## 3 Problem Description  
ATO over ERTMS can be applied in either mainline or urban railways. However, its main purpose is the application in the complex operation of mainline networks. ERTMS is the mandatory signalling system for European high-speed trains although it has been applied in a variety of systems. For this reason, this study will focus on high-speed railways.  
ATO over ERTMS specification establishes that the on-board equipment must calculate the set of driving commands to be followed in order to meet the timing points. Thus, the eco-driving defined is a mono-objective optimisation problem with the aim of minimising the energy consumption. The optimisation problem is subject to time constraints. Time constraints are derived from the timing points and impose several objective times in the middle and the end of the journey. According to the ATO over ERTMS specification, the objective times imposed by the timing points must be met within a certain time window or tolerance. Therefore, there is some flexibility in the timetable meeting.  
Thus, the eco-driving problem defined above can be formulated as:  
Find \(X\) which minimises  
\[EC(X) \tag{1}\]  
subject to  
\[\left|OT_{l}-t_{l}(X)\right|<\varepsilon_{l},\ \ \ \ \ l=1,\ldots,\ L \tag{2}\]  
where \(EC(X)\) is the energy consumption of solution \(X\), \(t_{l}(X)\) is the resulting passing time of solution \(X\) in the timing point \(l\) located at position \(s_{l}\), \(OT_{l}\) is the objective time for the point \(l\), \(\varepsilon_{l}\) is the tolerance allowed to the passing point \(l\) and \(L\) is the number of timing points including the arrival point at the station.  
Each possible solution \(X\) is the set of holding speed without braking commands and a final coasting command, collected in a command matrix \(C_{m}\) as was proposed in [82].  
A \(C_{m}\) is composed by \(n_{s}\) sections where the ATO applies certain driving command. The first \(n_{s}-1\) sections are holding speed without braking commands. Although holding speed without driving command was initially proposed to be manually applied, it also can be applied by an ATO. It is a more energy efficient way to negotiate steep downhills compared with the common holding speed commands. Furthermore, it ensures the comfortability of the speed profile limiting the sudden changes in train acceleration. The use of several holding speed commands is needed to meet the intermediate timing points because it is difficult to meet several intermediate time goals using only one holding speed command. Furthermore, depending on the profile of the line and the speed limitations, the minimum energy consumption solution could need a more complex solution instead of a single speed command [28]. The final driving command defines a coasting period before the final braking.  
The commands contained in \(C_{m}\) are subject to a minimum speed restriction for the holding speed without braking commands (\(vc_{min}\)) and a minimum position from which the train can start the final coasting command (\(sc_{min}\)). Thus, it is ensured that the train does not run at too low speed, which could be perceived as unpleasant by passengers.  
The problem defined can be classified as a CNOP. As was pointed out in the introduction, this paper proposes the application of Nature Inspired Techniques to the ATO over ERTMS eco-driving problem.  
Several algorithms have been applied in the literature to solve CNOPs such as Genetic Algorithm (GA), Particle Swarm Optimisation (PSO), Evolutionary Strategies (ES), or Differential Evolution(DE); but there is a preference for the application of Differential Evolution algorithm to solve CNOPs compared with other algorithms [73]. Therefore, DE has been selected to solve the ATO over ERTMS eco-driving problem. Section 4 explains in detail the characteristics of the DE algorithm proposed.