[Paper Review] Generalised single particle models for high-rate operation of graded lithium-ion electrodes: systematic derivation and validation

Giles Richardson, Ivan Korotkin+2 more|arXiv (Cornell University)|Jun 26, 2019

Advanced Battery Technologies Research36 references4 citations

TL;DR

This paper systematically derives a generalized single particle model (SPM) for lithium-ion batteries with graded electrodes, using asymptotic analysis to identify discrepancies from the porous electrode theory (PET) model at high C-rates. It introduces a correction term that significantly improves accuracy—reducing computation time by two orders of magnitude compared to PET—enabling fast, reliable simulation of graphite, NMC, and LFP electrodes up to 12C rates.

ABSTRACT

A derivation of the single particle model (SPM) is made from a porous electrode theory model (or Newman model) of half-cell (dis)charge for an electrode composed of uniformly sized spherical electrode particles of a single chemistry. The derivation uses a formal asymptotic method based on the disparity between the size of the thermal voltage and that of the characteristic change in overpotential that occurs during (de)lithiation. Comparison is made between solutions to the SPM and to the porous electrode theory (PET) model for NMC, graphite and LFP. These are used to identify regimes where the SPM gives accurate predictions. For most chemistries, even at moderate (dis)charge rates, there are appreciable discrepancies between the PET model and the SPM which can be attributed to spatial non-uniformities in the electrolyte. This motivates us to calculate a correction term to the SPM. Once this has been incorporated into the model its accuracy is significantly improved. Generalised versions of the SPM, that can describe graded electrodes containing multiple electrode particle sizes (or chemistries), are also derived. The results of the generalised SPM, with the correction term, compare favourably to the full PET model where the active electrode material is either NMC or graphite.

Motivation & Objective

• To systematically derive the single particle model (SPM) from the porous electrode theory (PET) model using formal asymptotic methods.
• To identify regimes where the standard SPM fails at high (dis)charge rates due to electrolyte spatial non-uniformities.
• To develop a corrected SPM with a physically motivated correction term that improves accuracy without sacrificing computational efficiency.
• To generalize the corrected SPM for electrodes with multiple particle sizes or chemistries (graded electrodes).
• To validate the corrected model against full PET simulations for NMC, graphite, and LFP electrodes across a range of C-rates.

Proposed method

• Derives the SPM from the PET model using formal asymptotic expansion based on the disparity between thermal voltage and overpotential change scales.
• Applies the method to a one-dimensional, pseudo-two-dimensional PET model with spherical electrode particles and nonlinear lithium diffusion in active materials.
• Identifies a correction term to the SPM by analyzing the solvability condition of the asymptotic expansion, capturing electrolyte concentration gradients.
• Incorporates the correction term into a generalized SPM formulation for multi-sized or multi-chemistry electrodes.
• Solves the corrected SPM by discretizing the electrode and electrolyte domains with N and 2N mesh points, respectively, yielding O(N²) complexity.
• Validates the corrected SPM against full PET simulations using experimental data for NMC, graphite, and LFP.

Experimental results

Research questions

• RQ1 In what regimes does the standard SPM fail to accurately predict high-rate behavior in lithium-ion electrodes?
• RQ2 How can the SPM be systematically corrected to account for electrolyte concentration gradients that arise at high C-rates?
• RQ3 To what extent can the corrected SPM accurately model graded electrodes with multiple particle sizes or chemistries?
• RQ4 How does the computational cost of the corrected SPM compare to the full PET model across different mesh resolutions?
• RQ5 Can the corrected SPM maintain accuracy for materials with flat voltage profiles, such as LFP, despite the asymptotic derivation not being strictly applicable?

Key findings

• The standard SPM shows significant discrepancies from the PET model at moderate to high C-rates (e.g., above 2C), primarily due to unaccounted-for electrolyte concentration gradients.
• The derived correction term reduces these discrepancies, improving agreement with the PET model across all tested materials (NMC, graphite, LFP) up to 12C.
• For graphite and NMC, the corrected SPM reproduces full discharge curves with high fidelity, even at 12C, despite the asymptotic derivation being based on small parameters.
• The corrected SPM achieves a 100x speedup over the full PET model: 1.5 seconds for N=50 mesh points vs. 170 seconds for the same resolution.
• The computational complexity of the corrected SPM scales as O(N²), compared to O(N⁴) for the full PET model, making it highly efficient for optimization and parameter estimation.
• The corrected SPM performs reasonably well for LFP, though accuracy is reduced in graded electrodes with two particle sizes, indicating limitations in the correction’s applicability to complex morphologies.