## Case study comparison of PSD of a wind farm with respect to one of its turbines during 12 minutes

A literature review on experimental data of power output PSD from wind turbines or wind farms can be found in (Mur-Amada & Bayod-Rujula, 2007), with a parameterization and analysis of the data from very different locations. (Apt, 2007) shows an interesting comparison of the spectrum of the wind power from a wide area.

In this sub-section, the analysis of a case based on (Mur-Amada, 2009) is presented. The similarity of the PSD at one turbine and at the overall output of a wind farm of 18 turbines is shown. If the fluctuations at every turbine are independent (i.e. the turbines behaves independently from each other), then the PSD of the wind farm is approximately the PSD of each turbine multiplied by the number of turbines and by the power flow efficiency. Each turbine experiments different turbulence levels and wind averages, so a representative turbine should be selected. The time lag between the variations measured in the farm and in the turbine depends on the farm layout. The phase information has been discarded because the phase of ergodic stochastic processes do not contain statistical information. Fig. 12 shows the power output of the wind farm and the scalled output of one turbine. Since the measured turbine is more exposed to the wind than others turbines, the ratio of the average power of the turbine to the farm is 14 (less than 18, the number of turbines in the farm). There is a clear reduction of the relative variability in the farm output and some slow oscillations between the turbine and the farm seem to be delayed. In fact, this section will show that the ratio of the fluctuations is about Vl8 because the measured fluctuations are mainly uncorrelated, the duration of the sample is relatively short (less than 12 minutes) and the wind does not show a noticeable trend during the sample.

If the turbines behave independently from each other and they are similar, then the PSD of the wind farm is the PSD of one turbine times the number of turbines in the farm and times a power efficiency factor. To test this hypothesis, the farm PSD is shown in solid black and the turbine PSD times 18 is in dashed green in Fig. 13, with good agreement.

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Fig. 12. Power output of the wind farm (in solid black) and the power of the turbine times 14.

Fig. 13 shows that the farm PSDp+(/) and the scaled turbine PSDp+(/ agree notably, showing that fluctuations up to 10-2 Hz are almost uncorrelated (frequency bellow 10-2 Hz is shown in the figure, but its value is biased by the window applied in the FFT and the relative short duration of the sample). However, the wind farm PSD is a bit lower than 18 times the turbine PSD, specially at the peaks and at f > 2fbiade (fbiade is the frequency of a blade crossing the turbine tower, about 1,54 Hz in this sample). On the one hand, this turbine experiences more cyclic oscillations, partly due to a misalignment of the rotor bigger than the farm average. On the other hand, this turbine produced an average of 1/14th of the wind farm power on the series #1 (see Fig. 12). This explain that PSD at f > 2fbiade is primarily proportional to power output ratio (the farm PSD is 14 times the turbine PSD). The real power admittance is shown in Fig. 14. The admittance is the ratio of the farm spectrum to the turbine spectrum of real power and it can be estimated as the square root of the PSD ratios. The level V18 has been added in dash-dotted red line to compare with the theoretical value of uncorrelated fluctuations.

In general terms, the assumption of uncorrelated fluctuations at frequencies higher than 10-2 Hz is valid: the admittance is approximately V18, the square root of the number of turbines in the farm. At f > 2fbiade, the admittance is more similar to V14 (the square root of the farm power divided by the turbine power). At f < 0,02 Hz, the admittance starts drifting from V18, indicating that oscillations at very low frequency are somewhat correlated.

There is a peak in Fig. 14 at 2 Hz < f < 2,5 Hz. The analyzed turbine may have comparative less fluctuations in such range than the other turbines in the farm (the measured turbine may have better adjusted rotor and blades, while others turbines may suffer from more vibration effects). But other feasible reason is a higher correlation degree between the turbines at such frequency band, probably induced by turbine control or voltage variations.

In short, real power oscillations quicker than one minute can be considered independent among turbines of a wind farm because the PSD due to fast turbulence and rotational effects scales proportionally to the number of turbines.

The former section has analyzed values logged with high time resolution (each grid cycle, 20 ms) but the duration was relatively short (a bit more than 10 minutes) due to storage limitations in the recording system. Ten-minute records with 20 ms time resolution allow studying fluctuations with durations between some tenths of second up to one minute However, this duration is insufficient for analyzing wind farm dynamics slower than 0.016 Hz with acceptable uncertainty.

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