## 3 Offline Optimization Algorithm  
### Approximately Equivalent Fuel Consumption Minimum Strategy  
Originating from the PMP algorithm, the ECMS algorithm was proposed by [PERSON] [5] and applied to the energy management strategy of HEVs. The core idea is to select an appropriate charge and discharge equivalent factor to transform the electricity consumption into the fuel consumption, minimize the instantaneous equivalent fuel consumption, and obtain an approximate global optimal solution. The mathematical model can be formulated as  
\[\left\{ \begin{array}{ll}\left[\dot{m}_{e}^{*},\dot{m}_{b}^{*}\right]=\arg \min_{T_{e},i_{cvt}}\left\{j_{cms}\right\}\\ \dot{j}_{cms}=\dot{m}_{e}+s\times\frac{m_{b}}{H_{L}}\\ T_{m}+T_{e}=\frac{T_{e}}{\{\dot{s}\}\times\dot{l}_{cvt}\times\dot{l}_{cvt}} \end{array} \right., \tag{11}\]  
where \(\dot{l}_{cms}\) is the instantaneous equivalent fuel consumption rate; \(soc\) is the battery capacity level; \(T_{e}\) and \(i_{cvt}\) are the engine torque and CVT speed ratio, respectively, and are system control variables; \(\dot{m}_{e}\), \(\dot{m}_{b}\), and \(\eta_{cvt}\) are obtained from the model of each component in the first section; \(H_{L}\) is the low heating value of gasoline; \(s\) is the equivalent factor; and \(\dot{m}_{equ}=s\times\left(\dot{m}_{b}/H_{L}\right)\) is the equivalent fuel consumption of the battery rate. The constraint conditions are as follows:  
\[\left\{ \begin{array}{ll}T_{e,\min}<T_{e}(t)<T_{e,\max}\\ T_{m,\min}<T_{m}(t)<T_{m,\max}\\ SOC_{\min}<soc(t)<SOC_{\max}\end{array} \right.. \tag{12}\][PERSON] and [PERSON] proposed the A-ECMS. By numerically fitting the fuel consumption rate of the engine and battery, the search space of the optimal control variable is reduced to shorten the optimization time. In this paper, A-ECMS is analyzed according to the simulation model in Chapter 1. Figure 2a shows the fitting curves of the instantaneous fuel consumption of the engine and Figure 2b shows the equivalent fuel consumption of the battery (the battery SOC is 0.5 and the equivalent factor is 1). It can be seen that, at a certain speed, the relationship between the instantaneous fuel consumption and the output torque of the engine can be divided into three sections. For HEVs, the actual working requirement of the engine is to avoid working in the low-load area (the first section). Therefore, the linear and quadratic functions are used to fit the second and third sections of the instantaneous fuel consumption curve of the engine. The intersection of the two curves was recorded as \(T_{opt}\). The battery equivalent fuel consumption curve can be divided into two sections. Each curve is composed of two quadratic functions, and the point where the motor output torque is zero is the connection point of the two quadratic curves. Equations (13) and (14) are fitting functions, where \(a_{i}\), \(b_{i}\), \(c_{i}\), and \(d_{i}\) (\(i=0,1,2\)) are the fitting parameters. According to Figure 2 and the definition of the convex function, it can be determined that, at a certain speed, the instantaneous fuel consumption of the engine is a convex function on \([T_{e,\min},T_{e,\max}]\), and the instantaneous equivalent fuel consumption of the battery is a convex function on \([T_{m,\min},T_{m,\max}]\). \[\dot{m}_{e}=\left\{ \begin{array}{cc}a_{0}\times T_{e}+a_{1},&T_{e,\min} \leq T_{e}\leq T_{opt}\\ b_{0}\times T_{e}^{2}+b_{1}\times T_{e}+b_{2},&T_{opt}\leq T_{e}\leq T_{e, \max}\\ 0,&T_{e}=0\end{array} \right. \tag{13}\]  
\[\dot{m}_{equ}= \begin{cases}c_{0}\times T_{e}^{2}+c_{1}\times T_{e}+c_{2},&T_{ m}\geq 0\\ d_{0}\times T_{e}^{2}+d_{1}\times T_{e}+d_{2},&T_{m}\leq 0\end{cases}  \tag{14}\]  
The goal of ECMS is to minimize the equivalent fuel at each instant. Because the instantaneous fuel consumption of the engine and the instantaneous equivalent fuel consumption of the battery are convex functions, the nature of the convex function shows that the minimum value can only be obtained in \([0,T_{r},T_{r}-T_{opt},U_{\max},U_{\max}]\), and among them,  
\[\left\{