TURKISH ATOMIC ENERGY AUTH O RITY
ÇEKMECE NUCLEAR RESEARCH AND
TRAINING
CENTER
I S T A N B U L - T U R K E Y
Technical Report N o: 36
APPLICATION OF BAYESIAN AN ALYSIS TO THE LOSSES OF OFFSITE POWER
FOR AK K U YU NPP SITE
Ulvi AD ALIO Ğ LU
Nuclear Engineering Department
December 1986
TURKISH ATOMIC ENERGY AUTHORITY
ÇEKMECE NUCLEAR RESEARCH AND TRAINING CENTER
ISTANBUL - TURKEY
Technical Report No: 36
APPLICATION OF BAYESIAN ANALYSIS TO
THE LOSSES OF OFFSITE POWER
FOR AKKUYU NPP SITE
Ulvi ADALIOĞLU
Nuclear Engineering Department
December 1986
SUMMARY
APPLICATION OF BAYESIAN ANALYSIS TO
THE LOSSES OF OFFSITE POWER FOR AKKUYU NPP SITE*
T h e d a t a on t h e o c c u r a n c e s and d u r a t i o n s of l o s s e s of o f f s i t e p o w e r f or A k k u y u N P P s i t e are a n a l y s e d by u s i n g
B a y e s i a n t e c h n i q u e . T h e m e a n f r e q u e n c y of l o s s e s a n d th e n o n r e c o v e r y p r o b a b i l i t i e s w i t h i n a t i m e i n t e r v a l a r e o b t a i n e d .
Ö Z E T
AKKUYU SANTRAL MAHALLİ İÇİN HARİCÎ GÜC
KAYBINA BAYES ANALİZİNİN TATBİKATI*
B a y e s a n a l i z t e k n i ğ i k u l l a n ı l a r a k A k k u y u s a n t r a l m a h a l l i i ç i n dış e l e k t r i k g ü c ü n ü n k a y b ı n a a i t t e k e r r ü r s a y ı l a n ve k a y ı p s ü r e l e r i n e ai t d a t a a n a l i z e d i l m i ş t i r . O r t a l a m a t e k e r r ü r s a y ı s ı ( f r e k a n s ı ) ve b e l l i b i r z a m a n a r a l ı ğ ı n d a e l e k t r i k g ü c ü n ü n g e l m e m e s i i h t i m a l l e r i h e s a p l a n m ı ş t ı r . * b a s e d on the w o r k s u p p o r t e d by I , A . E . A . u n d e r the c o n t r a c t N o .3 4 2 1 / R 2 / R B .
TABLE
OFCONTENTS
P a g e s 1. I n t r o d u c t i o n — --- --- --- I 2. M a t h e m a t i c a l B a c k g r o u n d — --- - — --- 1 2.1. B a y e s i a n A n a l y s i s --- --- ■-— 1 2.2. D i s t r i b u t i o n F u n c t i o n s --- 2 2 . 2 . 1 . P o i s s i o n D i s t r i b u t i o n --- 2 2 . 2 . 2 . G a m m a D i s t r i b u t i o n --- --- 3 3 . A p p 1 i ca t i on to A k k u y u S i t e — --- --- -— 3 3.1. D a t a on L O S P at A k k u y u --- --- 3 3.2. C u r v e F i t t i n g to D a t a --- 4 3 . 2 . 1 . F r e q u e n c y D i s t r i b u t i o n of L O S P --- 4 3 . 2 . 2 . P r o b a b i l i t y D i s t r i b u t i o n s of O f f s i t e P o w e r R e c o v e r y --- --- --- --- 9 4 . De s c r ip t ion of C o d e s ---- <--- --- --- -- 12 4.1. C o d e P O I S --- --- — --- --- --- 12 4 . 1 . 1 . I n p u t of P O I S — ---- - — --- 12 4 . 1 . 2 . D e s c r i p t i o n of S u b r o u t i n e s --- 13 4.2. C o d e G A M A --- --- --- — --- 14 4 . 2 . 1 . I n p u t of G A M A - - --- --- --- 14 4 . 2 . 2 . D e s c r i p t i o n of S u b p r o g r a m m e s --- - - 14 5. C o n c l u s i o n --- — --- --- --- - - --- 15 R e f e r e n c e s --- ---*--- --- --- 15 A p p e n d i x A - L i s t i n g of P O I S --- *--- 17 A p p e n d i x B - L i s t i n g of G A M A ---- --- ---■--- 20LIST
OFTABLES
P a g e s T a b l e 1- E n e r g y l o s s e s on 154 kV t r a n s m i s s i o n line b e t w e e n 1 8 .4 • 1 9 8 2 - 5 . 4 . 1 9 8 5 a r o u n d A k k u y u s i t e --- 5 T a b l e 2- E n e r g y l o ss e s on 38Ü kV t r a n s m i s s i o n line b e t w e e n 1 . 1 . 1 9 8 0 - 2 2 • 4 . 1 9 8 0 a r o u n d A k k u y u s i t e ---- --- 6 T a b l e 3- P o s t e r i o r d i s t r i b u t i o n of f r e q u e n c y of loss of o f f s i t e p o w e r for A k k u y u s i te ---- 8 T a b l e 4- P r o b a b i l i t y d i s t r i b u t i o n of r e c o v e r y t i me s for A k k u y u s i t e --- 11 T a b l e 5- P r o b a b i l i t i e s of f a i l u r e s to r e c o v e r o f f s i t e p o w e r f or s o m e r e a c t o r s i t e s --- 12 T a b l e 6- I np ut for P 0 1 S --- 12 T a b l e 7- I np ut for G A M A -- --- 14LIST
ofFIGURES
Fig, 1- Q ( A ) f u n c t i o n v e r s u s A --- 9 Fig. 2- Q (T ) f u n c t i o n v e r s u s t i m e --- 10I. INTRODUCTION
A n a d e q u a t e t r e a t m e n t of a s et of t e s t d a t a m u s t b e p e r f o r m e d , b o t h q u a l i t a t i v e l y a n d q u a n t i t a t i v e l y , f o r an e v a l u a t i o n of t h e i r r e l i a b i l i t y i m p l i c a t i o n s . T h e s t a t i s t i c a l i n f e r e n c e s n e e d b e s t e s t i m a t e v a l u e s of c e r t a i n p h y s i c a l l y i m p o r t a n t c o n s t a n t s t o g e t h e r w i t h s o m e c o n f i d e n c e i n t e r v a l s a s s o c i a t e d w i t h t h e s e v a l u e s . B o t h c l a s s i c a l a n d B a y e s i a n a n a l y s i s t e c h n i q u e s c a n b e u t i l i z e d to o b t a i n i n f o r m a t i o n f r o m t e s t d a t a . T h e B a y e s i a n a p p r o a c h h a s a d v a n t a g e t h a t it c a n b e u s e d f o r s m a l l to m o d e r a t e n u m b e r of t e s t d a t a . H o w e v e r it m a y a f f e c t a l l r e s u l t s b e c a u s e of t h e p r i o r a s s u m p t i o n s d u r i n g t h e i n i t i a l a s s e s s m e n t of a n a l y s i s . In t h i s r e p o r t t h e f r e q u e n c y of l o s s of o f f s i t e p o w e r a n d t h e n o n r e c o v e r y p r o b a b i l i t i e s , of o f f s i t e p o w e r w i t h i n c e r t a i n p e r i o d of t i m e s a r e o b t a i n e d b y a B a y e s i a n a n a l y s i s f o r A k k u y u 1 N P P . I n s t e a d of c o n s i d e r i n g w h o l e n a t i o n a l g r i d , l o c a l g r i d s y s t e m is a s s u m e d to p r o v i d e r e p r e s e n t a t i v e d a t a f o r t h e a n a l y s i s ( 1 ) . S m a l l p r o g r a m m e s a r e w r i t t e n f o r c u r v e f i t t i n g p r o c e s s e s in o r d e r to o b t a i n p r o b a b i l i t y d e n s i t y f u n c t i o n s a n d c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n s .2. MATHEMATICAL BACKGROUND
2 .1 . B a y e s i a n a n a l y s i s (2) T h e B a y e s t e c h n i q u e s u i t a b l e to a n a l y s e s m a l l to m o d e r a t e n u m b e r of t e s t d a t a a s s i g n s a p r i o r p r o b a b i l i t y d i s t r i b u t i o n f o r d a t a , a n d a p o s t e r i o r p r o b a b i l i t y d i s t r i 2
b u t i o n is o b t a i n e d . This p o s t e r i o r d i s t r i b u t i o n b e c o m e s p r i o r d i s t r i b u t i o n for the n e x t set of d a t a if the t e s t s a r e
p e r f o r m e d to u p d a t e the da ta . The c y c l e d e s c r i b e d r e p e a t s i t s e l f . Th e s e l e c t i o n of p r i o r d i s t r i b u t i o n is the c r i t i c a l p a r t in B a y e s i a n a p p r o a c h . S e v e r a l c a n d i d a t e d i s t r i b u t i o n s m a y be t e s t e d in o r d e r to f i n d b e s t r e p r e s e n t a t i v e of a v a i l a b l e da ta .
Le t f(A) be the p r i o r d i s t r i b u t i o n of the v a r i a b l e A, say, f a i l u r e f r e q u e n c y of an e v e n t . T h e c o n d i t i o n a l p r o b a b i l i t y P ( x ; n , A ) is d e f i n e d as the p r o b a b i l i t y of h a v i n g x f a i l u r e s in a g i v e n n u m b e r of t e s t s , n, fo r the g i v e n f a i l u r e f r e q u e n c y A. T he p o s t e r i o r d i s t r i b u t i o n p ( A ; n , x ) is g i v e n by p ( A ;n ,x) oo P ( x ; n , A) f ( A) / d A P ( x ; n , A ) f ( A ) o 2.2. D i s t r i b u t i o n f u n c t i o n s (1 ,3 ,4) B r i e f d e s c r i p t i o n s of s o m e c o n t i n u o u s p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n s w i l l be g i v e n h e r e . 2 . 2 . 1 . P o i s s o n d i s t r i b u t i o n Th is d i s t r i b u t i o n d e s c r i b e s p h e n o m e n a fo r w h i c h the a v e r a g e p r o b a b i l i t y of an e v e n t is c o n s t a n t a n d i n d e p e n d e n t of the n u m b e r of p r e v i o u s e v e n t s . F o r c o n s t a n t f a i l u r e rate , A, the m e a n n u m b e r of f a i l u r e s is t he p r o d u c t of A a nd the o b s e r v a t i o n t i m e t. Th e p r o b a b i l i t y of e x a c t l y r f a i l u r e s o c c u r r i n g in t i m e t is t h e n g i v e n by
3 P(r,t) ( X t ) rr -At e T he m e a n and v a r i a n c e of t are g i v e n by m 2 o 1/A 1 / A2 (2.1) (2.2) 2. 2. 2. G a m m a d i s t r i b u t i o n The G a m m a d i s t r i b u t i o n is the g e n e r a l c as e of c o n t i n u o u s P o i s s o n d i s t r i b u t i o n (or E r l a n g e n d i s t r i b u t i o n ) . The p r o b a b i l i t y d e n s i t y f u n c t i o n of G a m m a d i s t r i b u t i o n is , / . M r-1 -At f(t) = U r'C r T ---- X>0, r > ° (2>3)
w h e r e the n u m b e r r n e e d not be an i nt ege r.
The m e a n and v a r i a n c e of t are e x p r e s s e d by the eq uat ions
m = r / A
02 = r / A 2 (2.4)
3. APPLICATION TO AKKUYU SITE
3.1. D a t a on L O S P at A k k u y u
It was a l m o s t i m p o s s i b l e to o b t a i n the i n f o r m a t i o n ab ou t the o c c u r e n c e s of loss of o f f s i t e p o w e r for w h o l e
T u r k e y . H o w e v e r local d a t a w e r e e a s i l y c o l l e c t e d f r o m r e c o r d ings and a r e g i o n a l e v a l u a t i o n of f r e q u e n c y of L O S P w o u l d be good e n o u g h for an i n it i a l e s t i m a t e of a v e r a g e f r e q u e n c y of grid loss.
4 T h e A k k u y u N P P s i t e is g o i n g to be f e d by two d i f f e r e n t t yp e of, e n e r g y t r a n s m i s s i o n li n es . O ne is 154 k V l i n e a l r e a d y e x i s t s an d is s u p p l y i n g e n e r g y . T he o t h e r w i l l c o n s i s t s of s e v e r a l ties to the n e a r b y da m s or n a t i o n a l g r i d by 3 8 0 k V l i n e s . Th e a c q u i r e d d a t a ( 5 ) for the g r i d l o s s e s on t h e s e tw o k i n d s of t r a n s m i s s i o n l i ne s a re g i v e n in T a b l e s 1 an d 2. T o t a l n u m b e r of l o s s e s f or a p e r i o d of t o t a l 8 . 2 7 y e a r s e x p e r i e n c e d on t h e s e two p o w e r l i n e s is 205.
A s s u m i n g tha t the g r i d is 1 0 0% a v a i l a b l e , the f r e
q u e n c y of o f f s i t e p o w e r l o s s e s is f o u n d as 2 4 . 7 p e r y e a r . T h e a v e r a g e d u r a t i o n of l o s s e s is o b t a i n e d by a v e r a g i n g d a t a o v e r the n u m b e r of o c c u r e n c e s , th at is t = I t . f . / İ f £ (3.1) i i w h e r e t _• is the d u r a t i o n , f. is th e n u m b e r of o c c u r e n c e s . T h e 1 l i re s u 1 1 is 184 m i n u t e s . 3.2. C u r v e f i t t i n g of dat a 3 . 2 . 1 . F r e q u e n c y d i s t r i b u t i o n of L O S P A s s u m i n g a u n i f o r m p r i o r d i s t r i b u t i o n the p o s t e r i o r d i s t r i b u t i o n of f r e q u e n c i e s of g r i d l o s s e s is a c c e p t e d as a P o i s s o n d i s t r i b u t i o n f O ) (a t)rR - A T e ( 3 . 2 ) w h e r e T is the t o t a l o b s e r v a t i o n p e r i o d , R is the t o t a l n u m b e r of e v e n t s o c c u r r e d d u r i n g T.
5 T a b l e 1- E n e r g y l o ss e s on 154 kV t r a n s m i s s i o n line b e t w e e n 1 8 . 4 . 1 9 8 2 - 5 . 4 . 1 9 8 5 a r o u n d A k k u y u site D u r a t i o n p o w e r loss of > t İ N u m b e r of O c c u r e n c e s , f £ P e r c e n t C o n t r i b u t i o n 1 m x x x x x x 6.9 2 m x x x x x x 6 .9 3 m x x x x x x x x x x x 1 2 . 6 4 m x x x x x x x x x x x 1 2 . 6 5 m X X X 3.4 6 m x x x x 4.6 8 m X X X 3.4 9 m X C M i-H 10 m X 12 m X 15 m X 16 m X 18 m X 25 m X 27 m X 35 m X 39 m X 44 m X 4 6 m X 1 h 9 m X 2 h 19 m X 4 h 20 m X 4 h 48 m X 5 h 40 m X X 2.3 7 h 7 m X 7 h 9 m X 7 h 17 m X 7 h 30 m X 8 h 52 m X 8 h 53 m X 9 h 12 m X 9 h 13 m X 9 h 33 in X 10 h 7 m X X 2.3 10 h 11 m X 11 h 52 m X 12 h 14 m X 14 h 49 m X 18 h 40 in X 21 h 1 m X 21 h 13 m X 2 2 h 56 in X 27 h 27 m X 29 h 11 m X 29 h 42 m X 29 h 52 m X 33 h 32 m X 34 h 25 in X T o t a l 87 100
T a b l e 2- E n e r g y l o s s e s on 380 kV t r a n s m i s s i o n l i n e b e t w e e n 1.1.1980 - 2 2.4.1985 a r o u n d A k k u y u s i t e D u r a t i o n of N u m b e r of p e r c e n t p o w e r l o s s , t^ O c c u r e n c e s , f. C o n t r i b u t i o n 2 m X X 1.7 3 m X X X 2.5 4 m x x x x x 4.2 5 m x x x x x x x x 6.8 6 m x x x x x x x 5.9 7 m x x x x x x x 5.9 8 m X X X 2.5 9 m X X 1.7 10 m x x x x x 4.2 11 ra X X 1.7 13 TO X X X 2.5 14 m X X X 2.5 15 TO X X 1.7 16 m X 0.8 18 m X 0.8 19 m X X X 20 m X 21 m X 25 ID X X 26 m X 27 m X X 29 m X 30 m X 31 m X X 33 m X X 34 m X 35 m X X 36 m X X 38 m X 39 m X 40 m X 41 m X 43 TO X 44 m X 46 m X 49 m X 51 m X 52 m X 55 TO X 57 m X 1 h 4 ra X 1 h 5 m X 1 h 7 m X X 1 h 8 m X 1 h 9 m X 1 h 10 m X 1 h 15 m X 1 h 34 m X 1 h 42 m X 2 h 7 ra X 2 h 13 m X 2 h 57 m X 3 h 1 7m X 3 h 27 m X 3 h 29 m X 3 h 35 m X 3 h 59 m X 4 h 24 m X 4 h 36 m X 6 h 1 m X 6 h 7 m X 6 h 22 m X 6 h 54 m X 6 h 55 m X 7 h 58 m X 9 h 57 ra X 10 h 57 m X 11 h 16 m X 13 h 25 m X 26 h 22 m X T o t a l 118 100
F o r th e A k k u y u d a t a T = 8 . 2 7 y e a r s R = 205 e v e n t s (3.3) A c o m p u t e r p r o g r a m , c a l l e d P O IS, is w r i t t e n in o r d e r to o b t a i n f r e q u e n c y d i s t r i b u t i o n by E q . ( 3 . 2 ) . P r o g r a m c a l c u l a t e s 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 t i o n , F ( A ) , d e f i n e d as A F ( A ) = / d A f f ( A 1) (3.4) o an d c o m p l e m e n t a r y cumulative d i s t r i b u t i o n f u n c t i o n , Q ( A ) , w h i c h is g i v e n by Q(A) = 1 - F(A) (3.5) H a v i n g o b t a i n e d p r o b a b i l i t y d e n s i t y f u n c t i o n p r o g r a m P O I S a l s o c a l c u l a t e s m e a n v a l u e o f f r e q u e n c i e s by oo m = / dX X f (A) (3.6) o and the o t h e r m o m e n t s by oo u = / dA ( A - m ) n f(A) n = 2 , 3 , 4 , . . . (3.7) n o
Th e s e c o n d m o m e n t g i v e s the v a r i a n c e and the c o e f f i c i e n t s of s k e w n e s s and k u r t o s i s a re d e f i n e d by 2/ 3 y 3 / y 2 P2 y4 / u 2 ( 3 . 8 ) r e s p e c t i v e l y .
8 T a b l e 3 s u m m a r i z e s p o s t e r i o r d i s t r i b u t i o n f u n c t i o n as f u n c t i o n of f r e q u e n c y a nd c a l c u l a t e d m e a n v a l u e a n d v a r i a n c e an d 6 ^ an d ^ 2 c o e f f i c i e n t s . T a b l e 3 - P o s t e r i o r d i s t r i b u t i o n of f r e q u e n c y of loss of o f f s i t e p o w e r f or A k k u y u s i te
Cumulative Comple. Cum. Frequency Density Function Dist. Func. Dist. Func.
X(year f(X) F(X) Q(X)
0.1700 E+02 0.5678 E-06 0.4827 E-05 0 .1 0 0 0 E+01 0.1802 E+02 0.1899 E-04 0.1047 E-04 0.9999 E+00 0.1904 E+02 0.3289 E-03 0.1271 E-03 0.9998 E+00 0 .2 0 0 0 E+02 0.2827 E-02 0.1295 E-02 0.9987 E+00 0 .2 1 0 2 E+02 0.1643 E-01 0.9544 E-02 0.9905 E+00 0.2204 E+02 0.5893 E-01 0.4495 E-01 0.9550 E+00 0.2300 E+02 0.1314 E+00 0.1352 E+00 0.8648 E+00 0.2403 E+02 0.2076 E+00 0.3115 E+00 0.6885 E+00 0.2476 E+02 0.2292 E+00 0.4756 E+00 0.5244 E+00 0.2505 E+02 0.2267 E+00 0.5407 E+00 0.4593 E+00 0.2601 E+02 0.1801 E+00 0.7416 E+00 0.2584 E+00 0.2703 E+02 0.1039 E+00 0.8872 E+00 0.1128 E+00 0.2805 E+02 0.4478 E+01 0.9409 E+00 0.3906 E-01 0.2901 E+02 0.1576 E+01 0.9884 E+00 0.1164 E-01 0.3303 E+02 0.4076 E+02 0.9974 E+00 0.2601 E-02 0.3105 E+02 0.8322 E-02 0.9995 E+00 0.4693 E-03 0.3202 E+02 0.1514 E-03 0.9999 E+00 0.7647 E-04 0.3304 E+02 0.2037 E-04 0.9999 E+00 0.8423 E-05 0.3400 E+02 0.2558 E-05 0.1000 E+01 0
Means = 0.24785 E+02 Betal == 0.1240 E+00 Std. Dev. * 0.17356 E+01 Beta2 ■= 0.30827 E+01
< V I > 0 9 F i g . l s h o w s c o m p l e m e n t a r y 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 t i o n , Q ( A ) , v e r s u s f r e q u e n c y . 3 . 2 . 2 . P r o b a b i l i t y d i s t r i b u t i o n s of o f f s i t e p o w e r r e c o v e r y T he o b s e r v e d d a t a for r e c o v e r y t i m e s w e r e u s e d to o b t a i n e s t i m a t e s of m e a n and v a r i a n c e by the e q u a t i o n s t n = y . t i f i / ^ f i (3.9) i i o^ = y ( t ^ - m ) ^ f j _ / £ f i (3.10) i i T h e c a l c u l a t e d v a l u e s ar e m = a 184 m i n 3 9 1 . 8 ( 3 . 1 1 )
10
A g a m m a d i s t r i b u t i o n , g i v e n by Eq. (2.3), a r e t h e n c o n s t r u c t e d to fit the m e a n and v a r i a n c e t h r o u g h the use of E q s . ( 2 . 4 ) . T he p a r a m e t e r s r and X are f o u n d to be as r = 0 . 2 2 0 5 5 X = 1 . 1 9 8 6 3 x l 0 -3 (3.12) C o m p u t e r p r o g r a m , n a m e d G A M A , is the s e c o n d s m a l l c o d e w r i t t e n to o b t a i n p r o b a b i l i t y d e n s i t y d i s t r i b u t i o n , and 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 t i o n s by m e a n s of E q . ( 2 . 3 ) and E q s . ( 3 . 4 - 3 . 8 ) . T a b l e 4 s u m m a r i z e s d i s t r i b u t i o n f u n c t i o n s , an d m e a n v a l u e and s t a n d a r t d e v i a t i o n of r e c o v e r y t i m e s . T h e m e a n and s t a n d a r t d e v i a t i o n o b t a i n e d f r o m d i s t r i b u t i o n f u n c t i o n s are m = 1 8 5 . 5 m i n a = 3 2 1 . 0 F i g . 2 is the c o m p l e m e n t a r y 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 t i o n for r e c o v e r y t im es . T h e p r o b a b i l t y t h a t r e c o v e r y t a k e s m o r e t h a n a g i v e n t im e p e r i o d is f o u n d f r o m thi s C C D F . T a b l e 5 g i v e s so me of the p r o b a b i l i t i e s w h i c h ar e the Q(t) v a l u e s t o g e t h e r w i t h t h o s e g i v e n for so m e B W R t y p e p l a n t s (6 ). T < M I N U T E S ) Fig.2- Q(T) function versus time
11 T a b l e 4- P r o b a b i l i t y d i s t r i b u t i o n of r e c o v e r y t i m e s for A k k u - y u s i t e Recovery Time t(min.) Density Function f(t) Cumulative Dist. Func. F(t) Comple. Cumu. Dist. Func. Q(t) 0 .1 0 0 0 E+00 0.3820 E+00 0.1910 E-01 0.9809 E+00 0.7666 E+00 0.7802 E-01 0.1186 E+00 0.8814 E+00 0.9433 E+01 0.1092 E-01 0.3195 E+00 0.6805 E+00 0.2009 E+02 0.5922 E-02 0.4042 E+00 0.5957 E+00 0.3009 E+02 0.4311 E-02 0.4549 E+00 0.5450 E+00 0.6009 E+02 0.2426 E-02 0.5507 E+00 0.4492 E+00 0.9009 E+02 0.1707 E-02 0.6118 E+00 0.3882 E+00 0.1 2 0 1 E+03 0.1316 E-02 0.6568 E+00 0.3431 E+00 0.1501 E+03 0.1067 E-02 0.6925 E+00 0.3075 E+00 0.1841 E+03 0.8737 E-03 0.7255 E+00 0.2745 E+00 0.2401 E+03 0.6642 E-03 0.7683 E+00 0.2317 E+00 0.3001 E+03 0.5195 E-03 0.8037 E+00 0.1963 E+00 0.3601 E+03 0.4194 E-03 0.8318 E+00 0.1682 E+00 0.4201 E+03 0.3461 E-03 0.8548 E+00 0.1452 E+00 0.4801 E+03 0.2903 E-03 0.8739 E+00 0.1261 E+00 0.5401 E+03 0.2464 E-03 0.8901 E+00 0.1099 E+00 0.6001 E+03 0.2113 E-03 0.9038 E+00 0.9618 E-01 0.7001 E+03 0.1662 E-03 0.9226 E+00 0.7733 E-01 0.8001 E+03 0.1328 E-03 0.9376 E+00 0.6939 E-01 0.9001 E+03 0.1075 E-03 0.9496 E+00 0.5037 E-01 0 .1 0 0 0 E+04 0.8785 E-04 0.9594 E+00 0.4059 E-01 0 .1 1 0 0 E+04 0.7235 E-04 0.9674 E+00 0.3257 E-01 0.1 2 0 0 E+04 0.5997 E-04 0.9741 E+00 0.2595 E-01 0.1300 E+04 0.4998 E-04 0.9795 E+00 0.2044 E-01 0.1400 E+04 0.4185 E-04 0.9842 E+00 0.1584 E-01 0.1500 E+04 0.3517 E-04 0.9880 E+00 0.1198 E-01 0.1600 E+04 0.2967 E-04 0.9913 E+00 0.8729 E-01 0.1700 E+04 0.2511 E-04 0.9940 E+00 0.5984 E-02 0.1800 E+04 0.2130 E-04 0.9963 E+00 0.3658 E-02 0.1900 E+04 0.1811 E-04 0.9983 E+00 0.1682 E-02 0 .2 0 0 0 E+04 0.1544 E-04 0.1000 E+01 0 .0
Means = 0.1855 E+03 Betal == 0.7046 E+01 Std. Dev. = 0.3210 E+03 Beta2 == 0.1068 E+02
- 12 T a b l e 5 — P r o b a b i l i t i e s of f a i l u r e s to re c o v e r o f f s i t e p o w e r fo r s o m e r e a c t o r s i t e s Cases Millstone Point 1 Grand Gulf 1 Limerick G.S. Akkuyu 1 Failure to recover within 1 /2 h 0.43 0 .2 - 0.545
II 11 II It 1 h - - - 0.449 It II II II 2 h 0.24 - 0.365 0.343 II II It II 4 h - - 0.158 0.232 II II II It 15 h - 9xl0~4 0.0504 II It II II 20 h 0.05 — — 2.6xl0-2 It II II It 24 h - — lxlO- 8 1.42xl0~2 II II II It 28 h - 0 .1 - 6.5x10 8
4. DESCRIPTION OF CODES
4.1. C o m p u t e r c o d e P O I S It is d e v e l o p e d to c a l c u l a t e p o s t e r i o r d i s t r i b u t i o n of f r e q u e n c y of los s of o f f s i t e p o w e r as m o d e l e d b y a P O I S S O N p r o c e s s , a n d b y a s s u m i n g a u n i f o r m p r i o r d i s t r i b u t i o n . 4 . 1 . 1 . I n p u t of P OI S T h e i n p u t d a t a are g i v e n as f o l l o w s : T a b l e 6 - I n p u t for P O I S No of Card Columns ProgrammeVariable Format Description
1 1-12 TL F12.5 Observation period, T
1 13-24 R 15 Tota1 number of failures in ' 1 25-29 NA 15 Number of points to be found
13 T a b l e 6 - (C o n t .) No of Card Columns Programme
Variable Format Description
1 30-35 NN 15 Number of intervals for integration (S100)
2 1-12 XI E12.5 Beginning of frequency interval for distribution
2 13-24 X2 E12.5 End of frequency interval for distri bution
4 , 1 . 2 . D e s c r i p t i o n of S u b r o u t i n e s
M A I N p r o g r a m : POIS MAIN programme reads input cards, calculates Poisson probability density function at each point of NA. Then it calls Subroutine TINT to integrate density func tion within the interval between to conse cutive frequency points. It normalizes density function to obtain unit, total probability Cumulative distribution func tions are obtained. Finally it calls Subroutine MOMENT to calculate mean, vari ance, 3rc* and 4 ^ moments of frequency with respect to density functions, and the
coefficients 3^ and 3^*
S u b r o u t i n e T I N T : It integrates probability density function according to trapezoidal rule.
Subroutine MOMENT : It calculates moments of frequency with respect to density function.
S u b r o u t i n e S I M P : It uses Simpson’s rule for the calculation of moment s .
14 4.2. C o d e G A M A It c a l c u l a t e s the p r o b a b i l i t y d i s t r i b u t i o n of r e c o v e r y t i m e s by g a m m a d i s t r i b u t i o n c u r v e f i t t i n g . 4 . 2 . 1 . I n p u t of G A M A T h e i n p u t d a t a of G A M A are as f o l l o w s : T a b l e 7- I n p u t f or G A M A Card
No Columns Variable Format Description
1 1-5 NR 15 Observed No. of data
1 5-10 NA 15 Number of points to be found on distri bution curves (^5000)
1 10-15 NN 15 Number of intervals for integration t(<100)
2 1-15 F(J) E15.5 Frequency of duration 2 15-30 TX( J) E15.5 Duration of loss
NR n
~ Z L 1-12 XI E12.5 Beginning of frequency interval for
distribution
12-24 X2 El 2.5 End of frequency interval for distribu tion
4.2 . 2 . D e s c r i p t i o n of p r o g r a m m e
M A I N p r o g r a m m e : It is essentially same programme of POIS MAIN
e x c e p t t h a t i t f i r s t calculates the X and R p a r aine t e i s o f G amma f unc t ion f r om the
o b s e r v e d d a t a o f l o s s o f offsite power, and
Gamma f u n c t i o n .
The s u b r o u t i n e s T IN T, M O M E N T and S I M P a re d e s c r i b e d in s u b c h a p t e r 4 . 1 . 2 .
15
5. CONCLUSION
T he loss of g r i d p o w e r w h i c h was v e r y f r e q u e n t d u r i n g th e p a s t y e a r s in T u r k e y is one of the m o s t i m p o r t a n t a c c i d e n t i n i t i a t o r s in n u c l e a r p o w e r p l a n t s . T h e d a t a for the r e g i o n a l g r i d a r o u n d A k k u y u N P F s i te are a n a l y s e d a c c o r d i n g to the B a y e s i a n t e c h n i q u e . T h e p o s t e r i o r d i s t r i b u t i o n of L O S P f r e q u e n c i e s for a u n i f o r m p r i o r .d i s t r i b u t i o n and G a m m a d i s t r i b u t i o n of r e c o v e r y t i m e s are f o u n d . M e a n f r e q u e n c y of l o s s e s a n d n o n - r e c o v e r y p r o b a b i l i t i e s are o b t a i n e d . T h e B a y e s i a n a n a l y s i s u s e d are s h o w n th a t it g i v e s th e d e s i r e d i n f o r m a t i o n a b o u t the L O S P i n i t i a t o r . M o r e r e a s o n a b l e r e s u l t s , e s p e c i a l l y m e a n f r e q u e n c y of L O SP , n e e d m o r e r e l i a b l e s t a t i s t i c a l d a t a fo r the g ri d l o s s e s . But the n o n - r e c o v e r y p r o b a b i l i t i e s f o u n d for A k k u y u s i te are c o m p a r a b l e to t h o s e g i v e n f or s o m e r e a c t o r s i te s .REFERENCES
1- P r o b a b i l i s t i c R i s k A s s e s s m e n t , L i m e r i c k G e n e r a t i n g S t a t i o n , P h i l a d e l p h i a E l e c t r i c C o . , M a r c h 1981, A p p . A . 2- N a n c y R . M a n n , Ray E . S c h a f e r , N o z e r D .S i n g p u r w a 1 l a , " M e t h o d s for S t a t i s t i c a l A n a l y s i s of R e l i a b i l i t y and L i f e D a t a " , J o h n W i l e y , 1974. 3- D r .L .L e d e r m a n of IAEA, J a n . 1986 ( p r i v a t e c o m m u n i c a t i o n ) . 4- N o r m a n J . M c C o r m i c k , " R e l i a b i l i t y and R i s k A n a l y s i s " , A c a d e m i c P r e s s , 1981.16
-5- T u r k i s h E l e c t r i c i t y A u t h o r i t y , D e p . of L o a d D i s t r i b u t i o n s and Sales, M a y 1985 ( p r i v a t e c o m m u n i c a t i o n ) .
6- U . A d a l ı o ğ l u , B . G . G ö k t e p e , F i n a l r e p o r t of C o o r d i n a t e d R e s e a r c h P r o g r a m m e on "The D e v e l o p m e n t of R i s k C r i t e r i a for the N u c l e a r Fuel C yc l e " , u n d e r c o n t r a c t No.: 3 4 21 / R B , A p r i l 1984.
17
Appendix A - Listing öf POIS
1 n 4 5 7 « w 9 İO 11 12 ■ib 14 15 16 17 is 19 2 0 21 22 2 3 24
25
26 2 7 26 29 30 31 32 33 34 35 36C POSTERIOR D I STRIBUTION OF LOSS OF ÖFFSITBT P O W E R G AS MODELED BY A P O ISSON PROCESS
C
IMPLICIT REALMS (A-H,0-Z> COMMON TL* R# CM* NN D IMENSION X (5001), PX <5001 >.0(5001 >. F<5001 > EXTERNAL LIBAEMULATE CALL L I B S t S T A K L I S H tülB SIMULATE) READ (5.10) T L . R . NA. Nü 10 FORMAT ( 2E12. 5, 215 > W R I T E <6. 13) TL, R. NA'. N N
12 FORMAT</5X. 'LAMDA*-', Efc2;J9, SX.','R**4,€flS. 5-5*. 'N A * % 15 l'NN=',I5> N0=NA+1 IR=R R E A D (5, 11) XI. <2 11 FO R M AT <2E 12. 5) W R I T E(t* 14) XI.X2
14 FORMAT ( /5X, /Xl*/VEl2. 5, 5X. X 2 = U E12L5)
DX*<X2-X1)/NA X < 1 >«X1 DO 2 I~2, MO X < I ) « X U - 1 ; + D X 2 CONTINUE G WRITE <6,89' 89 F O R M A T (1 İX, ' I . 12X, X 10X. 'PX ' ) I F ' R . C-T 20 ) GO TO 81 c
G CALCULATION Or GAMMA FUNCTION WHEN IR SMALL INTEGER
XIN*1 DO 5 U = i , IE ■XIN*XIN*IJ: 5 CONTINUE GM-XlN C DO 3 Ur- 1 , N O FT ~7L'* X •; -J) P X ( U > =FT-> ,îlR* C E X F < - F T ) ,' C-M C W R i TE İ 6, 9 3 > U . \ < -J > ■ F X L > ■J CONTINUE 93 F O R M A T ' 1OX, 14 $ X •2El2;5> GO TO 191 81 CONTINUE C
C CALCULATION OF ; GAM M A F U N C T ION AND ) DISTRI BUT I ON
C TOGETHER WHEN IR LARGE INTEGER
DO 42 I — 1 .NO X IN-1 FT*TL-< X ( I > D O 5J. L - 1 - IF: X.i U - X I N * F T « D E X F F-F'V/r.-’) /h 51 CONTINUE P X < I )= X I N
54 55 56 57 56 59 60 61 62 63 64 65 66 67
66
69 70 71 72 73 74 75 76 77 76 79 60 83. 62 83 8 4 85 8 6 87 80 89 90 Q 1 92 93 94 95 96 97 95 9? 100 101 102 100 104 105 106 107 106 109 18 C WRITE ( fc* 93 > I. X<I).PX(I') 42 CONTINUE C 191 CONTINUE C W R I T E <6
, 90)90 FORMAT! / I IX, ' I S 11X, XXI V9X, 'XX2',BX, ' F ( l V )
1
=1
XX1=0. XX2=X(I> F(I>=PX(I>*X<I>/2. C W R I T E (6
, 91) I, X X I ,X X 2 , F (I > DO 4 1=2, NOxxi=x< :>-i)
XX.2=X< I ) CALL T I NT (XXI, X X2, F D I S ). F (I > =F •; I -1)+FDIS C W R I T E (6,91> I,XXI,XX2,F(I) 4 CONTINUE 91 F O R M A T(1
OX, 14. 5X. 3E12. 5) DO 105 1=1,NO P X ( I )=FX <I )/F (N O ; 105 CONTINUE DO 106 1=1,MO F ! l i = F ( D / F ( N ü ) Q < I ) = i- F >: I ) 106 CONTINUE W R I T E (6
,1 1 0
) 110 FORMAT(/10X,5H I •SX,5H T , 10X,5H PX , İ2X,5H F < I ) , 131
. 5H G<1
) ) DO 50 :--.= i,NQ W R I T E(6, 115) O'-., X(K> , PX(K>, F(K), Q(K) ) 50 CONTINUEİ İ 5 FORMAT ■ 12X, 15, 5XVE12 5, 5X, E12 5, 5X, E12. 5, 5X,-E12. 5)
GALL M O M E N T <X ,D X ,N O , P X ) STOP
END
S U B R O U 7INE TINTIXX.l, X X 2 ,YOR) IMPLICIT REALs
8
i A-H, 0~Z> COMMON 7L, R, CM. NH N T = N N + 1 IR=R .DDX=( ,■ X2
- X X1
» /NN K‘ ~ i YOR=0 FZ=0. Y=XX İ IF (R GT 20. ) 00 TO 15 11 CONTINUE FT=TL*Y PY=FT*«R*DEXP <-F T ) /CM V OR = YOR + (P Y+ PZ >*DDX •' 2 PZ=PY Y-=Y+DDX N = N+1
IF(N GT. NT)00
TQ13
. GO TO 11lie 111 112 1 1 3 114 115 1 1 6 117 116 1 1 9 120 121 122 1 2 3 1 2 4 12 5 12 6 127 126 129 13C 131 132 1 3 3 134 1 3 5 1 3 6 137 1 3 6 137 1 4 0 141 142 143 144 145 146 147 1 4 0 149 1 5 0 151 152 15 3 154 15 5 1 5 6 157 156 157 16 0 161 16 2 163 164 165 166 167 166 1 6 9 1 7 0 171 172 173 19 15 CONTINUE XIN=1. FT=TL*Y DO 18 1=1.IR XIN=XIN*FT*DEXP(- F T / R )/I 18 CONTINUE PY=XIN Y O R = Y O R + (P Y + P Z )*DDX/2. PZ=PY Y=Y+DDX N=M+1 I F (N . GT. NT) GO TO 13 GO TO 15 13 RETURN END SUBROUTINE M O M E N T (X ,DX,NO. P X ) IMPLICIT REAL*8 <A-H, Q-Z)
DIMENSION X (5001>.P X (5001).P Y 1 <5001), P Y2(5001>. PY3(5001) 1» PY4 < 5001)
CALL SIMP(PX, DX, NO# AMOO) DO 5 1=1.NO
P Y 1 (I>=X <I)*PX(I ) 5 CONTINUE
CALL SIMP(PY1* DX» NO# A M 0 1 ) DO 12 1=1.NO
PY2<I>=<X<I)-AM01)*#2-*PX(I ) P Y 3 (I ) = < X <I )- A M 0 1 >**3*PX<I> PY4(I)=(X(I>~AM01)**4*PX<I> 12 CONTINUE
CALL SIMP(PY2» DX# NO. AM02) CALL SIMP(PY3.DX. NO. AMG3) CALL SIMP(PY4.DX.NO# AMC4) B 1=AM03**2/(A M G 2 * # 3 ) B 2=A M 0 4 /(AM02**2) ST=DSQR T < A M 0 2 )
WRITE (6. 9) AMOO. AMO I'
9 F O R M A T (//5 X . 'TOPLAM IHIIMAL='. E12. 5.5X. 'ORTALAMA D E G E R = '.E l 2. 5> WRITE<6. 10) AMC2» AM03.AM04
10 FORMATC/5X. 'VARY A N S = E 1 2 . 5. 5X. 'SKEWNESS=',E12. 5, 15X. 'KURTOSIS=', El2. 5)
WRITE < 6 , 17) ST,Bl.B2
17 FORMAT ( /5X, 'STD. DEV. = '. E12. 5. 5X. 'BETA1= ', E12. 5, 5X# 'BETA2= ' 1, E 12. 5 >
RETURN
END
SUBROUTINE SIMPtPX.DX,NO,AINT) C INTEGRATION BY SIMPSON RULE
IMPLICIT REAL*B <A-H,0-Z> DIMENSION PX(5C01) N0G=(NG-1>/2 N01=N0ü-l PZC=0. PZ1=0. DO 10 1=1,N01 N I = 2 * H 1 PZC=PZC + P X < N I ) 10 CONTINUE DO 11 1=1,NOO NJ=2*I PZ1=PZ1+PX<NJ) 11 CONTINUE A I N T = (PX (1)+4.* P Z 1+2.*PZO+PX(NO>>*DX/3. RETURN END
20 A p p e n d i x B - L i s t i n g of G A M A
1
C C A L C U L A T I O N OF P R O B A B I L I T Y D I S T R I B U T I O N OF R E C O V E R Y n c. C T I MES DY G A M MA D I S T R I B U T I O N C U R V E FITTIN G3
IMPLICIT R E A L*8
<A-H,0-Z> 4 C O M M O N TL> R, GM, NN C w.* D I M E N S I O N TX(ICOO),X<5C01>, PX(5001),G< 5 0 0 1 ).F(5001) t W E X T E R N A L L I B $ E M U L A T E 7 C ALL L I B $ E S T A B L I S H ( L I B ^ E M U L A T E )e
W R I T E (6
.43) 9 43 F O R M A T ( / / 1 5 X , ' INPUT DATA '//) İC R E A D (5/8
) NR, NA, NN11
e
F O R M A T (315)1
£ W R I T E (6,32) N R , N A , N N13 32 F O R M A T (5X, 'OBSER. NO OF D A T A ® ', I5/5X, 'NO OF P O I N T S
14 1. ON DISTR. =', I5/5X, 'NO OF INTERVAL FOR INTEG. = ',
15 215/ )
16 R EAD ( 5, 7 > ( (F(U), TX(U> ), J=i, NR>
17 7 F O R M A T (4Eİ 5. 5) 1C W R I T E (
6
,33) ((F (J >, T X (J )),J® 1,N R ) 19 33 F O R M A T (10 X , ' O B S E R V ED D A T A '/<5 X , 4 E 1 5 5))20
R E A D (5, İl) XI, X221
11
F O R M A T (2E12. 5)22
W R I T E (6
, 142) XI. X223 142 FORMATS/5X, 'EE.GINN. OF VAR I A. INTER. =' , E12. 5, 5X,
24 1L 'END OF VAR IA INTER. = ', E12. 5)
25 W R I T E <
6
, 14)26 14 F O R M A T ( / // 1 5 X , ' OU T P U T DATA ',//)
27 c
2
C c C A L C U L A T I O N OF MEAN AND V A R I A N C E FROM O B S E R V E D29
c
DATA VALU ES 30 c 31 AF=0. 3 eC AME=0 33 DO 28 I = İ,NR 34 A F = A F + F( I ) 35 A M E = A M E + F(I ) * T X(I) 36 28 C O N TINUE 37 A M E = A M E / A F 30 AVAR=0. 39 DO 29 1=1,NR 40 A V A R = A V A R + F < I ) * < T X ( I>- A M E)**2 41 29 CONT I N U E 42 A V A R = A V A R / A F 43 W R I T E <6
, İ 13 ) AME, AVAR44 113 FORMAT < 5 X , 'MEAN® ' • Eİ2. 5, 5X, 'VARIANCE® ', E12. 5)
45 T L * A M E / A V A R 46 R = A M E * T L 47 W R I T E i t , 13) T L , R 43 13 F O R M A T ( /5X, 'LAMDA® ', El 2. 5, 5X, 'R=',E12. 5 > 49 NQ=WA+i 50 D X ®<X 2 - X 1>/NA 51 52 c C A L C U L A T I O N OF GAMMA FUN C T I O N 53 Z=R
54 55 Si-57 5G 59 6C 61 62 63 64 65 66 67 6 6 69 70 71 72 73 74 75 76 77 76 79 BC ei 82 83 84 85 86 87 86 89 90 91 92 93 94 95 96 97 96 99 1 0 0 101 1 0 2 103 1Ö4 105 106 107 1 0 8 109 n o 21 C 0 X 0 = 1 . 5748646*Z+. 9512363*Z**2-. 699 S5S 8 * Z * * 3 C 1+. 4 2 4 5 5 4 9 * Z * * 4 - . 1010678*Z**5 G X C - 1 . 577191652*Z+. 9 8 8 2 0 5 8 9 1 * Z**2-. 8 9705 693 7*Z** 3 1+. 9 1 8 2 0 6 8 5 7 * Z * * 4 - , 756704078*Z**5+. 4821 9 9 3 9 4 * Z ** 6 2-, 1 9 3 5 2 7 8 1 8 * Z * * 7 + . 0 3 5 8 6 8 3 4 3 * Z * * 8 GM = G X C / Z C X(1)=X1 DO 2 1=2,NO X (I)=X <I - İ )+DX 2 C O N T I N U E WRITE(6< 89) 89 F O R M A T < 1 IX, ' I ', 12X, X', 10X, 'PX') DO 3 J — 1,NO FT=TL*X<J) PX( J)=7L*FT**(R-1. > -s-DEXP ( -FT > /GM C WRITE(6, 93) J, X<J),FX(J) 3 C O N T IN U E C 93 F O R M A T < 10X, 14, 5X, 2E12. 5) C W R I T E (6, 90) C 90 F O R M A T ( / ! 1 X, 'I 11X, 'XXI ', 9X, ' XX2 ', 8X, 'F < I ) ' ) 1 = 1 XX1=0. X X 2 = X (I ) F<I )=PX(I>*X<I)/2. C W R I T E (6,91) I , X X 1, XX2, F < I > DO 4 1=2,NO X X i = X < I - l > XX2=X C I )
C ALL T I N K XXI, XX2, FDIS) F < I ) = F ( I - l )+ F DI S C WR I TE (6,- 91) I, XXI, XX2, F<I> 4 CON T IN U E C 91 F O R M A T ( 10X, 14, 5X, 3E12. 5) DO 105 1=1,NO P X < I )=P X <I )/F <N O ) 105 C O N T I N UE DO 106 i=i,NO F t I > =F { I )./ F (NO ) G(I)=i-F<I) 106 C O N T I N U E W R I T E (6,110) 11C FORMAT (/10X, 5H I , 8X, 5H T , 10X, 5H PX , 12X, 5H F(I),13X 1, 5H Q (I ) ) DO 50 K = 1 ,N 0 W R I T E <6, 115) <K, X(K), PX<K),F<K>. Q(K)) 50 C O N T I N UE
115 F O R M A T ( 12 X , 15, SX, E12. 5, 5X, 0 2 . 5, 5X, E12. 5, 5X, E12. 5) C ALL MOMENT ( X.- DX, NO, PX )
STOP END S U E R 0 U 7 I NE T IMT < X XI, X X2, Y O R ) IMPLICIT R E A L * 8 (A-H,0-Z) C O M M O N T L , R,GM, NM N T = N N + 1 DDX=( XX2-XX 1 ) /NN
110 111 112 113 114 115 116 117 118 119 12C 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 136 139 14C 141 142 143 144 145 146 147 140 149 150 151 152 153 154 155 156 157 158 159 160 161 162 162 164 165 _ 22 _ N*1 Y0R-0. PZ-0. Y*XX1 11 CONTINUE FT-TL*Y PY«TL*FT**(R-1. )*DEXP(-FT)/GM VOR-YOR+(PY+PZ)*DDX/2. PZ-PY Y-Y+DDX N-N+l IF(N .CT. NT) CO TO 13 GO TO 11 13 RETURN END
SUDROUTINE MOMENT (X.-DX.NO*PX) IMPLICIT REAL*6 (A-H. 0-Z)
DIMENSION X (5001>, PX<5001), PY1(5001). PY2(5001>.PY3(5001) 1, PY4(5001>
CALL SIMPIPX. DX.NQ, AMOO) DO 5 1*1.NO
PY1(I)*X(I)*PX(I) 5 CONTINUE
CALL SIMP (PYl,DX,NO. AM01> DO 12 1*1.NO
P Y 2 (I >*<X( I>-AM01>«*2*PX<I >
PY3 <I>*(X(I)-AMQ1)**3*PX(I) PY4<I)-(X <Il-AMOl)«*4*PX<l> 12 CONTINUE
CALL SIMP(PY2,DX.N0. AM02) CALL SIMP(PY3» DX. NO* AM03) CALL SIMPİPY4» DX, NO» AM04) B 1 *AM03**2/ ( AMC'2**3) B2»AM04/(AM02*«2) ST*DSGRT(AM02)
WRITE(6* 9) AMOC.AMQl
9 FORMAT(//5X*.'TOPLAM İHTİMAL*', E12. 5, 5X, 'ORTALAMA DEĞER*', E12. 5) WRITEC6, 10) AM02, AM03.AM04
10 FORMAT! /5X« 'VARYANS*'. E12. 5, SX, ' S K E W N E S S * E l 2. 5. ‘ 15X, 'KURTOSIS*', Eİ2. 5)
WRITE(6.17) ST.B1.B2
17 FORMAT</5X. 'STD. DEV. *', E12. 5, 5X. 'BETA1*', E12. 5. 5X, 'BETA2*'
1 .E 1 2. 5)
RETURN
END-SUBROUTINE SIMP(PX, DX, NO,AINT)
C INTEGRATION BY SIMPSON RULE
IMPLICIT REAL«6 (A-H. 0-Z) DIMENSION PX(5C01) NOG*(NO-1)/2 NOl«NOD-1 PZO-O. PZ1-0. DO 10 1*1, N01 NI*2*I+1 PZO-PZO+PX(NI) 10 CONTINUE DO 11 1*1,NOO NJ«2*I PZ1»PZ1+PX(NJ) 11 CONTINUE AINT*(PX(l)+4. «P1İ42. *PZO+PX(NG))*DX/3. RETURN END
