Description

We now have three options for modelling open-circuit potential hysteresis in PyBaMM: 1) "current sigmoid"; 2) "Wycisk"; and 3) "Axen", all of which have slightly different implementations.

Both "Wycisk" and "Axen" are single-state hysteresis models which contain a single ODE for a hysteresis state variable, h, of the form dh/dt = K(1-sgn(i))h, but again with slightly different implementations. The current sigmoid model is a zero-state model (the OCP just jumps between branches depending on the sign of the applied current), and doesn't track the variable h

We propose we unify these models by (see #4332 (comment)):

• Making sure all hysteresis models use the state variable -1<h<1 (in the current sigmoid model h will just jump from -1 to 1)

• Add a base one-state model that solves a general one-state equation:

$\frac{\partial h}{\partial t} = \gamma\cdot\left(\frac{i_\mathrm{surf}a_\mathrm{vol}}{\phi_\mathrm{act}Fc_\mathrm{max}}\right)\cdot \left(1 - \mathrm{sgn}(i_\mathrm{surf})h\right)$

for $i_\mathrm{surf}$ the interfacial phase-specific current density (anodic current density positive).

• Allow a scalar input or arbitrary functional input for $\gamma$, or allow $\gamma$ to be computed using the specific Wycisk formula with parameters $K$, $n$ and the OCP:

$\gamma = K \cdot \left(\frac{\frac{\partial q_\mathrm{vol}}{\partial U}}{\left.\frac{\partial q_\mathrm{vol}}{\partial U}\right|_\mathrm{ref}}\right)^{-n}$

where the evaluation at a reference point (implicitly at $x_\mathrm{Li}=x_\mathrm{Li,ref}$ which in turn defines $K$ as the decay rate at the reference stoichiometry) is to ensure that we take an arbitrary power only of a unitless quantity.

• Allow general (particle-wise) initialisation of the hysteresis state, not necessarily = 0.

Unifying the models will also allow us to more reasonably include hysteresis heating (see https://github.com/About-Energy-OpenSource/AEPyBaMM/blob/main/src/aepybamm/pybamm_tools.py#L91).

Motivation

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Possible Implementation

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Additional context

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