# Development of a Multiscenario Planning Approach for Urban Drainage Systems
## 2 Methodology
### Two-Phase Multi-Scenario-Based UDS Planning
#### 2.3.2 Phase II: Multi-Scenario-Based Planning Based on Component-Wise RC  
The multi-scenario-based planning model in Phase II was proposed to identify a compromise solution that could adequately withstand a set of scenarios. Phase II included two steps. In Step 1, the scenario-optimal solutions, i.e., \(X^{*}\) = [\(X^{1*}\), \(X^{2*}\),..., \(X^{i*}\)], obtained from Phase I, were analyzed to identify the common elements. For example, the potential diameter of a pipe that has a range of 0.76-1.22 m in the scenario-optimal solutions was limited in the identified range in seeking the compromise solution (Step 2). As with Phase I, Phase II was also solved using the SWMM linked with the HSA to seek a compromise solution.  
In Step 2, the multi-scenario-based planning model determined both the pipe sizes and the layout, to minimize the total system cost constrained by the allowable maximum RC. Note that the total system cost incorporates the penalty cost that is computed considering UDS performance. The proposed model in this study was formulated as:  
\[\text{Minimize}\,F\left(X|\mathbf{\omega}\right) \tag{8}\]  
and subject to  
\[G\left(X|\mathbf{\omega}\right)\,>\,0\text{ and } \tag{9}\]  
\[RC_{i}\,\leq\,RC_{\text{max}} \tag{10}\]  
where \(F\) and \(G\) represent abbreviated forms of Equations (1) and (2), respectively; \(\mathbf{\omega}\) is a vector of all scenarios; \(RC_{i}\) is the RC in scenario \(i\), which represents the sum of overpayments and supplementary payments in the system; and \(RC_{\text{max}}\) is the allowable maximum RC. The total RC in scenario \(i\) is expressed as  
\[\text{RC}_{i}\,=\,RC_{i}^{O}+RC_{i}^{S} \tag{11}\]  
where \(RC_{i}^{O}\) and \(RC_{i}^{S}\) are the overpayments and supplementary payments in scenario \(i\), respectively. In this study, the component-wise (e.g., pipe diameters and manhole depth) RC was computed by comparing the scenario-optimal solutions (Phase I) and the potential compromise solution generated in Phase II. The overpayments and supplementary payments in scenario \(i\) are expressed as  
\[\left\{\begin{array}{l}RC_{i}^{O}=\sum_{n=1}^{P}\left(RC_{i,n}^{O}\right)\\ RC_{i}^{S}=\sum_{n=1}^{P}\left|RC_{i,n}^{S}\right|\end{array}\right. \tag{12}\]  
where \(n\) represents the individual components, which can be links or nodes and \(P\) is the total number of components.  
The overpayment and supplementary payment in scenario \(i\) were calculated using the sum of each component \(n\). The overpayment and supplementary payment in scenario \(i\) for each component \(n\), with constrained conditions between the scenario-optimal solutions (Phase I) and the potential compromise solution generated in Phase II, are expressed as  
\[\begin{cases}RC_{i,n}^{O}&\text{if }X_{n}^{i*}-X_{n}^{comp}\,>0\\ RC_{i,n}^{S}&\text{if }X_{n}^{i*}-X_{n}^{comp}\,<0\end{cases} \tag{13}\]where \(X^{i*}\) is scenario-optimal solution in scenario \(i\), obtained from Phase I and \(X^{comp}\) is the potential compromise solution generated by Phase II.  
The overpayments and supplementary payments were computed by comparing the \(n\)th components (links or nodes) of \(X^{i*}\) and \(X^{comp}\).