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Review of Wind SimulationMethods For Horizontal-AxisWind Turbine Analysis

D. C. PowellJ. R. Connell

June1986

Preparedfor the U.S. Department of Energyunder Contract DE-ACW76RLO1830

Pacific Northwest LaboratoryOperated for the U.S. Deputmentof EnergybyBattelle Memorid Institute

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DISCLAIMER

Thisreportwasp r q ~ r e dsanaccountofwork sponsoredbyanagencyof theUnited StatesGovernment. Neither the United StatesGovernment nor anyagency thereof, nor any of their employees, makesany warranty, expressorimplied, or assumesany legal liabilityor responsibility for theaccuracy, com-pleteness, or usefulness of any information, apparatus, product, or promsdisclwed,or representsthat itsusewouldnotinfringeprivatelyownedrights.Referencehertin to any specific commercialproduct, process, or servicebytrade name, trademark, manufacturer, or otherwise, does not necessarilyconstitute or imply its endorsement, recommendation, or favoring by theUnited StatesGovernment or any agency thereof.The viewsandopinions ofauthorsexpressedkreindonotnecessarilystateorreflect thoseof theUnitedStates Government or any agency thereof.

PACIFIC NORTHWEST LABORATORY

operatedbyBATTELLEforthe

UNITED STATES DEPARTMENT OF ENERGY

under ContractDE-AC06-76RLO 1830

Printed in the United States of AmericaAvailable from

NationalTechnicalInformationServiceUnited States Departmentof Commcrcc

5 X S PonRoyal RoadSpringfield, Virginia 22161

NTISPrice Codes

Microfiche A01

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REVIEW OF WIND SIMULATION METHODS FORHORIZONTAL-AXIS WIND TURBINE ANALYSIS

D. C. PowellJ. R. Connel l

June 1986

Prepared forth e U.S. Department o f EnergyUnder Contract DE-AC06-76RLO 1830

Pa c i f i c Nor thwest Labora toryRic hla nd, Washington 99352

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SUMMARY

T h i s r e p o r t r ev ie ws t h r e e r e p o r t s on s i m u l a t i o n o f w in ds f o r use i n wi nd8

t u r b i n e f a t i g u e a n a l y s i s . The thr ee re po r t s are presumed t o represent the

s t a t e o f t h e a r t . The aut hors o f th e models and t he names giv en th e models i n

the p resen t r epo r t a re

1. R. M. Sundar and J. P. Sul l ivan, Purdue Universi ty (Purdue method)

2. P. S. Veers, Sandia Nat ional Laborator ies (Sandia method)

3. J. R. Connell and D. C. Powel 1, Paci f ic Nor thwest Laboratory (PNL method) .

The Purdue and Sandia methods sim ulat e co rr e l a t ed wind data a t two p o i n t s

r o t a t i n g a s on t h e r o t o r o f a h o r i z o nt a l- ax i s w i nd t u rb i ne . The PNL method a t

p resen t s i mu l ates on l y one po i n t , whi ch ro ta t es e i t he r as on a ho r i zon t a l - a x i s

w ind t u rb i n e b l ade o r as on a ve r t i c a l - ax i s w i nd t u rb i ne b l ade .

The s pec tra o f sim ulat ed dat a ar e present ed fr om t he Sandia and PNL models

unde r compa rab le i np u t cond i t i ons , and t he ene rgy ca l cu l a ted i n t he r o t a t i o na l

spikes i n the spec tra by th e two models i s compared. Al though agreement

between the two methods i s no t impress ive a t t h i s t ime , iniprovement o f th e

Sandia and PNL methods i s recommended as the best way to advance the sta te o f

t h e a r t . Phys ica l de f i c i enc i es o f t he models a re c i t e d i n t he rep o r t and t ech-

n i c a l recommendations ar e made f or improvement.

The repor t a l so rev iews two genera l methods fo r s imu la t ing s ing le - po i n t

data , c a l l e d the harmonic method and the wh it e noi se method. The harmonic

method, wh ich i s the bas is o f a l l th re e spe c i f i c methods rev iewed, i s recom-

mended over the wh i te no is e method i n s imu l a t in g w inds f o r w ind t u rb ine

a n a l y s i s.

iii

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ACK I ~OWLEDGMENT

The authors g ra te fu l l y acknowlege the he lpf ul reviews by Paul Veers .,

A. H. Mil ler , and George P. Tennyson, and the considera ble tasks o f t yp in g and

ed i t i ng per formed by Rosemary E l l i s and Betsy Owczarski , res pec t i ve l y. The

r ev ie w by V eers was p a r t i c u l a r l y i n f o r m a t i ve and c r i t i c a l i n p re p ar i ng t h i s

document.

Th i s work was funded by the U.S. Department of Energy under Contract

DE-AC06-76RLO 1830. The Pa ci f i c Northwest Labora tory i s opera ted f o r the U.S.

Depar tment o f Energy by B a t t e l l e Memor ia l I n st i t u t e.

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CONTENTS

SUMMARY .

NOMENCLATURE . .- .

1.0 INTRODUCTION .

2.0 BASICS OF SPECTRAL SIMULATION OF WIND I N A ROTATIONAL FRAME .

2.1 SIMULATION OF A STOCHASTIC PROCESS GIVEN ITS SPECTRUM .

2.1.1 The Harmon ic Method

2.1.2 The Whi te 11o ise Method .2.2 SIMULATION OF ROTATIONAL DATA .

3.0 REVIEW OF THE THREE METHODS .3 .1 THE PURDUE METHOD .

3.2 THE SANDIA METHOD .

3.2.1 Geometry.

3 .2 .2 E u l e r i a n Power Spe ct ra .

3.2.3 Cross-Spect ra Between Sampl ing Po in ts .

3.3 THE PNL METHOD.

3 . 3. 1 Ro t a t i o n a l T h e or y .

3 .3 .2 S imu la t i on .

4.0 COMPARISON OF SANDIA AND PNL METHODS4 . 1 INPUT PARAMETERS

4.1 .1 Cor respondence o f Length Sca les and CoherenceDecay Parameters .

4.2 COMPARISON OF PARTICULAR CASES.

5.0 CONCLUSIONS AND RECOMMENDATIONS -REFERENCES.

APPENDIX A - MATHEMATICAL DETAIL OF SIMULATION METHODS .

APPENDIX B - THE CONCEPT OF SPECTRAL DENSITY AS PROBABILITY DENSITY .

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FIGURES

Layout o f S imu la t ion Po in ts on Hor izon ta l-Axis WindTurb ine Rota t ion Disk

.Comparison o f u-Component by Kaimal and by Fros t .

Geometry o f r i n E qu at io n (3.17) .

Comparison of Spectra of Simulated Data - Sandia Modeland PHL Model, 4 Values o f Coherence .

Ratio s o f Harmonic Spi ke Variances Cal cul ated Usingth e Sandia Model t o 'C orr ec te d' Variances f rom the PNLMode1Comparison of Spectra of Simulated Data

-Sandia Model

Using Frost Spectrum and PNL Model.

TABLES

Variances i n t he Harmonic Spikes .

Rat ios o f Harmonic Spi ke Variances Calcu lated Usingth e Sandia Model t o ' Co rre ct ed ' Variances from th e PNLModel

v i i i

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COHDEC

2Cross s p e c t r d l am p l i t ude o f ( u j , u k ) a t f r equency n , (m/s) /HzFou r ie r wave ampl i tude a t harmon ic m y (m/s ) , Appendix AArctang ent , an angle whose tangent i s

......

?

Complex Fou r ie r c oe f f i c i e n t f o r har monic ni, Appendix A2

Cospectrum o f ( u j , u k ) a t f re qu en cy n , ( rn ls ) /Hz2

Complex cross- spectrum of [ u j , u k ) a t f r equency n , (m/s) /HzSquared coherence a t f requency n

Code name f o r coherence decay fa ct or

Code name f o r coh erence decay f a c t o r , same as COHDEC

Code name f o r ro ta t i on a l f requency, Hz

Ord in al number o f members o f a t im e se r i es , Appendix A

I n t e g r a l s c a l e o f u-component tu rbu lence i n a longwind d i re c t i on

I n t e g r a l s c a l e o f u-component t u r bu lence i n c r ossw ind d i r e c t i o n

Harmonic number, Appendix A

Cycl ic f requency, HzPac i f i c No r t hwes t Labo r a t o r y

Qua dra ture spectum of ( u u k ) a t f re q ue nc y nj 'Rad iu s o f r o t a t i o n , m

2 2Spec t r a l dens i t y a t f requency n , (m/s ) /Hz o r m /s ; i n t e g r a t e sf rom ze ro t o i n f i n i t y t o y i e l d t h e v ar ia nc e

Two-si de d s p e c t r a l d e n s i t y a t fr eq ue nc y n , h a l f o f S ( n ) f o r2

p o s i t i v e n , (m/s) /Hz; i n t e g r a t e s from mi nus i n f i n i t y t o p l u si n f i n i t y t o y i e l d t he va ri an ce

To t a l t im e span o f a t im e se r i es o r wind r eco r d , s

Mean wind speed, m/s

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The alon gwin d component of tu rb ul en ce , m/s; general ized namef o r a t ime ser ies , Append ix A.

zo Surface roughness 1ength, m1 P ,2P ,3P,. .

.

Rota t ion a l f requency, second, th i rd , ...

harmonics

A t D i s c r e t i z a t i o n o r sa mp li ng i n t e r v a l f o r a ti m e s e r i e s

rms o f u-component turbulence, m/sFrequency i n r a d i a n s l s

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1.0 INTRODUCTION

Computational analysis of wind turbine responses t o winds using an aerody-t

namic model often requires input of wind speed fluctuation in the form of time

series. If s elective variation of wind input charac ter ist ics i s required in

the investigation of wind turbine response t o the wind, there i s an obvious

advantage t o using mathenlatically simulated wind data whose character may be

varied readily. The al ternat ive t o mathematical simulation is costly measure-

ments from multianemometer arrays under the not-made-to-order conditions pro-

vided by nature.

The principal purpose of this report i s t o present an evaluation of wind

simulations now under consideration in the modeling of wind turbine response

t o the wind. This report a lso includes basic discussion of spectral simulation.

The Pacific Northwest Laboratory ( PNL ) has developed a simulation of wind

speeds that mimics the fluctuations in speed encountered around the disk of

wind turbine blade rotation (Connell 1982). Two simulations other than the PNL

simulation of the ro tationally sampled wind speed have been considered for use

in the U.S. Department of Energy's wind turbine design program: one developed

a t Purdue University (Sundar and Sullivan 1983) and one developed a t Sandia

National Laboratories (Veers 1984). The three wind simulations, designated as

the Purdue method, the Sandia method, and the PNL method, are described in

this report.

In these methods, simulation involves a transformation from frequency spec-

t ra t o wind speed varia tions as a function of time. The required starting

information for the simulations i s a spectrum of horizontal wind speed. Simula-

t i on consis ts of incorporating the spectral information and phase assumptions

into a wind-speed time series.

The PNL method explici tly includes a simulation of wind speed in a rotating

frame of reference, as would be experienced by a point on the blade of a

horizontal-axis wind turbine. The Purdue and Sandia methods. bo t h generate

Eulerian, or fixed-point, time se rie s a t many points in the y-z plane (the

plane of horizontal -axis wind turbine blade rotation). The Purdue and Sandia

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methods obtain ro ta tional time se r ies by a simple rot at ional sampl ing rou tine

applied to the simulated field of Eulerian wind speed. A11 three methods usesome spectral model of the wind speed as the basis for simulation. 1

The second sect ion of th i s report cons ist s of some basic remarks about

wind speed time se r ies simulation from a spec tral basi s. This sect ion can be

readily bypassed by those more interested in the resulting simulations and

their evaluation. The t hi rd sect ion i s a comparative review of a l l three

methods. The fo ur th sec tion compares output of the Sandia and PNL models. The

f i f t h sect ion contains concluding remarks and recommendations.

One res ervat ion ap pli es to al l of the methods described in th i s re por t.

Turbulence i s not exactly a s e t of fluc tua tions whose magnitudes are d i s t r i -

buted in a normal, or Gaussian, manner. I t i s now believed t o consist usually

of in te rm it te nt elements of intense turbulence embedded in longer periods of

re la ti ve ly quiescien t flow. This i s sometimes ca lled the Gaussian patch hypo-

thesis (panofsky and Dutton 1984). A satisfactory simulation from this pointof view i s one th at would incorporate the patchiness feat ure. To incorporate

patchiness , one must know from real data anal ys is the proportions of the time

series in each mode and one must be able to model the variance in each mode as

a funct ion of the more basic parameters, such as the mean wind speed, atmos-

pheric s t ab i l i t y , and surface roughness. Mechanically, t hi s type of simulationappears to be not too difficult, provided that one knows the required propor -

tions for such a model. B u t this information will have to come in the future.

Therefore the simulation methods described in this report are simple Gaussian.

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2.0 BASICS OF SPECTRAL SIMULATIONOF WIND IN A ROTATIONAL FRAME

i The general pr inci ples involved in th i s study may be separa ted i nt o two

problems. One i s th at of simulation of time se ri es of wind speed fl uc tuat ionsfor a stationary, stochastic process using the continuous wind speed variation

frequency spectrum. The other i s that of construc ting the spectral information

such tha t a time se ri es of horizontal wind speed a t a point ro ta ting a t con-

stant angular velocity in a circle in a vertical plane perpendicular to the

direction of the mean wind may be simulated. The path i s tha t of a point on

the blade of a horizontal-axis wind turbine. These problems are discussed

briefly in the next two subsections. They are treated separately and in

greater detail in Appendix A.

2.1 SIMULATION OF A STOCHASTIC PROCESS GIVEN ITS SPECTRUM

Two qui te di fferent methods of simulating a s to cha st ic process may be

encountered in the literature. For brevity they are called the harmonic method

and the white noise method. The former i s the ea sier one to work with, and,

from our point of view, the more advantageous.

2.1 .1 The Harmonic Method

Any time series of evenly spaced di screte data may be converted i nto aFourier series of discrete harmonics. If the origin al data s e t , which we shall

assume to be real, i . e . , without imaginary parts, has N points (degrees offreedom), the Fourier series will have N/2 harmonics, each with an amp1 itudedesignation and a phase designat ion such that when the N/2 harmonics and themean value ar e summed, the or igi na l data ar e ret rieved . In the Fourier se ri es

the original N degrees of freedom i n the data are converted to N/2 amplitudeand N/2 phase degrees of freedom. ( N i s assumed to be an even in te ge r. )

The power spectrum of the data i s formed by squaring the amplitudes andmultiplying by twice the total running time for the data. When t h i s power

spectrum i s used as a basi s fo r simulating the origi nal data, the N/2 degreesof freedom that are in the phase part are not included in the spectral informa-

tion and must be resupplied by some assumption. The usual assumption i s that

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the phase i s an uncorrelated random vari abl e from harmonic to harmonic. If

so , the r es ul ti ng simulated time se r ies wil l be of approximately Gaussian di s-

tributi on. I t will not be identical with the original se ri es , b u t will have

the same st at is ti c al cha rac ter ist ics i f the original s er ie s was a1so Gaussian. I

2.1.2 The White Noise Method

The power spectrum of the origina l data i s als o the basi s of the white

noise simulation. The spectrum i s converted t o a t ra ns fe r function by taking

the square root. (This conversion can be performed i f i t i s assumed that the

time length of the data sample i s uni ty , which can be done without l oss of

general i ty . ) The Fourier transform of the t rans fe r function i s then taken,

which yields the impulse response function. The final data are then created

by convolution integration of the impulse response function with Gaussian whitenoise . All t he degrees of freedom in the simulation are provided by the

Gaussian white noise. References are Bendat and Piers01 (1971) and Frost andMoulden (1977).

2.2 SIMULATION OF ROTATIONAL DATA

Two very d i f feren t methods for simulating ro ta tional da ta , the PNL and

Sandia methods, wi 11 be reviewed in more de ta i l . Connell (1982) derived anautocorrelation function f or wind speed a t a r ota ting point (s ee Section 3.0

of t hi s repo rt ). The autocorrel ation function was transformed in to the corre-

sponding spectrum of cycl i c frequencies of fluctuat ion. Therefore, the' PNL

method uses the spec tral transformation of th at autoco rre lat ion function. The

PNL simulation provides the phases of the cycl ic components of the simulated

time series by exercising a random number generator from which an even, random

distribution of phase angles between 0 and 27~resu l t s . This assumption doesnot neces sar ily produce the best des cri ption of real winds. The PNL simula-

tion only computes winds for the si ng le point th at i s revolving around a

ci rc le .

The Sandia and Purdue methods are quite different from the PNL method in

that they must compute wind values for many points throughout the plane within

the rotor disk a t each time in te rv al . Because the blade rotates in the y-zplane, the in tent of these methods i s f i r s t to generate suff ic ie nt spectral

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information that the wind field in the plane as a function of ( t , y , z ) is gener-ated. Following this relatively difficult step, the simulated rotationally

sampled data are easily selected from the whole field..

The three methods depend on numerical engineering that should produceuseful r es ul ts in sp i te of objections tha t may be rai sed on theoret ical grounds.

Because the scale of turbulence increases with height in the atmospheric bound-

ary laye r, real turbulence i s not homogeneous in a ve rt ic al plane. Therefore,

spectral description of turbulence in such a rotational plane involves diffi -

cul t ie s th at have not received de fi ni ti ve treatment in any of the models

reviewed.

The basic mate ri als that the Sandia and Purdue methods draw upon in di f-

fe rent ways are the simulation functions of several var iab les (and mu1 t i variateprocesses) by Shinozuka (see Shinozuka 1971 and Shinozuka and Jan 1972). The

work of of Shinozuka i s d i f f i cu l t t o comprehend because of certain greater

genera lit y in h is assumptions than i s required fo r present purposes. Further

discussion of the Shinozuka approach i s given in Appendix B. The direct des-

cr ip ti on and review of the thr ee simulation methods i s given in the next

section.

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3.0 REVIEW OF THE THREE METHODS

I n thi s section, three methods fo r simulating the rotational ly sampled

horizontal wind speed time series flowing parallel t o the axis of rotation of

a wind turbine are examined in some detail. These methods were developed by

1. R. M. Sundar and J . P . Sullivan of Purdue University (Purdue method)

2 . P . S. Veers of Sandia National Laboratories (Sandia method)

3 . J . R. Connell and D . C . Powell of the Paci fic Northwest Laboratory ( P N Lmethod).

All three use the harmonic method rather than the Gaussian white noise method

for simulating time series.

3.1 THE PURDUE METHOD

The method reviewed here i s as given by Sundar and Sullivan (1983). The

Purdue method, like the Sandia method, generates many time series of wind each

a t a specified location on the y-z plane, which i s the plane perpendicular t o

the assumed mean wind direction. That i s , each generates a wind field u ( t , y , z ) .The Sandia and the Purdue methods pick o u t rotational series from the total

wind field. Both methods are ult imately derived from previous work of

Shinozuka (1971) and Shinozuka and Jan (1972).

The Purdue model differs from,the Sandia model in that a three-dimensionalspectrum is required. Since none i s avai lable in the meteorological l i terature,

certain physical assumptions must be implied in order t o arrive at an assumed

form for the three-dimensional spectrum. The form given in the Purdue model

implies t h a t turbulence in the lateral direction and turbulence in the vertical

direction depend on turbulence in the longitudinal direction b u t are independent

of each other. This appears t o be a reasonable engineering assumption provided

that the scales of turbulence in the two late ral direct ions are not too differ-

ent, which holds in the model as parameterized.

There are two other aspects of the paper describing the model t h a t are dif-

f icu l t t o understand. The f i r s t equation from the paper i s

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where

'm. 33 = t h e mean win d speed i n m/s a t a h e i gh t o f 10 m (33 f t )CT = a ground roughness factor (not the surface roughness length zo)X = 4000 n/V i n m - l , a wave number.

The three-dimensional sp ect ra l equat ion, Sundar and Su l l i v an ' s Equat ion (4 ) ,

i s :

where a and B a re f u n c ti o n s of c o r r e l a t i o n i n t h e y and z d i r e c t i o n s . w i se v i d e n t l y 2 ~ n .

I n order t o use Equat ion (3.2) the parameters a and B must be determined.The development inv olv ed can be best t rac ed by st a r t in g w i t h Equat ion ( 2 ) i n

the Sundar and Su l l iv an paper , which def ines the cor re la t i on funct ion. Th is

e qu at i on i s w r i t t e n :

Cor = exp[-c n L/V]

by which the authors must mean

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where

L = the argument , the spat ia l l e n g t h t o wh ic h t he s p a t i a l c o r r e l a -t i o n i s a pp li ed

i = 1, 2, 3 f o r 1ongi t ud i na l , l a t e r a l , o r v e r t i c a l c o r r e l a t i ond i r e c t i o n s

n = spect ra l f requency

V = mean wind speed

C ( i ) = a d imens ionless constan t t o be dete rmined f o r each d i re c t io n .I f the tu rbu lence i s .assumed t o be i s o t ro p i c , the C ( i ) f o r two l a t e r a l d i r e c-t i o ns w i l l s t i l l d i f f e r from t h a t i n t h e l o n gi t u d i n a l di r e ct i o n.

The f or m on t h e r i g h t s i d e o f t h i s e q ua ti on i s d i f f i c u l t t o under stand.There i s no spect ra l f requency parameter o r sp ect r a l wave number paramete r i n

a c o r r e l a t i o n f u n c t i o n . A r ea so na bl e f or m f o r t h i s s p a t i a l c o r r e l a t i o n i s

C o r (L ) 1 ( i ) = ex p - L / L e n g t h ( i ) ( 3 .5 )

C o r ( L ) ] ( i ) = ex p - C ( i ) L / L e n g t h . (3 .6 )

I n Equat ion (3 .6) we assume t h a t th e Length i s a measured c or r e l a t io n leng th

and tha t C ( i ) i s a necessary ad j u s t i v e cons tan t . I n Equ atio n (3 .5) we assumetha t the ad jus tment has been incorpora ted in to ' L e n g t h ( i ) I . Equation (3.4)would be formal ly acceptab le if t h e n had a p a r t i c u l a r v a lu e, s ay t h a t a t

wh ich maximum energy occurs i n the spect rum, wh ich , i n c i de n t a l l y , i s t he

unknown we are after.

A

second source o f con fus ion i s the equa t ion g iven fo ra,

which reads:

a = C / ( 2 ~ r v ) (p er m) . ( 3 .7 )

Th i s equa t i on i mp l i es t ha t C must c ar ry a d imension o f p er second.

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Because of these d i f f icu l t ies and the large amount of computer t ime appar-

en t l y requ i red t o run the model , no fu r t he r comparison o f t he s imu lated re su l t s

wi t h s i mula t ion s from oth er models i s made.

3.2 THE SANDIA METHOD

3.2.1 Geometry

The Sandia method i s developed i n pol ar coord inates, the na tur a l system

fo r a r o ta t i n g bl ad e. A t ime se r ies u ( t , r , 8 ) i s s imulated, where r and 0 arethe usua l po la r coord ina tes : r and 8 a r e i n t h e y-z p lane , the ro t a t i ona l

p la ne o f a h o r i z o n ta l- ax is w ind tu rb ine . To se lec t a ro t a t i ona l t ime se r ies

f r om u , i t i s s u f f i c i e n t t o s i mu l ate u a t many d i f f e r e n t a n gl es f r om th eo r i g i n i n one r a d i a l c i r c l e . Ac tu al ly th e model can s imulat e u ( t , r , 0 ) i n asmany as th ree concent r ic rad ia l c i rc les . (The code i s ea s i ly mod i fied so t ha t

the s imu la t i on can be a r b i t r a r i l y made i n many rad i a l c i r c le s . ) The configu-

r a t i o n i s shown i n F i gu r e 1.

The for mal d if f er en ce between t he Sandia method and the Purdue method i s

t ha t Sandia replaces t he requirement f or a three-dimensional spectrum w i t h a

m a t r i x o f o n e-d imensional spec t ra where t he pr i nc ip a l d iagona l o f the matr ix

con sis ts o f t he power spectrum a t each observat ion po in t and the of f -d i . agona1elements are the cross- spec t ra app l icab l e between the po in ts . From th i s i n f o r -

mation u ( t ,r ,9 ) may be simu lat ed by inv ers e Four ier tra nsfo rmat ion , and one ormore ro t at i on al se r i es may be pick ed out, which the program does. The method

de r i ves from the work of Shinozuka ( 1971 ) , Shinozuka and Jan ( 1972 ) , andSmall wood (1982) .

3.2.2 Eu le ri an Power Spectra

The code calculates an Euler ian u-component power spectrum t o app ly a t

each sampling point on Figure 1. Because th i s spectrum i s a funct io n of hei ght

z and mean wind speed U ( z ) , the spect rum i s d i f fe re n t a t d i f fe ren t sampl ingpo in ts . Le t us assume t h a t we wish t o sim ulat e a ser ie s of N data a t each o f

M azimuths from th e wind tu rb in e hub and a t IVRAD radia l d is tances from the hub.A ro ta t io n ra te , ROT, and a sampl ing i n t er va l , DTy are given , where DT i s thet ime i t takes th e b lade t o pass from one az imuth t o the next. Thus,

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FIGURE 1. Layout o f S imu la t ion Po in ts on Hor izon ta l - Ax isWind Turb ine Ro tat ion Disk

DT = ~ / (N *ROT )(seconds) . ( 3 . 8 )

The spec t ra l dens i ty func t ions S(k , j m) are generated, where k var i es w i t hazimuth, j v a r i e s w i t h r a d i a l d i s ta n c e, and m va r i es w i t h spec t ra l ha rmon i cnumber:

k = l , Nj = 1, NRADm = 1, M, M = NDATA/2.

Each m cor responds t o a c y c l i c f requency:

n = m/(NDATA * DT), m = 1, M ( 3 . 9 )Note t h a t DT i s t he sampl ing i n te rv a l t h a t de te rm ines the maximum computablef requency fo r the spec t ra .

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The Sandia model uses a u-component power spectrum from Frost e t a l . (1978,

page 4.1). We wi 11 cal l t h i s the Frost spectrum. When th i s spectrum i s com-puted with the Kainial u-component spectrum for neutral stability (Kaimal eta l . 1972 ) , which i s considered st a te of the ar t by meteorologists, the re su lt 4i s as shown in Figure 2.

0CY-'0,

\

10.'

1 0

K a ~ m a l S ( n )Neut ra lt 0 Frost nS(n) I

FIGURE 2 . Comparison of u-Component by Kaimal and by Frost

The Frost spectrum matches the Kaimal spectrum a t low frequency. However,

the im ort an t frequency range for rot ati ona l sampling i s the higher f requen-c i e s ( a ' where the Frost spectrum yields values that are about 2.5 times thoseof the Kaimal spectrum.

( a ) I t may be added th a t th ere i s gre at er accuracy in the higher frequencypart of any state-of- the-ar t meteorological spectrum than there i s inthe lower frequency pa rt . Therefore, a user-created mathematical modelof boundary layer turbulence power spectra should match the recognizedstate-of- the-ar t spectrum on the high-frequency slope in preference tothe low-frequency slope when both cannot be conveniently be matched.

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The spec t ra l dens i t y f unc t i ons f o r t he 14 * NRAD po in ts p rovi de t he pr i n c i -pa l d iagona l o f t he spec t ra l ma t r i x . The c ross- spectra between the locat ions

prov ide the o the r members o f the mat r i x .

3.2 .3 Cross-

Spectra Between Sampling Points

To d e r i v e t h e f i n a l c o r r e l a t e d v e l o c i t i e s u ( t , r , 0 ) , t he re l a tedness o f t hewinds a t t h e d i f f e r e n t p o i n t s i n F ig ur e 1 a re expressed i n a c ross- spec t ra l

m a t r i x . The number o f columns and rows i n the ma tr ix i s t he produ ct of the

number o f az imuth ang les mu l t i p l i e d by the number o f r a d i i , i\l x WRAD. Le t usc a l l t h i s p roduct NN. Then the cross - spec t ra l ampl i tude ma t r i x i s

The Ajk a re g i ven by

where

I f Ajk rep resen ts the separa t ion d is tance be tween loca t ion j and loca t ion k ont h e p la n e o f r o t a t i o n , t h e coherenc e i s g i v e n by

coh jk(n) = exp - n ajk/u]where A i s sepa ra t i on d i s tance between p o i n t s j and k, and where D sho uld bej ki n t h e 10 t o 15 r ange f o r coherence ove r a l a t e r a l sepa ra t i on d i s tance . (See

Kristensen 1979.) On the d iagonal o f the mat r i x A, the coherence i s 1 and theA 's a re t he spec t ra l dens i t i e s S . The ampl i tudes a re then conver ted t o a t rans -

f e r f u n c t i o n m a t r i x , H, t h a t i s l ow er t r i a n g u l a r such t h a t

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Twhere H (n) i s the t ranspose of H(n) . Note th a t t he re i s a separa te H m a t r i x .Lfor each frequency n.

The next step i s t o form random phase desi gnat io ns e ~ p ( i @ ( n ) ~ )i n t o t h ed iagona l o f a mat r i x X where a l l of the diagonal elements are zero. Elements

o f t he p roduc t mat r i x , HX , ar e of th e form

where 0 i s an evenly di st r i b ut e d random va r i ab le between zero and 2n. Thef i na l v e l o c i t y v ec t or f o r each l oc a t i on ( r y e ) i s t hen c a l c u l a t ed by

V i ( t r ,0) = Inverse Four ier Transform , (3.15)

where (1) re fe rs t o a column vecto r of ones. From the complete f i e l d o fV ( t ,r ,0), t he ro ta t iona l se r ies a re eas i l y se lec ted .

The cor re la t ion o f t he separate ve loc i t y vec tors V ( t ) wi th each otherreduces with increased separat ion distance, because the cross

-

cor re la t ionsdepend on the coherence speci f ied, which var ies inversely wi th separat ion dis -

tance. For th e same reason, t he cr os s- cor re la t ion i s h ighes t a t l ow f requen-

c ies . Th is can be seen i n th e low phase differenc es between phase de si gn ati on s

a t the same frequency n when n i s small f o r two cor re la ted ve loc i t y vec to rs .

The Sandia model i s the onl y one of the thr ee reviewed here t o take a t

le as t p ar t i a l account of ve r t i c a l i nhomogenei ty o f t urbulence . Th is i s done

by using a height-dependent sp ec if ic at io n of th e power spectum a t each sampling

p o i n t i n t h e s im ul at io n.

3.3 THE PNL METHOD

The PNL method f i r s t computes a ro t a t i o n a l spectrum and uses th e harmonic

simu lat ion method t o s imulate one t ime ser ies . No extraneous data ( f o r unused

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l o c a t i o n s i n t h e d i s k ) a r e e i t h e r c omputed i n t h e s p e c t r a l domain o r s i mu l at e d

i n the t ime domain. A s i n g l e t i m e s e r i e s i s computed f o r a s i n g l e s e l e c t a b l e

r a d iu s o f r o t a t i o n ..

The PNL model has been extended t o apply t o d i f f er en t wind components i n

r e l a t i o n t o t he v e r t i c a l - a x i s t u r b i n e . A l s o , the s imu la ted t ime ser ies f romth e PNL model in cl ud e th e mean wind eff ec t.

3 .3 .1 Rot at i ona l Theory

The ro ta t i ona l t heo ry has been p rev i ous l y repo r ted i n de ta i l (Connel l 1982

and Powe l l e t a l . 1985) . On ly th e b r i e f es t summary o f mate r ia l f rom these

r e p o r t s w i l l b e g iv e n he re .

T h e b a s i c s t a t i s t i c a l t h e o r y f o r t u r b u l e n c e a s e x p e r i e n c e d b y a p o i n t

mov ing i n a ve r t i c a l p l ane pe rpend i cu l a r t o t he mean w ind d i r e c t i o n i n t he w i nd

tu rb ine layer has been g iven by Conne l l (1982) . The au toco r re l a t i on f unc t i on

o f u-component t u rbu l ence i n t he pa th of t he ro ta t i ng p o i n t i s g i ven by

where:

I = the gamma functiono2 = t he t u rbu l ence va r i ancer = the separa t ion d is tance , as descr ibed be low

L = a l e n g t h c h a r a c t e r i s t i c of t h e t ur b ul e nc e

K = the mod i f i ed Bessel fun ct io n o f the second k i nd o f f r ac t i on a lo rde r 113

x = the alongwind component of th e sepa rat ion distan ce.

To phy s i ca l l y unde rstand t he r, one must v - i s u a l i z e the two ends o f the seg-

ment de f in ing r as fo l l ows. The one end i s f i x e d t o an a r b i t r a r y p o i n t on t h e

b l ad e and t h us de s cr i be s a c i r c u l a r p a t h i n t h e d i s k of r o t a t i o n , p a r a l l e l t o

the mean wind d i re ct io n, as the b lad e ro t a te s. The ot he r end i s downstream of

i t s o r ig i n a l p o s i t i o n on t h e b la d e a t a d i s ta n c e x, o r U t where U i s mean w ind

speed and t i s t r a v e l t i m e . M a th e ma t ic a ll y t h i s i s ex pr es se d b y

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where p i s th e radius of rot ati on of the one end and R i s the ro tat ion ra te*

(see Connel1

1982). The two terms express the rotational and translation

e f f e c t s , re sp ec ti ve ly , as shown in Figure 3. The second term, ( u t 1 2 , dominatesthe f i r s t , [2p s i n ( ~ / 2 ) ] ~ ,as t becomes large.

Po is position of P at initial timent = 9 + 2 n nwhere n i s nurnber of revolutions in time t

FIGURE 3. Geometry of r in Equation (3.17)

I t i s eas il y seen t ha t t he correla tion function of Equation (3.16) may be

transformed into one w i t h time fo r t he argument i f we su bs ti tu te from Equation

(3.17) int o Equation (3.16) and furt her s ub st it ut e U t f o r x i n Equation (3.16).

These steps not only yield a time correlation function, b u t also incorpor-

ate the rota t ional feature by specifying r according to the rotational formula

in Equation (3.17). The rota tio nal fe at ur e i s passed onto the spectrum in thetransformation:

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The pr ob le m t r a n s f o r m a t io n co mp ut at io n i s f a c i l i t a t e d b y d e f i n i n g t h e f o l -

l ow ing d imens ion less var iab les . The f i r s t two r e l a t e tu rbu lence and machine

p r op e rt i es ; t h e l a s t i s t i me- r e l at e d t o b l ad e r o t a t i o n r a t e :

where

a = a r a d i a l d i s ta n c e f r om t h e c e n te r o f r o t a t i o n d i v i d e d by t h e

l a t e r a l s ca le o f u-

component turbulence; t h i s a i s n o t r e la te d t oa i n Equat i on (3.7)

U = the mean wind speedR = b l ad e r o t a t i o n r a t e i n r a d ia n s p e r s econd

B = t he pe r i o d o f r o t a t i on made d imension less by d i v i d i ng by t hep e r i o d U(2n /LX)-

T = t he c o r re l a t i on t ime made d imens ion less by mu l t i p l y i n g by t her o t a t i o n f re qu en cy i n r a di a ns .

The parameters L and L are two len g th sca les , i n p lace of the one th a tY

i s i n E q u at io n (3.16). T h i s i s a c t u a l l y a m o d i f i c at i o n of t h e th e or y o f

Equat ion ( 3 . 1 6 ) , m o t iv a t ed b y t h e f a c t t h a t t h e l e n g t h s c a le i n a i s a ss oc ia te dw i t h t h e ra d iu s, w hic h ex te nd s i n a l l l a t e r a l d i r e c t i o n s, w h i l e t h e l e n g t h

s ca l e i n f3 i s assoc ia ted w i t h th e lo ng i t ud in a l mean wind speed. Thus, i t i shypothes ized t h a t more than one len gt h sca le may be ph ys ic a l ly r e le van t .

We n o t e t h a t

Sub s t i t u t i on of Equa t ions ( 3 . 1 9 ) , ( 3 . 2 0 ) , and (3.21) i n t o (3 .16) y i e l ds

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where

The function p(?) i s a corre lat ion in the dimensionless time var iable ? interms of the dimensionless parameters a and B.

The co rr el ati on func tion may be transformed int o the spectrum S(n) asfollows:

where

3.3.2 Simulation

The harmonic method was used to simulate real-time data from the rota-

ti onal spectrum. The Gaussian white noise method was f i r s t used. On the aver-

age only 79% of the variance in the f i r s t fi ve harmonic spikes ( l P , 2P, 3P, 4P,5P) of the spectrum in Equation (3.24) was recaptured in the spectrum of the

corresponding spikes of the simulated data. Therefore, the method was dropped

in favor of the harmonic method.

In the next section we compare the spect ra of simulated time se ri es with

the spectra used as inputs t o the simulation.

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4.0 COMPARISON OF SANDIA AND PNL METHODS

In th i s sect ion we compare rot at io na l s pec tr a of simulated wind speed time

series data from the Sandia method and the PNL method. The Sandia model was

formed around the Kaimal spectrum; the PNL model about the von Karman spectrum.

However, this difference i s not expected to contribute significant differencesto the resu l t s .

4.1 INPUT PARAMETERS

The methods ar e not s t r i c t l y comparable because of th e di ff er en t physics

in the input parameters. The input parameters required to run either model are

hu b height

surface roughness length

mean wind speed a t hub height

blade rotation rate

radius of rotation of a point on the blade

sampling interval

number of data i n one time series.

The input parameters pecu li ar t o each model ar e

Sandia model - coherence decay factor for u-

component turbulencecoherence in lateral directions

PNL model - longitudinal length scale-- integral scale of u-componentturbulence in longitudinal direction Llateral length scale-- integral scale of u-componentturbulence in rotation plane LY '

The requirement fo r a longitud inal length sc ale ( L x ) comes from the au to -correlation function (see Equation ( 3 . 1 6 ) ) , which i s the bas is of the rot a-tional simulation. A von Karman autocorrelation function (von Karman 1948)resu l t s i f the ro ta t iona l fea ture i s shu t o f f . The length scale L, may betaken to be about 4 times the hub height. I f so, th is i s equivalent to using

z for a length scale, which the Kaimal and Frost spectra do directly. The

other length scale in the PNL model i s for the la te ral directi on ( L ) . Physi -Yc a l l y , this requirement corresponds to that for a lateral coherence decay

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factor in the Sandia model. In principle, the lateral length scale and the

la te ra l coherence decay fa ctor vary inve rse ly. The PNL model may also read in

a turbulence variance as part of the input. Turbulence variance i s obtained

for the Sandia model by integrating the spectrum.

Powell e t a1. (1985) report that variances in the harmonic spikes calcu-lated by the PNL model matched observed resu lts quite well when the la tera l

length scale was used in both roles.

4.1.1 Correspondence of Length Scales and Coherence Decay Parameters

The coherence decay fac to r i s used in conjunction with the assumption that

coherence i s an exponential function of frequency. Such an expression was

wri tten by Davenport (1961) for coherence in the x-

direction

where D i s the coherence decay fac to r, n i s frequency in Hz, Ax is separation

distance, and U i s the mean wind speed. D i s presumably dif fe re nt fo r each of

u , v , and w. Note that coherence decreases as D increases.

In fa c t , Equation (4.1) i s an empirical expression. There i s no the ore ti-

cal ju st if ic at io n fo r the exponential formulation. Therefore, the re1 ation sassumed between D and turbulence parameters may not be well founded.

Longitudinal coherence has been formulated with considerable sophistica-

tion (see Kristensen 1979). However, a good approximation holds that the decay

factor i s proportional to turbulence i nte ns it y, ou/U. Specifically, for theu-component the decay fact or i s about 15 times the i nt en sit y (s ee Chan e t a1.1983 and Powell 1974). For neutral s t ab i l i ty the re lat ion of i nt en si ty to

height may be given by

where zo i s the surface roughness length. Thus D is proportional t o thei nverse of 1n(z/zo) for 1ogari thmi c profi le wi nds.

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There i s no sat is fa ct or y power relat ion of inte gral scale s fo r the h o r i -zontal components to height. Panofsky and Du t t o n (1984) recommend againstusing the parameter altogether and state that results of different investi -

gators on th is subject are widely scattere d. Counihan (1975) gives a graphical

formulation of L,, which i s impossible to reduce to a power or logarithm of z.If one considers state-of- the-art spectra only, i t would appear th at turbulence

length scale s are lin ea r functions of height b u t best efforts at observation

show that the increase of L, and L with height i s sig nif ica ntl y le ss than theY

first power.

Therefore , we know of no way from the l i t e r a tu re to compare one turbulence

length L, with the corresponding coherence decay parameter. Panofsky andDutton (1984, p. 218) give a relation between L and coherence. B u t the i n t e -

Ygrat ion i s weighted so heavily on S ( n ) a t low frequenc ies as to be use lessunless one ar bi tr ar il y se ts a lower frequency lim it of integ ratio n other than

zero.

The Sandia and PNL models are derived from different modeling bases and

are not closely related theoretically. However, i t i s the output time s er ie s

and their properties that are most important to compare.

4.2 COMPARISON OF PARTICULAR CASES

The PNL model has already been shown to produce rotational spectra that

match qui te well those produced by cre ating rota tion al time seri es from a ve r t i-

cal plane array (VPA ) of anemometers a t Clayton, New Mexico, fo r the neu tra l

and unstable atmospheres (see Powel 1 e t a1. 1985). In the comparison be1 ow,one se t of input parameters i s used for the PNL model. These parameters are

Rotation ra te 0.6667 Iiz Mean wind speed 8 m/sRadius of ro ta ti on 20 m Surface roughness 0.01 m

Hub height 30 m Lx 120 mSampl ing in te rv al 0.125 s L~ 48 niSample length 1024 data

The spectrum of simulated data generated with these specifications was com-pared with s pect ra generated with the Sandia model when the coherence decay

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parameter was va r i ed f rom ru n t o run. The values used were 2 , 7.5, 20, and 50.

The ve rs ion o f th e Sandia code th a t was t ran sm i t t ed t o us by Sandia used 7.5

f o r coherence decay. Otherwise, th e parameters used i n the Sandia model were

the same as used i n th e PNL model , except fo r the le ng th s ca les t h a t are no t

par t o f the Sand ia mode l .

The resu l ts f rom the Sand ia mode l a re compared w i th the fou r resu l ts from

th e PNL model i n Fi gu re 4. The graphs are arranged i n order o f decreasing

coherence i n the Sandia model . These graphs clear ly show that the harmonicsp ikes become be t t e r def ine d i n th e Sandia ou tpu t as coherence decreases.

L ikewise , Powe l l e t a1. (1985) found that the harmonic spikes become betterd e fi n ed a s t h e l a t e r a l l e n g t h s c a l e, , decreases.L~

---- Sandia

COHDEC = 50

1o0N

I\"-Y 10-1E--C-V) 1 0 - 22 ~ 1 - ~

0.0 1 .O 2.0 3.0 4.0 0.0 1 O 2.0 3.0 4.0

FIGURE 4. Comparison of Spec tra of Sim ulat ed Data - SandiaModel and PNL Model, Fou r Values of Coherence

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The con t ras t between peak and va l l ey i s cons i derab ly g rea t e r i n the PNL

model spe ct r a than i n th e Sandia model spect ra , p a r t i c u l a r l y when the Sandia

model uses a l a t e r a l coherence decay f a c t o r i n t he range t h a t i s be l i eved t o

be co rre ct , say 10 t o 15. Al though we cannot be c e r t a i n which model i s more

n e a rl y c o r re c t , t h er e i s m at e r ia l i n t h e r e po r t by Powel 1 e t a1.

(1985) tha t

suppo rt s t he PNL model i n t h i s r espec t , bu t no t w i t h ou t qu a l i f i c a t i o n . On th e

negative side, when model-produced spectra were compared with observed spectra,

t h e d e f i n i t i o n o f t h e h a r m o n i c s p i k e s i n t h e t h e o r et i c a l spectrum i s s l i g h t l y

s t ronger than t h a t i n t he observed spectrum. On the po s i t i ve s ide, Connel l and

George (1984) s t a t e t h a t th e spectrum of measured f l a p w i s e bending moment cor -responds b et t er w i t h t h a t computed usi ng VPA ro t a t i o n a l l y sampled data when a

low pass f i l t e r i s used such th a t t he minimum between the harmonic sp ikes i s

l owe r . A l so on t he po s i t i ve s ide , t he f a l l o f f of t he peaks o f harmoni c sp i keswi th f requency more ne ar l y approaches the expected exponen t ia l f a l l o f f than i s

t he case w i t h t he f a l l o f f o f ha rmon ic peak sp i kes i n t he spec t ra f rom the

Sandia model. These peaks ma in ta in more constancy t han would be expected. I n

oth er words, the re appears t o be too much over tone st r eng th a t h i gh f requencies

i n the spectr a produced by th e Sandia model .

A no th er p e r s p e c t iv e i s p r o v i d e d by l o o k i n g a t t h e v a r i a nc e s c a l c u l a t e d b y

in tegrat ing a spectrum over each harmonic sp ike. These var iances f o r the one

PNL run and f o r t h e f o u r Sand ia runs a re l i s t e d i n Tab l e 1.

TABLE 1. Var iances i n t he Harmonic Spikes

Run Variances - Rotat ional Spectrum (m/s) 2T o t a l 1P 2P 3P 4P 5P COHDEC

PNL 0.8371 0.3592 0.1114 0.0547 0.0343 0.0244

VEK0004 0.5688 0.0487 0.0223 0.0152 0.0122 0.0099 2VEKOOOO 0.5457 0.0718 0.0425 0.0291 0.0253 0.0229 7.5

VEK0002 0.5323 0.0859 0.0648 0.0444 0.0430 0.0396 20VEK0003 0.5455 0.0942 0.0855 0.0619 0.0681 0.0608 50

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For comparison, these ratios were calculated between 'corrected' valuesof the PNL variances and the variances from the Sandia runs. The c~~r rec t i ngwas done by using the ra tios calculated (see Powell e t al . 1985) be.:ween themodeled harmonic spike variance and those observed for the quasi-neutral case

studied i n that report. These ratios were

Thus the corrected variances from the PNL run are

The ratios between the variances produced by the Sandia runs and theabove are given in Table 2 and in Figure 5.

TABLE 2. Ratios of Harmonic Spike Variances CalculatedUsing the Sandia Model t o 'Corrected" Variancesfrom the PNL Model

1P 2P 3P 4P 5P COHDEC

0.19 0.23 0.28 0.32 0.30 2.0

0.28 0.44 0.54 0.67 0.70 7.5

0.33 0.67 0.83 1.14 1.20 20.0

0.36 0.88 1.15 1.81 1.85 50.0

I t appears that the Sandia model may ser iously underestimate the energyin the 1P and 2P region, and that the variance estimates for the higher fre-quencies are reasonably good if a coherence decay parameter of abou-; 20 is used.

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Coherence Decay = 2.0- 0 Coherence Decay = 7.5Coherence Decay = 20.00Coherence Decay = 50.0-

---

I I0 1 2 3 4 5

H a r m o n ~ c1P, 2P, 3P. 4P. 5P

FIGURE 5. Ra ti os of Harmonic Spi ke Vari ances Ca lcu lat ed Using th eSandia Model t o ' Cor rec ted ' Variances from the PNL Model

As an af te rt ho ug ht , th e Sandia model was r u n once more with the Frost

spectrum and t he coher ence decay value of 7 .5 t h a t was provided by Veers. The

r e s u l t i sVariances (m l s ) ' (using Frost spectrum and COHDEC = 7.5)Tota l 1P 2P 3P 4P 5P

1.1013 0.1524 0.0888 0.0597 0.0523 0.0468

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Rat i os o f t hese va r iances t o co r rec ted PNL values are

The sp ec tr al comparison i s shown i n F igure 6.

I - PNL- - - Sandla

FIGURE 6. Comparison of Spectra of Simulated Data - SandiaModel Using Frost Spectrum and PNL Model

Th i s comparison i s obv io us ly bet te r than th at obta ined when the Kaimalspectrum, wh ich i s presumed t o be more accurate, i s used i n the Sandia model .Th is typ e o f di lemma can occur as lo ng as we do no t understand a l l the phys ics

t h a t should go i n t o these models. However, if we ar e t o have confidence i n

such models, i t seems th a t t he on l y d i re c t i o n approp r i a te t o t h e i r p resent

st at e o f development i s t o cont inue t o improve them by in co rp or at ing as much

o f th e r i g h t physics as we can, and hope a t l e as t f o r an eventual convergence

o f re su l t s fo r the r i g h t reasons when the output of d i f fe re nt models i s

compared.

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CONCLUSIONS AND RECOMMENDATIONS

Of the th ree models reviewed, two simulations of turbulence as seen by a

point on a rotating wind turbine blade have been found to be the most promising:

the Sandia and the PNL models. The Sandia model has the dist inct ion of simul-

taneously simulating rotati onal winds, as seen by two or more points on the

blade. The rota tiona l time se r ies in the Sandia model are se lected from simu-

la ti on of turbulence as a funct ion of time, r ad ia l dis tance from the hub, and

azimuth a t points in the ( r y e ) plane. The PNL model simulates one rotationalseries without simulating unused data, b u t does not yet include simultaneous

time seri es a t two or more poin ts.

The two models are not exact ly comparable in terms of basic physics

because they use different as well a s common input parameters. When compari-

sons of ro ta tiona lly sampled output time se ri es a re made, the r es ul ts a re si g-

ni fi ca nt ly di ff er en t and the Sandia model seems to be the le ss accurate. In

particular, the variance of the 1 P and 2P harmonic spikes of the spect ra of thesimulated r otationally sampled wind speed from the Sandia model seems def ic ient

Fur ther, the decrease of harmonic spike amplitude with increasing harmonic

number seems to be insufficient in the Sandia model.

An ideal model would be one that simulates required portions of a physi-

cally realizable flow field without using large amounts of computer time. Such

a model would generate r ea l i s t i ca l ly correla ted ro ta ti on al ly sampled wind

fluctu ation data a t two or more points having ch ar ac te ri st ic rotat ional spectra

a t two or more points without the generation of superfluous wind data.

Further development of simulations of the rotationally sampled wind speed

must concentrate on increased accuracy and on r ea l i s t i c phase and cor re la tion

rel ati onships between wind speeds a t di ff er en t points in the disk of the tur -

bine rotor.

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REFERENCES

Bendat, J . S . , and A. G . Piersol . 1971. Random Data: Analysis and MeasurementProcedures. Wiley- In terscience , New York, New York.

Chan, S. M . , D. Curtice, and S.-

K. Chang. 1983. Methods of Wind TurbineDynamic Analys is . AP-3259, Research Project 1977-1, Systems Control, Inc.,Palo A1 to , Cal i fornia .

Connell, J. R. 1982. "The Spectrum of Wind Speed Fluctuations Encountered bya Rotating Blade of a Wind Energy Conversion System. " Solar Energy 29:363-375.

Connell, J. R . , and R. L . George. 1983. "A New Look a t Turbulence Experiencedby a Rotating Wind Turbine." Presen ted a t Energy Sources TechnologyConference and Exhibition, January 30-February 3, 1983, Houston, Texas.American Society of Mechanical Engineers, New York, New York.

Counihan, J . 1975. "Review Paper-

Adiabatic Atmospheric Boundary Layers: AReview and Analysis of Data from the Period 1880-1972." Atmospheric Environ-ment 9:871-905.

Davenport, A. G. 1961. "Spectrum of Horizontal Gustiness near the Ground inHigh Winds." Qu ar te rl y Journal of th e Royal Meteorological Societ y 87:194-211.

Frost , W . , B. H . Long, and R . E . Turner. 1978. Engineering Handbook on th eAtmospheric Environmental Guidelines f o r Use i n Wind Turbine Generato rDevelopment. Final r epor t , NASA Con tract EX-76-A-29-1028, NASA TechnicalPaper 1359, llational Aeronautics and Space Administration, Washington, D. C.

Frost , W . , and T. H . Moulden, ed. 1977. Handbook of Turbulence. Plenum Pr ess,New York, New York.

Kaimal, J . C . , J . C. Wyngaard, Y . Izumi, and 0. R . Cote. 1972. "Spectral Char-ac te r i s t i c s of s u r f a c e - ~a y e rTurbulence." Qu ar te rl y Journal of' the RoyalMeteorological Society 98:563-589.

Kristensen, L. 1979. "On Longitudinal Spectral Coherence." Boundary-LayerMeteorology 16:145-153.

Panofsky, H . A . , and J . A . Dutton. 1984. Atmospheric Turbulence . John Niley& Sons, New York, New York.

Powell, D. C. 1974. Analyses of Para l le l and Orthogonal Wind Components AlongLines of Towers over Homogeneous Desert. P h . D . Thesis, University of Utah,Salt Lake City, Utah.

Powell, D . C . , J . R. Connell, and R . L. George. 1985. Ve ri fi ca ti on of Theo-retically Computed Spectra for a Point Rotating in a Vertical Plane.PNL-5440, Pacific Northwest Laboratory, Richland, Washington.

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Shinozuka, M. 1971. "Simulat ion o f Mul t i va r i a t e and Mul t id i mensional RandomProcesses." Journa l o f the Acoust ica l Soc ie ty o f America 49(1 , p a r t 2) :357-368.

Shinozuka, M. , and C.-M. Jan. 1972. " D i gi t al Sim ulat ion o f Random Processes

and I t s A p p l ica t i o ns ."

Journal o f Sound and Vi br at io n 25 : l l l - 1 2 8 .Smallwood, D. 0. 1982. "Random Vibration T e s t i n q o f a S i n q l e Test I tem wi th~ u l t i ~ i eInput Control System." I n proceed ings o f t he ~ n s t i t u t eo f E nv ir on-

mental Sciences. ed. J. Ehmann. DD. 42-49. In s t i tu t e o f Env ir onmentalSciences, ~ o u n t - p r o s p e c t ,~ l l i n o i s .

Sundar, R. M., and J. P. Su ll iv an . 1983. "Performance o f Wind Turbines i n aTurbulent Atmosphere. " Solar Enerqy 31 567-575.

Veers, P. S. 1984. Modeli ng St oc has ti c Wind Loads on Ve rt ic al - Axis Turb ines.SAND83-1909, Sandia National Laboratories, A1buquerque, New Mexico.

von Karman, T. 1948. "Progress i n the S t a t i s t i ca l Theory o f Tu rbulence ."Proceedi ngs o f t h e Na tio nal Acadeniy o f Science 34:530-539.

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APPENDIX A

MATHEMATICAL DETAIL OF SIMULATION METHODS

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APPENDIX A

MATHEMATICAL DETAIL OF SIMULATION METHODS

A. 1 HARMON1C METHOD

The harmonic method may bes t be i l l u s t r a t e d by a general example. Le t us

b e g i n w i t h t h e d i g i t a l r e p r e s e n t a t i o n o f u ( t ), 0 < t - T, which i s

Th i s t i me se r i es has N degrees o f f reedom. The power sp ectrum o f u ( k A t ) i sr e l a t e d t o t h e c omplex c o e f f i c i e n t s o f h arm onic a n a l y s i s o f u ( k ). These areg iven by

N

These c o e f f i c i e n t s s a t i s f y t h e r e l a t i o ns h i p s :

where the * denotes the complex conjugate,

and

N / 2va r i ance { u l = c(m) c* (m) .

m=1

Without lo s in g es se nt ia l g en er a l i t y, we may assume

and thereby make the rema inder o f the w r i t i n g eas ie r .

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The inve rse Four ie r t r ans fo rmat ion re la t i on i s g i ven by :

Th is reversa l o f the operat ion shows th at a l l the in format i on th at was i n the

rea l t ime se r ies u ( k ) (w i t hou t imag ina ry pa r t s ) i s a l so i n t he complex se r iesc (m) .

The harmonic concept o f Fo ur ie r tr an sf or mat io n may be more e a s i l y seen i f

we de fi ne t he amp1 i tu d e and phase o f each harmonic from t he c(m) ' s accord ingt0

where 0 i s one symbol f o r phase angle. Then u ( k ) i s th e sum o f harmonics i nterms o f the ampl i tude and phase o f each according t o

~ ( k )= ~ ( m ) o s k m / ~ e ( m ) ] .m=1 (A. 10)(a) Because the m i n c(m) may be po si t i ve o r negat ive, i t i s cus tomary i n

t h e o r e t i c a l w r i t i n g t o re co gn iz e a tw o-sided spectrum that i s an evenfunc t ion o f n and wh ich in teg ra tes t o y i e l d t he va r iance accord ing t o :

N/ 2Variance i u } = An x ~ 2 ( n ) .

m=1where S2(n) i s t he t wo

-

sided spectrum. The one-

sided and two-

sided spectraa re re la ted accord ing t o

The one-s ided spect rum i s the the vers ion genera l ly used i n phys ica lappl ica t ion s .

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The one-s ided spect rum i s r e la t ed t o the c ' s by

where n i s t he cy c l i c f requency, and t o th e variance by

N/ 2Variance { u } = An S(n ) .

m=1

The re la t i on o f t he ampl itudes t o t he spect rum i s

2S (n ) An = 0.5 A (m), n = m/T,

which may be written

The var iance i s g iven i n terms o f t he ampl i tudes by

Variance { u } = 0.5 [ r n = 1. N/Z] { ~ ~ ( r n ) l.

( A . 11)

(A. 12)

(A. 13)

(A. 14 )

A. 2 THE WHITE NOISE METHOD

The wh i te no ise method cons i s ts o f th e f o l lo w ing s teps . Begin wi th the

spectrum, S(n ) ; take the square ro o t o f S(n ) t o g e t a t r a n s fe r f u n c t i o n , H(n ) ;do the Fo ur ie r t ran sfor m t o get the impulse response func t io n, and convolve the

impulse response f un ct io n wi t h Gaussian whi te noise.

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Now l e t S2wh(n) be the two-sided spectrum of discre te Gaussian white noiseof un it variance. Then we can wr it e:

S2(n) = H ( n ) H * ( n ) S2wh(n) .Since the Gaussian white noise i s of un it variance,

S2wh(n) = A t ,

( A . 15)

( A . 16)

for a l l d i scre te n ,

[ ( n ) 1 = 1 / T , 2 /T , . .. 1 / 2A tso tha t t h i s spectrum used in Equation (A.16) gives a variance of uni ty. Then

Equation (A . 17) becomes

S2(n) = H ( n ) H * ( n ) A t . ( A . 17)If we remove a l l of the phase variation from the transfer function, the

tr an sf er function i s now rea l.

H(n) = H*(n) . ( A . 18)Then t he tr an sf er function i s numerically the square roo t of the spectrum.

That i s ,

which may be reduced numerically t o

because for the simulation purpose A t i s a f r ee parameter t ha t may be se t a t

uni ty.

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We n o t i c e t h a t H(n) and the amplitudes A (m ) o f F o u r i e r c o ef f i c e n t s a r eanalogous s ince both are t he square root s o f a d imensi onal ly ad justed product

i n c l u d i n g S ( n ) . The impulse response f un ct io n i s obt a ined by

h ( ~ )= Inverse Four ie r t rans form o f H(n ) o r

L e t us w r i t e

The f i na l ope ra t i on i s the convo lu t i on o f the impulse response fun c t i on

with Gaussian white noise. I f u s ( k ) ar e the sim ulat ed data, we have

where G ( ) i s Gaussian wh i te no i se o f u n i t var iance and P i s a t i me c u t- o f fpo i n t t ha t must be se lec ted a f t e r i nspec t ing the h ( ) func t i on . The subject ivedetermina t ion o f P i s a d isadvantage o f t h i s method th a t has no counterpar t i n

th e harmonic method. We programmed t h i s method and found th a t t he var ian ce o f

data s imulated by the whi t e noise method was in va r i ab ly l ess than the var iance

obta ined by in tegra t ing the spec t rum used fo r the s imu la t ion . We found no

advantages i n us ing th e whi te nois e method.

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APPENDIX B

THE CONCEPT OF SPECTRAL DENSITY AS PROBABILITY DENSITY

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APPENDIX B

THE CONCEPT OF SPECTRAL DENSITY AS PROBABILITY DENSITY

Shinozuka (1971) has an in t er es t in g concept o f a t i me se r ie s as the summa-t i o n o f u n i t ampl i tude harmonics a t randomly spaced frequencies where a proba-

b i l i t y dens i t y g ( n ) app l ies t o t he f requencies. The quo t i ent of spec tra ldens i t y ~ ( ' n )d i v i de d by t h e v a ri an ce i s t h i s p r o b a b i l i t y d e ns i ty . That i s ,

Because e i th e r s ide o f the equa t ion i n teg ra t es t o un i t y , the quan t i t y on the

r i g h t does i ndeed have tha t p rope r ty of a p r ob ab i l i t y dens i t y . Th i s i s impor-

t an t because s i mp l i f i ca t i on s may be a f fected i n spect ra l work by j u s t i f i e d

ap pl ic at io n o f pr ob a bi l i t y theory. For example, i n two d imensions we def ine

the spectrum such that

and S(nx ,ny ) /Var iance has some pro per t i es of a j o i n t prob abi l i t y dens i t y . Morei mp or ta nt , a j o i n t p r o b a b i l i t y d e n si t y P(x ,y ) i s the p roduct o f two i nd i v idua ld e n s i t i e s P ( x ) and P( y ) i f th e processes i n x and y are mut ual l y independent.Therefore, th e two-dimensional spectrum S(nx ,ny) i s the produc t o f two i n d i -v i d u a l sp e c t r a l d e n s i t i e s S( nx ) and S(ny ) if t h e va r i a t i o n o f t h e two o r i g i n a lprocesses i n the x and y d i r e c t i on s i s mu tua l l y i ndependen t; o t h e r w i s e not .REFERENCE

Shinozuka, M. 1971. "Simulation of Mult ivar iate and Mult idimensional Random

Processes." Journal o f the Acoust ica l Societ y o f America 49 (1, pa r t 2 ) :357-368.

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