PyBaMM EIS Data Generation: Impedance Features and AI Labels

Impedance spectra represent the definitive signature of battery physicochemical dynamics, compressing complex electrochemical phenomena—ranging from ultra-fast electron transport to macroscopic solid-state diffusion—into the frequency domain. For advanced battery management systems (BMS), Electrochemical Impedance Spectroscopy (EIS) is the foundation of physics-informed aging models. The high-frequency intercept bounds the ohmic resistance, mid-frequency arcs parameterize charge-transfer kinetics and Solid Electrolyte Interphase (SEI) capacitance, while the low-frequency Warburg tail dictates bounded diffusion limits. This article presents a rigorous methodology for synthesizing PyBaMM core EIS outputs into high-fidelity labeled tensors for deep learning.

The core challenge is not simply generating plausible Nyquist plots, but ensuring rigorous tractability. Operating points, thermodynamic state of charge (SOC), model formulations (SPM, SPMe, DFN), parameter spaces, and specific degradation mechanisms (e.g., loss of lithium inventory, active material isolation) must be mathematically coupled and documented.

1. The Mathematical Foundations of Impedance

The small-signal EIS response represents a linearized perturbation around a non-linear electrochemical operating point. The fundamental object is the complex impedance transfer function:

$$ Z(omega) = frac{tilde{V}(omega)}{tilde{I}(omega)} = Z_{re}(omega) + j Z_{im}(omega) $$

In the context of Equivalent Circuit Models (ECM), the transfer function for an $n$-RC network coupled with a Warburg element $Z_W$ is given by:

$$ Z_{ECM}(s) = R_Omega + sum_{k=1}^{n} frac{R_k}{1 + s R_k C_k} + Z_W(s) $$

where $s = jomega$. However, physics-based AI models demand features beyond simple curve fitting. We extract bounded metrics derived from the continuum physical chemistry equations (e.g., concentrated solution theory and Butler-Volmer kinetics):

• High-Frequency Ohmic Intercept ($R_Omega$): Extrapolated limit as $omega to infty$, dominated by electrolyte conductivity and current collector contact resistance.
• Charge-Transfer Arc Metrics: $- max (Z_{im})$, corresponding to the double-layer capacitance $C_{dl}$ and charge transfer resistance $R_{ct}$, modulated by the specific surface area of the porous electrode.
• Warburg Diffusion Tail ($sigma_W$): The asymptotic phase shift at $omega to 0$, describing the Li-ion intercalation into the graphite/NMC crystal lattice.

2. Frequency-Domain Synthesis in PyBaMM

Generating computationally rigorous EIS data requires solving the coupled Partial Differential Equations (PDEs) of the Doyle-Fuller-Newman (DFN) or Single Particle Model with electrolyte dynamics (SPMe). In PyBaMM, the pseudo-spectral collocation methods solve the linearized system directly.

import numpy as np
import pybamm

# Instantiate the physical model with differential surface formulations
model = pybamm.lithium_ion.DFN(options={
    "surface form": "differential",
    "particle shape": "spherical",
    "SEI": "solvent-diffusion limited"
})

# Define degradation parameters and nominal chemistry
params = pybamm.ParameterValues("Chen2020")
params["SEI kinetic rate constant [m.s-1]"] = 1e-15

# Initialize the EIS simulation engine
eis = pybamm.EISSimulation(model, parameter_values=params)

# Define a logarithmically spaced frequency vector (1 mHz to 10 kHz)
frequencies = np.logspace(-3, 4, 60)
solution = eis.solve(frequencies)

# Extract real and imaginary tensors
z_re = solution["Z_re [Ohm]"].entries
z_im = solution["Z_im [Ohm]"].entries
nyquist_tensor = np.column_stack((frequencies, z_re, z_im))

The specification "surface form": "differential" is mandatory. It ensures the differential algebraic equation (DAE) solver correctly processes the capacitance of the double-layer, which is structurally required for frequency-domain stability.

3. Operating Envelopes and Latent State Identifiability

A solitary Nyquist plot suffers from extreme parameter unidentifiability. Isomorphic impedance signatures can emerge from entirely different combinations of pore-wall flux limitations and solid-phase diffusivity. To break this symmetry, our datasets mandate state-of-charge (SOC) and thermal sweeps.

• soc_matrix: Defines the thermodynamic intercalation fractions $theta_p$ and $theta_n$.
• temperature_c: Actuates the Arrhenius dependencies governing transport and kinetic rates.
• protocol_id: Encodes the deterministic load profiles (e.g., DST, WLTP) dictating the hysteresis paths.

4. Bridging PINNs and BMS Hardware Constraints

Modern Battery Management Systems (BMS) operate under severe deterministic time constraints (e.g., RTOS on Cortex-M4/M7). Physics-Informed Neural Networks (PINNs) trained on these PyBaMM-generated EIS tensors must be distilled. You cannot run a DAE solver or a massive Transformer on an ASIL-D automotive microcontroller.

We use the high-fidelity PyBaMM EIS data to pre-train localized MLPs. The BMS then runs a lightweight Extended Kalman Filter (EKF) or an Unscented Kalman Filter (UKF) where the observation model $h(x_k)$ is replaced by a quantized, statically compiled neural network inference block. This bridges the gap between deep physical chemistry (SEI porosity reduction) and real-time linear algebra constraints.

5. Quality Assurance and Pipeline Generation

cd pybamm-ai-data-lab
python3 -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt
python src/run_all.py --backend pybamm --workers 8 --precision float64

For scientific integrity, never mix fallback surrogate interpolations into the primary training tensors. Solver tolerance failures in the DAE backend indicate stiff physical regimes (e.g., near 0% SOC at sub-zero temperatures) and must be explicitly labeled or masked, never silently imputed.

References

• PyBaMM EIS Simulation DAE Solvers
• Physics-based battery model parametrisation from impedance data