Distributed slack

[luke-kiernan]

Many power flow references do a 2-step approach:

1. Unknowns are just voltage angles (PQ and PV) and magnitudes (PQ only). Solve for those quantities.
2. Calculate our P (at REF) and Q (and PV and REF) using those values

We don't. We keep all 2n equations and 2n variables. That makes distributed slack a bit more complicated. And the code is pretty poorly documented.

If c_k are our proportional weights, we want

(P_{calc, k} - P_{spec, k})/c_k = (P_{inj, ref} - P_{spec, ref})/c_{ref}

Solve for P_{calc, k}:

P_{calc, k} = P_{spec, k} + c_k (P_{inj, ref} - P_{spec, ref}) /c_{ref}

So then we just replace our usual P_{calc, k}(V, θ) - P_{spec, k} = 0 equation at PV buses with

P_{calc, k}(V, θ) - P_{spec, k} - c_k (P_{inj, ref} - P_{spec, ref})/c_{ref} = 0

I believe that's what the code is doing, but I'm still working on going through it carefully.