## PSDr f

PSD,

Ueq,turbine

wind

Uwind.) is the mean wind during the sample, n is the average sensitivity of the power respect the wind and a and b are the dimensions of the wind farm according to Fig. 10. The decay constants for lateral and longitudinal directions are, Aiong and Am, respectively. For the Rutherford Appleton Laboratory, (Schlez & Infield, 1998) recommended Aiong w (15±5) vuwind / (Uwmd) and Am w (17,5±5) (m/s)-1 auwtnd, where auwtnd is the standard deviation of the wind speed in m/s. IEC 61400-1 recommends A w 12; Frandsen (Frandsen et al., 2007) recommends Aw 5 and Saranyasoontorn (Saranyasoontorn et al., 2004) recommends A w 9,7.

H2(f) is the quadratic coherence between the equivalent wind of the farm, relative to the turbine. H(f) measures the correlation of the phase difference between the equivalent wind of the farm relative to the turbine at frequency f. If H (f) is unity, the turbine phasors have

the same angle and the turbine fluctuations are synchronized at that frequency. If H(f) is zero, the phasors have uncorrelated arguments and hence, the turbine fluctuations are stochastically uncorrelated at that frequency. Hence, H(f) is the correlation level at frequency f of the fluctuations among the turbines, measured from 0 to 1. The transition frequency from correlated to uncorrelated fluctuations is obtained solving H2 (f) =1/4. Thus, the cut-off frequency of narrow wind farms with a « b is:

' bAlat

In the Rutherford Appleton Laboratory (RAL), Aut ~ (17,5±5)(m/s)-1 aUwind and hence fcut,lat ~ (0,42±0,12) (Uwmd) / (aUwind b). A typical value of the turbulence intensity auwind/ (Uwmd) is around 0,12 and for such value fcutllat ~ (3.5±1)/b, where b is the lateral dimension of the area in meters. For a narrow farm of b = 3 km, the cut-off frequency is in the order of 1,16 mHz. In Horns Rev wind farm, Aiat= (Uwind) /(2 m/s) and hence fcutllat ~ 13,66/b, where b is a constant expressed in meters. For a wind farm of b = 3 km, the cut-off frequency is in the order of 4,5 mHz (about four times the estimation from RAL).

In RAL, Along ~ (15±5) auwind / (Uwind) . A typical value of the turbulence intensity auwind / ( UWind) is around 0,12 and for such value Apng & (1,8±0,6).

For a significative wind speed of (Uwinj) ~10 m/s and a wind farm of a = 3 km longitudinal dimension, the cut-off frequency is in the order of 2,19 mHz.

In the H0vs0re wind farm, Along = 4 (about twice the value from RAL). The cut-off frequency of a longitudinal area with Along around 4 (dashed gray line in Fig. 11) is:

For a significative wind speed of (Uwind) ~ 10 m/s and a wind farm of a = 3 km longitudinal dimension, the cut-off frequency is in the order of 2,26 mHz. In accordance with experimental measurements, turbulence fluctuations quicker than a few minutes are notably smoothed in the wind farm output. This relation is proportional to the dimensions of the area where the wind turbines are sited. That is, if the dimensions of the zone are doubled, the area is four times the original region and the cut-off frequencies are halved. In other words, the smoothing of the aggregated wind is proportional to the longitudinal and lateral lengths (and thus, related to the square root of the area if zone shape is maintained).

In sum, the lateral cut-off frequency is inversely proportional to the site parameters Alat and the longitudinal cut-off frequency is only slightly dependent on Along, Note that the longitudinal cut-off frequency show closer agreement for H0vs0re and RAL since it is dominated by frozen turbulence hypothesis.

Fig. 11. Normalized ratio He(f) for transversal a « b (solid thick black line) and longitudinal a » b areas (dashed dark gray line for A¡ong = 4, long dashed light gray line for A¡ong = 1,8). Horizontal axis is expressed in either longitudinal or lateral adimensional frequency a Along f / < Urnnd) or b Alat f /< Uwnnd).

However, if transversal or longitudinal smoothing dominates, then the cut-off frequency is approximately the minimum of f ., , and f ,, . The system behaves as a first order

system at frequencies above both cut-off frequencies, and similar to a % order system between f and f .

Jcut,lat Jcut,long

Fig. 11. Normalized ratio He(f) for transversal a « b (solid thick black line) and longitudinal a » b areas (dashed dark gray line for A¡ong = 4, long dashed light gray line for A¡ong = 1,8). Horizontal axis is expressed in either longitudinal or lateral adimensional frequency a Along f / < Urnnd) or b Alat f /< Uwnnd).

However, if transversal or longitudinal smoothing dominates, then the cut-off frequency is approximately the minimum of f ., , and f ,, . The system behaves as a first order

system at frequencies above both cut-off frequencies, and similar to a % order system between f and f .

Jcut,lat Jcut,long

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