On the Use of Cramer-Rao Lower Bound for Least-Variance Circuit Parameters Identification of Li-ion Cells

TL;DR

This work tackles accurate identification of a ten-parameter Li-ion ECM from EIS data by formulating CNLS parameter estimation with a Gaussian measurement model and deriving $CRLB$-based limits via the Fisher Information Matrix. It introduces an automated initialization procedure leveraging spectral geometry and proposes an $E$-optimal design to select informative frequencies, thereby reducing parameter variances. Numerical results show the estimator is efficient and that CRLB-guided frequency adjustments decrease the confidence-ellipsoid volume by at least 25%, with substantial variance reductions for most parameters. The methods have practical impact for battery management and can extend to other electrochemical systems requiring precise impedance-based parameter identification.

Abstract

Electrochemical Impedance Spectroscopy (EIS) and Equivalent Circuit Models (ECMs) are widely used to characterize the impedance and estimate parameters of electrochemical systems such as batteries. We use a generic ECM with ten parameters grouped to model different frequency regions of the Li-ion cell's impedance spectrum. We derive a noise covariance matrix from the measurement model and use it to assign weights for the fitting technique. The paper presents two formulations of the parameters identification problem. Using the properties of the ECM EIS spectra, we propose a method to initialize ECM parameters for the Complex Non-linear Least Squares (CNLS) technique. The paper proposes a novel algorithm for designing the EIS experiments by applying the theory on Cramer-Rao Lower Bound (CRLB) and Fisher Information Matrix (FIM) to the identification problem. We show that contributions to the FIM elements strongly depend on the frequencies at which EIS is performed. Hence, the algorithm aims to adjust frequencies such that the most information about parameters is collected. This is done by minimizing the highest variance of ECM parameters defined by CRLB. Results of a numerical experiment show that the estimator is efficient, and frequency adjustment leads to more accurate ECM parameters' identification.

Figures (8)

• Figure 1: Conceptual accuracy contour plot of EIS instrument defining the areas $A_i, i = 1,\dots,m$ providing the values of the impedance relative magnitude error (undefined in the outer area $A_{\O}$ since $|\bar{Z}|$ exceeds the capability limits of an EIS instrument).
• Figure 2: Impedance spectrum of the Li-ion cell ECM. The parameters are indicated below each element.
• Figure 3: Equivalent MF region spectrum cleaned from the interference produced by $\bar{Z}_{HF}$ and $\bar{Z}_{LF}$ for different ratios of the time constants of $\bar{Z}_{arc,1}$ and $\bar{Z}_{arc,2}$ (cell's parameters $R_1$, $Q_1$, $\phi_1$, $R_2$ and $\phi_2$ true values are those reported in Tab. \ref{['tab:ECMparamValues']}, while parameter $Q_2$ changes its values to achieve a desired $\frac{\tau_2}{\tau_1}$ ratio).
• Figure 4: Bode plots obtained using the initial and estimated parameters (a-b) and minimum, maximum and mean errors of reconstructed EIS spectra compared to the true values (c-d).
• Figure 5: Contributions to the diagonal elements of FIM as a function of frequencies for every ECM parameter.