Minimal Path Sets Seismic Reliability Evaluation of Lifeline Networks with Link and Node Failures

M. B. Javanbarg,

1 C. Scawthorn

2, J. Kiyono

2 and Y. Ono

3

1 Postdoctoral Research Fellow, Graduate School of Engineering, Kyoto University,

Japan, P.O. Box 6158540; PH (+81)-75-383-7499; FAX (+81)-75-383-3253; email: javanbarg@quake.kuciv.kyoto-u.ac.jp 2 Professor, Department of Urban Management, Kyoto University, Japan , email:

cscawthorn@worldnet.att.net and kiyono@quake.kuciv.kyoto-u.ac.jp 3 Assistant Professor, Department of Urban Management, Kyoto University, Japan, email: ysk@quake2.kuciv.kyoto-u.ac.jp ABSTRACT

A heuristic minimal path sets method for analyzing seismic reliability of lifeline networks is proposed. For large-scale lifeline networks reliability computation can become prohibitive or inaccurate using the current enumeration methods. Although a few studies have considered both node and link failures, none of these methods has utilized the minimal paths method. This paper proposes a minimal path algorithm to evaluate reliability of relatively large lifeline networks with unreliable nodes and links. As the primary task, the method can deduce all minimal paths of a directed/undirected network. In the algorithm, minimal paths for both nodes and links are traced performing several procedures. The terminal-pair reliability of the network is then evaluated by performing an efficient Monte Carlo simulation as an alternative to exact reliability calculation methods. In terms of computational speed, the results obtained compare well with existing algorithms. The algorithm requires minimum memory storage and minimum user-defined data to represent the topology of a network. A series of case studies illustrate that the method can be used to evaluate reliability of the large-scale lifeline systems subject to node and link failures, efficiently. INTRODUCTION

Lifeline systems such as water, gas, transportation, communication and power systems are vital to daily life of citizens. The damages and losses resulted from malfunction of lifeline systems during several past earthquakes conclude that these systems are vulnerable. The performance of lifelines under seismic excitation can be measured performing a reliability analysis which has been an important research topic in earthquake engineering for past three decades.

In literature, two kinds of computations exist for network reliability: exact methods, and approximate simulations. The exact methods provide an exact reliability; in contrast, the simulation methods provide an approximate result. Moreover, two classes of exact methods have been utilized to compute the network reliability. The first class deals with the enumeration of all minimal paths or cuts.

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A path is minimal if it has no proper subpaths. Conversely, a cut is set of network components such that if these components fail, the system fails. If the probability of failure of nodes and links is known, the reliability expression can be calculated using different techniques, like inclusion-exclusion or sum of disjoint products (SDP) methods because this enumeration provides non-disjoint events. Numerous works related to this kind of analysis have been done in literature, (Ahmad 1988, Locks 1987). However, in minpaths and mincuts methods, as the number of links become large, the number of minimal paths and cuts will be large and SDPs or inclusion-exclusion expression for minpaths and mincuts will be increased too large to be stored and be processed.

In the second class, the algorithms are based on reducing process in network topology. The first process consists of reducing the network by replacing some special substructures by smaller ones. The replaced substructures can be elementary, such as two adjacent links (series-parallel reductions), or more complex as a subgraph, polygon-to-chain (Choi and Jun 1995), and delta-to-star reductions (Gadani 1981). The reductions allow us to compute the reliability of series-parallel networks in linear time; the reductions are recursively applied until resulting in a single edge. Nevertheless, for general networks, such substructure can not be found. In this case, the factoring process is applied. The idea is to choose a component, and decompose the problem into two sub-problems: the first assumes the component has failed, and the second assumes it is functioning (Jie and He, 2002). In factoring algorithm the number of factored reliability networks will increase exponentially with the number of links increases thereby computational time becomes prohibitive for large networks.

This paper proposes a minimal path enumeration algorithm mainly to deduce all minimal paths in directed/undirected large-scale networks. More, a Monte Carlo simulation (MCS) for evaluating reliability based on the obtained minimal paths is applied. The proposed algorithm can consider both link and node failures. The computational efficiency of the algorithm is illustrated by application to several networks with sufficient complexity. A case study including a water distribution network is also performed to show the applicability of the algorithm to a real-life system such as lifeline networks. LITERATURE REVIEW

Several methods are available in the literature to enumerate all simple paths corresponding to a terminal-pair.

The available methods for paths enumeration may be divided into three distinct categories: (1) exhaustive search techniques (Jensen and Bellmore, 1969); (2) matrix techniques (Misra, 1970 and Kim et al., 1972), and (3) heuristic techniques (Young et al, 1972, Allen et al, 1976, Jasmon and Kai, 1985, and Fotuhi-Firuzabad et al, 2004). In the search techniques, node by node exhaustive search method has been used to enumerate all the paths/cutsets of a graph. The matrix methods use the power of the connection/adjacency matrix to obtain simple paths of a reliability graph. These methods have the disadvantage of increased complexity with increasing size of the reliability graph. These methods using connection or adjacency matrices calculate either upper or lower diagonal entries or the whole matrix for all its higher powers to enumerate the paths of different cardinality. In most of existing heuristic algorithms, minimal paths are deduced from list of each element and its predecessors. The list of elements together with their predecessors, representing the connectivity of

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the network, are fed into the algorithms as input data and processed for minimal path sets.

Most of the available methods assume that the nodes are perfectly reliable and thus, have to be complemented or transformed with the methods that account for node failure (Rueger, 1986; Torrieri, 1994; Netes and Filin, 1996; Ke and Wang, 1997) MINIMAL PATH SETS ALGORITHM

The minimal paths enumeration algorithm we have proposed in this study is an extension of the existing heuristic algorithm originally developed by Al-Ghanim (1999) for generating minimal paths of directed networks considering link failure only. There are two developments in our algorithm compared to A-Ghanim’s method: first, our algorithm is applicable to both directed and undirected networks, and next the proposed algorithm can consider both link and node failures. Figure 1 presents the flow chart of the proposed algorithm. Considering Figure 2a and b, which respectively depicts a commonly used directed and undirected bridge network, the algorithm is explained in the sequel. Step 1. The only input data is a 2×m ( =m number of links) link array ),,( ijpjiE which

its elements e are links from each node i to node j with previously computed seismic

probability of failure ijp . However, if the nodes of network are imperfect, ),,( kpyxV

which shows the set of nodes ( nk ,...,2,1= ) with coordinates x and y , and node

probability of failure kp can be also considered as an additional data inventory. The nodes and links then can be numbered in any arbitrary manner. Step 2. In general, the connection/adjacency matrix A is developed to introduce the topology of network to any computer program for network analysis. For instance, the connection matrix for both directed ( dA ) and undirected ( uA ) networks presented in Figure 2 are defined as:

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Step 3. The connection matrix can then be converted to a two dimensional array

),( jiP called predecessor matrix which its first column holds all sequentially ordered network nodes or links with regard to directed and undirected network, respectively. For a given node (link) stored in row i the rest of columns in that row represents node (link) number precede node (link) i. For example, the directed predecessor matrix dP and

undirected predecessor matrix uP associated with the networks presented in Figure 1 are as follows.

TCLEE 2009: Lifeline Earthquake Engineering in a Multihazard Environment ©2009 ASCE 1110

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Step 4. For directed network, identifying the source node s and sink node t , the current path can be initialized as }},{,{ sntpath n= . Search and swap procedures can be applied

to predecessor matrix of nodes dP for deducing all the paths. Once a node become current, the search procedure backtracks through the predecessor matrix to the source node, and a path is traced. Then the swap procedure changes the predecessor for each sink node and traces another path. For instance, the first path is searched through matrix

dP is },,{ 1241 nnnpath = . Applying swap procedure the next path is traced as

},,{ 1342 nnnpath = . All the traced paths are then stored in a node minimal path matrix

dN .

For undirected network, identifying the source link successors te and sink link

predecessors te , current path can be initialized as }},{,{ smt eeepath = . Considering a modification, the same abovementioned search and swap procedures can be applied to predecessor matrix of links uP to trace all paths from sink to source. The modification is

that, during backtracking the matrix uP in each path, once a link becomes current an inclusion procedure checks whether either of its predecessors are included in predecessor set of previous link in the path. If so, that path is not stored and program continues to search and swap procedures. All the traced paths are stored in a link minimal path matrix uL . For instance, the corresponding minimal paths matrices to networks presented in Figure 2 are:

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Step 5. Utilizing the connection matrix dA , it is easy to obtain link minimal paths

matrix of directed network, dL . Similarly, the link minimal paths matrix uL can be

converted to node minimal path matrix, uN . Hence, applying few heuristic procedures, it is possible to deduce minimal path of a given network considering unreliable nodes and links. Eqs. 4 and 5 illustrate the minimal path matrices, dMPM and uMPM , respectively for both directed and undirected networks presented in Figure 2.

TCLEE 2009: Lifeline Earthquake Engineering in a Multihazard Environment ©2009 ASCE 1111

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RELIABILITY EVALUATION USING MONTE CARLO SIMULATION

In literature, the reliability computation methods for systems with known paths can be classified into four groups: (1) inclusion-exclusion algorithms based on the Poincare's form of the system structural function (Fong and Buzacott, 1987), (2) sum of disjoint products algorithms creating lists of orthogonal Boolean products whose Boolean sum composes the system structural function (Abraham, 1979), (3) binary decision diagram (BDD) algorithms based on gradual decomposition of the system structural function using Shannon formula and reduction techniques (Brayant, 1986, and Rauzy, 1996 ), and (4) non-standard algorithms (e.g. Malinowski, 2004). Both inclusion-exclusion and SDP are the preferred approach used for obtaining terminal-pair reliability when the size of network is small. However, as the size of the network increases, utilizing these methods may become impractical. Previously, Kuo et al. (1999) has successfully applied ordered binary decision diagram (OBDD) algorithm to reliability evaluation of a 2×100 lattice network. More recently, Javanbarg et al. (2009) also illustrated the application of OBDD to reliability evaluation of infrastructure networks. However, since the complexity of the later algorithm is a function of the source-terminal path length and the degree of outgoing links at all nodes, applicability of this new method to reliability evaluation of large-scale lifeline networks should be confirmed performing more case studies.

Because of high cost of reliability computation with known minimal paths by uses of the traditional methods, the simulation techniques have been used for reliability evaluation of large networks (Fishman, 1986). In this study, an existing MCS technique by Ramirez-Marquez and Coit (2005) is applied to evaluate the terminal-pair reliability with known minimal paths. At each run of MCS, a state vector is produced, in which two states can be considered for each node/link in the path: 0-state refers to failed state with probability of pq −= 1 and 1-state induces success state with success probability p . Once a state vector is obtained, it is compared to the set of minimal paths obtained

by the proposed minimal paths algorithm, to check if it can dominate at least one of the minimal paths corresponding to rows of dMPM or uMPM matrices. Reliability can then be evaluated by dividing the number of system successes to number of simulations.

TCLEE 2009: Lifeline Earthquake Engineering in a Multihazard Environment ©2009 ASCE 1112

Start

Read input data, V(x,y,pk) and E(i,j,pij)

Check the type of network

Undirected, cij=cji=e if link e connected from node i to j and cij=cji=0, otherwise

Directed, cij=e and cji=0, if link e connected from node i to j, and cij=cji=0, otherwise

Convert connection matrix to node predecessor matrix

Convert connection matrix to link predecessor matrix

Construct node minimal path matrix backtracking node predecessor matrix

Construct link minimal path matrix backtracking link predecessor matrix

Minimal path matrix(nodes and links)

Convert node minimal path matrix to link minimal path matrix

Convert link minimal path matrix to node minimal path matrix

Prepare connection matrix of order n×n, cij

End

Read source node s, sink node t, and initialize the

path={t, {nn}, s}

Read source link successor(s) es, sink link predecessor(s) et, and initialize the path={et, {em}, es}

Figure 1. Flow chart of the proposed minimal paths sets algorithm.

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Figure 2. Illustrative example: a) directed network b) undirected network.

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EXPERIMENTAL RESULTS

To examine the efficiency of the proposed algorithm, the directed networks, i.e. network numbers 1,2,3,4,5,6 and 7 depicted in Figure 3, were taken from Kuo et al. (1999) and Soh and Rai (2005) which recently have been also utilized as the case networks by Gebre and Ramirez-Marquez (2007). The algorithm developed was programmed in MATLAB 7.5. To compare the results with previous works a link failure probability of 0.5 was considered for each link entire the networks. Node failure was not accounted for; however, as discussed this can be easily implemented by the proposed algorithm. Table 1 lists the computational effort providing the number of minimal paths, run time for minimal paths enumeration, total run time for reliability evaluation and the reliability of each case network. The 2×100 lattice network (network number 6 in Figure 3) was considered as the computational limit by Kuo et al. (1999). The proposed method could analyze the ts − reliability of this lattice network within 1 minute. Additionally, the reliability analysis of a 3×100 lattice network (network number 9) was also performed. As seen, the reliability of such a relatively large network could be evaluated within a total run time less than 3 minutes. Moreover, the algorithm could analyze the 6×6 lattice depicted as network number 7, in which the minimal cutset algorithm presented by Gebre and Ramirez-Marquez (2007) was not able to analyze it. Furthermore, reliability of a 10×10 lattice network was also computed to illustrate the efficiency of the presented method.

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Figure 3. Experimental networks tested on algorithm.

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Table 1. Computational results of experimental networks. # Network

# Node

# Link

# Minimal paths

Run time for minimal paths (seconds)

Total run time for reliability (seconds)

Reliability

1 18 29 44 0.028 6.369 0.2966 2 11 18 36 0.027 4.268 0.2065 3 16 24 20 0.026 5.165 0.1778 4 14 21 15 0.026 4.516 0.2781 5 25 40 70 0.033 8.883 0.1184 6 200 298 100 1.723 59.330 0.0 7 36 60 252 0.065 15.649 0.0903 8 100 180 48,620 238.682 676.481 0.020 9 300 497 5050 76.297 169.978 0.0

CASE STUDY

To illustrate the applicability of the proposed method to a practical application as a case network of undirected network, it was applied to seismic reliability analysis of the Kobe city water supply system in Japan. Javanbarg and Takada (2007), and Javanbarg (2008), developed a probabilistic seismic reliability analysis for water supply systems by performing flow analysis with the aid of Monte Carlo simulation. In their study, a part of Kobe city water distribution network was examined as the case study and the results of the model were compared to the flow simulation results of the actual observed damage data during the 1995 Kobe Earthquake. In Figure 4, buried pipelines are overlaid with observed seismic intensity of the Kobe Earthquake. Pipeline damage locations with different failure modes are also depicted in Figure 4. Considering the actual damage locations and types of failure mode, they assigned correspondent amount of leakage to each link. Hydraulic analysis of seismically damaged network was deterministically performed and results of nodal reliability were found in terms of pressure availability at each demand node which we will present later in this paper.

Table 2 presents the pipe material type as well as probability of failure corresponding to each link. Considering failure probability of each link, Javanbarg and Takada (2007) also performed a MCS to probabilistically estimate the seismic damage entire the network. Hydraulic analysis of the seismically damaged network was also probabilistically performed by MCS. Nodal pressure availability was then computed as the reliability at each node.

Accordingly, the schematic network model for minimal path sets reliability analysis is presented in Figure 5. The results of nodal reliability using proposed model are compared with the results of both deterministic and probabilistic simulations models (Javanbarg & Takada 2007) as presented in Figure 6. The results of reliability analysis using OBDD (Javanbarg et al., 2009) are also presented in Figure 6. It illustrates that the minimal path sets algorithm results are very close to results of OBDD. However, as discussed before the efficiency of the OBDD reliability analysis for large-scale networks is still an open question.

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Figure 4. Pipelines damage in the Kobe Earthquake (Javanbarg and Takada 2007). Table 2. Failure probability of links in the case study network (Javanbarg and Takada 2007).

Link No. Pipe material Probability of failure ( iq )

1 CIP 0.717 2 DIP 0.219 3 DIP (seismic joint) 0.0 4 DIP 0.537 5 CIP 0.549 6 DIP 0.046 7 DIP 0.023 8 DIP 0.062 9 CIP 0.422 10 CIP 0.329 11 CIP 0.620 12 DIP 0.080 13 CIP 0.238 14 CIP 0.222 15 DIP 0.064 16 DIP 0.158 17 DIP 0.162 18 CIP 0.354 19 CIP 0.242 20 DIP (seismic joint) 0.0

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Figure 5. Schematic network model of the case study.

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Flow analysis based on actual damage data (Javanbarg & Takada, 2007)

Probabilistic flow analysis by Monte Carlo simulation (Javanbarg, 2008)

BDD Reliability (Javanbarg et al., 2009)

Minimal Pathsets (present work)

Figure 6. Comparison between the results of case study and previous works.

CONCLUSION

This paper presents a heuristic technique to enumerate minimal path sets of a relatively large-scale lifeline network as the primary task. As the second task, it utilizes an efficient and simple Monte Carlo simulation to evaluate the terminal-pair reliability based on the obtained minimal paths matrix in the first stage. The proposed algorithm can be applied to seismic reliability analysis of lifeline networks considering both node and link failures. The algorithm is easy to program. In terms of computational speed, the results obtained compare well with existing algorithms. The technique requires minimum memory storage and minimum user-defined data to represent the topology of a network and follows a modular design strategy.

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The presented algorithm is appropriate for analyzing the seismic reliability of lifeline networks in terms of connectivity measure. However, considering different failure modes and states for components of network the serviceability measure of lifeline systems under seismic environment may be much more of interest. Hence, the extension of the model to allow multistate reliability analysis of the capacitated networks is an emerging task. REFERENCES Abraham, J.A. (1979) “An improved method for network reliability.” IEEE Trans.

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