## 1 Introduction  
With the growing concern about greenhouse gases and air pollution on the environment, hybrid electric vehicles (HEV) are being considered as one of the alternatives towards clean mobility [1]. One of the advantages of vehicle hybridization is that the combination of internal combustion engines (ICE) and electric motors (EM) provides an additional degree of freedom to the powertrain that allows meeting the driver power demands more efficiently. In addition, the use of batteries and EMs allows energy recovery during braking [2]. Among mobility alternatives, urban buses present a high potential of electrification (including hybridization) due to their high mileage covered, known routes and strict requirements for air pollution control in cities. The previous reasons are leading European cities and manufacturers who invest efforts in electrified powertrains for buses [3] such as pure electric, fuel cell or HEV and in other sustainable alternatives such as hydrogen engines. HEV performance depends on powertrain configuration, and accordingly, there is an extensive literature discussing HEV powertrain architectures (parallel, series, power split, etc.) and sizing [4; 5; 6]. However, the additional degree of freedom of HEVs, i.e., the power split between battery and fuel tank, addressed by energy management strategies (EMS), also plays a key role in vehicle performance. For this reason, several approaches have been investigated in the literature. An extensive review of HEV energy management can be found in [7]. As expected, the literature points out that the performance and optimal EMS are closely related with driving factors such as driver behavior, road slope, traffic and weather conditions that can influence the energy distribution in HEV [8]. Another important aspect that can be extracted from the available literature is that while the EMS can be addressed by the application of heuristic techniques, there is room for optimal control theory, leading the EMSs to specifically address the minimization of the vehicle energy consumption [7]. In this group, some studies are focused on offline optimization, where the EMS of a given HEV in a pre-defined driving cycle is optimized. Among the techniques applied for offline optimization, dynamic programming (DP) [9] and methods based on Pontryagin's minimum principle (PMP) [10; 11; 12] are the most widespread. DP is widely applied in offline optimization problems; this method consists of subdividing the nonlinear dynamic problem into subproblems in a discretized time, and then proceeding backwards generating a cost-to-go function depending on the state value and control action chosen. Then, the optimal cost-to-go is calculated for every sample time and stored in a matrix. When the entire problem has been examined, the optimal solution is obtained by tracking the path with the lowest cost-to-go. The major issue of the previous techniques is the requirement of a priori knowledge of the driving cycle, something that only happens in homologological cycles or very particular vehicle applications. For this reason, DP is widely used as an optimal benchmark solution to compare with the results of other approaches, as well as to extract control parameters for other control strategies [13]. Several approaches have been investigated in the literature in order to overcome the driving cycle dependency to develop methods that can be applied for control purposes, leading to near-optimal results. Most of them are based on the equivalent consumption minimization strategy (ECMS), that is based on applying, in any time-step, the control action that minimizes a weighted function of the energy consumption from the fuel tank and the batteries [14]. Among them, adaptive-ECMS (A-ECMS) [15; 16] and adaptive-PMP (A-PMP) [17] provide online implementable solutions that consist in updates to the co-state, or the equivalence factor in the case of ECMS as driving conditions change, making possible to reach near-optimal solutions [7].