## 2 Literature Review
### Eco-Driving Approaches
The eco-driving problem has been studied from a multi-objective perspective where the objective is not to find a single optimal speed profile for a journey but a Pareto front of non-dominated solutions  
Multi-Objective Particle Swarm Optimisation (MOPSO) algorithm and Non-dominated Sorting Genetic Algorithm II (NSGA-II) have been applied and compared obtaining the Pareto front of speed profiles from which driving commands can be extracted to be programmed in sub-urban railway ATO [60]. Similarly, in [61] MOPSO is also applied to obtain efficient speed profiles for urban railways but including robustness criteria to the operation uncertainty such as mass variations and delays. In [62] the Pareto front of speed profiles is obtained using NSGA-II but taking into account train mass variations by means of fuzzy numbers. Non-dominated Sorting Genetic Algorithm III (NSGA-III) is used in combination with detailed simulation in [63] to obtain efficient speed profiles balancing not only the running time and the energy consumption but also the risk of suffering delays at the arrival. Compared with the studies mentioned above, the algorithms needed by the on-board ATO over ERTMS system must be capable of generating speed profiles that not only meet a target running time in the journey minimising energy consumption as usual, but also meet intermediate passing timing points. This introduces new constraints to the speed profile optimisation problem that can be defined in the form of target times with a tolerance. In the literature, similar problems have been studied where the train control is restricted by intermediate time targets. In [64], a method is proposed to modify optimal speed profiles in order to meet intermediate timing points. This method starts with an optimal speed profile obtained in the basis of Pontryagin's Maximum Principle that meets the arrival time to the next station. Then, a constructive algorithm modifies the initial speed profile adding new holding speed phases to meet the intermediate timing points. Although the method founds feasible speed profiles, the energy efficiency is not guaranteed. In [65], the problem of speed profile optimisation subject to intermediate time windows or target points is discussed. Time restrictions are usually located in junctions, passing stations, or positions where a faster train overtakes a slower one. Thus, intermediate target times are needed to meet the timetable. The train path envelope is proposed in this paper to define the train driving. Train path envelope defines a set of bounds to the feasible train driving that can be formed by intermediate target times and/or target speed ranges. These bounds can be obtained as a result of the planned timetable or provided by a traffic control and conflict resolution system as proposed in the ON-TIME project [66], where train path envelope is applied. The application of Pontryagin's Maximum Principle and dynamic programming are proposed in [65] to solve the eco-driving problem subject to a train path envelope. Although the optimal solution is obtained, the speed restrictions imposed in the intermediate target time position fix the speed of the train in that positions. By this way, the problem can be solved optimising independently each section between two target time positions. The problem of obtaining the optimal speed profile of two trains traveling in the same direction in a flat track was studied in [67]. The safe separation of the two trains is ensured not allowing them to occupy the same track section at the same time. With this objective, the track is divided into sections defined by signal positions, and section clearance times are specified. The clearance time of a section can be based on a timetable and defined as the latest allowed exit time for the leading train and the earliest possible entry time for the following train. As a result of this definition, clearance times establish target times in the middle of the journey for both trains.