WIND ENERGY SYSTEMS - Cd3wd

Chapter 2—Wind Characteristics 2–39
error between the linearized ideal curve and the actual data points of p(u i ). The process is
somewhat of an art and there may be more than one procedure which will yield a satisfactory
result. Whether the result is satisfactory or not has to be judged by the agreement between
the Weibull curve and the raw data, particularly as it is used in wind power computations.
The first step of linearization is to integrate Eq. 27. This yields the distribution function
F (u) which is given by Eq. 39. As can be seen in Fig. 15, F (u) is more nearly describable by a
straight line than f(u), but is still quite nonlinear. We note that F (u) contains an exponential
and that, in general, exponentials are linearized by taking the logarithm. In this case, because
the exponent is itself raised to a power, we must take logarithms twice.
ln[− ln(1 − F (u))] = k ln u − k ln c (52)
This is in the form of an equation of a straight line
y = ax + b (53)
where x and y are variables, a is the slope, and b is the intercept of the line on the y axis. In
particular,
y = ln[− ln(1 − F (u)]
a = k (54)
x = lnu
b = −k ln c
Data will be expressed in the form of pairs of values of u i and F (u i ). For each wind speed
u i there is a corresponding value of the cumulative distribution function F (u i ). When given
values for u = u i and F (u) =F (u i ) we can find values for x = x i and y = y i in Eqs. 55. Being
actual data, these pairs of values do not fall exactly on a straight line, of course. The idea
is to determine the values of a and b in Eq. 53 such that a straight line drawn through these
points has the best possible fit. It can be shown that the proper values for a and b are
w∑
w ∑
a =
x i y i
i=1
w
w∑
i=1
x i y i −
i=1
( ∑ ) w 2
x i
w∑
x2 i − i=1
w
i=1
=
w∑
(x i − ¯x)(y i − ȳ)
i=1
(55)
w∑
(x i − ¯x) 2
i=1
b =ȳ i − a¯x i = 1 w∑
y i − a w∑
x i (56)
w w
i=1 i=1
Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001