Production Costing Simulation in Thermal Power Systems Using the Mixture of Conditional Load Distribution Functions

Working Paper

Production Costing Simulation in Thermal Power Systems Using the

Mixture of Conditional Load Distribution Functions

J. Hoffer P. Dorfner

WP-90-80

^I

December 1990

International Institute for Applied Systems Analysis A-2361 Laxenburg Austria

Telephone: (0 22 36) 715 21 *0 Telex. 0 7 9 137 ~ i a s a a Telefax ( 0 2 2 36) 71313

Production Costing Simulation in Thermal Power Systems Using the

Mixture of Conditional Load Distribution Functions

WP-90-80 December 1990

'Computer and Automation Institute, Hungarian Acadeln!. ol' Sciences, P.O. Box 63, H-1502 Budapest, Hungary

2Hungarian Electricity Board

( M V M T ) ,

Budapest, Hu11gai.y

Working Papers are interim reports on work of the International Institute foi .41111li('cl Systems Analysis and have received only limited review. Views ^01. opinions expressccl herein do not necessarily represent those of the Institute or of its National R/Ieml,c~~.

Organizations.

El 11 ASA

International Institute for Applied Systems Analysis 0 A-2361 Laxenburg 0 Austr~a 8.

W.8 Telephone: (0 22 36) 715 2 1 * 0 Telex: 079 137 iiasa a Telefax ( 0 2 2 36) 71 Z 1 3

Foreword

T h e simulation of production costs for power systems is a key factor in the capacity expansion as well as some other questions raised in connection with planning for power systems. This paper develops a probabilitic model for production cost simulation that is able to handle uncertainty of both generating units and peak load forecast.

Alexander B. Kurzha.nski Chairmarl System and Decision Sciences Prograni

PRODUCTION COSTING SIMULATION I N THERMAL POWER S Y S T E M S USING THE MIXTURE O F CONDITIONAL LOAD D I S T R I B U T I O N FUNCTIONS

J. HOFFER

Computer and A u t o m a t i o n I n s t i t u t e , Hungarian Academy o f S c i e n c e s Budapest, P.O.Box 63, H-1502, Hungary

P. DOWNER

Htmgarian E l e c t r i c i t y Board (MVMT)

Budapest, P.O.Box 43, H-1251, Hungary

Abstract: I n t h e 1980's t h e curnulard. method became p o p u l a r i n r e l i a b i l i t y t y p e a l g o r i t h m s f o r p r o d u c t i o n c o s t e v a l u a t i o n , p a r t i c u l a r l y i n t h e e v a l u a t i o n o f loss-of-load p r o b a b i l i t y CLOLP), e n e r g y n o t s e r v e d (ENS) and e x p e c t e d e n e r g y g e n e r a t i o n (EEQ> o f a s e t o f g e n e r a t i n g u n i t s belonging t o an e l e c t r i c power s y s t e m . W e developed a p r o b a b i l i s t i c model which is a b l e t o handle t h e u n c e r t a i n t y o f b o t h g e n e r a t i n g units and peak load f o r e c a s t . In o r d e r t o model t h e load including peak load f o r e c a s t u n c e r t a i n t y w e use c o n d i t i o n a l p r o b a b i l i t y d i s t r i b u t i o n s . W e show t h a t t h e cumulant method is s t i l l applicable, as w e c a n compute all t h e moments o f t h e load d u r a t i o n c u r v e Cload d i s t r i b u t i o n > w i t h o u t d i s c r e t i z i n g t h e d e n s i t y f u n c t i o n o f peak load f o r e c a s t .

Keywords: e l e c t r i c power s y s t e m , p r o d u c t i o n c o s t s i m u l a t i o n ,

I. INTRODUCTION

For a t h e r m a l power s y s t e m t h e p r e d i c t i o n o f t h e e x p e c t e d eneI.gy g e n e r a t i o n o f t h e u n l t s , t h e loss-of-load probability and t h e e x p e c t e d value o f u n s e r v e d e n e r g y are i m p o r t a n t a s p e c t s b o t h i n s y s t e m o p e r a t i o n and planning.

F o r a more realistic and a c c u r a t e s i m u l a t i o n o f s y s t e m o p e r a t i o n , multiblock r e p r e s e n t a t i o n o f u n i t s ' f o r c e d o u t a g e w a s i n t r o d u c e d Csee C13, 133, 151, C63>. However i n t h e literature t h e maxlmum load l e v e l is usually f i x e d a t a given value by t h e load d u r a t i o n c u r v e . I t m e a n s t h a t t h e p r o b a b i l i t y w i t h which load e x c e e d s t h e given v a l u e is equal t o z e r o . With t h i s model < r e p r e s e n t a t i o n > o f t h e load d u r a t i o n c u r v e t h e peak load f o r e c a s t u n c e r t a i n t y and e x t r e m e load v a l u e s c a n n o t b e t a k e n i n t o a c c o u n t .

W e p o i n t o u t h e r e t h a t i n t h e p a p e r w e w r i t e a b o u t load d u r a t i o n c u r v e which is o f t e n called i n t h e llterature as i n v e r t e d , normalised load d u r a t i o n c u r v e . In t h e f i g u r e , s y s t e m load < t h e a r g u m e n t o f t h e f u n c t i o n > is shown o n t h e h o r i z o n t a l axis and p r o b a b i l i t y w i t h which load e x c e e d s t h e c o r r e s p o n d i n g load value ( d e p e n d e n t v a r i a b l e ) is shown o n t h e v e r t i c a l axis.

I t is v e r y i m p o r t a n t t h a t w e have an a c c u r a t e approximation f o r t h e d i s t r i b u t i o n of peak load v a l u e s < t h e t a i l of t h e load d u r a t i o n c u r v e > , as i t can h a v e I n f l u e n c e o n t h e maintenance scheduling plan. I t is obvious t h a t f e w e r c h a n g e s o f t h e load d u r a t i o n c u r v e , a r o u n d t h e minimum load value, d o e s n ' t modify t h e number o f u n l t s t o b e loaded, only t h e e x p e c t e d e n e r g y g e n e r a t i o n of some units changes. On t h e c o n t r a r y , i f w e p e r t u r b e t h e load d u r a t i o n c u r v e a r o u n d t h e

peak load value, keeping i t s o r i c i n a l s h a p e , t h e number of u n i t s t o be loaded c h a n g e s i n o r d e r t o m e e t t h e p r e s c r i b e d LOLP limit. This f a c t c a n s i g n i f i c a n t l y Influence t h e maintenance scheduling plan.

In t h e model of t h e p a p e r w e s u p p o s e t h a t t h e tail of t h e load d u r a t i o n c u r v e C1.e. t h e maximum value of load) c a n change. T h e load d u r a t i o n curve, f o r a given peak load value, i s a piecewise l i n e a r f u n c t i o n , joining t h r e e p a r t s . The first two p a r t s are t h e same f o r all v a l u e s o f peak load. The peak load value c a n change according t o a well-known d i s t r i b u t i o n

<e.g. uniform o r exponential). Above t h e f i r s t i n t e r v a l t h e load d u r a t i o n c u r v e is equal t o I. The second i n t e r v a l r e p r e s e n t s t h e expected b a s e load domain of t h e power s y s t e m , t h e t h i r d one s i m u l a t e s peak load f o r e c a s t u n c e r t a i n t y Csee Figure 1.3.

p r o b .

The p a r a m e t e r s o f t h e load d u r a t i o n c u r v e Cminlmum load value, expected maxfmum value, t h e probability of t h e e v e n t t h a t t h e load exceeds t h e previously mentioned value, p a r a m e t e r c s ) of t h e d i s t r i b u t i o n o f peak load f o r e c a s t ) c a n be defined by t h e e n e r g y planner.

1 -'

G

Using t h i s r e p r e s e n t a t i o n t h e e x a c t s h a p e of t h e load

"---"-' I

a

Figure 1.

d u r a t i o n c u r v e becomes unknown, b u t t h e cumulant method is s t i l l applicable, as w e c a n compute all t h e moments o f t h e r e s u l t i n g load d i s t r i b u t i o n .

A t t h e Hungarian E l e c t r i c i t y Board a p r o g r a m package has been implemented o n an IBM PC XT o r AT t o check t h e s y s t e m r e l i a b i l i t y level. The u s e r of t h e package i s enabled t o s i m u l a t e s e v e r a l a v a i l a b i l i t y - s i t u a t i o n s o f t h e power s y s t e m , by s e t t i n g units available o r unavailable, changing: t h e m a i n t e n a n c e scheduling plan, t h e p a r a m e t e r s o f t h e load d u r a t i o n c u r v e .

The r e a l i s t i c and easy-to-handle c h o i c e f o r t h e d i s t r i b u t i o n o f peak load f o r e c a s t is as follows:

-

u n i f o r m d i s t r i b u t i o n ( w i t h given m a x i m u m load o r w i t h given mean o f t h e maximum load>,

-

e x p o n e n t i a l d i s t r i b u t i o n (with g i v e n mean o f t h e maximum load o r w i t h given r i g h t e n d p o i n t o f t h e i n t e r v a l t h e p r o b a b i l i t y of which is g r e a t e r t h a n 0.999.

11. MOMENTS OF THE RESULTINO LOAD DUIUTION CURVE

L e t X b e t h e random v a r i a b l e r e p r e s e n t i n g t h e load, T be t h e random v a r i a b l e of peak load f o r e c a s t , p < t > be t h e probability d e n s i t y f u n c t i o n of T with existing: moments

<J-1,

...,

J > :

L e t LDC<x>, F < x > , and fCx> d e n o t e t h e load d u r a t i o n , t h e load d i s t r i b u t i o n and t h e load d e n s i t y f u n c t i o n , r e s p e c t i v e l y :

L e t LDCCxJt) b e t h e conditional load d u r a t i o n f u n c t i o n

<supposing T

=

t > and d e n o t e b y Xt t h e corresponding random variable. Then t h e load d u r a t i o n f u n c t i o n LDCCx) is t h e i n t e g r a l of load d u r a t i o n f u n c t i o n s LDCCx ( t ) depending on p a r a m e t e r t. Using t h e following: n o t a t i o n s :

w e have:

0

LDCCx)

-

^LDCtx It> p<t> dt,

-OD

(I,

F < x > -

J

F < x I t > p < t > d t ,

-(I,

(I,

f

-

f < x l t > p < t > d t .

'(I,

I t is o b v i o u s t h a t t h e e x p l i c i t f o r m o f t h e load d u r a t i o n c u r v e d e p e n d s o n t h e 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 < t > , and w e are seldom able t o t r a n s f o r m i t t o a q u i t e s i m p l e f o r m u l a .

In o r d e r t o use t h e cumulant method w e n e e d only t h e c u m u l a n t s o f t h e random v a r i a b l e r e p r e s e n t i n g t h e load.

Cumulants are polynomials of t h e c e n t r a l moments and, t h u s polynomials of t h e moments, as w e l l (see Kendall and Stuart 121). T h e r e f o r e w e need only t h e moments of t h e random v a r i a b l e o f t h e load:

By v i r t u e o f t h e Fubinl t h e o r e m (see Rudln 141) t h e o r d e r o f i n t e g r a l s c a n b e changed i n <2>, i f t h e k- t h moment

< k = 1 . K > o f t h e v a r i a b l e Xt easts and i t is f i n i t e f o r all p o s s i b l e t value; i.e. i f t h e r e v e r s e d i n t e g r a l is f i n l t e . This l a s t a s s u m p t i o n holds i n t h e c a s e s w e examine l a t e r . By changing t h e o r d e r o f t h e i n t e g r a l w e h a v e t h e following formula:

A s t h e i n n e r i n h g r a l is equal t o M<<>:

OD

M < f l >

- ^J

^pet) M<<> dt.

' O D

T h e r e are s e v e r a l c a s e s when i t is quite e a s y t o compute M&>. One _{o f t h e m is} as follows: HC<> is a p o l y n o d a l of t h e v a r i a b l e t a n d w e know all t h e needed moments o f t h e random v a r i a b l e T (see Example 1. a n d Example 2. below). I f

and

hold, t h e n

a0 n

k

M C ~ >

= S

p < t >

Z

cWtJ dt, -m

3-1

(I) n

k j

M t f l >

= Z

c k j t pCt> dt, al

3-1

n k ^(I)

-Q?

A s i l l u s t r a t i o n , c o n s i d e r t h e following examples:

E x a m p l e 1.: Xt is o f nofmal d i s t r i b u t i o n , t c a n b e e i t h e r t h e m e a n o r t h e s t a n d a r d d e v i a t i o n o f Xt. Denote t h e k - t h moment by mk and t h e v a r i a n c e of t h e d i s t r i b u t i o n by .'a Then

and

are valid, and b y using t h e above r e c u r s i o n - f o r m u l a , i t is e a s y t o see t h a t MC<> is t h e polynomial of e i t h e r t h e mean o r t h e s t a n d a r d deviation.

E x a m p l e 2.: L e t Xt be of uniform d i s t r i b u t i o n , and t one of t h e two e n d p o i n t s of t h e i n t e r v a l of possible values. If t h e i n t e r v a l in q u e s t i o n is Cr,sl, t h e n

which is a polynomial of e i t h e r r o r s.

E x a m p l e 3.: L e t xt be of exponential d i s t r i b u t i o n t h e p a r a m e t e r of which is t , T be of uniform d i s t r i b u t i o n on t h e i n t e r v a l Cr,sl C r , s are fixed). Then

f o r k

-

^1,

111. SPECIFICATION OF LDCCx I t ) AND T USED IN THE PROGRAM PACKAUE

In o u r model c o n s t r u c t i o n : let a and b b e t h e e n d p o i n t s of t h e i n t e r v a l where t h e load is simply uniformly d i s t r i b u t e d . L e t c <O<c<l> b e t h e p r o b a b i l i t y of t h e e v e n t of t h e load being g r e a t e r t h a n b o r equal t o i t . In o r d e r t o follow t h e n a t u r e of t h e p r a c t i c a l problem, s u p p o s e t h a t

p < t >

=

0 , f o r t

<

b .

W e c a n define t h e load d u r a t i o n f u n c t i o n LDC<x I t > as follows (see Figure I . ) :

f l 7 f o r x

<

^{a ,}

L D C < x I t >

=

^I + < x - a > < c - l > / < b - a > , f o r a S x

<

b , c

-

c < x - b > / < t - b > , f o r b < x < t ,

0 ⁷ f o r x Z t .

Then applying < I > w e have

f o r x

<

o , f o r o 5 x < b ,

I J

^tc

-

c < r - b > / < t - b > l p < t > dt, f o r x L b .

For a fixed t , f < x ( t > is as follows:

( 1 - c > / < b - a > , f o r a I x

<

^{b ,}

f < x I t )

=

f o r b I x

<

t ,

o t h e r w i s e .

Moments of Xt can be e x p r e s s e d as follows:

a,

M<$>

=

xk f < x l t > dx,

-a

k 1-c

M<<>

= J

^x

5~ & + J x -

^{& I}

t - b

.I-

J

r k dx

+ -

b- a t - b

Consequently t h e moments o f X c a n b e e x p r e s s e d by means of t h e moments of T. S u b s t i t u i n g i n <3> w e o b t a i n :

In t h e p r o g r a m package T is s u p p o s e d t o b e of e x p o n e n t i a l d i s t r i b u t i o n , t h e p o s s i b l e v a l u e s o f which are g r e a t e r t h a n b o r equal t o i t . L e t d <d>O> be t h e p a r a m e t e r of T. In t h i s case t h e d e n s i t y f u n c t i o n is

f o r t

<

b , p < t >

d e x p < - d < t - b > > , f o r t l b .

W e t h i n k t h a t t h e a s s w n p t i o n of e x p o n e n t i a l l t y is c l o s e t o t h e n a t u r e of t h e peak load d i s t r i b u t i o n . W e s u p p o s e t h a t T

could b e modellised w i t h u n l f o r m d i s t r i b u t i o n as w e l l , b u t w e h a v e no numerical e x p e r i e n c e f o r t h i s case y e t .

The e x p e c t e d value o f T is

and t h e r e a d e r can e a s i l y v e r i f y (using t h e method of p m t i a l i n t e g r a t i o n > t h e following r e c u r s i o n f o r m u l a

MTJ>

-

^bJ

⁺

^j

a ntrJ-'>,

^{f o r 312.}

This c o m p l e t e s t h e d e t a l l s of c o m p u t a t i o n o f M C P ) .

In o r d e r t o s p e e d up t h e c a l c u l a t i o n s w e u s e d a r e c u r s i o n f o r m u l a f o r t h e q u a n t i t i e s

and

as well. I t is o b v i o u s t h a t

v k

k+1 ^{V k Q} + b

,

W k

k+l

-

^{M C T}

^{> +}

^{'L, k b .}

S i n c e

w e o b t a i n

MXk>

^can b e e x p r e s s e d by means o f u a n d a, as well:

k k

IV. CONCLUSIONS

Following t h e m e t h o d d e s c r i b e d i n t h e p a p e r w e n e e d n o t d i s c r e t i z e t h e d e n s i t y f u n c t i o n p < t > , a n d i n this way t h e r e is n o n e e d for e v a l u a t i n g l o s s - o f - l o a d p r o b a b i l i t y , e n e r g y n o t s e r v e d a n d e x p e c t e d e n e r g y g e n e r a t i o n of units r e p e a t e d l y for all t h e impulse v a l u e s of peak l o a d forecast.

V. REFERENCES

C11 E l e c t r i c G e n e r a t i o n E x p a n s i o n A n a l y s i s S y s t e m 1-2. EPRI R e p o r t EL-2561, Palo Alto, California <1982>.

C21 M. K e n d a l l a n d A. Stuart T h e A d v a n c e d T h e o r y o f S t a t i s t i c s . D i s t r i b u t i o n T h e o r y , Vol. 1, C h a r l e s a r l f f l n &

Company Limi Led, London (1977).

C31 N.S. Rau, P. T o y and K.F. Schenk 'Expected Energy Production Costing by t h e Method of Moments' IEEE T r a n s . Power A p p a r a t u s and S y s t e m s Vol.PAS-99 NO.^., 1980, pp. 1908-1915.

C41 W. R u d i n R e a l a n d C o m p l e x A n a l y s i s , McQraw-Hill, London C1970).

C51 K.F. Schenk 'Cumulant Method i n Production Cost

Evaluation', IAEA L e c t u r e 30.5.6, Argorine National L a b o r a t o r y , Argonne, IlUnois C1985).

C63 W i e n A u t o m a t i c S y s t e m P l a n n i n g P a c k a g e <WASP>, User's Manual, IAEA, Vienna C1980).