Docs/kpg run/DC Optimal Power Flow (DC-OPF)

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

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%

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

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

Decision Variables

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

Parameters

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

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

Bus	Generator Capacity (MW)	Load (MW)
1	0–100	0
2	0–100	0
3	0	150

Transmission Lines

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

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.

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

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

→ Detailed Comparison

Next Steps

Full AC model: AC Optimal Power Flow →
Multi-period scheduling: Unit Commitment →
Compare all models: Solver Comparison →
Try it: KPG Run Getting Started →

← Unit Commitment (UC)AC Optimal Power Flow (AC-OPF) →