## 6 Results and Discussion
In the same way that the rollout strategy leads to a similar evolution of \(SoE\) as DP, the distribution of ICE operating points also approaches the DP solution. Figure 8: Comparison between the \(SoE\) and fuel consumption evolution obtained with DP assuming a priori knowledge of the route (optimal solution), the proposed online applicable method (roll out) and the standard online applicable method ECMS calibrated with a single loop. Figure 9: Probability distribution of the deviation between the optimal \(SoE\) (DP) and that obtained by the proposed online applicable method (rollout) and the standard online applicable method ECMS calibrated with a single loop. On the basis of the precedent discussion, the one-step look-ahead rollout outperforms the standard ECMS. This is justified by the fact that although both strategies have the same calibration requirements, i.e., both need the optimization of a reference cycle, ECMS lumps all the information from the reference cycle in a single parameter \(s\), while the rollout exploits all the cost-to-go from the reference cycle. In this sense, the control policy provided by the ECMS approach is based on a single constant without dependence (excepting the \(SoE\) correction) of the system state. On the contrary, the rollout is richer and considers the impact of the system states (\(SoE\) and \(ICE_{on}\)) and vehicle position in the control policy. Of course, this is performed at the expense of higher storage requirements, despite the dependence on the size of the problem, in the case at hand \(\mathcal{J}^{*}(x_{j},s_{l})_{ref}\) has size \(51\times 2\times 152\), representing a discretization in the \(SoE\) of 51 points between 0.3 and 0.7, considering two possible states for \(ICE_{on}\) (on and off) and a 100 m discretization of the 15.1 km loop distance. Regarding the applicability for control purposes, the standard ECMS and look-ahead rollout run faster than real time, and 875 and 409 times faster than real time, respectively, for the driving cycle of Figure 8 in a standard laptop (Intel(R) Core(TM) i7-8550U CPU @ 1.80 GHz 1.99 GHz with 8.00 GB RAM). The rollout algorithm requires more computation resources but is still far from the limit. Moreover, neither the rollout nor the ECMS require information about future conditions, so they have potential to be applied in control purposes. On the contrary, DP, despite running 390 times faster than real time, depends on a priori knowledge of the driving cycle, so cannot be used for control applications. Although the proposed rollout algorithm performs better than the standard ECMS, there is a noticeable gap between its performance and DP. Expressions (16) and (20) used for DP and rollout, respectively, highlight the two main simplifications that are the reason of such performance differences:  
* The integral term Equation (16) has a time interval (\(\delta t\)) of a single time-step (one-step look ahead). * The optimal cost to go from future state (\(x(t+\delta t)\)) to the end (\(\mathcal{J}^{*}(x(t+\delta t),t+\delta t)\)) is approximated by the cost-to-go computed in the reference loop. The first simplification is important to allow online application since power demands beyond the next time-step usually cannot be assumed. However, with the advent of vehicle to vehicle (V2V) and vehicle to infrastructure (V2I) technologies, one can expect an accurate extension of the look-ahead horizon. To analyze the impact of having information about future driving conditions on the proposed control algorithm, Figure 11 shows the fuel consumption and the deviation from the optimal \(SoE\) evolution of the rollout algorithm assuming perfect driving cycle knowledge in different horizons. One can observe that increasing the look-ahead horizon allows approaching the optimal \(SoE\) evolution and fuel consumption, but there is still a gap that cannot be closed. For this reason, look-ahead horizons beyond 250 s do not show a substantial improvement in the fuel consumption. The reason for that is the second simplification hypothesis.