## 3 High-Voltage Battery and Air Cooling System Modeling
### Throughput-Based Battery Capacity Fade Model
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline C-rate index for & \(c\) & [0.5; 2; 4; 6; 8; 10; 12; 14; 16; 18; 20] \\ pre-exponential factor & & [31,630; 21,681; 17,307; 12,934; \\ Pre-exponential factor as & \(B(c)\) & 13,512; 15,512; 12,099; 11,380; 13,656; \\ function of C-rate & & 16,342; 14,599] \\ Activation factor & \(A_{f}(c)\) & 3814.68\(-\)44.56\(\cdot c\) \\ Power-law factor & \(z\) & 0.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Battery ageing parameters for A123 26650 cell [23]. Battery High Temperature Sensitive Optimization Based HEV Energy and Thermal Management Calibration  
The optimization-driven calibration of the HEV on-line energy and thermal management strategies will be discussed in this section. Indeed, an adaptation of HEV energy and thermal management according to the ambient temperature value may allow finding the optimal trade-off between an enhanced fuel economy and satisfactory high-voltage battery lifetime. Table 4 summarizes variables and the calibration objective function for the optimization-based calibration of the HEV energy and thermal management. The calibration of HEV energy and thermal management is desired to be optimal and it must bring into play both \(T_{bat-cooling,ON}\) and \(T_{bat-cooling,OFF}\). Values for these parameters will depend both on the ambient temperature \(T_{amb}\) and on whether the _HVAC_ system is in operation or not. Moreover, adapting the rule-based HEV EMS according to the battery temperature and predicted ageing rate may improve the accuracy and effectiveness of the vehicle control system. This work particularly considers varying the battery \(SOC\) which determines the transition from the charge depleting in pure electric (\(EV\)) mode to blended charge-depleting (HEV) hybrid as a function of \(T_{amb}\). This value of battery \(SOC\) threshold was initially set to 0.30 for the baseline HEV energy management approach described in Section 2.1 and now it is indicated as \(SOC_{EV,OFF}\). Figure 6 illustrates the flowchart for simulating the HEV in a given driving mission according to the implemented energy and thermal management approaches. The three parameters that need calibration in this framework (\(T_{bat-cooling,ON}\), \(T_{bat-cooling,OFF}\), \(SOC_{EV,OFF}\)) are highlighted in Figure 6. When the HEV driver starts the driving mission, the battery temperature was assumed to be equal to the ambient temperature. The _HVAC_ system state was selected at the beginning of the driving mission, and it was considered to be constant throughout the journey. When the _HVAC_ system was set to be in operation, the cabin temperature was assumed to be 20 \({}^{\circ}\)C from the first instant of the driving mission while neglecting transient thermal phenomena in the cabin. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Parameter** & **Symbol** & **Bounds** \\ \hline \multirow{4}{*}{Calibration variables} & Upper battery pack & \(T_{bat-cooling,ON}\) & \multirow{2}{*}{[\(T_{amb}\)-40\({}^{\circ}\)]} \\  & temperature threshold for & & \\  & cooling system activation & & \\  & Lower battery pack & & \\  & temperature threshold for & & \\  & cooling system de-activation & & \\  & Lower battery \(SOC\) threshold & & \\  & for pure electric to hybrid & & \\  & electric operation transition & & \\  & Operative cost associated to & & \\  & function & the entire plug-in HEV & \(J_{calibration}\) (31) & \\  & lifetime & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Variables and objective function for the optimization based HEV energy and thermal management calibration. At each time instant, the terms impacting on the instantaneous battery power \(P_{bat}\) used in Equation (15) were updated.