Weibull and Rayleigh distributions

The Weibull and Rayleigh distributions are continuous probability distributions used to characterize the random data of wind speeds. These distribution functions are characterized by parameters and their purpose is to model the probability distribution of wind speeds.

To describe the wind resource, measurements of two variables such as wind speed and wind direction are recorded. Wind speed (magnitude) is measured using an anemometer. The wind speed magnitude data is processed to determine the wind potential of a given site. The data forms the probability distribution of wind speeds, which allows the calculation of the energy potential and the annual energy that can be produced by a wind turbine and a wind farm.

Weibull Distribution Function

Waloddi Weibull was a Swedish mathematician and engineer. He studied at the Royal Institute of Technology in Stockholm and received his doctorate from the University of Uppsala in 1932. His contributions are in the area of material fatigue and in statistics for his studies on the Weibull distribution. The probability distribution that bears his name was published in 1951 in the Journal of Applied Mechanics under the title Statistical Distribution Function of Wide Applicability. The Weibull distribution is immensely popular in reliability theory because it includes decreasing, constant, and increasing failure rate distributions. He received the Gold Medal of the American Society of Mechanical Engineers in 1972 and the Grand Gold Medal of the Swedish Academy of Engineering Sciences in 1978.

In most cases of energy interest, the probability distribution of wind speed approximates the Weibull probability density function. Since wind speed changes continuously, it is necessary to describe it statistically. The Weibull distribution can be used to model a wide range of applications in engineering, quality control, finance, and climatology.

Depending on the application, the Weibull distribution depends on three or two parameters that determine the shape, scale, and location of the distribution function.

v: Wind speed (m/s).
f(v): Weibull probability density function.
c: Scale factor (m/s), whose value is close to the annual mean velocity.
k: Shape factor, which characterizes the asymmetry or skewness of the function f(v).
u: Location factor.

The following cases are presented to observe how the parameters affect the Weibull distribution:

As a function of the shape factor “k” and with constant scale factor “c”

Figure 2 shows the Weibull (2-parameter) distribution for different shape parameters (k) and a constant scale parameter (c). The wind speeds are distributed with values from 0 to 20 m/s. It is observed that as “k” becomes larger, the variation of the hourly mean wind speed around the annual mean is small. For values of “k” less than or equal to 3, it can be seen that the variation is more significant.

As a function of the scale factor “c” and with constant shape factor “k”

Figure 3 shows the Weibull distribution (2 parameters) for different scale parameters (c) and a constant shape parameter (k). The wind speeds are distributed with values from 0 to 20 m/s. The value of “c” indicates how windy the place under study is, i.e. the higher the value of “c”, the higher the annual average wind speed.

As can be seen, the Weibull parameters are very important to approximate the distribution of wind speeds. The calculation of the Weibull parameters can be done in different ways depending on the wind data obtained. The greater the amount of wind speed data during a year, the more accurate the parameters can be obtained. It is advisable to take average wind speed data at short time intervals (10 minutes) during a year.

Rayleigh Distribution Function

John Rayleigh was a British mathematician and physicist. He was educated at Cambridge University, where he graduated in 1865. He was a professor of experimental physics and director of the Cavendish Laboratory in Cambridge. He became Chancellor of Cambridge University in 1908. He conducted research on light, color, electricity, resonance dynamics, and gas vibrations. In 1904 he won the Nobel Prize in Physics for his research on the density of a number of gases and for the discovery of argon, the first rare gas found in the atmosphere. He was also responsible for the establishment of electrical units of measurement.

The Rayleigh distribution is a special case of the Weibull distribution with a parameter of the form (k) equal to 2. It is simpler because it is a single-parameter function. The Rayleigh distribution is used in various applications, such as in the physical sciences to model wave behavior, wind speed, sound/light radiation, communications theory, medical imaging, and others.

v: Wind speed (m/s).
f(v): Weibull probability density function.
c: Scaling factor (m/s), value close to the annual mean velocity.

Figure 5 shows the Rayleigh distribution for different scale parameters (c). The wind speeds are distributed with values from 0 to 20 m/s. It can be observed that this distribution is more suitable when the scale parameters are higher than 5 m/s. Above this value, the difference between the Rayleigh distributions is not significantly different.

When should I use the Weibull or Rayleigh distribution?

The Weibull distribution can be applied to any value of the annual mean wind speed in the study area. A good database of wind speed data should be available for short time intervals, usually in 10-minute bands (ten-minute measurements) for a one-year study. If the study area has good wind resource conditions, the Rayleigh distribution will adequately describe the wind speed of the area.

Conversely, the Rayleigh distribution is less reliable if the mean annual wind speed of the study area does not exceed 4.5 m/s. In addition, low turbulence and low wind variability must be present in the study area for the wind speed distribution to correctly fit the Rayleigh distribution.

Finally, the calculation of the parameters of the Weibull and Rayleigh distributions is very important to obtain an adequate approximation of the probability distribution of the wind speeds. The estimation of the parameters can be found in the following article:

Calculation by the method of least squares

Reference

[1] Villarrubia López M. (2012). Ingeniería de la Energía Eólica. Facultad de Física, Universidad de Barcelona. Alfaomega Grupo Editor, S.A. de C.V.

[2] Munteanu, I., Bratcu, A. I., Cutululis, N.-A., & Ceanga, E. (2010). Optimal control of wind energy systems: Towards a global approach. Springer.