Multi-objective optimization design of integrated pump station based on NSGA - III

Multi-objective optimization design of integrated pump station based on NSGA - III | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Multi-objective optimization design of integrated pump station based on NSGA - III Rui Li, He Wang, Kunlun Xin, Tao Tao This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3906818/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 3 You are reading this latest preprint version Abstract Safe water supply and energy conservation are two goals that the water supply industry has always focused on. Integrated pump station adopts a dual mode water supply system which has adjustable water storage capacity and can utilize the pressure of inlet water effectively, hence, it is a new type of solution in optimizing the operation of urban water supply systems. In order to solve the problem of insufficient water supply pressure during peak hours in the railway station and municipal party committee area of S city, the author constructed a pipe network hydraulic model, conducted a systematic analysis of the water supply pipe network, and optimized the design of IPSs using NSGA-III combined with EPANET. Finally, the hydraulic model of the pipe network is used to verify the feasibility of the scheme. The results show that under the premise of ensuring safe water supply, energy conservation and water age optimization can be achieved simultaneously, and the fluctuation intensity of the total water supply from the water plants is effectively reduced. Energy consumption Water age NSGA-III Integrated pump station Multi objective optimization design Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1 Introduction With the development of urban economy and population, the scale of cities is constantly expanding, and the demand for urban water supply is increasing. The contradiction between supply and demand has brought new challenges to water supply enterprises. The main problems are: first, the water pressure at some points of Water Distribution System(WDS) is insufficient during peak hours; second, the pressure fluctuation in WDS is sometimes significant; third, the energy consumption of water supply equipment is high; fourth, the leakage loss of WDS is large; fifth, water stays in storage facilities for longer periods of time, which can lead to the risk of poor water quality. Optimal scheduling is an important means of WDS operation and management, which plays an important role in ensuring water supply requirements and energy conservation. In recent years, some cities have overused pressurized water supply equipment for local pressurization, resulting in excessive water pumping from municipal main pipes, which leads to low pressure in the pipe network during peak hours, and water supply in some areas is not guaranteed. These problems have become one of the typical problems faced by WDS in the process of urbanization. The pump station equipped with a traditional ground water storage tank (GWST) can contribute to regulating water supply quantity; however, this method cannot effectively utilize the pressure of inlet water, resulting in higher energy consumption. Hence, a new type of pump station called integrated pump station (IPS) is adopted in the case study, which combines the booster pump (BP) with water storage tank pump (WSTP). The BP extracts water from the municipal pipeline network, whereas the WSTP retrieves water from a storage tank. This research mainly discusses the optimization scheduling problem for this new type of pump station. The decision variables in a traditional optimal scheduling problem consist of the state combinations of pumps and valves. The main objective function is the total energy consumption of water supply, and other objective functions such as total energy cost(Odan and Ribeiro Reis et al., 2015; Fooladivanda and Taylor, 2017 ; Cimorelli and D Aniello et al., 2020), water age (Prasad and Walters, 2006 ; Al-Jasser, 2007 ), water quality (Mala-Jetmarova and Barton et al., 2015 ; Shokoohi and Tabesh et al., 2017 ) are also considered for multi-objective optimization. The constraints of optimal scheduling are numerous and complex, including pipe network hydraulic constraints (Bagirov and Barton et al., 2013 ), node pressure control constraints (Makaremi and Haghighi et al., 2017 ), pump operation constraints (Zhuan and Xia, 2013 ; Quintiliani and Creaco, 2019 ), water level constraints (Costa and Prata et al., 2016 ; Oikonomou and Parvania et al., 2018 ), etc. While the direct scheduling model is relatively simple, optimizing it in a large and complex WDS poses a significant challenge. This is attributed to the substantial number of pumps and valves, further complicated by their different categorizations. The optimal scheduling of WDS is a typical mixed integer optimization problem (Wu and Simpson et al., 2012 ), which contains a large number of non-convex and nonlinear constraints. Additionally, it has been demonstrated to be an NP-hard problem with extremely high computational complexity (Bagloee and Asadi et al., 2018 ; Zamzam and Dall Anese et al., 2018 ). The optimization algorithms for solving the optimization scheduling problem of WDS mainly include exact algorithm and heuristic algorithm. Exact algorithm models the optimization scheduling problem of WDS mathematically, and solves the problem using conventional deterministic mathematical models based on the problem's analytical features. The main methods include dynamic programming (DP) (Lansey and Awumah, 1994 ), linear programming (LP) (Price and Ostfeld, 2014 ), nonlinear programming (NLP) (Bonvin and Demassey et al., 2021 ), mixed integer nonlinear programming (MINLP) (Bragalli and D Ambrosio et al., 2012), mixed integer linear programming (MILP) (Liu and Barrows et al., 2020 ; Salomons and Housh, 2020 ), and hybrid solution (Vieira and Ramos, 2008 ). In addition to exact algorithm, another type of solving algorithm is the heuristic algorithm. Due to the large number of non-convex and nonlinear calculations caused by calculating pipe network adjustments and constraints, exact algorithms are difficult to obtain analytical solutions (Hooshmand and Jamalian et al., 2021 ). Therefore, heuristic algorithms are widely used to solve the optimization scheduling problem of WDS. The research on heuristic algorithms is very rich, with genetic algorithms (GA) (Mora-Melia and Iglesias-Rey et al., 2013 ; Gonzalez Perea and Angel Moreno et al., 2020 ) as the representative. The advantage of heuristic algorithms is that they do not require complex derivative calculations and initial values for decision variables. Compared with exact algorithms, these heuristic methods are more likely to obtain global optimal solutions. For small-scale pipe networks, using heuristic algorithms to solve optimal scheduling problem may have relatively high computational complexity. However, for larger-scale optimal scheduling problems, heuristic methods may be the only feasible solution method. Other commonly used heuristic algorithms include: fast non-dominated sorting genetic algorithm (NSGA-II) (Artina and Bragalli et al., 2012 ; Makaremi and Haghighi et al., 2017 ), NSGA-III (Tao and Yan et al., 2022 ), particle swarm optimization (PSO) (Patel and Goyal, 2016 ), ant colony optimization (ACO)(Afshar and Masoumi et al., 2015 ), and so on. Compared to PSO and ACO, genetic algorithms have better global search capabilities. However, NSGA-II can only handle low-dimensional optimization problems with a target dimension of ≤ 3, once the dimension increases, the non-dominated individuals in the population increase exponentially, making it difficult to distinguish between good and bad individuals based on Pareto dominance. NSGA-III algorithm is developed based on NSGA-II algorithm, and uses the reference point method to select individuals(Deb and Jain, 2013 ). NSGA-III is superior to NSGA-II in terms of algorithm robustness, solution quality, population diversity, and constraint scalability. Therefore, this research adopts NSGA-III algorithm. Overall, scholars have studied the optimal scheduling problem of WDS from different perspectives, some scholars such as (Kurek and Ostfeld, 2013 ) have conducted a comprehensive analysis, but it is only based on theoretical models. At present, there are limited real case studies available on the joint scheduling of water tank filling and pump frequency conversion. This research takes the WDS of S city in China as an example, utilizes NSGA-III algorithm to optimize the operation of the IPS. The results show that the IPS can play a role in energy conservation, water age optimization, and "peak shaving" while ensuring regional water supply demand. 2 Methodology 2.1 objective function In this research, two types of indicators are designed, energy consumption and water age, and the optimal scheduling model is constructed from the perspectives of hydraulics and water quality. 2.1.1 Energy consumption indicator While ensuring that the head of any node in WDS meets the minimum service head, the energy consumption of pump operation should be reduced as much as possible. The calculation method for the energy consumption of the water supply pump station is shown in Eq. ( 1 ). $$E=\sum _{t=1}^{N}\sum _{p=1}^{M}\frac{{\rho }\text{g}{\text{Q}}_{p}^{t} {\text{H}}_{p}^{t}\varDelta \text{t}}{{\eta }_{p}}$$ 1 Where the above variables are defined as below: E energy consumption of pump,(kw·h); 𝜌 specific gravity of water,(𝜌=1000 kg/m 3 ); G gravitational acceleration,(g = 9.81 m/s 2 ); t time t. \({\text{Q}}_{p}^{t}\) water supply flow rate of p th pump at time t,(m 3 /s); \({\text{H}}_{p}^{t}\) head of p th pump at time t,(m); $$\varDelta \text{t} \text{h}\text{y}\text{d}\text{r}\text{a}\text{u}\text{l}\text{i}\text{c} \text{c}\text{a}\text{l}\text{c}\text{u}\text{l}\text{a}\text{t}\text{i}\text{o}\text{n} \text{s}\text{t}\text{e}\text{p},(\varDelta \text{t} =3600\text{s});$$ \({\eta }_{p}\) the average efficiency of the pump station during the dispatching period is uniformly set to 100% by default; M total number of pumps; N total time of delay simulation,(N = 24) ; 2.1.2 Water age indicator While ensuring service pressure and energy consumption optimization, it is also necessary to ensure that the water age of the IPS is as short as possible to meet the water quality requirements. The average water age is calculated as shown in Eq. ( 2 ): $${\text{T}}_{\text{a}\text{v}\text{g}}^{i}=\frac{1}{N}\sum _{t=1}^{N}{\text{T}}_{t}^{i}$$ 2 where \({\text{T}}_{\text{a}\text{v}\text{g}}^{i}\) the average water age of i th IPS, (h); \({\text{T}}_{t}^{i}\) mixed water age of i th IPS at time t, (h). 2.1.3 Multi-objective function setting The fitness of different individuals in GA population is evaluated based on the total energy consumption of regional pump stations, the average water age of each IPS. Additionally, penalty function constraints are added separately to eliminate unsuitable solutions. The objective function of energy consumption indicator is shown in Eq. (3). F1= \(\sum _{i=1}^{m}{\text{E}}_{i}\) (3) where \({\text{E}}_{i}\) the total energy consumption of i th IPS, m the total number of IPSs. The objective function of water age indicator is shown in Eq. (4). F2(i) = \({\text{T}}_{\text{a}\text{v}\text{g}}^{i}\) (4) The comprehensive optimization target calculation is shown in formula (5). Minimize \(\text{F}={w}_{0}\text{F}1+\sum _{i=1}^{m}{w}_{i}\text{F}2\left(\text{i}\right)\) (5) where \({w}_{0}\) weight of the objective function F1; \({w}_{i}\) weight of the objective function F2(i); 2.2 Constraint conditions 2.2.1 Water level of water tank (1) water level range $${Y}_{{i}_{min}}\le {Y}_{i}\le {Y}_{{i}_{max}} (i=\text{1,2},\cdots {N}_{\text{Y}})$$ 6 where \({Y}_{{i}_{min}}\) minimum limit of water level, m; \({Y}_{{i}_{max}}\) maximum limit of water level, m; \({N}_{\text{Y}}\) The number of water tanks in all IPSs. As the outlet pipe of the water tank is positioned higher than the tank's bottom, complete drainage of the water in the tank is not possible. Therefore, in the following cases, the minimum limit of the water level \({Y}_{{i}_{min}}\) is taken as 0.2m, the maximum limit of the water level \({Y}_{{i}_{max}}\) is taken as 10.2m, and the maximum effective water depth is 10 meters. (2) 24-hour cycle water level (penalty function) Considering the periodic characteristics of the daily operation of the WDS, it is necessary to ensure that the water level in the water tank at 24:00 is the same as that at 0:00 at the same day. The expression is given in Eq. ( 7 ), and the penalty function is given in Eq. ( 8 ), which is added to F2(i). $${Y}_{{i}_{24}}={Y}_{{i}_{0}}(i=\text{1,2},\cdots {N}_{\text{Y}})$$ 7 $${\varDelta H}_{i}=\left|{Y}_{{i}_{24}}-{Y}_{{i}_{0}}\right|$$ 8 where \({Y}_{{i}_{0}}\) 、 \({Y}_{{i}_{24}}\) water level of water tank at 0:00 and 24:00, m; \({\varDelta H}_{i}\) The difference between the water level at 24:00 and 0:00 in i th tank, m. 2.2.2 Inlet and outlet water balance of water tank $$\sum _{n}{Q}_{i,n}^{t}=\frac{{S}_{i}}{\varDelta t}\left({Y}_{i}^{t}-{Y}_{i}^{t+\varDelta t}\right)$$ 9 where \({Q}_{i,\text{n}}^{t}\) the algebraic sum of the water volume of n nodes associated with the i th water tank at time t, i.e., the net water output, m 3 ; \({S}_{i}\) cross-sectional area of i th water tank, m 2 ; \({Y}_{i}^{t}\) water level of i th water tank at time t, m; \({Y}_{i}^{t+\varDelta t}\) water level of i th water tank at time t+∆t, m. 2.2.3 Hydraulic balance of the pipe network The hydraulic balance of the pipe network is shown in Eq. ( 10 ), which sums the flow rates of the pipe segments that flow into and out of node j. The flow rate into the node is set to a negative value, while the flow rate out of the node is set to a positive value. $$\left\{\begin{array}{c}{\sum }_{k\in {s}_{j}}(\pm {q}_{k})+{Q}_{j}=0\\ {h}_{k}={S}_{k}{q}_{k}^{2}-H\end{array}\right.$$ 10 where \(j\) total number of nodes; \(k\) total number of pipes; \({S}_{j}\) the association set of node j; \({q}_{k}\) flow of pipe, L/s; \({Q}_{j}\) flow of node j, L/s; \({h}_{k}\) head loss of pipeline, m; \({S}_{k}\) friction coefficient of pipeline; \(\text{H}\) pumping head, when no pump station is set, H is taken as 0, m. 2.2.4 Balance of water supply and demand $$\sum {Q}_{t}=\sum {D}_{t}$$ 11 where \(\sum {Q}_{t}\) total water supply, \({\text{m}}^{3}/\text{h}\) ; \(\sum {D}_{t}\) total water demand, \({\text{m}}^{3}/\text{h}\) . 2.2.5 Pressure of critical node (penalty function) The goal of pump boosting is to ensure that the pressure of critical nodes in WDS meet the minimum pressure requirement, and at the same time, does not exceed the minimum pressure too much for energy conservation reasons. Therefore, the maximum and minimum pressure limits for critical nodes should satisfy the Eq. ( 12 ): $${{H}_{{c}_{min}}\le H}_{c}\le {H}_{{c}_{max}} (c=\text{1,2})$$ 12 where \({H}_{{c}_{min}}\) minimum pressure limit for critical nodes, m; \({H}_{{c}_{max}}\) maximum pressure limit for critical nodes, m; c index of critical nodes. For the case where the pressure of critical node is below the minimum pressure limit or exceeds the maximum pressure limit, a penalty function f is set as shown in Eq. ( 13 ), which is added to F1. $$\text{m}\text{i}\text{n} \text{f} = \text{w}·\sum _{t=1}^{24}\left[\sum _{c=1}^{2}\left({H}_{{c}_{min}}^{t}-{H}_{j}^{t}\right) \text{o}\text{r} \sum _{c=1}^{2}({H}_{c}^{t}-{H}_{{c}_{max}}^{t})\right]$$ 13 where w the punishment coefficient, which is taken as 100 in the case study. 2.2.6 Pump operation mode There are two types of pumps for IPS, BP and WSTP, both of which are variable frequency pumps. The BP uses municipal water for direct boosting, which can utilize higher residual pressure; while the WSTP boosts the water from the storage tank, which can utilize lower residual pressure. Therefore, the BP is more energy-efficient. In order to better reduce energy consumption and reduce the number of times the pump is switched on and off, the following rules are applied in the case study: (1) The water tank only fills water during non-peak hours, while at the same time the storage pump is switched off;(2) The WSTP only works during peak hours, when the water tank stops filling water;(3) The BP can be switched on at any time during the period of 24 hours;(4) In theory, the speed ratio (r/r 0 ) of variable speed pumps is in the range of [0,1], where 0 represents 0Hz and 1 represents full frequency of 50Hz. However, the operation of variable frequency pumps at low frequencies can cause many hazards, such as severe motor heating, pump cavitation, vibration, and noise. Therefore, the specific speed ratio in the case is set to optimize in the range of [0.5,1], as shown in Eq. ( 14 ). $${0.5\le \text{r}/\text{r}}_{0}\le 1$$ 14 where r actual speed, r/min; r 0 rated speed, r/min. 2.3 Introduction to NSGA-III Algorithm This research uses the NSGA-Ⅲ algorithm to optimize decision variables, the flowchart of NSGA-Ⅲ algorithm is shown in Figure. 1. 3 Case study 3.1 Overview of the water distribution network S City belongs to a hilly area with a water supply area of 62 square kilometers, a pipeline length of 490 kilometers (100mm diameter or above), a water supply population of 800,000, a total water supply capacity of 380,000 m3/d, and an actual (average) water supply of 251,000 m3/d. The topology of WDS is shown in Figure. 2. Figure. 3 shows the 24-hour variation of water supply. From Fig. 3(a)(b)(c), we can see the fluctuation of water supply for the three water plants, and Figure (d) shows total water supply which reflects the daily fluctuation of water demand for users. The characteristics of water demand during the morning and evening peak hours are significant, with a valley value of approximately 5,000 ~ 7,000 m3/h, a peak value of 13,000 ~ 15,000m3/h, an average value of 10,000 ~ 11,000m3/h, and a maximum hourly peak coefficient of 1.4. The railway station (RS) and the municipal party committee (MPC) are two areas with high water demand, as shown in Figure. 2. Among them, the daily water demand of the RS area is 14,333m³ (about 5.7% of the total daily water supply), the daily water demand of the MPC area is 7,667m³ (about 3.1% of the total daily water supply). Due to the incomplete construction of urban pipe networks and pump stations, there is often a shortage of water supply pressure during peak hours (8:00–10:00, 21:00–23:00) in these two areas, and the water supply company often receives complaints from residents about the problem of water outage. Furthermore, the ground elevation fluctuation within the two regions is relatively large, and some users at high altitudes and high building floors have been always suffering from the water shortage. Improving the problem of insufficient pressure in local areas can generally be achieved in two ways. First, using a unified pressurized water supply model (i.e. By increasing the pressure of water from water plants) to solve the problem, but this approach can lead to excess pressure in most of the pipe network, high energy consumption, and high leakage loss. Second, adding local regulation and pressurization facilities, such as GWST pump station, BP station, and the IPS recommended in the case study. Comparatively speaking, the construction of an IPS to solve the problem of insufficient regional pressure and water supply security is theoretically a better choice, and is also the focus of this research. 3.2 Model construction and analysis The hydraulic model constructed in this project covers all pipelines with a diameter of 100mm or greater. The model data was selected on June 7, 2021, and the hydraulic model simulation time step was 1 hour, with a total duration of 24 time periods. After establishing the model, a verification assessment of model accuracy was conducted, and the model simulation results were consistent with the actual situation of the WDS. All pressure errors were within 1.0 meter, and flow errors were within 10%. Therefore, the model meets the analysis requirements. Using this model to analyze the current situation, the RS and MPC areas do experience insufficient pressure during peak hours. The minimum pressure of critical node in the RS area is around 8 meters (an 8-story building actually requires a water head of 36 meters), while the minimum pressure of critical node in the MPC area is around 12 meters (a 9-story building actually requires a water head of 40 meters). Therefore, water supply cannot be guaranteed. 3.3 Structural function design of IPSs According to the site survey, a reasonable location for the pump station was selected, and local pipeline network were modified to separate the transmission and distribution pipelines in the RS and MPC areas. Due to the limited available space and the unsuitability for excavation in the two areas, it is difficult to use conventional GWST. In this case, the IPS with a cylindrical water tank with a height of 11 meters is adopted. The main functions are as follows: (1) Adopting a dual mode water supply system of "BP + WSTP", the BP fully utilizes the residual water pressure at the inlet of pump, and the WSTP can also utilize the pressure of higher water level in the cylindrical water tank, achieving significant energy-saving effects. (2) By adopting intelligent water inlet and outlet control in the water tank, the optimal water age of the IPS can be achieved, which is conducive to ensuring downstream water quality and hygiene. (3) The water storage tank of the pump station can have a peak- shaving and valley-filling effect, which improves the ability to guarantee downstream water supply during peak periods, and significantly reduces the pressure fluctuation of the upstream pipeline network. Therefore, it is beneficial for the balanced operation of water plant pump stations. The structure diagram and model topology of the IPS are shown in Figure. 4. The IPS is simulated in the way of "water tank + pump" in the constructed pipe network model. In the model, two storage tanks are equivalently transformed into one water tank, and the inlet valve adopts a flow control valve (FCV) that can control the flow rate. The pump station's flow is designed based on the regional water demand, while the head of the pump station is designed according to the head demand at the critical node. The volume of the water tank is determined based on the water demand during the corresponding peak demand periods for a duration of 2 hours. The selection of parameters for the IPS at the RS and MPC are shown in Table 1 . Table 1 Selection of parameters for IPSs Water Supply Area Water tank Pump Number Diameter (m) Height (m) Volume (m 3 ) Q (m 3 /h) H (m) number category RS —— 836 27 1 BP 2 10 11 1700 836 45 1 WSTP MPC —— 448 24 1 BP 2 8 11 1000 448 44 1 WSTP 3.4 Operating mode design of IPS The operation mode of the pumps in the IPS includes three types: a. the BP works alone; b. the WSTP works alone; c. the BP and the WSTP work together. In order to minimize the energy consumption during operation and reduce the water age of the IPS, this scheme adopts plan c during peak hours and plan a during non-peak hours. There are two water tank filling modes: d. water filling throughout the whole period; e. water filling only during part of the period. In order to reduce the control complexity of the FCV, plan e is adopted for water tank filling. Water is stored during non-peak hours and supplied during peak hours, with two refilling and discharging cycles per day. In this case study, the design of the IPS operation mode is outlined in Table 2 . Table 2 Design of the IPS operation mode 3.5 Decision variables optimization based on NSGA-III algorithm 3.5.1 Optimization Model Setting Using the NSGA-III algorithm to optimize the operating frequency of the pump (BP and WSTP) and the inlet flow rate of the water tank, while ensuring pressure at critical nodes meets both maximum and minimum limit requirements, the goal of simultaneously optimizing the operating energy consumption and the water age of each IPS is achieved. With a simulation time step of 1 hour, an optimized operating plan is generated for the operating frequency of the pump and the inlet flow rate of the water tank within a day. 3.5.2 Genetic algorithm parameter settings For the optimization model mentioned above, the relevant parameters are as follows: Population: 100 Iteration times: 500 Probability of mutation: 0.01 The parameter F in differential evolution: 0.4 Recombination probability: 0.8 Normalized weight: w0 = 0.001, w1 = 0.1, w2 = 0.1 4 Results and discussion Through 500 iterations, the pareto front plot for the three objectives is shown in Figure. 5. The results marked with circles in the figure are the optimal solutions based on normalized weights. The results are as follows: F1 = 1921, F2(1) = 3.55, F2(2) = 5.38. F1 consists of two parts, where the actual energy consumption value is 1848 and the penalty function value is 73. The penalty function value is caused by the pressure of critical node slightly exceeding the maximum limit of set pressure. Optimization results of decision variables: The optimization results of pump speed ratio and water tank inflow are shown in Figure. 6. 4.1 Energy consumption analysis The energy consumption comparison data for different operating modes are shown in Table 3 . Compared with the local pressurization mode, the energy consumption of the unified pressurization water supply mode in water plants is too high, which is obviously an unfeasible mode. In the local pressurization modes, the mode of setting a GWST with pump pressurization has the highest \({\text{E}}_{\text{I}\text{P}\text{S}\text{s}}\) , which is the sum of energy consumption of all IPSs; if the energy consumption of water plants is considered, it also has the highest \({\text{E}}_{\text{t}\text{o}\text{t}\text{a}\text{l}}\) , which is defined as Eq. ( 15 ). $${\text{E}}_{\text{t}\text{o}\text{t}\text{a}\text{l}}={\text{E}}_{\text{I}\text{P}\text{S}\text{s}}+\sum _{i=1}^{n}{\text{E}\text{p}}_{i}$$ 15 where \({\text{E}\text{p}}_{i}\) is the energy consumption of water plant i. Although the booster mode has the lowest \({\text{E}}_{\text{I}\text{P}\text{S}\text{s}}\) , the \({\text{E}}_{\text{t}\text{o}\text{t}\text{a}\text{l}}\) of the booster mode even slightly exceeds that of the IPS mode. In addition, in the IPS mode, compared with conventional GWST, the head of the pump can be significantly reduced due to the higher available head of the cylindrical water tank. Based on experience, it is estimated that using a high cylindrical water tank can save about 10–20% energy compared to using a GWST. Overall, the comprehensive energy consumption of IPS mode is optimal. Table 3 Energy consumption comparison of different operating modes Energy consumption (KW·h) Water Plant A Water Plant B Water Plant C IPSs Total Unified pressurized water supply mode 4184986 1648833 61220 ———— 5895039 GWST mode 1974942 537543 450632 2670 2965787 Booster mode 1972242 540267 450319 1648 2964476 IPS mode 1968520 532403 450683 1848 2953454 Note: In this case, due to the fact that the pressure in the RS and MPC areas is mainly related to the pressure of water plant A&B, the unified water supply mode increases the water supply pressure of water plant A&B, resulting in a corresponding reduction in the water supply and energy consumption of water plant C. In the local pressurization mode, the water pressure of the three water plants is relatively balanced, and the water outflow and energy consumption are also relatively balanced. 4.2 Water age analysis By optimizing the inflow and outflow of the water tank, the water level variation of two areas is shown in Figure. 7(a). The water age variation of the water tank and the node with maximum water age (MWAN) in two areas are shown in Figure. 7(b). The results show that the average water age of the MWAN in the two areas is 7.87/7.79h respectively, and the maximum water age is 14.08/13.52h respectively, both of which meet the requirements and can ensure water quality safety. 4.3 Balance analysis of total water output of water plants The comparison of total water supply from the water plants before and after the construction of the IPSs is illustrated in Figure. 8, indicating that the IPS has played a certain role in peak-shaving and valley-filling. The fluctuation intensity of the total water supply from the water plants \({Q}_{V}\) is used as a metric for the efficiency of the water tank of the IPS to achieve "peak- shaving and valley-filling". \({Q}_{V}\) is calculated using the sample standard deviation formula, as shown in Eq. ( 16 ). Before the construction of the IPS, \({Q}_{V}\) was 2203 m3/h, and after the construction, \({Q}_{V}\) was 1904 m3/h. It can be seen that after the construction of the IPSs, the total water supply of the water plants is more balanced. If more regulation and storage facilities are used in the future, the total water supply curve of the water plant will gradually approach the average line. $${Q}_{V}=\sqrt{\frac{\sum _{t=1}^{N}{\left({Q}_{t}-\stackrel{-}{Q}\right)}^{2}}{N-1}}$$ 16 where \({Q}_{v}\) The fluctuation intensity of the total water supply from the water plants,m 3 /h; \({Q}_{t}\) The total outflow of the water plants at time t,m 3 /h; \(\stackrel{-}{Q}\) Average of total outflow within 24 hours,m 3 /h; 4.4 Pressure analysis of the critical nodes Through the construction of two regional IPSs, model simulation analysis shows that the pressure at the critical node of the RS/MPC reaches 36/40 meters respectively, meeting the water supply pressure requirements for the critical node in each area. By setting the pressure penalty function for the critical node, it is guaranteed that the pressure will not be overly redundant. In practical engineering, the pressure at the critical node can be maintained within a reasonable range through the application of end constant pressure control technology. 5 Conclusions This research proposes an IPS solution to solve the problem of insufficient pressure during peak hours in two local areas. The NSGA⁃Ⅲ multi-objective optimization algorithm is used to optimize the decision variables, and the optimal solution is verified by a hydraulic simulation using EPANET. The results demonstrate that energy conservation and water age optimization can be accomplished while ensuring a safe water supply, effectively reducing the fluctuation intensity of total water supply from the water plants. The relevant conclusions of this design case are as follows: 1. The unified pressurized water supply mode has high energy consumption and is not economical, which can cause pressure redundancy in most of WDS and increase the amount of water leakage. The mode of local pressurization is more economical and reasonable, which can make the spatiotemporal pressure of WDS more balanced. Among various modes of local pressurization, the IPS mode is optimal, which strikes a good balance between safe water supply and low energy costs. 2. By optimizing the operation mode of the water tank in the IPS, the average water age of the MWAN in the two areas is 7.87/7.79h respectively, and the maximum water age is 14.08/13.52h respectively, both of which meet the requirements of water quality. 3. The construction of IPSs can enhance the resilience of the WDS, reduce the fluctuation intensity of the total water supply from the water plants, which is 2203 m3/h and 1904 m3/h respectively before and after the construction of IPSs. It can be seen that the water plant's total supply is more balanced after the construction of IPSs. 4.The IPS designed in this case has the advantages of small footprint, good energy saving, adjustable storage, and controllable water age. Under the policy guidance of "energy conservation and dual carbon" and "urban resilience", it is likely to gain more market favor in the future, especially in densely populated cities with undulating terrain. NSGA-Ⅲ algorithm has the advantages of robustness, population diversity, and constraint scalability, and has good results for such multi-objective optimization problems. Declarations Author contributions Rui Li: Data collection, Modeling, Programming design, Methodology, Visualization, Initial draft writing; He Wang: Programming design, Visualization; Kunlun Xin: Funding acquisition, Supervision, Writing - Review and Suggestions; Tao Tao: Writing - Review and Suggestions. Acknowledgments We thank the water company for providing the basic information. We also thank the anonymous reviewers and editors for their comments and suggestions. Funding This work was financially supported by National Natural Science Foundation of China [grant numbers: 52270093]. Ethical Approval: Not applicable. Consent to Participate: All authors give their consent to participate. Consent to Publish: All authors give their consent to publish. Conflict of Interests: No potential conflict of interests was reported by the author(s). Data Availability Data will be made available on request. References Afshar, A. and F. Masoumi, et al. (2015). "Reliability Based Optimum Reservoir Design by Hybrid ACO-LP Algorithm." WATER RESOURCES MANAGEMENT 29 (6): 2045-2058. Al-Jasser, A. O. (2007). "Chlorine decay in drinking-water transmission and distribution systems: Pipe service age effect." Water research 41 (2): 387-396. Artina, S. and C. Bragalli, et al. (2012). 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"Boosting genetic algorithm performance in pump scheduling problems with a novel decision-variable representation." Journal of Water Resources Planning and Management 146 (5): 04020023. Costa, L. and B. D. Prata, et al. (2016). "A Branch-and-Bound Algorithm for Optimal Pump Scheduling in Water Distribution Networks." WATER RESOURCES MANAGEMENT 30 (3): 1037-1052. Deb, K. and H. Jain (2013). "An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints." IEEE transactions on evolutionary computation 18 (4): 577-601. Fooladivanda, D. and J. A. Taylor (2017). "Energy-optimal pump scheduling and water flow." IEEE Transactions on Control of Network Systems 5 (3): 1016-1026. Gonzalez Perea, R. and M. Angel Moreno, et al. (2020). "Decision Support System Based on Genetic Algorithms to Optimize the Daily Management of Water Abstraction from Multiple Groundwater Supply Sources." WATER RESOURCES MANAGEMENT 34 (15): 4739-4755. Hooshmand, F. and M. Jamalian, et al. (2021). "Efficient Two-Phase Algorithm to Solve Nonconvex MINLP Model of Pump Scheduling Problem." Journal of Water Resources Planning and Management 147 (8): 04021047. Kurek, W. and A. Ostfeld (2013). "Multi-objective optimization of water quality, pumps operation, and storage sizing of water distribution systems." Journal of environmental management 115 : 189-197. Lansey, K. E. and K. Awumah (1994). "Optimal pump operations considering pump switches." Journal of Water Resources Planning and Management 120 (1): 17-35. Liu, Y. and C. Barrows, et al. (2020). "Optimization framework to assess the demand response capacity of a water distribution system." Journal of Water Resources Planning and Management 146 (8): 04020063. Makaremi, Y. and A. Haghighi, et al. (2017). "Optimization of pump scheduling program in water supply systems using a self-adaptive NSGA-II; a review of theory to real application." Water Resources Management 31 : 1283-1304. Mala-Jetmarova, H. and A. Barton, et al. (2015). "Exploration of the Trade-Offs between Water Quality and Pumping Costs in Optimal Operation of Regional Multiquality Water Distribution Systems." Journal of Water Resources Planning and Management 141 (6): 04014077. Mora-Melia, D. and P. L. Iglesias-Rey, et al. (2013). "Design of Water Distribution Networks using a Pseudo-Genetic Algorithm and Sensitivity of Genetic Operators." WATER RESOURCES MANAGEMENT 27 (12): 4149-4162. Odan, F. K. and L. F. Ribeiro Reis, et al. (2015). "Real-time multiobjective optimization of operation of water supply systems." Journal of Water Resources Planning and Management 141 (9): 04015011. Oikonomou, K. and M. Parvania, et al. (2018). "Optimal demand response scheduling for water distribution systems." IEEE Transactions on Industrial Informatics 14 (11): 5112-5122. Patel, H. M. and R. V. Goyal (2016). Optimal Design of a Booster Chlorination System for a Drinking Water Distribution Network Using EPANET and PSO. World Environmental and Water Resources Congress 2016. Prasad, T. D. and G. A. Walters (2006). "Minimizing residence times by rerouting flows to improve water quality in distribution networks." Engineering Optimization 38 (8): 923-939. Price, E. and A. Ostfeld (2014). "Discrete pump scheduling and leakage control using linear programming for optimal operation of water distribution systems." Journal of hydraulic engineering 140 (6): 04014017. Quintiliani, C. and E. Creaco (2019). "Using additional time slots for improving pump control optimization based on trigger levels." Water Resources Management 33 : 3175-3186. Salomons, E. and M. Housh (2020). "A practical optimization scheme for real-time operation of water distribution systems." Journal of Water Resources Planning and Management 146 (4): 04020016. Shokoohi, M. and M. Tabesh, et al. (2017). "Water quality based multi-objective optimal design of water distribution systems." Water Resources Management 31 : 93-108. Tao, Y. and D. Yan, et al. (2022). "Multi-objective optimization of water distribution networks based on non-dominated sequencing genetic algorithm." PLOS ONE 17 (11): e0277954. Vieira, F. and H. M. Ramos (2008). "Hybrid solution and pump-storage optimization in water supply system efficiency: A case study." Energy Policy 36 (11): 4142-4148. Wu, W. and A. R. Simpson, et al. (2012). "Incorporation of variable-speed pumping in multiobjective genetic algorithm optimization of the design of water transmission systems." Journal of Water Resources Planning and Management 138 (5): 543-552. Zamzam, A. S. and E. Dall Anese, et al. (2018). "Optimal water–power flow-problem: Formulation and distributed optimal solution." IEEE Transactions on Control of Network Systems 6 (1): 37-47. Zhuan, X. and X. Xia (2013). "Optimal operation scheduling of a pumping station with multiple pumps." Applied Energy 104 : 250-257. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 30 Jan, 2024 Editor assigned by journal 28 Jan, 2024 First submitted to journal 27 Jan, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3906818","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":270357276,"identity":"08363dd1-84d4-4cf3-ac80-373d9b4f882d","order_by":0,"name":"Rui Li","email":"","orcid":"","institution":"Tongji University College of Environmental Science and Engineering","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Rui","middleName":"","lastName":"Li","suffix":""},{"id":270357277,"identity":"fa3838fe-158d-47d2-844f-fee51f9cd7be","order_by":1,"name":"He Wang","email":"","orcid":"","institution":"Tongji University College of Environmental Science and Engineering","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"He","middleName":"","lastName":"Wang","suffix":""},{"id":270357278,"identity":"84974682-c5ed-4429-9600-c636758ba082","order_by":2,"name":"Kunlun Xin","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyklEQVRIiWNgGAWjYDACHubGByCasRnEIU4LY7PBAVK1tEkcgHOI0WFw5mBb9YeKO3bN7QyMD962McibE9Ii2dvYduPAmWfJjc0MzIZz2xgMdzYQ0MLPz9h242Db4WSgX9ikedsYEgwOENDCBtRSANXC/psoLfy8jW0MQC12IFuYidIi2XOwWeLMmcMJjM2MzZJzzkkYbiCkxeBM8sEPFRWH7Q37Dx/88KbMRp6gLTCQuLGBsQFISxCpHgjs5YlXOwpGwSgYBSMNAACemEE2bQpjHQAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0001-5476-9374","institution":"Tongji University College of Environmental Science and 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1","display":"","copyAsset":false,"role":"figure","size":29101,"visible":true,"origin":"","legend":"\u003cp\u003eFlowchart of NSGA⁃Ⅲ algorithm\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3906818/v1/c753da42b1304809f434fb1f.png"},{"id":50546841,"identity":"3a910fa3-4b5d-46ab-91ac-237948af60c1","added_by":"auto","created_at":"2024-02-02 09:26:46","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":220757,"visible":true,"origin":"","legend":"\u003cp\u003eThe topology of WDS\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3906818/v1/9c776b9148cd75d82b6b19c4.png"},{"id":50546835,"identity":"c864abcf-e546-44cd-8c32-c3d3f05f738e","added_by":"auto","created_at":"2024-02-02 09:26:46","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":89011,"visible":true,"origin":"","legend":"\u003cp\u003e24-hour box line diagram of water supply\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3906818/v1/459f112f5a05e7e68488c7b6.png"},{"id":50546836,"identity":"c3261440-da81-4403-b316-1db2094a6f99","added_by":"auto","created_at":"2024-02-02 09:26:46","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":131602,"visible":true,"origin":"","legend":"\u003cp\u003eStructural diagram and model topology diagram of the IPS\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3906818/v1/50fb1fdacbbc54195df3a429.png"},{"id":50547274,"identity":"859e159f-cb98-4968-a6f0-0348422388bb","added_by":"auto","created_at":"2024-02-02 09:34:46","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":84454,"visible":true,"origin":"","legend":"\u003cp\u003ePareto front plot of F1,F2(1), F2(2) (Generations=500)\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-3906818/v1/6e96e4aa7c3d1b6eb96c4459.png"},{"id":50546839,"identity":"260a2edd-e726-4abd-8285-4ff4dd6423ad","added_by":"auto","created_at":"2024-02-02 09:26:46","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":65207,"visible":true,"origin":"","legend":"\u003cp\u003eOptimization results of pump speed ratio and water tank inflow\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3906818/v1/9704745345a907a44bd7757c.png"},{"id":50546837,"identity":"89ebf9c9-df28-4a22-9ad3-3899c2e1809e","added_by":"auto","created_at":"2024-02-02 09:26:46","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":94710,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Water level variation. (b) Water age of water tank and the MWAN.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-3906818/v1/8acc47790cdd0e7298f43bf1.png"},{"id":50546842,"identity":"64dffcc8-8922-4888-98ad-cee09be99db7","added_by":"auto","created_at":"2024-02-02 09:26:46","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":44790,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of total water supply from water plants before and after the construction of IPSs\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-3906818/v1/fea43708c599c29098c98161.png"},{"id":50547860,"identity":"e5bfbef7-d08d-496c-a558-ccd77aff1ff1","added_by":"auto","created_at":"2024-02-02 09:50:48","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1238882,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3906818/v1/da267aec-b792-427c-a2f8-577f94327f85.pdf"}],"financialInterests":"","formattedTitle":"Multi-objective optimization design of integrated pump station based on NSGA - III","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eWith the development of urban economy and population, the scale of cities is constantly expanding, and the demand for urban water supply is increasing. The contradiction between supply and demand has brought new challenges to water supply enterprises. The main problems are: first, the water pressure at some points of Water Distribution System(WDS) is insufficient during peak hours; second, the pressure fluctuation in WDS is sometimes significant; third, the energy consumption of water supply equipment is high; fourth, the leakage loss of WDS is large; fifth, water stays in storage facilities for longer periods of time, which can lead to the risk of poor water quality. Optimal scheduling is an important means of WDS operation and management, which plays an important role in ensuring water supply requirements and energy conservation.\u003c/p\u003e \u003cp\u003eIn recent years, some cities have overused pressurized water supply equipment for local pressurization, resulting in excessive water pumping from municipal main pipes, which leads to low pressure in the pipe network during peak hours, and water supply in some areas is not guaranteed. These problems have become one of the typical problems faced by WDS in the process of urbanization. The pump station equipped with a traditional ground water storage tank (GWST) can contribute to regulating water supply quantity; however, this method cannot effectively utilize the pressure of inlet water, resulting in higher energy consumption. Hence, a new type of pump station called integrated pump station (IPS) is adopted in the case study, which combines the booster pump (BP) with water storage tank pump (WSTP). The BP extracts water from the municipal pipeline network, whereas the WSTP retrieves water from a storage tank. This research mainly discusses the optimization scheduling problem for this new type of pump station.\u003c/p\u003e \u003cp\u003eThe decision variables in a traditional optimal scheduling problem consist of the state combinations of pumps and valves. The main objective function is the total energy consumption of water supply, and other objective functions such as total energy cost(Odan and Ribeiro Reis et al., 2015; Fooladivanda and Taylor, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Cimorelli and D Aniello et al., 2020), water age (Prasad and Walters, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2006\u003c/span\u003e; Al-Jasser, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2007\u003c/span\u003e), water quality (Mala-Jetmarova and Barton et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Shokoohi and Tabesh et al., \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) are also considered for multi-objective optimization. The constraints of optimal scheduling are numerous and complex, including pipe network hydraulic constraints (Bagirov and Barton et al., \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), node pressure control constraints (Makaremi and Haghighi et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), pump operation constraints (Zhuan and Xia, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Quintiliani and Creaco, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2019\u003c/span\u003e), water level constraints (Costa and Prata et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Oikonomou and Parvania et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), etc. While the direct scheduling model is relatively simple, optimizing it in a large and complex WDS poses a significant challenge. This is attributed to the substantial number of pumps and valves, further complicated by their different categorizations.\u003c/p\u003e \u003cp\u003eThe optimal scheduling of WDS is a typical mixed integer optimization problem (Wu and Simpson et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2012\u003c/span\u003e), which contains a large number of non-convex and nonlinear constraints. Additionally, it has been demonstrated to be an NP-hard problem with extremely high computational complexity (Bagloee and Asadi et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Zamzam and Dall Anese et al., \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). The optimization algorithms for solving the optimization scheduling problem of WDS mainly include exact algorithm and heuristic algorithm. Exact algorithm models the optimization scheduling problem of WDS mathematically, and solves the problem using conventional deterministic mathematical models based on the problem's analytical features. The main methods include dynamic programming (DP) (Lansey and Awumah, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1994\u003c/span\u003e), linear programming (LP) (Price and Ostfeld, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2014\u003c/span\u003e), nonlinear programming (NLP) (Bonvin and Demassey et al., \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2021\u003c/span\u003e), mixed integer nonlinear programming (MINLP) (Bragalli and D Ambrosio et al., 2012), mixed integer linear programming (MILP) (Liu and Barrows et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Salomons and Housh, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2020\u003c/span\u003e), and hybrid solution (Vieira and Ramos, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). In addition to exact algorithm, another type of solving algorithm is the heuristic algorithm. Due to the large number of non-convex and nonlinear calculations caused by calculating pipe network adjustments and constraints, exact algorithms are difficult to obtain analytical solutions (Hooshmand and Jamalian et al., \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Therefore, heuristic algorithms are widely used to solve the optimization scheduling problem of WDS. The research on heuristic algorithms is very rich, with genetic algorithms (GA) (Mora-Melia and Iglesias-Rey et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Gonzalez Perea and Angel Moreno et al., \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) as the representative. The advantage of heuristic algorithms is that they do not require complex derivative calculations and initial values for decision variables. Compared with exact algorithms, these heuristic methods are more likely to obtain global optimal solutions. For small-scale pipe networks, using heuristic algorithms to solve optimal scheduling problem may have relatively high computational complexity. However, for larger-scale optimal scheduling problems, heuristic methods may be the only feasible solution method. Other commonly used heuristic algorithms include: fast non-dominated sorting genetic algorithm (NSGA-II) (Artina and Bragalli et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Makaremi and Haghighi et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), NSGA-III (Tao and Yan et al., \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), particle swarm optimization (PSO) (Patel and Goyal, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2016\u003c/span\u003e), ant colony optimization (ACO)(Afshar and Masoumi et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2015\u003c/span\u003e), and so on.\u003c/p\u003e \u003cp\u003eCompared to PSO and ACO, genetic algorithms have better global search capabilities. However, NSGA-II can only handle low-dimensional optimization problems with a target dimension of \u0026le;\u0026thinsp;3, once the dimension increases, the non-dominated individuals in the population increase exponentially, making it difficult to distinguish between good and bad individuals based on Pareto dominance. NSGA-III algorithm is developed based on NSGA-II algorithm, and uses the reference point method to select individuals(Deb and Jain, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). NSGA-III is superior to NSGA-II in terms of algorithm robustness, solution quality, population diversity, and constraint scalability. Therefore, this research adopts NSGA-III algorithm.\u003c/p\u003e \u003cp\u003eOverall, scholars have studied the optimal scheduling problem of WDS from different perspectives, some scholars such as (Kurek and Ostfeld, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2013\u003c/span\u003e) have conducted a comprehensive analysis, but it is only based on theoretical models. At present, there are limited real case studies available on the joint scheduling of water tank filling and pump frequency conversion. This research takes the WDS of S city in China as an example, utilizes NSGA-III algorithm to optimize the operation of the IPS. The results show that the IPS can play a role in energy conservation, water age optimization, and \"peak shaving\" while ensuring regional water supply demand.\u003c/p\u003e"},{"header":"2 Methodology","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 objective function\u003c/h2\u003e \u003cp\u003eIn this research, two types of indicators are designed, energy consumption and water age, and the optimal scheduling model is constructed from the perspectives of hydraulics and water quality.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section3\"\u003e \u003ch2\u003e2.1.1 Energy consumption indicator\u003c/h2\u003e \u003cp\u003eWhile ensuring that the head of any node in WDS meets the minimum service head, the energy consumption of pump operation should be reduced as much as possible. The calculation method for the energy consumption of the water supply pump station is shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e).\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$E=\\sum _{t=1}^{N}\\sum _{p=1}^{M}\\frac{{\\rho }\\text{g}{\\text{Q}}_{p}^{t} {\\text{H}}_{p}^{t}\\varDelta \\text{t}}{{\\eta }_{p}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere the above variables are defined as below:\u003c/p\u003e \u003cp\u003eE energy consumption of pump,(kw\u0026middot;h);\u003c/p\u003e \u003cp\u003e\u0026#120588; specific gravity of water,(\u0026#120588;=1000 kg/m\u003csup\u003e3\u003c/sup\u003e );\u003c/p\u003e \u003cp\u003eG gravitational acceleration,(g\u0026thinsp;=\u0026thinsp;9.81 m/s\u003csup\u003e2\u003c/sup\u003e );\u003c/p\u003e \u003cp\u003e \u003cem\u003et\u003c/em\u003e time t.\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\text{Q}}_{p}^{t}\\)\u003c/span\u003e \u003c/span\u003e water supply flow rate of p\u003csup\u003eth\u003c/sup\u003e pump at time t,(m\u003csup\u003e3\u003c/sup\u003e/s);\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\text{H}}_{p}^{t}\\)\u003c/span\u003e \u003c/span\u003e head of p\u003csup\u003eth\u003c/sup\u003e pump at time t,(m);\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\varDelta \\text{t} \\text{h}\\text{y}\\text{d}\\text{r}\\text{a}\\text{u}\\text{l}\\text{i}\\text{c} \\text{c}\\text{a}\\text{l}\\text{c}\\text{u}\\text{l}\\text{a}\\text{t}\\text{i}\\text{o}\\text{n} \\text{s}\\text{t}\\text{e}\\text{p},(\\varDelta \\text{t} =3600\\text{s});$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\eta }_{p}\\)\u003c/span\u003e \u003c/span\u003e the average efficiency of the pump station during the dispatching period is uniformly set to 100% by default;\u003c/p\u003e \u003cp\u003eM total number of pumps;\u003c/p\u003e \u003cp\u003eN total time of delay simulation,(N\u0026thinsp;=\u0026thinsp;24) ;\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section3\"\u003e \u003ch2\u003e2.1.2 Water age indicator\u003c/h2\u003e \u003cp\u003eWhile ensuring service pressure and energy consumption optimization, it is also necessary to ensure that the water age of the IPS is as short as possible to meet the water quality requirements. The average water age is calculated as shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e):\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${\\text{T}}_{\\text{a}\\text{v}\\text{g}}^{i}=\\frac{1}{N}\\sum _{t=1}^{N}{\\text{T}}_{t}^{i}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\text{T}}_{\\text{a}\\text{v}\\text{g}}^{i}\\)\u003c/span\u003e \u003c/span\u003e the average water age of i\u003csup\u003eth\u003c/sup\u003e IPS, (h);\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\text{T}}_{t}^{i}\\)\u003c/span\u003e \u003c/span\u003e mixed water age of i\u003csup\u003eth\u003c/sup\u003e IPS at time t, (h).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e2.1.3 Multi-objective function setting\u003c/h2\u003e \u003cp\u003eThe fitness of different individuals in GA population is evaluated based on the total energy consumption of regional pump stations, the average water age of each IPS. Additionally, penalty function constraints are added separately to eliminate unsuitable solutions.\u003c/p\u003e \u003cp\u003eThe objective function of energy consumption indicator is shown in Eq.\u0026nbsp;(3).\u003c/p\u003e \u003cp\u003eF1=\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sum _{i=1}^{m}{\\text{E}}_{i}\\)\u003c/span\u003e\u003c/span\u003e (3)\u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\text{E}}_{i}\\)\u003c/span\u003e \u003c/span\u003e the total energy consumption of i\u003csup\u003eth\u003c/sup\u003e IPS,\u003c/p\u003e \u003cp\u003em the total number of IPSs.\u003c/p\u003e \u003cp\u003eThe objective function of water age indicator is shown in Eq.\u0026nbsp;(4).\u003c/p\u003e \u003cp\u003eF2(i) =\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{T}}_{\\text{a}\\text{v}\\text{g}}^{i}\\)\u003c/span\u003e\u003c/span\u003e (4)\u003c/p\u003e \u003cp\u003eThe comprehensive optimization target calculation is shown in formula (5).\u003c/p\u003e \u003cp\u003eMinimize \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{F}={w}_{0}\\text{F}1+\\sum _{i=1}^{m}{w}_{i}\\text{F}2\\left(\\text{i}\\right)\\)\u003c/span\u003e\u003c/span\u003e (5)\u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({w}_{0}\\)\u003c/span\u003e \u003c/span\u003e weight of the objective function F1;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({w}_{i}\\)\u003c/span\u003e \u003c/span\u003e weight of the objective function F2(i);\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Constraint conditions\u003c/h2\u003e \u003cdiv id=\"Sec8\" class=\"Section3\"\u003e \u003ch2\u003e2.2.1 Water level of water tank\u003c/h2\u003e \u003cp\u003e(1) water level range\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$${Y}_{{i}_{min}}\\le {Y}_{i}\\le {Y}_{{i}_{max}} (i=\\text{1,2},\\cdots {N}_{\\text{Y}})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({Y}_{{i}_{min}}\\)\u003c/span\u003e \u003c/span\u003e minimum limit of water level, m;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({Y}_{{i}_{max}}\\)\u003c/span\u003e \u003c/span\u003e maximum limit of water level, m;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({N}_{\\text{Y}}\\)\u003c/span\u003e \u003c/span\u003e The number of water tanks in all IPSs.\u003c/p\u003e \u003cp\u003eAs the outlet pipe of the water tank is positioned higher than the tank's bottom, complete drainage of the water in the tank is not possible. Therefore, in the following cases, the minimum limit of the water level \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y}_{{i}_{min}}\\)\u003c/span\u003e\u003c/span\u003e is taken as 0.2m, the maximum limit of the water level \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y}_{{i}_{max}}\\)\u003c/span\u003e\u003c/span\u003e is taken as 10.2m, and the maximum effective water depth is 10 meters.\u003c/p\u003e \u003cp\u003e(2) 24-hour cycle water level (penalty function)\u003c/p\u003e \u003cp\u003eConsidering the periodic characteristics of the daily operation of the WDS, it is necessary to ensure that the water level in the water tank at 24:00 is the same as that at 0:00 at the same day. The expression is given in Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e7\u003c/span\u003e), and the penalty function is given in Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e8\u003c/span\u003e), which is added to F2(i).\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${Y}_{{i}_{24}}={Y}_{{i}_{0}}(i=\\text{1,2},\\cdots {N}_{\\text{Y}})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${\\varDelta H}_{i}=\\left|{Y}_{{i}_{24}}-{Y}_{{i}_{0}}\\right|$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({Y}_{{i}_{0}}\\)\u003c/span\u003e \u003c/span\u003e、\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y}_{{i}_{24}}\\)\u003c/span\u003e\u003c/span\u003e water level of water tank at 0:00 and 24:00, m;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\varDelta H}_{i}\\)\u003c/span\u003e \u003c/span\u003e The difference between the water level at 24:00 and 0:00 in i\u003csup\u003eth\u003c/sup\u003e tank, m.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section3\"\u003e \u003ch2\u003e2.2.2 Inlet and outlet water balance of water tank\u003c/h2\u003e \u003cp\u003e \u003cdiv id=\"Equ6\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\sum _{n}{Q}_{i,n}^{t}=\\frac{{S}_{i}}{\\varDelta t}\\left({Y}_{i}^{t}-{Y}_{i}^{t+\\varDelta t}\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({Q}_{i,\\text{n}}^{t}\\)\u003c/span\u003e \u003c/span\u003e the algebraic sum of the water volume of n nodes associated with the i\u003csup\u003eth\u003c/sup\u003e water tank at time t, i.e., the net water output, m\u003csup\u003e3\u003c/sup\u003e;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({S}_{i}\\)\u003c/span\u003e \u003c/span\u003e cross-sectional area of i\u003csup\u003eth\u003c/sup\u003e water tank, m\u003csup\u003e2\u003c/sup\u003e;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({Y}_{i}^{t}\\)\u003c/span\u003e \u003c/span\u003e water level of i\u003csup\u003eth\u003c/sup\u003e water tank at time t, m;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({Y}_{i}^{t+\\varDelta t}\\)\u003c/span\u003e \u003c/span\u003e water level of i\u003csup\u003eth\u003c/sup\u003e water tank at time t+∆t, m.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section3\"\u003e \u003ch2\u003e2.2.3 Hydraulic balance of the pipe network\u003c/h2\u003e \u003cp\u003eThe hydraulic balance of the pipe network is shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e10\u003c/span\u003e), which sums the flow rates of the pipe segments that flow into and out of node j. The flow rate into the node is set to a negative value, while the flow rate out of the node is set to a positive value.\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\left\\{\\begin{array}{c}{\\sum }_{k\\in {s}_{j}}(\\pm {q}_{k})+{Q}_{j}=0\\\\ {h}_{k}={S}_{k}{q}_{k}^{2}-H\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(j\\)\u003c/span\u003e \u003c/span\u003e total number of nodes;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(k\\)\u003c/span\u003e \u003c/span\u003e total number of pipes;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({S}_{j}\\)\u003c/span\u003e \u003c/span\u003e the association set of node j;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({q}_{k}\\)\u003c/span\u003e \u003c/span\u003e flow of pipe, L/s;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({Q}_{j}\\)\u003c/span\u003e \u003c/span\u003e flow of node j, L/s;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({h}_{k}\\)\u003c/span\u003e \u003c/span\u003e head loss of pipeline, m;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({S}_{k}\\)\u003c/span\u003e \u003c/span\u003e friction coefficient of pipeline;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\text{H}\\)\u003c/span\u003e \u003c/span\u003e pumping head, when no pump station is set, H is taken as 0, m.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section3\"\u003e \u003ch2\u003e2.2.4 Balance of water supply and demand\u003c/h2\u003e \u003cp\u003e \u003cdiv id=\"Equ8\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\sum {Q}_{t}=\\sum {D}_{t}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\sum {Q}_{t}\\)\u003c/span\u003e \u003c/span\u003e total water supply, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{m}}^{3}/\\text{h}\\)\u003c/span\u003e\u003c/span\u003e;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\sum {D}_{t}\\)\u003c/span\u003e \u003c/span\u003e total water demand, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{m}}^{3}/\\text{h}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e \u003ch2\u003e2.2.5 Pressure of critical node (penalty function)\u003c/h2\u003e \u003cp\u003eThe goal of pump boosting is to ensure that the pressure of critical nodes in WDS meet the minimum pressure requirement, and at the same time, does not exceed the minimum pressure too much for energy conservation reasons. Therefore, the maximum and minimum pressure limits for critical nodes should satisfy the Eq.\u0026nbsp;(\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e12\u003c/span\u003e):\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$${{H}_{{c}_{min}}\\le H}_{c}\\le {H}_{{c}_{max}} (c=\\text{1,2})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({H}_{{c}_{min}}\\)\u003c/span\u003e \u003c/span\u003e minimum pressure limit for critical nodes, m;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({H}_{{c}_{max}}\\)\u003c/span\u003e \u003c/span\u003e maximum pressure limit for critical nodes, m;\u003c/p\u003e \u003cp\u003e \u003cem\u003ec\u003c/em\u003e index of critical nodes.\u003c/p\u003e \u003cp\u003eFor the case where the pressure of critical node is below the minimum pressure limit or exceeds the maximum pressure limit, a penalty function f is set as shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e13\u003c/span\u003e), which is added to F1.\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\text{m}\\text{i}\\text{n} \\text{f} = \\text{w}\u0026middot;\\sum _{t=1}^{24}\\left[\\sum _{c=1}^{2}\\left({H}_{{c}_{min}}^{t}-{H}_{j}^{t}\\right) \\text{o}\\text{r} \\sum _{c=1}^{2}({H}_{c}^{t}-{H}_{{c}_{max}}^{t})\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003ew the punishment coefficient, which is taken as 100 in the case study.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section3\"\u003e \u003ch2\u003e2.2.6 Pump operation mode\u003c/h2\u003e \u003cp\u003eThere are two types of pumps for IPS, BP and WSTP, both of which are variable frequency pumps. The BP uses municipal water for direct boosting, which can utilize higher residual pressure; while the WSTP boosts the water from the storage tank, which can utilize lower residual pressure. Therefore, the BP is more energy-efficient.\u003c/p\u003e \u003cp\u003eIn order to better reduce energy consumption and reduce the number of times the pump is switched on and off, the following rules are applied in the case study: (1) The water tank only fills water during non-peak hours, while at the same time the storage pump is switched off;(2) The WSTP only works during peak hours, when the water tank stops filling water;(3) The BP can be switched on at any time during the period of 24 hours;(4) In theory, the speed ratio (r/r\u003csub\u003e0\u003c/sub\u003e) of variable speed pumps is in the range of [0,1], where 0 represents 0Hz and 1 represents full frequency of 50Hz. However, the operation of variable frequency pumps at low frequencies can cause many hazards, such as severe motor heating, pump cavitation, vibration, and noise. Therefore, the specific speed ratio in the case is set to optimize in the range of [0.5,1], as shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e14\u003c/span\u003e).\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$${0.5\\le \\text{r}/\\text{r}}_{0}\\le 1$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003er actual speed, r/min;\u003c/p\u003e \u003cp\u003er\u003csub\u003e0\u003c/sub\u003e rated speed, r/min.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Introduction to NSGA-III Algorithm\u003c/h2\u003e \u003cp\u003eThis research uses the NSGA-Ⅲ algorithm to optimize decision variables, the flowchart of NSGA-Ⅲ algorithm is shown in Figure. 1.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"3 Case study","content":"\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Overview of the water distribution network\u003c/h2\u003e \u003cp\u003eS City belongs to a hilly area with a water supply area of 62 square kilometers, a pipeline length of 490 kilometers (100mm diameter or above), a water supply population of 800,000, a total water supply capacity of 380,000 m3/d, and an actual (average) water supply of 251,000 m3/d. The topology of WDS is shown in Figure. 2.\u003c/p\u003e \u003cp\u003eFigure. 3 shows the 24-hour variation of water supply. From Fig.\u0026nbsp;3(a)(b)(c), we can see the fluctuation of water supply for the three water plants, and Figure (d) shows total water supply which reflects the daily fluctuation of water demand for users. The characteristics of water demand during the morning and evening peak hours are significant, with a valley value of approximately 5,000\u0026thinsp;~\u0026thinsp;7,000 m3/h, a peak value of 13,000\u0026thinsp;~\u0026thinsp;15,000m3/h, an average value of 10,000\u0026thinsp;~\u0026thinsp;11,000m3/h, and a maximum hourly peak coefficient of 1.4.\u003c/p\u003e \u003cp\u003eThe railway station (RS) and the municipal party committee (MPC) are two areas with high water demand, as shown in Figure. 2. Among them, the daily water demand of the RS area is 14,333m\u0026sup3; (about 5.7% of the total daily water supply), the daily water demand of the MPC area is 7,667m\u0026sup3; (about 3.1% of the total daily water supply). Due to the incomplete construction of urban pipe networks and pump stations, there is often a shortage of water supply pressure during peak hours (8:00\u0026ndash;10:00, 21:00\u0026ndash;23:00) in these two areas, and the water supply company often receives complaints from residents about the problem of water outage. Furthermore, the ground elevation fluctuation within the two regions is relatively large, and some users at high altitudes and high building floors have been always suffering from the water shortage.\u003c/p\u003e \u003cp\u003eImproving the problem of insufficient pressure in local areas can generally be achieved in two ways. First, using a unified pressurized water supply model (i.e. By increasing the pressure of water from water plants) to solve the problem, but this approach can lead to excess pressure in most of the pipe network, high energy consumption, and high leakage loss. Second, adding local regulation and pressurization facilities, such as GWST pump station, BP station, and the IPS recommended in the case study. Comparatively speaking, the construction of an IPS to solve the problem of insufficient regional pressure and water supply security is theoretically a better choice, and is also the focus of this research.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Model construction and analysis\u003c/h2\u003e \u003cp\u003eThe hydraulic model constructed in this project covers all pipelines with a diameter of 100mm or greater. The model data was selected on June 7, 2021, and the hydraulic model simulation time step was 1 hour, with a total duration of 24 time periods. After establishing the model, a verification assessment of model accuracy was conducted, and the model simulation results were consistent with the actual situation of the WDS. All pressure errors were within 1.0 meter, and flow errors were within 10%. Therefore, the model meets the analysis requirements.\u003c/p\u003e \u003cp\u003eUsing this model to analyze the current situation, the RS and MPC areas do experience insufficient pressure during peak hours. The minimum pressure of critical node in the RS area is around 8 meters (an 8-story building actually requires a water head of 36 meters), while the minimum pressure of critical node in the MPC area is around 12 meters (a 9-story building actually requires a water head of 40 meters). Therefore, water supply cannot be guaranteed.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Structural function design of IPSs\u003c/h2\u003e \u003cp\u003eAccording to the site survey, a reasonable location for the pump station was selected, and local pipeline network were modified to separate the transmission and distribution pipelines in the RS and MPC areas. Due to the limited available space and the unsuitability for excavation in the two areas, it is difficult to use conventional GWST. In this case, the IPS with a cylindrical water tank with a height of 11 meters is adopted. The main functions are as follows:\u003c/p\u003e \u003cp\u003e(1) Adopting a dual mode water supply system of \"BP\u0026thinsp;+\u0026thinsp;WSTP\", the BP fully utilizes the residual water pressure at the inlet of pump, and the WSTP can also utilize the pressure of higher water level in the cylindrical water tank, achieving significant energy-saving effects.\u003c/p\u003e \u003cp\u003e(2) By adopting intelligent water inlet and outlet control in the water tank, the optimal water age of the IPS can be achieved, which is conducive to ensuring downstream water quality and hygiene.\u003c/p\u003e \u003cp\u003e(3) The water storage tank of the pump station can have a peak- shaving and valley-filling effect, which improves the ability to guarantee downstream water supply during peak periods, and significantly reduces the pressure fluctuation of the upstream pipeline network. Therefore, it is beneficial for the balanced operation of water plant pump stations.\u003c/p\u003e \u003cp\u003eThe structure diagram and model topology of the IPS are shown in Figure. 4. The IPS is simulated in the way of \"water tank\u0026thinsp;+\u0026thinsp;pump\" in the constructed pipe network model. In the model, two storage tanks are equivalently transformed into one water tank, and the inlet valve adopts a flow control valve (FCV) that can control the flow rate.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe pump station's flow is designed based on the regional water demand, while the head of the pump station is designed according to the head demand at the critical node. The volume of the water tank is determined based on the water demand during the corresponding peak demand periods for a duration of 2 hours. The selection of parameters for the IPS at the RS and MPC are shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cstrong\u003eTable 1\u0026nbsp;\u003c/strong\u003eSelection of parameters for IPSs\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"103%\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"11.11111111111111%\" rowspan=\"2\"\u003e\n \u003cp\u003eWater Supply Area\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"46.464646464646464%\" colspan=\"4\"\u003e\n \u003cp\u003eWater tank\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"42.42424242424242%\" colspan=\"4\"\u003e\n \u003cp\u003ePump\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"13.095238095238095%\"\u003e\n \u003cp\u003eNumber\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.476190476190476%\"\u003e\n \u003cp\u003eDiameter\u003c/p\u003e\n \u003cp\u003e(m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.714285714285714%\"\u003e\n \u003cp\u003eHeight\u003c/p\u003e\n \u003cp\u003e(m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.095238095238095%\"\u003e\n \u003cp\u003eVolume\u003c/p\u003e\n \u003cp\u003e(m\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.714285714285714%\"\u003e\n \u003cp\u003eQ\u003c/p\u003e\n \u003cp\u003e(m\u003csup\u003e3\u003c/sup\u003e/h)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.523809523809524%\"\u003e\n \u003cp\u003eH\u003c/p\u003e\n \u003cp\u003e(m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.095238095238095%\"\u003e\n \u003cp\u003enumber\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\"\u003e\n \u003cp\u003ecategory\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"11.34020618556701%\" rowspan=\"2\"\u003e\n \u003cp\u003eRS\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"47.422680412371136%\" colspan=\"4\"\u003e\n \u003cp\u003e\u0026mdash;\u0026mdash;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.278350515463918%\"\u003e\n \u003cp\u003e836\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.24742268041237%\"\u003e\n \u003cp\u003e27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.34020618556701%\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.371134020618557%\"\u003e\n \u003cp\u003eBP\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"13.095238095238095%\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.476190476190476%\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.714285714285714%\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.095238095238095%\"\u003e\n \u003cp\u003e1700\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.714285714285714%\"\u003e\n \u003cp\u003e836\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.523809523809524%\"\u003e\n \u003cp\u003e45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.095238095238095%\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\"\u003e\n \u003cp\u003eWSTP\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"11.34020618556701%\" rowspan=\"2\"\u003e\n \u003cp\u003eMPC\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"47.422680412371136%\" colspan=\"4\"\u003e\n \u003cp\u003e\u0026mdash;\u0026mdash;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.278350515463918%\"\u003e\n \u003cp\u003e448\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.24742268041237%\"\u003e\n \u003cp\u003e24\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.34020618556701%\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.371134020618557%\"\u003e\n \u003cp\u003eBP\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"13.095238095238095%\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.476190476190476%\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.714285714285714%\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.095238095238095%\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.714285714285714%\"\u003e\n \u003cp\u003e448\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.523809523809524%\"\u003e\n \u003cp\u003e44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.095238095238095%\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\"\u003e\n \u003cp\u003eWSTP\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Operating mode design of IPS\u003c/h2\u003e \u003cp\u003eThe operation mode of the pumps in the IPS includes three types: a. the BP works alone; b. the WSTP works alone; c. the BP and the WSTP work together. In order to minimize the energy consumption during operation and reduce the water age of the IPS, this scheme adopts plan c during peak hours and plan a during non-peak hours. There are two water tank filling modes: d. water filling throughout the whole period; e. water filling only during part of the period. In order to reduce the control complexity of the FCV, plan e is adopted for water tank filling. Water is stored during non-peak hours and supplied during peak hours, with two refilling and discharging cycles per day.\u003c/p\u003e \u003cp\u003eIn this case study, the design of the IPS operation mode is outlined in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \n\u003cp\u003e\u003cstrong\u003eTable 2\u0026nbsp;\u003c/strong\u003eDesign of the IPS operation mode\u003c/p\u003e\n\u003cp\u003e\u003cimg 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\" width=\"568\" height=\"178\"\u003e\u003c/p\u003e\n\u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Decision variables optimization based on NSGA-III algorithm\u003c/h2\u003e \u003cdiv id=\"Sec21\" class=\"Section3\"\u003e \u003ch2\u003e3.5.1 Optimization Model Setting\u003c/h2\u003e \u003cp\u003eUsing the NSGA-III algorithm to optimize the operating frequency of the pump (BP and WSTP) and the inlet flow rate of the water tank, while ensuring pressure at critical nodes meets both maximum and minimum limit requirements, the goal of simultaneously optimizing the operating energy consumption and the water age of each IPS is achieved. With a simulation time step of 1 hour, an optimized operating plan is generated for the operating frequency of the pump and the inlet flow rate of the water tank within a day.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section3\"\u003e \u003ch2\u003e3.5.2 Genetic algorithm parameter settings\u003c/h2\u003e \u003cp\u003eFor the optimization model mentioned above, the relevant parameters are as follows:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003ePopulation: 100\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eIteration times: 500\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eProbability of mutation: 0.01\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eThe parameter F in differential evolution: 0.4\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eRecombination probability: 0.8\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eNormalized weight: w0\u0026thinsp;=\u0026thinsp;0.001, w1\u0026thinsp;=\u0026thinsp;0.1, w2\u0026thinsp;=\u0026thinsp;0.1\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"4 Results and discussion","content":"\u003cp\u003eThrough 500 iterations, the pareto front plot for the three objectives is shown in Figure. 5. The results marked with circles in the figure are the optimal solutions based on normalized weights. The results are as follows:\u003c/p\u003e \u003cp\u003eF1\u0026thinsp;=\u0026thinsp;1921, F2(1)\u0026thinsp;=\u0026thinsp;3.55, F2(2)\u0026thinsp;=\u0026thinsp;5.38.\u003c/p\u003e \u003cp\u003eF1 consists of two parts, where the actual energy consumption value is 1848 and the penalty function value is 73. The penalty function value is caused by the pressure of critical node slightly exceeding the maximum limit of set pressure.\u003c/p\u003e \u003cp\u003eOptimization results of decision variables: The optimization results of pump speed ratio and water tank inflow are shown in Figure. 6.\u003c/p\u003e \u003cdiv id=\"Sec24\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Energy consumption analysis\u003c/h2\u003e \u003cp\u003eThe energy consumption comparison data for different operating modes are shown in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Compared with the local pressurization mode, the energy consumption of the unified pressurization water supply mode in water plants is too high, which is obviously an unfeasible mode. In the local pressurization modes, the mode of setting a GWST with pump pressurization has the highest \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{E}}_{\\text{I}\\text{P}\\text{S}\\text{s}}\\)\u003c/span\u003e\u003c/span\u003e, which is the sum of energy consumption of all IPSs; if the energy consumption of water plants is considered, it also has the highest \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{E}}_{\\text{t}\\text{o}\\text{t}\\text{a}\\text{l}}\\)\u003c/span\u003e\u003c/span\u003e, which is defined as Eq.\u0026nbsp;(\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e15\u003c/span\u003e).\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$${\\text{E}}_{\\text{t}\\text{o}\\text{t}\\text{a}\\text{l}}={\\text{E}}_{\\text{I}\\text{P}\\text{S}\\text{s}}+\\sum _{i=1}^{n}{\\text{E}\\text{p}}_{i}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{E}\\text{p}}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the energy consumption of water plant i.\u003c/p\u003e \u003cp\u003eAlthough the booster mode has the lowest \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{E}}_{\\text{I}\\text{P}\\text{S}\\text{s}}\\)\u003c/span\u003e\u003c/span\u003e, the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{E}}_{\\text{t}\\text{o}\\text{t}\\text{a}\\text{l}}\\)\u003c/span\u003e\u003c/span\u003e of the booster mode even slightly exceeds that of the IPS mode. In addition, in the IPS mode, compared with conventional GWST, the head of the pump can be significantly reduced due to the higher available head of the cylindrical water tank. Based on experience, it is estimated that using a high cylindrical water tank can save about 10\u0026ndash;20% energy compared to using a GWST. Overall, the comprehensive energy consumption of IPS mode is optimal.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eEnergy consumption comparison of different operating modes\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colspan=\"5\" nameend=\"c6\" namest=\"c2\"\u003e \u003cp\u003eEnergy consumption (KW\u0026middot;h)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWater Plant A\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eWater Plant B\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eWater Plant C\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eIPSs\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eTotal\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eUnified pressurized water supply mode\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e4184986\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1648833\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e61220\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026mdash;\u0026mdash;\u0026mdash;\u0026mdash;\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e5895039\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eGWST mode\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e\u003cb\u003e1974942\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e537543\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003e450632\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e2670\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e\u003cb\u003e2965787\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eBooster mode\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e\u003cb\u003e1972242\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e540267\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003e450319\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e1648\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e\u003cb\u003e2964476\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eIPS mode\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e\u003cb\u003e1968520\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e\u003cb\u003e532403\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003e450683\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e\u003cb\u003e1848\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e\u003cb\u003e2953454\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"6\"\u003eNote: In this case, due to the fact that the pressure in the RS and MPC areas is mainly related to the pressure of water plant A\u0026amp;B, the unified water supply mode increases the water supply pressure of water plant A\u0026amp;B, resulting in a corresponding reduction in the water supply and energy consumption of water plant C. In the local pressurization mode, the water pressure of the three water plants is relatively balanced, and the water outflow and energy consumption are also relatively balanced.\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec25\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Water age analysis\u003c/h2\u003e \u003cp\u003eBy optimizing the inflow and outflow of the water tank, the water level variation of two areas is shown in Figure. 7(a). The water age variation of the water tank and the node with maximum water age (MWAN) in two areas are shown in Figure. 7(b).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe results show that the average water age of the MWAN in the two areas is 7.87/7.79h respectively, and the maximum water age is 14.08/13.52h respectively, both of which meet the requirements and can ensure water quality safety.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec26\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Balance analysis of total water output of water plants\u003c/h2\u003e \u003cp\u003eThe comparison of total water supply from the water plants before and after the construction of the IPSs is illustrated in Figure. 8, indicating that the IPS has played a certain role in peak-shaving and valley-filling.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe fluctuation intensity of the total water supply from the water plants \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Q}_{V}\\)\u003c/span\u003e\u003c/span\u003e is used as a metric for the efficiency of the water tank of the IPS to achieve \"peak- shaving and valley-filling\". \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Q}_{V}\\)\u003c/span\u003e\u003c/span\u003e is calculated using the sample standard deviation formula, as shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ13\" class=\"InternalRef\"\u003e16\u003c/span\u003e). Before the construction of the IPS, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Q}_{V}\\)\u003c/span\u003e\u003c/span\u003e was 2203 m3/h, and after the construction, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Q}_{V}\\)\u003c/span\u003e\u003c/span\u003e was 1904 m3/h. It can be seen that after the construction of the IPSs, the total water supply of the water plants is more balanced. If more regulation and storage facilities are used in the future, the total water supply curve of the water plant will gradually approach the average line.\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$${Q}_{V}=\\sqrt{\\frac{\\sum _{t=1}^{N}{\\left({Q}_{t}-\\stackrel{-}{Q}\\right)}^{2}}{N-1}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({Q}_{v}\\)\u003c/span\u003e \u003c/span\u003e The fluctuation intensity of the total water supply from the water plants,m\u003csup\u003e3\u003c/sup\u003e/h;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({Q}_{t}\\)\u003c/span\u003e \u003c/span\u003e The total outflow of the water plants at time t,m\u003csup\u003e3\u003c/sup\u003e/h;\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\stackrel{-}{Q}\\)\u003c/span\u003e \u003c/span\u003e Average of total outflow within 24 hours,m\u003csup\u003e3\u003c/sup\u003e/h;\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec27\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Pressure analysis of the critical nodes\u003c/h2\u003e \u003cp\u003eThrough the construction of two regional IPSs, model simulation analysis shows that the pressure at the critical node of the RS/MPC reaches 36/40 meters respectively, meeting the water supply pressure requirements for the critical node in each area. By setting the pressure penalty function for the critical node, it is guaranteed that the pressure will not be overly redundant. In practical engineering, the pressure at the critical node can be maintained within a reasonable range through the application of end constant pressure control technology.\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Conclusions","content":"\u003cp\u003eThis research proposes an IPS solution to solve the problem of insufficient pressure during peak hours in two local areas. The NSGA⁃Ⅲ multi-objective optimization algorithm is used to optimize the decision variables, and the optimal solution is verified by a hydraulic simulation using EPANET. The results demonstrate that energy conservation and water age optimization can be accomplished while ensuring a safe water supply, effectively reducing the fluctuation intensity of total water supply from the water plants. The relevant conclusions of this design case are as follows:\u003c/p\u003e\u003cp\u003e1. The unified pressurized water supply mode has high energy consumption and is not economical, which can cause pressure redundancy in most of WDS and increase the amount of water leakage. The mode of local pressurization is more economical and reasonable, which can make the spatiotemporal pressure of WDS more balanced. Among various modes of local pressurization, the IPS mode is optimal, which strikes a good balance between safe water supply and low energy costs.\u003c/p\u003e\u003cp\u003e2. By optimizing the operation mode of the water tank in the IPS, the average water age of the MWAN in the two areas is 7.87/7.79h respectively, and the maximum water age is 14.08/13.52h respectively, both of which meet the requirements of water quality.\u003c/p\u003e \u003cp\u003e3. The construction of IPSs can enhance the resilience of the WDS, reduce the fluctuation intensity of the total water supply from the water plants, which is 2203 m3/h and 1904 m3/h respectively before and after the construction of IPSs. It can be seen that the water plant's total supply is more balanced after the construction of IPSs.\u003c/p\u003e \u003cp\u003e4.The IPS designed in this case has the advantages of small footprint, good energy saving, adjustable storage, and controllable water age. Under the policy guidance of \"energy conservation and dual carbon\" and \"urban resilience\", it is likely to gain more market favor in the future, especially in densely populated cities with undulating terrain. NSGA-Ⅲ algorithm has the advantages of robustness, population diversity, and constraint scalability, and has good results for such multi-objective optimization problems.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eAuthor contributions\u003c/p\u003e\n\u003cp\u003eRui Li: Data collection, Modeling, Programming design, Methodology, Visualization, Initial draft writing;\u003c/p\u003e\n\u003cp\u003eHe Wang: Programming design, Visualization;\u003c/p\u003e\n\u003cp\u003eKunlun Xin: Funding acquisition, Supervision, Writing - Review and Suggestions;\u003c/p\u003e\n\u003cp\u003eTao Tao: Writing - Review and Suggestions.\u003c/p\u003e\n\u003cp\u003eAcknowledgments\u003c/p\u003e\n\u003cp\u003eWe thank the water company for providing the basic information. We also thank the anonymous reviewers and editors for their comments and suggestions.\u003c/p\u003e\n\u003cp\u003eFunding\u003c/p\u003e\n\u003cp\u003eThis work was financially supported by National Natural Science Foundation of China [grant numbers: 52270093].\u003c/p\u003e\n\u003cp\u003eEthical Approval: Not applicable.\u003c/p\u003e\n\u003cp\u003eConsent to Participate: All authors give their consent to participate.\u003c/p\u003e\n\u003cp\u003eConsent to Publish: All authors give their consent to publish.\u003c/p\u003e\n\u003cp\u003eConflict of Interests: No potential conflict of interests was reported by the author(s).\u003c/p\u003e\n\u003cp\u003eData Availability\u003c/p\u003e\n\u003cp\u003eData will be made available on request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAfshar, A. and F. Masoumi, et al. 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Xia (2013). \u0026quot;Optimal operation scheduling of a pumping station with multiple pumps.\u0026quot; Applied Energy \u003cstrong\u003e104\u003c/strong\u003e: 250-257.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"water-resources-management","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"warm","sideBox":"Learn more about [Water Resources Management](https://www.springer.com/journal/11269)","snPcode":"11269","submissionUrl":"https://submission.nature.com/new-submission/11269/3","title":"Water Resources Management","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Energy consumption, Water age, NSGA-III, Integrated pump station, Multi objective optimization design","lastPublishedDoi":"10.21203/rs.3.rs-3906818/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3906818/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSafe water supply and energy conservation are two goals that the water supply industry has always focused on. Integrated pump station adopts a dual mode water supply system which has adjustable water storage capacity and can utilize the pressure of inlet water effectively, hence, it is a new type of solution in optimizing the operation of urban water supply systems. In order to solve the problem of insufficient water supply pressure during peak hours in the railway station and municipal party committee area of S city, the author constructed a pipe network hydraulic model, conducted a systematic analysis of the water supply pipe network, and optimized the design of IPSs using NSGA-III combined with EPANET. Finally, the hydraulic model of the pipe network is used to verify the feasibility of the scheme. The results show that under the premise of ensuring safe water supply, energy conservation and water age optimization can be achieved simultaneously, and the fluctuation intensity of the total water supply from the water plants is effectively reduced.\u003c/p\u003e","manuscriptTitle":"Multi-objective optimization design of integrated pump station based on NSGA - III","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-02-02 09:26:41","doi":"10.21203/rs.3.rs-3906818/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"","date":"2024-01-30T22:14:37+00:00","index":0,"fulltext":""},{"type":"editorAssigned","content":"","date":"2024-01-28T20:22:13+00:00","index":"","fulltext":""},{"type":"submitted","content":"Water Resources Management","date":"2024-01-27T22:17:53+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"water-resources-management","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"warm","sideBox":"Learn more about [Water Resources Management](https://www.springer.com/journal/11269)","snPcode":"11269","submissionUrl":"https://submission.nature.com/new-submission/11269/3","title":"Water Resources Management","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"e69293ad-a557-4a50-b19e-214b3c11b570","owner":[],"postedDate":"February 2nd, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2024-02-02T09:26:41+00:00","versionOfRecord":[],"versionCreatedAt":"2024-02-02 09:26:41","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3906818","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3906818","identity":"rs-3906818","version":["v1"]},"buildId":"7rjqhiLT3MXkJMwkYKINL","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}