A statistical model for battery ageing matched the laboratory data used to build it with a reported training root-mean-square error (RMSE) of 0.191%. RMSE is a summary of how far the model’s estimates were from the measured values in the training data. But no validation data were available for previously untried configurations. The result therefore shows how closely the model fit this experiment, not how accurately it can predict cells or charging conditions it did not see.
The study aimed to model repeated capacity-loss measurements and battery state of health, or SoH, while accounting for two kinds of variation: noise within a cell’s repeated profile and differences between cells. It was a controlled laboratory methods case study, and the supplied analysis does not establish performance under real-world electric-vehicle conditions.
A model for repeated battery tests
At the centre was a hierarchical model. For each ageing profile, it used a simple power law—an equation chosen to describe the pattern of capacity loss—and a single-knot cubic B-spline to represent how ageing parameters changed with charging current. The repeated-measurements structure separated measurement noise within a profile from variation among profiles, including cell-to-cell differences.
The researchers estimated the parameters with regularised iterative generalised least squares, or RIGLS, combined with ridge regularisation and BIC-based selection of fixed-point tuning parameters. The nonlinear model was handled with first-order Taylor linearisation. Confidence and prediction intervals were derived with a first-order delta-method expansion, with prediction intervals including both parameter uncertainty and measurement variability.
The interval widths were shown graphically, but the supplied analysis says they were not reported numerically in the text. That limits how precisely readers can judge the uncertainty around individual predictions from the published description.
A tightly controlled laboratory setup
The protocol tested two cells at each charging-current setting from 1 to 5 A, for ten cells in all. The test cells were randomly selected from the source batch.
Cells were aged at 25 degrees Celsius, discharged at a constant 2 A, and charged at currents ranging from 1 to 5 A. Capacity was checked every 50 cycles until failure. Because this design kept the main laboratory conditions fixed, it did not test varying environmental or usage conditions.
The dataset was also shaped by outlier handling. Cell 5 was deemed an outlier and excluded, so the pooled variance result was reported over nine cells. A preliminary nonlinear FAST-LTS procedure also flagged a cell-8 observation at (10.75, 3.80), which was set aside for the remainder of the analysis.
Strong fit, limited test of prediction
RIGLS converged after two iterations, and the full model’s training RMSE was 0.191%. The residuals—the differences between measured and fitted values—showed small systematic errors, meaning the low summary error did not amount to a perfectly pattern-free fit.
The abstract also reports SoH prediction accuracy of plus or minus 0.191% for SoH values from 0 to 20. That figure needs careful reading: the supplied analysis says the exact evaluation split is not described and that validation data were unavailable for previously untried configurations. It is therefore a reported result from the available analysis, not an independently demonstrated performance measure on new data.
The model also estimated how variation was distributed. Its pooled level-1 variance scale was 0.00772 over nine cells; level-1 refers to variation within the repeated measurements of a profile. The reported level-2 random-effects correlation matrix had an off-diagonal value of minus 0.743, while the standard-error matrix included entries of 0.0374 and 0.0233. The authors interpreted those standard-error values as evidence of small cell-to-cell variation, but these remain model estimates rather than a check on performance in a new experiment.
Where the evidence stops
Outlier decisions are another reason to keep the result in perspective. The supplied analysis reports no sensitivity analysis for the excluded observation or for cell 5, so it does not show how much the fitted estimates would change if those data were retained. The exclusion also reduced the analysed cell sample, which may affect estimates of between-cell variation.
The authors interpret the model as accurately reflecting the laboratory cyclic-ageing data and as accounting for both intra-cell and inter-cell variation. They also state that applying it to real-world electric-vehicle use would require modification. Those conclusions are limited by the absence of validation data beyond the configurations used in the study.
The analysis does not establish that changing charging current caused different ageing outcomes. It models profiles collected under charging currents from 1 to 5 A within a fixed laboratory protocol, so its evidence concerns how the model represents the observed profiles rather than a causal test of charging current.
The clearest next test would be independent validation on held-out cells and on charging-current configurations the model has not seen. The supplied analysis also identifies the need to test time-varying temperature, charging, discharge and customer-use conditions, and to examine whether larger experiments produce more stable between-cell covariance estimates. Until then, the 0.191% figure is best treated as a measure of fit to the reported laboratory data, not a demonstrated guarantee of battery or electric-vehicle performance.
Paper data and sources
Original title: A Repeated Measurements Approach to $SoH$ Battery Modelling of Cyclic Aged Data in a Laboratory Environment
Authors: Mark Cary, Charles Bokor
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text