## 2 Materials and Methods
### Electrical Model and Fitting of Impedance Measurements  
The analysis of the impedance spectra was performed with the open source fitting tool of ISEA available at [26]. The principles behind the fitting methodology are also described in [27]. The tool was adapted to include the comparison of spectra with superimposed DC-currents. The chosen equivalent electrical circuit, which was used for the fitting process, is depicted in Figure 2. The areas of impedance values represented by the EEC elements are marked within the initially measured spectrum of battery 2-014. The parameters for the inductive element (L), the inner resistance (\(R_{\mathrm{i}}\)), and one ZARC element as well as one RC-element were collected over time to see their development with operation time of the batteries. The ZARC element consists of the charge transfer resistance for the negative electrode \(R_{\mathrm{ct,NAM}}\) and a constant phase element (CPE), which includes \(C_{\mathrm{ct,NAM}}\) and \(\varphi_{\mathrm{ct,NAM}}\). Its impedance is described by Equation (1):  
\[\underline{Z_{\mathrm{ZARC}}}(\omega)=\frac{R_{\mathrm{ct,NAM}}}{1+(j\!\cdot\! \omega\!\cdot\!R_{\mathrm{ct,NAM}}\!\cdot\!C_{\mathrm{ct,NAM}})^{\varphi_{ \mathrm{ct,NAM}}}} \tag{1}\]  
The exponent \(\varphi_{\mathrm{ct,NAM}}\) describes a depression of the semi-circle which would appear in a Nyquist diagram. Such a depression is usually associated with a distribution of relaxation times of a process around a mean value. The RC-element here mainly describes the processes with higher time constants and consists of the charge transfer resistance of the positive electrode \(R_{\mathrm{ct,PAM}}\) and the capacitance \(C_{\mathrm{ct,PAM}}\). In this case, \(\varphi_{\mathrm{ct,NAM}}\) equals 1. In earlier studies the EEC of two separated electrodes was extended by Warburg elements based on the fundamental Randles circuit [8]. Such a Warburg element would describe the increased overpotentials at longer excitation durations due to diffusion processes. Here solely the faster reactions shall be parameterized as the parameters will be utilized in another step within a model which simulates these diffusion processes separately (as in [28]). Therefore, the EEC is not integrating Warburg elements for the two electrodes to describe the behavior at low frequencies. The shift between the first impedance spectrum and the one measured after 568 days is illustrated in Figure 2.  
One restriction, which needs to be mentioned, is that such an EEC can either be fitted to measurements of one single cell or to a 12 V block with 6 cells in series, as it was done in the conducted tests. In the latter case it is expected that the general behavior of these 6 cells is equal and can be described by a serial connection of cell EEC, which can be joined to one summarized EEC. In this way the battery behavior can be described, but a deviation of the cells in SoC and SoH will overlap and influence the signal. It is expected that the impedance of the full battery cannot represent the worst cell and thus available capacity might be overestimated ([11] see page 119 ff.). Additionally, for a full battery model the EEC is usually combined with further models describing for example the diffusion process within the cell [28]. Such a hybrid model would include processes in frequency and time domain for a better estimation of all overpotentials and thus the available capacity at different loads. However, for the aim of comparing the indications of field measurement devices and the impedance spectroscopy, the basic approach of value comparison for a full battery was chosen. For the analysis here, solely the change of parameters at full-state of charge and 25 \({}^{\circ}\)C after float charge will be evaluated. Additionally, measured impedances were validated with the _zHit_ method described in [29]. Measurement points of the impedance spectra were excluded if this criterion was not met. Especially the points at low frequencies violated this criterion.