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name: dc-power-flow description: "DC power flow analysis for power systems. Use when computing power flows using DC approximation, building susceptance matrices, calculating line flows and loading percentages, or performing sensitivity analysis on transmission networks."

DC Power Flow

DC power flow is a linearized approximation of AC power flow, suitable for economic dispatch and contingency analysis.

DC Approximations

1. Lossless lines - Ignore resistance (R ≈ 0)
2. Flat voltage - All bus voltages = 1.0 pu
3. Small angles - sin(θ) ≈ θ, cos(θ) ≈ 1

Result: Power flow depends only on bus angles (θ) and line reactances (X).

Bus Number Mapping

Power system bus numbers may not be contiguous (e.g., case300 has non-sequential bus IDs). Always create a mapping from bus numbers to 0-indexed array positions:

# Create mapping: bus_number -> 0-indexed position
bus_num_to_idx = {int(buses[i, 0]): i for i in range(n_bus)}

# Use mapping for branch endpoints
f = bus_num_to_idx[int(br[0])]  # NOT br[0] - 1
t = bus_num_to_idx[int(br[1])]

Susceptance Matrix (B)

Build from branch reactances using bus number mapping:

# Run: scripts/build_b_matrix.py
# Or inline:
bus_num_to_idx = {int(buses[i, 0]): i for i in range(n_bus)}
B = np.zeros((n_bus, n_bus))

for br in branches:
    f = bus_num_to_idx[int(br[0])]  # Map bus number to index
    t = bus_num_to_idx[int(br[1])]
    x = br[3]  # Reactance
    if x != 0:
        b = 1.0 / x
        B[f, f] += b
        B[t, t] += b
        B[f, t] -= b
        B[t, f] -= b

Power Balance Equation

At each bus: Pg - Pd = B[i, :] @ θ

Where:

• Pg = generation at bus (pu)
• Pd = load at bus (pu)
• θ = vector of bus angles (radians)

Slack Bus

One bus must have θ = 0 as reference. Find slack bus (type=3):

slack_idx = None
for i in range(n_bus):
    if buses[i, 1] == 3:
        slack_idx = i
        break
constraints.append(theta[slack_idx] == 0)

Line Flow Calculation

Flow on branch from bus f to bus t (use bus number mapping):

f = bus_num_to_idx[int(br[0])]
t = bus_num_to_idx[int(br[1])]
b = 1.0 / br[3]  # Susceptance = 1/X
flow_pu = b * (theta[f] - theta[t])
flow_MW = flow_pu * baseMVA

Line Loading Percentage

loading_pct = abs(flow_MW) / rating_MW * 100

Where rating_MW = branch[5] (RATE_A column).

Branch Susceptances for Constraints

Store susceptances when building constraints:

branch_susceptances = []
for br in branches:
    x = br[3]
    b = 1.0 / x if x != 0 else 0
    branch_susceptances.append(b)

Line Flow Limits (for OPF)

Enforce thermal limits as linear constraints:

# |flow| <= rating  →  -rating <= flow <= rating
flow = b * (theta[f] - theta[t]) * baseMVA
constraints.append(flow <= rate)
constraints.append(flow >= -rate)