Wake decay constant for the infinite wind turbine array ... · The infinite sum may be approximated...
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Wake decay constant for the infinite wind turbine arrayApplication of asymptotic speed deficit concept to existing engineering wake model
Rathmann, Ole; Frandsen, Sten Tronæs; Nielsen, Morten
Published in:EWEC 2010 Proceedings online
Publication date:2010
Document VersionPublisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):Rathmann, O., Frandsen, S. T., & Nielsen, M. (2010). Wake decay constant for the infinite wind turbine array:Application of asymptotic speed deficit concept to existing engineering wake model. In EWEC 2010 Proceedingsonline European Wind Energy Association (EWEA).
WAKE DECAY FOR THE INFINITE WIND TURBINE ARRAY
Application of asymptotic speed deficit concept to existing engineering wake model
Ole Rathmann, Sten Frandsen and Morten NielsenRisoe-DTU, National Laboratory for Sustainable Energy, y gy
Acknowledments:Dong, Vattenfall for providing dataEU Upwind, Danish Strategic Research Council, Energinet.dk (PSO) for sponsoring the work
Wake Decay for the Infinite Wind Turbine Array [1]Technical topic Wakes – ID265
WAKE DECAY FOR THE INFINITE WIND TURBINE ARRAY
Application of asymptotic speed deficit concept to existing engineering wake model
Outline• Background• Asymptotic speed deficit from boundary layer considerations• “WAsP Park” model details• Asymptotic speed deficit of the “WAsP Park” model• Adjustment of WAsP Park model• Comparative wind farm predictions• Conclusions
Wake Decay for the Infinite Wind Turbine Array [2]
Background
Very large wind farms: • Standard wake models seems to underpredict wake effects.
Recent investigations by Sten Frandsen [1, 2]: • The reason is the lack of accounting for the effect a large wind farm may have on The reason is the lack of accounting for the effect a large wind farm may have on
the atmospheric boundary layer, e.g. by modifying the vertical wind profile.
• In some way the effect of an extended wind farm resembles that of a change in y gsurface roughness: increased equivalent roughness length.
Idea:• While more detailed models are underway [3],
modify the existing WAsP Park engineering wind farm wake model to take this boundary-layer effect into account.
[1] Frandsen, S.T., Barthelmie, R.J., Pryor, S.C., Rathmann, O., Larsen, S., Højstrup, J. and M. Thøgersen ,Analytical modelling of wind speed deficit in large offshore wind farms. Wind Energ. 2006, 9: 39-54.[2] Frandsen, S: The wake-decay constant for the infinite row of wind turbine rotors. Draft paper (2009 ).[3] Rathmann O., Frandsen S, Barthelmie R., Wake Modelling for intermediate and large wind farms, Paper BL3.199, EWEC
Wake Decay for the Infinite Wind Turbine Array [3]
2007
Asymptotic speed deficit from boundary layer considerations (1)
When should a wind farm be considered as large/infinite?
(Hand drawing illustrating the initial idea)
Wake Decay for the Infinite Wind Turbine Array [4]
Asymptotic speed deficit from boundary layer considerations (2)
BL-Limited infinite wind farm
ln(z)
H
Geostrophic wind speed GHzzU => )(
Roughness z00
Uh
Outer layer
Friction velocity 0∗u
Hub height shear 2htUcρt =
Jump in friction velocity at hub-height due to rotor thrust: ρ(u*
eff)2 = ρ(u*i)2 + t
c = π/8 C /(s s ) t = ρC U 2U
Inner layer
0, zu i∗
Approximation: homogeneously distributed thrust ct
ct = π/8 Ct /(sr sf), t = ρCtUhsr and sf: dimensionless* WTG-distances (along- and across-wind) *by Drotor
Z<h: profile according to ground surface friction velocity u*i / roughness z0.Z>h: profile according to increased friction velocity u eff(= u ) / roughness z eff (=z )Z>h: profile according to increased friction velocity u*
eff(= u*0) / roughness z0eff (=z00).
Equivalent, effective surface roughness: ( )( )20 0exp / / ln( / )eff
H t Hz h c h zκ κ= ⋅ − +
Wake Decay for the Infinite Wind Turbine Array [5]
Asymptotic speed deficit from boundary layer considerations (3)
Approximate geostrophic drag-law lnu GG Afκ
∗∗
⎛ ⎞⎛ ⎞≈ −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
General hub-height wind speed:
0fzκ ⎜ ⎟⎝ ⎠⎝ ⎠
( ) GU hG
=⎛ ⎞
Free flow: Flow over wind farm:
*1 ln G A ih f
⎛ ⎞+ −⎜ ⎟⋅⎝ ⎠
0 1/ ln hi = 2 2
0 , tTot add add
ci i i i= + =
R l i d d fi i
00z 0 ,Tot add add κ
01 ln'
G ih f
⎛ ⎞+ ⎜ ⎟⋅⎝ ⎠Relative speed deficit ε: 1
1 ln' Tot
h fG i
h f
ε ⎝ ⎠− =⎛ ⎞
+ ⎜ ⎟⋅⎝ ⎠
Wake Decay for the Infinite Wind Turbine Array [6]
Asymptotic speed deficit from boundary layer considerations (3)
Comparison with wind farm (Horns Rev):sr ≈ sf ≈7, h=80m, DR = 60 m 0.85
0.9
C 0 4 0 0 01
Wake deficit about 50% of the BL-limiting value. Horns Rev wind farm NOT “infinite”
0.75
0.8
Spee
d ratio
Ct=0.4, z0=0.01
Ct=0.4, Z0=0.0002
Ct=0.8, z0=0.01
Ct=0.8, z0=0.0002
Horns Rev wind farm NOT infinite .
0.65
0.7
4 6 8 10 12 14 16
Data
4 6 8 10 12 14 16
Free wind speed (m/s)
Horns Rev Power density (W/m2) [4]Horns RevDistance for severewake interference
(kwake=0.075)
Actual extension
Power density (W/m ) [4]Horns Rev 2MW turb’s(observed)
Entire North Sea5 MW turb’s(Frandsen – BL-limited)
7.5 km 5 km 2.9 1.8
[4] Barthelmie,R.J., Frandsen,S.T., ., Pryor, S.C., Energy dynamics of an infinitely large offshore wind farm., Paper 124, European Offshore WInd 2009 Conference and Exhibition Stockholm Sweden (Sept 2009)
Wake Decay for the Infinite Wind Turbine Array [7]
European Offshore WInd 2009 Conference and Exhibition, Stockholm, Sweden (Sept. 2009).
WAsP Park Model Details
Wake Evolution and speed deficit [5,6]
Adir.overlap( =A1(R) )
Aref.overlap
( )2
( )( ) 01 1 ( ) " " " "type overlaptype ADV U C type dir refδ⎛ ⎞⎜ ⎟
Speed deficit from single wake:
( ) ( )( ) 001 0 ( )
0 01 1
1 1 , ( ) " .", " ."2
yp pypt RV U C type dir ref
D kX Aδ = − − =⎜ ⎟+⎝ ⎠
Resulting speed deficit at a downwind turbine:
( )2 ( .) 2 ( .) 2, ,
. '( ) ( )dir ref
turb i turb i turbi upw turb s
V V Vδ δ δ∈
= +∑[5] N.O.Jensen, A Note on Wind Generator Interaction, Risø-M-2411, Risø National Laboratory 1983.[6] I Katic J Højstrup N O Jensen A Simple Model for Cluster Effeciency Paper C6 EWEC 2006 Rome Italy 1986
Wake Decay for the Infinite Wind Turbine Array [8]
[6] I.Katic, J.Højstrup, N.O.Jensen, A Simple Model for Cluster Effeciency, Paper C6, EWEC 2006, Rome, Italy, 1986.
Asymptotic speed deficit of the “WAsP Park” model
Speed deficit for a turbine in an infinite wind farm Speed deficit the same for all turbines, thus also the turbine thrusts.Infinite (convergent!!) sum:
( ) ( ) ( )2
22 20 0
1
( ) ( ) ; ( ) ; 1 12
Rupwind j w j w t
j R
DV U N s x x CD kx
δ ε ε ε ε∞
=
⎛ ⎞= = = − −⎜ ⎟+⎝ ⎠
∑
The infinite sum may be approximated by
xj: Distance to upwind turbine row j. N(xj): number of turbines row j throwing wake on the rotor in focus.Uupwind: Wind speed immediately upwind of a turbine
Gpark
sr = sf = 7; h/DR=1
( ); , , / ,Parkr f R t
V G k s s h D CU
δ=
The infinite sum may be approximated by an infinite integral - a simple function G:
1
1.5
2
sr sf 7; h/DR 1
( )0
fupwindU ε
Since Uupwind = Uw = Ufree-δV: 0
0.5
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Wake expansion coefficient k
0; ( ; )1
appapp parkw
w wappFree w
V G layout kU
εδ ε ε εε
= = =+
Wake Decay for the Infinite Wind Turbine Array [9]
Adjustment of the “WAsP Park” model
Adjustment to match the BLbased asymptotic speed deficitFor “deep” positions the wake expansion coefficient k of the Park Model is modified to approach the BL based asymptotic speed deficit value k :approach the BL-based asymptotic speed deficit value kinf:
. ( ;[ , , , ]) ( , , , )infin park inf r f t BL based r f tV k s s h C V s s h Cδ δ −=
Wake expansion coefficient kOriginal Adjusted to match asymptotic speed defic it
The k-change applies when a wake overlaps with a downwind rotor (to both wakes involved), using a relaxation f t F
1 ( ) overlapadj adj adjj j inf j relax
w
Ak k k k F
A+ = + −
factor Frelax:
The change of the wake expansion cofficient is indicated.
Model paramters used in the following (basedModel-paramters used in the following (based on Horns Rev data):
kinitial:= 0.075 (recommended value for onshore!)Frelax = 0.2segment ”j” segment ”j+1”
Wake Decay for the Infinite Wind Turbine Array [10]
relax g j segment ”j+1”
Comparative wind farm predictions: Horns Rev (1)
Turbines: 2MW, DR = 80m, Hhub = 60mLayout: sr = sf = 7
Wake Decay for the Infinite Wind Turbine Array [11]
Comparative wind farm predictions: Horns Rev (2)
1
ed
Horns Rev - 8.5 m/s - 270o
DataPred. BL-AdjustedPred. Std. park
Wind direction: 270o +/- 3o
Wind speed: 8.5 m/s +/- 0.5 m/s
0.9el
. red
uced
win
d sp
ee
p
0 2000 4000 60000.7
0.8Re
Downwind distance [m]
1
d
Horns Rev - 12.0 m/s - 270o
Data - int.lineData - ext.linePred. BL-AdjustedP d Std k
0.9
el. r
educ
ed w
ind
spee
d Pred. Std. park
Wind direction: 270o +/- 3o
Wind speed: 12.0 m/s +/- 0.5 m/s
0 2000 4000 60000.7
0.8Re
Wake Decay for the Infinite Wind Turbine Array [12]
Downwind distance [m]
Comparative wind farm predictions: Horns Rev (3)
Wind direction:1
Horns Rev - 8.5 m/s - 222o (int.line)Data - int.linePred. BL-AdjustedPred. Std. park
1Horns Rev - 8.5 m/s - 222o (ext.line)
Data - int.linePred. BL-AdjustedPred. Std. park Wind direction:
222o +/- 3o
Wind speed:8.5 m/s +/- 0.5 m/s
0 8
0.9
Rel
. red
uced
win
d sp
eed
0 8
0.9
Rel
. red
uced
win
d sp
eed
Pred. Std. park
0 2000 4000 6000
Downwind distance [m]
0.7
0.8R
0 2000 4000 6000
Downwind distance [m]
0.7
0.8R
Wind direction:1
d
Horns Rev - 12.0 m/s - 222o (int.line)Data - int.linePred. BL-AdjustedPred. Std. park
1ed
Horns Rev - 12.0 m/s - 222o (ext.line)Data - int.linePred. BL-AdjustedPred. Std. park Wind direction:
222o +/- 3o
Wind speed:
12.0 m/s +/- 0.5 m/s0.8
0.9
Rel
. red
uced
win
d sp
eed
0.8
0.9
Rel
. red
uced
win
d sp
ee
0 2000 4000 6000
Downwind distance [m]
0.70 2000 4000 6000
Downwind distance [m]
0.7
Wake Decay for the Infinite Wind Turbine Array [13]
Comparative wind farm predictions: Nysted (1)
Turbines: 2.33 MW, DR = 82m, Hhub = 69mLayout: sr = 10.6, sf = 5.9
Wake Decay for the Infinite Wind Turbine Array [14]
Comparative wind farm predictions: Nysted(2)
1e
pow
erNysted -10.0 m/s - 278o
DataPred. BL-AdjustedPred. Std. park
Wind direction: 278o +/- 2.5o
Wind speed: 10.0 m/s +/- 0.5 m/s0.6
0.8
Rel
. red
uced
turb
ine
0 2000 4000 6000
Downwind distance [m]
0.4
1
1.2
pow
er
Nysted -10.2 m/s - 263o
DataPred. BL-AdjustedPred. Std. park
Wind direction: 263o +/- 2.5o
Wind speed: 10.2 m/s +/- 0.5 m/s0.8
Rel
. red
uced
turb
ine
p
0 2000 4000 6000
Downwind distance [m]
0.6
R
Wake Decay for the Infinite Wind Turbine Array [15]
Downwind distance [m]
Conclusions
• The adjustment of the wake expansion coefficient towards a value matching the BL-limited asymptotic speed deficit seems a valuable engineering approachengineering approach
• A value for the wake expansion coefficient close to that normally used for onshore – locations seems reasonable in this approach also for off-shore wind farmswind farms
• The model (relaxation factor) needs to be fine-tuned in order not to produce over estimations.Th d l d t b t t d it ti ith k ff t b t• The model needs to be tested on situations with wake effects between neighboring wind farms.
Wake Decay for the Infinite Wind Turbine Array [16]