## 4 Results and Discussion
### Analysis of the Use of the Peukert and Liebenow Equations for Lithium-Ion Cells
[PERSON]'s model [18] for calculating the remaining battery capacity has the form  
\[C_{t}=C_{m}-\sum_{i=0}^{t}I_{eff}(i_{i},T_{i})\Delta t,I_{eff}(i_{t},T_{t})=f_{ 1}(i_{t})f_{2}(T_{t})=\gamma(i_{t})^{\alpha}\bigg{(}\frac{T_{ref}}{T_{t}} \bigg{)}^{\beta} \tag{10}\]  
where \(C_{m}\) is an absolute capacity of a fully charged battery; \(C_{t}\), \(i_{t}\) and \(T_{t}\) are the battery remaining capacity, current and temperature in an instant of time \(t\), respectively; \(T_{ref}\) is the reference temperature for an investigated battery; \(\Delta t\) is the summing interval; and \(\alpha\), \(\beta\) and \(\gamma\) are empirical constants. In paper [18] the following values were used:  
\[T_{ref}=298K,\Delta t=1s \tag{11}\]  
In accordance with Equation (10), the equation  
\[\Delta C_{r}(i)=f_{1}(i)f_{2}(T)\Delta t \tag{12}\]  
shows that, as the discharge current increases, the removed effective capacity also increases. This correlation is in accordance with the commonly-used Peukert Equation (1) justification. In general, Equations (10) and (12) show that, when battery temperature decreases and load increases, the rate at which the remaining battery capacity is reduced also increases. In addition, dividing the entire battery discharge time into the sum of small time intervals \(\Delta\)t allows us to consider (under dynamic load) that at each time interval \(\Delta\)t the current and temperature are constant. This makes it possible to use the empirical equations found for constant currents in Equation (10), in particular the Peukert equation, as shown below. The model (10) was tested for commercial automotive-grade lithium batteries [18]. The maximal relative error received under diverse discharge dynamical modes did not exceed 5%. The authors of [19] showed that the battery discharge capacity \(C\) is related to the effective current \(I_{eff}(i,T)\) by the equation  
\[C=\frac{C_{m}}{I_{eff}(i,T)/i} \tag{13}\]  
Thus, the analytical model of [PERSON] [18] uses the Peukert equation of the form  
\[C(i,T)=\frac{A}{i^{n}}\bigg{(}\frac{T}{T_{ref}}\bigg{)}^{\beta},n=\alpha-1,A=C _{m}/\gamma \tag{14}\]  
In Equation (14), the first multiplier is the classic Peukert Equation (1) and the second multiplier takes into account the dependence of the discharge capacity on temperature. The research presented in this manuscript showed that the classic Peukert Equation (1) can be used to calculate the discharge capacity of a lithium-ion battery in the range of currents from 0.2\(C_{n}\) to currents at which the battery discharge capacity begins to rapidly decrease (Figure 3(1)). This is a rather wide range of discharge currents, where the Peukert Equation (1) corresponds very well to the experimental data (Table 4). However, if the battery operates at high dynamic load in an electric vehicle, it is possible that the discharge currents will go beyond this range. In this case, the remaining battery capacity will not be calculated correctly. Therefore, in this case, instead of the classic Peukert Equation (1), it is necessary to use the generalized Peukert Equations (7)-(9), which are valid for any discharge currents (Figure 2 and Table 2). From Equation (14), at a constant current \(I=i0\), we obtain an equation for the dependence of the discharge capacity on the temperature in the form  
\[C(T)=C_{mref}\bigg{(}\frac{T}{T_{ref}}\bigg{)}^{\beta},C_{mref}=A/i0^{n} \tag{15}\]  
In Equation (15) at \(T=T_{ref}\), the discharge capacity is \(C=C_{mref}\) as in Equation (6). Of course, Equation (15) with an optimal choice of the empirical constants \(C_{mref}\) and \(\beta\) can describe the experimental dependence of the discharge capacity on temperature (Figure 1) over a fairly wide temperature range [18]. However, Equation (15) contains two fundamental disadvantages. First, from dependency (15), it follows that the capacity \(C\) reduces to zero at \(T=0\). However, from the electrochemical point of view, it is understandable that the capacity \(C\) released by the battery reduces to zero much earlier.