DC Power Flow
DC power flow is a linearized approximation of AC power flow, suitable for economic dispatch and contingency analysis.
DC Approximations
- •Lossless lines - Ignore resistance (R ≈ 0)
- •Flat voltage - All bus voltages = 1.0 pu
- •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)