## 2 Equivalent Circuit Modeling of the Lithium-Ion Batteries
### Mathematical Description of the MSCC Charging Method  
The charging time, charging cost, and charging capacity of the MSCC charging method for the Thevenin ECM of the LiB were derived by analyzing the electrical relationships. Based on [PERSON]'s voltage law for the circuit shown in Figure 1, the terminal voltage can be expressed as  
\[V_{T}=V_{R_{0}}+V_{R_{p}}+V_{b} \tag{6}\]  
where \(V_{Cp}=V_{R_{p}}\). Figure 6 illustrates a conceptual diagram of the MSCC method, showing the typical battery charging voltage and current shapes corresponding to varying SOC. Previous work [23] demonstrated that a five-stage CC approach is efficient. Although increasing the number of current stages can reduce charging time, it also increases the  
Figure 5: Identified characteristic curves. (**a**) OCV vs. SOC. (**b**) \(R_{0}\), \(R_{p}\), and \(R_{eq}\) vs. SOC. (**c**) \(\tau\) vs. SOC.  
complexity and implementation costs. From Figure 5, the charged battery capacity of each state is given by  
\[\Delta Q_{n}=\int I_{n-1}dt=I_{n-1}\Delta t_{n}=C_{b}\Delta V_{n}, \tag{7}\]  
where \(I_{n-1}\) denotes the charging current of the previous stage and \(\Delta V_{n}\) represents the increased OCV of each state compared to the OCV of the previous stage. From (7), the charging time required in the charging stage can be calculated as follows:  
\[\Delta t_{n}=\frac{C_{b}\Delta V_{n}}{I_{n-1}}. \tag{8}\]  
Figure 6: CC-CV charging curves. (**a**) Charging voltage (**b**) Charging current. (**c**) Charging temperature.  
The OCV \(V_{Cb,n}\) in stage \(s\), can be expressed as  
\[V_{b,s}=V_{cut-off}-I_{c,n-1}R_{eq}(SOC_{n}),\ n=2,3,4,5, \tag{9}\]  
where \(V_{cut-off}\) denotes the cut-off voltage of each current transition point and \(I_{c,n-1}\) represents the charging current of each stage.  
In the MSCC charging method illustrated in Figure 7, the current at each stage is selected constantly like the CC mode of the CC-CV technique: in the CC stage, the voltage increases to the cut-off voltage, where once the voltage reaches the cut-off, the current decreases exponentially, like the current behavior in the CV mode, with the voltage drop determined by the following equation; in the next stage, the voltage increases again according to the selected CC value. The charge current rate at each stage is constrained to be lower than the previous stages for better charging efficiency, which is defined as the ratio of the discharge capacity to the capacity charged in the battery during the charging process.  
Based on (4), the voltage drop at each stage can be expressed as  
\[\Delta V_{1}=V_{cut-off}-I_{c,1}R_{eq}(SOC_{2})-V_{b,1}, \tag{10}\]  
\[\Delta V_{n}=I_{c,n-1}R_{eq}(SOC_{n})-I_{c,n}R_{eq}(SOC_{n+1}),\ \ n=2,3,4, \tag{11}\]  
\[\Delta V_{5}=V_{cut-off}-I_{c,5}R_{eq}(SOC_{6}), \tag{12}\]  
where \(I_{c,n}\) denotes the charging current of stage. The current amplitude of the final stage was calculated to charge the battery to 100% capacity as follows:  
\[I_{c,5}=\frac{V_{cut-off}-V_{b,5}}{R_{eq}(SOC_{6})}. \tag{13}\]  
Therefore, the total charging time can be obtained by summing the times of the MSCC charging stage as follows:  
\[CT_{t}=\sum_{n=1}^{5}\Delta t_{n}=\sum_{n=1}^{5}\frac{C_{b}\Delta t_{n}}{I_{c,n}}. \tag{14}\]  
Figure 7: Conceptual diagram of the MSCC charging method.  
The energy loss (EL) was calculated by integrating the ohmic loss of each CC stage as follows:  
\[L_{tot}=\sum_{n=1}^{5}\Delta L_{n}=\sum_{n=1}^{5}\int I_{n-1}^{2}R_{eq}(SOC)dt. \tag{15}\]