Extrapolating to hub height: methods and uncertainties

Wind turbine hub heights have steadily increased over recent decades, with modern onshore turbines commonly featuring hub heights between 80 m and 120 m, and offshore turbines often exceeding 150 m. Historically, meteorological masts (met masts) used for wind resource assessment were typically 60 m to 80 m tall, with anemometers placed at several levels up to the mast's maximum height. This creates a common scenario where the highest measured wind speed is still significantly below the proposed turbine hub height.

Extrapolation is therefore necessary to estimate the wind speed at the hub. This is not a simple linear scaling; the relationship between wind speed and height is non-linear and influenced by surface roughness, atmospheric stability, and terrain. Direct measurement at hub height is ideal but often impractical or uneconomical during the prospecting and development phases of a wind farm. Consequently, models like the power law and log law are employed to bridge this vertical gap, introducing a degree of uncertainty into the estimated wind resource and ultimately the projected energy yield.

Two primary models are used for vertical wind profile extrapolation: the power law and the log law.

Power Law

The power law is an empirical relationship, often favoured for its simplicity. It is expressed as:

S(h) = S₀ * (h / h₀)^α

where S(h) is the speed at height h, S₀ is the speed at a reference height h₀, and α is the shear exponent. The α value varies significantly with surface roughness and atmospheric stability. A commonly cited value for neutral atmospheric conditions over open terrain is α = 1/7 (approximately 0.14). However, α can range from less than 0.1 over smooth surfaces to over 0.3 in very rough terrain or stable atmospheric conditions.

Log Law

The log law (or logarithmic law) is derived from fluid dynamics principles and is theoretically more robust, especially in the surface layer (the lowest 10-100 m). It is expressed as:

S(h) = (u* / κ) * ln(h / z₀)

where u* is the friction velocity, κ (kappa) is the von Kármán constant (approximately 0.4), and z₀ is the roughness length. The roughness length is a physical characteristic of the surface, representing the height at which the wind speed theoretically becomes zero. Typical values range from 0.0002 m for calm water to 1 m for urban areas.

While the log law requires knowledge of u* and z₀ (which can be derived from measurements at two heights), it is generally considered more accurate for neutral stability conditions within the surface layer. For extrapolation over larger vertical distances, especially into the atmospheric boundary layer where stability effects become significant, both models have limitations.

Both the power law and log law in their basic forms assume neutral atmospheric stability. This condition occurs when the vertical temperature gradient is adiabatic, meaning there is no buoyancy-driven turbulence (convection) or suppression of turbulence (inversion). Neutral conditions are most common during strong winds or overcast skies.

However, the atmosphere is rarely perfectly neutral. Atmospheric stability significantly influences the wind shear:

• Unstable conditions (e.g., sunny days with light winds) lead to convective turbulence, mixing momentum vertically and reducing shear (lower α). The wind profile becomes flatter.
• Stable conditions (e.g., clear nights with light winds) suppress turbulence, leading to strong shear (higher α) and a steeper wind profile, often with a low-level jet.

To account for stability, modifications are applied to the log law, such as the Monin-Obukhov similarity theory. This involves adding a stability correction term Ψ(h/L) to the log law equation:

S(h) = (u* / κ) * [ln(h / z₀) - Ψ(h/L)]

where L is the Obukhov length, a measure of atmospheric stability. Calculating L requires measurements of temperature and humidity gradients, which are not always available from standard met masts.

Without stability corrections, extrapolations can significantly over- or underestimate wind speeds at hub height, particularly during periods of strong stability or instability. For example, under stable conditions, a simple power law might underestimate the wind speed at hub height, leading to an underprediction of energy yield.

When direct measurements are available at two different heights, h₁ and h₂, with corresponding wind speeds S₁ and S₂, the shear exponent α for the power law can be empirically determined. Rearranging the power law equation, we get:

α = ln(S₂ / S₁) / ln(h₂ / h₁)

Let's consider an example:

• Anemometer at h₁ = 40 m measures S₁ = 7.0 m/s.
• Anemometer at h₂ = 60 m measures S₂ = 7.8 m/s.

First, calculate the ratio of speeds and heights:

S₂ / S₁ = 7.8 / 7.0 ≈ 1.114 h₂ / h₁ = 60 / 40 = 1.5

Now, compute the natural logarithms:

ln(1.114) ≈ 0.108 ln(1.5) ≈ 0.405

Finally, calculate α:

α = 0.108 / 0.405 ≈ 0.267

This calculated α value of 0.267 is relatively high, suggesting a rougher surface or stable atmospheric conditions at the time of measurement. Once α is known, it can be used to extrapolate to a target hub height, for instance, h_hub = 100 m:

S_hub = S₂ * (h_hub / h₂)^α S_hub = 7.8 * (100 / 60)^0.267 S_hub = 7.8 * (1.667)^0.267 S_hub = 7.8 * 1.144 ≈ 8.92 m/s

This method provides a site-specific α but assumes α remains constant across the entire extrapolation range and over time, which is a simplification.

The accuracy of hub height extrapolation diminishes significantly as the vertical distance between the highest measurement and the target hub height increases. This is due to several factors:

1. Assumptions of the models: Both power law and log law are simplifications of complex atmospheric processes. They perform best within the surface layer and under neutral conditions. Extrapolating far beyond the measured range pushes these models beyond their ideal operating conditions.
2. Variability of shear: The shear exponent α or the roughness length z₀ are not constant. They change with wind speed, wind direction, atmospheric stability, and even season. Using a single, averaged α for long-term energy yield assessments can lead to systematic errors.
3. Terrain effects: Complex terrain (hills, valleys, coastlines) introduces localised flow phenomena such as speed-up effects, recirculation zones, and wakes, which are not well-captured by simple 1D vertical profiles. These effects can significantly alter the wind profile at hub height compared to what a simple extrapolation might suggest.
4. Measurement errors: Errors in the reference wind speed measurements (e.g., due to anemometer calibration, icing, or mast interference) propagate and are amplified during extrapolation.

For example, extrapolating from 60 m to 150 m means extending the profile by 90 m. If the assumed α is off by just 0.05 (e.g., 0.15 instead of 0.20), the estimated wind speed at 150 m can differ by several percentage points, leading to substantial errors in annual energy production (AEP) estimates. Industry guidance commonly attributes an uncertainty of 1–3% to extrapolation alone, potentially higher for large extrapolation ratios.

To reduce the uncertainties associated with extrapolation, Lidar (Light Detection and Ranging) technology has become increasingly prevalent in wind resource assessment. Lidar devices emit laser pulses and measure the Doppler shift of the backscattered light from aerosols in the atmosphere to determine wind speed and direction at multiple heights simultaneously.

Ground-based Lidars can measure wind speeds from as low as 10 m up to 200–300 m or more, covering the entire range of typical turbine hub heights. This capability offers several advantages:

• Direct measurement at hub height: Lidar can provide actual measurements at the proposed hub height, eliminating or significantly reducing the need for extrapolation.
• Detailed vertical profiles: Lidar systems provide high-resolution wind profiles, revealing the true shape of the shear and any complex features like low-level jets that might be missed by a few discrete anemometer levels.
• Validation of extrapolation models: Lidar data can be used to validate the shear exponents or roughness lengths derived from met mast data, or to refine the parameters used in numerical wind flow models.
• Reduced mast interference: Unlike anemometers on a mast, Lidar measurements are generally free from mast-induced flow distortion.

While Lidar systems are more expensive than traditional met masts, their ability to provide accurate, hub-height wind data over a wide range of heights often justifies the investment by reducing project financing risk and improving the confidence in energy yield predictions. They are particularly valuable for sites with complex terrain or very tall turbines.

The accuracy of hub height wind speed extrapolation has a direct and significant impact on the estimated Annual Energy Production (AEP) of a wind farm. Wind turbine power output is highly sensitive to wind speed, typically following a cubic relationship (Power ∝ Speed³).

Consider a turbine with a rated power of 3 MW. If the extrapolated hub height wind speed is underestimated by just 5% (e.g., 8.0 m/s instead of 8.4 m/s), the estimated power output could be underestimated by approximately 15% (since 8.0³/8.4³ ≈ 0.857). This seemingly small error in wind speed can translate into a substantial difference in AEP and, consequently, in project revenue and financial viability.

Worked Example: Assume a turbine operates at an average wind speed of 8.0 m/s at hub height. Its power curve indicates an output of 2.0 MW at this speed. If an inaccurate extrapolation leads to an estimated average speed of 7.6 m/s (a 5% underestimate), the power curve might suggest an output of only 1.7 MW. Over a year (8760 hours), this difference amounts to:

AEP_true = 2.0 MW * 8760 h = 17,520 MWh AEP_estimated = 1.7 MW * 8760 h = 14,892 MWh

This represents a difference of 2,628 MWh annually, which at a conservative electricity price of €80/MWh, translates to over €210,000 in lost revenue per turbine per year. For a multi-turbine wind farm, this error quickly escalates into millions of Euros.

Accurate extrapolation, supported by robust measurement campaigns and advanced modelling, is therefore critical for de-risking wind energy projects and ensuring realistic financial projections. The Wind Agent's use of modelled data at multiple heights aims to provide a more reliable basis for understanding wind at working levels.