**5. Eco-Driving Optimization Algorithm**

A multi-agent based optimization method called MOPSO (multi-objective particle swarm optimization algorithm) [78] is used to generate energy-efficient speed profiles [44]. Energy consumption/running time Pareto fronts are obtained for different values of the engine traction and brake efficiencies and the linear resistance of the catenary.

This nature-inspired algorithm imitates the behavior of a swarm of insects looking for food. These insects are considered particles that move iteratively within a search space. In this case, each particle corresponds to a speed profile and the running time and energy consumption have to be optimized. A fitness value is assigned to each particle. In biology terms, fitness stands for a degree of adaptability of an insect or how close it is to an abundant source of food. To optimize its fitness, each particle position is updated at every iteration by computing its velocity as a function of its past position with best fitness, *pbest*, and the global best position found by the whole swarm, *gbest*. The latter variable accounts for the information shared between the particle ensemble.

To initialize the algorithm, a set of particles with random positions and velocities is generated. At every iteration *i*, the non-dominated solutions, which are those particles for which there are not any calculated solutions with lower running time and energy consumption, are stored in an archive *A*. The non-dominated solutions in *A* are sorted in decreasing order by using a crowing distance (*CD*) operator [44]. This way, the low-density zones of the Pareto front are given more priority. *pbest* of each particle is updated and *gbest* is drawn randomly from the archive, giving priority to the solutions with higher crowding distance, which correspond to the solutions at the top of *A*.

The positions and velocities of the particles at the next iteration are generated by using Equations (27) and (28),

$$v\hat{p}\_{\hat{j}}(i) = wv\hat{p}\_{\hat{j}}(i-1) + c\_1r\_1(pbest\_{\hat{j}} - \hat{x}\hat{p}\_{\hat{j}}(i-1)) + c\_2r\_2(gbest - \hat{x}\hat{p}\_{\hat{j}}(i-1))\tag{27}$$

$$\mathfrak{x}\mathfrak{p}\_j(i) = \mathfrak{x}\mathfrak{p}\_j(i-1) + \mathfrak{v}\mathfrak{p}\_j(i) \tag{28}$$

where *xp*ˆ *<sup>j</sup>* , *vp*ˆ *<sup>j</sup>* are the position and velocity of the *j*-th particle; *w* is an inertia constant that weights the previous velocities; *c*<sup>1</sup> and *c*<sup>2</sup> are two social factor constants that weight the distance to *pbestj*; and *gbest*, *r*<sup>1</sup> and *r*<sup>2</sup> are random numbers drawn uniformly between 0 and 1.

After the position of all the particles in the current iteration has been updated, the new non-dominated solutions are included in the archive and all the newly dominated solutions are deleted. *pbest* is updated for all the solutions in the archive. The solutions in *A* are sorted in decreasing order of *CD*, and finally, the *gbest* is drawn randomly from *A*, giving a higher probability to those solutions with higher *CD*. This procedure is repeated until a certain number of iterations I is reached. Figure 1 shows the MOPSO algorithm flowchart.

**Figure 1.** MOPSO algorithm flowchart.

### **6. Case Study and Results**

In this section, the holding speed without braking with final coasting eco-driving strategy is compared against the standard holding speed driving, and the difference in energy consumption is quantified. In both cases, there is a single cruise speed and all the energy regenerated by the electrical brake is assumed to be returned to the power grid. The dependence of the energy consumption (measured at pantograph and at substations) on the engine efficiency and catenary linear resistance is analyzed. The nature-inspired optimization algorithm introduced in Section 5 is used for obtaining the Pareto front of speed profiles for the Talgo-Bombardier class 102 train running on two sections of Spanish high-speed lines. Track grades, curves, speed limits, tunnels, neutral zones and the positions of electrical substations are considered in the line models.

The first section analyzed is an 85.4 km long section between Calatayud and Zaragoza in the Madrid–Barcelona line with a 2 × 25 kV power supply system. The second section is a 38.9 km long line section between Puertollano and Ciudad Real in the Madrid–Sevilla line, with a 1 × 25 kV power supply system. The train has two 8 MW engines of 200 kN of maximum traction effort. The maximum traction curve is shown in Figure 2. The power consumed by the auxiliary systems is 325 kW and in any neutral zone the train has to regenerate enough energy to maintain these systems on, as presented in Equation (24). The running resistance curve is presented in Figure 3. The empty mass of the train is 324 t, and its length is 200 m. Furthermore, apart from the train's empty mass, 50 t-worth of passengers have been taken into account.

**Figure 2.** Maximum tractive effort curve for the considered Talgo-Bombardier class 102 train.

**Figure 3.** Running resistance for the Talgo-Bombardier class 102 train. It is modelled by using Equation (6).

Regarding the comfort constraints, the maximum and the service decelerations are 0.4 m/s2, the maximum acceleration is 0.67 m/s<sup>2</sup> and the maximum admissible jerk is, in absolute value, 0.7 m/s3. The power constant is considered to be *cos*ϕ = 1. The social factors *c*<sup>1</sup> and *c*<sup>2</sup> in the MOPSO algorithm are equal to 2 and the inertia constant *w* is 0.2.

## *6.1. Calatayud–Zaragoza Case Study*

Figure 4 shows the height profile of the line section between Calatayud and Zaragoza. As can be seen, the considered section is mostly downhill. There are three neutral zones, which are represented in magenta in the lower part of the figure.

**Figure 4.** Height profile of the Calatayud–Zaragoza section. Neutral zones are represented in magenta.

In this subsection, the impact of the efficiency ratio of the motors on the energy consumption is analyzed for both driving strategies. The traction and brake engine efficiencies are considered equal and constant within a range between 85% and 97%. The influence of the linear resistance of the catenary on the energy consumption is also studied. In the case of the Calatayud–Zaragoza section, the catenary configuration is 2 <sup>×</sup> 25 kV, and different values of its linear resistance, between 21·10−<sup>6</sup> <sup>Ω</sup>/<sup>m</sup> and 37·10−<sup>6</sup> <sup>Ω</sup>/m, are tested. The MOPSO algorithm was used to generate speed profiles for both standard and eco-driving strategies. For each driving strategy the non-dominated speed profiles in terms of energy consumption and running time define the Pareto front. Figure 4 shows Pareto fronts for the net consumption measured at the electrical substations for the extreme values of the considered interval for the efficiency ratio of the motors and for the linear resistance of the catenary. The Pareto fronts corresponding to the eco-driving strategy are represented in blue, while those corresponding to the standard driving strategy are represented in red.

As shown in Figure 5, the consumed energy decreases as the engine efficiency increases. The energy consumption increases for higher values of the linear resistance, since more regenerated energy is dissipated at the catenary. For the fastest speed profilesm the difference in energy consumption is significantly lower than for the rest. These speed profiles are close to the flat-out speed profile; thus, the final coasting phase is very short, and therefore has little relevance in terms of energy-saving, so both driving strategies produce similar results in terms of energy consumption. It can also be observed that the eco-driving strategy produces faster and less energy-consuming speed profiles than

the standard driving strategy, as all the solutions for the latter one are dominated by those produced by the former. Table 1 is focused on the analysis for the commercial running time (24 min), varying the efficiency ratio and the linear resistance. It shows the difference (in %) in net energy consumption at the electrical substations between both driving strategies. Results shown in this table are especially relevant for the infrastructure administrator.

**Figure 5.** Pareto fronts for the net consumption at the electrical substations for the considered extreme values of the engine traction and braking efficiency (85% and 97%) and linear resistance of the catenary (21·10−<sup>6</sup> <sup>Ω</sup>/m and 37·10−<sup>6</sup> <sup>Ω</sup>/m), for the Calatayud–Zaragoza section.

**Table 1.** Difference (in %) in net energy consumption at the electrical substations between the eco-driving and standard driving strategies for different values of the energy traction and braking efficiency and the linear resistance in the Calatayud–Zaragoza section for the commercial running time (24 min).


In Table 1, two trends can be observed. The difference in energy consumption decreases as the linear resistance of the catenary decreases. It decreases even more significantly when the efficiencies of the engine brake and traction are higher. The energy saving at the electrical substations between the eco-driving strategy and the standard driving is 2.47–4.69%.

Obviously, the measured energy consumption at the pantograph does not depend on the linear resistance of the catenary. Figure 6 shows the dependence of the energy savings (measured at the pantograph) on the energy efficiency.

**Figure 6.** Difference in energy consumption at the pantograph (in %) between the considered driving strategies for different values of the engine traction and brake efficiency in Calatayud–Zaragoza section.

It may be observed that for the considered engine efficiency interval, there is an approximately linear relation between the difference in energy consumption and the engine efficiency. The eco-driving strategy produces speed profiles with lower energy consumption than the standard driving strategy. Results shown in Figure 6 are especially relevant for train operators. The energy saving measured at pantograph of eco-driving with respect to standard driving is between 2.1% and 4.3%.

In Figure 7 the difference in energy consumption at the substations is analyzed for each linear resistance of catenary. Results are shown for different values of the engine efficiency and running time. A grey vertical dashed line is used to highlight the commercial running time, which is 24 min.

**Figure 7.** *Cont*.

**Figure 7.** Difference in energy consumption (in kWh) between the considered driving strategies at the electrical substations for the Calatayud–Zaragoza section for different values of the catenary linear resistance *r*ˆ.

Three different scenarios can be distinguished. Close to the flat-out running time, there is a large increase in the difference in energy consumption between the eco-driving and the standard driving strategies; this is due to the action of the final coasting phase in the former driving strategy. Next, there is a scenario in which the difference is approximately constant. For the slowest speed profiles, the difference in energy consumption is significantly lower, as can be seen in Figure 5, and the difference in energy consumption between the considered driving strategies becomes smaller as the running time increases, as the coasting phase will be shorter and therefore its impact will be less significant. If the dependence of the results on the energy braking and traction efficiency is considered, it can be observed that for a fixed running time the difference in energy consumption seems to depend linearly on the difference in energy efficiency. For the same value of the engine efficiency, the difference in energy consumption increases slightly with the linear resistivity of the catenary.

Last, the speed profiles obtained for the commercial time with the two considered driving strategies are presented. Table 2 shows the command vectors for these two speed profiles. Notice that the considered line section ends at kilometric point (k.p.) 306.7 at Zaragoza station (stopping point). The coasting phase in eco-driving starts 18 km before the stopping point.


**Table 2.** Command vectors for the commercial time speed profiles obtained with the two considered driving strategies for Calatayud–Zaragoza section.

In Figures 8 and 9, the speed profiles are shown. The train speed is represented in blue, the speed limits are represented by red lines and the track altitude is shown in green. The holding speed is represented by an orange dashed line.

**Figure 8.** Eco-driving speed profile for the commercial time in the Calatayud–Zaragoza section.

**Figure 9.** Standard driving speed profile for the commercial time in the Calatayud–Zaragoza section.

Figure 8 shows that in the holding speed regimes there are few points in which the train goes faster than the cruise speed in the eco-driving strategy, as there are few steep downhill grades. The most significant difference between the two speed profiles in Figures 8 and 9 is after position 288.7 km, in which the train is coasting in Figure 8, while in Figure 9 it is holding its speed until it reaches the braking curve. Near position 300 km the train has to brake due to the presence of a speed limit.

### *6.2. Puertollano–Ciudad Real Case Study*

Finally, results are shown for the second case study, in which eco-driving speed profiles were designed for the considered Talgo-Bombardier class 102 train in a 38.9 km long line section between Puertollano and Ciudad Real. This line section has a catenary configuration of 1 × 25 kV, and three different values of its linear resistance between 54·10−<sup>6</sup> <sup>Ω</sup>/m and 110·10−<sup>6</sup> <sup>Ω</sup>/m are considered in this study. In Figure 10 the height profile of the considered line section is represented in blue, while the two neutral zones are represented in magenta in the lower part of the figure. In general terms, this line section is flatter than the Calatayud–Zaragoza line section.

**Figure 10.** Height profile of the line section between Puertollano and Ciudad Real. Neutral zones are represented in magenta.

Again, speed profiles were generated for the two considered driving strategies by using the MOPSO algorithm. The obtained Pareto fronts are represented in Figure 11, in which they are sorted in terms of the net consumption at the substations. Results are shown for the extreme values of the considered engine efficiency interval, 85% and 97% and for catenary linear resistances 54·10−<sup>6</sup> <sup>Ω</sup>/m, <sup>70</sup>·10−<sup>6</sup> <sup>Ω</sup>/m and 110·10−<sup>6</sup> <sup>Ω</sup>/m. The standard and eco-driving speed profiles are represented in red and blue, respectively. The commercial time, which is 13 min, is highlighted by a vertical grey dashed line.

**Figure 11.** Pareto fronts for the net consumption at the electrical substations for the extreme values of the engine traction and braking efficiency (85% and 97%) and the three considered values of the linear resistance of the catenary (54·10−<sup>6</sup> <sup>Ω</sup>/m, 70·10−<sup>6</sup> <sup>Ω</sup>/m and 110·10−<sup>6</sup> <sup>Ω</sup>/m).

Results presented in Figure 11 are similar to those in Figure 5, although this time the difference in energy consumption between two solutions with the same running time is usually larger than in the Calatayud–Zaragoza case. For a fixed running time, the energy consumption is larger when the engine is more inefficient and when the catenary has a larger linear resistance. The speed profiles in the holding speed without a braking Pareto curve dominate those in the holding speed with a braking Pareto curve.

Table 3 shows the difference in energy consumption at the substations between the two considered driving strategies for the commercial running time in the Puertollano–Ciudad Real section. Results are shown for different values of the engine efficiency and the linear resistance of the catenary.



Once again, the lower the linear resistance is, the lower the difference in energy consumption is. Moreover, the difference in consumption between two speed profiles produced by using the two considered strategies is higher when the train engine is less efficient. The difference in energy consumption between the considered driving strategies for the commercial running time is between 9.70% and 12.55%.

Figure 12 shows the energy consumption measured at the pantograph for the commercial time speed profiles. As can be observed, for an engine efficiency of 85%, the eco-driving strategy can be used for achieving a 10.8% energy saving with respect to the standard driving strategy, while for an engine efficiency of 97% the energy saving between the two strategies is 8.3%.

**Figure 12.** Difference in energy consumption at the pantograph (in %) between the considered driving strategies for different values of the engine traction and brake efficiency in Puertollano–Ciudad Real section.

The difference in energy consumption between the considered driving strategies is analyzed in Figure 13 for the considered linear resistances of the catenary and engine efficiencies and for different running times.

The difference in energy consumption grows up to a maximum close to the commercial running time. The commercial running time (13 min, that is 780 s) is highlighted by means of a grey dashed line. The more-or-less constant scenario in Figure 7 is at this point concave; this indicates that the shape of the difference in energy consumption depends on the height profile of the track. This time, the difference in energy consumption is larger when considering different values of the linear resistivity, as there is a larger difference between those values. Again, the difference in consumption seems to decrease linearly with the engine efficiency.

**Figure 13.** Difference in energy consumption (in kWh) between the considered driving strategies at the electrical substations for the Puertollano–Ciudad Real section for different running times. Different values of the catenary linear resistance *r*ˆ and different values of the engine efficiency are considered.

In Table 4 the command vectors of the two driving speed profiles that satisfy the commercial running time are presented. The eco-driving final coasting phase is approximately 10 km long and its cruise speed is 17.4 km/h higher. Figures 14 and 15 show the commercial speed profiles obtained for the eco-driving and standard driving strategies, respectively. The train speed is represented in blue, the speed limits are represented by red lines and the track altitude is shown in green. The holding speed is represented by an orange dashed line.

**Eco-Driving Strategy Standard Driving Strategy** Cruise speed (km/h) 236.5 219.1 Final position of the cruise phase (km) 287.1 Braking curve

**Table 4.** Command vectors for the commercial time speed profiles obtained with the two considered driving strategies for the Puertollano–Ciudad Real section.

**Figure 14.** Eco-driving speed profile for the commercial time in the Puertollano–Ciudad Real line section.

**Figure 15.** Standard driving speed profile for the commercial time in the Puertollano–Ciudad Real line section.

In this case, the train coasts approximately in 25% of the total length of the line section. In comparison to Figures 8 and 9, in this section the role of the final coasting phase is more important. This explains the fact that the difference in energy consumption between the considered driving strategies in percentage is much higher for the Puertollano–Ciudad Real section.

### **7. Discussion**

The studies developed in this paper have been carried out with the objective of assessing the effectiveness of eco-driving under different electrical scenarios and to answer the question about whether or not eco-driving is useful in high-receptivity networks. The results obtained suggest that the answer to the question is that it is useful to apply eco-driving. Besides, it has to be taken into account that the value of energy saving because of efficient driving will depend on the efficiency of the electrical chain (catenary loses and motor efficiency) and on the running time.

The cases studied in Section 6 are two cases of an alternate current power supply system with high receptivity. These systems are, by nature, much more effective using regenerate energy than direct current systems. In alternate current systems, the energy generated by train braking can be used by other trains or sent back to the distribution network for other uses. However, there is part of the regenerated energy that will be lost anyway because of the catenary losses and the motor's efficiency. Saving part of the lost energy is the main benefit of eco-driving, and the results support this. It has been shown that, the lower the energy lost in the motor and catenary, the lower the difference between efficient non-efficient driving.

The results obtained in Puertollano–Ciudad Real show energy savings produced by eco-driving according to the typical values between 10% and 15% in the literature [68]. On the other hand, the results obtained in Calatayud–Zaragoza present values that are further reduced. One reason to explain this is the track profile. Although the average grade in both cases is downhill, the difference in altitude between the end and start of the journey is higher in Calatayud–Zaragoza. The presence of steeper downhills leads to lower energy consumption results and lower differences between different speed profiles.

Catenary losses can also explain why the energy reduction figures obtained by the eco-driving strategy in the Calatayud–Zaragoza stretch are lower than those obtained in the Puertollano–Ciudad Real stretch (the 2 × 25 kV system is more efficient than the 1 × 25 kV system because of lower rate of power losses).

The influence of the commercial running time cannot be neglected. In both cases, it can be observed that there are three scenarios of energy saving due to eco-driving depending on the running time. Close to the flat-out running time, the energy saving that can be obtained by efficient driving is zero because there is no possibility of modifying the driving to obtain the minimum running time. From there, there is a small interval of running time wherein the energy saving provided by eco-driving increases as the running time increases. After that, there is a zone of running time values in which the energy saving rate is approximately constant. This zone typically contains the commercial running time for the journey. Finally, for the largest running times, the difference in energy consumption between efficient and non-efficient driving is gradually reduced.

Therefore, a second order reason for the reduced energy saving figures in Calatayud–Zaragoza is that the commercial running time of this journey is higher than usual and it is located in the running time interval in which the energy savings provided by eco-driving are reduced.

Despite this, the minimum energy saving provided by eco-driving is relevant (2.5%) in a scenario of 97% motor efficiency. This motor efficiency is very high compared with its typical values, which are between 85% and 90%. Therefore, taking into account the typical motor efficiency, the minimum energy saving provided by eco-driving can reach more that 3.5%.

With these results, it can be said that high network receptivity to braking energy can mitigate the effect of eco-driving, but it is still significant.

Obviously, results can vary in a field-test because of simulation error. However, as explained in Section 2, the value of simulation error in energy consumption is 0.4%. This error can slightly quantitatively modify the results obtained but cannot qualitatively modify them and the trends obtained.
