# Multi-Objective Optimization for Plug-In 4 WD Hybrid Electric Vehicle Powertrain
## 6 Optimization Results and Analysis
(**a**) The three-dimensional Pareto solution set; (**b**) the term Pareto front graph between the acceleration time and electric energy consumption and (**c**) the term Pareto front graph between the acceleration time and fuel economy. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**The Rear-Drive Motor**} & \multicolumn{2}{c}{**ISG Motor**} \\ \cline{2-5}  & **0.7 \(\leq\eta_{\textit{mat}}\) \textless{} 0.8** & \(\eta_{\textit{mat}}\geq\) 0.8** & \(0.7\leq\eta_{\textit{sig}}\) \textless{} 0.8 & \(\eta_{\textit{sig}}\geq\) 0.8 \\ \hline NDS-1 & 0.74 & 0.26 & 0.54 & 0.46 \\ NDS-3 & 0.63 & 0.37 & 0.37 & 0.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the rear-drive motor and ISG motor’s efficiencies between NDS-1 and NDS-3. From Figure 12, in order to achieve one extreme goal, solutions of NDS-1, NDS-3 and NDS-4 sacrificed a lot of the other goals. Therefore, a best compromise solution should be found after obtaining the Pareto optimal front. The fuzzy min-max approach in [14] can be applied to find the best compromise solution. The \(p\)th objective function is represented by a membership function \(h_{p}\), defined as:  
\[h_{p}=\left\{ \begin{array}{ll}1&f_{p}\leq f_{p\min}\\ \frac{f_{p\max}-f_{p\min}}{f_{p\max}-f_{p\min}}f_{p\min}&<f_{p}<f_{p\max}\\ 0&f_{p}\geq f_{p\max}\end{array} \right. \tag{15}\]  
where \(p=1,2,\ldots,N_{obj}\), \(N_{obj}\) is the number of the objective functions for the member of the Pareto optimal front and \(f_{p\min}\) and \(f_{p\max}\) are the minimum and maximum values of the \(p\)th objective function of Pareto optimal set. Each member of the Pareto optimal front is evaluated using \(h\), defined as:  
\[h=\sum_{p=1}^{N_{obj}}h_{p} \tag{16}\]  
The best compromise solution is that having the maximum value of \(h\). The solution of NDS-2 is the best compromise solution after using the fuzzy min-max approach. Therefore, the solution of NDS-2 was chosen as the final optimal solution in this paper. The results for NDS-1, NDS-2, NDS-3, NDS-4 and the original scheme (OS) are shown in Table 4. The comparison between the original scheme (OS) and the final optimization scheme (NDS-2) is also shown in Table 4. The original scheme's powertrain parameters and control strategy parameters were obtained from the prototype vehicle manufactured by an automobile company. The original scheme's 100 km electric energy consumption, 100 km fuel consumption and acceleration time were achieved from a simulation using the uniform simulation model built in this study. According to this table, compared with the original scheme (OS), the final optimized scheme (NDS-2)'s 100 km electric energy consumption, 100 km fuel consumption and acceleration time were reduced by 1.21%, 6.18% and 5.49%, respectively. Figure 13: Effects of the engine’s maximum power on the fuel consumption and acceleration time. In this study, the Pareto optimal front obtained from the NSGA-II approach was compared with the weight approach. The weight approach is described as follows:  
\[\left\{ \begin{array}{l}J=w_{1}l_{ele}+w_{2}l_{fuel}+w_{3}l_{acc}\\ \sum\limits_{i=1}^{3}w_{i}=1\\ w_{i}\subset[0:0.25:1],\ i=1,2,3\end{array} \right. \tag{17}\]  
The weight approach transforms the multi-objective optimization into single-objective optimization by setting weight factors \(w_{1}\), \(w_{2}\) and \(w_{3}\). By varying these weight factors, different portions of the Pareto optimal front can be generated [14]. In this study, the varying step of all weight factors was 0.25, then, there were 15 combinations of weight factors for plug-in HEV parameter optimization. These weight factor combinations were computed one by one to obtain a group of optimal solutions. These optimal solutions were the Pareto optimal solutions searching by the weight approach.