## 2 Model and Methodology
### Vehicle Model  
A schematic representation of the BEV model developed for the present study is shown in Figure 1. Vehicle velocity was used as the only input signal to compute the traction or braking power required for the vehicle. A backward-facing approach was therefore employed for the analyses in which the power demanded by the road is back-propagated throughout the driveline, starting from the wheels and reaching the battery. Consistently, with traction or braking (i.e., battery regeneration) events, energy was considered to be flowing out of (red arrow) or into (green arrow) the battery, respectively. Accordingly, the resistant forces (and powers) were defined for each powertrain component (i.e., wheels, frontal final drive, gearbox) until the motor/generator (MG). The latter was responsible for providing the vehicle with the proper propulsion through the front axle (i.e., driven axle). Finally, the electric power required by the MG was considered as the real power demand of the battery. The resistant force \(F_{res}\) experienced by the vehicle can be calculated as the sum of rolling \(F_{rol}\), aerodynamic (i.e., air drag) \(F_{zero}\), inertia \(F_{in}\) and grade resistances \(F_{gr}\). The complete formula is:  
\[F_{res}=F_{rolling}+F_{zero}+F_{iner}+F_{grade}\ [N] \tag{1}\]  
\[F_{res}=m\cdot g\cdot K_{r}\cdot cos\alpha+\frac{1}{2}\cdot\rho\cdot v^{2}\cdot A _{f}\cdot C_{x}+M_{at}\cdot a\cdot v+m\cdot g\cdot sin\alpha\ [N] \tag{2}\]  
where \(m\) is the vehicle mass, \(g\) is the gravitational acceleration, \(K_{r}\) is the rolling resistance coefficient, \(\alpha\) is the road slope, \(\rho\) is the air density, \(v\) is the vehicle velocity, \(A_{f}\) is the vehicle frontal area, \(C_{x}\) is the drag coefficient and \(a\) is the vehicle acceleration. The apparent translating mass \(M_{at}\) is calculated as:  
\[M_{at}=m+\sum_{i}\frac{I_{i}\cdot\tau_{i}^{2}}{r_{d}^{2}}\ [kg] \tag{3}\]  
with \(r_{d}\) as the rolling radius, \(i\) representing the i-th rotating component in the vehicle (e.g., final drive), and \(I\) and \(\tau\) representing the inertia coefficient and the transmission ratio of the i-th rotating component, respectively. Once the resistant force is defined, the resistant power can be calculated as:  
\[P_{res}=F_{res}\cdot v\ [W] \tag{4}\]  
Moreover, the value of the resistant force can be used to estimate the different torque requests at any stage of the driveline, starting from the wheels turning in reverse to the MG. The torques at the wheels \(T_{w}\), final drive \(T_{fd}\) and gearbox \(T_{g}\) can be estimated as follows:  
\[T_{w}=F_{res}\cdot r_{d}\ \ [Nm] \tag{5}\]  
\[T_{fd}=\frac{T_{w}}{\eta_{fd}\cdot\tau_{fd}}\ [Nm] \tag{6}\]  
\[T_{g}=\frac{T_{fd}}{\eta_{g}\cdot\tau_{g}}\ [Nm] \tag{7}\]  
Figure 1: Scheme of the BEV model with the main energy flows. where \(r_{d}\) is wheel radius, \(\tau_{fd}\) and \(\eta_{fd}\) are the transmission ratio and the efficiency of the final drive, respectively, and \(\tau_{g}\) and \(\eta_{g}\) are the transmission ratio and the efficiency of the gearbox, respectively. Both the \(\tau_{g}\) and \(\eta_{g}\) depend upon the gearbox model involved in the study. In this paper, a continuously variable transmission (CVT) has been modelled and the transmission ratios were estimated as a function of the vehicle speed and resistant power. The actual values of the CVT transmission ratio were derived from signals recorded during the experimental driving missions. Specifically, starting from the vehicle speed and electric motor traces, the look-up table (LUT) shown in Figure 2 was generated. Of note, the considered range for the transmission ratio was between 2 and 4.5. The computation of the rotational speed at the wheels \(\omega_{w}\), final drive \(\omega_{fd}\), gearbox \(\omega_{g}\) and motor \(\omega_{m}\) is important for the evaluation of the power requested by the MG and hence to the battery.