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:

python

# 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:

python

# 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):

python

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):

python

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#

python

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:

python

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:

python

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