WIND ENERGY SYSTEMS - Cd3wd

Chapter 2—Wind Characteristics 2–40
In these equations ¯x and ȳ are the mean values of x i and y i ,andw is the total number of
pairs of values available. The final results for the Weibull parameters are
k = a
(
c = exp − b )
k
(57)
One of the implied assumptions of the above process is that each pair of data points is
equally likely to occur and therefore would have the same weight in determining the equation
of the line. For typical wind data, this means that one reading per year at 20 m/s has the
same weight as 100 readings per year at 5 m/s. To remedy this situation and assure that we
have the best possible fit through the range of most common wind speeds, it is possible to
redefine a weighted coefficient a in place of Eq. 55 as
a =
w∑
p 2 (u i )(x i − ¯x)(y i − ȳ)
i=1
(58)
w∑
p 2 (u i )(x i − ¯x) 2
i=1
This equation effectively multiplies each x i and each y i by the probability of that x i and
that y i occurring. It usually gives a better fit than the unweighted a of Eq. 55.
Eqs. 55, 56, and 58 can be evaluated conveniently on a programmable hand held calculator.
Some of the more expensive versions contain a built-in linear regression function so Eqs. 55
and 56 are handled internally. All that needs to be entered are the pairs of data points.
This linear regression function can be combined with the programming capability to evaluate
Eq. 58 more conveniently than by separately entering each of the repeating data points m i
times.
Example
The actual wind data for Kansas City and Dodge City for the year 1970 are given in Table 2.3.
The wind speed u i is given in knots. Calm includes 0 and 1 knot because 2 knots are required to spin
the anemometer enough to give a non zero reading. The parameter x i is ln(u i ) as given in Eq. 55. The
number m i is the number of readings taken during that year at each wind speed. The total number
of readings n at each site was 2912 because readings were taken every three hours. The function p(u i )
is the measured probability of each wind speed at each site as given by Eq. 20. Compute the Weibull
parameters c and k using the linearization method.
Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001