## 2 Methodology
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:  
\[\Delta H_{local}=K\frac{\
u^{2}}{2g} \tag{6}\]  
\[v=\frac{Q}{A} \tag{7}\]  
\[A=\frac{\pi D^{2}}{4} \tag{8}\]  
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]:  
\[\sum W_{I}-\sum W_{O}=De \tag{2}\]0  
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]:  
\[\sum W_{I}-\sum W_{O}=De \tag{2}\]1where \(El(Re)\) is the elevation at the reservoir and \(El(N)\) is the elevation at the node, \(N\). The minimum required pressure at all demand nodes is 25 m [4]. \[\sum W_{I}-\sum W_{O}=De \tag{2}\]2  
where \(H_{j}\) is the pressure head at node \(j\), \(Min\big{(}H_{j}\big{)}\) is the minimum necessary pressure, and \(J\) is the number of nodes. Figures 3-5 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 m\({}^{3}\)/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  
\[\sum W_{I}-\sum W_{O}=De \tag{2}\]3 Centralized radial water distribution network layout (WDN 1). 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  
\[\sum W_{I}-\sum W_{O}=De \tag{2}\]4 Looped water distribution network layout (WDN 2). \[\sum W_{I}-\sum W_{O}=De \tag{2}\]5 Hybrid water distribution network layout (WDN 3). system.