Wind resource and site assessment - Wind Power Generation and Wind Turbine Design - page 64

3.2 Weibull distribution

It is very important for the wind industry to be able to relatively simply describe
the wind regime on site. Turbine designers need the information to optimise the
design of their turbines, so as to minimise generating costs. Turbine investors need
the information to estimate their income from electricity generation.
One way to condense the information of a measured time series is a histogram.
The wind speeds are sorted into wind speed bins. The bin width is typically 1 m/s.
The histogram provides information how often the wind is blowing for each wind
speed bin.
The histogram for a typical site can be presented using the Weibull distribution
expressing the frequency distribution of the wind speed in a compact form. The
two-parameter Weibull distribution is described mathematically as

k − 1

⎛

k

⎞

k u u

⎛⎞ ⎛⎞

fu () = exp −

⎜⎟ ⎜ ⎜⎟ ⎟ (3)

⎝⎠ ⎝⎠

AA A

⎝ ⎠

where f ( u ) is the frequency of occurrence of wind speed u . The scaling factor A is
a measure for the wind speed while the shape factor k describes the shape of the
distribution. The cumulative Weibull distribution F ( u ) gives the probability of the
wind speed exceeding the value v and is given by the simple expression:

⎛

k

⎞

u

⎛⎞

Fu () exp = −

⎜ ⎜ ⎝⎠ ⎟ (4)

A

⎝ ⎠

Following graph shows a group of Weibull distributions with a constant mean
wind speed of 8 m/s but varying k factor. Note that high wind speeds become more
probable with a low k factor.
The Weibull distribution can degenerate into two special distributions, namely
for k = 1 the exponential distribution and k = 2 the Rayleigh distribution. Since
observed wind data exhibits frequency distributions which are often well described
by a Rayleigh distribution, this one-parameter distribution is sometimes used by
wind turbine manufacturers for calculation of standard performance fi gures for
their machines. Inspection of the k parameter shows that, especially for Northern
European climates, the values for k are indeed close to 2.0.
On a global scale, the k factor varies signifi cantly depending upon local climate
conditions, the landscape, and its surface (Fig. 8). A low k factor (<1.8) is typical
for wind climates with a high content of thermal winds. A high k factor (>2.5) is
representative for very constant wind climates, for example trade winds. Both
Weibull A and k parameters are dependent on the height and are increasing up to
100 m above ground (Fig. 9). Above 100 m the k parameter decreases.
The Weibull distribution is a probability density distribution. The median of the
distribution corresponds to the wind speed that cuts the area into half. This means
that half the time it will be blowing less than the median wind speed, the other half