## 3 Enhanced SoC-Balancing Strategy with Saturation
### Selection of the Current Limits in CDRs
For convenient analysis, auxiliary function \(f_{1}(x)\) is defined as  
\[f_{1}(x)=\left(\sqrt[3]{x-2}+2-x\right)^{-1},x=i_{\text{sum}}/i_{\text{statu}} \in(2,3] \tag{26}\]  
In fact, \(f_{1}(x)\) is positive, and it increases when \(x\) increases. Based on Inequation (16), the following relationship can be deduced:  
\[\frac{i_{\text{statu}}}{(w+2)i_{\text{statu}}-i_{\text{sum}}}<\frac{SoC_{10}}{ (w+1)SoC_{10}-SoC_{20}-SoC_{30}} \tag{27}\]  
Also, \(f_{2}(y)\) is defined as follows  
\[f_{2}(y)=\frac{y}{(w+1)y-SoC_{20}-SoC_{30}},y=SoC_{k0}\in(0,1] \tag{28}\]  
In fact, \(f_{2}(y)\) decreases while \(y\) increases, thus proving that  
\[\frac{SoC_{10}}{(w+1)SoC_{10}-SoC_{20}-SoC_{30}}<\frac{SoC_{20}}{wSoC_{20}-SoC _{30}} \tag{29}\]  
Combining Inequation (27) with Inequation (29), we attain the following: \(t_{0}<Q*SoC_{20}/i_{\text{statu}}\). In brief, the assumption in Inequation (24) is true as long as Inequation (16) is established. In series scenarios, the minimum \(i_{\text{statu}}\) is the same as that in parallel scenarios. However, there exists an upper limit \(i_{\text{statu}}\) for \(i_{\text{statu}}\). If \(i_{\text{statu}}\) exceeds \(i_{\text{statu}}\), a step change in the reference current will appear when some ESUs become desaturated. Furthermore, this leads to the deterioration of \(u_{\text{dc}}\) in actual systems. To improve power quality, this phenomenon ought to be avoided. According to Equation (7), \(CDR_{11}\) decreases and \(CDR_{23}\) increases after the balancing strategy is enabled. If \(CDR_{11}\) drops to \(i_{\text{statu}}/i_{\text{sum}}\) after \(CDR_{23}\) increases to \(i_{\text{sat}}/i_{\text{sum}}\), the unexpected step reference current is actually averted, which is expressed as  
\[\frac{i_{\text{satu}}}{i_{\text{sum}}}>\frac{SoC_{1}}{SoC_{1}+SoC_{2}+SoC_{3}} \tag{21}\]0  
Suppose that \(n\) is large enough to make each phase current constant in the beginning. If \(i_{\text{sat}}<i_{\text{sum}}-i_{\text{satu}}-i_{\text{sat}}<i_{\text{statu}}\), three currents are separately distributed as  
\[\frac{i_{\text{satu}}}{i_{\text{sum}}}>\frac{SoC_{1}}{SoC_{1}+SoC_{2}+SoC_{3}} \tag{21}\]1  
By substituting Equation (31) into \(CDR_{23}\), we have  
\[\frac{i_{\text{satu}}}{i_{\text{sum}}}>\frac{SoC_{1}}{SoC_{1}+SoC_{2}+SoC_{3}} \tag{21}\]2Combining Equations (3), (31), and (32) yields  
\[\frac{i_{\text{satu}}}{i_{\text{sum}}}>\frac{SoC_{1}}{SoC_{1}+SoC_{2}+SoC_{3}} \tag{21}\]3  
To obtain the critical \(i_{\text{satu}}\) in Equation (30), replace \(t_{2}\) with \(t_{1}\). After combination with Equation (31), \(CDR_{11}\) is rewritten as  
\[\frac{i_{\text{satu}}}{i_{\text{sum}}}>\frac{SoC_{1}}{SoC_{1}+SoC_{2}+SoC_{3}} \tag{21}\]4  
An analytical solution is hard to acquire, as Equation (34) is a high-order equation about \(i_{\text{satt}}\). Instead, tool software can be used to facilitate the solving procedure. Then, a numerical solution can be obtained, and that is exactly the value of \(i_{\text{satu}}\). When load change occurs, \(i_{\text{satt}}\) and \(i_{\text{satu}}\) require proper adjustment so that the equilibrium point is still accessible and the step current references are escapable. According to energy conservation, \(i_{\text{sum}}\) varies in the same direction and proportion when \(i_{o}\) suddenly changes. That is to say, the variation in \(i_{\text{sum}}\) directly reflects the variation in \(i_{o}\). Accordingly, \(i_{\text{satt}}\) and \(i_{\text{satt}}\) are rearranged as  
\[\frac{i_{\text{satu}}}{i_{\text{sum}}}>\frac{SoC_{1}}{SoC_{1}+SoC_{2}+SoC_{3}} \tag{21}\]5  
where \(i_{\text{sum0}}\), \(i_{\text{satt0}}\), and \(i_{\text{satt0}}\) are the initial values of \(i_{\text{sum}}\), \(i_{\text{satt}}\), and \(i_{\text{satt}}\), respectively. By using Equation (35) to update the two limits, \(i_{\text{satt}}/i_{\text{sum}}\) remains constant when \(i_{o}\) varies. By substituting Equation (35) into Equation (33), the required duration becomes \(i_{\text{sum0}}/i_{\text{sum}}\) times its original value.