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Modified Gaussian Process Regression Models for Cyclic Capacity Prediction of Lithium-ion Batteries

Kailong Liu, Xiaosong Hu, Zhongbao Wei, Yi Li, Yan Jiang

arXiv:2101.00035v1cs.LGeess.SY

TL;DR

The paper addresses cyclic capacity prediction for Li-ion batteries under varying temperature and DOD, where degradation is nonlinear and existing GPR kernels underuse ageing knowledge. It develops a condition-aware GPR structure with two modified kernels, finding that the knowledge-integrating Model B improves training accuracy over Model A and supports one-step and multi-step prediction. The approach is evaluated on NMC batteries under varied cycling conditions.

  • Problem

    Battery capacity prognostics are complicated by nonlinear degradation across operating conditions, while existing GPR studies insufficiently integrate electrochemical or empirical ageing knowledge into covariance functions.

  • Method

    The paper develops a cyclic-condition-aware GPR structure, Model A with an ARD-SE kernel, and Model B coupling Arrhenius and polynomial components in a compositional kernel.

  • Results

    Model B describes capacity-fading dynamics better than Model A, with maximum training ME, MAE, and RMSE reduced by 40.7%, 36.0%, and 37.8%, respectively.

  • Takeaways & Limitations

    The modified GPR approach provides satisfactory one-step and multi-step capacity predictions with reliable uncertainty quantification under varied cyclic conditions.

Abstract

from arXiv · show

This paper presents the development of machine learning-enabled data-driven models for effective capacity predictions for lithium-ion batteries under different cyclic conditions. To achieve this, a model structure is first proposed with the considerations of battery ageing tendency and the corresponding operational temperature and depth-of-discharge. Then based on a systematic understanding of covariance functions within the Gaussian process regression, two related data-driven models are developed. Specifically, by modifying the isotropic squared exponential kernel with an automatic relevance determination structure, 'Model A' could extract the highly relevant input features for capacity predictions. Through coupling the Arrhenius law and a polynomial equation into a compositional kernel, 'Model B' is capable of considering the electrochemical and empirical knowledge of battery degradation. The developed models are validated and compared on the Nickel Manganese Cobalt Oxide (NMC) lithium-ion batteries with various cycling patterns. Experimental results demonstrate that the modified Gaussian process regression model considering the battery electrochemical and empirical ageing signature outperforms other counterparts and is able to achieve satisfactory results for both one-step and multi-step predictions. The proposed technique is promising for battery capacity predictions under various cycling cases.

I. INTRODUCTION

Li-ion battery capacity degrades through cycling, making accurate capacity prognostics important for reliability and safety. Existing methods motivate modified GPR models that incorporate cyclic conditions and battery-ageing knowledge while retaining uncertainty quantification.

  • Battery degradation is nonlinear and depends on manufacturing and operating conditions, complicating cyclic capacity prognostics.
  • Model-based approaches can estimate cyclic capacity under certain conditions, but their performance depends directly on model parameters.
  • SVM and ANN methods provide deterministic capacity predictions but do not quantify prediction uncertainty.
  • GPR offers nonparametric, probabilistic modelling with uncertainty estimates for possible future battery-health values.
  • Existing GPR studies largely use conventional kernels and insufficiently combine battery ageing electrochemical or empirical knowledge with covariance functions.
  • The paper proposes a cyclic-condition-aware GPR structure and develops Model A with ARD-SE and Model B with Arrhenius and polynomial kernel components.

II. CYCLIC AGEING DATASET AND GPR MODEL STRUCTURE

The study uses NMC/graphite pouch-battery data collected across temperature and DOD conditions to examine cyclic capacity degradation. Four cases support training, while two additional cases test prediction effectiveness and robustness.

  • The dataset comes from commercial 21Ah Nickel-Manganese-Cobalt/graphite pouch batteries, with temperature, DOD, and FECs treated as stress factors.
  • Six cyclic conditions combine DOD levels of 50%, 80%, and 100% with operational temperatures of 35℃ and 45℃.
  • Deep cycling and higher temperatures, including Case 4 and Case 6, generally produce faster capacity degradation.

B. GPR model structure

The proposed GPR structure predicts future capacity from recent capacity history together with cyclic temperature and DOD. It supports both one-step prediction and recursive multi-step forecasting under fixed cyclic conditions.

  • Cyclic temperature and DOD are selected as inputs because temperature-dependent side reactions and DOD-dependent electrode stress affect capacity degradation.
  • The model combines previous and current capacity terms with cyclic temperature and DOD to capture ageing tendencies across operating conditions.
  • Training maps a history of capacity values plus cyclic temperature and DOD to the next capacity value.
  • Multi-step prediction recursively feeds each predicted capacity into the next input while keeping temperature and DOD fixed until the target step.

III. TECHNOLOGIES

The paper introduces two modified covariance functions within GPR and evaluates their performance indicators for cyclic battery-capacity prediction under varied conditions.

  • The technology section covers GPR fundamentals, two designed covariance functions, and the corresponding performance indicators.

A. Gaussian Process Regression

Gaussian process regression models nonparametrically represent outputs as Gaussian processes governed by mean and covariance functions. Conditioning on training data produces predictive means and variances, thereby incorporating uncertainty into regression outputs.

  • Gaussian-process formulation: GPR models place a Gaussian-process distribution over functions, with mean m(x) and covariance k(x, x′) specifying the process.The covariance function is also the kernel function, expressing relevance through input similarity.
  • Prior distribution: The prior distribution of regression outputs is Gaussian and includes a noise term σ_n.The noise term is incorporated through the covariance structure of the observations.
  • Prediction: After hyperparameter optimization, GPR obtains predictions by evaluating the conditional distribution p(y′|x′, x, y).Hyperparameters are optimized by minimizing the negative log marginal likelihood.
  • Prediction: The predictive mean ȳ′ gives the expected output, while cov(y′) represents the uncertainty range of the predictions.This probabilistic output distinguishes GPR from methods that provide only deterministic point predictions.

B. Model A’ with ARD-SE kernel

The paper modifies GPR covariance functions to handle nonlinear, multidimensional cyclic ageing inputs. Model A uses ARD-SE feature relevance, while Model B combines physically or empirically motivated kernel components for temperature, DOD, and capacity history.

  • Kernel design: The covariance function strongly influences GPR performance, so its selection must be tailored to the application.The paper notes that suitable covariance-function choice is case-specific.
  • Kernel design: The isotropic SE kernel is limited for highly nonlinear mappings involving capacity histories, temperature, and DOD as multidimensional inputs.Its single isotropic structure provides limited expressiveness for these inputs.
  • Model A: Model A modifies the SE kernel with ARD, assigning separate lengthscales so irrelevant capacity, temperature, or DOD features can be restrained.Large lengthscales suppress irrelevant features and produce a relatively sparse, explanatory feature subset.
  • Model B: Model B couples battery ageing knowledge into separate kernel components, using Arrhenius dependence for cyclic temperature and a polynomial form for DOD.The Arrhenius temperature component uses reciprocal temperatures, while the DOD component is non-stationary.
  • Model B: Previous and current capacities are represented with SE-kernel components so GPR can study ageing-trend correlations during training.These capacity terms form part of the model input alongside cyclic temperature and DOD.
  • Model B: Model B multiplies the capacity, temperature, and DOD components into a positive semi-definite compositional kernel that captures their interactions.The input vector contains capacity history, cyclic temperature, and cyclic DOD.
  • Evaluation: The models are evaluated using MAE, maximum absolute error, and RMSE to compare predicted capacities with experimental values.MAE reflects average absolute error, ME the maximum difference, and RMSE gives greater weight to larger absolute errors.

IV. PREDICTION RESULTS AND ANALYSIS

The prediction analysis evaluates modified GPR models for cyclic battery capacity under varied conditions, with Model A using an ARD-SE kernel to capture ageing trends and support one-step and multi-step prediction.

  • Case study 1 - Model A: Model A uses an ARD-SE kernel intended to extract relevant features for capacity prediction under different cyclic conditions.The analysis motivates improved kernels because a single isotropic SE function is limited for nonlinear, multidimensional inputs.
  • Case study 1 - Model A: Model A captures overall capacity ageing trends across cyclic temperature and DOD cases, with all training RMSE values below 0.15 Ah.The maximum uncertainty-bound difference is nearly 1 Ah, while Case 4 has the largest reported training errors.
  • Case study 1 - Model A: Model A tracks the testing degradation trend in one-step prediction, with RMSE below 0.1 Ah for both Case 5 and Case 6.The reported one-step ME values are 0.2410 Ah for Case 5 and 0.1689 Ah for Case 6.
  • Case study 1 - Model A: Model A performs iterative multi-step prediction by feeding each new estimate into the next step, causing uncertainty bounds to widen with prediction horizon.The widening reflects accumulated uncertainty during covariance propagation.

B. Case study 2 - Model B

Model B incorporates electrochemical and empirical ageing knowledge through an improved kernel and is evaluated for training, one-step, and multi-step capacity prediction.

  • Case study 2 - Model B: Model B describes capacity-fading dynamics better than Model A, with training maximum ME, MAE, and RMSE reductions of 40.7%, 36.0%, and 37.8%, respectively.For Model B, the corresponding maximum values are 0.1689 Ah, 0.0557 Ah, and 0.0790 Ah in Case 1.
  • Case study 2 - Model B: Model B captures one-step ageing trends for Cases 5 and 6, with all cases having ME within 0.15 Ah, MAE within 0.05 Ah, and RMSE within 0.06 Ah.These values are 11.2%, 25.0%, and 28.4% lower than Model A's corresponding results.
  • Case study 2 - Model B: Model B achieves satisfactory multi-step prediction accuracy for Cases 5 and 6, with Case 5 MAE and RMSE reductions of 77.3% and 75.9% versus Model A.For Case 6, Model B reports ME 0.2004 Ah, MAE 0.0512 Ah, and RMSE 0.0771 Ah, with at least 46.9% increased accuracy over Model A.

V. COMPARISON ANALYSIS AND DISCUSSION

The comparison analysis tests different input-term counts and benchmarks the modified models against conventional GPR using a single SE kernel.

  • V. COMPARISON ANALYSIS AND DISCUSSION: The study compares five input-capacity-term settings and conventional GPR with a single SE kernel on the same dataset.The comparisons assess input-term effects and the effectiveness of modified GPR-based models.

A. Comparative study with different number of input capacity terms

The input-term study balances degradation-history information against model complexity and overfitting, selecting two capacity terms for Model B.

  • A. Comparative study with different number of input capacity terms: Increasing capacity input terms can provide more degradation information but also increases model complexity and computational effort.The study also notes that too many terms may contribute to overfitting.
  • A. Comparative study with different number of input capacity terms: Five input-capacity-term settings, from one through five, are evaluated using Model B's multi-step prediction results.The comparison is designed to evaluate input-term effects and help prevent overfitting.
  • A. Comparative study with different number of input capacity terms: Two capacity input terms significantly reduce ME, MAE, and RMSE compared with one term, while adding more terms slightly increases the indicators.The study therefore sets the number of input capacity terms to two to preserve accuracy and restrain overfitting.

B. Comparison with SE-based GPR model

The modified GPR kernels are evaluated against a conventional single squared-exponential GPR model using training and multi-step testing indicators. Both modified models outperform SEGM, while Model B provides the strongest results by incorporating electrochemical and empirical ageing elements.

  • Experimental comparison: SEGM, Model A, and Model B are compared using ME, MAE, and RMSE on the same training and testing datasets.The comparison covers total-dataset training results and total-dataset multi-step prediction results.
  • Training results: 0.1689Ah ME, 0.0428Ah MAE, and 0.0589Ah RMSE are achieved by Model B on the total training dataset.These values represent decreases of 58.3%, 59.7%, and 55.1%, respectively, relative to Model A.
  • Training results: Model A and Model B both improve training performance over SEGM, with the modified GPR results described as nearly twice as good.SEGM records 0.6481Ah ME, 0.2290Ah MAE, and 0.2712Ah RMSE, whereas Model A records 0.4053Ah, 0.1061Ah, and 0.1313Ah.
  • Multi-step prediction: 0.2576Ah ME, 0.0652Ah MAE, and 0.0835Ah RMSE are obtained by Model B for multi-step cyclic capacity prediction.Compared with Model A, these correspond to decreases of 63.6%, 74.7%, and 72.3%, respectively.
  • Overall finding: The modified-kernel GPR models are considered promising for capacity prediction under various cyclic cases.Model B’s advantage is attributed to coupling electrochemical or empirical ageing elements into the covariance function.

VI. CONCLUDING REMARKS

The paper develops modified GPR approaches for predicting NMC lithium-ion battery capacity under varying temperature and DOD conditions. Model B couples ageing knowledge into a compositional kernel and achieves satisfactory predictions with reliable uncertainty quantification, outperforming the ARD-SE covariance function in training and testing.

  • The proposed approaches predict cyclic capacity for NMC-based lithium-ion batteries under varied operational temperature and DOD conditions.
  • Model B couples the Arrhenius law and a polynomial equation into a compositional GPR kernel to incorporate electrochemical and empirical battery-degradation knowledge.
  • Model B provides satisfactory capacity-degradation predictions and reliable uncertainty quantification across various cyclic conditions.
  • Model B’s mechanism-informed covariance function outperforms the ARD-SE covariance function in both training and testing phases.
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