## 3 Two-Step Parametrization Method
### Validation of Method  
This section shows the validation of the two-step parametrization method. Simulating the model with a plausible initial guess of the parameter vector \(\theta_{0}\) and using the measured inputs \(\mathbf{u}\) (described in Section 4 and depicted in Section 5 under parametrization data) yields the simulated model outputs \(\mathbf{y}\). The initial values \(\mathbf{x}(t_{0})\) are obtainable by assuming steady-state at the first sample instant. The data sequence is appropriately selected so that at sampling instant \(t_{0}\), the steady-state assumption holds. The simulated model outputs with additional Gaussian noise replace the measured outputs \(\mathbf{y}^{*}\) as reference data to validate the method. In this case, the proposed method has to deliver the used parameter \(\theta_{0}\) on average if it is an unbiased estimator. Conducting the two-step parametrization method described in Section 3 yields the parameters' total information, shown in Figure 2a,b. Due to the floating-point arithmetic computation, the result's accuracy depends on the used hardware [36]. Therefore, the parameters with a total information value less than the hardware accuracy (indicated by the yellow line) are not identifiable. The most significant thermodynamic parameters in descending order are \(k_{\text{ca,em,out}}\), \(k_{\text{ca,sm,out}}\), \(k_{\text{ca,em,in}}\), \(k_{\text{an,sm,out}}\), \(V_{\text{ca,cm}}\), \(k_{\text{cond}}\), \(k_{\text{an,em,in}}\), \(k_{\text{perm}}\), \(k_{\text{an,em,out}}\), and \(V_{\text{an,cm}}\). The least significant ones in ascending order are \(V_{\text{ca,sm}}\), \(V_{\text{an,em}}\), \(V_{\text{an,sm}}\), \(V_{\text{ca,em}}\), \(k_{\text{evap}}\), \(k_{\text{an,leak}}\), and \(\tau_{\text{m}}\). Only parametrizing the most significant thermodynamic parameters reduces the dimension of the sub-solution space by \(\frac{15}{25}=60\%\) compared to the global one. The most significant electrochemical parameters in descending order are \(E_{\text{ca,act}}\), \(E_{\text{an,act}}\), \(e_{2}\), \(K_{\text{ca}}\), \(K_{\text{an}}\), and \(R_{\text{c}}\). The least significant ones in ascending order are \(CD_{\text{an}}\), and \(CD_{\text{ca}}\). Only parametrizing the most significant electrochemical parameters reduces the dimension of the sub-solution space by \(\frac{19}{25}=76\%\) compared to the global one. The least significant parameters are kept constant at their initial values at all times. The two submodels with their most significant parameters are subject to parametrization. MATLAB R2020b's genetic algorithm optimizer parametrizes each submodel with respect to the most significant parameters \(\theta_{\text{ms}}\) 100 times [37]. Heuristic algorithms increase the probability of finding the global minimum in a high dimensional solution space compared to iterative methods. Every iteration randomly initializes the genetic algorithm optimizer and the Gaussian noise for the model outputs. The optimizer's population size is 200, the generation limit is 20, and the remaining options remain unchanged. Figure 2c,d illustrate the relative distribution of the estimated parameters \(\theta_{\text{ms,opt}}\). The conclusion is that the estimated parameters are equal to the used parameters \(\theta_{0}\) on average, validating the proposed method. The respective spreads of the estimates correlate with the total information. The more information exists, the smaller the spread. The correlation is not perfect but will converge to the Cramer-Rao bound (15) by increasing the population size, the generation limit, and the number of estimations. For comparison, Figure 2e depicts the relative distribution of the estimated parameters \(\theta_{\text{opt},i}\) with a usual \"single-step\" approach for the entire model. The genetic algorithm optimizer minimizes the objective function \(J\) (23) in a single step for the entire model without any further considerations.