Rayleigh Distribution

The range is determined by the scale parameter and the skewness and kurtosis are constants.  It is equivalent to a Weibull distribution with these parameters:

Scale weibull	√2* ScaleRayleigh
Shape weibull	2

The distribution is used in physics where it is used in the study of various types or radiation, such as sound and light and in signal processing.  The example below is based on its use as a model for wind speed and is often applied to wind driven electrical generation.

Profile

Parameters

Parameter	Description	Characteristics
scale	Minimum Value	A float > -∞ and < ∞

Range

The random numbers are greater zero to positive infinity.

Functions

As the scale factor is also the mode, the equations can also be written in terms of the mode.

Properties

Example

The Rayleigh distribution is used as a model for wind speed.  The model describes the distribution of wind speed over the period of a year.  The two graphics below show the distribution of daily average wind speed, one onshore and the other at an exposed offshore location, in both cases the quality of the fit is good.  This type of analysis is used for estimating the energy recovery from a wind turbine.

The following pages use the Rayleigh distribution in a sample calculation to estimate energy recovery from the wind:

Electrical Power from the Wind

Description of the data and explanation of the calculations

Distribution of wind speed by hours per year

During the curve fitting for this example, an number of points emerged.  First, the quality of the fit was dependent on the range of wind speed at a given location.  The quality of the fit was  lower for locations where the range was from say, 5 knots to 35 knots as would be expected in an offshore location, than where it was from zero to 25 knots, as in the onshore example above.  The effect of this was that the model would underestimate the energy available to a wind turbine.  This is apparent in the offshore example, where the model overestimates the number of calm days and under estimates the number of days with typical operating conditions.

Where the Rayleigh distribution is a poor model for a given location, it may be appropriate to fit the data to a Weibull distribution.  This is effectively changing the shape factor from an independent variable to a dependent one (the Rayleigh distribution is a Weibull one with the shape factor set to two).

Secondly, the nature of the data used for the curve fit affected results.  The "best" fits were obtained with daily averages over a one year period.  Un-averaged hourly data gave variable results.  This suggests that it is necessary to understand the averaging process.  A simple arithmetic mean would underestimate the energy available, thus it may be appropriate to use a form of geometric average which gives weight to the higher wind speeds (energy recovery is proportional to the square of the wind speed).

The height of the observations is also important.  The standard height for meteorological observations is 10m, whilst wind turbines are often mounted on towers which are 30 - 80 metres high to gain advantage of reduced friction with the earth's surface.

Finally, the best fit (i.e. lowest chi squared score) was derived using the matching moment equation for the scale factor (the mode), suggesting that the average is a reasonable estimator of the modal value.

Whilst the distribution of wind speed over a year is useful for determining the total energy available, it may also be necessary to take account of the variation over the year.  The graphic below is a box and whisker plot showing the monthly variation for the Offshore Nantucket example.

Parameter Estimation - Matching Moments

The matching moment equation for the scale factor is derived by rearranging the equation for the mean:

An attractive feature of the Rayleigh distribution is that the mode can be estimated from the mean.

Random Number Generation

Random number generation (referred to as r) for a Rayleigh distribution can be performed by transforming a continuous uniform variable in the range (0,1) i.e. greater than zero and less than one (referred to as u) with the distribution's inverse probability function:

r=g(u)

Using Basic style code, the function would be similar to:

r = sqr(-2 * scale  * scale * log(u))

Page Updated: 04-Apr-2008

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