## 5 Case Study
### Two Intermediate Timing Points  
In this subsection, the performance of the algorithms in a journey with two intermediate timing points is studied. These points are located at positions 250 km and 278 km. The passing times of the timing points are listed in Table 5 (given that the departure time is at 0:00:00) and, as in the previous example, are based on the commercial timetable. Figure 8: Speed profile (**a**) and position evolution with time (**b**) of the most efficient and most energy consuming solutions found by the algorithm for the case of one intermediate timing point. The DE algorithm and the GA are executed to obtain the speed profile of minimum consumption that fulfils these timing points. As in the previous case, three values for tolerance have been used: 1 s, 5 s, and 10 s.  
Figure 9 shows the evolution of the average value of the fitness of the best solution found by the GA throughout the iterations for the three time tolerance values. The figure is divided into the red infeasible area and the green feasible area. As in the previous case, the GA is not capable of finding a feasible solution in 24 iterations. The DE algorithm is capable of finding feasible solutions although the complexity of the problem has increased with one more restriction. Figure 10 shows the evolution of the mean fitness value of the best solution found by the DE at each iteration. In all of the 20 executions, the DE algorithm was capable of obtaining a feasible solution. The relaxation of constraints facilitates the search. Figure 10 shows that with a tolerance of 1 s the algorithm found feasible solutions at iteration 10, with a tolerance of 5 s, the feasible solutions are found at iteration 6, and with tolerance of 10 s they are found at iteration 5. Comparing these results \begin{table}
\begin{tabular}{c c} \hline \hline
**Location of Timing Point** & **Target Time** \\ \hline
250 km & 00:09:10 \\
278 km & 00:16:20 \\
306.7 km & 00:26:00 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Location and target of timing points in the case of two intermediate timing points. Figure 9: Evolution of the mean value of the fitness of the best solution found by the GA for the case of two intermediate timing points. with those obtained with one intermediate timing point, it can be seen that, with the new timing point, the algorithm needs more iterations to find feasible solutions. This is because the new restriction reduces the feasible region and it is harder to reach. Figure 11 is a zoom of the feasible region. It shows that solutions are improved during iterations reducing the energy consumption and it is stabilised before reaching 24 iterations in all the cases. That means that the number of iterations given to the algorithm is enough to converge to the optimal solution. Figure 11 presents, also, the effect of increasing the tolerance. As in the previous case, it is important to obtain not just a feasible solution, but the best possible one. For instance, in the case of 1 s of time tolerance, the most energy consuming solution found by the algorithm consumes 0.99 MWh, while the most energy efficient solution presents a result of 0.96 MWh. This is a difference of 3.1%, which cannot be neglected. This difference is lower compared to the one obtained in the one intermediate timing point case, because the higher the number of restrictions, the lower flexibility to drive the train and the more similar are the feasible speed profiles. Figure 12 shows the speed profile of the most efficient and most energy consuming solutions found by the algorithm. Both solutions meet the target times and these two speed profiles are more similar than those shown in the previous section in Figure 8. The main difference between these two solutions is that the efficient speed profile coasts at 285 km position, while the most energy consuming solution does not apply coast, and regulates speed.