Frandsen, Sten Tronæs

Nielsen, Morten

Rathmann, Ole

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Wake decay constant for the infinite wind turbine arrayApplication of asymptotic speed deficit concept to existing engineering wake modelRathmann, Ole Steen; Frandsen, Sten Tronæs; Nielsen, MortenPublished in:EWEC 2010 Proceedings onlinePublication date:2010Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (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 ARRAYApplication of asymptotic speed deficit concept to existing engineering wake modelOle Rathmann, Sten Frandsen and Morten NielsenRisoe-DTU, National Laboratory for Sustainable EnergyAcknowledments:Dong, Vattenfall for providing dataEU Upwind, Danish Strategic Research Council, Energinet.dk (PSO) for sponsoring the workWake Decay for the Infinite Wind Turbine Array [1]Technical topic Wakes – ID265WAKE DECAY FOR THE INFINITE WIND TURBINE ARRAYApplication of asymptotic speed deficit concept to existing engineering wake modelOutline• 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• ConclusionsWake Decay for the Infinite Wind Turbine Array [2]BackgroundVery 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 Wake Decay for the Infinite Wind Turbine Array [3]2007Asymptotic 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 ctt   t r f ,   t hsr and sf: dimensionless* WTG-distances (along- and across-wind)                *by DrotorZ<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 )       *  *0    0 00 .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: 0z( ) GU hG= ⎛ ⎞Free flow:                        Flow over wind farm: *1 ln A ih f+ −⎜ ⎟⋅⎝ ⎠ 0 1/ lnhi = 2 20 , tTot add add ci i i i= + =R l i d d fi i0z κ01 ln 'G ih f⎛ ⎞+ ⎜ ⎟⋅⎝ ⎠e at ve spee  e c t ε: 11 ln' TotG ih fε− = ⎛ ⎞+ ⎜ ⎟⋅⎝ ⎠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 0.850.9C 0 4 0 0 01Wake deficit about 50% of the BL-limiting value. Horns Rev wind farm NOT “infinite”0.750.8Speed ratiot= . , z = .Ct=0.4, Z0=0.0002Ct=0.8, z0=0.01Ct=0.8, z0=0.0002     .0.650.74 6 8 10 12 14 16DataFree wind speed (m/s)Horns Rev Power density (W/m2) [4] Distance for severewake interference(kwake=0.075)Actual extension  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]      , ,  . .WAsP Park Model DetailsWake 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:01 0 ( )0 01 1, . , .2t RD kX A= − − =+⎝ ⎠Resulting speed deficit at a downwind turbine:( )2 ( .) 2 ( .) 2, ,. '( ) ( )dir refturb i turb i turbi upw turb sV 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 1986Wake Decay for the Infinite Wind Turbine Array [8] . , . , . . ,      ,  ,  , , , .Asymptotic speed deficit of the “WAsP Park” modelSpeed deficit for a turbine in an infinite wind farm Speed deficit the same for all turbines, thus also the turbine thrusts.Infinite (convergent!!) sum: ( ) ( ) ( )222 20 01( ) ( ) ; ( ) ; 1 12Rupwind j w j w tj RDV U N s x x CD kxδ ε ε ε ε∞=⎛ ⎞= = = − −⎜ ⎟+⎝ ⎠∑The infinite sum may be approximated byxj: 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 turbineGparksr = sf = 7; h/DR=1( ); , , / ,Park r f R tV G k s s h D CU δ =       an infinite integral - a simple function G:11.52     0upwindεSince Uupwind = Uw = Ufree-δV: 00.50 0.02 0.04 0.06 0.08 0.1 0.12 0.14Wake expansion coefficient  k0; ( ; )1appapp parkww wappFree wV G layout kUεδ ε ε εε= = =+Wake Decay for the Infinite Wind Turbine Array [9]Adjustment of the “WAsP Park” modelAdjustment 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 :  -      inf. ( ;[ , , , ]) ( , , , )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 itThe k-change applies when a wake overlaps with a downwind rotor (to both wakes involved), using a relaxation f t F1 ( )overlapadj adj adjj j inf j relaxwAk k k k FA+= + −ac or relax: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 = 0.2segment ”j” segment ”j+1”Wake Decay for the Infinite Wind Turbine Array [10]  Comparative wind farm predictions: Horns Rev (1)Turbines:   2MW, DR = 80m, Hhub = 60mLayout: sr = sf = 7Wake Decay for the Infinite Wind Turbine Array [11]Comparative wind farm predictions: Horns Rev (2)1edHorns Rev - 8.5 m/s - 270oDataPred. BL-AdjustedPred. Std. parkWind direction: 270o +/- 3oWind speed: 8.5 m/s +/- 0.5 m/s0.9el. reduced wind spee0 2000 4000 60000.70.8ReDownwind distance [m]1dHorns Rev - 12.0 m/s - 270oData - int.lineData - ext.linePred. BL-AdjustedP d Std k0.9el. reduced wind speed re . . parWind direction: 270o +/- 3oWind speed: 12.0 m/s +/- 0.5 m/s0 2000 4000 60000.70.8ReWake Decay for the Infinite Wind Turbine Array [12]Downwind distance [m]Comparative wind farm predictions: Horns Rev (3)Wind direction:1Horns Rev - 8.5 m/s - 222o (int.line)Data - int.linePred. BL-AdjustedPred. Std. park1Horns Rev - 8.5 m/s - 222o (ext.line)Data - int.linePred. BL-AdjustedPred. Std. park  222o +/- 3oWind speed:8.5 m/s +/- 0.5 m/s0 80.9Rel. reduced wind speed0 80.9Rel. reduced wind speed0 2000 4000 6000Downwind distance [m]0.7.R0 2000 4000 6000Downwind distance [m]0.7.RWind direction:1dHorns Rev - 12.0 m/s - 222o (int.line)Data - int.linePred. BL-AdjustedPred. Std. park1edHorns Rev - 12.0 m/s - 222o (ext.line)Data - int.linePred. BL-AdjustedPred. Std. park  222o +/- 3oWind speed:12.0 m/s +/- 0.5 m/s0.80.9Rel. reduced wind speed0.80.9Rel. reduced wind spee0 2000 4000 6000Downwind distance [m]0.70 2000 4000 6000Downwind distance [m]0.7Wake 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 = 5.9Wake Decay for the Infinite Wind Turbine Array [14]Comparative wind farm predictions: Nysted(2)1e powerNysted -10.0 m/s - 278oDataPred. BL-AdjustedPred. Std. parkWind direction: 278o +/- 2.5oWind speed: 10.0 m/s +/- 0.5 m/s0.60.8Rel. reduced turbine0 2000 4000 6000Downwind distance [m]0.411.2powerNysted -10.2 m/s - 263oDataPred. BL-AdjustedPred. Std. parkWind direction: 263o +/- 2.5oWind speed: 10.2 m/s +/- 0.5 m/s0.8Rel. reduced turbine p0 2000 4000 6000Downwind distance [m]0.6RWake Decay for the Infinite Wind Turbine Array [15]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.Th d l d t b t t d it ti ith k ff t b t• e mo e  nee s o e es e  on s ua ons w  wa e e ec s e ween neighboring wind farms.Wake Decay for the Infinite Wind Turbine Array [16]

Wake decay constant for the infinite wind turbine array:Application of asymptotic speed deficit concept to existing engineering wake model

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Wake decay constant for the infinite wind turbine array:Application of asymptotic speed deficit concept to existing engineering wake model

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