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96 Fundamentals of Ocean Renewable Energy

where c and k are parameters for this distribution. In particular, c is the scale
parameter and k is the shape parameter. Both parameters are estimated based on
historical wind speed data. The scale parameter is related to the average wind
speed as follows
¯ u
c = or E(u) = cΓ(1 + 1/k) (4.15)
Γ(1 + 1/k)
in which Γ is the Gamma function. The shape parameter reflects the shape of
the PDF. If we take the derivative of the cumulative distribution, the PDF of the
Weibull distribution becomes
dF(u) u k−1
u k
f(u) =− = k exp − (4.16)
du c k c
Fig. 4.11 shows examples of the Weibull PDF, assuming various shape factors.
A special case of the Weibull distribution is called the Rayleigh distribution for
k = 2. This is the most common value that can be used to describe the hourly
variation of wind speed at many locations.
As mentioned before, given the historical wind speed data at a site, the best
values for k and c are selected. This will be demonstrated for a sample site in
the next section. Assuming that the PDF of wind speed at a site is evaluated,
the average theoretical power intensity can be also estimated. Referring to

FIG. 4.11 Weibull distribution corresponding to various shape parameters (k). The average wind
speed for all cases is 9 m/s. The corresponding values for scale parameter, c, are calculated based on
Eq. (4.15).