## 2 Overvoltage of a Battery
### Problem of Low Cell Balancing Current
#### 2.1.1 In the Case of Having Cell Balancing and in the Case of Not Having It  
The authors conducted simulations for four series-connected batteries with and without cell balancing. Bat #1, Bat #2, Bat #3, and Bat #4 were set to 60 Ah, 59.4 Ah, 60.6 Ah, and 60 Ah, respectively, assuming a battery tolerance of \(\pm 1\%\). Figure 1a shows the simulation model with and without cell balancing. Figure 1b shows the battery aging settings, where all aging cycle variables for the batteries are the same, except for the values related tocapacity, which differ according to the settings. The batteries in Figure 1 use the lithium-ion battery library provided by MATLAB R2019a, and the CC-CV battery charger library also provided by MATLAB R2019a was used. Figure 2 shows the Stateflow for the chart shown in Figure 1. Figure 2a shows the Stateflow applied to Charts #1, #2, #3, and #4, which demonstrates a simple voltage comparison cell balancing method commonly used in the industry. Cell balancing is activated when a voltage difference of 30 mV or more is detected among the batteries and stops when the voltage difference decreases to 10 mV or less. Figure 2b shows the Stateflow for Chart #5, which illustrates the conditions for charging and discharging. Charging was performed at 2 C and ended at 0.5 C in the constant voltage (CV) region, while discharging was performed at 2 C.  
Figure 3 shows the simulation results of Figure 1, where (a) shows the voltage results and (b) shows the SOC results. The difference between the cases with and without cell balancing cannot be easily observed due to a low cell balance. In Figure 3a, it was confirmed that the voltage of Bat #3 was the highest during charging and the lowest during discharging. This phenomenon was explained in the paper \"Effect of capacity variation in series-connected batteries on aging\" as occurring because the discharge and charge of relatively smaller batteries occur relatively quickly. Figure 1: Serial battery simulation. (**a**) Serial-connected model; (**b**) with cell balance model; (**c**) battery aging settings. During the constant current (CC) charging phase, if the ideal capacity of the batteries is the same, the voltage of each battery should be equally distributed, as shown in Equation (1). \[\mathrm{V_{batt1}}=\mathrm{V_{bat2}}=\mathrm{V_{bat3}}=\mathrm{V_{batt4}} \tag{1}\]  
However, if there is a deviation in the battery capacity, as shown in Figure 3, Equation (2) applies. In Equation (2), the initial voltage of each battery is different, there is no voltage limit during the CC phase, and the rise in voltage is determined by the internal resistance of each battery because the charging current is the same for all batteries. \[\mathrm{V^{\prime}_{batt1}}=\mathrm{V_{batt1}}+\mathrm{I_{CC}}\times\mathrm{R_{ batt1}},\mathrm{V^{\prime}_{bat2}}=\mathrm{V_{bat2}}+\mathrm{I_{CC}}\times\mathrm{R_{ bat2}},\mathrm{V^{\prime}_{bat3}}=\mathrm{I_{CC}}\times\mathrm{R_{bat3}}, \mathrm{V^{\prime}_{batt4}}=\mathrm{V_{bat4}}+\mathrm{I_{CC}}\times\mathrm{R_{bat4}}\]  
\[\mathrm{V^{\prime}_{battn}}\text{: each battery's charging voltage; }\mathrm{V_{battn}}\text{: each battery's initial charging voltage;} \tag{2}\]  
\(\mathrm{I_{CC}}\): steady-state current of the constant current region; \(\mathrm{R_{battn}}\): the DCIR for each battery's SOC. Figure 3: One cycle of simulation results. (**a**) Serial-connected voltage; (**b**) with cell balance voltage; (**c**) serial-connected SOC; (**d**) with cell balance SOC. Figure 2: Stateflow of chart in Figure 1a. (**a**) Charts #1, #2, #3, and #4; (**b**) chart #5. During the CV phase that follows, voltage redistribution occurs due to the voltage limit of the charging voltage and the different SOCs, as shown in Equation (3). \[