Multiobjective Optimization of Operational Responses for Contaminant Flushing in Water Distribution Networks Leonardo Alfonso, Ph.D.1; Andreja Jonoski2; and Dimitri Solomatine3 Abstract: Contamination emergency in water distribution systems is a complex situation where optimal operation becomes important for public health. In case of emergency corrective operational actions for flushing the pollutant out of the network are needed, which have to be fast and accurate. Under such a stressful situation, trial-and-error simulation experiments with the hydrodynamic and water quality models cannot be applied since significant number of model evaluations may be required to identify the optimal solution. This paper presents a methodology for finding sets of operational interventions in a supply network for flushing a contaminant by minimizing the impact on the population. The situation is treated as both single- and multiobjective optimization problem, which is solved by using evolutionary optimization approaches, in combination with the EPANET solver engine. The methodology is tested on a simple imaginary network configuration, as well as on a real case study for the city of Villavicencio in Colombia. The results prove the usefulness of the approach for advising the operators and decision makers. DOI: 10.1061/共ASCE兲0733-9496共2010兲136:1共48兲 CE Database subject headings: Water distribution systems; Water pollution; Optimization; Algorithms. Author keywords: Flushing; Operation; Optimization; Valves; Globe; COPA.
Background Problems related to water distribution networks 共WDNs兲 can be broadly attributed to two large classes: design and operational. Solutions of most problems can be simplified by using mathematical and optimization models. Optimization methods used may depend on the complexity of the problem and amount of time available for solving it. For example, in operation, time constraints may prevent the full exhaustive search of the optimal solutions, whereas in design such constraints could be relaxed. Typically, optimization problem do not allow analytical formulation 共since objective function calculation relies upon model runs兲, thus they can be solved only by using relatively inefficient randomized direct search methods, like a popular genetic algorithm 共GA兲 关see Goldberg 共1989兲 and Holland 共1975兲兴. A brief overview of the relevant research is given below. For solving design problems, during the past three decades a number of optimization techniques for minimizing the overall costs of a network have been suggested and tested. Simpson et al. 共1994兲 solved first the least cost design problem using GAs and 1 Research Fellow, UNESCO-IHE Institute for Water Education, P.O. Box 3015, 2601DA Delft, The Netherlands 共corresponding author兲. E-mail:
[email protected] 2 Senior Lecturer, UNESCO-IHE Institute for Water Education, P.O. Box 3015, 2601DA Delft, The Netherlands. E-mail:
[email protected] 3 Professor, UNESCO-IHE Institute for Water Education, P.O. Box 3015, 2601DA Delft, The Netherlands; and, Water Resources Section, Delft Univ. of Technology, The Netherlands. E-mail:
[email protected] Note. This manuscript was submitted on September 22, 2006; approved on June 3, 2009; published online on December 15, 2009. Discussion period open until June 1, 2010; separate discussions must be submitted for individual papers. This paper is part of the Journal of Water Resources Planning and Management, Vol. 136, No. 1, January 1, 2010. ©ASCE, ISSN 0733-9496/2010/1-48–58/$25.00.
later Savic and Walters 共1997兲 exposed the GAs for singleobjective optimization of pipe costs related to pipe size. Similar problem was explored by Abebe and Solomatine 共1998兲 who applied and compared two optimization approaches, adaptive cluster covering and GAs, enabling a choice between the accuracy and the required computer time. Hewitson and Dandy 共2000兲 included a water quality penalty cost within the overall cost function to ensure acceptable disinfectant levels in the network and optimized this single objective by means of the GAs approach. In order to prioritize budget investments in system expansions, Wu and Simpson 共1996, 2001兲 introduced the fast messy GA in the least cost design of pipe networks. Later, Wu et al. 共2005兲 presented a case study where the fast messy GA method was used for optimizing a network expansion, minimizing costs and maximizing benefits, using multiobjective approach. Reis et al. 共1997兲 considered the use of GAs for maximizing the leakage reduction by finding an optimal location of isolation valves, posing it as a single optimization problem. The water quality issue was also tackled by Tryby et al. 共2002兲 with the optimal location of boosters for secondary disinfection, formulated as a mixed integer linear programming problem. Other important multiobjective evolutionary-based optimal pipe network design are Farmani et al. 共2005兲, Prasad and Nam-Sik 共2004兲, and Prasad et al. 共2004兲. For solving operational problems, optimization methods have been applied to deal with different issues, such as leakage reduction 共Savic and Walters 1995; Pezzinga and Petitto 2005兲, both using GA for solving single-objective problems; pump scheduling 共Jowitt and Germanopoulus 1992; Pezeshk and Helweg 1996兲, by using linear programming and discrete adaptive search algorithms, respectively, for solving the single objective of minimizing energy costs; and network disruption for repairs 共Simão et al. 2004兲, using a logic process knowledge algorithm to find out the best set of valves to close when a disruption of service is needed. The real time reaction during emergency is, however, a specific operational problem for which the worldwide awareness has increased dramatically after the terrorists’ attacks of the past years
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关see, e.g., Ostfeld 共2006兲兴. This operational problem should be solved, in principle in two steps: 1. Identify the exact sources of contamination, moment of intrusion, and its duration, if possible; and 2. Determine the corrective actions minimizing the propagation of contaminant or, in more general terms, damage to public health. Approaches to solve Problem 1 are presented, e.g., by Laird et al. 共2005兲 and Preis and Ostfeld 共2006兲. The major results reported in this paper have been achieved already in 2006 in the framework of the masters study at the hydroinformatics core of the UNESCO-IHE Institute for Water Education 关Alfonso 共2006兲, available at http://www.ihe.nl/hi/MSc_abstracts/2006/0604%20Alfonso%20Segura%20Leonardo.htm兴. When the paper was submitted to ASCE in September 2006, to our knowledge there were no published approaches dealing with Step 2. During the reviewing period, Baranowski and LeBoeuf 共2008兲 presented a procedure to resolve Step 2. Their objective was to minimize the total contaminant concentration in all nodes for all times steps after the detection by manipulating pipe status and node demands. Their approach is characterized by the following: single-objective optimization 共rather than multiobjective兲, nonrealistic assumption that flushing can be done at any node and every pipe can be closed, lack of pressure-dependent demands, and pumps were not considered to be active elements in the flushing procedure. On the other hand, the writers introduced an interesting analysis of impact in terms of reaction time and explicitly used information from sensors which is more realistic than assuming various sources of pollution 共which is, however, more general兲. In January 2007, one of the reviewers of the present paper pointed to an earlier work by Ostfeld 共2006兲. During the final round of reviewing, in November 2008 we were glad to see the publication by Preis and Ostfeld 共2008兲 that also presented an approach similar to ours. Their approach, however, is less general due to several limiting assumptions: it assumes constant-flow hydrants for the flushing, does not consider occurrence of negative pressures, does not use pumps to assist flushing, and assumes that only one point in the network is polluted. It uses only the multiobjective approach, not considering the procedures for making the final choice of the optimal strategy. Yet another approach for dealing with Step 2 was introduced by Poulin et al. 共2008兲. This heuristic approach isolates the polluted water by closing proper valves and leaves one pipe to let clean water to come into the isolated area, which is flushed by hydrants. Heuristic rules are applied for choosing which valves to close, which, at the same time limit the extent of isolated areas, ensure the fast operational response and meet prespecified operational constraints. Although the approach assumes static demand analysis 共a strong assumption兲, it is interesting since it considers the way the operation teams carry out the actual response. The present paper addresses the problem of minimizing the damage from network contamination 共to be posed formally later兲 for situations when an operator would not have enough information to identify the sources of contamination with the adequate accuracy. For example, these are cases when there is information about the high concentration of a pollutant only from one monitoring node 共sensor兲. In such cases Problem 1 cannot be reliably solved 共until more information is collected兲 but Problem 2 still has to be solved quickly. This paper addresses Problem 2 and for modeling purposes it is simply assumed that the contaminant is introduced at the monitoring node. When contamination events take place in a WDN, the operator’s reaction must be quick and accurate 共to avoid contamination
propagation兲. However, under emergency conditions the process of decision making is very difficult and often stressing. From one side, the rational decision about the operation actions to be taken is not easy, even if a hydraulic and quality model of the system are available; from another side, such decision must be taken in a short period of time, with the associated stress. When the pollutant concentrations are extremely high and there are imminent life losses, the utilities must shut the entire system down. Nevertheless, a different approach can be adopted when less harmful contamination is found: isolating the contaminant by operating valves in order to reduce the affected area of the system and simultaneously open accessories like hydrants, vents, or drains are used to flush out the contaminated water. Switching pumps is also within the operational possibilities. However, the selection of the appropriate set of elements to be operated is a main issue because the hydraulic behavior and the water quality in looped systems are quite complex and because a wrong selection could make the situation worse. On the other hand, the restricted time for reaction, and the stress associated with the public health concern, makes it impossible to conduct experiments at that particular moment with a model in order to find a solution. It has to be pointed out that even with enough time for analysis and with a reliable model, a suitable solution may not come up easily, simply because of the fact that the number of the system elements to be operated may be very high. In this paper, the problem of operational responses is addressed by using currently available optimization methods together with the hydraulic and quality solvers for pressurized networks, by solving an optimization problem in a multiobjective formulation by the nondominated sorting GA II, NSGA-II 共Deb et al. 2002兲 and in a single-objective formulation with the criteria combined by a GA. Both optimization approaches are applied to the two case studies: a simple imaginary WDN and a real-world system.
Formulation of the Operational Response as an Optimization Problem We consider a WDN with pipes 共some of them represent valves兲, nodes 共some of them represent hydrants兲, and pumps. Removal of contaminant is supposed to occur when flushing is performed by “interventions” 共operational actions兲—opening or closing certain valves or hydrants or switching pumps. Such interventions are considered to be decision variables in this problem and their number is equal to the number of elements 共valves, pumps, or hydrants兲 to be operated. The problem is in identifying the values of decision variables 共i.e., interventions兲 that prevent the pollutant propagation 共reducing associated damages兲. At the same time the number of interventions, or network changes, which are associated with costs, are to be minimized as well. Figs. 1 and 9 共case studies兲 provide examples. Note that in this paper the terms “objective” and “criterion” are used interchangeably. Criteria „Objectives… The problem is considered to be a multicriteria 共multiobjective兲 and involves two main criteria: criterion C1 characterizing damage to public health associated with the network contamination and criterion C2 representing the costs associated with the operational effort required to set the network to a desirable condition. These criteria 共objective functions兲 are characterized below.
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Fig. 1. Distribution network of Case Study 1 with scenarios of pollution considered
Criterion C1 relates to the damage to public health associated with the network contamination 共measured by the total cost incurred by contamination兲. The value of this criterion depends on the existence of certain pollutant concentrations in the network and has to be minimized. There are various ways to formulate this criterion. Formulation 共1兲 considers the damage at any given node to be calculated as the function of the number of nodes polluted at least at one time step during the entire simulation period 共referred to as “number of polluted nodes,” npn兲. By a “polluted node” we understand a node with pollution concentration above a specified threshold. When minimizing C1, indirectly two key issues are addressed: first, reducing the pollution extent 共contaminated area兲 in the network and second, reducing the time of exposure of concentrations above the threshold. Formulation 共2兲 considers damage at any given node that is calculated as the sum of damages at each time step and the latter is calculated as a nonlinear “nodal damage function” of the concentration 共a version of the sigmoid function which has to be calibrated for every situation can be used兲. The total network damage could be defined as the sum of
the damages of each node in the network for all time steps of the simulation. Note that these formulations are crude and of course do not contain all the factors allowing for accurate assessment of the damage to public health. Moreover, it is clear that some nodes have bigger impact on public health than others if they become contaminated, e.g., supply nodes for schools, hospitals, and densely populated areas. However, in this paper, it is assumed that the population density is equally distributed over the nodes so all nodes are equally important in terms of impact. In principle we may introduce other formulations as well, for example, to consider the risk of contamination, understanding by risk the potential damage multiplied by the probability of the realization of this damage 共for example, associated with probabilities of contaminant intrusion at various nodes and times兲. However, it is not a trivial task to accurately assess these probabilities 关see, e.g., Ostfeld et al. 共2006兲兴 so full model of risk is not considered here. We performed experiments using formulations 共1兲 and 共2兲 for C1 but for the sake of testing the optimization methodology only the results using formulation 共1兲 will be presented, so C1 = npn.
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The reason to use this simplistic formulation is that it does not depend on the choice of damage functions, which is often very difficult to construct and which have influence on the optimization results. Of course, in real-life applications, each particular case needs studies to identify the appropriate class of damage functions assessing public health risk, which is a considerable problem in itself; 关see, e.g., National Research Council NRC 共2005兲兴, and this should be a topic of further studies. Criterion C2 共to be minimized兲 represents the costs associated with the operational effort required to set the network to a desirable condition, e.g., closing certain valves and/or opening hydrants for flushing the contaminant. There are also various ways to formulate this criterion, but in this paper we consider its value to be the number of the operational interventions 共oi兲 needed. In real-life applications the appropriate cost function that would reflect the actual costs associated with the oi should be used. Fig. 2. Definition of objectives for single- and multiobjective optimization
Constraints A number of constraints are considered: positive nodal pressures, topological checking to ensure network connectivity, and technical operational capacity to implement interventions. Approaches to Solving an Optimization Problem The posed multiobjective optimization problem can be solved using two approaches: • Multiobjective optimization: when a number of solutions are generated, such that each of them is better than at least one other solution on at least one objective 共so in this respect no solution can be formally preferred to another one兲, and it is up to the decision maker to make a selection between them; • Single-objective optimization: when several objectives are combined 共for example, as a weighted sum or by measuring the distance to the “ideal point”兲 into one composite objective and a solution is sought that minimizes this objective.
= decision space; k = number of objectives; and f = objective function. In the problem addressed, the costs C1 and C2 mentioned above are considered as separate objectives to be minimized independently. An example where a number of generated solutions are displayed in two-criteria space is presented in Fig. 2. The points represent states resulting from particular operational configurations of the network; the dark points represent the Paretooptimal set of solutions. Single-Objective Formulation The multiobjective problem can be transformed into a singleobjective one, by combining all the objectives Ci 共i = 1 , . . . , N兲 into one objective C. This can be done in a number of ways, for example, by weighing the N objectives C=
Methods and Tools
兺i=1 wiCi N
共3兲
where
兺i=1 wi = 1 N
Multiobjective Formulation The multiobjective optimization problem can be defined as the problem of finding a vector of decision variables that satisfies constraints and optimizes a vector function whose elements represent the objective functions. Since these objective functions are usually in conflict with each other, the term “optimize” means finding such a solution that would give the values of all the objective functions acceptable to the decision maker. Vector of decision variables 关Xⴱ兴 is optimal if there is no feasible vector of decision variables 关X兴 which would improve some objective without causing a simultaneous degrading in at least one other objective 关see, e.g., Tang et al. 共2005兲兴. Formally, for a minimization problem, 关Xⴱ兴 苸 ⍀ is Pareto optimal if for each 关X兴 苸 ⍀ and I = 兵1 , 2 , . . . , k其, either
共4兲
in the case of two criteria N = 2 and C = wC1 + 共1 − w兲C2, note that the criteria values have to be normalized to prevent the domination of objectives with high values over other objectives and, for this, additional study is needed to identify the possible minimum and maximum values of all objective functions. Another way of combining the objectives is by measuring the distance to the ideal point where all criteria take the minimum value. If such a point is at the origin of the criteria space 共corresponding to no contamination and no operational costs兲 then the composite objective is 共Fig. 2兲 C=
冑兺
N i=1
C2i
共5兲
For the problem considered N = 2 and f i共关X兴兲 ⱖ f i共关Xⴱ兴兲,
∀i苸I
共1兲
or there is at least one i 苸 I so that ⴱ
f i共关X 兴兲 ⬍ f i共关X兴兲
共2兲
where I = set of integers that range from one to the number of total objectives; 关X兴 and 关Xⴱ兴 = vectors of decision variables; ⍀
C = 冑C21 + C22
共6兲
where C1 relates to damage due to contamination and C2 relates to the cost of the oi required to set the network to a desirable condition. This objective function ensures damage and interventions in the network 共costs兲 receive the same weight. In the case studies below we used Eq. 共6兲.
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COPA MODULE
Calculate number of operational interventions, oi
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Optimal solution found? no
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Fig. 3. Interaction COPA optimization tool 共GLOBE/NSGAX兲
Tools Used and Experimental Setup Due to the fact that the objective function values are calculated by running a model, it is not possible to apply the gradient-based methods of optimization. In such cases the methods based on randomized search are typically applied. The advantage is also that such methods do not typically assume existence of a single extremum 共minimum兲 so they are referred to as global 共multiextremum兲 optimization methods. One of the ideas widely used in randomized search is the idea of evolutionary optimization 关see, e.g., Deb 共2001兲兴. For single-objective approach the global optimization methods implemented in GLOBE software 共Solomatine 1999兲 are used, and the optimization process is organized in a fashion similar to the one used by Abebe and Solomatine 共1998兲. For multiobjective approach the NSGA-II method 共Deb et al. 2002兲 implemented in the NSGAX software 共Barreto et al. 2006兲 is used. The function to be optimized is encapsulated in the executable program called changing operation in pollutant affectation 共COPA兲, which is presented below. GLOBE. GLOBE 共Solomatine 1999兲 is an optimization tool that can find the minimum of a function dependent on multiple variables, the value of which is given by an external program or a dynamic link library. It is possible to impose box constraints 共bounds兲 on the variables’ values. No special properties of the function are assumed. There are seven 共with variations—nine兲 algorithms of randomized search implemented so that the user can tune to his/her problem and that can be run in a batch for the same function. The algorithm used for this paper was GA. COPA module and interaction with GLOBE. The COPA module was developed as a console application in Borland Delphi and it runs the EPANET hydraulic and quality engine solver for distribution networks 共Rossman 2000兲. Note that to calculate objective functions once, COPA may need to run EPANET in extended period simulation mode and with short time step and long simulation time this could be a computationally intensive task. Given a new operational network state it calculates the objective functions C1 and C2 共and aggregates them into C兲. These outputs are stored
in a text file which can then be read by optimization tool 共GLOBE or NSGAX兲. The communication between the optimization tool, COPA and EPANET 共Fig. 3兲 employ the EPANET programmer toolkit. The role of an optimization tool is to generate an input vector 共characterizing the operational intervention and, hence, the new network status兲 and supply it to COPA module. Dimension of this vector is equal to the number of elements 共valves, pumps, or hydrants兲 to be operated, and its values are binary numbers 共0s and 1s兲 indicating the operational status of each element: 1 indicates that the corresponding element is open 共switched on兲 and 0—closed 共switched off兲. The following two sections cover the case studies, along with the obtained results and discussions.
Case Study 1 A simple WDN is considered 共Fig. 1兲. It consists of a system of 41 pipes with the same diameter, length, and Manning roughness 共0.20 m, 1,000 m, and 0.01, respectively兲, 25 junctions with zero elevation and 0.5 L/s of base demand which are affected by a typical consumption pattern. The reservoir provides the system with a constant head of 50 m. Valves are located in the pipes P2, P6, P10, P14, and P18 共all initially opened兲 while the hydrants, originally closed, are located in the nodes J9 and J13. The system is polluted by a conservative contaminant that is injected under three possible scenarios Sc1 共one node—J9兲; Sc2 共two nodes close to each other—J9 and J10兲, and Sc3 共two nodes far from each other—J9 and J15兲. In each scenario it is assumed that the location of the contamination sources become known after applying some available method 关see, e.g., Preis and Ostfeld 共2006兲兴, from which the location of the pollutant injection, the time ti when injection took place, and the concentration of c are known. The duration of the injection di, however, is an additional variable that requires to be assumed. The initial status for all valves is considered to be 1
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Single-objective optimization C→ min Solutions for t i=02:00, d i=3h, t r=4h
Number of polluted nodes,
npn
12 P2: 1 P6: 0 P10: 1 P14: 1 P18: 0 J17: 1 J13: 0 C=10.44
10 8 6 4
P2: 1 P6: 0 P10: 0 P14: 1 P18: 1 J17: 1 J13: 0 C=7.62
Sc 1: J9 polluted Sc 2: J9, J10 polluted
2
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1
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Fig. 4. Single-objective optimization solutions for Sc1, Sc2, and Sc3 of Case Study 1, presented in the npn-oi solution space
共opened兲 and for all hydrants to be equal to 0 共closed兲. A node is considered polluted when its concentration exceeds the threshold value ct. A conservative pollutant is also assumed. In EPANET flow through an open hydrant is simulated as an emitter, with pressure-driven demand Q given by Q = K冑P, where K is the emitter coefficient and P is the pressure drop across the emitter. In order to simulate the opening of a hydrant a pattern demand was introduced so that it starts flowing tr hours after the injection takes place. This time of reaction includes the time required for detection, simulation, personnel transportation, and operation of the accessories in question. Additionally, a minimal residual Pmin pressure in the network has been set as a constraint in the optimization problem. Certainly, there might be network configurations in which the pressure in one or more nodes dropped so low that back-siphonage effects with the associated water quality problems may be generated. For simplicity, parameters tr and Pmin are user defined. For all three scenarios the following parameters values were used: ti = 02: 00; c = 100 mg/ L; ct = 5 mg/ L; di = 3 h; tr = 4 h; K = 10 l / s / m0.5; and Pmin = 1 m. It must be noted that the considered problem is quite simple in terms of optimization: the number of evaluations required to cover the full solution space is only 27 = 128. Therefore, the GA was set to run with the relatively small population 共20兲 and the number of generations 共5兲 resulting in 100 evaluations 共COPA runs兲 which constitutes 78% of the solution space. Single-Objective Optimization Eq. 共6兲 is used to calculate the 共single兲 objective characterizing the quality of the solution from the two objectives C1 and C2. In this case there are seven variables, which can be either 0 or 1, and the network has the initial status 1111100 for the elements P2, P6, P10, P14, P18, J13, and J17. This means that initially all valves are opened and the two hydrants are closed. The solutions obtained for each scenario are shown in Fig. 4, where the operational status of the network for each scenario is shown together with their values for C. It must be added that for this case no normalization of C1 and C2 was carried out because, for this simple case, both npn and oi are numerically similar. This situation is different for bigger networks, as discussed in the next
section. 共It is worth mentioning that for the considered simple network the optimal solution found by GA coincides with the one obtained by exhaustive search.兲 In spite of the apparent simplicity of the problem for the first scenario, the solution found is not trivial. Indeed, for scenarios Sc1 and Sc2 an intuitive solution would be, probably, to close P6, P10, P14, and P18—in order to isolate the contaminated area— and to open J17 to flush the contaminant out. However, this solution requires five oi and the npn is nine, yielding a function value of C = 10.29, which, obviously, is far from being an optimal solution. This proves the difficulty of choosing the best configuration even in a simple network, with enough time available for trial-and-error analysis. Note that the mentioned intuitive solution can be obtained in the optimization process if a very low concentration value is considered as threshold. In this case, one portion of the network will be highly contaminated while the rest of it will be completely clean. This shows that the suggested methodology is flexible to be used under different scenarios. Solutions obtained for the first two scenarios are similar: pipes P6 and P10 are selected for closure and J17 is the hydrant to be opened. The only difference between them is that for Sc2 pipe P14 is also closed. The value of C is a bit higher in the second scenario due to the fact that two nodes are contaminating the network. The impact of the solutions for each scenario can be observed by comparing the resultant concentrations in the most affected 共closest to the source兲 nodes 共J8, J10, J11, and J12兲 with the concentrations when no oi are carried out. For Sc1, the status of the system when no oi are made is presented in Fig. 5共a兲. J10 and J12 are suffering the most as they are immediately after the pollution of 100 mg/L during 3 h occurring in J9. Fig. 5共b兲 shows the effects of closing P6, P10, and opening J17: even though node J10 has the same performance 共due to the reaction time of 4 h兲, in all mentioned nodes the pollution is under the threshold of 5 mg/L after 08:00. On the other hand, people demanding water from J12 are exposed only 1 h to a higher concentration, whereas in the original situation the exposure was lasting from 8:00 to 14:00 h. Figs. 6 and 7 show the same effects 共reduction of exposure time and concentrations兲 when analyzing the solutions obtained for scenarios Sc2 and Sc3, correspondingly. Multiobjective Optimization As mentioned before, the multiobjective problem was defined as minimization of the damage to public health C1, 关in this paper measured as npn, as explained in “Criteria 共Objectives兲”兴 and the associated operational costs C2 共measured as the number of oi兲 in the network as separate objectives. The algorithm was applied for the problem posed, using the same evolutionary parameters as in the case of the singleobjective optimization. In Fig. 8 Pareto-optimal solutions for each scenario are presented. The dashed ovals indicate that for a particular number of oi the same solution was found. It is interesting that for the three scenarios, one of the possible solutions is to make no interventions at all, pipe P6 is always included 共as in the single-objective optimization approach兲 and that opening the hydrant J13 is not considered in any of these solutions. Additionally it can be noted that if only one intervention is considered 共oi= 1兲, this action, which is to close P6 in all three scenarios, does not reduce significantly the npn value: it is reduced only by 1 if compared to the “do-nothing” option. This action isolates the pollutant and facilitates its dilution but since no flushing is involved, the pollutant remains in the network affecting almost the same number of nodes. Consider now the solution
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(a)
(a)
Pollutant concentration at selected nodes Solution for Sc2: P6, P10 and P14 closed; J17 open 4 interventions, 7 affected nodes 100 90
Node J8
Concentration (mg/L)
80 70
Node J9 (polluted)
60
Node J10 (polluted)
50
Node J11
40
Node J12
30 20 10
Threshold = 5mg/L
0 0
2
4
6
8
10
12
14
16
18
20
22
24
Time (h)
(b)
(b)
Fig. 5. Effect of the solution for Sc1, Case Study 1, on pollutant concentrations at selected nodes: 共a兲 do-nothing alternative; 共b兲 solution of Sc1, Fig. 4
Fig. 6. Effect of the solution for Sc2, Case Study 1, on pollutant concentrations at selected nodes: 共a兲 do-nothing alternative; 共b兲 solution of Sc2, Fig. 4
with oi= 2 for Sc1 and Sc2: it includes flushing through hydrant J17—an action quite consistent with an engineering judgment. These two actions already lead to a considerable reduction of npn. For the case of oi= 3, closing pipe P14 is now added to the set of previously mentioned interventions and the value of npn is halved. Note that for oi= 3 the same solution was obtained with the single-objective optimization approach for Sc1 when two objectives were aggregated 共Fig. 4兲 and therefore the analysis of pollutant concentrations in time presented in Fig. 5 is valid for this case as well. This analysis demonstrates the usefulness of having several solutions to be considered by a decision maker in the process of selection of the best one. The results obtained with both single and multiobjective optimization demonstrate that the use of the npn across all simulation time as objective function helps in reducing both exposure time and pollutant concentrations, as can be noted from Figs. 5–7. These variables affect directly the damage in public health, which, however, is not quantified explicitly in this paper.
Sector 11 of the Villavicencio supply network 共Aquadatos 2000兲 and the best options for operating the system have to be found. In EPANET model of the Sector 11 there are 247 junctions 共from which 30 are considered hydrants initially closed兲, 367 pipes 共from which 60 are considered to contain valves, all initially opened兲, and two pump stations, initially switched off 共Fig. 9兲. This means that 92 elements are possible to be operated and therefore this is an optimization problem with 92 variables. As in the first case study, a conservative contaminant is injected under three possible scenarios: Sc1 共one node—J2119兲; Sc2 共two nodes close to each other—J2119 and J2120兲, and Sc3 共two nodes far from each other—J2119 and J2164兲. For all three scenarios the following parameters values were used: ti = 02: 00; c = 230,000 mg/ L; ct = 0.3 mg/ L; di = 4 h; tr = 5 h; K = 10 l / s / m0.5; and Pmin = 1 m. With 92 binary variables, this problem needs a serious computational efforts. The total number of possible solutions 共search space兲 is 292 ⬇ 4.95ⴱ1027. With such an enormous search space no optimization algorithm will guarantee the convergence to a global optimum so we have adopted a two-phase procedure. First, after several experiments aimed at optimizing the convergence and accuracy of the algorithms, we have adopted the GA and NSGA-II parameters, according to Table 1. Then we ran GA and NSGA-II and found a number of feasible 共hopefully close-to-optimal兲 so-
Case Study 2—Villavicencio, Colombia A case study in Villavicencio, Colombia is considered. A source of pollution is supposed to be identified in a node of the hydraulic
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Table 1. GA Parameters Used for Single- and Multiobjective Optimization Single-objective GA Multiobjective NSGA-II Parameter
CS 1
CS 2
CS 1
CS 2
Selector FR FR FR FR Population size 20 100 20 100 Number of generations 5 50 5 50 Crossover probability 0.9 0.9 0.9 0.9 Mutation probability 0.1 0.1 0.1 0.1 Elitism BCK BCK BCK BCK Note: CS⫽case study; FR⫽fitness rank; and PCK⫽best chromosome kept.
5,000 COPA model runs and 89 min of running time on a 2.4GHz PC兲. The identified solutions are reported below. We realize that the choice of optimization algorithms’ parameters may need further attention 共we just used the order of values generally recommended in the literature兲 but due to the computational complexity of this exercise we leave this for further research.
(a) Pollutant concentration at selected nodes Solution for Sc3: P6 and P18 closed; J17 open 3 interventions, 9 affected nodes 100
Concentration (mg/L)
90 80
Node J8
70
Node J9
60
Node J10
50
Single-Objective Optimization
Node J11
40
Node J12
30 20 10
Threshold = 5mg/L
0 0
2
4
6
8
10
12
14
16
18
20
22
24
Time (h)
(b)
Fig. 7. Effect of the solution for Sc3, Case Study 1, on pollutant concentrations at selected nodes: 共a兲 do-nothing alternative; 共b兲 solution of Sc3, Fig. 4
Fig. 10 presents the solutions for the three scenarios. As in the first case study, Eq. 共6兲 is used to calculate the single-objective C from C1 and C2, again with C1 = oi and C2 = npn. In order to give equal weights to these criteria, both have been normalized during the optimization process by dividing them by the maximum possible values for each one. Note that for comparison purposes in Fig. 10 the nonnormalized criteria values are used. For this case study the solutions that require more than 15 oi have been neglected. There are two reasons for that: first, it represents a version of “operational capacity constraint” and, second, this allows for saving a considerable amount of computational time. It is possible to see that solutions for scenarios Sc1 and Sc2 are the same because the pollution spreading is similar in both cases.
lutions. We analyzed the sets of oi present in these solutions and figured it is worth considering further only 18 elements to be operated. With these 18 binary variables 共and much smaller search space with only 218 ⬇ 2.62ⴱ105 possible solutions兲 we ran GA and NSGA-II again with the same GA parameters 共requiring
J2119
Pump 1
Pollution source P1342
P1304 P1463
for scenarios
J1199
Sc1, Sc2, Sc3
J2118 P1461
J2120 Pollution source for scenario Sc2
P1408
Multi-objective optimization Solutions for t i=02:00, d i=3h, t r=4h
P2814
Number of polluted nodes , npn
18 16
Sc 1: J9 polluted
14
Sc 2: J9, J10 polluted Sc 3: J9, J15 polluted
12 P2: P6: P10: P14: P18: J17: J13:
10 8 6 4 2
1 0 1 1 1 0 0
P2: P6: P10: P14: P18: J17: J13:
P2: P6: P10: P14: P18: J17: J13:
1 0 1 1 1 1 0
1 0 0 1 1 1 0
J2164 Pollution source for scenario Sc3
Pump 2
P1439
0 0
1
2
3
4
Number of operational interventions, oi Pump station
Fig. 8. Pareto-optimal solutions of the multiobjective optimization for Sc1, Sc2, and Sc3, Case Study 1
Valve
Pollution source
Hydrant
Reservoir
Fig. 9. WDN of Sector 11, Villavicencio
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Single-objective optimization Solutions for t i=02:00, d i=3h, t r=4h P1304=0 P1439=0 J2118=1
12 10
P1408=0 P1439=0 P1500=0 P1428=0 J2118=1
C=0.16
8
C=0.27
6 Sc 1: J2119 polluted
4
Sc 2: J2119, J2120 polluted 2
Sc 3: J2119, J2164 polluted
0 0
1
2
3
4
5
Number of operational interventions,
Multi-objective optimization Solutions for t i=02:00, d i=3h, t r=4h
90 Number of polluted nodes , npn
Number of polluted nodes
npn
14
80
Sc 1: J2119 polluted
70
Sc 2: J2119, J2120 polluted Sc 3: J2119, J2164 polluted
60 50
P1408=0 P1439=0 P1500=0 P1428=0 J2118=1
40 30 P1304=0 P1439=0 J2118=1
20 10 0
6
0
1
2
oi
3
4
5
6
Number of operational interventions, oi
Fig. 10. Multiobjective optimization solutions for Sc1, Sc2, and Sc3 of Case Study 2, presented in the npn-oi solution space
Fig. 11. Pareto-optimal solutions of the multiobjective optimization for Sc1, Sc2, and Sc3, Case Study 2
The solution that gives the minimum value of C requires three oi and affects eight nodes. This improves the situation significantly if compared to the do-nothing alternative, as a npn reduction of 88% is obtained 共see Table 2兲. Moreover, the results for Sc3 also shows a good performance: with just five interventions the npn goes down to 13 共reduction of 84%兲. A number of interesting observations are worth mentioning 共see Fig. 9 and Table 2兲. First, the closure of pipe P1439 and the opening of J2118 共Fig. 9兲 are the unique oi that are common to the solution of all three scenarios. Second, no pump operation is considered in these solutions. Third, in order to avoid the pollutant propagation to the southern part of the network, P1408 becomes an important element in the solution for Sc3. For this scenario, the closure of P1500 is important: if it is not operated together with the other four elements, the npn value is increased up to 90, which is worse than the do-nothing alternative. It can be said that without the optimization tools it would be very difficult to find that the closure of P1500 共not a trivial choice!兲 leads to a considerable reduction in pollutant spread.
J2118, in order to isolate the pollution and flush it 共Fig. 9兲. This configuration yields npn= 83 with oi= 6, which, as our further analysis shows, is not optimal. Even more, this solution rises the npn value by 16% with respect to the do-nothing alternative, which is patently unacceptable. A detailed analysis of this solution shows that the closure of P1342 and P1461 indeed stops the pollution spread toward the east but, at the same time, these closures increment the velocities of the pipe nearby the pollution source with the consequent pollutant propagation to the west, where there are not enough valves available to completely stop it. This demonstrates how dangerous it is to follow intuitive solutions—which are also not easy to find in large networks. Referring to the results of optimization presented on Fig. 11 and Table 2, a number of interesting outcomes from our experiments that can be mentioned are: • If solutions with oi= 1 are considered, the following can be observed. In Sc1 and Sc2, the npn reduces by 40% just by closing pipe P1304 共Fig. 9兲, which is a nontrivial solution since it is far away from where pollutant has been injected. Although the reason for this is that pollutant is not spread because flow velocities in the area drop considerably, it is almost impossible to come up with this solution just by playing manually with the model. Similar situation occurs for Sc3, in which npn drops by 40% just by switching on the pump 1. • For Sc1 a solution with an npn= 1 was found, with oi= 4, in which closing P2814, P1439, and P1500, together with open-
Multiobjective Optimization In this case the objectives C1 = oi and C2 = npn are to be minimized as well. Before the results of optimization are analyzed, a possible “intuitive” solution for Sc1 is presented: closed pipes P1342, P1461, P2814, and P1463 and open hydrants J1199 and
Table 2. Solutions for Sc1, Sc2, and Sc3 for Single- and Multiobjective Optimization of Case Study 2 Multiobjective optimization 共solution for oi= 1兲
Single-objective optimization Element
Initial status npnSc1 = 69npnSc2 = 69npnSc3 = 84
PMP-1 J2118 P1304 P1408 P1439 P1500 P2814 Note: 0⫽closed/off
0 0 1 1 1 1 1 and 1⫽opened/on.
Sc1 C = 0.16npn= 8oi= 3
Sc2 C = 0.16npn= 8oi= 3
Sc3 C = 0.27npn= 13oi= 5
Sc1 npn= 42
Sc2 npn= 42
Sc3 npn= 56 1
1 0 0
1 0 0
1 0
0
0 0 0 0
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ing J2118 are the interventions to make. In this case, only the node J2119 共the pollution source兲 is contaminated. The minimum value for Sc2 is also obtained with the previously mentioned solution, yielding npn= 4, which represents a reduction of 94% with respect to the do-nothing alternative. • Due to the complexity of the pollution scenario Sc3 the number of oi is higher than is needed for Sc1 and Sc2: to reduce npn significantly 共from 84 to 13兲 five interventions are needed. The elements to operate in this case are J2118, P1439, and P2814 共selected in previous solutions兲, having in addition to modify P1408 and P1500. This yields npn= 13; note that this solution was also found as a result of single-objective optimization. • The solutions found for Sc1 and Sc2 with oi= 3 are the same that were obtained with the single-objective approach 共see Fig. 10兲 and therefore the analysis presented in Single-Objective Optimization applies as well. • It is very interesting that in all the solutions that yield the least value of npn, only one hydrant, J2118, was included, even for the complex case of Sc3. Now, possibly exhausted, the reader will be happy to move to the next section.
Conclusions and Recommendations From this study the following conclusions can be drawn: • With respect to earlier research reviewed in Background, the present work introduced new methods enhancing contamination management of WDNs. • Three basic factors are present in all the solutions found in both case studies: they all tend to isolate the contaminant, to flush it out, and/or to dilute it. • The results revealed that there exist combinations of oi that could be even more dangerous than the do-nothing alternative, in terms of network damage. For example, in some cases the use of pump stations for accelerating the flushing process is not always a good idea. • Although isolation, flush, and dilution are considered when trying to come up with an intuitive solution that solves the problem properly, this is a very difficult task. For the first case study the presented intuitive solution was relatively easy to obtain due to its simplicity but still that solution was not among the optimal solutions. For the second complex case study, a good intuitive solution 共such that at least reduces npn if compared to the do-nothing solution兲 could not be even suggested. • Results demonstrate that using the npn as an objective function characterizing the impact on public health is a simple and robust way to deal with the extension of the contamination, the time of exposure of the nodes to the pollutant, and its concentration values. Additionally, this concept provides a straightforward picture of the state of the network for the decision maker. • A multicriteria approach 共as opposed to single-criterion one兲 makes it possible to generate several solutions from which the most appropriate one can be chosen based on additional analysis; such involvement of a decision maker may improve the acceptance of the system by managers and practitioners. • Model-based optimization for real-life problems requires considerable computational effort and the adopted two-phase procedure helped to arrive to optimal 共or close-to-optimal兲
•
•
•
•
•
•
•
•
•
solutions even with PC-level computing power and quite standard optimization algorithms. The framework 共COPA—EPANET—optimization tools GLOBE and NSGAX兲 can be used for other case studies and it is flexible enough to incorporate various pollution scenarios. The following recommendations can be suggested: Since the damage function used in this paper was simplistic 共albeit robust兲, more sophisticated definitions would be worth formulating based on the following possible approaches 共not mutually exclusive兲: To separate the contamination impact into more objective functions, in particular, extension of contamination, pollutant concentration, and time of exposure; To formulate a suitable definition of contamination risk to be used as an objective function, in order to consider all the possible factors that can affect the public health. The objective function associated with the operational costs 共in this paper being simply the number of oi兲 could be modified to consider the travel time of the operators from the water supply station to the place where the interventions are required, the time needed by the operators for closing a particular type of valve, as well as the optimal path that must be taken by the operators to operate the network. For this purpose, the elements with the higher impact in the network should be the first to intervene. Some of the assumptions made in this paper could be relaxed or dropped and it is recommended to do so. For example: 共1兲 allow for scenarios with pollution occurring at different times; 共2兲 allow for situation with unknown pollution sources 关and develop methods including the approach suggested by Preis and Ostfeld 共2006兲兴; and 共3兲 allow for oi to be phased in time. It would be useful to develop the rules for selecting a limited set of elements to be changed since this will decrease the size of the optimization problem. For large networks, this methodology can be used to generate in advance the optimal interventions for various possible contamination scenarios. In this way they would be available immediately when a contamination event is detected. In this research, only the evolutionary optimization algorithms 共GA and NSGA-II兲 were employed. However, for large networks their running times may be prohibitively long. This prompts for the development of other more efficient random search methods requiring less model runs.
Notation The following symbols are used in this paper: C ⫽ composite objective; Ci ⫽ single-objective i; c ⫽ concentration of the pollutant at the source 共mg/L兲; ct ⫽ concentration of the pollutant that is considered harmful 共mg/L兲; di ⫽ duration of the pollutant injection 共h兲; K ⫽ emitter coefficient for pressure-dependent flow in hydrants 共l / s / m0.5兲; N ⫽ number of objectives; Pmin ⫽ minimal residual pressure allowed in the network 共m兲; ti ⫽ time at which pollution has been injected 共hh:mm兲;
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tr ⫽ time of reaction for operational response since ti 共h兲; wi ⫽ weight value for the objective i; 关X兴 ⫽ any vector of decision variables; and 关Xⴱ兴 ⫽ optimal vector of decision variables.
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