## 3 Gear Ratio Optimization Problem  
Assuming that the electric motor and battery parameters are determined, different gear ratio combinations can significantly affect the dynamic performance and energy consumption of the vehicle. Generally, for an EV with fixed-ratio transmission, larger gear ratio means better dynamic performance, but at high-driving-speed conditions the operating point of the motor may be out of high-efficiency zone, while a smaller gear ratio may decrease hill-climbing ability and accelerating performance. Taking the acceleration time from standstill to 100 km/h and the 100 km electric energy consumption under specific drive cycle as the power performance and economic indices, respectively, consistent with the above analysis, the two indicators may conflict with each other for the EV adopting the two-speed transmission and a global optimized solution cannot be obtained. From an optimization  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Gear State** & **Brake Status** & **Level Analogy** & **Gear Ratio** \\ \hline \multirow{3}{*}{\(1\)st Gear} & BK1:engaged & \multirow{3}{*}{\(i_{\text{g1}}=1+k_{2}\)} \\  & BK2:disengaged & & \\ \hline \multirow{3}{*}{\(2\)nd Gear} & BK1:disengaged & \multirow{3}{*}{\(i_{\text{g2}}=1+\frac{k_{1}k_{2}}{1+k_{1}+k_{2}}\)} \\  & BK2:engaged & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Gear states for the two-speed transmission.  
Figure 1: Layout of the two-speed transmission drivetrain: (**a**) Diagram of the electric power-train; (**b**) Cross-sectional view of the virtual prototype of the electric power-train (EM—Electric motor; BK1—First gear brake assembly; R1—First ring gear; C1—First carrier; S1—First sun gear; P1—First planet gears; BK2—Second gear brake assembly; R2—Second ring gear; C2—Second carrier; S2—Second sun gears; P2—Second planet gears).  
perspective, gear ratio optimization can be summarized as a multiobjective optimization problem as follows:  
\[\begin{array}{ll}\min_{x}&J(x)=\left[J_{acc}(x),J_{de}(x)\right]^{T}\\ \text{s.t.}&x\in\Omega,\end{array} \tag{1}\]  
where \(J_{acc}\) and \(J_{ele}\) denote the acceleration time from 0 to 100 km/h and the 100 km electric energy consumption, respectively. \(x\) is the decision vector and is chosen to be the first and second gear ratio \(i_{1}\) and \(i_{2}\) (final drive ratio included). \(\Omega\) is the feasible region where \(x\) should satisfy all constraints, like vehicle and components dynamic constraints.  
In this study, the NSGA-II approach is selected to solve the multiobjective optimization problem. The NSGA-II is a kind of genetic algorithm, in which the fast nondominated sorting and crowded distance sorting methods are introduced to generate Pareto-optimal solutions. More details about the framework of NSGA-II can be found in [23]. Figure 2 presents the flowchart of the gear ratio optimization procedure with the NSGA-II approach.  
The flow can be described as the following steps:  
Step 1: Generate initial population.  
Step 2: Evaluate the objective functions (i.e., the acceleration time and the electric energy consumption) of each individual and sort the population by fast nondominated ranking. It should be noted that, to get the best energy consumption economy, the 100 km electric energy consumption is  
Figure 2: Flowchart of the gear ratio optimization procedure with the Nondominated sorting genetic algorithm-II (NSGA-II) approach.  
obtained via fast Dynamic Programming (DP) for the certain gear ratio combination. The details of DP are introduced in the following Section 3.2.2.  
Step 3: Produce the child population via selection, crossover, and mutation operation.  
Step 4: Evaluate the objective functions of the child population as Step 2.  
Step 5: Keep the best parent-child individuals from the parent and child population.  
Step 6: Perform the above steps until the stopping criteria is met.