Wake Decay for the Infinite Wind Turbine Array
[1]
Application of asymptotic speed deficit concept to existing engineering wake model
Ole Rathmann, Sten Frandsen and Morten Nielsen
Risoe-DTU, National Laboratory for Sustainable Energy
Acknowledments:
Dong, Vattenfall for providing data
EU Upwind, Danish Strategic Research Council, Energinet.dk (PSO) for sponsoring the work
WAKE DECAY FOR THE INFINITE WIND TURBINE ARRAY
Technical topic Wakes – ID265
Wake Decay for the Infinite Wind Turbine Array
[2]
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
[3]
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 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 surface 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 2007
Wake Decay for the Infinite Wind Turbine Array
[4]
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
[5]
Asymptotic speed deficit from boundary layer considerations (2)
BL-Limited infinite wind farm
ct = π/8 Ct /(sr sf), t = ρCtUh2
sr and sf: dimensionless* WTG-distances (along- and across-wind) *by D rotor
Z<h: profile according to ground surface friction velocity u*i / roughness z0.
Z>h: profile according to increased friction velocity u*eff(= u*0) / roughness z0
eff (=z00).
Equivalent, effective surface roughness:
U h
U
ln (z )
H
G e o s tro p h ic w in d sp e e d GHzzU )(
O u te r la y e r
In n e r la y e r
R o u g h n e ss z 0 0
F ric tio n v e lo c ity 0u
0, zu i
H u b h e ig h t sh e a r 2htUcρt
Jump in friction velocity at hub-height due to rotor thrust: ρ(u*
eff)2 = ρ(u*i)2 + t
Approximation: homogeneously distributed thrust ct
2
0 0exp / / ln( / )effH t Hz h c h z
Wake Decay for the Infinite Wind Turbine Array
[6]
Asymptotic speed deficit from boundary layer considerations (3)
Approximate geostrophic drag-law
General hub-height wind speed:
Free flow: Flow over wind farm:
Relative speed deficit ε:
0
lnu G
G Afz
*
( )
1 ln
GU h
GA i
h f
0
0
1/ lnh
iz
2 20 , t
Tot add add
ci i i i
01 ln'
1
1 ln' Tot
Gi
h f
Gi
h f
Wake Decay for the Infinite Wind Turbine Array
[7]
Asymptotic speed deficit from boundary layer considerations (3)
Comparison with wind farm (Horns Rev):
sr ≈ sf ≈7, h=80m, DR = 60 m
Wake deficit about 50% of the BL-limiting value. Horns Rev wind farm NOT “infinite”.
0.65
0.7
0.75
0.8
0.85
0.9
4 6 8 10 12 14 16
Spee
d ra
tio
Free wind speed (m/s)
Ct=0.4, z0=0.01
Ct=0.4, Z0=0.0002
Ct=0.8, z0=0.01
Ct=0.8, z0=0.0002
Data
Horns RevDistance for severe wake interference
(kwake=0.075)
Actual extension
7.5 km 5 km
Power density (W/m2) [4]
Horns Rev 2MW turb’s(observed)
Entire North Sea5 MW turb’s(Frandsen – BL-limited)
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
[8]
WAsP Park Model Details
Wake Evolution and speed deficit [5,6]
wake expansion
coefficient k
A dir.o verlap ( = A 1(R ) )
A re f.o verlap
2
( )( ) 001 0 ( )
0 01 1
1 1 , ( ) " .", " ."2
type overlaptypet R
ADV U C type dir ref
D kX A
2 ( .) 2 ( .) 2, ,
. '
( ) ( )dir refturb i turb i turb
i upw turb s
V V V
Speed deficit from single wake:
Resulting speed deficit at a downwind turbine:
[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
[9]
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:
0
; , , / ,Parkr f R t
upwind
VG k s s h D C
U
The infinite sum may be approximated by an infinite integral - a simple function G:
Since Uupwind = Uw = Ufree-δV:
2
22 20 0
1
( ) ( ) ; ( ) ; 1 12
Rupwind j w j w t
j R
DV U N s x x C
D kx
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
0
0.5
1
1.5
2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Wake expansion coefficient k
Gpark
sr = sf = 7; h/DR=1
0; ( ; )1
appapp parkw
w wappFree w
VG layout k
U
Wake Decay for the Infinite Wind Turbine Array
[10]
Adjustment of the “WAsP Park” model
Adjustment to match the BL-based 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 kinf:
. ( ;[ , , , ]) ( , , , )infin park inf r f t BL based r f tV k s s h C V s s h C
Wa ke exp a nsion c oeffic ie nt k
O rig ina l Ad juste d to m a tc h a sym p to tic sp e e d d e fic it
The change of the wake expansion cofficient is indicated.
Model-paramters used in the following (based on Horns Rev data): kinitial:= 0.075 (recommended value for onshore!)
Frelax = 0.2
1 ( ) overlapadj adj adjj j inf j relax
w
Ak k k k F
A
segment ”j” segment ”j+1”
The k-change applies when a wake overlaps with a downwind rotor (to both wakes involved), using a relaxation factor Frelax:
Wake Decay for the Infinite Wind Turbine Array
[11]
Comparative wind farm predictions: Horns Rev (1)
Turbines: 2MW, DR = 80m, Hhub = 60m
Layout: sr = sf = 7
WT01
WT02
WT03
WT04
WT05
WT06
WT07
WT08
WT11
WT12
WT13
WT14
WT15
WT16
WT17
WT18
WT21
WT22
WT23
WT24
WT25
WT26
WT27
WT28
WT31
WT32
WT33
WT34
WT35
WT36
WT37
WT38
WT41
WT42
WT43
WT44
WT45
WT46
WT47
WT48
WT51
WT52
WT53
WT54
WT55
WT56
WT57
WT58
WT61
WT62
WT63
WT64
WT65
WT66
WT67
WT68
WT71
WT72
WT73
WT74
WT75
WT76
WT77
WT78
WT81
WT82
WT83
WT84
WT85
WT86
WT87
WT88
WT91
WT92
WT93
WT94
WT95
WT96
WT97
WT98
423 424 425 426 427 428 429 430
Easting (km) UTM Zone32
6147
6148
6149
6150
6151
6152
No
rth
ing
(km
)
222 deg.
270 deg.
Wake Decay for the Infinite Wind Turbine Array
[12]
Comparative wind farm predictions: Horns Rev (2)
Wind direction: 270o +/- 3o
Wind speed: 8.5 m/s +/- 0.5 m/s
0 2000 4000 6000
D ow nw ind d istance [m ]
0.7
0.8
0.9
1R
el.
red
uce
d w
ind
spee
dH orns R ev - 8 .5 m /s - 270 o
D ata
Pred. BL-Adjusted
Pred. S td. park
0 2000 4000 6000
D ow nw ind d istance [m ]
0 .7
0.8
0.9
1
Rel
. re
duce
d w
ind
spee
d
H o rns R ev - 12.0 m /s - 270 o
D ata - in t.line
D ata - ext.line
P red . B L-A djusted
P red . S td. park
Wind direction: 270o +/- 3o
Wind speed: 12.0 m/s +/- 0.5 m/s
Wake Decay for the Infinite Wind Turbine Array
[13]
Comparative wind farm predictions: Horns Rev (3)
Wind direction: 222o +/- 3o
Wind speed:
8.5 m/s +/- 0.5 m/s
0 2000 4000 6000
D ow nw ind d istance [m ]
0.7
0.8
0.9
1
Rel
. red
uced
win
d sp
eed
H orns R ev - 8 .5 m /s - 222 o (in t.line)D ata - in t.line
P red. BL-Adjusted
Pred. S td. park
0 2000 4000 6000
D ow nw ind d istance [m ]
0.7
0.8
0.9
1
Rel
. red
uced
win
d sp
eed
H orns R ev - 8 .5 m /s - 222 o (ext.line)D ata - in t.line
P red. BL-Adjusted
Pred. S td. park
Wind direction: 222o +/- 3o
Wind speed:
12.0 m/s +/- 0.5 m/s
0 2000 4000 6000
D ow nw ind d istance [m ]
0.7
0.8
0.9
1
Rel
. red
uced
win
d sp
eed
H orns R ev - 12 .0 m /s - 222 o (in t.line)D ata - in t.line
P red. BL-Adjusted
Pred. S td. park
0 2000 4000 6000
D ow nw ind d istance [m ]
0.7
0.8
0.9
1
Rel
. red
uced
win
d sp
eed
H orns R ev - 12 .0 m /s - 222 o (ext.line)D ata - in t.line
P red. BL-Adjusted
Pred. S td. park
Wake Decay for the Infinite Wind Turbine Array
[14]
Comparative wind farm predictions: Nysted (1)
Turbines: 2.33 MW, DR = 82m, Hhub = 69m
Layout: sr = 10.6, sf = 5.9
WT11WT21
WT31WT41
WT51WT61
WT71WT81
WT12WT22
WT32WT42
WT52WT62
WT72WT82
WT13WT23
WT33WT43
WT53WT63
WT73WT83
WT14WT24
WT34WT44
WT54WT64
WT74WT84
WT15WT25
WT35WT45
WT55WT65
WT75WT85
WT16WT26
WT36WT46
WT56WT66
WT76WT86
WT17WT27
WT37WT47
WT57WT67
WT77WT87
WT18WT28
WT38WT48
WT58WT68
WT78WT88
WT19WT29
WT39WT49
WT59WT69
WT79WT89
673 674 675 676 677 678
Easting (km) UTM Zone 32
6046
6047
6048
6049
6050
No
rth
ing
(km
)
Nysted
263
277
Wake Decay for the Infinite Wind Turbine Array
[15]
Comparative wind farm predictions: Nysted(2)
Wind direction: 278o +/- 2.5o
Wind speed: 10.0 m/s +/- 0.5 m/s
Wind direction: 263o +/- 2.5o
Wind speed: 10.2 m/s +/- 0.5 m/s
0 2000 4000 6000
D ow nw ind d istance [m ]
0.4
0.6
0.8
1
Rel
. re
duce
d tu
rbin
e po
wer
N ysted -10 .0 m /s - 278 o
D ata
Pred. BL-Adjusted
Pred. S td. park
0 2000 4000 6000
D ow nw ind d istance [m ]
0.6
0.8
1
1.2
Re
l. re
duce
d tu
rbin
e p
ower
N ysted -10.2 m /s - 263 o
D ata
Pred. BL-Adjusted
Pred. S td. park
Wake Decay for the Infinite Wind Turbine Array
[16]
Conclusions
• The adjustment of the wake expansion coefficient towards a value matching the BL-limited asymptotic speed deficit seems a valuable engineering 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 farms
• The model (relaxation factor) needs to be fine-tuned in order not to produce over estimations.
• The model needs to be tested on situations with wake effects between neighboring wind farms.
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