## 3 Data Analysis
### Voltage Dispersion between Cells  
The voltage dispersion between cells measures the variation of discharge between the different battery cells [15]. It is interesting, especially in batteries without BMS, as it can indicate battery malfunction that is not able to balance the discharge between the cells [16]. This indicator can also be used to compare the state of the different cells within the battery. To evaluate the variation of discharge in each cell, the difference between the end and start voltage is subtracted and compared to the difference between end and initial average  
Figure 6: Depth of discharge vs. cycle life. voltage of all the cells in that charge cycle. As the final voltage is 4.2 V in all cases, the equation is  
\[\mathrm{V_{Disp}=(4.2V-V_{input})-(4.2V-V_{input-avg})=V_{input-avg}-V_{input}} \tag{2}\]  
Figure 8 represents the evolution of the voltage dispersion of each individual cell for battery 5. A high variability of the voltage between the cells can be observed, even though no trend can be seen. No correlation between these values and the individual internal resistance of each cell was identified, although it is not ruled out that it may exist. To study the evolution of this parameter over the life of the battery, the total dispersion of the 10 cells in each cycle is studied. The values are transformed to dimensionless numbers based on each pack average voltage dispersion. Figure 9 presents the evolution of the voltage dispersion of battery 5. A slight upward trend throughout life might be observed, but it is not precisely clear because the values are extremely variable between cycles. This same observation can be made for the rest of the battery packs under study. To observe more clearly the trend of this evolution, a weighted correction of the values obtained was made, taking into account the three previous and subsequent values, so that the peaks in the graph are eliminated. For this purpose, Equation (3) is used:  
\[V_{corrected-i}=\frac{V_{i-3}+2(V_{i-2}+V_{i-1})+3V_{i}+2(V_{i+1}+V_{i+2})+V_{i+ 3}}{13} \tag{3}\]  
This weighted correction will continue to be used in this analysis. Figure 10 shows a clear upward trend in the voltage dispersion in the weighted description. Finally, a direct relationship can be obtained between the aging of the battery and the increase in voltage dispersion between cells. This makes sense, as one may observe that the aging of a battery could cause malfunctions that prevent the cells from balancing well during discharge. Figure 7: Depth of discharge vs. internal resistance growth. Figure 8: Voltage dispersion per cell (battery 5). Figure 9: Voltage dispersion battery 5. Figures 11 and 12 show the relationship of the average cell voltage dispersion of each battery to the number of cycles and the growth of internal resistance. It is observed that the relationship is concise, a greater dispersion of the voltage leads to the premature aging of the battery and, subsequently, to greater growth of its internal resistance. Due to this trend, voltage dispersion between cells is one of the parameters that will be considered in the final modelization. Figure 11: Voltage dispersion vs. cycle life. Figure 10: Weighted voltage dispersion of battery 5. It also seems interesting to analyze whether there is a direct relationship between voltage dispersion and discharged capacity, since the greater the discharge depth, the more the battery is stressed, and malfunctions could occur that cause the cells to discharge asynchronously. Figure 13 presents the evolution of the comparison between both parameters for battery 5 during their service life; the weighted correction of the voltage dispersion was applied. In addition, both parameters are presented in a dimensionless way based on each battery average values. There does not seem to be a clear relationship between them. The same analysis can be made for the rest of the battery packs. Figure 12: Voltage dispersion vs. internal resistance growth. Figure 13: Voltage dispersion (w.c.) vs. capacity discharged for battery 5.