## 1 Introduction  
Regarding the increasingly serious issues of energy shortage and environmental pollution, attention from governments, research institutions and automobile companies is shifting from traditional fuel vehicles to new innovative solutions [1, 2, 3]. To reduce air pollution and emissions of greenhouse gases, many governments incentivize the production of zero emissions vehicles (ZEVs), such as battery electric vehicles (BEVs) and fuel cell electric vehicles (FCEVs). The German Parliament has recently decided to ban internal combustion engine vehicles (ICEVs) by 2030 [4]. Many other countries have announced that they will proceed with a similar policy between 2025 and 2050 [5]. However, most of these bans do not include hybrid electric vehicles (HEVs) and plug-in hybrid electric vehicles (PHEVs), which include ICEs as a power source. While the shift from ICEVs to ZEVs has already started, it may take decades for customers to select ZEVs as a cost effective and convenient choice. Moreover, batteries still require decisive improvements to be considered as unique energy sources, vehicle cost should reduce and infrastructure should be globally adapted to the new changes. In the meantime, thermal efficiency of ICEs is expected to further improve and HEVs and PHEVs are expected to play a significant role. Hybrid electric vehicles represent the best tradeoff between traditional fuel vehicles and pure electric vehicles (EVs), thus becoming the perfect transitional stage. The fuel efficiency of HEV has traditionally improved through new developments in aerodynamics, engine technology, light-weight materials, and innovative concepts for powertrain component design [2; 6]. HEV uses two power sources: an internal combustion engine and an electric battery; thus, the fuel economy varies substantially depending on the power source used to meet the vehicle request. The introduction of hybrid powertrain has allowed a substantial step forward in the direction of energy saving. Indeed, the presence of one or more electric motors does not only provides a significant benefit in terms of efficiency performance by optimizing the engine operating point, but it also introduces the idea of regenerating part of the energy dissipated during braking [7; 8]. Hybrid powertrains are generally divided into three categories: series, parallel, and series-parallel (power-split), as defined by [PERSON] [9]. In series architecture, the engine output power is first converted into electric power which can be used to either charges the battery or to propel the wheels through the same electric motor [10]. The parallel configuration allows both the engine and the electric motor to deliver power in parallel to drive the wheels [11]. The series-parallel incorporates the characteristics of both the series and parallel configurations, but including an additional mechanical link compared with the series architecture and an additional generator compared with the parallel architecture. The power-split category represents the most attractive solution due to its multimode and flexible nature which guarantees an adaptive solution to every vehicle working condition and fully realizes the powertrain potential to achieve both better fuel economy and improved drivability [12; 13]. The major issue of series-parallel powertrain is represented by the multiple energy sources that require a complex power flow management and an optimized control strategy. Usually, the procedure to assess this problem consists of (1) an identification step to search for the whole set of powertrain modes, through the activation of dedicated clutches and brakes, and (2) the optimization of the working condition for each powertrain actuator. An automated methodology to categorize and identify all the possible configurations of a power-split powertrain with more than two Planetary Gear Systems (FGSs) is proposed by [PERSON] et al. [14; 15]. [PERSON] et al.