## 4 Simulation Results
### Siting and Timing of Three New PSs  
Figure 19 shows the optimal siting and timing considering three new PSs. The first PS is settled in Rozzano municipality, located in the extreme lower part of the map. It is worth noticing that the position of the PSs is close to the squares having the highest power demand, a direct consequence of the fitness function explained in Equation (7). Instead, the second and the third PSs surround the city center, where power density is high, filling the gaps between the existing PSs. Figure 18: Current PS service areas. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{3}{c}{**1\({}^{\circ}\) Iteration**} & \multicolumn{2}{c}{**2\({}^{\circ}\) Iteration**} & \multicolumn{2}{c}{**3\({}^{\circ}\) Iteration**} & \multicolumn{2}{c}{**4\({}^{\circ}\) Iteration**} \\ \hline \multirow{2}{*}{**PS Name**} & **Fitness** & **PS** & **Fitness** & **PS** & **Fitness** & **PS** & **Fitness** & **PS** \\  & **[km]** & **Timing** & **[km]** & **Timing** & **[km]** & **Timing** & **[km]** & **Timing** \\ \hline PS1 & 1.60 & 2 & 1.58 & 3 & 1.50 & 3 & - & 3 \\ \hline PS2 & 1.80 & 4 & 1.70 & 4 & 1.65 & 4 & 1.40 & 4 \\ \hline PS3 & 1.75 & 3 & 1.56 & 2 & - & 2 & - & 2 \\ \hline PS4 & 1.50 & 1 & - & 1 & - & 1 & - & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Example of the PSs timing process. The restricted number of PSs concerning the Milan and Rozzano area does not allow installing any PSs on the left-hand side of the map, whose squares remain far from the related PSs. The PS service areas are better highlighted in Figure 20. The new PS called \"1\" split the service area of the existing PS called \"SU\" almost by half, giving undoubted advantages in reducing the distance between squares and the related PSs. The cells' colors show that the algorithm does not fail in finding a feasible solution, which guarantees a power demand of the PSs lower than 75% of the transformers' nominal power. Nevertheless, five PSs out of fourteen have a loading greater than 50%, reporting potential criticalities in dealing with additional future power demand increases. It is worth noticing that the PS \"OV\" service area still has a significant extension. Regarding fitness, the average distance between the squares and the related PSs is 2.11 km, while the average weighted distance is 1.35 km. On the one hand, since the algorithm locates the new PSs closer to the power demand, with three new PSs instead of the seven PSs owned by a competitor DSO, a significant reduction of about 9% of the weighted distance can be obtained (current value is 1.48 km). On the other hand, the reduced number of PSs brings an average distance higher than the current one, i.e., 1.82 km. It is worth noticing that the average distance is not the fitness function of the proposed methodology. Its reduction, if it happens, is just a consequence of the average weighted distance optimization. Table 4 shows the main outputs of the methodology and reports the siting, timing of the new PSs, the best average distance, and the best average weighted distance compared  
Figure 19: Output of the simulation results considering three new PSs. Blue points are the new PSs, while gray is the existing ones. Figure 20: PS service areas of three incremental PSs. The colors show the loading of the PS transformers. to the current condition. The last column details the fitness variation while installing, in the order suggested by the algorithm, the three new PSs. It is nice to see that, with only two new PSs, the best average weighted distance dropped below the current value, which indicates the poor location of the current seven PSs owned by a competitor DSO. However, the loading indicated in Figure 20 suggests the need to install several more PSs. Regarding the optimization process, Figure 21 reports the best fitness through the 150 iterations and the global best fitness.