A Comparison of Standard Coherence Models for Inflow … · 2008-11-08 · Coherence Models for...

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Lance Manuel

Department of Civil Engineering,University of Texas at Austin,

Austin, Texas 78712 USA

Paul S. VeersWind Energy Technology Department,

Sandia National Laboratories,Albuquerque, New Mexico 87185 USA

A Comparison of StandardCoherence Models for InflowTurbulence With Estimates fromField MeasurementsThe Long-term Inflow and Structural Test (LIST) program, managed by Sandia NatLaboratories, Albuquerque, NM, is gathering inflow and structural response data omodified version of the Micon 65/13 wind turbine at a site near Bushland, Texas. Witobjective of establishing correlations between structural response and inflow, prestudies have employed regression and other dependency analyses to attempt toloads to various inflow parameters. With these inflow parameters that may be thougas single-point-in-space statistics that ignore the spatial nature of the inflow, no sigcant correlation was identified between load levels and any single inflow parameteven any set of such parameters, beyond the mean and standard deviation of thheight horizontal wind speed. Accordingly, here, we examine spatial statistics inmeasured inflow of the LIST turbine by estimating the coherence for the three turbucomponents (along-wind, across-wind, and vertical). We examine coherence specboth lateral and vertical separations and use the available ten-minute time series othree components at several locations. The data obtained from spatial arrays onmain towers located upwind from the test turbine as well as on two additional towereither side of the main towers consist of 291 ten-minute records. Details regardingmation of the coherence functions from limited data are discussed. Comparisonsstandard coherence models available in the literature and provided in the InternatiElectrotechnical Commission (IEC) guidelines are also discussed. It is found thaDavenport exponential coherence model may not be appropriate especially for modthe coherence of the vertical turbulence component since it fails to account for reducin coherence at low frequencies and over large separations. Results also show thMann uniform shear turbulence model predicts coherence spectra for all turbulenceponents and for different lateral separations better than the isotropic von Ka´rman model.Finally, on studying the cross-coherence among pairs of turbulence components basfield data, it is found that the coherence observed between along-wind and verticabulence components is not predicted by the isotropic von Ka´rman model while the Mannmodel appears to overestimate this cross-coherence.@DOI: 10.1115/1.1797978#

Keywords: Coherence, Inflow, Turbulence.

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I IntroductionThe present study was motivated by failure to find meaning

correlation between turbine loads~extremes and fatigue! andsimple inflow parameters that did not describe the spatial strucof the inflow. For example, in previous regression studies by Nson et al.@1# it was found that single-point-in-space statisticsinflow ~e.g., at the hub height! were insufficient for predictingwind turbine loads such as extreme edgewise~in-plane! or flap-wise ~out-of-plane! bending moments at the blade root. It wathus, thought to be useful to study the full spatial descriptionthe random inflow turbulence field. The availability of inflow da~as time series segments! at multiple locations through the Longterm Inflow and Structural Test~LIST! program offers a uniqueopportunity to estimate the spatial statistics of the inflow. It is obelief that future studies on the correlation of wind turbine loato inflow may be improved if such spatial statistics are taken iconsideration.

One means of identifying the frequency-dependent and rannature of interactions in the inflow at different locations and h

Contributed by the Solar Energy Division of the THE AMERICAN SOCIETY OFMECHANICAL ENGINEERS for publication in the ASME JOURNAL OF SOLAR EN-ERGY ENGINEERING. Manuscript received by the ASME Solar Energy Division, Ja2004; final revision, June, 2004. Editor: P. Chaviaropoulos.

Copyright © 2Journal of Solar Energy Engineering

Downloaded 08 Nov 2008 to 128.62.160.80. Redistribution subject to ASCE

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these might influence turbine loads is to study the coherencalong-wind (u), across-wind (v), and vertical (w) turbulencecomponents. For design purposes, prescribed models for eachbulence component, including power spectra and coherence stra, are required in order to perform simulations of the complinflow field. Computational aerodynamic analysis software uthis simulated inflow to compute the structural response that mbe used to determine design loads. Different theoretical turbulemodels utilized to construct full-field wind simulation may resuin significant differences in design loads for wind turbines. Vekamp@2# compared predictions of turbine tower and blade fatigloads based on inflow simulation with several turbulence modFor example, with two of the models studied—the Mann unifoshear model@3# where the spectral tensor for atmospheric surfalayer turbulence in neutral conditions based on the Navier-Stoequations as well as conservation of mass is modeled; andVeers model@4# where the three turbulence components, contrto the Mann model, are assumed independent—he concludedpredicted fatigue design loads were quite different. This findsuggests the importance of selecting appropriate turbulence mels in design. We are also therefore interested in assessingvalidity of the existing turbulence spectral models currently dfined in the International Electrotechnical Commission~IEC! stan-dard @5# as well as of other well-established coherence mod

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004 by ASME NOVEMBER 2004, Vol. 126 Õ 1069

license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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available in the literature by comparing them with coherenspectra of full-field wind turbulence components estimatedrectly from field measurements from the LIST program manaby Sandia National Laboratories, Albuquerque, NM.

Another motivation for the present study is that although seral field studies of the coherence structure of turbulence hbeen carried out in Europe~see, for example, Mann et al.@6#;Schlez and Infield@7#; Larsen and Hansen@8#!, comparatively farfewer such studies, especially with the appropriate heightsseparation distances that are useful for wind turbine applicatiand that take into account site-dependent turbulence length shave been carried out in the United States where the environmtal conditions and coherence structure of the inflow may beferent.

Recent findings from inflow measurements carried out in Erope by Larsen and Hansen@8# reveal that coherence of the alonwind turbulence component for different vertical separationscreases with increasing separation, with decreasing integral lescales, and with decreasing measuring altitudes. In contrastlateral separations, the same dependence of coherence on awas observed as was seen for vertical separations; howeveclear dependence of coherence on integral length scale anspatial separation was identified based on the measuremHowever, these conclusions are based upon limited data andlateral separation distances in that study were greater than 7~much larger than the separations that we will study!. Also, in thatstudy, only the along-wind coherence was considered. Here,available inflow data from the LIST program allow us to analycoherence spectra for all three turbulence components at diffevertical and lateral separations. In addition, we also studycross-coherence of each pair of distinct turbulence componenthe center of the rotor circle.

In the following, power spectra and coherence spectra forthree turbulence components are estimated using several oLIST data sets by employing the Welch’s modified Periodogr~block-averaging! spectral estimation method. To facilitate undestanding of the coherence as a function of primary inflow paraeters, we study these separately for different bins defined onbasis of hub-height mean and standard deviation values of hzontal wind speed. Since only limited data from the first phasethe LIST program were used and the estimations of the coherfunctions may be subject to large statistical uncertainty, errortistics of the estimated spectra defined in terms of bias, variaand confidence limits are discussed.

II LIST Data SetThe data sets employed to estimate spatial coherence sp

throughout this study were provided by Sandia National Labotories through the ongoing Long-Term Inflow and Structural T~LIST! program~Sutherland et al.@9#, Jones et al.@10#!. The LISTprogram has made available continuous time series of atmospinflow conditions as well as structural response data for a mfied Micon 65/13 wind turbine~referred to as the LIST turbine inthis study!. This measurement campaign is taking place atUnited States Department of Agriculture-Agricultural ReseaService~USDA-ARS! site in Bushland, Texas which is characteistic of a Great Plains site with essentially flat terrain andprimary wind direction at the site is from 215 deg with respectTrue North~see Fig. 1 for details of the LIST test site!.

The data were recorded as ten-minute segments, each of wcontains approximately 18,000 data points, at a sampling rat30 Hz ~implying a Nyquist frequency of 15 Hz!. Characterizationof the inflow in this study relies on an array of five sonic anemoeters mounted on three meteorological towers located apprmately 30 m in front of the LIST turbine as shown in Fig. 1. Tcenter tower is directly upwind of the LIST turbine. The other twtowers are one rotor-disk radius to the left and right of the centower. The five sonic anemometers are mounted as follows: atheight which is 23 m from the ground; at the top and bottom

1070 Õ Vol. 126, NOVEMBER 2004


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the rotor circle; and at positions left and right of the center of trotor circle ~see Fig. 2!. Two additional meteorological towerslocated in front of the sister turbines, approximately 38 m awfrom the center tower on the left and right, were constructedmeasure horizontal wind velocities at the hub height using canemometers. The additional inflow data from one of these ttowers are used to study the coherence of the along-wind tulence component at this greater lateral separation.

In preparation for the analysis, 291 of the available ten-mindata records were grouped into bins, depending on the horizomean wind velocity at hub heightUhub and the standard deviationof the same hub-height velocity,shub. This was done in order to

Fig. 1 Schematic view of the LIST test site showing the LISTturbine „Turbine B … and five meteorological towers

Fig. 2 Primary inflow instrumentation on the main meteoro-logical towers upwind of the LIST turbine. Hub height is 23 m;anemometers on the center tower are spaced 8.5 m verticallyapart; anemometers at hub height are spaced 7.7 m laterallyapart.

Transactions of the ASME


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obtain a representative sample of data sets for each bin soconclusions could be made about the dependence of coherenthese inflow parameters, regarded as primary by the wind encommunity. The number of available data sets for each binshown in Table 1. We recognize that turbulence standard deviadoes not affect coherence in a direct manner. A more appropinflow parameter for binning might have been the turbulencetegral length scaleL, as it is more directly related to the spaticoherence structure of wind fluctuations. However, we chosbin the LIST data set in this study on the basis of mean wspeed and turbulence standard deviation because such binnoften used to define wind turbine classes for design purposespecified in the IEC guidelines@5#.

All the bins that contained a sufficient number of samples wanalyzed; however, only results based on data sets from threeare discussed in the following. These selected bins corresponthe boldfaced entries in Table 1. For simplicity, these are refeto as Bins A, B, and C in the following. The range in values of tmeanUhub and the standard deviationshub of the horizontal windspeed at hub height for each of the three bins is shown in Tabalong with the number of data sets availableNg . In passing, wemention that with binning on the basis of mean wind speedturbulence length scale~instead of turbulence standard deviatio!for the data sets studied here, a very similar numbers of datain three bins~like the Bins A, B, and C, here! is found. As a result,apart from the physically more sound reasons for choosing toon length scale when studying coherence, no improved statissignificance advantage is derived from such binning.

The LIST inflow data records collected by several anemomeas discussed above provide a good opportunity to study the coence behavior of the three wind turbulence components fordifferent vertical separations of 8.5 and 17.0 m where the wspeed time series were measured by sonic anemometers locathree positions on the center tower mast. In addition, usingdata collected by three sonic anemometers located at hub hon the three center towers as well as one cup anemometer ofar-right tower, the inflow coherence spectra for three lateral serations can be analyzed. However, since the mean wind direcduring the measurement campaign was not perpendicular toplane containing the wind speed sensor array, the actual laseparations for coherence calculations are not equal to thezontal distances between any two sensors and hence needcorrected. This situation that results because the mean wind dtion is not perpendicular to the plane of the mast array is illtrated graphically in Fig. 3. Correction to account for this effeccarried out by multiplying the actual horizontal distance betwethe two sensors under consideration by the cosine of the a

Table 1 The number of ten-minute data sets in each bin char-acterized by the mean, Uhub , and standard deviation, shub , ofthe hub-height horizontal wind velocity.

Uhub ~m/s!

shub ~m/s! 7-9 9-11 11-13 13-15 15-17 .17

0.5-1.0 52 29 0 0 0 01.0-1.5 22 33 20 3 0 01.5-2.0 5 21 25 8 5 32.0-2.5 3 6 6 14 14 92.5-3.0 0 2 0 2 2 7

Table 2 Three bins selected for the numerical studies and theavailable number of ten-minute data sets, Ng .

Bin Uhub ~m/s! shub ~m/s! Ng

A 9-11 1.0-1.5 33B 11-13 1.5-2.0 25C 15-17 2.0-2.5 14

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between a normal to the plane of the sensor array and the hzontal mean wind direction for each ten-minute data record. Ththese corrected lateral separations are averaged over all theminute data sets in each bin. This finally leads to three differlateral separations for each bin that were used while reporinflow coherence spectra results. These lateral separations are13.1, and 32.5 m for Bin A; 6.7, 13.4, and 33.2 m for Bin B; an6.0, 12.1, and 29.9 m for Bin C as summarized in Table 3.

III Coherence Estimation: A ReviewA mathematical definition of the~magnitude-squared! coher-

ence function,g2( f ), of two stationary random processes,p andq, may be given as

g2~ f !5uSpq~ f !u2

Spp~ f !Sqq~ f !(1)

whereSpp( f ) andSqq( f ) are the one-sided~auto-!power spectraldensity functions ofp and q, respectively, at frequency,f ; andSpq( f ) is the one-sided complex-valued cross-power spectral dsity function between the two processes. Note that some reences refer to the coherence function as the root-coherence wis the absolute value of the normalized cross-power spectrum,ug( f )u5uSpq( f )u/ASpp( f )Sqq( f ). Hereinafter, we will refer tothe coherence function as the magnitude-squared coherencetion @defined by Eq.~1!# and we will use either of these two term~i.e., coherence or magnitude-squared coherence! interchangeably,unless specifically stated otherwise. It should be mentioned athat, for a homogeneous turbulence field, the quadrature spec~i.e., the imaginary part of the cross-power spectrum! will be rela-tively small when compared with the cospectrum~i.e., the realpart of the cross-power spectrum!. Hence, it is commonly as-sumed in wind turbulence simulation studies that the crospectral density function is real. As a result, the phase informaconveyed by referring to a phase spectrum is generally considless important than the magnitude-squared coherence or thecoherence for wind turbine studies. Accordingly, here, the phspectrum is excluded from our discussions. The coherenceg2( f ),is always greater than or equal to zero and by Schwarz’s ineq

Fig. 3 Illustration of the corrected lateral separation D com-puted from the horizontal distance between two sensors andthe angle between the mean wind direction and a normal to theplane of the mast array u

Table 3 Averaged angle between the mean wind direction andthe plane of the mast array as well as the vertical and lateralseparation distances available for estimating inflow coherencespectra for each bin.

Separation distances~m.!

Lateral Vertical

Bin Averaged angle (°) Small Intermediate Large Small Intermed

A 9.7 6.5 13.1 32.5 8.5 17.0B 28.9 6.7 13.4 33.2C 237.7 6.0 12.1 29.9

NOVEMBER 2004, Vol. 126 Õ 1071


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where the quantities with a caret indicate that each is estimfrom the limited data (Ng data sets for each bin!. Because of theuse of limited data and of records of finite length, the estimateEq. ~2! have unavoidable statistical errors. For example, an oous distortion will occur if one tries to compute the coherenfunction by using only one realization of each of the two prcesses,p andq. For such an estimation procedure, the compucoherence function will be identically equal to unity at all frquencies. In practice, each sampled time series is thereforeinto N subsegments to prevent this source of distortion. FoNnonoverlapping realizations of a stationary, Gaussian randomcess, the bias~Bias! and the variance~Var! of the coherence spectrum estimate at any frequency due to time-record truncationfects were derived~Carter@11#! and may be expressed as

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Note that Eqs.~3a! and ~4a! are approximate and result fromtruncating series expressions for the exact bias and varianccoherence estimates while Eqs.~3b! and ~4b! involve further ap-proximations that are applicable only whenN is large. The readeris referred to the work of Carter@11# and Kristensen and Kirkegaard@12# for exact expressions for the bias and variance of cohence estimates. It is recognized that allowing for overlappingsubsegments can further reduce the statistical variability ofestimates because of the greater number of degrees of freeused. Experimental investigation by Carter@11# where a Hanningdata window was utilized suggests that 50% overlapping isoptimal choice as a compromise between reduction in stanerror and computation time.

Besides conventional bias as predicted by Eq.~3a!, bias powerleakage effects may be introduced if the selected resolution bwidth is wide relative to the actual range of frequencies associwith a peak in the spectrum. This category of bias, whichknown as ‘‘resolution bias’’ may lead to large distortions nearpeak of the spectrum. In studying turbulence components,type of bias may be present due to the significant amount ofergy at low frequencies. Even though a theoretical expressiothis bias is not available, Jacobsen@13# concluded from numericasimulation studies that resolution bias effects will occur especiwhen the number of degree of freedom increases. This is typic



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associated with the number of~overlapping! subsegments useand when the coherence values are close to unity. Thus, ucertain circumstances, this resolution bias may dominate theventional bias@Eq. ~3a!# resulting from a finite number of statistical realizations. Jacobsen@13# did not report any systematic variance dependency on power leakage from his experimeindicating that the variance~or standard error! in coherence spectra may be estimated by Eq.~4a! alone.

Another source of bias in estimating coherence is due to psible time delay between the two signalsp and q that can beeasily detected either by simply observing the correspondcross-covariance function of the two time series or by observinlinear trend in the phase spectrum. This source of bias mighimportant when studying coherence in turbulence componeover large lateral separations.

IV Estimation ProcedureFor each bin defined in Table 2, the wind velocities in t

measured frame of reference are transformed into the stanmicrometeorological coordinate system, defined as along-w(u), across-wind (v), and vertical (w) components. Next, eachten-minute time series of interest is partitioned into a suitanumber of 50% overlapped subsegmentsNd , each of which isconsidered as a realization of the process for use in estimatiothe spectra. Hence, each bin class yields a total ofN5Nd3Ngrealizations. Then, for each zero-mean realization~the mean isremoved first!, a detrending~assuming a linear trend!, a prewhit-ening filter, and a Hanning data window multiplication are aplied, before raw power and cross spectra are finally compuusing the Fast Fourier Transform~FFT! algorithm. Next, thesespectra derived from the realizations are averaged in each bobtain representative power spectral density functions and crpower spectral density functions for use in Eq.~2! to compute thecoherence spectrum,g2( f ). This coherence estimate is then biacorrected using Eq.~3a!. Note that the prewhitening filter is applied here in order to reduce possible bias in power spectestimates in the low-frequency region due to leakage proble~Jenkins and Watts@14#; Schwartz and Shaw@15#!. The prewhit-ening filter coefficients, based on a simple moving average mowere determined such that the power spectrum of the filteredbulence signal is essentially ‘‘white,’’ resulting in less powerlower frequencies that ‘‘leaks’’ into the neighboring higher frquency region. By a trial-and-error procedure, the filter coecients were found to be such that the prewhitened signalyt , wasequal to 1.01xt21.00xt21 wherex is the original zero-mean, detrended signal. It is important to point out that detrending aprewhitening the inflow turbulence signal is essentially equivalto applying linear high-pass filters to the signal, which will naffect coherence spectrum estimates. This is because the ltransfer functions will apply to both the auto- and cross-specand will eventually cancel out based on how they appear inestimate of the coherence spectrum given by Eq.~2!. Trend re-moval and prewhitening procedures, however, do directly inence auto- and cross-spectrum estimates individually. Disreging either of these two procedures can lead to bias in auto-cross-spectra individually, especially in the lowest frequencygion ~negligibly at higher frequencies!. We accordingly employthese procedures here because even though they do not influcoherence estimates~our focus!, we will briefly discuss auto-spectra estimates based on data and comparisons with theormodel where such procedures help limit bias problems.

V Numerical StudiesSeveral issues related to coherence estimates based on the

field measurements are investigated. For simplicity, we will reto power spectral density function of the along-wind (u), across-wind (v), and vertical (w) turbulence components asu-, v-, andw-power spectra, respectively. Similarly, for lateral and verticseparations studied, we refer to coherence functions of the al



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A Choices of Subsegment Length. To achieve good esti-mates of coherence spectra, it is necessary to use an appronumber of overlapping subsegments that can provide satisfactsmall amounts of bias and standard error, without introducmuch spectral distortion as discussed before. A large numbesubsegments, which is associated with small individual subsment length, might limit confidence intervals on the coherenestimate but it might introduce unacceptable spectral distorand, thus, lose the fine structure of the coherence especiallythe low frequency region where the coherence value is largerational approach was adopted in this study whereby bcorrected~magnitude-squared! coherence spectra were estimatfor different numbers of subsegments for a ten-minute time seThese estimates were then compared until an optimal numbesubsegments was found. For the sake of illustration, we deschow this optimal number of subsegments was arrived at westimating theuu-coherence for a lateral separation of 13.4using the 25 data sets that were taken from Bin B (Uhub511– 13 m/s andshub51.5– 2.0 m/s). The same procedure mbe applied for other turbulence components and/or for vertseparations. Figure 4(a) shows estimateduu-coherence spectra~for 13.4 m lateral separation! based on the use of 2, 4, 8, 16, an32 subsegments in each ten-minute record. Figure 4(b) showsestimates of the coherence spectrum along with 90% confideintervals with the optimal number of subsegments, equal to 8 h

Fig. 4 „a… Estimates of uu -coherence spectra for a lateralseparation distance of 13.4 m using data from Bin B and differ-ent numbers of subsegments Nd , „b… estimate of theuu -coherence spectrum and 90% confidence intervals usingNdÄ8



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In this illustration, the number of subsegmentsNd that is lessthan or equal to 8 does not cause severe distortion that can rfrom subsegments that are too short. A total of 200 realizati(8325) is selected for estimating the coherence spectrum sthis number provides the smallest standard error~compared tolower values ofNd that have similar bias! and still has very slightspectral distortion~compared to higher values ofNd that havesystematically larger resolution bias!. Using Eqs.~3a! and ~4a!,90% confidence intervals were determined on the bias-correcoherence spectra. These are shown in Fig. 4(b). Note that thisconclusion on what is a suitable number of subsegments mighcase-specific and the appropriate or optimal number of subments might vary from one case to another. Note also thatchoice for the number of subsegments is not limited to intepowers of two. We selected these numbers as powers of 2 onlillustration purposes here.

In general, for the different separation distances that are usethis study, splitting the ten-minute time series into four to eig50% overlapped segments will create individual realizationsabout 2–4 m in length making it possible to resolve frequencielow as 0.004–0.008 Hz with reasonable statistical confidence~andhence, avoid the problem of resolution expected at the lofrequency peak of coherence spectra!. In passing, we point outthat for 50% overlapping, each realization will have length eqto 10 minutes divided by (Nd11)/2 and will be able to resolvefrequencies as low as (Nd11)/1200 Hz.

B Power Spectra. Power spectral density functions of thwind turbulence components were determined for the three BA, B, and C before using them to estimate coherence spectrathe sake of brevity, in Fig. 5, we show power spectra only for BB for the three turbulence components at the center point ofrotor circle mast. The estimated spectra show that the energthe along-wind turbulence component is slightly greater thanin the across-wind component in the low-frequency rangesignificantly greater than that in the vertical turbulence comnent. However, the power spectra for all three turbulence comnents asymptotically have the same slope and magnitude ininertial subrange~high-frequency range!. For the along-wind (u)turbulence component, the estimated power spectrum is compwith the Kaimal spectrum@16#. See the Appendix for a descriptioof the Kaimal along-wind turbulence spectrum model used. A sface roughness lengthz0 of 0.5 cm is used with the Kaimal modein the comparison~typical values ofz0 for mown grass terrain areabout 0.1–1.0 cm!. Comparison between the theoretical alonwind turbulence power spectrum and the estimated spectrum f

Fig. 5 Estimated power spectra of the along-wind „u …, across-wind „v …, and vertical „w … turbulence components for Bin Band comparison with the Kaimal along-wind turbulence spec-trum †16‡ assuming a surface roughness of 0.5 cm.



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Fig. 6 Estimates of uu -coherence spectra for different vertical separations „top row …; and for different lateral sepa-rations „bottom row … based on data from Bins A, B, and C

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C Coherence Estimates, Empirical Models, and Influenceof Separation. In this section, we study the inflow coherencethe along-wind turbulence component (uu-coherence! as esti-mated from the LIST field measurements and compare this msured coherence with predictions based on two commonly uempirical coherence models–the Davenport exponential m@17# and the IEC exponential model~Thresher et al.@18#; IEC/TC88 61400-1@5#!.

1 Davenport’s Exponential Coherence Model.Based on nu-merous experimental results, Davenport@17# hypothesized that thecoherence spectrum of the along-wind (u) turbulence componenfor different vertical separations could be described by an exnential function with a decay parameter,c, as follows:

g2~ f !5exp@2c~ f D/U !# (5)

where f represents the frequency of interest,D is the separationdistance, andU is the mean wind speed at a specified referenceaverage height. As can be seen from Eq.~5!, there are two keyassumptions inherent in Davenport’s empirical model. First,coherence function is assumed to be dependent only on the raexponential decay,c, and on the reduced frequency,f r ~equal tof D/U). This implies no direct dependence of theuu-coherencespectrum on the vertical separation distanceD as a separate variable; only an indirect dependence by virtue of the reducedquency in Eq.~5!. Second, the proposed coherence function in~5! approaches unity when the frequency approaches zero. Seresearchers have extended this exponential format to modecoherence spectra for all three turbulence components andboth vertical and lateral separations where the applicable deparameters in each case are estimated based on experimensults ~see, for example, Jensen and Hjort-Hansen@19#!. Thesemodels have been widely used and have been recommendmany references. For example, Simiu and Scanlan@20# recom-mend a decay parameter,c, of about 20 and 32 for the coherencspectra of the along-wind turbulence component for verticallateral separations, respectively.~Note that the values of 20 and 3given here are twice as large as what are actually mentione

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Simiu and Scanlan@20# because their decay parameters therefor the root-coherence function, not the magnitude-squared coence function, as we are discussing here.! Presently, we shall seethat the coherence estimates from the LIST measurements arin agreement with the Davenport model assumptions. It willseen that the first assumption of the Davenport model may novalid because the observeduu-coherence estimates are dependon the separation distance as an additional independent paramLater, we shall see that the Davenport exponential model~whichwas originally proposed only foruu-coherence and vertical separations! may not be simply extended so as to be used forvv- andww-coherence functions since the estimated coherence funcfrom measurements, especially forww-coherence do not approacunity as the frequency approaches zero.

Coherence spectra are estimated based on the along-windu)turbulence time series available at several spatially distributedcations. Bins A, B, and C provided 33, 25, and 14 ten-minsamples, respectively, as seen in Tables 1 and 2. The variousration distances in the vertical and lateral directions availableeach of the three bins are presented in Table 3.

For vertical separations of 8.5 and 17.0 m, the time series uto estimate the spectra were measured by sonic anemometecated at three positions on the center tower mast located appmately 30 m upwind of the LIST turbine: one at the top of trotor circle, one at the center~i.e., at the hub height of 23 m!, andone at the bottom of the rotor circle. Theuu-coherence estimate~for the two vertical separations! along with their 90% confidenceintervals are shown in Fig. 6~top row! for the three different bins.It is observed that, for all bins, and especially for Bins A andwhere there was a relatively large number of data sets avail~and thus smaller statistical uncertainty!, the uu-coherence evenwhen plotted against the reduced frequencyf r ~equal to f D/U)decreases as vertical separation increases from 8.5 to 17.0 m.clearly shows a dependence of theuu-coherence on vertical separation distance. Nevertheless, if we determine the decay rate ouu-coherence functions by fitting the Davenport model of Eq.~5!to the measured coherence over a range of reduced frequefrom 0 to 0.3, we find for Bin B, that the decay parameterc isapproximately 17 and 25, respectively, for vertical separations8.5 and 17.0 m. For these separations, the estimated decayare in the same range as the decay parameter of 20 recomme



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Table 4 Exponential decay rate, c, in the estimates uu -coherence spectra for different verticaland lateral separations based on fits to Davenport’s expolnential coherence model.

Davenport’s Exponential Model Parameter~c!

Lateral Separations Vertical Separations

Bin

Smallseparation

~6.0 to 6.7 m!

Intermediateseparation

~12.1 to 13.4 m!

Largeseparation

~29.9 to 33.2 m!

Smallseparation~8.5 m!

Intermediateseparation~17.0 m!

A 17 27 127 14 24B 19 28 48 17 25C 16 24 45 19 27

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by Simiu and Scanlan@20#. Still, the decay parameterc increaseswith separation distance~from 17 to 25 here! and the decay raterecommended in standard references may clearly not be validlarger separations. Note again that the dependence ofuu-coherence spectrum on the vertical separation distanceserved from measurements in this study contradicts Davenpexponential coherence model where this along-wind coherefunction when expressed in terms of reduced frequency issumed to be independent of separation.

For lateral separations~refer to Table 3 for the relevant separtions for each bin studied!, the ten-minute data sets used to esmate coherence spectra of the along-wind turbulence compowere measured by three sonic anemometers located at themain towers and a cup anemometer on the far right tower~see Fig.1!. All of these anemometers are at the hub height~23 m!. Noteagain that the lateral separations used here for each bin havecorrected from the actual horizontal distances between the pnent sensors to account for the direction of the mean wind duthe measurement campaign which was not, in general, perdicular to the mast array. Theuu-coherence estimates for differenlateral separations are shown in Fig. 6~bottom row! for the threedifferent bins. Again, when plotted against reduced frequency,f r ,the uu-coherence systematically decays faster with reducedquency at larger lateral separations. Note that a recent studLarsen and Hansen@8#, where they estimateduu-coherence~forlarger lateral separations than is the case here!, suggested thathere was no consistent dependence of coherence on spatialration. Exponential decay rates~based on fitting the measurecoherence spectra using Bin B data with the Davenport model! forseparations of 6.7, 13.4, and 33.2 m are 19, 28, and 48, restively, compared with about 32 in Simiu and Scanlan@20#. Thisagain suggests a faster decay ofuu-coherence with reduced frequency for increased lateral separations. A similar pattern isfound for Bins A and C as summarized in Table 4. Note that whcomparing estimated decay rates for theuu-coherence spectra inTable 4~at least for Bins A and B!, it is found that over compa-rable separation distances in the lateral and vertical direction,decay with reduced frequency is faster laterally than it is vecally. However, this difference is not significant in part dueuncertainty associated with coherence estimates based on aited number of data sets. For the sake of comparison, it shoulpointed out that Larsen and Hansen@8# also compared coherencdecay at various lateral and vertical separations, and concluthat the decay is faster vertically~contrary to what we found here!but the lateral separations studied by them were much larger~79and 170 m! than those considered here, and their vertical septions were not comparable to their lateral separations.

2 IEC Exponential Coherence Model.Our finding in theprevious section is in agreement with the work by several otresearchers~e.g. Kristensen and Jensen@21#; Mann et al.@6#! whoshowed, based on their experimental results, that Davenport’sherence assumptions may be invalid especially in situations wthe spatial separation becomes large since this simple, empmodel fails to account for the reduction in coherence at low fquencies and large separations. To account for such limitation

r Energy Engineering

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the Davenport model, Thresher et al.@18# proposed another exponential coherence model that is also empirically based~as is theDavenport model!. In this model, an additional term is present thinvolves the ratio of separationD to the coherence scale parameterLc and that allows for reduction in coherence with increaseseparation. This exponential model can be expressed as

g2~ f !5$exp~2a@~ f D/U !21~bD/Lc!2#1/2!%2 (6)

This model is currently recommended in the IEC guidelines@5#for wind simulation and the decay parametersa andb are to betaken as 8.8 and 0.12, respectively, while the coherence sparameterLc is approximately 56 m for a wind turbine with a huheight of 23 m~see IEC/TC88 61400-1@5# for details!. In thepresent study, we will refer to this modified exponential modelthe IEC exponential model. Similar to fits with the Davenpomodel, here we estimate the parametersa andb for the exponen-tial model in Eq.~6! from estimates of theuu-coherence spectraAgain, fits of the estimateduu-coherences to the model wercarried out for a range of reduced frequencies between 0 andFor illustration purposes, first we use data from Bin B alone. Fvertical separations of 8.5 and 17.0 m, the estimateduu-coherencespectra were fit to the IEC exponential model and the decayrametera was found to be 8.6 and 12.3, respectively, while tparameterb was about 0.00 and 0.02, respectively. Similarly, flateral separations of 6.7 and 13.4 m, the decay parametera was9.4 and 13.6, respectively, while the parameterb was about 0.05and 0.03, respectively. The fits based on these parametersshown in Figs. 7(a) and 7(b) for the vertical and lateral separation cases, respectively.

It is observed that the IEC exponential model~with the associ-ated parametersa andb) generally provides a good representatiof the uu-coherence spectra for Bin B. Similar good fits of thexponential model for Bins A and C were also found. The emated parameters for all three bins are summarized in TablNote that these fits shown in Figs. 7(a) and 7(b) and the param-eters summarized in Table 5 for the individual bins and sepations were based on using a subset of the collected data forbin and separation. Of greater interest is to employ all of the dand estimate the parametersa and b. Then, by using these estimated parameters, it would be interesting to see howuu-coherence spectra at different separations compared withexponential model using the overall estimated parametersa andb. For the vertical separations, over a range of reduced frequcies from 0.0 to 0.3, the overall least-squares fit parametersa andb for the exponential model foruu-coherence were 9.7 and 0.06respectively, based on data recorded for two separations, 8.517.0 m. For lateral separations, these overall parameters werand 0.13, respectively, based on data recorded over nine setions ranging from 6.0 m to 33.2 m. The overall parametersboth separation directions are included in Table 5. In the lateseparation case, these estimated parameters are fairly closevalues recommended in the IEC guidelines wherea andb are 8.8and 0.12, respectively. Fits based on the overall parametera59.7, b50.13) in the lateral separation case are shown in Figfor Bin B estimated coherence at three different separations



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similar plot was not developed for the vertical separation caseto the limited number~two! of separations available for analysiAs one might expect, Fig. 8 shows that predictions of the exnential model are not as good as when the parametersa and bused were fit for each separation distance and bin individuaThe IEC exponential model appears to work better at the smaseparations~6.7 and 13.4 m for Bin B!. However, at the largerseparation of 33.2 m, the model overpredicts coherence at redfrequencies between 0.05 and 0.15 but significantly underpredcoherence at lowest frequencies; the concave-down second cture in the IEC exponential model in the low-frequency regiappears to contradict what is observed in theuu-coherence esti-mated from data.

In summary, the Davenport exponential coherence mothough simple to use, may not be able to accurately describecoherence structure in the wind turbulence components par

Fig. 7 Exponential fits based on the IEC exponential model inEq. „6… for uu -coherence spectra with „a… vertical separationsof 8.5 and 17.0 m and „b… lateral separations of 6.7 and 13.4 musing data from Bin B



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larly for large separations because this model fails to accountreduction in coherence for low frequencies and large separatiThe IEC exponential model is an improvement foruu-coherencesince two parameters~instead of one in Davenport’s model! areutilized to account for dependence on both reduced frequencyseparation distance. Still, as shown in Fig. 8, the IEC coheremodel predictions foruu-coherence at large lateral separationsquite different from coherence estimates based on field measments especially at low frequencies. Because different coheremodels employed in wind simulation procedures for wind turbload calculations can sometimes lead to significant differencedesign loads, more accurate theoretical coherence models marequired for better predictions of the coherence structure that win turn, lead to more realistic wind turbine design loads. Two sutheoretical coherence models will be discussed later and compwith estimates based on the field data. Next, we comparecoherence structure in the different turbulent components asexamineuu-, vv- and ww-coherence spectra as estimated frothe LIST field measurements.

D Coherence Structure of Each Turbulence ComponentIn the previous section, we only discussed results related toherence spectra of the along-wind (u) turbulence component–i.e.theuu-coherence. Coherence spectra for all three turbulence cponents~i.e., uu-, vv-, andww-coherences! for two spatial sepa-rations are compared here. Figure 9 shows estimated coherspectra of each turbulence component for small and intermedlateral and vertical separations based on data in Bins A, B, an

Fig. 8 Exponential fits based on the IEC exponential model inEq. „6… for uu -coherence spectra with lateral separations of 6.7,13.4, and 33.2 m using data from Bin B. The decay parametersused in this plot „aÄ9.7 and bÄ0.13… were estimated by fittingthe model to all the measured uu -coherence spectra for all lat-eral separations and from all bins.

Table 5 Paramters, a and b, of the IEC exponential model fits to the measured vertical andlateral coherence spectra.

IEC Exponential ModelParameters (a,b) forLateral Separations

IEC Exponential ModelParamters (a,b) forVertical Separations

Bin

Smallseparation

~6.0 to 6.7 m!

Intermdiateseparation

~12.1 to 13.4 m!

Largeseparation

~29.9 to 33.2 m!

Smallseparation~8.5 m!

Intermediateseparation~17.0 m!

A 8.4, 0.06 12.2, 0.08 17.2, 0.17 7.0, 0.08 10.1, 0.08B 9.4, 0.05 13.6, 0.03 21.2, 0.03 8.6, 0.00 12.3, 0.02C 7.9, 0.02 11.4, 0.05 21.5, 0.03 9.4, 0.00 13.9, 0.01

All 9.7, 0.13 9.7, 0.06



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Fig. 9 Comparison of estimated coherence spectra for each turbulence component with „a… lateral separations „twoupper rows … and „b… vertical separations „two lower rows … based on data from Bins A, B, and C.

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For lateral separations, the wind velocity time series usedeach component was measured by sonic anemometers mounhub-height level of the three center tower masts. It is seen in9(a) that the across-wind (v) turbulence component is more coherent than the along-wind (u) and vertical (w) components forall bins. Comparing theuu- andww-coherence, it is seen for abins that theuu-coherence is larger than theww-coherence in thelow frequency range but lower at high frequencies. As expeccoherence in all three turbulence components decreases witcreased lateral separation at all frequencies. At the larger laseparations, it is seen that thevv-coherence is still the largest othe three components while theuu-coherence is higher than thww-coherence over a larger-reduced frequency range. Thesein Fig. 9(a) show that therelative coherence levels for the threturbulence components agree with those based on the von Ka´rman

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coherence model@22# where isotropy was assumed in deriving thenergy spectrum. When studying coherence spectra for diffevertical separations@see Fig. 9(b)], the vertical turbulence com-ponent appears to be the most coherent followed by the acrwind and along-wind components. In the isotropic model, thevv-andww-coherence depend on the direction of the separation.vv-coherence for lateral separations is the same asww-coherence for vertical separations and vice versa. This mexplain the change in relative importance of thevv- andww-coherence among the three spectra shown in Figs. 9(a) and 9(b). Note, however, that the relative levels of theuu- andww-coherence spectra for lateral separations are somewhat dent from the relative levels of theuu- and vv-coherences forvertical separations. This observation is not in agreement withisotropic model and thus suggests a lack of isotropy in the infl



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turbulence. To model the inflow coherence under these conditmore correctly, then, the isotropy constraint needs to be relaNext, we study more closely the coherence structure of eachbulence component by comparing the estimated coherence spbased on measurements with those predicted by two theoremodels–the isotropic von Ka´rman coherence model@22# and theMann uniform shear coherence model@3# where local isotropy isnot assumed.

E Comparing Coherence Structure of Each TurbulenceComponent With Theoretical Coherence Models. As hasbeen already discussed, the Davenport coherence model faaccount for the decrease in coherence at low frequencies anlarge separations while the IEC exponential coherence modeldictions for large separations do not adequately match coherestimates from field measurements. Hence, alternative theorecoherence models might be considered instead of empirical mels for use in predicting inflow coherence. Here, we considersuch theoretical coherence models that are available in the liture. These include~i! the isotropic von Ka´rman turbulence model@22# where isotropy is assumed in deriving the energy spectand ~ii ! the Mann uniform shear turbulence model@3# where theisotropic von Karman energy spectrum is assumed to be rapidistorted by a uniform, mean velocity shear. Predictions basedthese theoretical models will be compared with the measuredherence spectra for all turbulence components and for varlateral spatial separations. For the sake of comparison withempirical model, the IEC exponential model discussed previouwill also be included whenever theuu-coherence spectra are stuied. ~There is no corresponding IEC exponential coherence moavailable forvv- andww-coherence spectra in the IEC standar!



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The isotropic von Ka´rman coherence model and the IEC exponetial model can be conveniently described using available closform mathematical functions. Both these models are currentlycluded into the IEC standard@5# and are recommended for usewind turbine load calculations. The Mann uniform shear modon the other hand, requires greater computational effort sinceinfluence of shear or vertical mean speed gradient~assumed uni-form! from the surface on isotropic turbulence is included in tmodel. Because of its complex formulation, the Mann coherespectrum offers no simple analytical solution but rather requnumerical integration involving the sheared three-dimensionallocity spectral tensor,F i j ~wherei and j are equal to 1, 2, and 3for the along-wind, across-wind, and vertical turbulence comnents, respectively!. Expressions for the magnitude-squared cherence spectra based on the von Ka´rman and the Mann modelsare provided in the Appendix while that for the IEC exponentmodel was shown previously in Eq.~6!.

Notice that one requires the parametersa andb and the coher-ence scale parameterLc in order to obtain the IEC exponentiacoherence spectra. For the von Ka´rman model, information on theisotropic integral length scaleL is necessary while the Mann coherence model requires an isotropic scale parameterl ~which isproportional to the isotropic integral scaleL), and one additionalnondimensional shear distortion parameters to quantify the influ-ence of the shear due to the ground surface. Obviously, all of thparameters are site-dependent and their values can be estimonly from field measurements at the site under considerationmost situations, however, this site-dependent information isreadily available and, in such cases, these various parameteronly be approximated. Since it is our intention to assess the

Fig. 10 Comparison of the estimated uu -coherence spectra for lateral separations „based on data sets from three bins …

with the IEC modified exponential „aÄ8.8, bÄ0.12, L cÄ56 m…, the von Ka rman „LÄ56 m…, and the Mann „sÄ3.9, lÄ14 m… models



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with the von Ka´rman „LÄ56 m… and the Mann „sÄ3.9, lÄ14 m… models

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lidity of the existing coherence models recommended in the wturbine design code, all the parameters used here are takerectly from the code; they are not estimated from the field msurements. For example, numerical values for the required paetersa, b, Lc , andL are each taken from the IEC standard@5#. Aswas mentioned before, the recommended decay parametersa andb in the IEC guidelines are 8.8 and 0.12, respectively, and forparticular case where the hub height is 23 m, the suggested vfor bothL andLc is approximately 56 m~see IEC/TC88 61400-1@5# for details!. For the Mann model, the two parametersl andsmay be approximately obtained by performing a least-squareof the model’s coherence spectra to the Kaimal model@16# insteadof to estimated coherence spectra based on measurements.@23# carried out such least-squares fits from whichs was found to

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Figure 10 shows estimateduu-coherence spectra from measurements for Bins A, B, and C at three different lateral sepations together with spectra predicted by the three models. Wconsidering small lateral separations, it is seen from the top rowFig. 10 that all three models predict similar levels of coherenHowever, the IEC exponential model appears to provide the bprediction of the three models foruu-coherence, except in thevery low frequency range where the Mann model providesbetter prediction. The estimateduu-coherence from data is typically lower than that from the von Ka´rman turbulence model for

Fig. 12 Comparison of the estimated ww -coherence spectra for lateral separations „based on data sets from three bins …

with the von Ka´rman „LÄ56 m… and the Mann „sÄ3.9, lÄ14 m… models



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all three bins. For larger lateral separations~as shown in the twolower rows of Fig. 10!, the three models predict very differencoherence behavior from each other. The von Ka´rman and the IECexponential models appear to predict similar trends~though thevon Karman model predictions are always higher! where the spec-tra at low frequency region lose their exponential character anda result, do no approach unity at these low frequencies. The Mmodel, on the other hand, predicts much higher coherence gents~faster decays! in the low-frequency range. It is evident fromthe figures that especially for the larger spatial separations,Mann coherence model is able to capture higher coherence leand faster decays~with reduced frequency! in uu-coherence spectrum in the low-frequency region, consistent with field measuments for the higher wind speed bins~B and C!. The coherencebehavior predicted by the other two models is grossly differfrom what was observed in the field measurements.

Figures 11 and 12, respectively, show estimatedvv- andww-coherence spectra from measurements for Bins A, B, andsmall and intermediate lateral separations compared with pretions based on the von Ka´rman and Mann models. It is clear fromFig. 12 that theww-coherence spectra estimated from data doapproach unity at very low frequencies. In fact, at the intermedseparations of around 12–13 m, theseww-coherence function val-ues fall below 0.3 at low frequencies. This is an obvious condiction with one of the Davenport model assumptions mentioearlier. This, in turn, suggests that extending the original Davport coherence model of Eq.~5! ~initially proposed foruu-coherence with vertical separations! to use for other turbu-lence components, especially for the vertical turbulence (w) com-ponent is clearly not suitable. When comparing the estimatedvv-andww-coherence spectra from data with von Ka´rman and Mannmodel predictions, it is observed that the Mann coherence mpredicts smaller coherence values compared with the von Ka´rmanmodel for both turbulence components. More importantly, thetimated coherence spectra based on the LIST data sets appagree very well with the Mann model predictions over all sepations and for all three bins studied.

F Cross-coherence. Cross-coherence spectra based on tdistinct turbulence components (uv-, uw-, andvw-! recorded atthe same point in space~here, the center of the rotor circle withheight of 23 m! were estimated and the results are summarizeFig. 13. The figure shows no significant correlation betweenalong-wind (u) and the across-wind (v) turbulence componentnor between the across-wind (v) and vertical (w) turbulence com-ponents in any of the three bins. This finding is consistent withisotropic von Karman model. It is also consistent with the Manuniform shear coherence model where theuv- andvw-coherencespectra are theoretically zero at all frequencies, due to thesymmetric character of the velocity spectral tensor componF12 andF23 in these two models. This is different for the corr



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lation between the along-wind and vertical components whlarger estimates of cross-coherence were observed. This observis contradictory to the von Ka´rman coherence model where thderived uw-coherence is expected to be zero at all frequencNote that theuw-coherence is related to the square of the frictivelocity and, thus, to the influence of wind shear. The von Ka´rmanmodel assumes isotropy and does not take the wind shear effecaccount; this, in turn, leads to failure to correctly predict tuw-coherence estimated from the field data. On the other handMann model has a spectral tensor componentF13, which is sym-metric with respect to the wave number of the across-wind turlence component; thus, providing a nonzerouw-coherence spectrum as can be seen in Fig. 13. It is clear, though, that the Mmodel appears to overestimate theuw-coherence function for allthe three bins studied. This finding is in agreement with the workMann @3# where he reported that the model overestimauw-coherence, and suggested that a more complex model wherinhomogeneous rapid distortion theory~Lee and Hunt@24#! is in-cluded might provide better predictions for this cross-coherenThe present study suggests that it might be important to mouw-coherence~e.g., by using the Mann model! when simulatinginflow to derive design loads for wind turbines, especially if adtional studies can confirm that incorporating this cross-coherecould lead to significant changes in turbine design loads.

G A Note on Atmospheric Stability. It is important to men-tion that the coherence spectrum for turbulence components oinflow depends to some degree on atmospheric stability conditiTheoretical models~such as the isotropic von Ka´rman and theMann uniform shear coherence models! available in the literatureare usually derived based upon the assumption that neutral or satmospheric conditions exist. Since there were a limited numbedata sets available for this particular study, we did not discardof the measured wind speed time series, some of which might hbeen associated with unstable conditions. The resulting mixturdata sets of different stability conditions may account for somethe differences found between coherence spectra predictedempirical/theoretical models and estimates based on the measdata as discussed in this paper.

VI ConclusionsIn this study, we examined spatial statistics using the LIST p

gram’s measured inflow turbulence by obtaining estimates of poand coherence spectra using ten-minute segments of three conents of the wind velocity at several different locations. Coherespectra for different lateral and vertical separations were studiewere cross-coherence spectra between distinct turbulent comnents. Estimation errors associated with coherence spectrascribed by bias, variance, and confidence intervals were also

Fig. 13 Estimates of cross-coherence spectra at the center of the rotor circle for the turbulence components taken twoat a time based on data from Bins A, B, and C, compared with the Mann uniform shear model. „The isotropic von Ka ´rmanmodel predicts zero cross-coherences at all frequencies. …



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cussed. Results obtained for the three different bins of ddefined in Table 2 allow us to make the following general concsions:

• The along-wind coherence spectra for both vertical anderal separations, when expressed in terms of reducedquency, decays faster with increasing separation.

• The Davenport model exponential decay parameterfound to depend on the separation distance. The assumpthat the inflow coherence spectrum approaches unity atfrequency is inconsistent with observed coherence spectrathe across-wind and vertical turbulence components. Heextending the Davenport coherence model hypotheses forwith these turbulence components is not appropriate.

• The IEC modified exponential model predictions of talong-wind coherence for small lateral and vertical sepations matched the estimated coherence from data fairly wexcept at very low frequencies. There, this model predlower coherence than was estimated from data. At large srations and low frequencies, the IEC modified exponenmodel fails to describe the observed fast decay of coherewith reduced frequency.

• The relativecoherence levels of the along-wind, across-winand vertical turbulence components for the different lateseparations as estimated based on data were in accordwith the von Karman model. The different relative levels ocoherence found when considering vertical separationsindicate a lack of isotropy in the turbulence structure.

• Using a required model parameter value as recommendethe IEC standard, the von Ka´rman model generally overestimated coherence when compared to estimates from data.was found especially so when studying the coherences oacross-wind and vertical turbulence components, but wasseen when studying the coherence of the along-wind turlence component, and especially at large lateral separatio

• Overall, the Mann uniform shear coherence model basedparameters fit with the Kaimal turbulence spectra appearagree reasonably well with the estimated coherence spefrom data for all three turbulence components although itunderestimate the along-wind coherence for small lateseparations. In particular, the Mann model predicts lofrequency coherence better than other models studied, anmodel does especially well compared to the von Ka´rmanmodel for across-wind and vertical coherence predictions

• Estimated cross-coherence between the along-windacross-wind turbulence components as well as betweenacross-wind and vertical components is far less significthan that between the along-wind and vertical turbulencomponents. This cross-coherence cannot be predicted bisotropic von Karman model since the influence of the sheis not included in the model, while the Mann model generaoverpredicted the cross-coherence between the along-wand vertical turbulence components.

AcknowledgmentsThe authors gratefully acknowledge the financial support p

vided by Grant No. 003658-0272-2001 awarded through thevanced Research Program of the Texas Higher Education Conating Board. They also acknowledge additional support frSandia National Laboratories by way of Grant No. 30914. Tauthors are grateful to Sandia’s Dr. Herbert J. Sutherland forviding us with the field data from the LIST program and ththank him and Dr. William E. Holley for their helpful insights ansuggestions. Finally, the authors are pleased to acknowledgeanonymous reviewers who helped us with very useful commeand discussions related to this paper.



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Appendix

A Kaimal Spectrum Model for Longitudinal Wind Veloc-ity Turbulence „Kaimal et al. †16‡….

Su~ f !5105u

*2 ~z/U !

@1133~ f z/U !#5/3

wherez is the height above the ground in meters,u* is the shearvelocity, u* '0.4U/ ln(z/z0), and z0 is the surface roughness imeters.

B The Isotropic von Karman Coherence Model for Lat-eral Separations„von Karman †22‡….

g2~ f !5H 21/6

G~5/6! Fz5/6K5/6~z!21

2z11/6K1/6~z!G J 2

for along-wind turbulence component,

g2~ f !5H 21/6

G~5/6! Fz5/6K5/6~z!

13~2p f D/U !2

3z215~2p f D/U !2 z11/6K1/6~z!G J 2

for across-wind turbulence component,

g2~ f !5H 21/6

G~5/6! Fz5/6K5/6~z!

23~D/aL!2

3z215~2p f D/U !2 z11/6K1/6~z!G J 2

for vertical turbulence component,

where z52pA~ f D/U !21~0.12D/L !2, a5G~1/3!

Ap G~5/6!,

L is the isotropic turbulence integral scale,L53.5L, L is theturbulence scale parameter~IEC/TC88 61400-1@5#!, G~ ! is theGamma function, andKn( ) is the modified Bessel function oordern.Note that in the isotropic model, thevv- andww-coherence func-tions depend on the separation direction while theuu-coherencefunction does not. Thevv-coherence for vertical separations is thsame as theww-coherence for lateral separations and vice ver

C The Mann Uniform Shear Model „Mann †3‡…. TheMann spectral tensor componentsF i j (k1 ,k2 ,k3) are given by

F11~k1 ,k2 ,k3!5E~k0!

4pk04 @k0

22k1222k1~k31b~k!k1!z1

1~k121k2

2!z12#,

F22~k1 ,k2 ,k3!5E~k0!

4pk04 $k0

22k2222k2@k31b~k!k1#z2

1~k121k2

2!z22%,

F33~k1 ,k2 ,k3!5E~k0!

4pk4 ~k121k2

2!,



rmal

F12~k1 ,k2 ,k3!5E~k0!

4pk04 $2k1k22k1@k31b~k!k1#z2

2k2@k31b~k!k1#z11~k121k2

2!z1z2%

F13~k1 ,k2 ,k3!5E~k0!

4pk02k2 $2k1@k31b~k!k1#1~k1

21k22!z1%

t

nn

f

d

p,

u

a



F23~k1 ,k2 ,k3!5E~k0!

4pk02k2 @2k2@k31b~k!k1#1~k1

21k22!z2#,

and the Mann coherence spectrum for spatial separations noto the along-wind direction is given by

g i j2 ~ f !5

U E2`

1È2`

1`

F i j ~k1 ,k2 ,k3!e2ik2d2e2ik3d3dk2dk3U2

E2`

1È2`

1`

F i i ~k1 ,k2 ,k3!dk2dk3E2`

1È2`

1`

F j j ~k1 ,k2 ,k3!dk2dk3

wherei and j 51, 2, 3 for the along-wind, across-wind, and vetical turbulence components, respectively,d1 , d2 , d3 are the non-dimensional spatial separation vector components, defined ad i5Di / l , k1 ,k2 ,k3 are the nondimensional spatial wave numbedefined aski52p f l /U, l is an isotropic scale parameter propotional to the isotropic integral length scaleL,

k5Ak121k2

21k32,

k05Ak212b~k!k1k31@b~k!k1#2,

r-

srs,r-

z15C12~k2 /k1!C2 ,

z25~k2 /k1!C11C2 ,

C15b~k!k1

2$k121k2

22k3@k31b~k!k1#%

k2~k121k2

2!,

C25k2k0

2

~k121k2

2!3/2 arctanS b~k!k1Ak121k2

2

k022@k31b~k!k1#b~k!k1

D ,

E~k!51.453k4

~11k2!17/6, the nondimensional, von Ka´rman isotropic energy spectrum,

Na-

nc-bo-

tralark.e Es-ss.,

-

tral

the

81,e-on

res’ Re-

ls

Tur-

,’’

nceford

b~k!5s

k2/3A1F2S 1

3,17

6,4

3,2k22D ,

ands is the shear parameter, while1F2( ) is the hypergeometricfunction.

References@1# Nelson, L. D., Manuel, L., Sutherland, H. J., and Veers, P. S., 2003, ‘‘Statis

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@14# Jenkins, G. M., and Watts, D. G., 1968,Spectral Analysis and its Applications,Holden-Day Inc., San Francisco, CA.

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