# Improved Electrochemical-Mechanical Parameter Estimation Technique for Lithium-Ion Battery Models
## 4 Results  
### Parameters and Constraints  
Certain battery parameters can be measured or evaluated experimentally with relatively simple tests and can therefore be considered known or fixed during the estimation procedure, which helps to reduce the complexity of the simulation. In contrast, some electrochemical parameters--such as diffusion coefficients and reaction rates--require more advanced and complex experimental procedures for accurate identification, and their values reported in the literature often span a wide range. Other physical parameters, such as electrode thickness and the number of electrode layers, can only be determined after battery teardown. Therefore, alongside the comparison between electrochemical and electrochemical-mechanical parameter estimation, two distinct scenarios were analyzed:  
1. Battery Teardown--The battery has been disassembled, and parameters such as electrode thickness and the number of layers are known. In this case, the set of parameters (\(\theta_{\mathbf{t}d}\)) to be estimated comprises the diffusion coefficients, the reaction rates, and the active material fractions of the two electrodes, as shown in Equation (21): \[\theta_{\mathbf{t}d}=\left\{D_{s,p}\quad D_{s,n}\quad k_{p}\quad k_{n}\quad\zeta_{p }\quad\zeta_{n}\right\}\] (21)
2. No Battery Teardown--The battery has not been disassembled, and the thickness of the electrodes and the number of layers are unknown. The extended set of parameters (\(\theta_{\mathbf{n}o-\mathbf{t}d}\)) is presented in Equation (22): \[\theta_{\mathbf{n}o-\mathbf{t}d}=\left\{\theta_{\mathbf{t}d}\quad t_{p}\quad t_{n}\quad N\right\}\] (22) In this case, it is necessary to constrain the overall battery thickness computed by the model \(t_{bat}^{mod}\)--accounting for the separator thickness \(t_{s}\), the current collector thickness \(t_{p}^{cc}\), and \(t_{n}^{cc}\) (Equation (23))--to remain within a specified tolerance range around the experimental value \(t_{bat}^{exp}\), as shown in Equation (24). \[t_{bat}^{mod}=N(t_{p}+t_{s}+t_{n})+\frac{N}{2}(t_{p}^{cc}+t_{n}^{cc})+2t_{ case} \tag{23}\] \[\frac{|t_{bat}^{mod}-t_{bat}^{exp}|}{t_{bat}^{exp}}\leq tol_{thick} \tag{24}\]  
Then, the parameters sets \(\theta_{td}\) and \(\theta_{no-td}\) are estimated by fitting experimental curves obtained from constant current cycles with a 100% depth of discharge (DOD) at a \(C/2\) current rate. In both scenarios, the stoichiometric windows are calculated with differential voltage analysis (DVA) at low current rates. The stoichiometric indexes corresponding to \(SOC=100\%\) (\(x_{100}\) and \(y_{100}\)) are imposed as fixed initial values during discharge, while \(x_{0}\) and \(y_{0}\) are constrained around their corresponding experimental value with a certain tolerance (\(tol_{x_{0}}\) and \(tol_{y_{0}}\) respectively), as shown in Equation (25). \[\frac{|x_{0}^{mod}-x_{0}^{exp}|}{x_{o}^{exp}}\leq tol_{x_{0}}\quad\frac{|y_{0} ^{mod}-y_{0}^{exp}|}{y_{o}^{exp}}\leq tol_{y_{0}} \tag{25}\]  
However, the estimation cannot be considered complete: the diffusion coefficients are current-dependent [23], and the contact resistance \(R_{c}\)--which is another challenging parameter to assess--is voltage-insensitive at low current rates, namely, its value does not influence the voltage when the current is low. Therefore, the experimental data at 3C discharge current rate are then used to assess \(R_{c}\) and extrapolate the diffusion coefficient behavior \(D_{s,k}(i)\) by linearly fitting the estimated \(D_{s,k}(i_{C/2})\) and \(D_{s,k}(i_{3C})\). Among the other parameters--considered fixed in the optimization process--there are material properties, like the molar mass \(M_{m}\) and the crystal density \(\rho^{crystal}\) of the active materials' lattice structure; the corresponding maximum lithium concentration that can be obtained, as stated in Equation (3); electrolyte properties; solid-phase conductivity; and the active material particle radius. Electrolyte parameters have a minor influence on the results; thus, they are taken from the literature.