# Development of a Multi-Objective Optimal Design Approach for Combined Water Systems
## 2 Optimal Design of Combined Water Systems
### Objective Function and Constraints  
The existing UDS was designed with a design rainfall of 10-year frequency. However, because flood damage occurs due to rainfall exceeding the design frequency due to climate change, the optimal design of the UDS was performed by applying the design rainfall with a 30-year frequency. In addition, in order to minimize flooding damage for rainfall exceeding the design rainfall, flooding does not occur in the 30-year design rainfall, and flooding was minimized in the design rainfall with a frequency of 50 years, and the optimal design was performed. The design cost, which is the objective function of UDSs and WDSs, was conducted under the assumption that the design cost increases as the pipe diameter increases [18]. Minimizing the design cost assumed a linear increase in the design cost as the diameter increased. In addition, the design cost can be expressed as Equation (1) by multiplying the cost according to the diameter of the pipe by the length. \[Minimum\ design\ cost=\sum_{p=1}^{p}Cost\big{(}D_{p}\big{)}L_{p} \tag{1}\]  
where \(P\) represents the total number of pipes, \(Cost(D_{p})\) means the design cost according to the diameter of the pipe, and \(L_{p}\) means the length of the pipe. Therefore, the design cost means the sum of the design cost according to the diameter per unit length of the pipe multiplied by the length of the pipe. For the objective function of the stormwater drainage system, an analysis was performed by increasing the design precipitation intensity to improve the resilience of the existing design for the drainage system. Minimization of flood volume at the increased design precipitation intensity was considered, as shown in Equation (2). The greater the design cost minimization, the greater the pipes overloading and flood vulnerability. There  
Figure 5: Schematic diagram of multi-objective optimization for the optimal design of a combined water system and the black circle is each Pareto-optimal solution considering the objective function 1 and 2.  
fore, the objective function was configured using the tradeoff between design costs in which there was a high vulnerability to flooding and overflow at the increased design precipitation intensity. \[\textit{Minimize Flooding}=\sum_{n=1}^{N}\textit{ Flooding}_{n} \tag{2}\]  
where, \(N\) represents all the nodes in the stormwater drainage network, and \(n\) indicates a particular node. Flooding is the sum of the flooding volume at all times at a particular node. Another objective function was the resilience of the WDS. The resilience of the WDS refers to the ability to recover from various abnormal situations to the pressure of normal situations. Therefore, it means the amount of extra energy excluding the minimum pressure required by the system out of the total energy supplied by the entire WDS, and maximization of the resilience of the WDS was considered to increase the extra energy [19]. The resilience of the WDS is shown in Equation (3). \[\textit{Maximize Resilience}=\frac{\sum_{n=1}^{N}q_{n}\big{(}h_{n}-h_{\textit{ req}}\big{)}}{\sum_{n=1}^{NR}\big{(}Q_{r}\times H_{r}\big{)}-\sum_{n=1}^{N}\big{(}q_{n} \times h_{\textit{req}}\big{)}} \tag{3}\]  
where \(N\) is the number of nodes, \(q_{n}\) is the demand at the corresponding node, \(h_{n}\) is the pressure at the corresponding node, \(h_{\textit{req}}\) is the minimum pressure required by the WDS, \(NR\) is the number of water sources, \(r\) is the corresponding water source, \(Q_{r}\) is the flow rate supplied from the corresponding water source, \(H_{r}\) is the head of the corresponding water source. Therefore, the restoring force is the value obtained by dividing the surplus energy supplied to the nodes of the water distribution networks, which is given a value between 0 and 1. In this study, three constraints (i.e., no flooding under design rainfall, pipes continuity, and required nodal pressure) were considered.