2.2.4. Calculation of the DRT

Due to the long measurement time for low-frequency impedances, the number of time constants *m* usually exceeds the number of frequencies measured. In contrast to frequency domain data, the resolution in the time domain is comparatively high. The number of interpolation points can also be selected to be higher than *m* in order to use the additional information provided by the high-resolution pulse measurement. To solve the over-determined system, the Tikhonov regularization is applied again. Equation (6) is extented by the regularization term:

$$J = \left\| A \cdot \mathbb{R}\_{\text{vec}} - \left\| I \right\|^2 + \left\| I \cdot \lambda \mathcal{U} \right\|^2 \tag{18}$$

where *λ* is the regularization parameter and *I* is the *m* · *m* indentity matrix. The value of the regularization parameter is optimized with respect to the sum of the square errors of the reconstructed voltage signal. A non-negative least square algorithm is proposed to solve the optimization problem. Due to the restriction to positive results, only physically possible polarizations can be calculated. Using the determined polarizations *Rvec* and the predefined time constants *τ*, the spectrum can be reconstructed for different angular frequencies *ω*:

$$Z\_{DRT} = \sum\_{k=1}^{m} \frac{R\_{\text{vec}}(k)}{1 + j\omega \tau\_k} \tag{19}$$

If a measured or modelled spectrum is available, the sum of the squares of errors of the capacity-resistive part can be calculated and used as an additional selection criterion for the regularization parameter. This method for the evaluation of the parameter was already proposed by [15] for the DRT calculation of frequency domain data.

In Figure 2b, the DRT is given; this was derived from the voltage course during the relaxation period shown in Figure 2a. The DRT reveals three processes vizualized by the peaks. The area under the peaks corresponds to the polarization contribution of the process and the position of the peak to the time constant. The time constants as well as the polarizations of the identified processes are consistent with the parameters of the RCelements used in the battery model. Table 1 shows the calculated values and the original parameter set of the battery model as well as the relative deviation.


**Table 1.** Comparison of the parameter set of the battery model and the values derived by the DRT.

The relative error of the parameters with larger time constants decreases due to the diminishing influence of the superimposed noise signal. The processes with short time constants only contribute to voltage changes for a short time and their voltage contribution quickly drops to zero. The voltage drop is also most pronounced in processes with the long time constants at the beginning of the relaxation, but it lasts longer. Therefore, the effect of these processes is visible at more measuring points, which enables an increasing averaging of the noise. In Figure 3a, the reconstructed voltage signal is plottet together

with the simulated voltage course in order to prove the accuracy of the DRT. Figure 3b shows the reconstructed impedance, derived by using the calculated distribution function and Equation (19), alongside the impedance spetrum of the battery model.

**Figure 3.** (**a**) Comparison of simulated and reconstructed voltage during the relaxation period. (**b**) Comparison of modeled and reconstructed impedance spectrum.

Regardless of whether the real part or the imaginary part is considered, the relative deviation between the two spectra of Figure 3b is less than 1% for the frequency range [ *feval*,*min* ··· *feval*,*max*]. In general, the accuracy is higher for lower frequencies within the specified frequency range. Because of the added noise in the simulation, the relative error of the reconstructed voltage signal (Figure 3a) cannot be used as an indicator for the quality of the distribution function. However, the reconstructed voltage follows the mean value of the simulated signal, even with dynamic behavior in the first seconds. Despite the good reconstruction of the dynamics, the impact of the noise is substantially minimized, which proves that no overfitting has taken place.

Based on the sampling rate of commercially available BMS, the resolution of the simulated values was only 100 ms. Since the impedance of automotive sized cells is usually very low and hence the voltage response, a strong noise signal has been added compared to the excitation. Nevertheless, processes with time constants that are only slightly larger than the resolution can be identified and their contributions determined. The results therefore show that the method introduced is theoretically suitable for determining the DRT and the cell impedance and examining the dynamics and processes parameters of an LIB.

#### **3. Experimental**

In order to further investigate the applicability and accuracy of the introduced method under real conditions, an experimental study was carried out. To validate the introduced method, automotive LIBs were characterized with both EIS and pulse tests and the resulting DRT and derived process parameters were compared.

The automotive cells were then cyclically aged under various conditions. The changes in processes and their parameters were determined using both methods. Thus, the results obtained through time and frequency domain data can be directly compared and evaluated with regard to the traceability of degradation mechanisms.

For experimental investigations, large format pouch-type LIBs with a nominal capacity of 50 Ah were used. The anode of the examined cells are made of graphite and the cathode consists mainly of nickel–manganese–cobalt-oxide (NMC). All cells were taken from the same batch to minimize the impact of manufacturing tolerances.

#### *3.1. Experimental Validation*

For the experimental validation of the method introduced one of the cell was fully charged and subsequently discharged to determine the available cell capacity. In order to minimize the influence of the cell impedance, a constant-current–constant-voltage (CCCV) protocol was used for both charging and discharging. The CV phases were held until the current dropped below 50 mA. In the CC phase, 25 A were applied during charging and 50 A during discharging. According to the determined capacity the cell was then charged with a constant current of 25 A to certain SOC values (20/40/60/80%). After reaching the desired SOC level, the current was set to zero. The relaxation of the voltage was recorded in high resolution with a sampling rate of 2 MHz in the first six seconds after the load jump. After six seconds, the sampling rate was reduced down to 1 Hz and the voltage relaxation recorded for 4 h.

The pulse tests were followed by EIS measurements. Galvanostatic excitation with an amplitude of 3 A was applied. The frequency was varied in the range from 50 kHz to 0.5 mHz. For frequencies above 100 Hz, 16 measurements per decade were carried out, otherwise only eight were performed. According to Barai et al. [30] the relaxation phase of the previous pulse test is long enough to meet stationary conditions.
