Modeling Earthquake Vulnerability of Highway ... - Science Direct

Available online at www.sciencedirect.com

Electronic Notes in Discrete Mathematics 41 (2013) 319–326 www.elsevier.com/locate/endm

Modeling Earthquake Vulnerability of Highway Networks ˙ Ar¸sık 1 Idil Department of Industrial Engineering Ko¸c University Istanbul, Turkey

F. Sibel Salman 2 Department of Industrial Engineering Ko¸c University Istanbul, Turkey

Abstract In this study, we investigate the earthquake vulnerability of highway networks whose links are subject to failure. We propose a model called α-conservative failure model that aims to capture the dependency among link failures in the event of an earthquake. According to this model, we calculate a path-based accessibility measure to assess the expected weighted average shortest distance to serve a unit demand after the earthquake. We test the proposed link failure model on a case study of the earthquake vulnerability in Istanbul. Keywords: Network Vulnerability, Reliability, Dependent Link Failures, Disaster Risk of Highways

1 2

Email: [email protected] Email: [email protected]

1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.108

320

1

˙ Ar¸sık, F. Sibel Salman / Electronic Notes in Discrete Mathematics 41 (2013) 319–326 I.

Introduction

The assessment of the earthquake vulnerability of highway networks is essential for both structural strengthening decisions by highway administrators in the pre-disaster stage and efficiently carrying out post-disaster emergency activities by means of proper preparedness plans. In the event of an earthquake, network components such as bridges and viaducts are subject to failure. In this study, we assume that the nodes are reliable and postulate that both the radiation of seismic waves and the structural properties of network components create a dependency among network link failures. To represent this dependency structure, we suggest a link failure model which we call α-conservative dependency model. According to this dependency model, given the marginal survival probability of each link, we identify a joint probability distribution of the surviving links which characterize the surviving network. We calculate a path-based accessibility measure to evaluate the earthquake service level of the network. We test the model and calculate the measure for the Istanbul highway network which is subject to earthquake hazard.

2

Literature Review

To maintain societal welfare in general and to manage post-disaster emergency response activities effectively, the vulnerability of infrastructure networks has been evaluated in various studies. Sohn [7] evaluates the importance of highway network links under flood damage by introducing an accessibility index to embody the decreasing effect of distance and the volume of traffic influence on a highway network. Chang and Nojima [1] assess network performance in terms of network coverage and transportation accessibility in the post-disaster stage. Nagurney and Qiang [5] define a unified network performance measure which integrates the flow information and is applicable to different types of networks with either fixed or elastic demands on them. In order to evaluate the flow network reliability, Gertsbakh and Shpungin [3] estimate a topological characteristic of the network called destruction spectrum (D-spectrum) with an efficient Monte Carlo simulation. In the event of a disaster, the dependency among network components that are exposed to the same amount of impact has been taken into consideration in a limited number of studies on network reliability. Taylor et al. [8] introduce a vulnerability analysis for the independent link failure case. Taking into account the node failures which would cause the failure of links attached to them, they claim that their analysis would work in dependent


321

link failure cases. Sel¸cuk and Y¨ ucemen [6] define a spatial correlation between two elements to calculate the seismic capacity of a single component in the network. G¨ unne¸c and Salman [4] define dependent link failures for highway networks under seismic hazard. Given the marginal link failure probabilitites, they propose a link failure dependency model in which they define vulnerability-based (VB) and set-based (SB) dependency. According to this model, the links are divided into vulnerability sets based on deterministic peak ground acceleration (P GA) values. Links within the sets depend on each other with VB-dependency, while links in different sets fail independently with SBdependency. In a vulnerability set, if a stronger (lower failure probability) link fails due to an earthquake, they assume that the weaker links fail with a probability equal to 1. VB-dependency yields m+1 surviving network realizations for a network composed of one vulnerability set with m links. Our contribution to the literature is that we propose a new dependency model for link failures that captures earthquake vulnerability of highways. The model generates a family of joint probability distributions for the random network subject to link failures. The input parameter for the proposed model determines the degree of dependency. Furthermore, all possible realizations may have a positive occurance probability, but for many realizations this probability gets very small. Therefore, our proposed model provides computational advantage in sampling based approaches (see Sections 4 and 6).

3

Notation

We are given an undirected graph G = (N, E), where E is the set of edges/links and N = {N1 ∪N2 ∪N3 } is the set of nodes consisting of three disjoint sets. N1 represents the set of demand points, N2 the set of emergency response facility (supply) points, and N3 the set of junction points. Let |E| = m. A network realization x = (x1 , x2 , ..., xm ) is a binary vector representing the state of the links. If xi = 1, link i is operational, and if xi = 0, link i is non-operational. For each link i ∈ E, its marginal survival probability p(i) is given. If link j is in the dependency set of link i, nij = 1, and 0 otherwirse. Additionally, the casuality demand at node v ∈ N1 are denoted by wv . Πvy is the set of paths between node v ∈ N1 and node y ∈ N2 , and πkvy is the length of the k th path between node v∈ N1 and node y∈ N2 . We further define the following. Let S represent the set of all possible network realizations (outcomes) defined by x. The probability of network th realization s ∈ S is defined as p(s). Finally, avy path between sk = 1, if the k nodes v and y survives in network realization s and 0, otherwise.

322

4


Proposed Conservative Dependency Model

For the failure model we propose, we partition the links into mutually exclusive sets. The links in different sets are spatially separated according to their P GA levels and the earthquake vulnerability of the structures on them. Each spatially and structurally defined set consists of links that have dependency among them, that is a dependency set according to the definition in [4], so that links in different sets fail independently. Different from [4], in a dependency set, the failure of a strong structure makes the failure of weaker structures of the same kind more likely by decreasing the survival probabilities of weaker links by the factor 100(1-α)%, as defined below. Definition 4.1 Given two links i and j in the same dependency set with survival probabilities p(i) and p(j), we say links i and j have α-conservative failure dependency, if p(i) p(j) implies P(i survives | j fails)=(1-α)p(i). According to 4.1, links i and j within a dependency set do not fail independently. Their covariance, α[p(i)p(j)][1 − p(i)] which is greater than 0 confirms the dependency. Pearson Product-Moment Correlation Coefficient is equal to α p(i)p(j)[1 − p(i)][1 − p(j)]−1 . Thus, the correlation between two links in the same dependency set is proportional to α. As α increases, positive correlation between the links increases. According to the choice of α, the dependency model avoids unlikely network realizations and captures the likely realizations where the link failures are correlated by shared structural characteristics of structures on the links. For α=0, the problem reduces to the independent link failure case and α=1 is the totally dependent case. Our model is conservative in the sense that we only focus on the decrease in probability of weaker links with respect to the failure of stronger links but not vice versa. Unlike the VB-dependency [4], our model does not limit the number of realizations but it changes the probabilities of network realizations according to a control parameter, α. The probability of each distinct network realization is the product of the updated marginal probabilities. However, we note that it is computationally intractable to generate the joint probability distribution for large m. Therefore, we propose a sampling algorithm to generate a sample of network realizations and then utilize sample average approximation to estimate any probabilistic measure of network vulnerability. We give the pseudo-code, Algorithm 1 below where the number of replications is specified as R.


323

Algorithm 1 Step 1: Order network links such that p(i) ≥ p(i + 1). For r = 1 to R For i = 1 to m Step 2: Generate a network realization array x with size m. Step 3: Set xi ← 0, ξi ← 1 pα (i) ← 0, ∀i = 1, .., m. Step 4: Update the survival probability of link i as i−1 pα (i) = p(i) − p(i) ∗ α ξj nij . j=1

Step 5: Generate a random number ϕ between 0 and 1. If pα (i) ≥ ϕ, then xi ← 1 and ξi ← 0. EndIf EndFor EndFor In Step 1, we sort the links with respect to their marginal survival probabilities in descending order. In Step 2, in each replication, we generate an array of all zeros with length m. In Step 4, we update the marginal survival probabilities of each link according to the state of the stronger links in its dependency set. Finally, in Step 5, we update the state of each link.

5

Performance Measure

As we mention in Section 2, the network vulnerability and reliability studies can be conducted on flow networks with edge capacities. In this study, we propose a path-based accessibility measure which considers the edge distances instead of edge capacities. Since the number of possible paths between any two nodes is usually high for large networks, it is not tractable to enumerate and check them all. Therefore, we generate a set of k loopless paths between each demand point(D) and emergency facility (F ), where D ∈ N1 , F ∈ N2 , by a k-shortest path algorithm. When all the generated paths between a D − F pair fail, we use a Penalty Cost which can either be interpreted as a proxy to costs of the paths that are not taken into consideration, or the cost of an alternative and reliable mode of transportation. For each network realization s ∈ S and for each demand point v ∈ N1 , we represent the surviving shortest path to the closest facility, min{πkvy avy sk }, with dsv . Finally, we calculate the y,k

expected weighted average shortest distance to send one unit of demand in

324


the network, EW AD(N, E). EW AD(N, E)=

p(s)dsv wv wv

s∈S v∈N1

(1)

v∈N1

This accessibility measure incorporates disaster risk in terms of demand figures as well as the condition of highway network. The following lemma [2] shows that this measure satisfies three desirable proporties. Lemma 5.1 Let G = (N, E) be a random undirected graph with link failures according to α consevative link failure mode. The following proporties hold. (1) Scale Invariance: The measure, EW AD is invariant with respect to scale changes in demand values, wv , at demand points. (2) Monotonicity: Let E ∗ = E∪{e∗ }.Then, EW AD(N, E ∗ ) ≤ EW AD(N, E). (3) Membership in [Deterministic Upper Bound (DUB), Deterministic Lower Bound (DLB)]: DLB≤ EW AD(N, E) ≤ DUB. Proof. (1) Assume that the amount of demand at node v ∈ N1 varies by a factor β, where β ≥ 0 so that the new demand is wv∗ =βwv , ∀v ∈ N1 . Denote by EW AD∗ , the measure with the new demand vector w∗ is equal to EW AD∗ (N, E) = βEW AD(N, E). (2) Clearly, adding a new link to the network may decrease dsv ∀s ∈ S and ∀v ∈ N1 , but will not increase it. (3) In the network realization s, where all links are operational, avy sk = 1 ∀v ∈ N1 , ∀y ∈ N2 and ∀k ∈ Πvy . Then, dsv ∀v ∈ N1 is equal to the shortest path to the closest emergency response facility y ∈ N2 and EW AD is equal to DLB. In the network realization s, where all links fail, dsv ∀v ∈ N1 is equal to Penalty Cost which is also equal to DU B. Thus, EW AD can take values between DLB and DUB. 2 Even checking k paths between each D − F pair increases computational burden. Therefore, for each demand point, we select and work with the shortest l paths to any emergency facility points where l ≤ k ∗ |N2 |. This aprroximation reduces the run time significantly.

6

Computational Tests on the Case of Istanbul

Our network, which is based on the Istanbul highway network, consists of 83 edges and 60 nodes. Out of all nodes, 34 of them represent districts and 26

325


of them are the junction points in the highway. Among districts, we select 8 districts where the hospitals are densely populated as the emergency response facility points and identify the remaining districts as the demand points. We estimate the marginal survival probability of each link from the uniform distribution by mapping the earthquake vulnerability scores on the links and site dependent PGA levels. We estimate each demand value based on casualty rates and the population values given in the Istanbul Earthquake Masterplan, 2004. For path generation, we take k = 30 and l = 30. We divide the edges into 9 dependency sets. We identify the longest path from any demand point as 75.95 km. We set the penalty cost to 76 km which is DU B on EW AD. We calculate the deterministic lower bound as 20.47 km for the case in which all edges survive. We then calculate EW AD for R = 1, 000, 000. We code Algorithm 1 and VB-dependency in MATLAB and estimate the proposed measures by sample average approximation. We report the elapsed time (ET ) in seconds, estimated EW AD in kilometers, Pearson Variation Index (P V I) for five α values, the independent link failure case (Indep.) and the VBdependency(VB). We also report the lower bound (LB95% ) and upper bound (U B95% ) of 95% confidence interval on EW AD in the Table 1. α = 1 α = 0.75 α = 0.50 α = 0.25 α = 0.15 Indep.

VB

2382

2556

2737

2525

2053

1258

2538

EW AD 62.17

61.26

59.79

54.51

48.46

33.91

62.15

LB95%

62.15

61.24

59.77

54.48

48.44

33.90

62.13

U B95%

62.19

61.28

59.81

54.53

48.48

33.92

62.17

PV I

0.17

0.18

0.18

0.21

0.23

0.16

0.17

ET (sec)

Table 1 Comparative Computational Results for Istanbul Highway

We note that the replication number R is sufficient because the P V I comes out to be small for the proposed probability distributions. We observe that as α decreases from 1 to 0.15, EW AD decreases bacause the level of dependency decreases. In the case of α = 0, we see that EW AD is less than any other values with different α. VB-dependency yields similar EW AD to the α = 1 case, which is expected. Finally, EW AD = 33.91 km is a reasonable distance. However, we include the effect of dependency in link failures, which increases the EW AD up to 62.17 km according to α.

326

7


Conclusion and Future Work

In this study, we proposed a new link failure dependency model and a new performance measure for the accessibility of demand points from supply points on a vulnerable highway under earthquake risk. For future work, dissimilar path generation methods in the literature can be tested instead of using the k-shortest path algorithm.

References [1] Chang, S.E., and N. Nojima, Reliability of lifeline networks with multiple sources under seismic hazard, Natural Hazards. 21 (2000), 1–18. [2] De-Los-Santos, A., and G. Laporte, and J.A. Mesa, and F. Perea, Evaluating passenger robustness in a rail transit network, Transportation Research Part C: Emerging Technologies. 20 (2000), 34–46. [3] Gertsbakh, I., and Y. Shpungin, “Spectral Approach to Reliability Evaluation of Flow Networks,” Proceedings of the European Modeling and Simulation Symposium. (2012), 68–73. [4] G¨ unne¸c, D., and F.S. Salman, Assessing the reliability and the expected performance of a network under disaster risk, OR Spectrum. 33 (2011), 499– 523. [5] Nagurney, A., and Q. Qiang, Fragile networks: identifying vulnerabilities and synergies in an uncertain age, International Transactions in Operational Research. 19 (2012), 123–160. [6] Sel¸cuk, A.S., and M.S. Y¨ ucemen, Measuring post-disaster transportation system performance: the 1995 Kobe earthquake in comparative perspective, Transportation Research Part A: Policy and Practice. 35 (2001), 475–494. [7] Sohn, J., Evaluating the significance of highway network links under the flood damage: An accessibility approach, Transportation Research Part A: Policy and Practice. 40 (2006), 491–506. [8] Taylor, M.A.P., and M. D’Este, and S.V.C Sekhar, Application of Accessibility Based Methods for Vulnerability Analysis of Strategic Road Networks, Networks and Spatial Economics. 6 (2006), 267–291.