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Statistical Analysis of (Viscous Turbulent) Winds

Statistical Analysis of (Viscous Turbulent) Winds. P M V Subbarao Professor Mechanical Engineering Department I I T Delhi. Models for Random but Reliable Resource…. Spectral Structure of Wind Gust. d. Simplified Description of Turbulent Wind. Statistical Distribution of Turbulent Wind.

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Statistical Analysis of (Viscous Turbulent) Winds

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  1. Statistical Analysis of (Viscous Turbulent) Winds P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Models for Random but Reliable Resource…

  2. Spectral Structure of Wind Gust d

  3. Simplified Description of Turbulent Wind

  4. Statistical Distribution of Turbulent Wind

  5. Two Variable (speed-direction) PDF for Wind

  6. Directional Projection of Two Variable PDF for Wind Speed • A single variable PDF can be generated using wind speed as a primary variable. • This function shows probability of occurrence of wind of a given speed in all possible directions.

  7. A Mathematical Model for Wind Speed PDF A comprehensive model for Probability of occurrence of a given value of wind velocity magnitude, v is • where  is a scale parameter with the same units as the random • Variable; • a is a shape parameter; • 1 and 2 are parameters with the same units as the random variable; • 0 is the weight in the mixture of the Singly Truncated from below Normal distribution (0 0 1).

  8. Magnitude Projection of 2V PDF For Wind Direction This is a mixture of von Mises distributions N is the number of components of the mixture; The mean direction parameter, j 0 and The Concentration parameter, 0  j < 2 are parameters; j are nonnegative quantities that sum to one as given by I0()is the modified Bessel function of the first kind and order zero

  9. Probability Density for Direction –Velocity Distribution The probability density for an angular–linear distribution the circular variable , is defined as

  10. Statistical Nature of Wind at a given site @ 100m

  11. Weibull distribution • Waloddi Weibull invented the Weibull distribution in 1937 and delivered his hallmark American paper on this subject in 1951. • He claimed that his distribution applied to a wide range of problems. • He illustrated this point with seven examples ranging from the strength of steel to the height of adult males in the British Isles. • He claimed that the function "…may sometimes render good service." • He did not claim that it always worked. • Time has shown that Waloddi Weibull was correct in both of these statements.

  12. Simple Models for Distribution of Wind Speed • Two probability distribution functions are commonly used for wind speed. • The simpler of the two is the Rayleigh distribution which has a single parameter c. • The Weibull distribution shown below has two parameters k and c. • c = Empirical Weibull scale factor (m/s) • The Rayleigh distribution is actually a special case of the Weibull distribution with k = 2.

  13. Weibull distribution of Wind Speed • The Weibull distribution has two parameters k and c.

  14. Statistical Characteristics of Weibull distribution Average value: Most probable value of velocity The maximum wind speed

  15. The Available Wind Power • There are several methods that can be used to estimate available wind power at a site. • More popular method is used here. • Determine the yearly mean wind speed. • The contribution from instantaneous fluctuating wind velocity is assumed as:  The effective wind velocity is estimated as: The available wind power

  16. Simplified Formula for Available Wind Power Estimation of Weibull Parameters is an essential step for accurate estimation of local available Wind Potential.

  17. Estimation of Weibull Parameters • Various methods are available to estimate Weibull parameters • Standard deviation method (STDM) • Method of moments (MOM) • Least square regression method (LSRM) • Maximum likelihood method (MLM) • Energy pattern factor method (EPFM)

  18. Energy Pattern Factor Method (EPFM) • Energy pattern factor (EPF) is defined as; • the mean of sum of cubes of all individual wind speeds considered in a sample divided by the cube of mean wind speed of the sample. where v(ti) is the wind speed in meters per second, observed at ti n is the number of wind speed samples considered. Shape parameter can be calculated using EPF, which is expressed as

  19. Analysis of PDF • A moment of order n is a parameter of the probability density function (pdf) f ,defined as: The first raw moment or Mean Central moments are also defined as:

  20. Higher Order Moments The variance The Skewness The Kurtosis

  21. Wind Power Density Map at 50 m Altitude Wind Power Density W/m2

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