# SOC Estimation with an Adaptive Unscented Kalman Filter Based on Model Parameter Optimization
## 2 Establishment of Battery Model and Parameter Identification
### Establishment of Lithium-Ion Battery Model  
The measured terminal voltage response curve of the Sanyo lithium battery with a rated capacity of 2.6 AH at a certain state (SOC = 0.9 with a constant current discharge of 260 mAh at 0.5 C) at the end of discharge is shown in Figure 1.  
The polarization effect of a lithium battery is usually expressed equivalently by an RC loop and is equivalent to the first-order zero input response after the battery is discharged. Its terminal voltage can be expressed as an exponential term (\(IRe^{\frac{\tau^{t}}{t}}\)). Region 2 in Figure 1 is the disappearance process of the polarization effect. Exponential curves are used to fit the single index, double index and triple index coefficients of region 2, and the fitting results are shown in Figure 2.  
Figure 1: Terminal voltage response at the end of discharge. Figure 2: Coefficient fitting of exponential curves. It can be seen from Figure 2 that the reduced chi-squared statistics of the double and triple exponential fittings are smaller than that of the single exponential fitting. That is, the random error effect of the double and triple exponential fittings are smaller than that of the single exponential fitting. At the same time, the correction decision coefficient of the double and triple exponential fittings is larger than that of the single exponential fitting and is closer to 1, and the fitting effect is better. Therefore, the double and triple exponential fittings can more accurately reflect the polarization effect of the battery. By comparing the results of the double and triple exponential fittings, the sum of the squares of the residuals of the double exponential fitting is less than that of the triple exponential fitting, and the correction decision coefficient of the double exponential fitting is also closer to 1 than that of the triple exponential fitting. The results show that the double exponential fitting effect is better than the triple exponential fitting effect. The reason for this fitting result is that although the third-order RC loop can theoretically better reflect the dynamic characteristics of the battery, the third-order RC loop has one more RC loop than the second-order RC loop, which means that the third-order RC loop has two more unknowns than the second-order RC loop in the process of data fitting by computer. Therefore, the fitting effect of the third-order RC loop is not as good as that of the second-order RC loop, and it is proved that the second-order RC model is better than the others for a lithium battery. On this basis, considered comprehensively, the second-order RC equivalent circuit model, as shown in Figure 3, is adopted in this paper. The model consists of three parts:  
Voltage source: \(V_{\text{oc}}\) represents the open circuit voltage of the power battery. In this paper, the influences of temperature and state of health (SOH) on the OCV are not considered, and the functional relationship between \(V_{\text{oc}}\) and the battery SOC is studied under the same temperature and SOH conditions. Additionally, the effect of corrosion on the battery is neglected because the focus of this paper is on the equivalent circuit model. Ohmic resistance: \(R\) is used to represent the internal resistance of the battery, which can be determined by the sudden change in voltage after the end of discharge, as shown in region 1 of Figure 1.  
RC loop circuit: Two links of a resistor and a capacitor are superposed to simulate battery polarization, which is used to simulate the process of voltage stabilization after discharge. Region 2 of Figure 1 shows the change in voltage influenced by the RC loop circuit. As shown in Figure 3, the functional relationship of the equivalent circuit model is as follows:  
\[\left\{ \begin{array}{c}E(t)=iR+u_{s}+u_{p}+U(t)=F(SOC(t))\\ i=\frac{u_{s}}{R_{s}}+C_{s}\frac{du_{s}}{dt}\\ i=\frac{u_{p}}{R_{p}}+C_{p}\frac{du_{p}}{dt}\end{array} \right. \tag{2}\]  
For the discretization of Equation (2), the state equation is solved as:  
\[\left[