Weibull Distribution (Two Parameter)

The Weibull distribution is extensively used in reliability engineering, it has also been applied to many other disciplines including medicine, finance and climatology.  In the renewable energy industry it is used as a model for wind speed.

The distribution is named after the Swedish engineer Wallodi Weibull.

Profile

Parameters

Parameter	Description	Characteristics
scale	defines the range and practical maximum. Also known as the characteristic life.	A float > -∞ and < ∞
shape	Determines the profile of the distribution.	A float > 0

Range

The range is from greater than zero to positive infinity.  However, in practice the right tail is classed as thin/light or bounded.

Functions

Properties

Random Number Generation

Random number generation (referred to as r) for a Weibull distribution can be performed by transforming a continuous uniform variable in the range 0 to 1 (referred to as u) with the distribution's inverse probability function:

r=g(u)

which can be reduced to:

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

r = scale * (-log(rnd))^(1/shape)

Parameter Estimation - Maximum Likelihood

The maximum likelihood equations can be expressed in the form:

These can be solved using iterative methods such as successive bisection to obtain the shape and scale factors.

Parameter Estimation - Linear Regression

The cumulative probability function (F(x)) can be re-arranged in a form such that linear regression can be applied:

The rearranged F(x) looks like this:

F(x) can be expressed as:

Where n is the rank of the value: x (the smallest value will be rank 1 and so on) and N is the number of values in the dataset.

The shape factor drops directly out of the regression equation, whilst the scale factor has to be derived from the intercept:

As always, the process is better illustrated with an example.  When working with wind data, it is desirable to have at least a full year's data, say, in the form of hourly observations.  A year's worth of data is approximately 8,000 values, for illustative purposes a small subset has been contrived from 2 years of observations at a location in the south of Engand.

Example Calculation

To obtain the x and y columns needed for the linear regression calculation, the wind speed observations are sorted into ascending order.  This allows the values to be ranked and F(x) obtained from the median ranks formula.

	X			Y
Wind Speed (m/s)	ln(speed)	Rank	F(Speed)	ln(ln(1/(1-F(speed)))
2.0	0.693	3554	0.210	-1.442
3.0	1.099	6817	0.404	-0.659
4.0	1.386	10163	0.602	-0.081
5.0	1.609	12955	0.767	0.379
6.0	1.792	14853	0.880	0.753
7.0	1.946	15848	0.939	1.030
8.0	2.079	16361	0.969	1.251
9.0	2.197	16624	0.985	1.439
10.0	2.303	16763	0.993	1.616
11.0	2.398	16821	0.996	1.755
12.0	2.485	16860	0.999	1.970
13.0	2565	16869	0.999	2.122
14.0	2.639	16872	0.999	2.276

Many software packages can perform regression analysis.  A convenient way of doing it is to use the XY plot in MS Excel.  If a linear trend line is added to the plot the, estimates of the slope and intercept can be obtained directly.  The plot below is based on the X and Y columns from the above table.

The the slope is the shape factor (1.846 in this example), The scale factor is obtained from:

Scale = exp(-intercept/slope) = exp(2.6436/1.846) = exp(1.432) = 4.187

For clarity, the above calculations have been presented with the numbers truncated to 3 decimal places.

Seasonality

The average wind speed at a given location is a function of climate and terrain.  There can be significant seasonal variations in wind speed.  In general, the shape factor is more or less constant throughout the year, but there can large changes in the scale factor and the average wind speed.  The graph below shows the shape and scale factors together with the mean for a relatively benign location like Southern England where the average wind speed at 10m is around 4 m/s.

In a harsh environment like the North Sea, seasonality is much greater as illustrated by the graph below:

The graphs are not comparable as the observation height for the North Sea example is higher, probably more than 30m above the surface.

Shape  Factor

For many locations the shape factor is close to 2.0.  This makes the Rayleigh Distribution (a special case of the Weibull distribution where the shape factor is set to 2) a reasonable choice for first pass analysis of a given location.  From a limited study of several locations, it seems that offshore and flat terrains where the the average wind speed is high, have shape factors of 2.0 or greater, whilst complex terrains where the average wind speed is low have shape factors of less than 2.0.

The Weibull Distribution as a Wind Speed Model

The graphs below show the observed distribution of wind speed and the distribution suggested by the Weighbull model.  The parameters are:

	South of England	North Sea
Obs. Height	10m	>30m
Shape	1.85	2.17
Scale	4.18	10.59
Mean	3.72	9.38

In both cases, there is a good agreement between the model and the observations.

Missing Data

 Weather may not include low wind speeds (say, 0 - 2 m/s).  For example in Metar reports, zero might mean that the wind speed is less than 2 knots.  Where this is known to be the case, the dataset can be ranked as described above, but only those observations where the wind speed is greater than zero are included in the regression.

Comparison of Locations

The distribution of wind speed can vary withing a small area.  As a result, care must be taken when choosing data on which to base a model.  I live in a sheltered urban area where the average wind speed maybe less than 1 m/s, on the hills to the north, it might be around 5 m/s.