PaperlayerView exact revision on arXiv

Source-linked AI summary

An asymptotic derivation of a single particle model with electrolyte

Scott G. Marquis, Valentin Sulzer, Robert Timms, Colin P. Please, S. Jon Chapman

arXiv:1905.12553v2physics.chem-ph

TL;DR

The paper addresses computationally expensive DFN modeling and ad-hoc reduced-model derivations. It systematically derives the SPM and electrolyte correction, reporting good DFN agreement, reduced complexity, and greater accuracy than comparable models.

  • Problem

    Ad-hoc reduced-model derivations may omit important terms, motivating a systematic mathematical derivation.

  • Method

    The paper systematically derives the SPM, an electrolyte correction, and a combined leading- and first-order voltage expression.

  • Results

    The canonical SPMe gives good agreement with the DFN model while substantially reducing computational complexity and improving accuracy over comparable reduced models.

  • Takeaways & Limitations

    The derived model provides a more accurate reduced model than other models of similar computational complexity.

  • Takeaways & Limitations

    In one limit, solving concentrations in all particles in each electrode cannot be avoided.

Abstract

from arXiv · show

The standard continuum model of a lithium-ion battery, the Doyle-Fuller-Newman (DFN) model, is computationally expensive to solve. Typically simpler models, such as the single particle model (SPM), are used to provide insight for control purposes. Recently, there has been a move to extend the SPM to include electrolyte effects, which increase the accuracy and range of applicability. However, these extended models are derived in an ad-hoc manner, which leaves open the possibility that important terms may have been neglected, resulting in the model not being as accurate as possible. In this paper, we provide a systematic asymptotic derivation of both the SPM and a correction term that accounts for the behaviour in the electrolyte. Firstly, this allows us to quantify the error in the reduced model in terms of ratios of key parameters in the model, from which the range of applicable operating conditions can be determined. Secondly, in comparing our model with the ad-hoc models from the literature, we show that previous models have neglected a key set of terms. In particular, we make the crucial distinction between writing the terminal voltage in pointwise and electrode-averaged form, which allows us to gain additional accuracy whilst maintaining the same computational complexity as the existing models.

1 Introduction

The DFN model is accurate but computationally demanding, motivating simpler physics-based models. This paper systematically derives the SPM and an electrolyte correction, identifies applicability conditions, and reports improved accuracy at comparable complexity.

  • Motivation: The DFN model remains too computationally complex for some applications, despite sophisticated numerical methods.
  • Motivation: Simpler physics-based models are desired for battery-management and coupled-cell applications because speed, memory, and numerical convergence matter.
  • Research gap: Existing electrolyte-extended SPM approaches generally rely on ad-hoc assumptions, leaving their correction terms insufficiently justified.
  • Approach: The paper applies asymptotic methods to the DFN model to derive the SPM and an electrolyte correction, with errors estimated from input-parameter ratios.
  • Scope: Applicability depends on large electrical conductivity and fast electrolyte lithium-ion migration relative to discharge timescales.
  • Results: The reduced model outperforms comparable ad-hoc models while retaining the same computational complexity, aided by electrode-averaged terminal voltage.
  • Approach: At leading order the reduction recovers the SPM; at first order it adds an electrolyte-concentration PDE and an algebraic terminal-voltage correction.

2 Doyle-Fuller-Newman (DFN) model

The DFN model represents lithium-ion transport, electrochemical reactions, and electrical behavior across porous electrodes, separator, electrolyte, and particles. Its variables are defined over macroscopic battery thickness and, for particles, microscopic radial coordinates.

  • Physical structure: A lithium-ion battery comprises two electrodes, a porous separator, an electrolyte, and current collectors connected to active material particles.
  • Operation: During discharge, lithium leaves negative-electrode particles, moves through the electrolyte, and reacts at positive-electrode particle surfaces while electrons travel through the external circuit.
  • Model overview: The DFN model is the standard lithium-ion battery model and can be derived by volume averaging or multiple-scales methods.
  • Geometry and assumptions: Active-material particles are modeled as spheres with spherically symmetric behavior, using x* across battery thickness and r* within particles.
  • Variables: Potentials, current densities, concentrations, and molar fluxes are assigned region and phase subscripts distinguishing electrodes, separator, electrolyte, and solid material.
  • Constitutive data: The model uses electrochemical-reaction, electrolyte-concentration, active-material-concentration, open-circuit-potential, and electrolyte-diffusivity functions and parameters.
  • Parameterization: The parameterization corresponds to a graphite negative electrode, LiPF6 in EC:DMC electrolyte, and lithium-cobalt-oxide positive electrode.

3 Asymptotic reduction of DFN model

The DFN model is nondimensionalized and systematically reduced through a distinguished asymptotic limit, yielding the leading-order SPM and a first-order electrolyte correction. Electrode-averaged voltage expressions are central to retaining first-order accuracy without additional computational complexity.

  • 3.1 Dimensionless form of DFN model: The DFN model is first nondimensionalized using characteristic scales, timescales, and dimensionless parameters before asymptotic reduction.The resulting dimensionless model provides the basis for the reduction.
  • 3.1 Dimensionless form of DFN model: The reduction takes the limit of high electrode and electrolyte conductivity and rapid electrolyte lithium-ion migration relative to discharge.This distinguished limit is represented by Ce → 0, σk → ∞, and ˆκe → ∞, with σkCe and ˆκeCe held constant.
  • 3.2.1 Leading-order model: At leading order, electrolyte potential drops, solid-phase Ohmic losses, and electrolyte depletion vanish, while all particles in each electrode behave identically.Thus one representative particle per electrode is sufficient, producing the dimensionless SPM.
  • 3.2.2 First-order correction: The first-order correction captures electrolyte concentration variations, concentration overpotential, electrolyte Ohmic losses, and solid-phase Ohmic losses.The electrode-averaged corrections to the reaction currents are zero, and the voltage correction is obtained algebraically without additional PDEs.
  • 3.3 Combined voltage expression: The resulting model, SPMe(S), combines the SPM with a first-order PDE and algebraic correction to the terminal voltage for nonuniform electrolyte effects.The correction extends the reduced model beyond the leading-order approximation.

4 Canonical SPMe

The canonical SPMe combines particle and electrolyte dynamics, using either a quasi-steady electrolyte approximation or a transient correction. Its independent linear PDE structure yields an algebraic terminal-voltage calculation and supports parallel computation.

  • The SPMe(S) applies when electrolyte dynamics are quasi-steady because current varies more slowly than the electrolyte diffusion timescale.
  • Transient electrolyte effects after current steps require scaling time with the lithium-ion migration timescale.
  • On the migration timescale, electrode-particle concentrations remain constant while electrolyte exchange terms are negligible at leading and first order.
  • The composite model produces the correct result on both diffusion and discharge timescales, but its first-order electrolyte correction requires solving a PDE.
  • The canonical SPMe comprises independent linear PDEs for particle concentrations and electrolyte-ion concentration, with terminal voltage obtained afterward from an algebraic expression.
  • The independent PDE problems create a naturally parallel structure, while linearity facilitates numerical methods and simpler analytic solutions such as SPMe(S).

5 Model comparisons

The SPMe is compared with the DFN and SPM using matched numerical discretization across constant-current discharges and internal-state tests. It substantially reduces computational demands while improving voltage accuracy over the SPM, although errors increase near high-rate discharge endpoints and nonlinear OCV regions.

  • Model setup: The comparison evaluates DFN, SPM, and SPMe models under constant-current discharge using common finite-volume discretization and numerical methods.
  • Computational complexity: The DFN model requires 1120 internal states, whereas the SPMe requires just over 10% of the DFN memory for the specified discretization.
  • Computational complexity: The SPMe replaces the stiff DFN differential-algebraic system with three linear parabolic PDEs whose discretization produces a well-conditioned ODE system.
  • Voltage comparison: At 0.1 C, the SPM has 1.72 mV RMS voltage error, rising to 19.86 mV at 1 C, while the SPMe error is 3.04 mV at 1 C and 13.34 mV at 3 C.
  • Voltage comparison: Most SPMe voltage components agree well with DFN predictions, but endpoint errors at high rate arise largely from poor electrode-averaged OCV estimation when OCV is highly nonlinear.
  • Computational complexity: The SPMe generally improves SPM accuracy by an order of magnitude at similar computation time, with coarser spatial discretization often sufficient.
  • Internal states: SPMe and DFN internal states generally agree, including particle concentration profiles, but discrepancies remain in electrolyte concentrations and potentials at large C-rates.

6 Critical assessment of variations on the SPMe in the lit-

The paper compares canonical and ad-hoc SPMe variants, focusing on terminal-voltage formulations and their agreement with the DFN model. The canonical SPMe is consistently more accurate while retaining comparable computational complexity.

  • Model differences: Literature SPMe variants commonly replace electrode-averaged concentration overpotentials and electrolyte Ohmic losses with pointwise terms.Some models also neglect solid-phase Ohmic losses or assume constant exchange-current densities.
  • Model differences: Mixing electrode-averaged and pointwise voltage terms prevents the literature expression from ensuring O(C_e^2) accuracy.The canonical formulation maintains consistent averaging across the voltage components.
  • Model differences: Constant exchange-current densities have a clear disadvantage because reaction overpotentials strongly depend on lithium and lithium-ion concentrations.The Han et al. model removes this assumption, while retaining pointwise concentration and electrolyte-loss terms.
  • Model comparison: At low C-rates, all three models match well, with an RMS voltage error of just 1.72 mV.The comparison evaluates RMS voltage error relative to the DFN model across constant-current discharge rates.
  • Model comparison: The compared SPMe variants have similar computational cost, with each evaluation taking on average 0.07 s and requiring 120 stored states in the stated discretization.The discretization uses 30 points in each electrode, 20 in the separator, and 15 in each particle.
  • Model comparison: Across all discharge rates, the canonical SPMe outperforms the literature models and converges to the DFN solution faster.Its RMS errors reach about 0.01 V at higher C-rates, compared with about 0.1 V for literature models; it is consistently an order of magnitude more accurate than Perez and Kemper.

7 Dimensional model summary and conditions for applica-

The dimensional SPMe is presented with validity conditions derived from the asymptotic model. These conditions determine when its error and accuracy claims apply, while transient-free operation permits a simpler electrolyte treatment.

  • Dimensional model: The dimensional SPMe is obtained by reapplying the model scalings and combining the leading- and first-order electrolyte concentration equations.The resulting formulation includes dimensional voltage, reaction, concentration-overpotential, electrolyte-loss, and solid-loss expressions.
  • Validity conditions: Table 6 provides the conditions that ensure the validity of the dimensional model.When these conditions are met, the model error at a particular time has the asymptotic size stated in the paper.
  • Validity conditions: Accuracy decreases when the open-circuit potential is significantly nonlinear because its second derivative contributes to the error.The relevant term is the absolute value of the second derivative of the OCP in electrode k.
  • Validity conditions: When current variations occur on timescales longer than electrolyte diffusion, the same accuracy can be achieved by neglecting the electrolyte time-derivative term.This produces the dimensional equivalent of the SPMe(S).

8 Summary and further work

The paper systematically derives reduced battery models and identifies consistent electrode-averaged voltage terms as the key distinction from prior ad-hoc extensions. The resulting canonical SPMe improves accuracy and applicability while reducing computational demands relative to the DFN model.

  • Summary: The authors systematically derive simplified models from the DFN model and quantify their error through parameter groupings.This permits model applicability to be determined a-priori from the relevant parameter ratios.
  • Summary: The canonical SPMe extends the leading-order SPM by including higher-order effects, producing a model applicable over a larger range of operating conditions.The extension is motivated by retaining important terms from the underlying physics.
  • Summary: The canonical SPMe agrees well with the DFN model while substantially reducing memory requirements and computation time.These reductions are identified as desirable for BMS, parameter estimation, and optimization.
  • Summary: The canonical SPMe is more accurate than other reduced models with similar computational complexity.The comparison supports its use as a more accurate reduced model within the studied scope.
  • Summary: Writing the output voltage with electrode-averaged OCVs, overpotentials, and Ohmic losses is identified as a key requirement overlooked in previous literature.The systematic derivation also identifies discrepancies in SPMe predictions and the minimal extensions needed to correct them.
  • Further work: Applying asymptotic methods to additional physical effects can help reduced models retain important terms and avoid inconsistent combinations that reduce accuracy.Potential extensions include mechanical, thermal, and degradation mechanisms, but introducing them ad hoc may ignore interactions present in the DFN model.

C Dimensionless voltage from [14]

The appendix converts the voltage expression from Kemper et al. into dimensionless form for direct comparison. It makes explicit the assumed collector-potential interpretation and accounts for the opposite current-sign convention.

  • Voltage interpretation: The voltage from Kemper et al. is converted into dimensionless form, with details provided in Appendix C to define the compared model precisely.The underlying concentration equations are treated separately from the terminal-voltage expression.
  • Voltage interpretation: The voltage is assumed to be the solid-phase potential difference between the positive and negative current collectors.The individual voltage components are not explicitly given in the source model.
  • Voltage interpretation: The current sign is reversed when converting the model because Kemper et al. define current in the direction of positive-charge flow.The paper uses the opposite current convention.

D Tables

The paper’s tables specify dimensional and dimensionless parameters, physical timescales, model-error and computation-time comparisons, and conditions required for applying equation (48).

  • Table 1 lists dimensional model parameters with values taken from reference [27].
  • Table 2 summarizes timescales for physical processes in the battery model and defines C as the C-rate.
  • Table 3 reports typical dimensionless parameter values for a C-rate defined using a 24 Am−2 reference current density.The stated cell configuration uses initial stoichiometries of 0.8 in the negative electrode and 0.6 in the positive electrode, with a 3.2 V voltage cutoff.
  • Table 4 gives RMS voltage errors between reduced models and the DFN model for a finite-volume implementation with 30 points in each domain.
  • Table 5 compares average computation times across C-rates for the SPM, SPMe, and DFN models.The table considers 0.1 C, 0.5 C, 1 C, 2 C, and 3 C, with 20, 40, and 1 points per domain respectively for the listed models.
  • Table 6 states key conditions for applying equation (48), including several scale-separation inequalities considered true in practical situations.
Loading 1905.12553v2…