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Project Description

# Project Description - Wind Farm Investment Project"Windy...

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Page 1 of 6 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 cherry-pick 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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Page 2 of 6 For non-integer 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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Project Description - Wind Farm Investment Project"Windy...

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