## 2 Parallel HEV System Model  
The simplified topology of the studied parallel HEV is shown in Figure 1. The engine, motor and automated mechanical transmission (AMT) are located on the same axis without the configuration of an extra coupling device. Such a layout provides greater reliability in a compact space. For dealing with different operating conditions, the clutch can be engaged or disengaged to enable the mode transition. In the electric drive mode, the clutch is open and only the motor provides the demanded torque to drive the vehicle [24; 25]. In the hybrid drive mode, the engine drives the vehicle together with the motor, and the clutch is locked. Therefore, the clutch serves as a vital enabling actuator element for mode transitions of the parallel HEV. The specification of the main components of the studied parallel HEV is listed in Table 1.  
This study focused on the impact of both clutch-torque-induced disturbance and different dynamic characteristics between the engine and motor on the performance of the parallel HEV. Using the lumped parameter method [26], the differential equation for vehicle dynamic performance during the clutch slipping process can be depicted as:  
\[\left\{ \begin{array}{l}J_{e}\dot{\omega}_{e}+b_{e}\omega_{e}=T_{e}+T_{c}\\ J_{m}\dot{\omega}_{m}+b_{m}\omega_{m}=T_{m}-T_{c}-\frac{1}{i_{g}i_{0}}\left[k_{ s}\theta_{s}+b_{s}\left(\frac{\omega_{m}}{i_{g}i_{0}}-\omega_{w}\right) \right]\\ J_{w}\dot{\omega}_{w}+b_{w}\omega_{w}=k_{s}\theta_{s}+b_{s}\left(\frac{\omega_ {m}}{i_{g}i_{0}}-\omega_{w}\right)-T_{L}\end{array} \right. \tag{1}\]  
where \(J_{e}\) is the combined inertia of the engine and accessories, \(J_{m}\) is the combined inertia of the motor rotor and transmission on the motor side, and \(J_{w}\) is the inertia of the wheels. \(\omega_{e}\), \(\omega_{m}\) and \(\omega_{w}\) are the engine speed, motor speed and wheel speed, respectively. \(b_{m}\) is the motor internal friction coefficient, \(b_{e}\) is the engine friction coefficient, and \(b_{w}\) is the damping coefficient of tires. \(i_{g}\) and \(i_{0}\) are the gear ratios of the AMT and final drive, respectively. \(b_{s}\) and \(k_{s}\) are the equivalent damping coefficient and spring coefficient of the driveline, respectively. \(\theta_{s}\) denotes the torsional displacement of the driveshaft. \(T_{e}\) and \(T_{m}\) are the engine torque and motor torque, respectively. \(T_{c}\) and \(T_{L}\) are the clutch torque and vehicle load torque, respectively. In addition, the vehicle load torque can be calculated by:  
\[T_{L}=r_{w}\left(mgf_{r}\cos\theta+mg\sin\theta+\frac{C_{D}A\omega_{a}^{2}}{21. 15}+\delta m\frac{dv_{a}}{dt}\right) \tag{2}\]  
where \(g\) is the gravity acceleration, \(m\) is the gross mass, \(f_{r}\) is the rolling resistance coefficient, \(\theta\) is the road slope angle, \(A\) is the frontal area, \(C_{D}\) is the air drag coefficient, \(v_{a}\) is the vehicle velocity, and \(\delta\) is the modified coefficient of the rotating mass. Additionally, the torque transferred from the clutch can be divided into three parts, which is closely related to the friction under the action of the normal force between the two sides of clutch [27; 28; 29]. The clutch torque is calculated by:  
\[T_{c}= \begin{cases}\begin{array}{c}0,\text{unlock}\\ \frac{2N_{e}u_{e}\left(R^{3}-r^{3}\right)F_{e}\text{sgn}(\Delta\omega)}{3(R^{2 }-r^{2})},\text{slip}\\ T_{i},\text{ lock}\end{array}\end{cases}  \tag{3}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Components** & **Description** \\ \hline Engine & Diesel type, peak torque: 800 Nm, maximum power: 162 kW \\ Motor & Permanent magnet, peak torque: 850 Nm, maximum power: 95 kW \\ Gearbox & AMT, gear ratios: 6.71, 3.77, 2.26, 1.44, 1, 0.77 \\ Battery & Lithium iron phosphate, nominal voltage: 483 V, capacity: 110 Ah \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The main parameters of the studied parallel HEV. Figure 1: Simplified topology of the HEV architecture.