The fluctuations of a group of turbines can be divided into the correlated and the uncorrelated components.
On the one hand, slow fluctuations (f < 10-3 Hz) are mainly due to meteorological dynamics and they are widely correlated, both spatially and temporally. Slow fluctuations in power output of nearby farms are quite correlated and wind forecast models try to predict them to optimize power dispatch.
On the other hand, fast wind speed fluctuations are mainly due to turbulence and microsite dynamics (Kaimal, 1978). They are local in time and space and they can affect turbine control and cause flicker (Martins et al., 2006). Tower shadow is usually the most noticeable fluctuation of a turbine output power. It has a definite frequency and, if the blades of all turbines of an area became eventually synchronized, it could be a power quality issue.
The phase ^i(f) implies the use of a time reference. Since fluctuations are random events, there is not an unequivocal time reference to be used as angle reference. Since fluctuations can happen at any time with the same probability -there is no preferred angle ^i(f)-, the phasor angles are random variables uniformly distributed in [-n,+n] (i.e., the system exhibits circular symmetry and the stochastic process is cyclostationary). Therefore, the relevant information contained in i(f) is the relative angle difference among the turbines of the farm (Li et al., 2007) in the range [-n,+n], which is linked to the time lag among fluctuations at the turbines.
The central limit for the sum of phasors is a fair approximation with 8 or more turbines and Gaussian process properties are applicable. Therefore, the wind farm spectrum converges asymptotically to a complex normal distribution, denoted by CN (0, o~Pfarm (f)). In other words, Re[Pjarm(f)] and Im[Pf+rm(f)] are independent random variables with normal distribution.
Thus, the one-sided amplitude density of fluctuations at frequency f from N turbines,
\Pf+arm (f) | , is a Rayleigh distribution of scale parameter V pfarm(f) = (| f™ CO l)V2/^ , where angle brackets <•> denotes averaging. In other words, the mean of \P'jarm(f) is
( lPftrm(f)l) = Vn/2 V Pfarm(f) where V Pfarmif) is the RMS value of the phasor projection. The RMS value of the phasor projection &pfarm(f) is also related to the one and two sided PSD of the active power:
Put into words, the phasor density of the oscillation, |Ppfarm (f) , has a Rayleigh distribution of scale parameter apfarm(f) equal to the square root of the auto spectral density (the equivalent is also hold for two-sided values). The mean phasor density modulus is:
For convenience, effective values are usually used instead of amplitude. The effective value of a sinusoid (or its root mean square value, RMS for short) is the amplitude divided by V2. Thus, the average quadratic value of the fluctuation of a wind farm at frequency f is:
If the active power of the turbine cluster is filtered with an ideal narrowband filter tuned at frequency f and bandwidth Af, then the average effective value of the filtered signal is aPfarmfWAf and the average amplitude of the oscillations is ^\Pfarm(f)\)'^ Af = aPfarm(f)^lAf n/2 . The instantaneous value of the filtered signal Ppfarmf Af( is the projection of the phasor Pfarm,(f)'ej2it/ Af in the real axis. The instantaneous value of the square of the filtered signal, Pfarm f a/ (t), is an exponential random variable of parameter \= [afarm(f)A f ]-1 and its mean value is:
* ' Exp distribution J
For a continuous PSD, the expected variance of the instantaneous power output during a time interval T is the integral of apfarm(f) between Af = 1/ T and the grid frequency, according to Parseval's theorem (notice that the factor 1/2 must be changed into 2 if two-sided phasors densities are used):
(PfamV) = 1( JIT \fm(f>\df) = \ f (\fm(f)\)df = f C%,mf)df (10)
In fact, data is sampled and the expected variance of the wind farm power of duration T can be computed through the discrete version of (10), where the frequency step is Af = 1/ T and the time step is At= T/m:
If a fast Fourier transform is used as a narrowband filter, an estimate of &pfarm(f) f°r f = k Af is 2Af (| FFTk[Pfarm (iAi)}|2). In fact, the factor 2Af may vary according to the normalisation factor included in the FFT, which depends on the software used. Usually, some type of smoothing or averaging is applied to obtain a consistent estimate, as in Bartlett or Welch methods (Press et al., 2007).
The distribution of ^Pfarm() can be derived in the time or in the frequency domain. If the process is normal, then the modulus and phase of Pfarm(fk) are not linearly correlated at different frequencies f . Then ^Pfarm() is the sum in (11) or the integration in (10) of independent Exponential random variables that converges to a normal distribution with mean (Pfarm(t)) and standard deviation ^ P]arm()) .
In farms with a few turbines, the signal can show a noticeable periodic fluctuation shape and the auto spectral density &pfarm(f) can be correlated at some frequencies. These features can be discovered through the bispectrum analysis. In such cases, ^ Pfarm(t) can be computed with the algorithm proposed in (Alouini et al., 2001).
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