Wake Decay for the Infinite Wind Turbine Array [1] Application of asymptotic speed deficit concept to existing engineering wake model Ole Rathmann, Sten

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


[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


[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


[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)


[5]


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


[6]


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


[7]


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).


[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.


[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


[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:


[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.


[12]


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


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




[13]



Wind speed:

8.5 m/s +/- 0.5 m/s

0 2000 4000 6000


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


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 speed:

12.0 m/s +/- 0.5 m/s

0 2000 4000 6000


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


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


[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


[15]

Comparative wind farm predictions: Nysted(2)

Wind direction: 278o +/- 2.5o


Wind direction: 263o +/- 2.5o


0 2000 4000 6000


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


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


[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.

Documents

Wake Decay for the Infinite Wind Turbine Array [1] Application of asymptotic speed deficit concept to existing engineering wake model Ole Rathmann, Sten