The vector sum of the four phasor in Fig. 4 is another random phasor corresponding to the farm phasor, provided the farm network losses are negligible. If some conditions are met, then the farm phasor can be modelled as a complex normal variable. In that case, the phasor amplitude has a Rayleigh distribution. The frequency f = 0 corresponds to the special case of the average signal value during the sample.
c) One and two sided spectra notation
One or two sided spectra are consistent -provided all values refer exclusively either to one or to two side spectra. Most differences do appear in integral or summation formulas - if two-sided spectra is used, a factor 2 may appear in some formulas and the integration limits may change from only positive frequencies to positive and negative frequencies. One-sided quantities are noted in this chapter with a + in the superscript unless the differentiation between one and two sided spectra is not meaningful. For example, the onesided stochastic spectral phasor density of the active power at frequency f is:
In plain words, the one-sided density is twice the two-sided density. For convenience, most formulas in this chapter are referred to two-sided values.
d) Case study
Fig. 5 to Fig 8 show the power fluctuations of a wind farm composed by 27 wind turbines of 600 kW with variable resistance induction generator from VESTAS (Mur-Amada, 2009). The data-logger recorded signals either at a single turbine or at the substation. In either case, wind speed from the meteorological mast of the wind farm was also recorded. The record analyzed in this subsection corresponds to date 26/2/1999 and time 13:52:53 to 14:07:30 (about 14:37 minutes). The average blade frequency in the turbines was fbiad£~ 1,48 ±0,03 Hz during the interval. The wind speed, measured in a meteorological mast at 40 m above the surface with a propeller anemometer, was Uwind = 7,6 m/s ±2,0 m/s (expanded uncertainty).
The oscillations due to rotor position in Fig. 5 are not evident since the total power is the sum of the power from 26 unsynchronized wind turbines minus losses in the farm network. Fig. 6 shows a rich dynamic behaviour of the active power output, where the modulation and high frequency oscillations are superimposed to the fundamental oscillation.
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