*3.2. Aging Study*

As part of the aging study, the LIBs were cycled under certain operating conditions that are within the manufacturer's cell specifications. A total of three aging scenarios were carried out, with the average SOC value being varied between the individual scenarios. With the exception of the charging current and the state of charge, the other operating conditions like the depth of discharge (DOD) (30%) and the discharge current (1.6 C) were set to be the same for all scenarios. Table 2 gives an overview about the deviating operating conditions for each scenario.


**Table 2.** Different operating conditions.

The SOC range of each scenario was set in such a way that, according to [31], different degrees of capacity depletion and deterioration of the individual electrodes could be expected. The aging study thus enables an assessment of whether the sensitivity of the introduced method is sufficient to determine aging mechanisms qualitatively or even quantitatively.

Before starting the cycling, a checkup was carried out for each cell in order to determine the initial cell parameters. During the cycling of the cells, the long-term tests were occasionally interrupted and further checkups were carried out. The checkup procedure contains a CCCV capacity measurement. In addition, at 20/40/60 and 80% a pulse test and an EIS measurement are performed, following the procedure described in Section 3.1. However, the maximum sampling rate during the pulse test is limited to 10 Hz, which is more realistic for online applications such as battery management systems.

#### **4. Results**

#### *4.1. Experimental Validation*

In this section, the results of the pulse tests and the EIS measurements of the experimental validation are shown and evaluated.

Figure 4 depicts the recorded cell voltage during the relaxation at 20%. In addition to the originally measured signal, the moving average of the signal is given in order to be able to assess the accuracy of the reconstructed signal. It can be stated that the deviation between the reconstructed and the averaged signal over the entire range under consideration is less than 0.5 mV.

**Figure 4.** Pulse test: Voltage relaxation at 20 % SOC.

The high measurement resolution reveals the inductive behavior of the cell voltage. Due to the inductance, the voltage over the cell connectors and current collectors drop to negative values immediately after the load jump. For less than 0.1 ms, the voltage rises again as the voltage across the inductive components drops quickly. Therefore, the voltage measurements for *t* < 0.1 ms have been omitted, as electrochemical processes (time constants of usually larger than 1 ms) are of interest for this study. However, it is pointed out that the inductive behavior can also be simulated by expanding the matrix *A* (Equation (9)) by the mathematical description of several parallel circuits of ohmic resistance and inductance (RL-elements).

Equation (19) is used to calculate the impedance of the cell for the pulse test at 20% percent. For better comparability with the spectrum measured directly after the pulse test, Figure 5a,b show the imaginary and real parts of both spectra over frequency.

**Figure 5.** Comparison of measured and reconstructed impedance at 20 % SOC (**a**) Imaginary part of impedance. (**b**) Real part of impedance.

Figure 5a shows that the imaginary part coincides with frequencies below 50 Hz. Due to the limited measuring range, only impedances at frequencies above 0.5 mHz can be specified for the measured spectrum.

The plot of the real part of the impedance in Figure 5b provides similar insights. The course of the reconstructed spectrum deviates at high frequencies, but follows the measured spectrum at frequencies of ≤5 Hz.

The increasing deviations at >5 Hz can be explained by the Butler–Volmer kinetics. The current excitation at relaxation period was set to zero, while the excitation during the direct measurement of the impedance was set between −3 A and 3 A. According to the Butler–Volmer equation, the kinetics of charge transfer processes are not linear. Thus, due to increasing charge transfer resistances, the potential response does not decrease linearly with the current excitation [32]. In addition, it has been experimentally proven in some publications that the relaxation time, which differs between the pulse test and the EIS, influences the charge transfer resistances [30,32]. It is known that charge transfer processes at the anode [4,15] and at the cathode [5,33] occur at moderate frequencies of 1–100 Hz. The deviations at frequencies of 5–100 Hz between pulse test and EIS were therefore to be expected.

The ranges of the spectrum at frequencies above 100 Hz could not be adequately reconstructed because the optimization function is limited to resistive–capacitive elements. The impedance values measured at over 100 Hz were therefore discarded and not plotted in Figure 5.

Figure 6 shows the DRT obtained from time domain data and the DRT determined from frequency domain data. The resolution of the DRT from time domain data is twice as high. Therefore, the heights of peaks in one DRT cannot be compared directly with those in the other DRT. In order to compare the polarization contributions of the identified processes, the sum of all polarization contributions within a peak must be determined instead. When calculating the polarization contributions for the first (peak at smallest time constant) to fifth peak, the same orders of magnitude result for the respective peak for both DRTs. In addition, both DRTs show that the contribution of the first and fifth peaks are greatest, followed by the fourth and second peaks. The third peak has a comparatively small contribution in both DRTs.

**Figure 6.** DRT derived from frequency domain data (**a**) and from time domain data (**b**).

Both distributions reveal four relevant processes with time constants in the range of <sup>2</sup> · <sup>10</sup>−<sup>3</sup> <sup>s</sup> <sup>&</sup>lt; *<sup>τ</sup>* <sup>&</sup>lt; <sup>2</sup> · 100 s. According to *fT* <sup>=</sup> <sup>1</sup> <sup>2</sup>·*πτ* , these time constants approximately correspond to frequencies between 0.1 Hz and 100 Hz. Processes in this frequency range were assigned to charge transfer [4,5,15,33] and surface films of the electrodes [4,34]. Due to the Butler–Volmer kinetics, time constants and polarization contributions differ between the two distributions, which explains the discrepancy that occurs in the first four peaks.

For time constants >10 s, the DRT obtained from time domain data shows a comparatively better resolution of the processes. The DRT based on frequency domain data only indicates a single process for time constants between 10 and 1000 s. In contrast, the DRT obtained through time domain data reveals the existence of three different processes. In addition, time constants that are two orders of magnitude higher can be resolved. If the spectrum is recorded via EIS, the measurement takes more than a day to resolve large time constants. The evaluation of pulse data with the method introduced can therefore be advantageous for examining processes with large time constants such as solid state diffusion.
