 EXAMPLES
EXTREME WIND

# Example for extreme winds

Extreme winds can be calculated with the following methods:

• Block maxima method
• Block maxima:
• Annual maxima (with/without resampling)
• Monthly maxima
• Distributions:
• Generalized extreme value distribution (GEV)
• Gumbel distribution
• Weibull method
• Weibull tail distribution
• Weibull extended tail distribution
• Weibull-Gumbel distribution

For the annual maxima, WIND DATA SUiTE has developed a resampling method, which allows to resolve also several similar big maxima within one year.

Return levels can be calculated up to a return period of 1000 years.
All quantities will be calculated which are necessary for the statistical evaluations and examinations, as probability plots, quantile-quantile plots, empirical and calculated cumulative distribution functions and probability density functions, asymmetric (profile log-likelihood function) upper and lower bounds of the 95%-confidence intervals of the return levels, and goodness-of-fit tests (chi-square test, Kolmogorov-Smirnov test).
Extreme winds according to DIBt 2012 and according to IEC-61400-1 will be reported for comparison with the calculated values.

WDS can perform all calculations also wind direction sector specifically (for arbitrary wind direction roses) and seasonally (separably for all 12 months from January to December).

The following example shows extreme wind calculations with the diurnal maximal wind peak values of the weather station Bremerhaven at 10 m height above ground level within the time range from 1980 to 2015 (Data Basis: Deutscher Wetterdienst, own elements added). The extreme winds have been calculated with the following three methods:

• Annual block maxima with resampling, Gumbel distribution
• Annual block maxima with resampling, generalized extreme value distribution (GEV)
• Weibull method, Weibull-Gumbel distribution

Fig.1 Time series of the diurnal maximal wind peak values (clipping). The absolute maximum of the diurnal maximal wind peak values occurs with 42.2 m/s on 1993/01/13.

Fig.2 (Probability Plot) shows the calculated cumulative distribution functions of the Gumbel distribution of the annual maxima (GCDF Gumbel), of the generalized extreme value distribution of the annual maxima (GCDF GEV), and of the Weibull distribution of the time series (WCDF Weibull), depicted versus the respective empirical cumulative distribution functions (ECDF) of the measured data. Additionally depicted is the (y=x)-line.

Fig.3 (Quantile-Quantile Plot) shows the return levels of the return periods of the annual maxima, calculated with the Gumbel distribution (uT Gumbel), the generalized extreme value distribution (uT GEV), and the Weibull-Gumbel distribution (uT Weibull-Gumbel. This for comparison only since Weibull-Gumbel is not based on annual maxima but of the Weibull distribution of all measured data), depicted versus the annual maxima at the respective return periods. Additionally depicted is the (y=x)-line.

Fig.4 (Return Level Plot) shows the return levels of the return periods (2 to 100 years), calculated with the Gumbel distribution (uT Gumbel), the generalized extreme value distribution (uT GEV), and the Weibull-Gumbel distribution (uT Weibull-Gumbel), and their respective upper (uT+CI) and lower (uT-CI) asymmetric bounds of the 95%-confidence interval (for the Weibull distribution, this statistic is not possible), depicted versus the return periods.

Gain of insight with WDS: The maximal measured wind peak value of the maximal diurnal wind peak values has a wind speed of 42.2 m/s and occurs on 1993/01/13 (see Fig.1). This maximal measured wind peak value is greater than the return level of 100 years with the GEV distribution (41.5 m/s, but still within the 95%-confidence interval) as well as with the Weibull-Gumbel distribution (41.4 m/s). The shape parameter of the GEV distribution is negative, so the GEV distribution is a Weibull-type distribution and not a Fréchet-type distribution. With the Gumbel distribution, the wind peak value of 42.2 m/s has a return period of approx. 35 years (see Fig.4).
Fig.3 shows that the wind peak value of 42.2 m/s is not an outlier relating to the Gumbel distribution, but it is one relating to the GEV distribution and the Weibull-Gumbel distribution.
Fig.2 shows that the deviations from the empirical cumulative distribution are in total smaller with the GEV distribution than with the Gumbel distribution, but the sequence of the wind peak values in the time series (see Fig.1) confirms the assumption that the wind peak value of 42.2 m/s does not represent an outlier.
So here, the Gumbel distribution is the most realistic one. This is also confirmed by an additional calculation after eliminating the wind peak value of 42.2 m/s: This indeed then yields a maximal measured wind peak value of just 39.0 m/s only, but the distributions and statistical relations among each other remain the same in essence, except of somewhat smaller return levels (notedly smaller with GEV). Fig.1:
Time series of the diurnal maximal wind peak values (clipping) Fig.2:
Calculated cumulative distribution functions versus empirical cumulative distribution functions Fig.3:
Calculated return levels of the return periods of the annual maxima versus the annual maxima Fig.4:
Return levels of the return periods