# Multi-Objective Optimization for Plug-In 4 WD Hybrid Electric Vehicle Powertrain
## 7 Conclusions  
The plug-in 4 WD HEV's energy management control strategy based on the CD-CS mode was developed. Then, the multi-objective optimization's mathematical model, which aims at minimizing the electric energy consumption under the CD stage, the fuel consumption under the CS stage and the acceleration time from 0-120 km/h, was established. Finally, the multi-objective parameter optimization problem was solved using the evolutionary based NSGA-II approach. Some of the results were compared with the original scheme and the classical weight approach.  
From the term Pareto front graph obtained using the NSGA-II approach, it could be seen that the two objectives of 100 km electric energy consumption and 100 km fuel consumption were in conflict with the acceleration time from 0-120 km/h, respectively. It was impossible for them to achieve the optimum at the same time. Therefore, the best compromise solution (NDS-2) was chosen as the final optimized scheme. Compared with the original scheme, the final optimized scheme's 100 km electric energy consumption, 100 km fuel consumption and acceleration time were reduced by 1.21%, 6.18% and 5.49% respectively.  
Compared with the weight approach, the Pareto optimal solutions obtained using the NSGA-II approach were better distributed over the entire Pareto optimal front, as well as the best compromise solution was also better.  
NSGA-II is an evolutionary based multi-objective optimization algorithm, the main drawback of this approach is computationally burdensome. Therefore, pursuing a more efficient algorithm to solve plug-in 4 WD HEV's multi-objective parameter optimization problems is future research work.  
[PERSON] wrote the paper, built the mathematical model of the multi-objective optimization and analyzed the optimization results; [PERSON] built the dynamic model of the powertrain and developed control strategy based on CD-CS mode; [PERSON] provided suggestions on mathematical model building and paper writing; [PERSON] provided optimization algorithms and made revisions to the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 51678075; 51665020), Major Special Science and Technology Project of Hunan Province (Grant No. 2019 SK2171), Hunan Province Key Laboratory of Smart Roadway and Cooperative Vehicle-Infrastructure Systems (Grant No. 2017 TP1016) and Key Research and Development Project of Jiangxi Province (Grant No. 20192 BBEL50015).  
The authors declare no conflict of interest.  
The following abbreviations were used in this manuscript:  
\begin{tabular}{l l} CD & charge depleting \\ CS & charge sustain \\ CVT & a continuously variable transmission \\
4 WD & 4-wheel drive \\ PHEVs & plug-in hybrid electric vehicles \\ NSGA-II & Non-dominated Sorting Genetic Algorithms-II \\ A & the windward area, its unit is m\({}^{2}\) \\ \(a(X,k)\) & the acceleration at velocity step \(k\), its unit is m/s\({}^{2}\) \\ \end{tabular}  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \(P_{emax}\) & \(P_{mmax}\) & \(P_{\text{ing\,max}}\) & \(i_{f_{\theta}}\) & \(i_{no}\) & \(k_{up}\) & \(k_{low}\) & \(f_{ele}\) & \(f_{fuel}\) & \(f_{acc}\) \\ \hline NSGA-II approach & 72 & 73 & 30 & 5.26 & 7.42 & 0.71 & 0.78 & 12.24 & 5.01 & 8.6 \\ Weight approach & 87 & 60 & 25 & 5.89 & 7.06 & 0.51 & 0.53 & 11.82 & 5.27 & 8.8 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The best compromise solution obtained by using the NSGA-II algorithm and the weight approach.