## 2 Lithium-Ion Battery Modeling and Experimental Setup
### Method of Constructing Fusion OCV Curve  
The OCV curve is part of the cell model and is critical for SOC estimation. However, the polarization effect affects the accuracy of the OCV test results [15]. The incremental test and low-current test are often used to measure the OCV curve. [PERSON] and [PERSON] [15] compared their performance in terms of discharge rate, SOC interval, relaxation time. Both [PERSON] [15] and [PERSON], et al. [27] agree that the incremental test is more suitable than the low current test for determining the relationship between OCV and SOC. However, the OCV curve based on the incremental test is usually obtained by polynomial fitting [28; 29; 30]. There are few control points obtained by the incremental test, and the fitting curve cannot reflect the non-linear electrode features. The low current test can reflect the characteristics of the electrode, but it is difficult to avoid the polarization effect due to the discharge by constant current. Figure 4a shows the test results of the incremental and low current tests. The incremental test provides 20 control points that are not affected by the polarization effects. The low-current test provides more non-linear characteristic information of the electrode, reflecting the OCV variation with the DOD. However, the polarization effect is evident with the continued discharge. Figure 4b shows the voltage difference between the two test control points:  
\[V_{k}^{(D)}=V_{k}^{(I)}-V_{k}^{(L)}, \tag{3}\]  
Figure 3: Schematic of the discharge current. (**a**) Incremental OCV test. (**b**) Partial zoom view of incremental OCV test. (**c**) Dynamic discharge test. (**d**) Partial zoom view of the dynamic discharge test. where \(V_{k}^{(l)}\) is the voltage of the incremental test in the \(k\)th control point, \(V_{k}^{(L)}\) is the corresponding control point voltage of the low-current test, and \(V_{k}^{(D)}\) is the voltage difference. The influence of the polarization effect for the low current test is non-linear with the increase of DOD, as shown in Figure 4b. To weaken the polarization effect while retaining the non-linear characteristics of the electrode, we propose a high-precision OCV curve fusion method based on the low-current test and incremental test data. Firstly, the low-current test control points corresponding to the same discharge depth range are obtained according to the incremental test control points. Then, the voltage difference between the two test control points is calculated, and the ratio of the corresponding voltage difference is taken as the fusion coefficient. Finally, the first-order derivative of the low-current test is taken as the variable, and the final fusion OCV curve is obtained based on the fusion coefficient and the incremental test control points. We first obtain the mapping curve of DOD with OCV and then convert to get the mapping curve of SOC with OCV. The control points obtained by the incremental test can be expressed as the set \(A=\left\{\left(DOD_{k}^{(I)},OCV_{k}^{(I)}\right)\right|k=1,2,\cdots,N\right\}\), where \(OCV_{k}^{(L)}\) is the measured OCV at the \(k\)th measured DOD value \(DOD_{k}^{(I)}\), \(N\) is the number of control points. The first \(N-1\) measured DOD of the incremental test are obtained at equal DOD intervals, and the last DOD is obtained by the cutoff voltage, where the length might differ from the previous DOD intervals. The measurements of the low-current test can be expressed as the set \(B=\left\{\left(DOD_{k}^{(L)},OCV_{k}^{(L)}\right)\right|k=1,2,\cdots,M\right\}\), where \(OCV_{k}^{(L)}\) is the measured OCV at the \(k\)th measured DOD value \(DOD_{k}^{(L)}\), \(M\) is the number of the measurements, and \(M\) is far greater than \(N\). We divide set \(B\) into \(N-1\) sub-intervals according to the DOD of set \(A\), and then fuse each interval data.