# Research on Multi-Mode Braking Energy Recovery Control Strategy for Battery Electric Vehicles
## 2 Brake Energy Recovery Control Strategy
### Conventional Brake Energy Recovery Control Strategy
#### 2.3.1 Front and Rear Axle Brake Force Distribution
In summary, the expression for the ideal I-curve can be obtained as follows:  
\[F_{ur}=\frac{1}{2}[\frac{mg}{h}\sqrt{b^{2}+\frac{4 hL}{mg}F_{uf}}-(\frac{mgb}{ h}+2F_{uf})] \tag{16}\]  
(2)  
ECE regulation front brake force distribution requirement curve:  
The ECE R13 regulation is an essential regulatory standard for vehicle braking systems. It specifies the technical requirements and test methods for vehicle braking devices to ensure the vehicles' safety, reliability, and performance in terms of braking meet specific standards. The regulation provides detailed technical specifications and standards for designing, manufacturing, and testing vehicle braking systems. It aims to ensure the safety and performance of vehicles in braking meet specific standards and improve driving safety. It is a high reference for the regulation's significance. The ECE R13 braking regulations require that the front wheels utilize the position of the coefficient of adhesion curve to be above the rear wheels, which utilize the coefficient of adhesion curve; the vehicle front and rear wheel braking force coefficient of adhesion curves, as well as braking strength and road friction coefficients between the expressions, are as follows [25]:  
\[z\geq 0.1+0.85(\varphi-0.2) \tag{17}\]  
\[\left\{ \begin{array}{l}\varphi_{f}=\frac{F_{uf}}{F_{zf}}\leq\frac{(z+0.07)}{0. 85}\\ \varphi_{r}=\frac{F_{ur}}{F_{zr}}\leq\frac{(z+0.07)}{0.85}\end{array} \right., \varphi_{f}\geq\varphi_{r} \tag{18}\]  
where \(z\) is the braking strength, \(\varphi_{f}\) is the front wheel attachment coefficient, \(\varphi_{r}\) is the rear wheel attachment coefficient, and \(\varphi\) is the road friction coefficient. The equation for the front and rear wheel braking force curves for the ECE braking regulations can be obtained by association as follows:  
\[\left\{ \begin{array}{l}F_{uf}=\frac{(z+0.07)(b+zh)mg}{0.85L}\\ F_{ur}=\frac{(z+0.07)(a-zh)mg}{0.85L}\end{array} \right. \tag{19}\]  
The collation gives the ECE R13 regulatory limit curve equation as follows:  
\[\frac{(F_{uf}+F_{ur})^{2}h}{mgL}+\frac{(F_{uf}+F_{ur})(0.07h+b)+0.07 mgb}{L}-0. 85F_{uf}=0 \tag{20}\]  
3. The f-line set brake force distribution requirement curve:  
The f-line group is the vehicle with different coefficients of adhesion road braking; the rear wheels are not locked, while the front wheels are locked. On the front wheel braking force relationship curve, the front wheels should be locked to meet the formula [21]:  
\[F_{xbf}=\varphi F_{zf}=\varphi(\frac{mgb}{L}+\frac{F_{xbf}h}{L}) \tag{21}\]  
where \(F_{xbf}\) is the vehicle's front brake force and \(F_{xbf}\) is the vehicle's rear brake force, \(F_{xb}=F_{xbf}+F_{xbr}\), which, when brought into the above equation, gives the line group equation as follows:  
\[F_{xbr}=\frac{(L-\varphi h)F_{xbf}}{\varphi h}-\frac{mgb}{h} \tag{22}\]  
To synthesize the joint road surfaces and their coefficients of adhesion in daily driving, take \(\varphi=0.7\). In summary, to ensure the safety and stability of vehicle braking, the vehicle's front and rear axle braking force distribution should be carried out in the closed interval surrounded by the I-curve, the f-line group, and the ECE regulation restriction curve, as shown in Figure 4.  
As can be seen from Figure 4, when the braking intensity \(z\leq z_{A}\) is a slight braking force demand, the rear axle braking force \(F_{ur}\) is zero, and the braking force is all assigned to the front axle. At this point, one can obtain the vehicle braking front and rear axle braking force and braking strength of the relationship between these, which, when brought into the calculation, can be derived as \(z_{A}=0.125\):  
\[z=\frac{F_{uf}+F_{ur}}{G} \tag{23}\]  
When \(z_{A}<z\leq z_{B}\), at this time, for small and medium braking power demand, the calculation can be obtained as \(z_{B}=0.525\). When \(0.125<z<0.525\), the front and rear axle braking power along the curve of the AB section for the distribution, and the vehicle front and rear axle braking power should meet the following formula:  
\[\left\{