## 2 Lithium-Ion Battery Modeling and Experimental Setup
### Co-Estimation of SOC and Model Parameters Based on UKF
Compute the value of the one-step forecasting model at time \(k\) \(\textbf{X}_{k/k-1}^{(i)}=\overline{\textbf{X}}_{k-1}^{(i)},\textbf{u}_{k-1} \Big{]}\qquad k=0,1,2,\cdots,2n\) \(\hat{\textbf{X}}_{k/k-1}=\sum\limits_{i=0}^{2n}W_{i}^{(m)}\textbf{X}_{k/k-1}^ {(i)}\) \(P_{k/k-1}=\sum\limits_{i=0}^{2n}W_{i}^{(c)}\Big{[}\textbf{X}_{k/k-1}^{(i)}- \hat{\textbf{X}}_{k/k-1}\Big{]}\Big{[}\textbf{X}_{k/k-1}^{(i)}-\hat{\textbf{X }}_{k/k-1}\Big{]}^{T}+\textbf{Q}_{k-1}\) 4. Compute the one-step forecasting sample points \(\overline{\textbf{X}}_{k/k-1}^{(0)}=\hat{\textbf{X}}_{k/k-1}\) \(\overline{\textbf{X}}_{k/k-1}^{(i)}=\hat{\textbf{X}}_{k/k-1}+\gamma\big{(} \sqrt{P_{k/k-1}}\big{)}_{(i)}\qquad i=1,2,\cdots,n\) \(\overline{\textbf{X}}_{k/k-1}^{(i)}=\hat{\textbf{X}}_{



\[MAE=\frac{1}{M}{\sum_{i=1}^{M}}|x_{i}-\hat{x}_{i}|, \tag{11}\]

where, \(x_{i}\) is the reference value of SOC, \(\hat{x}_{i}\) is the estimation of SOC, \(M\) is the number of samples. ### Comparison of SOC Estimation under Different OCV Curve Construction

In this section, the SOC estimation results using the fusion OCV curve, the polynomial OCV curve based on incremental testing, and the OCV curve based on low-current testing are compared and analyzed, respectively. Note that their estimation methods all use the UKF method proposed in Section 3.2. Figure 7 shows the experimental results of a pulse constant current discharge test. Figure 7a shows the results of the measured terminal voltage, the estimated terminal voltage, and the estimated OCV of the proposed method, respectively. Figure 7b shows the terminal voltage estimation errors. It can be seen from Figure 7a,b that the estimated terminal voltage errors are only 0.005-0.02 V when the current suddenly appears and disappears, yet it is less than 0.001 V during the constant current discharge process. The zoom view in Figure 7a shows that the terminal voltage and OCV drop suddenly when the state switches from relaxation to discharge and then decreases slowly during the discharge; The terminal voltage rises rapidly when the state switches from discharge to relaxation, and slowly during the relaxation process. The estimated OCV converges to the reference value of the terminal voltage slowly. The above results show that the proposed method has an expected OCV estimation result and an excellent tracking performance of terminal voltage. Figure 7c is the SOC estimation results based on different OCV curve construction methods. It can be seen that the SOC estimation results based on the fusion OCV curve have the least estimation error during the discharge process, and can converge to the reference SOC value in the relaxation stage. The SOC estimation absolute error is shown in Figure 7d. It can be seen that the error based on the

Figure 7: Comparison of different OCV construction methods in pulse constant current discharge test. (**a**) Terminal voltage and OCV estimation result. (**b**) Estimation error of terminal voltage. (**c**) Comparison of SOC estimation result. (**d**) Comparison of SOC estimation error. fusion OCV curve is least, the absolute estimation error is basically controlled within 4%, and the maximum is 6%. The SOC estimation error based on the low-current test OCV curve increases with the increase of discharge time. Furthermore, the SOC estimation results cannot converge to the reference value in the relaxation stage, which is caused by the polarization effect. The SOC estimation error based on the increment test OCV curve does not accumulate with the discharge time, but the estimation results cannot converge to the reference value accurately in the relaxation stage. This is due to the fact that the increment test OCV curve cannot describe the OCV local non-linearity accurately. The numerical comparison of the estimated SOC error is shown in Table 2. It can be seen that the RMSE of SOC estimation results based on the fusion OCV curve is reduced by 1.09-2.96%, and the MAE is reduced by 1-3.08% compared with the other two methods.