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Identifiability and parameter estimation of the single particle lithium-ion battery model

Adrien M. Bizeray, Jin-Ho Kim, Stephen R. Duncan, David A. Howey

arXiv:1702.02471v2eess.SY

TL;DR

The paper addresses whether single-particle battery models are identifiable and whether their parameters can be credibly estimated for electrochemical modelling. It groups and estimates parameters using linearised-model analysis and frequency-domain EIS, finding six necessary parameter subgroups and emphasizing electrode OCV slopes and multiple DoDs for reliable estimation.

  • Problem

    Electrochemical-model parameters must be credible for battery-management state estimation, but their observability and reliable estimation remain challenging.

  • Method

    The paper groups and partially non-dimensionalises the SPM, analyzes transfer-function uniqueness, and estimates parameters by least-squares fitting of experimental and predicted EIS impedances.

  • Results

    Six parameter subgroups are necessary, while reliable estimation depends on nonzero electrode OCV slopes and complementary measurements across multiple DoDs.

  • Takeaways & Limitations

    A single-DoD impedance dataset cannot generally estimate both electrodes accurately; complementary DoDs should expose significant anode and cathode OCV gradients.

Abstract

from arXiv · show

This paper investigates the identifiability and estimation of the parameters of the single particle model (SPM) for lithium-ion battery simulation. Identifiability is addressed both in principle and in practice. The approach begins by grouping parameters and partially non-dimensionalising the SPM to determine the maximum expected degrees of freedom in the problem. We discover that, excluding open circuit voltage, there are only six independent parameters. We then examine the structural identifiability by considering whether the transfer function of the linearised SPM is unique. It is found that the model is unique provided that the electrode open circuit voltage functions have a known non-zero gradient, the parameters are ordered, and the electrode kinetics are lumped into a single charge transfer resistance parameter. We then demonstrate the practical estimation of model parameters from measured frequency-domain experimental electrochemical impedance spectroscopy (EIS) data, and show additionally that the parametrised model provides good predictive capabilities in the time domain, exhibiting a maximum voltage error of 20 mV between model and experiment over a 10 minute dynamic discharge.

NOMENCLATURE

The nomenclature defines the SPM’s electrochemical variables, transfer functions, capacities, resistances, and grouped parameter vectors, while motivating physics-based models for battery management.

  • The nomenclature includes lithium concentrations, stoichiometries, diffusion coefficients, particle coordinates, reaction quantities, capacities, and voltage-related variables.
  • It defines the SPM transfer functions, applied current, charge-transfer resistance, theoretical capacity, particle radius, temperature, voltage, and grouped parameter vector θ ∈R6.
  • SoC denotes remaining cell energy relative to full charge, while SoH is commonly defined using aged-cell capacity or resistance relative to a pristine cell.
  • Physics-based models are pursued because they can provide degradation insights and support health-aware management, but credible parameterisation for specific cells remains challenging.

Contributions

The paper studies whether SPM parameters can be identified in principle and estimated in practice, using grouped physics-based parameters and frequency-domain EIS data. It emphasizes OCV slopes and model validity as central conditions for credible estimation.

  • The study investigates grouped-parameter identifiability in the SPM and practical estimation from simulated and experimental frequency-domain EIS data.
  • The paper argues that identifiability analysis should precede estimation because fitting all parameters simultaneously can produce ill-conditioned optimization and physically meaningless estimates.
  • A flat electrode OCV curve produces parameter unidentifiability, making electrode OCV slopes central to lithium-ion battery model identifiability.
  • The SPM describes electrode thermodynamics, solid-phase lithium diffusion, and interfacial kinetics while neglecting electrolyte dynamics and assuming uniform electrode reaction rates.
  • The SPM is generally considered valid below 1C to 2C, although the paper reports reasonable linearised-SPM voltage errors with current peaks up to 6C.
  • Lithium diffusion is modeled by spherical Fickian dynamics, with current-related fluxes, nonlinear OCV measurement, and Butler-Volmer overpotential kinetics.

C. Identification of grouped parameters

The SPM is reformulated using dimensionless coordinates, stoichiometry, and physically meaningful parameter groups to expose its minimum parameterization. Six grouped parameters fully parameterize the model under stated initial-condition and OCV assumptions.

  • Dimensionless radial coordinates and stoichiometry are introduced to reformulate the over-parameterized SPM in terms of a minimum number of parameter groups.
  • Changing from concentration to stoichiometry and shifting by the initial stoichiometry sets the governing-equation initial condition to zero.
  • Three grouped parameters arise for each electrode, including a diffusion time constant, yielding six physically meaningful groups overall.
  • The grouped formulation uses the same diffusion-equation structure for both electrodes and retains the voltage measurement equation with electrode overpotentials.
  • The grouped parameters appear individually rather than as parameter products and have a one-to-one mapping with the original six parameter groups.
  • Assuming known initial electrode stoichiometries and known electrode OCV functions, the six grouped parameters are sufficient to fully parameterize the SPM.

III. STRUCTURAL IDENTIFIABILITY

Structural identifiability is assessed through a transfer-function formulation of the linearised SPM. Because nonlinear OCV and Butler-Volmer terms prevent direct application of the linear definition, the model is linearised around a fixed DoD point.

  • Six sufficient grouped parameters do not guarantee that all six can be identified from the battery current-voltage response.
  • For linear time-invariant models, structural identifiability is examined through uniqueness of the parameterized transfer function.
  • A model is globally identifiable with one solution, locally identifiable with finitely many solutions, and unidentifiable with infinitely many solutions to the identifiability equation.
  • The nonlinear SPM is linearised around a fixed DoD point because its voltage equation contains nonlinear OCV functions and Butler-Volmer kinetics.
  • Known measured OCV functions make linearisation suitable for estimating diffusion-submodel parameters and a linearised approximation of kinetics.

A. Diffusion model transcendental transfer function

The paper derives a transfer function for spherical particle diffusion by transforming the linear diffusion problem into the frequency domain. The resulting current-to-surface-stoichiometry transfer function is then written for both electrodes.

  • Derivation: The spherical diffusion model is converted into an equivalent transfer-function problem because its governing initial-boundary value problem is linear.The derivation uses a Laplace transform and applies the particle-centre and surface boundary conditions.
  • Surface response: The transfer function tracks surface stoichiometry rather than the full particle concentration field because surface stoichiometry enters the voltage equation.It is obtained by evaluating the transformed solution at the particle surface and dividing by the input current.
  • Electrode models: The cathode and anode diffusion transfer functions are subsequently expressed using the grouped model parameter vector.These electrode-specific transfer functions provide the diffusion components of the linearised SPM.

B. Linearisation of the voltage measurement equation

The nonlinear voltage measurement equation is linearised around an equilibrium stoichiometry under small-current operation. Its local sensitivities depend on current and on the cathode and anode OCV gradients.

  • Linearisation: The voltage equation is linearised using a first-order Taylor expansion around the initial stoichiometry.The approximation assumes small input-current amplitude and operation close to the initial depth of discharge.
  • Sensitivity terms: The linearised voltage relates voltage deviations to perturbations in input current and electrode surface stoichiometries.The relevant partial derivatives are evaluated at the reference point.
  • OCV dependence: The cathode and anode OCV gradients enter the linearised voltage equation as the sensitivities to surface stoichiometry.These gradients are evaluated at the chosen linearisation point.
  • Kinetics: The intercalation reaction kinetics contribute through a charge-transfer resistance parameter.This parameter represents the resistance arising from the electrode reaction kinetics.

C. Transfer function of the linearised SPM

The linearised SPM transfer function combines electrode diffusion, voltage sensitivities, and charge-transfer resistance. Its frequency response shows diffusion and capacitive regimes whose parameter sensitivity depends strongly on electrode OCV slopes.

  • Transfer function: The linearised SPM transfer function is obtained by transforming the linearised voltage equation and dividing voltage by input current.It describes the frequency-domain voltage response around the equilibrium point.
  • Identifiable parameters: Only the difference between the cathode and anode kinetic parameters appears in the charge-transfer resistance, so the kinetics are identifiable only as lumped Rct.Infinitely many kinetic-parameter pairs produce the same transfer function.
  • Identifiable parameters: The practical parameter-estimation problem is reduced to three independent parameters after incorporating known OCV-gradient information and grouping the model parameters.The identifiable vector contains the electrode diffusion time constants and charge-transfer resistance.
  • Frequency response: At low frequencies, the Nyquist response transitions from a 45° diffusion tail toward a capacitive vertical asymptote caused by finite-length particle diffusion.The low-frequency departure from semi-infinite diffusion is argued to provide informative data for estimating SPM parameters.
  • Frequency response: Larger diffusion time constants yield higher cell impedance, while faster diffusion shifts capacitive behaviour toward higher frequencies.The response is more sensitive to cathode than anode diffusion at the stated depth of discharge because the anode OCV curve is relatively flat.

D. Structural identifiability analysis

Structural identifiability is tested by asking whether equal transfer functions imply equal grouped parameters. The linearised SPM is identifiable in the general case, but flat or symmetrically matched OCV slopes create exceptions that require parameter ordering or additional operating points.

  • Identifiability criterion: Structural identifiability is assessed by requiring equality of transfer functions for almost all frequency-domain values to imply equality of parameters.The analysis applies this condition to the transfer function of the linearised SPM.
  • Parameter separation: The charge-transfer resistance is separated first because it is the only additive parameter independent of the Laplace variable.The remaining identifiability condition concerns the diffusion-dependent terms.
  • General case: The diffusion time constants are structurally identifiable in the general case when the transfer-function equality forces their corresponding grouped parameters to match.The result depends on the non-trivial frequency dependence of the diffusion function.
  • Exceptions: A flat electrode OCV function makes the model unidentifiable because it hides that electrode’s diffusion dynamics.This is the mechanism underlying the exceptional cases identified in the analysis.
  • Exceptions: When electrode OCV-gradient magnitudes are equal, ordering the diffusion time constants restores structural identifiability.Using data at several depths of discharge can also provide significant OCV slopes in both electrodes.

IV. FREQUENCY-DOMAIN PARAMETER ESTIMATION

The estimation algorithm fits the linearised SPM transfer function to frequency-domain EIS data by least squares, while separating purely resistive terms through linear regression. Because single-DoD data can be practically unidentifiable, estimation combines measurements across several DoDs.

  • The algorithm estimates selected SPM parameters by minimizing squared errors between measured and predicted real and imaginary impedances across frequencies.The identified vector may contain all model parameters or a subset such as the identifiable parameter vector.
  • Single-DoD estimation can be practically unidentifiable when an electrode OCV function is too flat, requiring impedance data from several DoDs.The multi-DoD loss is formed by summing the losses over the selected DoD levels.
  • Purely resistive contributions, including charge-transfer, ohmic-contact, and passivation resistances, are estimated separately by linear regression and removed from the frequency response.These terms shift the impedance response along the real axis and can therefore be separated from the remaining dynamics.
  • The charge-transfer resistance can be estimated by fitting a 45° line to the low-frequency diffusion response and extrapolating its real-axis intercept.The regression fixes the slope at 45° and estimates the real intercept, β0 = −R0.
  • Very low-frequency data must be discarded when OCV-related pseudo-capacitive effects invalidate the resistive regression.The data-reduction procedure removes the lowest-frequency point iteratively until the coefficient of determination reaches the chosen R2 threshold of 0.98.

V. RESULTS AND DISCUSSION

Synthetic EIS experiments show that diffusion time constants can remain ambiguous or unidentifiable at a single DoD, especially when electrode OCV slopes are similar or one is much smaller. Combining complementary DoDs produces an unambiguous parameter estimate.

  • The synthetic-data test used linearised SPM parameters for an LCO cell reported in the literature.The reference parameters were used to generate the synthetic electrochemical impedance data.
  • At a single DoD, the loss function may contain multiple minima because anode and cathode diffusion time constants can be interchanged.This ambiguity persists even when electrode OCV slopes are distinct, although one minimum may be narrow and barely visible.
  • When one electrode OCV slope is an order of magnitude smaller than the other, that electrode’s diffusion time constant becomes unidentifiable.The resulting loss-function minimum is elongated along the corresponding diffusion-time-constant axis.
  • At best, single-DoD data determine the two diffusion time constants without assigning them to specific electrodes.This occurs when the electrode OCV slopes are equal at the chosen DoD.
  • Combining EIS data at 5 %, 25 %, 75 % and 95 % DoD produces a single global minimum corresponding to the parameters used to generate the synthetic data.Complementary DoDs are preferred because each electrode can successively exhibit a large OCV slope while the other is negligible.

B. Parameter estimation using experimental data

Experimental EIS data are used to estimate the SPM’s resistive and diffusion parameters across depth-of-discharge conditions. The results show that parameter uncertainty and predictive accuracy depend strongly on the selected DoD points and OCV slopes.

  • Experimental setup: EIS measurements from 10% to 90% DoD show high-frequency charge-transfer behavior, a 45° diffusion slope, and low-frequency capacitive effects.The cell impedance was measured across 5 kHz to 200 µHz, with the high-frequency semicircle discarded for the simplified resistance treatment.
  • Estimation of high-frequency purely resistive terms: The purely resistive term R0 is estimated by linear regression after discarding high-frequency and low-frequency points affected by charge-transfer, capacitance, or DoD variation.The retained regression achieves a coefficient of determination R2 greater than 0.98, and R0 includes charge-transfer, passivation-layer, contact, and ohmic resistance.
  • Parameter estimation performance: Anode diffusion is highly uncertain at 10% DoD because the anode OCV slope is very small, whereas both electrode dynamics affect the response at 80% DoD.Even at 80% DoD, the loss function remains more elongated along the anode parameter axis, indicating greater anode uncertainty.
  • Parameter estimation performance: Combining experimental impedance data at 10% and 80% DoD yields a single loss-function minimum, but the fit generalizes poorly to 50% DoD.The RMS impedance prediction error is 47.88 mΩ at 50% DoD, compared with 6.66 mΩ at 10% DoD and 6.84 mΩ at 80% DoD.
  • Parameter estimation performance: The estimation assumes fixed, perfectly known OCV slopes, although their measurement and numerical differentiation are highly sensitive to millivolt-scale noise.The measured OCV may also differ from the actual cell OCV, creating an additional source of uncertainty.

C. Model validation in the time domain

The linearised SPM was validated against measured voltage during a nearly 10-minute dynamic discharge, with an adjusted cathode OCV slope improving agreement. The model achieved maximum and RMS voltage errors of 20 mV and 10 mV, respectively, despite separate frequency-domain parametrisation and linearised kinetics.

  • Model formulation: Unknown electrode theoretical capacity was eliminated from the linearised SPM through two changes of variable.This approach assumes a linearised model; nonlinear OCV would require knowledge of theoretical capacity.
  • Model formulation: The time-domain linearised SPM was solved using Chebyshev orthogonal collocation in MATLAB®.
  • Experimental validation: Over almost 10 minutes, the comparison covered discharge from 10 % to 20 % DoD using measured and simulated voltage responses.The measurements used the same Kokam cell type and equipment as the EIS experiments.
  • Model validation: A cathode OCV slope multiplied by 0.78 corrected an increasingly negative voltage offset in the initial predictions.The adjustment addressed an inaccurate assumed OCV slope over the tested DoD range.
  • Model validation: 20 mV maximum and 10 mV RMS voltage errors were obtained between the adjusted simulation and measured data.The validation remained reasonable despite separate frequency-domain parametrisation, different cells for OCV measurements, and linearised kinetics under current peaks up to 6C.
  • Identifiability implications: The study identifies six parameter subgroups for full parametrisation, while estimation at a given DoD identifies only three among them.This broader identifiability result frames the conditions under which the time-domain validation should be interpreted.
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