Design optimization of hilly water distribution network under different operation conditions

The water distribution network (WDN), a vital piece of urban infrastructure, may be costly in terms of both money and energy. Avoiding wasteful resource consumption is significant, both economically and environmentally. Building the water distribution networks accounts for 70% of the total cost (Ezzeldin & Djebedjian 2020). Furthermore, with a WDN, reliability is particularly important. Optimizing the WDN means lowering construction costs and increasing system safety. One of the most challenging tasks in municipal management is the construction of WDNs. The elevation varies greatly in mountainous urban areas, resulting in uneven pressure, which challenges the WDN design.

Several components of WDN play a vital role in transferring water from a reservoir or tank to users (e.g., pipes, pumps, valves). Among these parts, pipes account for a sizable portion of the construction cost. Because of this, the WDN design is focused on optimizing the pipe diameter selected from a range of discrete, commercially available dimensions. The main goal of WDN optimum design is to save money. It is, however, impractical to reduce costs mindlessly while ignoring contingencies (e.g., a pipe break or a sudden surge in demand node). As a consequence, lowering network costs should not be the main goal of the long-term piping operation. The design optimization of WDN is often considered a multi-objective optimization problem. The researchers opt for the cheapest option. Cost reduction is the researchers’ priority, followed by strengthening the water distribution network's dependability. Currently, researchers apply scalarization techniques to simplify multi-objective optimization problems (Palod et al. 2022).

Many WDN studies’ optimal methods have changed over the past three decades. During the 1970s, Alperovits & Shamir (1977) initiated the first serious discussion and analysis of WDN design optimization. The approach put forth by Alperovits & Shamir (1977) was later expanded upon by other academics, with Fujiwara & Khang (1990) extending it to non-linear programming. The use of optimization algorithms for optimal WDN design has been a hot research topic since the 1990s. Heuristic and metaheuristic optimization of WDN are gaining significant critical attention. Genetic algorithm (GA) is based on biological evolution principles such as heredity, variation, and crossover, which are used in nature to develop better individuals and populations.

(2021) Different Evolution (DE) is a population-based algorithm that differs from GA in terms of variety. By adding the weighted difference between the two vectors to the third vector, DE completes the variation process. Vasan & Simonovic (2010) used this algorithm to design WDN. Bilal et al. (2021) introduced a fuzzy C-means adaptive differential evolution (FCADE) that was shown to optimize WDNs with minimal computational effort and was indifferent to the size or complexity of WDNs. Zhao et al. (2018) proposed a systematic optimization technique for real-time WDN operation, in which the discrete-continuous driving variables in WDNs are handled by an adapted mixed-integer DE. Mirjalili et al. (2014) presented the Gray Wolf Optimization (GWO) in 2014. The use of the GWO algorithm for the optimal design of water supply networks has received little attention. The study offers a new algorithm that combines GWO and DE to quickly and accurately achieve the WDN optimization goal.

As the research progresses, WDN optimization is no longer limited to a single objective of pipe network cost but gradually evolves into a multi-objective function, such as reliability. Geem (2015) proposes a new fuzzy theory-based velocity reliability index for multi-objective optimization of water supply pipeline networks using a summation search to seek a water supply optimization scheme that minimizes cost and maximizes reliability. Palod et al. (2022) proposed a parameter-free method for generating Pareto preambles and not involving non-dominated ranking, combined with a Jaya operation model with cost and reliability as dual objective functions. And the model test results showed that the parameter-free property significantly improved the optimization efficiency. Mu et al. (2021) proposed a network optimization model containing multiple reservoirs, using NSGA-II to optimize the cost and reliability of the water supply network. The reliability was quantified using two functions, pressure uniformity and leakage uniformity, and was validated for five case pipe networks. In addition, Vertommen et al. (2021) have carried out a water supply network design that is optimal based on actual demand situations. Liu et al. (2021) use a variety of algorithms to reasonably optimize the accident conditions of both models for the single-objective model with economic objectives and the multi-objective model with financial and reliability goals. Surco et al. (2018) propose a scenario-based optimization method for water supply networks, thus considering the uncertainty of demand in water supply network design and evaluating the design performance under different water demand scenarios. Martínez (2010) introduces a method to estimate network reliability in normal and fault states based on making the cost analysis minimal, and compares the optimization results in terms of cost and reliability to provide a more in-depth analysis of the comparison between ring and branch networks. Marques et al. (2018) present a multi-objective optimization model for the flexible design of water supply networks with four objective functions involving financial and environmental issues and network reliability. And optimization results provide a thorough analysis of the trade-offs between the objectives and confirm the importance of considering the depreciation of all these four objectives.

The rest of the paper is structured as follows. Decision variables, constraint conditions, numerous operations, and objective functions are organized into four sections in section 2 of the proposed optimization model. Section 3 provides an overview of GWO and DE algorithm theory. In part 4, we go over the application of WDNs in greater depth. In the concluding section, the paper's conclusions and prospects are discussed.

The combination of the pipe network diameter layout (choice factors) is often optimized for the water supply network to produce the lowest possible pipe network cost (objective function). The minimal head at the nodes, the pipe's flow velocity, and the law of conservation of mass and energy are typically constraints on this problem. Additionally, the pipe diameter can only be chosen within a specific range of pipe diameters.

In academic research, water demand uncertainty is usually determined by stochastic methods, which are not applicable to the optimization of actual WDNs. In contrast, scenario-based optimal WDN design is more practicable and leads to superior optimization choice solutions of WDN. The amount of water required varies depending on the event, such as a fire or an accident. Focusing on the amount of water necessary for various circumstances is crucial to the design of WDNs. The optimal design of water supply networks in this research is based on water demand in various scenarios (the maximum daily maximum hourly operation model, average hourly operation model, firefighting operation model, accidental conditional operational model).

The performance of the hybrid algorithm is illustrated with a case study in town L. The WDN is approximately 28 km². Figure 5 shows the WDN with one reservoir, one tank, two pumps, 39 nodes, and 44 pipes.

Table 1 lists the basic information about the pipe network. With the local regulations of the town of L, h_min,j and h_max,j are set at 10 m and 60 m, respectively. The value of k in this study is fixed at 0.8. The fire occurrence site is considered the pipe network's lowest pressure point, i.e., node 8, and the fire-fighting demand is set to 15 L/s. The AOP states that normal water delivery cannot be maintained due to the collapse of the 33rd pipe. The water delivery network ensures a minimum of 70% water consumption under AOP. As a result, v_min,i is 0.15 m/s and v_max,i is 1.5 m/s.

According to local municipal construction and planning, T is taken to be 20, b is 6.5%, p is taken to be 2.8, and the rate of depreciation and investment repayment for WDN is 13.1%. η is taken to be 0.80, γ is taken to be 0.4. Table 2 shows the price of power at different times of day in Q city.

Because the elevation difference between the case nodes in hilly places produces excessive pressure in some nodes, ductile iron pipes with superior pressure resistance are employed to avoid pipe breaking. The H-W coefficient is set to 130 in this scenario. The 11 commercially available pipe diameters, their corresponding actual codes, and their cost per m are listed in Table 3. The total number of solution spaces (search space) is 11⁴⁴.

For comparison purposes, GWO, DE, and GWO-DE were selected to execute 30 independent optimization-seeking calculations on the fitness curve at the stated maximum number of iterations. The GWO-DE, GWO, and DE runs were all completed on a PC with an Intel Core i7-9700 k 3.0 GHz, 8.0 GB RAM, and a 64-bit operating system. The number of individuals is limited to 100, and the number of iterations is limited to 800.

From an analytical standpoint, it is also particularly critical to compare the level of data dispersion that addresses these options. Figure 7 shows box plots of the results obtained for 30 running times in terms of the total cost (a), reliability (b), cost per unit reliability (c), running time (d). The minimum value, maximum value, first quartile limit, median, third quartile limit, and maximum value of the data set are shown in Figure 7. The median is the horizontal line inside the rectangle. In contrast, the minimum and maximum values are the upper and lower horizontal lines outside the rectangle.

Meanwhile, the boxplot for cost per unit reliability (Figure 7(c)) shows that the solutions obtained by GWO-DE are lower than other algorithms. Therefore, the GWO-DE can achieve the same reliability at a lower cost than other algorithms, which may be preferred by businesses or governments seeking higher reliability at a lower cost. Figure 6(d) shows the differences in running times between the three algorithms. GWO generates two huge outliers in 30 independent runs. In contrast, GWO-DE generates four significant outliers in 30 independent runs. In 30 independent runs, GWO produces two huge outliers, whereas GWO-DE produces four significant outliers, indicating that it is exceedingly unstable regarding the algorithm running time, especially GWO-DE.

GWO-DE outperforms GWO and DE in terms of search accuracy and efficiency, as well as the capacity to avoid local optimum. The multi-operation models’ case optimization design improves the adaptability of the pipe network in various situations by optimizing both pipe diameter and pump optimization. In this paper, the hydraulic simulation model (EPANET) is combined with the multi-objective optimization model (GWO, DE, GWO-DE) to obtain the water supply optimization model under different algorithms based on the highest daily maximum hourly operating conditions. And the corresponding quantification of the water pressure in the pipe network for the three common daily operating conditions (AH operation model, FF operation model, A operation model) to adjust the pipe diameter of the case network accordingly. Thus, the unit reliability cost of the example network can be minimized, i.e., the most cost-effective water supply design can be obtained. It is found that the GWO-DE algorithm outperforms the other two algorithms both in terms of economy and optimization time. From the optimization results, there is a potential risk of leakage and water hammer accidents at some nodes of the mountain WDNs. The result suggests that setting pressure reduction valves in some pipe sections or zoning water supply to the mountain pipe network is worth considering. This paper has not yet conducted an in-depth study for this part. The research results of this paper provide theoretical support and quantitative analysis methods for safe water supply design in mountainous areas. Furthermore, the pipe network model presented in this paper is an idealized and abstracted model based on a real-life case study that is not representative of the actual scenario. The parameter values have not been thoroughly investigated.

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

The authors declare there is no conflict.