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Koopman Spectral Reduced-Order Modeling of Spherical Diffusion in Lithium-Ion Batteries
Jihoon Moon
TL;DR
Physics-based battery models provide internal electrochemical states but can be computationally expensive, and existing SPM reductions do not directly exploit the analytical spectrum of spherical diffusion. This paper derives an analytical Koopman spectral model for spherical diffusion and projects the current-dependent surface flux into a controlled state-space representation. The 80-mode model is nearly indistinguishable from the finite-volume reference and runs about 31 times faster.
Problem
Physics-based battery models can be computationally expensive, and existing SPM reductions generally do not directly exploit the analytical spectral structure of spherical diffusion.
Source: §1 · pp. 1–2Method
The model derives Koopman eigenfunctionals and eigenvalues from the self-adjoint diffusion operator and projects the current-dependent surface flux into modal coordinates.
Source: §1 · p. 2Results
The 80-mode model was nearly indistinguishable from the 400-control-volume FVM and ran approximately 31 times faster.
Source: §3.2 · pp. 7–8, §4 · p. 8Takeaways & Limitations
Modal order can be selected according to the required balance between accuracy and computational cost for real-time battery estimation and control.
Source: §3.2 · p. 8The reduction neglects electrolyte concentration gradients and assumes identical particle radii and a spatially uniform reaction rate within each electrode.
Source: §2.1 · p. 2
Abstract
from arXiv · showhide
Physics-based battery models provide internal electrochemical states for estimation and control, but solving the partial differential equations governing solid diffusion can be computationally expensive. This paper therefore develops an analytical Koopman spectral reduced-order model for the single particle model. The Koopman eigenfunctionals and eigenvalues are derived directly from the eigenfunctions of the self-adjoint zero-flux diffusion operator. The zero mode represents the volume average concentration, the nonzero modes describe decaying radial gradients, and projection of the current dependent surface flux yields a linear state-space model that reconstructs the average, surface, and full radial concentrations. Unlike data-driven Koopman models, the proposed formulation requires neither training data nor empirical lifting functions. Compared with a 400 control volume finite volume model (FVM) during a constant 1C discharge, the 80 Koopman mode model achieves negative and positive surface concentration RMSE values of $1.26\times10^{-5}\,\mathrm{mol/m^{3}}$ and $4.10\times10^{-6}\,\mathrm{mol/m^{3}}$, a terminal voltage RMSE of $8.42\times10^{-4}\,\mathrm{V}$. The models with 5 and 80 Koopman modes were approximately 225 and 31 times faster than the FVM model, respectively. These results demonstrate a physically interpretable and computationally efficient representation of battery diffusion dynamics.
1 Introduction
Battery models face a tradeoff between computational cost and access to internal electrochemical states, motivating reduced-order physics-based models. This work addresses the gap in existing reductions by analytically deriving Koopman spectral coordinates for spherical diffusion and evaluating the resulting model against a high-order finite-volume reference.
1 Introduction
- Physics-based electrochemical models describe lithium transport but can require substantial computation, while low-cost equivalent circuit models provide limited information about internal concentration dynamics.This tradeoff motivates reduced-order electrochemical models for real-time battery applications.Source: §1 · p. 1
- Existing single-particle-model reductions use rational approximation, spatial discretization, moment approximation, or parameter fitting rather than directly exploiting the spherical diffusion operator’s analytical spectrum.Prior approaches include transfer-function, second-order, Padé-based, and optimization-based models, with extensions incorporating degradation effects.Source: §1 · pp. 1–2
- For linear diffusion PDEs, Koopman eigenfunctionals can be constructed as linear functionals associated with spatial-operator eigenfunctions, linking diffusion eigenfunction expansions to Koopman coordinates.Koopman-based reduced representations and control formulations have also been developed for PDEs.Source: §1 · p. 2
- The paper analytically derives physically interpretable Koopman coordinates and eigenvalues from the diffusion operator, then projects current-dependent boundary flux into a controlled state-space model.The zero coordinate represents volume-average concentration; nonzero coordinates represent radial diffusion modes with analytical decay rates, and the model is compared with a high-order finite-volume reference for accuracy and efficiency.Source: §1 · p. 2
2 Methodology
The methodology formulates spherical solid diffusion in the SPM and derives Koopman modal coordinates from the zero-flux diffusion operator. It then projects current-dependent surface flux onto those coordinates to form a boundary-controlled model that reconstructs concentration outputs.
2.1 Single-Particle Solid Diffusion Model
- The SPM represents each electrode as a spherical active-material particle with identical particle radii and a spatially uniform reaction rate, while neglecting electrolyte concentration gradients.Volume-average concentration relates to lithium inventory and electrode state of charge; surface concentration enters the overpotential and open circuit potential.Source: §2.1 · pp. 2–3
2.2 Koopman Formulation of the Spherical Diffusion Dynamics
- The Koopman formulation defines modal coordinates from eigenfunctionals of the zero-flux diffusion dynamics, whose zero-input evolution follows the Koopman eigenvalues.The method derives the eigenfunctionals from the spatial eigenfunctions of the self-adjoint diffusion operator.Source: §2.2 · p. 3, §2.2.2 · p. 5
2.2.1 Spherical Diffusion Eigenfunctions
- With zero surface flux, separation of variables produces spherical diffusion eigenfunctions, and each nonuniform temporal mode decays exponentially.The dimensionless roots satisfy tan ζ_n = ζ_n, while the zero-eigenvalue mode is constant.Source: §2.2.1 · pp. 3–4
2.2.2 Koopman Eigenfunctionals and Modal Coordinates
- Self-adjointness makes the adjoint eigenfunctions coincide with the diffusion eigenfunctions, defining Koopman eigenfunctionals whose coordinates encode modal amplitudes.The zero modal coordinate is exactly the volume-average concentration.Source: §2.2.2 · p. 5
2.3 Boundary-Controlled Koopman Reduced-Order Model
- Projecting current-dependent surface flux onto retained Koopman coordinates yields a finite-dimensional state-space model with full radial, average, and surface concentration outputs.The model treats the eigenfunctionals and eigenvalues from zero-flux diffusion separately from the boundary input.Source: §2.2.2 · p. 5, §2.3 · pp. 5–6
3 Results and Discussion
The section evaluates Koopman models against a 400-control-volume FVM during a constant 1C discharge, measuring output accuracy, radial reconstruction, and runtime. Increasing modal order improves agreement, while the Koopman models run faster than the FVM.
3.1 Numerical Comparison Procedure
- The comparison uses a 400-control-volume FVM with matching parameters, initial conditions, and surface-flux inputs, testing NK ∈ {5, 10, 20, 40, 80} over a constant 1C discharge.Evaluation covers surface concentration and voltage time histories, radial profiles, and simulation runtime.Source: §3.1 · p. 6
3.2 Comparison Results
- For NK = 80, negative-electrode surface RMSEt is 1.26 × 10−5 mol m−3, positive-electrode surface RMSEt is 4.10 × 10−6 mol m−3, and terminal-voltage RMSEt is 8.42 × 10−4 V.The FVM and Koopman curves are nearly indistinguishable at this modal order.Source: Figure 1 · p. 7, §3.2 · p. 7
- The FVM takes 0.27 s; NK = 80 and NK = 5 take 0.0088 s and 0.0012 s, respectively—about 31× and 225× speedups.The nearly indistinguishable NK = 80 result and small low-order concentration errors support selecting modal order according to the required accuracy-cost balance.Source: §3.2 · pp. 7–8
- At NK = 80, radial RMSEr is 0.16 × 10−5 mol/m3 for the negative electrode and 0.05 × 10−5 mol/m3 for the positive electrode; errors shrink as NK rises.At lower modal orders, the largest deviations occur near particle centers.Source: §3.2 · p. 7, Figure 2 · p. 8
4 Conclusion
The conclusion presents an analytical Koopman spectral model for spherical solid-phase diffusion in the battery SPM. Against a 400-control-volume FVM, it reports accurate concentration and voltage predictions with substantially shorter runtimes, while identifying future validation and integration work.
4 Conclusion
- The model derives Koopman eigenfunctionals, eigenvalues, and spatial modes from the self-adjoint diffusion operator, with zero and nonzero coordinates representing average concentration and radial gradients.Projecting the current-dependent surface flux yields a linear state-space model that reconstructs average, surface, and full radial concentrations.Source: §4 · p. 8
- For the 80-mode model, negative- and positive-electrode surface concentration RMSE values are 1.26 × 10−5 mol m−3 and 4.10 × 10−6 mol m−3, with terminal-voltage RMSE of 8.42 × 10−4 V.The 80- and 5-mode models are approximately 31 and 225 times faster than the FVM, respectively.Source: §4 · p. 8
- Future work will address nonlinear transport properties, experimental validation under dynamic current profiles, and integration with estimator and controller frameworks.Source: §4 · p. 8