DC Optimal Power Flow determines the cost-minimizing generator dispatch while respecting transmission network constraints using a linearized (DC) approximation of AC power flow.
The DC approximation simplifies complex AC power flow equations through several assumptions:
Flat Voltage Profile
When: Voltage variations small (0.95-1.05 p.u.)
Small Angle Differences
Valid when: ,
Line Resistance Negligible
Valid when: (typical for transmission)
Reactive Power Ignored
Valid when: Focus on active power markets
Lossless Network
Consequence: May underestimate costs by 2-3%
Transmission planning (not distribution)
Market clearing (hourly/day-ahead)
Congestion analysis
Economic studies
Fast multi-scenario analysis
Voltage stability studies
Reactive power planning
Detailed loss allocation
Distribution networks (high R/X)
Validating feasibility
Trade-off: AC-OPF is much slower than DC-OPF
Symbol | Description | Example (KPG193) |
|---|---|---|
Set of generators | ||
Set of buses | ||
Set of transmission lines | ||
Generators at bus | Multiple gens per bus possible | |
Generator index | ||
Bus index | ||
Line index | ||
Sending bus of line | From-bus | |
Receiving bus of line | To-bus |
Variable | Unit | Description |
|---|---|---|
p.u. | Active power output of generator | |
rad | Voltage angle at bus | |
p.u. | Power flow from sending bus to receiving bus on line | |
p.u. | Power flow from receiving bus to sending bus on line |
Parameter | Unit | Description |
|---|---|---|
$/h | Generation cost function | |
p.u. | Generation limits | |
p.u. | Line reactance | |
p.u. | Thermal line limit | |
rad | Angle difference limits | |
p.u. | Demand at bus |
Minimize total generation cost:
Same as ED, but now dispatch must respect network constraints.
Stability constraint on voltage angles:
Physical meaning:
Limits stress on transmission equipment
Prevents instability
Typically: to
In practice: Often relaxed to (non-binding)
Generator capacity constraints:
Linearized power flow:
Where is the line susceptance.
Reverse direction:
Key insight: Power flows from higher angle to lower angle, proportional to angle difference.
Flow cannot exceed thermal rating:
Physical meaning:
Conductor heating limits
Equipment ratings
Safety margins
Binding constraints create congestion → price separation
Kirchhoff's Current Law at each bus:
Interpretation:
For every bus, the net power injection is exactly balanced by the local demand.
Must hold at every bus simultaneously.
One bus as reference:
Purpose: only angle differences matter, so a reference bus is selected and its voltage angle is fixed.
Choice: Typically slack bus (Bus 190 in KPG193).
Bus | Generator Capacity (MW) | Load (MW) |
|---|---|---|
1 | 0–100 | 0 |
2 | 0–100 | 0 |
3 | 0 | 150 |
Line | From Bus | To Bus | Reactance X (p.u.) | Flow Limit F̄ (MW) |
|---|---|---|---|---|
1–2 | 1 | 2 | 0.1 | 100 |
2–3 | 2 | 3 | 0.1 | 100 |
3–1 | 3 | 1 | 0.1 | 100 |
Without network (ED solution):
Gen 1: 100 MW (at max)
Gen 2: 50 MW
Cost: 10(100) + 20(50) = $2,000/h
With network (DC-OPF solution):
Line limits create congestion
Gen 1: 75 MW
Gen 2: 75 MW
Cost: 10(75) + 20(75) = $2,250/h (+12.5%)
Congestion cost: $250/h
$2,250/h - $2,000/h
Below is an interactive visualization showing how generation is dispatched with network constraints as demand at bus 3 changes in a 3-bus system.
Observations:
At low demand: No congestion, prices uniform
At high demand: Lines hit limits, price separation
Expensive generator at bus 2 must run due to constraints
Total cost > ED cost when congested
Security-Constrained DC-OPF (SCOPF) include N-1 contingencies.
Multi-Period DC-OPF couple DC-OPF with ramping.
Chance-Constrained DC-OPF enforce limits with a specified violation probability under uncertainty.
Feature | ED | UC | DC-OPF | AC-OPF |
|---|---|---|---|---|
Problem Type | LP/QP | MIP | LP/QP | NLP |
Network Model | ✗ | ✗ | ✓ DC (Linearized) | ✓ AC |
Time Periods | Single | Multiple (24+) | Single | Single |
Commitment | ✗ | ✓ Binary | ✗ | ✗ |
Transmission Limits | ✗ | ✗ | ✓ | ✓ |
Voltage Constraints | ✗ | ✗ | ✗ | ✓ |
Reactive Power | ✗ | ✗ | ✗ | ✓ |
Transmission Losses | ✗ | ✗ | ✗ | ✓ |
Solve Time | Fastest | Slow | Fast | Medium |
Full AC model: AC Optimal Power Flow →
Multi-period scheduling: Unit Commitment →
Compare all models: Solver Comparison →
Try it: KPG Run Getting Started →