## Abstract

Many operators of water distribution networks (WDNs) are unable to meet the increasing demand for water. Utility operators in such situations resort to rationing the supply as a partial solution to this problem; this, in turn, may lead to disproportionate allocation of water or inequity in supply. In this study, we propose a mixed integer non-linear program formulation and an efficient solution approach to minimize the inequity in supply, subject to hydraulic constraints and additional constraints on hours of supply and valve operation. Further, we show that the schedule can be obtained using a data-driven approach based on flow and level measurements, which eliminates the modelling effort and uncertainty associated with the use of hydraulic models. We demonstrate the proposed approaches through simulations of a real WDN, and experiments conducted on a topologically similar laboratory-scale network.

## HIGHLIGHTS

A technique is proposed for scheduling intermittent WDNs that can be implemented in the field.

The operational objective is equity in supply and constraints on implementation are incorporated.

The technique is demonstrated on a simulated and laboratory scale of a real network.

## NOMENCLATURE

Time interval between two event points in the scheduling horizon

Resistance to flow in arc . Computed as where

*l*denotes the length, denotes the pipe coefficient, and*d*denotes the diameter of the pipeThe demand of village

*i*for the scheduling horizonThe set of arcs (pipes/valves) in the system

The set of pipes in the system

The flow rate through arc in time interval

*t*(variable)The flow rate into village

*i*in system state*k*(parameter)- , ,…,
Groups of villages, the supply to which is controlled by the same valve operator;

*G*is the set of such groupsThe set of time intervals in which supply shall not be provided to village

*i*,Total hydraulic head of water at node

*i*in time interval*t*The set of source nodes in the system

The set of beneficiary villages in the system (STs in the laboratory-scale WDN)

The set of intermediate nodes in the system

The set of operational states for which flow rates are available

The maximum number of times the valve

*i*may be operated in a dayNumber of time intervals in the scheduling horizon of duration

The status of valve leading to village

*i*in state*k*. Assigned the value 1 if the valve is ON and 0 otherwise (parameter in problem )The status of valve leading to village

*i*in time interval*t*. Assigned the value 1 if the valve is ON and 0 otherwise (variable in problem )The status of valve leading to village

*i*immediately before the start of the scheduling horizonIndicator of whether state

*k*is active in time interval*t*. Takes a value 1 if the state is active, 0 otherwiseIndicator to whether the valve leading to village

*i*is operated (either switched ON or OFF) at the beginning of time interval*t*. Takes a value 1 if the valve is operated, 0 otherwiseRelative deviation between demand and supply in village

## INTRODUCTION

The increase in population and living standards across the globe has led to a steady increase in demand for water. Operators of a large number of water distribution networks (WDNs) are already unable to meet their requirements and many more will soon be facing this problem (Willuweit & O'Sullivan 2013). Rationing the supply is the natural (short-term) solution to this problem and this has led to *intermittent water supply* becoming a norm in several emerging economies (Vairavamoorthy *et al.* 2007; Christodoulou & Agathokleous 2012; Nyahora *et al.* 2020; Loubser *et al.* 2021; Vinturini *et al.* 2021). As one may expect, these altered schemes of operation bring up several challenges in network management such as supply inequity, time of supply, and operational constraints. We propose a novel scheduling strategy that aims to overcome these challenges, and at the same time, is easy to train and relatively unsophisticated to implement. It is worth emphasizing that the objective of this work is not to advocate the intermittent operation of networks, but to assist the efficient operation of intermittent systems, when continuous supply is not an option.

### Background

The class of intermittently operated WDNs addressed in this work – commonly referred to as multi-village schemes (MVSs) or regional rural water supply systems (RRWSSs) – have structural and operational features that differentiate them from continuously operated WDNs (Batish 2003; Bhave & Gupta 2006; Prasad & Sohoni 2011). In intermittent systems, water is usually transported from its source to overhead tanks (OHTs) with the help of pumps or by gravity. These OHTs are intermediate storage facilities from which water is distributed to the consumers downstream (Hooda & Damani 2019). The supply window usually lasts for only a few hours each day and the consumers are expected to collect and store water for use during the remaining hours (Ilaya-Ayza *et al.* 2018; Fraga *et al.* 2022). In contrast, continuous WDNs (the well-studied type) keep the system pressurized and supply water to consumers at all times of the day. Though the structural differences between these systems may be subtle, the differences between the corresponding mathematical models are often more pronounced. For example, for intermediate storage of water, continuous WDNs employ floating reservoirs that are both filled and withdrawn through pipes connected at the bottom. This makes the pressure in the system and the level in the reservoirs coupled variables. On the other hand, in intermittent WDNs, the intermediate OHTs are filled by pipes connected to their top and the withdrawal is made through pipes connected at their bottom (World Bank Group 2012). The pressure at the supply side (upstream of the intermediate storage tank) is now independent of the level of water in the downstream OHT, and therefore, the filling process is much easier to model (Kurian *et al.* 2018).

The limitation in supply time is the key to regulating supply. In intermittent WDNs, when supply is available, consumers withdraw at the maximum rate and store the water for use in the remaining hours of the day. As the available water is limited, it is the utility providers' responsibility to regulate supply intervals such that the distribution is *equitable*. Hence, equity in distribution becomes the primary objective of scheduling intermittent WDNs (Solgi *et al.* 2014).

Heuristic-based operational policies could result in inefficient operation of the network. WDNs are complex systems with a non-convex set of feasible flow rates, and varying demands. Standard operational procedures and heuristic schedules often fail in WDNs (Chinnusamy *et al.* 2018a), and it is not easy to adapt existing solutions to changes in demand. Despite the problem being computationally challenging, there is merit in identifying optimal operational schedules through mathematical optimization (Ghaddar *et al.* 2015; Bonvin *et al.* 2017).

Often, there are practical challenges in the optimal operation of WDNs in emerging economies. The primary challenge is the unavailability of reliable data to develop the hydraulic models needed for preparing the schedule. The topology and size of WDNs may change multiple times in the lifespan of a system, some of which may not be properly documented. Even for a network with a known network structure, the network parameters change over time due to scaling and corrosion in the pipes. The resulting prediction errors of the mathematical models could cause the schedules generated to be sub-optimal, or even worse – infeasible. The second set of challenges arises as constraints on implementation. Though there have been extensive efforts made towards the automation of WDNs, most of the systems in developing countries are still manually operated (Chinnusamy *et al.* 2018b). This introduces several constraints on valve operation as operators often need to travel long distance between locations. In this context, it is advantageous to use data-driven models derived from experiments performed in the field, and develop operational schedules that are easily computed and implemented.

### Literature review

Equity in distribution has been highlighted as a crucial aspect of intermittent water networks by several researchers (Ameyaw *et al.* 2013; Gottipati & Nanduri 2014; Gullotta *et al.* 2021). In these design problems, equity was quantified as the uniformity in demand satisfaction across the network. As zero supply could also, in principle, be termed as equitable supply, a constraint was introduced on the total water supplied (Ameyaw *et al.* 2013). We introduce equity in a scheduling problem (rather than a design problem) where equity is quantified as the variations in the relative deviation of supply from demand (defined later in problem ). This metric, while in essence having the same objectives as the works mentioned earlier, aims to achieve both demand satisfaction and supply uniformity with a single expression. This is particularly relevant in scheduling problems where the complete supply of water could be infeasible and hence cannot be incorporated as a hard constraint.

Solution methods for the operation and control problems in WDNs have been actively studied for many decades. Though there have been several mathematical programming-based approaches and metaheuristics proposed for solving this class of difficult problems, most of them focused on energy optimization in continuously operated networks (López-Ibáñez *et al.* 2008; Bagirov *et al.* 2013; Ghaddar *et al.* 2015; Shi & You 2016; Mala-Jetmarova *et al.* 2017). Intermittent water networks, even though constituting more than 40% of the WDNs in low- and middle-income countries (Mohan & Abhijith 2020), have been addressed rarely in these studies, potentially due to the structural and functional differences and challenges associated with modelling these systems (see Sarisen *et al.* (2022) and Sashikumar *et al.* (2003) for a review of the challenges associated with modelling intermittent water systems). On the other hand, the scientific literature on intermittent water networks is focused primarily on model development, or on design problems, and the operational problems have not been well addressed (Ingeduld *et al.* 2008; Ameyaw *et al.* 2013). Nevertheless, the operation of intermittent WDNs has been gaining attention recently with one of recent editions of the Battle of Water Networks (2022; BIWS) being around this theme (Sánchez-Navarro *et al.* 2021; Abdelazeem & Meyer 2022; Marsili *et al.* 2023). In Amrutur *et al.* (2016), the authors proposed a mathematical program to determine if continuous supply can be implemented in a system that was originally being operated intermittently. Ilaya-Ayza *et al.* (2017) proposed an optimization problem to reorganize the existing supply hours in an intermittent network to reduce the peak flow rates and distribute water equitably. Following an approach that is different from most of the existing scheduling formulations, we propose a technique in which hydraulic computations do not have to be performed every time the scheduler is executed as discussed in the following paragraphs.

Recently, we reported a technique for the optimal operation of WDNs using discrete valves for minimizing energy consumption (Kurian *et al.* 2018). In the general scheduling problem, the hydraulic models of WDNs are non-convex functions, resulting in a mixed integer non-linear optimization problem for scheduling operations. An equivalent but tractable mixed integer linear program was developed by first solving the hydraulic model for all possible discrete states of the network and using these results in the scheduling problem formulation. This was further extended to the problem of minimizing supply time and demonstrated experimentally (Chinnusamy *et al.* 2018a).

*et al.*(2022), Hooda & Damani (2019), and Ilaya-Ayza

*et al.*(2017) for examples of networks from three different countries, and Kumpel & Nelson (2016) for data highlighting the prevalence of such networks). Specifically, we present (i) a scheduling problem with the objective of maintaining equity in supply and incorporate constraints to model practical limitations and (ii) a data-driven modelling technique that makes use of flow measurements rather than detailed hydraulic models in determining an appropriate schedule. Figure 1 summarizes the key differences between the existing approaches for scheduling WDNs and the proposed approach for operating MVSs. We demonstrate the implementation of our technique through hydraulic simulations and through experiments.

The rest of the paper is organized as follows. Section 2 describes the system and the problem under consideration. In Section 3.1, we present a technique for decoupling hydraulic equations from the scheduling problem. Based on this, we describe a novel formulation for the scheduling problem in Section 3.2. The results of implementing the technique on a numerical model and a laboratory-scale network are given in Section 4. Finally, we conclude with the key takeaways in Section 6.

## SYSTEM DESCRIPTION AND PROBLEM STATEMENT

### System description

*et al.*(2023) describe a large network supplying to individual consumers). It is assumed that there is sufficient storage available at the villages, i.e., water would be withdrawn at the maximum rate possible in all instances when supply is available. Further, we formulate the problem under the assumption that the source is maintained at a constant head (Bhave & Gupta 2006).

The control elements available in the system are ON/OFF valves located upstream of the villages and the pumps if there are any. It is assumed that one valve is available for each village/node. We consider the valves to be of ON/OFF type rather than continuous type due to the cost and complexity associated with installing and maintaining continuous control valves. Though several existing networks use control valves (Amrutur *et al.* 2016; Marsili *et al.* 2023), these valves require an additional feedback loop equipped with a flow transmitter for operation. It is common for RRWSSs not to be instrumented to this extent and such systems are the subject of the present work. Several different mathematical models have been proposed for predicting the steady-state conditions (post-pressurization) and filling process (pressurization) of intermittent water networks (Abdelazeem & Meyer 2022). In the present work, we assume the network to be free of leaks, allowing us to work with the network under pressurized conditions alone, as reported in other recent works (Nyahora *et al.* 2020; Sánchez-Navarro *et al.* 2021; Abdelazeem & Meyer 2022). As the demand nodes are inlets to OHTs, no water drains from the pipes when the valves are closed. Transients in the network on operation of valves and pumps are also assumed to be negligible since the fluid is incompressible and the valves and pump settings are changed slowly (Sankar *et al.* 2015). We consider the system to be manually operated and not necessarily automated, i.e., given a schedule, valve operators travel between parts of the network and manually operate the valves assigned to them at the respective times. For this reason, sufficient time has to be allocated between operations of valves assigned to the same operator. These are additional requirements of manually operated WDNs and the framework proposed here is completely applicable to automated systems as well.

Overall, the technique is directly applicable to a widely prevalent class of networks referred to as RRWSSs given that any leaks present are relatively low/negligible.

### Problem description

The problem we address here is the operational planning of WDNs with system characteristics as described earlier in Section 2.1. Given the network details, available water, demands at the villages, and other operational constraints, the objective is to determine a schedule that would distribute available water equitably to the villages while satisfying constraints.

In an ideal scenario, any utility provider would want the supplied water to be the same as the demand of all villages. But, in practice, we might not be able to achieve this, for two reasons. The first reason is the scarcity of water at the source. If the available water is less than the demand, the utility providers are left with no choice but to ration the supply. The second reason is the carrying capacity of the network. In this case, enough water may be available at the source, but all of it cannot be supplied in the given time due to limitations offered by the infrastructure. In either case, the operational policy has to be such that the distribution is equitable. That is, if the villages are supplied with less than their requirement, it is desirable that the fractional reduction in supply is similar for all villages. We use the fractional deviations from desired demands as a measure of equity and minimizing such deviations becomes the primary objective of scheduling the system.

An operational policy would also have to satisfy certain practical constraints. The first one is the convenience of collection. Villages may at times lack common storage reservoirs, and in these cases, villagers wait for the supply hours and collect water from standpipes when it is available. Hence, it is incumbent on the utility provider to supply water at hours convenient for collection. Early morning hours or evenings are generally convenient for collection, and supply during night or noon makes collection a difficult task. The convenient hours may also be different for different villages. While scheduling the supply, these preferences have to be taken into consideration and supply has to be timed accordingly.

The second set of constraints is imposed by the system and the operators. Too many switches of valves and pumps in the system are not only inconvenient but also reduce the life of the instruments. Therefore, an upper limit has to be imposed on the number of times each of these devices is switched ON and OFF. Infrequent switches would also allow gradual changes in valve/pump positions, thereby reducing the transients and wear in pipes (Prescott & Ulanicki 2008). Further, in manually operated systems, a single operator may be tasked with operating multiple valves in the network. In these cases, the time required for travelling between the valves has to be considered while preparing the schedule. Valves leading to villages/nodes that are operated by the same operator and at a considerable distance from each other cannot be operated simultaneously. All these constraints are taken into consideration in the scheduling problem presented here.

### Problem formulation

*t*and 0 otherwise. Relative deviation for any consumer location is the ratio of deviation between demand and supply relative to the total demand. The operational objective of minimizing the deviation between demand and supply translates to minimizing the maximum of the deviation vector in the mathematical problem . The following additional definitions are used in the problem:where is the set of all villages receiving supply. The upstream and downstream nodes of the valve corresponding to village

*i*are given by and , respectively. is the set denoting the source node. is the set of intermediate nodes in network – nodes which are neither a source nor a demand point. is the set of all pipes, represented as directed edges (ordered pairs). is the set of time intervals in which village

*i*shall not be supplied water. is the group of valves controlled by the same operator.

*G*is the set of such groups.

In the formulation, the first three constraints, i.e., Equations (1)–(3), represent the hydraulic model of the WDN in consideration. This includes the flow conservation at the nodes, head loss in pipes modelled by Hazen–Williams equations and the boundary conditions on the pressure at the source and the demand nodes (). Since the valve states are decision variables of the optimization problem, the next two constraints (Equations (4) and (5)) relate the flow and pressure drop across the valve to the valve status. These equations imply that the flow across the valve is zero when it is closed, and the pressures immediately upstream and downstream of a valve are equal, when the valve is fully open. Equation (6) defines the relative deviation between demand () and supply for all villages present in set . Equation (7) prohibits supply to villages at hours inconvenient for collection. This constraint is particularly important if any village has standpipes installed and the supply therefore has to be at specific hours. Equations (8) and (9) impose the constraint that each valve operator can operate at most one valve in a time interval since a valve operator is assigned to a group of villages/nodes. Here, is the state of valve *i* at the beginning of the scheduling horizon (known parameter). Equation (10) specifies an upper bound () on the number of times valve *i* may be operated in a day.

It is well recognized that MINLPs in general and, specifically, scheduling problems in WDNs such as P are difficult to solve (Shi & You 2016; Kurian *et al.* 2018). In the subsequent section, we describe a method for efficient solution using appropriate reformulations.

## SOLUTION TECHNIQUE

Given the complexities in solving the problem P, we divide the entire process into two simpler steps. Section 3.1 describes the first step of decoupling hydraulics from scheduling. Based on this decoupling, Section 3.2 presents a tractable reformulation for the scheduling problem.

### Decoupling hydraulic simulation from optimization

Hydraulic equations, typically the flow conservation and the head loss equations, are commonly used for modelling and simulating WDNs. Given the network parameters and boundary conditions, solving the corresponding hydraulic equations gives the flow rates and pressures (head) at different nodes in any WDN. This type of hydraulic simulation is also known as pressure-driven analysis. In the scheduling problem described earlier (), Equations (1)–(3) essentially define the hydraulic model for the WDN. As evident from Equation (2), the hydraulic model of WDNs is a set of non-linear relations and contributes significantly to the complexity of the scheduling problem (Shi & You 2016). One way of dealing with this complexity is to develop a separate function to solve these equations, and invoke this function several times within the scheduling algorithm (Naoum-Sawaya *et al.* 2015). In this case, the hydraulic equations have to be repeatedly solved several times online to generate the schedule. We propose an alternative approach to completely decouple the hydraulic simulation from the optimization problem, which results in an efficient strategy suitable for online implementation (Kurian *et al.* 2018).

*a priori*, the solutions (flow rates through different pipes) can be passed on to the scheduling problem as parameters. The non-linear hydraulic equations are not solved during the process of solving the optimization problem and this reduces its complexity. It may be noted that in the present approach, solving hydraulic equations is a one-time procedure which can be performed off-line, and the obtained flow rates at the demand nodes for different states can be stored and used whenever a schedule has to be prepared. A schematic describing the approach is given in Figure 4. A similar strategy was used to solve an energy minimization problem (Kurian

*et al.*2018). This is clearly different from other existing approaches for decoupling hydraulics from the optimizer, where the hydraulic solver is generally a separate subroutine, invoked several times during every execution of the scheduling algorithm (Naoum-Sawaya

*et al.*2015).

### Scheduling problem reformulation

*t*and 0 otherwise. Further, another decision variable is introduced to track whether the settings of valve (ON or OFF state) changes at time interval (). The valve positions () and flow rates into demand nodes/village tanks () for different system states

*k*are parameters in the optimization problem –

*v*and

*f*were decision variables with slightly different definitions in P. There is no change in the objective function, namely, relative deviation between demand and supply.

As in the case of P, the objective of is to minimize the relative deviation between demand and supply. The constraints are modified to accommodate the changes in variables and parameters. The first constraint, i.e., Equation (13), ensures that only one state is chosen for a time interval. The next two constraints define the variable list and ensure that if the valve changes position from interval to *t* and 0 otherwise. Equation (14) is for the first interval, and Equation (15) is for the remaining intervals; is the known state of valve *i* at the beginning of the scheduling horizon. Equation (16) is similar to constraint 9 in problem P for restricting the valve operations assigned to a single operator. Equation (17) is the bound on the daily operations for a valve (similar to Equation (10)). Equation (18) is the constraint allowing supply to villages only at hours convenient for collection (similar to Equation (7)).

*a*is the travel time between valves and . The new constraint ensures that enough time is allocated to operators for commuting between villages to operate the valves.

The two-step solution technique described here is conceptually straightforward, computationally efficient and effortlessly solved using standard linear solvers. Further, the schedules prepared this way are easily implemented in RRWSSs with little or no automation of the network.

### Replacing the hydraulic simulation model with a data-driven approach

In Section 3.1, we assumed that a hydraulic model of the network is available for computing the flow rates corresponding to different states using a pressure-driven analysis. For RRWSSs that we address in this paper, it is quite possible that complete network information or its topology is unavailable, and any structural changes carried out to the network are also not documented. Furthermore, over the years, the performance of the network could deteriorate due to scaling within the pipes and degradation of the pumps. In such cases, it is not easy to obtain a reliable hydraulic model of the WDN, even if the topology of the network is known. As an alternative, we propose a data-driven approach using field measurements of flows from the network, which eliminates the need for obtaining a hydraulic model of the network.

Given the network flow information for all states, the task of the scheduling problem is to choose the valve and pump configurations or, equivalently, the distinct states that are to be operational during different intervals of the time horizon, typically a day. For a given state, the only output of a hydraulic simulation that is used by the scheduling problem is the flow rates at demand nodes or villages. As the number of states is finite, the flow rates at different demand nodes for different network states can also be obtained from routine operation of the network. Flow rates in several network states are generally measured during routine operation. If necessary, additional experiments can be conducted in the field to determine the flow rates received by villages in other network states and these measured values can then be used in the scheduling algorithm rather than the results of a hydraulic model. Using measured flow rates to prepare the schedule could go a long way in mitigating the effects of imperfect network and hydraulic models.

In the next section, we demonstrate the applicability of our proposed scheduling algorithm using simulation as well as experiments conducted on a laboratory-scale network under different conditions.

## CASE STUDIES

The scheduling technique described in the previous section was demonstrated on two networks. One of them was the hydraulic model of a real network which uses a model to generate the data required for the optimal scheduling problem. The other was a test bed developed in the laboratory and uses the data-driven approach. This section describes the studies conducted and their outcomes. The optimization problems were all formulated using AMPL (Fourer *et al.* 1987) and solved on the NEOS server (Czyzyk *et al.* 1998). The solution times for all problems discussed below were on the order of seconds or a few () minutes.

### Model-based case study: Osmanabad water network

The first network analysis was the Osmanabad water network shown in Figure 2 and originally presented in Bhave & Gupta (2006). The hydraulic model was solved using EPANET. As EPANET is primarily intended for demand-driven analysis (and not pressure-driven analysis), we modelled the demand nodes as OHTs with the appropriate height and no water, effectively resulting in a model of a pipe exiting the atmosphere. The resultant flow rates under each of the states are given in Supplementary Appendix Table A.4.

The objective function of the scheduling problem was the minimization of maximum relative deviation (minimize the maximum of the deviation vector ) from the demands given in Supplementary Appendix Table A.2. In order to implement a scheduled supply, it was assumed that there exists an ON/OFF valve in the pipe leading to each village. These could be operated suitably to supply the correct amount of water. The scheduling horizon considered here was 10 h and this was discretized into 24 equal intervals. Two different scenarios were considered. These are described in the following sections.

#### Scenario 1: Supply with constraints on demand and operation

In the first scenario, the objective was to supply water with the following additional restrictions:

No valve shall be switched ON more than two times, i.e., . This was intended to restrict the total valve operations in the network.

Only one of the three valves leading to villages 4, 5, and 6 shall be operated at the same time. It was assumed that the three valves were manually operated by the same operator.

Villages 2 and 3 shall be supplied only in time slots 3–12. This constraint was representative of collection convenience for villages supplied by a standpipe.

Following the procedure described in Section 3.1, simulations were carried out to identify the flow rate for all system states. As there were six ON/OFF valves in the system, there are a total of 64 states (i.e., ) including the trivial state where all valves were OFF. The states were numbered 1 to 64 with S1 being the state where all valves were OFF and S64 being the state where all valves were ON. The flow rates delivered to different villages in each state are given in Supplementary Appendix Table A.4. In the table, each row represents a state and the flow rate received by each village is given in the corresponding column.

On examining the schedule given in Figure 6, it is clear only about one-third of the cells are coloured. On average, about two out of the six villages receive supply in each interval. This is an indication that the network has ample carrying-capacity (which may not be fully utilized) and one could potentially meet the demand in a much lower time, with a small trade-off in objective function value.

#### Scenario 2: Supply with limited control

The second scenario was conceptualized from a situation observed in a rural water supply scheme in India. In some systems, one beneficiary village or node may be located at a substantial distance from the rest of the network, and therefore, the valve leading to the particular village is never operated. The remaining valves have to be operated keeping in mind the demands of the village located far off. Once the demand of all villages is met, the supply from the source is shut off and this stops the supply to the village located far off as well. To replicate this scenario, we imposed an additional constraint in the WDN that the valve leading to village 5 should always be kept open. The only possibility to shut off the supply to village 5 was to shut off the supply for the entire network. Mathematically, this was equivalent to removing all states with village 5 not receiving supply and any other village receiving supply. Other constraints from Section 4.1.1 were retained.

### Data-driven case study: laboratory network

The above-described formulations were also implemented on a laboratory-scale water network to demonstrate that the developed schedules are easily implemented on real systems. The laboratory system was topologically similar to the Osmanabad WDN. The tests conducted were also similar to the cases discussed earlier. The following sections give a description of the system and the studies.

#### Experimental set-up

Water was supplied to the OHT by a 0.5HP centrifugal pump from a large reservoir at ground level and operated using a relay. A control valve was installed downstream of the OHT to regulate flow and a flow transmitter (Burkert 8081 Ultrasonic Flow Sensor) was used to record the flow rate. Throughout this study, the control valve in the outlet of the OHT was kept completely open as we were only addressing systems with ON/OFF valves. The STs had solenoid valves (Burkert 6011) placed at their inlet as well as outlet. These valves could switch ON and OFF the inflow and outflow (drain) for each ST. The levels in all tanks (OHT and STs) were monitored by ultrasonic level transmitters (Baumer U500) installed at the top. All transmitters and actuators were interfaced to a computer using a National Instruments DAQ card and the devices were programmed and controlled using LabVIEW.

The complete set-up was representative of the Osmanabad WDN with the OHT corresponding to the source and the STs corresponding to storage available at beneficiary villages. The solenoid valves upstream of the STs were similar to the ON/OFF valves usually available for every village in an RRWSS. The demands given in Table 1 were assigned to the STs. The problem of scheduling valves in an RRWSS translated to the problem of scheduling solenoid valves regulating supply to the STs in the set-up. The objective of the operational planning was the distribution of available water equitably (minimize the maximum of the deviation vector ) in the given time horizon. As in the Osmanabad case, two different scenarios were considered as explained below.

Tank number . | Requirement () . |
---|---|

T1 | 6.40 |

T2 | 19.25 |

T3 | 11.35 |

T4 | 7.04 |

T5 | 7.04 |

T6 | 10.80 |

Tank number . | Requirement () . |
---|---|

T1 | 6.40 |

T2 | 19.25 |

T3 | 11.35 |

T4 | 7.04 |

T5 | 7.04 |

T6 | 10.80 |

#### Scenario 1: Supply with constraints on demand and operation

In the first scenario, a schedule was prepared to meet the demands given in Table 1. The total time available for supplying water was 3,000 s and this was discretized into 24 equal intervals of 125 s. The following restrictions were also imposed:

No valve shall be switched ON more than two times.

Only one of the three valves leading to STs T1, T2, and T3 shall be operated at the same time.

STs T4 and T6 shall be supplied only in time slots 6–10.

In order to maintain the level of water in the OHT constant at 42.5 () cm, a control program was implemented in LabVIEW to switch ON the centrifugal pump as and when the level in the OHT fell below 40 cm and switch OFF when the level reached 45 cm.

It may be noted that we did not use a hydraulic model for this experimental network and instead used a completely data-driven approach. Therefore, as described in Section 3.3, experiments were carried out to identify the flow rates into STs in different network states. The system with six valves had 64 () states in total. Each configuration was kept active for 3 min and the change in level in each ST during this interval was used to calculate the flow rate into the respective ST. After supplying water for 3 min, the system was kept idle for 20 s before the level measurements were made in order to ensure steady measurements. The flow rates obtained in each state are given in Supplementary Appendix Table A.5.

#### Scenario 2: Supply with limited control

The second scenario considered here was similar to the one described in Section 4.1.2. Here, all constraints from the previous scenario were imposed. In addition, the valve corresponding to T2 was constrained to be always ON. T2 would receive water whenever a supply was provided from the source. The only possibility of stopping the supply to T2 was to shut off the source.

These results demonstrate that it is possible to apply the proposed model-based formulations and data-driven approach to determine and implement equitable supply while satisfying the constraints on operation.

## DISCUSSION

The scheduling technique presented in this work is computationally efficient and straightforward to implement in small and aggregated WDNs, e.g., rural water supply schemes. Nevertheless, there still exist multiple avenues for improvements, some of which are currently being worked on.

The data requirement in the current approach could increase with the number of demand nodes. However, it must be noted that it does not scale with the number of edges or nodes. Further, it is encouraging to note that, given information on flow rates in a subset of states, the proposed technique could still prepare near-optimal schedules with the available data provided the known states are *efficient*. Techniques for identifying such efficient states have been proposed and experimentally validated (Adhityan *et al.* 2022). Another approach would be to group demand nodes with similar characteristics, e.g., elevation, demand pattern, etc., and prepare schedules for the reduced network.

In the present work, we assumed the system to be free of leaks, which, in practice, is not the case for several networks (Mohan Doss *et al.* 2023). Given enough information on the leak and how it varies with the network states, one could incorporate the information into the scheduling problem as constraints to prepare schedules resulting in reduced leaks. In addition, leaks could cause water to drain out of pipes when a part of the network is not being supplied. The filling process of these empty pipes could be a challenging problem for scheduling algorithms (Sashikumar *et al.* 2003; Sarisen *et al.* 2022). However, it is encouraging to note that scheduling problems do not require the entire intermediate information on how the pipes are filled. Given information on the filling times while transitioning between each pair of states (which can be computed once, and need not be repeated every time the scheduler is invoked), one could incorporate this into the scheduling framework with minimal effort. Such an update would make it even more desirable to work with a subset of states rather than the complete set.

The scheduling technique proposed here is designed for systems installed with ON/OFF valves. While these are common in rural water networks, several systems use a different type of actuator – continuous control valves – or a mix of the two (Amrutur *et al.* 2016; Bonvin *et al.* 2017). Adapting the technique for such systems is another task that is being worked on.

## CONCLUSION

In this paper, we presented a novel, two-step scheduling technique for operating a class of rural WDNs. The objective of this mathematical programming-based approach was the equitable distribution of available water with constraints on operation. Linear constraints were added to the scheduling problem to model several classes of practical restrictions. The problem was formulated such that detailed hydraulic or network models can be replaced by data-driven models if necessary. The resulting mathematical problems were solved with relative ease and the deviations between demand and supply were found to be very low for the networks tested here. Further, the method was demonstrated on a physical network to validate the solution procedure. The proposed method can be implemented in the field as long as measurements of steady-state flows under different network configurations are available.

Separating the scheduling problem of rural WDNs from the data acquisition step (model simulation/field experiments) and posing the latter as a one-time exercise that is not repeated, we presented a new paradigm for scheduling that works on much less information compared with traditional approaches. This was realized by exploiting the knowledge that the flow rates received by the demand nodes are what matters most to the scheduling routines, and that these mathematical problems are usually agnostic to intermediate pressures and flow rates in the network. With its negligible computational burden, simplicity in execution, and capability to include field measurements, we envisage the technique (and its future derivatives) to be a practicable solution for the management problems faced by several rural WDNs.

## ACKNOWLEDGEMENTS

We thank Prof. S. Murty Bhallamudi (IIT Madras) for the technical discussions that proved very helpful and his invaluable suggestions on improving the manuscript. We would also like to thank the anonymous reviewer for their constructive criticism, which helped substantially improve the contents of the manuscript. This work was partially supported by the Department of Science and Technology, Govt of India under the water technology initiative: DST/TM/WTI/DD/2K17/39, DST/TM/WTI/WIC/2K17/82(G)-WATER-IC for SUTRAM of EASY WATER, and DST/TM/WTI/2K13/144.

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*International Journal of Industrial and Systems Engineering*. doi:10.1504/IJISE.2023.10054305. (In press.)

*Sugave Water Scheme: Multi-Village Drinking Water Scheme Analysis*. CTARA, IIT Bombay, Mumbai, India. Available from: https://www.cse.iitb.ac.in/~sohoni/AS-MVS.pdf.