## 2 Materials and Methods
### Electrical Equivalent Circuit Model
With Equation (9) the voltage drops \(V_{RC,x}\) over the RC-elements are calculated for every RC-element. The parameter \(n\) is the counter for the time steps, while \(n-x\) specifies the values from previous time steps. For \(I_{x}\left(n-1\right)\) and \(I_{x}\left(n-2\right)\) the same value are used at first, which are the initial values for \(I_{t}\) and \(I_{b}\). \[V_{RC,x}\left(n\right)=\frac{V_{RC,x}\left(n-1\right)\cdot\left(\frac{2\cdot R _{ct,x}\cdot C_{dl,x}}{\Delta t}-1\right)+R_{ct,x}\cdot\left(I_{x}\left(n-1 \right)+I_{x}\left(n-2\right)\right)}{\frac{2\cdot R_{ct,x}\cdot C_{dl,x}}{ \Delta t}+1} \tag{9}\]  
With the determined \(V_{RC,x}\) the cell voltage \(V_{cell}\) is calculated with Equation (10), which was generated with the Kirchhoff's voltage law applied to the closed circuit \(S_{1}\) in Figure 3.  
\[I=I_{0}\cdot\left(exp\left(\frac{\alpha\cdot n\cdot F}{R\cdot T}\cdot\eta \right)-exp\left(-\frac{(1-\alpha)\cdot n\cdot F}{R\cdot T}\cdot\eta\right)\right) \tag{3}\]0  
At this point all values in the srEEC are calculated for one simulation step. At next, the local currents \(I_{t}\left(n\right)\) and \(I_{b}\left(n\right)\) are determined for next simulation step using the perviously calculated voltages \(V_{RC,x}\) (Equations (11) and (12)). \[I=I_{0}\cdot\left(exp\left(\frac{\alpha\cdot n\cdot F}{R\cdot T}\cdot\eta \right)-exp\left(-\frac{(1-\alpha)\cdot n\cdot F}{R\cdot T}\cdot\eta\right)\right) \tag{3}\]1  
\[I=I_{0}\cdot\left(exp\left(\frac{\alpha\cdot n\cdot F}{R\cdot T}\cdot\eta \right)-exp\left(-\frac{(1-\alpha)\cdot n\cdot F}{R\cdot T}\cdot\eta\right)\right) \tag{3}\]2With these new values the new local \(SoC_{x}\left(n\right)\) are determined as well (Equation (13)). Here to an half of \(C_{nom}\) is used for SoC, as the local SoC corresponds to an half of the electrode, respectively. \[I=I_{0}\cdot\left(exp\left(\frac{\alpha\cdot n\cdot F}{R\cdot T}\cdot\eta \right)-exp\left(-\frac{(1-\alpha)\cdot n\cdot F}{R\cdot T}\cdot\eta\right)\right) \tag{3}\]3  
The duration of one time step \(\Delta t\) is adapted to the dynamic change of the currents through the capacitances. Therefore these currents are determined with Equation (14) for the present simulation step and are compared with the values from previous step. \[I=I_{0}\cdot\left(exp\left(\frac{\alpha\cdot n\cdot F}{R\cdot T}\cdot\eta \right)-exp\left(-\frac{(1-\alpha)\cdot n\cdot F}{R\cdot T}\cdot\eta\right)\right) \tag{3}\]4  
The duration \(\Delta t\) is used to calculate \(V_{RC,x}\). If the changes of the currents between two steps are too big, Equation (9) is not valid anymore and the model becomes instable. The current changes are limited to \(0.02\,\mathrm{A}\). If the limit was exceeded, \(\Delta t\) is divided by 10 and the values for \(V_{RC,x}\) are calculated again for the same simulation step, until the differences are below the limit or \(\Delta t\) is equal \(1\times 10^{-5}\,\mathrm{s}\). A duration of \(1\times 10^{-5}\,\mathrm{s}\) for \(\Delta t\) ensures the stability of the model with the used parameters and the adaption of \(\Delta t\) reduces the simulation duration and generated data volume. The determined currents through the capacitances (Equation (14)) are used in the next simulation step to determine the currents through \(R_{ct,x}\), which are the differences between \(I_{x}\) and \(I_{Cdl,x}\left(n-1\right)\). The model does not simulate the local acid density on the basis of diffusion, generation and consumption of sulfate ions during charging and discharging, gravity and electrolyte mixing, like in the work of [PERSON] et al. or [PERSON] [19; 25] In the test cell a homogeneous SoC could be set before the measurements with stratification, so that the initial condition of the acid was well known for _Strat. X_. Furthermore, the duration of EIS measurements, which have to be simulated, was only \(14.5\,\mathrm{h}\), so that diffusion processes did not change the local acid densities significantly.