## 2 Materials and Methods
### Parameterization
The \(R_{ct}\) values together with the fitting curves are presented in Figure 11.  
\[R_{ct,neg,s} =6.47\times 10^{-3}\ \Omega+2.48\times 10^{-5}\ \Omega\%^{-1}\cdot SoC_{x}+2.9\times 1 0^{-4}\ \Omega\text{A}^{-1}\cdot I_{x}-5.18\times 10^{-7}\ \Omega\%^{-2}\cdot SoC_{x}^{2}\] \[\quad-1.68\times 10^{-4}\ \Omega\%^{-1}\text{A}^{-1}\cdot SoC_{x} \cdot I_{x}+1.15\ \Omega\text{A}^{-2}\cdot I_{x}^{2}+1.48\times 10^{-6}\ \Omega\%^{-2}\text{A}^{-1}\cdot SoC_{x}^{2}\cdot I_{x} \tag{17}\] \[\quad+1.53\times 10^{-5}\ \Omega\%^{-1}\text{A}^{-2}\cdot SoC_{x} \cdot I_{x}^{2}-1.72\times 10^{-4}\ \Omega\text{A}^{-3}\cdot I_{x}^{3}\]  
Equation (17) describes the polynomial function for the small-signal value of \(R_{ct,neg}\). This value was used to calculate the impedance spectrum during simulation. For the calculation of the voltages and current distribution in the srEEC, the large-signal values are required, as already described in Section 2.3. The large-signal value was determined from the integration of Equation (17) from 0 A to the present local currents \(I_{t}\) and \(I_{b}\). The measured SoC-dependency of the ohmic resistance is described by a polynomial function given in Equation (18). Figure 11 shows the determined resistance values over SoC as boxplot. \[R_{ohmic}=8.33\times 10^{-8}\ \Omega\%^{-2}\cdot SoC^{2}-1.70\times 10^{-5}\ \Omega\%^{-1}\cdot SoC+3.06\times 10^{-3}\ \Omega \tag{18}\]  
The resistance of the lead grid is not SoC-dependent and was separated from the measured ohmic resistance, like presented in the srEEC with \(R_{g,t/b}\) and \(R_{i,AM}\). From the design of the lead grid the grid resistance on the top and the bottom was estimated. The grid was 14.5 cm wide, 11 cm high and 1.85 mm thick. The grid bars were 2 mm wide in the middle part and 1 mm in the border area. With 11 bars of 2 mm and 8 bars of 1 mm and a specific resistance of lead of 0.208 \(\Omega\)mm\({}^{2}\)/m a resistance value of 0.187 m\(\Omega\) was calculated for the top part (\(R_{g,t}\)). For the bottom part with 2 bars of 2 mm and 13 bars of 1 mm the grid resistance \(R_{g,b}\) amounted to 0.33 m\(\Omega\). Figure 4: Illustration of the pre-processing and fitting of a discharge pulse for the parameterization of positive electrode. The SoC-dependency of the measured ohmic resistance was attributed to \(R_{i,AM}\). To parameterize this dependency of \(R_{i,AM}\) Equation (19) was solved with the Matlab function _vpsolve_. The equation describes the total impedance of the EEC in Figure 3 for high frequencies, when the RC-elements can be assumed as short-circuits. \[R_{ohmic}=2\cdot R_{g,t}+\frac{(2\cdot R_{g,b}+R_{i,AM})\cdot R_{i,AM}}{2\cdot R _{g,b}+2\cdot R_{i,AM}} \tag{19}\]  
After Equation (19) was solved for various values of SoC, the determined values for \(R_{i,AM}\) over SoC were used to parameterize the polynomial function of \(R_{i,AM}\) given in Equation (20). \[R_{i,AM,homogeneous}=1.66\times 10^{-7}\ \Omega\%^{-2}\cdot SoC^{2}-3.39\times 10^{- 5}\ \Omega\%^{-1}\cdot SoC+5.07\times 10^{-3}\ \Omega \tag{20}\]  
The SoC dependency of \(R_{ct,neg}\) in charging direction is covered by Equation (17). For discharging the determined \(I_{0,neg}\) of \(R_{ct,neg}\) over SoC can be described with a polynomial function (Equation (21)). \[I_{0,neg}=3.35\times 10^{-4}\ \mathrm{A\%}^{-2}\cdot SoC^{2}+\left(4.71\times 1 0^{-2}\ \mathrm{A\%}^{-1}-\frac{c_{full}-c_{ref}}{c_{ref}\cdot 100\%}\right) \cdot SoC+1.77\ \mathrm{A} \tag{21}\]  
According to the separation of acid concentration dependency of \(I_{0}\), described in Section 2.3, the parameterized factor of the second term in Equation (21) is adapted. \(c_{full}\) is the molar concentration at 100% SoC (\(4.972\,\mathrm{mol/L}\)) and \(c_{ref}\) equals to the molar concentration at 0% SoC (\(4.681\,\mathrm{mol/L}\)). \(I_{0,pos}\) of the positive electrode did not change significantly over SoC (see Figure 10), so that an average value of \(0.15\,\mathrm{A}\) was used over the whole SoC range.