Non-parametric optimization technique for water distribution in pipe networks

The water distribution system (WDS) forms an integral part of modern civilization. The water is supplied from the storage points to the consumer's end through interconnected pipes forming a complex pipe network. This network mainly consists of valves, pipes, pumps, nodes, and storage tanks. Due to the involvement of various components and its vast size, it is often considered as an expensive civil infrastructure and hence needs to be optimized. Babu & Vijayalakshmi (2012) from their studies found that the pipes in the network constitute almost 70% of the total network cost.

Pipe friction accounts for head-loss occurring in a pipe network, and this head-loss exhibits a nonlinear relationship with the discharge. Due to this non-linearity, the optimization of a water distribution network (WDN) is considered under the category of combinatorial non-deterministic polynomial-hard problems referred to as NP-hard, which is difficult to solve (Eusuff & Lansey 2003; Babu & Vijayalakshmi 2012). Thus, the optimization of an NP-hard problem serves as a standard benchmark for evaluating the various computational techniques. The optimal design of WDN can be executed either by heuristic techniques, classical optimization techniques, or evolutionary algorithms (EA). Linear programming (Alperovits & Shamir 1977) and non-linear programming (Fujiwara & Khang 1990) categorized as classical optimization techniques are inapt for discrete design problems and end up with a split pipe design, which is impractical to implement. However, over the past two decades, a drastic shift towards the application of EAs in various fields has been observed. EAs are preferred over other optimization techniques because of their capability to locate and surpass local optima for NP-hard discrete optimization problems. Additionally, they are easy to use and can be incorporated into any simulation model.

Monsef et al. (2019) compared various EAs for the optimal design of WDN. However, most of the evolutionary techniques previously applied for the optimal design of WDN like genetic algorithm (GA) (Savic & Walters 1997), simulated annealing (SA) (Cunha & Sousa 1999), harmony search (HS) (Geem et al. 2002), shuffled frog leaping algorithm (SFLA) (Eusuff & Lansey 2003), particle swarm optimization (PSO) (Suribabu & Neelakantan 2006), differential evolution (DE) (Suribabu 2010), and self-adaptive differential evolution (SADE) (Zheng et al. 2013) involve the use of algorithm-specific parameters like crossover, mutation, and selection in GA, self-cognitive constant, global cognitive constant and inertia weight in PSO and number of memeplexes, number of frogs per memeplex and submemeplex, and number of evolutions within a memeplex in SFLA, for their convergence to the optimal results (Zheng et al. 2013) and it is extremely important to state that merely incorporating these constants in the optimization model does not guarantee an optimal solution. The efficiency and robustness of an EA are significantly dependent and affected by the settings of algorithm parameters that need to be synchronized for different optimization problems. Tolson et al. (2009) reported that about 3–8 parameters are required to be fine-tuned for any EA. These are just algorithm-specific constants and do not include standard pre-specified controlling parameters such as the population size and number of iterations, which may also terminate the simulation run. Also, algorithm-specific parameters once tuned for an optimization problem cannot be directly incorporated into another as appropriate parameters of EAs are varied for different optimization problems, which are normally adjusted by a trial-and-error process. Determining the appropriate parameters by this trial-and-error process for any design problem is a complete study by itself; for example, a sensitivity study on the appropriate parameters for SFLA has been done by Eusuff & Lansey (2003). Suribabu (2010) conducted a similar study on DE for the optimal design of WDN and revealed that the results obtained by the DE algorithm are highly vulnerable to the parameters selected for the problem as well as it being extensively tedious to make the appropriate parameter change with each new WDN problem. Thus, it is extremely computationally expensive and time-consuming to adjudge the proper parameters for a newly given WDN. This unavoidable computational burden to tune the algorithm-specific parameters is sometimes higher than the actual computational efforts required to compute the optimal solution.

The previous decade has observed the growth of hybrid optimization techniques that involved the use of two or more techniques in their structure. Hybrid techniques show rapid convergence when compared with other techniques (Sedki & Ouazar 2012; Zheng et al. 2013). Babu & Vijayalakshmi (2012) developed a hybrid model by combining the properties of both PSO and GA. The hybrid model PSO–GA outperforms the PSO model, but indirectly calls for tuning of the parameters of both algorithms involved in the hybrid model and hence hybrid optimization techniques need more computational efforts than the single optimization techniques. This study focuses on the development of JayaNet, a Jaya algorithm-based model for solving discrete pipe-sizing problems. The application of Jaya in various fields of science and technology (Gao et al. 2017) has been observed but none of these applications have been to the optimization of WDN. Using such techniques for optimization eliminates the synchronization process and is thus found to be highly efficient.

The term indicates the tendency of the solution to move closer to the best solution and the term indicates the tendency of the solution to avoid the worst solution. is accepted if it gives better function value. All the accepted function values at the end of the iteration are maintained and these values become the input to the next iteration.

Originally, the Jaya algorithm was developed for continuous optimization problems. In the present work, the continuous domain of diameter is converted into a discrete set of available diameters by applying the bound constraint of the average point of corresponding two consecutive discrete values, and hence presents an analysis-based simulation–optimization model JayaNet. The same has been developed for the optimal design of WDN and the analysis of the WDN is carried out using a hydraulic network solver, EPANET 2 (Rossman 2000) which is an open-source code. The flowchart of the simulation optimization process is shown in Figure 1.

Calculation of fitness value is then followed by the identification of the best and the worst solutions from the current population set. The initial solutions are then updated using Equation (6). Each updated solution is compared with previous solutions at each iteration. The process of JayaNet is continued until the termination criterion is reached. A fixed number of iterations is considered as the termination criterion for JayaNet (Savic & Walters 1997; Tolson et al. 2009; Suribabu 2010).

To test the efficiency of this optimization technique, four well-known benchmark networks are considered: a two-loop network (Alperovits & Shamir 1977), the Hanoi network of Vietnam (Fujiwara & Khang 1990), the Kadu network (Kadu et al. 2008), and the Farhadgerd network of Iran (Moosavian & Lence 2018). The optimized results from JayaNet for the two-loop network and Hanoi network are compared with various other EAs based on the average number of function evaluations (NFE). However, for the Kadu network, such comparison is not possible, as all the previous studies reported the optimal cost only once or twice in all the trial runs (Barlow & Tanyimboh 2014) and hence it is compared for minimum NFE. Such results are not available for comparison in the case of the Farhadgerd network. The effect of population size on the minimum NFE required by JayaNet to reach the optimal solution is also compared for the two-loop network, Hanoi network, Kadu network, and Farhadgerd network.

This is a hypothetical network taken from Alperovits & Shamir (1977). The network has six demand nodes and eight links arranged in two loops along with a 210 m elevated reservoir. The schematic sketch of the network along with its hydraulic data is attached in the Supplementary Material. A minimum of 30 m pressure head is required for all the nodes in the network. Table 1 shows the commercially available pipe diameters and their cost per metre length for all the networks considered. A total of 14 commercial pipe diameters are available for this network. Although it is a small network, a complete solution struggles to find the global optimal network design from 14⁸ ≈ 1.48 × 10⁹ variants of network designs, which makes this network difficult to solve and thus it is involved in the present study.

Each pipe of the two-loop network is 1,000 m long with the Hazen–Williams coefficient of 130. The optimal cost of the network is reported as 419,000 units by many researchers (Savic & Walters 1997; Cunha & Sousa 1999; Geem et al. 2002; Eusuff & Lansey 2003; Sedki & Ouazar 2012). A comparison of JayaNet with other optimization techniques for the two-loop network is given in Table 2, for which 30 trial runs are performed. In the present optimization method, the global optimal solution is obtained in just 1,560 average NFE as compared with around 65,000 evaluations required by GA (Savic & Walters 1997), 4,750 evaluations by DE (Suribabu 2010) and 3,080 evaluations by PSO–DE (Sedki & Ouazar 2012). This NFE value is the least among the algorithms reported in Table 2. The 1,560 average NFE reported above is obtained for the population size of 20 and 500 iterations as the stopping-criterion. However, the minimum NFE obtained using JayaNet among 30 trials is 940 as compared with 2,200 obtained by Siew & Tanyimboh (2012) for ten trials.

Figure 2(a) shows the convergence curve for minimum NFE for the two-loop network. However, the minimum NFE required by the algorithm to find an optimal solution is found to be largely dependent on the population size selected. When the population size is set to 50 under the same stopping-criterion of 500 iterations, the minimum NFE required by the algorithm for convergence is 2,800, and 4,500 NFE for a population size of 100, which shows the significance of population size for the minimum NFE required to reach the optimal solution. The minimum NFE required by JayaNet to reach the global optimal solution for three different population sizes of 20, 50, and 100 is depicted by the bar chart in Figure 3(a); as the population size increases, the minimum NFE required by the algorithm for its convergence also increases. This is due to increased crowding of the solutions. Also, it is significantly important to state that the average NFE required by all the algorithms presented in the table except JayaNet is post-sensitivity study for appropriate parameters, and thus the reported average NFE by other researchers does not incorporate the computational effort required to tune the algorithm.

The Hanoi network (Fujiwara & Khang 1990) is a three-looped gravity-fed WDN consisting of 34 pipes, 32 nodes, and a reservoir kept at an elevation of 100 m (refer to the Supplementary Material for network layout). There are six possible commercial pipe diameters leading to a total search space of 6³⁴ ≈ 2.865 × 10²⁶, which is large and hence is suitable for testing the evolutionary methods. The total length of the network is 39.420 km. The minimum pressure head requirement for all the nodes is 30 m. All the hydraulic data of the network like demand at a node, nodal elevation, pipe length, etc are given in the Supplementary Material.

The comparison of the results obtained by JayaNet with other optimization techniques applied previously for the Hanoi network is given in Table 3. JayaNet required an average of 14,283 NFE to reach the optimal point, which is the least when compared with the other techniques presented in Table 3. It is seen that the computational convergence of JayaNet is better over other optimization techniques.

The optimal cost obtained by some algorithms in Table 3 is $6.056 × 10⁶ which is slightly less than the cost of $6.081 × 10⁶ obtained by JayaNet. It is due to different values of dimensionless unit conversion factor used (10.5088 in case of $6.056 × 10⁶ and 10.667 in case of $6.081 × 10⁶) in the Hazen–Willian headloss formula for network analysis. Figure 2(b) depicts the convergence for the minimum NFE required by JayaNet for the Hanoi network. The effect of population size on minimum NFE required by JayaNet is shown by the bar chart in Figure 3(b). It is seen that as the population size increases, minimum NFE to reach the optimal point also increases. Also, it would be wrong to interpret that selecting less population size is beneficial for quick convergence, as a certain degree of diversity is required to reach the optimal point.

This network (Figure 4) consists of two reservoirs, 34 pipes and 26 nodes arranged in nine loops, and is thus categorized as a complex network. In addition to this, there are ten commercially available diameters, leading to a total search space of 10³⁴ and hence it is difficult to solve using conventional optimization techniques. The available diameters and the corresponding unit cost for the network are given in Table 1. The hydraulic data of the network like nodal demand, pipe length, minimum nodal pressure requirement, etc. are given in the Supplementary Material. The network was first suggested by Kadu et al. (2008), who obtained the minimum cost of the network as Rs. 131.678 × 10⁶ using GA as the optimization technique. However, GA with search space reduction (Kadu et al. 2008) converges to Rs. 126.368 × 10⁶, which is more than the optimal cost obtained by JayaNet without a reduction in search space. The minimum cost obtained by JayaNet for this network is Rs. 126.058 × 10⁶, which is reported in Table 4. JayaNet obtained the above optimal cost for 39,680 minimum NFE. The convergence curve for the above mentioned optimal solution is shown in Figure 2(c). The near-optimal cost is obtained by CFOnet, as Rs. 126.535 × 10⁶, but is computationally very expensive, taking 259,476 NFEs. PSO converges in 22,000 NFEs but the obtained optimal solution is more than by the JayaNet. The optimal cost of Rs. 126.058 × 10⁶ of the Kadu network using JayaNet is obtained for a population size of 80 and 500 iterations. However, for the same 500 iterations and the population size of 20, JayaNet converges to Rs. 127.949 × 10⁶, which is less than various costs reported in Table 4. Barlow & Tanyimboh (2014) obtained the minimum cost of the Kadu network as Rs. 124.690 × 10⁶ in 142,000 minimum function evaluations when using a multi-objective optimization approach. The effect of population size on minimum NFEs for the Kadu network is depicted by the bar chart shown in Figure 3(c).

The Farhadgerd network is the most complex network considered in the present study. The network has 68 pipes, 53 nodes, and one reservoir with an elevation of 510 m. The minimum pressure head requirement for all nodes of the network is 20 m. There are nine possible pipe diameters, which leads to a total search space of 9⁶⁸ ≈ 7.74 × 10⁶⁸. The recorded optimal cost for the Farhadgerd network is nearly USD 18 million (Moosavian & Lence 2018). This network was developed by Moosavian & Lence (2018) and has been used here to demonstrate the efficiency of JayaNet as an optimization technique for large WDN. All the hydraulic data regarding the network is provided in the Supplementary Material and the layout of the network is shown in Figure 5.

Since it is a new benchmark network, comparative results with other optimization techniques are not available in the literature. Considering a population size of 100 and 500 iterations as the stopping-criterion, JayaNet took almost 48,400 NFE to obtain the optimal cost of 17.53 million USD, and pressure head at all nodes for this network cost was found to be greater than 20 m. Figure 2(d) shows the convergence of optimal results for minimum NFE. A large set of the population size of 100, 150 and 200 is considered for this network and the effect of population size on minimum NFE to locate the optimal solution is shown in Figure 3(d). The statistical analysis of all the networks is combined and presented in Table 5, which includes the obtained best mean cost solution, minimum and maximum network cost, standard deviation, and median as obtained by the JayaNet technique. Table 5 also gives the configuration of the machine that is used for running the simulation–optimization model, JayaNet. The simulation time for a single trial run for each network is also included. Also, the median obtained for all the networks is close to the minimum cost and this indicates the capability of JayaNet to search the optimal solutions (Kadu et al. 2008; Babu & Vijayalakshmi 2012; Mohammadi- Aghdam et al. 2015).

The efficiency and robustness of any EA are sensitive to its parameters. The appropriate parameter values for each WDN optimization are decided by the hit and trial method. This synchronization of parameters for each WDN significantly increases the computational efforts and time and thus reduces its reputation in the research community. The JayaNet method proposed in the present study has overcome the above limitation. In the proposed JayaNet, only the population size and penalty parameter λ are to be adjusted based on network size.

A total of four WDN with the number of pipes ranging from eight to 68 have been used to verify the effectiveness of the proposed JayaNet. For the two-loop and Hanoi networks, the minimum cost is same as for other EAs with fewer NFE, but for the Kadu network, the JayaNet performed better than other EAs, previously applied for optimization of this pipe network in terms of minimum optimal solution. In the case of the Farhadgerd network, such a comparison is not possible due to the unavailability of data as this network is relatively new. The NFE are found to increase with population size for all networks and the increase is more for large networks. The rate of convergence is seen to be dependent on size of network.

The JayaNet shall be applied for large networks and multi-objective optimization of networks in future work. Considering the facts mentioned above, it can be justified that the proposed JayaNet model performed better and is more efficient, i.e. it requires minimum NFE and thus takes less computational effort than other optimization techniques for the optimal design of WDN.

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