Impact of Network Configuration on Hydraulic Constraints and Cost in the Optimization of Water Distribution Networks

Abstract

This study introduces a novel approach for the multi-model analysis of complex water distribution networks (WDNs). The research focuses on designing and optimizing various WDN configurations while adhering to hydraulic constraints. Several key parameters and criteria are considered to achieve an efficient design. Additionally, different network layouts are evaluated, including looped and non-looped systems with varying numbers of reservoirs. Next, an analytical approach is developed to optimize the proposed WDNs, taking into account pipe type, length, and diameter, as well as nodal demands, elevations, pressure losses, and water velocities. Cost analysis reveals that a single-reservoir, non-looped WDN has the lowest cost (USD 26,892), while a two-reservoir, looped WDN has the highest (USD 30,861). The design inflows vary linearly, ranging from 0.0212 to 0.205 m³/s for a 0.3 m pipe diameter and from 0.0589 to 0.5694 m³/s for a 0.5 m pipe diameter. Further, a new approach based on the Coral Reef Algorithm (CRA) is developed and implemented to improve the technical and economic viability of the designed WDNs. The CRA effectively showcases its capacity to iteratively enhance network design by reducing overall costs significantly. Notably, higher demand multipliers yield even more efficient solutions, suggesting the algorithm’s adaptability to varying demand scenarios.

1. Introduction

The development of water energy systems [1] has emerged as a response to increasing energy demands and the rapid depletion of traditional energy sources [2]. For centuries, cities have faced significant challenges in water management [3]. In this context, Water Distribution Networks (WDNs) play a crucial role in urban infrastructure. A primary challenge lies in optimizing WDNs to achieve an optimal balance between performance and cost [4].

To effectively optimize WDNs, it is essential to consider various technical factors, including energy constraints, pipe characteristics, water flow rates, and pressure constraints [5]. Once a WDN has been designed and constructed, it can transport a specified amount of water to consumers while maintaining designated pressure ranges [6].

Various methods for optimizing WDNs and examining their configurations have been developed: Cassiolato et al. [4] minimized the cost of looped WDNs by integrating mixed-integer nonlinear programming (MINLP) with a deterministic approach. Their model efficiently determines pressure and velocity values. Sitzenfrei et al. [7] highlighted the lack of complex network analysis (CNA) in the literature regarding optimal WDNs and demonstrated how CNA can enhance WDN efficiency.

Winkler et al. [8] addressed pipe failure issues in WDNs using “boosted decision trees”, showing that their model serves as a viable alternative to traditional statistical models. Ines et al. [9] combined remote sensing with Genetic Algorithm (GA) optimization to explore water management solutions, developing a model that significantly enhanced regional crop productivity. Eusuff and Lansey [10] optimized WDN design using the Shuffled Frog Leaping Algorithm, resulting in optimal pipe sizes and demonstrating the flexibility of their method for designing pumps or other network components. Li et al. [2] proposed a multi-objective moth-flame algorithm model for optimizing water resource utilization in multi-reservoir systems, successfully achieving a set of Pareto solutions that balance trade-offs among various components.

Haghighi et al. [5] combined GA with Integer-Linear Programming (ILP) to optimize WDNs, finding their model computationally efficient. Keedwell and Khu [11] introduced a hybrid GA for designing WDNs, which provided more economically viable designs compared with non-heuristic approaches. Stefanizzi et al. [12] identified that WDNs often suffer from water leakages and proposed replacing Pressure Reducing Valves (PRVs) with Pumps as Turbines (PaTs), selecting optimal PaT configurations through their approach. Assad et al. [13] noted that many WDNs are in a deteriorated state, increasing vulnerability to natural and anthropogenic hazards. They introduced a new metric alongside an optimization framework aimed at enhancing the resilience of WDNs, demonstrating an increase in network resilience of nearly 20%. Creaco et al. [14] discussed real-time control of WDNs, emphasizing objectives such as service pressure regulation and energy production.

The application of Computational Fluid Dynamics (CFD) is also crucial for optimizing WDNs. In their study, Karpenko et al. [15] developed a CFD-based approach to analyze and enhance fluid flow characteristics in hydraulic systems, which can be adapted for WDN optimization. By employing numerical simulations grounded in Reynolds-averaged Navier–Stokes equations in three-dimensional space, the authors examined the effects of angular fitting connections on fluid dynamics, revealing complex hydrodynamic phenomena at local resistances. These phenomena are essential for accurately characterizing both linear and local resistances, significantly impacting the overall efficiency of fluid transport systems.

It is also imperative to highlight that machine learning and modern simulation approaches are revolutionizing the optimization of water distribution networks, turning traditional systems into intelligent, adaptive infrastructures. Such technologies are seen as the brains behind the operation, allowing the network to not just function but to think and respond in real time. In this context, Hajgató et al. [16] implemented deep reinforcement learning (DRL) to control pump operations. Their DRL agent learns optimal pump speeds based on instantaneous nodal pressure data, outperforming conventional methods with over 98% efficiency while doubling the speed. It is akin to having a seasoned operator who anticipates and adjusts to system demands instantly, enhancing both efficiency and responsiveness.

On another front, Hu et al. [17] tackled the persistent issue of leakage in urban water networks using multiscale neural networks combined with density-based clustering. By reducing complex pipe networks into manageable zones, they significantly improved leakage detection accuracy—by 78% compared with methods like SVM and KNN. This can be thought of as the transformation of a tangled web into organized neighborhoods, making it easier to pinpoint and address leaks promptly, thereby conserving water and reducing losses.

Furthermore, Nie et al. [18] explored the incorporation of big data and IoT sensors within underwater management systems. Their method utilizes real-time data collection through IoT to proactively monitor water usage and quality, employing a SCADA system for sustainable management. This is like giving the water network a sensory system, allowing it to detect and respond to issues before they escalate, promoting operational safety and sustainability in smart cities.

Despite the advancements in optimizing water distribution networks, there remains a notable gap in existing models that study the performance of WDNs with fixed nodes and inflows—specifically those with constant consumption rates for each node. This study aims to fill this gap by proposing a novel mathematical model tailored to analyze the performance of WDNs characterized by fixed nodes and consumption rates. The model incorporates various parameters, including cost, demand, head losses, node elevations, design pressure, and water velocity, to assess their effects on optimizing these proposed WDNs.

Additionally, this research introduces an optimization model that delves into the complex interplay between pressure and velocity within the network. By scrutinizing these dynamics, the study elucidates pathways to achieving optimal configurations that enhance operational efficacy and economic viability. The insights garnered from this analysis significantly advance the understanding of WDN behavior, offering substantive and practical implications for engineers and planners in the field. Ultimately, this work aims to inform more nuanced design and management strategies for urban water distribution systems, thereby contributing to the development of resilient and efficient infrastructures.

The remaining parts of the paper are as follows. The methodology is described in Section 2. Section 3 discusses the results and the conclusions are pointed out in Section 4.

2. Methodology

WDNs have been widely studied, specifically in recent times due to their complex infrastructure [12]. Their design can be formulated as a least-cost optimization with a selection of pipe sizes as the decision parameters [8]. The cost of a WDN is related to the pipe parameters, which include lengths and diameters. Therefore, the cost optimization model of a WDN can be defined as follows [4]:

$$C=\sum _{i=1}^N{C(D}_i)\times L_i|i=1,\dots ,N$$

(1)

where $C$ is the overall cost of the WDN, and ${C(D}_i)$ is the cost of pipe $i$ with the corresponding diameter $D_i$ and length $L_i$. $N$ is the number of pipes in a proposed WDN.

The difference between the inlet node water flow rate ($W_I$) and outlet node water flow rate ($W_O$) is equal to the node demand ($De$) [4]:

$$\sum W_I-\sum W_O=De$$

(2)

By considering the measurements obtained from water meters, the nodal demand can also be expressed as follows:

$$D_{a,t}=M_{a,t}\times D_{b,t}$$

(3)

where $D_{a,t}$ is the demand at the node $a$ and time $t$, $M_{a,t}$ is the demand multiplier, and $D_{b,t}$ is the base demand [19]. The multiplier takes into account any government policies regarding the investments on a larger macroeconomic scale.

Figure 1 shows the average hourly profile of residential consumers regarding the water demand according to the multiplier estimates [19]. Figure 2 demonstrates the average daily water consumption profile in the case of different cities in the United States together with the multiplier estimates according to the statistics obtained from the American Water Works Association (AWWA) [20].

The head losses in pipelines can be estimated according to the Hazen–Williams equation [19]:

$$∆H={\displaystyle \frac{10.67P_L}{{P_{RC}}^{1.852}\times D^{4.8704}}}\times {W_I}^{1.852}$$

(4)

where $∆H$ is head losses (meters of water column), $P_L$ is the pipe length (meters), $D$ is the diameter (meters), $W_I$ is the flow rate in the pipe (${\mathrm{m}}^3/\mathrm{s}$), and $P_{RC}$ is the pipeline roughness coefficient, which is dimensionless [20]. In this study, according to the previous results, the $P_{RC}$ is assumed to be 130 [4].

In addition, based on one-dimensional (1D) hydraulic models, a methodology for CFD-based pipeline network optimization in water distribution systems is developed to approximate the complex behavior of fluid flow in pipes. Equation (4), a widely adopted approach, is used to compute frictional head losses along each pipe segment. The Hazen–Williams equation is an empirical formula, which is particularly useful in design practices because it offers a balance between accuracy and computational simplicity when analyzing water distribution networks [20]. It solves the complex physics of turbulent flow into a form that is computationally efficient for design purposes.

In optimizing a network, candidate pipe diameters are evaluated by calculating the associated friction losses and then comparing these results against performance criteria such as minimum acceptable head loss and construction cost. The optimization is typically performed by sweeping through a range of potential diameters and selecting the design that meets hydraulic constraints while minimizing cost. This deterministic approach has been validated in numerous studies on network optimization, where engineers use these methods to achieve efficient and cost-effective designs [4].

The Reynolds number for each candidate diameter can be calculated according to the following equation:

$$Re={\displaystyle \frac{\rho \times V\times L}{\mu }}={\displaystyle \frac{V\times L}{\nu }}$$

(5)

where $\rho$ is density, $V$ is velocity, $L$ is length, $\mu$ is bulk viscosity, and $\nu$ is kinematic viscosity. The flow regime is determined by comparing Re with standard thresholds (Re < 2000: Laminar; 2000 ≤ Re < 4000: Transitional; Re ≥ 4000: Turbulent) [21].

The calculation of local resistance in water distribution networks consists of quantifying the energy losses at specific components, such as fittings and valves, which are not captured by friction loss alone. Therefore, the following equations are used:

$${∆H}_{local}=K{\displaystyle \frac{v^2}{2g}}$$

(6)

$$v={\displaystyle \frac{Q}{A}}$$

(7)

$$A={\displaystyle \frac{\pi D^2}{4}}$$

(8)

where the velocity $v$ is determined by the flow rate Q, and the cross-sectional area of the pipe, and K is the loss coefficient for each type of fitting [22].

The water speed can be estimated according to the following equation [4]:

$$V={\displaystyle \frac{4W_I}{\pi D^2}}$$

(9)

The pressure level in WDNs should reach a point that meets users’ demand requirements. When it falls below a considered limit, which may be a function of building height, the WDN cannot supply all users with the requested flows. As such, at nodes with low-pressure levels, the distributed flow becomes lower than the demand. On the other hand, high pressure can lead to negative impacts, such as an increase in leakage and pipe bursts and a reduction in infrastructure life. Hence, managing the pressure to maintain it close to a desired pressure threshold is essential.

The pressure at each node ($N$) can be calculated based on the following formula by considering the constraint in Equation (7) [4]:

$$P_r=-\sum ∆H+\left[El\left(Re\right)-El\left(N\right)\right]$$

(10)

where $El\left(Re\right)$ is the elevation at the reservoir and $El\left(N\right)$ is the elevation at the node, $N$. The minimum required pressure at all demand nodes is 25 m [4].

$$H_j\ge Min\left(H_j\right)|j=1,\dots ,J$$

(11)

where $H_j$ is the pressure head at node $j$, $Min\left(H_j\right)$ is the minimum necessary pressure, and $J$ is the number of nodes.

Figure 3, Figure 4 and Figure 5 depict the proposed system’s WDN layouts. Elevation at reservoirs is assumed to be 54 m [4]. As can be seen, the number of nodes and the overall inflow in all three structures are equal. There are fifteen pipes in the first and second WDNs and eighteen in the third WDN. The values shown in each diagram indicate water flows in m³/h unit.

The layout in Figure 3 features a single source distributing water through a branching network to 11 nodes. The design is simple and likely has lower installation costs than the other layouts. However, it may face challenges with pressure imbalances, especially at peripheral nodes. As demands vary across nodes, those farther from the source might receive insufficient flow during peak usage. Therefore, this layout offers less flexibility in handling sudden demand changes but could be suitable for areas with predictable usage patterns.

In the configuration shown in Figure 4, the 11 nodes are interconnected through closed loops, enabling bidirectional flow. This design excels at balancing pressure across nodes, even when faced with fluctuating demands. The looped structure provides redundancy, ensuring uninterrupted supply if a single pipe fails. While this layout offers superior reliability and adaptability to varying nodal demands, it comes with higher construction and maintenance costs due to the increased total pipe length. This design is particularly beneficial for areas with diverse or unpredictable water usage patterns.

The layout shown in Figure 5 combines elements of both radial and looped networks, creating a balanced approach to water distribution. It likely features localized loops in high-demand zones and radial branches in areas with lower or more stable demand. This hybrid design optimizes the trade-off between cost and reliability by strategically placing redundancies where they are most needed. It offers more flexibility than the radial layout in handling demand variations while being potentially more cost-effective than a fully looped system. However, it requires careful hydraulic modeling to ensure that critical pipes are not overloaded during peak demand periods.

The fixed total inflow (i.e., 1.15 m³/h) and nodes (i.e., 11 nodes) across all layouts provide several advantages for water distribution network design and analysis. This standardization allows designers to test how different layouts handle variable nodal demands while maintaining the same total supply, enabling a direct comparison of system performance under diverse usage scenarios. With a consistent node count, engineers can quantify each WDN’s resilience to pipe failures or contamination events, particularly highlighting the inherent risk mitigation of looped networks through flow rerouting. Additionally, this fixed framework offers insights into cost efficiency across different designs. For instance, radial systems may be more suitable for budget-constrained projects with predictable demand patterns, while hybrid systems present scalable solutions for growing urban areas with evolving water needs. Overall, this approach provides a controlled environment for evaluating how structural diversity in WDNs addresses real-world challenges such as demand variability and infrastructure resilience.

Table 1. Cost classifications of various pipes.

Pipe Number	Pipe Type	Approximate Average Cost (USD/m)	Merits	Demerits	Approximate Lifetime (Years)
1	Chlorinated polyvinyl chloride (CPVC)	2.467	(1) Inexpensive, (2) Less corrosion	(1) Possible leaks, (2) Lower quality	50–75
2	Cross-linked polyethylene (PEX)	3.947	(1) Less corrosion, leak, burst etc., (2) In-floor heating	(1) Less integration possibility	80–100
3	Copper	16.447	(1) Proven utilization history, (2) Compatible with building codes, (3) Recyclable, (4) UV resistant, (5) Unfailing in natural disasters	(1) Expensive, (2) Possibility of bursting, freezing, and corrosion	50–100

shows the classification of different pipes that can be used for WDNs [23]. Due to the specific features of PEX pipes (e.g., low cost, the necessary resistance, and stability), they are considered for the design of WDNs in the current study.

2.1. Flow Distribution

In this study, a heuristic approach was adopted to approximate the progressive reduction in flow along the network. Starting with the full inflow $Q_{total}$ at the source, the first pipe (with pipe ID 1) is assumed to carry the entire inflow, i.e., no reduction is applied $(1-\alpha \times \left(1-1\right)=1)$. For subsequent pipes, we assume a linear reduction in flow given by:

$$Q_{pipe}=Q_{total}\times (1-\alpha \times \left({pipe}_{ID}-1\right))$$

(12)

where $\alpha$ represents the assumed fractional reduction in the flow per successive pipe. A higher coefficient means that with each additional pipe, a larger fraction of the flow is removed or diverted (simulating higher losses or greater demand along the branch). Conversely, a lower coefficient indicates that the reduction in flow from one pipe to the next is less severe, so more of the original flow is maintained as the water is distributed through the network.

For WDN 1 (the centralized radial network) and WDN 2 (the looped network), the coefficient α is assumed to be 0.03, which implies that each additional pipe in the sequence receives 3% less of the original inflow than its predecessor, reflecting the expected cumulative effect of distributed demand and branch flow losses. For the hybrid network (WDN 3), the coefficient is reduced to 0.02:

$$Q_{pipe}=Q_{total}\times (1-0.02\times \left({pipe}_{ID}-1\right))$$

(13)

WDN 3 is designed as a hybrid network, combining elements of both radial and looped configurations. Unlike a purely radial network—where water flows from a single source through a sequence of pipes (resulting in significant cumulative flow reductions)—or a fully looped network, the hybrid design includes localized loops in high-demand zones. These loops provide alternate paths for water to reach peripheral nodes, effectively “preserving” higher flow rates downstream. In this scenario, the flow reduction per pipe is less drastic because the loops help maintain pressure and supply. Thus, when formulating the heuristic model for WDN 3, a lower coefficient (0.02) is selected to reflect that each pipe loses a smaller percentage of the total inflow compared with the other configurations.

In practice, determining the exact value of the coefficient would involve calibrating the model against detailed hydraulic measurements. The chosen values (0.03 for the radial/looped systems and 0.02 for the hybrid system) are approximations that capture the qualitative differences in network design: networks with additional supply paths (localized loops) can maintain higher flow downstream, justifying a lower reduction factor. This approach is common in preliminary network analyses where a balance is struck to capture the dominant trends in flow distributions.

2.2. Implementation of a CRA-Based Approach

The Coral Reef Algorithm (CRA) is a metaheuristic algorithm inspired by the natural process of coral reef formation, introduced by Salcedo-Sanz et al. in 2014 [24]. It is designed to solve complex optimization problems, such as minimizing costs or maximizing the technical efficiency, by mimicking coral reef growth, reproduction, and competition processes. The algorithm begins by initializing a population of random solutions, each represented as a “coral” in the reef, corresponding to potential configurations.

In this study, each solution’s fitness is evaluated by considering the cost as the objective function, formulated as:

$$f\left(D\right)={\sum }_{i=1}^N{c}_i·l_i$$

(14)

where D is the network, $c_i$ is the cost per unit length, $l_i$ is the length of the $i$-th pipe, and $N$ is the number of pipes.

The process then iteratively selects the top-performing solutions (e.g., the top half by fitness), mutates them to generate new configurations, and updates the population by combining parents and offspring. This cycle repeats until the maximum iterations are reached, making it adaptable to WDNs by tailoring the fitness function and constraints to specific needs, including flow rates and pipe diameters.

Upon completing the iterations, the CRA algorithm retrieves the best solution—the configuration with the lowest cost—and refines it using linear programming to enforce pressure constraints, ensuring hydraulic feasibility. The final cost is calculated, and the output provides optimized pipe lengths and costs, delivering a practical solution for network design. The flowchart in Figure 6 illustrates the developed CRA-based approach to optimize the proposed WDNs, systematically detailing the process from initialization to final results.

3. Results and Discussion

3.1. Preliminary Assessment

In this study, the designs of three studied WDNs are analyzed. The total incoming flow (i.e., 1.15 m³/h) and the number of nodes (i.e., 11 nodes) are considered to be constant and a mathematical model is developed to analyze the proposed WDNs. The optimum cost is calculated by considering the pipe lengths and their costs. Results regarding the analysis of the node demand and elevation, head losses, design inflows according to the water velocity, and pressure requirements based on the velocity are the next steps of the study.

Table 2. Pipe length information for each WDN.

WDN 1 (Pipes)	Pipe Length (m)	WDN 2 (Pipes)	Pipe Length (m)	WDN 3 (Pipes)	Pipe Length (m)
1	500	1	500	1	500
2	200	2	200	2	200
3	185	3	185	3	195
4	160	4	160	4	150
5	250	5	250	5	165
6	210	6	210	6	153
7	230	7	170	7	99
8	155	8	175	8	140
9	140	9	180	9	145
10	150	10	172	10	162
11	138	11	168	11	255
12	200	12	113	12	178
13	170	13	155	13	186
14	160	14	190	14	183
15	140	15	165	15	192
-	-	-	-	16	177
-	-	-	-	17	174
-	-	-	-	18	175

reports the information about the pipes employed in the proposed WDNs. According to Equation (1), the optimum costs are estimated to be USD 26,892, USD 26,937, and USD 30,861 for the first, second, and third WDNs, respectively.

3.2. Nodal Demand Analysis

One of the key design parameters of WDNs is associated with their nodal-based demands. As shown in Figure 3, Figure 4 and Figure 5, the overall demand is 1.15 m³/s. The nodal demands for all WDNs are calculated according to Equation (2) and shown in Figure 7.

3.3. Head Loss Trends

This part of the study determines the effect of WDNs’ pipe diameters on the head losses. Figure 8 demonstrates the head (pressure) losses based on variations in pipe diameter. As the pipe diameter increases, there is a decrease in head losses. To predict the head losses according to different pipe diameters, based on the Equation (4), it is found that 3-degree polynomial equations describe the trend of head losses with R squared values calculated to be 0.9958. It is also noted that WDNs 1 and 2 have a similar and close trend but WDN 3 behaves quite differently (in WDN 3, most values for the pressure losses are close to zero). Significantly lower pressure losses can occur when there are more water tanks for storing water and more pipes with small lengths and large diameters. On the other hand, it has become evident that when a looped WDN (WDN 1) changes to an unlooped WDN (WDN 2) with the same conditions, the pressure losses may rise slightly.

3.4. Node Elevation Estimation

To design a WDN, it is mandatory to determine the elevations at each node. By assuming the pipe diameter of 0.5 m, the elevations at each node are estimated and can be observed in Figure 9. Each WDN is denoted with a number (i.e., first WDN: WDN 1, second WDN: WDN 2, third WDN: WDN 3). From nodes 1 to 9, the elevations for all structures are equal. In the case of the first WDN, the node elevations vary from a minimum of 1.1718 m observed for node 4 to a maximum of 14.5025 m observed for node 10. This is also true in the case of the second and third WDNs; however, the maximum node elevations for these two WDNs (2nd and 3rd) occurred at node 10 with values of 24.5538 and 24.506 m, respectively.

3.5. Interplay of Design Inflows and Water Velocity

According to Ref. [4], the following range of water velocity is considered.

$$V_{min}\le V\le V_{max}|V_{min}=0.3 \mathrm{m}/\mathrm{s} \mathrm{and} V_{max}=3 \mathrm{m}/\mathrm{s}$$

(15)

Afterward, the required inflows for each node are calculated. This enables the WDN to operate under adjusted conditions of water velocity. Figure 10 depicts the estimation of the required design inflows for the system to operate under the effect of different pipe diameters. Based on a linear trend, the design inflows vary from a minimum of 0.0212 to a maximum of 0.205 m³/s in the case of a pipe diameter of 0.3 m, while they vary from a minimum of 0.0589 to a maximum of 0.5694 m³/s in the case of a pipe diameter of 0.5 m.

3.6. Estimation of the Design Pressures

Figure 11 demonstrates the design pressures according to the water velocities and pipe diameters. Node 1 situated in the three studied WDNs is selected and examined for the analysis. Exponential equations describe the trend of the design pressures as follows:

$$y=91.04\times e^{-0.057x}|R^2=0.6455$$

(16)

$$y=58.088\times e^{-0.022x}|R^2=0.8556$$

(17)

$$y=51.585\times e^{-0.009x}|R^2=0.9097$$

(18)

The analysis indicates that the error calculated for each model decreases as the pipe diameter increases. Therefore, at higher pipe diameters, there is more accuracy for obtaining the exponential models of pressure-velocity for a WDN. The models are applicable to providing necessary information about the design process for WDNs with different levels of pressure, velocity, and pipe diameter. In addition, these mathematical models play a dominant role in the layout, design, and operation of WDNs.

Analysis of the pressure and velocity can also be fulfilled according to polynomial regression with a high accuracy. Equations (19)–(21) describe the polynomial regression of pressure and velocity by considering the pipe diameters of 0.09, 0.12, and 0.2 m:

$$y=-0.016x^2-0.1054x+48.245|D=0.09 \mathrm{m}$$

(19)

$$y=-{0.0114x}^2-0.0753x+48.191|D=0.12 \mathrm{m}$$

(20)

$$y=-0.0063x^2-0.0415x+48.13|D=0.2 \mathrm{m}$$

(21)

R-squared for Equations (19)–(21) is calculated to be ≈0.999. To reach an optimization trade-off model for the prediction of pressure according to the velocity, we assume that Equations (19) and (21) are equal, which results in Equation (22):

$$y=0.0063x_1^2-0.016x_2^2+0.0415x_1-0.1054x_2+0.115$$

(22)

3.7. CRA-Based Optimization of the Cost

Figure 12 illustrates the convergence behavior of a developed Coral Reef Algorithm (CRA) applied to the optimization of studied water distribution networks, where the objective is to minimize cost while satisfying hydraulic constraints. The different lines correspond to varying demand multipliers (1×, 1.5×, and 2×), which scale the water demand at the network’s demand nodes. The downward trend of all lines demonstrates the CRA’s ability to iteratively improve the network design by reducing the overall cost. Notably, higher demand multipliers generally lead to lower overall costs than the base case (1×). This unexpected result might indicate that the optimization process can potentially find more efficient solutions for higher-demand scenarios. The convergence rate also varies, with steeper initial declines indicating faster improvements in the early stages of the optimization. This suggests that the CRA effectively explores the solution space and rapidly identifies promising designs before gradually refining the solution towards a local or global optimum.

3.8. Hydraulic and Economic Analysis

Figure 13 represents how effectively each type of pipe (Pipes 1, 2, and 3) is being utilized compared with its initial capacity. The utilization index is calculated by taking the optimized length of each pipe type and dividing it by its original length, resulting in a ratio that indicates the extent to which the pipe’s capacity is being used. A higher utilization index signifies that the pipe is being used more efficiently, whereas a lower index suggests underutilization. In addition, the cost contribution reflects how much each pipe type contributes to the overall costs of the system based on its optimized length and associated cost per unit length. This metric provides insight into which components are driving costs within the network, allowing for better decision-making regarding resource allocation and potential areas for cost savings.

Pipe 3 exhibits the highest utilization index, suggesting it is being utilized most efficiently relative to its initial capacity, followed by Pipe 2, and then Pipe 1, which shows the lowest utilization. The figure also depicts the cost contribution of each pipe type to the overall system cost. Notably, while Pipe 3 has the highest utilization, it also contributes the most significantly to the total cost (82.65%), indicating that despite being efficiently used, its length and/or cost per unit length are substantial drivers of the overall system expense. Pipe 2 contributes 13.22% to the total cost, while Pipe 1 contributes only 4.13%, despite having a non-negligible utilization index. This analysis highlights the trade-off between utilization and cost, providing valuable insights for optimizing network design by identifying potential areas for cost savings, such as potentially reducing the length or using less expensive material for Pipe 3, while ensuring sufficient capacity and utilization across all pipe types.

Figure 14 illustrates the impact of varying demand multipliers on both the total optimized length of pipes and the associated total costs within a water distribution network. The optimized pipe length remains relatively constant across all demand multipliers. However, the top portion of each bar, representing the total cost, increases significantly with higher demand multipliers. This indicates that while the total length of pipes used in the optimized solutions does not change drastically with increasing demand, the cost of those pipes increases substantially. This cost increase is likely due to the need for larger-diameter pipes or higher-grade materials to accommodate the increased flow demands while maintaining adequate pressure. This visualization clearly demonstrates the trade-offs between meeting higher demand and the associated cost implications, highlighting the importance of considering demand projections and cost-effectiveness in the design and optimization of water distribution networks. The diagram underscores that even when pipe length remains similar, the necessary changes in pipe characteristics to handle increased demand lead to considerably higher total costs.

Figure 15 presents a bar chart of the total head loss for the studied network configurations. For each network, the head loss is the sum of friction losses (from pipe length) and local losses (due to fittings). As can be seen, WDN 1 and WDN 2 have almost identical head loss values because of their nearly similar geometries and flow rate assignments. The head losses in WDN 2 and WDN 3 are also analogous, indicating that—even though WDN 3 incorporates some distinct structural features and additional cumulative hydraulic losses—its overall hydraulic performance aligns closely with that of WDN 2.

As shown in Figure 16, the pairwise correlation coefficients among the selected hydraulic and economic parameters are depicted. Red color indicates a strong positive correlation (close to +1), while blue indicates a strong negative correlation (close to −1). Colors closer to white indicate weak or no correlation (close to 0). A strong positive correlation between length and cost is apparent, which is expected since cost is modeled as proportional to the product of pipe cross-sectional area and length. Additionally, the correlation matrix reveals that the friction loss and local loss are highly correlated with the total head loss, indicating that both frictions along the pipes and losses at fittings drive the cumulative hydraulic losses. This insight helps in validating the model assumptions and highlights which variables exert the greatest influence on overall system performance.

Furthermore, the correlation analysis between the Reynolds number and parameters such as flow rate and friction loss elucidates the influence of flow regime on hydraulic performance. The observed moderate or weak correlations indicate that although turbulence contributes to energy dissipation, other factors—namely, pipe geometry and material properties—play a more dominant role in determining losses. These insights are critical for refining design strategies and ensuring that the optimization model faithfully represents the behavior of the water distribution network. By identifying variables that are strongly correlated with key performance indicators, such as total head loss, we can pinpoint the factors that most significantly impact system performance.

The behavior of the fluid flow plays a key role in the head loss calculations. Figure 17 demonstrates the Reynolds number for each pipe in the three networks. The Reynolds number is computed using the flow rate, pipe diameter, and water viscosity. The plot reveals variability in Reynolds numbers among pipes, which results from the flow rate assignment that decreases with pipe order. This variation is crucial because it confirms that while most pipes exhibit turbulent flow (Re > 4000), there is a spread in the local flow conditions—a key factor when verifying the validity of the head loss calculations.

4. Conclusions

WDNs are complicated structures consisting of various linked nodes and pipes, which transport water of suitable quality from sources to supply customers in a service area. WDNs, which play a substantial role in water supply systems, are considered as one of the major infrastructure assets of industrial society.

The optimization of WDNs supports the decision-making process by determining the optimal trade-off between costs and performance. A major challenge in WDNs is the network design. While the complex nature of WDNs has already been explored with complex network analysis (CNA), the literature is still lacking a CNA of optimal water networks when there are different WDN structures with fixed nodes and inflows.

In this study, an original approach for the optimum design of WDNs is presented. The following results are derived from the analysis:

• Variations of head losses based on the pipe diameters showed that a 2-degree polynomial equation with R squared being 0.9638 can describe the trend well.
• Based on a linear trend, the design inflows vary from a minimum of 0.0212 to a maximum of 0.205 m³/s in the case of a pipe diameter of 0.3 m, while they vary from a minimum of 0.0589 to a maximum of 0.5694 m³/s in the case of a pipe diameter of 0.5 m.
• The design pressures according to the water velocities and pipe diameters were estimated. For this purpose, one of the nodes of the WDN was selected and examined. A pressure-velocity model was considered. It was shown that two regressions can describe the model with high accuracy: exponential and polynomial regression. Polynomial regression outperformed the exponential regression, while in the exponential regression, as the pipe diameter increased, the error calculated for each model decreased.

The findings reveal that the cost of WDNs varies significantly based on their configuration, with a single-reservoir, non-looped system demonstrating the lowest cost at USD 26,892, while a two-reservoir, looped system incurs the highest cost at USD 30,861. The analysis of head losses and design inflows indicates that larger pipe diameters contribute to reduced head losses, thereby enhancing the overall efficiency of the network. Furthermore, the study establishes that the design pressures are closely linked to water velocities and pipe diameters, with polynomial regression models providing high accuracy in predicting these relationships.

The analysis and optimization approach described in this study provides a robust framework for real-world water distribution network design. By integrating fundamental hydraulic equations—such as the Hazen–Williams equation for friction losses and additional formulations for local losses and Reynolds number calculation—with heuristic flow distribution models, engineers can quickly evaluate network performance under different configurations. This method allows for a preliminary assessment of critical factors such as pressure losses, cost efficiency, and flow regimes. The developed model facilitates choosing the best combinations of pipes, nodes, and pumps by taking into account a number of design constraints, including pressure, pipe diameters, head losses, and velocity. This helps to create durable and effective urban water distribution systems. For practical applications, engineers should calibrate the heuristic coefficients using field measurements or detailed simulations to tailor the model to the specific characteristics of their systems. It is recommended that practitioners use these models to guide initial design choices, optimize pipe sizing and layout, and identify potential areas for system improvement before committing to more resource-intensive detailed CFD analyses for critical network segments.

The study’s limitations include its examination of dynamic real-world scenarios, such as the consequences of fluctuating water demand and unforeseen failures. The study’s characteristics of fixed nodes and consumption are another aspect; as a result, other possible intricate features ought to be investigated. To further refine WDN optimization techniques and increase their adaptability, future research should also concentrate on combining real-time data as well as existing water systems with sophisticated forecasting methods.