Michael Graham - Wind Energy Handbook

It is often useful to know the maximum gust speed that can be expected to occur in any given time interval. This is usually represented by a gust factor G , which is the ratio of the gust wind speed to the hourly mean wind speed. G is obviously a function of the turbulence intensity, and it also clearly depends on the duration of the gust – thus the gust factor for a one‐second gust will be larger than for a three‐second gust, because every three‐second gust has within it a higher one‐second gust.

Figure 2.8 Gust factors calculated from Eq. (2.46)

While it is possible to derive expressions for gust factors starting from the turbulence spectrum (Greenway 1979; ESDU 1983), an empirical expression due to Weiringa (1973) is often used because it is much simpler and agrees well with theoretical results. Accordingly, the t ‐second gust factor is given by

(2.46)

where I uis the longitudinal turbulence intensity. Figure 2.8shows the gust factors for several different turbulence intensities and gust durations calculated according to this expression.

In addition to the foregoing descriptions of the average statistical properties of the wind, it is clearly of interest to be able to estimate the long‐term extreme wind speeds that might occur at a particular site.

A probability distribution of hourly mean wind speeds such as the Weibull distribution will yield estimates of the probability of exceedance of any particular level of hourly mean wind speed. However, when used to estimate the probability of extreme winds, an accurate knowledge of the high wind speed tail of the distribution is required, and this will not be very reliable because almost all of the data that was used to fit the parameters of the distribution will have been recorded at lower wind speeds. Extrapolating the distribution to higher wind speeds cannot be relied upon to give an accurate result.

Fisher and Tippett (1928) and Gumbel (1958) have developed a theory of extreme values that is useful in this context. If a measured variable (such as hourly mean wind speed ) conforms to a particular cumulative probability distribution F ( ), that is, F ( ) → 1 as increases, then the peak values of hourly mean wind speed in a given period (a year, for example) will have a cumulative probability distribution of F N, where N is the number of independent peaks in the period. In the UK, for example, according to Cook (1982), there are about 100 independent wind speed peaks per year, corresponding to the passage of individual weather systems. Thus, if as for a Weibull distribution, the wind speed peaks in 1 year will have a cumulative probability distribution given approximately by . However, as indicated previously, this is unlikely to give accurate estimates for extreme hourly means, because the high‐wind tail of the distribution cannot be considered to be reliably known. However, Fisher and Tippett (1928) demonstrated that for any cumulative probability distribution function that converges towards unity at least exponentially (as is usually the case for wind speed distributions, including the Weibull distribution), the cumulative probability distribution function for extreme values will always tend towards an asymptotic limit

(2.47)

as the observation period increases. U ′is the most likely extreme value, or the mode of the distribution, while 1/ a represents the width or spread of the distribution and is termed the dispersion .

This makes it possible to estimate the distribution of extreme values based on a fairly limited set of measured peak values, for example, a set of measurements of the highest hourly mean wind speeds recorded during each of N storms. The N measured extremes are ranked in ascending order, and an estimate of the cumulative probability distribution function is obtained as

(2.48)

where m( ) is the rank, or position in the sequence (starting with the lowest), of the observation . Then a plot of against is used to estimate the mode U ′and dispersion 1/ a by fitting a straight line to the data points. This is the method due to Gumbel.

An illustration of the Gumbel method is provided in Figure 2.9, using some sample extreme wind data for a particular 29‐year period. The upper plot shows the sample of extreme values. The middle plot shows the estimated cumulative distribution obtained by ordering these values, with the dashed line showing the fitted distribution obtained using the Gumbel method. The lower plot shows how that fit is obtained, giving a mode U ′= 27.8 m/s with a dispersion of 2.52 m/s from the inverse slope.

Lieblein (1974) has developed a numerical technique that gives a less biased estimate of U ′and 1/ a than a simple least squares fit to a Gumbel plot.

Having made an estimate of the cumulative probability distribution of extremes F ( ), the M ‐year extreme hourly mean wind speed can be estimated as the value of corresponding to the probability of exceedance F = 1 − 1/ M .