## 3 High-Voltage Battery and Air Cooling System Modeling
### Throughput-Based Battery Capacity Fade Model  
A largely employed throughput-based capacity fade model was used to evaluate the high-voltage battery cyclic ageing. This model was calibrated for A123 26650 cells and it is based on the concept of battery charge throughput (Ah-throughput) [14]. Here, the Ah-throughput (\(Q_{EOL}\)) processed throughout the battery lifetime included the energy charged from the grid [16]. \(Q_{EOL}\) was associated to the overall quantity of charge (in ampere-hours) that a battery is able to process before reaching its end of life and it is calculable as:  
\[Q_{EOL}=\int_{0}^{t_{EOL}}\lvert I_{bat}\rvert dt \tag{20}\]  
where \(t_{EOL}\) is the time instant in which \(EOL\) occurs, which depends on the battery operating conditions. The \(EOL\) is considered to happen when the percentage value of battery nominal capacity faded (indicated by \(C_{fade,\%}\)) equals 20% (i.e., \(C_{fade,\%}=20\)). When the battery operates under constant conditions, \(Q_{EOL}\) and \(C_{fade,\%}\) could be evaluated using the equation proposed in [14] for A123 26650 cells. This formulation was inspired by the Arrhenius' gas equation [7] and it can only be employed for battery temperature ranging between 15 \({}^{\circ}\)C and 60 \({}^{\circ}\)C:  
\[C_{fade,\%}=B\cdot exp\left(-\frac{A_{f}}{T_{bat}}\right)\left(\frac{Q_{EOL, const}}{N_{p}\cdot 1\ \mathrm{Ah}}\right)^{z} \tag{21}\]  
where battery Ah-throughput was indicated as \(Q_{EOL,const}\) to specify that it refers to constant operating conditions. \(B\), \(A_{f}\), and \(z\) respectively indicate the pre-exponential, the activation, and the power-law factors. Both \(B\) and \(A_{f}\) are a function of the cell C-rate \(c\)[31]. Since these parameters refer to a single cell, \(Q_{EOL,const}\) was divided by the number of parallel branches \(N_{p}\) in order to scale to the whole battery pack. Table 3 reports the values of parameters used in the considered battery ageing model. The values of pre-exponential factor \(B\) as function of the C-rate were determined by conducting an ageing experimental campaign [23]. As \(B\) and \(A_{f}\) vary according to the battery C-rate, \(Q_{EOL,const}\) depends on both the operating conditions \(T_{bat}\) and C-rate, as illustrated in Figure 5. The range of the battery pack C-rate allowed as function of the considered plug-in HEV characteristics (i.e., battery pack capacity, \(MGP0\) maximum power, and \(MGP4\) maximum power) is highlighted in Figure 5. The larger capacity of a plug-in HEV battery slightly reduces the overall values of the C-rate, and this implies that the variation in \(Q_{EOL,const}\) might largely relate to thermal characteristics. Looking again at Figure 5, thermal phenomena are forecasted to have a remarkable influence, even when the battery temperature is contained within the ideal range of 15 \({}^{\circ}\)C to 35 \({}^{\circ}\)C.  
After calculating \(Q_{EOL,const}\) using Equation (21), the overall quantity of charge \(Q_{const}\) supplied by the battery under fixed values of C-rate and temperature became known. Then, it became possible to evaluate the current battery ageing state by comparing \(Q_{const}\) with \(Q_{EOL,const}\). However, since batteries employed in HEVs are characterized by varying operation over time, the contribution of each combination of the C-rate and \(T_{bat}\) towards the achievement of \(Q_{EOL}\) over time could be evaluated by integrating the instantaneous \(SOH\) variation:  
\[SOH=SOH_{0}-\int\limits_{0}^{t}\frac{|I_{bat}|}{Q_{EOL,const}(c,T_{bat})}dt \tag{22}\]  
where \(SOH_{0}\) is the initial state-of-health of the battery and it is equal to 1 at the beginning of battery life. Predicting the battery \(SOH\) variation over time as a function of C-rate and temperature is possible in this way thanks to the implemented numerical ageing model. Figure 5: Allowed lifetime Ah-throughput predicted by the numerical ageing model as a function of the cell temperature and C-rate.