Turbine Wake Model for Wind Resource Software

EWEC 2006 Wind Energy Conference and Exhibition
The wake area is thus:
π
A ( ) ( ( )) 2
w
x = Dw x − ACut − off
(11)
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where the last terms takes cut-off into account if the wake hits the ground surface.
Now, each time a wake passes a turbine enshrouded within it, it must experience an expansion
corresponding to the stream tube area expansion, ΔA T , ( eq.(6)). To take this into account we have
modified the wake diameter model further as
1/2
⎡ x ⎤
Dw( x) = DR
max ⎢β,
Γ+α ⎥ (12)
⎣ DR
⎦
Here Γ is a dimensionless number, with a value of zero at x=0, and increasing in jumps every
time a turbine is passed as
A ( x, Γ+ΔΓ) −A ( x, Γ ) =Δ A (13)
w w T
The work by Frandsen et al. [6] indicates that the expansion coefficient α should vary with the
thrust coefficient – in fact the indications go that it should be proportionally to it. However, in the
present work the coefficient α is treated as a constant model parameter.
3.2 Mosaic-tiles model
The concept of the mosaic speed deficit distribution is illustrated in fig.3. For a mosaic wake
speed distribution the balance equation (9) takes the rather complicated form:
1
n
n
T = A δ (1 −δ ) (14)
2 ∑ i
( k) ( k) ( k)
U 1 1 ( k )
0 i =
∑∑
ρ J J J
k = J
A 1
A 12
A 123
Wake 1 A 1
A 123
Wake 2
A 12
Wake 1
Wake 2
Wake 3
Wake 3
A 13
Affected rotor
Affected rotor
Ground
Ground
Figure 3. Examples of overlapping wake “mosaic-tiles” configurations.
Here J (k) denotes a tile associated with a certain sub-set of k wakes, and A and δ indexed with this
symbol denotes the tile area and the relative speed deficit within the tile. E.g., when considering
n=3 wakes, J (1) can take the values [1] , [2] and [3], while J (2) may take the values [1,2], [1,3],
[2,3].
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