Weibull distribution and wind power density

Wind speeds are not normally distributed; they are skewed towards lower values, with a long tail extending to higher speeds. This is because wind speed cannot be negative, and there is a physical limit to how strong winds can become in a given climate, while calm conditions (zero speed) are frequent. This inherent asymmetry means that a simple average speed alone does not fully describe the wind resource.

For example, a location might have an average wind speed of 6 m/s, but this could be achieved through many hours of 2 m/s winds and fewer hours of 15 m/s winds, or through a more consistent distribution around 6 m/s. The distribution's shape is crucial for understanding how often certain speeds occur, which directly impacts energy production or operational windows.

Wind resource assessment, particularly for wind energy projects, relies on understanding this distribution. The frequency of occurrence of various wind speeds determines the amount of energy that can be extracted. A distribution with a higher proportion of moderate-to-strong winds, even if the average is the same, will yield significantly more power than one dominated by light winds.

The Weibull distribution is a two-parameter probability distribution widely used to model wind speed data. It is defined by its scale parameter (c) and shape parameter (k).

• The scale parameter (c), measured in units of speed (e.g., m/s), is related to the average wind speed. A higher 'c' value indicates higher average wind speeds.
• The shape parameter (k) is dimensionless and describes the shape of the distribution. A 'k' value of approximately 2, commonly cited for many wind regimes, results in a Rayleigh distribution, a special case of the Weibull. Lower 'k' values (e.g., 1.5) indicate a wider spread of speeds, with more calm periods and more extreme gusts. Higher 'k' values (e.g., 3.0) suggest a narrower, more consistent range of speeds around the mean.

The probability density function f(v) for a wind speed v is given by:

f(v) = (k/c) * (v/c)^(k-1) * exp(-(v/c)^k)

This function describes the likelihood of observing a particular wind speed. For instance, a site with c = 7 m/s and k = 2.0 will have a different wind speed profile than a site with c = 7 m/s and k = 2.8, even though their average speeds might be similar. The former would exhibit a broader range of speeds, including more periods of both very low and very high wind, while the latter would show a more concentrated distribution around the mean.

The Weibull parameters (c and k) are typically derived from historical wind speed observations. This involves fitting the observed frequency distribution of wind speeds to the theoretical Weibull curve. Various methods exist for this fitting, including the method of moments, the maximum likelihood method, and graphical methods.

For example, if a year of hourly wind speed data is available, the data can be binned into speed classes (e.g., 0-1 m/s, 1-2 m/s, etc.). The frequency of observations in each bin forms the empirical distribution. Software tools then use algorithms to find the 'c' and 'k' values that best represent this empirical distribution with a theoretical Weibull curve.

It is important that the observed data used for fitting is representative of the long-term wind climate of the site. Short-term measurements or data from atypical years can lead to inaccurate Weibull parameters and, consequently, incorrect assessments of the wind resource. Commonly, at least one year of continuous, high-quality data is recommended for a robust fit, with longer periods (e.g., 3-5 years) preferred for critical applications like wind farm development.

While the Weibull distribution provides a detailed picture, the mean wind speed remains a fundamental metric. For a Weibull distribution, the mean wind speed (v_mean) can be calculated from its parameters c and k using the Gamma function Γ:

v_mean = c * Γ(1 + 1/k)

This relationship allows the mean speed to be estimated directly from the fitted Weibull parameters without needing to average all individual speed observations. However, for wind power, the relationship is not linear; it follows the cube law. The power available in the wind is proportional to the cube of the wind speed (v^3). This means that even small increases in wind speed lead to significant increases in available power.

Specifically, the kinetic energy of a parcel of air is 0.5 * m * v^2. If that parcel passes through an area A in time t, the mass m is rho * A * v * t, where rho is air density. Substituting this into the kinetic energy equation and dividing by t gives the power P:

P = 0.5 * rho * A * v^3

This cube law highlights why accurate wind speed distribution, especially at higher speeds, is critical for wind energy assessment. A site with a slightly higher mean speed or a 'k' value that skews the distribution towards higher speeds can be far more productive.

Wind power density is a measure of the available wind power per unit of swept area, typically expressed in Watts per square metre (W/m²). It is a key metric for assessing the quality of a wind resource, as it normalises the power by area, allowing for direct comparison between sites.

The average wind power density (P_avg) is calculated by integrating the cube of the wind speed over its probability distribution, multiplied by 0.5 * rho (where rho is air density, commonly taken as 1.225 kg/m³ at sea level and 15°C):

P_avg = 0.5 * rho * ∫(v^3 * f(v)) dv

For a Weibull distribution, this simplifies to:

P_avg = 0.5 * rho * c^3 * Γ(1 + 3/k)

Worked Example: Consider a site with Weibull parameters c = 7.5 m/s and k = 2.2. Assuming standard air density rho = 1.225 kg/m³.

First, calculate the Gamma function terms. Γ(1 + 1/2.2) = Γ(1.4545) ≈ 0.893 and Γ(1 + 3/2.2) = Γ(2.3636) ≈ 1.196.

Mean speed: v_mean = 7.5 * 0.893 = 6.6975 m/s.

Power density: P_avg = 0.5 * 1.225 * (7.5)^3 * 1.196 = 0.5 * 1.225 * 421.875 * 1.196 ≈ 308.2 W/m².

This value represents the average power that could be captured by a perfectly efficient turbine with a swept area of 1 m² at this location and height. Typical Irish coastal sites can exceed 500 W/m² at 80 m, while inland sites might be 200-400 W/m².

The Weibull distribution can also be used to estimate the frequency of calm hours, defined as periods when wind speed falls below a certain threshold, often the cut-in speed of a wind turbine (typically 3-4 m/s). This is crucial for energy yield assessments and operational planning.

The cumulative distribution function (CDF) F(v) gives the probability that the wind speed is less than or equal to v:

F(v) = 1 - exp(-(v/c)^k)

To find the percentage of calm hours below a threshold v_threshold, one simply calculates F(v_threshold). For example, if a turbine's cut-in speed is 3 m/s, and the Weibull parameters are c = 7.5 m/s and k = 2.2:

F(3) = 1 - exp(-(3/7.5)^2.2) = 1 - exp(-(0.4)^2.2) = 1 - exp(-0.126) ≈ 1 - 0.881 = 0.119

This indicates that approximately 11.9% of the time, the wind speed will be below 3 m/s at this site. This translates to about 0.119 * 8760 hours/year ≈ 1042 calm hours annually.

Understanding calm hour frequency helps assess the reliability of a wind resource and informs decisions on hybrid energy systems or storage solutions. Sites with very low 'k' values tend to have more calm hours and more high-wind hours, reflecting a less consistent wind regime.

While effective for describing the bulk of wind speed frequencies, the Weibull distribution has limitations, particularly when dealing with extreme wind events. It tends to underestimate the frequency of very rare, high-magnitude gusts and storm events.

Extreme value theory, using distributions like the Gumbel or Generalised Extreme Value (GEV) distribution, is more appropriate for analysing and predicting the return periods of extreme winds. These are critical for structural design and safety assessments, where a 50-year or 100-year return period gust speed is required.

Furthermore, the Weibull distribution assumes stationarity in the wind climate, meaning the underlying statistical properties do not change over time. While suitable for long-term averages, it does not capture short-term variability, seasonality, or the impact of climate change on wind patterns.

For operational decision-making, such as crane lifts or drone flights, real-time forecasts and exceedance probabilities from ensemble models are more relevant than long-term climatological distributions. The Wind Agent's exceedance fan and ensemble plume are designed to address these short-term, probabilistic needs.