7/29/2019 Numerical Simulation Of

1/3

N u m e r i c a l s i m u l a t i o n o fa n n u a l m a x i m u m w i n ds p e e d sR . V . M i l f o r dNational Building Research Institute, Council or Scientific and Industrial Research, PO Box395, Pretoria 0001, South Africa(Received May 1986; revised December 1986)N u m e r i c a l s i m u l a t i o n o f h o u r l y m e a n w i n d s p e e d d a t a i s u s e d t oi n v e s t i g a t e t h e a s s u m p t i o n o f s t a t i st i ca l i n d e p e n d e n c e u s e d w h e nd e r i v i n g e x t r e m e w i n d s p e e d s d i r e c t l y f r o m t h e p a r e n t d i s tr i b ut k ~ n .K e y w o r d s : n u m e r i c a l s i m u l a t i o n , w i n d s p e e d

I n a r e c e n t i n v e s t i g a t i o n x, a n n u a l m a x i m u m w i n d s p e e d sd e r i v e d d i r e c t l y f r o m t h e p a r e n t d i s t r i b u t i o n f u n c t i o nw e r e c o m p a r e d w i t h t h o s e o b t a i n e d f r o m a T y p e - Ie x t r e m e - v a l u e d i s t r i b u t i o n f i t t e d t o t h e a n n u a l m a x i m aa n d t o t h e s q u a r e o f t h e a n n u a l m a x i m a . I n d e r i v i n g th ea n n u a l m a x i m u m w i n d s p e ed f r o m t h e p a r e n t d i s t r ib u t i o nf u n c t i o n i t w a s a s s u m e d t h a t s u c ce s s iv e h o u r l y m e a n w i n ds p e e d s w e r e s t a t i s t i c a l l y i n d e p e n d e n t . T h i s a s s u m p t i o ny ie ld ed sa t i s f ac to r y r esu l t s , b u t i s in co n f l ic t w i th s im i la rs tu d ies r ep o r ted . 2 - 4I n t h e p r e s e n t s t u d y , n u m e r i c a l s i m u l a t i o n o f h o u r l ym e a n w i n d s p e e d d a t a i s u s e d t o i n v e s t ig a t e e x t r e m e w i n ds p e e d s d e r i v e d d i r e c t l y f r o m t h e p a r e n t d i s t r i b u t i o n . I np a r t i c u l a r , t h e a s s u m p t i o n o f s t a t is t i c a l i n d e p e n d e n c e i se x a m i n e d .S t a t i s t i c s o f h o u r l y m e a n w i n d s p e e d d a t aS t a t is t ic s o f S o u t h A f r i c a n h o u r l y m e a n w i n d s p e e d d a t ah av e b een r ep o r ted p r ev io u s ly , l ' s R e lev an t s ta t i s t i c s f o ro n e s t a t i o n , a t J a n S m u t s A i r p o r t ( J o h a n n e s b u r g ) , a r eg iv en h e r e . I n Figur e 1 t h e c u m u l a t i v e d i s t r i b u t i o nf u n c t i o n o f t h e p a r e n t d i s t r i b u t i o n f u n c t i o n i s g iv e n . T h eta i l o f th e d i s t r ib u t io n i s f i t t ed b y a Weib u l l d i s t r ib u t io no f t h e f o r m

Pv(v) = e ( ~ / 2 ' ~ ) ' (1)w h e r e Pv(v) i s th e p r o b ab i l i ty o f v b e in g ex ceed ed , an d= 2 .7 2 an d k = 1 .5 9 f o r Jan Sm u ts A i r p o r t . I n Figur e 2t h e a u t o c o r r e l a t i o n f u n c t i o n o f t h e w i n d s p e e d f l u c t u a -t ion s is g iven . T he f ir s t f ive coeff ic ien ts are 0 .81 , 0 .66 , 0 .55 ,0 .4 7 an d 0 .4 1 , co r r esp o n d in g to a t ime lag o f 1 , 2 , 3 , 4a n d 5 h o u r s r e s p e c t i v e l y . A d d i t i o n a l s t a t i s t i c s f o r J a nS m u t s A i r p o r t a r e : m e a n c y c l i n g r a t e vo = 2 .36 cycles /daya n d r o o t - m e a n - s q u a r e w i n d s p e e d a v = 2 .3 5 m / s .

I n Figur e 3 a c o m p a r i s o n i s m a d e o f p r e d i c t e d a n n u a lm a x i m u m w i n d s p e e d s t o g e t h e r w i t h t h e m e a s u r e da n n u a l m a x i m a a t J a n S m u t s A i r p o r t . T h e f o l l o w i n ge s t i m a t e s o f t h e a n n u a l m a x i m u m w i n d s p e ed a r e u s e d :T y p e - I i n v

Fv(v)=e -e . . . . . . ( 2 a )T y p e - I i n v2

P . ( v ) = e - e . . . . . . . (2b)a n d t h e d e r i v e d d i s t r i b u t i o n s

F~(v) = [1 - Pv(v)] 8760 (2c)

3 O2 O

E

5

F i g u r e I

O I 1 I I0 . 5 I 0 I I 0 2 1 0 3P r ob ab i l i t y P v ( v )

M e a s u r e d p a r e n t d i s t r i b u t i o n f u n c t i o n

I I10-510 7

0141-0296/88/01065-03/$03.00(t') 1988 Butterworth & Co ( P u b l i s h e r s ) L td E n g . S t r u c t . 1 9 8 8 , V o l. 1 0 , J a n u a r y 6 5

7/29/2019 Numerical Simulation Of

2/3

Numerical simulation of annual maximum wind speeds. R, V, Milford

A

1 . 0 0

0 . 7 5

0 . 5 0

0 . 2 ~

0

- 0 . 2 5 '0 I 0 0 2 0 0T i m e - l o g { h )

Figure 2 M e a s u r e d a u t o c o r r e l a t i o n f u n c t i o n

3 0

'~ 2oE

IO

Figure 3

0 I . I i i L i0 . 5 0 9 0 . 9 9 0 . 9 9 9P ro b a b i l i t y Fz (V )

A n n u a l m a x i m u m w i n d s p e e d s us i n g m e a s u r e d d a t a .. . . . . . T y p e - I i n v ( e q u a t i o n ( 2 a ) ) ; - - - , T y p e - I i n v 2 ( e q u a t i o n ( 2 b ) ) ;, d e r iv e d ( e q u a t io n ( 2 c ) ) ; - . - , d e r i ve d ( e q u a t io n ( 2 d ) ) .

Auto-regressive model l ing of hourly mean winds p e e dD e t a i l s o f a u t o - r e g r e s s i v e ( A R ) m o d e l l i n g o f c o r r e l a t e dr a n d o m v a r i a b l e s c a n b e f o u n d i n R e f e r e n c e 7. T h e b a s i so f A R m o d e l l i n g i s c o n t a i n e d i n t h e f o l l o w i n g e q u a t i o n :

Py r = Y q - B o ( X , - - . ~ ) - ~ , A , { y , _ , - ~ ) (3 )

/ =1

w h e r e y , i s t h e r t h c o r r e l a t e d r a n d o m v a r i a b l e w i thd i s t r i b u t i o n f u n c t i o n P y ( y ) a n d p r e s c r i b e d a u t o c o v a r i a n c ef u n c t i o n C r r ( z ) . g r i s a n u n c o r r e l a t e d w h i t e n o is e r a n d o mv a r i a b l e w i t h d i s t r i b u t i o n f u n c t i o n Px(x) . T h e p a r a -m e t e r s A i a r e o b t a i n e d f r o m t h e a u t o c o v a r i a n c e f u n c t i o nC . ( r ) . 7A R m o d e l l i n g i s s t r i c t l y o n l y v a l i d f o r G a u s s i a nr a n d o m v a r i a b le s , a s th e s u m o f G a u s s i a n v a r i a b l e sr e m a i n s G a u s s i a n . H o w e v e r , i t c a n b e a p p l i e d t o n o n -G a u s s i a n r a n d o m v a r ia b l es , a l t h o u g h s o m e d i s t o r ti o n o ft h e p r e s c r i b e d d i s t r i b u t i o n f u n c t i o n P x ( x ) o c c u r s .U s i n g t h e m e a s u r e d a u t o c o r r e l a t i o n f u n c t i o n f o r J a nS m u t s a n d a n a s s u m e d p a r e n t W e i b u l l d i s t r i b u t i o nf u n c t io n , s i m u l a te d h o u r l y m e a n w i n d s p e e d d a t a f o r 20 0y e a r s w a s g e n e r a t e d . T h e p a r a m e t e r s c~ a n d k o f t h ea s s u m e d p a r e n t d i s t r i b u t i o n ( f o r t h e u n c o r r e l a t e d d a t a )w e r e s e l e ct e d b y t r i a l a n d e r r o r , a n d r e s u l t e d i n a p a r e n td i s t r i b u t i o n o f t h e c o r r e l a t e d d a t a w i t h p a r a m e t e r sc t = 2 .72 m s - 1 an d k = 1 .59, show n in Figure 4. O n l y t h ef i rs t 1 00 h o u r s o f t h e m e a s u r e d a u t o c o r r e l a t i o n f u n c t i o nw a s u s e d . T h e f i rs t 1 0 0 h o u r s o f t h e a u t o c o r r e l a t i o nf u n c t i o n o f t h e s i m u l a t e d d a t a i s a l m o s t i d e n t i c a l w i t ht h a t s h o w n i n Figure 2 . A r e s u l t i n g r o o t - m e a n - s q u a r ew i n d s p e e d o f 2 . 3 5 m s - 1 w a s o b t a i n e d , a n d a m e a nc y c l i n g r a t e o f 2 . 92 c y c l e s / d a y . T h i s m e a n c y c l i n g r a t e i sn o t a s c l o se t o t h e m e a s u r e d 2 . 3 6 c y c l e s /d a y o b t a i n e d f o rJ a n Smuts a s one w ou ld l i ke , bu t i s s t i l l s a t i s fa c to ry fo ra r b i t r a r y w i n d d a t a .

a n dF, ,(v) = e -v + r (2d)

w h e r e v + ( v ) = ( 2 ) Vo ~ ~ e x p - 2 ~( 2 e )

I n E q u a t i o n s ( 2 a )- ( 2 d) , F o (v ) i s t h e c u m u l a t i v e d i s t r i b u -t i o n f u n c ti o n o f t h e a n n u a l m a x i m u m w i n d s p e ed . E q u a -t ion (2c ) i s de r ive d on the a s s umpt ion tha t s uc c e s s iveh o u r l y m e a n w i n d s p e e d s a r e s t a t i s t i c a l l y i n d e p e n d e n t( w i t h 8 7 6 0 = 3 65 2 4 , b e i n g t h e n u m b e r o f h o u r s p e rye a r ) . Equ a t io n (2d ), i n w h ic h v (v) i s t he a ve ra ge n um be ro f u p c r o s s i n g s o f v a n d T e q u a l s o n e y e a r , i s b a s e d o n aP o i s s o n a p p r o x i m a t i o n . D e t a il s o f t h e d e r i v a ti o n o fe q u a t i o n s ( 2 c) a n d ( 2 d) c a n b e f o u n d i n R e f e r e n c e s 1 a n d 2 .In Figure 3 i t is s e e n tha t t he f i t o f e qu a t io ns (2c ) a nd( 2 d) t o t h e o b s e r v e d d a t a i n t h e t a i l o f d i s t r i b u t i o n i ss a t i s f a ct o r y f o r J a n S m u t s A i r p o r t , b u t f o r m o s t o t h e rs t a t i o n s s u c h a g o o d f i t c o u l d n o t b e o b t a i n e d , s '6 I t i sa l s o s e en t h a t t h e T y p e - I e x t r e m e - v a l u e d i s t r i b u t i o n i n vo v e r e s t i m a t e s t h e a n n u a l m a x i m u m w i n d s p e e d w i t hi n c r e a s i n g r e t u r n p e r i o d a s c o m p a r e d t o e q u a t i o n s ( 2 b ) ,( 2 c ) a n d ( 2 d ) . T h i s p h e n o m e n o n h a s b e e n e x p l a i n e d i nR e fe re nc e s 3 a nd 1 .

Annual maximum wind speedsA n n u a l m a x i m u m w i n d s p e e d s a r e n o w i n v e s t i g a te d u s in gt h e s i m u l a t e d w i n d s p e e d d a t a . I n Figure 5 a c o m p a r i s o ni s m a d e b e t w e e n t h e m e a s u r e d a n n u a l m a x i m a i n t h e2 0 0 - y e a r p e r i o d a n d t h e b e s t fi t e q u a t i o n s f o r t h e T y p e - Iin v , t he Type - I i n v 2 , a n d t h e d e r i v e d d i s t r i b u t i o n

3O2O

Ev -. IOI

Figure 4

{ 3 I I I I I i0 . 5 I 0 - I I 0 2 I 0 - 3 1 0 -5 1 0 - 7P ro b o b i l i t y P v { v )

S i m u l a t e d p a r e n t d i s t r i b u t i o n f u n c t i o n

6 6 E n g . S t r u c t . 1 9 8 8 , V o l . 1 0 , J a n u a r y

7/29/2019 Numerical Simulation Of

3/3

30

E 20

a o I 0

Num er ica l simulat ion of annu al ma x imum win d speeds: R . V. Mi l ford3o

0 I i I I I0 . I 0 5 0 , 9 0 . 9 9 0 . 9 9 9P ro b o b i l i t y ' ~ v (~ ')

F igure 5 A n n u a l m a x i m u m w i n d s p e e d s u s in ~ s i m u l at e d d a ta .. . . . , T y p e d i n v ( e q u a t i o n ( 2 a ) ) ; - - , T y p e - I in v ( e q u a t i o n ( 2 b ) ) ; d e r i v e d ( e q u a t i o n ( 2 c ) )

2OE

~= 10

0 I I I I I0 .1 0 . 5 0 . 9 0 . 9 9 0 . 9 9 9

P ro b e b i l i ty Fv ( v )F igure 6 C o n v e r g e n c e o f d e r i v e d d i s t r i b u t i o n ( e q u a t i o n ( 4 ) ) t o -w a r d s s i m u l a t i o n d a t a f o r ~b = 0 . 0 1 , 0 .1 a n d 1 . 0

equa t ion (2c ). The P oi s son app roxim a t ion i s no t inc ludedin F i g u r e 5 as i t i s a lmost ident ical to the deriveddis t r ibu t ion of equ at io n (2c) . I t i s again veri f ied that th eTyp e- I d i s tr ibu t ion in v ove res t ima tes the annua l m aximawith increas ing re turn period. I t i s a lso seen that there isexce l len t agreement be tween eq ua t ion (2c) (and a l soequa t ion (2d)) and the Type - I d i s tr ibu t ion in v2, a l tho ughth i s compar i son i s on ly va l id for an in i t i a l d i s t r ibu t ionwith k -- 2.0.The va l id i ty of the a s sump t ion of st a t is t ica l i ndepend -ence of succes sive hou r ly me an w ind speeds can bees t ima ted b y equa t ing equa t ion s (2c) and (2d) . Rewr i t ingequa t ion (2c) a s

g v ( v ) = I 1 - P v ( / ) ) ] ~ 8 7 6 0 (4)i t fo l lows tha t ( ' ) 'l n [ - l n ~ ' .(v)] '-~ ~ I v k - (2~-~) ln(~b8760))]

(5a)which i s a Type- I d i s t r ibu t ion in v k . Rewri t ing equa t ion(2d) yields

1 k- - l n [ - - l n P v ( V ) ] ~ - - ( ~ ) I v k - - ( 2 ' ~ ) kk~Tv U k - 1x ln{(27r)" VoT ( ~ T ~ ) ( 2 - ~ ) } ] (5b,

which i s aga in a T ype- I d i s t r ibu t ion in v k . T h e m o d e t e rmin equa t ion (5b) i s s een to be dependent on the meanwind speed , bu t th i s dependence i s smal l. Subs t i tu t ing fora, k, v0 and av , us ing a mean va lue of v = 20 m s - 1 inequa t ion (5b) , and equa t ing the mode t e rms in equa t ions(5a) an d (5b) yield s ~b -~ 0.78. U sing ~b = 1.0 results in aconse rva t ive e s t ima te of the wind speed cor respo ndingto a g iven re turn pe r iod , bu t w i th an e r ror o f on ly abo ut2 % .The as sum pt ion of st a t is t ica l i ndepend ence i s a lsoinves t igated in F i g u r e 6 i n which the convergance ofequa t ion (4) towards the s imula ted da ta i s shown for~b = 1.0, 0.1 a nd 0.01 (o r ~bT = 8760, 876 an d 87). It isobse rved tha t l a rge e r rors occ ur for 4 ' mu ch l e ss than 1 .0 .

By compa r i son , i n R efe rence 2 a s imi la r, bu t n o tiden ti ca l, equ a t ion to eq ua t ion (1) was used for the in i ti a ld i s tr i b u t io n a n d t h e n u m b e r o f i n d e p e n d e n t o b s e r v a t io n swas then ob ta ined by equa t ing the re su l t ing equa t ionss imi la r to eq ua t ions (2c) and (2d) to a Ty pe- I ex t reme-value dis t r ibut ion in v ( i .e . not Vk) , resul t ing in ~ b- 0.1.Us ing the e s t ima tes for the mod e of an equiva len t Type- Idis t r ibu t ion in v, Reference 4 o btai ned ~b-~ 0.10, which,again, i s not cons is tent wi th the present s tudy. The resul tsof the presen t inves t iga t ion conf i rm the prev iou s a s sump-t ion tha t succes s ive hou r ly mea n w ind speeds can bet rea ted a s be ing s t a t i s ti ca l ly indep endent .D i s c u s s i o n o f r e s u lt s

The a s sump t ion of st a t is t ica l i ndepend ence o f succes sivehour ly mean wind speed obse rva t ions has been inves t i -ga ted us ing s imula ted wind speed da ta . The s imula tedwind speed da ta was gene ra ted us ing the measuredautocor re l a t ion func t ion a t a wea the r - record ing s t a t iona t J an Smu ts A i rpor t . On ly the f i rs t 100 hours o f theau tocor re l a t ion func t ion was used and hence long- t e rmseasonal effects are not s imulated. Notwiths tanding this ,the ana lys i s has conf i rmed tha t t he a s su mp t ion o f s t a t is -t i ca l i ndependence can be used when es t ima t ing theannua l maximum wind speed d i s t r ibu t ion func t ion .R e f e r e n c e s1 M i l fo rd , R . V. A nn ua l m ax im um w ind s peeds f rom in i t ia l d i s t r ibu -t ion func t ions . J Wind Engng Indus t r ia l Aer odynam ics , 1987 , 25 (2 ) ,163 1782 D a v e n p o r t , A . G . T h e d e p e n d e n c e o f w i n d l o a d s o n m e t e o r o l o g ic a lp a r a m e t e r s , Pr ec . I n t . Con fer ence on W ind Ef f ec t s on Bu i ld ings andStruct . , Ottawa, Canada, September 1967, 19--823 Cook , N . J. Tow ards a be t t e r e s t im a t ion o f ex t r em e w inds , J . W i n dEngng and Indus t r ia l Aer odynam ics , Mar ch 1982, 9 (3 ) , 295 -3204 G r igo r iu , M . Es t im ates o f des ign w ind f rom s ho r t r eco rds , J. S truct .D i v. , A S C E , M a y 1982, 1 0 8 ( S T 5 ) , 1 0 3 4 - 1 0 4 85 M i l fo rd , R . V . A nn ua l m ax im um w ind s peeds fo r S ou th A f r ica .The Civi l Engineer in South Afr ica, 1987, 25(1) , 15-196 M i l fo rd , R . V . Ex t r em e va lue ana lys i s o f S ou th A f r i can m e an hour lyw ind s peed da ta : I I . I n te rna l Repor t 85 /3 , Struct . a nd GeotechnicalEngng D iv . , Na t iona l Bu i ld ing Res ear ch In s t i tu te , CSIR , Pr e tor ia ,M a y 19857 S am aras , E ., S h inozu ka , M . and Ts u ru i , A . A R M A rep re s en ta t ion o f

r andom p roces s es , J . Engng Mech . D iv ., ASC E, M ar ch 1985, 111(3 )449-461

Eng . S t ruc t . 1988 , Vo l . 10 , Jan ua ry 87