Environmental Engineering Reference
In-Depth Information
Methods for Estimating Weibull Distribution Factors
There are several methods which can be used to estimate the Weibull factors
C
and
k
,
depending on the available wind statistics and the desired level of sophistication in data
analysis [Justus
et al.
1978]. These methods are (1) the
least-squares curve-fit
,
(2)
median
and quartile
,
(3)
annual average and standard deviation
,
(4)
annual average and fastest
mile
,
and (5)
variance vs. annual average trend.
The least-squares method requires an
observed
wind speed histogram
,
which is frequency data for
n
speed intervals or
bins.
A
duration curve is then constructed from the histogram and the results plotted in the
linearized form
y
=
y
0
+
m x
(8-8a)
x
= ln (
U
)
y
= ln [- ln (
F
/ 8,760)]
The data are then least-squares curve-fit with a line of slope
m
and intercept
y
0
,
from which
k
=
m C
= exp (-
y
0
/
m
)
(8-8b)
The median and quartile method is useful if a complete histogram is not available, since
it requires only the wind speeds at
F =
2,190, 4,380, and 6,579 hours. The third method
requires the annual average wind speed,
U
a
,
and the annual standard deviation from this
average, s
a
.
The fourth method uses the publication
Local Climatological Data
(from the
National Climatic Data Center in Ashville, North Carolina) which lists monthly mean wind
speeds and the monthly fastest mile,
U
max
(average speed associated with the most rapid one
mile run of wind), for many locations in the U.S. The parameter
k
is determined from
U
max
and then
C
can be calculated using Equation (8-7). Finally, Justus
et al.
[1976a] identified
a general trend between the annual
variance
of the wind distribution (equal to the square
of the standard deviation) and the annual average wind speed, so that a qualitative estimate
of the average wind speed is sufficient to make a rough estimate of the Weibull factors.
The references cited should be consulted for more details.
Wind Speed Distribution Data
The results of work by Justus
et al.
[1976b] on wind speed distributions have been
incorporated into a graphical and analytical format to provide for the rapid construction of
duration curves for geographic locations throughout the U.S. [Frost
et al.
1978]. Values of
C
and
k
,
adjusted to a 10-m reference elevation, are given for 138 geographical locations
in the U.S., based on data with averaging times of one minute [Justus
et al.
1976b] and two
minutes [Doran
et al.
1977]. It must be borne in mind that wind speed duration curves are
quite sensitive to location because of
surface roughness
conditions, and are valid only for
relatively flat terrain. Steady wind speed data include the influence of
atmospheric stability
and
terrain features
peculiar to the site at which they were measured, but the influence of
these factors is normally small enough under higher wind conditions that the available data
are a good representation of the wind for design purposes.
The factors in the Weibull distribution are elevation-specific and must be adjusted to
account for
wind shear.
One method for adjusting the
C
and
k
values for changes in
elevation uses power law equations [Justus
et al.
1976a, Spera and Richards 1979]. These
equations are valid for elevation corrections over a fairly wide range of surface roughnesses,
provided the terrain is fairly level. Wade and Walker [1988] confirmed the adjustment of
C
for elevation, but found it unnecessary to adjust
k.
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