## Abstract

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.

## HIGHLIGHT

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

## INSTRUCTION

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.

## RESEARCH METHODS

### Optimization model of WDN

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

*D*is the diameter of

_{i}*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

*Q*is inflow of a node,

_{in}*Q*is outflow of a node. The energy conservation constraint of each pipe section in the network must follow:where

_{out}*h*is the pressure drop of pipe

_{i}*i*. The pressure constraint of each node in the network must follow:where

*h*and

_{min,j}*h*are respectively the lower and upper bound of the pressure head of node

_{max,j}*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:where

*v*is the velocity of pipe

_{i}*i*,

*v*and

_{min,j}*v*are the minimal and maximal velocity of pipe

_{max,j}*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.

### Multiple operation models for water distribution networks

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).

*Q*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.

_{h}### Objectives functions

*TC*is defined with the following formula:where

*W*is construction cost of WDNs,

_{1}*W*is operation cost of WDNs.

_{2}*W*is defined with the following formula:where

_{1}*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:where f (·) is the cost function with two arguments of pipe diameter

*D*and pipe length

_{i}*L*;

_{i}*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 (

*γ*

_{i}) and the energy unevenness factor for the design year water supply (

*γ*) are equal. W

_{2}is defined with the following formula:where

*E*is electricity tariff prices for periods

_{i}*i*,

*T*is water supply time for electricity consumption period

_{i}*i*, Ss is pump station assembly in the water network,

*Q*is the flow rate of j pumping station,

_{ij}*is combined efficiency of*

*η*_{ij}*j*pumping station for electricity consumption period

*i*,

*Z*is Free head at control points,

_{c}*Z*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:

_{cm}*K*, 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:where

_{j}*H*is the pressure of node

_{j}*i*,

*h*,

_{j}*h*and

_{min,j}*h*are the real head, minimum head and serve pressure head, respectively, at node

_{req,j}*j*. In this paper, the system reliability is calculated by the weighted average method, i.e.where

*N*is the number of demand nodes in the network,

*W*is the weighting factor of node

_{j}*j*,

*Q*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:where

_{s}*λ*

_{1},

*λ*

_{2},

*λ*

_{3},

*λ*

_{4},

*λ*

_{5},

*λ*

_{6},

*λ*

_{7}are, respectively, the penalty parameter to be set on network size.

*v*,

_{i,max hour}*v*,

_{i,min,max hour}*v*,

_{i,max,max hour}*p*,

_{j,max hour}*p*,

_{j,min,max hour}*p*are respectively the actual, maximum and minimum values of

_{j,max,max hour}*i*pipe velocity and

*i*nodal pressure under maximum daily maximum hourly operation model.

*p*,

_{j,ave,ser}*p*are, respectively, the service pressure and actual pressure of

_{j,ave}*i*nodal pressure under the average hour operation model.

*p*and

_{j,req,fire}*p*are, respectively, the service pressure and actual pressure of

_{j,fire}*i*nodal pressure under the firefighting model.

*p*and

_{j,req,acc}*p*are, respectively, the service pressure and actual pressure of

_{j,acc}*i*nodal pressure under the accident model. The value of

*λ*

_{1},

*λ*

_{2},

*λ*

_{3},

*λ*

_{4},

*λ*

_{5},

*λ*

_{6},

*λ*

_{7}are all set to 10

^{6}. Hence the objective function adaptation value of the model l may be written as:

## SELECTED ALGORITHMS

### GWO

*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.

**is the distance between a certain gray wolf and the prey,**

*D**t*indicates the current iteration,

**and**

*A***are parameter vectors,**

*C*

*X*_{p}is the position vector of the prey, and

**is the position vector of a certain gray wolf. The calculation formulas of vectors**

*X***and**

*A***are as follows:where**

*C***and**

*r*_{1}**are random vectors in [0,1], a is given by the following equation:where G is the maximum number of iterations, the gray wolf population updates its position by using the positions of**

*r*_{2}*α*,

*β*,

*δ*.

### DE

**) are chosen randomly as**

*X*

*X*_{a}and

*X*_{b}, and their weighted differential (

*X*_{a}-

*X*_{b}) is then utilized to disturb the third individual. The process is known as a mutation, and it can be illustrated using the equation:

### GWO-DE for WDN design

*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.

### Improved GWO adaptive convergence factor

*a*is improved as follows:

*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:

### Improved DE convergence factor and crossover factor

*et al*., 2015) with the following equations.where freq is the adjustment parameter.

## APPLICATION

### Case study

The performance of the hybrid algorithm is illustrated with a case study in town L. The WDN is approximately 28 km^{2}. Figure 5 shows the WDN with one reservoir, one tank, two pumps, 39 nodes, and 44 pipes.

### Parameter determination

Table 1 lists the basic information about the pipe network. With the local regulations of the town of L, *h _{min,j}* and

*h*are set at 10 m and 60 m, respectively. The value of

_{max,j}*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*is 0.15 m/s and

_{min,i}*v*is 1.5 m/s.

_{max,i}Pipe or node number . | Node 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 number . | Node 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 |

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.

Period . | Time period . | Tariff (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 |

Period . | Time period . | Tariff (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 |

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 11^{44}.

Size . | Real code . | Cost (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 |

Size . | Real code . | Cost (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 |

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.

## RESULTS AND DISCUSSION

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.

^{6}in 30 independent GWO runs, whereas the DE generates two significant outliers of 6.223 × 10

^{6}, 6.067 × 10

^{6}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 × 10

^{6}, and other medians of the boxplot produced by other algorithms are 6.070 × 10

^{6}, 5.811 × 10

^{6}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.

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.

^{10},8.8 × 10

^{10}]. 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.

Pump number . | Operation model . | Flow . | Head . |
---|---|---|---|

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 number . | Operation model . | Flow . | Head . |
---|---|---|---|

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 number . | Operation model . | Flow . | Head . |
---|---|---|---|

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 number . | Operation model . | Flow . | Head . |
---|---|---|---|

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 number . | Operation model . | Flow . | Head . |
---|---|---|---|

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 number . | Operation model . | Flow . | Head . |
---|---|---|---|

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 |

## CONCLUSION AND FUTURE

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.

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

**11**, 341–359.

*Research and Application of Single-Object Multi-Senario Optimal Design in Water Distribution Networks*

## Author notes

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