182 IEEE Transactions on Power Systems, Vol. 5, No. 1, February 1990

A NEW PROBABILISTIC POWER PLOW AWiLYSIS METHOD

A. P. Sakis Meliopoulos School of E l e c t r i c a l Engineering

Georgia I n s t i t u t e of Technology At l an ta , Georgia 30332-0250

George J. Cokkinides Xing Yong Chao Department of E l e c t r i c a l Engineering School of E l e c t r i c a l Engineering

Universi ty of South Carol ina Georgia I n s t i t u t e of Technology Columbia, South Carol ina 29208 At lan ta , Georgia 30332-0250

Abstract

A s imula t ion method of t h e composite power system is proposed fo r t he purpose of eva lua t ing the proba- b i l i t y d i s t r i b u t i o n func t ion of c i r c u i t flows and bus vo l t age magnitudes. The method c o n s i s t s of two s t e p s . F i r s t , g iven the p r o b a b i l i s t i c electric load model, t he p r o b a b i l i t y d i s t r i b u t i o n func t ion of t he t o t a l genera- t i o n of gene ra t ion buses i s computed. Second, c i r c u i t flows and bus vo l t age magnitudes a r e expressed a s l i n e a r combinations of power i n j e c t i o n s a t gene ra t ion buses. This r e l a t i o n s h i p allows the computation of t h e d i s t r i b u t i o n funct ions of c i r c u i t flows and bus vo l t age magnitudes. The method inco rpora t e s major ope ra t ing p r a c t i c e s such a s economic d i spa tch and n o n l i n e a r i t i e s r e s u l t i n g from the power flow equat ions. Val idat ion of t h e method i s performed v i a Monte Carlo s imulat ion. Typical r e s u l t s a r e presented which i l l u s t r a t e t h a t t he proposed method matches very well r e s u l t s obtained with Monte Carlo s imulat ions. P o t e n t i a l a p p l i c a t i o n s of t h e proposed method a re : (1) composite power system r e l i a - b i l i t y a n a l y s i s and (2) i ransmiasion loss eva lua t ion .

Key Words

Power Flow Economic Dispatch S tochas t i c Load Model P robab i l i t y Di s t r ibu t ion Function (PDF) PDF of C i r c u i t Flow PDF. of Bus Voltage Monte Carlo Simulation

In t roduc t ion

T r a d i t i o n a l power flow a n a l y s i s t r e a t s t he e l e c t r i c load and the generat ing u n i t s of the system a s d e t e r m i n i s t i c a l l y known q u a n t i t i e s . This i s only t r u e fo r a l imi t ed nmber of s i t u a t i o n s , fo r example, i n a r e a l t ime environment where the e l e c t r i c load and gene ra t ion can be d i r e c t l y measured. In any o t h e r power flow app l i ca t ion , however, t h e r e i s unce r t a in ty a s soc ia t ed with t h e a v a i l a b i l i t y of gene ra t ing u n i t s and the e l e c t r i c load. In many a p p l i c a t i o n s , such a s r e l i a b i l i t y a n a l y s i s of t h e composite (gene ra t ion and t ransmission) power system and t ransmission loss evalu- a t i o n , use of t he t r a d i t i o n a l power flow formulat ion l eads t o an extremely l a r g e number of power flow cases fo r t he purpose of cap tu r ing a l l t he va r i ances of t he e l e c t r i c load and gene ra t ion d i spa tch schedules . In t hese cases , it i s appropr i a t e t o use methods which

89 SM 714-7 PWRS by the IEEE PozRr System Engineering Committee of the I E E E Power Engiileering Society for presentation a t the IEEE/PES 1989 Summer Meeting, Long Beach, California, July 9 - 14, 1989. made available for printing May 9, 1989.

A paper recommended and approved

Manuscript submitted September 1, 1988;

d i r e c t l y t r e a t the unce r t a in ty . Methods of power flow a n a l y s i s which recognize the unce r t a in ty o f t he gene ra t ion and e l e c t r i c load a r e r e f e r r e d t o a s p r o b a b i l i s t i c power flows.

The f i r s t no t ion of p r o b a b i l i s t i c power flow appeared i n the e a r l y 1970s. Borkowska, Allen e t a l . 114,151 have proposed a s impl i f i ed p r o b a b i l i s t i c load flow. Two assumptions were introduced: (1) t h e e l e c t r i c power system i s represented with a DC network model ( t h u s , t h e r e a c t i v e power flow i s neg lec t ed ) , and (2) the r e a l pa r t of t h e bus e l e c t r i c loads a r e independent random v a r i a b l e s . With these assumptions, a convent ional d e t e r m i n i s t i c power flow i s solved f i r s t , a s s m i n g net nodal loads equal t o t h e i r mean values . This s o l u t i o n determines t h e ope ra t ing point about which t h e load flow equat ions a r e subsequent ly l i n e a r i z e d . Within t h i s model, t h e gene ra t ion d i spa tch procedure i s modeled with an a r b i t r a r y func t ion which a l l o c a t e s the v a r i a t i o n of t h e t o t a l e l e c t r i c load t o the s p e c i f i c gene ra t ion buses. Since t h e v a r i a b l e s of t h e nodal e l e c t r i c load a r e assumed independent, t h e p r o b a b i l i t y dens i ty func t ions of t h e c i r c u i t flows can be computed with a series of convolut ions. La te r , t h i s bas i c method has been extended t o the AC network model [181.

The assumption of independence of t h e nodal e l e c t r i c loads i s u n r e a l i s t i c . Da S i l v a et a l . pro- posed a l i n e a r dependence model of e l e c t r i c loads [191. Using a l i n e a r i z e d power flow model, they proposed a method which combines Monte Carlo s imula t ion and convolut ions. Dopazo e t a l . [161 proposed a method which models t he c o r r e l a t i o n between the load a t any two buses. Their proposed method assumes t h a t c i r c u i t flows and bus vo l t age magnitudes a r e Gaussian d i g t r i b - uted and, thus, only the va r i ance must be computed. Monte Carlo s imulat ions i n d i c a t e t h a t it i s u n r e a l i s t i c t o assume Gaussian d i s t r i b u t i o n s of c i r c u i t flows and bus vo l t ages . For t h i s reason, Sauer and Heydt [ 2 0 ] have proposed the use of higher moments ( t h i r d and fou r th ) fo r accurate r e p r e s e n t a t i o n of 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 funct ions.

An e f f i c i e n t method fo r t r e a t i n g t h e c o r r e l a t i o n among bus loads and the generat ion d i spa tch procedure has been proposed i n [21]. The model assumes Gaussian d i s t r i b u t i o n of bus loads and a l i n e a r i z e d economic d i spa tch model. The c i r c u i t flows and bus vo l t ages a r e expressed a s a Linear combination of t h e bus loads only. The l i n e a r i z e d equat ions a r e u t i l i z e d t o de t e r - mine t h e moments of p r o b a b i l i t y d e n s i t y func t ion of c i r c u i t flows and bus vo l t ages . The i n c l u s i o n of t h i s model i n a r e l i a b i l i t y a n a l y s i s method r e s u l t e d i n more accu ra t e r ep resen ta t ion of t h e e l e c t r i c load a t reduced computational requirements [211. While t h i s approach models t h e economic r ed i spa tch of gene ra t ing u n i t s due t o e l e c t r i c load v a r i a t i o n s , it i s based on t h e l i nea r - ized power flow equat ions and the l i n e a r i z e d economic d i spa tch model. As such, i t s a p p l i c a b i l i t y i s l imi t ed . This paper p re sen t s a new approach fo r t h i s model which addresses th ree important aspects : (1) t he economic d i spa tch of gene ra t ing u n i t s , ( 2 ) t h e e f f e c t s of n o n l i n e a r i t i e s of t he power system model, and ( 3 ) t he unce r t a in ty a s soc ia t ed with t h e a v a i l a b i l i t y of generat ing u n i t s . Val idat ion of t he method v i a Monte Carlo s imulat ion is a l s o presented.

0885-8950/90/0200-0182/$01.00 @ 1990 IEEE

183

The equat ions descr ib ing t h e e lectr ic load model for a system- with n buses a r e :

0 l ( B ) z ( t ) =

z ( t ) =

P ( t > =

Model Descr ip t ion

The proposed model provides p r o b a b i l i s t i c charac- t e r i z a t i o n s of c i r c u i t flows and bus v o l t a g e magnitudes f o r a g iven e lectr ic load and genera t ion system model, S p e c i f i c a l l y , consider a power system a s is i l l u s t r a t e d i n Fig. l a . The following assumptions a r e made:

( 1 ) A p r o b a b i l i s t i c e lectr ic load model i s given. ( 2 ) The genera t ing u n i t parameters and forced

( 3 ) The t ransmiss ion system is known. outage r a t e s a r e known.

Under these assumptions, it i s d e s i r e d t o compute the p r o b a b i l i t y d i s t r i b u t i o n func t ion o f c i r c u i t flow S and bus v o l t a g e magnitude V i f o r each c i r c u i t 11 an% bus i. Major opera t ing p r a c t i c e s , such a s economic d i s p a t c h , must be considered.

The s t a t e d o b j e c t i v e i s achieved with a two s t e p model. In t h e f i r s t s t e p , t h e e l e c t r i c load and genera t ing system model is used t o c h a r a c t e r i z e t h e power i n j e c t i o n s , Y, a t the system buses a s random v a r i a b l e s . This i s i l l u s t r a t e d i n Fig. lb. The random v a r i a b l e s , Y, a r e i n genera l c o r r e l a t e d . Subsequently, a p r o b a b i l i s t i c power flow provides 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 f u n c t i o n of S V i from the p r o b a b i l i s t i c model of t h e i n j e c t i o n s Y. "

POWER SYSTEM

SLi

Figure 1. Schematic Representation of an Electric Power System.

The P r o b a b i l i s t i c Electric Load Model

The e lectr ic load model provides a p r o b a b i l i s t i c d e s c r i p t i o n of system bus loads and allows modeling of conforming or nonconforming bus loads. The assumptions of t h e p r o b a b i l i s t i c e l e c t r i c load model a re : (1) Bus e l e c t r i c loads a r e t y p i c a l l y s t r o n g l y c o r r e l a t e d . It i s , t h e r e f o r e , reasonable t o assume t h a t they a r e generated a s a l i n e a r combination of a small number o f independent s t o c h a s t i c processes . (2) The power f a c t o r of t h e e l e c t r i c load a t a s p e c i f i c bus i s c o n s t a n t .

S i ( t ) =

is an m-vector processes is an n r v e c t o r processes i s an nrvec tor processes i s an n-vector power)

of independent white no ise

of s t a t i o n a r y s t o c h a s t i c

of nonst a t ionary s t o c h a s t i c

of bus e l e c t r i c loads ( r e a l

a r e vec tor func t ions of a r b i t r a r y polynomials i s t h e backward opera tor is a cons tan t n x 1 v e c t o r i s an n x m mat r ix i s t h e complex e lectr ic load a t bus i i s a cons tan t f o r bus i; i t i s dependent upon t h e power f a c t o r of t h e load a t bus i.

The model descr ibed with Eqs. (1) and (2) (ARIMA model) h a s been e x t e n s i v e l y used t o r e p r e s e n t t h e e lectr ic load. For example, see References [ 2 4 , 2 5 1 . It is well known t h a t it i s capable of r e p r e s e n t i n g t h e periodic- i t i e s a s w e l l as t h e nons ta t ionary proper ty o f t h e e l e c t r i c load. The innovat ion introduced here i s t h e l i n e a r model A which t r a n s l a t e s t h e low order nonsta- t i o n a r y s t o c h a s t i c process v e c t o r v ( t ) i n t o the vec tor P ( t ) of t h e bus e lectr ic loads. The number of independent processes v ( t ) , m, For a system with conforming bus one. The t o t a l e lectr ic load i s bus loads :

Above equat ion provides t h e

is i n genera l low. loads , m i s equal t o t h e summation of a l l

m

t o t a l e l e c t r i c load, I I ( t ) , a t t i m e t a s a f u n c t i o n o f t h e s t o c h a s t i c process a r r a y v ( t ) . The model has been s t r u c t u r e d i n such a way t h a t t h e s t o c h a s t i c processes v ( t ) a r e normalized, i .e . t h e y assume v a l u e s i n t h e i n t e r v a l ( 0 , l ) . For p r o b a b i l i s t i c power flow a p p l i c a t i o n s , it i s necessary t o c h a r a c t e r i z e the t o t a l e lectr ic load a t a s p e c i f i e d f u t u r e t i m e or a t a s p e c i f i e d f u t u r e i n t e r v a l ( f o r example, one week, one month, one year i n t e r v a l ) . For a s p e c i f i e d t i m e i n t e r v a l , T, which s h a l l be r e f e r r e d t o a s t h e s imula t ion t i m e , t h e s t o c h a s t i c processes v ( t ) and E ( t ) a r e rep laced wi th random v a r i a b l e s V and L. Then Eq. (5) becomes

m L = a + 1 aiVi

O j = 1

The s t a t i s t i c s o f t h e random ' v a r i a b l e s V can be obtained from the ARIMA model (1) and ( 2 ) . From t h e known s t a t i s t i c s o f V, 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 func t ion , P (E) of L, a r e computed. The complementary d i s t r i b u t i o k func t ion , L o ( l l ) , o f t h e t o t a l load is def ined with

L ( 1 ) = 1.0 - FL(II) = Pr[L > I I ]

The complementary d i s t r i b u t i o n func t ion , Lo(II), depends on m independent random v a r i a b l e s v i , i = 1,2 , . . . ,m. It should be poin ted out t h a t t h i s model

184

i s a g e n e r a l i z a t i o n of the load d u r a t i o n curve used i n genera t ion system r e l i a b i l i t y ana lys i s . Observe t h a t f o r m = 1, above model i s e x a c t l y t h e load d u r a t i o n curve. S p e c i f i c a l l y , when m = 1, the d i s t r i b u t i o n func t ion , Lo@), i s a func t ion of one random v a r i a b l e only , vi , i . e .

In t h i s case the d i s t r i b u t i o n func t ion , L (I) versus I ( o r v i ) can be p l o t t e d y i e l d i n g the l%ad d u r a t i o n curve. A t y p i c a l func t ion , L (I), i n t h i s case i s i l l u s t r a t e d in Fig. 2.

I , - a + a v 0 1 1

I block 1. unit i

Figure 2. Illustration for the Computation of Unit Probability Distribution Function.

Generation System Model

A genera t ing u n i t , i, of c a p a c i t y c i MW is modeled with a set of c a p a c i t y s t a t e s , each s t a t e with a spec i - f i e d p r o b a b i l i t y . The p t o b a b i l i t y d e n s i t y f u n c t i o n of t h i s model is expressed with

mi

fx i (Xi) = k=O 1 PikG(Xi - di k)

where

mi i s t h e number of c a p a c i t y s t a t e s ( inc luding

dik a r e the c a p a c i t i e s o f t h e states (note , zero and f u l l c a p a c i t y of the u n i t )

dio = 0.0 and dimi = c i ) .

For m i = 1, t h i s model is t h e w e l l known up and down model. For c l a r i t y of presenta t ion , t h e method i s d iscussed i n terms of t h e up and down model of a uni t . Unit outages a r e independent. In a d d i t i o n t o t h i s model, each u n i t i s descr ibed with a product ion cos t func t ion versus uni t output . This func t ion can be a quadra t ic func t ion or a piecewise l i n e a r funct ion.

Generation System Simulation

The problem of genera t ion system s imula t ion is def ined a s follows. Given the p r o b a b i l i s t i c e l e c t r i c load model for the time period under cons idera t ion and a l ist o f a v a i l a b l e genera t ing u n i t s , s imula te t h e o p e r a t i o n of t h e system i n order t o compute the p r o b a b i l i t y d i s t r i b u t i o n func t ions o f the bus power i n j e c t i o n s and t h e i r c o r r e l a t i o n s . The process should account for the e f f e c t s of economic schedul ing func- t i o n s wi th in the time period considered and the random forced outages of the u n i t s .

Given the load and genera t ion models, t h e followir& product ion q u a n t i t i e s can be computed with t h e c l a s s i c a l p r o b a b i l i s t i c method [l]:

P r o b a b i l i t y of o p e r a t i o n of u n i t i: Pr[Unit i i n o p e r a t i o n ] = Pr[Unit i Output > 01. Expected value of produced energy from the u n i t . Expected value of c o s t o f o p e r a t i o n of u n i t i.

Refinements o f t h i s method have been developed over the years . The ref inements can be c l a s s i f i e d i n t o two groups. In t h e f i r s t group, t h e o b j e c t i v e of t h e ref inements is t o speed up t h e computerized procedure of the s imula t ion method. Very f a s t procedures have been developed based on t h e cumulant method [6-8, 10-131. In t h e second group, t h e o b j e c t i v e is t o improve the s imula t ion method of opera t ing p r a c t i c e s such as economic d i s p a t c h , maintenance, e t c . Procedures f o r s imula t ing incremental loading of u n i t s based on economic c r i t e r i a have been developed [3-5,9]. A l l t h e s e ref inements can be incorporated i n the proposed p r o b a b i l i s t i c power flow. For c l a r i t y o f presenta t ion , we s h a l l use t h e method descr ibed i n Ref. [91 t o present the p r o b a b i l i s t i c power flow. This method is b r i e f l y descr ibed a s follows. Consider n u n i t s o f t h e system opera t ing a t l e v e l s x1,x2,...,xn. I f un i t k is not i n opera t ion , then obviously Xk W i l l equa l 0 . Since t h e r e is a f i n i t e p r o b a b i l i t y t h a t any u n i t can be forced out , t h e output of u n i t i, x i , can be considered t o be a random v a r i a b l e with p r o b a b i l i t y of u n a v a i l a b i l i t y equal t o pi. We w r i t e

(7 )

&(Xi = 0) = qi (8)

where Xi i s a random v a r i a b l e represent ing the genera t ion of u n i t i. Assume t h a t t h e electric load equals I. For t h i s condi t ion , t h e apparent load Ia w i l l be

L a = & - x 1 - x2 - ... - xn

Since I , x , . . . D x a r e not d e t e r m i n i s t i c a l l y known, t h e above equlation “can be rep laced with i t s equiva len t equat ion i n terms of t h e corresponding random v a r i a b l e s

(9)

La = L - X I - x* - ... - xn ( 1 0 )

where L is a random v a r i a b l e represent ing the e l e c t r i c load and Xi i s a random v a r i a b l e represent ing the output of un i t i. Since the p r o b a b i l i t y d i s t r i b u t i o n func t ions of t h e random v a r i a b l e s L, XI,...DKn a r e known and s i n c e these random v a r i a b l e s a re independent, 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 func t ion o f t h e random v a r i a b l e La i s computed wi th a series of convolut ions.

I f we assume t h a t I > 0 ( t h a t is, load exceeds genera t ion) , then anothera u n i t should be brought i n t o o p e r a t i o n or one o r more o f t h e opera t ing u n i t s should i n c r e a s e t h e i r output . Assume t h a t un i t i is opera t ing a t xi and t h a t it is s e l e c t e d according t o a c r i t e r i o n t o respond t o any i n c r e a s e s i n the load. When the c r i t e r i o n is s e l e c t e d t o be the incremental product ion c o s t of t h e u n i t , then the descr ibed procedure s imula tes t h e economic d i s p a t c h p r a c t i c e . I n genera l , i f I > 0, t h e output of u n i t i w i l l increase from x i t o .“i + Ax., where Ax. is a small increment (1-5 MW). We s h a l l r g f e r t o this’ increment a s the block Ax.. It i s noted t h a t i f xi = 0, t h e increment Ax, may’not be small . In t h i s c a s e , u n i t i will be &ought i n t o opera t ion a t a l e v e l a t l e a s t equal t o minimum al lowable opera t ing l e v e l . With the descr ibed formulat ion and a p p l i c a t i o n o f b a s i c p r o b a b i l i t y theory , t h e expected energy t o be produced and cos t of opera t ion and requi red f u e l a r e computed as fol lows:

185

Step 1: Compute the p r o b a b i l i t y d i s t r i b u t i o n func t ion of random v a r i a b l e

L' = L + x . a i

L e t it be FL1(z). I f x i = 0, s k i p t h i s s t e p and assume FL, (z) = FL (2). (Note t h a t t h i s s t e p r e q u i r e s a deconvoluti8n.)

Step 2: Compute

x . +Ax 1 1

E(Axi) = (l-qi)T I ( l - F L I ( z ) ) d z z=x.

x . +Ax

+ (l-qi)T I ' - dfl:) ( l-FL(z))dz

i z=x

where

6(x i ) = 1 i f x . = 0 , & ( x i ) = 0 i f x . # 0

f ( z ) = production cos t func t ion o f u n i t i T = s imula t ion t i m e period ( i n hours) E ( A X ~ ) = expected energy t o be produced from

Axl

(11)

(12)

block

i ' C(Axi) = extec ted cos t of opera t ion of b lock Ax

P r o b a b i l i t y D i s t r i b u t i o n Function of Unit Output

This func t ion for u n i t i is defined with:

Pr[Xi < a ] = F (a) (13)

It i s computed by cons ider ing t h e c o n t r i b u t i o n s from i n d i v i d u a l blocks o f t h e u n i t i. A s an example, consider t h e loading of block j of u n i t i. Assume t h a t t h i s block i s loaded i n such an order t h a t it "sees" t h e equiva len t load E . The complementary p r o b a b i l i t y d i s t r i b u t i o n functior? o f t h e equiva len t load a i s i l l u s t r a t e d i n Fig. 2 (dot ted curve). The p r o b a b i f i t y d i s t r i b u t i o n func t ion of X. i s computed a s follows. Consider the following i d e n t i t y :

Gi

Pr[X. < a] = Pr[Xi a lEi ]Pr [Ei ]

- where Ei 1 s t h e event t h a t u n i t i i s a v a i l a b l e . t h e complimentary event of Ei. equation f o r block j of u n i t i y i e l d s

E . i s Appl ica t ion o f ahove

where a., a r e the l i m i t s of b lock j , u n i t i, and p , = PrfE.:i+l Equation (14) provides t h e c o n t r i b u t i o n 0) block ' j of uni t i t o 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 func t ion o f u n i t i.

P r o b a b i l i t y D i s t r i b u t i o n Function of Generation Bus Power

For t h e p r o b a b i l i s t i c power flow a n a l y s i s , of i n t e r e s t i s the p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n of t h e t o t a l genera t ion a t a bus. Consider genera t ion bus k which comprises t h e set o f genera t ing u n i t s M(k). The t o t a l g e n e r a t i o n i s denoted with the random var i -

Yk = 1 x . (15) a b l e Yk. Thus,

icM(k) '

(16) F ( y ) = Pr[Yk < yl 'k

where Fy ( y ) i s 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 func t ion

o f t h e v a r i a b l e Yk. It i s computed by summing up t h e c o n t r i b u t i o n s from a l l genera t ion blocks belonging t o u n i t s o f bus k. For t h i s purpose, consider t h e block j o f u n i t i which i s connected t o bus k. The contribu- t i o n of t h i s block depends on t h e a v a i l a b i l i t y of t h e genera t ing u n i t s o f bus k a l r e a d y loaded. For purposes of expla in ing t h e p e r t i n e n t equat ions , t h e following d e f i n i t i o n s a r e introduced:

k

M Set of genera t ing u n i t s p a r t i a l l y or f u l l y loaded

M' Subset of M comprising the genera t ing u n i t s not

M" Subset o f M comprising t h e genera t ing u n i t s

M(k) S e t of genera t ing u n i t s connected t o bus k La = L - 1 X Equivalent load "seen" by the u n i t s

before block j , u n i t i

connected t o genera t ion bus k

connected t o g e n e r a t i o n bus k, excluding u n i t i

U E M ' of genera t ion bus k

Ea An event defined as a s p e c i f i c combination of a v a i l a b l e / u n a v a i l a b l e u n i t s i n the set M". Each event E corresponds t o genera t ion z a t bus k equal t o a

2 = 1 x UEM"

Using t h e introduced n o t a t i o n , t h e c o n t r i b u t i o n of block j of u n i t i t o the func t ion Fyk(y) is computed with:

Note t h a t :

Pr[ ( 1

Pr[Ea] = Pr[ 1

Xu+Xi<y) IEaEi] = FL (z-) UEM" a

Xu = 21 UEM"

i Pr[Ei] = p

Upon mathematical manipulation and rep lac ing the summation with i n t e g r a t i o n y i e l d s :

Y- min(y z + a )

z=o a* a Pr[Yk<y] = pi I dF(z) dFL ( a )

min(y, z ) + qi f dF(z) , dFL (1 )

z=o a =O a

where:

F(z) is t h e cumulative p r o b a b i l i t y func t ion o f

U& MI'

t h e v a r i a b l e z = 1 Xu.

Equation (18) provides t h e c o n t r i b u t i o n o f block j , u n i t i, o f bus k t o 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 func t ion of Yk. The i n t e g r a l (18) is computed for each b lock o f a l l u n i t s connected t o bus k. Upon

186

completion, 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 func t ion of t he t o t a l gene ra t ion Yk a t bus k is known.

P r o b a b i l i s t i c Power F l o w Analysis

The s imula t ion method descr ibed so f a r provides t h e d e s c r i p t i o n of power i n j e c t i o n s t o system buses. Given t h i s in format ion , it is d e s i r a b l e t o compute the 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 c i r c u i t f lows o r bus vo l t age magnitudes. For t h i s purpose, a power flow s o l u t i o n i s obtained assuming the power i n j e c t i o n a t t h e system buses is equal t o t h e expected va lues of power i n j e c t i o n s at t h e va r ious buses. The expected va lues o f power i n j e c t i o n s a r e computed from the ca l cu la t ed d i s t r i b u t i o n s def ined with Eq. (16). Subsequently, a l i nea r i zed model of c i r c u i t flows and bus vo l t ages is developed i n terms of power i n j e c t i o n s a t gene ra t ion buses. This l i nea r i zed model inc ludes the e f f e c t s of e l e c t r i c load v a r i a t i o n s ince e l e c t r i c load changes a r e absorbed by gebe ra t ion changes. Thus, i n genera l , a c i r c u i t flow o r a bus vo l t age magnitude, which i s represented with a random v a r i a b l e U, i s expressed a s a l i n e a r combination of t he power in j ec - t i o n s Y a t the system genera t ion buses:

(19)

where: '

a = Known cons tan t c o e f f i c i e n t s Y k = Power i n j e c t i o n a t t he k th gene ra t ion bus

Yk = Expected va lue of power i n j e c t i o n at the

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 randan v a r i a b l e W is computed from the known p r o b a b i l i s t i c models of t h e power i n j e c t i o n s Yk. As a mat te r of f a c t , t he power i n j e c t i o n s Yk are expressed as the sum of un i t output a t bus k , y ie ld ing:

- k th gene ra t ion bus.

where: M(k) is t h e set of u n i t s connected t o bus k

X i

Xi

is t h e output of un i t i is the expected va lue of un i t i output .

- The p r o b a b i l i t y d i s t r i b u t i o n func t ion of t h e random v a r i a b l e W is computed a s a by-product of t he simula- t i o n procedure descr ibed i n t h e previous sec t ion . S p e c i f i c a l l y , f o r each loading of a block of a u n i t , t h e c o n t r i b u t i o n t o the p robab i l i t y dens i ty func t ion of t he v a r i a b l e W is computed. Consider, fo r example, t h e loading of b lock j , u n i t i i l l u s t r a t e d i n Fig. 2. Denote t h i s event as follows:

E2 = [ loading of b lock j , un i t i ]

The p r o b a b i l i t y of t h e event E2 equa l s dpji i l l u s t r a t e d i n Fin. 2.

The loading of block j , un i t i con t r ibu te s t o the var i - ab l e W by an amount equal t o a x a . < x. < a where ak. i s def ined with Eq. (l'5)i'ana a.,ia.+lj% def ined i n Fig. 2. The p r o b a b i l i t y of t h i J codt r ibu- t i o n i s equal t o pidpji, where pi is the p r o b a b i l i t y of a v a i l a b i l i t y of un i t i and d p j i is depic ted i n Fig. 2. In add i t ion , block j , un i t i con t r ibu te s zero t o t h e v a r i a b l e W with p r o b a b i l i t y 1-pi. The computation of t h e con t r ibu t ions t o the probab il it y d i s t r i b u t i o n func t ion of t he v a r i a b l e W is performed r ecu r s ive ly i n p a r a l l e l with the s imula t ion method descr ibed e a r l i e r .

In summary, t h e p r o b a b i l i s t i c power flow c o n s i s t s of so lv ing a usual power flow problem assuming t h a t t h e power i n j e c t i o n s a t t he system buses a r e the expected

va lues yk. Subsequently, c i r c u i t flows and bus vo l t age magnitudes a r e expressed a s a l i n e a r combination of t he power in j ec t ions . The l i n e a r i z a t i o n r e q u i r e s one forward and back s u b s t i t u t i o n [231.

Hand1 ing of Net work Non1 inea r i t ies

The proposed method has been extended t o account f o r n o n l i n e a r i t i e s r e s u l t i n g from the power system network model. For t h i s purpose, t he system e l e c t r i c load is p a r t i t i o n e d i n t o a number of segments. As an example, cons ider the independent random v a r i a b l e v1 Of t h e e l e c t r i c load model. Assming t h a t 5 = 3, t h e following events may be def ined

Ci = {0.33(i-l) < V1 < 0.33i} , i = 1,2,3

Each of t h e above events r e p r e s e n t s a range of e l e c t r i c load g iven by Eq. ( 5 ) . On t h e o the r hand, each of t h e above events has a p r o b a b i l i t y of occurrence:

Now t h e method descr ibed i n t h i s paper i s app l i ed c o n d i t i o n a l l y upon each event C; and the r e s u l t s added. Note t h a t f o r each event Ci, a d i f f e r e n t network oper- a t i n g cond i t ion w i l l be used f o r l i n e a r i z a t i o n , The o v e r a l l procedure i s i l l u s t r a t e d i n Fig. 3. In t h i s way, n o n l i n e a r i t i e s r e s u l t i n g from network models a r e taken i n t o account.

Monte Car lo Simulation

Va l ida t ion o f t h e proposed method wi th a c t u a l system measurements i s very d i f f i c u l t i f not impossible. A v i a b l e v a l i d a t i o n method is by means of Monte Car lo s imula t ion . The Monte Carlo s imula t ion i s based on e x a c t l y t h e same models of e l e c t r i c load , genera t ion , and t ransmiss ion system as t h e proposed method. In t h i s way, t h e r e s u l t s of t h e Monte Carlo method a r e d i r e c t l y comparable t o t h e r e s u l t s of t h e proposed method. Note t h a t i n t h e Monte Car lo method, t h e number of t r i a l s must be l a r g e f o r meaningful r e s u l t s . This requirement h inde r s t he a p p l i c a b i l i t y of t he Monte Car lo method t o l a r g e s c a l e power systems. For t h i s reason , t h e v a l i d a t i o n procedure has been l imi t ed t o small s i z e power systems.

The Monte Car lo s imula t ion has been appl ied t o the 24 bus LEEE R e l i a b i l i t y Test System [22 ] . Tests have been performed t o determine t h e number of t r i a l s requi red f o r meaningful r e s u l t s . The tests cons i s t ed of i nc reas ing the number of t r i a l s while observing the r e s u l t s . It has been observed t h a t when t h e number of t r i a l s exceeded 5,000, no apprec iab le changes occur t o t h e r e s u l t s . Based on these observa t ions , a l l subsequent Monte Car lo s imula t ions were performed with 10,000 t r i a l s .

Example Resul t s

The 24 bus IEEE R e l i a b i l i t y Test System (RTS) 1221 has been used a s the example system. No c i r c u i t outages were assumed. Generating un i t da t a ( c a p a c i t i e s and forced outage r a t e s ) and e l e c t r i c load v a r i a t i o n s were assumed t o be those def ined i n Ref. [221. Generating un i t cos t da t a were modified. S p e c i f i c a l l y , quadra t i c cos t c o e f f i c i e n t s were def ined a s i l l u s t r a t e d i n Table 1. The purpose of t h e modi f ica t ion was t o accentua te the e f f e c t s o f t h e economic d i spa tch process.

The s imula t ion of t h i s system fo r a period of one year has been considered. For s impl i c i ty , un i t maintenance has been neglec ted . The electric load s p e c i f i e d fo r t h e RTS i n Ref, 1221 i s a conforming

187

I I Select Number of Load Levels (Events C i ) b i = O

i = i + l < I Assume Load Level i

Compute Probabili ty of Event C i , pCl I P e r f o r m Generation Simulation.

Compute Expected Unit Output Level. Compute Bus Expected Generation

.Apply t o the Power Network - Expected Electric Load - Expected Bus Generation

.Solve Power Flow Problem Compute Quant i t ies of In te res t - Linearize Quant i t i t es of In te res t with respec t t o Bus Generation

1 Compute Probabili ty Density Funct ion

of Quant i t ies of In te res t . (Conditional Upon Event C )

1

I Compute Probabili ty Density Funct ion I of Q u a n t i t i e s of In te res t

Figure 3. Flow Chart of the Proposed Probabilistic Power Fbw Method.

load represented with a s i n g l e independent v a r i a b l e . The parameters of the e l e c t r i c load model for a study period o f one year have been computed a s follows: F i r s t , t h e hour ly chronologica l load d a t a were cons t ruc ted from the information proviced i n Ref. [221. These d a t a a r e described with the equation:

11 = 965.82 + 1884.3 v (MW)

where v i s a v a r i a b l e assuming va lues between 0 and 1. Using above equat ion , t h e chronological load d a t a 11 were transformed i n t o chronological d a t a of the v a r i a b l e v. From these d a t a , 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 func t ion of t h e v a r i a b l e v i s computed. Subsequently, the t o t a l e l e c t r i c load i s d i s t r i b u t e d t o system buses (conforming load) , y ie ld ing the following model of bus r e a l power:

P1 P2 P3 P4 P5 P6 P7 P8 P9 PI0 P13 P14 PI5 PI6 PI8 P19 P20

37 33 61 25 24 46 42 58 59 66 90 66

107 34

113 61 43

+

71 64

119 49 47 90 83

113 116 129 175 128 209 66

220 120 85

V

In a d d i t i o n , the complex power a t a bus of t h e RTS system i s :

Si = P . + j0.2Pi

Table 1. Generating Unit Data

Cost C o e f f i c i e n t s Unit S ize # of Units F.O.R. - a b c

12 20 50 76

100 155 I97 350 400

0.02 0.10 0.01 0.02 0.04 0.04 0.05 0.08 0.12

14 1 0

87 98

124 113 180 229

32.0 45.0

0. I 16.0 26.0 13.0 24.0 24.0

7.0

0.01 0.001 0.011 0.02 0.01 0.015 0.01 0.014 0.016

P r o b a b i l i t y d i s t r i b u t i o n func t ions o f c i r c u i t flows and bus v o l t a g e magnitudes have been computed with the proposed method and with Monte Carlo s imula t ion . Figures 4 and 5 i l l u s t r a t e t y p i c a l r e s u l t s . F igure 4 inc ludes 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 a c i r c u i t flow and a bus v o l t a g e magnitude computed using a s i n g l e segment r e p r e s e n t a t i o n of t h e e l e c t r i c load. Note t h a t the c i r c u i t flow matches reasonably well the Monte Carlo r e s u l t s , while t h e bus vol tage magnitude shows s u b s t a n t i a l d e v i a t i o n s from Monte Carlo r e s u l t s . The d i f f e r e n c e s a r e a t t r i b u t e d t o t h e n o n l i n e a r i t i e s of t h e power flow equat ions . F igure 5 i l l u s t r a t e s t h e d i s t r i b u t i o n s of flow and v o l t a g e magnitude f o r the same c i r c u i t and bus a s i n Fig. 4. The r e s u l t s of Fig. 5 were obtained by using a t h r e e segment r e p r e s e n t a t i o n of t h e e l e c t r i c load a s it has been d iscussed e a r l i e r , applying t h e method f o r each load segment s e p a r a t e l y , and adding the r e s u l t s . Note t h a t these d i s t r i b u t i o n s match the Monte Carlo r e s u l t s much b e t t e r . In genera l , r e p r e s e n t i n g t h e e l e c t r i c load with more segments w i l l provide b e t t e r r e s u l t s . However, it should be observed t h a t the amount of computation is propor t iona l t o the number of segments. Using t h r e e segments provides a good compromise between accuracy and speed. A l l the simula- t i o n r e s u l t s presented i n Figs. 4 and 5 have been obtained with a s t e p s i z e of 5 MW for the incremental economic d ispa tch .

Evalua t ion of Method E f f i c i e n c y

The proposed method c o n s i s t s o f t h r e e b a s i c computational procedures: (1) genera t ion system s imula t ion method, ( 2 ) s tandard power flow s o l u t i o n , and (3) l i n e a r i z a t i o n of q u a n t i t i e s of i n t e r e s t such a s bus vol tage magnitude, c i r c u i t flows, etc.

188

Monte Carlo Simulation

240 300 360 420 480 540 600

Flow in MVA 1.00

0.75

.- 2 - 0.50

D e a

0.25

0.00

---- Monte Carlo Simulation /’

Proposed Method /

I -

I I 1 I I 0.94 0.96 0.98 1.00 1.02 1.04 1.06

Voltage in pu

Figure 4. Probability Distribution Functions Using a Single Segment Load Model. (a) Circuit 14-16 Flow (b) Bus 6 Voltage

The performance of s tandard power flow algori thms I S well known and w i l l not be discussed here . The l i n e a r i z a t i o n procedure i s based on the c o s t a t e method presented i n Ref. [231. The performance of t h i s method has been well documented i n [231. S p e c i f i c a l l y , t h e computations required fo r t he l i n e a r i z a t i o n of a s p e c i f i c quan t i ty of i n t e r e s t ( fo r example, a c i r c u i t flow) a r e approximately equal t o t h a t of a forward and back s u b s t i t u t i o n with the t a b l e of f a c t o r s of t h e Jacobian ma t r ix of a s tandard Newton-Raphson power flow. To complete the performance eva lua t ion of t h e proposed method, i t i s s u f f i c i e n t t o provide d a t a on the performance of t h e gene ra t ion system s imula t ion method. The execu t ion t i m e of t h e gene ra t ion system s imula t ion method depends on two main parameters: (1) number of u n i t s and ( 2 ) s t e p s i z e fo r s imula t ing the incremental economic d i spa tch . Execution t imes of t he method versus s t e p s i z e and pa rame t r i ca l ly with the number of u n i t s i s given i n Fig. 6. The r e s u l t s have been obtained with two systems: (1) a 32 u n i t system ( the I E E E R e l i a b i l i t y Test System) and ( 2 ) a 62 u n i t , 4198 MW peak load system. The r e s u l t s has been obtained on an IBM PS /2 Model 80, 20 MHz.

Conclusions

A new p r o b a b i l i s t i c power flow a n a l y s i s method is proposed, capable of computing p r o b a b i l i t y d i s t r i b u t i o n func t ions of c i r c u i t flows and bus vo l t age magnitudes. The method i s based on a d e s c r i p t i o n of bus power i n j e c t i o n s a s random v a r i a b l e s . The computation of t h e

1.00

0.75

Monte Carlo Simulation Proposed Method

240 300 360 420 480 540 600

Flow in MVA 1.00

0.75

.- 2 - 0.50

D e n

0.25

0.00

/--- Monte Carlo Simulation i ----

- Proposed Method

I I I I I I 4 0.96 0.98 1.00 1.02 1.04 1.06

Voltage in pu

Figure 5. Probability Distribution Functions Using a Three Segment Load Model (a) Circuit 14-16 Flow (b) Bus 6 Voltage

300

200

t - 100 d 8 I 2 0 c

E 30 F c 20 .- c 3

8 x 10. w

62 Units, 4198 MW Peak Load

Peak Load (IEEE RTS)

tT I I I I 1 2.5 5.0 7.5 10.0 15.0 20.0

Step Size (MW) - Figure 6. Execution Tine of the Generation System

Simulation Method. (IBM PS/2 Model 80, 20 MHz)

s t a t i s t i c s of t h e bus power i n j e c t i o n t a k e s i n t o c o n s i d e r a t i o n t h e major opera t ing p r a c t i c e s of power systems such a s economic d ispa tch . Subsequently, c i r c u i t flows and bus vol tage magnitudes a r e expressed a s a l i n e a r combination of bus power i n j e c t i o n s . Their s t a t i s t i c s a r e computed from t h e s t a t i s t i c s o f t h e power i n j e c t i o n s . The proposed method has been v a l i d a t e d v i a Monte Carlo s imula t ion .

The computational requirements of t h e method a r e moderate. S p e c i f i c a l l y , they a r e comparable t o t h e sum of usual power flow a n a l y s i s and p r o b a b i l i s t i c production c o s t i n g [ l l .

The implementation o f t h e method i s s t r a i g h t - forward. As a mat te r of f a c t i t can be implemented with appropr ia te modi f ica t ions of a power flow algorithm and a p r o b a b i l i s t i c production c o s t i n g algorithm. P o t e n t i a l a p p l i c a t i o n s of t h e method a r e (1) r e l i a b i l i t y a n a l y s i s of power systems, and ( 2 ) t ransmiss ion loss evalua t ion .

Acknowledgements

The au thors g r a t e f u l l y acknowledge the National Science Foundation for the support of the work repor ted i n t h i s paper (Grant No. ECS-8715364).

References

H. Baleriaux, E. Jamoulle, Fr. Linard De Guertechin, "Simulation de 1' e x p l i o t a t i o n d' un parc de machines thermiques de production d ' e l e c t r i c i t e couples a des s t a t i o n s de pompage," Revue E ( e d i t i o n SRBE), vo l . 5, no. 7, pp. 3-24, 1967. R. R. Booth, "Power System Simulation Methods Based on P r o b a b i l i t y Analysis," IEEE Transac t ions on Power Apparatus and Systems, vo l . PAS-91, pp. 62-69, JanuaryIFebruary 1972. M.A. Saner et a l . . "A new genera t ion production - - cos t program t o recognize forced outages," Trans. Power App. Syst. , vo l . PAS-91, pp. 2114- '2124, SeptemberIOctober 1972. M.A. Sager, A . J . Wood, "Power system production c o s t calculations-Samule s t u d i e s recognizing - forced outages," I E E E Trans. Power App. Sys t . , vo l . PAS-92, pp. 154-158, January/February 1973. J. Zahavi, J. Vardi, B. Avi-Itzhak, "Operating c o s t c a l c u l a t i o n of an e l e c t r i c power genera t ing system under incremental loading procedure," I E E E Trans. Power App. Sys t . , vo l . PAS-96, pp. T85r 2 92. Januar v/Febr uarv 1 9 77. N. S. Rau, P. Toy, K. F. Schenk, "Expected Energy Production Costs by the Method of Moments," I E E E Trans. on Power Apparatus and Systems, vo l . 99, no. 5, pp. 1908-1915, September/October 1980. J. P. Stemel, R. T. Jenkins , R. A. Babb, W. D. Bayless, "Production Costing Using the Cumulant Method of Representing the Equivalent Load Curve," I E E E Trans. on Power Apparatus and Systems, vo l . PAS-99, no. 5, pp. 1947-1956, September/ Oc t ob er 1 980. J. P. Stremel, " S e n s i t i v i t y Study of t h e Cumulant Method of Calcu la t ing Generation System R e l i a b i l i t y , " I E E E Trans. on Power Apparatus and Systems, vo l . PAS-100, no. 2, pp. 771 -778, February 198 1. A.P. Meliopoulos, "Computer aided i n s t r u c t i o n o f energy source u t i l i z a t i o n problems," IEEE Trans. on Education, vo l . E-24, no. 3, pp. 204-209, August 1981.

191

201

189

K. F. Schenk, R. B. Misra, S. Vassos, "A New Method f o r the Evalua t ion of Expected Energy Generation and Loss of Load Probabi l i ty , " IEEE Trans. on Power Apparatus and Systems, vo l . PAS- 103. no. 2. UD. 294-303. February 1984.

~ . _ H. Duran, "On Improving t h e Convergence and Accuracy of the Cumulant Method of Calcu la t ing R e l i a b i l i t y and Production Cost ,If I E E E Trans. on Power Systems, vo l . PWRS-1, no. 3, pp. 121-126, August 1986. L. A. Sanabria, T. S. Dil lon , "An Error Correc- t i o n Algorithm for S tochas t ic Production Costing," I E E E Trans. on Power Systems, v o l . 3, no. 1, pp. 94-100, February 1988. G. Gross, N. V. Garapic, B. McNutt, "The Mixture of Normals Anproximat ion Technique for Equivlent .. Load Duration Curves," I E E E Trans. on Power Systems, vol. 3, no. 2, pp. 368-374, May 1988. B. Borkowska, " P r o b a b i l i s t i c load flow,'' IEEE Trans. on Power Apparatus and Systems, vo l . PAS-93, no. 3, pp. 752-759, May/June 1974. R.N. Allan, B. Borkowska, C.H. Grigg, "Probabil- i s t i c a n a l y s i s of power flows," Proc. of the I E E , vo l . 121, no. 12, pp. 1551-1556, December 1974. J.F. Dopaso, O.A. K l i t i n , A.M. Sasson, "Stochas t ic load flows," I E E E Trans. on Power Apparatus and Systems, vo l . PAS-94, no. 2 , DD. 299-309. MarchIAuril 1975. L L

R.N. Allan, C.H. Grigg, D.A. Newey, R.F. Simmons, " P r o b a b i l i s t i c power flow techniques extended and appl ied t o opera t iona l d e c i s i o n making," Proc. of t h e I E E , v o l . 123, no. 12, pp. 1317-1324, December 1976. R.N. Al lan , M.R.G. Al-Shakarchl, " P r o b a b i l i s t l c techniques i n AC load flow a n a l y s i s , " Proc. of t h e I E E , vo l . 124, no. 2, pp. 154-160, February 1977. A.M. Leite da S i l v a , V.L. A r i e n t i , R.N. Al lan , " P r o b a b i l i s t i c load flow cons ider ing dependence ~~

between input nodal powers," I E E E Trans. on Power Apparatus and Systems, v o l . PAS-103, no. 6, up. 1524-1530, June 1984. ._ P.W. Sauer, G.T. Heydt, "A genera l ized s t o c h a s t i c power flow algorithm," Paper A78-544-9, presented a t the 1978 IEEE/PES Summer Meeting, J u l y 1978. A.P. Meliopoulos, A.G. B a k i r t z i s , R. Kovacs, "Power system r e l i a b i l i t y e v a l u a t i o n using s tochas t ic - load flows," f E E E Trans. on Power Apparatus and Systems, v o l . PAS-103, no. 5, pp. 1084-1091, May 1984. ..

I E E E Committee Report , " I E E E R e l i a b i l i t y Test System," I E E E Trans. on Power Apparatus and Systems, vo l . PAS-98, pp. 2047-2054, 1979. A. P. MeliODoulOs. G. Contaxis. R. R. Kovacs, N . D. Reppen, N. Balu, "Power System Remedial Action Methodology," I E E E Trans. on Power Systems, vo l . PWRS-3, no. 2, pp. 500-509, May 1988. Comparative Models f o r E l e c t r i c a l Load Forecas t ing , Edited by Bunn and Farmer, John Wiley 6 Sons, 1985. S. Vemuri, W.L. Huang, and D . J . Nelson, "On-Line Algorithmis for Forecas t ing Hourly Loads of an E l e c t r i c U t i l i t y , " IEEE Trans. on Power Apparatus and Systems, vo l . PAS-100, no. 8, pp. 3775-3784, August 1981.

190

Biosketches

A. P. Sakis Meliopoulos, (M '76, SM '83) was born i n K a t e r i n i , Greece, i n 1949. He rece ived the M.E. and E.E. diploma from the National Technical Univers i ty of Athens, Greece, i n 1972; the M.S.E.E. and Ph.D. degrees from the Georgia I n s t i t u t e of Technology i n 1974 and 1976, r e s p e c t i v e l y . In 1971, he worked for Western E l e c t r i c i n A t l a n t a , Georgia. In 1976, he jo ined t h e f a c u l t y of E l e c t r i c a l Engineering, Georgia I n s t i t u t e o f Technology, where he is p r e s e n t l y an Associate Professor . He i s

a c t i v e i n teaching and research i n the general a r e a s of modeling, a n a l y s i s , and cont ro l of power systems. He has made s i g n i f i c a n t c o n t r i b u t i o n s t o power system grounding, harmonics, and r e l i a b i l i t y assessment of power systems. He is the author of the book, Power Systems Grounding and Trans ien ts Marcel Dekker, June )ph, Numerical Solu t ion Methods of Algebraic Equations, EPRI monograph series. DK. Meliopoulos i s a member of the Hel len ic Socie ty of Profess iona l Engineers and the Sigma X i .

George Cokkinides (IEEE member 1985) was born i n Athens, Greece, i n 1955. He obtained t h e B.S. , M.S., and Ph.D. degrees a t the Georgia I n s t i - t u t e of Technology i n 1978, 1980, and 1985, r e s p e c t i v e l y . From 1983 t o 1985, he was a research engineer a t the Georgia Tech Research I n s t i t u t e . Since 1985, he has been with the Univers i ty of South Caro l ina a s an Ass is tan t Professor

o f E l e c t r i c a l Engineering. His r e s e a r c h i n t e r e s t s inc lude power system modeling and s imula t ion , power e l e c t r o n i c s a p p l i c a t i o n s , power system harmonics, and measurement ins t rumenta t ion . DK. Cokkinides i s a member of the I E E E Power Engineering Socie ty and the Sigma X i .

Xing Yong Chao (IEEE s tudent member 1988) was born i n Nanjing, China, i n 1960. He received the B.S. degree from Shandong Poly technica l Univers i ty , China, i n 1982, and t h e M.S. degree from Nanjing Automation Research I n s t i t u t e of Minis t ry of Water Resources and E l e c t r i c Power of China i n 1985. From 1985 t o 1987, he was a r e s e a r c h engineer a t Nanjing Automation Research I n s t i t u t e .

Curren t ly , he i s pursuing h i s Ph.D. degree a t the Georgia I n s t i t u t e of Technology. H i s r esearch i n t e r e s t s inc lude power system r e l i a b i l i t y assessment, power system r e l a y i n g , and computer a p p l i c a t i o n s i n power systems.