Analysis of Generalized Peukert's Equations for Capacity Calculation of Lithium-Ion Cells

At first, the influence of temperature on the cells' released capacity was studied. The cells were cycled inside of a Binder MK240 climatic chamber (Binder GmbH, Germany). The measurements were performed in a training cycling mode. The obtained experimental results are presented in Fig. 3.

In this research,¹⁰ the lithium-ion cells' released capacity at different temperatures was described in the following empirical equation:

$$\begin{eqnarray}&&C={C}_{mref}K\displaystyle \frac{{\left(\displaystyle \frac{T-{T}_{L}}{{T}_{ref}-{T}_{L}}\right)}^{\beta }}{(K-1)+{\left(\displaystyle \frac{T-{T}_{L}}{{T}_{ref}-{T}_{L}}\right)}^{\beta }},\end{eqnarray} \tag{ 17 }$$

where С_mref is the top capacity released by a cell at temperature T_ref; T_L is a temperature, at which С = 0, i.e., the temperature close to the freezing temperature of the electrolyte at which all electrochemical processes stop; and T_ref is a reference temperature for the type of cells studied.

At T = T_ref, the multiplier connected with the temperature in Eq. 17 was equal to 1, i.e., it had no influence on the cell capacity. However, when T → ∞, the thermal multiplier in the Eq. 17 tended toward the value of parameter K. Thus, parameter K showed (theoretically) how much capacity С in Eq. 17 could increase with the cell temperature growth in comparison with capacity C_mref.

The optimal parameters for Eq. 17 were found using the least squares method using the Levenberg–Marquardt optimization procedure. The obtained optimal parameters are given in Table I.

In Fig. 3, it can be seen that in the temperature range 25 °C–55 °C, the capacity's relative deviation from its average value was less than 1%. Thus, in this temperature range, the temperature influence could be neglected when studying the dependence between the cells' released capacity and the discharge current. This statement was true for all the studied cells (Tables II–IV).

The cells were cycled inside of a Binder MK240 climatic chamber (Binder GmbH, Germany) at the temperature Т = 25°C. For the purposes of heat exchange and cooling intensity increase, when the cells were discharged by high currents, modified heat sinks were fastened to them (usually used for computer processor cooling). This fixation was done with the aid of heat-conducting paste MX-2 (ARTIC) and a specially manufactured clamp (Fig. 4). As a result, the temperature of the cells in all our experiments was well below 55 °C. None of the studied cells had protection, which why it was possible to discharge them using high currents.

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During the experiment, we constantly checked for possible aging of the cells before each measurement of each experimental point on any curve (Figs. 3 and 5–7). Before each change in the discharge current or element temperature, at least three training cycles were performed (see experimental section). However, the capacity C_t obtained in the training cycles was very close to the maximum capacity of the C_m cell, since the discharge current was small at 0.2 C_n. Therefore, by constantly measuring the C_t before each experimental point, we could control the aging process of the cells. All of our experiments used new cells. In addition, less than 100 charge–discharge cycles were required to obtain any curve in Figs. 5–7. This is probably why the capacitance C_t changed by less than 2% in all our experiments. This result indicated that aging processes were not significant in our experiments and did not affect the view of the curves in Figs. 5–7.

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It should be noted that in the coordinates (С/С_m, i/i₀ (or i/i_k)), the view of the curves in Figs. 5–7 did not change even with significant aging processes. This was because in the case of cell aging, the parameters С, С_m, i, and i₀ (i_k) decreased proportionally. We experimentally verified this repeatedly.

Initially, lithium-ion cells of different capacities, formats, and manufacturers with the cathode LiMn₂O₄ (i.e., IMR cells) were studied. Figure 5 provides the obtained experimental data for the studied cells (in the standardized coordinates). The absolute error of the measured experimental data was ΔC = ±0.071 Ah (calculated using the statistical method, see experimental section). Parameters C_m and i_k in Fig. 5 were taken from Table II. In this work, the optimal parameters for Eqs. 5–7 were found via the least squares method using the Levenberg–Marquardt optimization procedure. The obtained optimal parameters are given in Table II. In Figs. 5–7, the experimental data were approximated by Eq. 7, as this equation had the least relative error of approximation (as shown later).

Next, we considered lithium-ion cells of different capacities, formats, and manufacturers with the cathode LiCoO₂ (i.e., ICR cells). In Fig. 6, the obtained experimental data are given (in the standardized coordinates) for the studied cells. The absolute error of the measured experimental data was ΔC = ±0.068 Ah (calculated using the statistical method, see experimental section). Parameters C_m and i_k in Fig. 6 were taken from Table III.

Finally, lithium-ion cells of different capacities, formats, and manufacturers with the cathode LiNiMnCoO₂ were studied (i.e., INR cells). In Fig. 7, the obtained experimental data are given (in the standardized coordinates) for the studied cells. The absolute error of the measured experimental data was ΔC = ±0.071 Ah (calculated using the statistical method, see experimental section). Parameters C_m and i_k in Fig. 7 were taken from Table IV.

From the analyses in Tables II–IV, it was possible to come up with the following conclusions:

Firstly, generalized Peukert's Eqs. 5–7 could be used to evaluate the cells' released capacities, as the relative error of the experimental data approximation by these equations was less than 5%. As a rule, this error was acceptable for practical evaluations of battery residual capacity.⁹ However, it should be noted that the average relative error of experimental data approximation in Eqs. 5–7 decreased in the following sequence of Eqs. 5, 6 and 7. Thus, in all experiments, Eq. 7 corresponded best to the experimental data.

We compared the absolute error of the measured experimental data (approximately ΔC = ±0.07 Ah) and the maximum difference D_m between the experimental values of the capacity and those of Eqs. 5–7 at the same discharge currents for each cell studied (Tables II–IV). From this comparison, only in Eq. 7 was ΔC > D_m for all studied cells. Therefore, only Eq. 7 strictly corresponded to the experimental data within the experimental errors.

Secondly, from a theoretical point of view, the most interesting equation was Eq. 7. It had a statistical basis unlike Eqs. 5 and 6, which were purely empirical equations.

The lithium-ion cell discharge process is a phase transition; it runs between different phases of active mass on the cathode and anode. For example, in this study, for ICR lithium-ion cells during the charge–discharge cycle, the following phase transitions were observed regarding the active mass on the cathode and anode:

$$\begin{eqnarray}&&{{\rm{CoO}}}_{2}+{{\rm{Li}}}^{+}+{{\rm{e}}}^{-}\leftrightarrow {{\rm{LiCoO}}}_{2}\,\left({\rm{cathode}}\right)\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{{\rm{LiC}}}_{6}\leftrightarrow {{\rm{C}}}_{6}+{{\rm{Li}}}^{+}+{{\rm{e}}}^{-}\,\left({\rm{anode}}\right)\end{eqnarray} \tag{ 19 }$$

Often in physics,^28,29 phase transitions are described by the complementary error function (Eq. 4), which is based on the normal law of distribution. Undoubtedly, on the level of molecules and ions, the discharge is a statistical process. Thus, judging by the good coincidence of the experimental data with Eq. 7 (Figs. 5–7), it was possible to conclude that the cells' discharge was a statistical process subject to the normal law of distribution. This experimental fact was significant for a theoretical explanation of the charge–discharge process in lithium-ion batteries as well as a better understanding of the Peukert's equation.

Thirdly, in the normalized coordinates (С/С_m, i/i₀ (or i/i_k)), Eqs. 5–7 depended only on the single parameter n. Notably, according to the experiments conducted (Tables II–IV), parameter n depended on neither the cell capacity nor its format or manufacturer.

However, as the ratio i₀/С_m (or i_k/С_m) increased, parameter n also grew; this was observed for cells of any type and manufacturer (Tables II–IV). Thus, the ratio i₀/С_m (or i_k/С_m) represented an important characteristic of the lithium-ion cells. Notably, if for two cells, the ratio i₀/С_m (or i_k/С_m) coincided within the experimental error, the parameter n of these cells also coincided within the experimental error (Tables II–IV); hence, the corresponding curves of the generalized Peukert's equations also coincided in the standardized coordinates. For example, in Fig. 5, the curves coincided for the Eizfan IMR21700 and Imren IMR18650 cells, whereas in Fig. 7, the curves coincided for the Samsung INR18650–20R and Sony US18650VT3 cells. Nevertheless, those cells had different capacities and were made by different manufacturers.

We further analyzed the ratio i₀/С_m (or i_k/С_m):

$$\begin{eqnarray}&&{i}_{0}=S{j}_{0},\,{\unicode{x00421}}_{m}\,=\,Sl\varepsilon ,\end{eqnarray} \tag{ 20 }$$

where j₀ is the density of the discharge current i₀, S is the surface area of electrodes, l is the average effective thickness of positive and negative electrodes, and ε is the specific effective capacity of electrodes.

Then,

$$\begin{eqnarray}&&\displaystyle \frac{{i}_{0}}{{C}_{m}}=\displaystyle \frac{1}{l}\left(\displaystyle \frac{{j}_{0}}{\varepsilon }\right).\end{eqnarray} \tag{ 21 }$$

The expression in the brackets on the right in Eq. 21 was determined by a particular electrochemical system (e.g., IMR) of the lithium-ion cells. For cells of this particular electrochemical system, this expression remained the same. Therefore, for lithium-ion cells of the particular electrochemical system, any growth of the ratio i₀/С_m (or i_k/С_m) was connected with a decrease of the electrodes' average effective thickness, l. However, the decrease of the electrodes' thickness (at the same cell capacity) enabled the cells to be discharged by high currents, bringing us to starter-type cells.²⁹ Thus, the higher the ratio i₀/С_m (or i_k/С_m), the higher the working currents, by which the cells could be discharged. This assertion held true for all the studied cells (Tables II–IV).

According to the experiments conducted herein (Tables II–IV), along with the growth of the ratio i₀/С_m, the parameter n also grew in Eqs. 5–7. Consequently, parameter n was a growing function of the ratio i₀/С_m:

$$\begin{eqnarray}&&n=fun\left({i}_{0}/{C}_{m}\right)={\rm{fun}}\left(\displaystyle \frac{1}{l}\left(\displaystyle \frac{{j}_{0}}{\varepsilon }\right)\right).\end{eqnarray} \tag{ 22 }$$

Thus, for cells of a particular electrochemical system (e.g., IMR), along with the decrease of the electrodes' effective thickness (l), parameter n grew according to Eq. 22. In this case, in Eqs. 5–7 (in concordance with Eqs. 13–15), the inclination angle grew in the generalized Peukert's equations in the point i = i₀ (or i = i_k). Consequently, the starter-type cells' released capacity fell much faster than that of cells able to be discharged by small currents. This fact corresponded perfectly to the experimental data obtained (Figs. 5–7).

Additionally, it should be noted that from Eq. 21, for parameter i₀, we obtained the following equation:

$$\begin{eqnarray}&&{i}_{0}={C}_{m}\displaystyle \frac{1}{l}\left(\displaystyle \frac{{j}_{0}}{\varepsilon }\right),\end{eqnarray} \tag{ 23 }$$

Thus, for cells of a particular electrochemical system (e.g., IMR) and with the same capacity, the parameter i₀ grew as the electrodes' effective thickness decreased (Eq. 23). This corresponded well to the experimental studies (Tables II–IV).

It was possible to conclude that as the discharge current grew, the starter-type cells' released capacity was conserved virtually unchanged much longer than those only able to be discharged by small currents. The starter-type cells' released capacity fell much faster (Eqs. 13–15 and 22), which was in good agreement with the experimental data (Figs. 5–7).

In our opinion, such behavior of generalized Peukert's Eqs. 5–7 regarding the lithium-ion cells could be explained with due consideration of the distribution pattern of the electrochemical process along the depth of a porous electrode.

To describe the distribution of the discharge current j(x) along the depth of the porous electrode (for a one-dimensional electrode model) in the simplest case of the Butler–Volmer linear function ($i=\alpha \cdot \phi$) on the boundary of an electrode/electrolyte, we obtained the following equation:^33,34

$$\begin{eqnarray}&&j\left(x\right)=J\sqrt{\displaystyle \frac{\alpha \rho }{s}}\displaystyle \frac{\cosh \left(\sqrt{\rho s\alpha }(l-x)\right)}{\sinh \left(\sqrt{\rho s\alpha }l\right)},\end{eqnarray} \tag{ 24 }$$

where the electrode start is located at the point x = 0, J is the density of the external current, ρ is the specific resistance of the electrolyte in a porous electrode, s is the surface area of pores in an electrode volume unit, $l$ is the half-thickness of an electrode at a bilateral current supply, α is the specific conductivity for the activation process on the boundary electrode/electrolyte, and φ is the polarization inside of a porous electrode.

In the case of a uniform distribution of the discharge electrochemical process along the depth of the porous electrode, the entire active mass of the electrodes would be discharged uniformly, regardless of the discharge current. In this case, the cells' released capacity would be the maximum С_m and not depend on a discharge current.

However, at very high discharge currents (because of the cells' inner resistance), when switching on the element to discharge, the voltage at its terminals would be less than the lower cut-off voltage, i.e., the cells' released capacity would be equal to zero. Thus, in the case of a uniform distribution of the electrochemical process along the porous electrode depth, the function С(i) had to have a rectangular form (Fig. 8 (3)). However, in a real electrode, the electrochemical process decreased exponentially along the depth of the porous electrode (Eq. 24).

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From Eq. 24, it followed that

$$\begin{eqnarray}&&j(0)/j(l)=\,\cosh \left(\sqrt{\rho s\alpha }l\right).\end{eqnarray} \tag{ 25 }$$

Thus, as the electrodes' effective thickness l decreased, the distribution of the electrochemical process along the porous electrode depth tended toward a uniform distribution, whereas the function С(i) tended toward a rectangular form (Fig. 8 (1, 2, 3)).

In general, as the electrode thickness decreased (or as parameter i₀/С_m grew (Eq. 21)) for cells of a particular electrochemical system, the statistical dispersion fell for a state of discharge of the active mass along the depth of the porous electrode (Eq. 25). Consequently, the statistical dispersion σ in Eq. 4 fell for the entire cell discharge process. In Eq. 7, the statistical dispersion for the cell discharge process in the standardized coordinates was determined by parameter 1/n. Thus, as the electrode thickness decreased, parameter n in Eqs. 5–7 grew (for cells of a particular electrochemical system), which corresponded to the obtained experimental data (Tables II–IV).

The same conclusion followed from Eq. 22 as well, i.e., that as the electrodes' effective thickness decreased, parameter n (for cells of a particular electrochemical system) grew, and, along with it, the standard deviation σ in Eq. 4 fell according to Eq. 8. Consequently, in this case, the function С(i) tended toward the rectangular form (Fig. 8 (1, 2, 3)).