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Wind Farm Investment Project
"Windy Profits Incorporated" is offering investors a chance to purchase shares of the output from several proposed
wind turbine farms. Three turbines will be placed at one of three locations for a total of nine turbines.
Location A: Armadillo Acres
(Site A1, Site A2, Site A3)
Location B: Blackbear Badlands
(Site B1, Site B2, Site B3)
Location C: Cobra Creek
(Site C1, Site C2, Site C3)
You can purchase 10% of the output from either location
A
,
B
or
C
for $10,000.
Your wind farm investment will provide you a cash revenue stream for
perpetuity. In this project you must identify the best location to invest in. Wind
Profits does not allow you to cherrypick individual sites from each location.
In the wind turbine industry, it is common to model wind speed data using the
Weibull distribution, named after its inventor Waloddi Weibull. The
pdf
for this
distribution has two parameters.
f
(
x
)
=
!
"
x
#
$
%
’
(
)
1
e
)
(
x
/
)
for wind speed
x
> 0
Below is a plot in Excel of the
pdf
for a Weibull distribution (
α
= 2,
β
= 1) over
the range 0 <
x
< 10.
Nordex 2.5 Megawatt Prototype Wind Turbine
The parameter
is called the
shape
parameter but is also called the
slope
. When
= 1, this is just the exponential distribution. For sites in
Northern Europe, the shape parameter is typically near 2 when wind
speed is measured in meters per second. The parameter
is called the
scale
parameter and merely changes the scale of the
x
axis.
The cumulative distribution function or CDF is defined as follows. The
probability that the wind speed satisfies 0
≤
x
is given by the integral of
f
(
x
) and will be denoted by
F
(
x
).
F
(
x
)
=
f
(
x
)
dx
0
x
!
=
1
"
e
"
(
x
/
#
)
$
Statistics of the Weibull Distribution:
The Weibull
mean
is given by:
μ
=
"
1
+
1
$
%
’
(
)
where
Γ
denotes the
Gamma function which is related to the factorial function. For integer values we have
!
1
+
n
( ) =
n
!
As a check of
the mean formula, when
= 1, we know we just have the exponential distribution (with
λ
= 1/
) and the mean above
reduces to
=1/
as expected.
=
"
1
+
1
$
%
’
(
)
=
"
2
( ) =
In general, the Gamma function is defined via the integral:
!
z
( ) =
e
"
x
x
z
"
1
dx
0
#
$
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For noninteger values, you can access the function in Excel via a circuitous path. The Excel command
gammaln
gives the natural log of the gamma function. Therefore you can use the following trick to find the gamma function in
Excel:
!
z
( ) =
exp gammaln(
z
)
( )
Weibull Median
The Weibull
median
is given by:
Median
=
!
ln2
( )
1/
"
Weibull Variance
The Weibull
variance
is given by:
2
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 Fall '07
 RobinCarr
 Normal Distribution, Probability theory, Exponential distribution, probability density function

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