Reliability, Risk and Safety: Theory and Applications – Briš, Guedes Soares & Martorell (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-55509-8

A multiple-objective approach for the vulnerability assessment of infrastructure networks M. Claudio & S. Rocco Universidad Central de Venezuela, Caracas, Venezuela

J.E. Ramirez-Marquez Stevens Institute of Technology, NJ, USA

E. Daniel & A. Salazar Ecole Nationale Supérieure des Mines de Saint Etienne, France

ABSTRACT: One of the most used approaches to assess the vulnerability of critical infrastructures is that based on complex network concepts. Several authors have evaluated such vulnerability based of different performance functions derived from the complex network theory, like the characteristic path length, the network efficiency or the reliability efficiency, an extension of the efficiency concept that considers the reliability of the links. However, the assessment has been evaluated as a single objective problem, for example, determining which network component has the most important role in the systems, from a vulnerability point of view. In this paper we recognize that the Decision-Makers (DM) often require a set of solutions more suitable in attaining certain goals. So for example, the DM can benefit from understanding the trade-off between an event with lower performance and higher cost or an event with lower cost and higher performance. Currently to address this challenge, the DM must solve several problems using a single objective approach, by varying a group of constraints. By contrast, in the multiple-objective (MO) formulation, the determination of the Pareto Frontier (PF) is determined via a single problem. To characterize the PF, this paper solves the multiple-objective Deterministic Network Vulnerability Problem (MO-DNVP) where two or more objectives are optimized by using an evolutionary algorithm. Numerical examples illustrate the approach.

1

INTRODUCTION

single link or node, or a group of links or nodes), the importance of each event d ∈D is evaluated as:

The infrastructure enterprise is key to sustaining the economy and societal-well-being of a nation. Recent world events have shown critical infrastructures to be particularly sensitive to partial or complete incapacitation, due to internal or external sources of failures or attacks. For internal failure sources, reliability engineering and risk analysis have provided tools and procedures for estimating, preventing and handling undesired failure events that occur at random in complex systems. However, external sources of failures, with emphasis on intentional attacks, can be potentially catastrophic and constitute a new challenge due to the current involvement of “. . . malevolent intelligence directed towards maximum social disruption’’ [Apostolakis and Lemon, 2005]. One of the most used approaches to assess critical infrastructures is that based on complex network concepts. A critical infrastructure (e.g. a power system) is abstractly modeled as a network G of nodes interconnected by links. Given a performance function
(G) > 0 and a set of possible events D (for example the removal of a

where
(G,d) is the performance of the network G including the event d ∈D. Then, the event d* which maximizes I(d) is considered as the most important event. In this case V = I(d*) is defined as the vulnerability of the network under the class of damages D, V ∈ [0,1] (Crucitti et al, 2005). Several authors have used this procedure [Crucitti et al, 2005, Rosato et al, 2009, Nagurney and Qiang, 2007], based of different performance functions: some derived from the complex network theory (e.g, the network efficiency [Crucitti et al, 2005]) others derived from the physical system modeled (e.g. DC power flow [Rosato et al, 2009]) In general, the assessment of the importance of components has been evaluated as a single objective problem, for example determining which network component must be reinforced such as the vulnerability is minimized or which component must be damaged such as the vulnerability is maximized. However the assessment of this analysis has not considered the cost

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of performing the event d, even if it has been recognize as a limitation. It is important to note that the Decision-Makers (DM) often require a set of solutions more suitable in attaining certain goals. So for example, the DM can benefit from understanding the trade-off between an event da with lower performance and higher cost or an event db with lower cost and higher performance. Currently to address this challenge, the DM must solve several problems using a single objective approach, by varying a group of constraints. By contrast, in the multiple-objective (MO) formulation, the determination of the Pareto Frontier (PF) is determined via a single problem. To characterize the PF, this paper solves the multiple-objective Deterministic Network Vulnerability Problem (MO-DNVP) where two or more objectives are optimized by using an evolutionary algorithm. Numerical examples illustrate the approach. The remainder of the paper is organized as follows: Section 2 contains an overview of the Performance functions. Section 3 presents a synthesis of multiobjective optimization while Section 4 shows the proposed approach on different systems. Finally, Section 5 presents the conclusions. 2

PERFORMANCE FUNCTIONS

A critical infrastructure (e.g. an electric power system) is abstractly modeled as a network G of nodes interconnected by links. Let G (N, A) represents a network. N corresponds to the set of nodes and A represents the set of links. Each link (i,j) could have an attribute defined by kij , for example, the distance between i and j, or the link reliability. Let assume that it is possible to define a function
(G) > 0 which allows assessing G. For example, for electric power systems infrastructure, such function is generally related to the ability of the system to provide an adequate supply of electrical energy [Wood and Wollenberg, 1996; Billinton. and Li, 1994]. Different performance functions have been used: some derived from the complex network theory, like the characteristic path length or the network efficiency [Crucitti et al, 2005] or the reliability efficiency, an extension of the efficiency concept that considers the reliability of the links [Zio, 2007]. As previously mentioned the importance of each event d ∈ D is evaluated by I(d). For example, in [Crucitti et al, 2005] the authors use the global efficiency of G (defined from the minimum number of steps from node i to node j) to evaluate the vulnerability of three electric power systems in Europe under three different classes of damages D: the removal of sets of one, two or three links. 3

for mathematical models that have multiple objective functions to be optimized [Coello, 1999]. Unlike optimization models with a single objective function, the interest is on finding a set of solutions that describe how the improvement of a single objective function value impacts the value of the other objectives. This set is commonly known as the Paretooptimal set and each of its elements as a Pareto optimal solution. In this respect, a general formulation of a multi-objective problem is:

Vectors f (x) and g(x) describe objective functions to be maximized and minimized, respectively. Similarly, the first and second sets of constraints describe possible constraints on network performance and resources. In the context of the present work, the interest could be, for example, to maximize I(d) with minimum expenditures. A solution x∗ that satisfies the constraints, is called Pareto optimal if:

To solve the above optimization problem, a Multiple-Objective EvolutionaryAlgorithms (MOEA) is embraced here. MOEA is a term employed in the Evolutionary Multi-criteria Optimization field to refer to a family of evolutionary algorithms formulated to deal with MO. MOEA are able to deal with non-continuos, non-convex and/or non-linear objectives/constraints, and objective functions possibly not explicitly known (e.g. the output of Monte Carlo simulation runs). The development of these algorithms has successfully evolved, producing efficient algorithms like SPEA2 [Zitzler et al, 2002], PESA2 [Corne et al 2001] and NSGA-II [Deb et al, 2002] among others. The algorithm used in this paper (MO-PSDA) [RamirezMarquez and Rocco, 2008] offers a simple, intuitive and efficient approach to the solution of the MO problem, with a minimum number of tuning parameters. However, any MOEA algorithm could be used. 4

COMPUTATIONAL EXAMPLES

4.1 The Italian high-voltage (380 kV) electrical transmission network

MULTI-OBJECTIVE OPTIMIZATION

Multi-objective optimization has been proposed as an approach to solve the problem of finding solutions

The Italian high-voltage (380 kV) electrical transmission network can be represented by an undirected

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Figure 2. Pareto approximation front for example 1 10, 15 20 an 28 simultaneous link damages.

Figure 1. Pareto approximation front for example 1.

graph of 127 nodes and 171 lines [Rosato et al, 2009]. The network has been studied in [Crucitti et al, 2005] using the global efficiency. The authors evaluated all the single, double and triple link damages and estimated the effects of higher damages. In this first example we assess the vulnerability of the system by maximizing the importance I(d) and minimizing the number of damages. Figure 1 shows the Pareto set approximation found using the MO-PSDA implementation. Out of the total 2171 potential solutions for this problem, the Pareto set was identified by analyzing only a total of 5000 combinations. In the Figure, the first extreme point located at (0,0) represents the solution where no link is damaged. The point located at (5.02 %, 1) corresponds to the maximum importance in the network when considering the damage of a single line. The values for double and triple links damages are equal to those obtained in [Crucitti et al, 2005]. Finally the figure shows the results when considering up to 8 simultaneously link damages. Figure 2 shows the results for 10, 15 20 an 28 simultaneous link damages.

Figure 3. Network for example 2.

Figure 4. Pareto approximation front for example 2.

4.2 Reliability efficiency Figure 3 shows the network analyzed in [Zio, 2007] using the reliability efficiency as performance function (numbers in parentheses represent the link reliability). Figure 4 shows the Pareto set approximation found. In the Figure, the first extreme point located at (0,0) represents the solution where no link is damaged. The point located at (22.2 %, 1) corresponds the maximum importance in the network when considering the damage of a single link (link 3). Finally the figure shows the results when considering up to 6 simultaneously link damages. 4.3 A 52-node network Figure 5 shows the network considered in [Manzi et al. 2001), with 52 nodes and 76 undirected links. Each link has a reliability of 0.95 and a damage cost (an integer value) randomly selected in [1,5] units. This value represents the cost needed to damage the link.

In this example a comparison among two different criteria is presented: Number of links out and costof link damage. Figure 6 and 7show the Pareto set approximation found using the MO-PSDA implementation. Out of the total 276 potential solutions for this problem, the Pareto set was identified by analyzing only a total of 5000 combinations. Figure 6 presents the solutions based on minimizing the number of links out, while Figure 7 shows the solution based on minimizing the cost of damages. In both figures the first extreme point located at (0,0%) represents the solution where no link is damaged. Note that in Figure 6 the number of links out can be considered as the total damage cost, if the damage cost of each link is assumed equal to 1 unit. However the information provided by both figures is different. For example, consider the case for LINKS OUT=4=COST. From Figure 6 it’s clear

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new approach to assess critical infrastructure. Contrary to other single-objective approach, the MODNVP is able to cope with multiple objectives. Thus, the DM can benefit from understanding the trade-off between different situations, characterized by conflicting objectives, such as vulnerability and cost. To solve the optimization problem, a MultipleObjective Evolutionary Algorithms (MOEA), a family of evolutionary algorithms formulated to deal with MO problems. The efficiency of the procedure proposed has been illustrated by several examples. REFERENCES

Figure 5. Network for example 3.

Figure 6. Pareto approximation front for example 3: Minimizing the number of links out.

Figure 7. Pareto approximation front for example 3: Minimizing the cost of link damage.

that the point (4,15%) corresponds to the maximum importance in the network when considering the damage of 4 links, while the point (4,7.62 %) in Figure 7, corresponds to the maximum importance in the network when considering a total cost of 4 units.

5

CONCLUSIONS

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This work presented the multiple-objective Deterministic Network Vulnerability Problem (MO-DNVP), a

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