## 2 Fuzzy Energy Management Strategy (F-EMS) of HEVs
At this point, the engine works at optimum efficiency, and the residual torque is supplemented by the motor or used to charge the battery;  
(3) When the battery is low, the motor works as much as possible in the generator state. At this point, the vehicle is driven primarily only by the engine, the engine works near the optimal efficiency curve, and the remaining battery is charged. In summary, the fuzzy rules of the F-EMS of HEVs consist of a series of \"if..., then... \" fuzzy conditional statements in the following form:  
\[\left\{ \begin{array}{l}\text{Rule 1}:\text{If }X_{in}\text{ is }A_{1}\text{ and }Y_{in}\text{ is }B_{1},\text{ then }Z\text{ is }C_{1},\\ \dots\\ \text{Rule }i:\text{If }X_{in}\text{ is }A_{i}\text{ and }Y_{in}\text{ is }B_{i},\text{ then }Z\text{ is }C_{i},\\ \dots\\ \text{Rule }m:\text{If }X_{in}\text{ is }A_{m}\text{ and }Y_{in}\text{ is }B_{m},\text{ then }Z\text{ is }C_{m}. \end{array} \right., \tag{2}\]  
where Rule \(i\) represents the \(i\)-th rule, \(m\) is the total number of fuzzy rule, \(Z\) is the rule output linguistic variable, here referring to the actual given engine output torque \(T_{e}\), \(X_{in}\) and \(Y_{in}\) represent the regular input linguistic variables, here referring to the SOC of power battery and the total demand torque \(T_{r}\), and \(A_{i}\) and \(B_{i}\) are the fuzzy sets SOC {VL, L, N, H, VH} and \(T_{r}\) {VS, S, RS, M, RB, B, VB}, corresponding to the input variables \(X_{in}\) and \(Y_{in}\), respectively. \(C_{i}\) is the fuzzy set corresponding to the output variable \(T_{e}\) {VS, S, RS, RM, M, VM, RB, B, VB}, wherein VL, L, N, H, and VH respectively indicate battery SOC from low to high: very low, low, normal, high, very high, while VS, S, RS, RM, M, VM, RB, B, and VB respectively represent the demand torque and engine output torque from small to a large extent: very small, small, relatively small, medium-small, medium, medium-large, relatively large, large, very large. In summary, the control rules of the fuzzy controller for HEV are shown in Table 1.  
In the F-EMS, the membership function settings of the demand torque, the battery SOC, and the given torque signal of the engine are respectively shown in Figures 2-4. According to the basic principle of fuzzy control, as more experience is learned, the division of the control rules becomes finer, and the membership function become narrower and thinner, such that the scope of control can be divided more finely and the control be more sensitive; otherwise, the control will be rougher and more stable. In the F-EMS of HEVs, if different driving cycles can be identified online and the fuzzy control rules or membership functions can be adjusted and optimized accordingly, the control effect of the F-EMS can be improved. According to the fuzzy rules, the fuzzy output rules of the HEV fuzzy controller are shown in Figure 5 through fuzzy reasoning. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**SOC** & **Tr** & **VS** & **S** & **RS** & **M** & **RB** & **B** & **VB** \\ \hline VL & VS & VS & S & RS & RM & M & VM \\ L & VS & S & RS & RM & M & VM & RB \\ N & S & RS & RM & M & VM & RB & B \\ H & RS & RM & M & VM & RB & B & VB \\ VH & RM & M & VM & RB & B & VB & VB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Control rules set by the fuzzy controller. Figure 4: Engine output torque membership function. Figure 5: Fuzzy inference input–output map. Figure 3: Battery state-of-charge (SOC) membership function. Figure 2: Demand torque membership function.