Wind Energy Handbook. Michael Barton Graham

Wind Energy Handbook

2.6 shows a similar figure for a typical cut‐out wind speed of 25 m/s. All peaks have moved to higher frequency as expected, but the modified von Karman now matches IEC editions 3 and 4 better with the very small roughness length.

2.6.6 Asymptotic limits

Other spectra may also be used, but to comply with the IEC standard the high frequency asymptotic behaviour must tend to the following relationship:

(2.35) StartFraction upper S Subscript u Baseline left-parenthesis n right-parenthesis Over sigma Subscript u Superscript 2 Baseline EndFraction long right-arrow Underscript n equals infinity Endscripts 0.05 left-parenthesis StartFraction normal upper Lamda 1 Over upper U overbar EndFraction right-parenthesis Superscript negative 2 slash 3 Baseline n Superscript negative 5 slash 3

with Λ₁ defined as above: it is a function only of height above ground but differs between edition 2 and edition 3 for heights above 30 m. Expressing this as

(2.36) StartFraction upper S Subscript u Baseline left-parenthesis n right-parenthesis Over sigma Subscript u Superscript 2 Baseline EndFraction long right-arrow Underscript n equals infinity Endscripts normal upper A upper U overbar Superscript 2 slash 3 Baseline n Superscript negative 5 slash 3

the asymptotic parameter A can then be compared for different spectra, as in Figure 2.7. This shows how the DS 472 asymptote is similar to IEC edition 2 (both Kaimal and von Karman spectra in that standard have the same asymptote), but the asymptote becomes much lower for IEC editions 3 and 4 above 30 m height and is now more comparable with Eurocode. The ESDU modified von Karman spectrum is more difficult to characterise because the asymptote now varies also with wind speed, surface roughness, and geographical latitude. Figure 2.7 shows the results for 20 m/s wind speed and 50° latitude for two different roughness lengths. The asymptote can be made to match the IEC editions 3 and 4 specification, but only by choosing a very small roughness length – even smaller to match edition 2. However, if such a low roughness length is selected, the turbulence intensity implied by the ESDU model will be much smaller than that required by the standards. It is common practice when using the ESDU model to adjust the surface roughness at each wind speed until the turbulence intensity matches the standard, although this clearly makes little physical sense. Obviously it is not possible to adjust the roughness to match both the required asymptote and the required turbulence intensity at the same time. Compared to the physical model, therefore, the standards are probably conservative, which is surely to be expected. It can also be argued that the physical model is valid for flat terrain, while many wind farms are built in complex terrain where the turbulence intensities will indeed be higher, and the length scales shorter.

Figure 2.7 Some asymptotic limits

Note also that the IEC edition 3 and 4 standards further specify that upper S Subscript v Baseline left-parenthesis n right-parenthesis equals upper S Subscript w Baseline left-parenthesis n right-parenthesis equals four thirds upper S Subscript u Baseline left-parenthesis n right-parenthesis in the high frequency limit.

2.6.7 Cross‐spectra and coherence functions

The turbulence spectra presented in the preceding sections describe the temporal variation of each component of turbulence at any given point. However, as the wind turbine blade sweeps out its trajectory, the wind speed variations it experiences are not well represented by these single‐point spectra. The spatial variation of turbulence in the lateral and vertical directions is clearly important, because this spatial variation is ‘sampled’ by the moving blade and thus contributes to the temporal variations experienced by it.

To model these effects, the spectral description of turbulence must be extended to include information about the cross‐correlations between turbulent fluctuations at points separated laterally and vertically. Clearly these correlations decrease as the distance separating two points increases. The correlations are also smaller for high frequency than for low frequency variations. They can therefore be described by ‘coherence’ functions, which describe the correlation as a function of frequency and separation. The coherence C (Δr,n) is defined by

(2.37) upper C left-parenthesis normal upper Delta r comma n right-parenthesis equals StartFraction StartAbsoluteValue upper S 12 left-parenthesis n right-parenthesis EndAbsoluteValue Over StartRoot upper S 11 left-parenthesis n right-parenthesis upper S 22 left-parenthesis n right-parenthesis EndRoot EndFraction

where n is frequency, S₁₂(n) is the cross‐spectrum of variations at the two points separated by Δr, and S₁₁(n) and S₂₂(n) are the spectra of variations at each of the points (usually these can be taken as equal).

Starting from von Karman spectral equations, and assuming Taylor's frozen turbulence hypothesis, an analytical expression for the coherence of wind speed fluctuations can be derived. Accordingly for the longitudinal component at points separated by a distance Δr perpendicular to the wind direction, the coherence C_u (Δr,n) is:

(2.38)

Here A_j(x) = x^j K_j(x) where K is a fractional order modified Bessel function, and

(2.39) eta Subscript u Baseline equals normal upper Delta r StartRoot left-parenthesis StartFraction 0.747 Over upper L Subscript u Baseline EndFraction right-parenthesis squared plus left-parenthesis c StartFraction 2 pi n Over upper U overbar EndFraction right-parenthesis squared EndRoot

with c = 1. L_u is a local length scale that can be defined as

(2.40)

where Δy and Δz are the lateral and vertical components of the separation Δr, and ^yL_u and ^zL_u are the lateral and vertical length scales for the longitudinal component of turbulence. Normally f_u(n) = 1, but ESDU (1975) suggests a modification at low frequencies where the wind becomes more

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