## 2 Model and Methodology
### Vehicle Model
The equations used for the estimation of the rotational speed along the driveline are:  
\[\omega_{w}=\frac{v}{r_{d}}\ [rpm] \tag{8}\]  
\[F_{res}=F_{rolling}+F_{zero}+F_{iner}+F_{grade}\ [N] \tag{1}\]0  
\[F_{res}=F_{rolling}+F_{zero}+F_{iner}+F_{grade}\ [N] \tag{1}\]1  
Once \(T_{g}\) and \(\omega_{g}\) were evaluated, the motor torque \(T_{m}\) could be computed. In the present paper, a positive torque was considered as representative of traction phases, whereas a negative torque was considered as representative of braking phases. According to traction scenarios, the torque demand of the MG was considered to be equal to the torque at the gearbox level. On the other hand, a reduced MG torque demand was considered during braking conditions. In such cases, the negative torque request is split on the two axles. On the non-driven axle, the braking action is achieved only by the effect of mechanical brakes, whereas it is given by the electric motor working as a generator (i.e., regenerative  
\[F_{res}=F_{rolling}+F_{zero}+F_{iner}+F_{grade}\ [N] \tag{1}\]2 High-resolution matrix for transmission ratio evaluation. braking operating mode) on the driven axle. Therefore, the required motor torque can be computed as:  
\[T_{m}= \begin{cases}T_{g}&for\ T_{g}>0\\ T_{g}\cdot\delta&for\ T_{g}<0\end{cases}  \tag{11}\]  
where \(\delta\) is the percentage of weight on the driving axle. Hence, the mechanical power and the electric power can be evaluated as:  
\[P_{m,\ mech}=T_{m}\cdot\omega_{m}\ [W] \tag{12}\]  
\[P_{m,\ el}=\left\{ \begin{array}{c}P_{m,mech}\cdot\eta_{m}\\ \frac{P_{m,mech}}{\eta_{m}}\end{array} \right.\ [W] \tag{13}\]  
in which \(\eta_{m}\) represents the motor efficiency, \(P_{m,mech}\) is the motor mechanical power and \(P_{m,el}\) is the motor electric power. The latter was calculated by multiplying or dividing the mechanical power by the motor efficiency according to braking or traction conditions, respectively. The motor efficiency was calculated according to a downscaled version of the efficiency map of a reference MG made for a passenger car application. The original map is shown in Figure 3. The efficiency map allowed for evaluation of the losses at the motor level to be considered based upon its torque \(T_{m}\) and rotational speed \(\omega_{m}\). The efficiency assumed positive values when in the range of \(-277\) Nm and \(370\) Nm, whereas it was null outside this interval. The rotational speed was limited to 15,000 rpm. The downscaling operation was performed according to a linear which started from the definition of a scaling factor \(s_{f}\) between the actual and the reference motors:  
\[s_{f}=\frac{P_{m,mech}}{P_{m,mech}^{*}} \tag{14}\]  
Figure 3: Efficiency map of the reference motor-generator. in which \(P_{m,mech}\) and \(P_{m,mech}^{*}\) represent the mechanical peak power of the downsized and the reference motors, respectively. Assuming the motor speed range to be constant, the motor torque can be defined as:  
\[\frac{T_{m}\cdot\omega_{m}}{T_{m}^{*}\cdot\omega_{m}^{*}}=\frac{T_{m}}{T_{m}^{* }}=s_{f} \tag{15}\]  
Finally, the motor mass and its inertia were re-calculated following the same approach. At the end of the downscaling procedure, the power and torque limit curves were obtained, as shown in Figure 4. A distinction between peak and continuous MG outputs was involved in the study to include the effects of the BTMS control on the battery. In fact, a possible action of the BTMS is to reduce battery power after peak power/torque values have been demanded for a given amount of time. Such a condition is typically involved in the control of the BTMS for electrified vehicles to prevent overheating phenomena. To clarify, non-complex strategies are typically involved for cooling the battery pack of very light vehicles (e.g., L7). A common solution is to embed a passive air-cooling system. Nevertheless, given the target of this research, the authors have chosen to avoid such an assumption and include the possibility of the BTMS to cut the current and power of the electric sources.