A dependency bounds analysis method for reliability assessment of complex system with hybrid uncertainty
Introduction
Reliability assessment is a process of analyzing system and its components reliability. The process includes reliability modeling, data collection, data processing, component reliability analysis and system reliability analysis [1]. To obtain system reliability, failure logic relationship between components or subsystems generally needs to be represented. A common reliability model to describe component-system relationship is Bayesian network (BN) [2]. BN is used to present the joint probability distribution of variables and their associated structure via directed acyclic graph. It has the advantage of representing causal relationships among variables by conditional probabilities. In the literature, BN is widely used in reliability assessment. For example, with the incorporation of multiple failure sequences and correlations between component failures, Mahadevan et al. [3] developed a new methodology to apply BN to structure system reliability analysis. By using BN, Abaei et al. [4] proposed a novel methodology to conduct reliability analysis of moored floating structures. Tien et al. [5] proposed novel algorithms to reduce memory storage required to construct BN for reliability assessment. In view of the advantages and extensive study of BN, this paper applies it as basic framework.
Complex electromechanical systems are the foundation of modern engineering equipment. They have been widely used in various engineering fields, e.g. military, aerospace, medical instruments, etc. Due to complex system structures, a lack of knowledge, limited experimental conditions, complex working environment, human factors etc., uncertainties always exist in reliability assessment [6], [7], [8]. Generally, uncertainty can be classified into epistemic uncertainty and aleatory uncertainty [9]. In reliability assessment, the appropriate treatment of aleatory uncertainty and epistemic uncertainty is a topic of great importance and hence of widespread interest [10,11]. Uncertainty can generally be tackled by classical probabilistic methods. However, in the presence of imprecise data or lack of knowledge, they often require some assumptions. Thanks to imprecise probabilistic approaches, they have been widely developed to manage uncertainty. An interesting model named probability box (p-box) has been proposed by Ferson et al. [12]. It considers a family of probability distributions to deal with epistemic uncertainty.
Recently, Rocchetta et al. [13] built an efficient reliability assessment framework subjected to p-box and exploited an adapted Importance Sampling algorithm to efficiently propagate p-box. Liu et al. [14] developed a reliability analysis approach based on probability and p-box hybrid model for uncertain structures. The work of Simon et al. [11] using p-box to analyze hybrid uncertainty is very close to part of the proposed work, except dependency problem is considered simultaneously here.
A complex system is made up of components, and there are multiple failure forms for each component. Due to processing, transportation, installation, environmental impact, human factors etc., some failures among components are generally dependent. Obviously, failure dependence reduces system reliability and complicates reliability analysis. For instance, in 2010, a fire broke out on Deep Sea Horizon Platform in Gulf of Mexico. The cause of the accident was the failure of the Subsea BOP system. The failure of the Subsea BOP system further caused oil and gas to be ejected from the wellhead. Finally, the well control system was completely out of control. Failure dependence also widely exists in aerospace industry [15]. According to the statistics, around 70% of component failures in aero engines are caused by dependency. Similarly, for a general mechanical system, such as rolling bearing, if the manufacturing of bearing roller is defective, wear and corrosion can affect the performance of the bearing's inner and outer rings.
In the literature, many papers can be found where dependency problem is managed. Wang et al. [16] applied both explicit and implicit methods in performing reliability analysis of phased-mission systems with probabilistic common cause failures. Mi et al. [17] produced an extended universal generating function to handle common cause failures and epistemic uncertainty. As a new tool of dependency bounds analysis, copula provides new insights into system reliability [18]. Eryilmaz [19] presented a new way of modeling s-dependence between two multi-state components. Wang et al. [20] used time-varying copulas to construct a s-dependent competing risk model for systems. However, in most real applications, there is some prior information on the dependency of components, but the available information may be insufficient to determine dependent parameters. Obviously, the above methods seem not to be very suitable for non-deterministic dependency problem. Therefore, a model for non-deterministic dependency problem needs to be built.
Affine arithmetic is a useful tool that aims to attack non-deterministic dependency problem in interval calculations [21,22]. With consideration of dependency between inputs, affine arithmetic can produce interval estimates [23]. In the field of theoretical research, some papers can be referred where affine arithmetic is used for dependency problem. For instance, Messine [24] proposed three new expressions of affine forms to reduce dependency problem. Based on affine arithmetic, De Figueiredo et al. [25] used an interval-based uncertainty model to provide a solution for dependency problem. Although there are some papers in the theoretical field, the definition and application of affine arithmetic in reliability assessment are still a cumbersome task. This paper focuses on this interesting task.
Hence, under BN framework, this paper aims to propose a dependency bounds analysis method to further improves and completes reliability assessment frameworks. The proposed method combines p-box and affine algorithm to handle uncertainty and non-deterministic dependency problem simultaneously in reliability analysis. For the sake of illustration, the method is illustrated into two engineering cases. In order to show the advantages of the proposed method, it is compared with the Frechet inequalities [26] and 2-stage Monte Carlo method [27]. The main advantage of such an approach is that it can not only produce tighter output bounds than the ones obtained by the Frechet inequalities method, but also be more time-saving than 2-stage Monte Carlo method. The results indicate that the method proposed in this study is better in line with practical engineering cases. The logical outline of this paper is pictured in Fig. 1.
This paper is organized as follows. Non-deterministic dependency problem of a complex system with hybrid uncertainty is stated in Section 2. Section 3 reviews some basic concepts which will be used in this paper. Section 4 states our proposed method in detail. Section 5 concerns two cases to show the computing mechanism of our method. Section 6 draws up conclusions and future works.
Section snippets
Problem statement
As previously mentioned, uncertainty can be classified into epistemic uncertainty and aleatory uncertainty. Aleatory uncertainty arises because a system can potentially behave in many different ways [28]. Epistemic uncertainty arises from imperfect modeling, simplification, limited availability of database etc., and hence can be reduced by better information [29,30]. Usually, parametric uncertainty is a typical type of epistemic uncertainty [10]. It concerns uncertain parameters of components
P-box and dempster-shafer structure
A p-box is defined as a probability interval of lower and upper () cumulative distributions at any situation . Cumulative distribution Fp of a probability density f is defined as , where P( · ) is probability measure and for each p ∈ R. A p-box collects a set of cumulative distributions bounded by and on quantiles:
Distributional p-box and distribution-free p-box are two different types of p-box [32]. For the first type, its underlying distribution is well-known (e.g.
Basic concepts
Dependency problem is a principal aspect concerned in this paper. One flaw of p-box algorithm is the assumption that inputs are independent so that it is not applicable for failure dependence. Thanks to affine arithmetic, it is a useful tool to handle dependency problem. A random variable x in affine arithmetic is represented as an expression of the form [22]:where is noise symbol whose value is unknown, except that it is restricted to and
Case study 1: a double power turret system
This section focuses on distribution-free p-box with dependency. For the sake of illustration, this section is dedicated to the study of a double power turret system initially given in [44]. Power turret is the main functional part of computerized numerical control machine (CNC) to realize tool reserve and automatic tool changing. The system is divided into five subsystems: signal device, transmission, positioning device, sealing device and power head. For each subsystem, its reliability
Conclusion
In reliability assessment, managing hybrid uncertainty and dependency problem in a complex electromechanical system is an important task. P-box is a very useful tool since it can represent aleatory uncertainty, epistemic uncertainty, interval-valued probabilities and other representations. For dependency problem, affine arithmetic is a helpful approach to keep track of correlation between input variables. Based on the two methods, this paper proposes a dependency bounds analysis method to deal
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China under contract No. 51805073 and U1830207, the Sichuan Science and Technology Project under contract No. 2019JDJQ0015, and the China Postdoctoral Science Foundation under contract No. 2015M582536.
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