Courtney, Michael

Gottschall, Julia

Mann, Jakob

Sathe, A.

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Systematic error in the estimation of the second order moments of wind speeds bylidarsSathe, A.; Mann, Jakob; Gottschall, Julia; Courtney, MichaelPublished in:Detailed Program (online)Publication date:2010Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Sathe, A., Mann, J., Gottschall, J., & Courtney, M. (2010). Systematic error in the estimation of the second ordermoments of wind speeds by lidars. In Detailed Program (online) ISARS.Systematic error in the estimation of the second ordermoments of wind speeds by lidarsA. SATHE1, J. MANN2, J. GOTTSCHALL2, M. S. COURTNEY21L& R, Section Wind Energy, TU Delft, Kluyverweg 1, 2629 HS Delft, The Netherlands. E-mail: A.R.Sathe@tudelft.nl2Wind Energy Division, Risø DTU, Frederiksborgvej 399, 4000 Roskilde, Denmark1 INTRODUCTIONA theoretical model is developed to estimate thesystematic errors in the second-order moments ofwind speeds measured by lidars. The systematicerrors are the errors that arise due to the averagingeffect in the line-of-sight and the quite large circlein which lidars measure wind speed. Two typesof lidars are considered, the ZephIR developed byQinetiQ (Natural Power) as a continuous wave (CW)lidar and the WindCube developed by Leosphereas a pulsed lidar. The verification is carried out bycomparing the second order moments measuredby ZephIR and WindCube with that of the sonicanemometer placed at different heights on a mete-orological mast.One of the first turbulence studies by lidars wascarried out by [1], which was more a demonstrationof feasibility of turbulence measurements by lidarssince no measurements were available high up inthe atmospheric boundary layer for comparison.[2] presented a technique to analyse lidar data forturbulence measurements, which was done mainlybeyond 600 m due to coarse spatial resolution.[3] presented a detailed analysis of estimating therandom errors in the measurement of the meanwind speed by lidars using the theory of isotropicturbulence. [4] demonstrated the sensitivity of thestreamwise velocity variance to the spatial andtemporal averaging. [5] presented a method to uselidar data for the estimation of turbulent energydissipation rate from the point of view of studyingwake vortices from an aircraft. A comprehensivereview is given in [6] that covers different remotesensing techniques for turbulence measurementsincluding lidars. A review of the use of lidars forwind energy applications is also presented in [7].Recently, studies have been carried out to modelthe spatial averaging effects [8] and compare the3D turbulence measurements using three staringlidars [9]. [10] modelled the systematic errors byapproximating the conical scan as a length scale.[11] attempted to estimate the momentum fluxesusing lidars and model the unfiltered turbulencefrom the CW lidar. In this work averaging effectsdue to the full extent of conical scanning andline-of-sight averaging is considered.2 THEORYThe model in this work is developed for the velocityazimuth display (VAD) technique of lidar scanning.The line-of-sight velocity (also called as radial ve-locity 푣푟) is measured by the lidar at each azimuthangle 휃. The half cone opening angle 휙 is kept con-stant throughout the scan. The CW and pulsed li-dars work on the principle of backscattering of theemitted radiation and subsequent detection of theDoppler shift in the frequency of the emitted radia-tion. Mathematically, 푣푟 is given as the dot productof the unit directional vector and the velocity field atthe point of focus,(1)푣푟(휃) = 풏(휃) ⋅ v(푑푓풏(휃))where 푑푓 is the focus distance at which the windspeeds are measured, v = (푢, 푣, 푤) is the velocityfield evaluated at the focus point 푑푓풏(휃) and 풏(휃) isthe unit directional vector given as,(2)풏(휃) = (cos 휃 sin휙, sin 휃 sin휙, cos휙)In reality it is impossible to obtain the backscatteredradiation precisely from only the focus point andthere is always backscattered radiation of differentintensities from different regions in space along theline-of-sight. Hence, it is necessary to assign appro-priate weights to the backscattered intensity suchthat the weight corresponding to the focus point isthe highest. Mathematically the weighted averageradial velocity can be written as,(3)푣˜푟(휃) =∫ ∞−∞휑(푠)풏(휃) ⋅ v(푠풏(휃) + 푑푓풏(휃)) 푑푠where 휑(푠) is any weighting function and 푠 is the thedistance from the focus. For simplicity we assumethat 푠 = 0 corresponds to the focus distance. Ourmodel is based on the assumptions of homogeneityand neutral atmospheric conditions. The flow field isalso assumed to be frozen during one scan revolu-tion. It is further assumed that Eq. 3 with an appro-priately chosen 휑(푠) models the averaging well.2.1 Systematic Error in CW lidarThe CW lidar scans at many points on the azimuthcircle. For the ZephIR lidar considered in our casea typical scan consists of measurements of 푣푟 at 50points on the azimuth circle, and often three circlesare used to get a single velocity vector. If we as-sume that 푢 is aligned to the mean wind direction, 푣is perpendicular to the mean wind direction, 푤 lieson the vertical axis and the mean wind comes fromthe North then 푣˜푟(휃) can be expressed as,(4)푣˜푟(휃) = 퐴+퐵 cos 휃 + 퐶 sin 휃where the coefficients 퐴 = 푤 cos휙, 퐵 = 푢 sin휙 and퐶 = 푣 sin휙 and the sign ambiguity in 푣˜푟(휃) is ne-glected. The assumption that the mean wind comesfrom the North is only made for simplicity. For a lidarmeasuring at many points on the azimuth circle thechoice of the mean wind direction does not mattersince averaging over the entire circle is carried out.The values of the coefficients 퐴, 퐵 and 퐶 are foundusing least squares method by fitting the measuredvalues of 푣˜푟(휃) at several azimuth angles in Eq. (4).Thus the coefficients can be written as Fourier inte-grals,(5a)퐴 =12휋∫ 2휋0푣˜푟(휃) 푑휃(5b)퐵 =1휋∫ 2휋0푣˜푟(휃) cos 휃 푑휃(5c)퐶 =1휋∫ 2휋0푣˜푟(휃) sin 휃 푑휃The variance of any variable 푋 can be denoted as휎2푋 = ⟨푋 ′2⟩, where ⟨⟩ denotes ensemble averag-ing. Applying this definition to coefficients 퐴, 퐵 and퐶 (Eqs. 5a–5c) and using Eq. (3), the following ex-pressions are derived:⟨푤′2⟩ cos2 휙 =∫Φ푖푗(풌)훼푖(풌)훼∗푗 (풌) 푑풌 (6)⟨푢′2⟩ sin2 휙 =∫Φ푖푗(풌)훽푖(풌)훽∗푗 (풌) 푑풌 (7)⟨푣′2⟩ sin2 휙 =∫Φ푖푗(풌)훾푖(풌)훾∗푗 (풌) 푑풌 (8)⟨푢′푤′⟩ sin휙 cos휙 =∫Φ푖푗(풌)훽푖(풌)훼∗푗 (풌) 푑풌 (9)where ⟨푢′2⟩, ⟨푣′2⟩, ⟨푤′2⟩ are the variances of the 푢, 푣and 푤 velocity respectively, ⟨푢′푤′⟩ is the momentumflux, Φ푖푗(풌) is the three dimensional spectral veloc-ity tensor, 풌 = (푘1, 푘2, 푘3) is the wave vector, ∗ de-notes complex conjugation and 훼푖, 훽푖 and 훾푖 are theweighting functions given as,훼푖(풌) =12휋푒i푑푓푘3 cos휙∫ 2휋0푛푖(휃 + 휃0)푒i푑푓푘ℎ sin휙 cos 휃 푒−푙∣푘ℎ sin휙+푘3 cos휙∣ 푑휃(10)훽푖(풌) =1휋푒i푑푓푘3 cos휙∫ 2휋0푛푖(휃 + 휃0) cos(휃 + 휃0)푒i푑푓푘ℎ sin휙 cos 휃 푒−푙∣푘ℎ sin휙+푘3 cos휙∣ 푑휃(11)훾푖(풌) =1휋푒i푑푓푘3 cos휙∫ 2휋0푛푖(휃 + 휃0) sin(휃 + 휃0)푒i푑푓푘ℎ sin휙 cos 휃 푒−푙∣푘ℎ sin휙+푘3 cos휙∣ 푑휃(12)where 푘ℎ =√푘21 + 푘22 is the magnitude of the hori-zontal wavenumber vector, cos 휃0 = 푘1/푘ℎ, sin 휃0 =푘2/푘ℎ, and 푛푖(휃 + 휃0) is the component of the unitdirectional vector obtained from Eq. 2. The analyti-cal expressions for Φ푖푗(풌) are used from [12]. Eqs.(10–12) are numerically integrated along with Φ푖푗(풌)(Eqs. 6–9) to estimate the second-order momentsmeasured by CW lidars.2.2 Systematic Error in pulsed lidarsThe assumption made in section 2.1 that the meanwind direction comes from the North cannot bemade for WindCube since it measures at four az-imuth angles only, viz. North (N), East (E), South(S), West (W). In this case the coordinate system issuch that 푢 is aligned in the mean wind direction.Then 푢 and 푣 are given as,푢 = 푢푁푆 cos Θ + 푢퐸푊 sin Θ (13)푣 = 푢푁푆 sin Θ− 푢퐸푊 cos Θ (14)where 푢푁푆 and 푢퐸푊 denote wind speeds in theNorth-South and East-West directions respectivelyand Θ denotes the wind direction. From simple ge-ometrical considerations we can also write,푢푁푆 =푣˜푟푁 − 푣˜푟푆2 sin휙(15)푢퐸푊 =푣˜푟퐸 − 푣˜푟푊2 sin휙(16)푤 =푣˜푟푁 + 푣˜푟푆 + 푣˜푟퐸 + 푣˜푟푊4 cos휙, (17)where 푣˜푟푁 , 푣˜푟푆 , 푣˜푟퐸 , 푣˜푟푊 represent the weighted av-erage radial velocities in the North, South, East andWest directions respectively. Proceeding in a simi-lar manner as in section 2.1, we get expressions forthe second order moments in terms of the weight-ing functions 푎푖(풌), 푏푖(풌) and 푐푖(풌) for 푤, 푢 and 푣respectively. 푏푖 is given as,푏푖(풌) =12 sin휙[(푛푖푁푒i푑푓풌⋅풏푵 휑ˆ(풌 ⋅ 풏푵 )− 푛푖푆푒i푑푓풌⋅풏푺 휑ˆ(풌 ⋅ 풏푺)) cos Θ+ (푛푖퐸푒i푑푓풌⋅풏푬 휑ˆ(풌 ⋅ 풏푬)− 푛푖푊 푒i푑푓풌⋅풏푾 휑ˆ(풌 ⋅ 풏푾 )) sin Θ](18)whereˆdenotes Fourier transform. It is worth notingfrom Eq. (18) that 푏푖(풌) depends on Θ. The expres-sions for 푎푖(풌) and 푐푖(풌) are very similar to Eq. (18)except that the variation with wind direction for 푐푖(풌)is deduced from Eq. (14). 푎푖(풌) does not vary withΘ as seen from Eq. (17).3 DATASETSThe measurements are carried out at Høvsøre,which is the national wind turbine test center in Den-mark. At this site five wind turbines are placed tothe North of the met-mast (coordinates 56∘26’26” N,08∘09’03” E ). 20 Hz sonic measurements are usedto compute true second order moments. They aremounted on booms in the North direction. TheZephIR and WindCube take approximately 1 and 6 srespectively to complete one scan in the VAD tech-nique. They are placed at about 40 and 15 m to theNorth and West of the met-mast respectively. Thehalf cone opening angle for both lidars is 30∘. Toensure that the wake effects from the wind turbinesand met-mast do not influence the measurements,the wind direction sector that is chosen for the analy-sis is 230∘–300∘. Since the systematic error in the li-dar turbulence measurements changes with height,four heights were chosen, viz. 40, 60, 80 and 100 m.The period of analysis is between January 2009 andNovember 2009. The wind speeds between 4 and25 m/s only are used. The data are also filtered forperiods of rain.4 COMPARISON WITH MEA-SUREMENTS0.0 0.2 0.4 0.6 0.8 1.0 1.2050100150200ΣlidarΣheightabovegroundz@mDwuvCone onlyCone + LorFigure 1: Comparison of the systematic error byZephIR. The dotted line shows only cone averagingand the solid line shows the averaging due to coneand 휑, assumed LorentzianFigure 1 shows the comparison of the systematicerror of the second order moments in the CWlidar (ZephIR) modelled in section 2.1 with themeasurements. The model shows that the system-atic error increases (i. e. the ratio of the secondorder moment measured by CW lidar to the truesecond order moment decreases) with height for 푢,푣 and 푤. This is because the probe length of CWlidar increases quadratically with height and thediameter of the conical circle also increases. Thekink at 60 m in the model prediction is because thelength scale used in the estimation of Φ푖푗 increaseslinearly with height up to 60 m and is consideredconstant beyond. The data shows an increasein the systematic error up to 60 m and a slightreduction beyond. The circles are the medians ofthe measured ratios of standard deviations whilethe extremes of the horizontal lines are the 25%and 75% fractiles, respectively. The deviation ofthe model with the data beyond 60 m could bebecause the standard input parameters from [13]from chosen to compute Φ푖푗 . It would be quiteinteresting to obtain empirical values of the inputsto Φ푖푗 using the measurements at the site as in [14].Moreover, all atmospheric stability conditions areconsidered. Both, the model and the data show thatthe attenuation in 푤 is much larger than in 푢 and푣. This can be explained by considering that thelength scales for 푤 are much smaller than 푢 and 푣.0.4 0.6 0.8 1.0 1.2050100150200250ΣlidarΣheightabovegroundz@mD0°15°30°45° 0°15°30°45°uvwFigure 2: Comparison of the systematic error byWindCube. Variation with wind direction is alsoseen.Figure 2 shows the comparison of the systematicerror of the second order moments in the pulsedlidar (WindCube) modelled in section 2.2 withthe measurements. WindCube has a constantprobe length, and hence, the model shows that thesystematic error decreases with height. Beyond60 m the model predicts a slight increase in thesystematic error with height. This is because theattenuation due to the circle averaging is moredominant than the attenuation due to the probelength. The dependence of the systematic error onthe wind direction is also seen. The data shows agood agreement with the model for 푤 at 60, 80 and100 m. For 푢 there is an increase in the systematicerrors at all heights, whereas for 푣 there is a slightincrease up to 60 m and a decrease beyond.0 10 20 30 40 50 600.00.51.01.52.02.53.0half opening angle Φ @°DΣlidarΣz = 80 mConeCone+LorvuwFigure 3: The sensitivity of the systematic errors to 휙at 푧 = 80 m. The solid line shows only cone averag-ing and the dotted line shows the averaging due tocone and 휑, where 휑 is modelled using LorentzianFigure 3 shows the variation of the systematic errorwith respect to 휙. It decreases non-linearly with 휙.The influence of line-of-sight averaging (Lorentzian)is not considerable.5 CONCLUSION AND DISCUS-SIONSystematic error in the second order moments ismodelled for the CW and pulsed lidar. In general thesystematic error increases with increasing height forthe CW lidar, whereas it decreases with increasingheight for the pulsed lidar. The model agrees wellwith the data for WindCube for 푢 and 푤. Owing tothe smaller length scales for 푤 the systematic erroris quite large in comparison to 푢 and 푣. Further in-vestigation will focus on estimating the inputs to Φ푖푗empirically. The data will be separated into differentstability classes and a comparison will be made withthe model.References[1] W L. Eberhard, R E. Cupp, and K R. Healy.Doppler lidar measurements of profiles of tur-bulence and momentum flux. Journal of Atmo-spheric and Oceanic Technology, 6:809–819,1989.[2] T. Gal-Chen, M. Xu, and W L. Eberhard. Es-timation of atmospheric boundary layer fluxesand other turbulence parameters from Dopplerlidar data. Journal of Geophysical Research,97(D17):18,409–18,423, 1992.[3] V A. Banakh, I N. Smalikho, F. Ko¨pp, andC. Werner. Representativeness of wind mea-surements with a CW Doppler lidar in the at-mospheric boundary layer. Applied optics,34(12):2055–2067, 1995.[4] Y L. Pichugina, R M. Banta, and W A. Brewer.Vertical profiles of velocity variances and TKEusing Doppler-lidar scan data. In 17th Sympo-sium on Boundary Layers and Turbulence, SanDiego CA, 2005.[5] I. Smalikho, F. Kopp, and S. Rahm. Mea-surement of atmospheric turbulence by 2-휇m Doppler lidar. Journal of Atmosphericand Oceanic Technology, 22(11):1733–1747,2005.[6] D A M. Engelbart, M. Kallistratova, andR. Kouznetsov. Determination of the tur-bulent fluxes of heat and momentum in theABL by ground-based remote-sensing tech-niques (a review). Meteorologische Zeitschrift,16(4):325–335, 2007.[7] S. Emeis, M. Harris, and R M. Banta.Boundary-layer anemometry by optical remotesensing for wind energy applications. Meteo-rologische Zeitschrift, 16(4):337–347, 2007.[8] M. Sjo¨holm, T. Mikkelsen, J. Mann, K. Enevold-sen, and M. Courtney. Spatial averaging-effects on turbulence measured by acontinuous-wave coherent lidar. Meteorol-ogische Zeitschrift, 18(3, Sp. Iss. SI):281–287,2009.[9] J. Mann, J. Cariou, M. Courtney, R. Parmen-tier, T. Mikkelsen, R. Wagner, P. Lindelow,M. Sjo¨holm, and K. Enevoldsen. Compar-ison of 3D turbulence measurements usingthree staring wind lidars and a sonic anemome-ter. Meteorologische Zeitschrift, 18(2, Sp. Iss.SI):135–140, 2009.[10] R. Wagner, T. Mikkelsen, and M. Courtney. In-vestigation of turbulence measurements witha continuous wave, conically scanning lidar.Technical Report Risø-R-1682(EN), Risø DTU,2009.[11] J. Mann, A. Pen˜a, F. Bingo¨l, R. Wagner, andM S. Courtney. Lidar scanning of momentumflux in and above the surface layer. Journal ofAtmospheric and Oceanic Technology, 2010.DOI:10.1175/2010JTECHA1389.1.[12] J. Mann. The spatial structure of neutral at-mospheric surface-layer turbulence. Journal ofFluid Mechanics, 273:141–168, 1994.[13] IEC. IEC 61400-1. Wind turbines – Design Re-quirements. 2005.[14] A. Pen˜a, S-E. Gryning, J. Mann, and C B.Hasager. Length scales of the neutral wind pro-file over homogeneous terrain. 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Systematic error in the estimation of the second order moments of wind speeds by lidars

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Systematic error in the estimation of the second order moments of wind speeds by lidars

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