Pf Pf e Pt e2nf tdt LPt1

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The factor 1/VT is between unity -used for pulses and signals of bounded energy- and 1/ T -used in the Fourier coefficients of pure periodic signals-.

Fortunately, definition (1) has the advantage that the variance of P(f) is the two-sided auto spectral density, ^ P(f) |2 ^ = PSDp (f), which is independent of sample length T and it characterizes the process. P(f) will be referred as stochastic spectral phasor density of the active power or just the (stochastic) phasor for short.

Historically, the term "power spectral density" was coined when the signal analyzed P(t) was the electric or magnetic field of a wave or the voltage output of an antenna connected to a resistor R. The power transferred to the load R at frequencies between f-Af/2 and f+Af /2 was 2 Af-PSDp (f) /R -that is proportional to PSDp (f) and the frequency interval. If P(t) is the electric or magnetic field of a wave, then the power density at frequency f of that wave is also proportional to Af-PSDp (f).

In this chapter, P(t) represents the power output of a turbine or a wind farm. The root mean square value (RMS for short) of power fluctuations at frequencies between f-Af/2 and f+Af /2 is | P(f) I \l2-Af . Power variance inside the previous frequency range is PSDp (f )-Af . Hence, PSDp (f) in this chapter does not represent a power spectral density and this term can lead to misinterpretations. Therefore, PSDp (f) will be referred in this chapter as the auto spectral density although the acronym PSD (from Power Spectral Density) is maintained because it is widespread. Sometimes PSDp (f) will be replaced by ap (f) to emphasize that it represents the variance spectral density of signal P at frequency f. Fig. 3. shows the estimated PSD from 13 minute operation of a squirrel cage induction generator (SCIG) directly coupled to the grid (a portion of the original data is plotted in Fig. 1). The original auto spectrum is plotted in grey whereas the estimated PSD is in thin black

(linearly averaged periodogram in squared effective watts of real power per hertz). The trend is plotted in thick red, the accumulated variance is plotted in blue, and the tower shadow frequency is marked in yellow.

The instantaneous output of a wind farm or turbine can be expressed in frequency components using stochastic spectral phasor densities. As aforementioned, experimental measurements indicate that wind power nature is basically stochastic with noticeable fluctuating periodic components.

Wrmsrt2/Hz Averaged Periodogram from Power Spectrum ljOE+12-

ljOE+11 IjOE+IO ljOE+9 ljOE+8 ljOE+7 ljQE+6 ljOE+5 ljOE+4 ljOE+3

Fig. 3. PSDp+(f) parameterization of active power of a 750 kW wind turbine for wind speeds around 6,7 m/s (average power 190 kW) computed from 13 minute data.

The signal in the time domain can be computed from the inverse Fourier transform:

P(t) = Vt r P(f) ej2nftdf _ = 2Vt r P(f)cos [2nft + y(f) 1 df (2)

An analogue relation can be derived for reactive power and wind, both for continuous and discrete time. Standard FFT algorithms use two sided spectra, with negative frequencies in the last half of the output vector. Thus, calculus will be based on two-sided spectra unless otherwise stated, as in (2). In real signals, the negative frequency components are the complex conjugate of the positive one and a % scale factor may be applied to transform one to two-sided magnitudes. b) Spectral power balance in a wind farm

Fluctuations at the point of common coupling (PCC) of the wind farm can be obtained from power balance equations for the average complex power of the wind farm. Neglecting the increase in power losses in the grid due to fluctuating generation, the sum of oscillating power from the turbines equals the farm output undulation. Therefore, the complex sum of the frequency components of each turbine Pturune if) totals the approximate farm output, Prarm(f):

Wrmsrt2/Hz Averaged Periodogram from Power Spectrum ljOE+12-

Fig. 3. PSDp+(f) parameterization of active power of a 750 kW wind turbine for wind speeds around 6,7 m/s (average power 190 kW) computed from 13 minute data.

The signal in the time domain can be computed from the inverse Fourier transform:

—> turbines dP* —* turbines —> turbines . /

P farm(f) = £ Pturbinei(f) « £ V^rbine^ = £ VAubneO ^%f (3)

For usual wind farm configurations, total active losses at full power are less than 2% and reactive losses are less than 20%, showing a quadratic behaviour with generation level (Mur-Amada & Comech-Moreno, 2006). A small-signal model of power losses due to fluctuations inside the wind farm can be derived (Kundur et al. 1994), but since they are expected to be up to 2% of the fluctuation, the increase of power losses due to oscillations can be neglected in the first instance. A small signal model can be used to take into account network losses multiplying the turbine phasors in (3) by marginal efficiency factors n = BPjarm/dPturbme i estimated from power flows with small variations from the mean values using methodologies as the point-estimate method (Su, 2005; Stefopoulos et al., 2005). Typical values of r/i are about 98% for active power and about 85% for reactive power. In some expressions of this chapter, the efficiency has been set to 100% for clarity in the formulas. In some applications, we encounter a random signal that is composed of the sum of several random sinusoidal signals, e.g., multipath fading in communication channels, clutter and target cross section in radars, interference in communication systems, wave propagation in random media and channels, laser speckle patterns and light scattering and summation of random current harmonics such as the ones produced by high frequency power converters of wind turbines (Baghzouz et al., 2002; Tentzerakis & Papathanassiou, 2007). Any random sinusoidal signal can be considered as a random phasor, i.e., a vector with random length and angle. In this way, the sum of random sinusoidal signals is transformed into the sum of 2-D random vectors. So, irrespective of the type of application, we encounter the following general mathematical problem: there are vectors with lengths Pt =|P | and angles ^ = Arg(Pi), in polar coordinates, where Pi and ^ are random variables, as in (3) and Fig. 4. It is desired to obtain the probability density function (pdf) of the modulus and argument of the resulting vector. A comprehensive literature survey on the sum of random vectors can be obtained from (Abdi, 2000).

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Renewable Energy 101

Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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