## 2 The Second Order Dynamic Model of the Cell and the Model of the LiBP
### The Model of LiBP  
Suppose that for the LiBP formed by \(N_{s}\) cell modules connected in series, each cell module includes \(N_{p}\) cells connected in parallel; the notations of the quantities are describe in Figure 3.  
For the cell module, which has some cells with different initial SoCs connected in parallel, after a certain period of time, the SoC of the cells in the module will balance itself. Then, the cells will have the same SoC and a voltage drop on two electrodes. The model of a LiBP is converted into a model of the cell modules connected in series, as shown in Figure 3. Thus, the average SoC of the cell module is the SoC of the cells in that module after a certain period of time. Illustration of the process of self-balancing of currents through four cells, and the SoCs and voltages of cells in a cell module are shown in Figures 4 and 5, respectively, in which the initial SoCs of four cells are 0.9 (90%), 0.85 (85%), 0.8 (80%) and 0.75 (75%). The internal resistances of cells \(R_{0}\) are 0.0013 \(\Omega\), 0.0012 \(\Omega\), 0.0012 \(\Omega\) and 0.0011 \(\Omega\). After a period of 200 s, the SoCs of four cells are balanced by themselves; all four cells have the same SoC and the same voltage drop on two electrodes.  
Figure 3: The model of lithium-ion battery pack (LiBP).  
We suppose that all the cells in the pack are the same kind, and are represented by a second order dynamic model with parameters \(\eta,Q,\gamma,M_{0},M,R_{0},R_{1}C_{1},R_{2}C_{2},R_{1},R_{2}\). The relationship between open-circuit voltage (OCV) and SoC according to the working temperature of the cell has been determined experimentally [6]. Define the parameters of the LiBP as follows:  
\(\mathbf{SOC}_{0}\in R^{(N_{y}\times N_{p})}\), \(\mathbf{R}_{0}\in R^{(N_{y}\times N_{p})}\) are the matrices representing the initial SoC, the resistance \(R_{0}\) of the cells in the pack, respectively.  
The initial SoC (at the time \(k=0\)) of the cell module \(i^{th}\) is counted as  
\[\overline{z}_{0}^{(i)}=\frac{1}{N_{p}}\sum_{j=1}^{Np}\mathbf{SOC}_{0}(i,j),i=1, 2,...,N_{s} \tag{7}\]  
The internal resistance of the cell module \(i^{th}\) is  
\[\overline{R}_{0}^{i}=\frac{1}{\sum_{j=1}^{N_{p}}\frac{1}{\overline{\mathbf{R} _{0}(i,j)}}}\;i=1,2,...,N_{s} \tag{8}\]  
The capacity of the cell module \(i^{th}\) is  
\[Q^{(i)}=\sum_{j=1}^{N_{p}}Q=N_{p}Q\;i=1,2,...,N_{s} \tag{9}\]  
Figure 4: The currents through four cells.  
Figure 5: The changing of the cell module voltage and the states of charge (SoC) of four cells in the cell module.  
The LiBP now is modeled by \(N_{s}\) cell modules connected in series. Normally, we need \(N_{s}\) estimators to estimate the SoC of \(N_{s}\) cell modules, and then average SoC of LiBP is average of the estimated SoC of \(N_{s}\) cell modules. This method has a large calculation burden if \(N_{s}\) is large, as the estimation algorithm is then complex, and noise is considered. In this research, by only using two estimators, the average SoC of a pack is estimated by the first estimator; the second estimator is used to estimate the SoC differences for the cell modules from the average SoC of the pack. Finally, the SoC of the cell modules is the sum of the average SoC of the pack and the SoC differences, respectively.