# Research on Multi-Mode Braking Energy Recovery Control Strategy for Battery Electric Vehicles
## 3 Optimization of Control Strategies Based on Genetic Algorithms
### Genetic Algorithm for Fuzzy Controller Optimization  
Currently, there are three standard optimization methods for fuzzy controllers: one is to optimize the formulated fuzzy rules; the second is to optimize the fuzzy control input and output variables' affiliation functions; and the third is to optimize both simultaneously. In this paper, the second optimization method is used to iteratively optimize the affiliation functions of the input and output variables. The optimization process is shown in Figure 8.  
Figure 8: Fuzzy control optimization process.  
Selection of optimization variables  
This paper mainly selects the optimization variables based on the characteristic parameters of the fuzzy controller's affiliation function, which refer to each affiliation function's inflection points and boundary points. The input variables of the fuzzy controllers designed in the conventional brake energy recovery control strategy use ZMF-, GAUSSMF-, and SMF-type affiliation functions. Equation (39) shows the expression of the GAUSSMF-type affiliation function, which has two characteristic parameters of mean \(\sigma\) = 0.5 and standard deviation \(c\) = 0.2.  
\[f(x;\sigma,c)=e^{\frac{-(x-c)^{2}}{2\sigma^{2}}} \tag{39}\]  
Equation (40) and Equation (41) are expressions for ZMF-type and SMF-type affiliation functions, respectively, with two characteristic parameters, \(a\) and \(b\), for ZMF-type and two characteristic parameters, \(c\) and \(d\), for SMF-type.  
The ZMF-type affiliation function expression is as follows [29]:  
\[f(x;a,b)=\left\{\begin{array}{l}1,x\leq a\\ 1-2(\frac{x-a}{b-a})^{2},a\leq x\leq\frac{a+b}{2}\\ 2(\frac{x-a}{b-a})^{2},\frac{a+b}{2}\leq x\leq b\\ 0,x\geq b\end{array}\right. \tag{40}\]  
The SMF-type affiliation function expression is as follows:  
\[f(x;c,d)=\left\{\begin{array}{l}0,x\leq c\\ 2(\frac{x-c}{d-c})^{2},c\leq x\leq\frac{c+d}{2}\\ 1-2(\frac{x-d}{d-c})^{2},\frac{c+d}{2}\leq x\leq d\\ 1,x\geq d\end{array}\right. \tag{41}\]  
Since all three input variables of the fuzzy controller use ZMF-type, GAUSSMF-type, and SMF-type affiliation functions, respectively, there are six optimization parameters for each input variable, totaling 18 optimization parameters. The location of the optimization variables for each input affiliation function is selected, as shown in Figure 9.  
Fuzzy controller output variables using a ZMF-type, three GAUSSMF-type, and an SMF-type affiliation function for ten optimized variables. The output variable affiliation function optimized variable position selection, as shown in Figure 10.  
Figure 9: Input optimization variable position selection: (**a**) brake strength affiliation function optimization variables; (**b**) vehicle speed affiliation function optimization variables; (**c**) SOC affiliation function optimization variables.  
In summary, 28 optimization variables were selected from the input and output variables. After the optimization variables are chosen, the value range of each optimization variable is constrained to avoid excessive cross-overlapping between different fuzzy subset affiliation functions, and the value range of optimization variables is shown in Table 3.