## 3 Two-Step Parametrization Method
### Procedure  
The proposed method parametrizes the model in two consecutive main steps:  
1. Thermodynamic submodel 1. Parameter sensitivity analysis with respect to the thermodynamic parameters \(\theta_{\text{th}}\) yields a subset with the most significant parameters \(\theta_{\text{th,ms}}\), where \(\theta_{\text{th,ms}}\,\subseteq\,\theta_{\text{th}}\) holds. 2. Parametrization with respect to the most significant parameters \(\theta_{\text{th,ms}}\) yields the optimized parameters \(\theta_{\text{th,opt}}\). The least significant parameters \(\theta_{\text{th,ls}}\) are kept constant at their initial values. 2. Electrochemical submodel 1. Solve thermodynamic submodel using the optimized thermodynamic parameters \(\theta_{\text{th,opt}}\) and store the resulting model states \(\mathbf{x}\) for further usage. 2. Parameter sensitivity analysis with respect to the electrochemical parameters \(\theta_{\text{el}}\) yields a subset with the most significant parameters \(\theta_{\text{el,ms}}\), where \(\theta_{\text{el,ms}}\,\subseteq\,\theta_{\text{el}}\) holds. 3. Parametrization with respect to the most significant parameters \(\theta_{\text{el,ms}}\) yields the optimized parameters \(\theta_{\text{el,opt}}\). The least significant parameters \(\theta_{\text{el,ls}}\) are kept constant at their initial values. Combining both solutions lead to the full optimized parameter vector \(\theta_{\text{opt}}\,=\,\left[\theta_{\text{th,opt}},\theta_{\text{el,opt}} \right]^{\text{T}}\). Assuming that the result is near the optima, the entire model with its optimized most significant parameters \(\theta_{\text{ms,opt}}=\left[\theta_{\text{th,ms,opt}},\theta_{\text{el,ms,opt}} \right]^{\text{T}}\) can be further refined with iterative methods to approach the optima while the least significant parameters are kept constant \(\theta_{\text{ls}}=\left[\theta_{\text{th,ls}},\theta_{\text{el,ls}}\right]^{ \text{T}}\) at their initial values. The two-step parametrization method requires measurement data (according to the model's inputs and outputs), and a model separated into a submodel with ODEs and a submodel without it. The optimization goal is to minimize an objective function \(J\), which contains the weighted squared errors between the simulated and measured output signals and a regularization term [19]:  
\[J(\theta)=\sum_{k=0}^{n_{\text{L}}}(\mathbf{y}(t_{k},\theta)-\mathbf{y}^{*}(t_ {k}))^{\text{T}}\mathbf{Q}_{\mathbf{y}}(\mathbf{y}(t_{k},\theta)-\mathbf{y}^{* }(t_{k}))+(\theta_{0}-\theta)^{\text{T}}\mathbf{Q}_{\theta}(\theta_{0}-\theta) \tag{23}\]Here, \(\mathbf{y}(t_{k},\theta)\) denotes the model output at sampling instant \(t_{k}\) for \(k\in\{1,2,\ldots,n_{k}\}\), \(\mathbf{y}^{*}(t_{k})\) the measured output, \(\mathbf{Q}_{\mathbf{y}}\) the output weighting matrix, \(\theta_{0}\) a plausible initial guess of the parameter vector, and \(\mathbf{Q}_{\theta}\) the regularization matrix. The weighting matrix \(\mathbf{Q}_{\mathbf{y}}\) takes the individual weighting of each output and the different output magnitudes into account. The last term in the objective function \(J\) (23), also called regularization, penalizes the deviation of the parameter vector \(\theta\) from its initial guess \(\theta_{0}\) and takes the different parameter magnitudes into account. The regularization is also beneficial if the parameters' uncertainty differs, for example, if the approximate values for some parameters are derivable from literature. The goal is to minimize the objective function \(J\). In this case, the optimization problem is stated as follows:  
\[\theta_{\text{opt}} =\underset{\theta}{\arg\min}\;J(\theta) \tag{24}\] \[\text{with respect to}\] \[\theta_{i\text{min}} \leq\theta_{i}\leq\theta_{i\text{,max}}\;\text{for}\;i\in\{1,2, \ldots,n_{\theta}\}\]  
Solving the optimization problem yields the optimized parameter vector \(\theta_{\text{opt}}\). For optimal results, the parameter space has to be constrained.