Sum of partially correlated phasor densities of power from several turbines

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4.1 Sum of fully correlated and fully uncorrelated spectral components

If turbine fluctuations at frequency f of a wind farm with N turbines are completely synchronized, all the phases have the same value f) and the modulus of fully correlated fluctuations IP+

sum arithmetically:

If there is no synchronization at all, the fluctuation angles ^i(f) at the turbines are stochastically independent. Since Pi, uncorr (f) has a random argument, its sum across the wind farm will partially cancel and inequality (13) holds true.

farm, uncorr\

This approach remarks that correlated fluctuations adds arithmetically and they can be an issue for the network operation whereas uncorrelated fluctuations diminish in relative terms when considering many turbines (even if they are very noticeable at turbine terminals).

A) Sum of uncorrelated fluctuations

The fluctuation of power output of the farm is the sum of contributions from many turbines (3), which are mainly uncorrelated at frequencies higher than a tenth of Hertz. The sum of N independent phasors of random angle of N equal turbines in the farm converges asymptotically to a complex Gaussian distribution, Pfarm (f) ~ CN[0, &pfarm(f)], of null mean and standard deviation o-farm(f) = nNa^f), where &i(f) is the mean RMS fluctuation at a single turbine at frequency f and n is the average efficiency of the farm network. To be precise, the variance o-^f) is half the mean squared fluctuation amplitude at frequency f, a^ (f) = ± ( | Pturbinei (f) Q = ( Re2 [ PturUne i (f) ]) = (Im2 [ PturMne i (f) ]) . Therefore, the real and imaginary phasor components Re[Pfarm (f)] and Im[Pfarm (f)] are independent real Gaussian random variables of standard deviation aPfarm(f) and null mean since phasor argument is uniformly distributed in [-n,+n]. Moreover, the phasor modulus |P/arm (f) | has Rayleigh[aPfarm (f)] distribution. The double-sided power spectrum |Pfarm (f )| is an Exponential [ A = 2 ap>farm(f) ] random vector of mean ^|Pfarm (f)f^ = 2 apfarmf) = 2 PSDPfarm (f) (Cavers, 2003).

The estimate from the periodogram is the moving average of NaveT. exponential random variables corresponding to adjacent frequencies in the power spectrum vector. The estimate is a Gamma random variable. If the PSD is sensibly constant on Nav<rAf bandwidth, then the PSD estimate has the same mean as the original PSD and the standard deviation is -J Naver times smaller (i.e., the estimate has lower uncertainty at the cost of lower frequency resolution).

4.2 Sum of partially linearly correlated spectral components

Inside a farm, the turbines usually exhibit a similar behaviour for a given frequency f and the PSD of each turbine is expected to be fairly similar. However, the phase differences among turbines do vary with frequency. Slow meteorological variations affect all the turbines with negligible time lag, compared to characteristic time frame of weather systems (i.e., the phasors Pturbine{f) have the same phase). Turbulences with scales significantly smaller than the turbine distances have uncorrelated phases. Fluctuations due to rotor positions also show uncorrelated phases provided turbines are not synchronized.

\ turbine ^ ' l \ turb,corr\J ' t 1 \ turb,uncorr\J ' !

If the number of turbines N >4 and the correlation among turbines are linear, the central limit is a good approximation. The correlated and uncorrelated components sum quadratically and the following relation is applicable:

where N is the number of turbines in the farm (or in a group of close farms) and rj is the average efficiency of the farm network (typical values are about 98% for active power and about 85% for reactive power). Since phasor densities sum quadratically, (14) and (15) are concisely expressed in terms of the PSD of correlated and uncorrelated components of phasor density:

PSDJarJf) ~ {rlN )2 PSDturb, oJS) + VN PSDturb, unc0rr (f) (16)

PSDturb(f) — PSDturb, corr(f) + PSDturb, uncorr(f) (17)

The correlated components of the fluctuations are the main source of fluctuation in large clusters of turbines. The farm admittance J(f) is the ratio of the mean fluctuation density of the farm, ^ | P farm (f) | ^ , to the mean turbine fluctuation density, ^ | P+rbme(f) | ^ ■

Note that the phase of the admittance J(f) has been omitted since the phase lag between the oscillations at the cluster and at a turbine depend on its position inside the cluster. The admittance is analogous to the expected gain of the wind farm fluctuation respect the turbine expected fluctuation at frequency f (the ratio is referred to the mean values because both signals are stochastic processes).

Since turbine clusters are not negatively correlated, the following inequality is valid:

The squared modulus of the admittance J(f) is conveniently estimated from the PSD of the turbine cluster and a representative turbine using the cross-correlation method and discarding phase information (Schwab et al., 2006):

t2,a PSDPfarm(f) / ,T\2 PSDturb,corr(f) ,PSDturb,uncorr(f)

PSDPturb (f) PSDtuJf) PSDUf)

If the PSD of a representative turbine, PSDpturb(f), and the PSD of the farm PSDpfarm(f) are available, the components PSD, , (f) and PSD, , (f) can be estimated from (16)

and (17) provided the behaviour of the turbines is similar.

At f ^ 0,01 Hz, fluctuations are mainly correlated due to slow weather dynamics, PSD, , (f) ^ PSD, , (f) , and the slow fluctuations scale proportionally turbuncorrv' turb,corr^-J' ' £ £ J

pSDpfamn(f) - (nNY PSDturbcoJf) . At f > 0,01 Hz, individual fluctuations are statistically independent, PSD , (f) ^ PSD, , (f) , and fast fluctuations are partially attenuated,

' turbuncorr^'' turb,corr^J' ' 1 J PSDPfam(f) - VN PSDturb,uncorr(f) .

An analogous procedure can be replicated to sum fluctuations of wind farms of a geographical area, obtaining the correlated PSD (f) and uncorrelated PSD (f)

components. The main difference in the regional model -apart from the scattered spatial region and the different turbine models- is that wind farms must be normalized and an average farm model must be estimated for reference. Therefore, the average farm behaviour is a weighted average of individual farms with lower characteristic frequencies (Norgaard & Holttinen, 2004). Recall that if hourly or even slower fluctuations are studied, meteorological dynamics are dominant and other approaches are more suitable.

4.3 Estimation of wind farm power admittance from turbine coherence

The admittance can be deducted from the farm power balance (3) if the coherence among the turbine outputs is known. The system can be approximated by its second-order statistics as a multivariate Gaussian process with spectral covariance matrix 2p(f) . The elements of 2p(f) are the complex squared coherence at frequency f and at turbines i and j, noted as Yij (f). The efficiency of the power flow from the turbine i to the farm output can be expressed with the column vector rjp = [V\, V2,---, Vn ], where T denotes transpose. Therefore, the wind farm power admittance J(f) is the sum of all the coherences, multiplied by the efficiency of the power flow:

The squared admittance for a wind farm with a grid layout of niong columns separated diong distance in the wind direction and niat rows separated dat distance perpendicular to the wind

Uwind is:

2n(VjlKnf

The admittance computed for Horns Rev offshore wind farm (with a layout similar to Fig. 10) is plotted in Fig. 9. According to (S0rensen et al., 2008), it has 80 wind turbines disposed in a grid of niat = 8 rows and niong = 10 columns separated by seven diameters in each direction (diat = dtong = 560 m), high efficiency (n ~ 100%), lateral coherence decay factor Am ~ Uwind/(2 m/ s), longitudinal coherence decay factor Along « 4, wind direction aligned with the rows and Uwind « 10 m/ s wind speed.

4.4 Estimation of wind farm power admittance from the wind coherence

The wind farm admittance J(f) can be approximated from the equivalent farm wind because the coherence of power and wind are similar (the transition frequency between correlated and uncorrelated behaviour is about 10-2 Hz for small wind farms). According to (Mur-Amada, 2009), the equivalent wind can be roughly approximated by a multivariate

80 n

70 50

15 10

10 20 50 100 200 500 1000 2000 Frequency LJyclesLdayL

Fig. 9. Admittance for Horns Rev offshore wind farm for 10 m/s and wind direction aligned with the turbine rows.

Gaussian process with spectral covariance matrix SUgq(f). Its elements are the complex coherence of effective turbulence at frequency f and at turbines i and j, denoted by jtj (f ). In this case, the column vector r/ueq = [ ^nY should be interpreted as the relative sensitivity of the farm power respect the equivalent wind in each turbine. Therefore, the wind farm power admittance J(f) is the sum of the complex coherence of effective quadratic turbulence among turbines:

For the rectangular region shown in Fig. 10, the admittance is:

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