Energy-efficient train driving can lead to a significant reduction in both CO₂ emissions and energy consumption. A lot of research has been done on the topic of energy-efficient train control that minimizes total traction energy consumption. Furthermore, the running time supplements in the timetable determine the possibilities for energy-efficient train driving. However, research on the distribution of these supplements in order to maximize the potential for energy-efficient train driving is limited. This paper considers the multiple-section train trajectory optimization problem, which aims at finding the optimal distribution of running time supplements over multiple stops for a single train given a total scheduled running time in order to maximize the potential for energy-efficient train driving. In addition, we compare two different methods to compute the energy-efficient train control: an indirect solution method and a direct solution method. We applied both methods to a Dutch case study for an Intercity train between the stations Utrecht Central and Arnhem Central. The running time supplement distribution that minimizes the total energy use for a train has the same cruising speed on each section. Furthermore, the shorter the distance between stops, the larger the relative running time supplement. The indirect solution method generates results very quickly compared to the direct solution method and can be used for real-time control. The direct solution method is able to provide direct feedback on the necessary optimality conditions and does not require a specialized code to generate the optimal speed trajectory profile, therefore, it is useful for exploring new variants of the optimal control problem.

Energy-efficient train driving can lead to a significant reduction in both CO2 emissions and energy consumption. A lot of research has been done on the topic of energy-efficient train control that minimizes total traction energy consumption. Furthermore, the running time supplements in the timetable determine the possibilities for energy-efficient train driving. However, research on the distribution of these supplements in order to maximize the potential for energy-efficient train driving is limited. This paper considers the multiple-section train trajectory optimization problem, which aims at finding the optimal distribution of running time supplements over multiple stops for a single train in order to maximize the potential for energy-efficient train driving. In addition, we compare two different methods to compute the energy-efficient train control: an indirect solution method and a direct solution method. We applied both methods to a Dutch case study for an Intercity train between the stations Utrecht Central and Arnhem Central. The results indicate that the optimal running time supplement distribution means that the cruising speed in each section is the same. Furthermore, the shorter the distance between stops, the larger the relative running time supplement. The indirect solution method generates results very quickly compared to the direct solution method and can be used for real-time control. The direct solution method is able to provide direct feedback on the necessary optimality conditions and does not require a specialized code to generate the optimal speed journey profile, therefore, it is useful for exploring new variants of the optimal control problem.