Variability in wind power

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An example of short-term variations in wind speed at low height is given in Fig. 3.35. These fluctuations correspond to the region of frequencies above the spectral gap of Fig. 2.110. The occurrences of wind "gusts", during which the wind speed may double or drop to half the original value over a fraction of a second, are clearly of importance for the design of wind energy converters. On the other hand, the comparison made in Fig. 3.35 between two simultaneous measurements at distances separated horizontally by 90 m shows that little spatial correlation is present between the short-term fluctuations. Such fluctuations would thus be smoothed out by a wind energy conversion system, which comprises an array of separate units dispersed over a sufficiently large area.

The trends in amplitudes of diurnal and yearly wind speed variations are displayed in Fig. 3.36, as functions of height (Petersen, 1974). Such amplitudes are generally site-dependent, as one can deduce, for example, from Figs 3.26 and 3.28 for diurnal variations, and from Fig. 3.25 for seasonal variations. The diurnal amplitude in Fig. 3.26 diminishes with height, while the yearly amplitude increases with height. This is probably a quite general phenomenon when approaching the geostrophic wind, but the altitude dependence may depend on geographical position and local topography. At some locations, the diurnal cycle shows up as a 24 h peak in a Fourier decomposition of the wind speed data. This is the case at Ris0, Denmark, as seen from Fig. 3.37, while the peak is much weaker at the lower height used in Fig. 2.110. The growth in seasonal amplitude, with height is presumably determined by processes taking place at greater height (cf. Figs. 2.45 and 2.46) as well as by seasonal variations in atmospheric stability, etc.

Figure 3.35 (left). Short-term variation in wind speed. Simultaneous record at two locations 90 m apart (measuring height probably in the range 5— 10 m) (based on Banas and Sullivan, 1976).

Figure 3.36 (below). Height dependence of diurnal and yearly amplitude of oscillation of the wind speed at Ris0. The average estimate is based on a Fourier decomposition of the data, with no smoothing (Petersen, 1974).

Fourier Decomposition

Figure 3.36 (below). Height dependence of diurnal and yearly amplitude of oscillation of the wind speed at Ris0. The average estimate is based on a Fourier decomposition of the data, with no smoothing (Petersen, 1974).

Fourier Decomposition

The wind speed variance spectrum (defined in section 2.C in connection with Fig. 2.110) shown in Fig. 2.37 covers a frequency interval between the yearly period and the spectral gap. In addition to the 24 h periodicity, the am plitude of which diminishes with increasing height, the figure exhibits a group of spectral peaks with periods in the range 3-10 days. At the selected height of 56 m, the 4 day peak is the most pronounced, but moving up to 123 m, the peak period around 8 days is more marked (Petersen, 1974). It is likely that these peaks correspond to the passage time of typical meso-scale front and pressure systems.

In analysing the variability of wind speed and power during a certain period (e.g. month or a year), the measured data are conveniently arranged in the forms of frequency distributions and power duration curves, much in the same manner as discussed in section 3.1.5. Figure 3.38 gives the one-year frequency distribution of wind speeds at two Danish locations for a height of about 50m. The wind speed frequency distribution (dashed curve) at Gedser (near the Baltic Sea) has two maxima, probably associated with winds from the sea and winds approaching from the land side (of greater roughness). However, the corresponding frequency distribution of power (top curve) does not exhibit the lower peak, as a result of the cubic dependence on wind speed. At the Ris0 site, only one pronounced maximum is present in the wind speed distribution. Several irregularities in the power distribution (which are not preserved from year to year) bear witness to irregular landscapes with different roughness lengths, in different directions from the meteorological tower.

Wind Power Spectrum

Figure 3.37. Variance spectrum of horzon-tal wind speeds at Ris0 (0 = 56° N) at a height of 56 m (based on ten years of observation; Petersen, 1974). The spectrum is smoothed by using frequency intervals of finite length (cf. Fig. 2.110 of section 2.C, where similar data for a height of 7 m is given).

Despite the quite different appearance of the wind speed frequency distribution, the power distribution for the two Danish sites peaks at roughly the same wind speed, between 10 and 11 m s-1.

Power duration curves

On the basis of the frequency distributions of the wind speeds (or alternatively that of power in the wind), the power duration curves can be constructed, giving the percentage of time when the power exceeds a given value. Figures 3.39 and 3.40 give a number of such curves, based on periods of a year or more. In Fig. 3.39, power duration curves are given for four US sites which have been used or are being considered for wind energy conversion and for one of the very low-wind Singapore sites.

Height 50m. Gedser 1960/61. Height 56m,Ris0 1961

Height 50m. Gedser 1960/61. Height 56m,Ris0 1961

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Figure 3.38. Frequency distribution of wind speed (right-hand scale) and of power, for a height of about 50 m at two Danish sites. The middle curves give frequency distributions for the output of a typical wind energy generator (Sorensen, 1978).

Figure 3.38. Frequency distribution of wind speed (right-hand scale) and of power, for a height of about 50 m at two Danish sites. The middle curves give frequency distributions for the output of a typical wind energy generator (Sorensen, 1978).

In Fig. 3.40, power duration curves are given for the two different Danish sites considered in Fig. 3.38, as well as for a site on the west coast of Sweden, at three different heights. These three curves for the same site have a similar shape, but in general the shape of the power duration curves in Figs. 3.39 and 3.40 depends on location. Although the Swedish Ringhals curves have non-negligible time percentages with very large power, the Danish Gedser site has the advantage of a greater frequency of medium power levels. This is not independent of conversion efficiency, which typically has a maximum as function of wind speed.

Figure 3.40 (right). One-year duration curves of power in the wind at Scandinavian sites (including different heights at the Ringhals site) (based on data from Ljungström, 1975 (Ringhals); Petersen, 1974 (Riso); Jensen, 1962 (Gedser)).

Figure 3.39 (left). One-year duration curves of power in the wind, for a number of locations and heights (Swanson et al., 1975 (Plum Brook and Grandpa's Knob); Coste, 1976 (Waterford); Asmussen, 1975 (Ludington); Nathan et al., 1976 (Changi)).

Power Curve Order Magnitude

Figure 3.40 (right). One-year duration curves of power in the wind at Scandinavian sites (including different heights at the Ringhals site) (based on data from Ljungström, 1975 (Ringhals); Petersen, 1974 (Riso); Jensen, 1962 (Gedser)).

3.3 Ocean waves

The order of magnitude of the total energy in wave motion is about 10-3 of the total kinetic energy in the atmospheric wind systems, according to the rough estimate made in section 2.4.1 in connection with Fig. 2.86. The wave energy of about 10 kJ m-2 found as an annual average in the North Atlantic Ocean corresponds to the accumulated wind energy up to a height of about 200 m, according to Fig. 3.32. This implies that, although the amount of energy stored in waves is much smaller than the amount stored in wind, the wave energy may still be equivalent to the height-integrated fraction of wind energy accessible for practical use, at least at the current level of technology.

From the tentative estimates in section 2.4.1, the average turnover time for the energy in wave motion in the open ocean may be of the order of a few days. This time is consistent with average dissipation mechanisms, such as internal friction and weak wave-wave interactions, plus shoreline dissipation modes. The input of energy by the wind, on the other hand, seems to be an intermittent process which for extended intervals of time involves only slow transfer of energy between waves and turbulent wind components, or vice versa, and between wind and wave fields propagating in different directions (any angle from 0 to 2n). However, large amounts of energy may be transferred from wind to waves during short periods of time (i.e. "short periods" compared with the average turnover time). This suggests that the energy storage in waves may be varying more smoothly than the storage in wind (both waves and wind represent short-term stored solar energy, rather than primary energy flows, as discussed in connection with Fig. 2.86). As mentioned in section 2.3.2, the wave fields exhibit short-term fluctuations, which may be regarded as random. On a medium time-scale, the characteristics of the creation and dissipation mechanisms may make the wave energy a more "dependable" energy source than wind energy, but on a seasonal scale, the variations in wind and wave energy are expected to follow each other (cf. the discussion topic in section 6.5.3).

3.3.1 Wave spectra

The energy spectrum F(k) of a random wave field has been defined by (2.78). Since the wavelength (or wave number k) is very difficult to observe directly, it is convenient instead to express the spectrum in terms of the frequency, v = < (k) / 2n = 1/T,

The frequency is obtained from the period, T, which for a harmonic wave equals the zero-crossing period, i.e. the time interval between successive pas sages of the wave surface through the zero (average) height, in the same direction. The spectral distribution of energy, or "energy spectrum" when expressed in terms of frequency, F1(v ) = 2n F1(m ), is usually normalised to the total energy (Barnett and Kenyon, 1975), whereas the wavenumber-dependent spectrum F(k), defined in section 2.3.2, was normalised to the average potential energy. Thus f F1(m ) dm = Wotal = 2 f F(k) dk.

Ocean Energy Spectrum
Figure 3.41. Energy spectrum of waves estimated to be "fully developed", for the Atlantic Ocean (data have been grouped according to the wind speed at a height of 20 m) (based on Moskowitz, 1964).

Figure 3.42. Fetch-limited energy spectrum of waves in the southern part of the North Sea. The wind is blowing from the continental shore (westward from Helgoland) (based on Hasselmann et al., 1973).

Figure 3.42. Fetch-limited energy spectrum of waves in the southern part of the North Sea. The wind is blowing from the continental shore (westward from Helgoland) (based on Hasselmann et al., 1973).

Figure Wind Turbine Ship Collison

Figure 3.41 shows a set of measured energy spectra, F1, based on data from the Atlantic Ocean (Moskowitz, 1964). The wave fields selected were judged to correspond to "fully developed waves", and data corresponding to the same wind speed at a height of 20 m were averaged in order to provide the spectra shown in the figure. It is seen that the spectral intensity increases, and the fre quency corresponding to the peak intensity decreases, with increasing wind speed.

Based on the similarity theory of Monin and Obukhov (1954), Kitaigorod-skii (1970) suggested that F1 (ft ) g2pw1 V'5 (V being the wind speed) would be a universal function of ft V g1, both quantities being dimensionless. Based on the data shown in Fig. 3.41, which approximately satisfies Kitaigorodskii's hypothesis, Moskowitz (1964) suggested the following analytical form for the energy spectrum of fully developed gravity waves,

F1(ft ) = 8.1 x 10-3 pw g3 ft5 exp(-0.74 (V{z=20 m} ft/ g)-4).

The usefulness of this relation is limited by its sensitivity to the wind speed at a single height, and by the difficulty of determining whether a given wave field is "fully developed" or not.

If the wave field is "fetch-limited", i.e. if the wind has only been able to act over a limited length, then the energy spectrum will peak at a higher frequency, and the intensity will be lower, as shown in Fig. 3.42. Hasselmann et al. (1973) have generalised (3.27) to situations in which the wave field is not necessarily fully developed, according to such data.

Figure 3.43. Simultaneous spectra of energy (top) and its time-derivative (based on laboratory experiments by Mitsuyasu, 1968).

The position of the spectral peak will move downwards as a function of time owing to non-linear wave-wave interactions, i.e. interactions between different spectral components of the wave field, as discussed in connection with (2.80) (Hasselmann, 1962). This behaviour is clearly seen in the laboratory experiments of Mitsuyasu (1968), from which Fig. 3.43 shows an example of the time derivative of the energy spectrum, d F1 (v)/d t. Energy is transferred from middle to smaller frequencies. In other experiments, some transfer is also taking place in the direction of larger frequencies. Such a transfer is barely present in Fig. 3.43. The shape of the rate-of-transfer curve is similar to the one found in later experiments for real sea (Hasselmann et al., 1973), and both are in substantial agreement with the non-linear theory of Hasselmann.

Observations of ocean waves often exist in the form of statistics on the occurrences of given combinations of wave height and period. The wave height may be expressed as the significant height Hs or the root mean square displacement of the wave surface, Hrms. For a harmonic wave, these quantities are related to the surface amplitude, a, by a2 = 2 (Hrms)2 = H2 /8.

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Figure 3.44. Frequency distribution of wave heights and zero crossing periods for Station India (59°N, 19°W) in the North Atlantic (based on one year of observation by Draper and Squire (1967), quoted by Salter, 1974).

The period is usually given by the observed zero-crossing period. Figures 3.44 and 3.45 give examples of such measured frequency distributions, based on a year's observation at the North Atlantic Station "India" (59°N, 19°W, the data quoted by Salter, 1974), and at the North Sea Station "Vyl" (55.5°N, 5.0°E, based on data discussed in problem section 6.5.3). The probability of finding periods between 8 and 10 seconds and wave heights, a, between 0.3 and 1.5 m is quite substantial at the site "India", of the order of 0.3. Also the probability of being within a band of zero-crossing periods with a width of about 2 seconds and a centre which moves slowly upwards with increasing wave height is nearly unity (the odd values characterising the contour lines of Fig. 3.44 are due to the original definition of sampling intervals in terms of feet and Hs).

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Figure 3.45. Frequency distribution of significant wave heights and zero crossing periods for Station Vyl (55°N, 5°E) in the North Sea (based on one year of observation by Nielsen, 1977).

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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.

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