Parameters Calculation of Solar- and Wind-Electric Water Lifting Systems

NOT FOR QUOTATION

WITHOUT THE PERMISSION OF THE AUTHORS

P - CALCULATION OF SOLAR -

AND WIND-ELECTRIC WATER LIFTING

SETEXS

S.P. U r i a s ' e v G.K. K u t l i e v

Working P a p e r s are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have r e c e i v e d only limited review. Views or opinions e x p r e s s e d h e r e i n d o not necessarily r e p r e s e n t t h o s e of t h e Institute or of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

FOREWORD

This p a p e r d e s c r i b e s r e s u l t s of the application of s t o c h a s t i c programming to p a r a m e t e r s calculation of s o l a r - and wind-electric w a t e r lifting systems. I t pro- vides a n example of both a r e a l i s t i c problem with i n h e r e n t stochasticity, and a valuable test problem f o r t h e algorithms under development. I t a l s o gives some in- sights into t h e n a t u r e of solutions of c e r t a i n c l a s s e s of s t o c h a s t i c programming problems, and the justication f o r t h e consideration of randomness in decision models.

Alexander B. Kurzhanski Chairman System and Decision Sciences Program

AUTHORS

Senior r e s e a r c h scientist Stanislav Urias'ev from the Institute of Cybernetics Academy of Sciences of the Ukr.SSR (252207 Kiev, USSR).

Research scientist Geldu Kutliev from the Scientic Production Unit SOLNCE Academy of Sciences the Tur. SSR (744032 Ashchabad, USSR)

CONTENTS

1 Introduction

2 Stochastic Model of t h e Solar- and Wind-Electric Water Lifting System

3 Stochastic Quasi-Gradient Algorithm 4 Stochastic Quasi-Gradient Calculation 5 Modeling of WEP and SPP Output 6 Calculations Results

References

-

vii

-

PARAMEmxS CAU=ULATION OF SOLAR -

AND W I N D - ~ ~ C WATER

LIFTING SYSTEMS S.P. U r i a s ' e v a n d G.K. K u t Liev

1. INTRODUCTION

The development of e n e r g y and r e s o u r c e saving technologies is v e r y impor- t a n t because of s h o r t a g e in traditional e n e r g y supply r e s o u r c e s (gas, oil, coal). A combined use of t h e wind and s o l a r e n e r g y f o r power and water supply of small iso- lated consumers is one of t h e promising tendencies in t h e world economy. The effi- ciency of a joint use of t h e s e power s o u r c e s under conditions of d e s e r t and moun- tain regions f e a t u r e high power c h a r a c t e r i s t i c s of wind-driven and s o l a r plants which mutually complement e a c h o t h e r when combined.

The p u r p o s e of t h e p r e s e n t calculation is to determine optimum values of p a r a m e t e r s of s o l a r - and wind-electric water supply systems. These systems are in- tended for use in p a s t u r e and i r r i g a t e d regions of t h e Middle Asia, Kazakhstan and in s t e p p e regions of t h e USSR European t e r r i t o r y . The optimum p a r a m e t e r s a r e s e l e c t e d depending on t h e p r e s e n c e of n a t u r a l power r e s o u r c e s in t h e region and t h e p r e s e t water consumption schedule, with minimum costs of c a p i t a l c o n s t r u c t i o n and o p e r a t i o n of t h e system for t h e whole planning period.

This a r t i c l e follows t h e a r t i c l e by Kutliev and Urias'ev [I]. In [I] i t w a s pro- posed to use s t o c h a s t i c quasi-gradient algorithms for calculation of water lifting systems. The following main p a r a m e t e r s of t h e system are calculated: t h e wind- e l e c t r i c plant (WEP) blade-swept a r e a , t h e solar-power plant (SPP) c o l l e c t o r area and t h e water r e s e r v o i r volume.

The problem u n d e r consideration i s r e d u c e d t o t h e s t o c h a s t i c programming problem [Z] [3]. A s t o c h a s t i c quasi-gradient algorithm with adaptive s t e p size con- t r o l i s used f o r solving t h e problem [4], [5]. This method w a s included i n t o t h e ADO/SDS l i b r a r y s t o c h a s t i c optimization codes developed in IIASA and package N S O developed in

V.M.

Glushkov Institute of Cybernetics (Kiev, USSR).

The water lifing s y s t e m being s t u d i e d c o m p r i s e s a water s o u r c e , a WEP and a S P P with pump equipment, a w a t e r r e s e r v o i r a n d a water t r o u g h system. The water consumption s c h e d u l e i s p r e s e t and t h e lifted water i s s u p p l i e d from t h e water s o u r c e p r i m a r i l y f o r e v e r y d a y n e c e s s i t i e s a n d f o r watering of c a t t l e . In e n s u r i n g t h e watering s c h e d u l e a n d filling t h e accumulating water r e s e r v o i r , w a t e r i s f e d f o r i r r i g a t i o n of g a r d e n s a n d fields of melons or for i r r i g a t i o n of t r e e s , s h r u b - b e r y , g r e e n p l a n t a t i o n s e t c . I t i s assumed t h a t t h e water s o u r c e o u t p u t i s unlimit- e d .

By v i r t u e of t h e f a c t t h a t t h e inflow of t h e wind a n d s o l a r e n e r g y i s a random q u a n t i t y , t h e o u t p u t of t h e p o w e r p l a n t s , t h e amount of w a t e r in t h e r e s e r v o i r at a n y i n s t a n t of time as well as t h e v a l u e s of water d e f i c i t a n d e x c e s s are random values.

2. STOCHASTIC MODEL O F THE SOLAR-

AND

WIND-ELECTRIC

WATER LIFTING SYSTEM

L e t us c o n s i d e r t h e model of t h e w a t e r s u p p l y s y s t e m f o r some independent s t a t i o n using WEP a n d S P P . The modeling p e r i o d ( a s e a s o n , a y e a r ) i s subdivided i n t o d a i l y p e r i o d s k

=

1, 2,.

.

. N . The o u t p u t of t h e w a t e r lifing system i s d e t e r m i n e d by t h e p a r a m e t e r s

w h e r e a l i s t h e WEP blade-swept area, a 2 is t h e S P P c o l l e c t o r a r e a , a 3 i s t h e wa- t e r r e s e r v o i r volume.

Vector a m e e t s t h e n a t u r a l r e s t r i c t i o n s :

L e t us assume t h a t a l l random v a l u e s are defined in p r o b a b i l i t y s p a c e (R,

F ,

P).

F o r e a c h p e r i o d k ( k

=

1, 2 ,

. . .

, N ) , l e t us i n t r o d u c e t h e following designa- tions: b ( k ) i s demand for w a t e r , V ( k ) is wind velocity daily a v e r a g e value, E l ( k ) i s daily t o t a l s o l a r r a d i a t i o n , Q ( a l , a 2 , V ( k ) , E l ( k ) ) i s q u a n t i t y of e l e c t r i c power g e n e r a t e d by t h e p o w e r p l a n t s , W ( k ) i s t h e amount of water lifted b y t h e e l e c t r i c pump, x ( k ) i s t h e amount of w a t e r in t h e r e s e r v o i r , Z l ( k ) a n d Z 2 ( k ) are w a t e r de- f i c i t a n d e x c e s s , r e s p e c t i v e l y .

The state of t h e system p a r a m e t e r s at instant k is c h a r a c t e r i z e d by t h e fol- lowing relations:

Z 2 ( k )

=

max 10, x ( k

-

^{1 )}

+

W ( k )

-

^{b ( k )}

-

^{a 3 j , k}

=

1 ,

. . .

,

N

, where values x ( k ), W ( k ) are determined by relations:

where G l ( V ( k ) ) , G 2 ( E l ( k ) ) are specific e l e c t r i c outputs of t h e WEP and SPP, respectively. Value R l i s a positive constant depending on t h e water lifting level H.

Below, t h e methods will b e d e s c r i b e d f o r calculating t h e values and G 2 ( E l ( k ) ) , k

=

1,

. . .

,

N .

The t o t a l cost minimum i s a c c e p t e d as t h e optimization c r i t e r i o n :

where 8 are o p e r a t i o n e x p e n d i t u r e s f o r calculation p e r i o d , A are one-time invest- ments, E i s t h e r e l a t i v e efficiency coefficient (&

=

0 . 1 5 ) , E ( y b ) is t h e mathematical e x p e c t a t i o n of t h e cost of water deficit, E ( n 6 ) i s t h e mathematical expectation of p r o f i t due to water e x c e s s .

The s o l a r - and wind-power system is c h a r a c t e r i z e d by l a r g e initial investments and small o p e r a t i o n c o s t , of t h e o r d e r of 10% of t h e investments

Investments A r e q u i r e d f o r construction of t h e systems comprise t h e following components: A 6 , A,

-

investments r e q u i r e d f o r WEP and SPP with pump equipment:

AH

-

investments f o r e l e v a t e d water r e s e r v o i r , Al

-

investments f o r t h e head p a r t , foundation and watering system, A 2

-

e x p e n d i t u r e s f o r t h e equipment t r a n - s p o r t a t i o n , A 3

-

t h o s e f o r t h e system assembly and e r e c t i n g . T h e r e f o r e

L e t u s assume [ 6 ] t h a t A

+

A

, ⁺

^A

⁼

c o n s t . L e t us p r e s e n t v a l u e s Ab a n d A, in t h e following way

w h e r e c l a n d c , i s s p e c i f i c c o s t of

WEP

a n d

SPP,

r e s p e c t i v e l y . To c a l c u l a t e t h e c o s t of t h e e l e v a t e d w a t e r r e s e r v o i r , we use t h e following r e l a t i o n [ 7 ] :

Denote: d l ( k ) - s p e c i f i c c o s t of w a t e r d e f i c i t , d 2 ( k )

-

s p e c i f i c p r o f i t d u e to t h e use of e x c e s s i v e w a t e r . T h e r e f o r e :

Then f r o m ( 6 ) we o b t a i n

Denote c

=

0.165(A1

+

A ,

+

^A3

+

2 6 0 ) t h e n

I t i s r e q u i r e d to find v e c t o r a

=

( a l, a ,, a 3) which minimizes function ( 7 ) u n d e r conditions ( 1 ) - ( 5 ) .

Denote

Then p r o b l e m ( 7 ) , ( 1 ) - ( 5 ) c a n b e w r i t t e n in t h e following form:

p r o v i d e d t h a t

rp(k)

=

b ( k )

-

^{W ( k )}

-

max 10, m i n [ a g ,

-

rp(k - 1 ) j j , rp(0) = O , W ( k )

=

R I Q ( a l , a 2 , V ( k ) , E ( k ) ) ,

Q ( a l , a 2 , V ( k ) , E ( k ) )

=

G1(V(k))al

+

G 2 ( E ( k ) ) a z , a l 2 0 , a 2 2 0 , a 3 2 0

.

3. STOCHASTIC QUASI-GRADIENT ALGORITHM

For solving t h e problem (9)-(13) t h e s t o c h a s t i c quasi-gradient algorithm with adaptively controlled p a r a m e t e r s w a s used [4], [5]. For improving t h e p r a c t i c a l convergence rate of this algorithm t h e scaling p r o c e d u r e suggested by S a r i d i s [8]

w a s implemented. Scaling p r o c e d u r e contains changes taking into account t h e pro- jection o p e r a t i o n and a d a p t i v e s t e p size c o n t r o l

where s is t h e i t e r a t i o n number; IID i s o r t h o p r o j e c t i o n o p e r a t i o n on t h e set D

=

[ a : a r 0 , a* 2 0 , a g 2 0 j ; p s r 0 i s t h e positive s t e p size; HS is t h e scaling matrix;

CS

i s t h e s t o c h a s t i c quasi-gradient, i.e.

The scaling matrix i s calculated as follows

hi ( s

+

1 )

=

ahi ( s )

+

Xi ( s

+

1 ) ( 1

-

^{a ) ,}

where k ( s

+

I ) , 0 S k ( s

+

^{1 ) 5} n is t h e quantity of numbers i f o r which

t;

^+I(,;

-

^{xf + I )}

>

0 .

S t e p size p s i s given by following r e c u r s i v e r e l a t i o n s

The recommended v a l u e s of p a r a m e t e r s a r e

4. STOCHASTIC QUASI-GRADIENT CALCULATION

Function F ( a ) c o m p r i s e s of t w o components

w h e r e

H e r e , f ( a ) i s a smooth function. Function y ( a , o) with f i x e d w E R, g e n e r a l l y s p e a k i n g , i s non-smooth with r e s p e c t t o a . N e v e r t h e l e s s , if random v a l u e s W ( k ) , k

=

1,

. . .

, N h a v e p r o b a b i l i t y d e n s i t i e s t h e n t h e mathematical e x p e c t a t i o n smoothness t h e i n t e g r a n d .

T H E O R E M 1 If r a n d o m v a l u e s W ( k ) , k

=

1 , 2 ,

. . .

, N h a v e p r o b a b i l i t y d e n s i - t i e s f o r a l l a l 2 0, a z 2 0 other t h a n a l

=

a z

=

^0, t h e n f u n c t i o n F ( a ) defined b y (9)- (2.2) i s d w e r e n t i a b l e o n the set a l 2 0 , a z 2 0, a , 2 0 except for r a y al

=

a z

= 0 , a , r ⁰ a n d

d l ( k ) , if q ( k )

>

^{0 ,}

- l ( k )

= 6,

o t h e r w i s e ,

if q ( k )

+

a3

<

0 , o t h e r w i s e

.

PROOF On s u b s t i t u t i o n of r e c u r r e n t r e l a t i o n s

( l o ) ,

(11) a n d ( 1 2 ) , function y ( a , o) becomes d i f f e r e n t i a b l e o n a with p r o b a b i l i t y 1 f o r a n y a which belongs to t h e set a l r 0 , a 2 ² 0 , a 3 2 0 e x c e p t f o r t h e r a y a l

=

a z

=

0 , a 3 ² 0 . The g r a d i e n t of t h e function y ( a , o ) i n t h e points w h e r e i t i s d i f f e r e n t i a b l e h a s t h e form:

N

ya ( a , Q )

= C

[ q a ( k ) u l ( k ) + ( ~ ( k ) + a 3 ) a

U Z ( ~ ) I

k = 1

where q , ( k ) , u l ( k ) , u 2 ( k ) , k

=

I ,

. . .

^, N are defined by r e l a t i o n s (10)-(12), ( 1 8 ) , (19).

From r e l a t i o n ( 1 8 ) , ( 1 9 ) , ( 2 0 ) i t i s e a s y to s e e , t h a t t h e mathematical e x p e c t a - tion of v e c t o r function y, ( a , o ) e x i s t s s i n c e t h e following e s t i m a t e s hold;

a n d functions of right-hand s i d e s of t h e inequalities h a v e mathematical e x p e c t a - tion.

Denote unit v e c t o r s ei , i

=

1 , 2 , 3 . The d i f f e r e n t i a b i l i t y of function E y ( a , w ) follows from Lebeque t h e o r e m on limit t r a n s i t i o n inside t h e i n t e g r a l sign [ 9 ]

~ (

+

ah e i , w ) - y ( a , 0 )

( E y ( a , w ) ) , ,

=

lim E

h = E Y , , ( ~ , o ) ,

h - 0

i

=

1 , 2 , 3 s i n c e

y ( a

+

h e 2 , w )

-

Y ( a , w ) N k

1

_h ^{( 5 max} ( J d l ( k ) I + l d 2 ( k ) l ) R 1

C C

IG2(V(L))I ¹

k k = 1 L = 1

f o r a n y h E R . Functions in right-hand s i d e s of t h e inequalities h a v e mathematical e x p e c t a t i o n s . The l a s t t h r e e inequalities follow from e s t i m a t e s f o r l y q ( a , w ) ( ,

i

=

1 , 2 , 3 . The t h e o r e m i s p r o v e d .

From t h e t h e o r e m we obtain t h e following formula f o r s t o c h a s t i c quasi- g r a d i e n t calculation:

w h e r e q i ( k ) , us ( k ) , u: ( k ) are o b t a i n e d from ( 1 0 ) - ( 1 2 ) , ( 1 8 ) , ( 1 9 ) by s u b s t i t u t i n g a r g u m e n t a by a p p r o x i m a t i o n as on t h e s - t h i t e r a t i o n .

5.

MODELMG OF WEP AND SPP OUTPUT

E x p r e s s i o n f o r R 1 i s s e t as follows:

R 1

= -

⁹ (25

1

w h e r e

Z i

i s t h e dimensions c o n v e r s i o n c o e f f i c i e n t , T~ i s t h e efficiency of t h e gen-

e r a t e d e n e r g y t r a n s f e r t o t h e consumer, 772 i s t h e pump efficiency, H is t h e speci- fied head.

Random values G l ( V ( k )), k

=

1 , 2,.

. .

N are modeled according to t h e formula

[lo]# [Ill

where wind velocity daily a v e r a g e values are specified by continuous distributions with densities:

where r ( k ) is t h e number of t h e WEP o p e r a t i o n h o u r s in a day; j i s t h e wind e n e r - gy utilization f a c t o r ; q p , 77t is t h e efficiency of t h e double reduction g e a r and syn- chronous wind-driven e l e c t r i c g e n e r a t o r ; pb i s a i r density, v ( k ) i s t h e daily a v e r - a g e values of t h e wind velocity r e d u c e d to height h

=

12m; c,, c, are coefficients of v a r i a t i o n and asymmetry, r e s p e c t i v e l y ; AV i s t h e wind velocity variation i n t e r - val (AV

=

1 ) ;

6

i s t h e a v e r a g e wind velocity f o r t h e time p e r i o d under considera- tion; b , c , d , e are t h e p a r a m e t e r s of t h e wind velocity probability density func- tion.

F o r r ( k ) , similarly to [12], i t w a s found t h a t

Generation of wind velocity a v e r a g e daily values V ( k ) , k

=

1 , 2,

. . . ,

N i s car- r i e d o u t according t o ( 2 7 ) by t h e elimination method [13]. Random numbers a l , az uniformly distributed o v e r [0,

11

are g e n e r a t e d by r e f e r r i n g t o t h e random numbers g e n e r a t o r . When assuming and pax as t h e initial and t h e maximum dai- ly a v e r a g e wind velocities, w e obtain f o r given location

7)

=

a 2 B , B

=

max g ( v )

.

7J ECQ,-C

Then l e t us c o m p a r e t h e v a l u e s q a n d g ( v ) . If q

>

^g ( v ) t h e n t h e a v a i l a b l e numbers v a n d q are omitted a n d t h e p r o c e s s i s r e p e a t e d with new v a l u e s a l a n d a 2 . Other- wise, t h e v value i s a c c e p t e d as t h e V ( k ) . Then ~ ( k ) i s d e t e r m i n e d by r e l a t i o n s ( 2 8 ) a n d s p e c i f i c daily power o u t p u t by t h e e l e c t r i c p l a n t s in k-th d a y i s c a l c u l a t e d us- ing ( 2 6 ) .

To c a l c u l a t e G 2 ( k )

-

s p e c i f i c o u t p u t of t h e SPP f o r k

=

^1,

. . .

, N , used a r e t h e following r e l a t i o n s [14], 1151:

G(E

^{( k ) )}

=

F f ( ~ ) , G o ( k max 10, Kc ( k )

-

Kd ( k )

1

^,

Kc ( k )

=

K F ' " ( ~ )

+

a 3 ( ~ T a X ( k )

-

~ ? ' " ( k ) ) ,

-

K; ( k )

=

^{~ ( k} )Kt (Tc ( k )

-

^{T b ( k}

1)

fQo(k

)(Tale

w h e r e q c i s t h e t o t a l e f f i c i e n c y of s o l a r e n e r g y c o n v e r s i o n t o e l e c t r i c a l o n e ac- c o r d i n g t o Brayton thermodynamic c y c l e ; G ( E ( k ) ) is s p e c i f i c e f f e c t i v e h e a t drawn of t h e s o l a r e n e r g y c o l l e c t o r ; F' i s c o e f f i c i e n t of t h e h e a t withdrawal;

( G ) ,

i s t h e r e d u c e d e f f e c t i v e a b s o r p t i v e c a p a c i t y of t h e c o l l e c t o r ; G 0 ( k ) i s e x t r a - a t m o s p h e r i c flow of s o l a r e n e r g y ; Kc ( k ) i s t h e c l e a r n e s s c o e f f i c i e n t ; K; ( k ) i s t h e t h r e s h o l d value of t h e c l e a r n e s s c o e f f i c i e n t ; K P i n ( k ) a n d Kmad ( k ) are t h e minimum a n d t h e maximum limit of t h e c l e a r n e s s c o e f f i c i e n t , r e s p e c t i v e l y ; Kt i s t h e c o l l e c t o r coeffi- c i e n t of h e a t losses; Tc ( k ) is t h e h e a t - t r a n s f e r a g e n t t e m p e r a t u r e ; Tb ( k ) is t h e am- b i e n t a i r t e m p e r a t u r e ; f ( k ) i s t h e c o l l e c t o r o p e r a t i o n time;

r

i s t h e s l o p e ; L 1 ^{2 ,} l 3 a r e p a r a m e t e r s depending on t h e p l a c e l a t i t u d e p; a3 i s a random number uni- formly d i s t r i b u t e d o v e r [0, I].

On t h e b a s i s of a v a i l a b l e e x p e r i m e n t a l d a t a , t h e following a n a l y t i c a l r e l a t i o n - s h i p s f o r determining t h e values of p a r a m e t e r s 1 1 , 1 2 , l 3 a n d t h e values of K; ( k ) h a v e b e e n found with t h e a i d of i n t e r p o l a t i o n polynomials:

KL ( k )

=

KL ( 6 ( k ) ) = - 0.534

.

1 0 - ~ 6 ~ ( k )

+

0.8674

.

1 0 - ~ 6 ~ ( k )

w h e r e

F o r g i v e n p l a c e , we assume

Kcmax(k )

=

KcmaX

=

c o n s t , k

=

1,

. . . ,

N

.

Then t h e modeling a l g o r i t h m G 2 ( F ( k ) ) , k

=

1 ,

. . .

, N will b e p r e s e n t e d with t h e aid of f o r m u l a e ( 2 9 ) - ( 3 0 ) as follows.

S p e c i f i e d are t h e v a l u e s F ' , ( G ) ,

K P ~ * ,

^7, a n d t h e p l a c e l a t i t u d e 9. The v a l u e s of p a r a m e t e r s Ll, L 2 , 1 are d e t e r m i n e d f r o m r e l a t i o n s ( 3 0 ) . With k

=

1 , cal- c u l a t e d o n e a f t e r a n o t h e r a r e t h e v a l u e s Ki ( k ) , ~ ? ~ " ( k ) a n d G o ( k ) . Modeling of a3 i s p e r f o r m e d b y r e f e r e n c e to t h e random n u m b e r s g e n e r a t o r a n d t h e n K , ( k ) ,

G ( E ~ ( ~ ) )

^{a n d} G 2 ( E l ( k ) ) are c a l c u l a t e d . Similar o p e r a t i o n s are p e r f o r m e d f o r t h e n e x t v a l u e s of k , k

=

2,

. . .

, N .

6.

CALCULATIONS RESULTS

In p e r f o r m i n g c a l c u l a t i o n s , d a t a f o r Turkmen SSR t e r r i t o r y a n d C e n t r a l K a r a Kum r e g i o n h a v e b e e n used [ 6 ] , [ 7 ] . Data f o r modeling a v e r a g e daily wind veloci- t i e s a n d s p e c i f i c o u t p u t of WEP are p r e s e n t e d in Table 1 .

In c a l c u l a t i n g G l ( V ( k ) ) , k

=

1,

. . .

, N, t h e following v a l u e s h a v e b e e n u s e d i n addition to t h e Table d a t a : AV

=

1.0 m / s ;

TABLE 1 S e a s o n Quantity

b

C

d L v'(m / s) P ~ ( ~ B m 3 )

c,

c s

Winter S p r i n g 1.812 0.500 1.324 1.100 3.900 1.211 0.784 1.079

Summer Autumn

F o r modeling G 2 ( E 1 ( k ) ) , k

=

1,

. . .

,

N ,

t h e following v a l u e s h a v e b e e n a c c e p t e d : t h e p l a c e l a t i t u d e 9

=

38;

F' =

0.982;

( G ) , =

0.738; K y x

=

0.850 a n d q ,

=

0.03.

The c o n s i d e r e d c o n t r o l p e r i o d i s o n e y e a r (N

=

365). Daily w a t e r demand f o r two s h e p h e r d teams ( e a c h consisting of t h r e e men), two t o t h r e e camels a n d 850 s h e e p are determined by t h e following q u a d r a t i c r e l a t i o n :

w h e r e maximum value (15.4m 3, i s o b s e r v e d in t h e middle of July.

The p r o g r a m i s w r i t t e n in FORTRAN. The following p a r a m e t e r s values are used in t h e algorithm; a

=

2 , u

=

0.8, A

=

0.8, po

=

0.01, a

=

0.5. The initial ap- proximation i s as follows: a:

=

0 . 5 m 2 , a;

=

3 m 2 , a:

=

3 0 m 3 . The R 1 value i s d e t e r m i n e f r o m formula ( 2 5 ) . Calculations are p e r f o r m e d with t h e following values of t h e model p a r a m e t e r s : w

=

1 ; q 1

=

0.9; q 2

- -

^0.7;

^H =

5 m ; c = O ; cl = Z O O w b / m ; c 2 = 2 5 0 r u b / m ; d l ( k )

=

d l

=

20 r u b / m 3 ; d p ( k )

=

d 2

=

0 , k = 1 ,

. . . ,

N.

To p r o v i d e a s t a b l e o p e r a t i o n of t h e algorithm, i t s f e a s i b l e area D i s n a r - rowed:

D =

[ a : a 2 0.5, a 2 3 , a 2 30

1.

The computing e x p e r i m e n t s r e s u l t s show t h a t t h e s e r e s t r i c t i o n s are n o t a c t i v e t h e r e f o r e s u c h c o n t r a c t i o n d o e s not c h a n g e optimum point. In case of small values of p a r a m e t e r s a l , a 2 , a n d a 3 , sto- c h a s t i c quasi-gradient a s s u m e s v e r y l a r g e a b s o l u t e values which r e s u l t in a l o s s of computation p r o c e s s s t a b i l i t y . To eliminate s u c h e f f e c t , t h e f e a s i b l e area i s n a r - rowed.

The calculation r e s u l t s are p r e s e n t e d in Table 2.

The l a s t column p r e s e n t s a v e r a g e d e s t i m a t e s of t h e o b j e c t i v e function ob- tained in t h e following way: with f i x e d v a l u e s a l , a 2 , a 3 , t h e p e r f o r m a n c e of t h e p l a n t was simulated a h u n d r e d times a n d e s t i m a t e s of t h e o b j e c t i v e function h a v e

TABLE 2 Results of calculation of main parameters of a water supply system using WEP and SPP

Iteration Parameter values Averaged values

Nn.

- - -.

S a ; , _- m 2 a ; , m 2 a s , m 2 Total water Total water Function

deficit e x c e s s value

z : " z I ( k ) , m 3 z : 6 5 z 2 ( k ) , m 3 p ( a ) , r u b / vear

been calculated:

and t h e n t h e a r i t h m e t i c mean for t h e s e 100 random realizations are calculated.

The annual a v e r a g e t o t a l water deficit and s u r p l u s are calculated similarly. Such calculations are performed in t h e initial approximation point and a f t e r 20 and 40 i t e r a t i o n s and in t h e point which is believed to b e t h e b e s t approximations. Ac- cording to norms, a 90% water provision i s usually r e q u i r e d . Calculation r e s u l t s show t h a t t h e t o t a l water deficit f o r one y e a r i s 1 3 m 3 in t h e optimum version. When i t i s remembered t h a t t h e t o t a l water demand i s 4581m3, then i t follows t h a t t h e wa- ter supply system with optimum p a r a m e t e r s provides 99% of t h e water demand. To- t a l amount of e x c e s s water i s 1 0 times g r e a t e r t h a n t h e t o t a l water demand. If n e c e s s a r y , t h e e n e r g y s p e n t f o r lifting e x c e s s water c a n b e used to m e e t demand for e l e c t r i c e n e r g y .

Calculations performed f o r increasing water lifting level H show t h a t t h e op- timum p a r a m e t e r s values i n c r e a s e with increasing H (Table 3).

TABLE 3 Optimum p a r a m e t e r values of water lifing system using WEP and S P P f o r d i f f e r e n t H, f o r Central Kara Kum region, Turkmen So- viet Socialist Republic ( q p

=

38").

* * *

H ,

m a l l m Z a z , m Z a 3 , m3

In t h e s e calculations, t h e estimates of t h e p a r a m e t e r s optimum values ob- tained with one H value are used as initial approximation f o r t h e n e x t one. Conse- quently s a t i s f a c t o r y approximations of t h e optimum points are found f o r a r a t h e r small number of i t e r a t i o n s (about 40 to 45).

Calculations a l s o were made f o r t h e water supply system using only S P P as a s o u r c e of power.

F o r t h e system using only WEP w e have Table 5.

TABLE 4

*

^{2 L}

H ,

m a,, m a 3 , m 3

TABLE 5

L

*

H ,

m a , , m 2 a,, m 3

The calculation experiments r e s u l t s show t h a t stochastic quasi-gradient algo- rithms a r e effective means f o r finding optimum p a r a m e t e r s of s o l a r - and wind- e l e c t r i c water supply systems.

REFERENCES

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Ermoliev, Ju. M.: Stochastic programming methods. Nauka, Moscow, 1976, 240 PP.

Yastremski. A.J.: Stochastic models of mathematical economics, Vyscha Shkola, Kiev, 1983, 1 2 7 pp.

Urias'ev, S.P. : A stepsize r u l e f o r d i r e c t method of s t o c h a s t i c programming.

Kibernetica 1980, No. 6, pp. 96-98.

Mirzoakhmedov, F. and S.P. Urias'ev: Adaptive s t e p size c o n t r o l f o r s t o c h a s t i c optimization algorithm. Zhurn. vych. mat. i mat. fiziki, 1983, 6 , pp.

1314-1325.

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S a r i d i s , G .M

.

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Kolmogorov, A.H. and S.V. Fomin: Elements of function t h e o r y and functional analysis. Moscow, Nauka, 1981, 544 pp.

Seitkurbanov, S. and V.A. Sergeev: Wind-energetic conditions of Turkmenia.

Ashkhabad. Turkman NIINTI Gosplan TSSR, 1983, 48pp.

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Izv. AN TSSR. Ser. FTCh a n d GN. 1984, No. 3, pp. 21-25.

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56-60.