## 5 Case Study
### Benefits of the Proposed Optimisation Method Compared with Single Time Target Approaches  
The aim of the optimisation approach proposed in this paper is to find the speed profile with the minimum energy consumption that meets the arrival and the intermediate target times. The typical numerical methods proposed in literature fails when tackle this problem given that they are designed to minimize the energy consumption meeting only the arrival time.  
A solution to this problem could be to divide the journey in sections between each pair of target times. However, the solution obtain in each section is not valid for the whole journey. It can be seen in the following example.  
A journey with one intermediate timing point is studied. This timing point is located between the departure and the arrival (that is, between 221.3 km and 306.7 km), at position 264 km. The target times chosen for the timing points are taken from the commercial timetable and are listed in Table 3 given that the departure time is at 0:00:00.  
The journey can be divided in this case in two sections: the first section between the departure station and the intermediate point 264 km (with arrival time of 12 min and 3 s) and the second section between the point 264 km and the arrival point (with arrival time of 26 min).  
Each section of the journey is optimised independently and the results are shown in Figure 3. As can be seen, the speed profile of the first section and the speed profile of the second section do not match in their common point 264 km. The speed profile of the first section applies coasting up to 200 km/h when the train approximates to 264 km to save energy. However, the optimisation of the second section starts the train at the maximum speed possible to coast as much as possible and consuming the lowest energy possible.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Population Size** & **Scaling Factor ( F )** & **Crossover Rate (C,)** & **Number of Sections in C\({}_{m}\)(\(n_{s}\))** & **Number of Iterations** \\ \hline
80 & 0.5 & 0.9 & 4 & 24 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Tuned parameters of de algorithm.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Location of Timing Point** & **Target Time** \\ \hline
264 km & 00:12:03 \\
306.7 km & 00:26:00 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Location and target of timing points in the case of one intermediate timing point.  
The previous effect can be neutralized fixing the speed of the train when passing for the 264 km point as can be observed in [65; 66]. However, it is needed to choose carefully the passing speed to obtain the minimum energy consumption. It can be observed in the following example.  
Figure 4 and Table 4 show the results obtained optimizing each section of the previous journey independently, fixing the passing speed of the train at 264 km point with different values.  
Results indicate that there is a value of passing speed that leads to the minimum energy consumption. Therefore, obtaining the most efficient speed profile dividing the journey would require several levels of optimization to obtain the most efficient passing speeds and the most efficient speed profiles.  
Figure 4: Speed profiles obtained optimizing independently each section between target times fixing the passing speed at 264 km to 300 km/h (a), 235 km/h (b) and 200 km/h (c).  
Figure 3: Speed profile obtained optimizing independently each section between target times.  
On the contrary, the method proposed in this paper solve the problem jointly in a single level, including the intermediate time restrictions in the optimization process. This way, the method proposed is a much more efficient numerical method.  
In the following subsections, the results of the proposed method are shown for one and two intermediate target times.