Hub Height, Shear and a Hybrid Day

What is modelled

Wind slows down near the ground. The page uses the power-law profile for open meadow, a turbine whose power grows with the cube of the speed and swept area, and a solar array with a half-sine day. A night-time wind boost (typical of many sites) is added so that the turbine and the array visibly fill each other in over one day.

Equations

• Wind profile v(h) = v10 (h / 10 m)α with α ≈ 0.14 for open ground.
• Turbine P = ½ ρ A v³ Cp η with ρ = 1.225 kg/m³, Cp = 0.38, η = 0.92, A = π (D/2)².
• Hub-height gain P(h) / P(10 m) = (h / 10)3α.
• Solar power A η Gpk sin(π (t − t0) / D) over daylight D, so the day energy is A η Gpk 2 D / π.
• Average of the cube ⟨(1 + m cos)³⟩ = 1 + 3 m² / 2.

Checks run in the page

• The shear exponent recovered from the wind at two heights.
• The hub-height power ratio against the cube of the speed ratio.
• Simpson integration of the solar day against the closed form.
• The day-average turbine power with a swinging wind against 1 + 3 m² / 2.

Honest simplifications

The wind is steady and uniform in direction, with no turbulence, wakes, cut-in, rated power or curtailment; the shear exponent is a single number although it changes with stability and roughness; the solar day is a clean half sine with no clouds, temperature loss or sun angle on the panel.

• Press Low hub 30 m, then the 76 m preset, and compare the day energy of the turbine.
• Press Rough forest terrain: a large shear exponent makes a tall hub matter more.
• Raise the night-time boost and watch the day average climb above the steady-wind value.