# Lithium Battery SoC Estimation Based on Improved Iterated Extended Kalman Filter
## 2 Construction of Battery Model
### Equivalent Circuit Model Parameter Identification
#### 2.2.2 Second-Order Thevenin Model Parameter Identification  
In order to identify the other parameters in the equivalent circuit model, the HPPC experimental data at 90% SoC are studied as an example [27], where the response curve is shown in Figure 3.  
First, the parameter \(R_{0}\) is identified. According to the literature [28], the signal loaded instantaneously by the current is a high-frequency signal, which  
Figure 3: Experimental curve of HPPC at SoC = 90%. Figure 2: The OCV-SoC fitting curve. the capacitors of the two RC loops, so the voltage drop from point A to point B is mainly due to \(R_{0}\). And, when the current is withdrawn, since the voltage at the two ends of the capacitors does not change abruptly, but the voltage at \(R_{0}\) does [29], the voltage change from point C to point D is also due to \(R_{0}\). Thus, the value of \(R_{0}\) can be expressed according to Equation (5):  
\[R_{0}=\frac{|U_{A}-U_{B}|+|U_{C}-U_{D}|}{2\Big{|}I_{discharge}\Big{|}} \tag{5}\]  
where \(U_{A}\), \(U_{B}\), \(U_{C}\), and \(U_{D}\) represent the end voltage values at points A, B, C, and D, respectively, and \(I_{discharge}\) represents the HPPC experimental discharge pulse current. Next, parameters \(R_{1}\), \(C_{1}\), \(R_{2}\), and \(C_{2}\) are identified. During current loading, capacitors \(C_{1}\) and \(C_{2}\) are charged, and when the current is withdrawn, the electrical energy in the capacitors is consumed by resistors \(R_{1}\) and \(R_{2}\) in the two RC loops, respectively. The voltages at the ends of the capacitors are subsequently reduced, and the terminal voltages of the batteries then rise slowly [30]. Considering the DE section as a zero-input voltage response, the change in voltage can be expressed as follows:  
\[U_{t}=OCV-IR_{1}\mathrm{e}^{-\frac{t}{R_{1}C_{1}}}-IR_{2}\mathrm{e}^{-\frac{t }{R_{2}C_{2}}} \tag{6}\]  
In order to obtain the values of the circuit model parameters, the curves of the DE segment are fitted using the cftool toolbox in MATLAB, and the custom equation is selected to self-fit the fitted equation. The self-fitted equation is as follows:  
\[f(x)=\mathrm{OCV}(SoC)-b\mathrm{e}^{-dx}-c\mathrm{e}^{-cx} \tag{7}\]  
In order to make the fitting results as good as possible, the range of the variable values should be limited. Since the parameters \(b\) and \(c\) of the self-fitting equation represent the voltages at the ends of the two capacitors, respectively, the values of \(b\) and \(c\) must be greater than or equal to zero. The values of \(d\) and \(e\) represent the time constants of the two RC loops and should also be not less than 0. According to the above equation:  
\[ \begin{array}{l}R_{1}=\frac{b}{|I_{discharge}|},R_{2}=\frac{c}{|I_{discharge}|}\\ C_{1}=\frac{1}{R_{1}d},C_{2}=\frac{1}{R_{2}e}\end{array}  \tag{8}\]  
By performing the same identification at different SoC points, the parameter identification results for this battery equivalent circuit model can be obtained, as shown in Table 2.  
The results of [PERSON] et al. [31] show that the migration rate of ions slows down and the Ohmic internal resistance increases at lower temperatures, while the opposite is true when the temperature is high, according to Equation (7), which will lead to more obvious dynamic hysteresis characteristics of the battery. This will lead to a change in the identification results of the battery equivalence model. Therefore, in practical applications, HPPC experiments  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**SoC** & \(R_{0}\) & \(R_{1}\) & \(C_{1}\) & \(R_{2}\) & \(C_{2}\) \\ \hline
0.9 & 0.22 &  
must be carried out at specific temperature intervals to calibrate the battery equivalent model parameters at different temperatures, then the corresponding battery equivalent model parameters should be called according to the real-time temperature when estimating the SoC. This ensures that the estimation of the battery SoC is least affected by the temperature.