# Improved Electrochemical-Mechanical Parameter Estimation Technique for Lithium-Ion Battery Models
## 2 Materials and Methods
### Objective Function  
Once the electrochemical-mechanical behavior of the battery is modeled, the objective function for the parameter estimation can be set. Least-Square Objective Function (\(LSOF\)) was used to measure and minimize the difference between observed data points and the model's predicted values. This approach is very useful for curve fitting since it evaluates the sum of squared errors computed over all data points, preventing positive and negative errors from canceling out and ensuring that larger deviations are penalized more heavily, improving convergence.  
In a time-dependent curve fitting, considering \(u_{i}\) the generic variable at the \(i\)-th time step, the \(LSOF\) is computed, as presented in Equation (8).  
\[LSOF(\mathbf{\theta})=\sum_{i=0}^{n}(u_{i}^{mod}(\mathbf{\theta})-u_{i}^{exp})^{2} \tag{8}\]  
A multi-objective function approach is needed to evaluate the integration of the battery deformation in parameter estimation. Therefore, two LSOFs need to be computed in order to account for both the electrochemical measurements of the voltage profile and the mechanical measurements of the battery deformation.  
\[LSOF_{cc}(\mathbf{\theta})=\sum_{i=0}^{n}(V_{i}^{mod}(\mathbf{\theta})-V_{i}^{exp})^{2} \tag{9}\]  
\[LSOF_{m}(\mathbf{\theta})=\sum_{i=0}^{n}(\Delta t_{bat,i}^{mod}(\mathbf{\theta})- \Delta t_{bat,i}^{exp})^{2} \tag{10}\]  
Due to the different scales of the electrochemical objective function \(LSOF_{cc}\) and the mechanical one \(LSOF_{m}\), Equations (9) and (10) have been normalized with respect to the square of the minimum voltage and maximum deformation (measured at the end of the test), respectively. Additionally, the battery capacity \(C\) is controlled with the same quadratic approach, obtaining the objective functions reported in Equations (11)-(13) which constitute the multi-objective problem statement presented in Equation (14) as a vector of objective functions.  
\[f_{LSOF_{cc}}(\mathbf{\theta})=\frac{LSOF_{cc}(\mathbf{\theta})}{V_{min}^{2}} \tag{11}\]  
\[f_{LSOF_{m}}(\mathbf{\theta})=\frac{LSOF_{cc}(\mathbf{\theta})}{\Delta t_{max}^{2}} \tag{12}\]\[f_{\rm C}(\mathbf{\theta})=\frac{(C^{mod}(\mathbf{\theta})-C^{exp})^{2}}{(C^{exp})^{2}} \tag{13}\]  
\[\mathbf{F}(\mathbf{\theta})=\begin{Bmatrix}f_{LSOF_{\rm ex}}(\mathbf{\theta})\\ f_{LSOF_{\rm ex}}(\mathbf{\theta})\\ f_{\rm C}(\mathbf{\theta})\end{Bmatrix} \tag{14}\]  
Depending on the same set of parameters \(\mathbf{\theta}\) from the same multiphysics modeling, the objective functions cannot be minimized at the same time and independently. Therefore, the multi-objective problem is solved using a scalarization approach to reduce computational complexity and enable variations in the weighting of the objective functions with respect to the Pareto technique [31; 32]. Thus, a single objective function is obtained in Equations (15) and (16), where \(\mathbf{w}\) is the weights vector of the objective functions. Weighting factors are eventually applied to prioritize certain objective functions over others. The electrochemical-only parameter estimation \(w_{2}=0\) is used to ignore the mechanical objective function.  
\[f(\mathbf{\theta})=\mathbf{w}^{\rm T}\cdot\mathbf{F}(\mathbf{\theta}) \tag{15}\]  
\[f(\mathbf{\theta})=w_{1}f_{LSOF_{\rm ex}}(\mathbf{\theta})+w_{2}f_{LSOF_{\rm ex}}(\bm {\theta})+w_{3}f_{\rm C}(\mathbf{\theta}) \tag{16}\]