Investigation of Optimum Sustainable Designs for Water Distribution Systems from Multiple Economic, Operational, and Health Perspectives

Abstract

Optimizing the design of water distribution systems often faces difficulties due to continuous variations in water demands, pressure requirements, and disinfectant concentrations. The complexity of this optimization even increases when trying to optimize both the hydraulic and the water quality design models. Most of the previous works in the literature did not investigate the linkage between both models, either by combining them into one general model or by selecting any representative solution to proceed from one model to another. This work introduces an integrated two-step framework to optimize both designs while investigating the reasonable network configuration selection from the hydraulic design view before proceeding to the water quality design. The framework is mainly based on a modified version of the multi-objective particle swarm optimization algorithm. The algorithm’s first step is optimizing the hydraulic design of the network by minimizing the system’s capital cost while maximizing the system’s reliability. The second step targets optimizing the water quality design by minimizing both the total consumed chlorine mass and the accumulated differences between actual and maximum chlorine concentrations for all the network junctions. The framework is applied to Safi Network in Yemen. Three scenarios of the water quality design are proposed based on the selected decision variables. The results show a superior performance of the first scenario, based on optimized 24-h multipliers of a chlorine pattern for a flow-paced booster station, compared to the other scenarios in terms of the diversity of final solutions.

1. Introduction

Water experts face several challenges when trying to manage the available water resources. One of these challenges is to optimize the design and operation of water distribution systems (WDSs). Designing WDSs is classified as a discrete combinatorial non-deterministic polynomial-time hard (NP-hard) optimization problem. Another difficulty arises when trying to optimize more than one objective while satisfying prespecified constraints [1].

Two types of WDSs design can be identified; the hydraulic design [2,3,4,5,6] and the water quality design [7,8,9]. As a rule of thumb, several difficulties arise when trying to develop an integrated framework to optimize both designs of WDSs. In these types of optimization problems, the formulation of the optimization problem itself, side by side with selecting the appropriate optimization algorithm, poses stress on researchers and engineers to achieve the required numerical accuracy [5,10]. Moreover, the multi-objective criterion used in design frameworks should be carefully selected since it should cover some or all of the economic, reliability, and environmental aspects. For instance, the conflicting nature between network cost, hydraulic reliability, and water quality should be carefully investigated in optimizing WDS design [9]. The obtained solutions are expected to represent the trade-off characteristics among the studied objective functions. The variety of solutions obtained from the design frameworks provides the decision-makers with the flexibility to choose the appropriate design accordingly.

Moreover, the wide variety of formulations available in the literature for designing WDSs, and selecting some or all of the design objectives (e.g., reducing network cost, increasing network reliability, improving water quality, reducing carbon emissions) within the problem formulation lays additional computational burdens. Therefore, it is crucial to develop WDSs design frameworks to achieve the optimum system performance from different sustainable perspectives while using reasonable computational resources. In practice, one of the critical technical limitations is the lack of reliable methods to select the proper network configuration to proceed from the hydraulic design to the water quality design and to choose an adequate representation of the booster station, which injects chlorine in a prespecified place within the network, in the optimization model. Addressing these limitations would benefit researchers and engineers to develop integrated design frameworks for WDSs, specifically when coupling hydraulic and water quality design models into one design framework. Thus, a trial is made in this work to study the linkage between both models using a systematic criterion, as will be discussed later.

Therefore, the main objective of the present work is to develop an integrated framework to obtain a diverse set of reliable and feasible solutions that optimize WDSs hydraulic and water quality designs. The developed framework does not require any parameter tuning or penalty coefficients to investigate the most convenient network configuration from the hydraulic design solutions to proceed to the water quality design step. Another secondary objective is investigating the most convenient set of decision variables for the water quality design step among three different proposed scenarios to reach the most diverse set of solutions. To meet these objectives, the present work introduces a general integrated framework for designing WDSs based on a modified version of MOPSO algorithm [11]. This version of MOPSO algorithm outperforms the original one in terms of convergence and diversity of the final obtained set of non-dominated feasible solutions. The general framework incorporates a two-step WDSs design: the hydraulic and the water quality designs. In the hydraulic design step, the network cost is minimized while maximizing its hydraulic reliability. The total chlorine mass and the accumulated differences between the actual and maximum chlorine concentrations are minimized in the water quality design step. The framework has been applied to a real WDS network in Yemen.

2. Literature Review

In general, the WDSs hydraulic design aims to find the optimum size of pipes, best locations, elevations, sizes and locations of storage tanks, setting of valves, and pumps scheduling while satisfying the pressure head and flow velocity constraints. Many objective functions may be targeted in this design, such as minimizing the total network cost, minimizing the network operational cost, and maximizing the network hydraulic reliability [5,12]. On the other hand, water quality design may comprise finding the optimum placement of contamination sensors, the mass of the disinfectant (e.g., chlorine), locations, and diurnal disinfectant patterns for booster stations while maintaining allowable disinfectants concentration. Many objectives can be targeted here, such as minimizing the contamination detection time, minimizing the total mass of injected chlorine, minimizing the water age, and minimizing the risk associated with using disinfectants while assuring that the concentration of the disinfectant is within prespecified standard limits [9,13,14].

Several studies in the literature have attempted to combine both hydraulic design and water quality design models of WDSs in a single model. To name a few, In Ref. [15], the authors developed a framework for WDSs design based on minimizing the pumps’ operational cost and both turbidity and salinity deviations from standard values. Their framework is mainly based on optimizing the number of working pumps (i.e., ON/OFF scheduling of pumps). In Ref. [16], the authors optimized the design of WDSs in a multi-objective manner, incorporating both the hydraulic design (by optimizing the pump speed) and water quality design (by optimizing the chlorine dosage). They proposed three scenarios to minimize energy and chlorine costs while maximizing the network’s hydraulic reliability. In Ref. [17], the authors proposed another integrated multi-objective optimization framework for WDSs design. They optimized the construction cost and a water quality reliability index based on maximizing the disinfectant residuals and minimizing the water age throughout the network. They investigated the optimum pipe diameters, storage tank dimensions, and disinfectant dosage to achieve the earlier objectives. Most of the previous models combined both the hydraulic and water quality models into one single model. However, a variety of different solutions for network configurations may be identified if both models are split into two separate models, paving the way for decision-makers to select the most reasonable and practical configuration for the network among many different options.

Various multi-objective optimization algorithms have been utilized in WDSs optimization problems [1], but none of them have proved superior to others [5,10] since all the optimization algorithms have their pros and cons. For instance, the particle swarm optimization (PSO) algorithm [18], and its multi-objective version (MOPSO) [19] are frequently used to solve WDSs design problems and often lead to satisfactory results. Despite the rapid convergence rate of MOPSO and the small effort needed for its parameters tuning, enhancements are still required to prevent it from falling into local minima and reduce (or even eliminate) the effort required for its parameters tuning [20]. Several attempts were made to enhance MOPSO numerical performance [21,22] or incorporate parts of the PSO algorithm into other state-of-the-art hybrid algorithms such as AMALGAM [23] and GALAXY [6] algorithms.

3. Methodology

3.1. Modified MOPSO Algorithm

Kennedy and Eberhart [18] developed the single-objective PSO algorithm, which imitates the swarm intelligence concept. The algorithm starts with an initial population having a specific number of particles, $jmax$, which move towards the area of optimum solutions. Each particle in the population represents a set of decision variables. Thus, the decision vectors ($x_1$, $x_2$, …, $x_j$, …, $x_{jmax}$) correspond to the positions of all particles. The algorithm iterates to improve the particles’ positions by following a leader that moves with the most convenient speed and direction.

The objective function is computed for each particle, and both its personal best value ($p_j^{best}$) and the global best value over all particles ($G^{best}$) are stored. Improvement in each particle position is made by changing its personal speed and direction as follows

$$v_{i+1}=w_i{v}_i+C_1{r}_1\left(p_j^{best}-x_i\right)+C_2{r}_2\left(G^{best}-x_i\right),s.t.v_{max}\le v_{i+1}\le v_{min}$$

(1)

where, $v_{i+1}$ and $v_i$ are the new and the old particle’s velocity vectors, in two consecutive iterations $i+1$ and $i$, which should fall between $v_{max}$ and $v_{min}$, the upper and lower bounds on velocities, respectively, $w_i$ is the inertia coefficient at iteration $i$, $r_1$ and $r_2$ are two random numbers between 0 and 1, $C_1$ and $C_2$ are the cognitive and social parameters, respectively and $x_i$ is the particle’s position at iteration $i$. In most studies, $v_{min}$ is set equal to -$v_{max}$ while $w_i$ is calculated as follows:

$$w_i=0.5+\frac{1}{2\left(\ln \left(i\right)+1\right)}$$

(2)

Then, the position of each particle is updated as follows

$$p_{i+1}=p_i+v_{i+1}$$

(3)

where, $p_{i+1}$ and $p_i$ are the new and old particle’s positions at iterations $i$+1 and $i$, respectively.

Afterwards, the algorithm repeats the previous steps for a specific number of iterations, and $G^{best}$ represents the best-obtained solution at the last iteration.

Coello and Lechuga [19] developed the multi-objective PSO algorithm, MOPSO, by extending the single-objective PSO algorithm by adding a repository to store the set of non-dominated solutions at each iteration. At a new iteration, a leader is chosen by the roulette-wheel selection from the stored particles in the repository of the previous iteration. The repository has a specific size, which is equivalent to the population size in this work. The repository particles represent the Pareto front (PF) solutions at a given iteration. Therefore, the final non-dominated solutions are located on the PF obtained at the last iteration.

The optimization constraints are handled as in Ref. [2]. The algorithm selects the repository particles close to or within the search space’s feasible region, aiming at moving the whole non-dominated solutions set towards this feasible region. This method performs better at handling the constraints than the penalty method, which needs several trial runs to find the best penalty coefficients, which is computationally expensive and often fails to reach their optimum values.

The algorithm’s performance is assessed using the hypervolume metric, $H{V}_m$, developed by Ref. [24]. $H{V}_m$ measures the area of the objective space that the PF covers in every iteration in terms of a reference point. The reference point is selected so that the PF should be of a convex shape. The value of $H{V}_m$ should increase with the iterations until reaching its maximum value at the last iteration.

When applying the original MOSPSO to the optimization of WDSs, the algorithm usually suffers from premature convergence since it frequently fails to escape from local minima during the search process. Additionally, the original MOPSO algorithm consumes an additional computational budget to perform many trial runs to identify the best values of the PSO parameters, which are $C_1$, $C_2$, and $v_{max}$. Accordingly, a modified MOPSO algorithm was developed in Ref. [11] to enhance its performance in optimizing WDSs design. Several strategies were introduced to the original algorithm, such as self-adaptive PSO parameters [20], a new repository members selection method based on maximizing the overall hypervolume of the final PF generated at the end of each iteration, a regeneration-in-collision strategy to reduce the probability of particles’ collision within the objective space and to prevent its premature convergence [25], and an adaptive population size [26,27]. Readers may refer to Ref. [11] for more details. This modified version of MOPSO algorithm is used in the current work.

3.2. Problem Formulation

Designing WDSs is a complex optimization problem since it is a discrete combinatorial NP-hard problem. This complexity is increased when a multi-objective formulation of the problem is needed. In this work, the design of a WDS is divided into hydraulic design and water quality design phases. In all cases, EPANET2.0 [28] is used to simulate the network to get pressure heads and chlorine concentration for each junction and flow velocities and discharges for each pipe. In MOPSO population, each particle position corresponds to a specific network configuration. An EPANET-MATLAB toolkit is utilized to monitor EPANET2.0 from MATLAB [29]. Details about each design step are provided in the following sections.

3.2.1. Hydraulic Design

In this step, each particle decision vector consists of several cells representing the decision variables of this design step, which are all the pipes’ diameters in the network along with the storage tanks’ location, diameter-to-height ratio, the height of supports, and the pumping power value. A set of the available diameters in the market should be provided, from which all the pipes’ diameters should be selected. The size of any storage tank is calculated manually from the required balance between the daily pumped water and the overall daily demand. Pumps of variable speed type account for any daily changes in demand.

The objective functions to be optimized in this step are minimizing the annual network cost ($\mathrm{ANC}$) and maximizing the network resilience ($\mathrm{NR}$). The annual network cost is a function of the annual cost of pipes ($\mathrm{APC}$) along with the annual storage tanks cost ($\mathrm{ATC}$) and the annual pumping energy cost ($\mathrm{AEC}$), as shown in Equation (4).

$$min \mathrm{ANC}=\mathrm{APC}+\mathrm{ATC}+\mathrm{AEC}$$

(4)

$$\mathrm{APC}=\sum _{k=1}^{N_p}{C}_k\left(d_{mar}\right)\times L_k\times CRF,\hspace{1em}mar=1,\dots ,N_{av}$$

(5)

$$\mathrm{ATC}=\sum _{s=1}^{N_{st}}\left(x_0+x_1 {\forall }_s+x_2 {\forall }_s H{S}_s\right)\times CRF$$

(6)

$$\mathrm{AEC}=\sum _{p=1}^{N_{ps}}24\times 365\times P_{pu} C_e$$

(7)

where $N_p$ is the total number of network pipes, $C_k$ is the pipe unit cost per length $L_k$ of pipe $k$ with an available diameter in the market $d_{mar}$, $CRF$ is the capital recovery factor based on the interest rate and the expected lifetime of the network, $N_{av}$ is the number of available diameters in the market, $N_{st}$ is the number of storage tanks within the network, $x_0$, $x_1$, and $x_2$ are tank cost coefficients, ${\forall }_s$ is tank $s$ storage volume, $H{S}_s$ is the height of the supports of tank $s$, $N_{ps}$ is the total number of pumps, $P_{pu}$ is the pumping power of pump $pu$, and $C_e$ is the energy unit cost.

The network resilience index ($NR$) is developed by Todini [30] and updated by Prasad and Park [2]. $NR$ represents the surplus available power for all the network junctions that may confront any upcoming failure due to pipes’ bursts, contamination intrusion, etc. $NR$ also includes a uniformity coefficient ($U_m$) which is calculated for each junction $m$ to check the redundancy of the network pipes. More details on this index can be found in Ref. [2]. The network resilience can be represented by Equation (8).

$$maxNR=\frac{{\sum }_{m=1}^{N_{jun}}{U}_m{Q}_m\left(H_m-H_m^{req}\right)}{\left({\sum }_{r=1}^{N_{rs}}{Q}_r{H}_r+{\sum }_{pu=1}^{N_{ps}}\frac{P_{pu}}{\gamma }\right)-{\sum }_{m=1}^{N_{jun}}{Q}_m{H}_m^{req}} ,$$

(8)

where $NR$ is between 0 and 1, $N_{jun}$ is the number of network junctions, $Q_m$ is junction $m$ demand, $H_m$ is junction $m$ piezometric head, $H_m^{req}$ is junction $m$ minimum piezometric head, $N_{rs}$ is the number of network reservoirs and storage tanks, $Q_r$ is the reservoir/storage tank $r$ inflow/outflow, $H_r$ is the water elevation of the reservoir/storage tank $r$, and $\gamma$ is the unit weight of water.

The uniformity coefficient ($U_m$) is calculated as follows:

$$U_m=\frac{{\sum }_{t=1}^{N_{tp}}{d}_t}{N_{tp}\times max\left\{d_t\right\}}$$

(9)

where, $N_{tp}$ is the number of pipes connected to any junction $m$, and $d_t$ is the diameter of pipe $t$ in which junction $m$ is one of its ends.

The MOPSO algorithm is run to find the feasible non-dominated set of solutions that satisfy maximum and minimum pressure constraints, $P_{max}$ and $P_{min}$, respectively, for each junction during the simulation time. The MOPSO algorithm is also run to find the pay-off characteristic curve between $ANC$ and $NR$ by obtaining the final PF from several runs. The final PF particles, obtained from all the runs, are grouped together, and the same non-dominating sorting used in the algorithm is applied to them to get one final accumulated PF. Hereby, the decision-makers should select only one solution from the final accumulated PF for the network’s actual implementation. In this work, the median solution of the final accumulated PF is selected, and it is considered one of the best solutions in the hydraulic design step, which is then used in the water quality design step.

3.2.2. Water Quality Design

The injection of disinfectants, such as chlorine, in WDSs is usually performed using the required dosage at a constant rate into the feeding reservoir. However, in many networks where the demand varies with time and location and the reservoir is not at the center of the network, the chlorine decays spatially and temporally, leaving some end users with unsatisfactory levels of chlorine concentrations. Therefore, a booster injection station may be required in a specific place to inject an additional chlorine dosage using a specific injection pattern to overcome the fluctuations in chlorine concentration for all end users. Thus, in this design step, three different water quality scenarios are developed to choose the best among them. The decision vector of the first scenario for each particle consists of 27 values, which represent the 24-h multipliers ranging from 0.0 to 1.0 of a chlorine pattern for a flow-paced booster station, the location of one booster station connected to a random junction, and two base chlorine injection concentrations for both the booster station and the reservoirs. The chlorine pattern of any reservoir is assumed uniform, leaving only the injected chlorine by the booster station to confront any fluctuations of the chlorine requirements within the network. The decision vector for the second scenario is similar to that of the first scenario, except that the multipliers of the chlorine pattern for the booster station are either 0.0 or 1.0. Finally, the decision vector for the third scenario differs from the second scenario only in representing the chlorine pattern for the booster station since the decision vector in this scenario contains only the starting hour and the ending hour for an assumed uniform chlorine pattern of the booster station, while all the other decision variables are kept the same for all design scenarios.

The objective functions here are minimizing both the total daily chlorine mass injected within the network ($DCM$) and the accumulated difference ($ACD$) between the maximum record of the actual chlorine concentration and the maximum allowed chlorine concentration for all network junctions without violating the maximum and minimum permitted chlorine concentrations ($C{C}_{max}$ and $C{C}_{min}$, respectively). $DCM$ is evaluated by accumulating all the daily chlorine mass injected at booster stations and reservoirs as follows

$$minDCM={\sum }_{kc=1}^{N_c}{\sum }_{\mathsf{\Delta }t=1}^{T}C{C}_{kc}^{\mathsf{\Delta }t} Q_{kc}^{\mathsf{\Delta }t} \mathsf{\Delta }t$$

(10)

where, $N_c$ is the total number of booster stations and reservoirs in the network, $\mathsf{\Delta }t$ is the prespecified time step in which the results are often recorded, $T$ is a 24 h period in which the chlorine consumption reaches a steady state, and $C{C}_{kc}^{\mathsf{\Delta }t}$ and $Q_{kc}^{\mathsf{\Delta }t}$ are the actual chlorine concentration and the outflow of booster station/reservoir $kc$ in time step, $\mathsf{\Delta }t$, respectively.

The second objective ($min$ $ACD$) aims at minimizing the gap between the maximum chlorine concentration recorded along the 24 h selected period ($C{C}_m^{max}$) and the maximum allowed concentration ($C{C}_{max}$) for all network junctions to confront any upcoming failures in chlorine concentrations due to any fluctuations in demands and pressures. $ACD$ is calculated as follows:

$$minACD={\sum }_{m=1}^{N_{jun}}\left(C{C}_{max}-C{C}_m^{max}\right) {\beta }_m$$

(11)

where, ${\beta }_m$ is a weighting factor for each junction, $m$, and may be calculated by dividing the accumulated daily water quantity, ${\forall }_m^T$, supplied to each junction, $m$, by the total daily water quantity supplied to all the network, ${\forall }_{total}$.

As in the hydraulic design step, the MOPSO algorithm is run to find the feasible non-dominated set of solutions that form a PF in each run. The algorithm is run several times, and the final results from all the runs are accumulated to form one final PF for each scenario. Figure 1 summarizes all the steps of the developed framework, including the hydraulic and WQ design.

4. Case Study and Model Implementation

The developed optimization framework is applied to Safi WDS in Yemen. Figure 2 shows the layout of the town network with pipes and junction IDs in black color, and maximum daily base demands of junctions (l/s) and pipes lengths (m) in red and blue colors, respectively. The network consists of 25 junctions and 35 pipes with 8 available diameters in the market.

The network is fed from a ground tank, which acts as a reservoir. This reservoir is located northeast of the network and is accompanied by a pumping station. The pumping station is assumed to have only one variable speed pump with constant power. All junctions have the same diurnal demand pattern with different base demands except for one junction located far west of the network. This junction in the far west is a factory whose demand starts from 7 am till 7 pm with a uniform demand pattern.

A single storage tank balances the demand to satisfy the diurnal demand pattern fluctuations during any normal operational day. It is worth noting that separate trials were made to investigate the effect of inserting two storage tanks in the network following the same criterion used for a single tank design. However, it was found that most of the results indicated locating one large storage tank side by side with another relatively small storage tank, indicating the insignificance of inserting more than one large tank in the Safi network. More information about the roughness of pipes, allowable pressures of nodes, pipes costs, and design variables limits can be found in

Table A1. Values of the Safi network coefficients and constraints.

Parameter	Description	Value	Unit
Prmin	min. allowable pressure head	20	m
Prmax	max. allowable pressure head	60	m
Velmin	min. allowable flow velocity	0.30	m/s
Velmax	max. allowable flow velocity	1.50	m/s
HSmax	max. tank bottom elevation	50	m
Pmax	max. pump power	200	kw
Ck	Pipe unit cost (d = 80 mm)	60	USD/m
	Pipe unit cost (d = 100 mm)	70	USD/m
	Pipe unit cost (d = 150 mm)	90	USD/m
	Pipe unit cost (d = 200 mm)	130	USD/m
	Pipe unit cost (d = 300 mm)	180	USD/m
	Pipe unit cost (d = 400 mm)	260	USD/m
	Pipe unit cost (d = 500 mm)	310	USD/m
	Pipe unit cost (d = 600 mm)	360	USD/m
Nav	Number of available diameters	8	-
Ce	energy unit cost	0.15	USD/kwh
I	interest rate	8	%
N	lifetime	20	years
CHW	Hazen–Williams coefficient	140	-

in the Appendix A, while the demand pattern and elevations of network nodes are shown in Figure A1 in Appendix A.

The minimum and maximum allowed pressure heads, $P_{min}$ and $P_{max}$, are assumed to be 20 and 60 m, respectively, while the minimum and maximum allowed chlorine concentrations, $C{C}_{min}$ and $C{C}_{max}$, are assumed to be 0.2 and 0.6 mg/L, respectively. The tank cost coefficients, $x_0$, $x_1$, and $x_2$ are set equal to 35 $\times$ ${10}^4$, 150, and 3, respectively.

Each junction’s cyclic state for its chlorine concentration pattern for this network is obtained after running the model for one day. The evaluation of the objective functions for each design step (i.e., the hydraulic design step and the water quality step) is carried out only on the second day of simulation to ensure that a cyclic chlorine pattern is reached for all the junctions. The network resilience ($NR$) is calculated at 8 pm on the second day, which is the maximum daily demand hour.

5. Results and Discussion

The modified MOPSO algorithm runs 15 times for each design step, the hydraulic design step and the water quality design step. The final PFs obtained from all the runs are accumulated, ranked, and sorted to form one final PF per design step, as shown in Figure 3. Based on the first scenario of the water quality design step and utilizing the configuration of the median solution for the resultant hydraulic design PF, the final PFs comprise 282 and 200 non-dominated feasible solutions for the hydraulic design and the water quality design, respectively.

Moreover, to investigate the effect of the hydraulic configuration of the network on the water quality design results, other hydraulic configurations from the resulting hydraulic design PF are used to proceed to the water quality design using the first water quality design scenario. These configurations are the first extreme point, the 25% quartile point, the 75% quartile point, and the last extreme point of the final hydraulic design PF, as shown in Figure 3a. In Figure 3b, the resulting water quality PFs from the different hydraulic configurations indicate no direct correlation between the final PF of the hydraulic design and the resulting PFs of the water quality design. This could be attributed to the fact that no clear sequence was identified for the resulting water quality PFs. For instance, the water quality PF based on the 75% quartile point from the hydraulic design dominates all the other resulting PFs in terms of the studied objective functions for the water quality design step. Therefore, the selection of the appropriate network hydraulic configuration should be carefully investigated by designers and decision-makers to reach a competent overall network design.

Regarding the different proposed scenarios for the water quality design step, the median solution of the hydraulic design step PF is used to obtain the final water quality PF for each scenario, as shown in Figure 4. The final PF of the first scenario has 200 feasible solutions with $H{V}_m$ of 1.772. On the contrary, the second and the third scenarios have 279 and 845 feasible solutions with $H{V}_m$ of 1.36 and 1.65, respectively. The PF of the first scenario ranges from 0.031 mg/L to 0.236 mg/L for $ACD$ and from 4.326 kg to 7.823 kg for $DCM$, covering the widest range of objective functions values among all the obtained PFs from other scenarios. The second scenario produces the least diverse and convergent PF among all the other scenarios PFs. It only covers a range from 0.142 mg/L to 0.239 mg/L for $ACD$ and 4.508 kg to 6.246 kg for $DCM$. The third scenario PF almost has the same range as the first scenario. However, a considerable gap in this PF is noticed, ranging from 0.150 mg/L to 0.201 mg/L for $ACD$ and from 5.348 kg to 6.234 kg for $DCM$. All the PFs characteristics for the three scenarios are summarized in

Table 1. A comparison between the final PFs of each design scenario for the water quality design step.

	Design Scenario
ID	I	II	III
No. of feasible solutions	200	279	845
HVm	1.772	1.360	1.650
Upper PF point ACD (mg/L)	0.031	0.142	0.024
Lower PF point ACD (mg/L)	0.236	0.239	0.251
Upper PF point DCM (kg)	7.823	6.246	7.829
Lower PF point DCM (kg)	4.326	4.508	4.600

. The first scenario is the most computationally effective. However, it may be classified as the most impractical scenario since it relies on a non-uniform chlorine pattern of the booster station with wide multipliers variability during the one-day operation. Several trials were made to smooth the non-uniform chlorine pattern that resulted from the simulation. However, all the obtained results were found to be infeasible. This indicates that the chlorine concentration is quite sensitive to any change confirming the impracticality of this scenario. The second scenario appears more practical than the first, but its implementation remains challenging due to the difficulty of turning the boosted chlorine on and off. On the other hand, the third scenario is the most practical from the author’s perspective, as it is the most commonly implemented by water utilities, despite a significant PF gap. However, the third scenario kept producing infeasible solutions for 6 runs out of the total 15 runs used for obtaining the results in the water quality design step, indicating significant difficulties in converging to the feasible region of the studied search space compared to the first and second scenarios. Therefore, for simplification purposes, the water quality PF resulting from the first scenario based on the median solution configuration from the final hydraulic design PF is utilized to analyze the rest of this work.

The hypervolume metric, $H{V}_m$, progress curves for all the runs for each design step are presented in Figure 5. The $H{V}_m$ is drawn versus the number of function evaluations, $NFE$. $NFE$ represents how many times the algorithm evaluates the objective functions during any single run and also represents the consumed computational budget. An $NFE$ of 40,000 is used in any single run for the hydraulic and water quality designs. A mean curve is also drawn in black in the same figure to represent the mean performance of the algorithm.

The algorithm rapidly converges to the area of feasible solutions in the hydraulic design step since it begins to record increasing $H{V}_m$ values from the beginning of $NFE$. On the contrary, the algorithm consumes relatively more $NFE$ at the beginning of any run in the water quality design step to converge to the feasible solutions area compared to the hydraulic design step. The reason for this is that the search space of the water quality design step is larger than that of the hydraulic design step since the former has a wider set of available options for the values of its decision variables.

For the PF of the hydraulic design step, the annual network cost, $ANC$, ranges from $5.02\times {10}^5$ to $6.94\times {10}^5$ USD, which corresponds to a network resilience, $NR$, varying from 0.45 to 0.89, respectively. Almost 82% of the PF solutions place the storage tank on a small hill in the western part of the network, while the rest of the solutions place it at scattered points west of the network. The PF solutions propose a mean height of tank supports of almost 22.00 $\pm 3.40$ m, a mean ratio of the tank’s diameter to height of around 0.92 $\pm 0.19$, and a mean pumping power of 55.48 $\pm 2.78$ kw.

The hydraulic design PF’s median solution is representative of all the other solutions. In this solution, the annual network cost, $ANC$, is about $5.67\times {10}^5$ USD, and the network resilience, $NR$, is around 0.74. The storage tank in this solution is located above the small hill west of the network. The tank has a support height of about 25.25 m, a diameter of approximately 17.25 m, and the ratio of the tank’s diameter to storage height is about 0.80. The pump generates a constant power of roughly 55.50 kw, and the pipes configuration is given in

Table 2. Network pipes configuration of the median solution.

	Pipes
ID	35	1–2, 17	3–5, 15–16, 29	6–9, 13–14, 18–23, 25–28, 30, 32, 34	10–12, 24, 31, 33
Diameter (mm)	500	400	300	200	100

. The storage tank stores water in low-demand periods while discharging water into the network in high-demand periods.

For the water quality design step, the extreme points on the PF curve show that $DCM$ varies from as low as 4.33 to 7.82 kg, while the corresponding $ACD$ value varies from 0.24 mg/L to 0.03 mg/L. The water-quality PF curve exhibits almost a linear trend. All the PF solutions locate the booster station adjacent to the storage tank. The PF solutions introduce mean base concentrations of the booster station and reservoir of around 0.333 mg/L and 0.447 mg/L with standard deviations of $\pm 0.036$ mg/L and $\pm 0.090$ mg/L, respectively.

The median solution of the water quality PF has $DCM$ of 6.12 kg per day and $ACD$ of about 0.11 mg/L while locating the booster station to the closest junction near the storage tank. The network’s water balance and the booster station’s chlorine pattern are shown in Figure 6. The negative values in Figure 6 show that the storage tank is in the process of filling, while the positive ones indicate that it is emptying. It may be noticed that there are two peaks in the chlorine pattern for this solution, which oddly match the same timing of the two peaks in the demand pattern. However, after investigating the other PF solutions, it is found that this matching is not supposed to be generalized or considered a rule since there is no direct correlation between the water demands and chlorine concentrations.

Figure 7 represents the daily chlorine concentrations for the reservoir, a centralized point, the booster station, and the factory located far west of the network. However, none of the junctions’ concentrations violate the chlorine constraints, $C{C}_{min}$ and $C{C}_{max}$, along any hour during the selected simulation day. The configuration of the booster station (i.e., the chlorine pattern, the station location, and the base chlorine concentration) along with the constant chlorine concentration provided by the reservoir satisfy the chlorine requirements along the studied day while pushing the concentration towards the maximum allowed concentration, $C{C}_{max}$. This criterion of pushing the actual chlorine concentrations towards $C{C}_{max}$ alleviates the violations of chlorine constraints that may occur during unusual events (e.g., pipe bursts, pipes leakages, etc.) while maximizing the chlorine residuals at the far ends of the network.

Figure 8 shows the mass flow rate of the total chlorine dosage during the studied 24-h time period. The accumulation of both mass flow rate curves for the reservoir and the booster station using a time step of 15 min results in a total chlorine mass of 6.12 kg per day. Relatively larger fluctuations are observed in the mass flow rate curve of the booster station compared to a nearly smooth curve of the reservoir. This is evident since the nodes’ concentrations from the east to the center of the network, which have the same demand pattern, are affected by the reservoir concentration. On the contrary, the booster station mainly affects the rest of the network with partially higher fluctuations due to the factory node’s variable demands compared to any other ordinary node. Therefore, the reservoir produces a larger chlorine dosage rate with smoother fluctuations, while the booster station produces a smaller one with higher fluctuations.

6. Conclusions

This study aimed to investigate the optimum design of water distribution networks in terms of both hydraulic and water quality design aspects. Most previous works combined both designs into one optimization model, neglecting the effect of splitting the whole optimization process into two separate models, i.e., the hydraulic design and the water quality design models. The splitting process is necessary to investigate the network behavior in each model and obligatory when decision-makers require larger sets of practical and feasible solutions for the studied networks. Accordingly, a two-step design framework based on a modified version of MOPSO algorithm is developed to optimize the network’s hydraulic and water quality designs and is applied to a real network in Yemen. In the hydraulic design step, a Pareto set of non-dominated solutions in terms of the network annual cost and the network resilience is obtained. The proposed framework identifies the configurations of pipes diameters, locations, dimensions, and elevation of a single high-elevated tank, and the required pumping power for each solution. Different solutions from the Pareto set of the hydraulic design step are selected as initial configurations for the water quality design step, and the 75% quartile point produced the most convergent Pareto set among all the studied points.

In the water quality design, three different Pareto sets in terms of the overall chlorine mass injected within the network and the accumulated differences between the actual and the maximum chlorine concentrations for all the network junctions using three different design scenarios are obtained. The chlorine pattern, the location, and the base concentration of a single booster station, along with the base chlorine concentration for the feeding reservoir per solution in the Pareto set, are identified. The three scenarios differ only in formulating the chlorine pattern of the booster station. The final PF of the first scenario assumes initial random values between 0.0 and 1.0 for the chlorine pattern multipliers, the second scenario assumes chlorine pattern multipliers of only 0.0 or 1.0, and the third scenario assumes a uniform chlorine pattern with unknown starting and ending times. Each scenario of the three proposed scenarios has its pros and cons. Although the first scenario is impractical due to the large variations in the chlorine pattern of the booster station, it provides the most diverse PF among all the other studied scenarios. The second scenario is more practical than the first one since it provides a sort of ON/OFF injection of chlorine dosage during the daily operation. However, it produces the least diverse PF among all the other obtained PFs from other scenarios. The third scenario, which could be classified as the most practical scenario, has a large gap in its final PF, reducing the number of design options available for decision-makers to cover all the values of the studied objective functions. Therefore, there exist trade-off characteristics between the accuracy and practicality of each water quality design scenario.

Additionally, for the water quality design step, the booster station revealed its capability to confront any expected fluctuations in the chlorine concentrations, especially for any considerably far or relatively large demand points.

In both design steps, the hydraulic design step and the water quality design step, the modified MOPSO algorithm converges rapidly to the area of feasible solutions within the search space, except for some tiny delays in convergence for the water quality design step. These delays are caused by the wider search space of the water quality design problem compared to that of the hydraulic design problem. Generally, the developed framework is a powerful tool for designing water distribution systems and could be applied to any other network. No parameter tuning of the MOPSO algorithm nor penalty coefficient for constraints handling is needed for the developed framework.

The following points are the most important conclusions of this work:

• The PF solutions of the hydraulic design are crucial for obtaining the best PF solutions for the water quality design. They should be clearly investigated before selecting the most appropriate hydraulic design configuration before proceeding to the water quality design step;
• The splitting of the whole WDSs design process into two sequential steps, the hydraulic design and the water quality design, helps provide decision makers with a more diverse set of solutions compared to solving all the variables simultaneously;
• The chlorine booster stations’ number, location, and pattern are very important in confronting chlorine decay, especially for relatively far and large demanding points, and the chlorine pattern shape of each booster station has a trade-off characteristic between its accuracy and its practicality.

7. Study Contribution to Theory and Future Work

Despite the availability of various frameworks in the literature for optimizing WDSs design, the developed framework in this work contributes to the theory by investigating the linkage between the hydraulic and water quality designs of WDSs to obtain a diverse set of optimum, reliable, and practical designs to select from. Moreover, it provides a fast-convergent and easy-to-use tool that successfully reaches semi-optimum design solutions while using a relatively small computational budget.

However, the current work has some limitations that may be investigated later, such as comparing the results of the developed framework, which is mainly based on a modified MOPSO algorithm, which may be compared with that of other optimization algorithms, such as NSGA-II, GALAXY, etc. Moreover, the investigation of water chlorine concentration in the storage tank was neglected, the insertion of more than one storage tank or more than one chlorine booster station was not included, and the seasonal variations in water demands were not considered.

Regarding future works, more objective functions, such as minimizing water leakages, minimizing water age, etc., may be added to the framework to enhance the utility of the proposed framework. Additionally, the same design procedures and guidelines used in this work may be applied to other benchmarks and real WDSs to ensure the applicability of the developed framework.