AC Branch Pi-Model + Transformer Handling

Implement the exact branch power flow equations in

acopf-math-model.md

using MATPOWER branch data:

[F_BUS, T_BUS, BR_R, BR_X, BR_B, RATE_A, RATE_B, RATE_C, TAP, SHIFT, BR_STATUS, ANGMIN, ANGMAX]

Quick start

• Use

  scripts/branch_flows.py

  to compute per-unit branch flows.
• Treat the results as power leaving the “from” bus and power leaving the “to” bus (i.e., compute both directions explicitly).

Example:

import json
import numpy as np

from scripts.branch_flows import compute_branch_flows_pu, build_bus_id_to_idx

data = json.load(open("/root/network.json"))
baseMVA = float(data["baseMVA"])
buses = np.array(data["bus"], dtype=float)
branches = np.array(data["branch"], dtype=float)

bus_id_to_idx = build_bus_id_to_idx(buses)

Vm = buses[:, 7]  # initial guess VM
Va = np.deg2rad(buses[:, 8])  # initial guess VA

br = branches[0]
P_ij, Q_ij, P_ji, Q_ji = compute_branch_flows_pu(Vm, Va, br, bus_id_to_idx)

S_ij_MVA = (P_ij**2 + Q_ij**2) ** 0.5 * baseMVA
S_ji_MVA = (P_ji**2 + Q_ji**2) ** 0.5 * baseMVA
print(S_ij_MVA, S_ji_MVA)

Model details (match the task formulation)

Per-unit conventions

• Work in per-unit internally.
• Convert with

  baseMVA

  :
  • (P_{pu} = P_{MW} / baseMVA)
  • (Q_{pu} = Q_{MVAr} / baseMVA)
  • (|S|{MVA} = |S|{pu} \cdot baseMVA)

Transformer handling (MATPOWER TAP + SHIFT)

• Use (T_{ij} = tap \cdot e^{j \cdot shift}).
• Implementation shortcut (real tap + phase shift):
  • If

    abs(TAP) < 1e-12

    , treat

    tap = 1.0

    (no transformer).
  • Convert

    SHIFT

    from degrees to radians.
  • Use the angle shift by modifying the angle difference:
    • (\delta_{ij} = \theta_i - \theta_j - shift)
    • (\delta_{ji} = \theta_j - \theta_i + shift)

Series admittance

Given

BR_R = r

,

BR_X = x

:

• If

  r == 0 and x == 0

  , set

  g = 0

  ,

  b = 0

  (avoid divide-by-zero).
• Else:
  • (y = 1/(r + jx) = g + jb)
  • (g = r/(r^2 + x^2))
  • (b = -x/(r^2 + x^2))

Line charging susceptance

• BR_B

  is the total line charging susceptance (b_c) (per unit).
• Each end gets (b_c/2) in the standard pi model.

Power flow equations (use these exactly)

Let:

• (V_i = |V_i| e^{j\theta_i}), (V_j = |V_j| e^{j\theta_j})

• tap

  is real,

  shift

  is radians

• inv_t = 1/tap

  ,

  inv_t2 = inv_t^2

Then the real/reactive power flow from i→j is:

• (P_{ij} = g |V_i|^2 inv_t2 - |V_i||V_j| inv_t (g\cos\delta_{ij} + b\sin\delta_{ij}))
• (Q_{ij} = -(b + b_c/2)|V_i|^2 inv_t2 - |V_i||V_j| inv_t (g\sin\delta_{ij} - b\cos\delta_{ij}))

And from j→i is:

• (P_{ji} = g |V_j|^2 - |V_i||V_j| inv_t (g\cos\delta_{ji} + b\sin\delta_{ji}))
• (Q_{ji} = -(b + b_c/2)|V_j|^2 - |V_i||V_j| inv_t (g\sin\delta_{ji} - b\cos\delta_{ji}))

Compute apparent power:

• (|S_{ij}| = \sqrt{P_{ij}^2 + Q_{ij}^2})
• (|S_{ji}| = \sqrt{P_{ji}^2 + Q_{ji}^2})

Common uses

Enforce MVA limits (rateA)

• RATE_A

  is an MVA limit (may be 0 meaning “no limit”).
• Enforce in both directions:
  • (|S_{ij}| \le RATE_A)
  • (|S_{ji}| \le RATE_A)

Compute branch loading %

For reporting “most loaded branches”:

• loading_pct = 100 * max(|S_ij|, |S_ji|) / RATE_A

  if

  RATE_A > 0

  , else 0.

Aggregate bus injections for nodal balance

To build the branch flow sum for each bus (i):

• Add (P_{ij}, Q_{ij}) to bus i
• Add (P_{ji}, Q_{ji}) to bus j

This yields arrays

P_out[i]

,

Q_out[i]

such that the nodal balance can be written as:

• (P^g - P^d - G^s|V|^2 = P_{out})
• (Q^g - Q^d + B^s|V|^2 = Q_{out})

Sanity checks (fast debug)

• With

  SHIFT=0

  and

  TAP=1

  , if (V_i = V_j) and (\theta_i=\theta_j), then (P_{ij}\approx 0) and (P_{ji}\approx 0) (lossless only if r=0).
• For a pure transformer (

  r=x=0

  ) you should not get meaningful flows; treat as

  g=b=0

  (no series element).