2.2.2. Predefinition of Time Constants *τ*

A logarithmically uniform distribution is defined for the time constants [*τ*<sup>1</sup> ··· *τm*]. The minimum and the maximum time constants *τmin* and *τmax* are determined based on the maximum sampling rate *fs*,*max* = <sup>1</sup> <sup>Δ</sup>*tmin* and the duration of the relaxation process *tmax*. The Shannon theorem states that frequencies that are higher than half of the sampling rate cannot be evaluated without further processing of the measured signal.

$$f\_{\text{eval,max}} \le \frac{1}{2 \cdot \Delta t\_{\text{min}}} \tag{10}$$

Taking into account the transient frequency *ft* = <sup>1</sup> <sup>2</sup>·*π*·*<sup>τ</sup>* of a RC-element, the following relationship result for the smallest observable time constant *τeval*,*min* [29]:

$$\frac{1}{2 \cdot \pi \cdot \pi\_{\text{evul},min}} \le \frac{1}{2 \cdot \Delta t\_{\text{min}}} \tag{11}$$

$$
\omega \Rightarrow \tau\_{\rm evul,min} = \frac{\Delta t\_{\rm min}}{\pi} \tag{12}
$$

The largest evaluable time constant *τeval*,*max* is limited by the duration of the relaxation phase according to [19].

$$f\_{\text{eval},\text{min}} \ge \frac{4}{t\_{\text{max}}} \tag{13}$$

$$
\Rightarrow \tau\_{\text{real\,max}} = \frac{t\_{\text{max}}}{8\pi} \tag{14}
$$

The minimum and maximum time constants of the distribution *τmin* and *τmax* are chosen to be two decades smaller, respectively, andlarger than the evaluable time constants in order to increase the accuracy at both ends of the frequency dispersion:

$$
\tau\_{\rm min} = \frac{\tau\_{\rm cval, min}}{100} = \tau\_1 \tag{15}
$$

$$
\tau\_{\text{max}} = \tau\_{\text{eval}, \text{max}} \cdot 100 = \tau\_m \tag{16}
$$

This procedure was similarly proposed by [15] for the DRT analysis of frequency domain data. In order to obtain a smooth DRT curve within the evaluated frequency range, the number of RC elements or time constants *m* must be selected to be high enough. A number of one hundred *τ* per decade was used for the analysis of the DRT in this study.

#### 2.2.3. Pre-Processing of Measurement Data

Before calculating the DRT, the existing measurement data should be checked for usability and preprocessed. According to the Equation (12), the sampling rate must be high enough (or the timestep between two successive voltage measurements Δ*tmin* small enough) to enable a meaningful evaluation of the DRT down to the time constant *τeval*,*min*. Therefore, when selecting the the sampling rate beforehand, the minimum time constant relevant for the investigation should be taken into account. If the maximum sampling rate of the measurement device is lower, the DRT can only be evaluated for time constants that satisfy Equation (12). This consideration is particularly important for BMS, as they often have sample rates of 10 Hz or less.

The processes with low time constants only have measurable contributions at the beginning of the relaxation phase and quickly subside. Therefore, the sampling rate does not necessarily have to be constant over the entire measurement period and can be reduced in order to avoid large amounts of measurement data. With very high sampling frequencies and the evaluation of very small time constants, the speed of the current control needs to be considered. As long as the current has not dropped to zero, Equation (5) is not valid. These voltage measurements should therefore be discarded and only larger time constants evaluated. Apart from that, only the voltage values should be evaluated that were recorded at least one time step after the load drop. Since pure ohmic resistances cannot be modelled with Equation (5), the influence of internal resistance on the voltage response should not be taken into account.

The OCV *UOCV* must be subtracted from the voltage measurements *umeas*(*t*) before determining the DRT. Assuming a completely relaxed cell at the end of the relaxation phase, this can be achieved by subtracting the last measured voltage value *umeas*(*tmax*) from all measurements. A considerably long relaxation phase is therefore imperative.

$$
\mu\_{\text{processel}}(t) = \mu\_{\text{meas}}(t) - lL\_{\text{OCV}} = \mu\_{\text{meas}}(t) - \mu\_{\text{meas}}(t\_{\text{max}}) \tag{17}
$$

At the beginning of the relaxation process, all processes contribute to changes in cell voltage. As relaxation progresses, only processes with larger time constants are relevant. Following the measurement data should be interpolated on a logarithmic scale in order to weigh the single measurement points accordingly. Due to the low impedance of automotive

cells, the signal-to-noise ratio is usually rather low. As the cell becomes increasingly relaxed, the signal-to-noise ratio deteriorates further, as can be seen in Figure 2. Forming the moving average before interpolation smooths the measured values and can therefore improve the ratio.
