/b and c- axis micro- and/or macro-cracks in the Y-123 matrix. The crack density for samples containing Ag is observed to be significantly lower than for samples fabricated without Ag. A detailed study of the microstructures and superconducting properties of Ag-doped, nano-composite bulk YBCO is presented.     

A PSA-based vital area identification methodology development

This paper presents a vital area identification method based on the current probabilistic safety assessment (PSA) techniques. The vital area identification method in this paper is focused on core melt rather than radioactive material release. Furthermore, it describes a conceptual framework with which the risk from sabotage-induced events could be assessed.Location minimal cut sets (MCSs) are evaluated after developing a core melt location fault tree (LFT). LFT is a fault tree whose basic events are sabotage-induced damages on the locations within which various safety-related components are located. The core melt LFT is constructed by combining all sequence LFTs of various event trees with OR gates. Each sequence LFT is constructed by combining the initiating event LFT and the mitigating event LFTs with an AND gate. The vital area could be identified by using the location importance measures on the core melt location MCSs. An application was made to a typical 1000 MWe pressurized water reactor power plant located at the Korean seashore.The methodology suggested in the present paper is believed to be very consistent and most complete in identifying the vital areas in a nuclear power plant because it is based on the well-proven PSA technology.     

The work process analysis model (WPAM)

The work process analysis model-I (WPAM-I) along with its products developed in a previous paper (Davoudian, K., Wu, J.-S. & Apostolakis, G., Reliability Engineering and System Safety, 45 (1994) 85–105 are used as inputs to WPAM-II. The goal is to provide the link between organizational factors (or dimensions), work processes, and probabilistic safety assessment parameters in order to facilitate the quantification of the impact of organizational factors on plant safety. This is achieved by calculating new (organizationally dependent) probabilities for minimal cut sets so that each new probability contains in it, either explicitly or implicitly, the effect of the pertinent organizational factors. A sample case is presented demonstrating the application of WPAM to a specific minimal cut set. Finally, sensitivity analyses are performed in order to explore the effectiveness of organizational improvements as a risk management strategy.