Selection of Real-Coded Genetic Algorithm parameters in solving simulation–optimization problems for the design of water distribution networks

Proper design of water distribution networks (WDN) plays a very important role in distributing adequate quality and quantity of water to consumers (Kanakoudis 2004). Two types of simulation approaches are used to analyze WDNs, namely steady-state and extended-period simulation. In steady-state simulation, the flows and other attributes do not change over time; whereas, in the extended-period simulation, the variation of flows and other attributes are simulated over various time periods (Cesario 1995). The simulation of a WDN is a difficult process. The main task in the analysis of a WDN involves solving simultaneous nonlinear equations (Kessler & Shamir 1989; Eiger et al. 1994; Dandy et al. 1996) These equations include a continuity equation for every node, an energy equation, and the equation relating pipe flow and the head-losses. There are many useful and efficient computer programs available for WDN simulation. One of the most popular computer programs is EPANET (Rossman 1993). To determine the optimum parameters in a WDN, the traditional approach is to use the trial and error method by applying a simulation model. The trial and error approach is a cumbersome task because, for the determination of optimum parameters, it is necessary to adjust the variables based on the results from the simulation model until some pre-defined specifications are satisfied. The complexity of this process increases as the number of decision variables increases (Wu & Simpson 2001) One solution to these problems is to use simulation–optimization approach. The combined simulation–optimization application in a WDN is generally based on a surrogate method or direct linking of the simulation and optimization model. In the surrogate method (Broad et al. 2005, 2010), the response of the complex simulation model is captured by simple models like artificial neural networks (ANNs) and then these simple models will be linked with the optimization model. In direct linking of simulation–optimization approach for any application (Mohan et al. 2007; Batista do Egito et al. 2023), simulation models are directly linked with an optimization model.

Kessler & Shamir (1989) described that the nonlinear relationship between flow and head-loss and the existence of discrete variables, such as pipe market sizes, make the optimum network design exceedingly complex. The objective function is also nonlinear and non-convex. Many of the traditional optimization methods namely Linear Programming, Nonlinear Programming, and Dynamic Programming (Alperovits & Shamir 1977; Chiplunkar et al. 1986; Bhave 1988; Kessler & Shamir 1989; Lansey & Mays 1989; Eiger et al. 1994) cannot be used to determine the global optimum results of WDN problems. Nowadays, the global optimization techniques namely Genetic Algorithm (GA) (Goldberg 1989), Simulated Annealing (Cunha & Sousa (1999), Particle Swarm Optimization (PSO), and Ant Colony Optimization Algorithms are widely used for optimizing WDN. Recently many researchers have adopted other metaheuristic algorithms namely Grass Hopper Optimization Algorithm, Sparrow Search Algorithm, and Artificial Bee Colony Algorithm to a few optimization problems (Chen et al. 2023; Wang et al. 2023; Yao et al. 2023). The most commonly used metaheuristic approach is genetic algorithms for a wide variety of problems (Goldberg 1989; Simpson et al. 1994; Savic & Walters 1997; Gupta et al. 1999; Wu & Simpson 2002; Pramada et al. 2018a, 2018b; Sangroula et al. 2022; Surono et al. 2022). Savic & Walters (1997) developed a GA program called GANET for the least-cost design of WDNs. In their study, the combined simulation–optimization problem of the least-cost design of WDNs was formulated and it was shown that GA was suitable for solving WDN problems. Vairavamoorthy & Ali 2000 proposed a methodology for the optimal design of WDNs using GA and used Pipe Index Vector for controlling the GA search.

Determining the best design for a WDN has always been very problematic for water resources managers. In any simulation–optimization approach, the computational time requirement for very complex problems is one of the biggest issues. High-performance computing facilities and parallel computing can significantly reduce the computational time for complex problems. Jetmarova et al. (2017) stated that even with efficient optimization algorithms or with high-performance computing or parallel computing, WDN designs are still not computationally efficient. In these cases, choosing an optimization module with less function evaluation will help to reduce the computational time. Function evaluation is the average number of objective function/fitness function evaluations until the stopping criterion has been reached within each run. Recently, population-based search algorithms have been extensively used in the field of WDN design. Even though Real-Coded Genetic Algorithm (RCGA) is very effective in handling various engineering problems, there are only limited studies related to the selection of parameters of RCGA to get the optimum design of the WDN. The main objective of the present study is the selection of parameters of RCGA in a simulation–optimization framework for getting the global optimal results of WDN with a smaller number of function evaluations.

In this study, EPANET is used as the simulation module and RCGA as the optimization module.

A WDN consists of various components namely pipes, nodes, pumps, valves, storage tanks or reservoirs, etc. EPANET is used as the simulation model in this study. EPANET is a computer program that performs an extended-period simulation of hydraulic and water quality behavior within pressurized pipe networks (Rossman 1993). It has functionalities for tracking the flow in each pipe, the nodal pressure, the elevation of water in each tank, and the transport of chemicals through the network during the simulation period.

The cost comprises only of the pipe cost which varies with respect to the diameters of the networks in this study.

The constraints comprise the continuity equation at the nodes, the energy balance equation along a loop, and maintaining the minimum pressure head requirements at the node.

The above optimization model is solved using RCGA. GA is a search technique based on the principles of natural selection. Holland developed the GA original theory in 1975. GAs, which fall under the domain of evolutionary algorithms, simulate the process of natural selection. In the domains of engineering and computer science, they are effective global optimization approaches that address challenging issues in engineering. In order to find more accurate approximations of a solution, a population of potential solutions is incorporated into the search space. At first, a large number of the population are generated at random. The objective function of these populations is then assessed. If the termination conditions are not satisfied after one generation, the process of choosing individuals based on their level of fitness in the feasible domain produces a new set of approximations. This technique produces improved children (offspring). There are a few parameters required to run a genetic algorithm, namely population size, the mutation probability, and the crossover probability. The usual way to get these parameters is to do a lot of experimentation to find a set of values that solves a particular problem. A broad rule of thumb, to start with, is to use a mutation probability of 0.05, a crossover rate of 0.6, and a population size of about 50. (Goldberg et al. 2005) RCGA was first implemented by Wright in 1991 (Wright 1991). RCGAs do not use any coding (eg. binary coding) of the problem variables; instead, they work directly with the variables. RCGA is simple and straightforward compared to binary-coded GA.

The developed methodology for the optimal design of a WDN is applied to two benchmark problems such as a Two-Loop WDN reported by Alperovits & Shamir (1977) and GoYang WDN reported by Kim et al. (1994), and its performance is evaluated by comparing the results with past literature.

Two-Loop WDN was first reported by Alperovits and Shamir as the least-cost design of WDN. Thereafter, many researchers used this network to test their approach. In this study, network configuration, with relevant network data, was taken from Alperovits & Shamir (1977).

In the paper by Alperovits & Shamir (1977), costs are given in arbitrary units. When we apply to a specific case study, specific monetary units may be given depending on the country. To generalize the study, for both benchmark problems adopted in this study costs are given in arbitrary units (Table 2).

After doing the sensitivity analysis the final input parameters of RCGA are: population size = 60; number of generations = 100; crossover probability = 0.7; mutation probability = 0.01.

Similar to the Two-Loop network, sensitivity analysis was carried out for the GoYang WDN and the input parameters chosen are: population size = 150; number of generations = 200; crossover probability = 0.8; mutation probability = 0.01. tournament selection, blend crossover and random mutation operators are used in this study.

Research related to the least-cost design of WDNs is presented in this study. RCGA and EPANET were considered as the optimization module and simulation module, respectively. The developed model is applied to two benchmark WDN problems, i.e Two-Loop and GoYang networks. It was found that the RCGA approach is a good tool after parameter optimization for the least-cost design of WDN as the results obtained from the application of the RCGA methodology for the two benchmark networks proved that RCGA can give optimal solutions with less function evaluation. The number of function evaluations will increase with increasing complexity of the system. For very complex systems with large numbers of pipes, it is suggested to use RCGA as the optimization module. RCGA are useful when the search space is very large and there are a large number of parameters involved. In reallife, WDNs are large and the search space of decision parameters (pipe diameter) is therefore also large. The advantage of RCGA is that uncertainty in demand and optimal positioning of the tank/reservoir can be easily incorporated into the model. For future studies, a real case study can be used to check the computational efficiency and also to include multiple objectives. Also, the metaheuristic algorithms Grass Hopper Optimization Algorithm, Sparrow Search Algorithm, and Artificial Bee Colony Algorithm can be applied to the benchmark problems and their computational efficiency checked in the simulation–optimization framework.

V.H.S.K. did formal assessment and was involved in conceptualization, data collection, framework of methodology, model development, and software analysis. S.K.P. did formal assessment, conceptualized, performed methodology, collected resources, and was involved in supervision, initial draft writing, review, and editing.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.