# Optimization of Mode-Switching Quality of Hybrid Tractor Equipped with HMCVT
## 2 Transmission System of Hybrid Tractor
### Power Transmission System Modeling  
This engine model adopts the interpolation modeling method based on bench test data, which require little computation and can be quickly modeled [14]. By collecting bench data from a Weichai WP6.180E40 diesel engine and using data interpolation technology, a three-dimensional map of engine output speed, torque, and throttle opening was established. It is shown in Figure 3.  
The numerical model of engine output torque can be expressed as:  
\[T_{e}=f_{1}(\alpha,n_{e}) \tag{3}\]  
Figure 3: Engine output speed–torque model. Figure 2: Hybrid speed-coupling mode transmission route. where \(\alpha\) is the throttle opening; \(T_{e}\) is the engine torque (N-m); and \(n_{e}\) is the engine speed (r/min). As another power source of hybrid vehicles, the electric motor plays the role of output driving torque when the vehicle is in pure electric mode and hybrid mode [15]. In this paper, the electric motor's operation was modeled based on an experimentally determined motor performance map. This maps the steady-state relationship between motor speed, torque, and efficiency. Intermediate points were computed through data interpolation, as shown in Figure 4.  
Motor efficiency can be expressed as:  
\[P_{m}=\frac{T_{m}n_{m}}{9550}\eta_{m}^{j} \tag{4}\]  
where \(P_{m}\) is the motor efficiency (kW); \(T_{m}\) is the motor output torque (N-m); \(n_{m}\) is the motor speed (r/min); \(\eta_{m}\) is the motor efficiency; and \(j\) is \(-1\) when the motor is used as a driving motor. When the motor is used as a generator, \(j\) is 1. This paper uses the equivalent internal resistance model to integrate the battery into a model consisting of an ideal voltage source \(U_{o}\) and an internal resistance \(R_{o}\) connected in series, as shown in Figure 5[16]. According to [PERSON]'s law, the relationship between the output voltage and the output current of a battery is as follows:  
\[U_{b}=U_{o}-R_{o}I_{b} \tag{5}\]  
Figure 4: The electric motor output speed–torque model. Figure 5: Battery-equivalent model. According to [PERSON]'s second law, the output current of the battery can be obtained:  
\[\left\{ \begin{array}{l}P_{b}=U_{b}\cdot I_{b}=U_{o}I_{b}-I^{2}{}_{b}R_{o}\\ I_{b}=\frac{U_{o}-\sqrt{U_{b}{}^{2}-4P_{b}R_{o}}}{2R_{o}}\end{array} \right. \tag{6}\]  
where \(P_{b}\) is the output power of the battery (W); \(U_{o}\) is the open circuit voltage at the battery end (V); \(U_{b}\) is the output voltage of the battery (V); \(I_{b}\) is the battery output current (A); and \(R_{o}\) is the internal resistance of the battery (\(\Omega\)). The clutch adopted in this article was a wet clutch, which was mainly composed of a friction plate, a driving plate, a driven plate, an oil cylinder, a piston, and a return spring. The engagement process of the clutch was divided into a separation stage, a sliding stage, and a complete engagement stage, as shown in Figure 6.  
\(t_{2}\) to \(t_{3}\) is the clutch slipping stage. Currently, the clutch oil pressure continues to rise, the distance between the driving plate and the driven plate of the clutch becomes zero, the clutch enters the slipping stage, and the clutch begins to transmit friction torque \(T_{c}\). The friction torque can be expressed as follows:  
\[T_{c}=\frac{2}{3}\cdot\frac{R_{2}^{3}-R_{1}^{3}}{R_{2}^{2}-R_{1}^{2}}\cdot\mu \cdot z\cdot S\cdot P \tag{7}\]  
where \(R_{1}\) is the inner radius of the clutch friction plate (m); \(R_{2}\) is the outer radius of the clutch friction plate (m); and \(\mu\) is the friction factor of the friction plate. \(z\) is the number of friction surfaces of the clutch; \(S\) is the piston area (m\({}^{2}\)); and \(P\) is the oil pressure (Pa). SimulationX is a standard tool for evaluating the interactions between all components in technical systems [18]. SimulationX software supports the output of C code and can be revised and compiled according to user needs. C code can be used freely without license restrictions and can realize joint simulation with MATLAB [19]. Figure 7 shows the vehicle simulation model.