# Multi-Objective Optimization for Plug-In 4 WD Hybrid Electric Vehicle Powertrain
## 2 The Structure and Dynamic Model of the Powertrain
### The Dynamic Model of the Powertrain  
The powertrain dynamic model is mainly applied to the development of the energy management strategy and the evaluation of economy and dynamic performance. Therefore, the model was deduced by quasi-static modeling technology [40; 41] in this study. The total driving force at the wheel is described as:  
\[F_{t}(t)=mgf_{r}\cos(\alpha(t))+mg\sin(\alpha(t))+\frac{c_{d}A}{21.15}v(t)^{2}+m \frac{dv(t)}{dt} \tag{1}\]  
Figure 1: The plug-in 4-wheel drive (4 WD) hybrid electric vehicle powertrain.  
where \(m\), \(c_{d}\), \(A\) and \(f_{r}\) are the vehicle mass, the coefficient of air resistance, the windward area and the rolling resistance coefficient, respectively. \(v(t)\) and \(\alpha(t)\) are the vehicle speed and road gradient, respectively.  
During deceleration, \(F_{t}(t)\) is negative, and the front and the rear axle braking force's distribution follows the fixed factor \(\beta\). Assuming the front and the rear axle braking force are \(F_{fb}(t)\) and \(F_{rb}(t)\), respectively. The braking force are calculated by:  
\[\left\{\begin{array}{l}F_{t}(t)=F_{fb}(t)+F_{rb}(t)\\ F_{fb}(t)=\beta F_{t}(t)\\ F_{rb}(t)=(1-\beta)F_{t}(t)\end{array}\right. \tag{2}\]  
Then, the torque and power demand at the wheel is described as:  
\[\left\{\begin{array}{l}T_{req}(t)=F_{t}(t)\cdot R\\ P_{req}(t)=F_{t}(t)\cdot v(t)\end{array}\right. \tag{3}\]  
where \(R\) is the wheel's radius.  
The torque and angular speed relationship of powertrain elements can be described as:  
\[\left\{\begin{array}{l}(T_{e}(t)+T_{igg}(t))\cdot i_{cvt}\cdot i_{fo}+T_{m} (t)\cdot i_{ro}=T_{req}(t)\\ \omega_{e}(t)=\omega_{ig}(t)=\omega_{w}\cdot i_{cvt}\cdot i_{fo}\\ \omega_{m}=\omega_{w}\cdot i_{ro}\end{array}\right. \tag{4}\]  
where \(T_{e}\), \(T_{igg}\) and \(T_{m}\) are the output torque of the engine, ISG motor and the rear-drive motor, respectively. \(\omega_{e}\), \(\omega_{isg}\), \(\omega_{m}\) and \(\omega_{w}\) are the angular speed of the engine, ISG motor, the rear-drive motor and the wheel, respectively. \(i_{cvt}\), \(i_{fo}\) and \(i_{ro}\) are the gear ratio of CVT, the front final drive and the rear final drive, respectively.  
The electric power demanded or generated by the ISG motor and the rear-drive motor is described as:  
\[\left\{\begin{array}{l}P_{b}(t)=P_{bi}(t)+P_{bm}(t)\\ P_{bi}(t)=\left\{\begin{array}{l}P_{igg}(t)/\eta_{isg}\\ P_{igg}(t)\cdot\eta_{isg}\\ P_{m}(t)\cdot\eta_{m}\end{array}\right.\begin{array}{l}P_{isg}(t)\geq 0\\ P_{isg}(t)<0\end{array}\right. \tag{5}\]  
where \(P_{b}(t)\), \(P_{bi}(t)\), \(P_{bm}(t)\), \(P_{isg}(t)\) and \(P_{m}(t)\) are the battery terminal power, the electric power demanded or generated by the ISG motor, the electric power demanded or generated by the rear-drive motor, ISG motor's output power and the rear-drive motor's output power, respectively. \(\eta_{isg}\) and \(\eta_{m}\) are the efficiency of the ISG motor and the rear-drive motor, respectively. These efficiencies are related to the motors' working point and obtained by looking up tables, and the base motor/generator for the ISG motor is shown in Figure 2a with a maximum torque of 115 Nm and maximum speed of 6000 rpm, the base motor/generator for the rear-drive motor is shown in Figure 2b, with a maximum torque of 145 Nm and maximum speed of 9200 rpm.  
Then, the battery's state of charge (SOC) is calculated by:  
\[\left\{\begin{array}{l}I(t)=\frac{-V_{\text{oc}}+\sqrt{V_{\text{oc}}^{2}-4R_{ \text{int}}P_{\text{p}}(t)}}{2R_{\text{int}}}\\ SOC(t+1)=SOC(t)-\frac{I(t)\cdot\Delta t}{Q_{0}}\end{array}\right. \tag{6}\]  
where \(V_{\text{oc}}\) and \(R_{\text{int}}\) are the battery's terminal voltage and the battery's internal resistance, respectively. They are related to SOC and obtained by looking up tables according to SOC, as shown in Figure 3a \(Q_{0}\) is the nominal battery capacity and \(\Delta t\) is the sample time.  
The fuel consumption can be solved by looking up the table, and the base engine is shown in Figure 3b.