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Gaussian Process Regression for In-situ Capacity Estimation of Lithium-ion Batteries

Robert R. Richardson, Christoph R. Birkl, Michael A. Osborne, David A. Howey

arXiv:1712.02595v2stat.AP

TL;DR

Reliable online lithium-ion battery capacity estimation is needed for operation and sizing, but existing approaches can require accurate models or noisy IC/DV feature extraction. GP-ICE regresses directly on smoothed voltage-time data using Bayesian Gaussian processes, achieving approximately 2–3% capacity-estimation error in short galvanostatic tests within selected voltage ranges.

  • Problem

    Accurate online capacity estimation is important for reliable battery operation and sizing, while model-based approaches require accurate battery models and IC/DV approaches amplify noise through differentiation.

  • Method

    GP-ICE smooths galvanostatic voltage-time data, samples time values at equispaced voltages, and uses Gaussian-process regression to estimate capacity with uncertainty.

  • Results

    GP-ICE outperformed IC/DV peak tracking by a factor ∼2.5 in RMSPE and enabled approximately 2–3% capacity estimates from as little as 10 seconds of galvanostatic operation in selected voltage ranges.

  • Takeaways & Limitations

    Direct voltage-time regression supports flexible short diagnostic tests while providing uncertainty estimates and identifying informative voltage ranges for battery-management-system diagnostics.

Abstract

from arXiv · show

Accurate on-board capacity estimation is of critical importance in lithium-ion battery applications. Battery charging/discharging often occurs under a constant current load, and hence voltage vs. time measurements under this condition may be accessible in practice. This paper presents a data-driven diagnostic technique, Gaussian Process regression for In-situ Capacity Estimation (GP-ICE), which estimates battery capacity using voltage measurements over short periods of galvanostatic operation. Unlike previous works, GP-ICE does not rely on interpreting the voltage-time data as Incremental Capacity (IC) or Differential Voltage (DV) curves. This overcomes the need to differentiate the voltage-time data (a process which amplifies measurement noise), and the requirement that the range of voltage measurements encompasses the peaks in the IC/DV curves. GP-ICE is applied to two datasets, consisting of 8 and 20 cells respectively. In each case, within certain voltage ranges, as little as 10 seconds of galvanostatic operation enables capacity estimates with approximately 2-3% RMSE.

I. INTRODUCTION

Accurate online capacity estimation supports reliable battery operation and sizing, but model-based methods require accurate models and IC/DV methods amplify noise through differentiation. GP-ICE instead regresses directly on voltage-time data and uses Gaussian processes to provide calibrated uncertainty.

  • Accurate online capacity estimates support reliable battery operation, corrective action, and battery sizing for microgrids and hybrid energy storage systems.
  • Model-based capacity-estimation approaches require an accurate battery model, while high-fidelity models can face parameter-identifiability challenges.
  • IC/DV-based regression requires differentiating voltage-time data, which amplifies measurement noise and makes peak magnitudes especially sensitive.
  • GP-ICE operates directly on voltage-time data after smoothing it with a Savitzky-Golay filter and sampling time values at equispaced voltages.
  • Gaussian processes provide non-parametric Bayesian regression with model complexity adapted to the data and probabilistic uncertainty intervals for capacity estimates.

A. Overview

GP-ICE learns capacity from labelled, smoothed galvanostatic voltage curves and estimates an unknown cell’s capacity from time values sampled at equispaced voltages during a short constant-current test. Gaussian-process regression produces predictions and uncertainty estimates evaluated with RMSPE and calibration score.

  • Offline:: Training data comprise smoothed galvanostatic voltage curves from cells at varying states of health, each paired with a known capacity.The database may include different C-rates, depths of discharge, and temperatures during aging, while the recorded full charge cycle uses fixed current and ambient temperature.
  • Online:: Online estimation applies constant current for duration ∆t, smooths the measured voltage-time data, and samples time values at n equispaced voltages between Vl and Vh.The sampled n × 1 vector x* is used as the regression input for the unknown-capacity cell.
  • Online:: Fixed voltage sampling keeps test inputs within voltage regions represented by the training data, avoiding extrapolation beyond available training curves.
  • B. Regression: Gaussian-process regression maps the sampled time vector x* to the unknown capacity y* using a Matérn 5/2 kernel.
  • B. Regression: Performance is evaluated with RMSPE, while calibration score measures the frequency of actual capacities lying within a specified credibility interval.For Gaussian predictive distributions, ±2σ corresponds to a 95.4% credibility interval.

III. DATASETS

The study validates GP-ICE on two datasets: an in-house Oxford battery-aging dataset and an open-access NASA dataset. Table I summarizes dataset sample counts and capacity ranges.

  • The Oxford and NASA datasets provide the two validation sources used in this work.Oxford data come from in-house aging experiments, whereas NASA data come from an open-access NASA Ames Research Centre repository.
  • Table I reports the total number of voltage-time curves across cells and the maximum initial and minimum final capacities.

A. Oxford

The Oxford dataset comprises 8 commercial pouch cells aged under repeated urban-drive discharge and constant-current recharge, with periodic characterization curves used to track capacity over life.

  • 8 commercial Kokam pouch cells with 740 mAh nominal capacity were aged at 40 °C using urban-drive discharges and 2C recharges.
  • 519 charge curves were collected as galvanostatic voltage samples for the regression inputs.
  • Capacity was calculated by integrating the 1C charge curves, with end of life defined by terminal voltage dropping below 0 V during discharge.
  • Capacity evolution typically reached end of life at approximately 8,000 cycles, although one cell failed near 5,000 cycles.
  • A different cell underwent a sudden capacity drop around 5,000 cycles, creating a challenging regime change for capacity estimation.

B. NASA

The NASA dataset contains 20 cells subjected to five randomized loading regimes, producing path-dependent capacity fade. GP-ICE nevertheless estimates capacity with increasing measurement duration and calibrated uncertainty.

  • 20 LG Chem 18650 cells were grouped into five groups of four, each undergoing a different randomized loading procedure at room temperature.
  • 842 galvanostatic voltage curves were measured using periodic 2A charge-discharge characterization cycles, with discharge curves used as inputs.
  • Capacity evolution differed substantially across groups, demonstrating path dependence in capacity fade.
  • Larger measurement durations generally improved accuracy, while starting-voltage differences had a less consistent effect in the Oxford-style overall analysis.
  • At ∆t = 1450 s, capacity was accurately estimated even for an isolated capacity near 0.5 Ah.
  • Average calibration scores were CS0.67σ = 0.432 and CS2σ = 0.849, indicating slightly over-confident but still reasonable uncertainty estimates.

B. NASA dataset

The NASA dataset is more challenging because cells experienced diverse randomized cycling regimes. GP-ICE performance improves with measurement duration and depends strongly on the starting voltage.

  • The NASA dataset includes five cycling regimes and statistically similar but non-identical load profiles within each group, making capacity estimation more challenging.
  • A 10 s measurement achieved approximately 10% accuracy when the starting voltage was Vl = 3.7 V, although confidence intervals were larger.
  • Average calibration scores were CS0.67σ = 0.493 and CS2σ = 0.920, indicating only slight over-confidence.
  • Increasing measurement duration generally improved the capacity estimate across the NASA dataset.
  • Performance showed a cliff near Vl = 3.5 V; above this starting voltage, very good performance was achieved regardless of measurement duration.

C. Comparison with IC/DV

GP-ICE is compared with IC+DV peak tracking using the same GP regression model, with performance depending on dataset, voltage range, and test duration. Appropriately selected GP-ICE tests achieve lower RMSPE, while some settings favor IC+DV.

  • GP-ICE uses smoothed voltage-time inputs, whereas IC+DV uses the locations and magnitudes of the largest IC and DV peaks.Both approaches use the same GP model, isolating differences in input construction.
  • 0.49% RMSPE versus 1.11% for IC+DV gives Oxford GP-ICE a 2.26-fold reduction in the best case.The best case uses Δt = 1,450 s and V_l = 3.7 V.
  • NASA GP-ICE outperforms IC+DV at any tested duration, including 10 s, when the starting voltage is sufficiently high at V_l = 3.7 V.The comparison is conditional on the starting voltage.
  • GP-ICE performs worse than IC+DV for lower V_l in NASA and shorter Δt values in Oxford.IC+DV also requires a broad voltage range to capture both curve peaks, which can require a long and variable test.
  • Performance improves negligibly beyond n ≈4 inputs for Δt = 450 s, while Oxford shows minor gains up to n ≈10 for Δt = 1,650 s.The sensitivity analysis covers both Oxford and NASA datasets.

A. Selection of model inputs

GP-ICE selects time values at equispaced voltages from smoothed galvanostatic voltage data rather than relying on cell-specific IC/DV features. This design supports flexible voltage ranges but remains dependent on correlations that vary across cells and conditions.

  • A. Selection of model inputs: GP-ICE uses time values at equispaced voltages because capacity may correlate with portions of the curve beyond IC/DV peaks.The inputs span the selected lower and upper voltage limits.
  • A. Selection of model inputs: Sampling across V_l to V_u imposes no restriction on which voltage range the online test must encompass.The method uses whatever voltage range the test happens to include.
  • A. Selection of model inputs: Equispaced measurements are expected to reflect the overall voltage curve best for a fixed number of inputs n.The paper notes that other design choices could improve performance.
  • B. Physical explanation: Raw voltage measurements can infer capacity without requiring exact knowledge of the physical degradation mechanisms.The approach exploits observable effects of degradation on voltage-time data without encoding cell-specific mechanisms.
  • B. Physical explanation: GP-ICE may transfer across cell chemistries without modification because it does not require expected IC/DV peak locations or counts.Equivalent performance is not guaranteed because voltage-capacity correlations vary across cells and voltage ranges.
  • B. Physical explanation: Application to dynamic drive cycles may be possible when sufficiently long constant-current portions are available, but cell dynamics make this non-trivial.The paper frames this as a possibility rather than an established result.
  • D. Related work: GP-ICE uses Gaussian processes, Savitzky-Golay preprocessing, and selected smoothed inputs, unlike a recurrent-neural-network approach using sequential partial charge curves.GP-ICE also examines how performance varies with starting voltage and measurement duration.

VI. CONCLUSIONS

The paper introduces GP-ICE for capacity estimation from short galvanostatic voltage-time measurements. It combines raw-data regression, automatic input extraction, and Bayesian non-parametric uncertainty handling, achieving approximately 2.5-fold lower RMSPE than IC/DV peak tracking while leaving temperature and C-rate variation for future work.

  • GP-ICE estimates battery capacity from small portions of voltage-time data recorded under constant-current operation.
  • GP-ICE directly regresses on raw voltage data instead of interpreting galvanostatic voltage data as IC or DV curves.
  • GP-ICE automatically extracts inputs by smoothing voltage data and sampling voltages from the smoothed curve.
  • Bayesian non-parametric Gaussian-process regression adapts to data complexity, avoids over-fitting, and provides uncertainty estimates.
  • Approximately 2.5-fold lower RMSPE than IC/DV peak tracking was demonstrated for GP-ICE.The paper also reports flexibility, shorter diagnostic tests, and insight into informative voltage ranges.
  • Variable ambient temperatures and higher pre-specified C-rates require future work with relevant training data.

APPENDIX GAUSSIAN PROCESS REGRESSION

Gaussian process regression performs non-parametric regression by modeling the target function as a Gaussian-process sample rather than assuming a parametric form. Predictions are obtained from conditional Gaussian distributions, with covariance hyperparameters optimized through marginal likelihood.

  • A Gaussian process assigns jointly Gaussian distributions to function values at any finite collection of input points, with mean m(x) and covariance Kij = κ(xi, xj).
  • Gaussian process regression models f(x) as a sample from a Gaussian process instead of assigning it a parametric form with estimated parameters.
  • The Matérn covariance function uses smoothness hyperparameter ν = 5/2; larger ν implies smoother functions.The paper reports similar performance with other common kernels, including the Squared Exponential.
  • With a zero mean function, m(x) = 0 is used for convenience in the regression formulation.
  • Predictions at test indices are obtained by computing the conditional distribution given labelled training inputs and outputs.The conditional distribution is obtained analytically using standard Gaussian-conditioning rules.
  • Covariance hyperparameters θ = {σf, ρ} are optimized by minimizing the negative log marginal likelihood, balancing bias and variance to ameliorate over-fitting.The paper uses conjugate gradients from the GPML toolbox.

Offline

The offline procedure ages cells, periodically acquires full voltage curves under a prescribed constant current, and smooths those curves before subsequent processing. The GP-ICE flow diagram illustrates this workflow using data generated for illustration.

  • Repeated drive cycles are applied across a range of cells as part of the aging campaign.
  • Full voltage curves are periodically acquired from all cells using a pre-specified constant current.
  • The acquired full voltage curves are smoothed with a Savitzky-Golay filter.
  • The GP-ICE flow diagram uses data generated for illustration purposes.
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