Dynamic fault tree analysis using Monte Carlo simulation in probabilistic safety assessment

Traditional fault tree (FT) analysis is widely used for reliability and safety assessment of complex and critical engineering systems. The behavior of components of complex systems and their interactions such as sequence- and functional-dependent failures, spares and dynamic redundancy management, and priority of failure events cannot be adequately captured by traditional FTs. Dynamic fault tree (DFT) extend traditional FT by defining additional gates called dynamic gates to model these complex interactions. Markov models are used in solving dynamic gates. However, state space becomes too large for calculation with Markov models when the number of gate inputs increases. In addition, Markov model is applicable for only exponential failure and repair distributions. Modeling test and maintenance information on spare components is also very difficult. To address these difficulties, Monte Carlo simulation-based approach is used in this work to solve dynamic gates. The approach is first applied to a problem available in the literature which is having non-repairable components. The obtained results are in good agreement with those in literature. The approach is later applied to a simplified scheme of electrical power supply system of nuclear power plant (NPP), which is a complex repairable system having tested and maintained spares. The results obtained using this approach are in good agreement with those obtained using analytical approach. In addition to point estimates of reliability measures, failure time, and repair time distributions are also obtained from simulation. Finally a case study on reactor regulation system (RRS) of NPP is carried out to demonstrate the application of simulation-based DFT approach to large-scale problems.     

Reliability assessment for systems suffering common cause failure based on Bayesian networks and proportional hazards model

The Bayesian network (BN) is an efficient tool for probabilistic modeling and causal inference, and it has gained considerable attentions in the field of reliability assessment. The common cause failure (CCF) is simultaneous failure of multiple elements in a system under a common cause, and it is a common phenomenon in engineering systems with dependent elements. Several models and methods have been proposed for modeling and assessment of complex systems with CCF. In this paper, a new reliability assessment method is proposed for the systems suffering from CCF in a dynamic environment. The CCF among components is characterized by a BN, which allows for bidirectional reasoning. A proportional hazards model is applied to capture the dynamic working environment of components and then the reliability function of the system is obtained. The proposed method is validated through an illustrative example, and some comparative studies are also presented.     

Predictive maintenance of complex system with multi-level reliability structure

Onboard sensors, which constantly monitor the states of a system and its components, have made the predictive maintenance (PdM) of a complex system possible. To date, system reliability has been extensively studied with the assumption that systems are either single-component systems or they have a deterministic reliability structure. However, in many realistic problems, there are complex multi-component systems with uncertainties in the system reliability structure. This paper presents a PdM scheme for complex systems by employing discrete time Markov chain models for modelling multiple degradation processes of components and a Bayesian network (BN) model for predicting system reliability. The proposed method can be considered as a special type of dynamic Bayesian network because the same BN is repeatedly used over time for evaluating system reliability and the inter-time–slice connection of the same node is monitored by a sensor. This PdM scheme is able to make probabilistic inference at any system level, so PdM can be scheduled accordingly.     

A discrete-time Bayesian network reliability modeling and analysis framework

Dependability tools are becoming an indispensable tool for modeling and analyzing (critical) systems. However the growing complexity of such systems calls for increasing sophistication of these tools. Dependability tools need to not only capture the complex dynamic behavior of the system components, but they must be also easy to use, intuitive, and computationally efficient. In general, current tools have a number of shortcomings including lack of modeling power, incapacity to efficiently handle general component failure distributions, and ineffectiveness in solving large models that exhibit complex dependencies between their components. We propose a novel reliability modeling and analysis framework based on the Bayesian network (BN) formalism. The overall approach is to investigate timed Bayesian networks and to find a suitable reliability framework for dynamic systems. We have applied our methodology to two example systems and preliminary results are promising. We have defined a discrete-time BN reliability formalism and demonstrated its capabilities from a modeling and analysis point of view. This research shows that a BN based reliability formalism is a powerful potential solution to modeling and analyzing various kinds of system components behaviors and interactions. Moreover, being based on the BN formalism, the framework is easy to use and intuitive for non-experts, and provides a basis for more advanced and useful analyses such as system diagnosis.     

Exact combinatorial reliability analysis of dynamic systems with sequence-dependent failures

Many real-life fault-tolerant systems are subjected to sequence-dependent failure behavior, in which the order in which the fault events occur is important to the system reliability. Such systems can be modeled by dynamic fault trees (DFT) with priority-AND (pAND) gates. Existing approaches for the reliability analysis of systems subjected to sequence-dependent failures are typically state-space-based, simulation-based or inclusion-exclusion-based methods. Those methods either suffer from the state-space explosion problem or require long computation time especially when results with high degree of accuracy are desired. In this paper, an analytical method based on sequential binary decision diagrams is proposed. The proposed approach can analyze the exact reliability of non-repairable dynamic systems subjected to the sequence-dependent failure behavior. Also, the proposed approach is combinatorial and is applicable for analyzing systems with any arbitrary component time-to-failure distributions. The application and advantages of the proposed approach are illustrated through analysis of several examples.     

Coherence region of the Priority‐AND gate: Analytical and numerical examples

In recent years, the need for a more accurate dependability modelling (encompassing reliability, availability, maintenance, and safety) has favoured the emergence of novel dynamic dependability techniques able to account for temporal and stochastic dependencies of a system. One of the most successful and widely used methods is Dynamic Fault Tree that, with the introduction of the dynamic gates, enables the analysis of dynamic failure logic systems such as fault‐tolerant or reconfigurable systems. Among the dynamic gates, Priority‐AND (PAND) is one of the most frequently used gates for the specification and analysis of event sequences. Despite the numerous modelling contributions addressing the resolution of the PAND gate, its failure logic and the consequences for the coherence behaviour of the system need to be examined to understand its effects for engineering decision‐making scenarios including design optimization and sensitivity analysis. Accordingly, the aim of this short communication is to analyse the coherence region of the PAND gate so as to determine the coherence bounds and improve the efficacy of the dynamic dependability modelling process.     

Reliability analysis of the dynamic system for the Chen model through sequential order statistics

Recently, the two-parameter Chen distribution has widely been used for reliability studies in various engineering fields. In this article, we have developed various statistical inferences on the composite dynamic system, assuming Chen distribution as a baseline model. In this dynamic system, failure of a component induces a higher load on the surviving components and thus increases component hazard rate through a power-trend process. The classical and Bayesian point estimates of the unknown parameters of the composite system are obtained by the method of maximum likelihood and Markov chain Monte Carlo techniques, respectively. In the Bayesian framework, we have used gamma priors to obtain Bayes estimates of unknown parameters under the squared error and generalized entropy loss functions. The interval estimates of the baseline reliability function are obtained by using the Fisher information matrix and Bayesian method. A parametric hypothesis test is presented to test whether the failed components change the hazard rate function. A compact simulation study is carried out to examine the behavior of the proposed estimation methods. Finally, one real data analysis is performed for illustrative purposes.     

Investigating the reliability and criticality of the maintenance characteristics of a diving support vessel

Maintenance tasks and their application in the shipping industry have evolved significantly in the recent years. Particularly in the offshore industry, safety onboard, environmental protection and intensive operational activities necessitate the minimization of down‐time and the preservation of an excellent performance ratio. The first step of an innovative ship maintenance strategy, which is proposed by the authors and is based on criticality and reliability assessment, is presented herein using the FTA tool with time‐dependant dynamic gates so as to represent in an accurate and comprehensive way the interrelation of the components of a system. The paper also presents a review of the maintenance standards and procedures, such as the ALARP concept, the Key Programme 3‐Asset Integrity (KP3) initiative, the OREDA handbook as well as the RCM and RBI principles. As part of the reliability assessment, the Birnbaum and Criticality reliability importance measures are utilized to validate the results of the analysis. A case study of a diving support vessel (DSV) illustrates the application of this strategy. The main systems examined are: the vessel's power plant, propulsion, water system, lifting, hauling and anchoring, diving and finally the safety system. The reliability of the main systems and subsystems as well as of their critical components is identified and suggestions of how to improve the overall reliability of the various systems both at a component, system and managerial level are also proposed. Copyright © 2011 John Wiley & Sons, Ltd.     

Bayesian reliability analysis of a products of probabilities model for parallel systems with dependent components

In a Bayesian reliability analysis of a system with dependent components, an aggregate analysis (i.e. system-level analysis) or a simplified disaggregate analysis with independence assumptions may be preferable if the estimations obtained from employing these two approaches do not deviate substantially from those derived through a disaggregate analysis, which is generally considered the most accurate method. This study was conducted to identify the key factors and their range of values that lead to estimation errors of great magnitude. In particular, a copula-based Bayesian reliability model was developed to formulate the dependence structure for a products of probabilities model of a simple parallel system. Monte Carlo simulation, regionalised sensitivity analysis and classification tree learning were employed to investigate the key factors. The resulting classification tree achieved favourable predictive accuracy. Several decision rules suggesting the optimal approach under different combinations of conditions were also extracted. This study has made a methodological contribution in laying the groundwork for investigating systems with dependent components using copula-based Bayesian reliability models. With regard to practical implications, this study also derived useful guidelines for selecting the most appropriate analysis approach under different scenarios with different magnitude of dependence.     

Application of Bayesian Methods for Age‐Dependent Reliability Analysis

In this article, the authors present a general methodology for age‐dependent reliability analysis of degrading or ageing components, structures and systems. The methodology is based on Bayesian methods and inference—its ability to incorporate prior information and on ideas that ageing can be thought of as age‐dependent change of beliefs about reliability parameters (mainly failure rate), when change of belief occurs not only because new failure data or other information becomes available with time but also because it continuously changes due to the flow of time and the evolution of beliefs. The main objective of this article is to present a clear way of how practitioners can apply Bayesian methods to deal with risk and reliability analysis considering ageing phenomena. The methodology describes step‐by‐step failure rate analysis of ageing components: from the Bayesian model building to its verification and generalization with Bayesian model averaging, which as the authors suggest in this article, could serve as an alternative for various goodness‐of‐fit assessment tools and as a universal tool to cope with various sources of uncertainty. The proposed methodology is able to deal with sparse and rare failure events, as is the case in electrical components, piping systems and various other systems with high reliability. In a case study of electrical instrumentation and control components, the proposed methodology was applied to analyse age‐dependent failure rates together with the treatment of uncertainty due to age‐dependent model selection. Copyright © 2013 John Wiley & Sons, Ltd.     

Reliability assessment for failure-dependent and uncertain systems: A Bayesian network based on copula method and probability-box

Managing failure dependence of complex systems with hybrid uncertainty is one of the hot problems in reliability assessment. Epistemic uncertainty is attributed to complex working environment, system structure, human factors, imperfect knowledge, etc. Probability-box has powerful characteristics for uncertainty analysis and can be effectively adopted to represent epistemic uncertainty. However, arithmetic rules on probability-box structures are mostly used among structures representing independent random variables. In most practical engineering applications, failure dependence is always introduced in system reliability analysis. Therefore, this paper proposes a developed Bayesian network combining copula method with probability-box for system reliability assessment. There are four main steps involved in the reliability computation process: marginal distribution identification and estimation, copula function selection and parameter estimation, reliability analysis of components with correlations and Bayesian forward analysis. The benefits derived from the proposed approach are used to overcome the computational limitations of n-dimensional integral operation, and the advantages of useful properties of copula function in reliability analysis of systems with correlations are adopted. To demonstrate the effectiveness of the developed Bayesian network, the proposed method is applied to a real large piston compressor.     

A quantification algorithm for a repairable system in the GO methodology

The GO methodology is an effective method of system reliability analysis. It has been applied to non-repairable systems. This paper discusses the application of the GO method to a repairable system which is described by a Markov model and presents the quantification algorithm of the steady characteristics of the repairable system. The calculation formulas of the ordinary operators and the logical gates are derived and the steady reliability parameters of the system such as average operation probability and average failure frequency can be directly computed by the GO method. The result of an example shows that the algorithm is correct. The algorithm will be useful for the safety analysis of most engineering repairable systems.     

Reliability importance analysis of Markovian systems at steady state using perturbation analysis

Sensitivity analysis has been primarily defined for static systems, i.e. systems described by combinatorial reliability models (fault or event trees). Several structural and probabilistic measures have been proposed to assess the components importance. For dynamic systems including inter-component and functional dependencies (cold spare, shared load, shared resources, etc.), and described by Markov models or, more generally, by discrete events dynamic systems models, the problem of sensitivity analysis remains widely open. In this paper, the perturbation method is used to estimate an importance factor, called multi-directional sensitivity measure, in the framework of Markovian systems. Some numerical examples are introduced to show why this method offers a promising tool for steady-state sensitivity analysis of Markov processes in reliability studies.     

Reliability analysis of hierarchical computer-based systems subject to common-cause failures

The results from reliability modeling and analysis are key contributors to design and tuning activities for computer-based systems. Each architecture style, however, poses different challenges for which analytical approaches must be developed or modified. The challenge we address in this paper is the reliability analysis of hierarchical computer-based systems (HS) with common-cause failures (CCF). The dependencies among components introduced by CCF complicate the reliability analysis of HS, especially when components affected by a common cause exist on different hierarchical levels. We propose an efficient decomposition and aggregation (EDA) approach for incorporating CCF into the reliability evaluation of HS. Our approach is to decompose an original HS reliability analysis problem with CCF into a number of reduced reliability problems freed from the CCF concerns. The approach is represented in a dynamic fault tree by a proposed CCF gate modeled after the functional dependency gate. We present the basics of the EDA approach by working through a hypothetical analysis of a HS subject to CCF and show how it can be extended to an analysis of a hierarchical phased-mission system subject to different CCF depending on mission phases.     

Footprint of knowledge acquisition improvement in failure diagnosis analysis

Fault tree analysis (FTA) as an effective and efficient risk assessment tool are widely used to analyze the reliability of a complex system. In this context, FTA can properly improve the safety performance of the system by preventing an event which may lead to occurrence of a catastrophic accident. However, traditional FTA is still suffering from dynamic structure demonstration and importantly epistemic uncertainty processing. In this study, a novel methodology is introduced using Bayesian updating mechanism to deal with dynamic structure and 2‐tuple fuzzy set named as intuitionistic fuzzy numbers are employed to cope with subjectivity of uncertainty processing. Accordingly, the most critical system components which affect the system reliability are recognized by using an appropriate sensitivity analysis method. The proposed methodology is then applied on a real case study application (a brake fluid filling system) in order to examine the effectiveness and feasibility of the approach. The results illustrated that the new methodology can have enough benefits for diagnosing the systems' faults compared with listing approaches of safety and reliability analysis. In terms of empirical case study, “electromotor failure” was evaluated as the second most critical basic event in conventional‐based approaches, whereas in the novel methodology “high pressure liquefied material” was recognized as the second one.     

Aggregated Markov-based reliability analysis of multi-state systems under combined dynamic environments

Reliability of a system may differ greatly when operating under different environments. However, the existing works have either neglected the environment factor in system reliability analysis or considered this factor for binary systems or systems subject to a single environment (parameter). In this paper, we make contributions by modeling a multi-state system operating under hybrid dynamic environments affected by multiple environmental parameters. Different Markov chains with finite states are used to represent the random system behavior and dynamic environments, leading to an aggregated Markov process that models the overall system behavior. An effective approach based on state partitions and aggregations is suggested for assessing the system reliability indexes, including reliability, availability, multi-point availability, and environment-based reliability. A high-pressure homogenizer system is analyzed to demonstrate the proposed model and show the comparison of the reliability of system under fixed and dynamic environment.     

Sensibility of Bayesian inference methods for reliability prediction of ageing systems,case of Diesel locomotives

This paper addresses the problem of reliability analysis of in-service identical systems when a limited number of lifetime data is available compared to censored ones. Lifetime (resp. censored) data characterise the life of failed (resp. non-failed) systems in the sample. Because, censored data induce biassed estimators of reliability model parameters, a methodology approach is proposed to overcome this inconvenience and improve the accuracy of the parameter estimation based on Bayesian inference methods. These methods combine, in an effective way, system’s life data and expert opinions learned from failure diagnosis of similar systems. Three Bayesian inference methods are considered: Classical Bayesian, Extended Bayesian and Bayesian Restoration Maximisation methods. Given a sample of lifetime data, simulated according to prior opinions of maintenance expert, a sensibility analysis of each Bayesian method is performed. Reliability analysis of critical subsystems of Diesel locomotives is established under the proposed methodology approach. The relevance of each Bayesian inference methods with respect to collected reliability data of critical subsystems and expert opinions is discussed.     

A new efficient algorithm for computing the imprecise reliability of monotone systems

Reliability analysis of complex systems by partial information about reliability of components and by different conditions of independence of components may be carried out by means of the imprecise probability theory which provides a unified framework (natural extension, lower and upper previsions) for computing the system reliability. However, the application of imprecise probabilities to reliability analysis meets with a complexity of optimization problems which have to be solved for obtaining the system reliability measures. Therefore, an efficient simplified algorithm to solve and decompose the optimization problems is proposed in the paper. This algorithm allows us to practically implement reliability analysis of monotone systems under partial and heterogeneous information about reliability of components and under conditions of the component independence or the lack of information about independence. A numerical example illustrates the algorithm.     

Dynamic reliability analysis of gear transmission system based on sparse grid numerical integration and saddle-point approximation method

Gear systems are widely used in various mechanical transmission systems. This paper aims to develop an effective and practical method for dynamic reliability analysis of gear transmission system. The proposed method can comprehensively evaluate the dynamic reliability of gear transmission system by adopting the fourth-moment SPA method. First, a nonlinear dynamics model of a single-stage spur gear transmission system is established, which simultaneously takes into account the nonlinear backlash, time-varying meshing stiffness, and static transmission error. After that, a dynamic reliability model for the tooth surface contact fatigue failure of gear system is established with the uncertainty of the motion, structure, and material parameters using stress-strength interference (SSI) theory. To be specific, the sparse grid numerical integration (SGNI) method is applied to solve the statistical characteristic parameters of the dynamic reliability of the system. The probability distribution of the performance function is obtained with the fourth-moment SPA method. Test examples show that the results of the proposed method are consistent with the results obtained by the Monte Carlo simulation (MCS) and superior to the maximum entropy with fractional moments (ME-FM) method, which verifies the effectiveness of this approach. Finally, the dynamic reliability of the gear transmission system with respect to load times is evaluated.     

Mechanical system reliability analysis using a combination of graph theory and Boolean function

A new method based on graph theory and Boolean function for assessing reliability of mechanical systems is proposed. The procedure for this approach consists of two parts. By using the graph theory, the formula for the reliability of a mechanical system that considers the interrelations of subsystems or components is generated. Use of the Boolean function to examine the failure interactions of two particular elements of the system, followed with demonstrations of how to incorporate such failure dependencies into the analysis of larger systems, a constructive algorithm for quantifying the genuine interconnections between the subsystems or components is provided. The combination of graph theory and Boolean function provides an effective way to evaluate the reliability of a large, complex mechanical system. A numerical example demonstrates that this method an effective approaches in system reliability analysis.