Physics of Wind Parks - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 134

6.2 Analytical Model for Mean Wind Speed in Wind Parks
In the 1990s reasoning on nearly infinitely large wind parks was a purely academic
exercise. Now, with the planning of large offshore wind farms off the coasts of the
continents and larger islands such exercises have got much more importance
(Barthelmie et al. 2005 ; Frandsen et al. 2006 , 2009 ). In principle, two different
approaches for modelling the effects of large wind parks are possible: a bottom-up
approach and a top-down approach. The bottom-up approach is based on a
superposition of the different wakes of the turbines in a wind park. It requires a
good representation of each single wake (see Sect. 6.1 ) in a three-dimensional flow
model (Lissaman 1979 ; Jensen 1983 ) and a wake combination model. Reviews are
given in Crespo et al ( 1999 ) and Vermeer et al ( 2003 ). Numerically, this approach
is supported by large-eddy simulations (LES) today (Wussow et al. 2007 ; Jimenez
et al. 2007 ; Steinfeld et al. 2010 ; Troldborg et al. 2010 ).
The top-down approach considers the wind park as a whole as an additional
surface roughness, as an additional momentum sink or as a gravity wave generator
in association with a temperature inversion aloft at the top of the boundary layer
(for the latter idea see Smith 2010 ), which modifies the mean flow above it
(Newman 1977 ; Bossanyi et al. 1980 ; Frandsen 1992 ). Crespo et al. ( 1999 ) rates
this latter class of models—although they have not been much used so far at that
time—as being interesting for the prediction of the overall effects of large wind
farms. Many of these models still have analytical solutions which make them
attractive, although they necessarily contain considerable simplifications. Never-
theless, they can be used for first order approximations in wind park design. More
detailed analyses require the operation of complex three-dimensional numerical
flow models on large computers in the bottom-up approach.
Smith ( 2010 ) uses an analogy to atmospheric flow over a mountain range in order
to derive his considerations. His model includes pressure gradients and gravity wave
generation associated with a temperature inversion at the top of the boundary layer
and the normal stable tropospheric lapse rate aloft. The pattern of wind disturbance is
computed using a Fast Fourier Transform. The slowing of the winds by turbine drag
and the resulting loss of wind farm efficiency is controlled by two factors. First is the
size of the wind farm in relation to the restoring effect of friction at the top and bottom
of the boundary layer. Second is the role of static stability and gravity waves in the
atmosphere above the boundary layer. The effect of the pressure perturbation is to
decelerate the wind upstream and to prevent further deceleration over the wind farm
with a favourable pressure gradient. As a result, the wind speed reduction in Smith's
( 2010 ) approach is approximately uniform over the wind farm. In spite of the uniform
wind over the farm, the average wind reduction is still very sensitive to the farm
aspect ratio. In the special case of weak stability aloft, weak friction and the Froude
Number close to unity, the wind speed near the farm can suddenly decrease; a
phenomenon that Smith ( 2010 ) calls 'choking'. We will not follow this idea here.
Rather, a top-down approach based on momentum extraction from the flow will be
presented in more detail in this subchapter.