Wind profile and extrapolation to the turbine cube

A 3 MW wind turbine with 130 m rotor diameter installed in a 140 m tower has the cube at 140 m from the ground. Most of the measurement campaigns with mast anemometers measure at 60–100 m. The wind at 140 m is not the same as at 80 m — and the difference is not small or negligible: depending on the vertical profile of the site, the speed in the cube can be 5 to 15% higher than at the measurement height. These 5–15% enter the cube of the power equation and amplify in energy: the power varies with V3.

The extrapolation of the speed measured to the height of the cube is one of the major components of uncertainty in campaigns with conventional masts. The LiDAR solves the problem in another way: it measures at various heights simultaneously, even at the height of the cube or above — eliminating the extrapolation and replacing it by direct measurement.

The law of power

The most used model in practice to describe the speed increase with height is the power law:

V(z) = V_ref × (z / z_ref)^α

z      — target height (m)
z_ref  — reference height with known wind speed (m)
V_ref  — mean wind speed at reference height (m/s)
α      — wind-shear exponent (shear exponent), adimensional

The exponent α is the parameter that characterizes the vertical profile of the site. It is not a constant of nature — it varies with the roughness of the terrain, the atmospheric stability, the time of day, and the direction of the wind. Reference values for different surface covers:

In BORA, the instantaneous α exponent is calculated line-to-line from two levels of measurement of LiDAR:

α = log(V_high / V_low) / log(h_high / h_low)

# Example: V(80m)=8.0, V(120m)=8.5
α = log(8.5 / 8.0) / log(120 / 80) = log(1.0625) / log(1.5)
  = 0.0606 / 0.4055 = 0.149

# The mean α is then adjusted to the full profile via curve_fit:
V_cubo = V_ref × (z_hub / z_ref)^α_fit    (power_law)

The Logarithmic Law

The logarithmic law is derived from the theory of the atmospheric boundary layer under conditions of neutrality (zero heat flow, dominant mechanical mixture):

V(z) = V_ref × ln(z / z₀) / ln(z_ref / z₀)

z₀   — roughness length (m): effective height of surface obstacles
ln   — natural logarithm

The length of roughness z0 is a property of surface coverage — not wind speed. It is determined by the adjustment of the measured profile or by reference tables based on the type of coverage. For neutral conditions (predominant in strong winds), the log law tends to represent the real profile better than the power law, because it has a more solid physical basis.

In practice, the two laws produce close results for modest extrapolations (up to 30–40% above the reference height).For larger extrapolations — campaigns with a mast at 80 m and a cube at 160 m — the difference between models can be 2–4% extrapolated speed, which translates into 6–12% difference in estimated AEP.

α daytime variation: why the mean α is not sufficient

The shear exponent is not constant throughout the day. It responds to the stability of the atmospheric boundary layer, which varies with the heating and cooling cycle of the soil:

• Day (convective): the sun warms the soil → air rises → intense turbulent mixture → gradient of smoothed velocity → α greater (more mixture = steeper profile). Apparent paradox: more turbulence = α greater because the turbulent mixture transports moment between layers.
• Night (stable): radiative cooling of the soil → static stability → less turbulence → low level jet possible → α lower in the lowest layer, decoupling with upper layer.

The practical implication of daytime variation: if the campaign measures at multiple levels (LiDAR), the BORA can calculate the α of each individual record with the row-wise formula and apply this α instantaneous in extrapolation. If the campaign measures at only one level (conventional 80 m anemometer), the engineer needs to assume an α — and any single α produces systematic error in part of the hours.

Sectoral α: roughness depends on direction

A plateau site with dense vegetation to the north and caatinga open to the south will have α different to north wind (Z0 greater, α greater) and south (z0 lower, α lower). . shear_by_sector() BORA calculates the mean α for each of the 16 direction sectors, revealing whether the shearing is isotropic or systematically varying with the predominant direction.

The sectoral variation of the α impacts the projection of AEP if the dominant direction is α different from the mean. A site with predominantly east wind (sector E), with α=0.18 for this sector, but α average of 0.24 (by dense caatinga to the north), will produce a different P50 depending on which α is used. BORA uses the α adjusted to the full profile — not the average value of a pair of sensors — to maximize the representativeness.

The advantage of LiDAR: direct measurement without extrapolation

The central technical reason to use LiDAR instead of conventional mast in modern designs is precisely that. A LiDAR WindCube V2 or similar measures the configured speed gates — typically 40, 60, 80, 100, 120, 140, 160 m or more. For a cube turbine at 140 m:

• • 80 m mastThe extrapolation error is an uncertainty component of AEP that can be 3–8% σ, depending on the variability of α. It needs to be declared in the uncertainty budget (see NT.06).
• LiDAR with gate at 140 mThe uncertainty of extrapolation of height goes to zero — replaced by the uncertainty of measurement of LiDAR in that gate (minor, typically 1–2% of documented bias).

The fit of the full profile via • fit_wind_profile() It also serves as a QA: a low R² in adjusting the power law to the LiDAR profile indicates that the profile does not follow monotonous behavior — which can be low-level jet signal, thermal inversion, or ground effect that deserves research.

Extrapolation and uncertainty budget

For campaigns with mast, the height extrapolation uncertainty needs to appear explicitly in the budget. The component is proportional to the α uncertainty and the height ratio:

σ_extrapolation ≈ ln(z_hub / z_ref) × σ_α

# Example: z_hub=140, z_ref=80, σ_α=0.03 (uncertainty of 1σ in α)
ln(140/80) = ln(1.75) = 0.559
σ_extrap ≈ 0.559 × 0.03 = 0.017  → ~1.7% wind-speed uncertainty

# Propagate to AEP (V^3): multiply by ~3
σ_AEP_extrap ≈ 3 × 1.7% ≈ 5% contribution to total AEP uncertainty

This component needs to be added in square with the other components of the uncertainty budget of AEP (sensor calibration, MCP, interannual variability), according to the methodology discussed in NT.06In LiDAR campaigns with direct measurement to cube height, σ_extrap → 0 and the component is eliminated from the budget.

In practice, the uncertainty of extrapolation of height is one of the most frequently cited technical reasons by Independent Engineers to recommend LiDAR instead of mast when the cube is above 120 m. For campaigns with mast already carried out, the way to reduce this component is to document the α based on multiple pairs of sensors in the mast (if available) and compare with ERA5 or MERRA-2 for the same period — the analysis of reanalysis data at multiple pressure levels provides an independent estimate of the local wind profile, which serves as a reference for the α adopted in the extrapolation.

The long-term adjustment via MCP (discussed in NT.14) uses the speed already extrapolated to the height of the cube as a site variable. If the extrapolation to the cube is done before the MCP with an incorrect α, the α error spreads to the P50 of long-term velocity — and from there to the AEP P50 and to the DSCR of the lender. The technical chain is sensitive at first.