Source-linked AI summary
Lithium-ion battery thermal-electrochemical model-based state estimation using orthogonal collocation and a modified extended Kalman filter
A. M. Bizeray, S. Zhao, S. R. Duncan, D. A. Howey
TL;DR
The paper addresses state estimation for a high-fidelity P2D battery model whose computational demands challenge embedded BMS deployment. It combines Chebyshev orthogonal collocation with an EKF for differential-algebraic equations, recovering model states from current, voltage, and temperature measurements. State-estimate error falls below 1% in under 200 s despite 30% initial SOC error, while parameter estimation remains a challenge for practical implementation.
Problem
High-fidelity P2D models support spatially resolved battery-state representation but are computationally intensive for embedded BMSs, while empirical and simplified models have narrower validity.
Method
The paper spatially discretises the thermal-electrochemical P2D model with Chebyshev orthogonal collocation and applies an EKF that corrects model states using noisy voltage and temperature measurements for a given current input.
Results
The anode bulk SOC estimate error falls below 1% in less than 200 s from a 30% initial absolute error during a full 4C constant-current discharge.
Takeaways & Limitations
The approach rapidly recovers internal states of the fully spatially distributed P2D model and could support advanced battery-management functions using high-fidelity online information.
Takeaways & Limitations
Estimating model parameters from experimental data remains challenging because many parameters must be inferred from limited non-invasive measurements.
Abstract
from arXiv · showhide
This paper investigates the state estimation of a high-fidelity spatially resolved thermal- electrochemical lithium-ion battery model commonly referred to as the pseudo two-dimensional model. The partial-differential algebraic equations (PDAEs) constituting the model are spatially discretised using Chebyshev orthogonal collocation enabling fast and accurate simulations up to high C-rates. This implementation of the pseudo-2D model is then used in combination with an extended Kalman filter algorithm for differential-algebraic equations to estimate the states of the model. The state estimation algorithm is able to rapidly recover the model states from current, voltage and temperature measurements. Results show that the error on the state estimate falls below 1 % in less than 200 s despite a 30 % error on battery initial state-of-charge and additive measurement noise with 10 mV and 0.5 K standard deviations.
1. Introduction
Battery management systems must estimate unmeasurable battery states under demanding operating conditions, but conventional models trade computational efficiency for limited validity. This work motivates physics-based P2D state estimation using Chebyshev collocation and an EKF to improve practical high-fidelity estimation.
- Motivation: Battery-pack scaling creates safety and reliability challenges requiring BMS hardware and algorithms to monitor cells and estimate states such as SOC and SOH.Available measurements include voltage, current, and temperature, while the target states are unmeasurable directly.
- Limitations of existing models: Current BMSs commonly use low-order empirical models that are computationally efficient but valid only within their parametrised operating conditions.Their parameters have limited physical significance, making broader validity data-intensive and degradation prediction difficult or impossible.
- Physics-based alternatives: Physics-based models cover thermodynamics, reaction kinetics, and transport over wider operating ranges, but their computational intensity limits embedded BMS use.The P2D model is identified as a widely used physics-based lithium-ion battery model.
- Prior state-estimation approaches: Simplified P2D-derived models such as the SPM reduce computational demands but are inherently limited by the SPM’s low-current validity range.Prior state-estimation approaches include EKF, backstepping PDE estimators, and reduced-order models.
- Numerical approach: Spectral and orthogonal-collocation methods reduce P2D model order while maintaining accuracy and physical parameter significance, with Chebyshev bases improving high-current convergence.Prior work applied these methods to electrolyte or full-P2D equations, motivating the collocation approach used here.
- Contribution: The paper applies an EKF to a Chebyshev-collocated thermal-electrochemical P2D model, targeting lower computational intensity than other observers while estimating full-model states.The authors identify this as the first attempt to estimate the states of the full P2D model using an EKF.
2. Thermal-electrochemical model
The thermal-electrochemical P2D model represents coupled lithium transport, reaction, electrical potential, and cell-temperature dynamics across porous electrode and separator domains. Chebyshev orthogonal collocation discretizes the model into a smaller system while retaining high-current accuracy, supporting efficient state estimation.
- Model coupling: The model couples a P2D electrochemical description with a lumped thermal energy balance through heat generation and temperature-dependent properties.Electrochemical dynamics cover lithium transport, reaction kinetics, and thermodynamics; the thermal model describes cell-temperature evolution.
- Electrochemical model: The cell is divided into anode, separator, and cathode domains, with superimposed solid and electrolyte phases and spherical solid particles in the electrodes.The porous-electrode representation neglects exact electrode microstructure and uses spatial coordinates across the cell and radially within particles.
- Electrochemical model: The P2D equations govern solid and electrolyte lithium concentrations, solid and electrolyte potentials, and volumetric reaction current under algebraic electrical constraints.The applied current is the model input, while local ionic and electronic currents satisfy is + ie = iapp; the terminal voltage is obtained from solid-phase potentials.
- Thermal model: The thermal model represents total volumetric heat generation as reaction, reversible, ohmic, and contact-resistance contributions, with convective heat removal to coolant air.For the 18650 cell, the convective surface-area-to-volume ratio is Ac/Vc = 253 m^-1; uniform cell temperature may be inadequate for large cells.
- Thermal coupling: Temperature dependence is included for electrolyte transport, solid diffusion, reaction kinetics, and open-circuit potential using empirical or Arrhenius relationships.Open-circuit potential is approximated by a first-order Taylor expansion around a reference temperature, with entropy change coefficient ∂U/∂T.
- Numerical discretization: Chebyshev orthogonal collocation discretizes the electrochemical equations in spatial coordinates into a smaller ODE/DAE system than finite differences at similar accuracy.The discretized system can be integrated with a time-adaptive ODE/DAE solver, and separate collocation nodes are used across cell and particle domains.
3. Thermal-electrochemical model simulation results and discussion
Chebyshev orthogonal collocation closely matches the high-fidelity P2D reference while substantially reducing simulation time across constant-current and dynamic conditions. Accuracy depends on collocation resolution, especially in solid particles and at higher C-rates, while bulk SOC remains highly accurate.
- Validation against high-fidelity simulations: The collocation model agrees with COMSOL from 1C to 10C, with 10 mV RMS and 50 mV maximum voltage errors.The collocation solution is typically 30 times faster: about 1–10 s per discharge versus about 5 min with COMSOL.
- Resolution requirements: Higher C-rates require more collocation nodes because they produce sharper concentration gradients, particularly near solid-particle surfaces.The number of particle nodes Np strongly affects voltage accuracy.
- Resolution requirements: Maximum voltage error can be misleading at 1C because it arises near very low SOC, where the anode concentration causes a rapid voltage drop.At higher C-rates, that low anode concentration is not reached, so the maximum error can be smaller.
- State accuracy: Bulk SOC relative error falls below 1% with fewer than 10 particle nodes and below 0.1% with 15 nodes up to 10C.The result applies to both the anode and cathode bulk SOC.
- High-rate internal dynamics: At 10C, electrolyte depletion and strongly nonuniform reaction rates reveal internal variations that simpler single-particle models cannot represent.The electrolyte is almost depleted in less than 50 s, and short-time reaction rates at the electrode-separator interface can be an order of magnitude higher than at the current collector.
- Dynamic-input performance: Under the Combined ARTEMIS Driving Cycle, 1700 s of dynamic simulation required 285 s, averaging 168 ms per simulated second.The solver required more computation for high-amplitude dynamic current peaks than for constant-current discharge.
4. State estimation using a modified EKF
The paper adapts an extended Kalman filter for the nonlinear, algebraically constrained P2D battery model by repeatedly linearising its differential-algebraic equations. The resulting modified EKF uses model predictions and noisy voltage and temperature measurements to estimate internal states.
- The modified EKF estimates P2D battery states from noisy voltage, current, and temperature measurements.The filter corrects model states by minimising the discrepancy between predicted and measured voltage and temperature for a given current input.
- At each time step, the algorithm performs state and covariance time updates followed by measurement updates.The state is projected using the nonlinear DAE model, while the covariance and estimates are updated recursively using linearised dynamics and measurements.
- Linearisation about the current estimate converts the nonlinear DAE system into locally linear ODEs usable by a standard Kalman filter.Jacobian matrices describe the differential and algebraic equations, and the algebraic constraint is eliminated assuming a nonsingular gz matrix, equivalent to an index-1 DAE assumption.
- The P2D model is represented as a nonlinear stochastic differential-algebraic state-space system with process and measurement noise.The input current is assumed noise-free, while process and measurement noises are additive, zero-mean Gaussian, and temporally uncorrelated.
- The modified EKF includes the linearised algebraic constraint in both the state-transition and measurement models.The resulting matrices support the Kalman time update, gain calculation, and error-covariance update.
- The filter requires a consistent initial algebraic state and an initial covariance structure that accounts for spatial state correlations.The initial algebraic state is computed with MATLAB fsolve, and the covariance initialization is described as crucial for convergence performance.
5. State estimation results and discussion
Simulation experiments evaluate the modified EKF on constant-current discharge and a dynamic charge/discharge driving cycle using model-generated measurements. The estimates rapidly track measured outputs and recover internal states from incorrect initial conditions.
- The experiments use the thermal-electrochemical model to emulate experimental data because internal states are difficult to verify in real batteries.The EKF receives applied current and measured voltage and temperature responses generated by reference simulations.
- The reference simulations span constant-current discharge and CADC operation, while the EKF starts from equilibrium guesses across initial SOCs from 100% to 50%.The tests examine convergence under multiple initial conditions and dynamic current excitation.
- 10 mV and 0.5 K standard deviations define the voltage and temperature measurement noise, while the process-noise covariance is set to zero.These values are used to construct the EKF measurement-noise covariance matrix.
- Within the first few 5 s time steps, voltage and temperature estimates rapidly converge during the first 200 s of a 4C discharge.The voltage estimate falls below the 95% measurement-noise confidence interval within the first few time steps for all tested initial times, with similar temperature behavior.
- 30% initial anode-bulk-SOC error falls below 1% in less than 200 s, or 300 s when initialized at t0 = 250 s, during full 4C discharge.Similar convergence is reported for cathode bulk SOC because lithium is conserved in the cell.
- Under CADC, the anode bulk SOC error falls below 1% by 150 s, while voltage and temperature converge rapidly using a 5 s time step.The EKF requires 0.6 s of computation per second of simulation on average on the reported desktop computer.
- During CADC operation, the EKF recovers and accurately tracks electrolyte concentration at the anode and cathode current collectors from incorrect initial conditions.The paper highlights electrolyte saturation or depletion as relevant to performance limitations during high current peaks.
6. Conclusion
The paper combines Chebyshev orthogonal collocation with an extended Kalman filter to estimate internal states of a fully spatially distributed P2D battery model. Simulations show rapid recovery from substantial initial SOC error, while future work targets observability, identifiability, and parameter estimation.
- Chebyshev orthogonal collocation produces a smaller, lower-cost P2D model while preserving accuracy against a higher-order COMSOL model.The approach enables the model to be coupled with a state observer for internal-state estimation.
- The EKF estimates the fully spatially distributed P2D model using a time-varying linear approximation of its differential-algebraic equations.The paper identifies this as the first attempt, to its knowledge, to estimate these internal states with an EKF.
- Less than 200 s is required to recover model states despite a 30% error in initial SOC.The proposed approach is described as suitable for advanced embedded battery-management systems and coupling to degradation models.
- Future work will investigate model-state and parameter observability and identifiability and develop a parameter-estimation algorithm.
Appendix A Change of variable for the spherical particle diffusion model
The appendix introduces a change of variable for the solid-phase spherical particle diffusion model and states the associated boundary conditions.
- The solid-phase particle diffusion model is reformulated using the change of variable ¯cs = rcs.Under constant diffusivity, the diffusion equation is expressed in the transformed variable.
- Boundary conditions are specified for the transformed spherical-particle diffusion equation.
Continuity of electrolyte electric potential
The electrolyte electric potential is continuous across both electrode–separator interfaces.
- At x = La, the anode and separator electrolyte potentials are equal.
- At x = Lc, the separator and cathode electrolyte potentials are equal.
Continuity of the concentration profile
The electrolyte concentration profile is continuous across both electrode–separator interfaces.
- At x = La, the anode and separator electrolyte concentrations are equal.
- At x = Lc, the separator and cathode electrolyte concentrations are equal.
Appendix C COMSOL implementation of the thermal-electrochemical P2D model
The COMSOL thermal-electrochemical P2D implementation uses coupled PDE, ODE, and DAE interfaces on one-dimensional cell and two-dimensional electrode-particle geometries.
- The COMSOL model uses PDE Interfaces together with ODE and DAE Interfaces for equation-based modelling.The implementation follows a previously described COMSOL thermal-electrochemical P2D model.
- The macro-scale cell is represented by a one-dimensional geometry divided into anode, separator, and cathode regions.
- Local reaction rate jLi and solid-phase surface concentration csurf are coupled between geometries using COMSOL’s linear extrusion function.
- The one-dimensional geometry uses uniformly spaced meshes with 22, 8, and 21 elements in the anode, separator, and cathode domains, respectively.
- The two-dimensional electrode-particle geometries use triangular core elements and quadrilateral surface elements.The anode discretisation includes 890 triangular and 528 quadrilateral elements.
Symbols
The symbol list defines geometric, electrochemical, thermal, material, and transport variables used throughout the model.
- Ac and As denote cell and electrode surface area, respectively, while Vc denotes cell volume.
- ce is electrolyte concentration, b is the Bruggeman coefficient, and Rs is the radius of solid-phase particles.
- T, Tref, and T∞ denote temperature, standard-state reference temperature, and coolant temperature, respectively.
- U and V denote electrode open-circuit potential and cell voltage, while t+0 denotes the Li-ion transference number.
- αa, αc, δ, ϵe, ϵf, ϵs, η, and κ represent charge-transfer, thickness, volume-fraction, overpotential, and conductivity parameters.
- s and θ0 denote average and initial solid-phase stoichiometry, while ρ denotes cell bulk density.
- φe and φs denote electrolyte and solid-phase surface electric potentials, respectively.
- The appendix lists electrochemical P2D model parameters and thermal model parameters used in the simulations.