Windenergie 2 - Extreme Wind Statistics
29 October 2025, Po Wen Cheng
Questions
- Which kind of lidar can measure several distances at the same time? - pulsed lidar
- Which velocity component of the wind can the lidar measure? - line of sight wind speed
- What is the sign of high quality lidar measurement? - high carrier to noise ratio
- Which Type of lidar scan you would use for a power curve validation? - VAD scan
- What is the minimum number of line of sight velocities neccessary to reconstruct the 3 domensional wind speed vektor? - 3, but more in practice
- Mean value and median are the same - no
- How to optain Cumulative Distribution Function from Probability Density Function? - through integration
- Do you know what skewness and kurtosis of a distribution function means?
- skewness: is the distribution symmetrical or not - 0 is perfectly symmetrical
- kurtosis: is the how much value is distributed towards the middle or to the tails - gaussian distributian has kurtosis of 3. Higher that 3 means larger tail than gaussian distribution, extremes are more likely to happen
- Where in the hurricane can you find the maximum wind speed? - on the wall of the hurricane, intense updrafts
- If an extreme wind speed of 30 m/s has a probability of occurance of 0.01% every year and the extreme wind speed has occured in the last four years? - The chance that it will occur next year is 0.01% (assume the events are independent of each other)
Why consider extreme values?
Extreme value analyis does not describe the usual behaviour of a stochastic phenomena, but the unusual and rarely observerd events, uses are:
- wind speed in wind energy
- wave height in ocean engineering
- floods in hydraulics engineering
- earthquakes in structural engineering
- ...
Return period
$$ p_e = P(X > x) = \frac{1}{T} $$ $$ p_{ne} = P(X \leq x) = 1 - \frac{1}{T} $$
Return period of 100 years: Event occurs once on average in the period of 100 years. $p_e = 0.01$
Problem: only verly small amount of data for extreme events - harder to predict
solution: If the maximum values are not available for years but in different resolutions the formula must be adjusted accordingly
Return periods corresponding to the available data records:
annual maximum:
$$ p_{ne} = 1 - \frac{1}{50} = 0.9800 $$
monthly maximum:
$$ p_{ne} = 1 - \frac{1}{50 \cdot 12} = 0.9983 $$
weekly maximum:
$$ p_{ne} = 1 - \frac{1}{50 \cdot 52} = 0.9996 $$
daily maximum:
$$ p_{ne} = 1 - \frac{1}{50 \cdot 365} = 0.9999 $$
Assumption: the maxima are independent of each other.
Fisher-Tippett theorem: The maximum of a sample of independent and identically distributed random variables after proper renormalization converges to the Generalized Extreme Value (GEV) distribution.
$$ F(x) = exp(-[1+\xi(\frac{x-\mu}{\sigma})]^{\frac{-1}{\xi}}) $$
We use the Gumbal Distribution, where the shape parameter $\xi \rightarrow 0$
| extreme values | extreme values sorted | rank | $\small p_{ne} = 1 - \frac{rank(x)}{N+1} $ | reduced variable $ y = -\ln(-\ln(p_{ne})) $ |
|---|---|---|---|---|
| 22.10 | 28.22 | 1 | 0.92 | 2.53 |
| 22.21 | 27.23 | 2 | 0.85 | 1.79 |
| 19.29 | 24.53 | 3 | 0.77 | 1.34 |
| 18.89 | 24.02 | 4 | 0.69 | 1.00 |
| 17.30 | 23.31 | 5 | 0.62 | 0.72 |
| 15.89 | 22.21 | 6 | 0.54 | 0.48 |
| 19.12 | 22.10 | 7 | 0.46 | 0.26 |
| 28.22 | 19.29 | 8 | 0.38 | 0.05 |
| 24.02 | 19.12 | 9 | 0.31 | -0.16 |
| 23.31 | 18.89 | 10 | 0.23 | -0.38 |
| 27.23 | 17.30 | 11 | 0.15 | -0.63 |
| 24.53 | 15.89 | 12 | 0.08 | -0.94 |
- Measure extreme values over time, e.g. the last 12 years
- Rank the values from 1 to 12
- Calculate the empirical non exceedable probability for all values
- Calculate the reduces variable (trick) - Gumbel distribution predicts a linear function for the reduced variables
- Determine the distribution parameters graphically (slide 16)
- Use the graph to extrapolate extreme values for longer time periods
Method of Moments
Distributions can be caracterized by a number of parameters which are called moments. Statistical moments capture the key qualities of a distribution in numerical form:
- Mean (measure of central tendency, location)
- Variance (measure of dispersion or scatter)
- Skewness (measure of symmetry of asymmetry)
- Kurtosis (measure of peakedness and tail-heaviness)
Maximum likelihood estimation
Find the probability density function among all probability density functions that the model prescribes, which is most likely to have produced the observed data
Method of least squares
Try to find a function $y = m \cdot x + b$ that fits the data points best, meaning minimizing the sum of squared vertical deviations $d_i$
Notiz an mich selbst: Monte Carlo Method lecture anschauen
Book recommendation: black swan