Modelling constant phase elements with ModellingToolkit

New to Julia
1

Hi all,

I’m attempting to very slightly extend the example given [here](https://mtk.sciml.ai/dev/tutorials/acausal_components/) to include a constant phase element. The constant phase element simulation in the time domain can be modelled as a current dependent voltage source, with the voltage gain/drop given by the convolution integral

V_{CPE}(t)=\frac{1}{\Gamma(\alpha)Q}\int_0^t (t-u)^{\alpha-1}I(u)du.

This can be approximated (with constant dt) using

V_{CPE}(t_n)=\frac{\Delta t}{\Gamma(\alpha)Q}\sum_{k=1}^{n-1}(t_n-t_k)^{\alpha-1}I(t_k)

To model this using the framework in the link above, I need to find some way of tracking the history of both the time and current state variables for each CPE modelled. If I was working in Python, the function I’d pass to the ODE solver would be wrapped in a class that could record the state variables passed to it. However, this does not seem to be compatible with the Julia philosophy, so I was wondering what the preferred solution would be?

2

There’s https://github.com/SciML/DelayDiffEq.jl
that handles the state history for you.

3

And ModelingToolkit hasn’t set this up at the symbolic level at this point, but it’s planned to use DelayDiffEq.jl to do so.

4

Ah ok - does that mean it’s not possible to use a DDE with ModellingToolkit as present?

5

Yes, it’s not possible at present. Right now, directly use DelayDiffEq.jl

6

Apologies for being dense, but does that mean I won’t be able to use modelling toolkit to construct and reduce a DAE system?

7

Yes, the analysis would need to be adjusted to handle delays anyways.

8

Ok, after some thought, in the meantime I would like to try and implement this as a current-controlled voltage source. Is it possible to provide an equation to the symbolic toolkit (such as

eqs =[
v~ v+CCVS_gain()
])
and then provide the function to calculate CCVS_gain() from the current and time state variable history to ODAEProblem()?

Additionally, apologies if this is a foolish question, but if there are multiple @named elements with the same component (e.g. multiple capacitors with a C parameter), would those multiple instances of the C parameter be treated as the same when calling structural_simplify()?

9

You mean like v~ v+CCVS_gain(v(t-tau),t)? That’s exactly what’s missing in the symbolic form right now.

No, they are namedspaced. See the component-based modeling example: https://mtk.sciml.ai/dev/tutorials/acausal_components/