Complex equations

New to Julia
1

After >20 years working with MODELICA I tried to use Complex equations in MTK.

In a simple example an ohmic resistor and a inductor are connected in series to a constant voltage source. I used per unit system with space-phasors and variable speed of reference frame @parameters wReference.

using ModelingToolkit
using DifferentialEquations
using Plots

frequency = 50.0
Omegarated = frequency*2pi
JJ = 1.0im


function NewVars(eqName::String, CCX::Vector{Complex{Num}})::Tuple{Expr, Expr}
    vari::String = "newVariables = @variables "
    substi::String = ""
    for cx in CCX
        stcx = string(cx)[1:end-3]  # without '(t)'
        vari = string(vari,stcx*"₋re(t) ",stcx*"₋im(t) ")
        substi = string(substi,"$eqName = substitute($eqName,(real($stcx)=>$(stcx)₋re)); ")
        substi = string(substi,"$eqName = substitute($eqName,(imag($stcx)=>$(stcx)₋im)); ")
    end
    return (Meta.parse(vari), Meta.parse(substi))
end

@variables t rad(t)=0.0  w(t)

CX = @variables begin
    deri(t)::ComplexF64
    amps(t)::ComplexF64
    gridVolts(t)::ComplexF64
end


@parameters  x r wReference = 1.0

D = Differential(t)
# DiffC(cpx) =  Complex(D(real(cpx)),D(imag(cpx)))  # used for testing

#NOTE
#NOTE  this line was added in Symbolics - diff.jl
# (D::Differential)(x::Complex{Num}) = Num(D(real(x)))+Num(D(imag(x)))*im
#NOTE


eqs1 = [ D(amps) ~ deri*Omegarated
        gridVolts ~ x*(deri + wReference*amps*JJ) + r*amps
        w ~ D(rad)
        w ~ Omegarated*(1.0-wReference)
        gridVolts ~ Complex(cos(rad), sin(rad))
    ]

eqs=deepcopy(eqs1)

(Bvaria, Bsubst) = NewVars("eqs",CX)
eval(Bvaria) # create new variables
eval(Bsubst) # substitute

pr = 0.3; px = 0.1
u0 = [0.0, 0.0, 0.0]
p = [x => px, r=> pr, wReference => 1.0]

# exact solution
strom = (1.0+0im)/(pr+JJ*px)
println("exact solution: amps = $(strom)")


@named model = ODESystem(eqs)
sys = structural_simplify(model)

prob = ODEProblem(sys, u0, (0., 1.0/frequency), p)
sol = solve(prob, Tsit5())
plot(sol)

my approach

(D::Differential)(x::Complex{Num}

an additional line in diff.jl
(D::Differential)(x::Complex{Num}) = Num(D(real(x)))+Num(D(imag(x)))*im

substitute real(x), imag(x)

real(x), imag(x) are replaced by new @variables x_re x_im

Questions

  • This works with monolithic models. But how to do with component based models? (e.g. connect() and Flow )

backup alternative

To split Complex equations manually. This works, but Symbolics should do this for me.

Thanks

2

It should, that’s just not all complete yet. For now the splitting needs to be done manually, but it’s on the menu.