## 2 The Determination of the Battery Model  
### The SOC-OCV Relationship Curve  
SOC means the residual capacity of the battery, which is a very important parameter for electric vehicles, especially in the operation of electric vehicles, which is particularly important for the correct arrangement of the remaining quantity and time in the course of the travel. Its mathematical expression is \(SOC=\frac{Q_{r}}{Q_{N}}\times 100\%\). \(Q_{r}\) is the remaining capacity of the battery, and \(Q_{N}\) is the rated quantity of the battery under the same condition. However, since the value of \(Q_{r}\) is difficult to be determined directly, this formula is rarely used in actual calculations. The most commonly used formula to calculate the battery SOC is the ampere-hour integral (AH) formula:  
\[SOC=SOC_{0}+\frac{1}{Q_{N}}\int_{0}^{t}\eta ldt \tag{1}\]Among them Q\({}_{\text{N}}\) is the rated capacity of the battery, and \(\eta\) is an influence factor related to the temperature and the discharge rate of the battery (approximately equal to 1). \(I\) is the charge discharge current of the battery. The AH method is simple and easy to make calculations. It only needs to collect the real time current and obtains SOC by integrating the time with the current [18; 19; 20].  
OCV means the current battery capacity of the battery estimated by measured interruption voltage of the battery in the open circuit state. Since the battery has been static after a long time, its port voltage has a relatively stable function relationship with the SOC, so by OCV, a relatively true SOC estimate can be obtained [21]. Due to the accuracy of the OCV method, the SOC value estimated by this method will be used as the reference value of the proposed method. Therefore, it is necessary to find a method that can accurately find out the relationship between the battery OCV and SOC.  
Commercial lithium iron phosphate batteries (LIPB) of 3.2 V/2200 mAh were used as a research object, and the SOC-OCV relationship curve was measured using the following method (T = 25 \({}^{\circ}\)C) [22; 23; 24]:  
* The battery was discharged to the lower limit voltage;
* The battery was charged to the upper limit voltage with a constant charging current, and the charging voltage and the charging capacity were recorded;
* The battery was discharged at a constant discharge current, and the discharge was stopped when the discharge capacity was the same as the charge capacity in step B, and the discharging voltage and the discharging capacity were recorded; and
* Data of the charging voltage and the discharging voltage of the battery relative to the state of charge of the battery were respectively obtained;  
At the same state of charge, the open circuit voltage value took the intermediate value of the charging voltage and the discharging voltage, thereby obtaining a SOC-OCV relationship curve as was shown in Figure 1. A high-order polynomial function representing the OCV-SOC relationship was shown as follows:  
\[\begin{array}{ll}\text{{OCV}}&=4802.561\text{{SOC}}^{9}-22851.658\text{{ SOC}}^{8}+46094.256\text{{SOC}}^{7}\\ &-51274.985\text{{SOC}}^{6}+34272.375\text{{SOC}}^{5}-14077.056\text{{SOC}}^{4} \\ &+3484.326\text{{SOC}}^{3}-491.298\text{{SOC}}^{2}+35.357\text{{SOC}}+2.236\\ \end{array} \tag{2}\]