Single Particle Model - Battery Design

Single Particle Model (SPM) is a simplified but physics‑based model of a lithium‑ion battery that represents each electrode (anode and cathode) as one “average” spherical particle rather than thousands of particles distributed through a porous electrode.

The Single Particle Model assumes that:

Each electrode behaves like one representative spherical particle.
Lithium concentration varies only radially inside that particle.
Electrolyte concentration and potential gradients are neglected or lumped into a single resistance.
Reaction current is uniform across the electrode thickness.

This dramatically reduces model complexity while retaining the essential physics of lithium intercalation.

1. Solid‑phase diffusion

Lithium concentration $c_{s} (r, t)$ evolves inside the spherical particle according to Fick’s law:

$\frac{\partial c_{s}}{\partial t} = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} D_{s} \frac{\partial c_{s}}{\partial r})$

This captures:

Rate‑dependent behaviour
Voltage hysteresis from diffusion gradients
SOC‑dependent dynamics

2. Intercalation reaction kinetics

The surface flux is linked to current via Butler–Volmer kinetics.

3. Ohmic losses

Electrolyte and solid‑phase potential drops are either:

Lumped into a single resistance (classic SPM), or
Explicitly modelled in the SPMe extension (Single Particle Model with electrolyte).

When the SPM is valid

SPM assumptions hold best for:

Low to medium current densities
Thin or highly conductive electrodes
Cells with relatively uniform reaction distribution

What the SPM ignores

No spatial variation across electrode thickness
No electrolyte concentration gradients
No electrolyte potential gradients
No thermal effects (unless added separately)
No mechanical stress or SEI growth (unless extended)

These simplifications make the SPM much faster but limit accuracy at high C‑rates.

The Single-Particle Illusion: Why SPME Fails to Predict High C-Rate Bottlenecks

Single-Particle Model with Electrolyte (SPME) operates on a convenient mathematical simplification: it collapses the entire complex, porous electrode into a single representative particle. At low C-rates, this volume-averaged macroscopic assumption holds. But when a cell is pushed into fast-charging or high-power discharge, SPME begins to lie.

Summary

The Single Particle Model is a reduced‑order electrochemical model that treats each electrode as a single spherical particle, solving only solid‑phase diffusion and reaction kinetics. It is accurate for low–medium C‑rates and widely used in BMS algorithms, control, and parameter estimation because it is fast, stable, and still grounded in real battery physics.

Widely used because:

Computationally lightweight (ideal for BMS and real‑time control)
Physics‑based (unlike empirical models)
Good for parameter estimation
Useful for degradation modelling (when extended)

It is a sweet spot between:

DFN model (very accurate but computationally heavy)
Equivalent circuit models (fast but not physics‑based)

References

S. Santhanagopalan, Q. Guo, P. Ramadass, R.E. White, Review of models for predicting the cycling performance of lithium ion batteries, Journal of Power Sources, Volume 156, Issue 2, 2006, Pages 620-628
M. Guo, G. Sikha, R.E. White, Single-Particle Model for a Lithium-Ion Cell: Thermal Behavior, Journal of The Electrochemical Society, Volume 158, Issue 2, 2011, A122

See also: Battery Cell Modelling