WIND ENERGY SYSTEMS - Cd3wd

Chapter 2—Wind Characteristics 2–47
meantime it appears that either function may yield acceptable results, with the Weibull being
more accurate and the Rayleigh easier to use.
Although the actual wind speed distribution can be described by either a Weibull or a
Rayleigh density function, there are other quantities which are better described by a normal
distribution. The distribution of monthly or yearly mean speeds is likely to be normally
distributed around a long-term mean wind speed, for example. The normal curve is certainly
the best known and most widely used distribution for a continuous random variable, so we
shall mention a few of its properties.
The density function f(u) of a normal distribution is
f(u) = 1
[
]
σ √ 2π exp (u − ū)2
−
2σ 2
(63)
where ū is the mean and σ is the standard deviation. In this expression, the variable u is
allowed to vary from −∞ to +∞. It is physically impossible for a wind speed to be negative,
of course, so we cannot forget the reality of the observed quantity and follow the mathematical
model past this point. This will not usually present any difficulty in examining mean wind
speeds.
The cumulative distribution function F (u) isgivenby
F (u) =
∫ u
−∞
f(x)dx (64)
This integral does not have a simple closed form solution, so tables of values are determined
from approximate integration methods. The variable in this table is usually defined as
q = u − ū or u =ū + qσ (65)
σ
Thus q is the number of standard deviations that u is away from ū. A brief version of this
table is shown in Table 2.4. We see from the table, for example, that F (u) for a wind speed
one standard deviation below the mean is 0.159. This means there is a 15.9 % probability
that the mean speed for any period of interest will be more than one standard deviation below
the long term mean. Since the normal density function is symmetrical, there is also a 15.9
% probability that the mean speed for some period will be more than one standard deviation
above the long term mean.
Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001