Turbine Wake Model for Wind Resource Software

EWEC 2006 Wind Energy Conference and Exhibition
3.3 Semi-linear model.
In wake calculations, for a certain downwind turbine rotor, the relevant property is the mean
value of the speed deficit over the rotor-plane area A RD :
1
A
= ∫∫ δ( y, zdydz )
(15)
RD
A
ARD
RD
Now, the semi-linear model is based on the observation that for small speed deficits, δ«1, the total
balance equation is linear in δ. This means that the contributions to the averaging integral from
the different upwind turbines are additive in the sense, that one may consider the effect of each
upwind turbine by itself, and then finally add up these contributions. This leads to the simple
linear approximation:
A
() i
1 [ Aw ( Δxi,
D) ∩ARD]
RD
A
≈ T
RD 2 ( i)
i
ρ U0 i Aw ( Δ xi,
D)
∑ (16)
() i
Here [ Aw
∩ A
RD]
indicates the overlapping area between the cross sectional area of wake no “i”
and the area A RD of the considered rotor, while Δx i,D is the downwind distance from the turbine
from which wake “i” originates.
In order to get the exact solution of the total balance equation in case of only a single wake
enshrouding entirely the downwind rotor, while having the same solution for multiple wakes in
the limit of small speed deficits, we adopt the following approximation to be used in the semilinear
model:
A
() i
1 [ Aw ( Δxi,
D) ∩ARD]
RD
A
(1 − )
RD
A
≈ T
RD
2 ( i)
i
ρ U0 i Aw ( Δ xi,
D)
∑ (17)
4 COMPARISON OF SEMI-LINEAR MODEL WITH OFF-SHORE
WIND FARM DATA
The wake model has been tested against accessible wind farm data. As a basis, for the wake
expansion coefficient a value of α =0.7 was used. However, as indicated in the figures, the
parameter was in some cases varied to see if better agreement with data could be obtained in this
way.
4.1 Middelgrunden Wind Farm data
Model predictions were compared to measured wind data from the Middelgrunden off-shore wind
farm just east of Copenhagen. Two southerly wind directions were considered. The first direction
(173°) was selected to be along the connection line between the two middle turbines, WT10 and
WT11, to get maximum wake effect in the middle of the wind farm. The second one (186°) was
selected to be along the connection line between the two southernmost turbines – WT20 and
WT19 - to get maximum wake effect at the latter. These situations are both indicated in the figure
4. The actual wind directions were measured by the yawing angle of the southern-most turbine
(WT20), while the wind speeds were deduced from the turbine power productions.
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