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A CONSEQUENCE MITIGATION MODEL FOR WATER NETWORKS SUBJECT TO INTENTIONAL PHYSICAL ATTACKS Hyung Seok Jeong1, Jianhong Qiao2, Dulcy M. Abraham3, Mark Lawley4, Jean-Philippe Richard5, and Yuehwern Yih6 ABSTRACT This paper presents a mitigation model which can reduce or minimize adverse consequences when a water network suffers from significant water shortage resulting from physical destruction of critical facilities in water network. The model optimizes the water supply plan for water customers utilizing the information regarding the priority of water customers. Two different implementation engines based on Branch and Bound (BnB) and Genetic Algorithms (GA) are developed and evaluated using two representative water networks. BnB is a general search method for solving discrete and combinatorial optimization problem, which guarantees a global optimal solution while GAs are heuristic search algorithms in which the global optimality of the solution is not assured. A hydraulic network solver, EPANET2.0 is linked with these two optimization tools to test the hydraulic feasibility of candidate solutions. The performance of the two different engines is evaluated in terms of the quality of solutions and computational efficiency. The findings of this study indicate that the proposed optimization model can successfully reduce the consequences while meeting hydraulic constraints in water networks. In addition, as the size of the network increases, the use of the GA engine is preferred because it takes significantly less computation time than the BnB approach, while providing reliable levels of accuracy. KEY WORDS Water Infrastructure Security, Homeland Security, Disaster Mitigation, Intentional Attacks, Branch and Bound, Genetic Algorithms
1
Doctoral Candidate, School of Civil Engineering, Purdue University, West Lafayette, IN 47907, Phone +1 765/494-0642, FAX 765/494-0644,
[email protected] 2 Doctoral Candidate, School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, Phone +1 765/491-0935, Fax 765/494-1299,
[email protected] 3 Associate Professor, School of Civil Engineering, Purdue University, West Lafayette, IN 47907, Phone +1 765/494-02239, FAX 765/494-0644,
[email protected] 4 Associate Professor, School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, Phone +1 765/494-5415, Fax 765/494-1299,
[email protected] 5 Assistant Professor, School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, Phone +1 765/494-5166, Fax 765/494-1299,
[email protected] 6 Professor, School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, Phone +1 765/4940826, Fax 765/494-1299,
[email protected]
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INTRODUCTION Protecting water infrastructure systems from intentional attacks and developing strategies to make the water networks robust, resilient, and more immune to intentional attacks has been one of the top priorities in water industry since the September 11, 2001 terrorist events. Physical attacks by intentional attackers have been the most common attack scenarios in history. Among various mitigation strategies for water networks against physical attacks, installation and enhancements of preventive measures around the water network can be very limited due to the extensive size of the system and difficulty in determining the cost effectiveness of these preventive measures to the event of extremely low probability. An effective consequence minimization strategy after an attack can be efficient in reducing the risk of water infrastructure against intentional physical attacks. In this paper, a mitigation model is described. This model, with sufficient computation, minimizes the adverse consequences of water shortage resulting from physical destruction of critical facilities in the water network. The model takes as input the residual water network (the portion of the network still functional after the attack) and a demand level and priority rating for each node. It then uses the hydraulic simulator EPANET 2.0 (Rossman 2000) to compute pressures at each of the network’s nodes. If any node exhibits inadequate pressure, then the overall demand cannot be met. Thus, demand across the network must be adjusted so that all demand node pressures can be brought to an acceptable level. This implies that some demand will not be met and adverse consequences will result. Adverse consequences resulting from water shortage at a particular node depend on the priority level of that node as well as the degree of shortage. Thus, the objective of the model is to identify a feasible demand pattern for the residual network that minimizes the consequences of water shortage. CONSEQUENCE MINIMIZATION MODEL Given an intentional attack, the consequence minimization problem consists of finding a hydraulically feasible demand pattern for the residual network that minimizes the negative impact of water shortages on the customer population. The model is as follows:
Min
wu f u (d u )
objective function
(1)
Q = Gu d u
for customer demand node u V
(2)
Q = Sv
for supply node v
(3)
u V
Qe
e e E:h ( e )=u
e E:t ( e )=u
Q
e e E :t ( e ) = v
e e E :h ( e ) = v
K e Qe = hu
(
hv
)
K e Qe Ae Q + Be Qe + C e = hu hmin hu - Elevu 2 e
hv
V
for e = (u,v)
E st. e has no pump
(4)
for e = (u,v) u V
E st. e has a pump
(5) (6)
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where, V E du fu(du) wu Qe Gu Sv Ke hu Ae, Be, Ce Elevu hmin
node set of the skeletonized residual network edge set of the skeletonized residual network {0, ½,1} proportion of water demand satisfied for customer node u V negative impact of reducing nominal demand to proportion du at node u V priority weight of customer node u V flow rate on pipe e E demand at customer node u V base capacity at supply node v V head friction loss coefficient head pressure of node u V characteristic parameters of the pump on pipe e elevation at node u V predefined nodal pressure requirements7 Hazen-Williams parameter ( = 1.852)
Variables are du for u=1…N, hu for u=1…N and Qe for e E. The du variables represent the proportion of nominal demand of each customer node to be satisfied. Water companies cannot directly control the quantity of water used at a customer node unless water supply is completely discontinued. However, in emergency situations, voluntarily participation of water customers to overcome the disaster can be anticipated. Drought management programs available in most states in the U.S. indicate that a reduction of at most 50% of water consumption can be achieved by modifying customer’s water consumption behavior (National Drought Mitigation Center, 2004). Based on this observation, in this paper, it is assumed that the demand of each customer node can be managed at one of three levels: 0 (water supply discontinued, no demand satisfied), ½ (half demand satisfied) and 1 (all demand satisfied). Once the decisions have been made (i.e., the du set to one of the three values), the flow and pressure characteristics of the residual network, represented as Qe and hu, respectively, are computed using hydraulic equations. Constraints (2) and (3) represent the mass conservation equations in the residual network with demand levels reduced according to the du variable settings. Constraints (4) and (5) represent the energy conservation equations. Finally constraints (6) insure that the effective pressure at each node is sufficient to provide water. The objective function (1) quantifies the consequences of the water shortage resulting from intentional attack by summing the consequences of the water shortage at each node of the network. For each node, the impact function, f: {0, ½,1} [0,1], takes as input the proportion of base demand satisfied and returns a number ranging from 0 to 1 indicating the impact severity (with 0 being no negative impact and 1 being maximum negative impact). The impact function is specific to each node of the skeletonized network. For example, hospitals might be severely affected by even small reductions in normal consumption, 7
We set hmin to 68.9 kN/m2 (10 psi) since it is typically recommended as the absolute minimum pressure for fire suppression (Shinstine et al. 2002).
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whereas agricultural areas might not suffer any consequences if water supply is temporarily discontinued. In this paper, the three impact functions are described and used, as represented in Figure 1. A customer with impact function f1 is highly sensitive to initial reductions in water use since the initial slope is relatively steep. f2 reflects a linear impact, while customers with impact function f3 are least affected by demand reductions.
Figure 1: Impact Functions The priority weight wu of customer node u reflects the relative importance of that node, with 0 being the lowest assignable priority and 1 the highest. Not all water customers are equally important. If at all possible, water must be supplied to critical facilities such as hospitals, emergency centers, etc. Developing priority weights for nodes will typically be a policy decision made by community leaders. The weight could also be chosen to reflect the impact on the local economy, the total number of people seriously affected by the outage, the impact on emergency services, the expected number of deaths incurred, or some combination of these metrics. Thus, the objective in this study is to find a customer demand pattern that minimizes consequences, equation (1), while maintaining hydraulic feasibility, equations (2)-(6). The model is a nonconvex nonlinear mixed integer optimization problem, and thus, by nature, is very difficult to solve. OPTIMAL AND HEURISTIC SOLUTION APPROACHES In this section, three methods for finding solutions to the consequence mitigation model of the previous section are described. BRANCH-AND-BOUND APPROACH The first method discussed is a Branch-and-Bound (B&B) algorithm. It is exact in the sense that, if the search completes, it finds the best possible solution for the demand alternatives given above. It consists of a solution representation (search tree), search strategy, and node elimination method (pruning).
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The B&B algorithm recursively divides the solution space into mutually disjoint subspaces, represented by a search tree. In this case, every level of the search tree corresponds to a particular customer, and every node at a given level represents one of the three demand options available for that customer. Figure 2 provides an example of search tree for a water system with three customers. In this section, the term node refers to the search tree not the skeletonized water network. A path starting from the root to a node at level i in the search tree represents a partial solution in which the demand levels for customers 1…i are fixed, while the others are free. If customer i is at the leaf level, then the corresponding path(r,i) is a complete customer demand pattern in which all customer demands are fixed. An optimal solution to the problem can be found by inspecting all of the 3N leaves of the tree where N is the total number of customers in the water network. However, it is typically unnecessary to develop the whole tree to find optimal solutions. To accelerate the discovery of an optimal solution, B&B directs the search procedure in the part of the tree that contains good solutions (search strategy) and avoids searching useless parts of the tree (pruning).
Figure 2: Search-Tree Structure for B&B The B&B algorithm creates and examines the nodes according to a probabilistic best first search strategy. The search/branching process keeps an OPEN set containing all unexpanded nodes encountered so far in the search. OPEN can be initialized with either the root node or solutions obtained from other heuristics presented later in the section. The algorithm sorts OPEN based on a measure called decision deviation. In a water network, a customer with high demand and low weight is a better candidate for demand reduction than a customer with low demand and high weight. Therefore, for each customer u in the water supply network, the anticipated demand decision for customer u (Ru*) can be computed Ru* = (Ru
Rmin ) / (Rmax
Rmin )
(7)
where Ru = wu / Gu , Rmax = max{Ru: u = 1…N} and Rmin = min{Ru: u = 1…N}.
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Ru* is a number between 0 and 1. Note that if Ru* is close to 1, then Ru is close to Rmax, which indicates that the ratio wu / Gu is large relative to the other nodes. This ratio is large when wu is large and/or Gu is small, and thus if Ru* is close to 1, then customer u is a poor candidate for demand restriction. In this case, du is set to 1 (full supply for customer u). If the ratio is relatively small, then either wu is small and/or Gu is large, and thus Ru* is closer to 0. In this case, customer u is a good candidate for demand restriction, and du is set to either 0 or ½ (restricted supply for customer u). Consider a node v in the search tree. Let c(v) be the associated customer level, which is 1…N, and let d(v) be the associated demand value, which is either 0, ½, or 1. The decision deviation for node v is defined as (v) = | d(v) – Rc(v)*|
(8)
and the path deviation for Path(r,v) as
(v) =
(i ) / Path (r , v)
(9)
i Path ( r ,v )
where |Path(r,v)| is the path cardinality. For any node v in OPEN, the path deviation is the average absolute difference between the fixed demands and the anticipated demands of the fixed customers. Intuitively, nodes with small path deviation values, if expanded, would be more likely to provide low consequence solutions. To avoid spending too much time in local search, limited randomness is introduced so that nodes with small deviation have high probability to be chosen but not with certainty. In the tree expansion process, since leaf nodes and expanded nodes will not be further explored, the algorithm deletes them from OPEN through updating. The search strategy can be further improved if parts of the search tree that do not contain optimal solutions are never inspected. Three pruning criteria can be used at any node of the tree to avoid further expansion of these nodes. Let x* be the best feasible solution found so far in the search and let z* be its value. Consider any node v of the search tree. An expanded optimistic solution can be obtained from that node by setting the demand of all free customers to 100%. Let z(v) be the consequence level of the expanded optimistic solution at v. If z* z(v), the best solution in the subtree rooted at v has a consequence level higher than z*. Therefore, it is unnecessary to explore any node of the subtree rooted at v (pruning by bound). Also, if the expanded optimistic solution is feasible and z(v)
