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DC Optimal Power Flow (DC-OPF)

DC Optimal Power Flow determines the cost-minimizing generator dispatch while respecting transmission network constraints using a linearized (DC) approximation of AC power flow.

flowchart TB
    A[System Demand by Bus] --> B[DC-OPF]
    C[Generator Costs & Limits] --> B
    D[Network Topology & Limits] --> B
    
    B --> E{Linear Program}
    
    E --> F[Generator Dispatch pi]
    E --> G[Locational Prices λb]
    E --> H[Line Flows fl]
    E --> I[Congestion Analysis]

DC Approximation

AC vs. DC Power Flow

The DC approximation simplifies complex AC power flow equations through several assumptions:

graph TB
    subgraph "AC Power Flow (Exact)"
        A1["P = V²G - VV'(G cos θ + B sin θ)"]
        A2["Q = -V²B - VV'(G sin θ - B cos θ)"]
        A3[Complex voltages]
        A4[Reactive power]
        A5[Losses]
    end
    
    subgraph "DC Power Flow (Approximation)"
        D1["P = B(θ - θ')"]
        D2[Simplified linear]
        D3[Flat voltages V=1]
        D4[No reactive Q]
        D5[Lossless]
    end
    
    A1 -.Linearize.-> D1
    A3 -.Assume.-> D3
    A4 -.Ignore.-> D4
    A5 -.Ignore.-> D5

DC Approximation Assumptions

  1. Flat Voltage Profile
  • When: Voltage variations small (0.95-1.05 p.u.)
  1. Small Angle Differences
  • Valid when: ,
  1. Line Resistance Negligible
  • Valid when: (typical for transmission)
  1. Reactive Power Ignored

Valid when: Focus on active power markets

  1. Lossless Network

Consequence: May underestimate costs by 2-3%

When DC Approximation Is Valid:

  • Transmission planning (not distribution)
  • Market clearing (hourly/day-ahead)
  • Congestion analysis
  • Economic studies
  • Fast multi-scenario analysis

When to Use AC-OPF Instead:

  • 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

→ Learn about AC-OPF

Mathematical Formulation

Complete DC-OPF Model

Sets and Indices

SymbolDescriptionExample (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

Decision Variables

VariableUnitDescription
p.u.Active power output of generator
radVoltage 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

Parameters

ParameterUnitDescription
$/hGeneration cost function
p.u.Generation limits
p.u.Line reactance
p.u.Thermal line limit
radAngle difference limits
p.u.Demand at bus

Formulation Explanation

(1a) Objective Function

  • Minimize total generation cost:
  • Same as ED, but now dispatch must respect network constraints.

Constraint Explanation

(1b) Angle Difference Limits

  • Stability constraint on voltage angles:

  • Physical meaning:

    • Limits stress on transmission equipment
    • Prevents instability
    • Typically: to

  • In practice: Often relaxed to (non-binding)

(1c) Generation Limits

  • Generator capacity constraints:

(1d) & (1e) DC Power Flow Equations

  • Linearized power flow:

  • Where is the line susceptance.

  • Reverse direction:

  • Key insight: Power flows from higher angle to lower angle, proportional to angle difference.

(1f) & (1g) Thermal Line Limits

  • Flow cannot exceed thermal rating:

  • Physical meaning:

    • Conductor heating limits

    • Equipment ratings

    • Safety margins

    • Binding constraints create congestion → price separation

(1h) Nodal Power Balance

  • 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.

(1i) Reference Angle

  • 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).

Three-Bus Example

Network Topology

Buses

BusGenerator Capacity (MW)Load (MW)
10–1000
20–1000
30150

Transmission Lines

LineFrom BusTo BusReactance X (p.u.)Flow Limit F̄ (MW)
1–2120.1100
2–3230.1100
3–1310.1100

Generator costs:

Solutions

  • 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

Interactive Visualization

DC-OPF Chart

Below is an interactive visualization showing how generation is dispatched with network constraints as demand at bus 3 changes in a 3-bus system.

3-Bus DC-OPF Chart

Demand at Bus 3: 150 MW

Network Flows

Bus 1Bus 2Bus 3L1 1→2: 16.7 MWL2 2→3: 66.7 MWL3 3→1: -83.3 MW

Dispatch & Angles

Gen 1 @ Bus 1100 MW
Gen 2 @ Bus 250 MW
Total Cost$2000.00/h
θ₁ (slack)0.000
θ₂-1.6667 rad
θ₃-8.3333 rad

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

Extensions

  • 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.

Solver Comparison

FeatureEDUCDC-OPFAC-OPF
Problem TypeLP/QPMIPLP/QPNLP
Network Model✗✗✓ DC (Linearized)✓ AC
Time PeriodsSingleMultiple (24+)SingleSingle
Commitment✗✓ Binary✗✗
Transmission Limits✗✗✓✓
Voltage Constraints✗✗✗✓
Reactive Power✗✗✗✓
Transmission Losses✗✗✗✓
Solve TimeFastestSlowFastMedium

→ Detailed Comparison

Next Steps