## 4 Case Study Development and Results
There are two ways to deal with this issue, either consider the city shape as it is or develop a square (or rectangle) that surrounds the city shape. In the first case, the solution space is reduced to the city shape; however, a large number of constraints should be added to model the city limits. In the latter case, the solution space increases, but the simplicity of the algorithm in terms of constraints is retained. As part of this analysis, the solution space of the basic scenario was extended, considering solution space dimensions ranging from \(-100\) to \(100\) in both directions (4 times larger than the original area) and \(-200\) to \(200\) in both directions (16 times larger than the original). The demand points and size remained as originally (within the \(-50\) to \(50\) range) while different initial station allocations were considered, from all stations being at point (0, 0) to randomly assigned within the large box. Figure 10 provides an indicative solution for the case of an area ranging from \(-100\) to \(100\), in which 10 charging stations are to be deployed, while Table 2 summarizes some results of the process regarding the fitness value of the total distance traveled. Not surprisingly, the algorithm, after some exploration of the whole space, has always converged to a solution very similar to the initial one and, interestingly, within the same computation time. This can highlight the GA converging capability regardless of the space size and the initial station placement. Figure 3:0 Indicative charging station deployment results for Case Study 5: (**a**) 5 times higher demand in specific points; (**b**) 10 times higher demand in specific points (normal demand points in dark blue color, increased demand points in light blue color, charging stations in orange color). Case Study 7: Station Allocation for EV Travel Distance (Cost) Minimization for Large Service Area Size and Number of Stations  
Regarding the size of the area for which the analysis is performed, the following aspects need to be considered: depending on the city (or neighborhood) size, the number of stations can vary from a few to many. In fact, this decision has also to do with the demand level, in the sense that if the demand is low and dispersed, it is not cost-effective to build a large number of stations. Instead, the users should travel a longer distance to reach a station. Within the present analysis, a relatively small number of stations have been considered (up to 25). This is very realistic for neighborhoods and small-to-medium-size cities, e.g., 10 by 10 km (thus every distance unit in the above analysis corresponds to 100 m). To evaluate the scalability performance of the algorithm, larger station numbers have been further considered (within the same plan structure and demand size and distribution). In particular, 50, 100, and 200 stations were examined. The total traveling distances for 50, 100, and 200 stations are 411, 197, and 105, respectively. Nevertheless, since the number of demand generated spots is 100, there is no need to go over that number in terms of stations built. The indicative results of this analysis are shown in Figure 3:1 \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number of Stations** & **Field Range** & **Total Distance** \\ \hline \multirow{3}{*}{10} & \(-\)50 to 50 & 2042 \\  & \(-\)100 to 100 & 2054 \\  & \(-\)200 to 200 & 2053 \\ \hline \multirow{3}{*}{20} & \(-\)50 to 50 & 1170 \\  & \(-\)100 to 100 & 1161 \\ \cline{1-1}  & \(-\)200 to 200 & 1165 \\ \hline \end{tabular}
\end{table} 
Table 2: Optimization distance results for Case Study 6. Figure 10: Indicative charging station deployment example for a \(-\)100 to 100 field range: (a) initial random solution; (b) final optimal solution (demand points in blue color, charging stations in orange color). A similar analysis assumes that there are 50 (100) demand spots of 4 (2) EVs each. The same number of stations is considered, with the expectation being that one station is placed exactly at each demand spot.