## 4 Results and Discussion
### Analysis of the Use of the Peukert and Liebenow Equations for Lithium-Ion Cells  
Now, consider the possibility of using the Peukert Equation (1) and the Liebenow Equation (2) for the lithium-ion cells. Neither Equation (1) nor Equation (2) can describe the experimental curves shown in Figure 2 over the entire variation range of the discharge current. For Equation (1), as the discharge current decreases, the cell released capacity tends to infinity, which does not make physical sense. Besides, at medium currents, the experimental curves are convex (Figure 2), while both Equations (1) and (2) give only concave curves (at positive values of all constants). As shown in Figure 2, the experimental curves are concave only starting from inflection point of the function \(C(i)\) and keep this form to infinity, as well as at small discharge currents. Hence, it is only possible to use Equations (1) and (2) in these intervals of the discharge currents. Let us check the possibility of using Equations (1) and (2) in these particular intervals of the discharge current values (Figure 3). Figure 3: Comparison between experimental data for lithium-ion cells of different types and Equations (1), (2) and (9). Number 1 marks the Peukert and Liebenow equations at small discharge currents while Number 2 at high discharge currents. The optimal parameters of Equations (1) and (2) for the lithium-ion cells in the interval of the discharge current values from the inflection point of the experimental curve to the highest currents used in experiments are presented in Table 3, and for small discharge currents in Table 4.  
At small discharge currents, the function \(C(i)\) is concave in the range from zero to the value of the discharge current at which function \(C(i)\) begins to decrease sharply (Figure 3). This range is different for different cells. It depends on the type of electrodes used, the design of the cells, their capacity, etc. We conducted experimental studies in the range of discharge currents from 0.2C\({}_{n}\) to the maximum currents \(i_{\max}\). The research results are presented in Table 4.  
Equations (1) and (2) were obtained based on experimental investigations of the lead-acid batteries [30; 31]. For lead-acid batteries, the Peukert equation is correct in the broad range of the discharge currents. Only at very small discharge currents, the Peukert equation does not correspond to the experimental data as the cell released capacity (in this equation) tends to infinity, which does not make physical sense. However, at all working discharge currents for lead-acid batteries, the Peukert equation corresponds to the experimental data quite well [30; 31]. At low discharge currents for lead-acid batteries, the Liebenow equation corresponds well to experimental data [31]. Thus, the Peukert equation \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Manufacturer and Model** & \(\mathbf{C_{n}\) **(Ah)1** & \(A\) & \(n\) & \(i_{max}\)**(A)** & \(\delta^{2}\)**(\%)** \\ \hline \multicolumn{6}{c}{Equation (1)} \\ \hline TrustFire IMR16340 & 0.65 & 0.612 & 0.015 & 3 & 0.49 \\ Aucooma IMR18650 & 2.8 & 2.839 & 0.013 & 15 & 0.79 \\ Keeppower IMR26650 & 4.2 & 4.073 & 9.06*10\({}^{-3}\) & 15 & 0.67 \\ Samsung INR18650-13Q & 1.3 & 1.325 & 0.015 & 10 & 0.60 \\ Sony US18650 VT3 & 1.5 & 1.504 & 6.73*10\({}^{-3}\) & 15 & 0.28 \\ \hline \multicolumn{6}{c}{Equation (2)} \\ \hline TrustFire IMR16340 & 0.65 & 0.625 & 0.015 & 3 & 0.18 \\ Aucooma IMR18650 & 2.8 & 2.862 & 4.04*10\({}^{-3}\) & 15 & 0.33 \\ Keeppower IMR26650 & 4.2 & 4.099 & 2.94*10\({}^{-3}\) & 15 & 0.15 \\ Samsung INR18650-13Q & 1.3 & 1.334 & 4.58*10\({}^{-3}\) & 10 & 1.44 \\ Sony US18650 VT3 & 1.5 & 1.509 & 1.78*10\({}^{-3}\) & 15 & 0.42 \\ \hline \hline \end{tabular}  
* Nominal cell capacity. 2 Relative error of experimental data approximation by Equations (1) and (2).  
\end{table} 
Table 4: Optimal parameters of Equations (1) and (2) for lithium-ion cells at small currents, in the range of experimental discharge currents from 0.2 Cn to imax.