Mathematical modelling of a nordic hydrological system, and the use of a simplified run-off model in the stochastic optimal control of a hydroelectrical power system

OF A S I M P L I F I E D R U N - O F F M O D E L IN THE S T O C H A S T I C O P T I M A L C O N T R O L OF A H Y D R O E L E C T R I C A L P O W E R S Y S T E M

M a g n e F i e l d ,

S t e i n L o c k w o o d M e y e r ,

S v e r r e A a m ~,

T h e E n g i n e e r i n g R e s e a r c h F o u n d a t i o n at the U n i v e r s i t y of T r o n d h e i m , The N o r w e g i a n I n s t i t u t e of T e c h n o l o g y (SINTEF), D i v i s i o n of A u t o m a t i c C o n t r o l , T r o n d h e i m , N o r w a y . U n i v e r s i t y of T r o n d h e i m , T h e N o r w e g i a n I n s t i t u t e of T e c h n o l o g y , D i v i s i o n of H i g h V o l t a g e S y s t e m s , T r o n d h e i m , N o r w a y . T h e N o r w e g i a n R e s e a r c h I n s t i t u t e of

E l e c t r i c i t y S u p p l y (EFI), T r o n d h e i m , N o r w a y .

i. I N T R O D U C T I O N

In the p l a n n i n g of l o n g - t e r m h y d r o e l e c t r i c p o w e r p r o d u c t i o n , a k i n d of s t o c h a s t i c o p t i m i z a t i o n w h e r e s o m e p a r t i c u l a r a s s u m p t i o n s on the b o u n d a r y c o n d i t i o n s are i m p l i c i t e l y p r e s e n t , are w i d e l y u s e d in N o r w a y ("The w a t e r v a l u e m e t h o d " , b a s e d o n the i n c r e m e n t a l c o s t p r i n c i p l e ) .

U s u a l l y the c a l c u l a t i o n is a l s o b a s e d o n the a s s u m p t i o n of no t i m e - c o r r e l a t i o n in the s t o c h a s t i c p a r t of the r u n - o f f , i.e. the w h i t e n o i s e a s s u m p t i o n . To g e t an i d e a of the e f f e c t of s u c h a s i m p l i f i c a - tion, it is of g r e a t i n t e r e s t to i n v e s t i g a t e the i m p o r t a n c e of

c o l o u r e d n o i s e in the r u n - o f f , i.e. the e f f e c t of d y n a m i c a l s t a t e s in the s y s t e m w h i c h g o v e r n s the r u n - o f f to the p r i m a r y c o n t r o l l e d h y d r o - e l e c t r i c w a t e r r e s e r v o i r s to be c o n t r o l l e d .

T h e f i r s t s t a g e in s u c h a p r o j e c t is the h y d r o l o g i c a l m o d e l - b u i l d i n g . S u c h a m o d e l m a y h a v e s e v e r a l p u r p o s e s , as:

a. A n a i d in the s i m u l a t i o n a n d b e t t e r u n d e r s t a n d i n g of the d y n a m i c s of h y d r o l o g i c a l s y s t e m s .

T h e m a i n p a r t of this w o r k w a s d o n e w h i l e the a u t h o r s w e r e w i t h the D i v i s i o n of A u t o m a t i c C o n t r o l at T h e N o r w e g i a n I n s t i t u t e of

T e c h n o l o g y .

b. R i v e r f l o w p r e d i c t i o n .

c. S i m u l a t i o n s for s u b - o p t i m a l h y d r o e l e c t r i c p o w e r s y s t e m s p l a n n i n g and p r o d u c t i o n .

d. In the c o m p u t a t i o n of s t o c h a s t i c o p t i m a l c o n t r o l laws of p o w e r p r o d u c t i o n .

In the c a s e a it is o b v i o u s l y p r e f e r a b l e to h a v e a m o d e l w h i c h is p h y s i c a l l y based, w h i l e this is n o t n e c e s s a r y for i n s t a n c e in the case d. In the l a t t e r case, a s i m p l e a b s t r a c t m o d e l w h i c h p o s e s s e s the m a i n d y n a m i c s is a p p r o p r i a t e , p a r t l y b e c a u s e of u n a v o i d a b l e u n - c e r t a i n t y in the long r a n g e all the same, and p a r t l y b e c a u s e of the d i f f i c u l t i e s e n c o u n t e r e d w h e n a p p l y i n g too c o m p l e x m o d e l s in o p t i m i -

zation.

2. H Y D R O L O G I C A L M O D E L L I N G

2.1. P r o c e s s c h a r a c t e r i s t i c s and the m u l t i l e v e l a p p r o a c h .

T h r e e k i n d s of m o d e l s of an I H D - r e p r e s e n t a t i v e b a s i n are p r e s e n t e d , w h e r e d i f f e r e n t d e g r e e s of c o m p l e x i t y are s u g g e s t e d . A l l of t h e m has a m u l t i l e v e l s t r u c t u r e . The f i r s t l e v e l c o n s i s t s of lumped,

i n t e r c o n n e c t e d n o n l i n e a r r e s e r v o i r s , w h e r e the w a t e r c o n t e n t s a r e the d y n a m i c a l s t a t e v a r i a b l e s . The s e c o n d level c h a n g e s some p a r a m e t e r s in the m o d e l w h e n the s t a t e s e x c e e d c e r t a i n d e f i n i t e v a l u e s , and in d e p e n d e n c e of some p a r a m e t e r s g o v e r n e d on the t h i r d level. F i n a l l D the t h i r d l e v e l g o v e r n s some of the p a r a m e t e r s a c c o r d i n g to t h e t e m p e r a - ture h i s t o r y . T h i s is n e c e s s a r y in N o r w a y b e c a u s e of the a l t e r n a t i n g c l i m a t i c c o n d i t i o n s .

C o n s i d e r a h y d r o l o g i c a l basin, as s h o w n in f i g u r e 1. The h y d r a u l i c i n p u t s / o u t p u t s are p r e c i p i t a t i o n (v2, n o t shown), c h a n n e l f l o w (qs) , g r o u n d w a t e r f l o w (qg) and e v a p o t r a n s p i r a t i o n (qe' n o t shown). The n o n - h y d r a u l i c i n p u t s or d i s t u r b a n c e s as t e m p e r a t u r e , w i n d and sun r a d i a t i o n a r e a l s o i n f l u e n c i n g the h y d r o l o g i c a l s y s t e m to a g r e a t e r or s m a l l e r extent.

It is d i f f i c u l t to m a k e a r e a s o n a b l y s i m p l e and g e n e r a l m o d e l of such a d i s t r i b u t e d - p a r a m e t e r s y s t e m l i k e a h y d r o l o g i c a l b a s i n .

A w i d e l y u s e d a p p r o a c h in f l o w s y s t e m s , for i n s t a n c e in c h e m i c a l e n g i n e e r i n g , is to a p p l y p h y s i c a l l u m p i n g of the system. H e n c e , w e s u b d i v i d e the b a s i n into p a r t i a l b a s i n s w h e r e the w a t e r s t o r a g e p a r t s

of the m o d e l are c o n s i d e r e d as s t i r r e d tanks. In this way, the sub- b a s i n s c a n m o r e e a s i l y b e a d a p t e d to g e n e r a l , p h y s i c a l l y b a s e d ,

m a t h e m a t i c a l m o d e l s . It is a s s u m e d t h a t the l u m p i n g is d o n e s u c h t h a t an a c c e p t a b l e a c c u r a c y in the d e s c r i p t i o n is o b t a i n e d for the a p p l i c a - t i o n in q u e s t i o n .

A t y p i c a l p a r t i a l b a s i n is s h o w n in f i g u r e 2. T h e c o m p o n e n t s vi, w h i c h t o g e t h e r w i t h qs and q g are c o n s i d e r e d to be the m a i n i n p u t s / o u t p u t s ( i n f l o w s / o u t f l o w s ) of the s y s t e m , are m e a s u r e d , qe(out) is the total e v a p o t r a n s p i r a t i o n E m 3 / d a y 3 . T h e v e c t o r Z is the

m e a s u r e m e n t v e c t o r ([ = Z ( ~ ) ) , w h i l e ~(out) is the o u t p u t s (outflows) f r o m the m o d e l .

O b s e r v e t h a t the f l o w s qs(in) and q g ( i n ) i n g e n e r a l m a y c o n s i s t of s e v e r a l c o n t r i b u t i o n s . F i r s t l y , we a s s u m e that the p a r t i a l b a s i n is s u f f i c i e n t l y h o m o g e n e o u s s u c h t h a t m e a n v a l u e s c h a r a c t e r i z i n g the d i s t u r b a n c e s , the s u r f a c e and the soil ( p r e c i p i t a t i o n , e v a p o t r a n s p i r a - tion, t e m p e r a t u r e etc.) @re g o o d a p p r o x i m a t i o n s . S e c o n d l y , w e a s s u m e t h a t the b a s i n is an u n c o n t r o l l e d , n a t u r a l b a s i n w i t h soil, i.e. u r b a n b a s i n s , g l a c i e r s and a r e a s w i t h n a k e d m o u n t a i n s o n l y are n o t c o n s i d e r e d . T h i s f o r m s the b a s i s of the p h y s i c a l l u m p i n g in the m o d e l - b u i l d i n g . T h e idea is of c o u r s e n o t n e w in h y d r o l o g i c a l m o d e l b u i l d i n g ; p h y s i c a l a p p r o x i m a t i o n a n d r e p r e s e n t a t i o n of u n d e r g r o u n d r e s e r v o i r s by t a n k

h a v e b e e n u s e d w i t h s u c c e s s E2~.

m o d e l s

T h e c r u s t of f r o z e n e a r t h and the s n o w d u r i n g the w i n t e r s e a s o n c o m p l i - c a t e a N o r d i c m o d e l , s i n c e the t e m p e r a t u r e and its h i s t o r y (the

t e m p e r a t u r e is in f a c t a s t a t e v a r i a b l e in a p o s s i b l y e n l a r g e d m o d e l of n a t u r e in this respect) is of i m p o r t a n c e for the d i s c h a r g e f r o m the b a s i n . A n o t h e r p r o b l e m is h o w the i n f i l t r a t i o n p r o g r e s s e s , b e c a u s e i n f i l t r a t i o n is n o t m e a s u r e d s y s t e m a t i c a l l y by the h y d r o l o g i s t s .

C o n s i d e r i n g the time a s p e c t , we are i n t e r e s t e d in a m o d e l e n c o m p a s s i n g the m o s t i m p o r t a n t l o n g - t e r m p r o p e r t i e s , s i n c e its p o t e n t i a l use is for e c o n o m i c a l d i s p a t c h of h y d r o e l e c t r i c p o w e r at long sight.

H o w e v e r , it o u g h t to h a v e a c e r t a i n d e g r e e of a c c u r a c y w i t h r e s p e c t to e s t i m a t e d r u n - o f f , s u c h t h a t p r e d i c t i o n e r r o r s i m p o r t a n t to the e c o n o m i c a l d i s p a t c h are r e a s o n a b l y w e l l m i n i m i z e d . E x p r e s s i o n s like this, a n d ' H e g r e e of a c c u r a c y " w i l l be g i v e n s p e c i a l a t t e n t i o n e l s e - w h e r e [5~.

It is s e e n t h a t the n a t u r e m a y be c o n s i d e r e d to f u n c t i o n like a

m u l t i l e v e l system. T h e c o m p l e t e s t r u c t u r e is i l l u s t r a t e d in f i g u r e 3.

In this p a p e r the ist level w i l l be r e p r e s e n t e d by a d y n a m i c a l w a t e r b a l a n c e system, w h i c h is a s s u m e d to be n o n l i n e a r and lumped.

Its s i m p l i f i e d m a t h e m a t i c a l r e p r e s e n t a t i o n in c o n t i n u o u s f o r m is the v e c t o r d i f f e r e n t i a l e q u a t i o n

= ~(~, v2(k),v-~, ~(in)' ~1(k)' £2,~) (i)

and

~(out) = ~(~' ~(in)' 21 (k) , B2' ~) (2)

X = ~(~, ~(in)' ~ 1 ( k ) ' ~2, ~) (3)

H e r e v3 is the m e a n e v a p o r a t i o n d u r i n g the s p r i n g and the summer, v2 is p r e c i p i t a t i o n , ~(in) is the i n f l o w v e c t o r and p 1 ( k ) , P 2 are p a r a m e t e r v e c t o r s s t e e r e d f r o m the h i g h e r levels of the model. ~i is p i e c e w i s e c o n s t a n t in time, and is c h a n g e d d i s c r e t e l y in time w i t h fixed i n t e r v a l s . ~ is the u n k n o w n p a r a m e t e r v e c t o r (to be d e t e r m i n e d ) , and finally, ~ is the state v e c t o r , c o m p r i s i n g the v o l u m e s of w a t e r in the tanks of the model. ~(out) is the o u t f l o w v e c t o r , b e i n g a d i r e c t f u n c t i o n of the p a r a m e t e r s , i n f l o w and states, and [ is the m e a s u r e m e n t v e c t o r .

The second level c o n s i s t s of a s y s t e m g o v e r n i n g s t a t e - d e p e n d e n t p a r a m e t e r s P2,

P2 = P2 (X, Pl (k)) (4)

On the third level, the "seasons" are u s e d as "states", and these are g o v e r n e d by the t e m p e r a t u r e (vl) h i s t o r y , the latter b e i n g an input to the model. On this level, c e r t a i n t e m p e r a t u r e - d e p e n d e n t p a r a m e t e r s

~i are d i r e c t l y g i v e n by the s e a s o n v e c t o r ~0,

p~ (k) = Pl (P0 (k)) (5)

w h e r e a s the t r a n s i t i o n s of [0 are g i v e n by a H u f f m a n table, w h i c h f o r m a l l y m a y be w r i t t e n as

P0 (k+l) = h(vl(k) , P0 (k)) (6

The c o m p o n e n t s of £i and ~2 are of "on-off" type (zero and one).

A d i a g r a m i l l u s t r a t i n g the p o s s i b l e t r a n s i t i o n s of " s e a s o n s " is g l v e n in f i g u r e 4. The H u f f m a n table a p p r o x i m a t e s the d y n a m i c s and h y s t e r e s i s of the s e a s o n a l t r a n s i t i o n s . The c o m p o n e n t s of £0 are the "season",

a c o u n t i n g p a r a m e t e r to r e g i s t r a t e the T M E A N - d a y s p e r i o d and the i n t e g r a t e d t e m p e r a t u r e (in o r d e r to c a l c u l a t e its m e a n v l M E A N o v e r T M E A N days).

2.2. P a r a m e t e r o b s e r v a b i l i t y .

A l l p a r a m e t e r s of a p r a c t i c a l h y d r o l o g i c a l m o d e l c a n n o t be d e t e r m i n e d f r o m s i m p l e o b s e r v a t i o n s and s e l e c t i v e m e a s u r e m e n t s of s p e c i f i c p h y s i c a l p a r a m e t e r s . It is a l s o c l e a r t h a t s i n c e a h y d r o l o g i c a l m o d e l is a s i m p l i f i e d o n e of a d i s t r i b u t e d p r o c e s s , e v e n e x a c t k n o w l e d g e of p h y s i c a l p a r a m e t e r s is less v a l u a b l e , s i n c e such p a r a - m e t e r s in g r e a t e r or s m a l l e r e x t e n t w i l l l o o s e t h e i r p h y s i c a l i n t e r p r e t a t i o n in the a p p r o x i m a t e m o d e l . H e n c e , m a n y p a r a m e t e r s of the m o d e l h a v e to be a d j u s t e d o n the b a s i s of m e a s u r e d i n p u t / o u t p u t t i m e s e r i e s for the basin. T h e o u t p u t m e a s u r e m e n t s w i l l n o r m a l l y be r e l a t i v e l y f e w in n u m b e r c o m p a r e d to the n u m b e r of u n k n o w n p a r a m e t e r s , a n d the q u e s t i o n of s t a t e and p a r a m e t e r o b s e r v a b i l i t y [_]D] of n o n l i n e a r m o d e l s c o m e s h e a v i l y into the p r o b l e m o f s e n s i b l e m o d e l b u i l d i n g . T h i s q u e s t i o n has b e e n n e g l e c t e d in h y d r o l o g i c a l m o d e l b u i l d i n g . Of c o u r s e a y e s / n o a n s w e r to the o b s e r v a b i l i t y q u e s t i o n is v a l u a b l e . H o w e v e r , for p r a c t i c a l d e s i g n of a m o d e l , i n f o r m a t i o n a b o u t h o w

o b s e r v a b l e the m o d e l is, is e q u a l l y i m p o r t a n t . I n f o r m a t i o n a b o u t this m a y for i n s t a n c e be o b t a i n e d f r o m the c o v a r i a n c e of the p a r a m e t e r e s t i m a t i o n e r r o r of an e s t i m a t i o n a l g o r i t h m E ~ , E ~ " T h i s p r o b l e m w i l l n o t be t r e a t e d in this paper.

2.3. M o d e l A.

F o r the f i r s t level, this v e r s i o n is s h o w n in f i g u r e 5. (Level IA.) S t a t e v a r i a b l e s a n d p a r a m e t e r s c a n as a r u l e be g i v e n a h y d r o l o g i c a l e x p l a n a t i o n , b u t this w i l l n o t be d o n e in d e t a i l here. H o w e v e r , in b r i e f w e h a v e as states:

xl: L a n d - s u r f a c e w a t e r s t o r a g e (water, ice, snow), x3: R e s e r v o i r s t o r a g e (lakes), r e f e r r e d to the d i s c h a r g e

t h r e s h o l d l e v e l

x4: A c c e s s i b l e soil m o i s t u r e

x 5 / x 6 : The p a r t of the g r o u n d w a t e r v o l u m e w h i c h d o e s n o t / d o e s i n t e r a c t w i t h the r e s e r v o i r s t o r a g e .

The p a r a m e t e r s K i (i = 1,2,...) m u l t i p l i e d by the v o l u m e s x i c o n t r i b u t e to the r a t e of c h a n g e of the v o l u m e s . H e n c e , a K i is in p r i n c i p l e the

i n v e r s e of a t i m e - c o n s t a n t . T h e s e p a r a m e t e r s d e p e n d on a n u m b e r of p h y s i c a l p a r a m e t e r s like area, c r u s t in the soil, the s p e c i f i c y i e l d of the soil, the s p e c i f i c h y d r a u l i c c o n d u c t i v i t y , h y d r a u l i c i n c l i n a t i o ~ d e p t h to b e d r o c k and the r o u g h n e s s a n d v e g e t a t i o n of the s u r f a c e .

T h e d i m e n s i o n l e s s p a r a m e t e r s G i (i = 1 , 2 , . . . , 7 ) are d i f f i c u l t to d e t e r - m i n e a p r i o r i , b u t t h e y are m a i n l y d e p e n d e n t on area. T h e p a r a m e t e r s A i c a n be d e t e r m i n e d d i r e c t l y f r o m a t o p o g r a p h i c a l map, s i n c e t h e y d e p e n d on a r e a only. T h e Q i - p a r a m e t e r s a r e d i m e n s i o n l e s s d i s t r i b u t i o n p a r a m e t e r s .

As is c l e a r f r o m f i g u r e 2, the m e a s u r e m e n t s in this s y s t e m are the g r o u n d w a t e r level, w a t e r stage in the r e s e r v o i r and the d o w n s t r e a m f l o w r a t e f r o m the r e s e r v o i r . H o w e v e r , the l a t t e r is p a r t l y r e l a t e d to the w a t e r stage. The m o d e l on l e v e l IA is thus g i v e n by 5 n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n s , 3 o u t p u t f l o w s g i v e n as f u n c t i o n s p f 5 s t a t e s and 3 inputs, and f i n a l l y 3 m e a s u r e m e n t s .

On the s e c o n d level (Level 2A) the v a l u e of the p a r a m e t e r v e c t o r

~2 = (BI,B3,B4,B6) is d e p e n d e n t of the s t a t e v e c t o r x and the p a r a m e t e r T v e c t o r ~I = (FI,F2,F3). T T h e c o m p o n e n t s of ~2 c h a n g e t h e i r v a l u e s w h e n the c o m p o n e n t s of x e x c e e d c e r t a i n t r e s h o l d v a l u e s , the "D"- p a r a m e t e r s .

On the t h i r d l e v e l (Level 3A), p o s s i b l e t r a n s i t i o n of the " s e a s o n " is d o n e e v e r y T M E A N days. W e f o u n d t h a t the r e p r e s e n t a t i o n of eq. (6) b y a H u f f m a n t a b l e w a s m o r e c o n v e n i e n t for the p r o b l e m a t h a n d t h a n a c u m b e r s o m e f o r m u l a t i o n w i t h d i s c r e t e - t i m e e q u a t i o n s c o n t a i n i n g l o g i c a l e x p r e s s i o n s . T h e m o t i v a t i o n for this l e v e l of the m o d e l , is the i n e r t i a in the t e m p e r a t u r e - d e p e n d e n t " p a r a m e t e r s " R a p i d t e m p e r a t u r e v a r i a t i o n s a f f e c t the h y d r o l o g i c a l s y s t e m v e r y little: The s p e c i f i c heat,

m e l t i n g and e v a p o r a t i o n h e a t of w a t e r are large, and s n o w is a good insulator, too. T h i s a l s o m e a n s t h a t the v a l u e and the d u r a t i o n of a p o s i t i v e t e m p e r a t u r e g r a d i e n t m u s t be l a r g e r to g e t the s y s t e m s w i t c h f r o m "winter" to "spring", t h a n t h o s e r e q u i r e d for a s w i t c h f r o m

"autumn" to w i n t e r " . T h e s e p h e n o m e n a are r e p r e s e n t e d by h y s t e r e s i s f u n c t i o n s . The e v a p o t r a n s p i r a t i o n is l a r g e r in the " s p r i n g " t h a n in the "autumn", b e c a u s e of the i n c r e a s i n g t e m p e r a t u r e and s i n c e l a r g e r a r e a s are c o v e r e d by w a t e r in the spring.

In this way, l e v e l 3A r e p r e s e n t s a p p r o x i m a t e l y the c o m p l e x d y n a m i c s of f r e e z i n g and m e l t i n g in the n a t u r e . A f i r s t o r d e r d i f f e r e n t i a l

e q u a t i o n d e s c r i b e s a p p r o x i m a t e l y the m e l t i n g (decay o f xl).

P a r a m e t e r o b s e r v a b i l i t y of m o d e l A.

If the S c h o e n w a n d t c r i t e r i o n for local o b s e r v a b i l i t y ~ is used, o b s e r v a b i l i t y c a n e a s i l y be t e s t e d for the m o d e l , s i n c e the m o d e l is p i e c e w i s e a n a l y t i c in the s t a t e s . A t e s t c a n be m a d e for e a c h of t h e s i t u a t i o n s o c c u r r i n g w i t h r e s p e c t to r e s e r v o i r l e v e l s v e r s u s the t h r e s h o l d v a l u e s . It is t h e n n o t s u r p r i s i n g t h a t the m o d e l A is n o t o b s e r v a b l e . T h e r e are 14 c o m p l e t e l y u n k n o w n p a r a m e t e r s and 5 s t a t e v a r i a b l e s to be e s t i m a t e d . In a d d i t i o n , it is to be n o t e d t h a t w e h a v e a s s u m e d that all the p a r a m e t e r s on the 3rd level can be f a i r l y w e l l rated, a n d t h a t the u n k n o w n " r e f e r e n c e v a l u e " HI (which is t h a t p a r t of the g r o u n d w a t e r r e s e r v o i r a s s u m e d n o t to i n f l u e n c e the d i s - c h a r g e f r o m it, see f i g u r e 5) c a n be r a t e d a p r i o r i .

T h e c o n c l u s i o n is t h a t the m o d e l has to be s i m p l i f i e d in o r d e r to g e t a m o d e l of a c o m p l e x i t y w h i c h m a t c h e s the a m o u n t of i n f o r m a t i o n g o t in this b a s i n .

It m a y a l s o be o b s e r v e d t h a t m o d e l IA is s i m p l e r t h a n the n o w w e l l - k n o w n S t a n f o r d W a t e r s h e d M o d e l [2].

2.4. M o d e l B.

F o r this v e r s i o n , the l e v e l s 2B and 3B are the same as 2A and 3A r e s p e c t i v e l y .

T h e ist level, level IB, is s h o w n in f i g u r e 6, and is a s i m p l i f i e d v e r s i o n of l e v e l IA. T h e p a r a m e t e r s and s t a t e s of this m o d e l c a n h o w e v e r to a less e x t e n t t h a n for m o d e l A be g i v e n a p h y s i c a l i n t e r - p r e t a t i o n , a p a r t f r o m the f a c t t h a t x s t i l l c o n t a i n s the " a v a i l a b l e "

w a t e r r e s o u r c e in the b a s i n . In p a r t i c u l a r , it is to be n o t e d t h a t the i n f i l t r a t i o n is n o t d e s c r i b e d by a d i f f e r e n t i a l e q u a t i o n in m o d e l B. G5 (= 1 - G6) e n c o m p r i s e s in o n e c o n s t a n t the s p e c i f i c h y d r a u l i c c o n d u c t i v i t y , s u r f a c e r o u g h n e s s a n d h y d r a u l i c i n c l i n a t i o n . A s s u m e n e w t h a t xl c a n be e s t i m a t e d f r o m m e a s u r e m e n t s of v2, or by a m e a s u r e - m e n t y4 u s i n g s n o w p i l l o w s . A s s u m e a l s o t h a t as m a n y of the p a r a m e t e r s as p o s s i b l e are r a t e d a p r i o r i w i t h g o o d a c c u r a c y , this i n c l u d e s all p a r a m e t e r s o n l e v e l 2-3. It t h e n t u r n s o u t t h a t the f o l l o w i n g s t a t e s and p a r a m e t e r s m u s t be e s t i m a t e d :

x2, x3, K4 (or K5), G3 (= 1 - G4), G5 (= 1 - G6) and GS.

If v2 ~ 0 or xl ~ 0 one c a n p r o v e by a p p l y i n g the S c h o e n w a n d t

o b s e r v a b i l i t y c r i t e r i o n t h a t m o d e l B is l o c a l l y o b s e r v a b l e in any state,

p r o v i d e d the w i n t e r s e a s o n is n o t p r e s e n t . T h i s a l s o a p p l i e s if A L 2 = 0 s u c h t h a t I T = (Yl, Y2)- D u r i n g the w i n t e r , it t u r n s o u t t h a t G5 is n o t o b s e r v a b l e .

S u c h p e c u l i a r i t i e s of a h y d r o l o g i c a l m o d e l m u s t be t a k e n into a c c o u n t if a s e q u e n t i a l s t a t e / p a r a m e t e r e s t i m a t o r is c o n s t r u c t e d , s i n c e n o n - o b s e r v a b l e p a r a m e t e r s w i t h i n c e r t a i n time i n t e r v a l s s h o u l d n o t be a d j u s t e d . T h i s w i l l n o t c a u s e a n y t r o u b l e to us, s i n c e b a t c h e s t i m a t i o n is used, s u c h t h a t the b e s t c o n s t a n t - v a l u e d set of p a r a - m e t e r s is found.

2.5. M o d e l C.

In o r d e r to c o m p a r e m o d e l B w i t h a s i m p l e r v e r s i o n w i t h r e s p e c t to the 3rd level, m o d e l C c o n t a i n s L e v e l IB and L e v e l 2A. On the t h i r d level, the H u f f m a n t a b l e is n o t i n c l u d e d , and " s e a s o n s " a r e m a d e d i r e c t l y d e p e n d e n t o n v l M E A N .

U n d e r the same c o n d i t i o n s as p u t o n m o d e l B, t h i s m o d e l is o b s e r v a b l e .

2.6. A d a ~ t i o n of the p a r a m e t e r s .

In o r d e r to g e t some f e e l i n g of the p r o b l e m s e n c o u n t e r e d in this f i r s t i n v e s t i g a t i o n , a s i m p l e b a t c h e s t i m a t i o n of the p a r a m e t e r s a n d s t a t e s w a s tried. A l t h o u g h it is o b v i o u s t h a t some of the p a r a m e t e r s d e p e n d on the c l i m a t i c c o n d i t i o n s in a m u c h m o r e s u b t l e w a y t h a n in the m o d e l s here, it is of i n t e r e s t to g e t an idea of h o w w e l l such l u m p e d m o d e l s c o u l d be f i t t e d to the m e a s u r e m e n t data. S i n c e m o d e l A is n o t o b s e r - v a b l e , the u n k n o w n p a r a m e t e r s and s t a t e s of the m o d e l s B and C w e r e a d a p t e d to m e a s u r e m e n t s f r o m a p a r t of the I H D - r e p r e s e n t a t i v e b a s i n

" S a g e l v a " . T h i s p a r t of the b a s i n , w h i c h is i l l u s t r a t e d in f i g u r e 7, is a small basin, b u t u n f o r t u n a t e l y not v e r y h o m o g e n e o u s .

The w e l l - k n o w n p r i n c i p l e of m a n y p a r a m e t e r e s t i m a t i o n s c h e m e s is s h o w n in f i g u r e 8, w h e r e ~ r e p r e s e n t s the four u n k n o w n p a r a m e t e r s (of m o d e l B) to be e s t i m a t e d . As a d j u s t m e n t s t r a t e g y a s i m p l e h i l l - c l i m b i n g m e t h o d h a s b e e n a p p l i e d ( " o n e - a t - a - t i m e " ) o v e r a d a t a i n t e r v a l of 2 y e a r s w i t h v e r y c h a n g i n g c l i m a t i c c o n d i t i o n s . (In a l a t e r w o r k ~ ] , a S I M P L E X s e a r c h m e t h o d i n c l u d e d in a b a t c h e s t i m a t i o n p r o g r a m for the U N I V A C 1108 ~ w a s used, b e i n g c o n s i d e r a b l y m o r e e f f i c i e n t . ) T h e loss f u n c t i o n a l to be m i n i m i z e d for o p t i m a l p a r a m e t e r v a l u e s w a s t a k e n as

~ 2 ( l Y l m ( t ) Yl(t) i + 8 lY2m(t) - Y2(t) l)dt tl

(7)

R e s u l t s f r o m a " b a l l i s t i c " s i m u l a t i o n f o r c i n g the m o d e l B w i t h the i n p u t d a t a o v e r 1 year, a r e s h o w n in f i g u r e 9. T is the m e a n t e m p e r a - ture d u r i n g 15 d a y s , a n d vl is p r e c i p i t a t i o n p e r day. xi, i = 1,2,3, are s i m u l a t e d w a t e r s t o r a g e s in the b a s i n , r e s p e c t i v e l y l a n d - s u r f a c e w a t e r s t o r a g e , g r o u n d w a t e r s t o r a g e and r e s e r v o i r s t o r a g e , yl is s i m u l a t e d g r o u n d w a t e r level, w h i l e y2 is s i m u l a t e d r e s e r v o i r w a t e r s t o r a g e level, y m i, i = 1,2, are the c o r r e s p o n d i n g m e a s u r e d levels.

W i t h the p a r a m e t e r s o b t a i n e d f r o m the e s t i m a t i o n , s o - c a l l e d r e c e s s i o n ("dry w e a t h e r " - ) c u r v e s w e r e s i m u l a t e d . T h e s e are s h o w n in f i g u r e i0.

H e r e qs is s u r f a c e d i s c h a r g e f r o m the g r o u n d w a t e r s t o r a g e . T h e y are b o t h s i m u l a t e d a c c o r d i n g to the t e m p e r a t u r e h i s t o r y shown. In a d d i t i o n p a r t s of r e c e s s i o n c u r v e s b e i n g c h a r a c t e r i s t i c of e a c h s e a s o n are p l o t t e d : qss d e n o t e s p u r e s u m m e r s u r f a c e d i s c h a r g e , q s a p u r e a u t u m n s u r f a c e d i s c h a r g e , and q s w c o r r e s p o n d i n g l y for the w i n t e r season.

S i m i l a r l y , e s t i m a t i o n a n d s i m u l a t i o n s w e r e p e r f o r m e d for m o d e l C, b u t the r e s u l t s w e r e less r e l i a b l e t h a n for m o d e l B u n d e r u n n o r m a l w i n t e r c o n d i t i o n s .

T h e c o n c l u s i o n is t h a t for a N o r d i c h y d r o l o g i c a l m o d e l it s e e m s

n e c e s s a r y w i t h s o m e k i n d of s e q u e n t i a l c o n t r o l of t e m p e r a t u r e - d e p e n d e n t p a r a m e t e r s , w h i c h a l s o in an a p p r o x i m a t e w a y t a k e s c a r e of the

d y n a m i c s of m e l t i n g and f r e e z i n g u n d e r d i f f e r e n t c o n d i t i o n s . It s e e m s w o r t h w h i l e to m a k e f u r t h e r i n v e s t i g a t i o n s on the b a s i s of a m o d e l h a v i n g a s t r u c t u r e like m o d e l B.

3. S T O C H A S T I C O P T I M I Z A T I O N OF H Y D R O E L E C T R I C P O W E R D I S P A T C H

3.1. S y s t e m d e s c r i p t i o n .

In the l o n g t e r m p l a n n i n g for the e c o n o m i c a l d i s p a t c h of h y d r o e l e c t r i c power, the o p t i m i z a t i o n i n t e r v a l o v e r w h i c h the g i v e n p e r f o r m a n c e f u n c t i o n a l is to be m i n i m i z e d (or m a x i m i z e d ) , u s u a l l y is in the r a n g e of a f e w m o n t h s to a b o u t o n e year. B e c a u s e of u n c e r t a i n t y in the f u t u r e r u n - o f f into the r e s e r v o i r s , a r e a s o n a b l e goal is to m i n i m i z e the e x p e c t e d v a l u e of the f u n c t i o n a l . Hence, w e w i l l h a v e to c o n s i d e r a s y s t e m m o d e l w h e r e the e n v i r o n m e n t a l m o d e l r e p r e s e n t i n g the r u n - o f f c o n t a i n s s t o c h a s t i c s t a t e v a r i a b l e s . S e e f i g u r e Ii, w h e r e w e h a v e

a. a m a t h e m a t i c a l p r o c e s s m o d e l for the p r o d u c t i o n s y s t e m , w i t h c o n t r o l v e c t o r ~ a n d s t a t e s (volumes) ~i,

b. a l u m p e d s t a t e v a r i a b l e m o d e l for the e n v i r o n m e n t (state v e c t o r x2), y i e l d i n g the r u n - o f f ~(~z) to the r e s e r v o i r s . T h e i n p u t to this m o d e l is an e x p e c t e d m e a n f u n c t i o n v o p l u s a w h i t e n o i s e s e q u e n c e Av w i t h a g i v e n d i s t r i b u t i o n (the p r e c i p i t a t i o n v = v O + Av).

In a d d i t i o n , t h e r e are g i v e n d a t a for the p o w e r d e m a n d , w h i c h p o s s i b l y a l s o m a y be d e c o m p o s e d like the p r e c i p i t a t i o n , in a m e a n v a l u e f u n c t i o n p l u s a s t o c h a s t i c term.

In N o r w a y it is u s u a l to d i v i d e the o p t i m i z a t i o n i n t e r v a l into sub- i n t e r v a l s of o n e w e e k , and u s e the s o - c a l l e d "water v a l u e m e t h o d " b a s e d on the i n c r e m e n t a l c o s t p r i n c i p l e . (A d e s c r i p t i o n of the b a s i c p r i n - c i p l e m a y be f o u n d in [9].) A n a n a l y s i s of t h i s a p p r o a c h w i l l s h o w t h a t the r u n - o f f is c o n s i d e r e d as p u r e s t o c h a s t i c (white noise) a r o u n d a d e t e r m i n i s t i c f u n c t i o n of time. C o n s i d e r i n g for i n s t a n c e f i g u r e 10, it is o b s e r v e d - e s p e c i a l l y d u r i n g the w i n t e r s e a s o n - t h a t s u c h an

a p p r o x i m a t i o n is less a c c u r a t e r e l a t i v e to the f i n e n e s s of the t i m e d i s c r e t i z a t i o n the s m a l l e r this d i s c r e t i z a t i o n i n t e r v a l is. T h e r e is c o n s i d e r a b l e d y n a m i c s in the r u n - o f f , w h i c h m a y be e x p r e s s e d by the a u t o c o r r e l a t i o n f u n c t i o n (in the l i n e a r case), or m o r e g e n e r a l l y , by a set of ist o r d e r d i f f e r e n t i a l e q u a t i o n s .

T h e d y n a m i c s w i l l s h o w up in the e v o l u t i o n of the p r o b a b i l i t y d i s t r i - b u t i o n , as s k e t c h e d in f i g u r e 12, w h i c h s h o w s the " s t a t i o n a r y "

p r o b a b i l i t y d i s t r i b u t i o n of Ar as a f u n c t i o n of time. In the linear, G a u s s i a n case, the e v o l u t i o n of the p r o b a b i l i t y d e n s i t y is u n i q u e l y g i v e n by the d i f f e r e n t i a l e q u a t i o n for the c o v a r i a n c e E { A r Z ( t ) } . To be m o r e s p e c i f i c , the c o m p l e t e s y s t e m m a y be f o r m u l a t e d as

!I (t) = fl (Xl (t), r(_x 2 (t)) , u(t) , t) _~2(t) = f 2 ( x 2 ( t ) , v(t), t)

(8)

(9) x(t) ¢ X (xiCt) e Xi) , uCt) ¢ U.

3.2. D i s c u s s i o n of the r u n - o f f m o d e l .

F o r l o n g - t e r m o p t i m i z a t i o n p r o b l e m s of the k i n d d i s c u s s e d h e r e it is o b v i o u s t h a t u n c e r t a i n t y is v e r y p r o n o u n c e d , as o b s e r v e d f r o m f i g u r e 12.

T h e r e s e e m s to be no p r a c t i c a l r e a s o n - at l e a s t for r e a s o n a b l y

h o m o g e n e o u s or s m a l l b a s i n s - to w o r k w i t h h i g h e r o r d e r r u n - o f f m o d e l s .

A n a b s t r a c t , Ist o r d e r l i n e a r m o d e l w i t h a t i m e - v a r i a b l e p a r a - m e t e r ( " t i m e - c o n s t a n t " ) e s t a b l i s h e d , say, o n the b a s i s of i n i t i a l

c o n d i t i o n r e s p o n s e s ( " r e c e s s i o n curves") of a m o r e c o m p l e x m o d e l like the r e s p o n s e s of f i g u r e i0, has the f o r m

xz(t) = - a ( t ) x 2 ( t ) + v(t) (I0)

w h e r e

v(t) = v0(t) + Av(t)

We m a y t h e n a s s u m e a l i n e a r r e l a t i o n s h i p b e t w e e n the e n v i r o n m e n t a l s t a t e x 2 of eq. (i0) and the r u n - o f f r,

r(t) = k - x z ( t ) = r0 (t) + Ar(t) (ii)

S u b s t i t u t i n g i n t o eq. (i0), w e h a v e

r(t) = - a ( t ) r ( t ) + k(v0(t) + &v(t)) (12)

T h e r e c e s s i o n f u n c t i o n is g i v e n b y the u n f o r c e d s o l u t i o n of eq. (12), t

- f a ( 8 ) d O

r(t) = r ( O ) - e o (13)

By l e t t i n g a(t) be a f u n c t i o n of time, it is p o s s i b l e to take i n t o a c c o u n t the e x p e c t e d m a i n s e a s o n a l c h a n g e s in the c l i m a t i c c o n d i t i o n s . A s e n s i b l e a p p r o x i m a t i o n is to a p p l y t h r e e d i f f e r e n t v a l u e s for a, t h e s e v a l u e s r e s p e c t i v e l y r e f e r r i n g to the w i n t e r season, the s n o w - m e l t i n g p e r i o d and the p e r i o d w i t h o u t snow, s n o w - m e l t i n g and frost.

1 is d e p e n d e n t o n the b a s i n , a n d is t y p i c a l l y b e t w e e n T h e time c o n s t a n t

I0 and 90 days, h a v i n g its l a r g e s t v a l u e d u r i n g the w i n t e r .

D u r i n g the s n o w m e l t i n g p e r i o d , the w a t e r f r o m the m e l t e d s n o w w i l l u s u a l l y be a d o m i n a t i n g p a r t of the r u n - o f f . A m a i n p a r t of this f l o w w i l l be d i s c h a r g e d into the r e s e r v o i r s f r o m the s u r f a c e .

In this work, no a t t e m p t is d o n e to m a k e use of an o p t i m a l a d a p t i o n of a(t) to the b e h a v i o u r of the b a s i n in q u e s t i o n .

It is q u i t e o b v i o u s t h a t i n e r t i a in the r u n - o f f d y n a m i c s is of g r e a t e r a n d g r e a t e r i m p o r t a n c e the s m a l l e r the r a t i o b e t w e e n r e s e r v o i r v o l u m e and i n t e g r a t e d r u n - o f f to the r e s e r v o i r t h r o u g h o n e y e a r is. F o r i n s t a n c e , if a r e s e r v o i r can a c c u m u l a t e on an a v e r a g e the r u n - o f f t h r o u g h 2-3 y e a r s (without d i s c h a r g e f r o m the r e s e r v o i r ) , it is o b v i o u s t h a t a d y n a m i c a l r u n - o f f m o d e l , c h a r a c t e r i z e d by a t i m e - c o n s t a n t of a b o u t a m o n t h , w i l l h a v e a l m o s t no e f f e c t on the e c o n o m i c a l d i s p a t c h

of such a system.

3.3.

A d y n a m i c a l d e s c r i p t i o n of the s t o c h a s t i c p a r t of the r u n - o f f i m p l i e s two e s s e n t i a l d i s t i n c t i o n s for the e c o n o m i c a l d i s p a t c h p r o b l e m , c o m - p a r e d to a r u n - o f f w h i c h is n o t c o r r e l a t e d in time.

a. I n s t e a d of u s i n g the " s t a t i o n a r y " d i s t r i b u t i o n of the r u n - o f f and p o s s i b l y c o n s i d e r it as w h i t e n o i s e , the d y n a m i c a l e v o l u t i o n of the r u n - o f f and its p r o b a b i l i t y d e n s i t y f r o m a g i v e n i n i t i a l c o n d i t i o n , is t a k e n care of ( p o s s i b l y w i t h a g i v e n u n c e r t a i n t y in the i n i t i a l

c o n d i t i o n ) .

~. S i n c e w e w o r k w i t h the e x p e c t e d e v o l u t i o n of the e n v i r o n m e n t a l states, t h e s e f u n c t i o n s and t h e i r a s s o c i a t e d d e n s i t y f u n c t i o n s are per d e f i n i t i o n g i v e n for the w h o l e o p t i m i z a t i o n i n t e r v a l . As is w e l l known, this w i l l in a c o n t r o l p r o b l e m r e s u l t in a r e a l i z a b l e " f e e d f o r w a r d "

c o u p l i n g f r o m the e n v i r o n m e n t a l s t a t e s to the c o n t r o l v e c t o r . F u r t h e r , t h e r e w i l l be a c o u p l i n g f r o m the r e s e r v o i r v o l u m e s to the c o n t r o l v e c t o r , w h i c h is the " f e e d b a c k p a r t " of the c o n t r o l law. (Of course, in a n o n l i n e a r p r o b l e m , t h e s e p a r t s c a n n o t be s e p a r a t e d , b u t the p r i n c i p l e is s t i l l there.) See f i g u r e 13.

To a p p l y s o l u t i o n by S t o c h a s t i c D y n a m i c P r o g r a m m i n g (S.D.P.), the s y s t e m e q u a t i o n s are u s e d in t h e i r t i m e - d i s c r e t e form. W i t h a d i s - c r e t i z a t i o n i n t e r v a l T, w e h a v e for a s i n g l e r e s e r v o i r ,

x I ((k+l)T) = x I (kT) - u(kT) + kx2 (kT) (14)

and for the e n v i r o n m e n t a l m o d e l

(k+l) T -a ((k+l) T-T]

X 2 ( ( k + l ) T ) = e -aT xz(kT) + [ e (Vo(T)+~V(T)) (15)

kT

If v(t) is c o n s i d e r e d c o n s t a n t w i t h i n the i n t e r v a l (kT, (k+l)T], a n d Av is t a k e n as a d i s c r e t e - t i m e w h i t e n o i s e s e q u e n c e , the l a t t e r e q u a t i o n s i m p l i f i e s to

i,. -aT,

x 2 ((k+l)T) = e - a T x 2 (kT) + ~ t ± - e ) (v 0 (kT) + Av(kT)) (16) To s i m p l i f y the notation, we w i l l in the s e q u e l u s e xi(k) for xi(kT) etc.

T h e o b j e c t i v e f u n c t i o n for the o p t i m a l c o n t r o l of t h e s y s t e m is as f o l l o w s . In N o r w a y it is c o m m o n l y a s s u m e d t h a t the m a r g i n a l i n c o m e s / e x p e n d i t u r e s d e p e n d e n t on the d i s p a t c h are a g i v e n f u n c t i o n

PF(Up(k) - u(k)), w h e r e PF is p r i c e per e n e r g y u n i t (ore/kWh). up(k) is p o w e r as o r d e r e d by c o n t r a c t f r o m c u s t o m e r s w i t h i n the o p t i m i z a - tion i n t e r v a l , and u(k) is the actual p o w e r p r o d u c t i o n . (GWh/month.) T h i s f u n c t i o n is o f t e n g i v e n as a s t a i r c a s e f u n c t i o n like the o n e in f i g u r e 14. T h e r e is h o w e v e r u n c e r t a i n t y in the f u t u r e p o w e r prices, so it m i g h t h a v e b e e n s e n s i b l e to take this u n c e r t a i n t y into c o n s i d e r a t i o n . In S.D.P. this can be d o n e w i t h o u t any d i f f i c u l t i e s , b u t w i t h an

i n c r e a s e in c o m p u t a t i o n time. In the e x a m p l e here, h o w e v e r , the s m o o t h c u r v e as shown on f i g u r e 14 has b e e n u s e d w i t h o u t u n d e r t a i n t y on it.

T h e e x p e n d i t u r e w i t h i n an i n t e r v a l Ek, k+l~ is Up (k) -u (k)

W k = f PF (~) d~ (17)

o

The o p t i m a l c r i t e r i o n is to m i n i m i z e the e x p e c t e d e x p e n d i t u r e s d u r i n g the o p t i m i z a t i o n i n t e r v a l (0,N) ,

N-I

E{J} = E{ ~ W k ( U p ( k ) - u ( k ) ) } (18)

k=0

As data, the f u n c t i o n s Up(.) and Vo(.) and the 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 p(~v) of ~v are given.

S i n c e the m a i n p u r p o s e h e r e is to o b t a i n a f e e l i n g of the i m p o r t a n c e of d y n a m i c a l m o d e l l i n g of the e n v i r o n m e n t of a h y d r o e i e c t r i c p o w e r s y s t e m for the e c o n o m i c a l d i s p a t c h , s t r a i g h t f o r w a r d S.D.P.

[13

t e x t b o o k of A o k i [i~. An a d v a n t a g e in such a p p l i c a t i o n s as this u s i n g D.P., is t h a t the state space is c o n s t r a i n e d b e c a u s e of m a x i m u m and m i n i m u m r e s e r v o i r v o l u m e s . Also, m a x i m u m / m i n i m u m v a l u e s for the r u n - o f f states m a y be r a t e d f a i r l y well. C o m p l i c a t e d o p t i m i z a t i o n c r i t e r i a i m p l y no d i f f i c u l t i e s . The m o s t s e r i o u s d r a w - b a c k s are the w e l l - k n o w n d i m e n s i o n a l i t y p r o b l e m and l o n g c o m p u t a t i o n time. The

s t o r a g e r e q u i r e m e n t s for r e a s o n a b l y l o w - o r d e r s y s t e m s (max. 4-5) m a y be s o l v e d by a p p l y i n g a m i x t u r e of d i f f e r e n t k i n d s of e x t e n s i o n s of

o r d i n a r y D.P. t e c h n i q u e s E7~, E8~.

3.4. E x a m p l e .

C o m p u t a t i o n of o p t i m a l c o n t r o l s for the f i r s t m o n t h in an o p t i m i z a t i o n i n t e r v a l of five m o n t h s in a c e r t a i n y e a r has b e e n d o n e u s i n g d a t a for a small p o w e r s t a t i o n in the m i d d l e of N o r w a y , n a m e d " J u l s k a r e t " . T h e

d a t a of the p r o d u c t i o n s y s t e m are:

P o w e r station.

M a x i m u m s t o r a g e c a p a c i t y : 60 m i l l . m 3

M e a n h e i g h t d i f f e r e n c e b e t w e e n p o w e r s t a t i o n a n d the r e s e r v o i r : M e a n e n e r g y c o n v e r s i o n : 4.17 m i l l . m 3 ÷ 1 G W h

M a c h i n e i n s t a l l a t i o n : 8 MW.

i00 m

T h i s g i v e s the c o n s t r a i n t s

0 ~ xl (k) ~ 14.4 (GWh)

0 ~ u1(k) ( 5.6 (GWh/month)

Up(k) is g i v e n in the f o l l o w i n g t a b l e (dim Up = G W h / m o n t h ) :

M o n t h : 1 2 3 4 5

Up: 2.7 1.9 4.6 4.4 3.9

T h e r u n - o f f s y s t e m .

T h e t o t a l p r e c i p i t a t i o n b a s i n for the s t a t i o n is A = 1 4 9 . 5 K m 2 = 149.5 x 106 m 2. The t i m e - c o n s t a n t for the r u n - o f f is e s t i m a t e d to T I = ~ = 1.2 m o n t h s o n the b a s i s o f a r e c e s s i o n c u r v e . 1 F o r s i m p l i c i t y , a -I is a s s u m e d c o n s t a n t . W e a s s u m e r = k ' x 2 = x2. T h e r u n - o f f e q u a t i o n w i t h d i m E x ~ = m 3, d i m Ev] = m / m o n t h , is

T T

AT1 TI

x2(k+l) = e T1x2(k) + 4 . i 7 ; ~ 0 6 ( i - e ) (Vo(k) + Av(k)) or

x2(k+l) = 0.434 x2(k) + 2 4 . 8 ( V o ( k ) + Av(k))

w h i c h is a s s u m e d v a l i d t h r o u g h o u t the o p t i m i z a t i o n i n t e r v a l . R e a l i s t i c v a l u e s of x2(k) are a s s u m e d to b e w i t h i n 0 ~ x2(k) ~ i0. T h e d e n s i t y f u n c t i o n p(Av) is e s t i m a t e d on the b a s i s of p r e c i p i t a t i o n t h r o u g h 40 years. T h e d a t a are n o t g i v e n here, b u t to g e t an i m p r e s s i o n of the s p r e a d , t h e v a r i a n c e 02 Av is g i v e n in the f o l l o w i n g t a b l e , w h e r e a l s o Vo(k) is t a b u l a t e d :

M o n t h k i

1 0 3 . Vo(k) 43

OAr(k) 2 90

2 39 94

3 4 5

41 34 37

87 57 61

P e r f o r m a n c e c r i t e r i o n : ...

F o r the o b j e c t i v e f u n c t i o n the s m o o t h c u r v e PF(Up(k) - u(k)) in f i g u r e 14 is used.

T h e r e s u l t s w o u l d be r a t h e r u n i n t e r e s t i n g in p r a c t i c e if the t e r m i n a l s t a t e xl (N) is n o t c o n s i d e r e d in the o p t i m i z a t i o n p r o b l e m , s i n c e this w o u l d i m p l y a p o l i c y w h i c h aims at e m p t y i n g the r e s e r v o i r t o w a r d s the e n d of the o p t i m i z a t i o n i n t e r v a l . M a n y k i n d s of c r i t e r i a t a k i n g the e x p e c t e d f i n a l s t a t e i n t o a c c o u n t c o u l d be t h o u g h t of. F o r i n s t a n c e , an a n a l y s i s of the p r i n c i p l e of the p r o c e d u r e u s e d in [9], s h o w s t h a t w i t h i n the a s s u m p t i o n of l i n e a r i t y in the p r o c e s s e q u a t i o n s , the p o l i c y

is to a i m at r e p r o d u c i n g the r e s e r v o i r v o l u m e a f t e r one y e a r [ ~ . A r e a s o n a b l e p o l i c y m i g h t be to let the e x p e c t e d f i n a l s t a t e x1(N) h a v e a s e n s i b l e v a l u e b a s e d o n e x p e r i e n c e for t h a t m o n t h of the season.

A m o r e d i r e c t , a n d in f a c t an e q u i v a l e n t a p p r o a c h , is to i n c l u d e a w e i g h t i n g o n the f i n a l s t a t e in J, w i t h s u c h a w e i g h t i n g t h a t t h e e x p e c t e d f i n a l s t a t e has a r e a s o n a b l e value. H e n c e , we use as an o p t i m a l c r i t e r i o n

E{J'} = E { J + dxl (N)} (19)

w h e r e J is g i v e n b y eq. (17) - (18).

R e s u l t s .

It is i n t e r e s t i n g to f i n d the v a r i a t i o n in the o p t i m a l p o w e r p r o d u c t i o n Uopt(0) of the f i r s t m o n t h as a f u n c t i o n of the i n i t i a l c o n d i t i o n x2(0) in t h e r u n - o f f m o d e l . T h e r e s u l t s are s h o w n in f i g u r e 15 for t h r e e d i f f e r e n t i n i t i a l s t o r a g e s xl (0) in the p o w e r s t a t i o n r e s e r v o i r a n d d = 3. As e x p e c t e d , the i n i t i a l s t a t e xz(0) has a c o n s i d e r a b l e e f f e c t on the o p t i m a l p o l i c y . The e x p e c t e d final s t a t e E { x I ( N ) }

( a p p l y i n g the e x p e c t e d r u n - o f f and p i c k i n g the c o n t r o l f r o m the c o m - p u t e d t a b l e s of o p t i m u m s t o c h a s t i c c o n t r o l s ) is 7.4 G W h at d = 3, and 8.2 G W h at d = 6. T h e two d i f f e r e n t v a l u e s of d g a v e no d i f f e r e n c e in the o p t i m u m c o n t r o l f o r the f i r s t stage. H o w e v e r , at d = 0,

U o p t ( 0 ) = 3.2 at xl(0) = 100% (14.4 GWh). T h e c o n t r o l p o l i c y for the f i r s t stage is r a t h e r i n s e n s i t i v e to the w e i g h t i n g f a c t o r on xl (N), as l o n g as the e x p e c t e d f i n a l s t a t e has a r e a s o n a b l e v a l u e for the m o n t h in q u e s t i o n . T h i s is m a i n l y an e f f e c t of the u n c e r t a i n t y of the f u t u r ~ and a l s o i n d i c a t e s t h a t it s h o u l d n o t b e n e c e s s a r y to u s e l a r g e r o p t i - m i z a t i o n i n t e r v a l s than, say, h a l f a year, in o r d e r to c o m p u t e the o p t i m a l c o n t r o l for t h e f i r s t m o n t h .

A n i n t e r e s t i n g c o m p a r i s o n is to c o m p u t e the o p t i m u m c o n t r o l if Ar is p u r e s t o c h a s t i c (white) w i t h a p p r o x i m a t e l y the same p r o b a b i l i t y

d e n s i t y as t h a t one w h i c h c a n be e s t i m a t e d f r o m the r u n - o f f o b s e r v a t i o n ~ It is n o t s u r p r i s i n g t h a t the c o m p u t e d v a l u e in this case, U o p t =

4 G W h / m o n t h , at x1(0) = 100% c o r r e s p o n d s to a v a l u e (see f i g u r e 15) w h i c h is c l o s e to the m e a n in the r u n - o f f for t h a t m o n t h .

Of c o u r s e , the n u m e r i c a l v a l u e s o b t a i n e d h e r e s h o u l d n o t be u s e d in a g e n e r a l d i s c u s s i o n of the g o o d n e s s of a p p r o x i m a t i o n b y u s i n g a n o n - d y n a m i c r u n - o f f d e s c r i p t i o n in the c o m p u t a t i o n of the e c o n o m i c a l d i s p a t c h for a n y h y d r o e l e c t r i c p o w e r system. H o w e v e r , the e x a m p l e c l e a r l y s h o w s t h a t the p r o b l e m s h o u l d b e g i v e n a t t e n t i o n .

4. C O N C L U S I O N S

R e s u l t s on s i m p l e b a t c h p a r a m e t e r e s t i m a t i o n of a h y d r o l o g i c a l s y s t e m h a v e b e e n p r e s e n t e d in the f i r s t part. T h e n u m b e r and k i n d of m e a s u r e - m e n t s j u s t i f y the s y n t h e s i s of a r a t h e r c r u d e m o d e l only. T h i s

c o n c l u s i o n has b e e n d r a w n on the b a s i s of o b s e r v a b i l i t y a n a l y s i s . H e n c e , it is n o t s u r p r i s i n g t h a t the g o o d n e s s of fit w i l l v a r y s o m e - w h a t d e p e n d e n t o n the season, a n d t h a t the s i m p l e m o d e l has d e f i c i e n - c i e s l i k e i n a c c u r a t e r e s e r v o i r level d u r i n g the w i n t e r and the spring, a n d too l o w g r o u n d w a t e r level d u r i n g the l a t e a u t u m n . H o w e v e r , it s h o u l d be k e p t in m i n d t h a t t h e ~ e r r o r s in the f i t t i n g w i l l d i s t r i b u t e on e a c h v a r i a b l e a c c o r d i n g to the w e i g h t i n g f a c t o r s in the loss f u n c t i o n a l [5~.

In the last s e c t i o n , w i t h r e s p e c t to the a p p l i c a t i o n of a h y d r o l o g i c a l m o d e l in the s t o c h a s t i c o p t i m i z a t i o n of a h y d r o l o g i c a l p o w e r system, it h a s b e e n d e m o n s t r a t e d t h a t the use of a d y n a m i c a l r u n - o f f m o d e l m a y be n e c e s s a r y in the c o m p u t a t i o n of the o p t i m a l c o n t r o l . A l t h o u g h it is o p e n for d i s c u s s i o n h o w c o m p l e x s u c h a m o d e l s h o u l d be, it is l i k e l y t h a t s i g n i f i c a n t i m p r o v e m e n t s in the c o n t r o l p o l i c y c a n be a t t a i n e d b y r e p r e s e n t i n g the m o s t i m p o r t a n t d y n a m i c s of the e n v i r o n m e n t a l s y s t e m in a s i m p l e f i r s t - o r d e r , s t o c h a s t i c m o d e l w i t h t i m e - v a r y i n g p a r a m e t e r s .

[ ~ Aoki, M.: Optimization of Stochastic Systems; topics in discrete- time systems. Academic Press, 1967.

8 Crawford, N.H., Linsley, R.K.: "Digital Simulation in Hydrology:

Watershed Model IV". Technical Report No. 39, Department of Civil Engineering, Stanford University, California, USA.

[ ~ Fjeld, M.: Dynamic Programming and Its Application to the

Dispatch Problem in Hydroelectric Power Systems. Report 72-90-S, The Engineering Research Foundation at the University of Trondheim, The Norwegian Institute of Technology (SINTEF), Division of

Automatic Control, Trondheim, Norway. (In Norwegian.)

4] Hertzberg, T.: MODTLP, a General Digital Computer Program for Fitting of General, Nonlinear, Dynamic Models to Experimental Data. The Norwegian Institute of Technology, The Chemical

Engineering Laboratory, N-7034, Trondheim, Norway. (In Norwegian.) 5] Holmelid, A.E.: State and Parameter Estimation of Hydrological

Systems. Thesis 1973. The Norwegian Institute of Technology, Division of Automatic Control, N=7034, Trondheim, Norway.

(In Norwegian.)

[6] Jacobson, D.H., Mayne, D.Q.: Differential Dynamic Programming.

American Elsevier Publ. Co., 1970.

7] Larson, R.E.: A Survey of Dynamic Programming Computational Procedures. IEEE Trans. on Autom. Control, AC-12, No. 6, pp. 767-774 (December 1967).

8] Larson, R.E.: State Increment Dynamic Programming. Elsevier 1968.

[9] Lindquist, J.: Operation of a Hydrothermal Electric System: A Multistage Decision Process. AIEE Journal, April 1962, pp. 1-7.

~ 0 ] Schoenwandt, U.: On observability of Nonlinear Systems. Preprints of the IFAC Symposium on Identification and Process Parameter Estimation, Prague, June 1970.

A C K N O W L E D G E M E N T

T h e authors wish to express their appreciation to the staff at the Division of Hydraulic Engineering at the Norwegian Institute of

Technology for giving data from the IHD representative basin "Sagelva"

at our disposal, and their helpfulness in various other questions concerning hydrology.

PB(1) I PB(2) 1 PB(3) ~ PB(4)

PB(5)

qg(1) qs(1) qg(2) qs(2) qg(3) qs(3) "- qg(4) ~s(~)qg(5 ) qs (5)

PB(6) I PB(7)

qg(6) qs(6) qg(7) \

~ ~ P B(8) q s ( 7 ~

qg(out) (8) qs (out) (8)

i , : Boundary of basin,along surface and sub-surface divide.

I Q m

: Boundary of basin,along (surface) divide.

Boundary of partial basin,along surface and sub-surface divide.

- Boundary of partial basin,along (surface) divide.

Fig. i. A large (hydrological) basin.

T J 3

qg(ouz) ,~..,.,.~*" (PB).

qs(out)

lllllllli"

AS A L l AL2

I

I

-U-

: External boundary of basin,along divide.

= Internal boundary of baeinlalong divide.

= Channel flow.

= VePtlcel section through soil moisture- and groundwater-zone.Only drawn where the divide is not also a sub-surface divide.

= Area of reservoir.

= Area of landifrom where overland flow runs into reservoir°

= Area of landifrom where overland flow rune into channel downstream reservoir.

= Meteorological station,with temperature recorder (vl).

= Recording precipitation gauge (v2).

= Evaporation pan ] are measuring evaporation (v3AS) and average evapo-

? evapotranspiration]

E v a p o t m a n s p i r o m e t e r J transpiration coefficient (EL : [ evaporation ~ )"

= Recording groundwater level (yl).

= Recording water stage gaugelin reservoir (y2) op downstream reservoir (y3).

= Outlet or measuring weim,whePe The function q(y2) Or q(y3) is known.

Fig. 2 . A typical partial basin.

Subsystem governing thermal states

("seasons") according to history

of vl. Informatio~ about "season"~

time and ~ is contained in ~0"

TemperatuDe-dependent parameters

(~l) are demived from "season".

Subsystem governing state-

dependent parameters (~2)

accordln E to ~ , ~0 and

£i"

Hydraulic subsystem

governing the dynamical

states (~) according to

, v2 , v3 , K(in) and ~ . LEVEL 3

LEVEL 2

LEVEL i

... p a( out ) S ...

Synchronism,!,,,

TMEAN days

between each

change of

"season".

,,,~/

3. The basin sketched as a hierarchical system~ Fig. 4. "Season" diagram .

qe(out)

v2 ( x2 = x5 + x6 ) ( yl : GB.x5 + Hi ) ( y2 = X3/A2 ) ~.,~.~., . [,.TF--q F--- (K4"F2)

~qg(in)

~

qs(in) +,

]

... ; ~ ~ ID3 ['--~.A2 • F3 I(K6) -- .... -I ... + qg(out) ~ qs(out) ( = q(y3) )

qe(ouz) v'~.AI.EL.F3 +

v2, l,.Data preparation (K4-F2) qg(out)

( yl = GB.x2 + ( y2 • x3/A2 ) ( y4 = Gg.xl ,bu~ only in the ter.)

... ~y2~ B3~

>

m

,-,F

('D

> o #

g .~.,...,,~ ~ ,,,.IIIIIIH|N IIIk \

,3 o

° ~

o

> z

(v3) Noise

Input ( vl , v2 , ~(in) ) i

J I Measurement

v I HYDROLOGICAL SYSTEM

C~ m)

u

~0(k+l) = ~(vl(k) , ~0(k) ) I Simulated

~l(k ) = ~i ( £0(k ) ) L m e a s u r e m e n t

M O D E L

£2 = ~2 ( ~ ' ~i (k))

S(out) = ~(~,S(in),~l(k),~2,~)

(O(OuT))~°utP ut

A~

ERROR CRITERION

1

ADJUSTMENT

S T I ~ T E ~ Y

Fi~. 8. SimulaYion and adjustment plan.

Y (m) D.5 0,0

y2 (~o%~ 3)

/

rJ ~' \, t", ,~",,

ot.J.~.i,kl ,.L.i,~l,,.,,k

,,

6 2 6 4 2 0 - 2 - 4 - 6

q ( 103.m3/day ) ~ECESSION- ("D~Y WEATHER-") CURVES 30 60 90

time ( days ) i 120 150 ~ (°C) MEAN TEMPERATURE THROUGH 15 DAYS 30 90 i .... !

r [

.... 150 time ( days ) "winter"

T

I V-NOISE l RUN-OFF ^{MODEL X 2}

I

t t-RESERVOIR INFLOW

.Es.vo.

I

POWER STATION I STORAOE:

il / v ^TIME

F i g , 11. Process and e n v i r o n m e n t a l model. Fig. 12. "Stationary" and conditional evolution of the probability density.

V-NOISE RUN-OFF

MODEL

f - RESERVOIR INFLOW RESERVOIR

AN D POWER STATION

PF

~lrelkWh] i lS.

10

P._FF (U,-U)

S.

3.

.lo ;o ~ 3"o l- ~ , I 0 0 %

Fig. 13. Principle Of control system solution. F i g . 14. Cost per energy u n i t .

Us

~ Xll0), 33%

XilO) .INITIAL STORAGE-CONDITION

2:s ~o T~S ,io lOW.; b x=(o)

INITIAL RUN-OFF STATE

Fig. 15. Optimum control for the Ist month as a function of the initial values in the states.