The Average Output Power of a Wind Turbine in Turbulant Wind

8/6/2019 The Average Output Power of a Wind Turbine in Turbulant Wind

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E L S E V IE R J . Wind Eng. Ind. Aerodyn. 51 (1994) 287-302

T h e a v e r a g e o u t p u t p o w e r o f a w i n d t u r b i n e i n at u r b u l e n t w i n d

A . R o s e n * , Y . SheinmanFaculty o f Aerospace Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel

(Received April 4, 1992; accepted in revised form June 28, 1993)

A b s tract

T u r b u le n c e h a s a n im p o r t a n t i n f lu e n c e o n t h e a v e r a g e o u tp u t p o we r o f a w in d t u r b in e t a k e no v e r a c e r t a in p e r io d o f tim e . T h e w in d d y n a m ic s is c o u p l e d t o t h e t u r b in e d y n a m ic c h a r a c t e r -i s t ic s and resu l t s in a fa i r ly compl ica ted behav io r . Thus , the common "s ta t ic" mode l o fc a l c u l a t in g t h e a v e r a g e p o we r , wh ic h is b a s e d o n t h e t u r b in e p o we r c u r v e a n d t h e a v e r a g e w in dspeed , ma y resu l t in inc reas ing e r ro rs . Th is pap er p resen ts th ree d i f fe ren t m ode ls fo r ca lcu la t ingth e a v e r a g e o u tp u t p o w e r , t a k in g i n to a c c o u n t t h e d y n a m ic c h a ra c t e r is t i c s o f t h e p h e n o m e n o n .T h e s e m o d e l s i n c lu d e d i r ec t t im e i n t e g r a t i o n u s in g a c c u r a t e w in d d a t a a n d a d e t ai l ed d y n a m icm o d e l o f t h e t u r b in e , a q u a s i - s t e a d y a p p r o a c h wh ic h is m u c h s im p le r t o a p p ly a n d t a k e s i n toa c c o u n t t h e w in d d y n a m ic s , a n d a n im p r o v e d e ff ic ie nt m o d e l t h a t a l s o in c lu d es t h e i n f lu e n c e o fth e d y n a m ic c h a r a c te r i st i c s o f th e t u r b in e . T h e l a s t im p r o v e d m o d e l i s b a s e d o n a s t u d y o f t h etu r b in e r e s p o n s e t o a s i n u s o id a l g u s t . A l l m o d e l s a r e c o m p a r e d w i th f i e ld m e a s u r e m e n t s i no r d e r t o s t u d y t h e ir a c c u r a c y . T h e c o m p a r i s o n e x h ib i ts t h e im p o r t a n c e o f i n c lu d in g a ll t h edynamic e f fec ts in the ca lcu la t ions .Key words: Ou tp u t p o w e r ; W in d t u r b in e ; T u r b u l e n t w in d ; T im e in t e g r a ti o n ; Dy n a m ic m o d e l ;Qu a s i - s t e a d y a p p r o a c h ; I m p r o v e d e f f i c i e n c y m o d e l

1 . I n t r o d u c t i o n

T h e p o w e r w h i c h i s p r o d u c e d b y a h o r i z o n t a l a x i s w i n d t u r b i n e i n t h e ca s e o fa c o n s t a n t w i n d s p e e d V , P s ( V ) , is g i v en b y t h e f o l l o w i n g e q u a t i o n :

e s ( V ) = K C p V 3 , ( l a )K = p A r , ( l b )

w h e r e p i s t h e a i r m a s s d e n s i t y a n d A , t h e d i s c a re a . C p is t h e p o w e r c o e f f i c i e n t o f t h et u r b i n e a n d i s i n g e n e r a l a f u n c t i o n o f V . T h e s u b s c r i p t s i n d i c a t e s t h a t t h e " s t a t i c "*Corresponding author.0167-6105/94/$07.00 1994 Elsevier Science B.V. All rights reservedSSDI 0 1 6 7 - 6 1 0 5 ( 93 ) E 0 0 4 3 - X


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2 9 0 A. Rosen, Y. Sheinman/J Wind Eng. Ind. Aerodyn. 51 (1994) 287 302I n o r d e r t o o b t a i n t h e e x p r e s s i o n f o r P q s in the case o f a s inuso ida l gus t ,

E q . (9 ) i s s u b s t i t u t e d i n t o E q . ( 8 ). F o r c o n v e n i e n c e t h e r e s u l t is n o n d i m e n s i o n a l i z e da f t e r d i v i d i n g i t b y P s(_ V ). I n w h a t f o l lo w s a n o n d i m e n s i o n a l o u t p u t p o w e r w i l l b em a r k e d b y a n u p p e r t il de . I f p o w e r s o f ( A / V _ ) g r e a t e r t h a n t h e s e c o n d a r e n e g l e c t e d ,t h e n t h e f o l lo w i n g e x p r e ss i o n f o r t h e n o n d i m e n s i o n a l a v e r a g e q u a s i - st e a d y p o w e r ( inthe case o f a s in uso id a l gus t ) , Pq~(_V, A) , i s ob ta ine d ( s imple t r ig nom et r i c re la t ion s a rea l s o u s e d d u r i n g t h e d e r i v a t i o n ) :

1 A2 CpMP q s ( V , A ) = 1 -~ 2 V 2 C p o ' ( 1 0 )A s l o n g a s t h e f r e q u e n c y f i s s m a l l i t i s e x p e c t e d t h a t a q u a s i - s t e a d y a p p r o a c h

wi l l g ive a fa i r ly acc ura te des c r ip t ion o f the sys tem beha v io r . B u t a s f i s inc rease dt h e n t h e d y n a m i c c h a r a c t e r is t i c s o f t h e w i n d t u r b i n e w i ll s t a r t t o i n f l u e n c e i t sre sponse . I t is r easona b le to a s sum e tha t in th i s case CpM an d Cpe wi l l de pe nd on th ef e q u e n c y f

I n o r d e r t o i n v e s t i g a t e t h e i n f l u e n c e o f t h e d y n a m i c c h a ra c t e r i s ti c s o f t h e t u r b i n e o ni ts r e s p o n s e , t h e d y n a m i c m o d e l o f t h e V e s t a s V 2 5 t h a t w a s d e s c r i b e d i n [ 2 , 3 ] i s u se d .T h e o u t p u t p o w e r i n t h e c a s e o f a s i n u s o i d a l g u s t ( a s g iv e n b y E q s . (2 ) a n d ( 9 )) isc a l c u l a t e d . T h e n u m e r i c a l s i m u l a t i o n is c o n t i n u e d f o r a r e la t i v e ly l o n g t i m e i n o r d e r t oa r r i v e a t a p u r e s t e a d y - s t a t e r e s p o n s e , a f t e r t h e t r a n s i e n t r e s p o n s e t o t h e i n i t i a lc o n d i t i o n s h a s d i sa p p e a r e d . D u r i n g t h is s t e a d y st a t e r e sp o n s e t h e w i n d t u r b i n e o u t p u ta l s o e x h i b i ts a p e r i o d i c b e h a v i o r w i t h a b a s i c f r e q u e n c y f , n a m e l y a b a s i c t i m e p e r i o d1 I f F r o m t h i s s te a d y - s t a t e r e sp o n s e a t i m e i n t e r v a l T s i m is c h o s e n s u c h t h a t Ts i m = / '/ / f~w h e r e n is a c e r t a i n in t e g e r n u m b e r t h a t i n d i c a t e s t h e n u m b e r o f c o m p l e t e b a s i cp e r i o d s w i t h i n T s i m . I n o r d e r t o c a l c u la t e t h e a v e r ag e o u t p u t p o w e r a l o n g t h i s p e ri o do f t i m e t h e o u t p u t p o w e r , P s i m ( f , _V , A , t ) , i s i n t e g r a t e d o v e r t h e t i m e p e r i o d T s i m a n dt h e n t h e r e s u l t is d i v i d e d b y T s i m . T h e n o n d i m e n s i o n a l v a l u e o f t h is a v e r a g e p o w e r isde no ted _Pslm(f , _V, A) an d i s de f ine d m a th em at ic a l ly a s

Tsima ~s im f , V , A ) - - _ P sim ( J~ V , A ) 1 f_ p s ( _ V ) - - T s i m P s ( _ V ) P s i m ( f , V, A, t)dt. ( 1 1 )

oG ra ph s o f -_P~im(f, V , A) a s a fun c t ion o f the f requ ency f , fo r va r io us va lues o f the

ave rag e w ind spee d _If an d a gu s t am pl i tu de o f A = 1 m/s , a re p re sen ted in F ig . 1 .W hi le d i s cu s s ing th i s f igure , i t i s a l so co nve n ien t (besides re fe r r ing to the f re que ncy f )t o a l s o r e fe r to t h e g u s t d u r a t i o n T~, w h i c h i s t h e t i m e b e t w e e n t w o c o n s e q u e n t p o i n t s ,w h e r e v ( t ) beco m es eq ua l to _V. In the case o f a s in uso id a l gus t , T~ i s equ a l to ha l f o f thet i m e p e r i o d w h i c h i s ( i / f ) .

F ro m Fig . 1 i t i s c l ea r tha t fo r f < 0 .0017 H z (T l > 30 s ) the f u n c t i o n s _ /~ sim (J~ V , A)o b t a i n c o n s t a n t v a l u e s t h a t d o n o t c h a n g e w i t h f T h i s a p p l ie s t o a l l t h e v a r i o u sa v e r a g e w i n d s p e ed s . T h u s , i n t h i s r e g i o n ~ i m ( f , _V, A ) is i n d e p e n d e n t o f t h e d y n a m i cc h a r a c t e r i s ti c s o f t h e s y s t e m , n a m e l y t h i s i s t h e r e g i o n w h e r e a q u a s i - s t e a d y m o d e l i n ggives accu ra te resul ts . Th e valu es o f _Psim(f, _V, A) in th is reg ion sh ou ld be id ent ic a lt o t h e r e s u l ts o f t h e q u a s i - s t e a d y a p p r o a c h t h a t w a s d e r i v e d i n R e f. [ 1 ] . I n o r d e r


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A. Rosen, Y. Sheinman/J. Wind Eng. Ind. Aerodyn. 51 (1994) 287-302 2911,15' A = I ~ V=7m / s e e1.1. . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . V_=8 / s e ed 7 m /$ ........V = 9 m / s e e

~. 1.05-~J V=10m / s e eu.! V = I I m / s e eI ........................ ....................... ........................ ........................ ........................ ....................... ....12 m/s ~ . - . . . . .

I "~10 o le0 . 9 1 . . . . . . . . . . . . . . . . .0.01 0.1f , F r eque nc y [ H z ]Fig. l . l ~ s im ( J ~ ~ A) as a function of the frequenc y , for various values of the average wind velocity(7 m /s ~< _V~< 12 m/s) and a sinusoidal gust having an amp litude A = 1 m/s.

0.440 . 4 2 ................................................................................................................................................o ,(?.326 ~ 8 9 l o 1'I 12V [ m / s ec ]

Fig. 2. The pow er coefficientcurve of the Vestas V25.

t o che ck t he con s i s t en cy o f t he ca l cu l a t i ons , -_Pq,(_Y, I~ ) w as a l so ca l c u l a t e d us i ngE q . (6 ). F o r t h a t p u r p o s e t h e p o w e r c o e f f i c ie n t c u r v e (C p) o f t h e t u r b i n e w a s o b t a i n e d ,a s s h o w n i n F i g. 2 , u s i n g t h e c o m p l e t e n u m e r i c a l m o d e l o f t h e V e s t a s V 25 . T h e n , a f t era p p l y i n g a s p li n e s m o o t h i n g a l g o r i th m , t h e f ir st a n d s e c o n d d e r i v a t i v e s o f C p


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292 A. Rosen, Y. Sheinman/J. Wind Eng. Ind. Aerodvn. 51 (1994) 287 ? ~( w i th r e s p e c t t o V ) w e r e c a l c u l a te d . T h e c a l c u l a t e d v a l u e s w e r e s u b s t i t u t e d i n t o E q . (7 )a n d t h e n i n t o E q . ( 6 ) . T h e v a l u e s o f _P q, t h a t w e r e o b t a i n e d a s a r e s u l t o f t h iss u b s t i t u t io n s h o w e d e x c e l le n t a g r e e m e n t w i t h t h e a p p r o p r i a t e v a l u e s o f ~.,im (./i If, A )( f r o m F i g . 1 ) f o r l o w f r e q u e n c i e s ( f < 0 . 0 0 1 7 H z ) .

T h e c u r v e s i n F i g . 1 e x h i b i t l o c a l e x t r e m u m p o i n t s ( in t h e v a l u e o f ~ sim ( j~ IV, A ) a sa f u n c t i o n o f f ) a t a f r e q u e n c y f ~ 0 .0 6 2 5 H z ( T~ ~ 8 s). I n a d d i t io n , t h e r e a r e l e ssp r o n o u n c e d l o ca l e x t r e m u m s a t f ~ 0 . 0 4 H z a n d f ~ 0 .0 2 5 H z ( Tl ~ 12 .5 s a n dT I ~ 2 2 s , r e s p e ct iv e l y ). T h e e x t r e m u m p o i n t s a t f ~ 0 .0 6 2 5 H z a r e m a i n l y r e s u l ts o ft h e p i tc h c o n t r o l s y s t e m . A s i n d i c a t e d i n R ef . [ 3 ] , th e c o m p l e t e a c c u r a t e d e t a il s o f t h ep i tc h c o n t r o l s y s t e m w e r e n o t s u p p l i e d b y t h e m a n u f a c t u r e r a n d , th e r e fo r e , th e y w e r ee s t i m a t e d a f t e r a c a r e f u l i n s p e c t i o n o f t h e o p e r a t i o n o f t h e t u r b i n e a t th e B e i t - Y a t i rs i t e i n I s r a e l . T h e r e f o r e , c e r t a i n d e v i a t i o n s b e t w e e n t h e a c t u a l p i t c h c o n t r o l s y s t e ma n d t h e m o d e l m a y e x i s t .A s e x p e c t e d , a t h ig h f r e q u e n c i e s t h e t u r b i n e f a il s t o r e s p o n d t o t h e f a s t v a r i a t i o n s i nt h e w i n d s p e e d a n d , t h e r e f o r e _ P s i m ( f , _ IV ,A ) a p p r o a c h es u n i t y .

A c l ea r t r en d w h i ch i s sh ow n i n F i g . 1 i s t ha t fo r _V < 9 m / s , P s~ m (f, _V, A ) is g r ea t e rt ha n P j _V ) (na m e l y " _P sim ( f _V, A ) i s g r e a t e r t ha n un i t y ) , w h i l e fo r _V~ > 1 0 m / s ano p p o s i t e t r e n d is s h o w n . T h e r e a s o n f o r th a t b e h a v i o r i s c l e a r ly s h o w n i n F ig . 2 , w h e r ea m a x i m u m i n C p d e f i n e s t w o d i f fe r e n t re g i o n s , w h e r e C p e i t h e r in c r e a s e s o r d e c r e a s e sw i t h V . W h i l e th e m a x i m u m p o i n t i n F ig . 2 o c c u r s a t V = 8 .4 3 m / s , th e t r a n s i t i o n i nF i g . 1 o c c u r s b e t w e e n 9 m / s a n d 1 0 m / s . E x a m i n a t i o n o f E q . (7 ) i n d i c a t e s t h a t t h isd i f f e r e n c e i s e x p e c t e d s i n c e 8 (_ V ) i s n o t d i r e c t l y p r o p o r t i o n a l t o C 'p a n d t h u s 8 (_ V ) d o e sn o t b e c o m e n e g a t i v e w h e n C 'p c h a n g e s s i g n.

E x a m i n a t i o n o f _P .,im (f, V , A ) , a s p r e s e n t e d i n F i g . 1 , i n d i c a t e s t h a t t h e r e i s a s t r o n gd e p e n d e n c e o f t h is v a r i a b l e o n t h e f r e q u e n c y f . T h i s d e p e n d e n c e is m i s s i n g i n E q . (1 0).I n o r d e r t o o v e r c o m e th is d r a w b a c k , a c o r r e c t io n f u n c t io n , d e n o t e d G l ( f , _V), s a d d e dt o E q . (1 0 ) i n o r d e r t o a c c o u n t f o r t h e d y n a m i c c h a r a c t e r i s t i c s o f t h e t u r b in e . T h i se x t e n s i o n r e s u l t s i n a c o r r e c t e d v a l u e o f t h e a v e r a g e p o w e r , _ P c ( f , _V, A ) , w h i c h i sd e f i n e d b y t h e f o l l o w i n g e q u a t i o n

1 A2 CpM CpM_ P ~ ( . f , ~ , A ) = 1 + 2 V --2C p~G I(f,,__ V ) = 1 + 1 , 2 ~ , o G 1 ( V _ ) . (12)In o rd e r t ha t ~ c ( f _V, A ) w i l l ag r ee w i t h ~ s i m ( f , _V, A ) a s p r e s en t ed i n F i g . 1 , G l ( f , _V)

i s d e f i n e d a s f o l l o w s :_Psim(f _V, A ) - Ps(_V) (13)G I ( J ; V ) = -P qs ,sim ( - ' A ) - P s ( V ) '

w h e r e P q ~,s im ( V , A ) is th e q u a s i - s t e a d y v a l u e o f t h e a v e r a g e o u t p u t p o w e r .I n l i g h t o f th e a b o v e d i s c u s s i o n P q~ ,~ im (V , A ) is d e f i n e d a s t h e a v e r a g e o u t p u t p o w e r a tl o w f r e q u e n c i e s ( in t h e c a s e o f t h e V e s t a s V 2 5 f < 0 . 0 1 7 H z o r T~ > 3 0 s), w h e r e t h ei n f l u e n c e o f d y n a m i c e f f e c ts is n e g l ig i b l e . I f t h e v a l u e o f P q~ ,s im ( V , A ) is k n o w n , t h e nCpM(V)/Cpo(V) an b e c a l c u l a t e d u s i n g E q . ( 1 0 )

CpM = 2 f eq s , slm(_V A) 1 ] v 2C ,o L P~(_V) A-~" (14)


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A. Rosen, Y. Sheinman/J. Wind Eng. Ind. Aerodyn. 51 (1994) 287-302 2 93Cu rves of G l(f , _V) as funct ion s o ff , for var ious va lues o f _V, are p resen ted in Ref .

[14] . T he func t ion Cpu/3Cpo (w hich acc ordin g to Eq. (21) be lo w is equal to 8(_1/)) i sa lso presented in Ref . [14] . Al l the funct ions are given for the Vestas V25.

In p r inc ip l e (CpM/Cpo)or G I~, _V) are a lso func t ions o f the gu st am pl i tud e A .Never the le ss , inves t iga t ion h as s how n tha t fo r t he p rac t i ca l r ange o f A, t he in f luence o fthe ampl i tude on these func t ions i s ve ry sma l l and i t can be thus neg lec t ed .

3 . A " r e a l " w i n d

I t i s c lear tha t a " rea l" gust cannot be represented as a s ingle s inusoidal gust .Ins t ead a r ea l gus t inc ludes a wide spec t rum of f requenc ie s and am pl i tudes . T hepur pos e o f t he p resen t Sec t ion is to p resen t ho w the re su l ts o f t he p rev iou s Sec t ion canbe app l i ed in o rde r to ana lyse a wind tu rb ine in a " rea l " (gus ty ) wind .

Sh inozuka [4 ] used the fo l lowing d i sc re t e r epresen ta t ion in o rde r to desc r ibea "rea l" gust ;N su = ~ (2S,(f~ )Af) cos(2 nfi t + 0s), (15)

wh e r e S u ( f ) i s th e P o w e r Sp e c t r a l De n s i t y ( PSD) f u n c ti o n o f t h e w i n d o n t h e s it e a n dNs is the nu m be r of f reque ncy bins. 0 i s a rand om pha se angle , wh ere 0 < 01 < 2r~. A fis the w idth of each f req uen cy bin , whilef~ i s the re prese nta t ive f re quen cy of the i th bin .

Sh ino zuka [5 ] show ed tha t fo r Ns > /50 a goo d spec t ra l r epresen ta t ion o f a typ ica lwind i s ob ta ined .The abo ve w ind represen ta t ion i s wide ly used , e spec ia lly in cases where the dy nam ic

cha rac te r is t i c s o f t he wind a re o f i n te re st . F or example , Su ndar and Su l l ivan [6 ] usedth i s r epresen ta t ion in o rde r to run a s imple s imula t ion o f a wind tu rb ine tha t ope ra t e sin a gusty wind.

There a re va r ious m ode l s o f Su( f ) , and i t is beyo nd the sco pe o f t h is pa pe r to rev iewa ll t he ex i s ting mode l s . The m ode l t ha t wi ll be used in the p resen t i nves t iga t ion w aspresen ted by K r i s t ensen e t a l. I -7 ] and D rage t [8 ] and i t appea rs in Ref . [14] fo r t heBei t -Yat i r s i te .

Sub s t i t u t ion o f Eq . (15) in to Eq . (8) and us ing s imple t r igonom et r i c r e l at ions ,l ead to the fo l lowing express ion fo r non d imen s iona l ave rage quas i - s t eady ou tpu tpo w er ~q~(lS ', S, , t ) (which i s obta in ed af ter the dim ension al ou tpu t p ow er i s d ivide d byP~(v)) :

xCp M ~ 2S~(.]ii)Af-_Pq,(_V, S, ) = 1 + 2 Cp ~ ,=Z, _ ~ . (16 )I t can b e sho w n [9 ] t ha t i f Eq . (15) i s used to desc r ibe the gus t , t hen the in t ens ity o ftu rbu lence , accord ing to the bas i c de f in i ti on b y Eqs . (4 ,5 ), beco me s

I N , s ) A i ]


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A. Rosen, Y. Sheinman/J. Wind Eng. Ind. Aerodyn. 51 (1994) 287-302 2 9 5E q . (1 1 ) f o r a s i n u s o i d a l g u s t a n d i t i s a s t ra i g h t f o r w a r d m a t t e r [ 3 ] t o e x t e n di t f o r a n y p e r i o d T ,, o v e r w h i c h a " r e a l w i n d " i s b l o w i n g . I n t h i s c a s e th e d e -t a il e d a c c u r a t e d y n a m i c c h a r a c te r i st ic s o f t h e w i n d a n d t h e t u r b i n e a r e t a k e n i n toa c c o u n t

T a_ P s im ~ --- T a a e s i m ( t ) d t ( 2 3 )

oP sim ( t) is o b t a i n e d w h e n t h e a c t u a l m e a s u r e d w i n d s p e e d , v ( t ) , i s u s e d a s a ni n p u t o f a d e ta i le d d y n a m i c m o d e l o f t h e w i n d t u r b i n e . A s a l re a d y w a s d o n e i nt h e p r e v i o u s S e c t i o n s , i t i s c o n v e n i e n t t o n o n d i m e n s i o n a l i z e t h e l a s t t h r e e a v e r a g eo u t p u t p o w e r v a lu e s , d i v id i n g t h e m b y P s ( f f) . I f t h e n o n d i m e n s i o n a l v a l u e is s m a l le rt h a n u n i t y t h e n i t i s c l e a r t h a t t h e " s t a ti c " a p p r o a c h i s o v e r p r e d i c t i n g t h e p r o d u c e de n e r g y [ 3 ] .

I n o r d e r t o i n v e s t i g a te t h e a c c u r a c y o f t h e c a l c u l a te d v a l u e s o f t h e a v e r a g e p o w e r ,t h es e v al u e s a r e c o m p a r e d w i th a n o n d im e n s i o n a l iz e d m e a s u r e d a v e r a g e p o w e rv a l u es , w h i c h a r e d e f i n e d b y t h e f o l lo w i n g e q u a t i o n

T aP _ m e ~ = T , I ~ ( V _ ) P m e s ( t ) d t , (24 )

0

w h e r e P in es(t) is t h e m o m e n t a r y m e a s u r e d v a l u e o f t h e a c t u a l o u t p u t p o w e r .T h u s t h e i n v e s t i g a t i o n i n c l u d e d t h e f o l l o w i n g s t a g e s :(1 ) T h e w i n d s p e e d a n d o u t p u t p o w e r o f t h e V e s t a s V 2 5 t u r b i n e i n B e i t - Y a t i r w e r e

m e a s u r e d o v e r 1 53 h [ 3 ] .(2 ) T h e t o t a l m e a s u r i n g t im e w a s d i v i d e d i n t o t i m e - s e g m e n t s o f 6 0 0 s e a c h , n a m e l y

9 1 8 d i f f e r e n t s e g m e n t s .( 3) F o r e a c h t i m e - s e g m e n t _V, a v a n d I v w e r e c a l c u l a t e d ( E q s . 4,5 ).(4 ) F o r e a c h t i m e - s e g m e n t P ~(_ V) w a s c a l c u l a t e d u s i n g E q . ( l a ) .( 5) F o r e a c h t i m e - s e g m e n t _Pm ~s w a s c a l c u a l t e d u s i n g E q . ( 24 ).( 6) T h e d y n a m i c m o d e l o f t h e t u r b i n e [ 2 ] w a s r u n f o r t h e e n ti r e 1 53 h a n d ~ sim

c a l c u l a t e d ( f o r e a c h t i m e - s e g m e n t ) u s i n g E q . (2 3).(7 ) T h e f u n c t i o n 8 ( V ) w a s o b t a i n e d b a s e d o n E q s . ( 14 ) a n d (1 9). T h e n -P qs w a sc a l c u l a t e d f o r e a c h t i m e - s e g m e n t b y u s i n g E q . ( 1 8 ) .

(8 ) F o r e a c h t im e s e g m e n t ( I . . . d l ) w a s c a l c u l a t e d u s i n g E q . (2 2). I n t h e s e c a l c u l a t i o n sS , ( f ) a n d t h e f u n c t i o n s G I( _V ,f ) w e r e a p p l i e d . T h e n E q . (2 1 ) w a s u s e d i n o r d e r t oc a l c u l a t e ~ c f o r e a c h t i m e - s e g m e n t .

T h e a b o v e d e s c r i b e d p r o c e d u r e r e s u l t e d i n a l a r g e a m o u n t o f d a t a . I n o r d e rt o a n a l y z e t h e s e d a t a , a l l t h e t i m e - s e g m e n t s w e r e d i v i d e d i n t o i n t e n s i t y o ft u r b u l e n c e b i n s . T h e i n t e n s i t y o f t u r b u l e n c e o f i n t e r e s t v a r i e s i n t h e r e g io nI v = 0 . 0 2 5 + 0 . 15 . S i n c e a b i n w i d t h o f 0 . 0 2 5 i s c h o s e n , t h e r e a r e f i v e i n t e n s i t y o ft u r b u l e n c e b i n s .F o r e a c h b i n , e v e ry a v e r a g e o u t p u t p o w e r v a l u e ( ~ . . . . ~ i m , ~ c a n d P _ q ,) isp l o t t e d a s a f u n c t i o n o f _IV. T h e r e s u l t s a r e q u i t e s c a t t e r e d a s s h o w n f o r e x a m p l e


10/16

2 9 6 A. Rose n , Y . She inm an /d . Win d Eng . l nd . Ae roc (v n . 51 (1994) 28 7 3021

0 . 9 9 . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . ..0 . 9 8 ........................' ...................- . - - - . - - ~ . = . . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . ..

l i . =

0 . 9 7 .........................,,r,_=.........,......,,..................................................................0 . 9 8 ...................................... m ..............- = -- . . - - - . - - - . i - - - . - i .........................................." - . . . . . . - - ~ .. .

~ ~ ~ - i = =o . g s .........................., , . . - , - ...................................................................._ ' 1 " i ' " .....0 . 9 4 = . i , ............................................................., . ~ . . .. .. .. .. ... . . . . . L . . ~ L . .. .. .. .. .. .. .. .. .. .. .. ..0 . 9 3 ................... ..............................................................m ........................................................................

i

0 . 9 2 ..................- - n - ..........................................................................................................................................m

m0 ,9 1 ..................................................................................................... ~ ........................................................

0 . 9 7 8 i I b 1'I 1 2 l aV [ m / s e c l( a )

10 . 9 9 ...................................................................................................................................................

0 ,9 8 ...........................................................................................................................0 . 9 7 ...................................................................................................... ..................................0 , 9 5 . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . .

=~ 0 . 9 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 I

0 . 9 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 . 9 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 . 9 2 . . . . .

2 1 ...................................................................................i J + , L7 8 9 1 0 11 12 13V ( m / N c )( b )

F i g . 3. ~ a s a f u n c t i o n o f V i n t h e b i n I~ = 0 . 1 + 0 . 1 2 5 : ( a ) b e f o r e a v e r a g i n g , ( b ) a f t e r a v e r a g i n g .

i n F i g . 3a , w here ~m** fo r t he b i n I , = 0 .1 + 0 . 125 i s p r e se n t ed . I n o rd e r t o be ab l e t oa n a l y z e t h e r e s u lt s , t h e s c a t t e r o f e a c h c u r v e i s r e d u c e d b y a v e r a g i n g e a c h c u r v e i nv e l o c i ty b in s o f a w i d t h o f 0 .5 m /s . T h i s w i d t h i s c h o s e n a c c o r d i n g t o t h e E u r o p e a ns t a n d a r d [ 1 0 ] a n d t h e A S M E r e c o m m e n d a t i o n [ 1 1 ] . T h e re s u lt s a f te r t h e a v e r a g i n ga r e s h o w n i n F i g . 3 b .


11/16

A. Rosen, Y. Sheinman/J. Wind Eng. Ind. Aerodyn. 51 (1994) 287-302 297D u r i n g e a c h a v e r a g i n g p r o c e d u r e t h e s t a n d a r d d e v i a ti o n , w h i c h i s a s s o c i a te d w i t h

e a c h a v e r a g e d c u r v e , is p l o t t e d a s a f u n c t i o n o f V . T h e r e s u l t s i n d i c a t e t h a t r e l a ti v e l yl a rg e s c a t te r a n d t h u s l a rg e s t a n d a r d d e v i a t i o n v al u e s a r e o b t a i n e d a t c o m b i n a t i o n s o flow va lu es o f _V an d Iv (_V < 8 m /sec , Iv = 0 + 0 .06). In th ese cases the w ind prof i le i su s u a l l y u n s t a b l e 1 -1 2] a n d t h u s u s i n g t h e w i n d s p e e d a t t h e h u b a s a r e p r e s e n t a t i v ev a l u e o f t h e e n t i r e d is c a r e a ( se e t h e c o m m e n t a f t e r E q . ( 2) ) m a y b e i n a c c u r a t e .

In Fig . 4 cur ve s o f _P . . . P_sim,_Pc an d _Pqs, as fu nc tio ns of _V, are pr es en ted for the fivei n t e n s i t y o f t u r b u l e n c e b i n s.

A c a r e f u l i n v e s t i g a t i o n o f F ig . 4 i n d i c a t e s t h a t i n g e n e r a l t h e r e i s a g o o d a g r e e m e n tb e tw e e n ~/~rnesan d P_~im. Also the ag ree m en t -_Pc an d _Pines s be t te r th an the ag ree m en tbetween -_Pq,a n d ~ mes " I n o r d e r t o f u r t h e r i n v e s t i g a t e t h e a g r e e m e n t between -_emcsa n dthe re s t o f t he c u rve s , a s t a t i s t i c a l t e s t is u se d . I t is a tw o t a i l pa i re d T- t e s t [13 ] .

T h i s t e s t d e s i g n c al ls fo r m a t c h i n g o r p a i r i n g t h e s u b j e ct s i n t o a n u m b e r o f p a i rs s ot h a t t h e i n d i v i d u a l s i n e a c h p a i r a r e a s m u c h a l i k e a s p o s s i b l e . I f t h e r e i s a s o u n db a s is f o r p a i r i n g , t h e i d e a i s t h a t w i t h i n e a c h p a i r a v e r y p r e ci s e c o m p a r i s o n ( o f i tsi n d i v i d u a ls ) is ac h i e v ed . T h e s e p r e c is e c o m p a r i s o n s c a n b e c o m b i n e d . T h e o u t c o m eo f t h e t e st p r o v i d e s a n e x a m i n a t i o n o f t h e h y p o t h e s i s : " I s t h e m e a n o f t h e d i f fe r -e n c e s b e t w e e n e a c h p a i r e q u a l t o z e r o i n a c o n s t a n t s i g n if i ca n c e l ev e l? " . W h e n t h et e s t o u t c o m e i s l e ss t h a n t h e s i g n i f i c a n c e l e ve l, t h e h y p o t h e s i s w i ll b e a c c e p t e d ,m e a n i n g t h a t t h e m e a n o f t h e d if f er e n c es w i ll n o t b e z er o . F o r a v a l u e h i g h e r t h a n t h e

Iv=0.025-0.05

1 ~ - . . ~ s ! ~ . ~ . ....................................................................................................................................

............................I.................................................................................................................................Iii ! ; \ i i ........................................................................ . .................................................

7 8 9 10 11 12 13V (m/see)( a )Fig. 4 . p . . . . _P, , ,, , ~o an d "_P,, as fun ct ion s of V__, for the f ive bins of in tensity of turbulence:(a) Iv = 0 .025 . -' 0 .05.


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298 A, Rosen. Y. Sheinman/J, Wind Eng. Ind. Aerodyn. 51 (1994) 287. 302

I v = 0 . 0 5 - 0 . 0 7 51 . 0 6 . . . . . . .1 . 0 4 - - ~ . ~ r u e s P i n e s

~ , . . E s i m1 . 0 2

o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . _ o j

0 . 9 4 6 " ~ 8 ~ ) 1 ' o 1 ' 1 1 ' 2 1 3_ v (m / s e c )(b )

I v = 0 . 0 7 5 - 0 . 11 . 0 6 p q s1 .0 4 ................ lk :~ :~ - ...................................................................................... _P m e\ Y ~~ ...................................................................................................... P s i r r

. . . . . . . i

" i 0 . 9 8 - ~ ~ " '

o . 9 = - I P = ~ "0 . 9 s 6 8 9 1 'o 1 '1 1 '2 1 3V ( m / se c )

(c )F i g . 4 . ( c o n t in u e d ) . ( b ) I , = 0 . 0 5 + 0 . 0 ? 5 , ( c ) I ~ = 0 . 0 7 5 + 0 . 1 .

s i g n if i c an c e l ev e l, t h e h y p o t h e s i s w i ll n o t b e a c c e p t e d ( t h e m e a n o f t h e d i f f e re n c e s w il lb e z e r o ) . I n t h i s c a s e t h e t w o g r o u p s a r e s i m i l a r w i t h i n t h e i r p a i r s . T h e s i g n i f i c a n c el e v e l w h i c h w a s u s e d i n t h i s e x a m i n a t i o n i s 9 5 % .

T h e r e s u l ts o f t h e t e s t a r e p r e s e n t e d i n T a b l e 1. T h e ~ , v a l u e s t h a t s u p p o r t t h eh y p o t h e s i s ( a n d t h u s i n d i c a t e a g r e e m e n t ) a r e m a r k e d b y ( + ), w h i l e z , v a l u e s t h a t d o


13/16

A. Rosen, Y. Sheinman/J. W ind Eng. Ind. Aerodyn. 51 (1994) 2 87 -3 02 2 9 9~=0 . 1 - 0 . 1 2 5

1 ~ !iiiiiiiiiiiiii[ 1 t............ . % ~ .. . . . . .::::::::::::::::::::::......................................................................._ s , mS . . % ~ . . . . . . . :S ....,; ......_Pc ~ Pq s

~ o 0 . 9 8 - t " " ; ;................................................................................. ~ ..... ..... ..... ::::::................................................... -'/,~.......

o.~ rl s ~ 815 9 9 1 s 1 'o 1 6 . s 1 '1 l i . 5 1 2V ( m / s e e )

( d )

I v = 0 . 1 2 5 - 0 . 1 51.02T ~| _P qs ~ . . . . .. . . . .. . . . . .. . ~ . . ..\1 . o 1 1 . . . . . . .. . . . . . . . . . .. . . . .. . . . . .. . . . .. i i ~ : . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ - -- - - - ~ . . . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . .. . . . . .. . . .

i o . o ~.. . . . . .. . . . . .. . . ~ . . . . . . . . / / . ................................ \ .......................................o . 0 ~ . . . . . . . . . . . ~ ~ ......................

0 . 9 2 .7 . 5 8 8 ; 5 ; 9 ~ 5 l r 0 113,5 11_V m/sec)( e )

F i g . 4 ( c o n t i n u e d ) . ( d ) I ~ = 0 . 1 - 0 . 1 2 5 , ( e ) I ~ = 0 . 1 2 5 + 0 . 1 5 0 .

n o t s u p p o r t t h e h y p o t h e s i s ( i n d ic a te d i s a g r e e m e n t ) a r e m a r k e d b y_ ( - ). T h e r e s u lt si n d i c a t e v e r y c l e a r l y t h a t t h e b e s t a g r e e m e n t i s o b t a i n e d b e t w e e n _P~=~ a n d _ P,i~ - T h ea g r e e m e n t b e t w e e n _ P~=~a n d _P i s o f l o w e r q u a l i t y , b u t s ti ll t h e h y p o t h e s i s i s a c c e p t e da t t h e 9 5 % s i g n i f i c a n c e l e v e l. T h e h y p o t h e s i s i s a c c e p t e d i n t h e _ P ~= , a n d _Pq~c o m p a r i -s o n o n l y i n o n e ( I~ = 0 . 0 2 5 - 0 .0 5 ) o f t h e f i v e i n t e n s i t y o f t u r b u l e n c e b i n s .


14/16

300 A . R o s e n , Y . S h e i n m a n / J . W i n d E n g . I n d ~ A e r o d v n . 5 1 ( 1 9 9 4 ) ~ 8 7 3 0 ,Table 1The results of a statistical two tail p aired T-test that exam ines the ag reement betwee n ffme~an d ~ m, ~ P_q~1, r, values for the com parison between:bin P ~ an d ~im _/~sim rid _/~c ~ s an d ~qs0.025 0.05 { ) 0.1837 ( + ) 0.2184 ( ~0.19590.05 0 .07 5 ( + ) 0.0932 ( ) 0.1534 ( ~0.01260.075 + 0.10 ( + ) 0.0870 ( + ) 0.0551 ( -- ~0.00020.10 + 0.125 ( + ) 0.1483 ( + ) 0.0783 ( ~0.00020.125 + 0.15 ( + ) 0.4107 [ + ) 0.0555 ( ~0.0001

T h e r e s u l ts o f t h e s t a ti s t ic a l t e s t s u p p o r t t h e i n it ia l o b s e r v a t i o n t h a t t h e m o s ta c c u r a t e m e t h o d o f c a l c u l a t i n g t h e a v e r a g e p o w e r i s b a s e d o n a d ir e c t c o m p l e t es i m u l a t i o n (_P~im). N e v e r t h e l e s s , t h e i m p r o v e d m e t h o d (_P~) t h a t t a k e s i n t o a c c o u n t t h ed y n a m i c c h a r a c t e r is t i c s o f t h e w i n d a n d t h e t u r b u l e n c e g i ve s v e r y g o o d r e s u l ts w h i c ha r e m o r e a c c u r a t e t h a n t h e r e s u l t s o f t h e q u a s i - s t e a d y m o d e l (_Pq~) o r t h e s t a t icm o d e l ( P ~ ) .

5 . C o n c l u s i o n s

F o u r d i f f er e n t m e t h o d s o f c a l c u l a t in g t h e a v e r a g e o u t p u t p o w e r ( o ve r a c e r t a i n ti m ed u r a t i o n ) o f a w i n d t u r b i n e , i n a tu r b u l e n t w i n d , w e r e p r e s e n te d :( a) T h e " s t a t i c " m e t h o d (_ Ps) r e q u i r e s a k n o w l e d g e o f t h e p o w e r c o e f f i c ie n t c u r v e

o f t h e t u r b i n e a n d t h e a v e r a g e w i n d s p e e d o v e r t h e s a m e p e r i o d o f t im e . T h i sm e t h o d d o e s n o t t a k e i n t o a c c o u n t t h e d y n a m i c c h a r ac t e ri s ti c s o f t h e w i n d o r t h et u r b i n e .

( b) T h e q u a s i - s t e a d y m e t h o d (_Pq~) u s e s t h e p o w e r c o e f f i c i e n t c u r v e , t h e a v e r a g e w i n ds p e e d a n d t h e i n t e n s it y o f t u r b u l e n c e. T h i s m e t h o d a c c o u n t s f o r t h e d y n a m i cc h a r a c t e r is t i c s o f t h e w i n d , b u t n o t t h e t u r b in e .

( c) T h e i m p r o v e d m e t h o d (_P c) u s e s th e p o w e r c o e f f i c ie n t c u r v e , t h e a v e r a g e w i n ds p e e d a n d a c o r r e c t e d i n t e n si t y o f t u r b u l en c e . T h i s i m p r o v e d m e t h o d c o n s i d e r s t h ed y n a m i c c h a r a c t e r is t i c s o f t h e w i n d a n d t h e t u r b in e . T h i s is a c h ie v e d b y d e f i n i n ga " n e w " e x t e n d e d i n t e n s i ty o f t u r b u l e n c e t h a t d e p e n d s o n t h e d y n a m i c c h a r a c t e r -i s ti c s o f t h e t u r b i n e .

( d) A d i r e c t a c c u r a t e c a l c u l a t i o n o f th e a v e r a g e o u t p u t p o w e r (_P~im ). T h i s v a l u eis o b t a i n e d a f t e r a v e r a g i n g t h e r e s u lt s o f a d i re c t s i m u l a t i o n o f t h e t u r b i n ep o w e r p r o d u c t i o n w h e n i t is e x p o s e d t o t h e s a m e w i n d s p e ed t h a t w a sm e a s u r e d . F o r t h i s s i m u l a t i o n a d e t a il e d d y n a m i c m o d e l o f t h e t u r b i n e isu s e d .I t w a s s h o w n t h a t w i n d t u r b u l e n c e h a s a n i m p o r t a n t i n fl u e nc e o n th e a v e r a g e

o u t p u t p o w e r . U s i n g t h e s t a ti c v a l u e o f t h e a v e r a g e p o w e r m a y r e s u l t in in c r e a s i n gi n a c c u r a c y .


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A. Rosen, Y. Sheinman/J. Wind Eng. Ind. Aerodyn. 51 (1994) 287-302 301T h e m o s t a c c u r a t e p r e d i c t i o n o f t h e a v e r a g e p o w e r i s o b t a i n e d w h e n a d ir e c t

s i m u l a t i o n o f t h e w i n d t u r b i n e p o w e r p r o d u c t i o n is c a r r ie d o u t . T h i s m e t h o d is t im ec o n s u m i n g a n d r e q u i re s a d a t a ( d e t a il e d v(t)) t h a t u s u a l l y d o e s n o t e x is t. A n y e f f o r t t oo b t a i n t h is d a t a u s u a l l y r e q ui re s v a r i a t io n s i n t h e c o m m o n m e a s u r i n g s y s te m a n dd e a l in g w i th e n o r m o u s a m o u n t s o f d a t a .

T h e n e w i m p r o v e d m o d e l f o r c a l c u la t i n g t h e a v e r a g e p o w e r _ Pc, t h a t w a s d e r i v e d i nt h e p a p e r , o f f e r s a v e r y e ff ic i e nt m e t h o d o f c a l c u l a t i n g t h e a v e r a g e p o w e r , t a k i n g i n t oa c c o u n t t h e d y n a m i c c h a r a c t e r is t i c s o f t h e w i n d a n d t h e t u r b i n e , a n d t h e i r c o u p l in g .T h i s m o d e l i s b a s e d o n d e f i n i n g a " c o r r e c t e d " i n t e n s it y o f t u r b u l e n c e t h a t t a k e s i n t oa c c o u n t t h e w i n d t u r b i n e d y n a m i c c h a r ac t e ri s ti c s . I t w a s s h o w n t h a t f o r t h is p u r p o s ea c o r r e c t i o n f a c t o r c a n b e de f in e d , w h i c h d e p e n d s o n t h e f r e q u e n c y a n d t h e a v e r a g ew i n d s p e ed . T h e d e p e n d e n c e o f t h e c o r r e c t i o n f a c t o r o n t h e g u s t a m p l i t u d e is s m a l la n d c a n b e n e g le c te d . T h i s c o r r e c t i o n f a c t o r is c o m b i n e d w i t h t h e P S D o f t h e w i n d o ns it e, t o y i e l d t h e c o r r e c t e d i n t e n s i t y o f t u r b u l e n c e .

T h e p r e s e n t i m p r o v e d m e t h o d c a n b e u se d i n o r d e r t o i m p r o v e t h e a c c u r a c y o f t h ep r e d i c t i o n o f t h e e n e r g y t h a t w i ll b e p r o d u c e d b y a w i n d t u r b i n e w h i c h i s l o c a t e d a ta c e r t a i n s i t e [ 1 4 ] .

AcknowledgementT h e r e s e a r c h t h a t l e a d t o th i s p a p e r w a s f i n a n c e d b y t h e I s r a e l M i n i s t r y o f E n e r g y

a n d I n f r a s tr u c t u re . T h e a u t h o r s w o u l d l ik e t o th a n k M r s . G o o d m a n f o r t y p i n g t hi sm a n u s c r i p t .

References[1] C.J. Christensen, J.B. Dragt, N.V.D. B org, O. Carlson, I .R. Derdelinx, R. Hunter, D . Infield, M.A.Lodge, J. van Meel, E. Lyesen, K. K ieft, J .P. M olly and U.S. Paulsen, Accuracy of Power CurveMeasurements, RISO-M -2632 Report, Riso National Lab., DK-4000, Roskilde, Denm ark, Novem ber1986.[2] Y. Sh einm an and A. R osen , A dynamic m odel for performance calculations of grid-connectedhorizontal axis wind turbines. P art I: Description of the model, W ind En g. 15(4) (1991) 211-228.[3] A. Rosen and Y. Sheinman, A dynamic model for performance calculations of grid-connectedhorizontal axis wind turbines. P art II: Validation, W ind E ng . 15(4) (1991) 229-239.[4] M. S hinozuka and C .M . Jan, Digital simulation of random processes and its applications, J. SoundVibr., 25 (1972) 111-128.[5] M. Shinozuka, Simulation o f multivariate multidimensional random processes, J . Accous. Soc. Am.,49 (1971) 357-367.[6] R.M . Sundar and J.P. Sullivan, Performance of wind turbines in a turbulent atmosphere, SolarEnergy, 31 (1983) 567-575.[7] L. Kristensen, H.A. Panofsky and S.D. Smith, Lateral coherence of longitudinal wind components instrong winds, Boundary Layer Meteorol., 21 (1981) 199.I - 8 ] J.B . Dragt, T he spectra wind spe ed fluctuations met by a rotating b lade and resulting load fluctu-ations, Proc. European W ind Energy Conf. Hamburg, Germany, 22-2 6 O ctober, 1984, pp. 4534 58.[9] E. L. Houghton and N.B . Carruthers, W ind Forces on buildings and structures: An Introduction,Edward Arnold Publishing, London, 1976.


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30 2 A. Rosen, Y. Sheinman/J. Wind L)~g. Ind. Aerodyn. 51 (1994) 287 302

[ 1 0 ] A . C u r v e r s a n d T . F . P e d e r s e n , R e c o m m e n d a t i o n fo r E u r o p e a n W i n d T u r b i n e S t a n d a r d o n P e r f o r-m a n c e D e t e r m i n a t i o n , E C N R e p o r t E C N - 8 9 - 1 6, N e t h e r l a n d s E n e r g y R e s e a rc h F o u n d a t i o n , P e t t en ,T h e N e t h e r l a n d s , 1 9 8 9 .

[ 1 1 ] A n o n , M e a s u r e m e n t s U n c e r t a i n t y , P a r t 1 , I n s t r u m e n t s a n d A p p a r a t u s , S u p p l e m e n t t o A S M E -Pe r f o r m a n c e t e s t c o d e s , A N SI / A SM E PT C 1 9. 1- 1 9 85 .[ 1 2 ] D .L. E l li o t, Wi n d S h e a r fo r L a r g e Wi n d T u r b i n e G e n e r a t o r s a t Se l e c t e d T a l l T o w e r S i te s, PN L - 4 8 9 5 ,

Pa c i f i c N o r t h w e s t L a b o r a t o r y , R i c h l a n d , WA , 1 9 8 4 .[13] J .L . Fo lks , Ideas o f Sta t i s t ic s , Wi ley, New York , 1981.[ 1 4 ] Y . Sh e i n m a n a n d A . Ro s e n , A d y n a m i c m o d e l o f t h e in f l u e n c e o f t u r b u l e n c e o n t h e p o w e r o u t p u t o f

a wind tu rb ine , J . W ind . Eng . Ind . Aerody n . , 39 (1992) 329 341 .

The Average Output Power of a Wind Turbine in Turbulant Wind

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