Welcome back to the second book.
We will focus now on the analytical expressions used for wind data assessment,
and on how to fit the parameters needed for them.
As we saw in the first part,
deriving analytical expressions to fit the probability density function
is way to reduce the number of parameters to characterize
the probability density function and the estimate wind power resource.
We announced the Weibull and Rayleigh distributions
as the most commonly used statistical functions used for wind power studies.
And in this block, we will have a closer look at them
and see some common used method to fit their parameters.
We will start with the Weibull distribution
that has become a widely used standard in wind energy application due to its simplicity.
It is described by the equation on the screen,
where phi is the probability of the wind being at speed V.
It depends on too easily estimated parameters.
k is known as the shape parameter and is positive and dimensionless.
The higher the value of k, the higher the median wind speed.
As we can see in the figure, it settles the shape of our function,
location with lots of low wind speeds,
as well as some very strong winds would have a shape value k below two,
whereas locations with fairly consistent wind speeds
around the median would have a shape value of three.
c is known as the scale parameter.
It is also positive and has velocity dimension.
Looking at the example, we can see why c is called the scale parameter.
Multiplying c by two stretches the scale two times,
but keeps the area under the curve to one, as it is a normalized distribution.
The Weibull distribution has become a widely used standard
in wind energy application due to its simplicity,
and there are simple analytical expressions for the moments as will be shown later.
It is a reference used in wind energy softwares,
such as Wind Atlas Analysis and Application Program, WAsP,
and it is included in regulations such that on wind turbine powered performance testing.
The cumulative distribution function of the wind speed V, evaluated at a given value V0,
is the probability that the wind speed will take a value
less than or equal to the given value V0.
It is expressed as the integrand from zero to phi zero.
In other words, it gives the area under the probability density function from zero to V0.
In the case of the Weibull distribution,
the cumulative distribution function also takes an analytical expression
which you can see on your screen.
The cumulative distribution function is often used to quantify the goodness of fit
of the Weibull distribution with respect to the observed probability density function,
as will be shown later.
The Rayleigh distribution is a particular case of Weibull distribution
with shape parameter k equals two.
This is convenient as in most locations around the world
the value of k is approximately two.
This reduces the expression of the PDF on one single parameter.
[speaking in French] The Rayleigh.
The Rayleigh distribution is a particular case of a Weibull distribution
with shape parameter k equals two.
This is convenient as in most locations around the world
the value of K is approximately two.
This reduces the expression of the probability density function to one single parameter.
Weibull distribution with its two parameters leads to very simple analytical expressions
for the moments which are key to easily estimate
the shape and scale parameters k and c.
So, to adjust the k and c parameters,
we will need to evaluate some of the function moments.
For example, the first moment, which corresponds to the mean wind speed,
is the integrand from zero to infinity of the product of the wind speed, V,
by the probability density function, Phi.
A simple mathematical manipulation of the expression
of the Weibull distribution leads to the Gamma function.
The steps are detailed here, but we will focus on this final expression.
Now, taking the deduced expression for the mean value
and the definition of the Gamma function plus its property relating Gamma
of Z plus one to Gamma of Z,
we can obtain an analytical expression of the mean wind speed expressed
as the product of the scale parameter
by the Gamma function of one plus one of the shape parameter.
And the scale parameter can be obtained from this expression.
It can be either computed numerically or tabulated to facilitate its calculation.
The properties of the Weibull distribution
and the Gamma function lead to a simple expression for higher moments as well,
expressed by this formula.
Other methods are used depending of the available information we might have.
Let's have a look at some of the most commonly used methods.
When only the mean wind speed is known, the Rayleigh distribution is the one to be used.
It is, in general, accurate enough and often used in wind atlas.
The procedure is the same.
We evaluate c from the mean value V and we use the Gamma function.
When more than one moment are known,
as for instance the mean wind speed in the standard deviation,
then the method of moment must be used.
We have illustrated the method combining the first moment, the mean wind speed,
and the second moment, the power of two of the standard deviation,
to compute the shape parameter k.
This method of estimation of distribution parameters
was introduced by Karl Pearson in 1894.
The Power Density method is also a method of moments.
It ensures that the energy content of the fitted Weibull distribution
equals the energy content of the observed histogram.
Wind energy is the kinetic energy of an air mass.
For a constant wind speed V crossing a normal area A during a given period T,
kinetic energy contained for the air mass
will be the one expressed here in the first formula.
The wind power density for a unit of area, and per unit of time,
is defined then as the kinetic energy per unit of area and unit of time which,
in its integral continuous form, it is related to the third statistical movement.
Taking the analytical expressions of the moments and defining the energy pattern factor,
we obtain an expression for the shape parameter k.
This method ensures the best estimation of wind energy potential,
but does not ensure the maximum likelihood with the observed distribution.
It is usually used when the mean wind speed and the wind energy density are known.
To produce, for instance a wind atlas,
the best parameter estimates are obtained using this method.
We finally include the analysis of the WAsP fitting method
which is the most used in the industry.
The WAsP method computes these parameters from the first and third moments
as well as the probability of exceeding the mean wind speed one minus Phi of V,
which must be estimated from the data.
The method focuses on the right-hand tail of the Weibull distribution
which is an important part of the distribution in terms of energy.
This is why the WAsP method is preferred amongst the wind energy industry.
In this method, k and c are calculated by solving these equation on the screen.
Popular methods also include the usual maximum likelihood estimate.
The principle of the maximum likelihood estimate, originally developed
by Ronald Fisher in 1912, states that the desired probability distribution
is the one that makes the observed data most likely, which means
that one must seek the value of the distribution parameters
that maximize the likelihood.
It is a method which is similar to the graphical method but overcomes its shortfalls,
such as the absence of confidence interval estimate
or the ambiguous choice of the initial parameter estimation.
The procedure is simple.
We build the cumulative distribution function F
and isolate the exponential, and we then apply the natural logarithm twice on each side.
Plotting the values and fitting them to a straight line,
we will obtain the Weibull parameters k and c.
The comparison of the wind speed distributions in the observation data
is made on the cumulative density function for wind speed.
We compute a distance score between the cumulative density function
of the tested distribution F
and the observed empirical cumulative density function F^hat_n.
The distance score is referred as D_n squared,
and is given by this equation on the screen,
where n is the number of wind speed measurements and omega is a weight function.
In the case of the Cramer-Von Mises statistics, the weight is constant,
so that the center of the distribution actually dominates the equation.
Here the center of the distribution is not a single point,
but the region around the median, mean, or maximum of the distribution.
Conversely, the Anderson-Darling score puts weight on the tails of the distribution.
The tail corresponds to the part of the distribution exceeding,
for example, the 90th percentile, that is, the extreme wind speeds.
In this case, the weight function omega takes the expression shown on the table.
To put even more emphasis on the extreme winds,
a right tail Anderson-Darling of second degree exists
which consist in replacing the exponent of the denominator of omega by two.
In the wind energy context, the wind turbines must work as often as possible.
This makes the center of the wind speed distribution the center of our interest,
corresponding to the distribution range around the most probable value.
Therefore in the following, only the Cramer-Von Mises score will be shown.
This figure shows a map of the Cramer-Von Mises scores
for the Weibull distribution in some sites in France.
The green dots show sites where the Weibull distribution fits accurately the observation,
whereas the red dots shows inaccurate fits.
For this data set the Weibull distribution is suited
to model the center of the wind speed distribution when the score is smaller
than an empirical threshold value of two.
This is the case of the Northern France site.
In Southern France the measured wind speed histograms deviate significantly
from the Weibull distribution because of the complex topography and,
as in this region, valley flow alternate with weaker and more isotropic wind flow.
I hope this lecture helps you to better understand how wind speed statistics is modeled.
And in the next video, we will see how it impacts wind energy assessment
and wind production evaluation.
I hope to see you on the next video.
Thank you.
