Optimal Operational Scheduling of Available Partially Closed Valves for Pressure Management in Water Distribution Networks

Abstract

This paper investigates an effective and practical method to define optimal operational scheduling of available Partially Closed Valves (PCVs) for pressure management in Water Distribution Networks (WDNs). The aim is to determine which of the valves and how much does it is opened every hour of the day to maximize the Network Pressure Reliability Index (NPRI) of WDNs. The Particle Swarm Optimization (PSO) algorithm is applied to find which of valves needs adjustment by using the Pipe Closure Index (PCI) in a calibration process, and the Ant Colony Optimization (ACO) algorithm is used to find the setting of valves. The developed models are prepared by linking the optimization algorithm with the hydraulic simulator (EPANET) in the MATLAB. A sample network and real WDN are used to validate the proposed method. The comparison of the results for the real WDN indicates that by using the proposed method, the average reliability index increases up to 32.6%, and the average leakage rate decreases up to 31.7%. So the proposed method is effective in increasing the reliability and decreasing the leakage rate of the network.

Introduction

A WDN that supplies drinking water from the reservoirs to the demand nodes is made up of pipes, valves, pumps, tanks, etc. The duty of the WDN is to provide the consumer demands in the appropriate quality, quantity and pressure. It is not easy to achieve this aim due to the network topography, the temporal and spatial variability of nodal demand and the complexity of the network connections and also the reactions between the various substances contained within the water and the internal wall of the pipes. One of the priorities of the WDN managers is to reduce the water leakages, which depredate up to 30–40% of all the water that enters the network (Avila et al. 2019). To minimize water leakages, pressure management is one of the effective and cheapest techniques that are possible by optimal operational scheduling of valves.

Pressure management can be achieved through the pressure reducing valves (PRVs) scheduling (De Paola et al. 2017; Samir et al. 2017; Gungor et al. 2019; Avila et al. 2019), Throttle Control Valves (TCVs) or Partially Closed Valves (PCVs) scheduling (Araujo et al. 2006; Gencoglu and Merzib 2017; Do et al. 2018), Flow Control Valves (FCVs) scheduling (Ali 2015; Creaco and Pezzinga 2018), application of Variable Speed Pumps (VSPs) (Hashemi et al. 2013; Bonvin et al. 2017; Page et al. 2019), application of Pumps operating As Turbines (PATs) (De Marchis and Freni 2015; Patelisa et al. 2016; Venturini et al. 2017), using pump and PRVs (Gupta and Kulat 2018; Dini and Asadi 2019), simultaneous VSPs and FCVs scheduling (Khatavkar and Mays 2019), coupling the PRVs with the PATs (Darvini and Soldini 2015; Lydon et al. 2017), coupling PRVs and Boundary Valves (BVs) that open and close according to a pre-determined time schedule (Wright et al. 2015; Nerantzis et al. 2019), optimal location of hydropower turbines (Corcoran et al. 2016). All of these methods are typically used to reduce leakage rate, improve network efficiency and recover excess head energy in the WDN and also reduce the operational costs of companies. However there is need for the extra costs to purchase, install and operate the required equipment (PRVs, FCVs, etc.) in the network for various methods implemented to pressure management that some companies can’t provide these costs. Hence, it restricts the use of a large number of this equipment in the WDN. Given the hundreds of available PCVs in the real WDNs, there is no physical limit to the prime number of these valves. Furthermore, the use of available valves can be useful in reducing the mentioned costs.

Despite a more PCVs in the real WDNs, it is impossible to set all of them for pressure management in WDNs. Therefore, it is necessary to reduce the search space of valves location is based on the idea that certain pipes in the network are more significant than others. By regarding this condition, appropriate locations for the PCVs can be identified on the network that will be used during the optimal setting of valves. In previous researches, the authors suggested several feasible methods to find suitable locations of valves that will be found during the optimization and calibration process, whereas, only the physical properties (Araujo et al. 2006) or the hydraulic and physical properties of the pipes (Arulraj and Rao 1995; Ali 2015; Dini and Asadi 2019) are considered to find the selected values. This process usually reduced the search space of pipe selection, but there are weaknesses in these methods like the complex calculations or inappropriate accuracy that needs to be remedied. Therefore, it is necessary to expand novel methods for selecting very important pipes to operational scheduling of valves.

In the previous researches, the optimal location and setting of valves were found with the various algorithms such as the Genetic Algorithm (GA) (Ali 2015; Araujo et al. 2006; Do et al. 2018; Gupta and Kulat 2018; Khatavkar and Mays 2019), the Particle Swarm Optimization (PSO) algorithm (Gungor et al. 2019; Dini and Asadi 2019), the Harmony Search (HS) algorithm (De Paola et al. 2017), the Simulated Annealing (SA) algorithm (Avila et al. 2019) and the Ant Colony Optimization (ACO) algorithm (Hashemi et al. 2013). All of these algorithms can be applied well to different issues in the WDN. But the major difference between them is that they have the ability to solve discrete or continuous optimization problems and also they can be used conversely in these problems. It is clear the calibration of the WDN in the valve selection process is a continuous optimization problem and the PCV valves setting is a discrete optimization problem. By the way, it can be noted that the PSO algorithm is basically developed for continuous problems, while the ACO algorithm is generally expanded for discrete problems.

A large part of the total revenues of Iranian water companies is spent on paying staff salaries and necessary corrective actions in WDNs and there are not enough funds to supply extra costs for other operational programs by buying new equipment. The first aim of this research is to improve the operational condition of the network by using the available equipment. So the optimal operational scheduling of available PCVs is considered for pressure management of WDNs. The second aim of this research is to determine the optimal location and setting of PCVs that is need to be adjusted. For this purpose, a new step by step methodology is developed with maximizing the reliability index by applying the valves selection index. The NPRI index proposed by Dini and Tanesh (2017b) that is defined as a fuzzy number, is used as a reliability index. To choose the best PCVs, a new pipe selection index is introduced and the PSO algorithm is applied to find them in the calibration process. Also, the ACO algorithm is used to find the setting of PCVs. It is developed in the MATLAB code that the PSO and ACO algorithms are used to set optimization problem and the hydraulic model of the WDN is implemented in the EPANET software. The methodology is applied to a sample network and real WDN.

Methodology

The flowchart of the methodology is shown in Fig. 1. There are five steps in this flowchart that demonstrate an overview of the methodology. In the first step, all parameters of the WDN and PSO and ACO algorithms are defined. In the second step, the diameter of pipes is recalculated by using the PSO algorithm within a calibration process by maximizing the NPRI index. In the third and fourth steps, the PCI value of pipes is calculated and some pipes with the PCI value higher than the limited value are selected. In the fifth step, the setting of selected PCVs is found by taking the ACO algorithm with maximizing the NPRI index.

Fig. 1
figure1

Flowchart of the methodology

Hydraulic Model

The hydraulic model of the WDN carried out through the extended period simulation (EPS) is generated in the EPANET software (Rossman 2000). It is performed in two various methods. The first one is the Demand Driven Simulation Method (DDSM) that is assumed nodal outflows are fixed and always available at the nodes. This method is employed to simulate the nodal demand of the network. In the extended period simulation model, variations of demand in the nodes are calculated by applying the pattern demand coefficient. The second one is the Head Driven Simulation Method (HDSM) that is assumed, there is a relationship between nodal outflows and nodal pressures. This method is used to simulate nodal leakage of the network that, is calculated by Eq. (1). (Araujo et al. 2006).

$$ {q}_j=\left(C\ \sum \limits_{k=1}^M0.5\ {L}_{ik}\right)\ {P_k}^{\beta } $$
(1)

where qk: is the leakage flow at node k (l/s), Pk: is the service pressure at node k (m), C: is the discharge coefficient of the orifice which depends on the shape and the diameter of pipes (l/(s.m(β + 1))), Lik: is the pipe length between nodes i and k (m), M: is the number of pipes connected to the node k and β: is the nodal pressure exponent (β = 1.18).

Pipe Closure Index

The formulation of the PCI index that is proposed firstly in this paper is given by Eq. (2).

$$ {\mathrm{PCI}}_{\mathrm{i}}=\left(1-\frac{1}{T}\sum \limits_{t=1}^T\frac{D_{io}^t}{D_i}\right)\ast \frac{\frac{1}{T}{\sum}_{t=1}^T{Q}_i^t}{\sum \limits_{j=1}^N{q}_j^t} $$
(2)

where \( {D}_{io}^t \): is the optimal diameter of pipe i at time t (mm), that is calculated in the calibration process. Di: is the real diameter of each pipe (mm), \( {Q}_i^t \): is the flow rate of pipe i at time t (l/s),\( {q}_j^t \): is the nodal demand of node j at time t (l/s), N: is the number of nodes and T: is the simulation period time (T = 24 h).

Network Reliability

In this section, the NPRI index, proposed by Dini and Tabesh (2017b) is applied to evaluate the reliability of WDNs. This index has been used in various researches (Dini and Tanesh 2019; Dini and Asadi 2019) that act like a fuzzy number. It gives the performance value of 0–1 to a pressure value of 0–100. Figure 2 shows the utility function of the NPRI index for each node. Also, the network utility function is presented in Eq. (3).

$$ {\mathrm{NPRI}}_{\mathrm{t}}=\frac{\sum \limits_{\mathrm{j}=1}^{\mathrm{NN}}{\mathrm{Q}}_{\mathrm{j}.\mathrm{t}}^{\mathrm{req}}\left(\mathrm{NPRI}\left(\mathrm{j}.\mathrm{t}\right)\right)}{\sum \limits_{\mathrm{j}=1}^{\mathrm{NN}}{\mathrm{Q}}_{\mathrm{j}.\mathrm{t}}^{\mathrm{req}}} $$
(3)

where NPRIt: is the network pressure reliability index at time t, NPRI(j,t): is the nodal pressure reliability of node j at time t, \( {\mathrm{Q}}_{\mathrm{j}.\mathrm{t}}^{\mathrm{req}} \): is the required nodal demand of node j at time t (l/s) and NN: is the number of nodes.

Fig. 2
figure2

Utility function of nodal pressure reliability index

PSO Algorithm

The PSO algorithm first proposed by Eberhart and Kennedy (1995) is a mathematical method that optimizes a problem by possessing a population of candidate solutions and moving these particles around the search space according to the classic formula over the particle position and velocity. Each particle’s travel is affected by its local and global best-known position. The position and velocity of each particle are improved by the Eqs. (4) and (5).

$$ {\mathrm{x}}^{\mathrm{i}}\left(\mathrm{t}+1\right)={\mathrm{x}}^{\mathrm{i}}\left(\mathrm{t}\right)+{\mathrm{v}}^{\mathrm{i}}\left(\mathrm{t}+1\right) $$
(4)
$$ {\mathrm{V}}^{\mathrm{i}}\left(\mathrm{t}+1\right)=\mathrm{W}{\mathrm{V}}^{\mathrm{i}}\left(\mathrm{t}\right)+{\mathrm{C}}_1{\mathrm{r}}_1\left({\mathrm{x}}^{\mathrm{i}.\mathrm{best}}\left(\mathrm{t}\right)-{\mathrm{x}}^{\mathrm{i}}\left(\mathrm{t}\right)\right)+{\mathrm{C}}_2{\mathrm{r}}_2\left({\mathrm{x}}^{\mathrm{g}.\mathrm{best}}\left(\mathrm{t}\right)-{\mathrm{x}}^{\mathrm{i}}\left(\mathrm{t}\right)\right) $$
(5)

where xi(t + 1): is the position of each particle may be described by the vector xi (xi = [\( {x}_1^i \), \( {x}_2^i \), \( {x}_3^i \)..., \( {x}_{Ndv}^i \)], Ndv is the number of decision variable at time step (t + 1), xi(t):is the position of each particle at time step t, Vi(t + 1):is the velocity of each particle at time step (t + 1), Vi(t): is the velocity of each particle at time step (t), xi. best(t): is the best known position of each particle at time step t, xg. best(t): is the best known position of all particles in the search space at time step t, C1: is the acceleration coefficient that gives the importance of personal best value and C2: is the acceleration coefficient that gives the importance of social best value, r1, r2: are random numbers generated from a uniform distribution in [0, 1] and W: is the inertia weight parameter of the particle velocity.

ACO Algorithm

The ACO algorithm first proposed by Dorigo et al. (1996) is inspired by the pieces of knowledge and observation on ant colonies. One of the most interesting behaviors of the ants occurs in finding food, and particularly how to find the shortest path between nest and food resources which, shows the intelligent behavior of the ants. Artificial ants are defined to simulate the movement of real ants to find the best answer. The probability function of the ACO algorithm (Dini and Tabesh 2017a) is as Eq. (6).

$$ {P}_{ij}\left(k,t\right)=\frac{{\left[{T}_{ij}(t)\right]}^{\alpha }{\left[{U}_{ij}(t)\right]}^{\beta }}{\sum_{j=1}^N{\left[{T}_{ij}(t)\right]}^{\alpha }{\left[{U}_{ij}(t)\right]}^{\beta }} $$
(6)

where Pij(k, t): is the probability of the k-th ant situated on node j at time step t, to choose an outgoing edge i, Tij(t): is the pheromone intensity present on the edge i on node j at time step t: Uij(t) is the desirability factor present on the edge i on node j at time step t, and α, β: are the parameters controlling the relative importance of pheromone intensity and desirability for each ant’s decision. The pheromone intensity function is as Eq. (7):

$$ {T}_{i.j}\left(t+1\right)=\rho {T}_{i.j}(t)+\Delta {T}_{i.j}(t) $$
(7)

where ρ: is the pheromone persistence factor representing the pheromone evaporation rate, ∆Ti. j(t): is the pheromone addition on edge i on node j at time step t. The evaporation of the pheromone levels enables the colony to forget poor edges and increases the probability of good edges being selected, Ti. j(t + 1): is the pheromone intensity present on the edges i on node j at time step (t + 1).

Case Study

The first case study is used in this research is Jowitt and Xu (1990) network that contains 37 pipes, 26 nodes, and three reservoirs. This is a widely disseminated example for optimal location and setting of valves in several referenced literature that already used by other authors (Araujo et al. 2006; De Paola et al. 2017; Dini and Asadi 2019). The second case study is a real WDN that is the third zone of Maragheh Network in East Azerbaijan Province, Iran. It is a network with a large number of parallel pipes in any path, which requires a large number of valves to control the network pressure. The simplified network has 263 pipes, 168 nodes, and one reservoir.

Result and Discussions

First Case Study

In this section, the corrected network of Jowitt and Xu (1990) that is applied first by Dini and Asadi (2019) is used. It is clear each of the pipes may be damaged during the operational period, and it is necessary to disrupt the flow of water to repair them. Therefore, it is assumed the network has 37 pipes with 37 valves. It is unreasonable to do the optimal operational schedule for all of them. So by using the PCI index as a pipe selection index, the most important pipes with higher PCI values are selected. For this purpose, first, the diameter of all pipes is found by using the PSO algorithm with maximizing the NPRI index in the calibration process. Then the PCI value of each pipe is calculated. In this step, 24 pipes of 37 pipes are selected, but it is not reasonable to do the optimal operational schedule for this number of pipes, because it is an extremely large number compared with all pipes of the network. To reduce the number of selected pipes, they are grouped under six categories according to their priorities such as pipes connected in series or parallel status. Table 1 shows the selected pipes and PCVs in each category. Based on the results of Table 1, the PCV valves of pipes 1, 15, 18, 25, 27 and 31 have been selected as a candidate to be adjusted their optimal settings. Figure 3 shows the location of PCV valves in the network.

Table 1 Selected pipes and PCVs for the sample network
Fig. 3
figure3

Selected locations of PCVs according to PCI index

To adjustment of the selected PCVs, it is assumed that the completely closed and fully open modes were set to zero and 100% respectively. Also, 19 partially closed modes with an interval of 5% were set between them (0, 5, 10, 15, 20 …). The ACO algorithm is used to set the selected valves. Figure 4 shows the optimal setting of the PCV valves every hour of the day. It is clear, the valve PCV6 is completely closed all day long. Also, the valve PCV5 is completely closed all day long except 9 to 10 am, in which maximum water consumption happens during that time while, the valve PCV1 is in a fixed open mode of 15% every hour of the day. The other valves such as PCV2, PCV3 and PCV4 typically have different settings every hour of the day. In the meantime, the valves PCV2 and PCV4 are changed slowly while the valve PCV3 is changed drastically. The valve PCV3 is completely closed from 3 to 6 am when, the minimum water consumption happens during that time.

Fig. 4
figure4

Optimal setting of the PCV valves

Figure 5 shows, the NPRI index and the leakage rate of the sample network every hour of the day. It could be clear, the NPRI index increases and the leakage rate decreases in the network with optimized PCVs compared to the network without PCVs. In other words, the average NPRI index and leakage rate in the network without PCVs are about 74.1% and 29.2 l/s, while, in the network with optimized PCVs, they are about 94.8% and 22.6 l/s respectively. Therefore, after optimal scheduling of the selected PCVs, the average NPRI index of the network increases more than 27.7% and the average leakage rate of the network decreases more than 22.5%. A comparison of the results of this study with the paper of Dini and Asadi (2019) shows that the number of PCV valves of this study is more than the number of PRV valves of that paper, while, the location of similar valves is the same. But the performance of this study is better than that paper. For example, it is clear that, the average NPRI index and the average leakage rate of this study are about 94.8% and 22.6 l/s, while they are about 93.6% and 23.0 l/s in that paper respectively. On the other hand, in this study, the average reliability and leakage rate of the network have been improved up to 1.25 and 1.7% respectively compared to the paper of Dini and Asadi (2019), that, shows the efficiency of the proposed method.

Fig. 5
figure5

Leakage rate and NPRI index with and without PCVs

Second Case Study

Maragheh WDN typically consists of five zones. Figure 6 shows the third zone of this network that has been simplified by excluding dispensable pipes. The WDN model was obtained by hydraulic and water quality calibration of the network (Dini and Tabesh 2014, 2017a). To find the location of selected PCVs, first, the optimal diameter of pipes is found by using the PSO algorithm with maximizing the NPRI index. Then the PCI value of each pipe is calculated. Table 2 shows the selected pipes and PCVs in each category for the sample network. Based on the results of Table 2, 51 pipes with the highest PCI index and 12 PCVs have been selected. Figure 6 shows the location of PCVs in the real WDN. All of 12 selected valves except PCV2, PCV6 and PCV11 are closed all day long. It is because of possible parallel pipes in most paths at the network. Therefore, by closing some of the pipes, other parallel pipes provide water demands. These programs reduce the pressure and improve the reliability and leakage rate of the network. Figure 7 shows the optimal setting of PCV2, PCV6 and PCV11 every hour of the day. It is clear that, the valve PCV2 is completely closed all day long except 10 am to 3 pm and 8 pm, in which the maximum water consumption happens during that time. Also, the PCV6 and PCV11 have different opening status in a day, except 2 to 6 am, in which the minimum water consumption happens during that time.

Fig. 6
figure6

Layout of Maragheh WDN

Table 2 Selected pipes and PCVs in Maragheh WDN
Fig. 7
figure7

Optimal setting of the PCV valves

Figure 8 shows, the NPRI index and the leakage rate of the real WDN every hour of the day. It is clear that, the average NPRI index and leakage rate of the network without PCV valves are about 50.7% and 24.0 l/s, while, they are about 67.2% and 16.4 l/s respectively in the network with optimized PCVs. In other words, by doing optimal operational scheduling of the selected PCVs in the third zone of Maragheh WDN, the average NPRI index of the network increases more than 32.6% and the average leakage rate of the network decreases more than 31.7%. Therefore, the proposed method, along with economical cost-saving, has a significant improvement in the reliability and leakage rate of the network.

Fig. 8
figure8

Leakage rate and NPRI index with and without PCVs

Conclusion

This paper presented a new method to determine which of the available valves and how much does it is opened every hour of the day. It is done by maximizing the NPRI index and using a new PCI index. The methodology was based on the PSO and ACO algorithm that is written in the MATLAB code by linking the hydraulic simulator of EPANET 2.0 software. For verifying the proposed method, the sample network (Jowitt and Xu 1990 network) and the third zone of Maragheh WDN were applied.

A comparison of the results showed that, the number of PCV valves in this paper was more than the number of PRV valves of Dini and Asadi (2019) paper for the sample network, while, the location of similar valves was the same. But the average reliability and leakage rate of the network in this study up to 1.25 and 1.7% respectively better than the Dini and Asadi (2019) paper. Also the average reliability and leakage rare of the real WDN with optimized PCV valves improved more than 32.6 and 31.7% respectively in comparison with the network without PCV valves. Therefore, the proposed method has a significant impact on leakage reduction and reliability increment along with economical cost-saving.

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Dini, M., Asadi, A. Optimal Operational Scheduling of Available Partially Closed Valves for Pressure Management in Water Distribution Networks. Water Resour Manage 34, 2571–2583 (2020). https://doi.org/10.1007/s11269-020-02579-4

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Keywords

  • ACO
  • EPANET
  • MATLAB
  • NPRI
  • PCV
  • PSO
  • Operational scheduling
  • WDN