# Optimal Fuel Consumption Modelling, Simulation, and Analysis for Hybrid Electric Vehicles
## 2 HEV Powertrain Modelling  
HEVs usually have two power sources: a combustion engine and an electrical motor with a large battery to run the vehicle. A typical HEV has a parallel HEV scheme for which the internal combustion engine (ICE) and the main electrical motor (EM2) are installed parallelly and independently, so that the HEV can run in four driving modes (DMs). At a low speed and/or low load, only the main EM2 propels the vehicle (pure electric mode or DM1). At a higher speed and/or higher load, the ICE is started by a starter/generator electric motor (EM1) and propels the vehicle (pure ICE mode or DM2); EM1 is turned off or becomes the generator to charge the battery. At a very high speed or very high load, the vehicle is run by both EM2 and ICE (hybrid mode or DM3). Lastly, at an extreme high speed and/or extreme high load, EM2, EM1, and ICE together run the vehicle (extreme hybrid mode or DM4). The vehicle battery can be charged offline from a plug-in cable or online by an EM1 generator when necessary. Hyundai launched the latest version of the Sonata Hybrid in 2021. This is a typical sedan middle-class parallel HEV. This HEV consists of 1 ICE with 4 cylinders of multiple point injections, a displacement of 2.4 litters, maximum power of 156 kW at 6000 rpm, and a peak torque of 256 N m; 1 main EM2 with the maximum power of 35 kW and maximum torque of 205 N m; 1 starter/generator EM1 of 8 kW and a maximum torque of 43 N m; the main battery Li-ion with a capacity of 6 Ah; an automated 6-speed gearbox; and an automated clutch to connect or disconnect the main motor EM2 to the ICE. The vehicle curb weight is 1569 kg. The HEV powertrain scheme is drawn in Figure 1.  
The HEV parallel powertrain scheme in Figure 1 can be modelled as a mechanical powertrain model as shown in Figure 2. The combustion engine, ICE, and the starter/generator motor, EM1, can be modelled as a lumped inertia, \(J_{1}\); \(M_{ICE}\) and \(M_{EV1}\) are the torques of ICE and EM1, correspondingly. \(\theta_{1}\) and \(\omega_{1}\) are the angular position and velocity of shaft 1, respectively. Similarly, the main electric motor, EM2, can be also modelled as a lumped inertia, \(J_{2}\); \(M_{EV2}\) is the torque on EM2. \(\theta_{2}\) and \(\omega_{2}\) are the angular position and the angular velocity of shaft 2, respectively. The automated gearbox is modelled as a mechanical transmission with a ratio, \(i\), via a damping system with \(k_{\theta}\), \(k_{\theta}\), and \(k_{\alpha}\) as the position, velocity, and acceleration damping coefficient, correspondingly. The third part of the vehicle powertrain is also modelled as a lumped inertia, \(J_{3}\), including shaft 3, the differential gear, and rear wheels. \(\theta_{3}\) and \(\omega_{3}\) are the angular position and the velocity of shaft 3, respectively. Additionally, \(r\) is the vehicle wheel rolling radius. In our model, shaft 3 is connected directly to the rear wheels and the vehicle speed is \(v=r\times\omega_{3}\) and \(\omega_{3}\) is the vehicle driven wheel angular velocity. The vehicle rolling resistance is calculated from the air density, \(\rho\); air drag coefficient, \(c_{w}\); vehicle crossing area, \(A\); vehicle wheel radius, \(r\); vehicle friction-resistant coefficient, \(f_{r}\);  
Figure 1: HEV parallel powertrain scheme. Figure 2: Modelling HEV’s powertrain. the natural gravity, \(g\); vehicle mass, \(m\); and polynomial coefficients on the vehicle velocity, \(a_{0}\), \(a_{1}\), and \(a_{2}\), accordingly. Then, vehicle rolling resistance torque, \(M_{g}\), can be calculated as:  
\[M_{g}=\Big{(}\frac{\rho}{2}c_{w}A(r\omega_{3})^{2}+f_{r}mg\Big{)}r+a_{0}+a_{1} \omega_{3}+a_{2}\omega_{3}^{2} \tag{1}\]  
In Equation (1), several other road conditions, such as the road dynamics, road increasement and environment conditions, can be added as disturbances leading to some reduction or increasement in the vehicle rolling resistance torque.