AC Optimal Power Flow (AC-OPF)

AC Optimal Power Flow determines the cost-minimizing generator dispatch while satisfying the complete AC power flow equations, voltage constraints, and reactive power limits.

AC vs. DC Power Flow

Key Differences

When AC-OPF is Required

• Use AC-OPF when:

  • Validating dispatch feasibility

  • Voltage stability critical

  • Reactive power planning

  • Accurate loss calculation

  • Generator capability curves matter

  • Detailed operational studies

• Use DC-OPF when:

  • Faster speed is critical

  • Voltage is not a concern

  • Large-scale studies (many scenarios)

  • Market clearing considering active power only

Mathematical Formulation

Complete AC-OPF Model

Sets and Indices

Decision Variables

• Key difference from DC: Voltage magnitudes are now variables!

Parameters

Network Parameters

Generator Parameters

System Parameters

Formulation Explanation

(1a) Objective Function

• Minimize total generation cost:

Constraint Explanation

(1b) Active Power Balance

• Kirchhoff's Current Law (real part):

• Physical meaning: For every bus, the net active power injection is balanced by the local active power demand.

• Nonlinearity in AC power flow

• Compare to DC: DC used simplified equation

(1c) Reactive Power Balance

• Kirchhoff's Current Law (imaginary part):

• Physical meaning:

  • Reactive power must also satisfy a nodal power balance at each bus.

(1d) & (1e) Generator Limits

• Active power limits:

• Reactive power limits:

• Important: Real generators have coupled P-Q capability curves (more complex than box constraints).

(1f) Voltage Magnitude Limits

• Operational voltage range:

• Limits in KPG 193:

  • 345 kV, 765 kV: 0.95 ≤ V ≤ 1.05 p.u.

  • 154 kV: 0.90 ≤ V ≤ 1.10 p.u.

• Why limits matter:

  • Equipment designed for nominal voltage

  • Low voltage → Poor motor performance, dimming lights

  • High voltage → Insulation stress, equipment damage

(1g) Line Flow Limits

• Apparent power (thermal) limit:

• More accurate than DC: Considers both P and Q flows

• Line flow calculation:

• Where is conjugate of line current.

(1h) Reference Angle

• Slack bus angle:

• Same as DC-OPF: angles are relative.

AC Power Flow Equations

Complex Power Formulation

Expanded Form

• For line from bus to bus :

  • Active power flow:

• Reactive power flow:

• Where:

  • - Conductance

  • - Susceptance

  • - Line charging (half at each end)

Why Nonlinear?

• Products of variables:

  • (bilinear term)

  • , (trigonometric)

• Consequence:

  • No global optimum guarantee

  • Multiple local optima possible

  • Convergence not guaranteed

  • Sensitive to starting point

Reactive Power and Voltage

Relationship

• Key principle: Reactive power supports voltage

Voltage Control

• Methods to control voltage:

  1. Generator excitation (increase Q output)

  2. Shunt capacitors (inject Q)

  3. Shunt reactors (absorb Q)

  4. Transformer taps (change voltage ratio)

• In AC-OPF: Optimize generator Q dispatch

Voltage Stability

Voltage Collapse

• Phenomenon: Insufficient reactive power → voltage decay

• Indicators in AC-OPF:

  • Voltage limits binding

  • Reactive limits binding

  • High losses

Transmission Losses

Loss Calculation

• Power losses in line:

Where:

• = Line current

• = Line resistance

• = Power flows

• = Voltage magnitude

• Key observations:

  • Losses are nonlinear in P and Q

  • Losses increase with square of power flow

  • Low voltage → higher losses

Extensions

• Security-Constrained AC-OPF with N-1 contingencies

• Optimal Reactive Dispatch: Given P dispatch, optimize Q for voltage profile

Solver Comparison

→ Detailed Comparison

Next Steps

• Compare all models: Solver Comparison →

• Learn simpler models: DC-OPF → or ED →

• Multi-period scheduling: Unit Commitment →

• Try it: KPG Run Getting Started →