Lecture 8

Lecture 8 - Lecture 8: Info on The Wind Farm Team Project...

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Page 1 of 16 L ecture 8: Info on The Wind Farm Team Project The Weibull distribution is important in the modeling of wind speed distributions. The probability density function f ( x ) for this distribution is: f ! , " ( x ) = x # $ % ( ) 1 e ) ( x / ) for x > 0 Here x denotes the wind speed while α and β are constants that characterize the wind distribution at a particular turbine site. 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. You should expect in the range from 1.5 to 3 for actual wind sites. The parameter is called the scale parameter and merely changes the scale along the x -axis as you can see in the following identity and plot. f , ( x ) = 1 # f ,1 x $ % ( ) Wind Turbine The corresponding cumulative distribution function is simply the integral of the density function: F , ( x ) = f , ( t ) dt = t = 0 x # 1 $ e $ ( x / ) Below is a plot produced in Maple of the pdf f , ( x ) for a Weibull distribution with = 2 while has the values 1, 2, 4, 8 over the range 0 < x < 15.
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Page 2 of 16 For each plot, α = 2 but β is respectively 1, 2, 4 and 8 The above plot shows the effects of changing the scale parameter. Each time is doubled, the plot is scaled down vertically by ½ but stretched horizontally by 2, thus preserving the total area under each curve as unity. The graph below shows the cumulative distribution functions for the same four cases. The dots indicate the median values where the curve F ! , " ( x ) equals ½ . The median is the wind speed where this occurs. When is doubled, the median also doubles. After all, is just a scale parameter. = 1 = 2 = 4 = 8 Medians
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Page 3 of 16 Median, Quartiles and Percentiles for Wind Speed Data To fit the data at your team's wind farm sites, you will need to implement the gamma function since this function appears in several of the statistical quantities you will calculate, such as the mean and the variance. The Weibull mean is given by: μ = ! " 1 + 1 # $ % ( ) The Weibull median is given by: Median = ln2 ( ) 1/ " The Weibull variance is given by: 2 = 2 # 1 + 2 $ % ( ) * + # 1 + 1 % ( ) * 2 % ( ) * Derivation of Median and Quartiles: 1. First Quartile: The expression for the first quartile Q 1 can be derived from the CDF as follows. The first quartile is defined by the condition F ( x ) = 1/4. One quarter of the points are less than this value and three quarters are above this value. Rearranging:
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Lecture 8 - Lecture 8: Info on The Wind Farm Team Project...

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