Højstrup, J.

Jensen, N.O.

Katic, I.

Online Research Database In Technology

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Vol. 1Publication date:1987Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Katic, I., Højstrup, J., & Jensen, N. O. (1987). A Simple Model for Cluster Efficiency. In W. Palz, & E. Sesto(Eds.), EWEC'86. Proceedings. Vol. 1 (pp. 407-410). Rome: A. Raguzzi.European Wind Energy AssociationConference and Exhibition7-9 October 1986, Rome - Italy.C6A SIMPLE MODEL FOR CLUSTER EFFICIENCYI.KATIC. J.HØJSTRUP. N.O.JENSENHowever, they uge detailed and rather complicateddescriptions for the wake profile. Hence, theyare more suitable for theoretical studies thanfor actual planning purposes with real turbines.SUM~1ARYA theoretical model, for the calculation ofthe output from wind farms, is described. ThemodelOtakes into account the characteristics ofthe turbines, and is intended to be used for op-timizing the cluster configuration for a givensite, where the distribution of wind speeds andvelocities are known. When compared to similar,more sophisticated modeIs, the results are gen-erally in agreement. Full-scale measurements ofwake interactions between the two Nibe turblnesare presented and the results discussed. Modelcalculations, carried out for the Nibe machines,show the importance of the increased turbulentmixing downstream of a turbine.l. INTRODUCTIONA wind turbine extracts energy from the windand consequently retards the wind field behindthe turbine. When we 6et sufficiently far down-stream, the wind speed will have returned to itsori~inal value. Another turbine positioned in thewake of the first ane will produce less energyfor a given free-stream wind speed. An example ofthis is illustrated in Fig. l. This showsmeasured power curves for the Nibe A-turbine inundisturbed flow and with the B-turbine runningupstream at a distance of five rotor diameters.Obviously, the power reduction is significant andstrongly dependent on the wind direct ion. Whenplanning clusters of wind turbines this powerreduction must be accounted for in arder to beable to estimate the yearly production for thedifferent cluster configurations possible at agiven site. Several factors lIIUSt be included ifthe power reduction for a cluster is to be cal-culated. These are:Fib' l. Output versus wind-speed curve for theNibe A-turbine when operating in wakeof the B-turbine.- distribution of wind speed and direction- characteristics of the wind turbines- characteristics of the wake of the turbines.Most previous investi~ations of the power reduc-tion for different cluster confi6urations neglectthe influence of topo~raphy and prevailing winddirections as well as the turbine characteristics.2. DESCRIPTION OF MODELThe model presented here is a further de-veloped version of a wake decay model proposedby N.O.Jensen [1]. Like most other models forwind farm calculation, it is based on the de-script ion of a single wake, in terms of an in-itial velocity deficit, and a wake decay coeffi-cient. However, it differs from most similarmodels in the following ways:1) The wake velocity profile is described in avery idealized way, by letting the wind velo-city be constant inside the wake.I.KatiE, J.Højstrup, N.O.JensenDepartment of Meteorology and Wind EnergyRis~ National LaboratoryDK-4000 RoskildeD E NMARK2) The actual characteristics of the turbines inthe farm can be incorporated in the model,thus the different behaviour of stall- andpitch-rebulated machines can be modelled.4073) In combination with the wind atlas method, themodel can estimate the ave rage yearly outputof a wind farm, if a sufficiently detailed mapof the area around the farm is available, andif the frequency distribution of the wind ineight direct ion sectors is known.The resulting velocity by summation of two wakesis thus found by:(1-V!U)2 z (1-Vi!U)2 + (1-V2!U)2 .) The program can be run on a personal computerwith a reasonable execution time.2.1 Wake descriptionThe wake behind a turbine is assumed to havea start diameter equal to the turbine diameter,and to sVread linearly as a function of downwinddistance. This simplification means that the wakevelocity cannot be found very accurately at alldownwind positions, but by adjusting the spreadangle to flt data at distances larger than aboutfour diameters, only the calculation of the near-wake zone will involve large errors. As wind tur-bines are seldom put closer to~ether than thisdistance, it is not necessary to make accuratecalculations here.Inside the wake the velocity is considetedconstant, instead of using the commonly seenGaussian distribution. This simplification ismade because the aim uf the model is to give anestimate of the energy content in the wind fieldseen by the downwind turbines, rather than to de-scribe the velocity field accurately.Here, VI and Vz represent the two ori~inal wakevelocities.Because tlle v~locity deficits are squared,the total velocity ueficit in the wake from aline, or from several rows of turbines, willquickly reach ,an equilibrium level, typicallyaf ter 3-4 turbines or rows. Tllis seems to be in~ood agreement with wind tunnel results [Z]. Thesame formula is used to calculate the groundeffect by vlacing an imabe wind turbine beneaththe real turbine. 'Because the thrust coefficient of a turbinenormally is dependent on the wind speed, the in-itial velocity deficit will alBa be, and thus thecalculated relative velocity deficit will only bevalid for ane wind speed. Together with thepower versus wind speed characteristics, it meansthat the farm efficiency can be totally differentfor different wind speeds. For example, if thespeed af ter the first row of turbines is reducedto below the cut-in speed, the efficiency isclose to zero. In contrast, for a wind speedhi;;her than the rated speed, the efficiency canexceed ane if the power curve of a single turbinehas a negativ~ slope af ter the rated speed. Whendesigning wind farms it is therefore importantto estimate the occurrence of the whole ranbe ofoperatin~ wind speeds and directions.The method of calculiition for a wind farmwith the Ris~ method can be divided into the fol-lowin.. st~ps.1) The type of wind turbine is chosen, andradius, hub heibht, power and thrust curvesare biven as data for the probram.2) The ~eometry of the cluster is 6iven in termsof coordinates for each turbine. It is pos-sible to align the coordinate system alongthe main axis of the cluster and later to turnit into the required geographical orientation.x3) Now a preliminary calculation can be made fordifferent wind speeds and -directions. toassure that the wake interaction does not giverise to an unacceptably small efficiency forthe cluster.Fig. 2. Schematic view af wake descriptian4) Parameters for the Weibull distribution of thesite are given for each 45-degrees directionsector toE,ether with the frequency of the windoccurring from that sector.With symbols defined in Fig. 2, a balance of mo-mentum gives:2D2U + (Dw2 -D2)U ~ D Vr w 5) The annual averabe output is calculated bystepping throu~h all wind speeds and direc-tions. This is the most time-consuming calcu-lationj on an Olivetti M24 PC with math-processor it takes about 15 minutes to calcu-late a cluster of 25 turbines.The wake velocity is found by the expression,V/U - l - Z,(l + ZkX/1»2 ,a is defined as the initial velocity deficitl-Ur/U but can also be expressed asa = (l - {1=Ct )/ZAs an' output from the last calculation comes thepruduction of a turbine at a ~iven site if therewere not wake interaction, and the productionfrom an averabe turbine in the park.. with wakestaken into account. The ratio between these iscalled the farm efficiency.where Gt is the thrust coefficient of the turbine.Hence. the velocity deficit of the wake at agiven position X is:1 - V/U - (1- {~)/(I + 2kX/D)2 .The problem of interacting wakes is solved by as-suming the kinetic energy deficit of a mixed waketo be equal to the sum of the energy deficits foreach wake at the calculated downwind position.3. COMPARISO!~ WITH OTHER MODELSSome comparisons have been made, using testexamples calculated by more sophisticated modersfrom TNO, Holland and CEGB, Great Britain [3].Although there are some differences at408certain wind directions, the overall results arethe same. The differences can be explained mainlydue to the simple wake profile of the Risø model,which gives an on-off like characteristic behav-iour of the wake effect when the wind directionchan6es.When comparing the modeis it is a problem how todetermine the wake decay constant, which is in-fluenced by a lot of factors, such as ambientturbulence level, turbine-induced turbulence, andatmospheric stability. However, the overall out-put for typical wind farm calculations is rela-tively insensitive to minor changes of the decayconstant. This is because a small constant givesa large power reduction, but only in a narrowzone while a large constant gives a smaller re-duction in a wider zone. The result is sensitiveto the value chosen for calculation in specificdirections only; when all directions are con-sidered, the importance of the wake decay con-stant vanishes.-"-"""".- .".'~~:::.-:::..-~... ,..,.,:;-.;t..-'/ ~"-::: ./- ": O~-..~~. ,o' ,o' . : oI 'o.I~ \',:0"-TNO "INDTUNNEl. NESUI.T,TNO PNOGR'N 'NII.I."I.ISS'N'N PRO"R,.CERI. PROGR,. 'E..,COISO PROGR,. 'P'RN'. '. .:;;r ':. . ,,-.: ~.~.~.. .,: . :. '. ,,. ,. . . . ". ,. ,. . . t ? . .. '. . . .. . . !.5OS0. . . . . . .. . . . .. . ..4. NIBE WAKE MEASUREMENTSDa~a from the Nibe site have been collectedto investif,ate the behaviour of the wake from thetwo 630-kW machines.The data used here were obtained frolD adata set of approximately 10,000 one-minute aver-ages taken in wake interaction situations at Nibedurinb the first half of 1986. Instrumentationand site were described in [4]. FiE,ure 4 showsthe arrangeUlent of masts and turbinesj however,mast O does not exist but is shown to illustratethe free boundary-layer velocity prof ile.The data are selected with both turbine srunnin~, and the direction sector is t5 debreesfrom the line connecting the turbines.The incoming wind speed is calculated byusing the B-turbine as the anemometer, and let-ting the speed at hub height be representative.From this speed the whole profile is calculated.The wind speed iso-curves shown on the figure areonlya rough estimate of the actual length sec-tion of the wake, because only a few point s aremeasured per curve. It is interesting to notethat the wakes from the two machines seem to bealmost similar, even though the incoming winØspeed on the A-turbine is much smalle r than onthe B-turbine. This indicates that the wake decayconstant is increasing af ter the A-machine be-cause of increased mixing due to a higher tur-bulence level.The problem of increased turbulence down-stream of a turbine is also illustrated inFif,' 5, which shows the velocity profiles at thefour masts, tooether with calculated wind speedsat hub hei6ht.If the same decay constant of 0.075 is usedbefare and af ter the A-turbine, it seems that themixin~ rate af ter the A-machine is underesti-mated, a constant of 0.11 fits much better here.However, the measurements are all done within4 diameters from the turbines, and may thereforenot be expected to fit the model very precisely.Also, the A-turbine differs from the B-turbine by having stays on the rotor, which mayinduce same additional turbulence.I' 'DIS~INCIDENT "'ND DI.ECT'DN DE..EE'o' 3. Comparison of the Kis\- model with others6 7.54 52.5DiametersFig. 4. Iso-curves for wind speed in the wake of Nibe turbines40970 ACKNOWLEDGEMENTSThe data from Nibe used here has arisenfrom a collaborative project (described in[4}) with, and is used with the permission of theCentral Electricity Generating Board, U.K., whoreceived financial assistance for the enhancementof the Nibe measuring system from the Commissionof the European Communitues, the Department ofEner~, U.K. and the National Board for Scienceand Technology, lreland.50:g 40...~.~ 30Cl:I:2010oBIBLIOGRAPHY1] N.O.Jensen "A Note on Wind Generator lnter-action" - Ris," - 19832.0 4.0 6.0 8.0 10.0~ + e ø Wind speed (m/s)I i i i i1I..t o 1 2 3 4Fig. 5. Measured velocity profiles and computedcentre~line velocities [~] S.A.Jensen and l'.lngham "Lokale l~iveresindflydelse p" vindm~lers energiproduktion"- Danish Maritime lnstitute - 19845. CONCLUSIONUsing the model presented for wind farm cal-culations. oood abreement is found with IlIore com-vlete models and with the few measurements avail-able.[3] J.F.Ainslie "Comparison of Wake ModeIs forComputing lnteractive Effects Between WindTurbines" - CEBG - 1982[4] D.T.Swift-Hook. J.H~jstruV. D.N.McIntosh.1).J.Nilborrow. and G.J.Taylor "Nibe WakeMeasureIllents vroJect in Denmark'" -- In Pro-ceedings of EWEC'84 - ed. W.Palz - publish-ed by H.S.Stevhens and Associates - 1985Until more full-scale measurements arecarried Gut it is difficult to conclude anythin~about the accuracy of the computations.Since so many asvects concerning wakes needto be clarified, it does not seem worthwhile toextend the model at present.The PC-prooram developed is believed to be apractical tool for wind farm developers as lon6as ane is only interested in the mean enerbY pro-duction. Other aspects, like additional loads dueto increased turbulence, or economics, must betreated separately.

A Simple Model for Cluster Efficiency

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A Simple Model for Cluster Efficiency

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