Field calibration of cup anemometers

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Field calibration of cup anemometers
Kristensen, L.; Jensen, G.; Hansen, A.; Kirkegaard, P.
Publication date:
2001
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Citation (APA):
Kristensen, L., Jensen, G., Hansen, A., & Kirkegaard, P. (2001). Field calibration of cup anemometers.
Denmark. Forskningscenter Risoe. Risoe-R No. 1218(EN)

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Risø–R–1218(EN)
Field Calibration of Cup Anemome-
ters
Leif Kristensen, Gunnar Jensen,
Arent Hansen, and Peter Kirkegaard
Risø National Laboratory, Roskilde, Denmark
January 2001
(cid:0)
Abstract Anoutdoorcalibrationfacilityforcupanemometers,wherethesignalsfrom
10 anemometers of which at least one is a reference can be can be recorded simulta-
neously, has been established. The results are discussed with special emphasis on the
statisticalsignificanceofthecalibrationexpressions.Itisconcludedthatthemethodhas
the advantage that manyanemometers can be calibrated accurately with a minimum of
workandcost.Theobviousdisadvantageisthatthecalibrationofasetofanemometers
maytakemorethanonemonthinordertohavewindspeedscoveringasufficientlylarge
magnituderangeinawinddirectionsectorwherewecanbesurethattheinstrumentsare
exposedtoidentical, simultaneouswindflows. Anothermainconclusion isthat statisti-
cal uncertainty must be carefully evaluated since the individual 10 minute wind-speed
averagesarenotstatisticallyindependent.
ISBN87–550–2772–5;ISBN87–550–2773–3(Internet)
ISSN0106–2840
Print:DankaServicesInternationalA/S,2001
Contents
1 Introduction 5
2 CalibrationSetup 6
3 StatisticalConsiderations 6
3.1 GainandOffsetbyOrthogonalFitting 8
3.2 StatisticalUncertainties 11
4 DataAnalysis 20
5 Conclusion 24
Appendices 28
A RegressionwithCurvature 28
B MonteCarloSimulations 29
C DiscreteSampling 32
D IntermittentSampling 34
Acknowledgements 40
References 41
Risø–R–1218(EN) 3

1 Introduction
Themostaccurateinstrumentformeasuringthemean-windspeedhasso-farbeenthecup
anemometer.Itwasinventedin1846bytheIrishastronomerT.R.Robinson(Middleton
1969, Wyngaard 1981) and the original instruments had four cups. Until the end of the
1920s much research went into experimenting with the number of cups and the arm
lengths. Brazier (1914) and Patterson (1926) found that shorter arms improvedthe lin-
earity of the calibration and that a three-arm cup rotor is optimal with respect to sen-
sitivity and suppression of the unevenness in the rotation (“wobbling”). A modern cup
anemometerisshowninFig.1.
Figure1.TheRisøanemometer.Thediameteroftheconicalcupsinthethree-cupplast
rotor is 7 cm. The body is made of anodized aluminum. The height of the instrument
from thebottom tothecenter oftherotor is24.7cm. Theoutputsignal, generatedby a
magneticallyactivatedswitch,isatrainofelectricpulses,twoforeachrotorrevolution.
By counting pulses over a certain period one obtains a number corresponding to the
mean-windspeedforthisperiod.
Thecupanemometernowappearstobethepreferredinstrumentformeasuringthemean-
windspeedinthewind-energycommunity,mainlybecauseofthelinearity,theaccuracy
andthestabilityofthecalibration.Butalsothefactthatthisinstrumentisomnidirectional
and easy to mount makes it attractive for routine measurements. There has been some
discussionabouttheimportanceoftheso-called“overspeeding”whichiscausedbythe
asymmetric response to instantaneous increases and decreases in the wind speed and
which,incidentally,isanecessarypropertyforthecupanemometertostartrotatingwhen
exposed to a wind, i.e. to function at all (Kristensen 1998). Some of those who believe
thatoverspeedingisa problem havepreferred touse a propeller-vaneinstrumentwhich
isconsideredsymmetricinitsresponsetooppositewinddirections.Theproblemishere
thatthepropellersignaldependsontheinstantaneousanglebetweenthewinddirection
andthepropelleraxis.Thevaneoftheinstrument,whichtriestoaligntheaxiswiththe
winddirection,willalwayslagbehind.Theresultisthattherewillalwaysbeasystematic
errorofthemeasuredmeanwind.Thiserrorisingeneralexacerbatedbythefactthatthe
propellerwillperformitsownmotionsinceitisseldommounteddirectlyoverthevertical
axis of the vane (Kristensen 1994). Also, on basis of the studies by Kristensen (1998),
Kristensen(1999),andKristensen(2000)theadverseeffectsoftheasymmetricresponse
of the cup anemometer seem exaggerated. Thusthe choice to use cup anemometers for
routinemeasurementsisasoundone.
Inthefollowingwediscussanoutdoorcalibrationsetupforanumberofcupanemome-
ters whichare simultaneouslyexposedtothe samewind. Inprinciplewecanletone of
Risø–R–1218(EN) 5
(cid:1)
theanemometersbe“thestandard”andthenintercalibratebycomparingtheoutputsig-
nals.Thisstandardanemometerisassumed tohavebeen calibratedina“certified” way
accordingtoanapproved,well-described methodbymeansofagoodwindtunnel with
littleturbulenceandaflatprofile.Itcouldbearguedthatananemometermountedoutdoor
isexposedtowindflowswhichchangesinstantaneouslyinbothwindsp(cid:2)eedandwinddi-
rectionwhereastheflowinawindtunnelisalwaysinthesamedirection.However,since
the cup anemometer allegedly measures the instantaneous component we really com-
pare mean signals from these outdoor anemometers through which the same “length of
air” haspassed, just aswe would havedonewhen using a windtunnel. Theonly reser-
vationone can haveis that the non-ideal angular response to verticalwind components
mayproduceabiasduetothefluctuatingverticalwindcomponent,but,asdemonstrated
byKristensen (1994),this can relativelyeasily bequantified if necessary.Here itis not
consideredaseriousproblem.
First we describe the calibration site. Then we discuss the statistical data analysis and,
finally,weillustratethemethodbyanalyzingrealdatafromthecalibrationinstallation.
2 Calibration Setup
The calibration setup is located behind the beach at the southwestern part of the Risø
peninsula, just north of the pier, as Fig. 2 shows. The dis(cid:3)tance(cid:3)to the water line is ap-
proximately12mandth(cid:3) eorientationofthecalibrationboomwherethetenanemometers
aremountedabout10moverthesurroundingterrainis19 –199 .Thismea(cid:4)nsth(cid:3) atwind
from(cid:3) the direction 289 will have travelled over a water fetch of about 7 km before it
simultaneouslyreachesalltheanemometer.Acceptingadirectionsectorof 45 around
289 ,thewaterfetchwillbeatleast3km.Inotherwords,thesitehasbeenchosentobe
well-exposedforwindsfromthepredominantwinddirectioninDenmark.
The calibration boom is mounted on the top of two steel lattice masts with triangular
crosssectionswiththesidelength0.25m.Fig.3showsasketchandaphotographofthe
measuringsetup.
ThedatarecordingiscarriedoutwithanAanderaadataloggerwith12ten-bitchannels.
Numberingthesefrom1to12,channel1isthetemperaturechannelandchannel12the
directionchannel.Channelsfrom2to11areusedfortheanemometeroutputs.Toprevent
(cid:5)
the signal from the Risø anemometer with two counts per revolution from causing the
10-bit channels to overflow during the counting period of 10 minutes, the numbers in
the registers are scaled down by the factor 25 32. Usually we have two positions for
reference anemometers. The rest of the positions are test positions. Table 1 gives the
“names”ofthetenpositions.
Thiscalibrationconfiguration,whereuninterrupted,consecutive10-minuteaverageshave
beenmeasured,hasbeenoperatingsince1996.
3 Statistical Considerations
(cid:6)
Thecalibrationprocedureimpliesthatwecomparetwoalmostidenticalresponsestothe
same signal. For symmetry reasons we adopt a method where the mean square of the
Actually,averagedoveronefullrotorrotation.
6 Risø–R–1218(EN)
Figure2.MapoftheRi(cid:3)søpeninsula.Thecalibrationsiteisshowninalittlemoredetailin
thecircularblow-up.Thedirectionperpendiculartothecalibrationboom,whichparallel
tothecoastline,is289 .Whenthewindisfromthisdirectionthemutualflowdisturbance
betweentheanemometersisatminimum.
(cid:3) (cid:3)
Figure3.CalibrationsetupatRisø,intheleftframedrawntothescale1:190.Theori-
entation of the boom with the ten cup anemometersis 19(cid:3) –199 . The lower boom is a
serviceboom usedfor mounting theinstruments. Thewind direction ismeasuredat the
end of a1.8 m long boom,pointing inthe direction 319 . Thethermometer ismounted
onaboomat2movertheground.Thetwosupportingmastsaretriangularlatticemasts
withthesidelength25cm.
Risø–R–1218(EN) 7
Table1.Positionnamesandchannelnumbers.Thenorthernmostandsouthernmostposi-
tionsare“TestA1”and“TestA2”,respectively.
Position Channel
TestA1 11
TestB1 10
Ref.1 9
TestC1 8
TestD1 7
TestD2 6
TestC2 5
Ref.2 4
TestB2 3
TestA2 2
perpendiculardistanceofthepointstotheregressionlineisminimized.Subsequentlywe
discussthestatisticalimplicationsoftheadoptedmethod.
3.1 Gain and Offs(cid:7) et by Orthogon(cid:8)al(cid:9) F(cid:10)(cid:11)(cid:9)ittin(cid:5) g(cid:12)(cid:13)(cid:12)(cid:13)(cid:12)
Wewanttofitastraightline toanumberofpoints x y i 1 N.
i i
(cid:5)
Theequation(cid:14)forthelineis
y ax b (1)
(cid:5)
or,invectorf(cid:14)orm,(cid:9)(cid:16)(cid:15) (cid:17) (cid:17)
r r q t ¥ q ¥ (2)
0
(cid:8) (cid:9) (cid:10)
wherer isapointonthelineandt aunitvectorinthedirectionoftheline.
0
The two signals x y are both representing wind velocities from the same type of
i i
anemometer.Nowweminimizethemeansquaredsumf oftheperpendiculardistances
difromt(cid:5)hepointsto(cid:12) theline,viz.
(cid:18)
f 1 (cid:229)N d2 (3)
N i (cid:8) (cid:9) (cid:10) (cid:7)
i 1
Wemustthereforefirstdeterminethedistanced fromapoint x y totheline .
i i i
(cid:5)
Intermsofthe(cid:14)unitv(cid:12)ectorsiand jdescribingtheCartesiancoordinatesystemwehave
r x i y j (cid:7) (4)
0 0 0
(cid:5)
Theunitvectort alon(cid:14)g andtheorthogonalunitvectornaregivenby
t cosa i sina j (5)
8 Risø–R–1218(EN)
(cid:19)
(cid:5)
and (cid:15) (cid:14) (cid:9)
n sina i cosa j (cid:7) (6)
wherea istheanglefromthex-axisto .
(cid:5)
Thesigneddis(cid:14)tanced fromagivenpoint
i
r (cid:7)xi y j (7)
i i i
(cid:5)(cid:21)(cid:20) (cid:5)
totheline is(cid:15) (cid:22)(cid:24)(cid:23) (cid:15) (cid:8) (cid:15) (cid:10) (cid:14)(cid:25)(cid:8) (cid:15) (cid:10) (cid:12)
d r r n x x sina y y cosa (8)
i i 0 i 0 i 0
(cid:27)
ThedefinitionsofthequantitiesweuseareillustratedinFig.4.
7
#"
; $ " &(cid:21)’)(+*-,/.103254
(cid:31)
9
(cid:29) (cid:25)! (cid:30)
7 8
:
%
6 6 (cid:26)
(cid:28)
(cid:5)
Figure 4.Illustration of the calculation of the distance d of one point to the regression
i
linewithoffsetbandslopea tana .
Equation(cid:5) (3)nowbecomes
(cid:15) (cid:8) (cid:15) (cid:10) (cid:14)(cid:25)(cid:8) (cid:15) (cid:10) = (cid:12)
(cid:18) <
f 1 (cid:229)N x x sina y y cosa 2 (9)
N i 0 i 0
i 1
We want to dete>rmi?ne the values of r and a that minimize f . In fact, there are only
0
twoindependentparametersofwhichtheanglea mustbeone.Sinceweexpecta tobe
diff(cid:5)erentfromn p 2,wecanfixeitherofx ory andusetheotherasafittingparameter.
0 0
Letusfixx .Inprinciple,thereisnoboundsonitsvalue,butitseemspracticaltochoose
0
x xwheretheaveragesymbolstandsfortheoperation
0 (cid:5) (cid:12)
(cid:18)
1 (cid:229)N
z z (10)
N i
i 1
Risø–R–1218(EN) 9