Network equilibrium laboratory

Braess Paradox

An extra connection can make a network worse. On a road, drivers choose their own routes. In a grid, an operator chooses generation, but the power flows still have to obey the network.

Traffic penalty

+15min

Grid savings from removal

$9.1k/hr

Equilibrium type

Worseafter shortcut

Grid status

Congestedtie connected

Original network

Drivers split between two equal routes at equilibrium.

65 min

Top route

2000

Bottom route

2000

Travel time

Shortcut added

Individual choices shift the system into a worse equilibrium.

80 min

Shortcut users

4000

Old-route users

Travel time

The shortcut makes everyone slower.At this demand, drivers individually prefer the shortcut route, but the resulting equilibrium overloads both variable-delay segments.

Traffic controls

Number of cars4000

lightclassic caseheavy

Model

Upper old route: T = flow(S-A) / 100 + 45

Lower old route: T = 45 + flow(B-E) / 100

Shortcut route: T = flow(S-A) / 100 + flow(B-E) / 100

Wardrop equilibrium: all used routes have equal minimum travel time.

What to notice

Tie connected

The low-impedance tie creates loop flow and may bind its thermal limit.

Congested

Cost / hour

$14.1k

G1 dispatch

360

G2 dispatch

140

Tie removed

Removing the troublesome connection can restore cheaper dispatch.

Optimal

Cost / hour

$5.0k

G1 dispatch

500

G2 dispatch

Removing the line lowers operating cost.The connected network forces expensive generation to relieve the tie-line constraint.

Grid controls

Try this: raise the tie-line limit until both networks cost the same. Reset, then lower G1’s maximum output until the connected grid becomes infeasible.

System load500 MW

G1 cost$10/MWh

G2 cost$75/MWh

G1 maximum output500 MW

Tie-line limit100 MW

Equations and assumptions

Minimize: C = c₁P₁ + c₂P₂

Generation: 0 ≤ P₁ ≤ G1 max, 0 ≤ P₂ ≤ 1200 MW

Balance: P₁ + P₂ = Load

DC flow: Fᵢⱼ = (θᵢ − θⱼ) / Xᵢⱼ

Feasibility: |Fᵢⱼ| ≤ line limit, with the connected and removed topologies solved separately.

A lossless DC approximation with fixed voltage magnitudes. Line reactances are relative weights (1 on the main lines, 0.2 on the tie); a common scale is absorbed into the angles. Voltage limits, reactive power, and contingencies are outside this example.

Dispatch result