AC Branch Pi-Model + Transformer Handling

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).