A. C. Hansen Visiting Associate Professor,

Department of Mechanical and Industrial Engineering,

University of Utah, Salt Lake City, Utah 84112

T. E. Hausfeld Staff Engineer,

Wind Energy Research Center, Solar Energy Research Institute,

Golden, CO 80401

Frequency-Response Matching to Optimize Wind-Turbine Test Data Correlation Pre-averaging is often applied to wind turbine test data to improve correlation be­tween wind speed and power output data. In the past, trial and error or intuition have been used in the selection of pre-averaging time and researchers and institu­tions have differed widely in their pre-averaging practice. In this paper a standardiz­ed method is proposed for selection of the optimum pre-averaging time. The method selects an averaging time such that the test data are low-pass-filtered at the same fre­quency as the response frequency of the test wind turbine/anemometer system. A theoretial method is provided for estimation of the wind system transfer function as a function of the anemometer location, rotor moment of inertia, the stiffness of the connection between the rotor and the electrical grid, hub height, rotor speed and wind speed. The method is based in proven theory, repeatable, easy to use and ap­plicable to a wide range of wind turbines and test conditions.

Results of the transfer function predictions are compared with the measured response of two wind systems. Agreement between the predicted and measured response is completely adequate for the purposes of the method. Example results of calculated averaging times are presented for several wind turbines. In addition, a case study is used to demonstrate the dramatic effects of test design and data analysis methods on the results of a power coefficient measurement.

Introduction

Test data obtained from wind turbines in the natural wind environment are essential to the understanding and improve­ment of wind energy systems. Most often, tests are intended to correlate some aspect of turbine response (power output, structural loads, yaw behavior, etc.) with the wind input to the system. This is complicated by the fundamental difficulty that it is impossible to directly measure the instantaneous ambient wind that is experienced by an operating wind turbine rotor. Such a measurement is impossible because induced velocity ef­fects make it necessary to measure winds at some distance from the rotor (typically 2-3 rotor diameters) and turbulence in the planetary boundary layer will cause significant changes in instantaneous wind speeds over those distances. The poor correlation between winds that can be measured and actual winds at the rotor results in an inevitable loss of correlation between wind data and turbine response (and the appearance of scatter in the data). For most system parameters there will be a further loss of correlation resulting from the inertial lag of the system in response to a change in the wind input. Thus there are two basic mechanisms by which the wind measured during a test will be imperfectly correlated with the system response: 1) poor correlation between the measured wind and the actual wind experienced by the entire rotor, and 2) poor correlation due to inertial lag of the system responding to changes in the ambient wind speed.

Contributed by the Solar Energy Division for publication in the JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received by the Solar Energy Division, November, 1985.

The Method-of-Bins is the generally accepted technique for analysis and summary of test data [1, 2]. Correlation is im­proved by time-averaging of the data prior to entry into the Method-of-Bins. Time-averaging mitigates the effects of both poor point-to-point wind correlation and inertial lag. The wind correlation is improved by averaging out high frequency fluctuations in wind speed and the inertial lag is masked if pre-averaging times exceed the system response time constant.

Thus it has become standard practice in the wind energy in­dustry to use pre-averaging of the raw data before applying the Method-of-Bins. To date, however, the selection of a pre-averaging time has been arbitrary and there are major dif­ferences in the averaging times used or advocated by various organizations. For instance, the International Energy Agency proposed test guidelines [3] call for use of a ten minute pre-averaging time for all wind turbines. The American Wind Energy Association proposed Performance Test Standard [4] recommends use of pre-averaging times ranging from a few seconds to one minute, the selection being at the discretion of the user.

There are costs associated with selection of a pre-averaging time that is either too short or too long. If the time is too short then scatter in the test data may result in misinterpretation or loss of important information in the ' 'noise.'' If the averaging time is too long then important information may be averaged out and test durations (and hence costs) will be unnecessarily long. One example of information lost as a result of averaging is the following: If a ten minute pre-average is used while

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I

testing a small turbine, complete start-up and shut-down se­quences could occur during the collection of one data point. The wind speed and power output data point so obtained would obviously have no value and important information about the control behavior would be lost.

It is clear that a method for selection of pre-averaging times must be developed to permit the maximum amount of useful information to be gleaned from test data. The method must be repeatable, applicable to a wide variety of wind turbines and test conditions, based in proven theory, and easy to use. Such a method is proposed in this paper.

The purpose of the method is to allow selection of a pre-averaging time that will maximize the amount of information that can be derived from raw test data. This is different from minimizing the "scatter" in the data. If minimum scatter is the only objective of a data analysis technique then lengthen­ing the pre-averaging time will achieve the objective. The ob­vious (and absurd) ultimate result of long pre-averaging would be reduction of an entire test record to one data point. There would be no scatter but the data would be virtually worthless for engineering evaluation of the wind turbine.

Three Wind Turbine Response Functions

The wind turbine acts as a low-pass filter to wind speed fluc­tuations that are measured at some distance from the turbine rotor. The test system (wind turbine, test anemometry and in­tervening atmospheric turbulence) has a response function which, in the frequency domain, is commonly called the filter shape function or transfer function. Pre-averaging is also a means of low-pass filtering. The proposed method of averag­ing time selection is based upon the premise that the transfer function of the pre-averaging filter should closely match that of the test system. If pre-averaging filters more severely than the test system then useful information will be lost. If pre-averaging allows higher frequency fluctuations to pass than the test system then meaningless and confusing scatter will be present in the test data. To select the optimum pre-averaging time, then, one must first determine the transfer function of the test system and then choose the pre-averaging time with a similar transfer function shape and identical cut-off frequen­cy. A means for estimating the test system transfer function will be presented in the following paragraphs.

For purposes of analysis the test system can be considered the composite of three subsystems. The first two subsystems are concerned only with the wind. The first relates the wind at the point of measurement to the wind at the location of the rotor hub (in the absence of induced effects). The second relates the instantaneous wind at the rotor hub to the spatial average of the wind over the rotor disc. The third subsystem relates the wind experienced by the rotor disc to the output of the turbine system. The particular case that will be analyzed here is the electrical power output of the wind system as a function of the scalar wind speed. Other parameters could be analyzed in a similar fashion.

Figure 1 shows a schematic of a test configuration. We will denote the frequency domain transfer function from the wind at the point of measurement to the wind at the hub as Hx. This will be called the anemometer displacement transfer function. The transfer function between the wind at the hub and the in­stantaneous wind averaged across the rotor will be called the disc averaging transfer function, denoted by H2. Finally the transfer function between the disc-averaged wind speed and the power output of the wind system will be the inertial response function, denoted by H3. The response function of the complete system is the product of the response functions of the various subsystems (for linear systems). That is,

\H(f)\2=\H1(f)\2x\H2(f)\

2x\H3(f)\2. (1)

The transfer function relating the wind at one point to the

Fig. 1 Sketch of the turbine test geometry

wind at a different point is presented by Panofsky and Dutton [5].

\HX {/) 12 = Coh(/) = exp ( - 6f(Lx + 2L,)/V] (2) In this equation, Kis the ambient wind speed, Lx and Ly are the longitudinal and lateral distances between two points at the same elevation as shown in Fig. 1, and / i s the frequency (Hz) of the wind fluctuation. Equation (2) is valid for strong, neutrally stable winds in regions where the long-term-average winds are uniform. For short vertical separations this equation can be used with acceptable accuracy. (In such a case Ly is the cross wind distance between the two points.) Note that the decay of coherence or correlation is twice as fast for lateral separation of the points as for longitudinal separation. Thus it is important to keep the measurement anemometry upwind of the turbine and at hub height as much as possible.

The transfer function between the disc average wind speed and the hub wind speed (H2) can be found using the same ex­ponential coherence relationship used in deriving equation (2). An approximation to the disc average can be made by using four points spaced 90 deg apart around a circle with a radius 75 percent of the rotor radius (the 0.75 R location is often con­sidered representative of the entire blade in rotary-wing analyses). Details of the calculation are presented in Appendix A. The result is,

\H2(f)\2 = l/3{exp[-(12+16.5R/H)(0.75Rf/V)] +

2 e\p[-(l2+U.67R/H)(0.53Rf/V)]} (3) In this equation R is the rotor radius and if is the hub height. This result is similar to that of Madsen and Frandsen [6] and Frandsen [7] but the present result is derived for a circular rather than a rectangular area and includes the influence of tower height. Note from equation (3) the best correlation is obtained for small rotor radii, large hub heights, and/or high wind speeds.

The response of a wind turbine system to changes in the disc wind speed is obviously complex and can involve dozens of variables. For the present purposes however it is important to keep the analysis straightforward and at a level of sophistica­tion comparable to that of the turbulence coherence in the previous paragraphs. The most important features of a tur­bine, for present purposes, are: 1) Is the turbine of (virtually) constant or variable speed? and 2) What is the rotating inertia of the system? The "stiffness" of the power train (or the change in rotor speed required per unit change in power out­put) and the inertia of the rotor system are the dominant variables influencing the time constant of the system. All other

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Table 1 Characteristics and test conditions of the University of Massachusetts wind furnace and the modified Northwind HRl Characteristic

Wind Speed Rotor Speed Rotor Diameter Moment of Inertia Hub Height Generator Stiffness Anemometer Lx Anemometer Lv

UMass Wind Furnace

14.2 mph 79.3 rpm

32.5 ft 255. lb-ft-s2

60 ft 0.21 kW/rpm

60 ft Oft

Northwind HRl

15 mph 150 rpm 16.0 ft

11.6 lb-ft-s2

50 ft 0.007 kW/rpm

50 ft 10 ft

variables, such as blade and tower structural stiffness, details of the generator construction, masses of individual sub­systems, etc., will be of secondary importance when attempt­ing to estimate the response time of the entire system. Therefore, in this analysis the simplest approach will be taken. For more detailed analyses of turbine system dynamics the reader is referred to [8] and [9].

Consider a power train/rotor system with polar moment of inertia / . The generator can be approximated as a linear system where power output changes are proportional to rota­tional speed changes, with a proportionality constant of k. That is,

k = dP/dU (5)

where P is the electrical power output and fl is the rotor speed. For small changes in speed, the "stiffness" k can be con­sidered constant. Conventional induction generators have a stiffness of the order of (rated power)/(3 percent-5 percent of rated speed). Variable speed generators have much lower stiff­ness. If the generator is a synchronous alternator the electrical connection will be extremely stiff and then the elastic stiffness of the power train shafts, dampers and gearboxes must be con­sidered as an important part of the overall rotor-to-grid stiffness.

If the available rotor power changes by an amount p, changes in the rotor speed and the power output will result. That is,

p = IQ(doi/dt)+kw. (6)

In this equation o> is the small change in rotational speed and Q is the average speed. This equation represents a linear, first-order system with a transfer function given by [10].

l i / 3 ( / ) l 2 = l / [ l + (27rr/)2] (7)

where the time constant, T, is

T=IQ/k. (8)

Of course, the critical parameter in defining the transfer func­tion is the time constant which depends upon the power train stiffness and the rotating mass moment of inertia. Constant speed systems will be very stiff and have a short time constant. Small, lightweight systems will have a low moment of inertia and a correspondingly fast response time.

Predicted System Response Functions and Comparison With Test Results

System response functions have been measured for two wind systems. Manwell and Kirchhoff measured the response of the University of Massachusetts Wind Furnace [8] and the transfer function of a slightly modified Northwind HRl was measured at the SERI Wind Energy Research Center. These data make it possible to partially test the validity of the predic­tions of the previous section. Table 1 lists key characteristics of the Wind Furnace and HRl.

Figure 2 shows predicted response functions for the UMass Wind Furnace. Each of the subsystem transfer functions is

0 '

CO

• o

D ) - 2 0 . CO

" - - 5 5 V N

> ^ System \ Response \

)

uispiauemer \ ^ Disc

\ v \ - Inertia

\ \ \ 1.0 .001 .01 0.1

Frequency-Hz Fig. 2 Predicted transfer functions of the Wind Furnace wind turbine

CD "O

i

a> 3

co

.001 .01 0.1

Frequency-Hz 1.0

Fig. 3 Predicted and measured transfer functions of the Wind Furnace

CD

c roco­co

-30

Displacement

System Response

Disc Average

Inertia

.001 .01 0.1 1.0

Frequency-Hz Fig. 4 Predicted transfer functions of the modified Northwind HR2

shown together with the composite function. Note the displacement and disc average filtering are of comparable magnitudes and considerably more important than the inertia filter. This is somewhat surprising for a variable speed system such as the Wind Furnace. Figure 3 compares the predicted system transfer function with the measured response. The solid line is the predicted results and the circles are measured data points. The agreement is quite good, both qualitatively and quantitatively. The rate of roll-off of the test data is greater than that of the predicted curve, but the cut-off fre­quency and overall trends are predicted with acceptable ac­curacy. (Recall that current debate over the correct pre-averaging time spans two orders of magnitude. The present results can narrow the debate significantly.)

Figure 4 shows predicted response functions for the HRl. For this small, variable speed system the inertia filter slightly dominates the wind filters. Larger or constant speed systems (that is, most systems on the market today) will show response dominated by the displacement and disc average filters. Figure 5 shows a comparison of predicted and test results. Again, the agreement is qualitatively and quantitatively acceptable.

These comparisons indicate the proposed method of predic­ting system response is adequate to the task at hand. Of course, the predictions could be improved by including more

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" Table 2 Calculated pre-averaging times for selected wind tur­bines and test conditions MACHINE

Northwind HR2 UMass Wind Furnace jay Carter 25 ESI 80 Mod-2

CONDITIONS FAVORABLE

157s 14. 12. 33.

129.

CONDITIONS UNFAVORABLE

277s 43. 40. 65.

414.

\ detail and modeling more features of the turbulent boundary - layer and the wind turbine system. But the most essential r features are included in the present model while keeping the " computing and data input requirements to a minimum.

T~ Selection of Pre-Averaging Time

After the transfer function of the wind turbine and anemometry system is estimated, the next step is selection of the pre-averaging time that will apply essentially the same response function to the test data. Recall that the optimum in­formation transfer will occur when the data analyzed has the same bandwidth as the wind turbine test system. Pre-averaging for time T is a form of low-pass filtering with a transfer function computed in Appendix B:

\H(f) I2 = 0.5[l-cos(27r/r)]/(7r/T)2. (9)

This filter has a cut-off frequency (response down 3 dB) of

/ c u l =0 .443 / r . (10)

Once the turbine transfer function is known, its cut-off fre­quency can be determined. The pre-averaging time T is then selected such that the turbine and pre-averaging cut-off fre­quencies are the same.

Table 2 shows the preaveraging times estimated for a variety of wind turbines. In all of these examples it was assumed the test anemometer was three rotor diameters from the hub. "Favorable conditions" assume the anemometer is directly upwind of the hub and the wind speed is 11.2 m/s (25 mph). "Unfavorable conditions" assume the anemometer is 60 deg off the wind turbine axis and the wind speed is 6.7 m/s (15 mph).

These averaging times are, as expected, longer for the larger wind turbines. They are also quite similar to averaging times that have actually been used for these machines. Such times have traditionally been selected by trial and error during the data analysis process. Notice they are all significantly less than the ten minute pre-averaging recommended by the IEA.

A Case Study

Performance data were collected on the modified North-wind HR2 at the SERI Wind Energy Research Center. The data were digitized and reduced into a rotor power coefficient versus tip speed ratio format. Shaft torque and angular veloci­ty were measured to provide rotor power values. An anemometer located 50 ft. upwind of the rotor measured wind speed used to calculate the power available in the wind. This power available was corrected to measured barometric pressure and ambient temperature. All data channels were digitized at a rate of 20 Hz. The Method-of-Bins was used to provide rotor power coefficient values for integer tip speed ratio bins. Figure 6 shows the results of the analysis when no pre-averaging is used. Note the power coefficients exceed the Betz limit over a wide range of tip speed ratios. (The curve shows power coefficients up to 0.8. Values up to 4.8 were measured but not plotted in the interest of detail in the region of greatest interest.) Though the instruments reported ac­curately, the data anlaysis gave totally misleading results. Of course, it is the lack of correlation between instantaneously

CD • o

~i - § - •

c o> m 2

-10-

-20

-30-.001

o \ ^Predicted

o \

Measured

.01

\ \ o \

\ o

0.1

Frequency-Hz Fig. 5 Predicted and measured transfer functions of the Northwind HR2

Tip Speed Ratio Fig. 6 Measured performance of the modified Northwind HR2

c o

if= <i> o O CD

O

0.7

0.6

0.5 ••

0.4 ••

0.3

0.2 ••

0 . 1 - -

0.0

Raw Data -* Pre-averaged -»• Boom anemom.

4 5 6 7 8 9 10 11 12

Tip Speed Ratio Fig. 7 Measured performance of the Northwind HR2 after pre-averaging

sampled wind speed and power output values that so greatly distorts the results.

A correction was applied to the data to compensate for the power absorbed by an increase in rotor speed or the power released by a decrease in rotor speed. This inertia correction was applied to each consecutive 20 Hz data sample. The sec­ond curve in Fig. 6 shows the slight influence of this correc­tion. Clearly an inertia correction alone cannot yield mean­ingful results.

Pre-averaging the data for times calculated using the method proposed in this paper significantly altered the Cp-TSR curve as shown in the curve labeled "preaveraged" in Fig. 7. The curve now presents more plausible results and much less scatter in the results as well. The raw data curve of Fig. 6 is reproduced here as well as well for comparison. But pre-averaging is not a solution for a test which was not design­ed as well as possible. The wind speed data measured 50 ft. from the turbine rotor and used in the previous figures were

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•

Fig.8 Photograph of the HR2 test machine (showing the anemometerboom)

accompanied by wind measurements at either end of ahorizontal boom attached to the machine. This boom, shownin Fig. 8, allowed wind speed to be measured nearly in therotor plane, one radius outboard of the blade tip. The twowind measurements were averaged to provide a better estimateof the wind at the rotor disc. The data were corrected for iner­tia effects and pre-averaged in accordance with the methodoutlined above. The result is presented in the "BoomAnemom." curve of Fig. 7. Better positioning of anemometryallows shorter pre-averaging times, and higher frequencymeasurement. Remember that Fig. 7 uses data from the sametest as Fig. 6. The positive, in fact essential, effects of goodanemometer placement and pre-averaging are evident.

(This example shows the sensitivity of a power coefficientmeasurement to the method of data analysis. Because of thisparticular sensitivity to wind speed, it is recommended that Cpmeasurements be made indirectly rather than directly asabove. First a power curve should be measured as a functionof wind speed. Then the Cp should be calculated from thefinal power curve. The Cp was calculated for each data pointin the above case study only to demonstrate the effects of testdesign.)

Alternatives to Pre-Averaging

Pre-averaging is a form of low-pass filtering. It offers thegreat advantage of simple and intuitive application. But it alsohas some disadvantages. The filter shape of a pre-averagingfilter is different from that of the typical wind turbine. Thismeans that the filters can only be matched at one point (the- 3 dB point is used in this work). A mismatch of filter shapeswill result in less than optimal transfer of correlated data. Thefirst author has investigated use of a non-recursive digital lowpass filter with an exponential roll-off quite similar to that ofthe wind turbine system. Such a filter is more time consumingand difficult to apply. At present it is not clear that thebenefits of a better filter match justify the cost. This matter isunder continuing investigation.

Perhaps more important, pre-averaging as it is used today,applies the same averaging time to all channels of data. This isa major disadvantage when analyzing load response data. (It isnot a problem if wind speed and power output data are the on­ly channels under investigation.) If a test is designed to in­vestigate structural loads, i.e., blade root bending or shafttorque, the application of pre-averaging to all channels of datawill destroy much cyclic load information of value. The solu­tion is to apply different low-pass filters to each channel ofdata. For instance, the wind speed might be pre-averaged tomatch the system power response; the wind direction andmachine yaw channels could be filtered at intermediate fre-

250 I Vol. 108, AUGUST 1986

quencies to remove uncorrelated wind direction fluctuationsbut retain rapid yaw events; and the strain gage channels couldbe filtered at relatively high frequencies (say, for example,twice the highest natural frequency of interest). Such a methodof data pre-processing has been employed by the authors indata analysis at the SERI Wind Energy Research Center.However, a detailed discussion of this technique is beyond thescope of this paper.

Conclusions

A method has been presented for determination of the ap­propriate pre-averaging to be used in processing wind turbinetest data. The averaging time is selected to give the time-seriesdata the same bandwidth as the wind turbine test system. Ithas been shown that the loss of correlation between measuredwind speed and power output is a result of three effects: 1) themeasured wind at a point is not perfectly correlated with thewind that would be observed at the hub location (in theabsence of the rotor induced velocities), 2) the wind at the hublocation is not perfectly correlated with the instantaneouswind averaged over the rotor disc area, and 3) inertial lag ofthe mechanical/electrical system results in loss of response torapid wind speed fluctuations. Essential features of thesephenomena can be characterized by the wind speed, the lateraland longitudinal distance to the test anemometer, the hubheight above ground, the rotor and power train moment of in­ertia and speed, and the stiffness of the electromechanical linkbetween the wind turbine rotor and the electrical grid.

Equations are presented for estimation of the transfer func­tion of each of these phenomena. The resulting systemtransfer functions, though calculated using highly simplifiedtheories, compare well with measured transfer functions fromtwo full-scale wind turbines. It is shown that the major causeof poor correlation is the (necessary) anemometer distancefrom the rotor and the resulting inability to measure the windspeed that is experienced by the rotor. Though the inertial laghas often been cited as the cause of poor correlation, in fact itis the dominant cause only for small, variable speed turbines.

The reader is cautioned that the method assumesequivalence of vertical and lateral displacements in the estima­tion of the wind transfer functions. Such an assumption maynot be valid for very large wind turbines or during periods ofatmospheric thermal stability or instability. In such a situationthe wind turbulence correlations must be represented in amore detailed manner.

Appropriate averaging times are calculated for several windsystems. Pre-averaging times between fifteen seconds and sixminutes are predicted for the full range of systems (from a 5 mrotor up to the Mod 2). In no case was a time of ten minutes,as recommended by some proposed test practices, ap­propriate. Use of excessive averaging will cause excessive testdurations and loss of meaningful correlated data as well.

When designing wind turbine tests it is important tominimize the crosswind distance to the anemometer becausecrosswind distance is twice as detrimental to the correlation aslongitudinal distance. In addition, use of small, constantspeed turbines is recommended for research testing in the at­mosphere. The wind speed measurements for small turbineswill have the best possible correlation with the turbineresponse, making it possible to derive the most detailed infor­mation from the tests.

References

1 Akins, R. E., "Performance Evaluation of Wind Energy ConversionSystems Using the Method of Bins - Current Status," Sandia LaboratoriesReport SAND77-1375, 1977.

2 Hansen, A. C., "Effects of Turbulence on Wind Turbine Performance,"Transportation Engineering Journal ofASCE, Vol. 106, No. TE 6, 1980.

3 Frandsen, S., Trenka, A. R., and Pedersen, B. M., "Recommended Prac-

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tices for Wind Turbine Testing (Power Performance Testing)," International Energy Agency.

4 "Performance Rating Document," Draft consensus standard prepared by the American Wind Energy Association, Alexandria, VA.

5 Panofsky, H. A., and Dutton, J. A., Atmospheric Turbulence, Models and Methods for Engineering Applications, Wiley-Interscience, New York, 1984.

6 Madsen, P. H., and Frandsen, S., "Pitch Angle Control for Power Limitation," Proceedings of the European Wind Energy Conference, Ham­burg, Germany, 1984.

7 Frandsen, S., and Christensen, C. J., "On Wind Turbine Power Measurements," Proceedings of the Third International Symposium on Wind Energy Systems, Technical University of Denmark, Aug. 1980.

8 Manwell, J. F., and Kirchhoff, R. H., "Wind Energy from Turbulence: Constant Tip-Speed-Ratio Operation," University of Massachusetts report, 1984.

9 Hinrichsen, E. N., and Nolan, P. J., "Dynamics of Single- and Multiple-Unit Wind Energy Conversion Plants Supplying Electric Utility Systems," Power Technologies, Inc., report R45/81. U.S. Department of Energy Report DOE/ET/20466-78-I, 1981.

10 Bendat, J. S., and Piersol, A. G., Random Data: Analysis and Measure­ment Procedures, Wiley-Interscience, New York, 1971.

A P P E N D I X A

Derivation of the Disc Average Transfer Function

Nomenclature Coh = Coherence function dy = distance between points /' and j f = frequency (Hz) H = hub height R = Rotor radius Sj = Power spectral density of u, = \X,\2

S/j = Cross spectral density of «,- and Uj = X*Xj Uj(t) = longitudinal wind speed at point i V = average wind speed at hub height -̂ 7 (/) = Fourier transform of «,-X* = Complex conjugate of Xt

Overbar denotes time average

The Disc Average Wind Speed Power Spectral Densi­ty. The PSD of the average wind speed can be approximated as the average of the PSD's at four points on the disc. We select four points at a radius equal to 0.75 R, equally spaced around the disc. The PSD is then

Sdisc = ' ( * ! +X2 +X3 +X,)* (* , +X2 +X3 +X4)

noting that SIJ=Xi*Xj= (Xj*Xi)* = SJi*

Sdisc = 1/16 (5, + S2 + S3 + 54 + (S12 + 512 •) + (S13 + Sa *) +

(5,4 + 514 *) + (S2i + S2i *) + (S24 + S24 *) + (S34 + S34 •))

The Cospectrum can be expressed as

C(,= (S„ + V ) / 2

Assume the statistics are identical at all four points, i.e.,

>->i = S2 = S3 = 54

and the points 1 and 2 are separated by the same distance as points 1 and 41. Then, if one assumes sufficiently small separa­tion distances such that lateral and vertical correlations are identical,

Q 2 = Q4 - C23= C34

and C13 = C24

such that Sdisc = 1/4{S1+2CI2 + Cii]

This result expresses the PSD of the disc-averaged wind speed in terms of the spectra and cospectra at and between points on the disc. Note that if the wind is perfectly correlated over the entire disc, (i.e., St = C12 = C13) then the disc PSD is simply the point PSD.

Panofsky and Dutton [5] give the cospectra in terms of the coherence function:

C12 = V t C o h ^ S ^ ) x cos012

= V(Coh12)S! x cos012 when S, = 5 2 .

and the coherence is given as

Coh12 = exp( - (12 +11.67 (R/H))(\ -06\Rf/V)),

Coh13 = exp( - (12 + 16.5(R/H))(1.5R//V)),

6l2 = 9.19/R2/(VH),

and 0n=201 2

Considerable simplification can be achieved to speed calcula­tion of the transfer function with negligible error by approx­imating the above expressions as

Sdisc/S, = \H2(f) I2 = l /3 (exp[ - (12+ l6.5R/H)(OJ5Rf/V)]

+ 2exp[- (12+ n.67R/H)(0.53Rf/V)])

This is equation (3) that is used to approximate the disc average wind speed transfer function.

A P P E N D I X B

The Response Function of a Pre-Averaging Filter

Consider a time series u(t). It has an associated time series of time averaged values U(t), where, for averaging time T,

U(t) = l/T\' u{Z)dH Jt-T

The Fourier Transform of the time-averaged signal is

F{f) = l/T\" \ u(^)d^xp(-i2irft)dt J - 0 0 Jt-T

Using elementary properties of Fourier Transforms, this can be expressed in terms of the Fourier Transform (F) of the raw signal u(t) as follows:

F(f) =F(f) [ l/(i2irfT)(l -exp(- ilirfT))

The transfer function associated with time averaging is then

I H(f) 12 = l/(/27r/7)2 (1 - exp( - HirfT))2

= 0.5[1 - cos(2;r/T)]/(7r/r)2 •

This is the result used in equation (9). The transfer function represents a low-pass filter with a cut-off frequency ( - 3 dB) found by solving for the frequency (/) for which

I T Y ( / ) I 2 = 0 . 5 0

The solution, found by iteration, is

/ = 0 . 4 4 3 / r

This is the result used for matching the turbine response and pre-averaging transfer functions.

Journal of Solar Energy Engineering AUGUST 1986, Vol. 108/251

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