# Development of a Multiscenario Planning Approach for Urban Drainage Systems
## 2 Methodology
### Two-Phase Multi-Scenario-Based UDS Planning
#### 2.3.1 Phase I: Scenario-Optimal Planning Solution  
The purpose of Phase I optimization was to simultaneously determine the pipe sizes and layout in order to minimize the total system cost and satisfy a set of predefined constraints. The total system cost is the sum of pipe construction cost (PipeCC), manhole depth cost (ManholeDC), and penalty cost (PenaltyC), as defined by  
\[\text{Minimize }F=\text{ PipeCC}\left(X\right)\;+\;\text{ManholeDC}\left(X \right)\;+\;\text{PenaltyC}\left(X\middle|\omega^{i}\right) \tag{1}\]  
and subject to  
\[G\left(X\middle|\omega^{i}\right)\;>\;0 \tag{2}\]  
where \(X\) is a set of decision variables (i.e., pipe diameter, layout, and manhole depth); \(\omega^{i}\) is \(i\)th future scenario, which is a combination of different rainfall patterns and nodal inflow distributions; and \(G\) is a set of constraints that can be checked based on the UDS simulation under \(\omega^{i}\). The PipeCC is expressed as  
\[\text{PipeCC}\;=\sum_{j=1}^{N}C_{c}\Big{(}D_{j}\Big{)}\times L_{j} \tag{3}\]  
where \(C_{c}\) is the construction cost according to the unit cost of a pipe of a certain diameter per unit length (USD/m), \(L_{j}\) is the length of the \(j\)th pipe (m), \(D_{j}\) is \(j\)th pipe's diameter (m), and \(N\) is the total number of pipes. The ManholeDC is computed according to  
\[\text{ManholeDC}\;=\sum_{k=1}^{M}M_{c}\times MD_{k} \tag{4}\]  
where \(M_{c}\) is the unit cost of a manhole (USD/m), \(MD_{k}\) is the depth of \(k\)th manhole (m), and \(M\) is the total number of nodes.  
To simultaneously consider UDS stability and the total system cost of the optimal solution, the penalty cost term was appended to the objective function, because the performance of the UDS (e.g., flooding due to UDS) designed by the proposed model must be considered. PenaltyC is a penalty cost that is activated if any of the constraints is not satisfied, but it also includes a function that represents the UDS performance by computing system flooding volume. PenaltyC is formulated as:  
\[\text{PenaltyC}\;=\alpha\cdot\sum_{k=1}^{M}FV_{k} \tag{5}\]  
where \(\alpha\) is a penalty constant and \(FV_{k}\) is the flooding volume at node \(k\) (m\({}^{3}\)/s).  
The constraints included in \(G\) in Equation (2) were (1) pipe-size hierarchy and (2) cover depth. First, the pipe diameters in a UDS should be designed such that the diameter of the downstream pipe (\(D_{down}\)) is greater than or equal to that of the upstream pipe (\(D_{up}\)), because it is more likely in practice that a greater volume of water is eventually drained downstream. The constraint for pipe diameters is expressed as follows:  
\[D_{up}\;\leq\;D_{down} \tag{6}\]  
The cover depth constraint limits the vertical Euclidean distance between the ground surface and the top of the pipe, i.e., cover depth. In other words, the cover depth at node \(k\) should be greater than or equal to the minimum cover depth allowed (\(C_{\text{min}}\)):  
\[C_{k}\;\geq\;C_{\text{min}} \tag{7}\]  
Note that the manhole depth was decided based on cover depth, manhole offset, and pipe diameters.  
The scenario-optimal planning model defined by Equations (1) to (7) calculates PenaltyC using the storm water management model (SWMM) [25] linked with the harmony search algorithm (HSA) [26]. The former is a hydraulic-hydroologic simulation model, whereas the latter is a metaheuristic optimization algorithm. The scenario-optimal planning solutions obtained in Phase I, by considering economic investment and system performance with flooding volume, were the best alternative for each individual scenario and together served as the basis when identifying the compromise solution in Phase II that performed well across all multiple scenarios.