Multi-Swarm Optimizer Applied in Water Distribution Networks
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Desalination and Water Treatment 161 (2019) 1–13 www.deswater.com September doi: 10.5004/dwt.2019.24146 Multi-swarm optimizer applied in water distribution networks Douglas F. Surcoa,b,*, Diogo H. Macowskia,b, João G. L. Coralc, Flávia A.R. Cardosoa, Thelma P.B. Vecchia, Mauro A.S.S. Ravagnanib aDepartment of Civil Engineering, Federal University of Technology, Via Rosalina Maria dos Santos, 1233 CEP 87301-899 Campo Mourão, Paraná, Brazil, emails: [email protected] (D.F. Surco), [email protected] (D.H. Macowski), [email protected] (F.A.R. Cardoso), [email protected] (T.P.B. Vecchi) bDepartment of Chemical Engineering, State University of Maringá, Av. Colombo, 5790 CEP 87020-900 Maringá, Paraná, Brazil, email: [email protected] cOperational Development, Umuarama Regional Unit, Companhia de Saneamento do Paraná, Sanepar, Av. Presidente Castelo Branco, 3525 CEP 87501-050 Umuarama, Paraná, Brazil, email: [email protected] Received 16 September 2018; Accepted 20 March 2019 abstract In the present work, a new approach is presented for the optimization of multi-modal nonlinear programming problems with constraints or a nondifferentiable objective function. The model is applied in the optimization of water distribution networks (WDN). An algorithm is proposed to solve the problem, based on multi-swarm optimization (MSO) with multiple swarms that work corpo- rately – a master swarm and several slave swarms – named multi-swarm corporative particle-swarm optimizer (MSC-PSO). There are discrete and continuous decision variables and the problem can be treated as a mixed discrete nonlinear programming (MDNLP) one. The combinations of the algo- rithm search parameters are obtained in a simple manner, allowing viable and promising solutions. A benchmark problem from the literature is studied, in which the installation costs of a WDN are to be minimized with a computational time of 50 s. The implementation of the algorithm is proven to be efficient, with reduction in pipe installation cost up to 1.08% when compared with results from the literature. The algorithm is also implemented in a primary network, installed in the town of Esperança Nova, Paraná, Brazil, with reduction of 4.28% in the total cost when compared to the current WDN in operation. The computational time for this case was 69 s. Keywords: Multi-swarm optimization algorithms; Particle-swarm optimization; Water distribution network; Optimization; MDNLP 1. Introduction demand nodes, as well as the minimum and maximum velocities in the pipes, must be considered. The optimization of a water distribution network (WDN) In the 1970s, several researchers [1–3], among others, is a multimodal nonlinear programming problem with studied WDN optimization using deterministic methods a nondifferentiable objective function and with real and such for solving linear programming (LP), nonlinear pro- discrete decision variables. The analytical solution to the gramming (NLP), and dynamic programming (DP) problems optimization problem is highly complex, requiring simul- formulations, with adaptations used in order to work around taneous analysis of the mass conservation equation in constraints or differentiability and continuity problems of the each node and of the energy conservation equation in each objective functions. In the case of the deterministic methods, network loop. Furthermore, the minimum pressures in the the solution is strongly dependent on the initial proposition. * Corresponding author. 1944-3994/1944-3986 © 2019 Desalination Publications. All rights reserved. 2 D.F. Surco et al. / Desalination and Water Treatment 161 (2019) 1–13 In the last four decades, however, heuristic methods have Multi-Swarm Optimization is an extension of the PSO been used due to their efficiency in dealing with the afore- algorithm. Interacting with several swarms, it is adequate for mentioned problems. Nevertheless, the use of hydraulic multimodal problem optimization. Chen et al. [22] presented simulators allows complex mathematical treatments to be a model inspired by mutualism, which is the occurrence avoided. of information exchange between the particles of a certain One of the most successful heuristic methods in the swarm as well as between particles from a swarm and the SI (swarm intelligence) group is PSO (particle swarm best ones from other swarms. The authors used 17 mathe- optimization). First introduced by Kennedy and Eberhart matical benchmark functions to prove the efficiency of the [4], the method is based on patterns found in the individual model. and social behaviors of certain natural species that require In the present work, a new approach is presented for communication between individuals for the survival of the the optimization of nonlinear programming problems of a group. This behavior is observed in several animal species, multi-modal nature, with constraints or a nondifferentiable some examples being schools of fish, flights of birds, and objective function, where variables are mixed (discrete and swarms of bees. Each particle (bird, bee, fish) represents a continuous numbers). The model is then used to optimize potential solution to the system, mathematically represented WDN at small and large scales. An algorithm based on MSO by a vector, xi, whose N components are the decision vari- is proposed in order to solve the problem. A new model is ables of the problem. At a certain instant, when the particle proposed, named Multi-Swarm Corporative Particle Swarm reaches its best position (best evaluation of the objective func- Optimizer (MSC-PSO). An initialization strategy for the par- tion), its coordinates are named Personal Best (Pbest) and are ticles is also employed for better distribution throughout the represented by vector pi. The best position reached by any search space, increasing the particles search of exploration. particle in the group is named Global Best (Gbest) and is Results obtained for a case study from the literature shown represented by vector g. The future location of particle i in the applicability of the proposed approach in finding better the search space will be directed by these vectors pi and g. optima when compared to previous evolutive algorithms. The particles of the PSO algorithm move towards a global or local optimum. When the leader particle (Gbest) stops 2. PSO algorithm at a local optimum, the group converges prematurely. This causes the search for the global optimum solution to be chal- The coordinates of particle i are represented by the lenging, especially in highly nonlinear multimodal problems. components of vector xi = (xi,1,…,xi,j,…,xi,N) in N-dimensional According to Xu et al. [5], PSO convergence is quicker space. When component xi,j is a real number (continuous), L U than other evolutionary algorithms [6], such as genetic it is limited by xj (lower limit) and xj (upper limit), that is, L U algorithms (GA), developed by Holland [7] inspired by the xi,j ∈ [xj ,xj ]. When xi,j is a discrete number, it belongs to SET theory of evolution; ant colony optimizations (ACO), devel- the set xj = {xj1, xj2,…,xjND}, which has ND elements. oped by Dorigo [8], based on ant foraging; simulated anneal- According to Kennedy and Eberhart [4], the new position ing (SA), developed by Kirkpatrick et al. [9], based on metal of particle i at (t + 1) is represented by Eq. (1): annealing; shuffled complex evolution (SCE), developed by Duan et al. [10]; harmony search (HS), the algorithm that xtij,,()+ 11= xtij()++vtij, () (1) mimics the musical harmony phenomenon, created by Geem et al. [11] inspired by the improvisation process of musicians; where vi,j(t + 1) is the velocity of component j of particle i, memetic algorithms (MA), developed by Dawkins, [12], given by Eq. (2): inspired in GA with na individual learning procedure able to refine local search and the shuffled frog leaping algorithm vt()+ 1 = vt()+−cr px()tc+−rg xt() (2) (SFLA), developed by Eusuff and Lansey [13] was inspired ij,,ij 11()ij,,ij 22()ji,j in MA and PSO in global optima search. where c and c are, respectively, the cognitive and social Being one of the most used optimization heuristic methods, 1 2 acceleration coefficients and r and r are uniformly particle swarm optimization (PSO) was used Trigueros et al. 1 2 distributed random numbers in the interval (0,1). The use of [14] and Ravagnani et al. [15] for the optimization of reuse Eq. (2) requires prior knowledge of the values of variables water networks. Several studies presented solutions to the p (component j of vector Pbest) and g (component j of WDN optimization problem using PSO, such as [16] and [17]. i,j j vector Gbest). The velocity of the particles is limited, that is, Surco et al. [18] presented a modified PSO algorithm, poten- v ∈ [v ,v ], where v and v are the predetermined lower tializing the particles for better exploration of the search i,j L U L U and upper limits. space and described the influence of the parameters in the Kennedy and Eberhart [23] implemented the PSO algorithm optimization process. for discrete binary variables by introducing Eq. (3): Considering other evolutive algorithms applied on WDN optimization, Zheng et al. [19] used ACO with parameters < adaptive strategies in searching better results. Reca et al. [20] 1 if rand() Sv()ij, developed a GA jointly with search space reduction meth- xij, = (3) 0 if rand ≥ Sv ods by limiting pipe diameters. El-Ghandour and Elbeltagi () ()ij, [21], used five evolutionary algorithms (GA, PSO, ACO, MA and SFLA) to solve the WDN optimization and concluded where S(vi,j) is a sigmoid limiting transformation represented that PSO has the best performance in achieving the better by Eq. (4) and rand( ) is a uniformly distributed random results. number in the interval (0,1). D.F. Surco et al. / Desalination and Water Treatment 161 (2019) 1–13 3 1 SM = 0 GbestG< best Sv()ij, v (4) 1+ e ij, Φ=05.