Application of Several Meta-Heuristic Techniques to the Optimization ...

Water Resour Manage (2008) 22:1367–1379 DOI 10.1007/s11269-007-9230-8

Application of Several Meta-Heuristic Techniques to the Optimization of Real Looped Water Distribution Networks J. Reca · J. Martínez · C. Gil · R. Baños

Received: 29 September 2006 / Accepted: 12 November 2007 / Published online: 13 December 2007 © Springer Science + Business Media B.V. 2007

Abstract The optimization of looped water distribution systems is a complex problem as the pipe flows are unknown variables. Although many researchers have reported algorithms for minimizing the network cost applying a large variety of techniques, such as linear programming, non-linear programming, global optimization methods and meta-heuristic approaches, a totally satisfactory and efficient method is not available as yet. Many works have assessed the performance of these techniques using small or medium-sized benchmark networks proposed in the literature, but few of them have tested these methods with large-scale real networks. The aim of this paper is to evaluate the performance of several meta-heuristic techniques: genetic algorithms, simulated annealing, tabu search, and iterated local search. These techniques were first validated and compared by applying them to a medium-sized benchmark network previously reported in the literature. They were then applied to a large irrigation water distribution network that has been proposed in a previous work to assess their performance in a practical application. All the methods tested performed adequately well, compared with the results found in previous works. Genetic algorithm was more efficient when dealing with a medium-sized network, but other methods outperformed it when dealing with a real complex one. Keywords Water distribution system · Pipe networks · Optimization · Heuristics

J. Reca (B) · J. Martínez Department of Rural Engineering, University of Almería, Almería, Spain e-mail: [email protected] C. Gil · R. Baños Department of Computer Architecture and Electronics, University of Almería, Almería, Spain C. Gil e-mail: [email protected]

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Notation The following symbols are used in this paper: C c cp F h hr k L n nd ne ng nl Pcros Pmut Pr PS Tcr Tini Tstop z α,a,b

Hazen–Williams roughness coefficient Pipe cost per unit length Penalty multiplier Fitness function Simulated nodal pressure head Required nodal pressure head Darcy–Weisbach roughness coefficient Pipe length Number of nodes Number of possible diameter sizes Number of evaluations Number of generations Number of links Crossover probability Mutation probability Perturbation rate Population size Cooling rate Initial temperature Stop condition temperature Memory length of the tabu list Hazen–Williams parameters

Subscripts: i j

Pipe diameter index Network node index

1 Introduction Looped water pipe distribution networks are traditionally used in many applications, such as urban and industrial water supply. Nowadays, they are also being introduced in some kinds of irrigation water distribution systems, where reliability can offset the increase in the cost of the network, such as in greenhouse horticultural systems. The optimization of looped water distribution systems is a complex problem, as the pipe flows are unknown variables. Mathematically, it is a non-linear, constrained, and multi-modal problem included in the class of complex combinatorial problems known as NP-hard (Gupta et al. 1993), which implies that it is not feasible to obtain the optimal solution in a polynomial time. As result of the extensive analysis performed by many authors in last decades, a large number of methods have been applied. First attempts to solve this problem were based on linear programming techniques. The classical work by Alperovits and Shamir (1977) proposed the so-called

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successive linear programming gradient method. The linear programming based gradient procedure has since been adapted and improved (Quindry et al. 1981; Fujiwara and Khang 1990). Several non-linear optimization models have been developed. Lansey and Mays (1989) proposed a non-linear programming technique (the generalized reduced gradient method) combined to a network simulator for solving the problem. A similar approach was proposed by Varma et al. (1997), but using another non-linear method (successive quadratic programming). Eiger et al. (1994) combined a global Branch and Bound algorithm and duality theory. Other global optimization methods have been successfully applied to this problem (Abebe and Solomatine 1998; Sherali et al. 1998). Heuristic methods (Glover et al. 1993) can be defined as simple procedures that provide satisfactory, but not necessarily optimal solutions to complex problems in a quick and easy way. One of the most promising and commonly used methods is Genetic Algorithm (Goldberg 1989) which is based on the rules of evolution and natural selection. Recently, many authors have proposed genetic algorithms for water distribution network design (Montesinos et al. 1999; Vairavamoorthy and Ali 2000; Reca and Martínez 2006). Other meta-heuristic methods, simulated annealing (Cunha and Sousa 1999), harmony search method (Geem et al. 2001) and the ant colony optimization method (Maier et al. 2003), have also been recently applied to this problem. Reca and Martínez (2006) developed the GENOME model, a Genetic Algorithm based model. They analyzed the performance of this algorithm by applying it to several benchmark networks and to a complex real-sized irrigation water distribution network. They concluded that the model performed well, especially when it was applied to small or medium-sized benchmark networks. Nevertheless, the performance of these methods should be improved when solving large-scale networks for practical application. The aim of this work is to evaluate the performance of several additional metaheuristics for the optimal design of large-scale looped water distribution networks. The methods implemented include genetic algorithms, simulated annealing, tabu search, and iterated local search. These methods will first be tested and compared by applying them to a medium-sized benchmark network. They will then be applied to a large irrigation water distribution network (Reca and Martínez 2006). In order to compare these methodologies to previous results, a least-cost optimization problem with pipe diameters as decision variables has been stated, while pipe layout, connectivity and demands are imposed.

2 Methodology 2.1 Problem Formulation The mathematical formulation of the optimal design of a looped water distribution network has been set up in many previous works (Savic and Walters 1997; Montesinos et al. 1999). The objective is to minimize the network investment cost. Further, the problem is constrained by the physical laws of mass and energy conservation. Also, minimum pressure requirements for users, minimum and maximum

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flow velocities and pipe size restrictions are imposed. For this reason, the fitness of a certain solution (F) is calculated as the sum of the cost of the pipes making up the network plus a penalty function applied to take into account nodal pressure head deficits (see Eq. 1). F=

nd 

ci Li + c p

i=1

n 

   max hr j − h j , 0

(1)

j=1

where: nd is the number of possible diameter sizes, ci the cost of the pipe of diameter i per unit length, Li the total length of pipe i in the network, c p the penalty multiplier, n the number of nodes, hr j the required pressure head in node j, and hj the actual pressure head computed by the hydraulic solver for node j. In these experiments, a very large penalty value (c p =100,000) was used in order to discard solutions with pressures below the requirements. 2.2 Programming Environment A new computer model called MENOME (meta-heuristic pipe network optimization model) has been developed with the aim of optimizing the design of looped water distribution networks. The MENOME model is an extension of the previously developed GENOME model (Reca and Martínez 2006). While the GENOME model was a Genetic Algorithm based optimizer, the MENOME computer program includes several additional meta-heuristic optimizers mentioned above. It also integrates a hydraulic network solver, a graphical user interface and database management module. The MENOME interface and the meta-heuristic algorithms have both been programmed in the Visual-Basic programming language. The model uses the well known, robust and tested network solver EPANET (Version 2.00.07; Rossman 2000). It performs extended period simulation of hydraulic and water quality behavior within a pressurized pipe network. It employs the gradient method proposed by Todini and Pilati (1987) for solving the mass and energy conservation equations. Using the Programmer’s Toolkit, the network solver EPANET is incorporated into the MENOME model. Two input data files are needed to run the model: the network configuration and the pipeline database. Network configuration is described in a standard EPANET file format and the pipeline database is given in a relational database that must include two fields: pipe inner diameter and cost per unit length. 2.3 Meta-Heuristic Approaches The meta-heuristic methods adapted to solve this problem are: genetic algorithm and three local-search methods that use simulated annealing, tabu search, and iterated local search. More details about these algorithms are provided below. 2.3.1 Genetic Algorithm, GA Genetic algorithms (GA; Goldberg 1989) are stochastic search procedures based on the evolutionary mechanisms of natural selection and genetics. GAs mimic the very effective optimization model that has evolved naturally for dealing with large, highly complex systems. GAs have been successfully used for their flexibility and robustness to solve many NP-hard problems (Garey and Johnson 1979) arising in different

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areas of science and engineering, including water distribution systems (Montesinos et al. 1999; Vairavamoorthy and Ali 2000). In a GA each solution is codified into a chromosome-like structure. Each chromosome has an objective function value, called fitness. A set of chromosomes together with their associated fitness is called population. The genetic algorithm starts with a random initial population of individuals and later follows an iterative reproductive process. The population size (PS) and the number of generations (ng ) through which this population is to evolve are both input parameters in the model. After the initial population is created and evaluated, the genetic operators are applied iteratively over successive generations in order to progressively improve the population. The breeding process comprises three simple operators: selection, crossover and mutation. The selection operator chooses parents from the population in such a way that favors the fitter individuals. The parents are recombined to form offspring individuals that inherit the characteristics of their parents using the crossover operator. Later, the mutation operator, consisting of randomly replacing a certain gene, is applied to promote population variability. Crossover and mutation operators are applied using a given probability, Pcros and Pmut , respectively. More details about the implementation of the GA can be found in Reca and Martínez (2006). 2.3.2 Simulated Annealing, SA The basic local-search approach corresponds to the so-called hill-climbing (also hill-descending) algorithms. These algorithms are based on examining a monotone sequence of improving solutions until a local optimum is found. Hill-climbing algorithms always stop at the first local optimum. To avoid this drawback, several alternatives have been proposed in the literature. One of them is simulated annealing (SA; Kirkpatrick et al. 1983), which is a stochastic relaxation technique based on the analogy to the physical process of annealing a metal. When a solid is heated, the particles take random configurations. The temperature is then slowly decreased, allowing particles to reach a state of minimal energy. In contrast with GAs, standard SA uses a single solution during the optimization process. In SA, better neighboring solutions are always accepted, whereas worsening solutions are accepted with a certain probability, which is dependent on a parameter, called temperature. The initial temperature (Tini ) diminishes in the successive iterations based on a factor termed cooling rate (Tcr ). This temperature is included within the Metropolis function (Metropolis et al. 1953), and it acts simultaneously as a control variable for the number of iterations of the algorithm and as a probability factor for a definite solution to be accepted. A decrease in the temperature implies a reduction in the probability of accepting movements which worsen the cost function. The process finishes when the temperature falls bellow a given threshold, Tstop , usually close to zero. Some studies on other optimization problems (Gil et al. 2002) have shown that slow annealing schemes often outperform the quality of the solutions. 2.3.3 Iterated Local Search with Simulated Annealing, ILSSA Iterated local search (ILS; Ramalhino et al. 2002) explores the search for local minima with respect to some given embedded heuristic, henceforth called localsearch. ILS achieves this heuristically as follows: given the current solution, a change or

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perturbation that leads to an intermediate state is applied. Localsearch is then applied to this intermediate solution. If the new solution passes an acceptance test, it becomes the next element of the walk; otherwise, the search will be restarted using the initial solution. This ILS procedure should lead to good biased sampling as long as the perturbations are neither too small nor too large. If they are too small, one will often fall back to the initial situation, and few new solutions will be explored. On the contrary, if the perturbation rates are too large, the new solution will be almost random, there will be no bias in the sampling, and we will recover a random restart type algorithm. The local search method used within ILS is SA. The perturbation applied between iterations consists of randomly modifying a percentage of pipes. This percentage is defined as perturbation rate Pr , which oscillates in the interval [0–1]. A value of perturbation rate close or equal to 1 is equivalent to a multi-start method.

2.3.4 Mixed Simulated Annealing and Tabu Search, MSATS Some studies, e.g. Talbi (2002), have shown that hybridizing two or more methods can outperform the quality of the solutions. When the search space is explored, the use of a limited neighborhood, as in hill-climbing/descending, can cause the appearance of cycles. An interesting way to avoid this drawback is to use tabu search (TS; Glover et al. 1993). TS uses a neighbourhood search procedure to iteratively improve a solution using a tabu list which contains the solutions visited in previous z steps in an attempt to avoid cycles in the search. Therefore, the adaptation of the mixed simulated annealing tabu search (MSATS) procedure (Gil et al. 2002) it is proposed. This meta-heuristic, which combines SA and TS, has been successfully tested in another combinatorial problem. These results led to the conclusion that the use of tabu search in combination with simulated annealing improves the results obtained when only SA or TS are applied separately. At each iteration of MSATS, admissible changes are applied to the current solution allowing transitions that increase the cost function as in simulated annealing. When a change increasing the cost function is accepted, the reverse change is forbidden during several iterations in order to avoid cycles in the search. The restrictions in the admissible movements are implemented by using a short term memory function which determines how long a tabu restriction will be enforced and the admissible changes at each iteration. This method has also been adapted to the design of looped water distribution networks.

2.4 Test Problems The two water distribution networks that have been used to evaluate the performance of the methods described above are: a well-known benchmark network typically used to evaluate this kind of algorithm (Hanoi network) and the irrigation network proposed by Reca and Martínez (2006). The configuration of these networks is described in what follows.

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[13]

11 [11] 10 34 [31]

35 [32]

26

[25]

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[16]

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14 [15]

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9 [9]

[10]

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[30] [18]

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31 24 [29]

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[3]

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[6]

[5] 4

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[21] 22 [22]

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Fig. 1 Layout of the Hanoi network

2.4.1 Hanoi Network The pipeline network for water supply in Hanoi (Vietnam) was proposed by Fujiwara and Khang (1990). This network consists of 32 nodes, 34 pipes and 3 loops (see layout in Fig. 1). The network has no pumping station as it is fed by gravity from a reservoir. The constant water level is 100 m. The minimum pressure-head requirement at all nodes is fixed at 30 m. A set of six available commercial-diameter pipes is used. Under these circumstances, there exist 634 = 2, 8651 1026 possible configurations. In this network the Hazen–Williams equation is also computed using the roughness coefficient C = 130. The values of the other parameters of the Hazen– Williams equation are the defaults of the EPANET 2.0 network analysis software (α = 4.277, a = 1.852, b = 4.5871). 2.4.2 Balerma Network The Balerma irrigation network is an adaptation of the existing irrigation water distribution network in the Sol-Poniente irrigation district, located in Balerma, province of Almería (Spain). This is a multi-source network, containing a total of 443 demand nodes (hydrants), fed by 4 source nodes. It has 454 pipes and 8 loops. The layout of the network is shown in Fig. 2. The Darcy–Weisbach equation has been proposed to calculate head-losses. The pipeline database is composed of ten commercial PVC pipes with nominal diameters ranging from 125 to 630 mm. There are 10454 possible configurations of this system. Absolute roughness coefficient, k, is equal to 0.0025 mm. The minimum pressure limitation is 20 m above ground level

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Fig. 2 Layout of the Balerma network

for each node. A more detailed description of this network can be found in Reca and Martínez (2006). Additional materials describing the network configuration and the pipeline database used in the optimization can be downloaded from the auxiliary materials site of the American Geophysical Union (AGU). 2.5 Parameter Settings To compare the results of different meta-heuristics, the stop criterion in the experiments cannot be fixed to a number of iterations, since population-based methods (GA) would require more runtime than non-population-based ones (SA, MSATS, ILSSA). Given this circumstance, the best way to guarantee the equality of conditions is that all the methods perform the same number of evaluations of the fitness function. That number of evaluations, ne , should depend on the complexity of the network. The search space is a function of the number of links nl and the number of possible pipe diameters nd . The following logarithmic expression has been adopted to establish a ratio between the number of evaluations in two networks with different search space. 





n · ln nd ne = l ne nl · ln nd

(2)

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Table 1 Parameters used in the empirical executions (Hanoi network) Methods

Parameters

Values

GA

PS ng Pcros Pmut Tini Tcr Tstop Tini Tcr Tstop Pr Tini Tcr Tstop z

100 f(ne ) 1.00, 0.95, 0.90, 0.85, 0.80, 0.75, 0.70, 0.65, 0.60, 0.55 0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45 500, 250, 150, 100, 75, 50, 25, 10, 5, 2 f(ne ,Tini ,Tstop ) 0.01 500, 250, 150, 100, 75, 50, 25, 10, 5, 2 f(ne ,Tini ,Tstop ) 0.01 0.25 500, 250, 150, 100, 75, 50, 25, 10, 5, 2 f(ne ,Tini ,Tstop ) 0.01 5

SA

ILSSA

MSATS

The resulting fitness function evaluations are 26,457, and 454,000 for the Hanoi and Balerma networks, respectively. In order to avoid the possible randomness of the search process due to the use of different initial solutions, they have all been obtained taking the largest diameter pipes for all the links in the networks. On the other hand, a parametric analysis has also been performed in order to achieve accurate configurations of each metaheuristic. Each meta-heuristic has been executed ten times, using different parameter configurations for the Hanoi network. In the GA, each one of these ten runs uses different crossover and mutation probabilities. On the other hand, the methods that apply simulated annealing use different annealing schedule. Table 1 summarizes the configurations used in each case. Due to the large computational time, the sensitivity analysis of the Balerma network has not been carried out. For this reason, the best parameters configuration obtained for the Hanoi network was used for the Balerma network.

3 Results and Discussion Table 2 shows the average and minimum cost found for the Hanoi network by each method. It can be seen that the minimum cost is obtained by GA (6173421

Table 2 Results obtained in Hanoi network (after ten different runs)

GA SA ILSSA MSATS

Average cost

Minimum cost

Best configuration

6,575,682 6,483,950 6,510,647 6,538,453

6,173,421 6,333,207 6,308,024 6,352,526

Pcros =0.95; Pmut =0.05 Tini =25 Tini =50 Tini =50

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9.5e+06 9e+06

cost

8.5e+06 8e+06 7.5e+06 7e+06 6.5e+06 6e+06 0

5000

10000

15000 evaluations

20000

25000

30000

Fig. 3 Comparison of GA, SA, and MSATS in the Hanoi network

monetary units) while the other methods provide slightly worse results. Nevertheless, SA provides the lowest average cost. Figure 3 displays the tendency of each algorithm using their best configuration parameters. ILSSA is not showed because it is not a sequential algorithm. This figure shows that not all the methods converge to the same result, although they are close to the best one which was obtained by GA. The minimum cost obtained with the meta-heuristic methods included in MENOME is slightly higher than the one found in previous works. The best solution found in literature using the EPANET 2.0 version was given by Eusuff and Lansey (2003) using the Shuffled Frog Leaping Algorithm (6,073,000 monetary units), only 1.63% lesser than the cost found by MENOME. Nevertheless, as previously explained, this work aims to compare the performance of several meta-heuristic methods with the same number of evaluations, rather than to improve the solutions found in previous works. In fact, a best solution was found in a previous work with our model GENOME (6,081,127 monetary units; Reca and Martínez 2006). The reason for this result is that in this previous work, the population size and the number of generations were greater. A more detailed comparison with results found in previous works can be found in Reca and Martínez (2006). Due to the very large computational time, the optimization of the Balerma Irrigation network was carried out using the best parameters configuration found for the Hanoi network. Table 3 shows the minimum cost obtained by each method. In this case, the minimum cost is obtained by MSATS (3,298,268 monetary units), while the other methods obtain worse results. Figure 4 shows the tendency of each algorithm except ILSSA. As can be seen, MSATS and SA converge faster than GA and provide better solutions.

Meta-heuristic techniques for WDN optimization Table 3 Results obtained in Balerma network (best configurations)

GA SA ILSSA MSATS

1377 Minimum cost

Configuration

3,737,600 3,475,740 4,310,016 3,298,268

Pcros =0.95; Pmut =0.05 Tini =25 Tini =50 Tini =50

It should be noted that this is the first time that most of these meta-heuristic methods have been evaluated using this network. These results can only be compared with the solution found with the GENOME model in a previous work (2,302,423 monetary units). Nevertheless, this solution was found after 20,000 generations and with a population number of 500 individuals, i.e. over 10 million function evaluations, much more than those performed in this work. The worst meta-heuristic in this network was ILSSA. The reason is that ILSSA is iterated ten times, but using a faster annealing scheduling than SA and MSATS, in order to perform the same number of evaluations. When dealing with large networks, the best meta-heuristic method is MSATS, which outperformed the rest. It not only gave better results, but it also converged faster than GA. These results can offer very useful guidelines for solving realistic loop water distribution networks. Future research will be directed at extending the problem to a multi-objective formulation considering not only the cost as the objective to minimize, but also other design safety aspects, such as the control and environmental reliability of the network.

1.4e+07 GA SA MSATS 1.2e+07

cost

1e+07

8e+06

6e+06

4e+06 0

5000

10000

15000

20000

25000

30000

evaluations

Fig. 4 Comparison of GA, SA, and MSATS in the Balerma network

35000

40000

45000

50000

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4 Conclusions This paper analyzes the performance of several heuristic methods for the optimal design of real looped water distribution systems for practical purposes. A new computer model named MENOME has been developed and tested. It integrates different search methods, as genetic algorithms, simulated annealing, tabu search, and iterated local search. In terms of the quality of the solutions, results denote that all the methods here adapted obtain good solutions in the test problems evaluated. In the Hanoi network the best result is obtained by the Genetic Algorithm, and it is close to the best results obtained by the other methods. Finally, in the larger network the best results are obtained by MSATS, while the other methods obtain solutions with higher cost. These results show that although the meta-heuristic methods used performed well in medium-sized networks, when they are applied to large-scale networks, a higher number of evaluations and greater computational time are needed to achieve an accurate result. Acknowledgements This work was supported by the Spanish Ministry for Science under contracts CTM2007-66639/TECNO and TIN2005-00447. The authors appreciate the support of the “Structuring the European Research Area” program, RII3-CT-2003-506079, funded by the European Commission.

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