## 4 Simulation and Result Discussion of the MOO Problem
### Optimization Decisions and Results Discussion  
Taking the individual with comprehensive regenerative braking performance as the final optimization solution, the optimal variable solution was [\(\mu_{\varphi,J}\), \(\mu_{\varphi,J}\), \(\mu_{w,J}\), \(\mu_{w,J}\)]T = [0.11, 0.53, 0.28,\(2\)-\(0.65\)]T. The results of the brake strength thresholds \(Z_{a\_l}\) and \(Z_{b\_h}\) after optimization are summarized in Table 4.  
As shown in Figure 12, the points with \"*\" represent the regenerative braking work-point after optimization, and the points without \"*\" represent the regenerative braking work-point before optimization. It can be seen that \(Z_{a\_l}\) and \(Z_{b\_h}\) at points \(A\) and \(B\) remain unchanged before and after optimization on high TAC. After optimization, points \(A\) and \(B\) move forward to \(A^{*}\) and \(B^{*}\) on medium TAC and points \(A\) and \(B\) move backward to \(A^{*}\) and \(B^{*}\) on low TAC. In addition, changes of the \(W\) function before and after optimization is shown in Figure 13.  
The analysis of the simulation results is described as follows: (1) In the case of \(\mu_{max}\in(0,\mu^{*}_{w\_h}]=(0,0.28]\), the wheel had a risk of slipping due to the poor TAC. The control parameters and \(W\) function did not change after optimization (\(W^{*}=W=0\)), the braking work-points were on the \(I\) curve to ensure the braking safety, and the front and rear wheel braking torques were provided by hydraulic torque. (2) In the case of \(\mu_{max}\in(\mu^{*}_{w\_h},\mu_{w\_l}]=(0.28,0.4]\), the threshold value \(\mu_{w\_l}\) \"drill down\" after optimization (\(W>W\)), which made the regenerative braking torque \"enter ahead\" to increase the regenerative braking energy. As shown in Figure 12a, under the conditions of low PTAC, the braking work-points \(C(W_{c5})\) and \(D(W_{c5})\) on the \(I\) curve before optimization would switch to \(C^{*}(W_{c1})\) and \(D^{*}(W_{c3})\) after optimization. (3) In the case of \(\mu_{max}\in(\mu^{*}_{w\_h},\mu_{w\_c}]=(0.4,0.54]\), threshold switching points \(A\) and \(B\) moved backward after optimization, and the regenerative braking ratio did not change or became larger at the same braking strength. (4) In the case of \(\mu_{max}\in(\mu_{w\_cr},\mu_{w\_h}]=(0.54,0.65]\), the threshold \begin{table}
\begin{tabular}{l c c c} \hline
**Parameter \(Z_{a\_l}\)** & **High Adhesion** & **Medium Adhesion** & **Low Adhesion** \\ \hline Before optimization & 0.400 & 0.400 & 0.0000 \\ After optimization & 0.400 & 0.346 & 0.0757 \\ \hline Parameter \(Z_{b\_h}\) & High adhesion & Medium adhesion & Low adhesion \\ \hline Before optimization & 0.600 & 0.400 & 0.0000 \\ After optimization & 0.600 & 0.533 & 0.1318 \\ \hline \end{tabular}
\end{table} 
Table 4: Results of \(Z_{a\_l}\) and \(Z_{b\_h}\) after optimization. Figure 12: Change of the work-point before and after optimization: (a) \(\mu_{max\_l}=0.3\); (b) \(\mu_{max\_m}=0.6\); and (c) \(\mu_{max\_h}=0.9\). Figure 13: Change of the \(W\) function before and after optimization. value \(\mu_{w_{h}}\)\"goes up\" after optimization (\(W^{*}<W\)), which accompanied by the threshold switching points \(A\) and \(B\) moved forward after optimization. Under the demand of high braking strength, the risk of locking and slipping was prone to occur. It can \"sacrifice\" smaller braking energy in exchange for better braking performance to ensure braking safety after parameter optimization. As shown in Figure 12b, under the demand of low braking strength, the braking work-points \(E(W_{c1})\) before optimization would switch to \(E^{*}(W_{c4})\) after optimization, and under the demand of medium braking strength, the braking work-points \(G(W_{c4})\) and \(I(W_{c4})\) before optimization would switch to \(G^{*}(W_{c4})\) and \(I^{*}(W_{c5})\) after optimization. (5) In the case of \(\mu_{max}\in(\mu^{*}_{w_{h}},1)=(0.65,1)\), the threshold switching points \(A\) and \(B\) remained unchanged after optimization (\(W^{*}=W=1\)), as shown in Figure 12c.