Weighted distribution of relaxation time analysis of battery impedance spectra using Gaussian process regression for noise estimation

Electrochemical impedance spectroscopy (EIS) is one of the most widely deployed methods to characterise electrochemical systems such as batteries, fuel cells or electrolyzers. The distribution of relaxation times (DRT) represents a technique to simplify EIS data by deconvolution with a suitable kernel, while with equivalent circuit modelling (ECM) a user-selected function is fitted to characterize the investigated system. Ideally, the residuals of a DRT fit should represent random white noise without systematic residuals, hence no useful data is lost by this analysis step. Thereby DRT can provide the number of distinguishable features based solely on the EIS data, without a priori knowledge of the response of the investigated system. It is demonstrated that such a 'lossless' DRT inversion is possible if the local noise amplitude is considered, which requires a weighted DRT procedure and a method to estimate the frequency dependent noise amplitude. A noise estimate to determine the necessary weights was obtained using multiple EIS acquisitions of the same battery at identical state-of-charge. Alternatively, it is shown that Gaussian process regression (GPR) is capable of estimating an equivalent weighting matrix from a single data set as a prerequisite for automatized weighted DRT inversion without user intervention. The obtained DRT spectrum is then used for the selection of an equivalent circuit model, its initial parametrization, and setting of constraints. The robustness and reliability of this technique is tested numerically using a simple digital twin model. Eventually, by means of the investigated battery it is discussed that using a combination of DRT and ECM, a more physically relevant description of processes in an electrochemical system can be achieved.