Computation of Electric Energy Exchange between two Power Systems

W O R K I N G P A P E R

COMPUTATION OF ELECTRIC ENERGY EXCHANGE BETWEEN TWO POWER SYSTEMS

Tibor TerGyhnszky

Pi1

Major

November 1988

W

P-88- 106

I n t e r n a t i o n a l I n s t i t u t e for Appl~ed Systems Analys~s

COMPUTATION OF ELECTRIC ENERGY EXCHANGE BETWEEN TWO POWER SYSTEMS

Tibor ~ e r ~ t y d n s z k ~

*

Pdl Major

November 1988 WP-88-106

* * * ~ i n i s t r ~ of Industry, Budapest

Institute for Electric Power Research, Budapest

Working Papers are interim reports on work of t h e International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of t h e Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE F O R APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

FOREWORD

The expected energy exchange between cooperating systems is an important infor- mation supporting capacity expansion planning for electric power systems. A model based on engineering considerations and a program system on IBM/PC-XT or A T compatibles has been developed for the stochastic analysis of the electric energy exchange between two interconnected power systems. Data required for the analysis are expected generation and load in the individual systems, and interties data. This work has been carried out in the frame of the IIASA Contracted Study "Modeling of interconnected power systems".

Alexander B. Kurzhanski Chairman System and Decision Sciences Program

CONTENTS

1 Introduction

2 Main Characteristics of Electric Energy Exchange 3 Statistical Analysis of Electric Energy Exchange 4 Problem Formulation and Solution Methodology

4.1 A Mathematical Model of Cooperating Power Systems 4.2 Elements of the Power System

4.2.1 Generators

4.2.2 Consumer-Demand

4.2.3 Scheduled Export-Import Based on Long Term Agreements 4.2.4 Tie-Lines

4.3 T h e Computational Procedure

4.3.1 Computation of the Aggregated Capacity-Availability Histogram of Generating Units

4.3.2 Computation of Deficiency-Excess Histogram 4.3.3 Computation of Exchange Power

4.3.4 Computation of Exchange Power for Real Connection 5 Realization on IBM/PC

5.1 Overview of t h e Program System 5 . 2 Running the Program

6 Possibilities for Further Development References

Figures

The u t i l i z a t i o n of power s y s t e m s h a s t o b e a d o p t e d t o management o b j e c t i v e s . The main t h r u s t of e l e c t r i c power s y s t e m o p e r a t i o n a r e t h e s e c u r i t y and c o s t o f o p e r a t i o n and q u a l i t y of s u p p l y . One of t h e ways t o f u l f i l l t h e management o b j e c t i v e s i s t h e c o o p e r a t i o n among e l e c t r i c power s y s t e m s .

The p u r p o s e o f t h i s p a p e r i s t o p r e s e n t methods f o r t h e c o m p u t a t i o n o f e l e c t r i c e n e r g y e x c h a n g e between c o o p e r a t i n g s y s t e m s . The s t o c h a s t i c n a t u r e of t h e a v a i l a b i l i t y of g e n e r a t i n g u n i t s a n d t r a n s m i s s i o n l i n e s and t h e randomness i n consumer demand a r e t a k e n i n t o a c c o u n t .

The a p p r o a c h employed h e r e i s b a s e d on a p a p e r by T.

T e r s t y h s z k y C11 and on t h e p a p e r s C2, 3. 41.

The main f e a t u r e s d i s c u s s e d a r e 1 i s t e d be1 ow:

- s t a t i s t i c a l a n a l y s i s of t h e random e f f e c t s i nf 1 u e n c i ng el e c t r i c power e x c h a n g e ,

-

s t o c h a s t i c m o d e l i n g o f t h e i n t e r c o n n e c t e d s y s t e m s c o n s i s t i n g of submodel s f o r t h e i n d i v i d u a l components.

-

c o m p u t a t i o n a l p r o c e d u r e s f o r t h e s o l u t i o n of t h e model.

-

p o s s i b i 1 i ti es f o r f u r t h e r d e v e l opment

.

The c a p a b i l i t y o f c a l c u l a t i n g t h e e x p e c t e d e l e c t r i c e n e r g y e x c h a n g e between c o o p e r a t i n g s y s t e m s c a n b e u t i l i z e d b o t h f o r s h o r t - t e r m p r o d u c t i on p l a n n i n g a n d 1 ong-term c a p a c i t y e x p a n s i o n p l a n n i ng. For a s h o r t - t e r m h o r i z o n b e t t e r u t i l i z a t i o n o f t h e e x i s t i n g equipment c a n b e a c h i v e d . I n t h e c a s e o f l o n g - t e r m p l a n n i ng. a l a r g e amount of a1 t e r n a t i v e s f o r t h e development o f t h e e n e r g y s y s t e m a r e a n a l y z e d . compared a n d a n a p p r o p r i a t e v a r i a n t s e l e c t e d . I n t h i s c a s e t h e e n e r g y e x c h a n g e i s c a l c u l a t e d f o r t h e i n d i v i d u a l a l t e r n a t i v e s

.

2 M A I N CHARACTERISTICS OF ELECTRIC ENERGY EXCHANGE

I n most c o u n t r i e s t h e e l e c t r i c power i n d u s t r y h a s r e a l i z e d t h e econorni c and t e c h n i c a l a d v a n t a g e s o f c o o p e r a t i o n . The e x c h a n g e o f el e c t r i c power became p a r t o f t h e i n t e r n a t i on21 t r a d e , t o o . But i n c o n t r a s t w i t h o t h e r commodities e l e c t r i c i t y h a s t h e v e r y s p e c i f i c c h a r a c t e r a s i t c a n ' t b e s t o r e d i n l a r g e q u a n t i t i e s . T h i s i m p l i e s t h a t i n a n emergency s i t u a t i o n Ccaused e . g . by f o r c e d o u t a g e s 3 t h e r i s k o f b r e a k i n g t h e c o n t i n u i t y o f s u p p l y i n c r e a s e s c o n s i d e r a b l y . I n t h i s c a s e t h e s u p p l y c o n t i n u i t y f o r c u s t o m e r s c a n b e e n s u r e d by power t r a n s f e r f r o m o t h e r power systems. The q u a n t i t y of e l e c t r i c power exchanged among t h e c o o p e r a t i n g c o u n t r i e s o r u t i l i t i e s i s t h e r e s u l t o f a compromise between a c o n c e r n f o r el e c t r i c power i ndependence. t h e economi c a d v a n t a g e s o f c o o p e r a t i o n , and t h e t e c h n i c a l p r o b l e m s a r i s i n g f r o m d i f f e r e n t s t r a t e g i e s o f f r e q u e n c y c o n t r o l .

The o v e r a l l amount o f e l e c t 1 c e n e r g y e x c h a n g e f o r t h e y e a r 1985 c a n b e s e e n on F i g u r e s 1 and 2 f o r t h e Western and E a s t e r n European c o u n t r i e s . r e s p e c t i v e 1 y Csee a l s o C 5 . 6 1 3 . The r el at i ve i mpor t ance of el ectr i c power exchange for

s e l e c t e d European c o u n t r i e s i s shown i n F i g . 3 C71.

Two d i a g r a m s a r e p r e s e n t e d t o show t h e s t o c h a s t i c c h a r a c t e r of t h e power e x c h a n g e :

F i g 4. shows t h e a g g r e g a t e sum of s i m u l t a n e o u s l o a d s of e l e c t r i c power t r a n s f e r c o n n e c t 1 o n s a c r o s s s t a t e b o r d e r s i n t h e W e s t European UCPTE s y s t e m , o v e r a 6 y e a r s p e r i od f o r a s p e c i f i e d t i m e of a month.

The power e x c h a n g e c a n b e s e e n i n d e t a i 1s on F i g . 5, f o r t h e c a s e o f t h e H u n g a r i a n i n t e r - t i e s . T h i s f i g u r e shows t h e 1 o a d p a t t e r n o f one-horrr a v e r a g e s c h e d l . ! e d e x p o r t-i mpor t on 1 o n g - t e r m a g r e e m e n t , i n c o m p a r i s o n w i t h t h e a c t u a l l o a d which c o n t a i n s a l s o t h e s h o r t - t e r m random power e x c h a n g e p e r f o r m e d on t h e b a s i s of mutual a s s i s t a n c e between t h e Hungarian Power System and t h e I n t e r c o n n e c t e d Power System o f CMEA.

C o n s i d e r i n g t h e unrni s t a k a b l y u n d e t e r m l n i st1 c n a t u r e of e l e c t r i c power e x c h a n g e a s t o c h a s t i c m o d e l i n g a p p r o a c h i s a n a p r o p p r 1 a t . e way o f d e a l i n g w i t h t h e problem. The model and comput at1 o n a l methods p r e s e n t e d i n t . h i s p a p e r a r e d e v e l o p e d for. t h e e s t i m a t i o n of power e x c h a n g e b a s e d b o t h on l o n g t e r m a n d s h o r t t e r m a g r e e m e n t s

.

3. STAT1 SZlCAL ANALYSIS OF ELECTRIC ENERGY EXCHANGE.

The b a s e l o a d o f t i e - l i n e s between c o o p e r a t i n g power s y s t e m i s f i x e d by l o n g t e r m a g r e e m e n t s . The a c t u a l l o a d o f l i n e s however, d i f f e r s f r o m t h i s a s s i g n e d t r a n s f e r l e v e l as a r e s u l t of c h a n g e s i n o p e r a t i n g c o n d i t i o n s s u c h a s f o r c e d o u t a g e of u n i t s .

A s t h e power e x c h a n g e i s o f a s t o c h a s t i c n a t u r e . i t.s amount c a n b e c o n s i d e r e d as a random v a r i a b l e . To p r o v i d e a b a s i s f o r model -development a s t a t 1 st1 c a l a n a l y s i s of t r a n s f e r r e d power between a 1 a r g e s y s t e m h a v i n g i n s t a l 1 e d c a p a c i t y of a p p r o x i m a t e l y 70000 MW and a s m a l l s y s t e m w i t h 4000 MW w a s c a r r i e d o u t .

To e n s u r e s u f f i c i e n t l y l a r g e s a m p l e s i z e s a c t u a l power d a t a of t r a n s f e r s f o r d a i l y h i g h l o a d p e r i o d s i n o n e y e a r h a v e b e e n t a k e n i n t o a c c o u n t . CFig. 6 . The a n a l y s i s w a s r e p e a t e d f o r 5 c o n s e c u t i v e y e a r s .

The h y p o t h e s i s t o b e t e s t e d w a s t h e f o l l o w i n g : The power e x c h a n g e i s a n o r m a l l y d i s t r i b u t e d random v a r i a b l e . The assumpi on u n d e r 1 y i ng o u r i n v e s t i g a t i o n w a s t h a t t h e d i s t r i b u t i o n f u n c t i o n b e l o n g s t o t h e P e a r s o n f ami 1 y o f p r o b a b i l i t y d i s t r i b u t i o n s C 8 1 .

R e s u l t s of t h e a n a l y s i s c o n f i r m e d i n a l l c a s e s c o n s i d e r e d t h a t t h e s h o r t - t e r m e l e c t r i c power e x c h a n g e i s n o r m a l l y d i s t r i b u t e d , s i n c e t h e c o r r e s p o n d i n g c o n s t a n t s i n t h e d i f f e r e n t i a1 e q u a t i o n i d e n t i f y i ng Pear s o n * f aml 1 y w e r e f o u n d t o b e a p p r o x i m a t e l y z e r o .

For t h e s a k e o f a more a c c u r a t e e v a l u a t i o n . i t i s a p p r o p r i a t e t o r e p r e s e n t d a t a i n r e l a t i v e u n i t s . T h e mean v a l u e r e p r e s e n t i n g t h e s y s t e m s ' c o o p e r a t i o n f 1 u c t u a t e s i n p r i n c i p l e a r o u n d zero. t h e r e f o r e i t i s r e a s o n a b l e t o e x a m i n e t h e s h o r t t e r m power e x c h a n g e i n p e r u n i t r e p r e s e n t a t i o n w i t h r e s p e c t t o t h e s t a n d a r d d e v i a t i o n Csee F i g . 7 3 .

I n o r d e r t o v e r i f y t h e h y p o t h e s i s c o n c e r n i n g t h e d i s t r i b u t i o n o f t h e power e x c h a n g e a c h i - s q u a r e - t e s t w a s c a r r i e d o u t . w h i c h c o n f i r m e d t h e h y p o t h e s i s o f normal i t y .

4. PROBLEM FORMULATI ON AND SOLUTION METHODOLOGY

4 . 1 A MATHEMATICAL MODEL OF COOPERATING POWER SYSTEMS

A power s y s t e m i s d e s i g n e d i n s u c h a way. t h a t m o s t l y i t c a n s a t i s f y t h e consumer demand, moreover i t h a s a r e s e r v e f o r t h e case o f f a i l u r e s . s o u s u a l l y it. i s p o s s i b l e t o g i v e a s s i s t a n c e f o r t h e c o o p e r a t i n g p a r t n e r i f i t i s i n t r o u b l e , a n d v i c e v e r s a .

I n t h e model p r e s e n t e d b o t h e n e r g y s y s t e m s a r e r e d u c e d t o a n o d e ; e a c h w i t h i t s own g e n e r a t i o n a n d l o a d a n d w i t h n o i n t e r n a l t r a n s m i ssi o n 11 mi t a t i o n s . T h e i r e n e r g y c o n s u m p t i o n s

.

t h e a v a i l a b 1 e power p l a n t c a p a c i t i e s a n d t h e t r a n s m i s s i o n c a p a c i t y o f t h e t i e - l i n e s y s t e m a r e c o n s i d e r e d t o b e random v a r i a b l e s C21.

For t h e e v a l u a t i o n o f t h e e x c h a n g e o f e n e r g y b e t w e e n t h e c o o p e r a t i n g s y s t e m s t h e f o l l o w i n g v a r i a b l es w i 11 b e i n t r o d u c e d :

-

^{y i} a n d y d e n o t e t h e random v a r i a b l e s o f t h e a g g r e g a t e d z

a v a i l a b l e c a p a c i t y o f g e n e r a t i n g u n i t s i n t h e f i r s t a n d s e c o n d s y s t e m , r e s p e c t i v e l y .

g Cz3 a n d g Cz3 d e n o t e t h e i r d e n s 1 t . y f u n c t i o n s , GiCz3 a n d

i 2

GzCz3 t h e d i s t r i b u t i o n f u n c t i o n s .

-

6* a n d d Z d e n o t e t h e random v a r i a b l e s o f t h e consumer demand i n t h e f i r s t a n d s e c o n d s y s t e m . r e s p e c t i v e l y .

-

L e t s b e t h e d e t e r m i n i s t i c v a r i a b l e o f t h e s c h e d u l e d e x c h a n g e of e n e r g y p r e d e f i n e d a c c o r d i n g t o l o n g t e r m a g r e e m e n t . I n t h e model t h i s t y p e o f e n e r g y e x c h a n g e i s t a k e n i n t o a c c o u n t as a demand. Let p a n d p d e n o t e t h e

i 2

e n e r g y demand m o d i f i e d by s p i = & + s

1

pz = 6 - s

Z

c

1 3

L e t h C and hzCx3 b e t h e i r d e n s i t y f u n c t i o n s . H i C 3

i

and HzCx3 t h e d i s t r i b u t i o n f u n c t i o n s .

-

^c a n d c a r e random v a r i a b l e s of d e f i c i e n c y / e x c e s s o f

1 Z

e n e r g y f o r b o t h s y s t e m s

I f t h e c . < O Ci=1,23 t h e n t h e r e i s d e f i c i e n c y of e n e r g y

L

i n t h e i t h s y s t e m , i f c.=O t h e n t h e demand a n d t h e g e n e r a t e d e n e r g y a r e b a l a n c e d , o t h e r w i s e t h e r e i s e n e r g y e x c e s s i n t h e s y s t e m .

L e t f Cx3 and f Cx3 b e t h e i r d e n s i t y f u n c t i o n s . FiCz3

1 2

and F Cz3 t h e d i s t r i b u t i o n f u n c t i o n s .

2

c a n d c c a n b e assumed t o b e i n d e p e n d e n t random

1 Z

v a r i a b l e s b e c a u s e t h e y mai n l y depend on i n t e r n a l c h a r a c t e r i s t i c s o f t h e i n d i v i d u a l s y s t e m s .

-

L e t

fr

d e n o t e t h e random v a r i a b l e of e n e r g y e x c h a n g e f l o w i n g t h r o u g h t h e t r a n s m i s s i o n l i n e s c o n n e c t i n g t h e s y s t e m s . L e t FCz3 d e n o t e t h e d i s t r i b u t o n f u n c t i o n of

f r .

The v a l u e of

fr

c a n v a r y between t h e maxi ma1 demand of t h e f i r s t s y s t e m m u l t i p l i e d b y -1, a n d t h e maximal demand o f t h e s e c o n d s y s t e m . A p o s i t i v e

fr

means e n e r g y f l o w i n g f r o m t h e f i r s t s y s t e m t o t h e s e c o n d one.

I n o u r model w e assume t h e f o l l o w i n g a s s i s t a n c e p o l i c y : W i t h i n a s y s t e m n o consumer l i m i t a t i o n s a r e imposed d u e t o d e f i c i e n c y i n t h e o t h e r s y s t e m . A s s i s t a n c e i s o n l y g i v e n when e n e r g y r e s e r v e i s a v a i 1 a b l e .

T h e r e a r e o t h e r p o s s i b i l i t i e s f o r a n a s s i s t a n c e p o l i c y . e. g . t h e s y s t e m s c o u l d s h a r e t h e c o s t s of damage a n d l i m i t a t i o n of consumers i n a p r e d e f i n e d r a t i o . However w e t h i n k t h a t t h i s p o l i c y i s n o t a f e a s i b l e o n e i n t h e normal e c o n o m i c a l , a c c o u n t i n g r el a t i o n s between d i f f e r e n t c o u n t r i e s .

According t o t h e a s s i s t a n c e p o l i c y assumed a b o v e t h e e x c h a n g e of e n e r g y i s d e t e r mined by t h e f 01 1 owing e x p r e s s i o n :

mi nC E _:

.

^-E ₂ ³ i f r l > O and s2<0

f

= m a d c*. ^-E 3 i f c l < O a n d c2>0 C 33

2

0 o t h e r w i se

The main p r o b l e m a d d r e s s e d i n t h i s p a p e r i s t h e c o m p u t a t i o n of t h e d i s t r i b u t i o n of v a r i a b l e

c .

The d i s t r i b u t i o n f u n c t i o n of

c

i s t h e f o l l o w i n g f u n c t i o n .

PC maxcc:. - c 2 3 < z . c*<o, c2>0 3 i f 110 C 43 I t i s e a s y t o see t h a t t h i s f u n c t i o n c a n b e e x p r e s s e d i n t h e si mpl e f o r m g i ven be1 ow:

Using t h e d i s t r i b u t i o n f u n c t i o n s of E* a n d c2 t h e f u n c t i o n c a n b e r e w r i t t e n a s f o l l o w s

F u n c t i o n FCz3 h a s a n o n z e r o jump a t z=O o f t h e f o l l o w i n g magni t ude

Fo = 1 + 2F C 03F2C 03 - F C 03

-

FzC03

1 i

c

^{7 3}

e x p r e s s i n g t h e p r o b a b i l i t y o f t h e e v e n t t h a t no e n e r g y e x c h a n g e o c c u r s . T h i s i s t h e c a s e when e i t h e r b o t h s y s t e m s c a n p r o d u c e t h e s u f f i c i e n t amount of e n e r g y t o s a t i s f y t h e i r consumers o r b o t h s y s t e m s h a v e e n e r g y d e f i c i e n c y .

As t h e d i s t r i b u t i o n f u n c t i o n h a s a jump t h e r e d o e s n o t e x i s t a c o r r e s p o n d i n g d e n s i t y f u n c t i o n . To c h a r a c t e r i s e p r o b a b i 1 i t i e s of t h e d i f f e r e n t amounts of e n e r g y e x c h a n g e t h e f 011 owing f u n c t i o n w i l l b e i n t r o d u c e d and c a l c u l a t e d .

-

For z < 0 i t e x p r e s s e s . i n t h e same way a s u s u a l h i s t o g r a m s . t h e p r o b a b i l i t i e s o f t h e amount. of e n e r g y t r a n s m i t t e d f r o m s y s t e m 2 t o s y s t e m 1 .

-

^{For z=O} i t s v a l u e i s t h e p r o b a b i l i t y of n o power e x c h a n g e .

- For z > 0 i t e x p r e s s e s t h e p r o b a b i l i t i e s o f t h e amount of e n e r g y t r a n s m i t t e d f r o m s y s t e m 1 t o s y s t e m 2 .

T h i s f u n c t i o n w i l l b e c a l l e d e x c h a n g e p r o b a b i l i t y f u n c t i o n and d e n o t e d by fCz3. As i t p l a y s s a m e r o l e as h i s t o g r a m s w e w i l l a l s o r e f e r t o i t a s e x c h a n g e h i s t o g r a m . Using t h e f o r m u l a g i v e n below f o r FCz3 a n e x p l i c i t f o r m c a n b e o b t a i n e d by d i f f e r e n t i a t i o n :

For c o m p u t a t i o n of t h e e x c h a n g e h i s t o g r a m f C z > i t i s n e c e s s a r y t o d e t e r m i n e t h e d e n s i t y f u n c t i o n s f Cz3 and

i

f 2 C z > . S i n c e s a n d s a r e t h e sum of random v a r i a b l e s

i 2

as g i v e n a b o v e , t h e y c a n b e computed by c o n v o l u t i o n i n t e g r a l s . The c o r r e s p o n d i n g d i s t r i b u t i o n f u n c t i o n s c a n

b e o b t a i n e d a f t e r w a r d s by i n t e g r a t i o n . I n t h e c o m p u t a t i o n s h i s t o g r a m s a r e u s e d as empi r i c a l d e n s i t y f u n c t i o n s .

The main s t e p s o f t h e c a l c u l a t i o n a r e t h e f o l l o w i n g :

- C o m p u t a t i o n o f t h e a v a i l a b i l i t y h i s t o g r a m o f g e n e r a t i ng u n i t s f o r b o t h e n e r g y s y s t e m s , which means c a l c u l a t i o n of t h e h i s t o g r a m s c o r r e s p o n d i n g t o g Cz3

: a n d g C 23.

2

-

C o m p u t a t i o n of t h e d e f i c i e n c y - e x c e s s h i s t o g r a m s f Cz3

i

and f Cz3 o n t h e b a s i s o f t h e a v a i l a b i l i t y h i s t o g r a m s

2

g Cz3 a n d g2Cz3. o f g e n e r a t i n g u n i t s , a n d t h e d a i 1 y

i

v a r i a t i o n o f consumer demand. The consumer demand i n c l u d e s s c h e d u l e d e x p o r t-i mpor t

.

- Knowing t h e d e f i c i e n c y - e x c e s s h i s t o g r a m f o r b o t h s y s t e m s , t h e p r o g r a m c a l c u l a t e s t h e p r o b a b i l i t i ec, of d i f f er e n t a m o u n t s o f e n e r g y e x c h a n g e b e t w e e n t h e t w o s y s t e m s . I n t h i s way t h e h i s t o g r a m o f t h e e n e r g y e x c h a n g e i s c a l c u l a t e d . I n t h i s s t e p t h e c o o p e r a t i n g l i n e s y s t e m i s c o n s i d e r e d t o b e a n i d e a l c o n n e c t i o n w i t h o u t c a p a c i t y 1 i m i t a t i o n s a n d w i t h o u t a n y f a i 1 ur.e or o u t a g e .

-

I n t h e l a s t . s t e p t r a n s m i s s i o n l i n e s y s t e m l i m i t a t i o n s a n d p o s s i b i l i t i e s o f f a i l u r e a r e t a k e n i n t o a c c o u n t . The c o m p u t a t i o n o f t h e power e x c h a n g e p r e s u p p o s e s t h e model i ng o f t h e g e n e r a t i n g u n i t s , t h e consumer. -demand, t h e s c h e d u l e d export -i mpor t , as w e 1 1 as t h e model o f t h e t i e-1 i n e s y s t e m . T h e s e components a r e d e s c r i b e d a s f o l l o w s .

4 . 2

ELEMENTS

OF

THE

POWER SYSTEM 4 . 2 . 1 GENERATORS

The r e q u i r e d d a t a f o r t h e c a p a c i t y - a v a i l a b i l i t y c h a r a c t e r i s t i c s o f t h e g e n e r a t i n g b l o c k s a r e summarized as f 01 1 o w s

.

-

maximum c a p a c i t y o f t h e i n d i v i d u a l b l o c k s CPi i n CMWI3,

10

-

s e l f - c o n s u m p t i o n of t h e b l o c k s C S , i n [ % I > ,

1

- f a i l u r e p r o b a b i l i t y of t h e b l o c k s Cpi i n C113.

Den0t.e by Ni t h e c a p a c i t y of t h e 1 - t h b l o c k d e c r e a s e d by i t-s s e l f c o n s u m p t i o n . where i d e n o t e s t h e i n d e x of t h e b l o c k .

4 . 2 . 2 CONSUMER-DEMAND

The consumer demand i s d e s c r i b e d by t h e random v a r i a b l e s b*

and b2 which a r e assumed t o b e n o r m a l l y d i s t r i b u t e d . The t i m e h o r i z o n i s s u b d i v i d e d i n t o p e r i o d s . The e x p e c t e d v a l u e s a n d s t a n d a r d d e v i a t i o n s a r e c o n s i d e r e d t o b e c o n s t a n t a l o n g e a c h t i m e - p e r i o d . s o t h e e x p e c t e d d a i l y l o a d - c u r v e i s a s t e p w i s e c o n s t a n t f u n c t i o n of t i m e . T h e r e f o r e t h e consumer demands a r e model e d b y

-

t h e i r s t e p w i s e c o n s t a n t f u n c t i o n e x p e c t e d v a l u e s CDk i n CMWl3.

-

t h e i r s t a n d a r d d e v i a t i o n s C c r inC113.

k

where k d e n o t e s t h e t i m e - i n t e r v a l i n d e x .

The d e v i a t i o n cr i s g i v e n w i t h r e s p e c t t o t h e peak Dmx. i n

m a x

p e r c e n t a g e of t h e peak v a l u e . The d e v i a t i o n s f o r t h e i n d i v i d u a l p e r i o d s a r e c a l c u l a t e d by t h e f 01 1 owl ng f o r mu1 a :

T h i s e q u a t i o n e x p r e s s e s t h e t e n d e n c y , t h a t a l o w e r l o a d i s a s s o c i a t e d w i t h a 1 o w e r d e v i a t 1 on.

The d e t e r m i n a t i o n of t h e consumer-demand i s b a s e d on t h e f o r e c a s t i n g of d a i l y load c u r v e s . The c a l c u l a t i o n of t h e e n e r g y e x c h a n g e i s u s u a l l y c a r r i e d o u t f o r a l o n g e r t i m e - i n t e r v a l . e . g . f o r o n e y e a r . Our program a l l o w s f o r s e l e c t i n g s i x s e a s o n s w i t h o p t i o n a l l e n g t h s . a n d i t i s p o s s i b l e t o s e l e c t d i f f e r e n t d a i l y - l o a d c u r v e s t o a l l of t h e s e a s o n s . F i g 9. shows a n example of a t y p i c a l d a i l y l o a d c u r v e .

4.2.3 SCHEDULED EXPORT-IMPORT BASED ON LONG TERM AGREEMENTS

The s c h e d u l e d expor t-i mpor t e n e r g y e x c h a n g e , d e n o t e d above by s , i s modeled as a consumer-demand. I t i s assumed t o be d e t e r m i n i s t i c , a n d i t s v a r i a t i o n w i t h i n t h e t i m e h o r i z o n i s c o n s i d e r e d as a s t e p w i s e c o n s t a n t f u n c t i o n . T h e r e f o r e t h e s c h e d u l e d e x p o r t-i mpor t i s model e d by a s t e p w i se c o n s t a n t f u n c t i o n w i t h v a l u e s

Ek

i n C M W I , where k d e n o t e s t h e t i m e - i n t e r v a l i n d e x .

As f o r t h e consumer -demand model i t i s h e r e a l s o p o s s i b l e t o select d i f f e r e n t c u r v e s , a c c o r d i n g t o t h e s e a s o n s .

In b o t h e n e r g y s y s t e m s t h e g e n e r a t i n g u n i t s must s e r v e t h e 21 g e b r a i c sum of t h e consumer -demand a n d t h e s c h e d u l e d e x p o r t -i mpor t

.

4 . 2 . 4 T I E - L I N E S

The d a t a r e q u i r e d f o r t h e d e s c r i p t i o n of t h e c o o p e r a t i n g l i n e s a r e t h e f o l l o w i n g .

-

m a x i m u m t r a n s f e r c a p a c i t i e s CL i n [ W l 3 ,

i

-

f a i l u r e p r o b a b i l i t i e s 1 inC133.

where i d e n o t e s t h e i n d e x of c o o p e r a t i n g l i n e . 4 . 3 THE COMPUTATI ONAL PROCEDURE

4 . 3 . 1 COMPUTATION OF THE AGGREGATED CAPACI TY -AVAI LAB1 L I TY HI STOGRAM OF GENERATI NG UNI TS

The p r o b a b i l i t y d i s t r i b u t i o n of t h e a v a i l a b i l i t y of g e n e r a t o r s i s a d i s c r e t e d i s t r i b u t i o n , s o i t s d e n s i t y i s c a l c u l a t e d i n t h e form of a h i s t o g r a m . L e t w e c o n s i d e r o n e o f t h e power s y s t e m s i n d i v i d u a l l y .

L e t

ei,

i = l , . - - , n b e t h e f o l l o w i n g random v a r i a b l e s d e s c r i b i n g u n i t a v a i l a b i l i t y f o r t h e i t h u n i t

w i t h p r o b a b i l i t y pi w i t h p r o b a b i l i t y 1-p.

where n i s t h e number o f g e n e r a t i n g u n i t s i n t h e s y s t e m . 1 2

Denote by 8. t h e p a r t i a l sums of Bi's

The c a l c u l a t i o n of t h e a g g r e g a t e d c a p a c i t y a v a i l a b i l i t y h i s t o g r a m w i l l b e c a r r i e d o u t i n t h e f o l l o w i n g way:

h

For Bi = Bi t h e h i s t o g r a m i s known

h

Knowing t h e h i s t o g r a m f o r 8 . C 1 I i I n - 1 3 t h e h i s t o g r a m

h L

f o r

'i+i c a n b e computed by c o n v o l u t i o n si n c e

h h

-Bi+Bi+i. U s i n g t h i s p r o c e d u r e t h e h i s t o g r a m f o r 8

'i+i- n

i s t h e a g g r e g a t e d c a p a c i t y a v a i l a b i l i t y h i s t o g r a m g C d .

Denote d t h e l e n g t h of t h e s u b i n t e r v a l s i n t h e h i s t o g r a m , and

i s t h e number of t h e s u b i n t e r v a l s i n t h e h i s t o g r a m . The f i r s t s t e p :

i f a < x < b o

0

o t h e r w i se

where a =d e n t i erCNi/d3 a n d

0

bo=d CentierCN /d3+13

i

g e n e r a l s t e p : L e t a l r e a d y b e C i l machines i n t h e s y s t e m . Then t h e n e x t machine Cnumber i + 1 3 i s a d d e d t o t h e s y s t e m by t h e f o l l o w i n g c y c l e . w i t h

m=k-1

,

k-2..

. .

. I .

Cx3 o t h e r w i s e C133

where

a n d

'~+2.k

c

^{X )} =gi+i,l(x)

C N is t h e c a p a c i t y , d e c r e a s e d by t h e selfconsumption; and p

i i

t h e f a i l u r e p r o b a b i l i t y 3 .

P e r f o r m i n g t h e g e n e r a l s t e p f o r e v e r y m a c h i n e i n a n y o r d e r , t h e f u n c t i o n g C x l , i . e . t h e c a p a c i t y - a v a i l a b i l i t y h i s t o g r a m o f t h e g e n e r a t o r s i s o b t a i n e d .

The c o m p u t a t i o n o f t h e e x a c t d i s t r i b u t i o n o f random v a r i a b l e s

Y1 a n d

y2 C a g g r e g a t e d c a p a c i t y a v a i 1 a b i 1 i t y 3 i s c o m p u t a t i o n a l l y i n f e a s i b l e f o r s y s t e m s o f r e a l i s t i c s i z e b e c a u s e o f t h e c o m b i n a t o r i a l e x p l o i s o n . I t i s c l e a r t h a t t h e c o m p u t a t i o n a l e f f o r t n e e d e d t o g e t t h e r e s u l t s h e a v i l y d e p e n d s o n t h e l e n g t h o f s u b i n t e r v a l s i n t h e h i s t o g r a m .

F i g . 11. shows a c a p a c i t y - a v a i l a b i l i t y h i s t o g r a m , c a l c u l a t e d by t h e p r o g r a m , w h i l e F i g . 12. shows t h e d i s t i b u t i o n f u n c t i o n . 4 . 3 . 2 COMPUTATI ON OF DEFI CI ENCY -EXCESS H I STOGRAM

For b o t h s y s t e m s , t h e e n e r g y p r o d u c e d by t h e m a c h i n e s must s e r v e t h e a l g e b r a i c sum o f t h e e n e r g y demanded by t h e c o n s u m e r s a n d t h e s c h e d u l e d e x c h a n g e o f e n e r g y b e t w e e n t h e t w o s y s t e m s . S i n c e t h e comsumption i s s u p p o s e d t o b e n o r m a l l y d l s t r i b u t e d , t h e r e f o r e t h e sum o f c o n s u m p t i o n a n d s c h e d u l e d e x p o r t / i m p o r t . d e n o t e d a b o v e by y l a n d

y, a r e n o r m a l l y d i s t r i b u t e d , a s w e l l . The e x p e c t e d v a l u e i s Mk=Dk+E a n d t h e

k s t a n d a r d d e v i a t 1 o n u

k'

D e n o t i n g by h Cx3 t h e d e n s i t y f u n c t i o n o f t h e random v a r i a b l e ,

k

h a v i ng t h e p a r a m e t e r s d e f i n e d a b o v e , t h e f 01 l o w i ng c o n v o l u t i o n i n t e g r a l g i v e s t h e d e f i c i e n c y - e x c e s s d e n s 1 t y f u n c t i o n :

k = 1 , 2 , w h e r e G C y 3 d e n o t e s t h e d i s t r i b u t i o n f u n c t i o n o f t h e k

c a p a c i t y - a v a i l a b i l i t y o f g e n e r a t o r s .

T h i s c a l c u l a t i o n c a n b e p e r f o r m e d f o r b o t h power s y s t e m s , s o w e h a v e t h e f u n c t i o n s which d e s c r i b e t h e p r o b a b i l i t y o f e n e r g y d e f i c i e n c y or excess f o r b o t h s y s t e m s .

4 . 3 . 3 COMPUTATION OF EXCHANGE POWER

The e x c h a n g e h i s t o g r a m i s c a l c u l a t e d a c c o r d i n g t o t h e e x p r e s s i o n C 8 3 . T h i s e x p r e s s i o n r e f e r s t o a g i v e n t i m e i n t e r v a l which i s a s u b i n t e r v a l i n t h e d a i l y l o a d c u r v e . The e x c h a n g e h i s t o g r a m f o r a g i v e n p e r i o d o f t i m e , e . g . f o r a d a y or f o r a y e a r , c a n b e d e t e r m i n e d b y summation o f t h e h i s t o g r a m s f o r t h e s e s u b i n t e r v a l s .

4 . 3 . 4 COMPUTATION OF EXCHANGE POWER FOR REAL CONNECTION

L e t mi i = l

, . . . ,

k d e n o t e t h e f 01 1 owing random v a r i a b l es

,

d e s c r i b i n g t r a n s m i s s i o n c a p a c i t y a v a i 1 a b i 1 i t y f o r t h e i t h ti e-1 i n e .

w i t h p r o b a b i l i t y 1 w i t h p r o b a b i l i t y 1-li

w h e r e k i s t h e number o f t r a n s m i s s i o n l i n e s i n t h e t i e - l i n e s y s t e m .

The method f o r t h e d e t e r m i n a t i o n o f a g g r e g a t e d t r a n s m i s s i o n c a p a c i t y a v a i l a b i l t y h i s t o g r a m i s v e r y s i m i l i a r t o t h e p r o c e d u r e a p p l i e d f o r t h e a g g r e g a t e d c a p a c i t y a v a i 1 a b i 1 i t y h i s t o g r a m f o r t h e g e n e r a t i n g u n i t s . D e n o t e pCx3 t h e h i s t o g r a m o f t h e a g g r e g a t e d c a p a c i t y a v a i l a b i l i t y , P C 3 i t s d i s t r i b u t i o n f u n c t i o n , a n d QCx3=1-PCx3.

F i g 10. shows a n e x a m p l e o f t h e d i s t r i b u t i o n o f t h e a v a i l a b l e t r a n s f e r - c a p a c i t y o f t h e c o o p e r a t i n g - l i n e s y s t e m .

Taken i n t o a c c o u n t t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e c a p a c i t y o f t h e c o o p e r a t i n g l i n e s y s t e m , i . e . t h e p o s s i b i l i t y o f u n e x p e c t e d b r e a k -down or o u t a g e s , t h e h i s t o g r a m o f t h e e x c h a n g e

power which c a n b e t r a n s m i t t e d on a n i d e a l l i n e s y s t e m C 8 3 i s m o d i f i e d a c c o r d i n g t o t h e f o l l o w i n g r e l a t i o n s :

-f l m a x

F i g . 13. shows a n e x a m p l e o n t h e h i s t o g r a m s o f e x c h a n g e power. F i g . 13/a shows t h e h i s t o g r a m f o r c o o p e r a t i o n lines w i t h u n l i m i t e d c a p a c i t y . F i g . 1 3 1 0 shows t h e c a s e cf' r e z l c o n n e c t i o n s . w h i l e F i g . 13/c shows t h e n o t r e a l i s e d e x c h a n g e power r e s u l t i n g f r o m 1 i m i t . a t i o n s or f a i l u r e of t h e t r a n s m i s s i o n s y s t e m .

5. REALIZATION ON I BM/PC

5.1 OVERVIEW OF THE PROGRAM SYSTEM

The a l g o r i t h m o u t l i n e d a b o v e h a s b e e n i m p l e m e n t e d o n a n IBM../PC. To p r o v i d e a c o n v e n i e n t u s e r i n t e r f a c e w i t h mu1 t . - w i ndow f aci 1 i t - y a n d g r a p h i c a l r e p r e s e n t a t i o n o f r e s u l t s g r a p h i c s t - o o l k i t s a r e u t i l i z e d .

T y p i c a l a p p l i c a t i o n s o f t h e s y s t e m r e q u i r e a series o f r u n s w i t h i n p u t d a t a w h i c h a r e v a r i a n t s o f a b a s e case d a t a s e t . D a t a s t r u c t u r e s a r e d e s i g n e d t o s u p p o r t t h i s k i n d of u s a g e . I n p u t d a t a f o r t h e d i f f e r e n t p a r t s o f t h e model a r e s t o r e d i n d i f f e r e n t d a t a - f i l e s . a n d t h e s y s t e m i s endowed w i t h c o n v e n i e n t d a t a r e t r i e v a l

,

modi f i c a t i o n a n d s t o r a g e f a c i l i t i e s . The r e s u l t of t h e c o m p u t a t i o n s a p p e a r s i n a g r a p h i c f o r m o n t h e s c r e e n , which c a n b e c o p i e d t o a dot.

m a t r i x p r i n t e r

.

The program-system c o n s i s t s of t h r e e programs. The a c t u a l c o m p u t a t i o n s a r e p e r f o r med by t h e program COOPER. whi 1 e r e s u l t s a r e d i s p l a y e d by t h e programs GENER a n d EXCHANGE.

Running t i m e of t h e program l a r g e 1 y d e p e n d s o n t h e s i z e o f t h e problem. For s y s t e m s w i t h 10000 MW g e n e r a t i o n c a p a c i t y and a s s u m i n g a 2 4 - p e r i o d s s u b d i v i s i o n f o r t h e d a i l y l o a d c u r v e s , t h e r u n n i n g t i m e i s a r o u n d 10 m i n u t e s o n a n IBM/PC AT w i t h a r i t h m e t i c c o p r o c e s s o r a n d t u r b o c a r d s . C o l l e c t i n g d a t a i s a c r u c i a l p o i n t i n t h e r e a l - l i f e a p p l i c a t i o n s of t h e s y s t e m . U s u a l l y u s e r s h a v e d a t a f o r t h e i r own c o u n t r y or u t i l i t y , b u t o b t a i n i n g c o r r e c t d a t a f r o m t h e c o o p e r a t i n g p a r t n e r i s n o t a1 ways a n e a s y t a s k .

5 . 2 R U N N I N G THE PROGRAM

The f i r s t a c t i o n t o b e p e r f o r m e d when u s i n g t h e s y s t e m i s t h e l o a d i n g of t h e g r a p h i c s d r i v e r s . T h i s c a n b e p e r f o r m e d simp1 y by e x e c u t i n g t h e f i l e s WM and GRAPHICS. The f i r s t of t h e s e s u p p o r t s w i ndow-mani p u l a t i o n s , t h e s e c o n d p r o v i d e s a p r i n t - s c r e e n c o p y u t i l i t y .

The g e n e r a l p a t t e r n o f d a t a i n p u t or m o d i f i c a t i o n i s t h e f o l l o w i n g : D a t a a r e t o b e t y p e d i n t o s p e c i f i e d f i e l d s on t h e s c r e e n . d a t a i n p u t i n t o a s p e c i f i c f i e l d i s c l o s e d by p r e s s i n g <RETURN>

,

c l o s i ng i n p u t f o r a p a r t i c u l ar s c r e e n c a n b e p e r f o r med by p r e s s i ng t h e

<

F10> f u n c t i on-key.

The main c o m p u t a t i o n a l p r o c e s s e s a r e p e r f o r m e d by t h e program COOPER. which d i s p l a y s d u r i n g r u n t i m e t h e r e q u i r e d u s e r i n t e r a c t i o n s a n d i n f o r m a t i o n on t h e c u r r e n t s t a g e o f c o m p u t a t i o n s .

The a c t i o n s t o b e p e r f o r m e d i n a t y p i c a l u s e r s e s s i o n a r e l i s t e d below.

1/ E x e c u t e t h e f i l e s

WM

a n d GRAPHICS, a n d a f t e r w a r d s COOPER. by s i m p l y t y p i n g t h e i r names o n s u b s e q u e n t s y s t e m p r o m p t s .

2 / P r e s s <FlO>

.

^{a menu} a p p e a r s f o r m o d i f i c a t o n s of g e n e r a l p a r a m e t e r s of t h e i n t e r c o n n e c t e d s y s t e m s . 3/ P e r f o r m modi f i c a t i o n s

,

i f needed.

4/ P r e s s F1O t o c l o s e i n p u t .

5/ A prompt a p p e a r s a s k i n g whether t h e new d a t a set i s t o b e s a v e d . S a v e d a t a set i f n e c e s s a r y .

6 / P r e s s < F 1 0 > , a menu a p p e a r s f o r m o d i f i c a t i o n s i n g e n e r a t o r d a t a Ccapaci t y , s e l f -consumpti on and f a i l u r e p r o b a b i l i t y 3 o f t h e f i r s t power s y s t e m .

7/ P e r f o r m m o d i f i c a t i o n s , i f needed.

8 / P r e s s F1O t o c l o s e i n p u t .

9/ A prompt a p p e a r s a s k i n g whether t h e new d a t a set. i s t o b e s a v e d . S a v e d a t a set i f n e c e s s a r y .

11/ C a r r y o u t s t e p s 6

-

9 f o r t h e s e c o n d s y s t e m .

1 2 / P r e s s < F 1 0 > , a menu a p p e a r s on m o d i f i c a t i o n s i n t h e s c h e d u l i ng of consumer demand and e x p o r t - - i mpor t

,

as w e l l a s i n t h e i r maxima.

13/ P e r f o r m m o d i f i c a t i o n s , i f needed.

1 4 / P r e s s F10 t o c l o s e i n p u t .

15,' A prompt a p p e a r s a s k i n g whether t h e new d a t a set i s t o b e s a v e d . S a v e d a t a set i f n e c e s s a r y .

16/ P r e s s <F10> f o r p o s s i b l e m o d i f i c a t i o n s i n t i e - l i n e c a p a c i t y - d i s t r i b u t i on. T r a n s m i t t a b l e power s h o u l d b e s p e c i f i e d i n d e c s r e a s i n g o r d e r of p r o b a b i l i t y .

17/ P e r f o r m m o d i f i c a t i o n s , i f needed.

18/ P r e s s F10 t o c l o s e i n p u t .

19/ A prompt a p p e a r s a s k i n g whether t h e new d a t a set i s t o b e s a v e d . S a v e d a t a set i f n e c e s s a r y .

20/ P r e s s < F 1 0 > , t h e c o m p u t a t i o n a r e s t a r t e d .

21/ A f t e r r u n n i n g t h e program COOPER, e x e c u t e t h e f i l e

GENER

by t y p i n g i t s name. T h i s program d i s p l a y s t h e a v a i l a b i l i t y h i s t o g r a m s a n d d i s t r i b u t i o n s f o r b o t h s y s t e m s .

22/ To d i s p l a y t h e h i s t o g r a m s and d i s t r i b u t i o n s of t h e e x c h a n g e power f o r t r ansmi ssi on 1 i n e s o f u n l i mi t e d c a p a c i t y , w i t h real c o n n e c t i o n s , a s w e l l a s t h e n o t r e a l i s e d e x c h a n g e power. e x e c u t e t h e f i l e EXCHANGE by t y p i n g i t s name.

6. POSSI B I LI TI ES OF FUFI"l"El? DEVELOPMENT

I n t h e y e a r 1987 t h e p r o g r a m s y s t e m f o r c a l c u l a t i o n o f e x c h a n g e power b e t w e e n t w o c o o p e r a t i n g s y s t e m s h a s b e e n d e v e l o p e d . W e i n t e n d t o m a k e f u r t h e r i m p r o v e m e n t s o n t h i s model i n c l u d i n g t h e f 01 1 o w i n g i t e m s :

-

To g e t a m o r e a c c u r a t e l o a d - m o d e l , t h e d a i l y l o a d c u r v e s s h o u l d b e i n c o r p o r a t e d i n t o t h e model b y t a k i n g i n t o a c c o u n t t h e i r d e p e n d e n c y o n t h e type o f t h e d a y C f i r s t workday a f t e r a h o l i d a y , g e n e r a l workday, s a t u r d a y , h o l i d a y , e t c . 3 .

-

^The p r e s e n t p r o g r a m w o r k s w i t h t h e p r o b a b i 1i t y d i s t r i b u t i o n f u n c t i o n o f t r a n s m i s s i o n c a p a b i l i t y b e t w e e n t h e t w o s y s t e m s . I t would b e d e s i r a b l e t o d e v e l o p a method f o r t h e c o m p u t a t i o n o f t h i s d i s t r i b u t i o n f r o m t h e t r a n s m i s s i o n c a p a c i t i e s a n d b r e a k down p r o b a b i 1 i ti es o f t h e i n d i v i d u a l t i e-1 i n e s

.

- The e n e r g y p r o d u c e d by t h e g e n e r a t i n g u n i t s i s

d e c r e a s i n g a c c o r d i n g t o t h e s c h e d u l e d mi n t e n a n c e . Usual 1 y t h e s t o c h a s t i c a n a l y s i s o f e x c h a n g e power i s p e r f o r m e d f o r l o n g t e r m p r e d i c t i o n s C 5

-

20 y e a r s > , s o t h e m a i n t e n a n c e s c h e d u l e i s n o t known when t h e a n a l y s i s i s c a r r i e d o u t . S i n c e t h e c a p a c i t y - a v a i 1 a b i l i t y o f t h e g e n e r a t i n g u n i t s d e p e n d s on t h e sel e c t i on o f g e n e r a t i n g u n i t s h a v i n g d i f f e r e n t f a i l u r e p r o b a b i l i t i e s , i t i s w o r t h w h i l e t o i n c o r p o r a t e a s i m u l a t i o n o f m a i n t e n a n c e s c h e d u l i ng.

The p o s s i b i 1 i ties o f d e v e l o p m e n t o u t 1 i n e d a b o v e r e f e r t o f u r t h e r d e v e l o p m e n t s o f t h e t w o - p o i n t model. W e i n t e n d t o p e r f o r m f u r t h e r r e s e a r c h t o d e v e l o p a s t o c h a s t i c model f o r t h e a n a l y s i s o f i n t e r c o n n e c t e d s y s t e m s c o n s i sti ng o f m o r e t h a n t w o s y s t e m s .

P r e k o p a 1 9 1 d e v e l o p e d a g e n e r a l m a t h e m a t i c a l programming model f o r c a p a c i t y e x p a n s i o n p l a n n i n g o f t r a n s p o r t a t i on type n e t w o r k s . P r e k o p a a n d B o r o s [ I 0 1 d e v e l o p e d a n a l g o r i t h m f o r

t h e c o m p u t a t i o n of LOLP f o r a d i s c r e t e moment i n c o o p e r a t i n g s y s t e m s . W e w i l l i n v e s t i g a t e t h e p o s s i b i l i t i e s of i n t e g r a t i n g t h i s t e c h n i q u e i n t o our model c o n s i s t i n g of more t h a n t w o e n e r g y s y s t e m s .

REFERENCES

1 . T e r s t y h s z k y . T: M o d e l l i n g of Random l o a d s of i n t e r - t i e s UN Seminar on Comparison on Models of P l a n n i n g and O p e r a t i n g E l e c t r i c Power Systems. Moscow 1 9 8 7 . EP/SEM.

12/R. 23.

2 . Bach. I

-

Hadik. 2. : C a l c u l a t i o n of t h e E l e c t r i c a l Energy Exchange Between Cooper a t i ng S y s t e m s . Acta Techni k a Academi ae Sci e n t i a r i um Hungar i c a e

.

^{Tomus 70C3-43.}

pp. 313-328 C18713.

3 . E s t e r h a s . S

-

T e r s t y a n s z k y . T.: A n a l y s i s of o n e hour a v e r a g e power f 1 ows between s y s t e m s w i t h t h e p r o b a b i 1 i t y t h e o r y . Acta Techni k a Academi ae Sti e n t i a r i um Hungar i c a e . Tomus 84C 3-43

.

pp. 281 -291 C 19773.

4. Hadik. Z

-

T e r s t y h s z k y . T . : M o d e l l i n g l o a d s of a c c i d e n t a l c h a r a c t e r of s y s t e m i n t e r c o n n e c t i o n s . E l e k t r o t e c h n i k a 71C19783 No 11-12. C i n Hungarian3

5. P r e l i m i n a r y c h a r a c t e r i s t i c s of t h e I n t e r c o n n e c t e d Power Systems of t h e CDU memeber s

.

P r a g u e May. 1 9 8 6 .

6. UCPTE 1983-1984. R a p p o r t Annuel. Roma 1984.

7. Annual B u l l e t i n of E l e c t r i c Energy S t a t i s t i c s f o r Europe.

UN. N e w York. 1 9 8 6 .

8. F e l l e r . W.: An i n t r o d u c t i o n t o p r o b a b i l i t y t h e o r y and i t s a p p l i c a t i o n s . John Wiley and S o n s . N e w York - London

-

Sydney 1 9 8 6 .

9. P r e k o p a . A. : Network p l a n n i n g u s i n g t w o - s t a g e programming under u n c e r t a i n t y . i n : Recent R e s u l t s i n S t o c h a s t i c Pr ogr ammi ng

.

Pr o c e e d i n g s Ober wol f a c h 1 9 7 9 . Spr i n g e r

.

1980. pp. 216-237.

1 0 . P r e k o p a . A.

-

B o r o s . E . : On t h e p r o b a b i l i t y of t h e e x i s t e n c e of a f e a s i b l e f l o w i n a t r a n s p o r t a t i o n n e t w o r k . RUCTOR R e s e a r c h R e p o r t August 19, 1986. The S t a t e U n i v e r s i t y of N e w Yersey. RUTGERS.

unit:

G'dh

N

D

WGC SLAVIA

PORTU

Fig.1. Electric energy exchange among the West-Zuropean

countries in 1985.

GERMAN FED .REP.

GDR

unit: _GWh

USSR

YUGOSLAVIA

Fig.2. Electric energy exchange among the East-European

countries in 1985.

1985.

USSR

I I

SWITZERLAND

PERCENT

Fig.3. Electric energy exchange in several European countries and its proportion related to gross electric energy consumption.

30 PORTUGAL

ITALY HUNGARY F.R.G.

FRANCE D A I l I 3 P r n

CZECHOSLOVAKIA BULGARIA

BELGIUM AUSTRIA

I

TWh 30 20 10 0 10 20

P / / / F

I

I 1

I

I I 1

- d a i l y maximum -

^-.

- n i g h t minimum

Fig.4. Aggregate sum o f s i m u l t a n e o u s l o a d o f e l e c t r i c

power f l o w a c r o s s s t a t e b o r d e r s i n UCPTE system.

PERCENT ^{MONDAY SATURDAY} 130

120 110 100 90 80

7 0

r

I I

HIGH LOAD1

- ^{ANALYSIS: 5 YEARS}

Fig. 6. Load curve in Hungary

december 1986.

- 1st year

* _{2nd year}

+ ^{3rd year}

,-

4th year

- -- - _{5th year}

Fig.7. Distribution functions of short-term

power exchange.

Fig.8. Checking hypothesis for distribution.

Theoretical and empirical density function.

P max

Fig.9. Daily load curve

Fig.10. Capacity of a transmission line system

30

F i g . 1 1 . C a p a c i t y - a v a i l a b i l i t y h i s t o g r a m o f g e n e r a t i n g u n i t s .

F i g . 1 2 . C a p a c i t y - a v a i l a b i l i t y d i s t r i b u t i o n o f g e n e r a t i n g u n i t s .

31

F i g . l 3 / a Demanded e x c h a g e o f e n e r g y

F i g . l 3 / b . R e a l i z e d e x c h a n g e o f e n e r g y

F i g . l 3 / c . Not r e a l i z e d e x c h a n g e o f e n e r g y F i g . 1 3 . E n e r g y e x c h a n g e h i s t o g r a m s

32