## 4 Results
Likewise, node 29 is still key for both EVs, with a prominent location to successfully complete routes at the depot, as shown in Figure 5. The EVs involved in the LMDP are depicted in Figure 6. Multiple visits in the LMDP pattern, which are translated in recharge events, from different EVs at charging stations can also be evidenced, with a 50% of reduction in the charging operation. Figure 4: Details of routes for \(Q_{LMDP}\) = 20. Figure 3: Details of routes for \(Q_{CVRP}\) = 30. To demonstrate the sensitivity of the mathematical model, some of the power distribution network nodes are banned for charging station installation. We took the case for \(Q_{CVRP}\) = 50 and \(Q_{LMDP}\) = 28 and fix the variable \(U_{g}=0\) to represent this ban, which is tagged in Table 1 with an asterisk. See in Figures 7 and 8 the routes performed by the EVs in the CVRP and LMDP patterns, considering the shadow area that contains the power network nodes not allowed for installing charging stations. Figure 5: Details of routes for \(Q_{CVRP}\) = 50. Figure 6: Details of routes for \(Q_{LMDP}\) = 28. Comparing the results from Figures 5 and 6 with respect to Figures 7 and 8, the routing cost for the CVRP is increased when some of the power network nodes are not allowed for charging station installation because this restriction makes the solution relocate the charging station that results in a larger route for the EV. Notice that a charging station had been installed at node 31; furthermore, incorporating the charging station install ban, this was relocated to node 24 as depicted in Figure 7. This situation can be seen with other charging stations that were relocated along the EV routes. Additionally, the number of charging stations that were relocated along the EV routes is increased when the charging station installation is increased. Figure 4.0 Details of routes for \(Q_{LMDP}=28\) considering the banned power network nodes. Figure 4.1 Details of routes for \(Q_{CVRP}=50\) considering the banned power network nodes. of charging stations is less, the increase in energy losses is reduced, and computational time reflects a significant decrease in comparison with the case of banned power network nodes. Although the objective function value is not decreased, the increment in this value is about 2%. This results in restricting power network nodes for installing charging stations, providing an improvement in finding a suitable solution, as part of a heuristic that can be initially run before executing the CPLEX solver. Figure 9 presents an exponential tendency on the delta in energy losses with respect to the benchmark case solution, considering the change in battery capacity. Particularly, for the experiments \(Q_{CVRP}=50/Q_{LMDP}=28\) and \(Q_{CVRP}=60/Q_{LMDP}=32\), there is a decrease of 18% and 27%, respectively, in delta of energy losses, when the ban is applied for some of the electrical nodes. From the mathematical model standpoint, this is translated into a solution space curtailment, forcing the installation of charging stations at more attractive locations in terms of value for the objective function, which results in a suitable practice to initially run a heuristic as mentioned before.