Co-Optimization of Valve Placement and Chlorine Dosing in Water Distribution Systems

Abstract

The primary objective of water distribution systems (WDSs) is to ensure a high-quality water supply. Chlorine, commonly used as a primary disinfectant in WDSs, requires precise control of its concentration to safeguard public health. However, the complex network structure and its highly nonlinear dynamics inherent in a WDS pose significant challenges in chlorine management. This study proposes an optimization approach to tackle these challenges by leveraging different kinds of valves to distribute water flow within the network, thereby realizing both chlorine and pressure management. A simplified chlorine propagation model is introduced, based on which three optimization problems are formulated and solved to compute optimal management strategies. The first one is a nonlinear programming (NLP) which minimizes source dosing across the WDS by an optimal distribution of the water flow in the pipelines, leading to a lower bound for the chlorine management. The second one is a mixed-integer nonlinear programming (MINLP) problem to localize isolation valves (IVs) and achieve a realistic and practical solution. The third one extends the MINLP framework to integrate pressure reducing valves (PRVs), optimizing the placement of both IVs and PRVs to enable a multi-objective approach that minimizes chlorine dosing while regulating the system pressure. The results of a benchmark demonstrate that the integrated use of IVs and PRVs significantly enhances the performance of both chlorine concentration and pressure regulation, offering an efficient solution for WDS management.

1. Introduction

Chlorine is the most widely used disinfectant in WDSs, and the level of chlorine concentration in water distribution systems (WDSs) represents one of the most important criteria for the assessment of drinking water quality. While some European countries have demonstrated that residual disinfectants may not be necessary with appropriate safeguards in place [1], maintaining an appropriate residual chlorine concentration remains a key metric of water quality in global WDS evaluations. Maintaining an appropriate residual safeguards public health by inactivating waterborne pathogens. As a result, water utilities worldwide are required to comply with national water quality regulations to ensure specified chlorine levels [2]. For instance, the United States Environmental Protection Agency suggests a maximum residual disinfection level of 4 mg/L for chlorine [3]. The World Health Organization (WHO) suggests the residual chlorine concentration in tap water should be between 0.2 and 0.5 mg/L [4]. To tackle this issue, various methods, including the placement of booster stations and hydraulic controls [5,6,7], have been proposed. These methods aim to optimize chlorine distribution in the network by managing the injection strategy and the flow dynamics.

Ensuring effective disinfection necessitates the maintenance of a minimum chlorine level to guarantee a robust disinfecting capacity in the network. Conversely, an excess of chlorine can give rise to various issues, including water with an undesirable taste, accelerated pipe corrosion, and amplifying the formation of disinfection by-products (DBPs). In raw water, the contact of natural organic matters with chlorine produces DBPs, including trihalomethanes, haloacetic acids, and chlorophenols, which are linked with various health issues [6,7,8,9,10,11,12]. Thus, maintaining a residual chlorine concentration within a reasonable range is imperative for the operations of WDSs.

However, in spite of the disinfectant measures, maintaining a consistent chlorine concentration throughout the network remains a challenging task. Chlorine undergoes complex reactions with natural organic matter in bulk water and with the biofilm on the pipe surfaces. These chemical and biological reactions can be affected by various factors such as pipe material and age, water temperature, pH value, corrosion, and biofilm formations [10,13,14]. Consequently, the concentration of chlorine at consumer taps often diminishes compared to that at the initial dosing point, a phenomenon known as chlorine decay which compromises the efficacy of chlorine as a purifier [2,11].

Factors influencing variations in chlorine concentration and water quality are inherently complicated. A lot of studies have been made on modeling reactions within pipes to better understand and manage these complexities. In [13], a mass transfer-based model was developed for predicting chlorine decay in WDSs. This model considers first-order reactions of chlorine occurring both in the bulk flow and at the pipe wall. Notably, this model has been incorporated into the widely used simulation tool EPANET [15]. The development of the EPANET multi-species extension [10] further extended the capability to model the reaction and transport of multiple interacting chemical species in WDSs. Furthermore, the study in [16] first proposed a two-component second-order chlorine decay model based on the concept of competing reacting substances. Subsequent work in [17] further refined this approach. To estimate the parameters in the second-order chlorine decay model, Bayesian statistical analysis incorporating Monte Carlo Markov chain with Gibbs sampling has been used in [18]. In [19], another significant development was contributed with their variable rate coefficient model derived from kinetic equations for concurrent bimolecular second-order reactions. The two-reactant model was proposed through comprehensive calibration against multiple decay test datasets and successful validation predictions [20]. However, in [21], the authors pointed out maintained that for practical water distribution network modeling, first-order kinetics assumptions for bulk and overall decay remain sufficiently accurate.

In [10], a simulation framework for modeling the scale-adaptive hydraulic and chlorine decay under steady as well as unsteady state flows was introduced. Results from this study demonstrated that dynamic hydraulic conditions have a significant impact on water quality deterioration, leading to a rapid loss of the disinfectant residual. In addition, investigations into the relationship between chlorine decay kinetics and key water parameters (e.g., natural organic matter, water temperature, chlorine demand) were conducted to more accurately predict chlorine decay in WDSs [22]. More recently, to improve the prediction of chlorine residuals within WDSs, full-scale WDS data were employed to explore the effectiveness of a hydraulic model combined with process-based chlorine residual and data-driven models [23]. Their study highlighted significant nonlinearities and intricate relationships between operational parameters and water quality, underscoring the complexity of accurately modeling the chlorine behavior in water systems.

Moreover, various studies have been made to control the chlorine concentration in WDSs and one widely employed approach is the use of booster stations. Booster stations can effectively inject disinfectant at selected locations within the network, aiming at a uniform spatial and temporal distribution of chlorine residuals. Optimization approaches were proposed to determine optimal locations for booster stations and injection strategies [5,24,25]. Furthermore, optimal scheduling of pumping and chlorine booster was studied to simultaneously minimize the pumping costs and maximize the injected chlorine dose to maximize the system protection [26]. In the study of [7], the objective was to minimize the average zone pressure while penalizing deviations from a target chlorine concentration, demonstrating a holistic approach to water quality management.

An alternative approach to managing chlorine concentration in WDSs is hydraulic control, which encompasses pump operations, tank levels, and valve settings [7]. The objective is to minimize the traveling time from water sources to demand nodes, thereby mitigating the disinfectant decay reactions and ensuring compliance with the chlorine residual specifications. Optimal pump operation in WDSs was investigated in [27], taking water quality considerations into account. The work in [28] introduced a method for real-time optimal valve operation and booster disinfection. In response to contamination incidents, valve placement and real-time scheduling were utilized for contamination isolation in WDSs [29,30]. Moreover, valves were strategically placed to maximize the self-cleaning capacity of WDSs under normal operations. This involves isolating certain areas using isolation valves (IVs) and manipulating pressure control valves, particularly during peak demand hours [31,32].

Previous studies have also introduced various frameworks considering chlorine dosage within WDSs, aiming at optimizing dosing schedules by minimizing the total dose amount while ensuring an evenly distributed chlorine residual throughout the network [33,34]. Model predictive control was used to adjust chlorine dosage in real-time [35]. Furthermore, a heuristic approach was developed for emergency scenarios, enabling the prediction of appropriate initial chlorine doses to ensure a rapid and effective response to emergency situations [22].

In addition to chlorine management, pressure should also be carefully managed in WDSs to maintain a balance between an adequate water supply and preventing increased water loss as well as pipe bursts caused by excessively high pressure [7,36,37,38]. To address these challenges, PRVs are widely used as an effective measure for pressure control in WDSs [7,37,38,39]. The placement and operation of PRVs have been optimized through various approaches. More recently, an approach was proposed to simultaneously optimize pressure and water age, in a multi-objective optimization framework [40].

In the meantime, from a methodological point of view, optimization has been extensively applied in addressing diverse challenges within WDSs. In studies such as [7,29,38,41], multi-objective optimization problems and mixed-integer nonlinear programming (MINLP) were used for optimal design and operation of WDSs. These include minimizing pumping costs [27,42], compromising water quality [28], responding to contamination events [30], determining optimal valve placements [31,43], and identifying suitable locations for chlorine booster stations [7]. Recent studies have introduced local energetic/surplus indices (LSI) for node- and pipe-level diagnostics in WDSs, which complement optimization-based approaches and can inform valve placement and objective design [44]. The outcomes of these investigations demonstrate the effectiveness of optimization methods for enhancing the performance of WDSs.

In summary, many previous studies have achieved disinfectant residual compliance via boosters and pump/tank operations; valve-based strategies have been explored. However, a joint, budget-constrained placement and operation of IVs and PRVs to meet a specified residual band without boosters—under explicit hydraulic feasibility, pressure limits—remains underexplored. Utilities often avoid new boosters; the practical question is whether a compliance can be achieved by redistributing flows with a limited number of valves and what the minimum source dose should be, as well as the trade-off with system pressure.

This study targets that gap by formulating and solving appropriate optimization problems. We address three practical questions: (i) Given a WDS and a residual band, what is the minimum source dosing required to meet compliance? (ii) Can compliance be achieved by re-distributing flows using a limited number of IVs? (iii) Can PRVs be co-optimized with IVs to reduce dosing while maintaining pressure constraints?

We first use a steady-state, first-order, complete-mixing surrogate for chlorine transport to enable tractable network optimization. Then, two kinds of optimization problems are formulated to keep the chlorine concentration within a specified range. In the first, we assume ideal per-link flow control; while not realistic, this provides a lower bound on the required source dose (NLP). The second considers a set of IVs that can be manipulated to distribute flows and thus chlorine (MINLP). Furthermore, we explore simultaneous minimization of chlorine dosing and system pressure through the placement of both IVs and PRVs, offering a more holistic approach to WDS management. We evaluate the approach on a benchmark network to demonstrate applicability.

The remainder of the paper follows. Section 2 introduces a model for describing the chlorine propagation inside the network driven by water flows governed by the hydraulics. It then describes the formulation of the nonlinear programming (NLP) and MINLP optimization problems for chlorine management, where IVs and PRVs are considered, respectively. Section 3 gives the results of a benchmark for verifying the proposed approach. The paper is concluded briefly in Section 4.

2. Materials and Methods

2.1. Modeling the Chlorine Propagation

To study chlorine management for a WDS, we need a model to describe the chlorine distribution in the network. In this section, we will introduce a simplified model to describe the chlorine propagation driven by the hydraulic dynamics of the network. We consider a general WDS consisting of:

• $n_0$ water sources (e.g. reservoirs or tanks);
• $n_n$ nodes (e.g. consumer household or industry);
• $n_p$ pipes connecting the water sources and nodes;
• $n_l$ pipe loops in the WDS.

Correspondingly, the WDS can be conceptualized as a directed graph with $n_n+n_0$ vertices and $2n_p$ links, similar to those considered in [40,41].

2.1.1. Chlorine Propagation Model

Typically, the advective transition of chlorine in the water along a pipe is described as a one-dimensional transport reaction model, where molecular diffusion and hydraulic dispersion are neglected [44,45]:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathsf{\partial }{C}_j(x,t)}{\mathsf{\partial }t}}+u_j{\displaystyle \frac{\mathsf{\partial }{C}_j(x,t)}{\mathsf{\partial }x}}& =R\left(C_j(x,t)\right),\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill C_j(x,0)& =C_{j0}\hfill \end{array}$$

(1b)

Equation (1a) is a partial differential equation (PDE), where $C_j\left(x,t\right)$ is the chlorine concentration along pipe j at distance x and time t, $u_j$ is the water flow velocity in the pipe j, R is the reaction rate as a function of the concentration which depends on the bulk and wall reaction [15]. Equation (1b) denotes the initial chlorine concentration in the pipe.

In this study, we assume that both bulk and wall reactions follow the first-order decay kinetics, respectively. This assumption implies that $R\left(C_j(x,t)\right)=-(k_b+k_w)C_j(x,t)=-r{C}_j(x,t)$, where $k_b$ and $k_w$ are the bulk and wall reaction coefficients, respectively, and r is the total reaction rate coefficient [46].

It is well recognized that optimizing the PDE model is challenging. However, the chlorine management under study is commonly a task to achieve an optimal steady-state operation for the network, i.e., chlorine is to be distributed inside the network but its concentration is time-independent. Therefore, Equation (1a) is reduced to:

$$u_j{\displaystyle \frac{d{C}_j\left(x\right)}{dx}}=-r{C}_j\left(x\right)$$

(2)

For a given flow velocity $u_j$, this equation can be solved as:

$$C_j\left(x\right)=C_j\left(0\right)e^{-\left({\displaystyle \frac{r}{u_j}}\right)x}$$

(3)

where $C_j\left(0\right)$ represents the chlorine concentration at the inlet of the pipe. If the pipe is the downstream pipe of a reservoir, this value corresponds to the chlorine dosing concentration. Equation (3) shows that the chlorine concentration decays along the pipe. Since the position within the pipe is given by $x\left(t\right)=u_{j}t$, the concentration at a given position within the pipe is determined by the time it takes for the water to travel from the inlet of the pipe to that position. Thus, Equation (3) facilitates the calculation of the chlorine concentration at each node (i.e., the outlet of the pipe) in the WDS, depending on the initial value, the reaction coefficient r, and the flow time t. The initial value and reaction coefficient r are predefined according to the knowledge of the system and the reaction kinetics. The water flow time $t_j$ to the pipe outlet is calculated by:

$$t_j={\displaystyle \frac{L_j}{u_j}}={\displaystyle \frac{\pi {\left(\frac{d_j}{2}\right)}^2}{q_j}}{L}_j$$

(4)

where $L_j$ and $d_j$ represent the length and diameter of pipe j, respectively. The flow rate $q_j$ is the only variable affecting the flow time and, consequently, the chlorine concentration in the network. In this study, valves are employed to regulate water flow rates in the pipes, thereby determining the distribution of the chlorine concentration in the network.

From Equations (3) and (4), the chlorine concentration at the outlet (i.e., at a node) of pipe j is:

$$C_j(L_j,t_j)=C_j\left(0\right)e^{-r{t}_j}$$

(5)

In addition, it is common that a node in a WDS has multiple inflows, as illustrated in Figure 1. To calculate the chlorine concentration at a node, we assume that the water entering a node blends entirely and instantaneously. Therefore, as shown in Figure 1, the chlorine concentration at node $n+1$ is calculated as a flow-weighted sum of the chlorine from the inflow pipes as follows:

$$C_{n+1}={\displaystyle \frac{{\sum }_{j=1}^n{q}_j{C}_j(L_j,t_j)}{{\sum }_{j=1}^n{q}_j}}={\displaystyle \frac{{\sum }_{j=1}^n{q}_j{C}_j{e}^{-r{t}_j}}{{\sum }_{j=1}^n{q}_j}}$$

(6)

From Equation (6), it is evident that the chlorine concentration at a node depends on the chlorine levels at upstream nodes and the flow rate in the connecting pipes. In this way, given a chlorine dosing at a source position, its propagation across the network can be readily calculated from node to node. Therefore, we can optimize the chlorine distribution by manipulating the flow rates in the pipes for chlorine management.

Our goal of study is to develop a planning-level decision tool. We adopt the standard EPANET bulk+wall first-order framework in steady state for three reasons: (i) it is widely used and accepted for network-wide optimization [13,21]; (ii) it captures the dominant sensitivity of residuals to travel time and dosing; and (iii) it enables tractable MINLP for valve placement. We acknowledge that dispersion and incomplete mixing at junctions can matter under certain hydraulics. To address this, we validate all optimal plans in EPANET’s water quality engine, showing our decisions are stable within practical ranges. We view more detailed kinetics as a valuable extension but outside the scope of the planning question we target.

2.1.2. Hydraulic Model

The manipulation of flow rates in the pipes of a network should comply with the hydraulic laws. The mass conservation in the whole system is expressed as follows:

$$NQ=D,$$

(7)

where the matrix N denotes the node-pipe adjacency. It has a dimension of $n_n\times n_p$, where each element $N_{ij}$ indicates the connection between node i and pipe j, specifically:

$$\begin{array}{c}\hfill N_{ij}=\left\{\begin{array}{cc}& 1\mathrm{if}\mathrm{flow}\mathrm{in}\mathrm{pipe}j\mathrm{flows}\mathrm{into}\mathrm{node}i,\hfill \\ & 0\mathrm{if}\mathrm{pipe}j\mathrm{is}\mathrm{not}\mathrm{connected}\mathrm{to}\mathrm{node}i,\hfill \\ & -1\mathrm{if}\mathrm{flow}\mathrm{in}\mathrm{pipe}j\mathrm{flows}\mathrm{out}\mathrm{of}\mathrm{node}i.\hfill \end{array}\right.\end{array}$$

(8)

In Equation (7), $Q={\left[q_1,q_2,\dots ,q_{n_p}\right]}^T$ represents the flow rates in the pipes, and $D={\left[d_1,d_2,\dots ,d_{n_n}\right]}^T$ denotes the individual demand at each node in the WDS.

Energy conservation is described loop-wise as follows:

$$\begin{array}{c}\hfill W{H}_f=0\end{array}$$

(9)

where the matrix W has a dimension of $n_l\times n_p$, with its element expressed as:

$$\begin{array}{c}\hfill W_{lj}=\left\{\begin{array}{cc}& 1\mathrm{if}\mathrm{flow}\mathrm{in}\mathrm{pipe}j\mathrm{is}\mathrm{clockwise}\mathrm{to}\mathrm{loop}l,\hfill \\ & 0\mathrm{if}\mathrm{it}\mathrm{is}\mathrm{not}\mathrm{present}\mathrm{in}\mathrm{loop}l,\hfill \\ & -1\mathrm{if}\mathrm{it}\mathrm{is}\mathrm{counter-clockwise}\mathrm{to}\mathrm{loop}l.\hfill \end{array}\right.\end{array}$$

(10)

In Equation (9), $H_f={\left[h_{f1},h_{f2},\dots ,h_{f{n}_p}\right]}^T$ represents the head losses on each pipe in the WDS. Equation (9) states that the directional sum of head losses around any loop in the WDS equals zero. The head loss on each pipe is calculated using the Hazen–Williams equation [15]:

$$h_{f,j}=L_j{\displaystyle \frac{10.67q_j^{1.852}}{{\chi }_j^{1.852}{d}_j^{4.8704}}}$$

(11)

where $L_j$ is the length of the pipe, ${\chi }_j$ stands for the Hazen–Williams coefficient, and $d_j$ denotes the inside diameter of the pipe, while $q_j$ signifies the flow rate through the pipe.

It is important to note that there is another way to describe energy conservation as follows [40,41,47,48]:

$$\begin{array}{cc}\hfill Q(-N^T(P+E)-H_f)& \ge 0\hfill \\ \hfill -N^T(P+E)-H_f-Mz& \le 0\hfill \end{array}$$

(12)

where N, Q, $H_f$ are defined as in Equations (7) and (9). The vectors P and E, each with dimension $n_n$, depict the pressure and elevation of the nodes in the WDS, respectively. M is a diagonal matrix with dimension of $n_p\times n_p$, expressing the extra head loss on the valves. And z is a $n_p$ dimensional 0–1 vector indicating the existence of a valve or not on the corresponding pipe. In comparison to Equation (9), Equation (12) captures the energy conservation across all pipes with the capability to describe the effect of valves on the flow rate.

In addition, to account for flow direction changes in the WDS, all matrices and vectors are expanded to have $2n_p$ columns or rows. This expansion will be explained in more detail later in the optimization problem formulation.

More importantly, the primary distinction between the two energy conservation models, Equations (9) and (12), lies in their scope and application. The loop-wise model focuses on conserving energy within each loop in the WDS, so that the flow in tree-structured areas can be determined by the mass conservation. This approach simplifies the computation but is limited in its ability to address scenarios where valves are present in the tree-structured areas, since it does not account for energy losses on the valves.

In contrast, the pipe-wise model (Equation (12)) provides a more comprehensive framework by ensuring energy conservation for each pipe, including the energy loss associated with valves placed on the pipe. This approach requires the head loss across the valves to be predefined, allowing the model to account for the impact of valves on the overall system energy balance. In this study, both IVs and PRVs are considered.

2.2. Optimization Problem Formulations

In this subsection, we will develop three optimization problems for chlorine management for WDSs. The first one, a NLP problem, aims to identify an ideally optimized chlorine dosing strategy and control the residual chlorine concentration by regulating the flow rates on each pipe, based on the loop-wise model. The second one, a MINLP problem, focuses on managing chlorine concentration by determining the IVs and adjusting the chlorine dosing at the water reservoirs. In addition, we propose an extension of the MINLP problem that enables simultaneous management of both chlorine distribution and system pressure by using PRVs.

2.2.1. The NLP Problem

Our objective is to minimize the residual chlorine throughout the WDS while ensuring that the chlorine concentration at each node remains within the specified range. The decision variables in the problem are the flow rates in individual pipes. Although control of the flow rate in each pipe is not realistic, the solution to this problem provides a lower bound of the chlorine dosing, valuable information for further studies on chlorine management. The NLP problem is defined as follows:

$$\begin{array}{ccc}\hfill & min\sum _{k=1}^{n_0}{C}_{S,k}\hfill & \hfill \end{array}$$

(13a)

$$\begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& NQ=D\hfill & \hfill \end{array}$$

(13b)

$$\begin{array}{ccc}\hfill & W{H}_f=0\hfill & \hfill \end{array}$$

(13c)

$$\begin{array}{ccc}\hfill & C_{min}\le C_i\le C_{max},\forall i=1,\cdots ,n_n\hfill & \hfill \end{array}$$

(13d)

$$\begin{array}{ccc}\hfill & 0\le q_j\le q_{max},\forall j=1,\cdots ,n_p\hfill & \hfill \end{array}$$

(13e)

where the objective function Equation (13a) aims to minimize the total chlorine dosing at the sources of the WDS. $C_{S,k}$ is the chlorine concentration at the source position k. The equality constraints (13b) and (13c) ensure the mass and energy conservation, as outlined in Equations (7) and (9). Here, special attention should be paid to the fact that water flow in loops can be bidirectional. To address this, the matrix N, with dimensions $n_n\times 2n_p$ in (13b), is defined as $N=[N_1,N_2]$, where $N_1$ is the adjacency matrix, as defined in Equation (8), and $N_2$=$-N_1$ to take flow direction reversal into account. D is a vector of size $n_n$ representing the demand at each node. Q is the flow rate vector with a dimension of $2n_p$, in which the elements $Q_j$ and $Q_{j+n_p}$ correspond to the same pipe but with different flow directions. The value at $Q_i$ represents the flow rate in the initial direction of the system; if the flow direction reverses, $Q_i$ becomes 0, and $Q_{i+n_p}$ takes a positive value.

The inequality constraints (13d) and (13e) ensure the chlorine concentration at each node and flow rate in each pipe within the specified upper and lower bound for the WDS operation, respectively. The chlorine concentration $C_i$ will be calculated by the node-to-node model (see Equation (6)).

2.2.2. The MINLP Problem

In practice, IVs, also known as on-off valves, are essential components of WDSs and are widely used to switch off specific pipes [49]. In this study, we employ the available on-off valves as optimization variables to manipulate the water flow distribution in the network for chlorine management, by defining the following MINLP problem:

$$\begin{array}{cc}\hfill & min\sum _{k=1}^{n_0}{C}_{S,k}\hfill \end{array}$$

(14a)

$$\begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& NQ=D\hfill & \hfill \end{array}$$

(14b)

$$\begin{array}{ccc}\hfill & Q(-N^T(P+E)-H_f)\ge 0\hfill & \hfill \end{array}$$

(14c)

$$\begin{array}{ccc}\hfill & -N^T(P+E)-H_f-M_{iv}\left(I{z}_{iv}\right)\le 0\hfill & \hfill \end{array}$$

(14d)

$$\begin{array}{ccc}\hfill & \sum _{j=1}^{n_p}{z}_{iv,j}=n_{iv},\hfill & \hfill \end{array}$$

(14e)

$$\begin{array}{ccc}\hfill & p_{min}\le p_i\le p_{max},\forall i=1,\cdots ,n_n\hfill & \hfill \end{array}$$

(14f)

$$\begin{array}{ccc}\hfill & C_{min}\le C_i\le C_{max},\forall i=1,\cdots ,n_n\hfill & \hfill \end{array}$$

(14g)

$$\begin{array}{ccc}\hfill & 0\le q_j\le q_{max}(1-z_{iv,j}),\forall j=1,\cdots ,n_p\hfill & \hfill \end{array}$$

(14h)

$$\begin{array}{ccc}\hfill & z_{iv}\in {\{0,1\}}^{n_p}\hfill & \hfill \end{array}$$

(14i)

where the matrix N, Q, D, and $H_f$ are the same as previously defined. The vectors P and E each have a dimension of $n_n$, denoting the pressure and elevation of nodes in the WDS, respectively. The matrix $M_{iv}$ is a diagonal matrix of size $2n_p\times 2n_p$, representing the additional head loss imposed by the IVs when they are closed. The elements of $M_{iv}$ are assigned by a large value to model the head loss when the corresponding IV is closed. $z_{iv}$ is a binary vector of dimension $n_p$, where each element indicates whether an isolation valve is used to close the corresponding pipe. A value of 1 indicates that it is closed, while a value of 0 indicates it is open. I is a block-diagonal matrix consisting of two identity matrices, each of size $n_p\times n_p$, stacked vertically. This structure expands the $n_p$ dimension vector $z_{iv}$ into a $2n_p$-dimension space, allowing accounting for bi-directional flows. This is necessary because, if an isolation valve is closed, it closes both flow directions in the pipe.

The objective function (14a) is the same as that of the NLP problem (see Equation (13)). Equation (14b) ensures the mass conservation. When considering IVs, energy conservation is enforced through the pipe-wise constraints (14c) and (14d). Equation (14e) ensures a user-defined number of IVs to be used.

The inequality constraints (14f)–(14h) impose bounds on pressure, chlorine concentration, and flow rate at each node and in each pipe, ensuring these variables to remain within their specified ranges. Equation (14h) also guarantees zero flow in pipes where IVs are closed. Finally, Equation (14i) defines $z_{iv}$ as a binary variable vector.

2.2.3. The Extended MINLP Problem

In the NLP and MINLP problems defined above, minimizing the chlorine dosing is the only objective. However, recent research has shown the potential to simultaneously minimize the water age and system pressure by PRVs [40]. Inspired by this study, we now propose an extended MINLP framework that integrates both IVs and PRVs to target both chlorine and pressure management. Specifically, we consider a certain number of PRVs to be installed in the WDS, i.e., their positions and settings will be optimized for the management. Combining PRVs with IVs enables a balanced and economically viable strategy for optimizing the interactions between pressure and chlorine concentration in the network. In this context, the objective function is now defined as:

$$min\alpha \sum _{k=1}^{n_0}{C}_{S,k}+(1-\alpha )\sum _{i=1}^{n_n}{p}_i$$

(15)

where $\alpha$ is a weighting factor that balances the dual objectives of minimizing chlorine dosing and total system pressure across the network. The constraints primarily follow those in the MINLP defined in the last subsection (see Equation (14)), with specific modifications due to adding the PRVs. To account for the additional pressure reduction introduced by the PRVs, constraint (14d) is updated as follows:

$$-N^T(P+E)-H_f-M_{iv}\left(I{z}_{iv}\right)-M_{prv}{z}_{prv}\le 0$$

(16)

where $z_{prv}$ is a binary vector of $2n_p$ variables, representing the placement of PRVs. Again, it accounts for bi-directional flows for each link within the system. The head loss at PRVs, represented by $M_{prv}$, has been previously defined in studies such as [38,40] based on the maximum and minimum possible hydraulic heads. However, in this work, we consider the limitations specific to the PRVs, i.e., they have a fixed pressure-reducing capacity. Consequently, $M_{prv}$ is assumed to be a constant for all PRVs in this study.

To prevent placing both IVs and PRVs on the same pipe, the following constraint is imposed:

$$0\le I{z}_{iv,j}+z_{prv,j}\le 1\forall j=1,\cdots ,n_p$$

(17)

The total number of available PRVs, denoted as $n_{prv}$, is constrained as:

$$\sum _{j=1}^{2n_p}{z}_{prv,j}=n_{prv}$$

(18)

Similarly to constraint (14i), the element of $z_{prv}$ are binary variables, i.e.,

$$z_{prv}\in {\{0,1\}}^{2n_p}$$

(19)

This extended MINLP problem provides a flexible framework for simultaneously optimizing the placement of IVs and PRVs, as well as determining the optimal settings for PRVs, to minimize both chlorine dosing and operational pressure within the WDS. By adjusting the weighting factor $\alpha$, the optimization can prioritize either pressure reduction, chlorine dosing, or a compromise between these two objectives. According to the application under consideration, the number of IVs and PRVs are predefined using $n_{iv}$ and $n_{prv}$, providing flexibility to accommodate various WDS configurations and operational goals. To solve the above defined problems, both the NLP and MINLP formulations are implemented using GAMS 43 [50]. The NLP problem is solved using the interior point solver IPOPT [51], while the MINLP is solved using the BONMIN solver [52]. IPOPT employs an interior-point filter line-search for the NLP, whereas BONMIN solves the MINLP via branch-and-bound with IPOPT handling the continuous relaxations. The optimization results are verified by simulation of the network using EPANET 2.2 [15] and the EPANET-MATLAB Toolkit [53].

3. Results and Discussion

Due to restricted access to operational water-quality datasets arising from privacy and critical-infrastructure security policies, we demonstrate the effectiveness of the proposed approach on the published benchmark network shown in Figure 2 [54]. The network consists of 37 pipes connecting 22 nodes and includes 3 reservoirs identified by the IDs 23, 24, and 25. The lower bound of the pressure at each node is set to 20 m. The flow rate in each pipe is restricted in the range of 0 to 150 LPS. The chlorine decay reaction coefficient r in Equation (2) is set at 1.0. This benchmark is chosen due to its loop structure, which is particularly suitable for testing the proposed approach. This is because in this looped topology, closing selected links redistributes flows through alternative paths, preserving supply while enabling targeted chlorine management. Before detailing each formulation, we summarize the main outcomes on the benchmark: (1) initial dosing of 0.6 mg/L fails to meet the 0.2–0.5 mg/L band; (2) the NLP lower bound requires only 0.2847 mg/L and halves the average residual (0.4550→0.2296 mg/L); (3) with two PRVs and three IVs, $\alpha =1$ yields a feasible plan with a source dose of 0.2996 mg/L and all node residuals within 0.2–0.5 mg/L; (4) reducing $\alpha$ to 0.9 lowers total system pressure from 816.42 to 729.77 m (≈10.6% reduction) while maintaining minimum nodal pressure ≥20 m, at the cost of a slight increase in source dosing to 0.3023 mg/L—clearly illustrating the dosing–pressure trade-off.

For the benchmark, we consider a scenario in which the chlorine concentration at each node should be within the range of 0.2 to 0.5 mg/L, which is the range recommended by WHO [4]. After multiple simulations using EPANET, we observed that, without optimization and extra control, it is impossible to determine a chlorine dosing that satisfies this stringent chlorine constraint by trial-and-error, i.e., it is difficult to maintain both the lower and upper bound throughout the network. We note that this is also the current status of the industrial practice. As shown in the following, our optimization approach can address this problem: both the NLP and MINLP problems are solved to hold this constraint by adjusting the flow distribution within the network.

3.1. Solution of the NLP Problem

The NLP results are presented in

Table 1. Chlorine Concentration in Benchmark WDS under NLP Model.

Chlorine Concentration (mg/L)	Initial	GAMS (Optimization)	EPANET (Simulation)
Reservoirs (IDs: 23, 24, 25)	0.6	0.2847	0.2847
Minimum	0.2020	0.2	0.1923
Maximum	0.5807	0.2818	0.2818
Avg Residual Chlorine	0.4550	0.2296	0.2305
Residual Chlorine Reduction (%)	0%	49.55%	49.35%

, highlighting the effectiveness of optimizing chlorine dosing and ensuring compliance with the predefined chlorine concentration specification. Specifically, the results illustrate an ideal chlorine dosing and residual chlorine scenario, achievable when the flow in each pipe is precisely controlled. For a comparison to a conventional operation as an initial case, we define a chlorine dosing with 0.6 mg/L at the reservoirs and the EPANET simulation shows a minimum chlorine concentration (0.202 mg/L) and a maximum concentration (0.5807 mg/L) which is above the upper bound (0.5 mg/L), making it infeasible. The corresponding average residual chlorine in the network is high at 0.4550 mg/L, as shown in Table 1. The NLP minimizes the reservoir dosing to 0.2847 mg/L which is the lowest value reachable for this WDS and thus represents the lower bound for the chlorine management. In the WDS, the chlorine concentration is indeed within the target range (0.2–0.5 mg/L). The average residual chlorine is reduced significantly to 0.2296 mg/L, representing a 49.55% reduction. Using the optimal decision as input, the EPANET simulation confirms the optimization results with minimal discrepancy (as seen in the last column of Table 1), which shows the accuracy of the NLP model.

3.2. Solution of the MINLP Problem

By utilizing IVs to distribute water flows, the MINLP provides an effective and practical approach to water quality management.

The optimization results with different numbers of IVs are shown in Figure 3. It can be seen that, with the IVs as decision variables, the reservoir dosing concentration drops significantly compared with the initial case, but it is always higher than the lower bound value (i.e., 0.2847 mg/L by the NLP). The minimum chlorine concentration across the network is consistently maintained around the predefined lower bound of 0.2 mg/L and the maximum value chlorine concentration also decreases significantly, dropping from 0.5807 mg/L initially to a range of 0.2990–0.4111 mg/L with up to six IVs. The average residual chlorine in the system decreases substantially with valve implementation, initially at 0.4550 mg/L, reduces to 0.3204 mg/L with one IV, and further to 0.2584 mg/L with six IVs. This highlights the effectiveness of using IVs for chlorine management.

Table 2. Locations of IVs in the Benchmark WDS to Reduce Residual Chlorine Concentrations.

Number of IVs	Locations of IVs (Pipe IDs)
1	31
2	31, 21
3	33, 31, 21
4	17, 31, 21, 20
5	16, 29, 10, 31, 20
6	16, 22, 27, 28, 29, 30

shows the pipe IDs corresponding to the locations of IVs in the network. It is evident that certain pipes are frequently selected for closure, such as 31 and 21. However, in general, the positions of the IVs vary depending on the number of IVs present.

The results shown in Figure 3 indicate that increasing the number of IVs improves the chlorine distribution and thus reduces residual chlorine concentrations. However, a comparison of scenarios with 1–4 IVs and 5–6 IVs reveals that adding more valves does not always guarantee better results. This suggests the existence of a certain number of IVs, beyond which additional placements yield diminishing returns in the system performance. This phenomenon can be explained by the structure of the network. In certain areas of the WDS, closing a single valve may have little to no effect if it does not sufficiently alter the flow distribution within that area. The benefits of IVs become apparent only when a certain number of valves are closed to fully isolate the area of the network. Therefore, in this example, scenarios with 2–4 IVs and 5–6 IVs achieve similar results, respectively.

3.3. Solution of the Extended MINLP Problem

In this case, we explore scenarios in the benchmark where PRVs and IVs are localized simultaneously, offering the possibility of managing both chlorine and pressure in the network. The $\alpha$ value in Equation (15) is predefined as 1.0 to minimize chlorine dosing alone and 0.9 to simultaneously minimize chlorine dosing and operational pressure. For this study, the lower bound of the network pressure is set at 20 m.

The optimization results are summarized in

Table 3. Performance Metrics for Simultaneous Placement of PRVs and IVs in the WDS.

Objective	PRVs (No.)	IVs (No.)	Reservoirs Dosing (mg/L)	Min Chlorine (mg/L)	Max Chlorine (mg/L)	Avg Residual Chlorine (mg/L)	Min Pressure (m)	Max Pressure (m)	Total Pressure (m)
Minimize Chlorine dosing ($\alpha =1$)	1	0	0.4296	0.2000	0.4198	0.3486	34.5219	43.3157	849.0499
	1	1	0.4122	0.2000	0.4099	0.3357	34.5151	43.2296	848.0437
	1	2	0.3596	0.2000	0.3497	0.2871	31.0949	42.2399	827.8520
	1	3	0.3001	0.2000	0.2940	0.2516	31.1153	42.4825	841.9206
Minimize Chlorine dosing and Pressure ($\alpha =0.9$)	1	0	0.4936	0.2000	0.4818	0.4022	29.4085	40.7529	806.8855
	1	1	0.3675	0.2000	0.3542	0.3027	31.0753	43.1549	846.6835
	1	2	0.4680	0.2000	0.4511	0.3715	24.9102	40.5722	741.4982
	1	3	0.3194	0.2000	0.3175	0.2934	31.0878	42.4924	823.5606
Minimize Chlorine dosing ($\alpha =1$)	2	0	0.3338	0.2000	0.3239	0.2690	31.1524	40.3501	810.9467
	2	1	0.4357	0.2000	0.4122	0.3568	31.1136	43.2178	847.8847
	2	2	0.2981	0.2000	0.2904	0.2641	32.2709	40.3661	786.5580
	2	3	0.2996	0.2000	0.2938	0.2641	31.1251	40.3228	816.4152
Minimize Chlorine dosing and Pressure ($\alpha =0.9$)	2	0	0.4838	0.2000	0.4779	0.4078	28.7497	40.7384	781.7898
	2	1	0.4655	0.2000	0.4550	0.3921	21.6023	33.3100	630.6005
	2	2	0.4134	0.2000	0.4107	0.3345	31.2451	41.9138	805.2318
	2	3	0.3023	0.2000	0.3003	0.2603	22.8751	40.9555	729.7666

, highlighting the impact of different numbers of PRVs and IVs in the benchmark WDS. Key performance metrics include reservoir chlorine dosing, minimum and maximum chlorine concentrations, the average residual chlorine at each node, and the minimum, maximum, and total pressures within the system.

It can be seen that, from the chlorine management perspective, the lower bound concentration is consistently maintained across all scenarios, demonstrating that both IVs and PRVs effectively ensure chlorine levels meet the specification. From the pressure management perspective, the lower bound of operational pressure (20 m) is always maintained, confirming the effectiveness of the approach in ensuring adequate pressure throughout the WDS.

When the objective is solely to minimize chlorine dosing ($\alpha =1$), the results shown in Table 3 indicate that, with only one or two PRVs, the chlorine dosing concentrations are lower compared to using only IVs, as seen in the previous subsection. This improvement arises because PRVs allow continuous flow adjustments, whereas IVs function in a binary (open/closed) manner. Moreover, when PRVs and IVs are combined, increasing the number of IVs progressively reduces chlorine dosing for the same number of PRVs. Similarly, scenarios with two PRVs consistently achieve chlorine dosing lower than those with only one PRV, although the pressure-related parameters remain largely unchanged.

When the objective includes both chlorine dosing and pressure reduction ($\alpha$ = 0.9), Table 3 shows broadly consistent trends: increasing the numbers of PRVs and IVs generally improves performance. Using $\alpha$ = 0.9, the objective balances dosing and total pressure; in comparison to the case of $\alpha$ = 1, solutions typically achieve lower total pressure at the expense of a slightly higher source dose. In general, substantial pressure reductions can be achieved while maintaining the required lower bounds for chlorine and pressure. For example, with 2 PRVs and 1 IV, $\alpha$ = 0.9 reduces total pressure from 847.9 m to 630.6 m (−25.6%) while slightly increasing the source dose from 0.4357 to 0.4655 mg/L (+6.8%); with 2 PRVs and 3 IVs, dosing is nearly unchanged (0.2996 → 0.3023 mg/L, +0.9%) for a notable pressure decrease (816.4 → 729.8 m, −10.6%); with 1 PRV and 2 IVs, the trade-off is steeper (0.3596 → 0.4680 mg/L, +30.1% vs. 827.9 → 741.5 m, −10.4%). The locations of PRVs and IVs are summarized in

Table 4. Locations of Simultaneous Placement of IVs and PRVs in the Benchmark WDS.

Objective	PRVs (No.)	IVs (No.)	Locations of PRVs (Pipe IDs)	Locations of IVs (Pipe IDs)
Minimize Chlorine dosing ($\alpha =1$)	1	0	31
	1	1	36	19
	1	2	1	31, 20
	1	3	10	12, 19, 31
Minimize Chlorine dosing and Pressure ($\alpha =0.9$)	1	0	28
	1	1	10	3
	1	2	20	31, 21
	1	3	18	12, 35, 31
Minimize Chlorine dosing ($\alpha =1$)	2	0	9, 21
	2	1	10, 31	17
	2	2	24, 18	10, 31
	2	3	8, 21	17, 10, 31
Minimize Chlorine dosing and Pressure ($\alpha =0.9$)	2	0	13, 28
	2	1	1, 11	31
	2	2	36, 5	17, 27
	2	3	28, 36	12, 27, 35

.

Overall, combining IVs with PRVs in a WDS significantly enhances the flexibility of both chlorine and pressure management. However, the effectiveness of these measures depends on careful consideration of the number of valves to achieve optimal system performance. Balancing chlorine dosing and pressure requires appropriate tuning of the weighting parameter.

4. Conclusions

This study presents a comprehensive approach to optimizing chlorine dosing and pressure regulation in WDSs. The NLP model is designed to minimize the chlorine concentration with flow control in each pipe, providing a baseline for understanding the chlorine behavior in the network. For practical management, we then formulate a MINLP model, introducing IVs as a feasible alternative. This model effectively minimizes chlorine dosing while satisfying the concentration constraints. In both optimization problems, a simplified chlorine propagation model and hydraulic dynamics are employed to describe the chlorine distribution in the network.

Building on this, the MINLP model is extended to incorporate PRVs to address the dual objectives of chlorine and pressure management. This multi-objective framework not only enhances chlorine distribution efficiency but also ensures hydraulic stability, thus mitigating risks such as leaks and pipe bursts associated with high pressure. The results of the benchmark show the influence of the weighting factor between chlorine dosing and operational pressure, alongside the impact of varying numbers of PRVs and IVs. It is observed that the performance is significantly affected by the chosen objective and valve configurations. Adjusting the weighting factor shifts the optimization focus between chlorine management and pressure control, while increasing the number of valves generally improves the performance. These findings provide valuable insights for water utility managers and engineers, offering practical guidelines for network management. Although this study does not optimize monetary cost explicitly, the reduction in source dosing is typically correlated with chemical usage cost, implying commensurate economic benefits.

One essential future work is to improve the convergence of the complicated MINLP problem, especially when IVs and PRVs are considered simultaneously. The placement of both requires binary variables, but their different functionalities lead to difficulties in finding the optimal solution. We experimented with various initial guess values to achieve convergence in the benchmark. Furthermore, the reported MINLP solutions do not claim global optimality; they correspond to feasible locally optimal solutions for the benchmark. Scaling to larger systems may require additional decomposition or heuristics to facilitate convergence. Another future work can be the consideration of bacterial contamination in the reaction model. Furthermore, when taking water tanks into account, extending the study to dynamic systems could be meaningful future work.