An extended recursive decomposition algorithm for dynamic seismic reliability evaluation of lifeline networks with dependent component failures

doi:10.1016/j.ress.2021.107929

Reliability Engineering & System Safety

Volume 215, November 2021, 107929

https://doi.org/10.1016/j.ress.2021.107929 Get rights and content

Highlights

•
An algorithm is proposed to evaluate dynamic reliabilities of lifeline systems.
•
Complex and large-sized systems with dependent failures can be analyzed.
•
Joint probabilities of paths and cuts of systems can be effectively calculated.
•
Most reliable paths of the systems with dependent failures can be singled out.
•
Faster convergence rate is obtained by selecting subgraphs to be decomposed.

Abstract

An extended recursive decomposition algorithm (e-RDA) is proposed to evaluate the dynamic seismic reliabilities of the complex and/or large-sized lifeline network systems with dependent component failures. To effectively analyze statistical dependence of failures of system components, multivariate extreme value response distributions of the components and joint occurrence probabilities of the (disjoint) shortest paths and cuts of systems, a multivariate Gumbel Copula is developed and introduced into the original recursive decomposition algorithm (RDA). Based on the established joint occurrence probability formula of the paths and cuts, two techniques that may accelerate convergence of the upper and lower reliability bounds of complex and/or large-sized systems are developed and embedded into the RDA. Illustrative examples are presented to demonstrate the accuracy, effectiveness and use of the extended RDA for the node weight (only node-type components may fail), edge weight (only line-type components may fail) and general weight (both node- and line-type components may fail) lifeline network systems.

Introduction

The fast and accurate evaluation of the seismic reliabilities of complex and/or large-sized lifeline network systems, such as electrical power networks, gas transmission networks, water supply networks, etc., is of important interest for assessing seismic risk and enhancing seismic resilience of a society. A key issue of the reliability evaluation is to calculate the node-pair connectivity probabilities of the systems under the actions of random earthquake loads. When the considered earthquake loads are modeled by the random processes, the lifeline systems can be considered as the random vibration systems. In this situation, the failures of the system components should be analyzed by using the random vibration theory, e.g., the first passage probability analysis of their nonlinear random seismic responses, and are usually statistically dependent due to the consistent or spatially correlated random earthquake excitations. In the present study, this kind of node-pair connectivity probability evaluation of lifeline network systems is named as the dynamic reliability evaluation of lifeline network systems.

To evaluate the network system reliabilities, one can use the naïve Monte Carlo (MC) simulation method [1]. This method first generates $K$ samples, $x^{k} = (x_{1}^{k}, \dots, x_{m}^{k}), k = 1, \dots, K$ , of the m-dimensional component state vector from the $m K$ independent samples drawn from a uniform random number generator and vector, $p = (p_{1}, \dots, p_{m})$ , of component operating probabilities, and then calculate the number $\hat{K}$ of vectors $x^{k}, k = 1, \dots, K$ for which the structure function of the network $Φ (x^{k}) = 1$ by the deterministic network connectivity analysis. Then an unbiased estimator for the network reliability $R$ is $\hat{R} = \hat{K} / K$ and its variance is bounded above by $R (1 - R) / K$ . Reduction of the variance of the MC simulation results can be obtained by several standard MC sampling techniques [2], [3], [4]. However, the main limitations of the MC methods are that they usually require a large number of simulations to achieve an acceptable level of convergence when the reliabilities of system components or systems are high. In addition, the sampling technique is usually difficult to estimate conditional probabilities. Hence the MC methods were often used in the early reliability analysis of lifeline systems [5,6].

Besides the simulation methods, the non-simulation methods have been also developed in the last decades to evaluate the network system reliabilities, such as the exact algorithms, heuristic algorithms, recursive decomposition algorithm (RDA), recursive compounding method, etc. The exact algorithms usually need to examine all states or generate all paths and/or cuts of the considered networks [7], [8], [9]. Essentially the exact calculation of the network reliabilities is a non-deterministic polynomial (NP) hard problem, hence the exact algorithms are quite inefficient for the complex and/or large-sized lifeline network systems. If the component state variables in the expressions of the system reliability given by the path-and-cut method are substituted by the corresponding component reliabilities, then the Esary-Proschan upper and lower bounds are produced and the true value of the network reliability lies between these bounds [10]. As the numbers of the paths and cuts increase, there will be an intersection of the upper and lower bounds, which offers the approximation to the true system reliability value. Based on this principle, the heuristic algorithms have been developed [11,12]. The shortcoming of the heuristic algorithms is that they cannot determine with what precision the reliability is estimated. In order to overcome the shortcomings of the simulation methods, exact algorithms and heuristic algorithms, the RDA was developed [13], [14], [15], [16]. The RDA is on the basis of the techniques of the generation and disjoint operation of the shortest paths and cuts, evolution and decomposition of the graphs as well as the approximation of the upper and lower reliability bounds. Since it was proposed, the RDA has been continuously extended and widely used in the seismic reliability assessment of complex and/or large-sized lifeline systems [17], [18], [19], [20]. The recursive compounding method successively compounds the system components from the node to the destination node until one single super component representing the connectivity of the source and terminal nodes is obtained [21,22]. This method is suitable for the middle-sized statistically correlated lifeline network systems, which the joint failure probabilities of the system components can be described by the multivariate Normal distributions or Normal Copula.

The above-mentioned RDAs or recursive compounding method were developed to evaluate the static seismic reliabilities of the large- or middle-sized lifeline systems with statistically independent or dependent component failures. However, as far as we know, there are no the efficient methods for evaluating the dynamic seismic reliabilities of complex and/or large-sized lifeline network systems with dependent component failures. The analysis of the seismic reliabilities of structures is a quite complex issue, which involves the definition of failure modes, solution of structural responses, estimate of the probabilities of failure, etc. [23,24]. In addition, it also involves various epistemic and aleatory uncertainties such as uncertainty of the mechanical behavior of building material, uncertainty of earthquake load and uncertainty of calculation method [25,26]. If the failures of lifeline components are defined as the first passage of their responses over the critical threshold levels [27,28] and only uncertainty of earthquake load is considered, then the evaluation of the dynamic seismic reliabilities of lifeline network systems will require estimation of the (joint) first passage probabilities of the (nonlinear) random responses of the lifeline components to the random earthquake excitations. In the last decades, many efficient methods have been developed for estimating the probabilities, such as the stochastic averaging method [29], probability density evolution method [30], moment method [28,31], subset simulation method [32], tail equivalent linearization method [33], asymptotic sampling method [34] and extreme value theory-based simulation methods [35], [36], [37], [38]. Simultaneously, multivariate extreme Copulas have been also developed for estimating the joint failure probabilities of system components [39], [40], [41], which can be used to estimate the joint first passage probabilities of the nonlinear random responses of lifeline components. The challenging issue is how to combine the existing computational methods of network reliabilities and structural dynamic reliabilities to develop an efficient algorithm for evaluating the dynamic seismic reliabilities of complex and/or large-sized lifeline network systems with dependent component failures, especially for the general weight (both node- and line-type components may fail) lifeline systems. The present study aims to solve this issue.

The rest of this paper is organized as follows. Section 2 provides the structure function-based expressions for the original RDA, which can facilitate the derivation of the formula for calculating the joint occurrence probabilities of disjoint shortest paths and cuts. Section 3 derives the expressions of the expectations of the structure functions of disjoint shortest paths and cuts based on the developed Gumbel Copula model, whereby the joint occurrence probabilities of the paths and cuts of systems can be calculated. Section 4 describes the techniques of selecting the most reliable paths and subgraphs to be decomposed, which may accelerate convergence of the upper and lower reliability bounds of systems. In Section 5, the dynamic seismic reliabilities of two illustrative network systems are analyzed to demonstrate the accuracy, effectiveness and use of the extended RDA for the node weight (only node-type components may fail), edge weight (only line-type components may fail) and general weight lifeline network systems. Finally, Section 6 summarizes the present study and provides main conclusions.

Section snippets

Node and edge weight networks

Consider first a node weight lifeline network system, i.e., only the node-type components such as the electrical equipment items of transformer substations may fail and all the line-type components such as the buses or electricity lines of transformer substations are perfectly operative. The network can be described by a directed or undirected graph $G (V, E)$ where $V$ and $E$ respectively denote the sets of the nodes and edges in the graph. Suppose the $m$ nodes have binary states, that is, operative

Expectations of the structure functions α_j(x,y) and β_j(x,y)

As mentioned earlier, in the present study the failure of a lifeline component is defined as the state that its nonlinear random seismic response first exceeds in magnitude the critical threshold level in a reference duration $T$ . The dynamic seismic reliabilities of the node- and line-type lifeline components can be then expressed as [27,37,38]: $r_{i} = E (x_{i})$ $= 1 - P {\exists τ \in [0, T] : X_{i} (τ) > b_{i}}$ $= P {max_{0 \leq t \leq T} X_{i} (τ) \leq b_{i}}$ $= F_{Z_{i}} (b_{i})$ and $r_{j} = E (y_{j})$ $= 1 - P {\exists τ \in [0, T] : Y_{j} (τ) > b_{m + j}}$ $= P {max_{0 \leq t \leq T} Y_{j} (τ) \leq b_{m + j}}$ $= F_{Z_{m + j}} (b_{m + j})$ for $i = 1, \dots, m$ and $j = 1, \dots, n$ ,

Selection of the most reliable paths and subgraphs to be decomposed

In order to accelerate convergence of the lower and upper reliability bounds of lifeline network systems, a selective RDA has been developed [18]. This method uses the most reliable path of the original graph and subgraphs as well as the subgraph which has the greatest contribution to the system reliability and failure probability in the recursive decomposition procedure. However, the selective RDA is developed on the basis of the assumption of independent component failures and only suitable

Simple directed network

This example investigates the accuracy and effectiveness of the extended RDA in evaluating the dynamic seismic reliabilities of node, edge and general weight lifeline network systems with independent and dependent component failures. Fig. 3 shows a directed network system with 6 nodes and 11 edges, in which the source and terminal nodes are node 4 and node 6, respectively. The dynamic reliabilities of the system components $u_{i} = F_{Z_{i}} (b_{i}), i = 1, \dots, 17$ and values of the parameter $θ$ of the Gumbel Copula

Conclusions

An extended RDA is proposed in the present study to effectively evaluate the dynamic seismic reliabilities of complex and/or large-sized node, edge and general weight lifeline network systems with dependent component failures. The extended RDA uses a multivariate Gumbel Copula with the margins are GEV distributions to model the multivariate extreme value responses of system components to random earthquake loads, whereby the joint occurrence probabilities of the (disjoint) shortest paths and

CRediT authorship contribution statement

Jun He: Conceptualization, Methodology, Software, Data curation, Writing – original draft, Investigation, Validation, Writing – review & editing.

Declaration of Competing Interest

The author declare that has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The present work has been supported by a grant of the National Natural Science Foundation of China (No. 51978397), which is gratefully acknowledged.

References (45)

I. Vanzi
Seismic reliability of electric power networks: methodology and application
Struct Saf
(1996)
WH Kang et al.
Evaluation of multivariate normal integrals for general systems by sequential compounding
Struct Saf
(2010)
P Castaldo et al.
Seismic reliability-based design of hardening and softening structures isolated by double concave sliding devices
Soil Dyn Earthq Eng
(2020)
AD Kiureghian et al.
Aleatory or epistemic? Does it matter?
Struct Saf
(2009)
P Castaldo et al.
Safety formats for non-linear finite element analysis of reinforced concrete structures: discussion, comparison and proposals
Eng Struct
(2019)
J He et al.
First passage times of stationary non-Gaussian structural responses
Comput Struct
(2007)
S.K Au et al.
Estimation of small failure probabilities in high dimensions by subset simulation
Probabilist Eng Mech
(2001)
K Fujimura et al.
Tail-equivalent linearization method for nonlinear random vibration
Probabilist Eng Mech
(2007)
C. Bucher
Asymptotic sampling for high-dimensional reliability analysis
Probabilist Eng Mech
(2009)
M Grigoriu et al.
Reliability of dynamic systems in random environment by extreme value theory
Probabilist Eng Mech
(2014)

J. He et al.

Estimate of small first passage probabilities of nonlinear random vibration systems by using tail approximation of extreme distributions

Struct Saf

(2016)

S. Eryilmaz

Multivariate copula based dynamic reliability modeling with application to weighted-k-out-of-n systems of dependent components

Struct Saf

(2014)

Á Rózsás et al.

The effect of copulas on time-variant reliability involving time-continuous stochastic processes

Struct Saf

(2017)

M. Hofert

Sampling Archimedean copulas

Comput Stat Data An

(2008)

HL Frisch et al.

Monte Carlo estimates of percolation probabilities for various lattices

Phys Rev

(1962)

K Kumamoto et al.

Dagger sampling Monte-Carlo for system unavailability evaluation

IEEE Trans Reliab

(1980)

MC Easton et al.

Sequential destruction method for Monte Carlo evaluation of system reliability

IEEE Trans Reliab

(1980)

T Elperin et al.

Network reliability estimation using graph evolution models

IEEE Trans Reliab

(1991)

HM Hwang et al.

Seismic performance assessment of water delivery systems

J Infrastruct Syst

(1998)

F. Moskowitz

The analysis of redundancy networks

Trans Am Inst Electr Eng Part I

(1958)

KK. Aggarwal

Symbolic reliability evaluation using logical signal relations

IEEE Trans Reliab

(1978)

D. Torrieri

Calculation of node-pair reliability in large networks with unreliable nodes

IEEE Trans Reliab

(1994)

Cited by (7)

Directed network-based connectivity probability evaluation for urban bridges
2024, Reliability Engineering and System Safety
Compared with an undirected network, a directed network can describe urban traffic flow and traffic infrastructure, such as ubiquitous one/two-way streets and single/double-deck bridges, more accurately. As the most important network performance indicator, the connectivity probability for a directed network is difficult to calculate owing to the NP-hard problem. This study examined the modelling and connectivity probability analyses of directed networks for urban bridges. To establish a directed bridge network, a directed topological graph was constructed based on actual crossroads and the direction of bridges and streets. Then, the failure probability, which is determined from single/double-deck bridge inspection results, was assigned to the directed edges to form an extended directed bridge network. For connectivity probability analysis, Monte Carlo simulations and an improved Dijkstra algorithm were employed for state combination enumeration and connectivity judgement of the directed bridge network. The all-terminal connectivity probability was approximately evaluated by summing the probability of the state combinations leading to the connected bridge network. The relative importance of directed edges was determined by combining the failure probability of the edge and the impact of its failure on network connectivity. By applying it to an urban city with 299 bridges, the effectiveness and accuracy of the proposed method was demonstrated, thereby providing scientific support for the evaluation, maintenance, and evolution of the urban bridge network.
An approach for reliability optimization of a multi-state centralized network
2023, Reliability Engineering and System Safety
Modern technological advances require unprecedented reliability improvements in engineering systems. This is especially true for the numerous complex structures used as infrastructure in modern industries such as communication and power systems. The structures of systems used in these industries are generally designed as centralized networks, mainly configured in a star structure, in which the end nodes are physically connected to a central one. The principal objective of the current study is to develop a novel approach for reliability analysis and design optimization of such networks, in which all nodes might consist of more than one multi-state heterogeneous component. The performance of each node will then be evaluated as affected by different activation strategies with switch failure also considered. Moreover, it is possible for both central and end nodes in star networks to share their surplus performance, in which case transmission losses must be considered. The universal generating function (UGF) method will be exploited to develop the relevant model. Another aspect of this study involves investigating the optimal component allocation, corresponding activation strategies, and resulting optimal system reliability. Finally, the model will be validated using a numerical example and a real case study.
Connectivity probability evaluation of a large-scale highway bridge network using network decomposition
2023, Reliability Engineering and System Safety
The existing NP-hard problem makes it difficult to evaluate the connectivity probability of large-scale networks, causing great barriers to evaluating and ensuring network safety. In this study, the connectivity probability for a highway bridge network composed of 1772 bridges is evaluated using network decomposition. First, the multilevel k-way graph partition is employed recursively to decompose the network into several approximately equally-sized subnetworks and minimum edge-cuts in series. Then, the decomposed network connectivity probability could be achieved in two steps: subnet and simplified network evaluations. In the subnet evaluation step, the subnet states with and without edge-cuts are judged respectively. Different from the existing connectivity binary definition as connected or disconnected, three states are redefined using adjacent matrices for each subnet: subnet with and without edge-cuts both connected (CCS), both disconnected (DDS), subnet with edge-cuts connected while without edge-cuts disconnected (DCS). For DDS would inevitably lead to the disconnection of the bridge network, such state wouldn't be enumerated in the following step for efficiency, while the CCS and DCS would be further represented by the terminal nodes. In the simplified network evaluation step, the highway bridge network connectivity can be thoroughly represented by treating terminal nodes and edge-cuts as a serial simplified network whose connectivity probability could be calculated by analysing the limited states. The connectivity probability evaluation shows high efficiency and accuracy in the investigated bridge network.
An analytical model for reliability assessment of the rail system considering dependent failures (case study of Iranian railway)
2022, Reliability Engineering and System Safety
Citation Excerpt :
Meango and Ouali developed a model that integrated failure interactions as successive random events and used an interrelated draw concept to model the dependent failure modes as consequences of extreme shocks [32]. He J. proposed an extended recursive decomposition algorithm (e-RDA) to evaluate the dynamic seismic reliabilities of the complex and/or large-sized lifeline network systems with dependent component failures [33]. The degree of dependence between two components or subsystems is indicated by the dependence coefficient according to the type of failure.
Reliability plays an important role in the stable and safe operation of transportation systems. The rail system is one of the vital sectors of transportation that has affected the process of economic and social development. Therefore, accurate predicting of reliability is essential when designing, operating, and maintaining a rail system. This paper proposes an analytical model to estimate the dependent failure rate for the Iranian railway system. This model aims to accurate assessment and predict the reliability of the whole railway system by considering both common cause failure and interactive failure. This model developed by using of the two-variable Taylor expansion approach to estimate the dependent failure rate. The common cause coefficient and interactive coefficient have been estimated by the expert estimation method with the IEC62061 checklist and design structure matrix (DSM) respectively. The proposed model is implemented on the Iranian railway system and the results are analyzed. This paper verifies the model using case studies in the form of different scenarios and compared them with other models. The research results show an improvement in the accuracy of reliability prediction compared to other models.
An adaptive subset simulation algorithm for system reliability analysis with discontinuous limit states
2022, Reliability Engineering and System Safety
Citation Excerpt :
Our method chooses the number of samples per level and the respective conditional probability adaptively to ensure that an adequate number of samples fall in the subsequent intermediate domain. Compared with other non-sampling based methods (e.g., [31–35]), the proposed method facilities using advanced deterministic network analysis algorithms considering complex network dynamics like cascading failure. On the other hand, owing to its sampling nature, the aE-SuS algorithm may require a large number of simulations to achieve an acceptable result.
Many system reliability problems involve performance functions with a discontinuous distribution. Such situations occur in both connectivity- and flow-based network reliability problems, due to binary or multi-state random variables entering the definition of the system performance or due to the discontinuous nature of the system model. When solving this kind of problems, the standard subset simulation algorithm with fixed intermediate conditional probability and fixed number of samples per level can lead to substantial errors, since the discontinuity of the output can result in an ambiguous definition of the sought percentile of the samples and, hence, of the intermediate domains. In this paper, we propose an adaptive subset simulation algorithm to determine the reliability of systems whose performance function is a discontinuous random variable. The proposed algorithm chooses the number of samples and the intermediate conditional probabilities adaptively. We discuss two MCMC algorithms for generation of the samples in the intermediate domains, the adaptive conditional sampling method and a novel independent Metropolis–Hastings algorithm that efficiently samples in discrete input spaces. The accuracy and efficiency of the proposed algorithm are demonstrated by a set of numerical examples.
Seismic reliability evaluation of spatially correlated pipeline networks by quasi-Monte Carlo simulation
2024, Structure and Infrastructure Engineering

View all citing articles on Scopus

View full text

An extended recursive decomposition algorithm for dynamic seismic reliability evaluation of lifeline networks with dependent component failures

Highlights

Abstract

Introduction

Section snippets

Node and edge weight networks

Expectations of the structure functions αj(x,y) and βj(x,y)

Selection of the most reliable paths and subgraphs to be decomposed

Simple directed network

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

Struct Saf

Struct Saf

Soil Dyn Earthq Eng

Struct Saf

Eng Struct

Comput Struct

Probabilist Eng Mech

Probabilist Eng Mech

Probabilist Eng Mech

Probabilist Eng Mech

Struct Saf

Struct Saf

Struct Saf

Comput Stat Data An

Monte Carlo estimates of percolation probabilities for various lattices

Phys Rev

Dagger sampling Monte-Carlo for system unavailability evaluation

IEEE Trans Reliab

Sequential destruction method for Monte Carlo evaluation of system reliability

IEEE Trans Reliab

Network reliability estimation using graph evolution models

IEEE Trans Reliab

Seismic performance assessment of water delivery systems

J Infrastruct Syst

The analysis of redundancy networks

Trans Am Inst Electr Eng Part I

Symbolic reliability evaluation using logical signal relations

IEEE Trans Reliab

Calculation of node-pair reliability in large networks with unreliable nodes

IEEE Trans Reliab

Expectations of the structure functions α_j(x,y) and β_j(x,y)