# Battery Sizing for Mild P2 HEVs Considering the Battery Pack Thermal Limitations
## 2 Vehicle Performance Modelling
### Longitudinal Vehicle Dynamics  
In this subsystem, the required traction force to overcome the resisting forces (i.e., aerodynamic and rolling resistance forces) is calculated as:  
\[F_{w}=M\cdot dv/dt+M\cdot g\cdot f_{r}+\frac{1}{2}\cdot\rho_{air}\cdot C_{x} \cdot A_{f}\cdot v^{2} \tag{1}\]  
where \(M\) is the vehicle mass; \(v\) and \(dv/dt\) are longitudinal speed and acceleration; \(f_{r}\) is coefficient of rolling resistance; \(C_{x}\) is aerodynamic drag coefficient; \(\rho_{air}\) is air density; \(A_{f}\) is vehicle frontal area; \(F_{W}\) is the longitudinal force on the wheels. The resisting force due to road inclination is not considered in the analysis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Unit** & **Variable** & **Value** \\ \hline \multicolumn{4}{c}{Vehicle} \\ Nominal mass & kg & \(M\) & 2041 \\ Frontal Area & m\({}^{2}\) & \(A_{f}\) & 2.4207 \\ Aerodynamic drag coefficient & - & \(C_{x}\) & 0.316 \\ \multicolumn{4}{c}{1 st—3.49; 2 nd—1.99; 3 rd—1.45;} \\ \multicolumn{4}{c}{1 st—1.5 th—0.71; 6 th—0.6} \\ \multicolumn{4}{c}{1 st—1.5 th—0.71; 6 th—0.6} \\ \multicolumn{4}{c}{1 st size} \\ \multicolumn{4}{c}{1 st size} \\ Passenger Capacity & & & P255/50 VR20 \\ \multicolumn{4}{c}{1 st} \\ \multicolumn{4}{c}{1 st item} \\ \multicolumn{4}{c}{1 st item} \\ SAE Net Torque @ rpm & Nm & 310 @ 2000 \\ Fuel System & - & Gasoline Direct Injection \\ SAE Net power @ rpm & kW & 169 @ 5000 \\ Displacement & L & 2.5 \\ \multicolumn{4}{c}{1 st item} \\ \multicolumn{4}{c}{1 st item} \\ Maximum power & kW & 27 \\ Maximum torque @ rpm & Nm & \(T_{em.max}\) & 65 @ 4000 \\ \multicolumn{4}{c}{1 st item} \\ Battery (Sanyo NCR18650 GA Lithium-ion cell) & \\ Nominal voltage & V & 3.6 \\ Nominal capacity & Ah & \(C_{tot}\) & 3.0 \\ Minimum battery SOC & \% & SOC\({}_{low}\) & 60 \\ Maximum battery SOC & \% & SOC\({}_{light}\) & 80 \\ Operating temperature & \({}^{\circ}\)C & \(-\)20–60 \\ Ambient temperature & \({}^{\circ}\)C & \(\theta_{amb}\) & 20 \\ Battery Pack Configurations & - & 1486p and 14s12p \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mazda CX9 2016 [31].  
Figure 2: The scheme of backward simulator.  
The total torque required on the wheels \(T_{W}\), includes the drag torque in the axle bearings \(T_{loss}\) and torque due to inertia of the wheels \(J_{W}\), they are added to the torque of the traction force on wheels \(F_{W}\):  
\[T_{w}=F_{w}\cdot R_{w}+4\cdot J_{w}\cdot\frac{dv}{dt}\cdot\frac{1}{R_{w}}+T_{ loss}. \tag{2}\]  
The wheel rolling radius \(R_{W}\) of the tire is calculated as:  
\[R_{w}=\varepsilon\cdot\ \left(\frac{D_{rim}\cdot 25.4}{2}+W\cdot\frac{AR}{100} \right). \tag{3}\]  
The coefficient \(\varepsilon\) takes into account the tire deflection, which for radial tires is in the range \(\varepsilon=0.92-0.98\)[35]. The nominal wheel radius can be obtained from the tire markings, i.e., the rim diameter \(D_{rim}\), nominal width \(W\) and aspect ratio \(AR\).  
Moreover, the angular speed of the wheel shaft is then found from the rolling radius and the speed \(v\):  
\[\omega_{w}=\frac{v}{R_{w}}. \tag{4}\]