# Online SoC Estimation of Lithium-Ion Batteries Using a New Sigma Points Kalman Filter
## 2 Modeling and Parameter Identification  
### Acquisition of SoC-OCV Curve  
The battery used in the test is a 18,650 cell and the basic parameters of the battery are shown in Table 1.  
The accurate calibration of the SoC-OCV relationship curve is the first step to accurately estimating the SoC. It is mentioned in more literature that the interval static method as the most suitable way to obtain the SoC-OCV curve. The instruments involved in this study to obtain the SoC-OCV mapping relationship mainly include the Arbin BT2000 Battery Test System and Temperature Chamber.  
The charge and discharge tests of the battery were carried out by the interval static method at a temperature of 25 \({}^{\circ}\)C. The steps are shown in Figure 1.  
We obtained a total of 11 sets of data for each of \(OCV_{\mathit{c}ht}\) and \(OCV_{\mathit{dis}}\). After taking the average of \(OCV_{\mathit{c}ht}\) and \(OCV_{\mathit{dis}}\) under the same SoC value, the results are shown in Table 2.  
The \(SSE\) refers to the sum of squares due to error. This statistical parameter calculates the sum of the squares of the error between the fitted data and the corresponding points  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & \begin{tabular}{c} **Capacity** \\ **Rating** \\ \end{tabular} & **Nominal Voltage** & \
\begin{tabular}{c} **Upper Cut-Off** \\ **Voltage** \\ \end{tabular} &
\begin{tabular}{c} **Lower Cut-Off** \\ **Voltage** \\ \end{tabular} \\ \hline
18,650 & 2000 mA h & 3.6 V & 4.2 V & 2.5 V \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of battery samples.  
Figure 1: Schedule of the charge and discharge tests.  
of the original data. The closer the \(SSE\) value is to 0, the better is the model selection and fitting, leading to successful prediction capabilities.  
\[SSE=\sum_{i=1}^{n}\omega_{i}(y_{i}-\hat{y_{i}})^{2} \tag{1}\]  
where, \(\omega_{i}\) is the weight, \(y_{i}\) is the actual value and \(\hat{y_{i}}\) is the predictive value.  
The adjusted R-square refers to the coefficient of determination, which is between 0 and 1 for this equation. The closer the coefficient is to 1, the stronger the explanatory power of the variables of the equation is for \(y\), and the model fits the data more accurately.  
The \(RMSE\) refers to the root mean square error, also called the standard deviation of the regression system. The closer its value is to 0, the better is the model selection and fitting, and the more successful is the data prediction.  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\omega_{i}(y_{i}-\hat{y_{i}})^{2}} \tag{2}\]  
In this work, three indicators of \(SSE\) adjusted R-square and \(RMSE\) were used to compare the fit of the \(SoC\)-\(OCV\) mapping curve at different orders, as shown in Table 3.  
Table 3 indicates that when the fitting order is 7, the quality of fit is the best. The fitting formula of \(OCV\) related to \(SoC\) is as follows (Equation (3)) and the obtained \(SoC\)-\(OCV\) relationship curve is shown in Figure 2.  
\[\begin{array}{l}U_{OCV}=-30.52\times SoC^{7}+115.94\times SoC^{6}-169.67\times SoC ^{5}+115.55\times SoC^{4}\\ -31.71\times SoC^{3}-0.85\times SoC^{2}+2.13\times SoC+3.30\end{array} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fitting Order** & \(SSE\) & **Adjusted R-Square** & \(RMSE\) \\ \hline
4 & 0.0007965 & 0.9981 & 0.01152 \\
5 & 0.0002463 & 0.9993 & 0.007019 \\
6 & 0.0001321 & 0.9995 & 0.005746 \\
7 & 0.00007738 & 0.9996 & 0.005079 \\
8 & 0.00005769 & 0.9996 & 0.005371 \\
9 & 0.00005408 & 0.9992 & 0.007354 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of the goodness of fit of the SoC-OCV curve.  
Figure 2: SoC-OCV mapping curve.