## 2 HEV Modelling and Control
### HEV Architecture and Model  
The HEV architecture chosen for this study is a full HEV parallel P2, having the EM located after the ICE right before the automated manual transmission. The parallel P2 HEV exhibits an interesting trade-off between energy consumption and ease of production owing to a limited difference with respect to a conventional ICE-powered vehicle [13]. This HEV architecture enables three ways of providing torque to the wheels: (1) pure electric mode and (2) pure thermal mode, i.e., when either the EM or the ICE solely deliver the total amount of power requested, respectively; (3) torque split mode, which is when both of the power sources contribute to providing the torque. A particular torque split mode relates to battery charging for which the ICE provides extra power compared with the value required to overcome the total requirement of the wheels in order to supply charging power to the battery. All the main components of this HEV powertrain architecture are illustrated in Figure 1. A gear ratio between the ICE and the transmission allows increasing the torque delivered by the former. A clutch is embedded that is disengaged when driving in pure electric mode. Moreover, the main data used for virtually simulating a compact size HEV are found in Table 1.  
It is worth mentioning that, concerning the vehicle mass, a variation \(\Delta m\) has been considered as a function of ICE size, EM size, and the number of gears. Therefore, starting from the baseline value (i.e., a five-speed transmission, a 90 kW ICE, and a 30 kW EM), \(\Delta m\) has either been subtracted or added depending on the sizes of components for the candidate HEV powertrain design. The formulas used for this purpose are found below and have been extracted from [14]:  
\[\Delta m_{i}=\Delta m_{transm,i}+\Delta m_{ICE,i}+\Delta m_{EM,i}, \tag{1}\]  
\[\Delta m_{transm,i}=(5-z_{i})\cdot 0.12\cdot ICE_{maxpow.base}, \tag{2}\]  
\[\Delta m_{ICE,\ i}=1.75\cdot\left(ICE_{maxpow,base}-ICE_{maxpow,i}\right), \tag{3}\]  
\[\Delta m_{EM,i}=0.53\cdot\left(EM_{maxpow,base}-EM_{maxpow,i}\right), \tag{4}\]  
where \(z_{i}\), \(ICE_{maxpow,i}\), and \(EM_{maxpow,i}\) are the number of gears, the ICE, and the EM peak powers of the i-th HEV candidate, respectively; \(\Delta m_{transm,\ i}\), \(\Delta m_{ICE,\ i}\), and \(\Delta m_{EM,i}\) are instead the variation of mass due to the diverse number of gears and the different sizes of ICE and EM, respectively. Concerning the modelling of the vehicle, a quasi-static approach was chosen that neglects transients to lower the computational cost. This method exploits the vehicle speed profile over time to compute the power request at the input shaft of the transmission. Equations (5) and (6) are used at each timestep for the aforementioned purpose:  
\[T_{wheel}=\left(A+B\ v+Cv^{2}+m\ a+m\ g\ sin(\alpha)\right)\cdot r_{dyn}, \tag{5}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Curb weight & 1162 kg & C & 0.41 N/(m/s)\({}^{2}\) \\ \(r_{wheel}\) & 0.28 m & Battery Pack Energy & 2.1 kWh \\ A & 104.49 N & \(ICE_{maxpow,base}\) & 90 kW \\ B & 2.43 N/(m/s) & \(EM_{maxpow,base}\) & 30 kW \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Hybrid electric vehicle (HEV) data. Figure 1: Parallel P2 full hybrid electric vehicle (HEV) architecture. \[T_{inp}=\frac{T_{wheel}}{\left(\tau_{diff}\cdot\tau_{gear}\right)\cdot\eta_{gear}^{ sign\ (Twheel)}}\, \tag{6}\]  
where \(T_{wheel}\) and \(T_{inp}\) are the torques required at the wheel and at the input shaft of the transmission, respectively. \(\
u\) is the target velocity of the vehicle and \(m\) is the vehicle mass, while \(a\), \(g\), and \(\alpha\) are the target acceleration, the gravity acceleration, and the road slope, respectively. \(A\), \(B\), and \(C\) are the coast down coefficients; \(\tau_{diff}\) and \(\tau_{gear}\) are the gear ratios of the differential and of the engaged gear; while \(\eta_{gear}\) is the transmission efficiency, which here is set to 90%.