PaperlayerView exact revision on arXiv

Source-linked AI summary

Physics-Informed Neural Networks for Prognostics and Health Management of Lithium-Ion Batteries

Pengfei Wen, Zhi-Sheng Ye, Yong Li, Shaowei Chen, Pu Xie, Shuai Zhao

arXiv:2301.00776v2eess.SPcs.AIcs.LG

TL;DR

Li-ion battery PHM lacks a general, flexible way to fuse empirical or physical degradation information with monitoring data. The paper uses PINN to combine a semi-empirical PDE or DeepHPM-discovered dynamics with a surrogate neural network, with uncertainty-based adaptive weighting; the methods are verified on LFP/graphite batteries and adaptive weighting reduces reported RUL errors.

  • Problem

    Existing empirical and physical battery models provide degradation information, but general and flexible methods for fusing that information with data-driven models are lacking.

  • Method

    The paper develops a PINN fusion scheme using a semi-physical semi-empirical PDE, DeepHPM-discovered dynamics when prior knowledge is limited, and uncertainty-based adaptive loss weighting.

  • Results

    Adaptive weighting reduces RUL RMSE from 48.81 to 45.86 cycles in case A and from 65.52 to 56.29 cycles in case B for PINN-DeepHPM.

  • Takeaways & Limitations

    PINN provides a flexible framework for combining dynamic degradation information with data-driven surrogate models for Li-ion battery PHM.

  • Takeaways & Limitations

    The study uses a vanilla neural network, and the effects of more powerful architectures such as RNNs and CNNs remain insufficiently studied.

Abstract

from arXiv · show

For Prognostics and Health Management (PHM) of Lithium-ion (Li-ion) batteries, many models have been established to characterize their degradation process. The existing empirical or physical models can reveal important information regarding the degradation dynamics. However, there are no general and flexible methods to fuse the information represented by those models. Physics-Informed Neural Network (PINN) is an efficient tool to fuse empirical or physical dynamic models with data-driven models. To take full advantage of various information sources, we propose a model fusion scheme based on PINN. It is implemented by developing a semi-empirical semi-physical Partial Differential Equation (PDE) to model the degradation dynamics of Li-ion batteries. When there is little prior knowledge about the dynamics, we leverage the data-driven Deep Hidden Physics Model (DeepHPM) to discover the underlying governing dynamic models. The uncovered dynamics information is then fused with that mined by the surrogate neural network in the PINN framework. Moreover, an uncertainty-based adaptive weighting method is employed to balance the multiple learning tasks when training the PINN. The proposed methods are verified on a public dataset of Li-ion Phosphate (LFP)/graphite batteries.

I. INTRODUCTION

The paper addresses flexible fusion of degradation-model information and monitoring data for Li-ion battery PHM. It proposes PINN-based fusion using semi-physical dynamics, DeepHPM discovery, and adaptive loss weighting.

  • I. INTRODUCTION: Model fusion combines complementary information because physics-, experience-, and data-driven models encode different forms of degradation knowledge.The paper motivates fusion through the distinct strengths of available model categories and the information they provide.
  • I. INTRODUCTION: PINN fuses empirical or physical degradation dynamics with data-driven models in a flexible PHM framework for Li-ion batteries.The paper positions PINN as a formalized approach for integrating dynamic models with monitoring-derived information.
  • I. INTRODUCTION: The proposed scheme develops a semi-physical semi-empirical PDE, uses DeepHPM when prior dynamics are limited, and adaptively weights multitask losses.The dynamic model is based on the Verhulst model and generalized using observable charging and discharging features.
  • I. INTRODUCTION: The methods are verified on a public experimental cycling dataset of LFP/graphite batteries.The evaluation targets Li-ion phosphate/graphite battery prognostics and health management.
  • I. INTRODUCTION: Prior degradation modeling spans high-complexity physical models, semi-empirical models, and data-driven PDE-discovery approaches such as DeepHPM.The related work contrasts physically detailed models with empirical approximations and learned governing dynamics.

B. Fusion of the Dynamics in Deep Learning

The paper reviews ways to integrate degradation dynamics into deep learning and focuses on continuous-time PINN fusion. It formulates battery capacity loss with dynamic models while noting practical limits of detailed physical parameterization and simplified assumptions.

  • B. Fusion of the Dynamics in Deep Learning: PINN constraint learning incorporates physical dynamics through regularization and uses automatic differentiation to obtain exact partial derivatives for continuous or discrete models.This distinguishes PINN from approaches that approximate differential terms with specially designed filters.
  • B. Fusion of the Dynamics in Deep Learning: The paper focuses on continuous PINN fusion of Verhulst and DeepHPM dynamics for settings with partial or absent prior domain knowledge.Verhulst supplies an established dynamic model, whereas DeepHPM discovers dynamics from data.
  • B. Fusion of the Dynamics in Deep Learning: Battery degradation modeling commonly neglects calendar aging because cycle aging contributes most to capacity and life loss on the considered timescales.The assumption follows the contrast between minute-scale cycling and calendar-aging effects observable over months or years.
  • B. Fusion of the Dynamics in Deep Learning: The formulation represents percentage capacity loss as a function of a virtual continuous cycle-time variable and uses discrete observations at monitoring cycles.This provides the temporal basis for expressing the degradation trend with dynamic equations.
  • B. Fusion of the Dynamics in Deep Learning: Detailed electrochemical models can be impractical because their composite parameters depend on numerous cell-specific quantities that are difficult to specify.The cited formulation includes parameters related to electrode geometry, crack propagation, and electrode porosity.
  • B. Fusion of the Dynamics in Deep Learning: The simple exponential degradation model may fail to fit batteries exhibiting accelerated degradation during early cycles.Its rate parameter is co-determined by state of charge, depth of discharge, and cell temperature.

B. Improved Verhulst Dynamic Model

The improved Verhulst model bounds capacity loss, incorporates initial SEI-related loss, and extends degradation dynamics to health indicators and time to represent cell heterogeneity.

  • B. Improved Verhulst Dynamic Model: The model assumes positive degradation parameters and constrains initial loss relative to the initial capacity-loss state.The supplied conditions include r > 0, 0 < u(t) < K, and 0 < C ≤ u0.
  • B. Improved Verhulst Dynamic Model: The model introduces an initial capacity-loss parameter C because SEI formation before use is not governed by the degradation dynamics.Capacity loss is bounded by K, while C represents initial loss from SEI formation.
  • B. Improved Verhulst Dynamic Model: The degradation rate declines as capacity loss approaches the upper bound K, constraining the increasing loss rate.The model uses the Verhulst logistic form, with 0 < u(t) < K.
  • B. Improved Verhulst Dynamic Model: Cell-to-cell heterogeneity is represented through different degradation-model parameters under identical operating conditions.The univariate time-based formulation models distinct cells through parameter differences.
  • B. Improved Verhulst Dynamic Model: Health indicators x are added alongside time t, yielding a PDE that models degradation rates across feature-space health states.The health-indicator vector can contain monitoring data or designed representative features.

C. Data-Driven Dynamic Model

The data-driven dynamic model generalizes degradation as a nonlinear PDE and uses DeepHPM to approximate unknown governing dynamics when explicit models are difficult to define.

  • C. Data-Driven Dynamic Model: The generalized PDE represents degradation evolution through a nonlinear function G of health indicators, time, the state, and its partial derivatives.The formulation is ut − G(x, t, u, ux, uxx, uxxx, . . . ; Θ) = 0.
  • C. Data-Driven Dynamic Model: The generalized PDE permits more flexible relations involving time, the degradation state, and partial derivatives than the specific Verhulst implementation.The Verhulst model is described as one specific implementation of the generalized dynamic form.
  • C. Data-Driven Dynamic Model: DeepHPM replaces the explicit dynamic function with a neural-network approximator when monitoring data are noisy or the dynamics are difficult to specify.Its trainable parameters Θ approximate G beyond a particular family of basis functions.

IV. METHODOLOGY

PINN combines a dynamic degradation model with a surrogate neural network and automatic differentiation to learn both the hidden solution and dynamic-model parameters from observations and governing constraints.

  • IV. METHODOLOGY: The PINN formulation avoids requiring direct access to the involved partial derivatives by using the surrogate network as a differentiable solution approximator.The surrogate network is parameterized by Φ and supports the model-fusion formulation.
  • IV. METHODOLOGY: The surrogate network approximates the hidden solution, while the dynamic model and automatic differentiation impose governing-system structure.The dynamic model may be explicit or approximated by DeepHPM.
  • IV. METHODOLOGY: PINN training identifies both surrogate parameters Φ and dynamic-model parameters Θ by minimizing mean squared error losses.The parameter-identification problem is a high-dimensional inverse problem.
  • IV. METHODOLOGY: The loss combines observation fitting with terms enforcing the PDE structure and its first-order time derivative.Lu measures regression error, while Lf and Lft impose PINN structure; observations may target SoH or RUL.

B. Framework of PINN

The framework compares a vanilla neural network with PINN variants using either the Verhulst model or DeepHPM, while adaptive weighting balances their multiple training losses.

  • B. Framework of PINN: The framework includes a vanilla NN baseline, PINN-Verhulst, and PINN-DeepHPM as alternative learning architectures.The latter two replace the explicit dynamic model with either the Verhulst equation or DeepHPM.
  • B. Framework of PINN: The PINN-DeepHPM framework substitutes DeepHPM for the Verhulst model and correspondingly changes the partial-derivative inputs to the dynamic model.The frameworks share a similar structure, and the derivative inputs are selected according to the application.
  • B. Framework of PINN: The surrogate and DeepHPM networks use fully connected layers with hyperbolic-tangent activation under the stated default architecture.The same basic structure is used for the involved surrogate networks and DeepHPM unless otherwise noted.
  • C. Weight Coefficients Tuning in Training PINN: Adaptive weighting learns coefficients for the data-fitting and physics-informed losses instead of relying on manual tuning.The regularized loss uses λu, λf, and λft, with a penalty on their product.
  • C. Weight Coefficients Tuning in Training PINN: Training the adaptively balanced objective is intended to balance multiple losses, and the dynamic model may be explicit or DeepHPM-based.The training procedure is summarized in Algorithm 1.

V. DATASET DESCRIPTION AND PREPROCESSING

The study uses 124 commercial LFP/graphite cells cycled to failure under varied fast-charging protocols and trains proposed models with observed inputs, outputs, and dynamic-model information.

  • V. DATASET DESCRIPTION AND PREPROCESSING: 124 commercial LFP/graphite cells from three A123 Systems batches were cycled to failure under dozens of fast-charging protocols.The cells have a nominal capacity of 1.1 Ah.
  • V. DATASET DESCRIPTION AND PREPROCESSING: The training objective combines data, physics-residual, and derivative losses with adaptive coefficients and a logarithmic weighting term.
  • V. DATASET DESCRIPTION AND PREPROCESSING: The training algorithm takes data D = {X, t, u} and hyperparameters, then outputs a surrogate neural network and dynamic model.
  • V. DATASET DESCRIPTION AND PREPROCESSING: During each epoch, automatic differentiation computes temporal and spatial derivatives used to form the dynamic-model residual before updating network, model, and weighting parameters.
  • V. DATASET DESCRIPTION AND PREPROCESSING: Charging protocols encode staged current changes from 0% to 80% SoC, followed by CC-CV charging to the 3.6 V cutoff.

B. Feature Extraction

Feature extraction combines point, interval, and trend descriptors from battery charging and discharging profiles to characterize degradation and relate measurements to SoH.

  • B. Feature Extraction: Extracted health features are designed to characterize each cell’s degradation process and correlate strongly with SoH.
  • B. Feature Extraction: Figure 5 illustrates the extracted features across operation cycles for cell #124.
  • B. Feature Extraction: Trend features model profile variation across cycles, while point and interval features capture selected profile values and durations.
  • B. Feature Extraction: The quadratic trend model estimates ω and b from discharge-capacity changes between 2.7 V and 3.3 V, with ε representing measurement error.
  • B. Feature Extraction: Additional descriptors include incremental-capacity extrema and variance, average temperature, internal resistance, and charging time, followed by moving-average smoothing.

C. Standardization

Standardization addresses scale differences across input and output channels while preserving the derivative information required by automatic differentiation.

  • C. Standardization: Standardization is applied because substantial variation among input and output channels can adversely affect model performance.
  • C. Standardization: The authors state that standardization keeps output-to-input derivatives unchanged for automatic differentiation.
  • C. Standardization: The procedure standardizes training samples and labels, with monitoring time treated as an input and standardized accordingly.
  • C. Standardization: General neural-network settings and case-specific optimal settings are organized in Tables I and II.

VI. CASE STUDY

The case study evaluates SoH estimation and RUL prediction under multiple train/test splits, using RMSE and RMSPE and comparing baseline, PINN-Verhulst, and PINN-DeepHPM models.

  • VI. CASE STUDY: The study forms cases A, B, and C with different training and test-cell assignments to evaluate generalization across battery cells.
  • VI. CASE STUDY: RMSE and RMSPE quantify prediction error, with RMSPE included because SoH and RUL can have distinct scales.
  • VI. CASE STUDY: 0.42% RMSPE is achieved by the data-driven baseline in case A, while PINN-Verhulst with AdpBal reaches 0.49% and PINN-DeepHPM reaches 0.43% without AdpBal.PINN-Verhulst with simply summed losses gives 1.41% RMSPE in case A.
  • VI. CASE STUDY: In case B, PINN-DeepHPM with AdpBal gives the lowest reported RMSPE at 0.42%, followed by PINN-Verhulst with AdpBal at 0.44%.The baseline, PINN-Verhulst (Sum), and PINN-DeepHPM (Sum) report 0.56%, 0.56%, and 0.51%, respectively.
  • VI. CASE STUDY: AdpBal improves fused-model performance in case B, while its weighting coefficients vary similarly across five training rounds.
  • VI. CASE STUDY: The average training times in case B are 87.6 s for the baseline, 122.7 s for PINN-Verhulst, and 126.0 s for PINN-DeepHPM.

B. RUL Prognostics

DeepHPM-based PINN RUL prognostics improved substantially in case B after adaptive weighting, while gains were smaller or negative in other cases. The section also reports limitations related to model complexity and neural-network architecture.

  • B. RUL Prognostics: 56.29 cycles was the case B RMSE for PINN-DeepHPM with adaptive balancing, reducing error 14.09% from the summed-loss variant.The corresponding baseline and PINN-DeepHPM (Sum) RMSEs were 64.48 and 65.52 cycles.
  • B. RUL Prognostics: 6.04% was the case A RMSE reduction from adaptive weighting, whereas case C improved 6.7% with DeepHPM but deteriorated after adaptive balancing.Case A RMSEs were 46.38 cycles for the baseline, 48.81 for PINN-DeepHPM (Sum), and 45.86 after AdpBal; case C decreased from 15.21 to 14.19 cycles before AdpBal worsened performance.
  • B. RUL Prognostics: The RUL experiments verify that model-fusion benefits generalize from SoH estimation, but the magnitude and direction of improvement depend on the case and weighting strategy.The reported pattern is minor improvement in case A, major improvement in case B, and deterioration after adaptive balancing in case C.
  • C. Discussion: Higher-complexity electrochemical models may improve estimation, but parameter specification remains challenging and only a vanilla NN architecture is used here.The paper identifies automatic parameter discovery for complex models and the limited structure of the vanilla NN as open concerns.
  • VII. Conclusion: The authors identify complex physical phenomena, convergence, stability, boundary conditions, network design, and optimization as future PINN challenges.They also propose studying temporal or spatial gradient propagation in RNNs, CNNs, and their variants.

APPENDIX A PERFORMANCE OF DIFFERENT SETTINGS ON THE

The appendix evaluates how baseline network architecture, RUL settings, and DeepHPM input combinations affect validation performance across battery cases. It organizes these studies in tables covering SoH, RUL, and DeepHPM configurations.

  • SoH estimation: Validation RMSPE studies vary hidden-layer counts and neurons per layer for baseline SoH estimation in cases A and B.Tables VI and VII report these settings and use the vanilla NN baseline.
  • RUL prognostics: Validation RMSE studies vary hidden-layer counts and neurons per layer for baseline RUL prognostics in cases A, B, and C.Tables VIII, IX, and X report the corresponding case-specific architecture evaluations in cycles.
  • DeepHPM settings: DeepHPM input combinations are evaluated on validation data using PINN-DeepHPM with simply summed losses.Table XI reports SoH metrics in percent and RUL metrics in cycles after the NN structures are selected according to the appendices.
Loading 2301.00776v2…