## 2 Materials and Methods
### Electrical Equivalent Circuit Model
The change of the exchange current over the concentration or the SoC was assumed to be linear, as the OCV SoC relationship was modeled as linear as well (see Equation (2)). This linear change of \(I_{0}\) was subtracted from the measured \(I_{0}\) values over SoC. The modified SoC dependency describes now the change of \(I_{0}\) caused only by transformations in the active mass and is given in Section 2.4. The concentration dependency is modeled using Equation (6) with the present molar concentration \(c\) and \(I_{0,ref}\), the exchange current at present SoC and at \(c_{ref}\), the molar concentration at 0% SoC. \[I_{0}=I_{0,ref}(c=c_{ref})\cdot\left(1+\frac{(c-c_{ref}}{c_{ref}}\right) \tag{6}\]  
Equation (6) is applied on \(I_{0}\) of the positive and the negative electrode in discharging direction. For \(R_{ct,neg}\) in charging direction this separation of the concentration dependency was not made, as the measured increase of the charge-transfer resistance in charging direction is significantly higher than changes generated by the acid concentration. For the resistances \(R_{i,AM,t}\) and \(R_{i,AM,b}\) a separated acid concentration dependency had to be introduced as well. During parameterization measurements the conductivity of the acid in the test cell increased from low to high SoC [8]. However, the parameterization measurements presented a decrease of the conductivity (see Section 2.4). With 20% the decrease is lower than measured by Theleon lead-acid starter-batteries (37% for 20 \({}^{\circ}\)C) [23]. So the decrease of acid concentration in the test cell extenuated the ohmic resistance increase towards low SoC and the increase is mainly generated by the accumulation of lead-sulfate crystals and the change of porosity. Therefore, a separate consideration of the concentration dependency of \(R_{i,AM}\) was implemented. The rate of the electrolyte resistance on \(R_{i,AM}\) could not be measured separately, but it was assumed that the electrolyte resistance provides the main part in \(R_{i,AM}\). Equation (7) describes this dependency with \(R_{i,AM,homogeneous}\) as the resistance corresponding to the present SoC and determined with Equation (20), \(dr_{i,am}\) as the relative change of specific resistance per g/cm\({}^{3}\) density change (here 3 \((\mathrm{g\,cm^{-3}})^{-1}\), based on data from [PERSON] [8]), \(\rho_{ref}\) as reference density (here 1.28 g cm\({}^{-3}\)) and \(\rho_{x}\) as the present densities in the top and bottom of the cell. \[R_{i,AM}=R_{i,AM,homogeneous}(SoC)\cdot\left(1+dr_{i,am}\cdot(\rho_{ref}-\rho x)\right) \tag{7}\]  
In case of stratification this equation is used to adapt \(R_{i,AM}\) in the top and the bottom of srEEC to the local acid densities. The influence of these separations of the acid concentration dependence from the measured SoC dependency is presented in Figure S1 provided as supplementary material. The SoC-dependency of the double-layer capacitances are described with simple polynomial functions (provided in Section 2.4) and no separation of the acid concentration dependency was considered, as the capacitances does not influence the current distribution in the steady-state. The simulation using this srEEC starts with the calculation of initial values for \(R_{i,AM,x}\) and \(C_{dl,x}\) based on initial local SoC. The capacitances are not charged at the beginning, therefore they can be assumed as short circuit and the voltages \(V_{RC,x}\) is equal to zero. The current \(I_{t}\) (0) is determined with Equation (8) and \(I_{b}\) (0) is determined with Equation (12). The current \(I_{cell}\) is the set DC-current during EIS. \[I_{t}\left(0\right)=\frac{I_{cell}\left(R_{g,b}+R_{i,AM,b}\right)+V_{0,b}-V_{0, t}}{R_{g,b}+R_{i,AM,b}+R_{i,AM,t}} \tag{8}\]  
In the next step the \(R_{ct,x}\) values are determined using the Butler-Volmer equation or Equation (17) for charging currents with \(I\) as the currents, which flow through the resistances. At the beginning these currents are zero.