# Parallel Hybrid Electric Vehicle Modelling and Model Predictive Control
## 2 Modelling of the Parallel HEV  
In parallel hybrid electric vehicles, both combustion engine and main electric motor are installed in parallel and work in independent configurations. The vehicle can run and switch in four driving modes: pure main electric motor (EV1) at a low speed and/or low load; pure combustion engine (EV2) at a high speed and/or high load; both main electric motor and combustion engine (EV3) at a very high and/or very high load; and a combination of the main electric motor, generator motor, and combustion engine (EV4) at extreme high speed and/or load. In 2021, Hyundai introduced a new version of Sonata Hybrid, the middle size family passenger HEV that combines updated technologies for a typical parallel electric vehicle, as shown in Figure 1.  
This parallel HEV consists of one internal combustion engine (ICE) with four cylinders with multiple point injection, a volume of 2.4 litters, a maximum power of 156 kW at 6000 rpm, and a peak torque of 265 Nm; one electric motor starter (EM2) with a maximum power of 8 kW and a maximum torque of 43 Nm; the main electric motor (EM1) with a maximum power of 35 kW and a maximum torque of 205 Nm; the battery HEV Li-ion with a capacity of 6.1 Ah; a transmission gearbox with fully automated 6 speeds; and a friction clutch engagement. The vehicle curb weight is 1569 kg. This Sonata Hybrid vehicle is used to simulate our system modelling and test the new MPC scheme with softened constraints. The schematic architecture of the 2021 Hyundai Sonata Hybrid in Figure 1 can be modelled with a simple drivetrain and is shown in Figure 2. The first part of this mechanical structure consists of an internal combustion engine (ICE) and the electric starter/generator motor (EM2) can be grouped into one inertia \(J_{1}\), including the left clutch disk, the sharp 1, EM2, and ICE. \(J_{1}\) is modelled as one rigid inertia. \(M_{ICE}\) and \(M_{EV1}\) are the torques on the ICE and EM2, respectively. \(\theta_{1}\) and \(\omega_{1}\) are the angular position and the velocity of the sharp 1, respectively. Similarly, \(J_{2}\) is modelled as the lumped rigid inertia of the main electric motor EM1 and the right clutch disk. \(\theta_{2}\) and \(\omega_{2}\) are the angular position and velocity of the sharp 2, respectively. The third powertrain part connecting the gearbox and the vehicle's driven wheels can be modelled by a gear ratio \(i\) via a torsional spring and damper with \(k_{\theta}\), \(k_{\beta}\), and \(k_{\alpha}\) as the stiffness, damping, and acceleration coefficient, respectively, of which the acceleration has not been studied before. The third part, the lumped inertia \(J_{3}\), consists of the rest of the vehicle, including the gearbox, differential gear, shaft 3, and the driven wheels. \(\theta_{3}\) and \(\omega_{3}\) are the angular position and velocity of shaft 3, respectively. \(rr\) is the rolling radius of the vehicle's wheels. Figure 1: The 2021 Hyundai Parallel HEV. In this paper, all vehicle dynamic formulas and constraints were taken from the technical book in [1]. The vehicle resistance torque is the approximation of the air density \(\rho\), air drag coefficient \(c_{w}\), the vehicle crossing area \(A\), the wheel rolling radius \(rr\), vehicle friction resistant coefficient \(f_{r}\), natural gravity \(g\), vehicle mass \(m\), and the polynomial coefficients of \(a_{0}\), \(a_{1}\) and \(a_{2}\) The vehicle rolling resistance torque \(M_{v}\) can be calculated as:  
\[M_{v}=\left(\frac{\rho}{2}c_{w}A(r\omega_{3})^{2}+f_{r}mg\right)r+a_{0}+a_{1} \omega_{3}+a_{2}\omega_{3}^{2} \tag{1}\]  
In Equation (1), the additional road conditions, such as the road dynamics, the road increase, and other environment conditions, can be added as disturbances that lead to some reduction of or increase in the vehicle rolling resistance torque. Changes of vehicle velocity, depending on the road conditions as well as the vehicle dynamic constraints between the vehicle speed and vehicle steering wheel, are referred to in [1].