# Research on Equalization Strategy of Lithium Battery Pack Based on Multi-Layer Circuit
## 3 Multi-Layer Equilibrium System Model
### Cell Balancing Constraints  
In order to ensure the safety of the equalization process and prevent overcharge, over-discharge and overheating of the battery, it is necessary to impose constraints on the equalization process. The constraints imposed in this study are current and SOC. Equalizer current constraint: Excessive equalizing current causes serious heating of the equalizer and affects the service life of the equalizer. It is necessary to limit the equalizing current within a certain range. The current constraint can be expressed as:  
\[0\leq\mathrm{I_{m,i_{m}}}\leq\mathrm{I_{m,max}} \tag{6}\]where \(\rm I_{m,max}\) is the maximum value of equalization current. Battery current constraint: Current has a significant impact on battery life and performance. Higher current will result in increased battery heating, which can result in major safety issues. As a result, the charge and discharge currents of the battery must be kept within a particular range to ensure the battery's safe and reliable performance. Although the exterior current of the battery pack remains constant during equalization, the current passing through each battery varies due to the equalization current's influence. The discharge equalization current and charge equalization current will not occur at the same time since the equalizer works in BCM and DCM. To ensure that the working current of the battery is within the allowable range, the charge and discharge current of the battery is constrained respectively. The equalizing current flowing through the battery can be expressed by the following formula:  
\[\left\{ \begin{array}{l}\rm I_{1,i,c}=\sum\limits_{m=1}^{n}I_{m,i_{m}}\ \text{if}(I_{m,i_{m}}\geq 0)\\ \rm I_{1,i,d}=\sum\limits_{m=1}^{n}I_{m,i_{m}}\ \text{if}(I_{m,i_{m}}\leq 0) \end{array} \right. \tag{7}\]  
where \(\rm I_{1,i,c}\) and \(\rm I_{1,i,d}\) are the charge equalization current and discharge equalization current of battery i, respectively. The maximum value of battery current is the sum of charge equalization current and external current, and the minimum value of battery current is the sum of discharge current and external current. The battery current can be expressed as:  
\[\left\{ \begin{array}{l}\rm I_{i,c}=I_{ex}+I_{1,i,c}\\ \rm I_{i,d}=I_{ex}+I_{1,i,d}\end{array} \right. \tag{8}\]  
The current constraint can be formulated by:  
\[\left\{ \begin{array}{l}\rm I_{-max}\leq I_{i,c}\leq I_{i+max}\\ \rm I_{-max}\leq I_{i,d}\leq I_{i+max}\end{array} \right. \tag{9}\]  
where \(\rm I_{+max}\) represents the upper limit of charging and \(\rm I_{-max}\) represents the upper limit of discharging current. To ensure the normal operation of the battery, the maximum charging and discharging current of the battery should be within the constraint range. Battery SOC constraint: When the battery is overcharged and discharged, the performance and service life of the lithium battery will be affected to varying degrees, and even lead to safety accidents in serious cases. Therefore, the SOC of the battery must be constrained, the constraint relationship is as follows:  
\[\rm SOC_{i,min}\leq SOC_{i}\leq SOC_{i,max} \tag{10}\]  
where \(\rm SOC_{i,min}\) and \(\rm SOC_{i,max}\) represent the upper and lower limits of battery SOC in the group, respectively. This part builds a multi-layer equalization circuit model, analyzes the equalization current and battery current throughout the equalization process, and analyzes the external current, battery SOC and current, and equalization current limitations. According to Equations (6)-(9), the battery's equalization current can change depending on the external current, limiting the battery's maximum operating current. To shorten the equalization time, Equation (5) can obtain the optimal equalization current. The fluctuation of battery equilibrium current with internal resistance and SOC is only considered in Reference [36], and the maximum operating current of the battery was not taken into account. When the battery pack's external current is too high, the current loaded on the battery is easily exceeded.