## 3 High-Voltage Battery and Air Cooling System Modeling
### HEV Fuel Economy and Battery Lifetime over Retained Driving Mix  
An appropriate objective function needs definition to guide the optimization-based calibration of the HEV energy and thermal management parameters, i.e., \(T_{bat-cooling,ON}\), \(T_{bat-cooling,OFF}\), and \(SOC_{EV,OFF}\). The minimization of HEV operative costs was considered in this work by simulating a realistic driving mix. Eight driving missions were retained including four standard drive cycles such as the worldwide harmonized light-vehicle test procedure (WLTP), the federal test procedure (FTP75), the RTS aggressive cycle (RTS95), and the highway federal test procedure (HWFET). Moreover, four real-world driving missions recorded by the authors by means of the global positioning system (GPS) were considered including up-hill (RWC01), 2.7-h highway (RWC02), down-mountain (RWC03), and downhill (RWC04) driving conditions. Time series for vehicle speed and net road altitude over time for all the eight driving missions retained are illustrated in Figure 11 in Appendix A, while Table 5 reports driving statistics. The overall vehicle lifetime in terms of mileage was divided while considering the eight driving missions steadily repeated. Each journey was associated with a coefficient \(\alpha_{freq}\) in Table 5 that represents the percentage of vehicle lifetime which was assumed to be traveled in the corresponding driving mission.  
The predicted HEV fuel economy, HEV electrical energy economy, and battery lifetime over the entire vehicle lifetime considering the introduced driving mix could be expressed as:  
\[Fuel_{avg}=\sum_{k=1}^{8}Fuel_{k}\cdot\alpha_{freq,k} \tag{26}\]  
\[Electricity_{avg}=\sum_{k=1}^{8}Electricity_{k}\cdot\alpha_{freq,k} \tag{27}\]  
\[batt_{life,avg}=\sum_{k=1}^{8}\frac{L_{cycle,k}}{SOH_{0}-SOH_{end,k}}\cdot\alpha _{freq,k} \tag{28}\]  
where \(Fuel_{k}\) and \(Fuel_{avg}\) are the HEV fuel economy values expressed in liters per 100 km related to the given driving mission \(k\) and the entire vehicle lifetime, respectively. \(Electricity_{k}\) and \(Electricity_{avg}\) are the plug-in HEV electrical energy economy values expressed in kilowatt-hours per 100 km related to the given driving mission \(k\) and the entire vehicle lifetime, respectively. \(L_{cycle,k}\) stand for the mileage in kilometers of the given driving mission \(k\), while \(SOH_{end,k}\) is the battery \(SOH\) at the end of the driving mission \(k\) according to the HEV simulated following the discussed energy and thermal management strategies.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Driving** & **WLTP** & **FTP75** & **RTS95** & **HWFET** & \begin{tabular}{c} **RWC01–** \\ **Uphill** \\ \end{tabular} & \begin{tabular}{c} **RWC02–Long** \\ **Highway** \\ \end{tabular} & \
\begin{tabular}{c} **RWC03–Down** \\ **Mountain** \\ \end{tabular} &
\begin{tabular}{c} **RWC04–** \\ **Downhill** \\ \end{tabular} \\ \hline Length \(L_{cycle}\) [km] & 23.3 & 17.8 & 12.9 & 16.5 & 17.8 & 296 & 27.4 & 16.7 \\ Time [s] & 1800 & 1878 & 887 & 765 & 1031 & 9792 & 2345 & 1123 \\ Max speed [km/h] & 131.3 & 91.2 & 134.4 & 96.3 & 112.7 & 135.4 & 84.9 & 103 \\ Average speed [km/h] & 46.5 & 34.1 & 52.5 & 77.6 & 62.1 & 108.8 & 42 & 53.5 \\ Average acceleration & 0.4 & 0.5 & 0.8 & 0.2 & 0.5 & 0.2 & 0.5 & 0.6 \\ Average deceleration & \(-\)0.4 & \(-\)0.6 & \(-\)0.8 & \(-\)0.2 & \(-\)0.6 & \(-\)0.2 & \(-\)0.6 & \(-\)0.6 \\ Max altitude—min altitude [m] & 0 & 0 & 0 & 0 & 278 & 523 & 682 & 225 \\ Final altitude— initial altitude [m] & 0 & 0 & 0 & 0 & 235 & 341 & \(-\)633 & \(-\)148 \\ Journey frequency \(\alpha_{freq}\) [\%] & 8.33 & 16.67 & 16.67 & 16.67 & 8.33 & 8.33 & 8.33 & 16.67 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Statistics for retained driving missions.