## Abstract

Intermittent water supply (IWS) is one of the effective methods to manage the consumption of urban water networks under water scarcity conditions. However, it is essential to minimize unfair water distribution in this method by defining a proper strategy. This study utilized the EPANET pressure-dependent hydraulic analysis and the gray wolf optimization algorithm to achieve maximum volumetric reliability under different scenarios in a district of the Hamedan urban water distribution network in Iran. The volumetric reliability of the network was evaluated in the IWS condition regardless of justice constraints, with the justice constraint, and by considering the leakage in the IWS network with the justice constraint. The first scenario demonstrated that the reliability decreased by an average of 4.6% for every meter of water level reduction in the tank. The second scenario revealed that the objective function was negligibly affected by the variation of the justice constraint; however, fluctuation of the water level in the tank significantly affected the volumetric reliability. In the third scenario, the objective function value was significantly impacted by leakage, ranging from 0 to 0.3 (representing the absence and presence of leakage in 30% of the nodes, respectively), resulting in an average decrease of about 17%.

## HIGHLIGHTS

A pressure-dependent hydraulic analysis in the intermittent water supply (IWS) under various scenarios.

Volumetric reliability assessment in an IWS network.

Evaluating the justice index in the IWS network.

Investigating the effect of leakage on the performance of the equitable IWS.

Development of simulation-optimization model in IWS.

## INTRODUCTION

Water shortage and technical problems in the network can disturb the provision of the required drinking water. Such problems can be caused by climatic changes, natural disasters, conflicts, intentional or accidental water pollution, or even insufficient hydraulic capacity of the network (Solgi *et al.* 2015, 2020; Bozorg-Haddad *et al.* 2016; Mohammadi *et al.* 2020; Nyahora *et al.* 2020). Access to water resources and the amount of water needed are affected by the mentioned factors, and meeting the current needs of consumers is one of the main challenges faced by urban water supply systems (Mokssit *et al.* 2018).

There are different solutions for addressing water crises. For example, water can be supplied to consumers continuously using recycled water or with lower-than-optimal pressure or supplied through rationing, which may include intermittent water supply (IWS). IWS is a common practice in many cities, particularly in developing countries (Soltanjalili *et al.* 2013; Bozorg-Haddad *et al.* 2016). Distribution networks with continuous water supply systems provide the water needs of consumers permanently, and the network is under pressure 24 h a day. On the other hand, IWS refers to distribution networks where the available water is insufficient to meet the needs of all consumers. In such cases, water is delivered to consumers for less than 24 h a day, and some consumers may have their water supply cut off at certain times (Bozorg-Haddad *et al.* 2016; Taylor *et al.* 2019). To evaluate the behavior of the system, we need to develop a hydraulic model for the IWS network. In a comprehensive study, Sarisen *et al.* (2022) review different methods for IWS hydraulic modeling. They found that for accurate modeling of IWS, some aspects of the system, such as water losses, pressure-deficient conditions, and the filling and emptying process, should be considered.

Totsuka *et al.* (2004) stated that three conditions create intermittent supply. The first case occurs when the available water (untreated) is insufficient to meet consumers' demand, resulting from absolute water scarcity. The second case occurs when existing economic facilities do not allow for an increase in the supply of purified water or expansion of the network, resulting in water scarcity caused by economic factors. The third case occurs when the network supply can be done continuously, despite repairs of leaks and operational works. However, weak technical management in the network causes technical scarcity, resulting in IWS (Haider *et al.* 2019; Taylor *et al.* 2019).

One of the main problems associated with IWS is unfair water distribution, a social issue. Certain parts of the network may have to wait longer for water than others located closer to the source. This unfairness is more noticeable when the network is filled with more demand than supply. Consumers closer to the source will benefit, while consumers farther away will face issues due to the longer time it takes for water to reach them. Additionally, demand in intermittent supply systems is pressure-dependent. Consumers tend to store large amounts of water in their personal tanks when the pressure is sufficient, decreasing water pressure during supply hours and disadvantaging those located further away from the source. Intermittent operation of urban water distribution networks is undesirable; however, it may be unavoidable depending on specific conditions (Solgi *et al.* 2020). As a result, IWS should be considered a last resort during a water shortage, and its use should be prevented through proper planning and appropriate measures during critical times (Ilaya-Ayza *et al.* 2017).

The level of network performance under different operating conditions can be evaluated by an index called reliability, which is an essential aspect of the design and operation of any system. A higher value of the reliability index in the distribution network during critical conditions indicates a higher level of trust in the network's performance under these circumstances. This means that fewer customers will be affected during abnormal conditions. In other words, reliability is an inherent characteristic of every system and should be considered an essential design and management parameter. Goulter (1995) used the reliability criterion as an indicator to check the extent to which the water distribution network fulfills its duties of providing the desired flow with adequate pressure and quality, even during abnormal conditions such as pipe breaks. Therefore, reliability can be defined as the probability of meeting a certain percentage of needs within a certain period. Reliability refers to the network's ability to provide consumers with their desired quantity and quality of water, both under normal and abnormal conditions (Nyahora *et al.* 2020). Gottipati & Nanduri (2014) stated that the reliability of a water supply system should consider all possible factors, such as changes in demand, water availability at the source, failures of components like pumps, treatment units, valves, and pipes, and the interplay of these factors in the system's performance.

IWS can complicate the operation of the water distribution network and lead to consumer dissatisfaction. Therefore, it is crucial to incorporate the principles of justice and fairness in the network. From this point of view, Solgi *et al.* (2015) developed an optimization model that incorporates the principles of justice and fairness in the network. The objective function of the optimizer model considered these two principles and was solved using the honeybee mating optimization algorithm (HBMO) along with a hydraulic simulation model. The results showed that this model could optimize the scheduling of IWS while preserving the principles of justice and fairness in all network nodes, even under severe water shortage conditions. Nyahora *et al.* (2020) developed a multi-objective optimization model using a genetic algorithm to maximize justice and reliability. They concluded that fairness and reliability could be practical and guiding objectives and criteria for decision-making.

In water distribution networks with intermittent supply conditions, water pressure can decrease significantly during supply hours, particularly in the early hours. Therefore, enhancing reliability in various conditions to improve consumer satisfaction is an important goal of this study. The article aims to create a simulation-optimization model for a real distribution network under different water shortage scenarios while considering justice in the system and the effect of leakage on network reliability which has not been studied in previous research.

## METHODS

### Hydraulic model

*et al.*2010). The pressure-flow equation used in this study is Equation (1), presented by Shirzad

*et al.*(2013), which is one of the most recent equations in this field:where is the available head at node

*j*; and are the available and required flow rates at node

*j*.

### Optimization model

The gray wolf optimization (GWO) algorithm introduced by Mirjalili *et al.* (2014) was used to solve the optimization problem of this study. This algorithm was derived from the group life and hunting mechanism of gray wolves. The group life of wolves has a precise and orderly social hierarchy. There are four groups which are called *α*, *β, δ*, and *ω* (from top to bottom). The *α* wolf is the group leader and is responsible for all decisions and management of the herd. The *β* wolf is at the second level and assists the *α* wolf in making decisions and other group activities. The third level is occupied by the delta *δ* wolf pack, which is under the command of the *α* and *β* wolves and dominates the *ω* wolf, which is responsible for the safety and integrity of the entire pack and typically obeys all wolves of higher levels.

In GWO, the wolves in the first three levels are considered the best answers in the search space, and the rest of the wolves must update their position in the search space based on these solutions.

GWO is competitive with other popular metaheuristic algorithms, such as differential evolution (DE), genetic algorithms (GA), and particle swarm optimization (PSO) algorithms (Masoumi *et al.* 2021, 2022).

*t*represents the current iteration,

*A*and

*C*are coefficient vectors towards the best position defined by Equations (4) and (5), respectively.

*X*

_{p}and

*X*represent the position vector of the prey and the gray wolf, respectively.where

*a*is a coefficient that linearly decreases from 0 to 2 throughout iterations, and

*r*

_{1}and

*r*

_{2}are random vectors within the range [0, 1].

*ω*)) are forced to update their positions according to the better positions found by the best search agents (the best solutions). Equations (6)–(12) formulate this process:

*X* and *X*_{(t+1)} are the variable positions in the current and next state, and *X _{α}*,

*X*, and

_{β}*X*are the positions of

_{δ}*α*,

*β,*and

*δ*wolves, respectively.

*NCNode*and

*NH*are the number of consumption nodes of the network and the number of hydraulic time steps during the shortage period, respectively. Also, and are the water volume of supply and demand at node

*i*and in the hydraulic time step

*h*.

*et al.*2015):where is the number of consumption nodes in the distribution network. and are the total supply at nodes

*i*and

*i′*in the intermittent supply period (unit of time or volume), respectively. and are the total demand of nodes

*i*and

*i′*in the period of intermittent supply (unit of time or volume).

### Leakage–pressure relationship

*i*) can be calculated as the following equation (Giustolisi

*et al.*2008):where is the constant coefficient of leakage (depends on pipe characteristics such as age and external factors such as environmental conditions, traffic load, external stresses, corrosion, etc.).

*P*is the pressure at the orifice, and

_{i}*N*presents the pressure power (based on the variety of the leak hole and pipe specifications, such as material and rigidity, its value changes from 0.5 to 1.5).

## CASE STUDY

### Specifications of distribution network

### Considered scenarios

The objective of this research was to maximize the volumetric reliability of the water supply network in three different scenarios as follows:

- 1.
Regardless of the justice constraint and without considering leakage in the network.

- 2.
Using justice constraint without any leakage in the network. We used three compliance ratings of 25, 50, and 75%.

- 3.
The same justice constraints as the second scenario are accompanied by leakage in the network. We randomly assigned the leakage to 10, 20, and 30% of the nodes.

Sixteen modes of rationing were considered for IWS. In other words, 16 states were defined where water was cut off during periods when the water consumption coefficient was zero. Different permutations of omitting the consumption at different hours were used. The rationing period was divided into 3-h intervals from 8 a.m. to 8 p.m. The objective is to allocate the rationing among users to maximize the network's reliability. Choosing each rationing pattern for each node depends on which pattern is deemed appropriate by the optimization algorithm. Table S1 in the supplementary information shows the 16 states of the rationing pattern.

## RESULTS AND DISCUSSION

The method proposed in this research was applied to the desired network in different scenarios, and the results of the objective function in the considered scenarios are presented in the following tables. Table 1 shows the results of the objective function value of volumetric reliability, where the justice index is regarded as zero under varying water levels in the tank during IWS conditions. As shown in the table, the volumetric reliability decreases by an average of 4.6% for every 1-m decrease in the water level in the tank.

Water level (m) . | Reliability . | Water level (m) . | Reliability . |
---|---|---|---|

17 | 0.9760 | 14 | 0.9086 |

16 | 0.9621 | 13 | 0.8707 |

15 | 0.9395 | 12 | 0.8207 |

Water level (m) . | Reliability . | Water level (m) . | Reliability . |
---|---|---|---|

17 | 0.9760 | 14 | 0.9086 |

16 | 0.9621 | 13 | 0.8707 |

15 | 0.9395 | 12 | 0.8207 |

Table 2 shows the values of the objective function in the second scenario. The justice constraint is observed in the system with varying justice indices of 0.25, 0.5, and 0.75 at different water levels in the tank. As shown in Table 2, the change in the justice constraint does not significantly affect the value of the objective function. However, this does not necessarily mean that the status of the decision variables is the same in all justice constraint situations. Instead, the status of the decision variables may differ, even though the output or objective function remains relatively constant.

Water level in tank (m) . | Justice Index . | Reliability . | Water level in tank (m) . | Justice Index . | Reliability . |
---|---|---|---|---|---|

17 | 0.25 | 0.9659 | 16 | 0.25 | 0.9596 |

0.5 | 0.9650 | 0.5 | 0.9563 | ||

0.75 | 0.9695 | 0.75 | 0.9617 | ||

15 | 0.25 | 0.9391 | 14 | 0.25 | 0.9074 |

0.5 | 0.9328 | 0.5 | 0.8861 | ||

0.75 | 0.9391 | 0.75 | 0.9074 | ||

13 | 0.25 | 0.8705 | 12 | 0.25 | 0.8172 |

0.5 | 0.8520 | 0.5 | 0.8154 | ||

0.75 | 0.8705 | 0.75 | 0.7997 |

Water level in tank (m) . | Justice Index . | Reliability . | Water level in tank (m) . | Justice Index . | Reliability . |
---|---|---|---|---|---|

17 | 0.25 | 0.9659 | 16 | 0.25 | 0.9596 |

0.5 | 0.9650 | 0.5 | 0.9563 | ||

0.75 | 0.9695 | 0.75 | 0.9617 | ||

15 | 0.25 | 0.9391 | 14 | 0.25 | 0.9074 |

0.5 | 0.9328 | 0.5 | 0.8861 | ||

0.75 | 0.9391 | 0.75 | 0.9074 | ||

13 | 0.25 | 0.8705 | 12 | 0.25 | 0.8172 |

0.5 | 0.8520 | 0.5 | 0.8154 | ||

0.75 | 0.8705 | 0.75 | 0.7997 |

In order to verify the model's performance, the above calculations were repeated for increased demand using the maximum hourly consumption coefficient (1.75) and the maximum daily consumption coefficient (1.25) of the network. The demand increase factor was calculated by multiplying these two values (2.19), and once again, assuming that only half of this increase occurs (1.59), the optimization was performed. The results are presented in Table 3.

. | Water level in tank (m) . | Justice Index . | Reliability . | Water level in tank (m) . | Justice Index . | Reliability . |
---|---|---|---|---|---|---|

Demand increase factor = 2.19 | 17 | 0 | 0.8210 | 16 | 0 | 0.7808 |

0.25 | 0.8194 | 0.25 | 0.7762 | |||

0.5 | 0.8182 | 0.5 | 0.7733 | |||

0.75 | 0.8071 | 0.75 | 0.7653 | |||

15 | 0 | 0.7340 | 14 | 0 | 0.6899 | |

0.25 | 0.7303 | 0.25 | 0.6878 | |||

0.5 | 0.7272 | 0.5 | 0.6792 | |||

0.75 | 0.7219 | 0.75 | 0.6686 | |||

13 | 0 | 0.6405 | 12 | 0 | 0.5901 | |

0.25 | 0.6397 | 0.25 | 0.5882 | |||

0.5 | 0.6387 | 0.5 | 0.5838 | |||

0.75 | 0.6257 | 0.75 | 0.5829 | |||

Demand increase factor = 1.59 | 17 | 0 | 0.9225 | 16 | 0 | 0.8962 |

0.25 | 0.9137 | 0.25 | 0.8856 | |||

0.5 | 0.9136 | 0.5 | 0.8836 | |||

0.75 | 0.9113 | 0.75 | 0.8755 | |||

15 | 0 | 0.8588 | 14 | 0 | 0.8528 | |

0.25 | 0.8553 | 0.25 | 0.8157 | |||

0.5 | 0.8505 | 0.5 | 0.8009 | |||

0.75 | 0.8456 | 0.75 | 0.7959 | |||

13 | 0 | 0.7623 | 12 | 0 | 0.7136 | |

0.25 | 0.7618 | 0.25 | 0.7105 | |||

0.5 | 0.7541 | 0.5 | 0.6905 | |||

0.75 | 0.7416 | 0.75 | 0.6858 |

. | Water level in tank (m) . | Justice Index . | Reliability . | Water level in tank (m) . | Justice Index . | Reliability . |
---|---|---|---|---|---|---|

Demand increase factor = 2.19 | 17 | 0 | 0.8210 | 16 | 0 | 0.7808 |

0.25 | 0.8194 | 0.25 | 0.7762 | |||

0.5 | 0.8182 | 0.5 | 0.7733 | |||

0.75 | 0.8071 | 0.75 | 0.7653 | |||

15 | 0 | 0.7340 | 14 | 0 | 0.6899 | |

0.25 | 0.7303 | 0.25 | 0.6878 | |||

0.5 | 0.7272 | 0.5 | 0.6792 | |||

0.75 | 0.7219 | 0.75 | 0.6686 | |||

13 | 0 | 0.6405 | 12 | 0 | 0.5901 | |

0.25 | 0.6397 | 0.25 | 0.5882 | |||

0.5 | 0.6387 | 0.5 | 0.5838 | |||

0.75 | 0.6257 | 0.75 | 0.5829 | |||

Demand increase factor = 1.59 | 17 | 0 | 0.9225 | 16 | 0 | 0.8962 |

0.25 | 0.9137 | 0.25 | 0.8856 | |||

0.5 | 0.9136 | 0.5 | 0.8836 | |||

0.75 | 0.9113 | 0.75 | 0.8755 | |||

15 | 0 | 0.8588 | 14 | 0 | 0.8528 | |

0.25 | 0.8553 | 0.25 | 0.8157 | |||

0.5 | 0.8505 | 0.5 | 0.8009 | |||

0.75 | 0.8456 | 0.75 | 0.7959 | |||

13 | 0 | 0.7623 | 12 | 0 | 0.7136 | |

0.25 | 0.7618 | 0.25 | 0.7105 | |||

0.5 | 0.7541 | 0.5 | 0.6905 | |||

0.75 | 0.7416 | 0.75 | 0.6858 |

As the amount of leakage is not constant and depends on pipe characteristics and water pressure, various formulas have been developed to estimate it. One of the most straightforward formulas for calculating leakage is Equation (15). However, since water pressure in urban networks can vary hourly, the amount of leakage also changes accordingly. Consequently, it is impossible to directly incorporate the variable leakage amounts into the model using software, as the leakage rate is also affected by changes in pressure. Given the dynamic nature of leakage, it is necessary to introduce the leakage rate into the model by accessing the program code, which represents an innovative approach in this research. The pressures obtained from the previous results without considering leakage are used to calculate the leakage, and the resulting leakage from the initial pressure is added to the total demand. In this research, for instance, 10, 20, and 30% of the nodes are randomly selected, and the probability that the selected nodes have the highest pressure increases. Additionally, the coefficient *K* used in the leakage calculation is randomly chosen between 0.05 and 0.5. Table S2 presents the objective function values for the desired network, considering the system's leakage.

Table S2 shows that as the leakage coefficient increases from zero to 0.3, the objective function value decreases by an average of approximately 17%, indicating a significant impact of leakage on the system.

The leakage calculations were performed for the network with increased demand factors of 2.19 and 1.59, and the results are also presented in Table S2. This table shows that leakage rate changes significantly impact the objective function value. The results presented in this table indicate that increasing the consumption coefficient from 1 to 1.59 and 2.19 has a significant impact (ranging from 15 to 43%) on the value of the objective function in all scenarios.

## CONCLUSIONS

A simulation-optimization model was developed in this study to provide the IWS schedule in the studied network during water shortage conditions. The model aims to achieve maximum reliability while addressing justice by considering hydraulic and social constraints. Various scenarios were analyzed and evaluated to understand the impact of different factors on the objective function value. The findings showed that increased demand and water level decreases led to a significant decrease in reliability value. Additionally, as justice in the network increased (25, 50, and 75%), the objective function value slightly decreased.

Moreover, a comparison was made between the current conditions of the desired network and the scenarios with increased network demand while observing the justice condition. The results showed that the reliability decreased as the demand in the network increased. Studying the effects of leakage in networks, especially in networks with longer lifespans, on the water available to consumers and the network's reliability is crucial. Therefore, a scenario called the presence of leakage in the network was defined, with different percentages of leakage in the nodes. The volumetric reliability was evaluated for leakage in 10, 20, and 30% of the nodes, with and without the justice constraint. The justice constraint was applied with 25, 50, and 75% compliance rates in the network. The results indicated that the objective function value was higher in the case of no leakage compared to the corresponding value in the case of network leakage.

This research underscored the significance of IWS and highlighted the importance of considering various constraints to optimize water distribution network reliability and justice. However, it would be better if we could consider economic and mechanical constraints in the optimization model, which can be regarded as the future work of this study.

## DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Reliability and Risk in Water Resources*

**14**,