## 3 Gear Ratio Optimization Problem
### Objective Functions
#### 3.2.2 The Electric Energy Consumption
\(\bullet\) **The DP calculation process**  
According to [PERSON]'s optimality principle [25], the global optimization problem can be decomposed into a series of simple minimization problems as follows:  
* Cost calculation at \(N-1\) step \[J_{N-1}^{*}\left(x(N-1)\right)=\min_{u_{G}(N-1)\in\{-1,0,1\}}L\left(x(N-1),u_{ G}(N-1),N-1\right).\] (14)  
Figure 3: Battery parameters: (**a**) The relationship between \(V_{oc}\) and state-of-charge (SOC) under charging or discharging status. (**b**) The relationship between \(R_{int}\) and SOC under charging or discharging status. 2. Intermediate calculation step for \(k=N-2\) to 0 \[J_{k}^{*}\left(x(k)\right)=\min_{u_{G}(k)\in\{-1,0,1\}}L\left(x(k),u_{G}(k),k\right) +J_{k+1}^{*}\left(x(k+1)\right),\] (15) where \(J_{k+1}^{*}\left(x(k+1)\right)\) represents the minimum objective function from the state variable at the \((k+1)_{th}\) step to the state variable at the \(N_{th}\) step. The DP calculation process consists of the backward calculation and forward calculation parts. First, starting from the \((N-1)_{th}\) step, based on Equations (14) and (15), the minimum objective function for each state variable at the previous step can be derived in an iterative way and the optimal control policy corresponding to this state variable is stored. In the forward calculation process, starting from the initial time and state variable, the optimal control variable at \(k_{th}\) step can be obtained from the optimal control policy stored in the above backward calculation process. Combining Equations (8) and (9), the state variable at the next step on the optimal path can be calculated and the above process is repeated until the final step is reached. In the optimization procedure shown in Figure 2, the population's update requires the computation of objective functions for each individual. The DP process needs to be invoked sequentially to calculate the 100 km electric energy consumption, so a fast solution method for DP problem is crucial. If the state variable space is meshed according to \(SOC_{max}\) and \(SOC_{min}\) during backward recursion, as the square region of the solid line envelope in Figure 4, some region of the state space cannot be reached during the specific drive cycle, which creates the unnecessary computational burden. In this paper, provided that the initial value of the SOC and drive cycle are known, the reachable region of SOC at any time step \(k\) can be calculated in advance, which can help reduce the search scope of the state variable space. The reachable regions of SOC at \(k_{th}\) step and \((k+1)_{th}\) step are the gray regions shown in Figure 4.  
The SOC range at \((k+1)_{th}\) step can be calculated by combining Equations (8) and (9):  
\[\left\{ \begin{aligned}  SOC_{high}(k+1)&=SOC_{high}(k)-\frac{V_{oc}(k)- \sqrt{V_{oc}^{2}(k)-4R_{int}(k)P_{batt,min}(k)}}{7200R_{int}(k)Q_{C}}\\ SOC_{low}(k+1)&=SOC_{low}(k)-\frac{V_{oc}(k)-\sqrt{V_ {oc}^{2}(k)-4R_{int}(k)P_{batt,max}(k)}}{7200R_{int}(k)Q_{C}}\\ SOC_{high}(0)&=SOC_{low}(0)=SOC_{ini},\end{aligned} \right. \tag{16}\]  
where \(SOC_{ini}\) is the initial value of the SOC; \(SOC_{high}(k+1)\) and \(SOC_{low}(k+1)\) denote the maximum and minimum value of SOC estimated at \((k+1)_{th}\) step, respectively; the open circuit voltage and  
Figure 4: Determine the SOC reachable region during the Dynamic Programming (DP) calculation process. internal resistance for \(SOC(k)\) at \(k\) step can be obtained from Figure 3; \(P_{batt,min}\) and \(P_{batt,max}\) are the minimum and maximum battery power and can be expressed as follows:  
\[P_{batt,max/min}=\frac{P_{vch}}{\eta_{tot}^{sgn(P_{vch})}\eta_{M}^{sgn(P_{vch})}}, \tag{17}\]  
where \(P_{vch}\) is the vehicle required power based on specific drive cycle, \(\eta_{M}\) is the motor efficiency at \(k\) step and varies as a function of motor torque and speed.