Comparison of Five Evolutionary Algorithms for Calibration of Water Distribution Networks

It is necessary to perform the model calibration process to effectively use hydraulic models and improve the performance of water distribution networks (WDNs) in the design and operation stages. Several methods are proposed for the calibration of WDNs including 1) trial-and-error procedure models; 2) explicit models or hydraulic simulation models; and 3) implicit models or optimization models.Trial-and-error schemes were implemented to update unknown model parameters by solving the water network equations (Walski, 1983; Bhave, 1988). These models are only suitable for small problems due to the low convergence rate. Explicit models involve solving an extended set of continuity and head-loss equations in which the number of calibrated parameters are equal to the number of measurement parameters (Ormsbee and Wood, 1986; Boulos and Wood, 1990; Boulos and Ormsbee, 1991 and Ferreri et al., 1994). In implicit calibration methods, an objective function is formulated and solved by an optimization model. These methods have been investigated by a majority of the previous research (Kapelan et al., 2007; Dini and Tabesh, 2014; Do et al., 2016; Xie et al.; 2017). Recent and numerous studies indicate that evolutionary algorithms are efficient in solving complex and real-life WDNs.In this paper, five optimization algorithms, gray wolf optimization (GWO), invasive weed optimization (IWO), the genetic algorithm (GA), the imperialist competitive algorithm (ICA) and the simulated annealing algorithm (SA) are compared for the simultaneous calibration of pipe roughness coefficient and water demand coefficient in WDNs. For a closer look at the performance, these algorithms are evaluated in terms of two new criteria including, the success rate and the efficiency rate are used.

A brief description of each algorithm is presented. Evolutionary algorithms are combined with static and dynamic models of WDNs under EPANET software using a MATLAB code. The objective function is the minimization of the mean absolute percentage error (MAPE) between simulated nodal pressure and pipe flow and their corresponding measured values. The performance of these evolutionary algorithms are evaluated in terms of some criteria include statistical analysis, the optimum solution obtained, the number of objective function evaluations, the success rate, and the efficiency rate. The success rate represents the quality of the solution obtained for a specific problem and the efficiency rate indicates the performance of the algorithm and it is a neutral tool to compare the performance the different optimization algorithms applied to solve the same problem.These EAs are applied to three popular standard mathematical benchmark functions including Sphere, Rastrigin and Rosenbrock, a benchmark water distribution network and a real-life network located in the north of Iran.

The performance of five optimization algorithms was assessed by applying several mathematical test functions and a benchmark and real WDNs. The results of the application to mathematical test functions show that in most cases GWO has the best performance. After that, the five algorithms applied to benchmark WDN. Results show that GWO outperformed the other algorithms in both the success rate and the efficiency rate. The success rate for the GWO, IWO GA, ICA and SA were 60%, 20 %, 20%, 0% and 0%, respectively. The efficiency rate for the GWO, IWO, GA, ICA and SA were 16.93, 4.37, 10.27, 0 and 0, respectively.The GWO algorithm required fewer objective function evaluations to converge to the final solution. In contrast, the IWO algorithm required more objective function evaluations to reach the final solution.For the real WDN, the objective function (MAPE) obtained from the GWO algorithm improved by about 23% , 30%, 9% and 41% compared to the two GA, IWO, ICA and SA algorithms, respectively.The computational time is calculated by considering the average time for 10 computations. The comparison results indicated that the GWO had the least time consumption for all cases. This demonstrated the better performance of the GWO algorithm in searching all the problem space and its ability to avoid getting stuck in local optima. The results showed that early convergence in both GA, SA and IWO algorithms causes the optimization process to be incomplete.

Calibration of a water distribution network is beneficial for the operation and control of the water system. In this paper, simultaneous calibration of pipe roughness coefficient and water demand coefficient in WDNs was performed based on five evolutionary algorithms including The GWO, IWO, GA, ICA and SA. The performance of each algorithm is evaluated by using two new criteria including, the success rate and the efficiency rate. The success rate shows the quality of the solution obtained for a specific problem and the efficiency rate represents the performance of the algorithm. The results show that the GWO outperformed the other evolutionary algorithms in terms of the success rate, the efficiency rate, and the rapid convergence to the best solution.