## 4 Results and Discussion
### Analysis of the Use of the Peukert and Liebenow Equations for Lithium-Ion Cells
As the battery temperature decreases, the available capacity decreases due to increase of internal resistance of the battery and retardation of the chemical metabolism of the batteries effectively hindering the chemical reaction rate [13]. Thus, the capacity \(C\) released by the battery is equal to zero at least at temperatures \(T_{L}\) not less than an electrolyte congelation point. Second, with increasing temperature, the discharge capacity \(C\) of the battery in Equation (15) also increases unlimitedly. As the battery temperature increases, the internal resistance of the battery decreases and the chemical metabolism of the battery grows, thus effectively increasing the capacity of the battery [13]. Nevertheless, it is evident that the capacity cannot increase constantly with temperature growth. At reasonably large temperatures, active materials of electrodes start degrading, as does the electrolyte. Hence, there must exist a maximum temperature limit above which the capacity released by the battery ceases to increase. Equation (15) does not take these two factors into account, while Equation (6) does. Therefore, from a theoretical point of view, Equation (6) more correctly describes the electrochemical processesoccurring in the battery when their temperature increases. Thus, Equation (6) can more accurately and over a wider range reflect the change in the discharge capacity of the batteries depending on the temperature. Thus, the conducted studies showed that, taking into account Equations (6)-(9), the analytical model of [PERSON] [18] can be significantly improved. Indeed, if instead of the classic Peukert equation, a more correct (for lithium-ion batteries) generalized Peukert equation (for example, Equation (9)) is used, and, instead of Equation (15), the more correct Equation (6) is used, then, for the generalized Peukert equation, accounting for the temperature, we obtain from (14) the equation:  
\[C(i,T)=\left(\frac{C_{mref}}{erfc(-B/n)}erfc\left(\frac{i-B}{n}\right)\right)K \frac{\left(\frac{T-T_{L}}{T_{ref}-T_{L}}\right)^{\beta}}{(K-1)+\left(\frac{T-T _{L}}{T_{ref}-T_{L}}\right)^{\beta}} \tag{16}\]  
In Equation (16), the discharge capacity is \(C=C_{mref}\) at \(T=T_{ref}\) and \(I=0\). Then, from Equation (16), taking into account Equation (13), for the effective current \(I_{\mathit{eff}}\left(i,T\right)\) (10) of the Hausmann [18] model, we obtain an improved equation. \[I_{eff}(i,T)=i\frac{erfc(-B/n)}{erfc\left(\frac{i-B}{n}\right)}\left(1+\left( \frac{T_{ref}-T_{L}}{T-T_{L}}\right)(K-1)\right) \tag{17}\]  
Thus, the use of Equations (7)-(9) and (6) (which most accurately reflect the electrochemical processes in lithium-ion batteries during their discharge) allows us to improve the analytical model by [PERSON] [18]. This statement is also true for other analytical models of lithium-ion batteries containing the Peukert equation [20; 21; 40]. In conclusion, we note that, from the theoretical point of view, the most interesting is Equation (9). Equation (9) possesses its statistical basis (5) (complementary error function and therefore its parameters have statistical meaning) unlike Equations (7) and (8), which are just empiric equations. The lithium-ion cells discharge process is a phase transition: it is the transition from phases of active mass of electrodes corresponding to cell's charged state to phases of the active mass of electrodes corresponding to the cell's discharged state. For example, in this study, for IMR lithium-ion cells during the charge-discharge cycle, the following phase transitions were observed regarding the active mass on the cathode and anode:  
\[2 MnO_{2}+Li^{+}+e^{-}\leftrightarrow\mathit{LiMn}_{2}O_{4}(\mathit{ Cathode}) \tag{18}\]  
\[C_{t}=C_{m}-\sum_{i=0}^{t}I_{eff}(i_{i},T_{i})\Delta t,I_{eff}(i_{t},T_{t})=f_{ 1}(i_{t})f_{2}(T_{t})=\gamma(i_{t})^{\alpha}\bigg{(}\frac{T_{ref}}{T_{t}} \bigg{)}^{\beta} \tag{10}\]0  
In physics [38], phase transitions are often described by the complementary error function (5), which is based on the normal law of distribution.