## 2 Model and Methodology
### Estimation of the Residual Driving Range for a Battery Electric Vehicle
Within the two different scenarios, the reduction in battery capacity over time differs, as shown in Figure 9. According to the data extracted from [26,27], battery capacity showed a nearly flat trend in the case of DoD80 (blue curve), whereas a steeper reduction was observed for DoD100 (red curve), as the termination condition was found at 400 cycles. In the case of DoD100, a peak occurred in the early stages of battery life, and was related to over-discharging phenomena which were responsible for a very fast capacity fading (i.e., roughly \(-20\%\) in 200 cycles). To clarify, the termination of both curves was correlated to the number of cycles at which 20% of the battery capacity was depleted. Using the same approach as for \(\lambda_{t_{bp}}\), the values of \(\lambda_{SoH}\) were considered in the range (0.8, 1). Once the available capacity of the battery was determined, a correct evaluation of the battery SoC was performed by producing a good estimation of the current flowing into the battery (see Equation (20)). To achieve this, the power required by the road (i.e., the traction/braking power) must be coupled with any further power demand. For a BEV, auxiliary loads assume a relevant role and therefore their inclusion in the model  
Figure 8: Trend of the battery life cycles as a function of the depth of discharge. Figure 7: Capacity variation as a function of the battery pack temperature. is needed [12]. Specifically, the larger amount of auxiliary power can be referred to cabin heating and cooling operations (i.e., HVAC). Within the present model, a reversible vapor compressor-heat pump (VC-HP) layout was considered as the vehicle's HVAC system. This choice was made due to the higher reliability shown by the VC-HP relative to other systems (e.g., evaporative cooling integrated thermal management) [28], which has increasingly spread throughout the automotive market [29]. According to the R32 coolant [29] and based upon the model suggested in [30], the curve shown in Figure 10 was built considering HVAC power consumption \(P_{HVAC}\) as a function of the ambient temperature, assuming the optimal cabin temperature is 24 \({}^{\circ}\)C (\(T_{cabin}\)). The model equations are:  
\[P_{HVAC}=\left\{ \begin{array}{cl}0&\text{turned off}\\ P_{heat}(T_{cabin}-T_{air})&[W]&\text{heating mode}\\ P_{cool}(T_{air}-T_{cabin})&\text{cooling mode}\end{array} \right. \tag{23}\]  
Figure 10: HVAC consumption in cooling and heating modes. Figure 9: Capacity fading in terms of battery depth of discharge equal to 80% (DoD80) and 100% (DoD100). As highlighted in Figure 10, power consumption has always been considered as a positive value. A baseline of 50 W was used to account for the fans' load. Moreover, human metabolic heat was taken into account, given the relatively small volume of an L-7 quadriceps cabin. The formulation of the metabolic power is:  
\[Q_{met}=\dot{met}*BSA= \begin{cases}147\;[W]&\text{driver}\\ 95\;[W]&\text{passenger}\end{cases}  \tag{24}\]  
in which \(Q_{met}\) is the metabolic power, _met_ is the metabolic rate and \(BSA\) is the body surface area. To clarify, a fixed BSA based on the European average value (1.73 m\({}^{2}\)) [31] was considered whereas two different metabolic rates accounted for the driver and the passenger (85 W/m\({}^{2}\) for the driver and 55 W/m\({}^{2}\) for each passenger) [32]. The heat produced by human bodies was added when the system was under cooling conditions, whereas it was subtracted under heating mode. No other thermal loads were considered in the study (e.g., sun irradiation) since they represent a minor impact on overall HVAC consumption [30]. As a final contribution, the sum of the loads coming from any other vehicle auxiliary (e.g., headlights, electric wipers, electric power window, etc.)