# Multi-Objective Optimization for Plug-In 4 WD Hybrid Electric Vehicle Powertrain
## 4 Mathematical Model of the Multi-Objective Optimization  
When establishing the objective function, scholars mainly take fuel consumption, equivalent fuel consumption or use the cost under certain driving cycles as the objective function of the economy index. In fact, PHEVs usually run in the CD and CS stage. In the CD stage, the vehicle is basically driven by the battery, the emission and fuel consumption in this stage are approximately zero, so it is reasonable to select electric energy consumption as the objective function of economy index in this stage; in the CS stage, energy consumption mainly comes from the engine, so fuel consumption should be chosen as the objective function of the economy index in this stage. The objectives of vehicle economy and power performance usually conflict with each other. Therefore, in the optimization  
vector, which includes the maximum power of the engine \(P_{\rm{emax}}\), maximum power of the rear-drive motor \(P_{\rm{mmax}}\), maximum power of the ISG motor \(P_{\rm{_{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ }}}}}}}}}}}} \left({{{\rm{\rm{\rm{\rm{\rm{\rm{{\rm{{\rm{{\rm{\rm{{\rm{{\rm{}}}}}}}}}}}}}}} \right)}}}}\), speed ratio of the front final drive \(i_{fo}\), speed ratio of the rear final drive \(i_{ro}\) and control strategy parameters (\(k_{up}\) and \(k_{low}\)), \(X_{j}\) is the decision variable, \(a\) and \(b\) are the lower and upper limits of each decision variable, respectively. The value of a and b is given in the \"Range\" column of Table 2. \(N_{\rm{max}}\) and N\({}_{\rm{imin}}\) are the upper and lower limits of the components' speed, respectively. \(T_{\rm{max}}\) and \(T_{\rm{imin}}\) are the upper and lower limits of the components' output torque, respectively. In order to obtain the above objective functions' values, different simulations are needed. When the electric energy consumption function is simulated, the vehicle runs in the CD stage through the SOC constraint. The 100 km electric energy consumption under the CD stage is calculated by:  
\[\left\{ \begin{array}{l}{f_{ele}(X)=\frac{{100}}{{86005}}\sum\limits_{k=0}^{N _{tot}}{P_{b}(X,k)}}\\ \\ {\rm{Vehicle~{}is~{}running~{}in~{}the~{}CD~{}stage,~{}and~{}the~{}SOC~{}constraint~{}is~{} described~{}as}}\\ \\ {0.25\leq SOC(k)\leq 0.95}\\ \\ {0.25\leq SOC(N_{tot})\leq 0.35}\\ \\ \end{array} \right. \tag{10}\]  
where \(k\) represents the \(k\)th simulation step, one simulation step was set to one second in this paper. \(N_{ele}\) is the total step for the electric energy consumption simulation, so \(k=1,2,\ldots,N_{ele}-1\). \(P_{b}\) is the battery power, its unit is kw and \(S_{1}\) is the driving mileage in the CD stage, its unit is km. \(SOC(0)\) is the initial SOC value. \(SOC(k)\) is the SOC value in the kth simulation step. \(SOC(N_{ele})\) is the SOC value at the end of the simulation. When the fuel consumption function is simulated, the vehicle runs in the CS stage through constraining the final SOC to be equal to the initial SOC. The 100 km fuel consumption under the CS stage is calculated by:  
\[\left\{ \begin{array}{l}{f_{fund}(X)=\frac{{100}}{{\rho 5^{2}}}\sum\limits_{k=0}^{N _{fund}}{{\dot{m}}_{f}(X,k)}}\\ \\ {\rm{Vehicle~{}is~{}running~{}in~{}the~{}CS~{}stage,~{}and~{}the~{}SOC~{}constraint~{}is~{} described~{}as}}\\ \\ {0.25\leq SOC(k)\leq 0.95}\\ \\ \end{array} \right. \tag{11}\]  
where \({\dot{m}}_{f}\) is the instantaneous fuel consumption, it can be obtained by looking up the table by the engine speed and engine torque, its unit is g/s. \(S_{2}\) is the driving mileage in the CS stage. \(\rho\) is the density of gasoline. \(N_{fuel}\) is the total steps for the fuel consumption simulation.