15. Wind statistics and the Weibull distribution.

It is a matter of common observation that the wind is not steady and in order to calculate the mean power delivered by a wind turbine from its power curve, it is necessary to know the probability density distribution of the wind speed. For those unfamiliar with statistics, this is simply the distribution of the proportion of time spent by the wind within narrow bands of wind speed.

As an example, the figure below shows a histogram of the frequency of different hourly wind speeds at 1 knot intervals (0.52 metres/second) from a weather station in Plymouth, UK. The data consists of three years of observations.

The overall mean wind speed is 10.2 knots (5.26 metres/second) and, because the histogram is skewed, it should be noted that this is not the most commonly occuring wind speed which is somewhat less than this at around 5 knots (2.58 metres/second) .

The basic measure of the unsteadiness of the wind is the standard deviation (or root mean square) of the speed variations. For the above data, the standard deviation is 6.28 knots (3.24 metres/second) so that the ratio of the standard deviation to the mean speed is 0.62 - and this almost certainly representative of the unsteadiness of the wind everywhere in the UK. However, the value used in the calculation of mean power is normally set at 52% which corresponds to a particular form of the wind distribution known as the Rayleigh distribution - see below.

In order to calculate the mean power from a wind turbine over a range of mean wind speeds, a generalised expression is needed for the probability density distribution. An expression which gives a good fit to wind data is known as the Weibull distribution. In non-dimensional form, this can be written as

In order to fit wind speed data to this equation, we need a value for the shape factor k. This is often obtained by some form of fitting procedure to the measured probability distribution but this is unnecessarily complicated. One of the simplest measures of the unsteady component of a random variable is the standard deviation or root mean square of the variable (σ) and, for the Weibull distribution, this can be shown to be given by

The figure below shows the variation of the non-dimensional standard deviation with the shape factor, k.

Thus, knowing the standard deviation, the shape factor k can be obtained. The simplest way of doing this is through a simple curve fitting procedure. A good fit to the relationship which is indistiguishable from the exact relationship on the scale of the above graph is k versus standard deviation fit

Having determined k, it is necessary to compute the value of the Gamma function for (1+1/k) so that it can be used in the equation for the probability density distribution. The Gamma function is rather complex but we need to know its values only over a rather narrow range of values of (1+1/k) from 1.173 to 2.014 corresponding to wind standard deviations of 20% and 100% respectively of the mean speed. The figure below shows the exact Gamma function compared with a simple polynomial fit.

The figure below shows a comparison between the Plymouth Mountbatten weather station and the Weibull equation for the measured standard deviation of σ/U = 0.62 which, from the results shown in the above figure, gives a shape factor, k, of 1.667 and a value of Γ(1+1/k)=Γ(1.6)=0.8943 (cf. exact value=0.8935).

A further check on the goodness of fit can be obtained from the third moment of the probability density distribution. For the Weibull distribution, it can be shown that

Rather fortuitously, this is exactly the value obtained from the experimental data.

A particular form of the Weibull distribution is referred to as the Rayleigh distribution and occurs when k=2. This is equivalent to a standard deviation of 52% of the mean wind speed and is taken as the default value in the WindPower program. However, in the program, this ratio can easily be scrolled from 20% to 100% and the effect of this on mean power instantly displayed.

It should be said that the present procedure for obtaining the Weibull shape factor, k, is a lot simpler and more direct than the rather complex non-linear fitting procedures often used and, moreover, it enables k to be obtained directly from the standard deviation - which is an easy quantity to measure.

Wind probability density distribution in urban areas.

Whilst the standard deviation of the wind speed fluctuations relative to the mean wind speed in rural areas or offshore sites is likely to be reasonably constant, the effect of the buildings in an urban area may have a strong influence on the characteristics of the wind as discussed in webpage 6. It is not only the mean wind speed that will be reduced but the standard deviation of the wind speed fluctuations may be increased too.

The two figures below show the probability distributions of wind speed in two urban sites. In the first figure, a small wind turbine is mounted next to a building with a screen of nearby trees. The mean wind speed is only 2.14 metres/second and it is a very turbulent site with a wind speed standard deviation that is 87% of the mean wind speed. The Weibull shape factor is 1.149. Clearly, this is not a good site on which to position a wind turbine.

The second example is a small wind turbine mounted on the roof of a seven storey apartment block. Although the turbine is about 30 metres above ground level, the mean wind speed is still only 4.46 metres/second which is too low for the turbine to be a really economic proposition. In a rural site in the UK, the wind speed at this height would be significantly higher - probably in the 7-8 metres/second. On the other hand, the standard deviation of the wind speed variations has now fallen to 59% of the mean wind speed and this is now quite close to the standard deviation of 62% used as the default value in the WindPower program.

Southorn Court

Reference The data used in the above figures were taken from a report Warwick Wind Trials Project available from www.warwickwindtrials.org.uk. The data were re-analysed using the method outlined in the first part of this webpage. Permission to use the two small photographs was kindly given by Encraft - www.encraft.co.uk.

Power output profile of wind turbines.

One of the problems of wind power is that the power output is highly variable and reliant on the vagaries of the weather. However, whilst the power output from a turbine cannot be predicted for a particular time, it is possible to estimate the proportion of time that the turbine produces different levels of power. The figure below is a sketch of a power curve. If we wanted to know the proportion of time that power would be produced between two limits P_u and P_l, it would correspond to the percentage of time that the wind speed lay in range u_u and u_l. For the Weibull distribution, it can be shown that the proportion of time T_i that the wind speed lies between two such limits is

where U_m is the mean wind speed.

One of the more obvious points of interest about wind power is the time when the wind speed is below the cut-in speed and so no power is being produced at all. The zero-power time, T₀, can be obtained from the above equation with u_l=0 and u_u equal to the cut-in wind speed u_cutin.

In the WindPower program, these formulae are used through a graphical procedure to estimate the percentage of time that a wind turbine will produce power between adjustable limits. The default case is the percentage of time that the turbine produces no power because the wind speed is below the cut-in speed.