Generalized Peukert Equations Use for Finding the Remaining Capacity of Lithium-Ion Cells of Any Format

In many studies, for predicting the remaining capacity of batteries belonging to different electrochemical systems, various analytical models based on the Peukert equation are used. This paper evaluates the advantages and disadvantages of the most famous generalized Peukert equations. For lithium-ion batteries, the Peukert equation cannot be used for estimation of their remaining capacity over the entire range of discharge currents. However, this paper proves that the generalized Peukert equations enable estimation of the capacity released by lithium-ion batteries with high accuracy. Special attention is paid to two generalized Peukert equations: C = Cm/(1 + (i/i0)n) and C = Cmerfc((i-i0)/n))/erfc(-i0/n). It is shown that they correspond to the experimental data the best.


Introduction
Currently, batteries are an integral part of diverse appliances, both for household purposes and special intention. Batteries with various electrochemical systems are used: lead-acid, alkaline, lithium-ion, etc. Among small-format batteries, lithium-ion cells prevail [1,2]. Most recently, lithium-ion cells have been used increasingly in machinery requiring large-format batteries. This is primarily related to the intensive development of electric vehicles (xEVs) [2,3]. The advantage of lithium-ion batteries over batteries of other electrochemical systems is due to the high specific capacity of these batteries and their reliability in operation.
In order to monitor the remaining battery capacity and the battery life and to optimize the battery performance, the practice drives up a demand for reliable and efficient battery models.
The most accurate models of batteries can be obtained based on electrochemical laws describing processes inside the batteries [4][5][6]. However, mathematical models constructed in this way are very sophisticated. As a rule, they require a measurement of many internal battery parameters that are often impossible or very difficult to measure. It is impossible to measure, for example, various parameters inside a porous electrode. In addition, powerful computers are required to solve those math models. Hence, these models cannot be resolved by the computers onboard aircraft or electric vehicles [7]. Another drawback of those models is the fact that for practical battery use, it is often necessary to measure any parameters without disassembling the battery. Batteries able to meet this requirement can only be built on the basis of analytical modeling [7][8][9][10][11]. The analytical models are constructed with the use of empirical equations, and parameters of these equations are found statistically by experimental data approximations.
Analytical models are often used as part of fundamental electrochemical models in cases when poorly studied processes are modeled. The analytical models, for example, are used as part of the fundamental electrochemical models when processes such as hydrogen accumulation in electrodes [12,13] or the thermal runaway of batteries are to be taken into account [14][15][16].
Any battery model must, first of all, reliably calculate the state of charge (SoC) of the battery, because the operability of the entire system depends on the SoC.
Many methods for evaluating the SoC are now known. The simplest method for evaluating the SoC is based on the open circuit voltage [17]. However, this method in the dynamic mode gives an error of up to 20% [18]. This method cannot be used at all for lithium-iron-phosphate batteries, because they have a flat discharge curve [19].
Better results for evaluating the SoC of a battery can be obtained using models based on the Kalman filter [20,21]. These models give a better estimation of the battery SoC than the previous method [20]; however, in the dynamic mode, these models give an error of up to 10% (according to our research).
Currently, battery SoC assessments are most often performed based on two methods. The first is using a stress profile. Secondly, the ampere-hours in a given discharge cycle are counted, which are then subtracted from the ampere-hours that the battery has released in the previous discharge cycle. However, this combined method in the case of dynamic loading also has a number of disadvantages, noted in a prior article [1].
To estimate the residual capacity of batteries in analytical models [7,8,22], the Peukert equation is often used. The Hausman model [7], in our opinion, is the most promising of this group of models.
The Peukert equation was developed for lead-acid batteries. This equation was the first to describe a battery's released capacity, dependent on the discharge current [22]. However, the Peukert equation is also used in various analytical models for batteries of other electrochemical systems [23]. For example, in certain models [7,8], the analytical method is used to estimate the remaining capacity of lithium-ion batteries.
The Peukert equation can be written in the following form: where C is the battery discharge capacity, i is the discharge current; and A and n are empirical constants. The Peukert equation (Equation (1)) is not applicable for very small discharge currents or medium discharge currents. According to Equation (1), at i→0, the released capacity tends to infinity, which is not possible for any battery. At small discharge currents from zero to 0.2C n , Peukert's equation no longer corresponds to the experimental data. At medium discharge currents (from the first inflection point of the experimental curve to the second inflection point of this curve), the experimental function C(i) is convex. However, the function C(i) for the Peukert equation is always concave (for n > 0).
Thus, Peukert's equation (Equation (1)) is not applicable for all possible discharge currents. This fact greatly limits the applicability of the Hausmann model [7] for estimating the residual capacity of lithium-ion batteries.
The generalized Peukert equations studied in this article are applicable for any discharge currents for Ni-Cd batteries [24]. They are also applicable, at any discharge currents, for small-sized lithium-ion batteries [25].
In this study, we investigated the applicability of generalized Peukert equations for batteries of any format and various manufacturers, including large-format lithium-ion batteries with LiFePO4 (LFP) cathodes.
The purpose of this paper is to prove the applicability of generalized Peukert equations at any discharge currents for lithium-ion batteries of any format and manufacturer. The solution to this problem will allow the very promising Hausmann model [7] to be used at any discharge current for any lithium-ion battery. This research is most relevant for electric vehicles (xEVs), because they use large-format lithium-ion batteries.

Theory
In the analytical models [7,8,26,27], the remaining battery capacity is calculated with aid of the Peukert equation. Often, lithium-ion batteries are operated in dynamic mode, when the discharge currents can change rapidly. In this case, the most promising model is the Hausmann model [7]. It has the form: where T t , i t , and Cr t , are the temperature, current and remaining capacity of battery at time moment t, respectively; C m is the maximum battery capacity; ∆t is the time interval for summation; T ref is the temperature value taken as the reference temperature for the specific battery; and α, β, and γ are empiric constants. In Equation (2), the entire discharge time is divided by the sum of small time intervals ∆t. If a time interval is small enough (for example, ∆t = 1s), then within this time interval, the current and the temperature can be considered constant. Hence, for description of the dependence between the capacity and the discharge current, it is possible to use the Peukert equation or other equations found for direct currents. A previous study [8] has shown that between the battery capacity C and the function I eff (i, T), the following dependence exists: Therefore, in the Hausmann model [7], the capacity is defined according to the following empiric equation: This is why, in Equation (4), the capacity dependence on the discharge current (the first multiplier) is defined by the Peukert equation (Equation (1)), whereas the capacity dependence on the temperature is determined by the second multiplier.
However, as it was noted in the Introduction, the Peukert equation is not applicable for either very small discharge currents of medium currents. Thus, this study investigated the applicability of other generalizations of the Peukert equation for lithium-ion batteries. For example, for this purpose, the following equations were studied [8]: where i is the discharge current, C is the battery capacity; and A, B, and n are empiric constants. Additionally, for calculating the battery capacity, other methods [28,29] exist, usually a combination of Equations (5) and (6).
Additionally, we investigated the complementary error function: often, the complementary error function [30] describes phase transitions. However, the process of battery discharge is phase transition in the electrodes: where i 0 is the average value of a statistical variable i, and σ is the standard deviation.

Equation Parameters
Equation (8) AMP20M1HDA ( The cells were charged in concordance with their operating instructions (by a constant current of 0.5C n up to the voltage value of 4.2 V (or 3.6 V for LFP cells), then by constant voltage up to the current value of 0.025C n (C n is the nominal cell capacity)).
Discharging was fulfilled by constant currents (down to the voltage value of 2.7 V (or 2.0 V for LFP cells)) in currents ranging from 0.2C n up to the current value at which C ≈ C n /10.
An electrochemical ZENNIUM workstation with a PP241 potentiostat was used for research. In experiments with high discharge currents, an electronic load (custom made) with a maximum input current of 430 A was used, connected to a ZENNIUM workstation. An LM35 temperature sensor was used to measure temperature.
For our experimental research, the following algorithm was used. First, the cells were cycled ten times in order to stabilize the layer SEI (solid electrolyte interphase). The cycling was performed in the training mode (charging was conducted as described above, whereas discharging was performed by the direct current 0.2C n , up to the voltage value of 2.75 V (or 2.0 V for LFP cells)). After that, the stability of the cell parameters was checked; for this purpose, three cycles were performed. If the spread of battery capacity values in these cycles was less than 5%, further measurements were made with the cell; otherwise, the cell was cycled again and again until the spread of the values decreased to 5% or the cell was replaced with a new one.
Secondly, to improve the reliability of the obtained results, cell capacity at a certain discharge current was calculated as the average of three measurements at a given discharge current value (if the spread for the measured capacity was less than 5%). Otherwise, training cycles were performed, or the cell was replaced with a new, more stable version.
Thirdly, in order to eliminate any impact of previous charge-discharge cycles (via residual effects), training cycles were performed before each measurement until the capacity spread in three consecutive charge-discharge cycles of the battery was less than 5%. Figure 1 shows a scheme of the measurement algorithm.
Energies 2021, 14, x FOR PEER REVIEW 5 of 9 Figure 1. The scheme of the measurement algorithm. max∆C is the maximum difference for the three measured capacities at a certain discharge current ik. n is the number of charge/discharge cycles at a certain discharge current ik.
Additionally, a statistical dispersion of cell capacities exists associated with the process of their manufacture, storage conditions, etc. Therefore, in order to reduce this kind of value dispersion, we standardized the measured cell capacities with their maximum capacity, Cm. Notably, the search for the Cm value of a cell under study was also conducted by experiments ( Table 1). The presentation of experimental data in standardized coordinates (i/Cm, C/Cm) ( Figure 2) has a clear advantage. In these coordinates, the experimental data reflect only the electrochemical processes in batteries and do not take into account the random dispersion of the parameters of identical batteries associated with their production, storage conditions, etc. This procedure made it possible to construct the empirical curves more reliably.
To minimize the influence of temperature on the battery capacity dependence on the discharge current, the measurements were performed at a temperature of T = 25 °C inside the thermal chamber Binder MK240 (BINDER GmbH, Germany). In addition, specially made radiators were used to increase heat removal from the battery. Each was attached to a cell with the aid of a heat-conducting paste and a clamp. Additionally, as shown by our preliminary experiments, in the range from 25 °C to 50 °C, the capacity released by a cell is virtually independent of temperature. This fact was also investigated and confirmed elsewhere [10,25].
The cells under study could be discharged with any currents because they had no protection.

Results and Discussion
For convenience, Equations (5)-(7) can be written so that they have the same em- Figure 1. The scheme of the measurement algorithm. max∆C is the maximum difference for the three measured capacities at a certain discharge current i k . n is the number of charge/discharge cycles at a certain discharge current i k .
Additionally, a statistical dispersion of cell capacities exists associated with the process of their manufacture, storage conditions, etc. Therefore, in order to reduce this kind of value dispersion, we standardized the measured cell capacities with their maximum capacity, C m . Notably, the search for the C m value of a cell under study was also conducted by experiments ( Table 1). The presentation of experimental data in standardized coordinates (i/C m , C/C m ) ( Figure 2) has a clear advantage. In these coordinates, the experimental data reflect only the electrochemical processes in batteries and do not take into account the random dispersion of the parameters of identical batteries associated with their production, storage conditions, etc. This procedure made it possible to construct the empirical curves more reliably.
To minimize the influence of temperature on the battery capacity dependence on the discharge current, the measurements were performed at a temperature of T = 25 • C inside the thermal chamber Binder MK240 (BINDER GmbH, Germany). In addition, specially made radiators were used to increase heat removal from the battery. Each was attached to a cell with the aid of a heat-conducting paste and a clamp. Additionally, as shown by our preliminary experiments, in the range from 25 • C to 50 • C, the capacity released by a cell is virtually independent of temperature. This fact was also investigated and confirmed elsewhere [10,25].
The cells under study could be discharged with any currents because they had no protection. within the experimental error. Consequently, from both the electrochemical (LMO) and the structural points of view, those electrode cells are identical, and in addition, the same electrolyte is used. However, these cells are made by different manufacturers and have a different format.

Figure 2. Approximation of experimental data using Equation (10) for different lithium-ion cells.
Cm values were taken from Table 1 for each cell. Table 1. Optimal parameters of generalized Peukert equations (Equations (8)-(10)) for the studied lithium-ion cells.

Manufacturer and Model Equation Parameters
Cn (Ah) 1 Cm (Ah) i0 (A) n δ (%) 2 Equation ( However, if the cell electrodes are different in their electrochemical or structural properties [30], then the experimental functions f(i) = C(i/Cm)/Cm of those cells must differ. In Figure 2, for the cells Efest IMR14500, XHData 18650, Efest IMR18500 and AMP20M1HDA, the experimental functions f(i) = C(i/Cm)/Cm differ. Thus, these cells are  Table 1 for each cell.

Results and Discussion
For convenience, Equations (5)-(7) can be written so that they have the same empirical parameters: In this case, all three equations (Equations (8)-(10)) satisfy the ratio C(0) = C m , where C m is the maximum experimental cell capacity. Figure 2 shows the measured experimental data. In Figure 2, the experimental data are approximated by applying Equation (10), because it had the smallest relative approximation error in all our experiments ( Table 1).
The optimal parameters for Equations (8)-(10) were found with the use of the least squares method and the Levenberg-Marquardt optimization procedure, i.e., the method of nonlinear regression was used within the framework of a computer statistical program. When calculating the parameters of Equations (8)- (10), this method also calculated various parameter estimates: errors, significance of the parameters, etc. The parameters and errors for the parameters found in this way are presented in Table 1.
It should be noted that for the correct error calculation of the determined parameters, statistical methods must be used. In this case, not only was the accuracy of the instruments taken into account, but also the inevitable statistical dispersion of parameters and inevitable statistical processes in batteries.
When using the measurement scheme (Figure 1), several measurements were performed at each discharge current, in a random order (and not sequentially at a certain discharge current). Before each measurement, the battery was brought to the same initial state by performing training cycles (see the "Experimental Methodology" Section) and monitoring the capacity in each cycle. The cycles were performed until, in three consecutive cycles, the capacity began to differ by less than 5% (usually it differed by less than 1%). This ensured the same initial state before each measurement (pиc. 1).
If the average value was found for three (or more) measurements at a certain discharge current, then this value would be the most reliable and accurate experimental value at this discharge current. The mean is least influenced by various random factors (as opposed to a single measurement).
Therefore, approximating these experimental mean values with an empirical equation yields the most reliable experimental curve and the most reliable parameter values for the empirical equations. In this case, the final results will be much less affected by various inevitable statistical dispersions of parameters and inevitable statistical processes, as opposed to single experimental values. However, when evaluating the errors, all experimental measurements were taken into account.
Of course, there are other research methods and schemes. However, over the many years of our experimental research, we have made sure that this scheme of experimental research is the most statistically substantiated and the most reliable in determining the parameters of empirical equations and their errors. Figure 2 shows the curves for which cell capacities and discharge currents are standardized by their maximum capacity. Standardization reduces the studied curves of specific cells to the curves of cells of unitary capacity. Hence, in these coordinates, the cells must have the same function f(i) = C(i/C m )/C m in the case when their electrolytes and electrodes are structurally and electrochemically the same. It is seen in Figure 2 that for the cells Efest IMR14500 and TrustFire IMR16340, the functions f(i) = C(i/C m )/C m coincide within the experimental error. Consequently, from both the electrochemical (LMO) and the structural points of view, those electrode cells are identical, and in addition, the same electrolyte is used. However, these cells are made by different manufacturers and have a different format.
However, if the cell electrodes are different in their electrochemical or structural properties [30], then the experimental functions f(i) = C(i/C m )/C m of those cells must differ. In Figure 2, for the cells Efest IMR14500, XHData 18650, Efest IMR18500 and AMP20M1HDA, the experimental functions f(i) = C(i/C m )/C m differ. Thus, these cells are different, either structurally (different thickness of the electrodes, different additives to the active mass of the electrodes, etc.) or electrochemically (different type of cathodes: LMO, LFP, etc.).
The conducted studies show that Equations (8)-(10) approximate the experimental data with a relative error of less than 4% (at any discharge currents). This is enough for any practical calculation. Thus, Equations (8)-(10) are accurate to naturally describe the electrochemical processes in lithium-ion cells during their discharge. Therefore, they can be used for state of charge (SoC) estimations of cells in different cell models. However, according to Table 1, the accuracy of approximation of the experimental data by Equations (8)- (10) increases in the following sequence: Equation (8), (9), (10). Specifically, Equation (10) corresponds to the experimental data the best.

Conclusions
The conducted studies showed that Equations (8)-(10) can be used effectively for practical calculations at any discharge currents. However, among Equations (8)-(10), the most preferable are the generalized Peukert equations, i.e., Equations (9) and (10). They had the lowest relative approximation error for the experimental data (Table 1). Thus, this study showed that in the Hausmann model (2)-(4), to correctly estimate the remaining capacity in lithium-ion cells (at any discharge currents), it is necessary to replace the Peukert equation (Equation (1)) with either Equation (9)