## 2 Materials and Methods
### Electrical Equivalent Circuit Model
\(I_{0}\) is the exchange current, \(\alpha\) is the symmetry factor between charging and discharging reaction, \(n\) is the number of electrons involved in the reaction, \(F\) is the Faraday constant, \(R\) is the gas constant and \(T\) is the temperature in Kelvin [24]. The Butler-Volmer equation provides the current rate \(I\) through the charge-transfer resistance. \[I=I_{0}\cdot\left(exp\left(\frac{\alpha\cdot n\cdot F}{R\cdot T}\cdot\eta \right)-exp\left(-\frac{(1-\alpha)\cdot n\cdot F}{R\cdot T}\cdot\eta\right)\right) \tag{3}\]  
The charge-transfer resistance \(R_{ct}\) is the quotient of \(\eta\) and \(I\). When \(I\) is measured or determined from a system of equations, which is the case here, Equation (3) needs to be resolved for \(\eta\). This was done numerically with the Matlab function _vapsolve_. Due to the nonlinearity of the Butler-Volmer equation, a distinction between the small-signal and the large-signal value of \(R_{ct}\) was made here. The small-signal value \(R_{ct,ss}\) is measured during EIS  
Figure 3: Spatially-resolved equivalent electrical circuit (srEEC) with two layers to simulate an inhomogeneous current distribution. with a small amplitude of sinusoidal current and corresponds to the derivative of the Butler-Volmer equation resolved accordingly to \(\eta\) (see Equation (4)). For the calculation of the current distribution in the srEEC the large-signal value \(R_{ct,ls}\) is required (Equation (5)), which is equal to the integration of the Butler-Volmer equation resolved accordingly to \(\eta\) over current \(I\)[16]. \[R_{ct,ss}=\frac{\delta\eta\left(I\right)}{\delta I} \tag{4}\]  
\[R_{ct,ls}=\frac{\eta}{I} \tag{5}\]  
Equation (3) could be parameterized for the positive electrode in charging and discharging direction, but for the negative electrode only in discharging direction. In charging direction the low concentration of Pb\({}^{2+}\)-ions, caused by a low dissolution rate of PbSO\({}_{4}\) crystals, limits the charge-transfer process on the negative electrode and induced an increase of the charge-transfer resistance. The dissolution rate is affected by the surface area volume ratio of the crystals and not by any over-potentials [16]. In discharging direction the SO\({}_{4}^{-}\)-ions are relevant for the reaction and because of their high concentration the charge-transfer process is not limited. The dissolution rate dependent concentration of Pb\({}^{2+}\)-ions for charging could be modeled and integrated in the Butler-Volmer equation, as it is described by [PERSON] and [PERSON] [23; 25] However, this would make the model complex and computationally intensive, so that the measured SoC and current dependency of \(R_{ct,neg}\) in charging direction was described with a surface polynomial function (Equation (17)). Because of the inhomogeneous current distribution the SoC levels in the top and the bottom become differently, so that the SoC-dependency of the single elements are included in the model. The dependency on SoC is induced by the changing conductivity and density of the acid, the reduction of porosity and the augmented accumulation of lead-sulfate cystals on the surface of the electrodes towards low SoC. The exchange current \(I_{0}\) includes the activity of the SO\({}_{4}^{-}\)-ions in the electrolyte as factor. For the activity the molar concentration of the electrolyte is used approximately [24]. During the test procedure with stratified electrolyte the molar concentration varied between 4.24 mol L\({}^{-1}\) and 6.25 mol L\({}^{-1}\), so that due to the concentration change \(I_{0}\) increased by 47% from low to high concentration. In the parameterization measurements the acid concentration increased only by 6% from 4.681 mol L\({}^{-1}\) at 0% SoC to 4.972 mol L\({}^{-1}\) at 90% SoC. The small concentration change over SoC during parameterization is caused by the large volume of the electrolyte (1200 mL) in the test cell. Because of this small concentration change the dependency of \(I_{0}\) on molar concentration was separated from the measured SoC dependency.