# Improved Electrochemical-Mechanical Parameter Estimation Technique for Lithium-Ion Battery Models
## 4 Results
### Teardown Configuration  
In this configuration, the electrode thickness and the number of layers are known via SEM analysis conducted after battery teardown, as illustrated in Figure 3 [16]. The estimation procedure requires initial guess values as a starting point for the parameters to be estimated. The initial value of the active material fraction can be evaluated from the electrode capacity at low discharge rates--in which the stoichiometric window is known--from the relation expressed in Equation (26). On the other hand, the diffusion coefficients and reaction rates are selected based on values commonly reported in the literature, which often span several orders of magnitude. In particular, [PERSON] et al. [37] and [PERSON] et \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Parameter** & **Symbol** & **Cathode** & **Separator** & **Anode** & **Unit** \\ \hline \hline \multirow{2}{*}{\begin{tabular}{l} \(\mathrm{Molar\ mass}\) \\ Crystal density \\ Maximum concentration \\ Electrolyte diffusivity \\ Solid-phase conductivity \\ Transference number \\ Bruggeman coefficient \\ Particle radius \\ \end{tabular} } & \(M_{m,k}\) & \(157.76\)\({}^{c}\) & \(-\) & \(79\)\({}^{c}\) & g/mol \\ \multirow{2}{*}{\begin{tabular}{l} \(\mathrm{Crystal\ density}\) \\ Maximum concentration \\ Electrolyte diffusivity \\ Solid-phase conductivity \\ Transference number \\ \end{tabular} } & \(\rho_{k,k}^{crystal}\) & \(3600\)\({}^{l}\) & \(-\) & \(2260\)\({}^{l}\) & kg/m\({}^{3}\) \\ \multirow{2}{*}{\begin{tabular}{l} \(\mathrm{Maximum\ concentration}\) \\ Electrolyte diffusivity \\ Solid-phase conductivity \\ Transference number \\ \end{tabular} } & \(228.820\)\({}^{c}\) & \(-\) & \(28.600\)\({}^{c}\) & mol/m\({}^{3}\) \\ \multirow{2}{*}{\begin{tabular}{l} \(\mathrm{Electrolyte\ diffusivity}\) \\ Solid-phase conductivity \\ Transference number \\ \end{tabular} } & \(1.3\cdot 10^{-10}\)\({}^{l}\) [34] & \(1.3\cdot 10^{-10}\)\({}^{l}\) [34] & \(1.3\cdot 10^{-10}\)\({}^{l}\) [34] & m\({}^{2}\)/s \\ \multirow{2}{*}{\begin{tabular}{l} \(\mathrm{Solid\ phase\ conductivity}\) \\ Transference number \\ \end{tabular} } & \(\sigma_{s,k}\) & \(6\)\({}^{l}\) [23] & \(-\) & \(100\)\({}^{l}\) [35] & S/m \\ \multirow{2}{*}{\
\begin{tabular}{l} \(\mathrm{Transference\ number}\) \\ Bruggeman coefficient \\ Particle radius \\ \end{tabular} } & \(t_{+}\) & \(0.363\)\({}^{l}\) [23] & \(-\) & \(-\) \\ \multirow{2}{*}{
\begin{tabular}{l} \(\mathrm{Bruuggeman\ coefficient}\) \\ Particle radius \\ \end{tabular} } & \(R_{k}\) & \(0.03\)\({}^{m}\) & \(-\) & \(10\)\({}^{m}\) & \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fixed parameters: \(l\), from the literature; \(c\), calculated; \(m\), measured. al. [34] reported representative values for the diffusion coefficients of LFP and graphite, respectively, while [PERSON] et al. [23] provided a comprehensive review of the ranges of reaction rate constants for both electrodes. The parameter ranges were taken from these references, and the initial guesses were chosen from the central order of magnitude of these ranges, as presented in Table 3, together with the final estimated values. Then, an additional analysis was performed by varying the initial values to assess the impact on the estimation results, as described in Section 4.5. \[C_{p}=\frac{c_{s,p}^{max}\Delta yFNAt_{p}\zeta_{p}}{3600}\quad C_{n}=\frac{c_{s, n}^{max}\Delta xFNAt_{n}\zeta_{n}}{3600} \tag{26}\]  
Figure 4 shows the voltage profile and the thickness change obtained at the end of the optimization process with the two estimation approaches in the teardown scenario. Furthermore, Table 4 shows the corresponding Root Mean Square Errors (RMSEs) as a sum of the RMSEs computed at C/2 and at 3C: the electrochemical-mechanical (ECM) approach leads to more accurate prediction of the battery behavior, presenting significantly lower errors; in fact, the electrochemical (EC) approach converges faster due to the simplicity of the objective function but leads to less accurate estimate.