**1. Introduction**

Railways are considered an inexpensive, fast and safe transport mode. Furthermore, this transport mode is energy-efficient and its contribution to global warming is less severe compared with others [1]. Despite that, its increasing activity throughout the world [2] makes it necessary to continue working on the energy efficiency of rail transport to achieve a sustainable transport industry without losing sight of service quality improvements.

Eco-driving, also named speed profile optimization or train trajectory optimization, is one of the most important measures with which to achieve significant energy reductions in the energy consumption of railway operations. It consists of obtaining a way to drive a train on a journey to fulfil a target running time with minimal energy consumption. Eco-driving has the advantage of being a short-term action that requires low investments, while other measures, such as improving the infrastructure [3–12] or rolling stock [13–17], usually require a significant investment and long/mid-term actions.

The eco-driving problem has been studied since 1968, when Ichikawa applied the Pontryagin's maximum principle to a greatly simplified train dynamics model, to derive the optimal control of a train [18]. Since then, numerous researchers have contributed to the eco-driving field with their proposals.

Control theory [18,19] shows that the optimal speed profile for a train running from two stations in a flat track consists of a sequence of four regimes: maximum traction, cruising, coasting and maximal braking. If the line geometry is more complex [20,21] or the train has regenerative braking [22], the most energy-efficient speed profile results as a smart combination of these four regimes.

In the literature, different optimization techniques have been proposed to find the right combination of the efficient driving regimes. These techniques can be divided into two groups [23]: analytical and numerical methods.

Typically, analytical methods apply the Pontryagin maximum principle to obtain the optimality necessary conditions, and using these results, apply different techniques to find the optimal speed profile. These techniques are: constructive algorithms [20–22,24–26], dynamic programming [27–31], sequential quadratic programming [30,32] and the Lagrange multiplier method over the discretized problem [33]. Other analytical methods are based on transforming the optimal control problem into a non-linear problem and solving it directly [34,35].

Analytical methods are, in most cases, fast procedures that can produce the optimal solution. However, the complexity of the problem must be reduced to apply these methods, due to the requirements for obtaining the analytical solution. This leads to simplifications in the train and line models and in the operational restrictions that the solution must comply.

On the contrary, the use of numerical methods does not demand simplifications in the train and line model, and any model to observe the requirements of passengers' comfort or operation characteristics can be included. In recent years, these methods have received growing attention aimed toward solving the eco-driving problem, because of the improvement of computational performance; in most cases, they are more computationally expensive than analytical procedures.

Different numerical methods have been applied to optimize the energy consumption of train driving: direct search algorithms [36,37], brute force [38], Monte Carlo simulation [39], artificial neural networks [40,41] and nature inspired algorithms [29,42–47]. Among these techniques, nature inspired computational intelligence is one of the most common methods applied to solve the speed profile optimization problem. These techniques provide a framework that can be easily implemented and are independent of the specificities of the problem. Moreover, they can be used in combination with complex train dynamics models that can be easily substituted by other different models when the characteristics of studied railway line change.

Many nature inspired computational intelligence techniques have been proposed in the literature to solve different problems [48]. When looking in detail at the algorithms applied to solve the train eco-driving problem, the following are present: the genetic algorithm (GA) [29,42,47,49–54], the multi-population genetic algorithm (MPGA) [55,56], GA with fuzzy parameters [57–60], differential evolution [46], ant colony optimization [29,61,62], simulated annealing [45,63], the indicator based evolutionary algorithm (IBEA) [64], the non-dominated sorting genetic algorithm II (NSGA-II) [43,44], multi-objective particle swarm optimization (MOPSO) [44,65] and dynamic versions of NSGA-II and MOPSO [66,67].

Apart from the specific technique applied to obtain the efficient train driving, the literature shows consistently that the use of eco-driving provides important energy savings [68]. Typically, trains on mainline railways are manually driven and the whole driving application is made by means of driver advisory systems (DASs). Several works have reported applications of eco-driving in DASs [69–73], wherein energy saving between 7% and 22% have been obtained. On the other hand, nowadays many urban railways are usually driven automatically by means of automatic train operation (ATO) systems. In these systems, efficient driving commands must be programmed to perform eco-driving speed profiles. Several authors reported applications of eco-driving in ATO systems, wherein energy savings between 6% and 18% have been obtained [44,65,74–76].

Savings obtained thanks to eco-driving are mainly due to the substitution of the use of braking by coasting periods. Besides, acceleration phase can optimized to reach higher energy efficiency. Nowadays, most trains are equipped with regenerative braking that allows recovering the kinetic energy during deceleration phases. In DC systems, regenerated energy cannot be totally used because it is necessary to have another train in the system consuming the amount of energy regenerated (except for trains which are equipped with on-board accumulation devices [77]). However, the use of reversible substations [5,6,8,9] allows them to use all the energy regenerated in a DC system by sending the regenerated energy not used by other trains to the utility grid. In modern high-speed systems the electrification is AC and substations allow the flow of energy in both directions (consumption or generation); thus, as in the previous case, it is possible to use all the regenerated energy produced (except electrical losses).

In this context of high use of regenerated energy, a question arises: is it worthwhile to minimize the energy consumption of a train, or will any driving over the same running time provide similar energy consumption?

The research objectives of this work are, on one hand, to assess the effectiveness of eco-driving under different scenarios of regenerated energy receptivity, and on the other hand, to see whether there is a scenario wherein eco-driving presents no relevant efficiency, providing an answer to the previous question.

The studies presented in this paper are based on real data from Spanish high-speed lines. The studied lines have been chosen because they are AC electrified, which offers high regenerated energy usage. The lines selected are Madrid–Barcelona, which is fed by a 2 × 25 kV power supply system, and Madrid–Sevilla, which is fed by a 1 × 25 kV power supply system. In the case study, two driving strategies are compared: an optimized eco-driving strategy and a standard driving strategy. The standard driving strategy consists of maintaining a holding speed during the journey to meet the running time. On the other hand, the efficient eco-driving consists of two driving commands: a holding without braking command and a coasting command before the braking up to the arrival station.

A MOPSO (multi objective particle swarm optimization) algorithm [78] combined with a detailed train simulation model [47] has been used in this piece of research to generate not only the efficient driving but also the standard driving for different target running times. This algorithm has the advantage of generating in a single run a set of non-dominated solutions, also known as Pareto front. A non-dominated solution is one that cannot be improved at the same time in all the problem's objectives. In the eco-driving problem, the objectives are the minimization of energy consumption and running time. As these objectives are conflicting, the result of the algorithm is Pareto front of solutions, where each solution is a speed profile with the minimum energy consumption for each possible running time. This way, MOPSO allows one to compare results easily in a wide range of running time values. MOPSO algorithm has demonstrated its suitability to the eco-driving problem in [44,66,67].

The usable regenerated energy that a train can produce depends mainly on two factors: the efficiency ratio of the train motor (that measures the efficiency of the train as a kinetic energy accumulator), and the transmission losses produced at catenary [30,79,80]. For this reason, the influences of different realistic values of these factors are analyzed in this paper to provide insights into the benefits provided by eco-driving on these types of lines.

This article is organized as follows: Section 2 describes the simulation model used to evaluate the different speed profiles. Section 3 describes the different driving strategies studied. The manual driving model is described in Section 4. Section 5 introduces the nature-inspired algorithm used to obtain the energy-efficient speed profiles. The results obtained and the discussion are shown in Section 6, and Section 7 presents the conclusions of this article.
