General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

Users may download and print one copy of any publication from the public portal for the purpose of private study or research.

You may not further distribute the material or use it for any profit-making activity or commercial gain

You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Downloaded from orbit.dtu.dk on: Dec 16, 2020

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]