# Online SoC Estimation of Lithium-Ion Batteries Using a New Sigma Points Kalman Filter
## 3 Proposed Algorithm Design and Flow  
The new proposed algorithm is based on the Stirling interpolation, and UT transformation to determine Sigma points set 1 and Sigma points set 2 to start the first cycle, as shown in Figure 5. The two sets of the sampling points have the same mean and covariance as the system state distribution. After updating the time and measurement, the state estimation, \(\overline{x}c_{k}\), was compared with \(\overline{x}u_{k}\) to update \(\overline{x}c_{k}\). When the termination conditions were met, the algorithm entered the second cycle. Based on the Stirling interpolation, Sigma point set 3, were determined. Then, after the time and measurement update, the posterior mean and covariance of the system state were calculated. \begin{table}
\begin{tabular}{c c c c c} \hline \(R_{0}\) (\(\Omega\)) & \(R_{1}\) (\(\Omega\)) & \(C_{1}\) (F) & \(R_{2}\) (\(\Omega\)) & \(C_{2}\) (F) \\ \hline
0.07009 & 0.002101 & 120.8 & 0.03927 & 1327.0 \\ \hline \end{tabular}
\end{table} 
Table 4: Parameter identification results of ECM. Figure 4: Current excitation curve under DST condition. The charge and discharge time of the battery under a certain working condition was set to \(N\) seconds. S1: Initialization. Figure 5: Flow chart of the proposed algorithm. (1) The SoC-OCV relationship curve and the 2-RC ECM model parameters, \(R_{0}\), \(R_{1}\), \(R_{2}\), \(C_{1}\), and \(C_{2}\), were prepared. (2) The initial value of SoC was guessed and the initial covariance matrix \(P_{0}\) at the first second was set. (3) The noise, \(\omega(k)\), and the measuring equation noise \(\
u(k)\) were set. (4) The given interval length of the central difference transform was set to \(h\) = 1.6, and the state vector dimension was set to \(L\) = 3. S2: Determination of weight. The weights, \(W_{c1}\), \(W_{m}\), and \(W_{c}\), were calculated as follows:  
\[W_{c1}=\frac{1}{4\times h^{2}} \tag{22}\]  
\[W_{m}(1)=\frac{\lambda}{L+\lambda} \tag{23}\]  
\[W_{c}(1)=\frac{1}{L+\lambda}+1-\alpha^{2}+\beta \tag{24}\]  
When \(jj\) goes from 2 to \(2L+1\),  
\[W_{m}(jj)=\frac{1}{2\times(L+\lambda)} \tag{25}\]  
\[W_{c}(jj)=\frac{1}{2\times(L+\lambda)} \tag{26}\]  
And \(\alpha\) = 0.01, \(\beta\) = 2, \(\lambda\) = 3 \(\times\)\(\alpha^{2}-L\). S3: In the first cycle of SoC estimation, time \(t\) goes from 2 to \(M\) seconds. (1) Determination of the Sigma points sets. a. The Stirling interpolation was used to determine the Sigma points set 1. \[\chi c_{k-1}=\left[\overline{\chi}c_{k-1},\overline{\chi}c_{k-1}+h\sqrt{P_{k- 1}},\overline{\chi}c_{k-1}-h\sqrt{P_{k-1}}\right] \tag{27}\]  
b. The UT transformation was used to determine the Sigma points set 2. \[\chi u_{k-1}=\left[\overline{\chi}u_{k-1},\overline{\chi}u_{k-1}+\sqrt{(L+ \lambda)P_{k-1}},\overline{\chi}u_{k-1}-\sqrt{(L+\lambda)P_{k-1}}\right] \tag{28}\]  
(2) Time update and calculation of the prior mean and covariance. The Cholesky decomposition of the covariance was performed. Figure 4:0  
Figure 4:1  
Figure 4:2  
\[ \begin{array}{l}Pc_{k|k-1}=\sum\limits_{i=1}^{L}W_{c1}(i)\Big{\{}\Big{[}A( \overline{\chi}c^{i}{}_{k|k-1}+h\sqrt{P_{k-1}})+Bu(t-1)\Big{]}-\Big{[}A( \overline{\chi}c^{i}{}_{k|k-1}-h\sqrt{P_{k-1}})+Bu(t-1)\Big{]}\Big{\}}\\ \Big{\{}\Big{[}A(\overline{\chi}c^{i}{}_{k|k-1}+h\sqrt{P_{k-1}})+Bu(t-1)\Big{]} -\Big{[}A(\overline{\chi}c^{i}{}_{k|k-1}-h\sqrt{P_{k-1}})+Bu(t-1)\Big{]} \Big{\}}^{T}+\sum\omega\end{array}  \tag{32}\]  
\[\chi u^{i}{}_{k|k-1}=A\overline{\chi}u_{k-1}+Bu(t-1) \tag{33}\]  
\[\overline{\chi}u_{k|k-1}=\chi u^{i}{}_{k|k-1}=\sum\limits_{i=0}^{2L}W_{m}(i) \chi u^{i}{}_{k|k-1} \tag{34}\]\[Pu_{k|k-1}=\sum\limits_{i=0}^{2L}W_{c}(i)\Big{[}\chi u^{i}{}_{k|k-1}-\overline{x} u_{k|k-1}\Big{]}\Big{[}\chi u^{i}{}_{k|k-1}-\overline{x}u_{k|k-1}\Big{]}^{T}+ \sum\omega \tag{35}\]  
\[\text{where, }A=\left( \begin{array}{ccc}1&0&0\\ 0&e^{-\frac{\Delta t}{1}}&0\\ 0&0&e^{-\frac{\Delta t}{2}}\end{array} \right),B=\left[ \begin{array}{c}-\frac {\eta\Delta t}{Q_{N}}\\ R_{1}(1-e^{-\frac{\Delta t}{1}})\\ R_{2}(1-e^{-\frac{\Delta t}{2}})\end{array}