# Active Equalization of Lithium-Ion Battery Based on Reconfigurable Topology
## 2 Double Layer Equalization Model
### Outer Layer: Power Distribution through Buck-Boost Converters  
As shown in Figure 5, the outer layer equalization topology is composed of \(m\) series-connected battery groups connected to \(m\) Buck-Boost converters, and the outputs of \(m\) converters are connected in series to supply stable power to the load.  
There are \(m\) converters connected in series with the same output current, so the converter with larger output voltage \(V_{cj}\) (\(j=1,2\ldots\ m\)) has larger output power. According to the working principle of the Buck-Boost converter, the relationship between the output voltage and the input voltage of the \(j\)th converter is given below  
\[V_{cj}=\frac{t_{\mathrm{on}}}{t_{\mathrm{off}}}V_{\mathrm{g}j}=\frac{t_{ \mathrm{on}}}{T-t_{\mathrm{on}}}V_{\mathrm{g}j}=\frac{\alpha_{j}}{1-\alpha_{j} }V_{gj} \tag{3}\]  
where \(t_{\mathrm{on}}\) is the on time of the MOSFET, \(t_{\mathrm{off}}\) is the off time of the MOSFET and \(T\) is the switching period of the MOSFET. \(\alpha_{j}\) can be calculated based on (4)  
\[\alpha_{j}=\frac{V_{cj}}{V_{gj}+V_{cj}} \tag{4}\]  
Figure 4: Discharge voltage curve.  
The output voltage of the converter can be controlled by adjusting the duty ratio \(\alpha_{j}\) of the converter. We arrange the battery groups in descending order of \(SOC_{gj}\) (\(j=1,\,2\,\dots\,\,m\)), which is the average SOC of the \(j\)th battery group. The converters connected to the battery groups with higher average SOCs are set with higher duty ratios and output higher voltages and powers; the converters connected to the battery groups with lower average SOCs are set with lower duty ratios and output lower voltages and powers. The relationship between the output voltages \(V_{\text{out}j}\) (\(j=1,\,2\,\dots\,\,m\)) can be expressed as  
\[V_{\text{c1}}\geq V_{\text{c2}}\geq\dots\geq V_{\text{cm}} \tag{5}\]  
\[V_{\text{c1}}+V_{\text{c2}}+\dots+V_{\text{cm}}=V_{\text{out}} \tag{6}\]  
\(k_{j}\) is the proportion of the output voltage of the \(j\)th converter of the total output voltage.  
\[k_{j}=\frac{V_{\text{c}j}}{V_{\text{out}}} \tag{7}\]  
The relationship between \(k_{j}\) (\(j=1,\,2\,\dots\,\,m\)) can be expressed as  
\[k_{1}\geq k_{2}\geq\dots\geq k_{m} \tag{8}\]  
\[k_{1}+k_{2}+\dots+k_{m}=1 \tag{9}\]  
From (4) and (7), the relationship between \(k_{j}\) and \(\alpha_{j}\) can be inferred as follows:  
\[\alpha_{j}=\frac{k_{j}\cdot V_{\text{out}}}{V_{gj}+k_{j}\cdot V_{\text{out}}} \tag{10}\]  
From (3) and (10), we can control the output voltage and power of the \(j\)th converter by adjusting \(\alpha_{j}\) by calculating the appropriate \(k_{j}\) according to the requirement of balancing speed, \(V_{\text{g}j}\), and \(V_{\text{out}}\). Until the average SOCs of all groups are equal (The average SOCs do not have to be completely the same. The average SOCs can be considered equal if the difference between any two average SOCs is less than \(\epsilon_{2}\)), all battery groups are set to discharge at the same power as (11).  
\[k_{1}=k_{2}=\dots=k_{m}=\frac{1}{m} \tag{11}\]  
The equalization of the outer layer is completed.  
Figure 5: Topology of outer layer.