**1. Introduction**

The use of energy storage systems is essential for the transition to renewable energies. Due to the unsteady energy supply from renewable energies such as wind power and photovoltaics, energy storage devices are required to compensate for fluctuations in output and to reliably provide energy at all times. For the electrification of the transport sector, energy storage systems are required in order to move vehicles independently of an external energy supply and without burning fossil fuels. In battery electric vehicles in particular, LIBs are by far the most frequently used battery technology due to their high energy and power density.

Large batteries are required in electric vehicles (EVs) and stationary energy storage devices, which are therefore the main cost factor in these applications. Accordingly, the requirements for the service life of the LIBs are high and a high degree of reliability is a prerequisite.

Therefore, non-invasive investigation methods are used to achieve a comprehensive understanding of the underlying degradation mechanisms of LIBs to enable system optimizations with regard to the service life. In addition, the non-invasive methods are required for the online diagnosis of LIBs and to identify potential risks during operation in order to ensure safe operation.

In current studies, the continous determination of the changes in kinetic parameters of LIBs made it possible to draw conclusions about progressive aging and degradation

**Citation:** Goldammer, E.; Kowal, J. Determination of the Distribution of Relaxation Times by Means of Pulse Evaluation for Offline and Online Diagnosis of Lithium-Ion Batteries. *Batteries* **2021**, *7*, 36. https://doi.org/ 10.3390/batteries7020036

Academic Editor: Kai Peter Birke

Received: 26 February 2021 Accepted: 2 April 2021 Published: 1 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

mechanisms [1–5]. Compared to recording of the open-circuit-voltage (OCV) or to directly determining the capacity, measuring the dynamic behavior of LIBs is usually less timeconsuming [1].

EIS as a non-invasive technique is widely used for aging studies in order to measure the impedance directly and thus to determine the kinetic parameters of the LIBs [6–9]. Aging processes such as lithium plating [3] and the formation of the solid electrolyte interface (SEI) [4] could be investigated using EIS.

The derivation of the DRT on the basis of the EIS measurement data as well as the further DRT analysis is a common method for the investigation of LIBs and other batteries. In [10–15] methods are investigated and described to obtain the DRT from impedance data. In several publications, the number of dominant electrochemical processes of LIB as well as their time constants and polarization contributions could be quantified by analyzing the DRT of the measured spectra [14–17]. Changes in the time constants and polarization contributions during the aging of the LIBs under investigation can also be determined using the DRT, as recently shown by Sabet et al. [5].

The main disadvantage of EIS is the long measurement time, especially for low frequencies. The measurement time ranges from several hours for frequencies in the millihertz range to several days for frequencies in the microhertz range. Therefore the bandwidth and resolution of the impedance measurement data available for the DRT is limited [15]. Additionally, only small signal amplitudes are permitted to ensure that the steady state condition is not violated during the EIS. The long measuring times and low signal amplitudes cannot be implemented under operating conditions and with commercially available measuring electronics [18], which is why DRT analyzes have so far been limited to laboratory tests. But even under laboratory conditions, when measuring lower frequencies, maintaining the steady state and current drift are becoming increasingly critical [19].

Due to the general availability in online applications, the characterization of cell impedance based on pulse data is of great interest and the subject of some current studies (e.g., [20,21]). The time domain data of current and voltage curves can be evaluated in the event of pulse excitations in order to determine the impedance of the LIB and other battery types. The calculation on the basis of time domain data is generally faster than the direct measurement of the impedance, since several frequencies are excited simultaneously with excitations in the time domain [19]. Thus, on the basis of time domain data, a higher frequency resolution and wider bandwidth of the impedance measurements can be achieved, which is of particular interest with regard to the DRT.

An overview of the options for calculating the impedance using time domain data is given in [22]. In [19,23], the time domain data resulting from pulse measurements were brought into the frequency domain using a fast Fourier transform (FFT) in order to calculate the low frequency impedance. However, the excitation of a pulse is nonperiodic. A window function must therefore be used for the transformations, which in turn generates an offset of the signal [18]. To avoid additional sources of error, as an alternative to the FFT or Laplace transformation, a pre-selected equivalent circuit model (ECM) can be parameterized using the time domain data [19]. The impedance is obtained by transforming the ECM into the frequency domain. The ECM is chosen in a manner that the impedance behavior of the battery can be reproduced. Therefore, the model must be selected individually for each battery type, cell chemistry and the electrochemical processes to be expected [24]. In addition, the impedance of the batteries can change significantly with aging, state of charge (SOC) as well as temperature and thus also the dynamic behavior in the time domain. For this reason, prior knowledge of the electrochemical processes involved and of the aging behavior and further dependencies is required in order to select a suitable model.

The DRT can be derived from the calculated impedance. If the selected ECM consists only of RC elements (parallel connection of resistance and capacitor) and an internal resistance (Thévenin model), the DRT is given directly by the model in the time domain and a transformation to frequency domain is not necessary. Analytical methods have already

been proposed for the online parameterization of a Thévenin model [20,25] but the number of RC elements is limited and too small to be able to derive a DRT with adequate resolution. The sufficient number of RC elements can be characterized by iterative numerical optimization methods such as genetic algorithms and method of least squares. Compared to the analytical methods, however, the computationally intensive iterative calculations are disadvantageous for online applications [25] and the number of RC elements is limited again due to the computing power available [26]. The initial values to be determined also require prior knowledge of the electrochemical processes and their approximate time constants and polarization contributions.

In summary, it can be stated that based on the analysis of the DRT, aging mechanisms have already been examined in numerous investigations. The DRT is determined according to the state of the art on the basis of the impedance. The direct measurement of the impedance by means of EIS cannot be implemented in current online applications without adaptations and at low frequencies, even under laboratory conditions, it is time-consuming and subject to inaccuracies. The indirect determination of the impedance based on the evaluation of time domain data leads to additional inaccuracies or requires an individual definition of a model for different cell chemistries and types. A direct derivation of the DRT based on time domain data, which can also be used online, would therefore be desirable.

In this study a method is proposed and described to derive the DRT directly from the measured voltage course during cell relaxation. The method is validated experimentally by comparing the results with a DRT obtained from EIS measurements. In an aging study, the introduced method and a subsequent DRT analysis are used to identify the time constants and polarization contributions of the electrochemical processes of LIBs. The method is shown to be sufficiently sensitive to quantify the changes in these parameters during aging without the need for EIS equipment or other devices not available in online applications. The study demonstrates that the DRT derived from time domain data offers significantly more insights into processes with large time constants respectively low-frequency impedance with less expenditure of time than a DRT derived from EIS measurements. In addition, a correlation between a polarization contribution of one of the processes with a large time constant and a degradation mechanism could be established. Thus, the proposed method offers a time-effective estimate of the kinetic parameters of a LIB, provides insights into battery degradation and is offline and online applicable.
