{"Bibliographic":{"Title":"Turbulence statistics for design of wind turbine generators","Authors":"","Publication date":"1980","Publisher":""},"Administrative":{"Date created":"08-17-2023","Language":"English","Rights":"CC 0","Size":"0000095700"},"Pages":["QC\n851\n.B6\nno.3\nDecember 1980\nU.S. DEPARTMENT OF COMMERCE\nBoulder, Colorado\nNOAA/ERL Wave Propagation Laboratory\nTURBULENCE STATISTICS\nFOR DESIGNOF\nWIND TURBINE GENERATORS\nPreprint of Report to DOE\nD\nBROBE","QC\n851\nB6\nno.3\nTURBULENCE STATISTICS\nFOR DESIGN OF\nWIND TURBINE GENERATORS\nPreprint of Report to DOE\nJ.C./Kaimal\nJ.E. Gaynor\nD.E. Wolfe\nReport Number Three\nDecember 1980\nNOAA\nBoulder Atmospheric Observatory\nAND ATMOSPHERIC\nAMOUNT NOAA ACTIVITION\nU.S. Department of Commerce\nNational Oceanic and Atmospheric Administration\nEnvironmental Research Laboratories\nOF\nLIBRARY\nOCT 6 1981\nA NOAA publication available from NOAA/ERL, Boulder, CO 80303.\n81 2874\nN.O.A.A.\nU. S. Dept. of Commerce","NOTICE\nThis report was prepared as an account of work sponsored\nby an agency of the United States Government. Neither\nthe United States nor any agency thereof, nor any of their\nemployees, makes any warranty, expressed or implied, or\nassumes any legal liability or responsibility for any\nthird party's use or the results of such use of any in-\nformation, apparatus, product or process disclosed in\nthis report, or represents that its use by such third\nparty would not infringe privately owned rights.\nPrepared by the Wave Propagation Laboratory,\nNOAA/ERL, Boulder, Colorado 80303 for the\nUnited States Department of Energy under\nInteragency Agreement No. DE-A106-79ET23115\nii","CONTENTS\nNomenclature\niv\nAbstract\n1\n1.0 INTRODUCTION\n1\n2.0 THE BOULDER ATMOSPHERIC OBSERVATORY\n3\n3.0 DESCRIPTION OF DATA\n8\n4.0 OUTLINE OF ANALYSIS\n20\n5.0 EFFECT OF HEIGHT VARIATION AND CHOICE OF FILTER\nON SPEED AND DIRECTION STATISTICS\n25\n6.0 ANALYSIS OF WIND ACCELERATION\n40\n6.1 Standard Deviation\n41\n6.2 Skewness\n41\n6.3 Kurtosis\n41\n6.4\nFrequency Distributions\n42\n6.5 Variations with Height\n42\n7.0 ANALYSIS OF GUST 0 PARAMETERS\n60\n8.0 ANALYSIS OF GUST 1 PARAMETERS\n77\n9.0 CHARACTERISTIC MAGNITUDE ANALYSIS\n94\n10.0 CONCLUSIONS\n100\nREFERENCES\n101\niii","NOMENCLATURE\nWind speed\nS\nWind direction\nD\nLongitudinal, lateral, and vertical wind components\nu, V, W\nHeight above ground\nZ\nCyclic frequency (Hz)\nn\nS (n)\nSpectral estimate\nWavelength ( = u/n)\nA\nLower limit of inertial subrange\nA i\nWavelength at spectral peak\nM M\nLowpass filter time\nT l\nHighpass filter time\nTh\nDifferencing interval\nAt\nDifferenced variable X where X = S or D\nAX\nIntegral length scale\nL\n[ IF\nFiltered time series\no(X)\nStandard deviation of X\no rms (X)\nRoot-mean-square of X\n+A0, -A0 0\nGUST 0 amplitudes\n+I0, -T. 0\nGUST 0 times\nGUST 1 amplitudes\n+A1, -A 1\n+T1, -T1\nGUST 1 times.\niv","TURBULENCE STATISTICS FOR DESIGN OF\nWIND TURBINE GENERATORS\nJ. C. Kaimal, J. E. Gaynor and D. E. Wolfe\nABSTRACT\nCharacteristics of moments and probability distributions\nfor two high-wind episodes observed at the Boulder Atmospheric\nObservatory are examined in depth. The two episodes represent\nentirely different stability conditions. Statistics for the\nGUST 0 and GUST 1 models for different heights and bandpass\nfilters are the main focus.\n1.0 INTRODUCTION\nA great deal of attention has been directed in recent years to the\ndevelopment of energy technologies that depend on sources not as easily\ndepleted as the fossil fuels in use today. The main focus of such atten-\ntion has been on such sources as the wind, the sun, and vegetation. As\na source of energy, the wind offers a clean, virtually inexhaustible,\nthough not always predictable, supply that can be harnessed; with each\nincrease in price of fossil fuel, its conversion to electrical power be-\ncomes more cost-effective.\nWind Energy Conversion Systems (WECS) under development cover a\nrange of sizes and output capacities, from small systems suitable for\nfarm and rural use to large megawatt systems designed for use in exist-\ning utility grids. Not surprisingly, the costs of fabrication and\nmaintenance rise sharply with size, so operating efficiency and fatigue\nlife become matters of primary concern in the design of the larger WECS\nsystems. Of fundamental importance is an understanding of a system's\nresponse to spatial and temporal fluctuations in the wind.","A WECS is typically a wind turbine generator with rotor blades\nmounted either vertically or horizontally. Larger systems tend to be\nconventional propeller types with horizontal axes. They operate in a\nregion of the atmosphere's boundary layer where wind shears and tur-\nbulence intensities can be very large. Spatial and temporal fluctua- -\ntions in the velocity field cause bending moments and vibrations that\naffect machine performance and reliability. Furthermore, as the blades\nrotate through air moving at different speeds and directions they are\nsubjected to varying loads. All of these factors have to be considered\nin any description of the wind field to which the wind turbine is ex-\nposed (See, e.g., Connell, 1979; a detailed discussion of all the\nfactors can be found in the same proceedings volume.)\nDevelopment of gust models to characterize the wind field has\nproceeded along two distinct lines. One approach uses statistics that\ncan be derived from accepted turbulence definitions. The frequency of\nan event such as exceedance of the velocity fluctuation or velocity\nchange with time (Cliff and Fichtl, 1978; Huang and Fichtl, 1979) beyond\na prescribed limit is estimated indirectly from turbulence spectra and\nlength scales. The other approach uses the statistics of discrete\nevents defined on the basis of arbitrary criteria such as the time\ninterval between zero crossings or successive positive and negative\npeaks (Ramsdell, 1975; Powell, 1979). The most promising of these\nmodels, GUST 0 and GUST 1, proposed by Powell (1979) involve a velocity\namplitude and a corresponding time scale. A detailed comparison of\nvarious gust models is given by Powell and Connell (1980).\nThe Boulder Atmospheric Observatory (BAO) with its ability to\ncollect and process turbulence data over a depth of 300 m for long\nobservation periods has been useful for verifying the statistics used in\nsome of the proposed gust models. In this report we examine in depth\nthe characteristics of moments and probability distributions for two\nhighwind episodes observed at the BAO.\n2","Figure 2.1--Instrumented -- 300-m tower at the Boulder Atmospheric Observatory.\n2.0 THE BOULDER ATMOSPHERIC OBSERVATORY\nThe BAO is located on gently rolling terrain 25 km east of the\nfoothills of the Colorado Rocky Mountains. This research facility\noperated by the Wave Propagation Laboratory of NOAA is designed to\nprovide high-quality measurements of atmospheric turbulence for boundary\nlayer studies and for calibration and comparison of atmospheric sensors.\nInstallations at the site include a 300-m instrumented tower (Fig. 2.1),\na variety of remote sensors, and a computer system that controls the\nacquisition and processing of data. The contour map of the immediate\nsurroundings (Fig. 2.2) reveals small-scale undulations superimposed on\na mean slope downward from south to north. The rolling nature of the\nterrain is highlighted by the exaggerated vertical scale in the three-\ndimensional terrain profile of Fig. 2.3.\n3","105°00'\nBoulder Creek\n4950\nRte. 52\n5000\n5050\n5100\n105° 00' 12\" W\nBAO\n40° 02' 54\" N\nTower\nElev. 5174'\nBldg.\n40°03'\nErie\nCounty Rd. 8\n7\n5250\n5300\n5150\nScale\n(mile)\n0\n0.5\n1.0\nFigure 2.2--Topographical map of the area around the Boulder Atmospheric\nObservatory. Height contours are in feet.\nBAO Tower\n1576 in\nI25\nCounty Rd.8\nRt.52\nCounty Line Rd.\n1576 m\nN\n76m\nERIE\n1.6km (1mile)\nFigure 2.3--Three-dimensional terrain profile of area shown in Fig. 2.2.\n4","Figure 2.4--View of instruments on the SSE boom during the September\n1978 \"PHOENIX\" experiment. The three-axis sonic anemometer is shown\nwith the fast-response temperature probe mounted in the vertical array.\nThe propeller-vane anemometer and the slow-response quartz thermometer\n(in the aspirated radiation shield) are mounted farther back on the\nboom.\nThe BAO tower is a guyed open-lattice structure of galvanized\nsteel. It has a constant triangular cross section with 3-m spacing\nbetween the legs. The eight fixed levels of instrumentation on the\ntower are at heights 10, 22, 50, 100, 150, 200, 250, and 300 m. Sensors\nare mounted on two retractable 4-m booms at each level. Each boom is\nattached to a large hinged support with fine adjustments for precise\nleveling.\nThe eight levels are instrumented identically. A three-axis sonic\nanemometer mounted at the end of the SSE boom (Fig. 2.4) measures the\nmean and turbulent fluctuations of the wind along three orthogonal axes.\nA fine platinum wire probe attached to the sonic vertical array measures\nfluctuations in temperature. Mean temperature is measured by a quartz\nthermometer housed in an aspirated glass shield. The propeller-vane\nanemometer shown on the SSE boom in Fig. 2.4 was moved to the NNW boom\nbetween the first and second high-wind episodes discussed here. A\ncooled-mirror hygrometer (not shown) is mounted on the NNW boom for\n5","measuring dew point temperatures. Further details of the instrumenta-\ntion can be found in Kaimal (1978a).\nThe tower sensors of particular significance to this study are the\nsonic and the propeller-vane anemometers. The sonic anemometers include\na mix of EG&G Model 198-2 two-axis probes and Ball Brothers1 Model 125-\n198 two-axis and Model 125-197 single-axis probes. The sonic anemom-\neter probes have a path length of 25 cm; their outputs are sampled at a\n10-Hz rate. The propeller-vane anemometer is the \"ruggedized\" R.M.\nYoung2 Propvane Model 8002. Its polystyrene propeller has a distance\nconstant of 2.5 m and a working range from 1 to 54 m/s. The speed and\ndirection outputs from this sensor are sampled once per second.\nThe sonic anemometers on the tower use a fixed orthogonal array.\nIts two horizontal axes are aligned along and perpendicular to the boom\n(see Fig. 2.4); the vertical axis is mounted on the outer end of the\narray pointing away from the tower. Since the orientation of the\nanemometer axes is fixed with respect to the tower, the horizontal wind\nmeasurements are affected when the wind blows parallel to one of the axes.\nAn approximate functional form for the underestimation caused by\nthe transducers is given by Kaimal (1980). Only the horizontal wind\ncomponents are corrected for this error, since vertical wind inclination\nangles are seldom large enough to justify correcting the W component.\nEach data point sampled is corrected in real time using an algorithm\nthat determines the magnitude of the correction from the ratio of the\nwind components measured along the two horizontal axes. The maximum\ncorrection for the winds along either axis is 13%, decreasing linearly\nwith angle to zero at 75 deg. 3 The net effect of this correction is a\n1The three-axis version of this array and associated electronics are now\navailable commercially from Applied Technology, Inc., Boulder, Colorado.\n2R. M. Young Company, 2801 Aeropark Drive, Traverse City, Michigan.\n3This correction is a function of the transducer diameter-to-path-length\nratio and may vary with probe design. The ratio in this probe is 1:25.\n6","7% to 8% increase in variance spread uniformly over the entire spec-\ntral bandwidth.\nAcquisition of data from the sensors is controlled by a PDP 11/34\ncomputer at the BAO site. Real-time computations of means, variances,\ncovariances, and Obukhov lengths are made for each consecutive 20-min\nperiod and recorded both on a line printer and a digital magnetic tape\nunit. The raw data from the sensors are also recorded on tape. To\nminimize tape storage only 10-s averages and 10-s grab samples (last\nsample in each 10-s period) are transmitted to Boulder for archiving.\nThe high-frequency information lost by not saving the entire time series\nis preserved, however, in the form of smoothed spectral estimates,\nblock-averaged to yield approximately seven logarithmically spaced\nestimates per decade. This spectral information is also transmitted\nalong with other data for use in reconstructing the high-frequency end\nof spectra computed later from the 10-s averaged data points.\nThe information transmitted through phone lines feeds into a larger\nmultiuser computer system centered on a PDP 11/70 in Boulder, where it\nis stored temporarily on disk for immediate use and later (a few days to\na week) archived on magnetic tape.\nThe archiving scheme is designed for easy and rapid retrieval of\ndata. Standard programs are available for inspection of the data. Some\nprint numerical summaries, and others produce graphs. The search pro-\ncedure for archived data is facilitated by a descriptor file in the\ncomputer. The descriptor, which consists of low-resolution (20-min\naveraged summary) data from all channels, with accompanying defect codes\nindicating the quality of information in each channel, is stored per-\nmanently in the computer. Details of the interactive access to the BAO\ndata are given by Lawrence and Ackley (1979).\nThe data sets used in this report were prepared from the raw data\ntapes where the full time resolution of the system is retained. All the\nanalyses described were performed on the PDP 11/70 computer using pro-\ngrams and subroutines specially developed for this project.\n7","3.0 DESCRIPTION OF DATA\nThe two cases chosen for this study are fairly typical of the\ndownslope wind storms observed along the Front Range of the Rocky\nMountains. Several such high-wind episodes occur throughout the year,\nalthough they tend to be more frequent during the winter months. The\ntwo cases designated A and B in this report were associated with widely\ndifferent static stabilities. The mean profiles of wind speed, wind\ndirection, and temperature in Fig. 3.1 show how different conditions\nwere for the two cases.\nCase A occurred on 11 September 1979 during the PHOENIX Experiment\nconducted at the BAO. High winds persisted for over 24 h starting at\nabout 1000 MST. Four hours of this episode were selected for analysis.\nFigure 3.2 shows the 20-min averaged speeds and directions for two\nheights for the entire period. (Data points represent averages for the\nfollowing 20 min. ) The temperature lapse rate stayed near-neutral during\nthis period.\nCase B, which occurred during the early hours of 5 December 1979,\nwas a much shorter wind storm than the one observed in September. The\nwind speeds were higher, and, despite the strong mixing that the winds\nproduced, a strong inversion persisted below 100 m (see Fig. 3.1). The\nspeed and direction plots in Fig. 3.3 reveal a highly nonstationary\nperiod which includes a 20-deg shift in wind direction1 and a near-\nsinusoidal rise and fall in wind speed.\nThe synoptic conditions associated with the two wind storms were\nnot unusual for the time of year they occurred. Case A represents a\ntypical chinook produced by an intensification of east-west pressure\ngradient over the mountains, which often accompanies the passage of a\ncold front. Upper-air flow was from the southwest. The pressure\nThe direction shift occurred gradually over the 20-min period, so the\nshift reflected in Fig. 3.2 is a fair representation of the actual rate\nof change in direction.\n8","350\nCase A\n300\n11 Sept 1978\n250\nTemperature\nWind\nSpeed\n200\n150\nNeutral\nDirection\n100\n50\n0\n276 278 280\n17\n18\n19\n12\n14\n16\n18\n20\n.\nT (°C)\nS (m/s)\nD (deg)\n350\nNeutral\nCase B\n300\n5 Dec 1979\nTemperature\n250\n200\nWind\nSpeed\n150\nDirection\n100\n50\n0\n13\n11\n12\n18\n20\n22\n24\n26\n280 282 284\nD (deg)\nT (°C)\nS (m/s)\nFigure 3.1--Mean wind speed, wind direction, and temperature profiles\nfor two high-wind episodes observed at the BAO. Cases A and B represent\ndifferent but fairly typical conditions associated with high winds along\nthe Front Range of the Rocky Mountains. (Case A duration: 1600-2020\nMST; Case B duration: 0040-0400 MST. )\n9","30\nCase A\n25\n20\n150\n15\n10m\n10\n5\n1600\n1700\n1800\n1900\n2000\nTime (MST)\n290\nCase A\n280\n10m\n150\n270\n260\n1600\n1700\n1800\n1900\n2000\nTime (MST)\nFigure 3.2--Plots -- of 20-min averaged wind speeds and directions at 10-m\nand 150-m levels for Case A. Each data point represents average for the\nsucceeding 20-min interval.\n10","Case B\n30\n150 m\n25\n20\n10m\n15\n10\n0300\n0400\n0100\n0200\nTime (MST)\n300\nCase B\n290\n150 m\n280\n10m\n270\n0400\n0100\n0200\n0300\nTime (MST)\nFigure 3.3 -- -Plots of 20-min averaged wind speeds and directions at 10-m\nand 150-m levels for Case B. Each data point represents average for the\nsucceeding 20-min interval.\n11","pattern moved very slowly westward, causing the storm to persist for a\nlong period. By contrast, Case B was more typical of a bora, which is a\ncold downslope wind of short duration. The winds in this case were caused\nby an upper air jet dipping down as it crossed the mountains. A surface\nlow had formed in eastern Colorado, and the jet was drifting southward\nrapidly. Upper air winds were predominantly from the northwest, and ac-\ncount for the cold temperatures and stable stratification near the ground.\nThe data set for Case A consists of sonic anemometer outputs sam-\npled 10 times a second from the four lowest levels: 10, 22, 50, and\n150 m. (The 100-m measurements were not used because of suspected noise\ncontamination in one of the axes ) For Case B, only the propeller-vane\ndata are used since the combination of cold temperatures and high winds\nintroduced noise spikes in the sonic anemometer data. Compatibility\nbetween the propeller-vane and sonic anemometer statistics was establish-\ned by comparing results from both types of instruments. When subjected\nto the same filtering, the variances, skewness, and kurtosis from the\ntwo instruments showed very little difference. The compatibility was\nfurther enhanced by subjecting the sonic anemometer measurements to a\n1-s nonoverlapping average to produce a time series similar to that\nobtained from the propeller-vane anemometer.\nSonic anemometers and propeller-vanes provide different types of in-\nformation. The former yield wind components, and the latter yield wind\nspeed and direction. Since the wind turbine generators are designed to\nface the mean wind, speed and direction are obviously the wind parameters\nthey respond to. We therefore converted the sonic anemometer horizontal\nwind components to speed and direction for all the analysis described\nhere. It is generally recognized that the spectral characteristics of\nspeed and direction very closely approximate those of the longitudinal\n(u) and lateral (v) wind components. Powell and Connell (1980) report\nwind speeds to be only 3% greater than the u-component variance and the\ndirection variance to be within 0.5% of that obtained by dividing the\nv-component variance by the square of the mean horizontal wind speed.\n12","The spectra of speed and direction measured at 10 m and 150 m are\nshown in Fig. 3.4 (Case A) and Fig. 3.5 (Case B) Because of the higher\nsampling rate in the sonic anemometers, the spectra extend a decade\nhigher in Case A than in Case B. The spectral behavior in this region\nis fairly predictable, so the high-frequency end in Case B may be\napproximated with reasonable certainty. The dashed lines in Fig. 3.5\nfollow the -5/3 power law predicted by Kolmogorov for the inertial sub-\nrange. (In our spectral representation, this power law appears as a\n-2/3 slope.) Included in Fig. 3.4 are the vertical velocity (w) spectra\nfor 10 m and 150 m; this information is available from the sonic anemom-\neters for Case A but not for Case B.\nThe inertial subrange represents the region of the spectrum where\nturbulent energy is neither produced nor destroyed, but handed down\nfrom the larger to the smaller eddies. In this region, all three com-\nponents of velocity follow the -5/3 power law. Also, the turbulent\nfield is locally isotropic (i.e., no preferred direction for eddies on\nthat scale) with vanishing covariance between the different velocity\ncomponents. The limiting wavelength on the low-frequency side is con-\ntrolled by stability and the height above ground. The spectral maximum,\nfarther down on the frequency scale, represents the region where energy\nis being produced by mechanical and buoyant forces, while farther up\nthe scale, beyond the range of our spectral computations, energy is dis-\nsipated in the form of heat. Where turbulent energy is being produced,\nthe spectrum slope tends to be shallower than in the inertial subrange,\nwhereas in the region where energy is being dissipated, the slope tends\nto be steeper. The higher spectral energy in the W component, compared\nwith the speed spectrum within the inertial subrange, is a consequence of\nisotropy. In our one-dimensional spectral representation (i.e., , spatial\ncut of the turbulent field along the streamwise direction), the inertial\nsubrange energy in the crosswind components (w and v) will appear 4/3\nas high as the energy in the streamwise component (u). To the extent\nthat the speed spectrum approximates the u spectrum, this condition is\nsatisfied with respect to W in Fig. 3.4 at wavelengths A < 0.5z, where A\nis the wavelength (=u/n) and Z is the height above ground.\n13","1\n10\nCase A\nSpeed and W Spectra\n0\n10\nS10\nS\n150\n10-1\nW 150\nW10\n2\n10\n102\nCase A\nDirection Spectra\n1\n10\nD\n10\nD150\n10°\n10-\n10 1\n10-4\n10-3\n10-2\n10-1\n10°\nn (Hz)\nFigure 3.4 -- --Logarithmic spectra of wind speed, W, and direction at 10-m\nand 150-m levels for Case A.\n14","1\n10\nCase B\nSpeed Spectra\n10°\nS10\nS\n150\n10-¹\n10-2\n102\nCase B\nDirection Spectra\nD10\n101\n9\nD\n150\n10°\n10-1\n10-2\n10-4\n10-1\n1\n10-\n10°\n10\nn (Hz)\nFigure 3.5 -- -Logarithmic spectra of wind speed and direction at 10-m and\n150-m levels for Case B.\n15","The limiting wavelength A. for the inertial subrange normally ap-\nproaches 0.5z only in an unstable boundary layer (Kaimal et al., 1972).\nAs the atmosphere becomes stably stratified, A. becomes a function of\nstability as well (Kaimal, 1973). Another feature of the spectra in\nFig. 3.4 that suggests a slightly unstable lapse rate is the constancy\nin the speed spectral peak with height. The wavelength at the spectral\nmaximum, ^m' remains constant at about 1.8 km between 10 and 150 m.\nIn\nconvectively unstable boundary layer, for u and V scales with the\na\nheight of the boundary layer (Kaimal, 1978b), but in the absence of a\nconventional boundary layer in Case A it is difficult to verify that\nrelationship. The wavelength of the peaks observed here is typical for\na daytime boundary layer roughly one 1 km deep.\nIn contrast to the indications provided by the speed spectrum, the\npeaks of the W and direction spectra in Fig. 3.4 behave as if the atmo-\nsphere is slightly stable (almost neutral). Here, A for W is smaller\nm\nthan the usual value of 5.9z observed in an unstable atmosphere, and\nfor direction varies with height as in a slightly stable layer. We\nA\nm\nfind ^m = 2z and 5z, respectively, for W and direction.\nAn interesting point to note in Fig. 3.4 is the presence of a spec- -\ntral gap centered around 10- 3 Hz in both speed and direction. This gap\nindicates a separation between the scales associated with boundary layer\nturbulence and the larger mesoscale features of the wind field. The\nupward swing in the spectra below 10-3 Hz is produced by the downward\ntrend in the time series (Fig. 3.2).\nThe trends are even more pronounced in Case B (Fig. 3.3), and their\neffect on the spectra extend to 10-2 Hz in Fig. 3.5. Consequently, the\nspectral gap is not as obvious as in Case A. However, the spectral\npeaks for at least the 10-m level can be identified and the relationship\nbetween A and Z examined. We find ^m = 10z and 3z respectively for\nm\nm\nspeed and direction, corresponding to values one might expect in moder-\nately stable layers (z/L ~ 0.2) under quieter conditions. Actual values\nof z/L ranged from 0.05 to 0.2 during the period.\n16","No particular significance is attached to the peaks and valleys\nat the low-frequency end of the spectra in Fig. 3.5. These features\nare reproduced almost identically at all levels between 10 and 150 m.\nThey reflect the finer details of the spectral content in the trends\nand cannot be attributed to computational uncertainties in the spectrum\nanalysis.\nFor the reader interested in relating spectral behavior to the ap-\npearance of the actual time series, we present the first 20 min of the\nspeed and direction fluctuations (deviations from the 20-min mean) in\nFigs. 3.6 and 3.7. In Case B, the amplitudes of the speed fluctuations\nare distinctly smaller than in Case A even though the direction ampli- -\ntudes are larger, not an unusual occurrence in stable layers. However,\nthe amplitudes of the speed fluctuations do increase with mean wind\nspeed and approach their maximum values an hour later.\nThese descriptions of the two cases are intended to serve as back-\nground for interpreting the distributions and moments presented in this\nreport. The spectral properties of the wind field, by themselves, ap- -\npear to be of limited value in WECS design, although idealized spectral\nforms have been used for calculating the frequency of occurrence of gust\nevents through Rice's equation (Powell and Connell, 1980). Bandpass\nfiltering appropriate to the response characteristics of the device is\nthe key to deriving parameters needed for the design of wind turbine\ngenerators. Our analysis therefore includes results for a range of\nfilter options. These ranges can be identified in the spectral plots.\nNot included in this report are the effects of wind shear on the\npropeller blades. Direction shears do not appear to be very signifi-\ncant, but the vertical speed gradients (5 m/s per 100 m for Case A and\n7 m/s per 100 m for Case B) must seriously alter the spectral content of\nthe wind field sensed by each propeller blade as it rotates. This pro-\n-\nblem is being addressed by Connell (1980) and Doran and Powell (1980)\n17","600\n600\nFigure 3.6 - - Plots of 1-s averaged data points of wind speed and direction\n400\n400\nfluctuations at 10 m for the first 20 min from Case A (1600-1620 MST). .\nTime (s)\n200\n200\n0\n0\n600\n600\n400\n400\nTime (s)\n200\n200\n0\n0\n+10\n+5\n0\n-5\n-10\n+20\n+10\n0\n-10\n-20","600\n400\n400\n(s)\nTime\n200\n200\n600 00\n400\n200\n200\n+20","4.0 OUTLINE OF ANALYSIS\nThe data set prepared for this study consists of wind speed and wind\ndirection measurements from three heights (10, 50, and 150 m) in the form\nof discrete data points spaced 1 S apart. In Case A the original 10-Hz\nsamples were block-averaged (see Chapter 3 for details) to achieve high-\nfrequency smoothing comparable with the smoothing inherent in the 1-Hz\npropeller-vane data used in Case B. With the three heights chosen, we\ncover the wind field near the ground, at hub level, and near the top of\nthe surface shear layer.\nFrom these data several more time series were generated. The first\nset of operations performed involves bandpass filtering to simulate the\nresponse of the WECS to fluctuations in the wind speed and direction.\nThree passbands, each a decade wide, simulate the response of systems\nwith slightly different characteristics In Fig. 4.1 these filters are\nrepresented by designations 30/3, 50/5, and 100/10, where the first\nnumber in each group represents the half-power point on the highpass\nside of the filter and the second number the half-power point on the\nlowpass side. The numbers are the widths, in seconds, of the running\nmeans used in the highpass and lowpass filters, respectively. First,\nthe original time series are averaged with a moving average filter of\nwidth Il to remove frequencies higher than 1/T o Hz (where Il = 3, 5, or\n10 s). The new time series thus created were then differenced from a\nmoving average of width Th to remove frequencies smaller than 1/Th Hz\n(where Th = 30, 50, or 100 s) The resulting time series are used for\ncalculating the statistical properties of the WECS response. Doran and\nPowell (1980) give algorithms for this type of digital filtering.\nThe rationale for bandpass filtering to simulate WECS response stems\nfrom the assumption that fluctuations of very long and very short periods\nhave little, if any, effect on the performance of the system. Spatial\naveraging of the wind fluctuations by the propellers diminishes WECS sen-\nsitivity to small-scale eddies, and the ability of the system to adjust\nto slow changes in wind speed and direction through blade pitch control\n20","Analysis Sequence for High Wind Data\nNo Filter\nSonic Anemometer\nData (0.1 Hz)\n30/3 Filter\nAcceleration\n11 Sept 1978\nGust 0\n50/5 Filter\nProp-Vane\n100/10 Filter\nGust 1\nData (1.0 Hz)\n5 Dec 1979\nAmpl. Pos.\nAmpl. Neg.\nMoments of\nTime Pos.\nFiltered Data\nTime Neg.\nMoments of\nDistribution\nFigure 4.1--Flow chart showing analysis procedure for Case A and Case B\ndata.\nand rotation into the wind diminishes its sensitivity to the large eddies.\nDoran and Powell (1980) note that the gust statistics are relatively\ninsensitive to highpass filtering for filter times longer than 100 S\n(probably the result of the spectral gap between 10- 2 and 10-3 Hz). For\nmoderately strong winds the averaging distance for a large propeller\ncorresponds to roughly 3 S, which also happens to be the smallest conven-\nient filter width for lowpass filtering. Our three bandpass filters are\ntherefore confined to a window between 3- and 100-s periods. Centered\nwithin this window is the basic 50/5 filter of Powell and Connell (1980).\nEach of the four sets of time series yields its own probability\ndistributions, means, standard deviations, skewness, and kurtosis. In\naddition, the time series are converted to gust parameters appropriate\nto three gust models: 1) the acceleration (speed and direction differ-\nence) model, 2) the GUST 0 model, and 3) the GUST 1 model. Probability\n21","distributions for each of the gust models are computed as well. The\nresult is an enormous array of plots and tables. A major challenge the\nauthors face is condensing the information in a way that makes it both\naccessible and useful to design engineers.\nThe acceleration model uses five distinct differencing intervals:\nAt = 1, 3, 5, 10, and 30 S. Although all At intervals can be analyzed,\nonly intervals equal to or larger than the lowpass interval T yield\nuseful information; the moving average effectively filters out differ-\nences across intervals of shorter duration. The differenced time series\navailable for analysis are listed in Table 4.1\nTable 1--Differenced time series available for analysis\nDifferencing Interval (s)\n10\n30\nFilter\n1\n3\n5\nX\nX\nnone\nX\nX\nX\n30/3\nX\nX\nX\nX\n50/5\nX\nX\nX\n100/10\nX\nThe GUST 0 and GUST 1 models define gust events in terms of gust\namplitudes and characteristic times. Definitions of amplitudes and\ntimes differ for the two models (see Fig. 4.2 ).\nThe GUST 0 model can be completely specified in terms of a positive\npeak amplitude +A 0' and the time interval +T 0 between zero crossings on\neither side. It can also be specified in terms of a negative peak\namplitude -A0 and the corresponding time interval -To. Both definitions\nare expected to yield similar statistics. Powell and Connell (1980)\ncombine both positive and negative values into one set for their analysis.\n22","GUST 0 Definition\nTo\n+A0\n+To\n-A0\nData sample restricted to\nTo > 3s for 30/3 filter\n> 5 s for 50/5 filter\n> 10 s for 100/10 filter\nGUST 1 Definition\n-A1\n+A1\nT1\n+T 1\nData sample restricted to\nT1 > 3s for 30/3 filter\n> 5s for 50/5 filter\n> 10 s for 100/10 filter\nFigure 4.2--Definitions of GUST 0 and GUST 1 parameters.\nThe GUST 1 model can also be specified in terms of a positive\namplitude +A1 which is the peak-to-peak amplitude between adjacent\nminima and maxima, and a time interval +T1 between the minima and the\nmaxima; the positive sign indicates a positive rate of change in the\nvariable. A comparable definition can be made in terms of amplitude\n23","-A and time -T1 where the negative sign indicates a negative rate\nof change in the variable.\nThus, for each of these two discrete gust models, we have two sub-\nsets of events, defined in terms of positive and negative changes to\ndescribe the properties of the wind field. In preparing the subsets\nfrom the bandpass-filtered data, we imposed a lower limit for the time\nintervals, so that only To's and I1's larger than 'e' the width of the\nlowpass moving average, were considered. This indirectly placed a lower\nlimit on the A0''s and A1's as well, since small amplitudes are more\noften than not associated with small time intervals. The parameters\nneeded for WECS design are the root-mean-square values of the amplitudes\nand times. These are calculated from the means and standard deviations\nof A's and T's and listed as tables in subsequent chapters of this\nreport.\n24","5.0 EFFECT OF HEIGHT VARIATION AND CHOICE OF FILTER\nON SPEED AND DIRECTION STATISTICS\nBefore we investigate the statistical properties of parameters\nmentioned in the previous chapter and their variation with height and\nfiltering, it is important to understand how the basic time series\nthemselves behave. Table 5.1 gives easy access to that information and\npermits direct comparisons between Case A and Case B.\nThe biggest variation, not surprisingly, is introduced by filter-\ning. Between the unfiltered and filtered time series, the standard\ndeviation drops a factor of 3 in wind speed and a factor of 2 in wind\ndirection, the skewness drops to insignificant levels, and the kurtosis\nrises above 3, reaching values as high as 8.4 in Case B.\nThe drop in the standard deviation is obviously a consequence of\nremoving contributions from large segments of the energy-containing\nregion of the spectra. The smaller skewness values in the filtered\ndata (indicative of a more symmetrical shape in the probability dis-\ntribution) and the larger kurtosis values (indicative of a more peaked\ndistribution) result from the exclusion of long-period fluctuations in\nthe filtered time series.\nVariations between filtered time series are not particularly signi-\nficant. (The only exception to this observation is the behavior of the\n150-m direction kurtosis in Case B; we interpret this as a reflection of\nthe strong static stability present at the time. ) On examining the\nvariability with height, we find a small but systematic decrease in the\nstandard deviations, but no such pattern can be found in the skewness or\nkurtosis. The 150-m direction kurtosis in Case B is an exception to\nthat rule, but that level was clearly above the top of the surface\ninversion where the structure of the fluctuations can be expected to\ndiffer from the structure within the inversion.\n25","100/10\n1.22\n1.14\n0.98\n1.94\n2.69\n1.84\n0.21\n-0.04\n-0.10\n0.13\n0.00\n-0.04\n3.15\n3.99\n4.24\n4.60\n4.62\n4.57\n50/5\n1.29\n1.17\n0.91\n3.13\n2.21\n1.90\n0.21\n-0.11\n-0.13\n0.04\n0.06\n0.05\n3.34\n4.44\n4.86\n4.50\n5.16\n7.16\nfilters\nCase B\nTable 5. - -Variation in the statistical properties of speed and direction with\nheight and choice of filtering. (All dimensions are in SI units.)\n30/3\n1.33\n1.17\n0.88\n3.66\n2.46\n1.95\n0.17\n-0.15\n-0.16\n-0.01\n0.11\n0.13\n3.38\n4.41\n5.44\n4.13\n5.12\n8.39\nNone\n4.23\n4.88\n5.01\n7.60\n5.95\n5.15\n-0.08\n-0.48\n-0.63\n-0.16\n-0.36\n-0.19\n2.59\n2.92\n2.44\n3.12\n2.86\n3.21\n100/10\n1.29\n1.22\n1.09\n3.53\n2.82\n2.45\n0.07\n0.02\n-0.14\n-0.02\n0.09\n-0.22\n3.56\n3.46\n3.57\n3.51\n3.66\n3.80\nfilters\n50/5\n1.23\n1.08\n0.98\n3.62\n2.92\n2.57\n0.01\n-0.03\n-0.13\n0.01\n0.06\n-0.11\n3.53\n3.63\n3.81\n3.81\n3.55\n3.64\nCase A\n30/3\n1.18\n0.97\n0.88\n3.76\n2.97\n2.58\n-0.05\n-0.08\n-0.10\n-0.02\n0.05\n-0.03\n3.60\n3.77\n3.88\n3.77\n3.51\n3.63\nNone\n3.52\n3.68\n3.16\n7.34\n5.71\n5.04\n0.46\n0.41\n0.23\n0.11\n0.20\n0.17\n2.66\n2.55\n2.52\n3.44\n3.27\n2.98\nHeight\n10\n50\n150\n10\n50\n150\n10\n50\n150\n10\n50\n150\n10\n50\n150\n10\n50\n150\nDirection\nDirection\nDirection\nVariable\nSpeed\nSpeed\nSpeed\nStatistical\ndeviation\nparameter\nStandard\nSkewness\nKurtosis","The frequency distribution plots presented in Figs. 5.1-5.12 offer\na detailed view of how the large skewness and kurtosis values manifest\nthemselves. For comparison, the true Gaussian distribution for the same\nvariance is superimposed on the observed distribution. (Multiplication\nfactor in parenthesis (e.g., , X 10) converts numbers on the abscissa to\nSI units ) The asymmetry in the unfiltered time series is immediately\napparent. That the distributions for Case A and Case B are skewed in\nopposite directions is not significant, being merely a consequence of\nthe asymmetry in the long-term trend with respect to the overall mean\nvalue. Interestingly enough, the asymmetry in the frequency distribu-\ntion is much less pronounced in the direction fluctuations than in the\nspeed fluctuations. The absence of such asymmetry in the filtered\ndistribution suggests that medium- and small-scale fluctuations are\ndistributed evenly across both sides of their computed running means.\nThus the assumption of a stationary Gaussian process, made in many gust\nmodels, has validity only for the filtered data, not for the unfiltered\ntime series.\n27","(x 10)\n(x 10)\n2.00\n4.00\n5.1 --Distributions of speed and direction at 10 m (no filter). . Bin sizes distri- are\nCase B. No filter; Z = 10 m\n1.00\n2.00\nFigure AS = 0.4 m/s, - - AD = 0.8 deg. Superimposed for comparison are the ideal Gaussian\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-1.00\n-2.00\n-2.00\n-4.00\n6.00\n4.00\n2.00\n2.00\n0.00\n6.00\n4.00\n0.00\n(x 10)\n(x 10)\n2.00\n4.00\nCase A. No filter; Z = 10 m\n1.00\n2.00\nbutions for the same variance.\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-1.00\n-2.00\n-2.00\n-4.00\n4.00\n2.00\n6.00\n0.00\n6.00\n4.00\n2.00\n0.00","(x 10)\n2.00\n(x 10)\n4.00\nFigure 5.2--Distributions of speed and direction at 50 m (no filters) AS = 0.4 m/s;\nNo filter; Z = 50 m\n1.00\n2.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\nCase B.\n-1.00\n-2.00\nM\n-2.00\n-4.00\n2.00\n4.00\n2.00\n0.00\n6.00\n0.00\n6.00\n4.00\n(x 10)\n4.00\n(x 10)\n2.00\nA. No filter; Z = 50 m\n2.00\n1.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\nAD = 0.8 deg.\n-2.00\n-1.00\nCase\n-4.00\n-2.00\n0.00\n2.00\n2.00\n0.00\n6.00\n4.00\n6.00\n4.00","2.00\n(x 10)\n(x 10)\n4.00\nFigure 5.3--Distributions of speed and direction at 150 m (no filters) . AS = 0.4 m/s;\nB. No filter; Z = 150 m\n1.00\n2.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-1.00\n-2.00\nCase\n-2.00\n-4.00\n6.00\n4.00\n2.00\n0.00\n4.00\n6.00\n2.00\n0.00\n2.00\n4.00\n(x 10)\n(x 10)\nA. No filter; Z = 150 m\n1.00\n2.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\nAD = 0.8 deg.\n-1.00\n-2.00\nCase\n-4.00\n-2.00\n4.00\n2.00\n0.00\n6.00\n4.00\n2.00\n0.00\n6.00","(x 10)\n(x 10)\n1.00\n2.00\n5. 4--Distributions - of speed and direction at 10 m (30/3 filter). AS = 0.2 m/s;\n30/3 filter; Z = 10 m\n0.50\n1.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase B.\n-1.00\n-2.00\n0.40\n0.20\n0.80\n0.60\n0.40\n0.00\n1.20\n1.00\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n(x 10)\n2.00\n(x 10)\n1.00\nA. 30/3 filter; Z = 10 m\n1.00\n0.50\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\nAD = 0.4 deg.\n-1.00\n-0.50\nCase\nFigure\n-2.00\n-1.00\n1.00\n0.60\n0.40\n0.20\n0.00\n0.20\n1.20\n0.80\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40","1.00\n(x 10)\n(x 10)\n2.00\nAS 0.2 m/s;\n30/3 filter; Z = 50 m\n0.50\n1.00\nFigure 5.5--Distributions of speed and direction at 50 m (30/3 filter) . =\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase B.\n-1.00\n-2.00\n1.20\n1.00\n0.88\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.00\n(x 10)\n(x 10)\n2.00\n30/3 filter; Z = 50 m\n0.50\n1.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\nAD = 0.4 deg.\n-0.50\n-1.00\nCase A.\n-1.00\n-2.00\n0.20\n1.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.00\n1.20\n0.80\n0.60\n0.40\n0.20\n0.00","1.00\n(x 10)\n2.0\n(x 10\nFigure 5. --Distributions - of speed and direction at 150 m (30/3 filter) . AS = 0.2 m/s;\n30/3 filter; Z = 150 m\n0.50\n1.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase B.\n-1.00\n-2.00\n0.60\n0.40\n0.20\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.00\n(x 10)\n(x 10)\n1.00\n2.00\nCase A. 30/3 filter; Z = 150 m\n0.50\n1.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\nAD = 0.4 deg.\n-1.00\n-0.50\n-2.00\n-1.00\n0.40\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.20\n0.00\n1.20\n1.00\n0.80","2.00\n(x 10)\n1.00\n(x 10)\nm/s;\nB. 50/5 filter; Z = 10 m\n5. 7 --Distributions - - of speed and direction at 10 m (50/5 filter) . AS = 0.2\n1.00\n0.50\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-1.00\n-0.50\nCase\n-2.00\n-1.00\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.00\n0.88\n0.60\n0.40\n0.20\n0.00\n1.20\n1.20\n2.00\n(x 10)\n1.00\n(x 10)\nA. 50/5 filter; Z = 10 m\n1.00\n0.50\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\nAD = 0.4 deg.\n-1.00\n-0.50\nFigure\nCase\n-2.00\n-1.00\n1.00\n0.40\n0.20\n0.00\n0.00\n1.20\n0.80\n0.60\n1.20\n1.00\n0.88\n0.60\n0.40\n0.20","1.00\n(x 10)\n2.00\n(x 10)\nFigure 5. 3--Distributions of speed and direction at 50 m (50/5 filter). . AS = 0.2 m/s;\n50/5 filter; Z = 50 m\n0.50\n1.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase B.\n-1.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.00\n(x 10)\n2.00\n(x 10)\nCase A. 50/5 filter; Z = 50 m\n0.50\n1.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\nAD = 0.4 deg.\n-0.50\n-1.00\n-1.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00","2.00\n(x 10)\n1.00\n(x 10)\nAS 0.2 m/s;\n50/5 filter; Z = 150 m\n1.00\n0.50\nFigure 5. -Distributions of speed and direction at 150 m (50/5 filter) . =\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-1.00\n-0.50\nCase B.\n-2.00\n-1.00\n0.80\n0.40\n0.20\n0.00\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.60\n1.20\n(x 10)\n2.00\n(x 10)\n1.00\n50/5 filter: ; Z = 150 m\n1.00\n0.50\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\nAD = 0.4 deg.\n-1.00\n-0.50\nCase A.\n-2.00\n-1.00\n0.60\n0.40\n0.20\n0.00\n0.20\n0.00\n1.20\n1.00\n0.80\n1.20\n1.00\n0.80\n0.60\n0.40","(x 10)\n(x 10)\n1.00\n2.00\n100/10 filter; Z = 10 m\nFigure 5.10--Distributions of speed and direction at 10 m (100/10 filter) .\n0.50\n1.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase B.\n-1.00\n-2.00\n0.60\n0.40\n0.20\n0.00\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n1.20\n1.00\n(x 10)\n2.00\n(x 10)\nAS = 0.2 m/s; AD = 0.4 deg.\n100/10 filter; Z = 10 m\n1.00\n0.50\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-1.00\n-0.50\nCase A.\n-2.00\n-1.00\n0.40\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.20","(x 10)\n1.00\n(x 10)\n2.00\n100/10 filter; Z = 50 m\n0.50\n.\n1.00\nFigure 5.11 - - Distributions of speed and direction at 50 m (100/10 filter)\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase B.\n-1.00\n-2.00\n1.00\n1.20\n0.80\n0.60\n0.40\n0.20\n0.00\n1.26\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.00\n(x 10)\n2.00\n(x 10)\nA. 100/10 filter; Z = 50 m\nAS = 0.2 m/s; AD = 0.4 deg.\n0.50\n1.00\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase\n-1.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00","(x 10)\n2.00\n1.00\n(x 10)\n100/10 filter; Z = 150 m\nFigure 5.12--Distributions of speed and direction at 150 m (100/10 filter) .\n1.00\n0.50\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-1.00\n-0.50\nCase B.\n-2.00\n-1.00\n0.60\n0.40\n0.20\n0.00\n1.00\n0.80\n0.60\n0.20\n0.00\n1.20\n1.00\n0.80\n0.40\n1.20\n2.00\n(x 10)\n1.00\n(x 10)\nAS = 0.2 m/s; AD = 0.4 deg.\n100/10 filter; Z = 150 m\n1.00\n0.50\nDirection (deg)\nSpeed (m/s)\n0.00\n0.00\n-1.00\n-0.50\nCase A.\n-2.00\n-1.00\n0.20\n0.00\n0.80\n0.60\n0.40\n1.00\n1.20\n0.20\n0.00\n0.80\n0.60\n0.40\n1.20\n1.00","6.0 ANALYSIS OF WIND ACCELERATION\nAll gust models define discrete gust events in terms of some aspect\nof turbulence. The time rate of change of the wind speed and wind direc-\ntion are convenient parameters for defining gust events since they are a\nmeasure of the severity of the event. The Cliff-Fichtl (1978) and Huang-\nFichtl (1979) models use speed difference information to create discrete\ngust events. Such models are sometimes referred to as gust-rise models,\nbut in the case of the former model that definition incorrectly implies a\npreference for a positive speed change. Their main usefulness is in\ncalculating the probabilities of extreme events, and they have also been\nfound useful for fatigue calculations (Powell and Connell, 1980).\nIn both the Cliff-Fichtl (CF) and Huang-Fichtl (HF) models the wind\nspeed time series is replaced by the speed difference computed across a\ntime interval At. In the CF model, the discrete events are the actual\nvalues of speed differences measured across At. In the HF model the dis-\ncrete events are the positive slope crossings of the speed difference\nfunction at level X. In contrast with the CF model, At in this model is\nmade as small as possible to make the difference function appear con-\ntinuous.\nAnother point of contrast between the models is their basic fre-\nquency factor. For the CF model this factor is a constant equal to\n1/ (20t), which is the Nyquist frequency for the differenced time series.\nFor the HF model the frequency factor is calculated through an expres-\nsion based on Rice's equation involving the filtered variance of the\ndifferenced time series and its power spectrum.\nThe choice of At for differencing is obviously a critical factor\nsince an improper choice of At could result in the underestimation of\nthe severity of the gust environment. In this chapter we examine the ef-\nfect of varying At in the unfiltered and the three filtered time series.\nThe plots in Figs. 6.1-6.6 show the standard deviations, skewness, and\nkurtosis for five At values: 1, 3, 5, 10, and 30 S. All three heights\n40","(10, 50, and 150 m) are represented. In addition, the frequency distri-\nbutions for At = 5, 10, and 30 are provided in Figs. 6.7-6.15. - The\nfollowing comments can be made regarding the response of the statistical\nparameters to the various operations performed on the time series.\n6.1 STANDARD DEVIATION\nThe disparity between the filtered and the unfiltered standard\ndeviations is not as large as in the undifferenced speed and direction\ntime series. This is because differencing is inherently a high-pass\nprocess; the low-frequency contributions that contribute greatly to the\nhigh standard deviations and cause departures from Gaussian behavior in\nthe skewness and kurtosis in the undifferenced time series are absent in\nthe differenced time series.\nThe unfiltered standard deviations are smallest for At = 1 s, but\nrise rapidly between At of 1 and 5 S and more gradually between 5 and\n30 S. The 30/3 and 50/5 time series show a distinct maximum at 10 s,\nwhereas the 100/10 time series show the same rate of increase with At as\nthe unfiltered data. If one considers the 50/5 filter the basic filter\nfor WECS design, the optimum choice for At is 10 S.\n6.2 SKEWNESS\nThe skewness values are not noticeably affected by filtering.\nThey are small, and become even smaller for speed as At increases.\nShown on an expanded scale, the direction skewness behaves more er -\nratically, but the values are so small one may consider them zero for\nall practical purposes.\n6.3 KURTOSIS\nThe kurtosis values show a definite decrease with increasing At,\nbut, in contrast to the behavior of the standard deviations and the\nskewness, there is little similarity between the kurtosis plots of Case\n41","A and Case B. In Case A, both the filtered and unfiltered kurtoses drop\nfrom numbers close to 5 at At = 1 S to about 3.5 at At = 30 S. In Case\nB the kurtosis values are larger, especially for direction and show a\nstrong dependence on choice of filter.\n6.4 FREQUENCY DISTRIBUTIONS\nOnly the 50/5 filtered distributions are shown here, but they amply\ndemonstrate differences between the kurtosis Case A and Case B. In the\nstrongly stable stratification of Case B, one can expect a higher pre- -\nference for smaller fluctuations in the speed and direction signals, and\nthis preference to be further accentuated in the differenced signals.\n6.5 VARIATIONS WITH HEIGHT\nThe sensitivity to height variation in the statistical properties\nof these differenced time series is much more pronounced than in the\noriginal time series. The standard deviations decrease with height for\nall choices of At. The kurtosis increases sharply with height; the\nkurtosis of 8.5 found in the 50/5 filtered data with At = 10 S repre-\nsents a significant departure from Gaussian.\nPowell and Connell (1980) note that the theoretical usefulness of\nany of the gust models requires that the filtered turbulence or gust\ndata be approximately Gaussian. The degree to which the kurtosis de- -\nparts from 3 has serious implications for the theoretical model. How-\never, large kurtosis is often the result of stationarity in the signal.\nDifferencing of the signal, by removal of the large low-frequency\noscillations in the signal, increases the kurtosis. We believe the\ndeparture of the kurtosis from 3 is not an important factor in the\nengineering design so long as the filter passband approximates the WECS\nresponse.\n42","Departures from Gaussian behavior notwithstanding, certain con-\nsistent patterns emerge as one normalizes the standard deviations of\nthe differenced data with the standard deviations of the correspond-\ning undifferenced filtered time series. As seen in Table 6.1 the ratios\nstay surprisingly constant from one level to another and from one case\nto the other. The only significant variations are those introduced by\nfilter choice and differencing interval. Denoting the standard devia-\ntion of the undifferenced filtered time series by [o(X)] 'F' where X is\nthe variable and F denotes filtered data, and the standard deviation of\nthe differenced time series by [o(Ax)] we can express the relationship\nbetween the two for the 50/5 filtered data differenced across 10 S (our\noptimum differencing interval) as\n1.47 for Case A speed\no(AX)\n(6.1)\n1.52 for Case B speed\n=\n1.52 For Case A cirection\nF\n1.51 for Case B direction\nThus, the standard deviation of the differenced signal can be approxi-\nmated with a high degree of confidence from the standard deviation of\nthe basic filtered time series.\nIn the CF and HF models, the rms value of the differenced data\n(which is the same as the standard deviation in the absence of a mean)\nis related to the rms value of the undifferenced data through a function\nf involving the wind speed S, the integral length scale L, and At:\nof\no(AS)\nV2\nE(S,\nL,\nAt)\n(6.2)\no(s)\n.\nThe form of f is different for the two models, but our results in\nTable 6.1 suggest that the effect of varying speed and integral length 0\nscale (as between Case A and Case B) is not very significant.\n43","Differencing Interval (s)\n1.44\n1.43\n1.58\n1.41\n1.38\n1.55\n1.41\n1.44\n1.54\n1.42\n1.40\n1.56\n1.40\n1.39\n1.60\n1.39\n1.39\n1.57\n30\nCase B\nTable 6. 1 - - -Normalized standard deviations for speed and direction differences (SI units)\n1.61\n1.50\n1.32\n1.63\n1.52\n1.28\n1.67\n1.53\n1.22\n1.57\n1.52\n1.37\n1.56\n1.50\n1.35\n1.65\n1.51\n1.28\n10\n1.46\n1.26\n1.40\n1.45\n1.23\n1.41\n1.18\n1.51\n1.50\n1.35\n1.45\n1.24\n5\nDifferencing Interval (s)\n1.42\n1.46\n1.62\n1.41\n1.45\n1.63\n1.39\n1.44\n1.61\n1.40\n1.41\n1.60\n1.42\n1.42\n1.59\n1.42\n1.40\n1.58\n30\nCase A\n1.63\n1.49\n1.21\n1.65\n1.46\n1.14\n1.65\n1.47\n1.17\n1.60\n1.49\n1.26\n1.64\n1.52\n1.29\n1.54\n1.34\n1.67\n10\n1.43\n1.18\n1.38\n1.05\n1.35\n1.08\n1.47\n1.27\n1.45\n1.24\n1.43\n1.21\n5\nFilter\n100/10\n100/10\n100/10\n100/10\n100/10\n100/10\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\nHeight\n10\n50\n150\n10\n50\n150\ndifference\ndifference\nParameter\nDirection\nSpeed","Case A, Z = 10 m\nCase B, Z= 10 m\n4\n4\n3\n3\n2\n2\n1\n1\n0\n0\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\n+2\n+2\n+1\n+1\n0\n0\n-1\n-1\n-2\n-2\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\n5\n5\n4\n4\n3\n3\nFilter\nNone\n30/3\n2\n2\n50/5\n100/10\n1\n1\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\nAt (s)\nAt (s)\nFigure 6. 1 - - Speed difference statistics at 10 m shown as a function of\ndifferencing interval. Standard deviations are in m/s.\n45","Case B, Z = 50 m\nCase A, Z = 50 m\n4\n4\n3\n3\n2\n2\n1\n1\n0\n0\n30\n40\n10\n20\n10\n20\n30\n40\n0\n0\n+2\n+2\n+1\n+1\n0\n0\n-1\n-1\n-2\n-2\n40\n10\n20\n30\n40\n10\n20\n30\n0\n0\n5\n5\n4\n4\n3\n3\nFilter\nNone\n2\n30/3\n2\n50/5\n100/10\n1\n1\n10\n20\n30\n40\n20\n30\n40\n0\n10\n0\nAt (s)\nAt (s)\nFigure 6.2 -- -Speed difference statistics at 50 m shown as a function of\ndifferencing interval. Standard deviations are in m/s.\n46","Case A, Z = 150 m\nCase B, Z = 150 m\n4\n4\n3\n3\n2\n2\n1\n1\n0\n0\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\n+2\n+2\n+1\n+1\n0\n0\n-1\n-1\n-2\n-2\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\n7\n7\nFilter\n6\n6\nNone\n30/3\n50/5\n100/10\n5\n5\n4\n4\n3\n3\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\nAt (s)\nAt (s)\nFigure 6.3 -- -Speed difference statistics at 150 m shown as a function of\ndifferencing interval. Standard deviations are in m/s.\n47","Case B, Z = 10 m\nCase A, Z= 10 m\n10\n10\n8\n8\n6\n6\n4\n4\n2\n2\n30\n40\n10\n20\n30\n40\n0\n10\n20\n0\n+0.4\n+0.4\n(Expanded scale)\n+0.2\n+0.2\n0\n0\n-0.2\n-0.2\n-0.4\n-0.4\n20\n30\n40\n10\n20\n30\n40\n0\n10\n0\n5\n5\n4\n4\n3\n3\nFilter\nNone\n30/3\n2\n2\n50/5\n100/10\n1\n1\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\nAt (s)\nAt (s)\nFigure 6. 4 - -Direction - difference statistics at 10 m shown as a function\nof differencing interval. Standard deviations are in degrees.\n48","Case A, Z = 50m\nCase B, Z = 50 m\n10\n10\n8\n8\n6\n6\n4\n4\n2\n2\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\n+0.4\n+0.4\n(Expanded scale)\n+0.2\n+0.2\n0\n0\n-0.2\n-0.2\n-0.4\n-0.4\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\n7\n7\nFilter\n6\nNone\n6\n30/3\n50/5\n100/10\n5\n5\n4\n4\n3\n3\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\nAt (s)\nAt (s)\nFigure 6.5 -- Direction difference statistics at 50 m shown as a function\nof differencing interval. Standard deviations are in degrees.\n49","Case B, Z = 150 m\nCase A, Z = 150 m\n10\n10\n8\n8\n6\n6\n4\n4\n2\n2\n10\n20\n30\n40\n0\n10\n20\n30\n40\n0\n+0.4\n+0.4\n(Expanded scale)\n+0.2\n+0.2\n0\n0\n-0.2\n-0.2\n-0.4\n-0.4\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\n11\n11\nFilter\nNone\n9\n9\n30/3\n50/5\n100/10\n7\n7\n(Expanded scale)\n5\n5\n3\n3\n0\n10\n20\n30\n40\n0\n10\n20\n30\n40\nAt (s)\nAt (s)\nFigure 6. 6--Direction difference statistics at 150 m shown as a function\nof differencing interval. Standard deviations are in degrees.\n50","(x 10)\n2.00\n1.00\n(x 10)\nFigure 6. 7 - - -Distributions of speed and direction differences at 10 m with 50/5 filter\nAt = 5 S\nAt = 5 S\n50/5 filter; Z = 10 m\n1.00\n0.50\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-1.00\n-0.50\nCase B.\n-2.00\n-1.00\n0.80\n0.60\n0.40\n0.20\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.00\nand At = 5 S. AS = 0.2 m/s; AD = 0.4 deg.\n(x 10)\n2.00\n(x 10)\n1.00\nAt = 5 S\nAt = 5 S\nCase A. 50/5 filter; Z = 10 m\n1.00\n0.50\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-1.00\n-0.50\n-2.00\n-1.00\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80","(x 10)\n1.00\n(x 10)\n2.00\nFigure 6.8 - -Distributions - of speed and direction differences at 50 m with 50/5 filter\nAt = 5 S\nAt = 5 S\n50/5 filter; Z = 50 m\n0.50\n1.00\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase B.\n-1.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.20\n0.40\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\nand At = 5 S. . AS = 0.2 m/s; AD = 0.4 deg.\n(x 10)\n(x 10)\n1.00\n2.00\nAt = 5 S\nAt = 5 S\nCase A. 50/5 filter; Z = 50 m\n0.50\n1.00\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-0.50\n-1.00\n-1.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00","(x 10)\n(x 10)\n2.00\n1.00\nFigure 6.9 - - Distributions of speed and direction differences at 150 m with 50/5 filter\nCase B. 50/5 filter; Z = 150 m\nAt = 5 S\nAt = 5 S\n1.00\n0.50\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-1.00\n-0.50\n-2.00\n-1.00\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.20\n1.00\nand At = 5 S. . AS = 0.2 m/s; AD = 0.4 deg.\n(x 10)\n2.00\n(x 10)\n1.00\nCase A. 50/5 filter; Z = 150 m\nAt = 5 S\nAt = 5 S\n1.00\n0.50\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-1.00\n-0.50\n-2.00\n-1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n1.00\n0.00\n1.20\n0.80\n0.60\n0.40\n0.20","(x 10)\n1.00\n(x 10)\n2.00\nFigure 6.10 - - Distributions of speed and direction differences at 10 m with 50/5 filter\nAt = 10 S\nAt = 10 S\n50/5 filter; Z = 10 m\n0.50\n1.00\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase B.\n-1.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\nand At = 10 S . AS = 0.2 m/s; AD = 0.4 deg.\n1.00\n(x 10)\n2.00\n(x 10)\nAt = 10 S\nA. 50/5 filter; Z = 10 m\nAt = 10 S\n0.50\n1.00\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase\n-1.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\nX","(x 10)\n1.00\n2.00\nFigure 6.11 - - Distributions of speed and direction differences at 50 m with 50/5 filter\nAt = 10 S\n50/5 filter; Z = 50 m\nAt = 10 S\n0.50\n1.00\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-0.50\n-1.00\nCase B.\n-1.00\n-2.00\n0.20\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.00\n1.20\n1.00\nand At = 10 S. AS = 0.2 m/s; AD = 0.4 deg.\n(x 10)\n2.00\n1.00\nAt = 10 S\nAt = 10 S\nCase A. 50/5 filter; Z = 50 m\n1.00\n0.50\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-1.00\n-0.50\n-2.00\n-1.00\n0.80\n0.60\n0.40\n0.20\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n1.20\n1.00\n0.00","1.00\n(x 10)\n2.00\n(x 10)\nFigure 6.12 - - -Distributions of speed and direction differences at 150 m with 50/5 filter\nCase B. 50/5 filter; Z = 150 m\nAt = 10 S\nAt = 10 S\n0.50\n1.00\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-0.50\n-1.00\n-1.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\nand At = 10 S. . AS = 0.2 m/s; AD = 0.4 deg.\n1.00\n(x 10)\n2.00\n(x 10)\nCase A. 50/5 filter; Z = 150 m\nAt = 10 S\nAt = 10 S\n0.50\n1.00\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-0.50\n-1.00\n-1.00\n-2.00\n0.60\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.40\n0.20\n0.00","2.00\n(x 10)\n(x 10)\n1.00\nFigure 6. 13 - - Distributions of speed and direction differences at 10 m with 50/5 filter\nAt = 30 S\nAt = 30 S\nB. 50/5 filter; Z = 10 m\n1.00\n0.50\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-1.00\n-0.50\nCase\n-2.00\n-1.00\n0.60\n0.20\n0.00\n0.80\n0.40\n0.20\n0.40\n0.00\n1.20\n1.00\n1.00\n0.80\n0.60\n1.20\nand At = 30 S . AS = 0.2 m/s; AD = 0.4 deg. .\n2.00\n(x 10)\n(x 10)\n1.00\nAt = 30 S\nAt = 30 S\nCase A. 50/5 filter; Z = 10 m\n1.00\nDirection difference (deg)\n0.50\nSpeed difference (m/s)\n0.00\n0.00\n-1.00\n-0.50\n-2.00\n-1.00\n0.20\n0.00\n0.60\n0.40\n1.00\n0.80\n0.20\n1.20\n0.00\n0.80\n0.60\n0.40\n1.20\n1.00","1.00\n(x 10)\n(x 10)\n2.00\nFigure 6.14 - - Distributions of speed and direction differences at 50 m with 50/5 filter\nAt = 30 S\nAt = 30 S\nCast B. 50/5 filter; Z = 50 m\n0.50\n1.00\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-0.50\n-1.00\n-1.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\nand At = 30 S. AS = 0.2 m/s; AD = 0.4 deg.\n1.00\n(x 10)\n2.00\n(x 10)\nAt = 30 S\nAt = 30 S\nCase A. 50/5 filter; Z = 50 m\n0.50\n1.00\nDirection difference (deg)\nSpeed difference (m/s)\n0.00\n0.00\n-0.50\n-1.00\n-1.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n8","(x 10)\n2.00\n1.00\n(x 10)\nFigure 6.15 - - Distributions of speed and direction differences at 150 m with 50/5 filter\nB. 50/5 filter; Z = 150 m\nAt = 30 S\n1.00\nDirection difference (deg)\n0.50\nSpeed difference (m/s)\n0.00\n0.00\n-1.00\n-0.50\nCase\n-2.00\n-1.00\n0.60\n0.40\n0.20\n0.00\n0.40\n0.20\n1.20\n1.00\n0.80\n0.00\n1.00\n0.80\n0.60\n1.20\nand At = 30 S. . AS = 0.2 m/s; AD = 0.4 deg.\n2.00\n(x 10)\n1.00\n(x 10)\nA. 50/5 filter; Z = 150 m\nAt = 30 S\n1.00\nDirection difference (deg)\n0.50\nSpeed difference (m/s)\n0.00\n0.00\n-1.00\n-0.50\nCase\n-2.00\n-1.00\n0.20\n0.00\n0.80\n0.60\n0.40\n1.00\n0.20\n0.00\n1.20\n0.60\n0.40\n1.00\n1.20\n0.80","7.0 ANALYSIS OF GUST 0 PARAMETERS\nThe GUST 0 definition of Powell and Connell (1980) contains the two\nelements essential to the definition of a discrete gust event: an\namplitude definition and a time definition. These events, designated A\n0\nand T are the maximum amplitudes and the time intervals between zero-\ncrossings bracketing those peaks respectively. In studies reported so\nfar, positive and negative values of A and T 0 have been grouped for\nanalysis. Positive and negative gusts have been treated as equivalent.\nThis assumption is clearly not valid for the unfiltered data because of\nthe degree of skewness in the original time series, but it is not an\nunreasonable one for filtered data. The statistical results presented\nin Figs. 7.1-7.3 are therefore confined to +A 0 and +T 0 Figures 7.4-7.9\noffer a direct comparison of the positive and negative gust distribution\nevents in the 50/5 filtered data. The scatter plots of amplitudes and\ntime given in Figs. 7.10-7.13 provide yet another representation of\ntheir positive and negative distributions. The symmetry in the gust\nbehavior is confirmed in these figures.\nThe GUST 0 statistics plotted in Figs. 7.1 and 7.2 show standard\ndeviations of the gust amplitudes decreasing with height, and those of\nthe gust times increasing with height. Sensitivity to filtering is much\nmore pronounced in the gust times than in the gust amplitudes. The\neffect of height variation in the GUST 0 kurtosis is not as systematic\nas in the standard deviations. In general, the tendency is for the\nkurtosis to increase with height. The variation is somewhat less sys-\ntematic in the cross correlation coefficients of Fig. 7.3, but the\ntendency of the cross correlation to decrease with height is apparent.\nIn the theoretical development of the GUST 0 model it is assumed\nthat the A0''s and To's have distributions that approach Gaussian.\nPowell and Connell (1980) expect such a distribution for data subjected\nto a bandpass filter approximately a decade wide. If the bandwidth is\nsignificantly smaller or larger than a decade, they expect both the gust\namplitudes and gust times to lose their Gaussian character. These\n60","expectations are satisfied in their experimental results. Their data,\nwhich include both positive and negative values, with no distinction\nof sign, fit a half-Gaussian distribution reasonably well. No limiting\nvalue is imposed on the amplitudes or the times. In our analysis we\ntreat positive and negative values separately but impose the condition\nthat the lowpass filter time. Even though indications of a\nfull Gaussian distribution are strong in our distribution plots, the ab-\nsence of values close to zero (imposed partly by the limit on TO and\npartly by the lowpass cutoff in the pass band) makes exact comparisons\ndifficult.\nAnother significant point of difference between our results and\nthose of Powell and Connell (1980) is the magnitude of the Ao To\ncorrelation coefficients. Our correlations are almost a factor of 2\nsmaller. The scatter in Fig. 7.3 confirms this point. A correlation\ncoefficient between 0.7 and 0.8 would have produced a tighter grouping\nof the data points.\nAs mentioned earlier the relevant parameters for WECS design are\nnot the standard deviations but the rms values of the A0''s and To's.\nConversion from one to the other requires knowledge of their mean values.\nFor example, ,\n(7.1)\nwhere and o(+AO) denote rms and standard deviation of +A0,\nand the overbar denotes mean value.\nIn Table 7.1 we present the average values of A0''s and To's for\nCases A and B. The corresponding rms values are given in Table 7.2.\nFor a Gaussian distribution the ratio of the rms to average value should\nbe VT/2 (~1.25). The numbers in the two tables almost satisfy this\ncondition.\n61","-6.87\n-11.18\n-21.11\n-6.97\n-11.49\n-20.61\n-7.76\n-12.48\n-22.94\n0\n-5.44\n-8.92\n-16.99\n-18.70\n-5.81\n-9.96\n-6.93\n-11.43\n-20.58\n-T\n-1.74\n-1.72\n-1.77\n-1.50\n-1.58\n-1.63\n-1.08\n-1.18\n-1.33\n-4.89\n-4.54\n-4.16\n-3.11\n-3.08\n-2.89\n-2.34\n-2.45\n-2.62\n0\n-A,\nCase B\n6.55\n10.59\n19.03\n7.07\n11.42\n20.79\n7.84\n12.39\n21.95\n5.47\n8.96\n16.20\n5.68\n9.52\n18.54\n6.84\n11.72\n22.32\n0\n+T\n1.80\n1.87\n1.89\n1.42\n1.53\n1.52\n1.04\n1.13\n1.26\n4.96\n4.69\n4.15\n3.22\n3.11\n2.93\n2.30\n2.46\n2.56\n+A0\n0\nTable 7. 1 -- -Average GUST 0 amplitudes and times (SI units)\n-7.55\n-12.96\n-23.81\n-8.08\n-12.96\n-23.84\n-8.29\n-12.97\n-23.81\n-6.41\n-10.62\n-20.42\n-7.04\n-11.11\n-19.53\n-7.58\n-11.44\n-19.07\n0\n-T\n-1.53\n-1.67\n-1.70\n-1.25\n-1.42\n-1.65\n-1.09\n-1.25\n-1.48\n-5.00\n-5.01\n-5.12\n-3.81\n-3.88\n-3.96\n-3.27\n-3.44\n-3.57\n0\n-A,\nCase A\n7.42\n12.19\n21.30\n8.37\n13.34\n24.08\n8.53\n13.90\n23.93\n6.60\n10.69\n20.12\n7.08\n11.06\n19.97\n7.64\n11.96\n19.87\n0\n+T\n1.49\n1.67\n1.76\n1.22\n1.35\n1.66\n1.07\n1.25\n1.39\n4.91\n5.05\n4.98\n3.86\n3.90\n4.00\n3.25\n3.35\n3.35\n+A0\n0\nFilter\n100/10\n100/10\n100/10\n100/10\n100/10\n100/10\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\nHeight\n10\n50\n150\n10\n50\n150\nParameter\nDirection\nSpeed","7.73\n12.62\n23.50\n7.82\n12.96\n23.24\n8.71\n13.85\n26.07\n6.05\n9.84\n18.94\n6.51\n11.14\n21.07\n7.86\n13.00\n23.39\nrms\n(-A )\n0\n1.94\n1.91\n1.93\n1.80\n1.85\n1.87\n1.35\n1.44\n1.58\n5.52\n5.09\n4.60\n3.68\n3.59\n3.34\n2.96\n3.02\n3.05\nrms\no\nCase B\n(+I)\n0\n7.30\n11.75\n20.96\n7.98\n12.69\n23.63\n8.82\n13.94\n24.72\n6.10\n9.81\n18.00\n6.35\n10.63\n20.91\n7.74\n13.33\n25.19\nrms\nTable 2--Root-mean-square GUST 0 amplitudes and times (SI units)\no\no rms (+A) 0\n2.05\n2.07\n2.09\n1.68\n1.77\n1.75\n1.28\n1.36\n1.49\n5.60\n5.25\n4.66\n3.85\n3.63\n3.34\n2.94\n3.04\n3.01\n8.46\n14.67\n26.22\n9.06\n14.40\n26.51\n9.31\n14.43\n26.81\n7.29\n11.92\n22.71\n7.84\n12.41\n22.04\n8.48\n12.65\n20.87\no rms (+AO) 0 o rms (+T 0 o rms (-A-) 0 rms\n1.77\n1.89\n1.93\n1.46\n1.63\n1.87\n1.30\n1.47\n1.71\n5.72\n5.64\n5.70\n4.36\n4.39\n4.38\n3.81\n3.93\n4.05\nCase A\n8.34\n13.60\n23.71\n9.40\n15.08\n26.83\n9.68\n15.60\n27.03\n7.43\n11.90\n22.91\n7.91\n12.37\n22.58\n13.34\n8.49\n22.42\n1.71\n1.88\n2.01\n1.41\n1.57\n1.89\n1.26\n1.44\n1.59\n5.61\n5.68\n5.57\n4.42\n4.42\n4.47\n3.75\n3.83\n3.75\nFilter\n100/10\n100/10\n100/10\n100/10\n100/10\n100/10\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\nHeight\n10\n50\n150\n10\n50\n150\nParameter\nDirection\nSpeed","Case B, Speed AO\nCase A, Speed +A0\n9\n9\n10 m\n50 m\n7\n7\n150 m\n5\n5\n3\n1.5\n3\n1.5\n1.0\n1.0\nD\n0.5\n0.5\n0\n0\n100/10\n30/3\n50/5\n100/10\n30/3\n50/5\nFilters\nFilters\nCase A, Speed +To\nCase B, Speed + +To\n9\n9\n7\n7\n5\n5\nDO\n15\n3\n15\n3\n@\n10\n10\n5\n5\n0\n0\n30/3\n50/5\n100/10\n30/3\n50/5\n100/10\nFilters\nFilters\nFigure 7.1 - - GUST 0 speed +A and speed +T kurtosis (upper part of\n0\n0\nframe) and standard deviation (lower part) as a function of bandpass\nfiltering. Standard deviations are in m/s and degrees.\n64","Case A, Direction +A0\nCase B, Direction + +AO\n13\n13\nO 10 m\n50 m\n150 m\n10\n10\n7\n7\n3.0\n4\n3.0\n4\n2.5\n2.5\n2.0\n2.0\n1.5\n1.5\n30/3\n50/5\n100/10\n30/3\n50/5\n100/10\nFilters\nFilters\nCase A, Direction +To\nCase B, Direction +To\n9\n9\n7\n7\n5\n5\n80\n3\n15\n3\n15\n10\n10\n5\n5\n0\n0\n30/3\n50/5\n100/10\n30/3\n50/5\n100/10\nFilters\nFilters\nFigure 7.2 -- GUST 0 direction +A\nand direction +T 0 statistics as a\n0\nStandard deviations are in m/s and\nfunction of bandpass filtering.\ndegrees.\n65","Case A, Speed (+Ao. +To)\nCase B, Speed (+AO. +To)\n1.2\n1.2\n10 m\n50 m\n150 m\n1.0\n1.0\n0.8\n0.8\n0.6\n0.6\n0.4\n0.4\n0.2\n0.2\n0\n0\n30/3\n50/5\n100/10\n30/3\n50/5\n100/10\nFilters\nFilters\nCase A, Direction (+A0. +To)\nCase B, Direction (+A0, +To)\n1.2\n1.2\n1.0\n1.0\n0.8\n0.8\n0.6\n0.6\n0.4\n0.4\n0.2\n0.2\n0\n0\n30/3\n50/5\n100/10\n30/3\n50/5\n100/10\nFilters\nFilters\nFigure 7.3--GUST -- 0 amplitude-time cross correlation coefficients as a\nfunction of bandpass filtering.\n66","8.00\n3.00\n(x 10)\nB. 50/5 filter; Z = 10 m\nGUST 0\nGUST 0\nm.\n10\n4.00\n4 - - GUST 0 speed A 0 and speed I'O distributions for 50/5 filter at\nSpeed amplitude (m/s)\n1.00\nSpeed time (s)\n8.00\n-1.00\n-4.00\nCase\n-8.00\n-3.00\n1.28\n0.86\n0.80\n0.40\n1.68\n0.40\n0.00\n1.60\n1.28\n0.00\n8.00\n3.00\n(x 10)\nA. 50/5 filter; Z = 10 m\nGUST 0\nGUST 0\n= 0.16 m/s; AT 0 = 0.6 S. .\n4.00\nSpeed amplitude (m/s)\n1.00\nSpeed time (s)\n0.00\n-1.00\nFigure 7.\n-4.00\nCase\n-3.00\n-8.00\n1.60\n1.20\n0.80\n0.40\n0.00\n1.20\n0.80\n0.40\n1.60\n0.00\n%\n67","(x 10)\n3.00\n8.00\nGUST 0\nB. 50/5 filter; Z = 50 m\nGUST 0\n5--GUST 0 speed A 0 and speed I' distributions for 50/5 filter at 50 m. .\n4.00\nSpeed amplitude (m/s)\n1.00\nSpeed time (s)\n0.00\n-1.00\n-4.00\nCase\n-3.00\n-8.00\n0.80\n0.40\n1.20\n0.80\n0.40\n0.00\n1.60\n1.20\n0.00\n1.60\n3.00\n(x 10)\n8.00\nGUST 0\nCase A. 50/5 filter; Z = 50 m\nGUST 0\n= 0.16 m/s; AT 0 = 0.6 S.\n4.00\nSpeed amplitude (m/s)\n1.00\nSpeed time (s)\n0.00\n-1.00\nFigure 7.5\n-4.00\n0\n-3.00\n-8.00\n0.40\n0.00\n0.40\n0.00\n1.20\n0.80\n1.60\n1.60\n1.20\n0.80","(x 10)\n3.00\n8.00\nFigure 7.6--GUST - - 0 speed A0 0 and speed T 0 distributions for 50/5 filter at 150 m.\n50/5 filter; Z = 150 m\nGUST 0\nGUST 0\n4.00\nSpeed amplitude (m/s)\n1.00\nSpeed time (s)\n0.00\n-1.00\n-4.00\nCase B.\nMA\n-3.00\n-8.00\n0.80\n0.40\n0.00\n0.40\n0.00\n1.60\n1.20\n1.20\n0.80\n1.60\n3.00\n(x 10)\n8.00\nGUST 0\nA. 50/5 filter; Z = 150 m\nGUST 0\nAA = 0.16 m/s; AT 0 = 0.6 S.\n4.00\n1.00\nSpeed amplitude (m/s)\nSpeed time (s)\n0.00\n-1.00\n-4.00\n0\nCase\n-3.00\n-8.00\n0.40\n0.00\n1.20\n0.80\n0.40\n0.00\n1.60\n1.20\n0.80\n1.60","2.00\n(x 10)\n3.00\n(x 10)\nFigure 7. 7 - - GUST 0 direction A 0 and direction T 0 distributions for 50/5 filter at 10 m.\nCase B. 50/5 filter; Z = 10 m\nGUST 0\nGUST 0\n1.00\nDirection amplitude (deg)\n1.00\nDirection time (s)\n0.00\n-1.00\n-1.00\n-2.00\n-3.00\n1.50\n1.00\n0.50\n0.00\n2.50\n2.00\n1.50\n1.00\n0.50\n0.00\n2.00\n(x 10)\n3.00\n(x 10)\nCase A. 50/5 filter; Z = 10 m\nGUST 0\nGUST 0\n1.00\nDirection amplitude (deg)\n1.00\nAA = 0.4 deg; 0 = 0.6 S .\nDirection time (s)\n0.00\n-1.00\n-1.00\n0\n-2.00\n-3.00\n1.00\n1.50\n0.50\n0.00\n2.50\n2.00\n1.50\n1.00\n0.50\n0.00","(x 10)\n3.00\n(x 10)\n2.00\nFigure 7. 8--GUST 0 direction A 0 and direction T 0 distributions for 50/5 filter at 50 m.\nGUST 0\nGUST 0\n50/5 filter; Z = 50 m\n1.00\nDirection amplitude (deg)\n1.00\nDirection time (s)\n0.00\n-1.00\n-1.00\nCase B.\n-3.00\n-2.00\n2.00\n1.50\n1.00\n0.50\n0.00\n2.50\n1.50\n1.00\n0.50\n0.00\n(x 10)\n(x 10)\n3.00\n2.00\nGUST 0\nCase A. 50/5 filter; Z = 50 m\nGUST 0\n1.00\nDirection amplitude (deg)\nAA = 0.4 deg; AT 0 = 0.6 S.\n1.00\nDirection time (s)\n0.00\n-1.00\n-1.00\n0\n-3.00\n-2.00\n0.50\n2.00\n1.50\n1.00\n0.00\n0.50\n0.00\n2.50\n1.50\n1.00\n%","2.00\n(x 10)\n3.00\n(x 10)\nB. 50/5 filter; Z = 150 m\nGUST 0\nGUST 0\nFigure 7.9--GUST - 0 direction A 0 and direction I / 0 distributions for 50/5 filter at\n1.00\nDirection amplitude (deg)\n1.00\nDirection time (s)\n0.00\n-1.00\n-1.00\nCase\n0\n-2.00\n-3.00\n1.50\n1.00\n0.50\n0.00\n2.50\n2.00\n1.50\n1.00\n0.50\n0.00\n2.00\n(x 10)\n3.00\n(x 10)\n150 m. AAO = 0.4 deg; = 0.6 S.\nCase A. 50/5 filter; Z = 150 m\nGUST 0\nGUST 0\n1.00\nDirection amplitude (deg)\n1.00\nDirection time (s)\n0.00\n-1.00\n-1.00\n-2.00\n-3.00\n1.50\n1.00\n0.50\n0.00\n2.50\n2.00\n1.50\n1.00\n0.50\n0.00","Case A. No filter; Z = 50 m; GUST 0\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n0.80\n1.00\n1.20\n0.00\n0.20\n0.40\n0.60\n-0.20\n0.00\n-0.80\n-0.60\n-0.40\n-1.20\n-1.00\n(x 10)\nSpeed amplitude (m/s)\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n3.00\n1.00\n2.00\n0.00\n0.00\n-2.00\n-1.00\n-3.00\n(x 10)\nDirection amplitude (deg)\nFigure 7.10--Plots of GUST 0 amplitudes vs. times for Case A at 50 m (no\nfilter).\n73","Case B. No filter; Z = 50 m; GUST 0\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-1.20\n-1.00\n-0.80\n-0.60\n-0.40\n-0.20\n0.00\n0.00\n0.20\n0.40\n0.60\n0.80\n1.00\n1.20\nSpeed amplitude (m/s)\n(x 10)\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-3.00\n-2.00\n-1.00\n0.00\n0.00\n1.00\n2.00\n3.00\nDirection amplitude (deg)\n(x 10)\nFigure 7.11--Plots of GUST 0 amplitudes vs. time for Case B at 50 m (no\nfilter)\n74","Case A. 50/5 filter; Z = 50 m; GUST 0\n4.00\n4.00\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n4.00\n6.00\n8.00\n0.00\n0.00\n2.00\n-8.00\n-6.00\n-4.00\n-2.00\nSpeed amplitude (m/s)\n(x 10)\n4.00\n4.00\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-0.50\n0.00\n0.00\n0.50\n1.00\n1.50\n2.00\n-2.00\n-1.50\n-1.00\nDirection amplitude (deg)\n(x 10)\nFigure 7.12--Plots -- of GUST 0 amplitudes vs. times for Case A at 50 m\n(50/5 filter).\n75","Case B. 50/5 filter; Z = 50 m; GUST 0\n4.00\n4.00\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-8.00\n-6.00\n-4.00\n-2.00\n0.00\n0.00\n2.00\n4.00\n6.00\n8.00\nSpeed amplitude (m/s)\n(x 10)\n4.00\n4.00\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-2.00\n-1.50\n-1.00\n-0.50\n0.00\n0.00\n0.50\n1.00\n1.50\n2.00\nDirection amplitude (deg)\n(x 10)\nFigure 7.13--Plots of GUST 0 amplitude vs. time for Case B at 50 m (50/5\nfilter)\n76","8.0 ANALYSIS OF GUST 1 PARAMETERS\nIn the GUST 1 model, the definitions of gust amplitude and gust\ntime (designated A. and I1) differ from those of GUST 0 in that the gust\namplitude is the peak-to-peak amplitude and the gust time is the time\ninterval between the two peaks (see Fig. 4.2). A positive sign indi-\ncates positive rate of change in the variable and a negative sign a\nnegative rate of change. An advantage of this model is its insensi-\ntivity to the low-frequency trends in the data. The disparity between\nthe filtered and unfiltered data in the statistical results is there-\nfore not so great as in the GUST 0 results. The main disadvantage of\nthe model is the need for an arbitrary specification of a minimum speed\nor time change that will qualify as a GUST 1 event. Otherwise, the\nsmallest scale fluctuations will be the only ones recognized. For our\ndefinition of a GUST 1 event we impose the condition that T1 be equal\nto or larger than the lowpass filter time used in the bandpass filter.\nThe sensitivity of the model to the tolerance interval is a factor that\nhas to be considered when interpreting the statistical results in\nFigs. 8.1-8.3. Distribution plots showing positive and negative events,\nsimilar to those presented for GUST 0, are shown in Figs. 8.4-8.9;\nthe\nscatter plots for the same events are in Figs. 8.10-8.13.\nComparing the GUST 1 statistics of Figs. 8.1 and 8.2 with those in\nFigs. 7.1 and 7.2, we find in the former set a greater sensitivity to\nheight change. It is also apparent that the standard deviation of A1 is\nroughly twice as large as the standard deviation of A0, while the reverse\nis true with the standard deviations of T1 and I0. Despite these differ-\nences, the A1 T1 cross correlations behave much like the A TO cross\ncorrelations in Fig. 7.3.\nHere, as in the GUST 0 model, the relevant parameters for WECS\ndesign are the rms values of the gust amplitudes and the gust times.\nThe time averages used for converting the standard deviations into rms\nvalues are presented in Table 8.1. The calculated rms values are given\nin Table 8.2.\n77","-4.34\n-4.29\n-6.45\n-11.21\n-6.61\n-11.53\n-4.51\n-6.93\n-11.53\n-3.68\n-5.64\n-10.02\n-3.74\n-5.85\n-10.34\n-4.09\n-6.55\n-11.34\n-T1\n1\n-2.62\n-2.72\n-3.12\n-2.10\n-2.26\n-2.52\n-1.45\n-1.66\n-1.93\n-8.01\n-8.10\n-9.06\n-4.97\n-5.15\n-6.06\n-3.40\n-3.75\n-4.08\n-A\n1\nCase B\n3.94\n6.05\n10.37\n4.02\n6.05\n10.56\n4.26\n6.61\n11.24\n3.70\n5.81\n10.26\n3.78\n5.83\n10.23\n4.10\n6.46\n11.23\n+T,\n1\nTable 8.1 -- -Average GUST 1 amplitudes and times (SI units)\n2.65\n2.84\n3.09\n2.16\n2.57\n2.90\n1.49\n1.71\n1.93\n8.08\n8.41\n8.72\n4.97\n5.21\n5.76\n3.39\n3.75\n3.98\n+A\n1\n-4.66\n-7.21\n-11.98\n-4.71\n-7.60\n-12.64\n4.95\n-7.63\n-12.66\n-4.02\n-6.22\n-10.82\n-4.30\n-6.61\n-11.02\n-4.43\n-6.77\n-11.11\n-T1\n1\n-2.08\n-2.32\n-2.59\n-1.61\n-1.94\n-2.24\n-1.45\n-1.74\n-2.07\n-7.35\n-7.84\n-8.73\n-5.49\n-6.06\n-6.48\n-4.55\n-5.07\n-5.66\n1\n-A.\nCase A\n4.29\n6.55\n10.99\n4.54\n7.04\n11.75\n4.63\n7.11\n12.09\n4.00\n6.24\n10.74\n4.32\n6.53\n10.77\n4.52\n6.91\n10.97\n1\n+T,\n2.23\n2.51\n2.88\n1.72\n2.06\n2.44\n1.50\n1.76\n2.19\n7.45\n8.02\n9.06\n5.64\n6.00\n6.57\n4.60\n5.18\n5.62\n+A\n1\nFilter\n100/10\n100/10\n100/10\n100/10\n100/10\n100/10\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\nHeight\n10\n50\n150\n10\n50\n150\nParameter\nDirection\nSpeed","(-T1)\n4.59\n6.72\n11.45\n4.65\n6.92\n11.14\n4.88\n7.29\n11.94\n3.83\n5.73\n10.26\n3.90\n5.99\n10.60\n4.37\n6.88\n11.70\nrms\no\n)\n1\n2.98\n3.01\n3.34\n2.51\n2.60\n2.84\n1.85\n2.03\n2.31\n8.08\n8.16\n9.33\n5.09\n5.31\n6.49\n3.74\n4.29\n4.98\n(\nrms\no\nCase B\no rms (+T 1 )\n4.15\n6.22\n10.60\n4.26\n6.28\n10.79\n4.56\n6.90\n11.60\n3.86\n5.92\n10.37\n3.96\n5.96\n10.48\n4.37\n6.74\n11.55\nTable 8.2 --Root-mean-square GUST 1 amplitudes and times (SI units)\no rms (+A, 1 )\n3.05\n3.18\n3.34\n2.62\n2.94\n3.22\n1.90\n2.09\n2.27\n8.15\n8.48\n8.85\n5.11\n5.36\n6.19\n3.71\n4.21\n4.82\n(-A 1 ) o rms (-T1 1 )\n5.06\n7.63\n12.53\n5.12\n8.13\n13.25\n5.42\n8.18\n13.26\n4.26\n6.48\n11.22\n4.61\n6.89\n11.28\n4.74\n7.10\n11.44\n2.44\n2.63\n2.86\n1.94\n2.25\n2.54\n1.77\n2.06\n2.37\n7.48\n8.05\n9.22\n5.74\n6.37\n6.92\n4.85\n5.51\n6.29\nrms\nCase A\no\n)\n1\n(+T\n4.59\n6.87\n11.31\n4.92\n7.44\n12.18\n4.97\n7.52\n12.66\n4.22\n6.46\n11.00\n4.62\n6.80\n11.00\n4.88\n7.27\n11.23\nrms\no\no rms (+A. 1 )\n2.67\n2.90\n3.23\n2.09\n2.41\n2.75\n1.85\n2.10\n2.50\n7.57\n8.20\n9.36\n5.87\n6.29\n6.94\n4.95\n5.66\n6.11\nFilter\n100/10\n100/10\n100/10\n100/10\n100/10\n100/10\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\nHeight\n10\n50\n150\n10\n50\n150\nDirection\nParameter\nSpeed","The rms values and averages in the tables confirm what the scatter\nplots of Figs. 8.10-8.13 show: that gust amplitudes tend to be larger\nand the corresponding gust times smaller in GUST 1 than in GUST 0. The\neffect of our gust time limit, IT11 > 'l' in the analysis is therefore\nparticularly severe in the distribution plots of Figs. 8.1-8.9.\nAlthough the +10's and -To's cover the range between T l and Th' the\n+11's and -T1's are confined to a narrow range between I and 2.5\ne Tests for Gaussian distribution will have little meaning under\nthose conditions.\nIt appears from the full set of distribution plots (only a few are\nshown here) that the highpass filter has no direct effect on the upper\nrange of T1's observed. Only the unfiltered data show a distribution\nthat is nearly Gaussian, but I1's are limited to +10 S. An appropriate\ncombination of lowpass filtering and amplitude tolerance specification\ncould, conceivably, yield more realistic distribution curves.\n80","Case A, Speed +A1\nCase B, Speed +A1\n9\n9\n10 m\n50 m\n7\n7\n150 m\n5\n5\n2.0\n3\n2.0\n3\n1.5\n1.5\n1.0\n1.0\n0.5\n0.5\n30/3\n50/5\n100/10\n30/3\n50/5\n100/10\nFilters\nFilters\nCase A, Speed +T1\nCase B, Speed +T1\n14\n14\n11\n11\n8\n8\n0\n15\n5\n15\n5\n10\n10\n5\n5\n0\n0\n30/3\n50/5\n100/10\n30/3\n50/5\n100/10\nFilters\nFilters\nFigure 8.1 -- GUST 1 speed +A and speed +T1 1 statistics as a function of\nbandpass filtering. Standard deviations are in m/s and degrees.\n81","Case A, Direction +A1\nCase B, Direction + A1\n13\n13\n10 m\n50 m\n10\n150 m\n10\n7\n7\n4.0\n4\n4.0\n4\n3.5\n3.5\n3.0\n3.0\n2.5\n2.5\n30/3\n50/5\n100/10\n30/3\n50/5\n100/10\nFilters\nFilters\nCase A, Direction +T1\nCase B, Direction +T1\n14\n14\n11\n11\n8\n8\n2.5\n5\n2.5\n5\n2.0\n2.0\n1.5\n1.5\n1.0\n1.0\n30/3\n50/5\n100/10\n30/3\n50/5\n100/10\nFilters\nFilters\nFigure 8.2 -- GUST 1 direction +A\nand direction +T statistics as a\n1\n1\nfunction of bandpass filtering.\nStandard deviations are in m/s and\ndegrees\n82","Case B, Speed (+A1. +T1)\nCase A, Speed (+A1. +T1)\n1.2\n1.2\no 10 m\n50 m\n150 m\n1.0\n1.0\n0.8\n0.8\n0.6\n0.6\no\n0.4\n0.4\no\n0.2\n0.2\n0\n0\n30/3\n50/5\n100/10\n30/3\n50/5\n100/10\nFilters\nFilters\nCase B, Direction (+A1, +T1)\nCase A, Direction (+A1, +T1)\n1.2\n1.2\n1.0\n1.0\n0.8\n0.8\n0.6\n0.6\no\n0.4\n0.4\nD\n0.2\n0.2\n0\n0\n100/10\n30/3\n50/5\n100/10\n30/3\n50/5\nFilters\nFilters\nFigure 8.3--GUST 1 amplitude-time cross correlation coefficients as a\nfunction of bandpass filtering.\n83","(x 10)\n8.00\n2.00\nGUST 1\nGUST 1\nB. 50/5 filter; Z = 10 m\nFigure 8. 4 - - GUST 1 speed A 1 and speed I1 distributions for 50/5 filter at m.\n4.00\n10\n1.00\nSpeed amplitude (m/s)\nSpeed time (s)\n0.00\n0.00\n-4.00\n-1.00\nCase\n-8.00\n-2.00\n1.60\n1.20\n0.80\n0.40\n0.00\n4.00\n5.00\n3.00\n2.00\n1.00\n0.00\n8.00\n(x 10)\n2.00\nGUST 1\nCase A. 50/5 filter; Z = 10 m\nGUST 1\nAA = 0.16 m/s; AT1 = 0.4 S.\n4.00\n1.00\nSpeed amplitude (m/s)\nSpeed time (s)\n0.00\n0.00\n-4.00\n-1.00\n-8.00\n-2.00\n1.60\n1.20\n0.80\n0.40\n0.00\n5.00\n4.00\n3.00\n2.00\n1.00\n0.00","(UTq red %)\n(UTq red %)\n(or x)\n(01 x)\nthe\nthe\n85","(x 10)\n8.00\n2.00\nFigure 8. 6 -- GUST 1 speed A1 1 and speed T1 distributions for 50/5 filter at 150 m.\nB. 50/5 filter; Z = 150 m\nGUST 1\nGUST 1\n1.00\n4.00\nSpeed amplitude (m/s)\nSpeed time (s)\n0.00\n0.00\n-4.00\n-1.00\nCase\n-8.00\n-2.00\n4.00\n3.00\n2.00\n1.00\n0.00\n1.60\n1.20\n0.80\n0.40\n0.00\n5.00\n2.00\n(x 10)\n8.00\nCase A. 50/5 filter; Z = 150 m\nGUST 1\nGUST 1\nAA1 = 0.16 m/s; AT1 1 = 0.4 S.\n1.00\n4.00\nSpeed amplitude (m/s)\nSpeed time (s)\n0.00\n0.00\n-1.00\n-4.00\n-2.00\n-8.00\n2.00\n1.00\n0.00\n4.00\n3.00\n0.40\n5.00\n1.60\n1.20\n0.80\n0.00","2.00\n(x 10)\n(x 10)\n2.00\nFigure 8. -- -GUST 1 direction A 1 and direction T1 distributions for 50/5 filter at\nGUST 1\nCase B. 50/5 filter; Z = 10 m\nGUST 1\n1.00\nDirection amplitude (deg)\n1.00\nDirection time (s)\n0.00\n0.00\nA\n-1.00\n-1.00\n-2.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n5.25\n3.50\n7.00\n1.75\n0.00\n2.00\n(x 10)\n2.00\n(x 10)\n10 m. AA1 = 0.4 deg; AT1 1 = 0.4 S.\nGUST 1\nGUST 1\nCase A. 50/5 filter; Z = 10 m\nDirection amplitude (deg)\n1.00\n1.00\nDirection time (s)\n0.00\n0.00\n-1.00\n-1.00\n-2.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n7.00\n5.25\n3.50\n1.75\n0.00\n%","2.00\n(x 10)\n2.00\n(x 10)\nFigure 8.8 - - GUST 1 direction A. and direction T1 distributions for 50/5 filter at\nGUST 1\nGUST 1\nB. 50/5 filter; Z = 50 m\nDirection amplitude (deg)\n1.00\n1.00\nDirection time (s)\n0.00\n0.00\n-1.00\n-1.00\nCase\n-2.00\n-2.00\n1.75\n0.40\n5.25\n3.50\n0.00\n1.00\n0.80\n0.60\n0.20\n1.26\n0.00\n7.00\n(x 10)\n(x 10)\n2.00\n2.00\n50 m. AA1 = 0.4 deg; AT1 = 0.4 S.\nGUST 1\nA. 50/5 filter; Z = 50 m\nGUST 1\nDirection amplitude (deg)\n1.00\n1.00\nDirection time (s)\n0.00\n0.00\n-1.00\n-1.00\nCase\n-2.00\n-2.00\n3.50\n1.75\n1.20\n1700\n0.20\n0.80\n0.60\n0.40\n0.00\n7.00\n5.25\n0.00\n8\n8","2.00\n(x 10)\n(x 10)\n2.00\nGUST 1\n50/5 filter; Z = 150 m\nGUST 1\nFigure 8.9 - - GUST 1 direction A1 1, and direction T1 distribution for 50/5 filter at\nDirection amplitude (deg)\n1.00\n1.00\nDirection time (s)\n0.00\n0.00\n-1.00\n-1.00\nCase B.\n-2.00\n-2.00\n1.20\n1.00\n0.80\n0.60\n0.40\n0.20\n0.00\n7.00\n5.25\n3.50\n1.75\n0.00\n(x 10)\n2.00\n(x 10)\n2.00\n150 m. AA = 0.4 deg; AT, = 0.4 S.\nGUST 1\nCase A. 50/5 filter; Z = 150 m\nGUST 1\nDirection amplitude (deg)\n1.00\n1.00\nDirection time (s)\n0.00\n0.00\n-1.00\n-1.00\n-2.00\n-2.00\n3.50\n1.75\n0.20\n0.00\n7.00\n5.25\n0.00\n1.20\n1.00\n0.80\n0.60\n0.40","Case A. No filter; Z = 50 m; GUST 1\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-1.20\n-1.00\n-0.80\n-0.60\n-0.40\n0.20\n-0.20\n0.00\n0.00\n0.40\n0.60\n0.80\n1.00\n1.20\nSpeed amplitude (m/s)\n(x 10)\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-3.00\n-2.00\n-1.00\n3.00\n0.00\n0.00\n1.00\n2.00\nDirection amplitude (deg)\n(x 10)\nFigure 8. 10--Plots of GUST 1 amplitudes VS. times for Case A at 50 m (no\nfilter)\n90","Case B. No filter; Z = 50 m; GUST 1\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-1.20\n-1.00\n-0.80\n-0.60\n-0.40\n-0.20\n0.00\n0.00\n0.20\n0.40\n0.60\n0.80\n1.00\n1.20\n(x 10)\nSpeed amplitude (m/s)\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n3.00\n-3.00\n-2.00\n-1.00\n0.00\n0.00\n1.00\n2.00\nDirection amplitude (deg)\n(x 10)\nFigure 8.11--Plots of GUST 1 amplitudes vs. times for Case B at 50 m (no\nfilter)\n91","Case A. 50/5 filter; Z = 50 m; GUST 1\n4.00\n4.00\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-8.00\n-6.00\n-4.00\n-2.00\n0.00\n0.00\n2.00\n4.00\n6.00\n8.00\n(x 10)\nSpeed amplitude (m/s)\n4.00\n4.00\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-2.00\n-1.50\n-1.00\n-0.50\n0.00\n0.00\n0.50\n1.00\n1.50\n2.00\nDirection amplitude (deg)\n(x 10)\nFigure 8.12 -- Plots of GUST 1 amplitudes vs. times for Case A at 50 m\n(50/5 filter)\n92","Case B. 50/5 filter; Z = 50 m; GUST 1\n4.00\n4.00\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n-8.00\n-6.00\n-4.00\n-2.00\n0.00\n0.00\n2.00\n4.00\n6.00\n8.00\nSpeed amplitude (m/s)\n(x 10)\n4.00\n4.00\n3.00\n3.00\n2.00\n2.00\n1.00\n1.00\n0.00\n0.00\n2.00\n-2.00\n-1.50\n-1.00\n-0.50\n0.00\n0.50\n1.00\n1.50\n0.00\nDirection amplitude (deg)\n(x 10)\nFigure 8. 13 -- Plots of GUST 1 amplitudes vs. times for Case B at 50 m\n(50/5 filter)\n93","9.0 CHARACTERISTIC MAGNITUDE ANALYSIS\nFor model applications it is often necessary to characterize the\ngust parameters with appropriate turbulent (or mean) properties of the\nflow. Here we follow the approach of Powell and Connell (1980) and\nnormalize the rms gust amplitudes by the standard deviation of the\noriginal filtered time series. (For bandpass filtered data the defini- -\ntions of the standard deviation and the rms value become equivalent.)\nThe rms gust times are normalized by the appropriate gust time average.\nThe normalized values in Tables 9.1-9.4 show remarkable consistency.\nThe following approximations can be made on the basis of these results:\n(9.1)\n(9.2)\n[\n12\n(9.3)\n(9.4)\nF\nThe approximations are valid for both speed and direction.\nAs pointed out by Powell and Connell,\n[\n(9.5)\nand\n(9.6)\n94","for a Gaussian distribution. Our larger ratio in (9.1) can be attributed\nto the gap centered around zero in the A 0 distribution (Figs . 7.4-7.9).\nRemoval of values close to zero has the effect of raising the rms value.\nPresumably, with the gap filled in, the ratio will drop to 1.3. We have\nno comparable theoretical expectation for the ratio in (9.2), but can\nassume from the same reasoning that the ratio would be slightly lower\nfor a Gaussian distribution. In any case we can deduce from our results\na relationship between GUST 0 and GUST 1 amplitudes:\n(9.8)\n.\nFor the gust times we have a smaller ratio in (9.2) than the 1.25\nrequired for a Gaussian distribution. Here we have two factors working\nin tandem. The increase in the rms value (in the numerator) due to\nthe gap in the frequency distribution is offset by a larger increase in\nthe average gust time (in the denominator), causing a net reduction in\nthe ratio. A relationship similar to (9.7) can be offered for the gust\ntimes\nrs15omo\n(9.8)\n.\n95","1.46\n1.48\n1.58\n1.54\n1.58\n1.64\n1.53\n1.58\n1.61\n1.51\n1.63\n1.71\n1.50\n1.62\n1.72\n1.52\n1.59\n1.66\n-A0\n0\nCase B\n1.54\n1.60\n1.71\n1.44\n1.51\n1.54\n1.45\n1.49\n1.52\n1.53\n1.68\n1.73\n1.57\n1.64\n1.72\n1.51\n1.60\n1.64\n0\nF\n+A\nTable 9. 1 -- -Normalized rms GUST 0 amplitudes [o (A0)/o(x)\n1.54\n1.50\n1.51\n1.51\n1.53\n1.48\n1.50\n1.57\n1.52\n1.56\n1.61\n1.47\n1.50\n1.55\n1.48\n1.53\n1.65\n0\n1.5\n-A,\nrms\nCase A\n1.45\n1.53\n1.56\n1.45\n1.45\n1.55\n1.43\n1.47\n1.46\n1.49\n1.57\n1.58\n1.49\n1.51\n1.59\n1.45\n1.49\n1.53\n+A0\nFilter\n100/10\n100/10\n100/10\n100/10\n100/10\n100/10\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\nHeight\n10\n50\n150\n10\n50\n150\namplitude\namplitude\nParameter\nDirection\nSpeed","1.11\n1.13\n1.13\n1.11\n1.12\n1.13\n1.13\n1.12\n1.14\n1.11\n1.10\n1.11\n1.12\n1.12\n1.13\n1.13\n1.14\n1.14\n-To\nCase B\n1.11\n1.10\n1.13\n1.13\n1.13\n1.12\n1.11\n1.09\n1.12\n1.12\n1.13\n1.13\n1.14\n1.13\n1.11\n1.11\n1.14\n1.13\n+TO\nTable 9. 2 --Normalized - - rms GUST 0 amplitudes [o (TO)\nrms\n1.12\n1.13\n1.10\n1.12\n1.11\n1.11\n1.12\n1.11\n1.13\n1.12\n1.12\n1.11\n1.11\n1.12\n1.13\n1.12\n1.11\n1.09\n-TO\nCase A\n1.12\n1.12\n1.11\n1.12\n1.13\n1.11\n1.13\n1.12\n1.13\n1.13\n1.14\n1.12\n1.13\n1.11\n1.12\n1.11\n1.12\n1.13\n0\n+T\nFilter\n100/10\n100/10\n100/10\n100/10\n100/10\n100/10\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\nHeight\n10\n50\n150\n10\n50\n150\nDirection\nParameter\ntime\ntime\nSpeed","2.24\n2.33\n2.74\n2.15\n2.22\n2.49\n2.10\n2.23\n2.36\n2.21\n2.61\n3.47\n2.07\n2.40\n3.35\n1.92\n2.26\n2.71\n-A1\nCase B\n2.29\n2.47\n2.74\n2.24\n2.56\n2.82\n2.16\n2.30\n2.32\n2.23\n2.71\n3.29\n2.08\n2.43\n3.19\n1.90\n2.22\n2.62\n+A1\nF\nTable 9. 3 - --Normalized - rms GUST 1 amplitudes [o (A0)/o(x)\n2.07\n2.14\n2.22\n2.00\n2.08\n2.08\n2.01\n2:10\n2.17\n1.99\n2.22\n2.61\n1.93\n2.18\n2.45\n1.87\n2.14\n2.57\n1\n-A\nrms\nCase A\n2.26\n2.36\n2.50\n2.15\n2.23\n2.25\n2.10\n2.14\n2.29\n2.01\n2.27\n2.65\n1.98\n2.15\n2.46\n1.91\n2.20\n2.49\n1\n+A\nFilter\n100/10\n100/10\n100/10\n100/10\n100/10\n100/10\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\nHeight\n10\n50\n150\n10\n50\n150\namplitude\namplitude\nParameter\nDirection\nSpeed","1.04\n1.04\n1.07\n1.04\n1.02\n1.07\n1.05\n1.04\n1.08\n1.05\n1.02\n1.02\n1.04\n1.02\n1.03\n1.07\n1.05\n1.03\n1\n-T\nCase B\n1.04\n1.05\n1.03\n1.02\n1.06\n1.04\n1.02\n1.07\n1.03\n1.04\n1.02\n1.01\n1.05\n1.02\n1.02\n1.07\n1.04\n1.03\n1\nTable 9. 4 - - Normalized rms GUST 1 amplitudes [o (TO) 0\n+T\nrms\n1.09\n1.06\n1.05\n1.09\n1.07\n1.05\n1.09\n1.07\n1.05\n1.06\n1.04\n1.04\n1.07\n1.04\n1.02\n1.07\n1.05\n1.03\n1\n-T.\nCase A\n1.05\n1.03\n1.06\n1.04\n1.02\n1.07\n1.08\n1.06\n1.04\n1.07\n1.06\n1.05\n1.07\n1.04\n1.02\n1.08\n1.05\n1.02\n1\n+T\nFilter\n100/10\n100/10\n100/10\n100/10\n100/10\n100/10\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\n30/3\n50/5\nHeight\n10\n50\n150\n10\n50\n150\nDirection\nParameter\ntime\ntime\nSpeed","10.0 CONCLUSIONS\n1)\nThe statistics computed for the two wind storms represented by\nCase A and Case B exhibit a high degree of internal consistency. When\nnormalized, the gust parameters become insensitive to differences in\nstatic stability, height, and choice of bandpass filter.\n2)\nBandpass filtering is essential for approximating Gaussian distri-\nbution in the original speed and direction time series, more so for\nthe speed than for the direction. The basic 50/5 bandpass filter of\nPowell and Connell appears to be a proper choice for the types of\nanalyses performed here.\n3) A consistent relationship exists between the standard deviation\nof the speed (and direction) differences and the standard deviation\nof the original filtered speed (and direction) time series. For the\n50/5 filtered data, at the optimum differencing interval of 10 s, the\nratio of the two standard deviations is 1.5.\n4) Normalized gust statistics for GUST 0 and GUST 1 models show\nconsistent and constant relationships that are directly applicable to\nWECS design.\n5) Further work is needed to determine the best approach for eliminat-\ning the gaps in the GUST 0 and GUST 1 distributions.\n6)\nBecause of the internal consistency in their statistical proper-\nties, the two cases studied here constitute an ideal data set for\nvalidation of other gust models.\n100","REFERENCES\nCliff, W.C. and G.H. Fichtl, 1978. Wind Velocity-Change (Gust Rise)\nCriteria for Wind Turbine Design. PNL-2526, Pacific Northwest\nLaboratory, Richland, Wash.\nConnell, J.R., 1979. Overview of wind characteristics for design and\nperformance. Proceedings: Conference and Workshop on Wind Energy\nSiting 1979, Portland, Oregon (June 19-21). American Meteorolog-\nical Society Boston, Mass., pp. 5-12.\nConnell, J.R., , 1980. Turbulence spectrum observed by a fast rotating\nwind turbine blade. PNL-3426, Pacific Northwest Laboratory,\nRichland, Wash.\nDoran, J.C. and D.C. Powell, 1980. Gust Characteristics for WECS Design\nand Performance Analysis. PNL-3421, Pacific Northwest Laboratory,\nRichland, Wash.\nHuang, C.H. and G.H. Fichtl, 1979. Gust-Rise Exceedance Statistics\nfor Wind Turbine Design. PNL-2530, Pacific Northwest Laboratory,\nRichland, Wash.\nKaimal, J.C., , J. C. Wyngaard, Y. Izumi and O.R. Coté, 1972. Spectral\ncharacteristics of surface layer turbulence. Quart. J. Roy.\nMeteorol. Soc., 98, 563-589.\nKaimal, J.C., 1973. Turbulence spectra, length scales and structure\nparameters in the stable surface layer. Bound.-Layer Meteorol. 4,\n289-309.\nKaimal, J.C., , 1978a. NOAA instrumentation at the Boulder Atmospheric\nObservatory. Preprints: 4th Symposium on Meteorological Obser-\nvations and Instrumentation, 1978, Denver, Colorado, American\nMeteorological Society, Boston, Mass., , pp. 35-40.\n101","Kaimal, J.C., 1978b. Horizontal velocity spectra in an unstable\nsurface layer. J. Atmos. Sci., 35, 18-24.\nKaimal, J.C., 1980. BAO sensors for wind, temperature and humidity\nprofiling. Chapter 1 in The Boulder Low-Level Intercomparison\nExperiment - Preprint of WMO Report, J.C. Kaimal et al., Eds.,\nNOAA/NCAR Boulder Atmospheric Observatory Report No. 2, pp. 1-6.\nLawrence, R.S. and M.H. Ackley, 1979. Interactive access to the BAO\ndata. Chapter 17 in Project PHOENIX: The September 1978 Field\nOperation, W.H. Hooke, Ed., NOAA/NCAR Boulder Atmospheric Ob-\nservatory Report No. 1., pp. 267-281.\nPowell, D.C., 1979. Wind fluctuations described as discrete events.\nProceedings: Conference and Workshop on Wind Energy Characteristics\nand Wind Energy Siting 1979, Portland, Oregon (June 19-21).\nAmerican Meteorological Society, Boston, Mass., pp. 71-79.\nPowell, D.C. and J.R. Connell, 1980. Definition of Gust Model Concepts\nand Review of Gust Models. PNL-3138, Pacific Northwest Laboratory,\nRichland, Wash.\nRamsdell, J.V., 1975. Wind and Turbulence Information for Vertical and\nShort Take-off and Landing (V/STOL) Operations - Results of\nMeteorological Survey. Report No. FAA-RO-75-94, Dept. of Trans-\nportation, Federal Aviation Administration Systems Research and\nDevelopment Service, Washington, D.C.\n102\n*\nU.S. GOVERNMENT PRINTING OFFICE: 1981 - 777-002/1263 Region No. 8","OF\nCOMMISS\n3\n8398\nwater\nSTATES\nOF\nA\nthe\nDDDDD"]}