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In order to optimize the water supply system for various operation models, this study uses a hilly water supply project as an example. In order to analyze the water supply network for optimization, this study uses two objective functions: cost and dependability. The designer can then select the best optimization strategy for his needs. The design of a real-world water supply network, a component of a water supply network in a seaside town in China, was done using this methodology. Based on the findings, it is crucial to improve the design of the water supply network under various operating models to get a more reliable design solution at a lower design cost.

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  • water distribution network; design; optimization; different operation model.

design, different operation model, optimization, water distribution network
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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.

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Optimization model of WDN

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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.

Decision variables

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The pipe diameter chosen from a group of discrete pipe diameters is the decision variable in this problem, which the equation can describe:
(1)
(2)
where Di is the diameter of i pipe, S is a collection of commercially available discrete diameters, n is the total number of commercially available discrete diameters. In multi-objective function optimization, a numerical encoding approach is required a specific transformation method from solution space into search space that the algorithm can handle. Binary encoding, real encoding, floating point encoding, symbol encoding, and mixed encoding are the five methods up to now. It is now well understood that the coding method influences the algorithm's convergence speed. Although binary encoding is straightforward to implement cross-variance, it has the drawback of exceeding encoding redundancy when there are too many possibilities of pipe diameters. Vairavamoorthy & Ali (2000) pioneered real coding, which has since become a popular method for WDN optimization. Real coding is avoided for redundancy. Real coding is used in WDN design optimization because of coding redundancy.

Constraint conditions

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The optimization procedure of the objective function must follow mass conservation and energy conservation from the standpoint of rationality and actuality. The mass conservation constraint of each node in the network must follow:
(3)
where Qin is inflow of a node, Qout is outflow of a node. The energy conservation constraint of each pipe section in the network must follow:
(4)
where hi is the pressure drop of pipe i. The pressure constraint of each node in the network must follow:
(5)
where hmin,j and hmax,j are respectively the lower and upper bound of the pressure head of node j. The square root of the pipe diameter and the design flow rate are inversely proportional when determining the flow rate (Yan & Liu 2014). The link between flow rate and pipe diameter and construction and operation expenses is depicted in Figure 1. Construction expenses can often be reduced by using economical flow rates or pipe sizes. The flow rate is set within a particular range, allowing for the most cost-effective objective function. The velocity constraint of each pipe in the network can follow:
(6)
where vi is the velocity of pipe i, vmin,j and vmax,j are the minimal and maximal velocity of pipe i. Here, in order to analyze the water network and to obtain hydraulic values such as flow rate, pressure head, and flow velocity, a popular simulator EPANET is used.
Figure 1
Different costs in relation to pipe diameter and flow rate.
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Different costs in relation to pipe diameter and flow rate.

Figure 1
Different costs in relation to pipe diameter and flow rate.
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Different costs in relation to pipe diameter and flow rate.

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Multiple operation models for water distribution networks

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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 MHMD (maximum hourly maximum daily) operation model refers to the hour with the highest water demand on a certain day of the year. The MDMH operation model used to build WDNs, accounts for a small percentage of the network's running time, resulting in a waste of resources and energy. Future eventualities (a burst or break in a section of pipe, a fire at a node, etc.) must be considered while designing WDNs. The most typical scenario of WDNs is the average hourly (AH) operation model, which accounts for most of the network's operational time. Under this operation model, the water demand of the node is a certain percentage of the MHMD water demand, calculated as follows:
(7)
where k is the specific percentage, the value of k is set to between 0.7 and 0.9 (Zhang 2008), Qh is the water demand under MHMD operation model. When a fire breaks out at a network node, the water demand at that node skyrockets, and the firefighting operation model describes the network's operational conditions. The pipe network can temporarily ignore the water demand of other nodes under this operation paradigm to ensure that water for firefighting is the primary concern. The MHMD water demand plus the firefighting water demand at the fire point equals the water demand at each node in firefighting conditions. The accident operation model refers to a failure of the main water supply segment. The water supply capacity of the entire network is reduced under this operation paradigm, as a piece of pipe ceases to deliver water until it is serviced and the regular water supply is restored.

Objectives functions

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While this optimization focuses on gravity-fed systems such as the two-loop and Hanoi network, designing a realistic mountainous metropolitan WDN is not as straightforward as it appears. Pumping stations usually increase water pressure to meet customers’ nodal water supply needs. Hence, researchers cannot afford to ignore the operating costs associated with the pumping station. The expenses in this study are annualized costs meaning that the WDN design cost is quantified as the annual needed cost. The total cost (TC) is divided into two categories: construction cost and operation cost. The research introduces the reliability (R) model for evaluating the flow and pressure of WDNs. The dual objectives of the optimization WDNs models are TC and R. This study employs the weighting method to normalize the dual objective function to ensure fairness between the objectives. This paper's design optimization aims to reduce costs while increasing reliability. The objective function of this paper is, therefore, the TC of operating the water supply network per unit of R, which the following equation can express:
(8)
TC is defined with the following formula:
(9)
where W1 is construction cost of WDNs, W2 is operation cost of WDNs. W1 is defined with the following formula:
(10)
where b is the benchmark yield, T is the payback period of pipe network construction, C is the investment cost of pipe network construction, p is annual depreciation and overhaul cost rates for pipe networks, and the value of p is set from 2.5 to 3.0. The following formula defines C:
(11)
where f (·) is the cost function with two arguments of pipe diameter Di and pipe length Li; M is the total number of pipes i. The current power rate in most Chinese cities is based on a tiered system. This paper assumes that the energy variation factor for water supply for each electricity consumption period (γi) and the energy unevenness factor for the design year water supply (γ) are equal. W2 is defined with the following formula:
(12)
where Ei is electricity tariff prices for periods i, Ti is water supply time for electricity consumption period i, Ss is pump station assembly in the water network, Qij is the flow rate of j pumping station, ηij is combined efficiency of j pumping station for electricity consumption period i, Zc is Free head at control points, Zcm is difference between the elevation of the control point and the pumping station elevation. Pragmatically, the delivered water demand of the node is closely related to this node's pressure. The relationship between node demand and node pressure is expressed in the following equation:
(13)
Kj, which is a constant, is the nodal resistance coefficient. Hence, this paper treats nodal flow and nodal pressure as the same indicator in discussing reliability. The R of node j introduced is computed as follows:
(14)
where Hj is the pressure of node i, hj, hmin,j and hreq,j are the real head, minimum head and serve pressure head, respectively, at node j. In this paper, the system reliability is calculated by the weighted average method, i.e.
(15)
where N is the number of demand nodes in the network, Wj is the weighting factor of node j, Qs is the flow of the network. The minimization of the design AC and the maximization of R are in direct opposition to one another. The penalty function is considered to impose a penalty on solutions that violate the constraints of the objective function. The penalty function is given as follows:
(16)
(17)
where λ1, λ2, λ3, λ4, λ5, λ6, λ7 are, respectively, the penalty parameter to be set on network size. vi,max hour, vi,min,max hour, vi,max,max hour, pj,max hour, pj,min,max hour, pj,max,max hour are respectively the actual, maximum and minimum values of i pipe velocity and i nodal pressure under maximum daily maximum hourly operation model. pj,ave,ser, pj,ave are, respectively, the service pressure and actual pressure of i nodal pressure under the average hour operation model. pj,req,fire and pj,fire are, respectively, the service pressure and actual pressure of i nodal pressure under the firefighting model. pj,req,acc and pj,acc are, respectively, the service pressure and actual pressure of i nodal pressure under the accident model. The value of λ1, λ2, λ3, λ4, λ5, λ6, λ7 are all set to 106. Hence the objective function adaptation value of the model l may be written as:
(18)
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GWO

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The natural population of gray wolves is divided into four social classes, namely α wolves, β wolves, δ wolves, ω wolves. The hierarchy of the gray wolf is shown in Figure 2. α wolves, the leader of the pack, manage and control the whole pack and govern the pack's behavior. β wolves, the candidate of the α, help α in decision-making and provide feedback to α. δ Wolf, the subordinate of the pack, is responsible for the security and logistics of the pack and rules ω. The optimal solution is defined as α, the suboptimal solution, and the third optimal solution are β and δ, respectively, for the purpose of constructing the social ranking of wolves in GWO. The hunting behavior of the gray wolf is divided into three stages: stalking, approaching the prey; surrounding and harassing the prey until it stops moving; attacking the prey.
Figure 2
Hierarchy of gray wolf.
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Hierarchy of gray wolf.

Figure 2
Hierarchy of gray wolf.
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Hierarchy of gray wolf.

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The mechanism by which wolves update their position is presented in Figure 3. The mathematical model of the surrounding behavior is as follows:
(19)
(20)
where D is the distance between a certain gray wolf and the prey, t indicates the current iteration, A and C are parameter vectors, Xp is the position vector of the prey, and X is the position vector of a certain gray wolf. The calculation formulas of vectors A and C are as follows:
(21)
(22)
where r1 and r2 are random vectors in [0,1], a is given by the following equation:
(23)
where G is the maximum number of iterations, the gray wolf population updates its position by using the positions of α, β, δ.
(24)
(25)
(26)
Figure 3
Position update mechanism in GWO.
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Position update mechanism in GWO.

Figure 3
Position update mechanism in GWO.
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Position update mechanism in GWO.

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DE

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Since its introduction by Storn & Prince (1997), DE has proven to be quite effective for various optimization problems, including pipe optimization problems with discrete pipe sizes, due to its good search capability and robustness. Unlike GWO, DE updates the offspring positions by mutation, crossover, and selection. Two individuals from the population (X) are chosen randomly as Xa and Xb, and their weighted differential (Xa-Xb) is then utilized to disturb the third individual. The process is known as a mutation, and it can be illustrated using the equation:
(27)
The weighting factor FR controls difference vector contraction, which is generally set at [0,2]. The intermediate produced by the mutation is swapped with the parent individual for the element in the crossover process. The crossover operation is implemented using the following equation for the j-th dimension of the i-th individual:
(28)
where cr is a random number between [0, 1], CR is the probability of crossover operations occurring, sd represents a random dimension. The selection process employs a greedy mechanism, in which the next generation is the parent if the offspring's fitness exceeds that of the parent, and the next generation is the offspring if the parent's fitness exceeds that of the offspring. The following equation is used to carry out the selection operation:
(29)

GWO-DE for WDN design

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GWO is a new intelligent optimization system that mimics grey wolves’ natural social rank and group hunting mechanism. GWO has the advantages of a simple structure, rapid convergence, and fewer adjustment parameters; however, local optima and premature convergence are common when solving high-dimensional composite function optimization problems. DE possesses the characteristics of global search capability high robustness, and the capacity to tackle complex function optimization problems easily. The main disadvantage of DE is its lack of local search capability, and over-reliance on parameter setting settings. In this paper (Zhang et al. 2017), a hybrid optimization technique for GWO-DE is introduced. The GWO-DE hybrid algorithm adopts a simple hybrid strategy: the algorithm search process is divided into two phases, the first stage uses the improved grey wolf optimization algorithm to discover the optimal solution, and the second stage uses the adaptive difference algorithm to find the optimal solution. Meanwhile, both phases use a greedy method to update the offspring's and ensure that population evolution continues in the right direction. Figure 4 depicts the WDND hybrid algorithm.
Figure 4
Optimizing WDN design.
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Optimizing WDN design.

Figure 4
Optimizing WDN design.
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Optimizing WDN design.

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Figure 5
The water distribution network.
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The water distribution network.

Figure 5
The water distribution network.
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The water distribution network.

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Improved GWO adaptive convergence factor

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The convergence factor of the original GWO algorithm decreases linearly with the increase in the number of iterations, which greatly affects the searchability of the algorithm. A nonlinear adaptive convergence factor is introduced to balance the local search ability and global search capability of the algorithm for improving this deficiency (Li 2021). a is improved as follows:
(30)
The appropriate expansion of the weight of α can effectively accelerate the convergence. Meanwhile, the increment of the random perturbation term can prevent the function from falling into the optimal local solution (Li 2021). X is improved as follows:
(31)

Improved DE convergence factor and crossover factor

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The FR and CR use the adaptive improvement approach adopted by (Draa et al., 2015) with the following equations.
(32)
(33)
where freq is the adjustment parameter.
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Case study

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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.

Parameter determination

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Table 1 lists the basic information about the pipe network. With the local regulations of the town of L, hmin,j and hmax,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, vmin,i is 0.15 m/s and vmax,i is 1.5 m/s.

Table 1

Nodal demand and pipe length for case

Pipe or node numberNode elevation (m)Node serve pressure head (m)The water demand at nodes (L/s)Pipe length (m)
1 115 – −618.49 880 
2 96.72 28 37.25 602 
3 108.24 28 23.16 1,194 
4 113.58 28 12.83 932 
5 118.72 28 14.56 789 
6 120.85 28 15.97 2,619 
7 135.38 12 8.39 936 
8 110.51 12 10.87 603 
9 108.28 12 15.72 2,271 
10 92.78 28 15.97 378 
11 89.42 28 27.82 920 
12 110.97 12 18.64 1,949 
13 92.82 28 25.51 984 
14 86.68 28 16.25 582 
15 73.34 28 27.82 1,401 
16 58.87 28 28.51 819 
17 54.34 28 29.47 352 
18 51.63 28 19.45 328 
19 55.54 28 22.87 550 
20 60.63 28 24.56 410 
21 75.85 28 27.13 743 
22 80.87 28 16.45 446 
23 83.47 28 20.41 823 
24 88.57 28 14.44 507 
25 81.24 28 18.54 928 
26 100.62 28 14.27 810 
27 113.28 28 13.61 378 
28 118.62 12 8.94 734 
29 131.54 12 7.64 1,613 
30 111.31 28 9.74 581 
31 117.26 28 11.34 953 
32 128.54 12 10.54 491 
33 136.48 12 9.72 452 
34 146.45 12 7.95 979 
35 156.28 12 6.47 1,300 
36 157.37 28 6.82 1,008 
37 164.19 12 10.27 1,300 
38 148.16 12 8.59 2,100 
39 115 – 0 3,300 
40 115 – 0 1,200 
41 120.5 – – 1,100 
42 – – – 960 
43 – – – 1,171 
44 – – – 100 
Pipe or node numberNode elevation (m)Node serve pressure head (m)The water demand at nodes (L/s)Pipe length (m)
1 115 – −618.49 880 
2 96.72 28 37.25 602 
3 108.24 28 23.16 1,194 
4 113.58 28 12.83 932 
5 118.72 28 14.56 789 
6 120.85 28 15.97 2,619 
7 135.38 12 8.39 936 
8 110.51 12 10.87 603 
9 108.28 12 15.72 2,271 
10 92.78 28 15.97 378 
11 89.42 28 27.82 920 
12 110.97 12 18.64 1,949 
13 92.82 28 25.51 984 
14 86.68 28 16.25 582 
15 73.34 28 27.82 1,401 
16 58.87 28 28.51 819 
17 54.34 28 29.47 352 
18 51.63 28 19.45 328 
19 55.54 28 22.87 550 
20 60.63 28 24.56 410 
21 75.85 28 27.13 743 
22 80.87 28 16.45 446 
23 83.47 28 20.41 823 
24 88.57 28 14.44 507 
25 81.24 28 18.54 928 
26 100.62 28 14.27 810 
27 113.28 28 13.61 378 
28 118.62 12 8.94 734 
29 131.54 12 7.64 1,613 
30 111.31 28 9.74 581 
31 117.26 28 11.34 953 
32 128.54 12 10.54 491 
33 136.48 12 9.72 452 
34 146.45 12 7.95 979 
35 156.28 12 6.47 1,300 
36 157.37 28 6.82 1,008 
37 164.19 12 10.27 1,300 
38 148.16 12 8.59 2,100 
39 115 – 0 3,300 
40 115 – 0 1,200 
41 120.5 – – 1,100 
42 – – – 960 
43 – – – 1,171 
44 – – – 100 
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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.

Table 2

Time-of-use electricity price of Q city

PeriodTime periodTariff (RMB/kW·h)
Peak periods 8:00–11:00 0.9203 
14:30–21:00 
Valley periods 12:00–13:00 0.6226 
23:00–7:00 
Other periods Other time periods 0.3249 
PeriodTime periodTariff (RMB/kW·h)
Peak periods 8:00–11:00 0.9203 
14:30–21:00 
Valley periods 12:00–13:00 0.6226 
23:00–7:00 
Other periods Other time periods 0.3249 
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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.

Table 3

Pipe sizes, corresponding real codes and their cost

SizeReal codeCost (RMB)
DN100 1 533 
DN150 2 681 
DN200 3 715 
DN250 4 819 
DN300 5 887 
DN400 6 1,077 
DN450 7 1,154 
DN500 8 1,213 
DN600 9 1,432 
DN700 10 1,691 
DN800 11 1,801 
SizeReal codeCost (RMB)
DN100 1 533 
DN150 2 681 
DN200 3 715 
DN250 4 819 
DN300 5 887 
DN400 6 1,077 
DN450 7 1,154 
DN500 8 1,213 
DN600 9 1,432 
DN700 10 1,691 
DN800 11 1,801 
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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.

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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.

As shown in Figure 6(a), the GWO generates a prominent outlier of 7.286 × 106 in 30 independent GWO runs, whereas the DE generates two significant outliers of 6.223 × 106, 6.067 × 106 in 30 independent runs of DE, respectively. The result indicates that GWO, DE are likely to fall into local extremes. The median of the boxplot for cost per reliability produced by GWO-DE is 5.520 × 106, and other medians of the boxplot produced by other algorithms are 6.070 × 106, 5.811 × 106 respectively. The boxplot of GWO-DE is in the lowest position, indicating that the overall quality of the solutions obtained by GWO-DE is better than the other two algorithms. The boxplot for the total cost (Figure 7(a)) reveals that GWO-DE produces lower-cost solutions than other algorithms, whereas the boxplot for R (Figure 7(b)) shows that DE and GWO-DE provide higher-cost solutions.
Figure 6
Comparison of the dispersion of 30 optimization schemes of the GWO, DE with GWO-DE for TC (a), R (b), TC/R (c), Time (d).
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Comparison of the dispersion of 30 optimization schemes of the GWO, DE with GWO-DE for TC (a), R (b), TC/R (c), Time (d).

Figure 6
Comparison of the dispersion of 30 optimization schemes of the GWO, DE with GWO-DE for TC (a), R (b), TC/R (c), Time (d).
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Comparison of the dispersion of 30 optimization schemes of the GWO, DE with GWO-DE for TC (a), R (b), TC/R (c), Time (d).

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Figure 7
Comparison of GWO-DE, GWO, and DE for optimal solution evolution.
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Comparison of GWO-DE, GWO, and DE for optimal solution evolution.

Figure 7
Comparison of GWO-DE, GWO, and DE for optimal solution evolution.
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Comparison of GWO-DE, GWO, and DE for optimal solution evolution.

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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.

In addition to the foregoing, a comparison of the best adaptation values of the three algorithms over 30 runs was present in Figure 7. In order to display the overall convergence trend, the convergence results of the later fitness curves cannot be clearly distinguished. In the early stage of iteration (zoomed area of the front part of Figure 7), both GWO and GWO-DE converge relatively slowly compared to the others. In the later stage of iteration, DE and GWO-DE show decent local search capabilities, and GWO-DE obtained the fitness-optimal solution. The convergence variation of three algorithmic adaptation curves for the adaptation degree is shown in the later inset between [8.4 × 1010,8.8 × 1010]. In particular, GWO-DE effectively improves the optimization accuracy and convergence speed of the algorithm in a short period of time, making up for the lack of local search capability of DE while providing DE with more computational resources for further optimization. For the pipe network, in this case, GWO-DE achieves the lower TC and the upper R, which demonstrates the combination of GWO-DE can further improve the performance of WDN optimization. According to the analysis of adaptation in the case, the excessive fitness was caused by the small flow rate of the pipe and the fact that some nodes did not reach the node service water pressure under the AH operation model. Figure 8 shows the optimization results of the case pipe network by the algorithm. After comparison, it is found that GWO-DE can optimize the pipe diameter at the end of the pipe network better than GWO and DE. As a result, Figure 9 compares the case node pressure under different operation models: when the pipe network is in the MHMD operation model and FF operation model, some node pressure does not meet the node service water pressure (Figure 9 dashed line), while when the pipe network is in AH operation model and A operation model, some nodes of GWO optimization have a tendency to be higher than the node maximum water pressure, from the node pressure balance analysis, the pipe network optimization effect: GWO-DE > DE > GWO. The nodes with high pressure are concentrated in the low-lying areas of the case pipe network, and it is known that the low elevation of the nodes is the main reason for the high pressure of the nodes here, which is very easy to occur leakage and water hammer accidents. Furthermore, the flow rate and head of the pumps (result see Tables 3,4–6) can be obtained under various operating conditions for better pump selection, which is beneficial for those governments and companies.
Table 4

The flow rate and head of the pumps obtained by GWO

Pump numberOperation modelFlowHead
45 MOMD operation model 157.97 96.36 
AH operation model 143.746 102.722 
FF operation model 159.388 95.697 
A operation model 138.372 104.968 
46 MOMD operation model 521.71 37.58 
AH operation model 425.922 42.835 
FF operation model 526.699 37.279 
A operation model 361.525 45.77 
Pump numberOperation modelFlowHead
45 MOMD operation model 157.97 96.36 
AH operation model 143.746 102.722 
FF operation model 159.388 95.697 
A operation model 138.372 104.968 
46 MOMD operation model 521.71 37.58 
AH operation model 425.922 42.835 
FF operation model 526.699 37.279 
A operation model 361.525 45.77 
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Table 5

The flow rate and head of the pumps obtained by DE

Pump numberOperation modelFlowHead
45 MOMD operation model 173.83 88.567 
AH operation model 158.594 96.071 
FF operation model 174.427 88.26 
A operation model 155.861 97.344 
46 MOMD operation model 541.516 36.363 
AH operation model 452.326 41.493 
FF operation model 551.869 35.708 
A operation model 379.61 44.994 
Pump numberOperation modelFlowHead
45 MOMD operation model 173.83 88.567 
AH operation model 158.594 96.071 
FF operation model 174.427 88.26 
A operation model 155.861 97.344 
46 MOMD operation model 541.516 36.363 
AH operation model 452.326 41.493 
FF operation model 551.869 35.708 
A operation model 379.61 44.994 
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Table 6

The flow rate and head of the pumps obtained by GWO-DE

Pump numberOperation modelFlowHead
45 MOMD operation model 175.907 87.491 
AH operation model 160.457 95.191 
FF operation model 176.445 87.21 
A operation model 156.941 96.844 
46 MOMD operation model 508.981 38.341 
AH operation model 420.22 43.114 
FF operation model 517.119 37.858 
A operation model 352.647 46.137 
Pump numberOperation modelFlowHead
45 MOMD operation model 175.907 87.491 
AH operation model 160.457 95.191 
FF operation model 176.445 87.21 
A operation model 156.941 96.844 
46 MOMD operation model 508.981 38.341 
AH operation model 420.22 43.114 
FF operation model 517.119 37.858 
A operation model 352.647 46.137 
View Large
Figure 8
Comparison of the network configurations of the optimal solutions derived from GWO (a), DE (b), and GWO-DE (c) methods.
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Comparison of the network configurations of the optimal solutions derived from GWO (a), DE (b), and GWO-DE (c) methods.

Figure 8
Comparison of the network configurations of the optimal solutions derived from GWO (a), DE (b), and GWO-DE (c) methods.
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Comparison of the network configurations of the optimal solutions derived from GWO (a), DE (b), and GWO-DE (c) methods.

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Figure 9
Comparison of nodal pressures under different operating conditions.
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Comparison of nodal pressures under different operating conditions.

Figure 9
Comparison of nodal pressures under different operating conditions.
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Comparison of nodal pressures under different operating conditions.

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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.

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

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The authors declare there is no conflict.

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Author notes

†

These authors contributed equally to this work and should be considered co-first authors.

© 2022 The Authors
This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits copying, adaptation and redistribution, provided the original work is properly cited (http://creativecommons.org/licenses/by/4.0/).

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