Matrix-based System Reliability Method Research Articles

Probabilistic modeling of disrupted infrastructures due to fallen trees subjected to extreme winds in urban community

Tree failures due to strong winds in urban areas cause extensive direct and indirect economic and environmental loss, including disrupting adjacent infrastructures, such as buildings, underground pipelines, roads and overhead powerlines. To effectively improve the resilience of a community subjected to extreme wind events through prevention, response and recovery, it becomes critical to rationally assess the risks of wind-induced tree failures and the disruptions to different types of infrastructures due to fallen trees. An integrated probabilistic methodology to model the performance of disrupted infrastructures is developed for fallen urban trees subjected to extreme winds in a typical community. Firstly, the finite element modeling of the trees subjected to wind loads is conducted and based on which the windthrow fragility curves of several typical urban tree species are developed. Secondly, a probabilistic framework is developed based on the fragility results to characterize the disrupted scenarios and further predict the disruption probability of some critical infrastructures due to fallen trees. The matrix-based system reliability (MSR) method is introduced to assess the transportation network performance. The proposed framework and MSR method are demonstrated in detail on studying the overhead powerline and transportation network of a small urban community in the city of Fort Collins, Colorado. In the demonstrative example, the probabilities of powerline disruption, road closure, and origin–destination disconnection and travel time reliability under different wind conditions are predicted. Finally, mitigation efforts such as crown thinning of trees are discussed to reduce possible risks of disrupting the infrastructures.

Multi-state system reliability analysis of HVDC transmission systems using matrix-based system reliability method

This paper presents a new reliability analysis method for High Voltage Direct Current (HVDC) transmission systems based on the Matrix-Based System Reliability method. The proposed method can compute the failure probability of HVDC transmission systems by use of efficient matrix-based procedures. Unlike conventional system reliability methods whose practicability strongly depends on the system size and the complexity of its multiple states, the proposed method can describe any general system event in a simple matrix form and therefore provides a straightforward way of handling the system events and estimating their probabilities. The main qualities of the proposed method are demonstrated by the reliability assessment of two multi-terminal HVDC (MTDC) transmission systems with multiple derated states. The performance of the proposed method and the results obtained were compared with two traditional assessment methods: Capacity Outage Probability Table (COPT) and Monte Carlo simulation. In this comparative analysis it is shown that the proposed method is a competitive alternative for reliability analysis of MTDC systems in terms of simplicity and efficiency.

Reliability growth analysis of k-out-of-N systems using matrix-based system reliability method

The advent of ever more complex systems in extensive areas of industries hampers efficient and accurate analysis of reliability and effective reliability-based decision making. Such difficulties may arise from the intricate formulation of system failure events, statistical dependence between component failure events, and the convoluted quantification of underlying probabilities of basic events. So-called k-out-of-N systems, which survive or succeed when at least k components are available among the total of N components, give rise to a high level of complexity. This type of systems are commonly introduced to secure a proper level of redundancy in operating engineering systems, but the intricate definition of the system events may elude the system reliability analysis. It is noted that such k-out-of-N systems are often tested and corrected over a certain period of time before their official usage or release in order to assure the target reliability of the system. For the purpose of reliability prognosis based on the data collected from the test period, reliability growth models (RGMs) have been widely used in software and hardware engineering. However, RGMs have been applied mostly to individual components, not at the system level. Furthermore, in complex systems such as k-out-of-N system, it is challenging to relate the reliability growth of components with that of the system. To address this need, in this paper, the matrix-based system reliability (MSR) method is extended to k-out-of-N systems by modifying the formulations of event and probability vectors. The proposed methods can incorporate statistical dependence between component failures for both homogeneous and non-homogeneous k-out-of-N systems, and can compute measures related to parameter sensitivity and relative importance of components. The reliability growths of components represented by RGMs are incorporated into the proposed system reliability method, so that the trend of system reliability growth can be effortlessly evaluated and predicted. Two numerical examples are introduced in this paper to demonstrate the proposed method and its applications: (1) hypothetical systems each consisting of series, parallel and k-out-of-N subsystems, and (2) a simplified high speed train system modeled by multiple k-out-of-N subsystems. Two types of RGMs, i.e. non-homogeneous Poisson process (NHPP) and Duane models are employed in these examples.

System Reliability Based Multi-Objective Design Optimization of Bridges

Continuous progress in structural reliability theory and methods, in addition to the evolving advances in optimization solution techniques, drives the steady development of reliability-based structural design process. The integration of system reliability in the design optimization of structures, especially bridges, improves the allocation of available, and usually scarce, budgets. However, the complexity in the determination of system reliability of large structural systems renders the consideration of system reliability in design optimization computationally prohibitive. This issue is even exacerbated when discrete design variables are considered. Furthermore, the optimization of structural systems typically requires using linked-discrete design variables. The recently developed matrix-based system reliability method (MSR) enables the computation of system reliability efficiently and accurately through convenient matrix calculations. The non-dominated sorting genetic algorithm with controlled elitism (NSGA-II) has proven its effectiveness and efficiency in numerous multi-objective optimization applications. The objective of this paper is to propose an approach for the system reliability-based multi-objective design optimization of bridges using the MSR method and NSGA-II and considering linked-discrete design variables. The approach is illustrated by redesigning an actual bridge in order to compare the results of the proposed approach with those of conventional bridge design.

Development of Bayesian Networks for the multi-hazard fragility assessment of bridge systems

This article proposes an approach for the derivation of multi-hazard fragility functions, through the use of system reliability methods and Bayesian Networks. A bridge system is broken down into its constitutive components to isolate specific failure mechanisms and damage states at the component level. At the system level, the probability of occurrence of failure modes (i.e. various configurations of component damage states) is estimated thanks to a Bayesian analysis. These system fragility functions can then be directly related to harmonized functionality levels in order to get accurate predictions of downtime or traffic reduction. The applicability of the Bayesian Network formulation is compared to the matrix-based system reliability method, in terms of accuracy and computation time, while modeling strategies are proposed in the case of large systems with complex failure modes or multi-state components. Finally, the proposed approach is applied to a bridge system that is exposed to multiple hazard events (earthquakes, ground failures and floods): using the Bayesian framework, four functionality loss levels can be predicted with fragility surfaces that are expressed as a function of peak ground acceleration and flow discharge, taking into account multi-hazard interactions at the vulnerability level (i.e. cumulated damage events).

Application of matrix-based system reliability method in complex slopes

Complex slopes are characterized by large numbers of failure modes, cut sets or link sets, or by statistical dependence between the failure modes. For such slopes, a systematic quantitative method, or matrix-based system reliability method, was described and improved for their reliability analysis. A construction formula of event vector cE was suggested to solve the difficulty of identifying any component E in sample space, and event vector c of system events can be calculated based on it, then the bounds of system failure probability can be obtained with the given probability information. The improved method was illustrated for four copper mine slopes with multiple failure modes, and the bounds of system failure probabilities were calculated by self-compiling program on the platform of the software MATLAB. Comparison in results from matrix-based system reliability method and two generic system methods suggests that identical accuracy could be obtained by all methods if there are only a few failure modes in slope system. Otherwise, the bounds by the Ditlevsen method or Cornell method are expanded obviously with the increase of failure modes and their precision can hardly satisfy the requirement of practical engineering while the results from the proposed method are still accurate enough.

Post-hazard flow capacity of bridge transportation network considering structural deterioration of bridges

The flow capacity of a transportation network can be reduced significantly if its constituent bridges are damaged by natural or man-made hazards. For rapid risk-informed decision making on hazard mitigation and response, it is therefore essential to have a capability to predict the post-hazard flow capacity of the network efficiently and accurately. However, this is a challenging task due to the uncertainty in hazards and structural damage, and the complex nature of the network flow analysis. Moreover, the bridge structures may experience significant deterioration over their life cycle, which requires time-varying network reliability analysis. This paper proposes a new non-sampling-based approach to estimate the time-varying post-hazard flow capacity of a bridge transportation network considering structural deterioration of bridges. The proposed approach evaluates the probabilities of structural damage scenarios efficiently using the matrix-based system reliability method and rapidly computes the corresponding flow capacities using a maximum flow capacity analysis algorithm. The matrix-based framework facilitates the integration of these results to obtain the probabilistic distributions and statistical moments of the network flow capacity. It also enables computing various measures useful for risk-informed decision making, such as the conditional mean and standard deviation of flow capacity given observed structural damage, and component importance measures. In the proposed approach, probability calculation and network flow analysis are performed separately, which renders time-varying post-hazard flow capacity analysis efficient. The proposed approach is demonstrated by a numerical example based on the Sioux Falls network under multiple bridge-deterioration scenarios simulating the progress of deterioration.

Further development of matrix-based system reliability method and applications to structural systems

In efforts to estimate the risk and reliability of a complex structure or infrastructure network, it is often required to evaluate the probability of a ‘system’ event, i.e. a logical function of multiple component events. Its sensitivities with respect to design parameters are also useful in decision-making processes for more reliable systems and in reliability-based design optimisation. The recently developed, matrix-based system reliability (MSR) method can compute the probabilities of general system events including series, parallel, cut-set and link-set systems, and their parameter sensitivities, by use of efficient matrix-based procedures. When the component events are statistically dependent, the method transforms the problem into an integral in the space of random variables which cause the statistical dependence, termed as the common source random variables (CSRVs). One can identify CSRVs by fitting a generalised Dunnett-Sobel (DS) model to a given correlation coefficient matrix. This article introduces two further developments of the MSR method: First, for efficient evaluation, it is proposed that the integral in the CSRV space can be performed using the first- or second-order reliability methods. Second, a new matrix-based procedure is developed to compute the sensitivity of the system failure probability with respect to the parameters that affect the correlation coefficients between the components. In addition, an extensive parametric study is performed to investigate the effect of the error in fitted generalised DS model on the accuracy of the estimates by the MSR method. The further developed MSR method is demonstrated by two examples: system reliability analysis of a three-storey Daniels system structure, and finite element reliability analysis of a bridge pylon system.

Single-Loop System Reliability-Based Design Optimization Using Matrix-Based System Reliability Method: Theory and Applications

This paper proposes a single-loop system reliability-based design optimization (SRBDO) approach using the recently developed matrix-based system reliability (MSR) method. A single-loop method was employed to eliminate the inner-loop of SRBDO that evaluates probabilistic constraints. The MSR method enables us to compute the system failure probability and its parameter sensitivities efficiently and accurately through convenient matrix calculations. The SRBDO/MSR approach proposed in this paper is applicable to general systems including series, parallel, cut-set, and link-set system events. After a brief overview on SRBDO algorithms and the MSR method, the SRBDO/MSR approach is introduced and demonstrated by three numerical examples. The first example deals with the optimal design of a combustion engine, in which the failure is described as a series system event. In the second example, the cross-sectional areas of the members of a statically indeterminate truss structure are determined for minimum total weight with a constraint on the probability of collapse. In the third example, the redistribution of the loads caused by member failures is considered for the truss system in the second example. The results based on different optimization approaches are compared for further investigation. Monte Carlo simulation is performed in each example to confirm the accuracy of the system failure probability computed by the MSR method.

Multi‐scale system reliability analysis of lifeline networks under earthquake hazards

AbstractRecent earthquake events evidenced that damage of structural components in a lifeline network may cause prolonged disruption of lifeline services, which eventually results in significant socio‐economic losses in the affected area. Despite recent advances in network reliability analysis, the complexity of the problem and various uncertainties still make it a challenging task to evaluate the post‐hazard performance and connectivity of lifeline networks efficiently and accurately. In order to overcome such challenges and take advantage of merits of multi‐scale analysis, this paper develops a multi‐scale system reliability analysis method by integrating a network decomposition approach with the matrix‐based system reliability (MSR) method. In addition to facilitating system reliability analysis of large‐size networks, the multi‐scale approach enables optimizing the level of computational effort on subsystems; identifying the relative importance of components and subsystems at multiple scales; and providing a collaborative risk management framework. The MSR method is uniformly applied for system reliability analyses at both the lower‐scale (for link failure) and the higher‐scale (for system connectivity) to obtain the probability of general system events, various conditional probabilities, component importance measures, statistical correlation between subsystem failures and parameter sensitivities. The proposed multi‐scale analysis method is demonstrated by its application to a gas distribution network in Shelby County of Tennessee. A parametric study is performed to determine the number of segments during the lower‐scale MSR analysis of each pipeline based on the strength of the spatial correlation of seismic intensity. It is shown that the spatial correlation should be considered at both scales for accurate reliability evaluation. The proposed multi‐scale analysis approach provides an effective framework of risk assessment and decision support for lifeline networks under earthquake hazards. Copyright © 2009 John Wiley & Sons, Ltd.

System reliability and sensitivity under statistical dependence by matrix-based system reliability method

A matrix-based system reliability (MSR) method has been recently proposed to compute the probabilities of general system events efficiently by simple matrix operations. The proposed matrix-based framework describes both a system event and the likelihood of its component events by vectors that are obtained by efficient matrix-based procedures. The probability of the system event is computed by the inner product of the two vectors. Therefore, the method is uniformly applicable to any type of system events including series, parallel, cut-set and link-set systems. In the case when one has incomplete information on component probabilities and/or on the statistical dependence between components, the matrix-based framework enables us to obtain the narrowest bounds on the system probability by linear programming. Various importance measures and conditional probabilities are also efficiently estimated by the proposed method. This paper presents the MSR method and further develops it in terms of statistical dependence and parameter sensitivity of system reliability. First, a method is developed to use the MSR method for systems with statistically dependent components. The correlation coefficients between the basic random variables or the component safety margins are represented by a Dunnett–Sobel class correlation matrix to identify the source of the statistical dependence and to make use of the matrix-based procedure developed for independent components. Second, a new matrix-based procedure is proposed to calculate the sensitivities of system reliability with respect to parameters. This paper demonstrates the MSR method and these further developments by two numerical examples of structural systems. First, the system fragility of a bridge structure is computed based on the analytical fragility models of the bridge components and the correlation coefficients between the seismic demands at different components. In the second example, the MSR method is used to estimate the probability of the collapse of a statically indeterminate structure subjected to an abnormal load. The sensitivities of the probability with respect to the means and standard deviations of uncertain member capacities are estimated for an optimal upgrade of the structural system.

Matrix-based system reliability method and applications to bridge networks

Using a matrix-based system reliability (MSR) method, one can estimate the probabilities of complex system events by simple matrix calculations. Unlike existing system reliability methods whose complexity depends highly on that of the system event, the MSR method describes any general system event in a simple matrix form and therefore provides a more convenient way of handling the system event and estimating its probability. Even in the case where one has incomplete information on the component probabilities and/or the statistical dependence thereof, the matrix-based framework enables us to estimate the narrowest bounds on the system failure probability by linear programming. This paper presents the MSR method and applies it to a transportation network consisting of bridge structures. The seismic failure probabilities of bridges are estimated by use of the predictive fragility curves developed by a Bayesian methodology based on experimental data and existing deterministic models of the seismic capacity and demand. Using the MSR method, the probability of disconnection between each city/county and a critical facility is estimated. The probability mass function of the number of failed bridges is computed as well. In order to quantify the relative importance of bridges, the MSR method is used to compute the conditional probabilities of bridge failures given that there is at least one city disconnected from the critical facility. The bounds on the probability of disconnection are also obtained for cases with incomplete information.