an algorithm for optimum control of static var compensators to meet phase-wise unbalanced reactive...

8/7/2019 An algorithm for optimum Control of Static VAR Compensators to meet Phase-wise unbalanced reactive power dem…

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E l e c t r ic P o w e r S y s t e m s R e s e a r c h , 1 1 (1986 ) 129 - 137 129

A n A l g o r it h m fo r O p t i m u m C o n t r o l o f S t a t ic V A R C o m p e n s a t o rs t o M e e t

P ha s e -wi s e Unba l a nced Rea c t i v e P o wer Dema nds

D. THUKARAM, B. S. RAMAKRISHNA IYENGAR and K. PARTHASARATHY

D e pa r tm ent o f E lec t r ica l Engineer ing , I nd ian I ns t i tu te o f Sc ience , Bangalore 560 012 ( Ind ia)

(Received July 18, 1986)

SUMMARY

The operation of thyristor-controlled static

VA R compensa tors (SVCs) at various conduc-

tion angles can be used advantageously to

meet the unbalanced reactive pow er demand sin a system. However, such operation intro-

duces harmonic currents into the AC system.

This paper presents an algorithm to evaluate

an optim um combination of the phase-wise

reactive power generations from SVC and

balanced reactive power supply from the

AC sys tem, based on the defined perfor-

mance indices, namely, the telephone influ-

ence factor ( T I F ) , the total harmonic current

factor ( I T ) and the distortion factor (D).

Results of the studies conducted on a typical

distribution system are presented and dis-

cussed.

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

P h a s e- w i se u n b a l a n c e d r e a c ti v e p o w e r d e-

m a n d i n p o w e r s y s t e m s i s d u e t o l a rg e s in g le -

p h a s e l o a d s a n d a l s o t o t h e l a r g e a n d f l u c -

t u a t i n g i n d u s t r i a l l o a d s s u c h a s e l e c t r i c a r c

f u r n a c e s , r o l l i n g m i l l s , e t c . T h e s e l o a d s u n -

b a l a n c e t h e s y s t e m a n d l e ad t o w i d e f lu c t u a -

t i o n s i n t h e s u p p l y v o l t a g e a n d e f f e c t s l ik e

i n c a n d e s c e n t l i g h t f l i c k e r , t e l e v i s i o n p i c t u r e

d i s t o r t i o n , d i s t u r b a n c e i n e l e c t ri c c o n t r o l

c i rc u i ts a n d c o m p u t e r e q u i p m e n t , e t c ., w h i c h

a r e u n d e s i r a b l e t o c o n s u m e r s . T h e s e t y p e s

o f h e a v y i n d u s t r i a l l o a d s a r e n o r m a l l y c o n -

c e n t r a t e d i n o n e p l a n t a n d a r e s e r v e d f r o m

o n e n e t w o r k t e rm i n a l ; t h e y c a n t h e r e f o r e

b e h a n d l e d b e s t b y a l o c al c o m p e n s a t o r c o n -

n e c t e d t o t h e s a m e t e rm i n a l . T h e r e a r e tw o

m a i n r e a s o n s f o r c o m p e n s a t i n g l ar g e f lu c -

t u a t i n g l o a d s : ( 1 ) , t h e A C s y s t e m i s t o o

w e a k t o m a i n t a i n t h e t e r m i n a l v o l t a g e w i t h i n

t h e a c c e p t a b l e v a r ia t io n s , a n d ( 2 ) it is n o t

e c o n o m i c a l , o r p r a c ti c a l , t o s u p p l y t h e r e a c -

t iv e p o w e r d e m a n d f r o m t h e A C s y s te m .

S h u n t c o m p e n s a t o r s a r e g e n e r a ll y u s e d to

r e d u c e o r c a n c e l t h e p h a s e - w i se u n b a l a n c e dr e a ct iv e p o w e r (V A R ) d e m a n d a n d t o m in i-

m i z e t h e r e a c t i v e p o w e r d r a w n f r o m t h e A C

s u p p l y l in e s. S t a t ic V A R c o m p e n s a t o r s a r e

p r e f e r r e d o v e r t h e t r a d it i o n a l V A R c o m -

p e n s a t o r s s u c h a s s a t u r a b l e r e a c t o r s a n d

s w i t c h e d c a p a c i t o r s o w i n g t o a d d i t i o n a l

a d v a n t a g e s l i k e f a s t r e s p o n s e , h i g h r e l i a b i l i t y ,

f l e x i b i li t y a n d l o w m a i n t e n a n c e c o s t . S e v e ra l

t y p e s o f S V C s w i t h d i f f e r e n t o p e r a t i n g f e a -

t u r e s c a n b e r e a l iz e d b y u s i n g v a r io u s p o w e r

c o n v e r s i o n c o n c e p t s a n d t h y r i s t o r c i r c u i t s .

T h e o p e r a t i o n o f t h y r i s t o r - c o n t r o l le d c o m -

p e n s a t o r s a t v a r io u s c o n d u c t i o n a n g le s c a n

b e u s e d a d v a n t a g e o u s l y t o m e e t t h e u n -

b a l a n c e d r e a c t iv e p o w e r d e m a n d s i n a s y s t e m .

H o w e v e r , s u c h o p e r a t i o n i n t r o d u c e s h a r m o n i c

c u r r e n t s i n t o t h e A C s y s t e m . I n s u c h ca s e s

i t b e c o m e s n e c e s s a r y e i th e r t o m i n i m i z e

h a r m o n i c g e n e r a t io n i n te r n a ll y o r p r o v i d e

e x t e r n a l h a r m o n i c f i l t e r s .

F r a n k a n d L a n d s t r o m [ 1 ] h av e p r e s e n t e d

a p o w e r f a c t o r c o r r e c t i o n m e t h o d f o r b al -

a n c e d a n d u n b a l a n c e d l o a d s. T h e y a d v o c a t e d

t h e u s e o f t h y r i s t o r - c o n t r o l l e d d e l t a - c o n -

n e c t e d c a p a c i t o r s i n s e ri e s w i t h r e a c t o r s .

T h e y a ls o s u g g e s te d t h a t t h e s e r e a c t o r s b e

u s e d a s f i lt e r s f o r t h e h a r m o n i c c u r r e n t s .

B a r t h o l d et al. [ 2 ] s u g g e s t ed a m e t h o d o f

d e t e r m i n i n g t h e e f f e c t o f h a r m o n i c s i n t e r m s

o f d e f i n e d p e r f o r m a n c e i nd i ce s . E n g b e r g et

al. [ 3 ] p e r f o r m e d t h e h a r m o n i c a n a ly s is o f

p h a s e c u r r e n t s i n a d e l t a - c o n n e c t e d p h a s e -

c o n t r o l l e d r e a c t o r . T h e m a g n i t u d e s o f v a r io u s

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

o f t h e f ir in g a n g l e o f t h e t h y r i s t o r s. H a m m e d

0378- 7796/ 86/$3 .50 © Elsevier Sequoia/Pri nted i n T h e N e t h e r l a n d s



1 3 0

and Mathur [4] propo sed a new configura-

tion for the reactors to reduce the harmonic

generation. They suggested a parallel combi-

nation of two phase-controlled reactors in

series with a fixed reactor. It is shown that,

by using this arrangement for the reactors

in a static VAR compensator, magnitudesof certain order harmonics can be reduced,

thereb y eliminating the need of filters.

Gyugyi e t a l . [5] examined the theoretical

foundati ons of thyristor-controll ed shunt

compensation. They established conditions

for unbalanced load compensation and volt-

age stabilization with the use of symmetrical

components, and suggested a scheme which

employs a fixed capacitor in parallel with a

thyristor-controlled inductor for the real-

ization of variable susceptances. The harmon-

ics generated by controlling the thyristorswitches were kept out of the line currents

by placing the fixed capacitor in series with a

filter network that draws the same funda-

mental current at the system frequency and

provides low impedance shunt paths at the

harmonic frequencies. Further, Gyug yi and

Tayl or [6] suggested two m eth ods for mini-

mizing the harmonics generated internally by

the thyristor-controlled reactors. One method

uses n reactor banks, each with 1 In of the

total rating, the reactor banks being con-

trolled sequ entiall y, tha t is, only one of then reactors is delay-angie controlled and each

of the remaining n -- 1 reactors is either fully

ON or fully OFF. The other m eth od uses two

identical delta-connected thyristor-controlled

reactor banks, one operated from the wye-

connected secondary windings, the other

from the delta-connected windings of a sup-

ply transformer. They also reported that the

harmonic cancellation is theoreticall y possible

by operating three, four, or more delta-

connect ed thyristor-controlled reactor banks

with appropriately phase-shifted voltages.This paper presents an algorithm to evalu-

ate an optimum combination of the phase-

wise reactive power generations from SVC

and balanced reactive power supply from the

AC system based on the defined performanc e

indices TIF (telephone influence factor), IT

{total harmo nic curre nt fac tor) and D (distor-

tion factor). The approach results in mini-

mization of the effect of harmonics in the

AC system, thereby reducing the burden on

the external harmonic filter. Results of the

studies conducted on a typical distribution

system for different loading and compensa-

tion c onditions are presented to illustrate the

algorithm.

2. MATHEMATICALMODEL

2 . 1 . C o m p e n s a t o r r e q u i r e m e n t s

Two types of static VAR compensa-

tors, namely, the fixed capacitor-thyristor-

controlled reactor (FC-TC R) and the thy-

istor-switched capacitor-thyristor-controlled

reactor (TSC -TCR) are considered for the

analysis. The block schematic arrangement

of a typical SVC is shown in Fig. 1. The

compensator essentially functions as a vari-

able reactance {capacitive and inductive

impedances). In order to establish the basiccompensation requirements it is assumed

that the phase-wise load demands are un-

balanced and time invariant. A series of such

steady-state loads at discrete time instants,

appropriately close to each other, can also

be emp loy ed to represent time-varying loads.

With this assumption, the compensator

requirement is to generate/absorb unbalanced

reactive power which, when combined with

the load demand, will represent balanced

load to the supply system. Consider a system

as shown in Fig. 2, where bus 1 representsthe AC system source node and bus 2 repre-

sents the load bus, with a static VAR com-

pensator connected at that bus. Let the

phase-wise load dem and be PLa +jQ La ,

PLb +jQLb and PLc +jQ Lc. Assuming the

phase-wise load seen by the source (bus 1)

after compe nsat ion to be PLa + jQS~, PLb +

jQSb and PLc + jQSc, the phase-wise voltages

V a / 6 a , V b / 6 b and V c / 5 e at the load bus (bus

2) are given by

[EL] ~bc = [ES] abc -- [ZS]~bc[I] abc (1)

where

I a = (PLa -- j Q S a ) / V ~ / - - 6 ~

I b = ( P L b - - j Q S b ) /V b L - - 6 b

I c = (PL¢ -- j Q S c ) / V d - - 6 ~

The nonlinear complex set of eqns. (1) can

be solved for load bus voltages using a proce-

dure similar to three-phase load flow analysis.

The phase-wise reactive power balance equa-

tions at the load bus are



131

-~ '-w . ] , / PH AS E A

" 1~ ~ P H A S E B

. j j P H A S E C

Q Cb~Q Rb '

QCQ aCc '

Fixed Capac i to r s l

T h y r is t o r S w i t c h e d C a p a c i t o r s

O R ' Q R c

Thyr is tor Cont ro l led R eacto r s

Fig. 1. Block schematic arrangem ent of an SVC for phase-wise reactive compensatio n.

B U S( ! )

>-

d

•~ I I I I I

R b q ~ P t . b + J Q s b

&o

B U S

®

0 r , u

PI .a+ jQi . a i

P L b + J Q L b

T C R

Fig. 2. Line diagram of the system.

I Ra

ta

i.a

x a

co

Fig. 3. De lta~ onne cted reactances of the thyristor-

controlled reactor.

[QS] abe + [QC] ab~ = [QR] abc+ [QL] ~bc (2)

For a given phase-wise unbalanced reactive

power dem an d [QL] abe , setting balanced

values for [QC] ~bc of the FC- TSC and

[QS] ~bc of t he source, the unba lan ced reac-

tive power [QR] abe absorptions of th e T CR

can be ob tain ed fr om the set of eqns. (2).

Considering the compensator as variable

delta-connected reactances as shown in Fig.

3, QRa, QRb and QRc are absorbed in the

unsym metr ica l reac tances Xab, Xbc and xca

of the TCR. Referring to Fig. 3, the fol-

lowing relation can be obtained:

[QR] abe = [AI [B ] (3)

where

[B] = [Bab , Bbc , Boa] t

with Bah = 1/Xab, Bbc = 1/Xbc, Bc~ = 1/Xc~

and [A ] is a 3 × 3 matr ix w ith

A(1, 1)= VaVa- VaVb COS(6a- 65)

A(1, 3) = Va V a - - Va V c cos(6 a -- 6c)

A(2, 1) = Vb Vb -- Vb Va COS(6b -- 6a)

A(2, 2) = Vb Vb -- Vb Vc COS(6b -- 6¢)

A(3, 2) = VcV¢ -- VcVb COS(6¢ -- 6b)

A(3, 3) = V¢V¢ -- VcV~ cos(6c -- 6~)

A(1, 2) = A(2, 3) = A(3, 1) = 0.0



1 3 2

F o r t h e p o s s i b l e v o l t a g e c o n d i t i o n s a t

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

r e a c t i v e p o w e r a b s o r p t i o n s , t h e v a l u e s o f t h e

d e l t a - c o n n e c t e d r e a c t a n c e s o f t h e c o m p e n -

s a t o r c a n b e o b t a i n e d f r o m t h e s o l u ti o n o f

e q n s . ( 3 ) .

2 . 2 . R e a l i z a t i o n o f v a r ia b l e r e a c t a n c e s

T h e v a r i a bl e r e a c t a n c e s o f th e c o m p e n -

s a t o r a r e a c h i e v e d b y d e l a y i n g t h e c l o s u r e

o f t h e t h y r i s t o r s w i t c h b y a n a n g l e a ( 0 <

a < 7 r/2 ). T h e u n s y m m e t r i c a l f i ri n g o f t h y -

r i st o r s c a n b e u s e d a d v a n t a g e o u s l y t o o b t a i n

t h e u n s y m m e t r i c a l d e l t a - c o n n e c t e d re a c -

t a nc e s . C o n s i d e ri n g o n l y t h e f u n d a m e n t a l

c o m p o n e n t , t h e u n s y m m e t r i c a l f ir in g a n g le s

a l , ~ 2 a n d a3 c o rr e s p o n d i n g t o t h e d e l t a

rea cta nce s Xab , Xbc a n d X ca c a n b e o b t a i n e d

b y s o l v i n g t h e e q u a t i o n s

Xab = X0b/ [1 - - 2 a l / l r -- s i n ( 2 o q ) / r r ] ( 4 a )

X b c - -- -X ° c / [ 1 - - 2 a 2 /z r - - s i n ( 2 a 2 ) / n ] ( 4 b )

X c a = X c 0 a / [ 1 - - 2aa/Tr - - s in (2o la ) /Tr ] (4c )

w h e r e X ° b , Xb0c a n d X c°a a r e t h e r e a c t a n c e s f o r

f u ll c o n d u c t i o n o f t h y r i s t o r s c o r r e s p o n d i n g

t o z e r o f i r i n g a n g l e s , t h a t i s, ~ 1 = a 2 = a 3 =

0 . 0 .

2 .3 . M e a s u r e m e n t o f h a r m o n i c e f f e c ts

T h e e f f e c t o f h a r m o n i c s i n t h e s y s t e m isg e n e r a ll y m e a s u r e d b y t h e c a l c u l a t io n o f

d e f i n e d p e r f o r m a n c e i n di c e s. T h e p re s e n t

s t a t e o f t h e a r t s u g g e st s t h e i n v e s t i g a t i o n o f

t h e t h r e e f a c t o r s [ 2 ] T I F , I T a n d D . T I F

p r o v id e s a m e a s u r e o f t e l e p h o n e i n t e r f e re n c e

t h a t m a y r e s u lt f r o m t h e p r e s e nc e o f ha r-

m o n i c v o l t a g e s a d j a c e n t t o t h e p o i n t o f

c o n n e c t i o n o f t h e S V C . IT p ro v i d e s a m e a -

s u re o f t h e T I F t h a t m a y r e s u lt f r o m h a r -

m o n i c c u r r e n ts i n j e c t e d i n t o t h e A C s y s t e m

b y t h e c o m p e n s a t o r , a n d D is a m e a s u r e o f

A C s i n u s o i d a l v o l t a g e d i s t o r t i o n . T h e p e r-f o r m a n c e i n d i c e s T I F , I T a n d D a r e c o m -

p u t e d b y t h e e q u a t i on s

T I F = [ ~= ( T I F ) h 2 ]

1 / 2

D = ( I h / I f )

h = 2

w h e r e

( T I F ) h = ( I h / I f )W h

( I T ) h = I h W h

T h e f u n d a m e n t a l a n d h a r m o n i c c o m p o -

n e n t s o f t h e l in e c u r r e n t s a re o b t a i n e d a s a

d i f f e r e n c e o f t h e c o r r e s p o n d i n g b r a n c h

c u r r e n t s ; I t a n d I h a r e g i v e n b y

Y mI f - - - (G~ 2 + H f2) 1 /2 s i n ( c o t - - ¢ - - 0 ~ )

2 1 r w L

Ih -

n c a L

w h e r e

- - - ( G h 2 + H h 2 ) 1 /2 s i n [ h ( c o t - - 4 ) - - O h ]

Gf = 3~ -- 43" -- 2 si n(23' ) -- 2~ -- sin(2/~)

Ht = V~[Ir - - 2¢ - - s in(2~)]

V h =

s i n [ ( h + 1)3'I s i n [ ( h -- 1)3"]

h + l h - - 1

-

2 si n 3' co s(h 3") I tsin[(__h+ _1)~]

h 2 t h + l

_ s i n [ ( h - - 1) /3 ] _ 2 s i n / 3 c o s ( h f l ) t

h - - 1 h

+ X / ~ t s i n [ ( h _ - - l ) ~ ] _ s i n [ ( h - - 1)~3]

2 ~ - # + 1 h - - 1

_ 2 s i n f i c o s ( h f i ) t

h

O f = t a n - l ( H ~ / G f )

Oh = t a n - l ( H h / G h )

h = h a r m o n i c o r d e r , ( 6 k -+ 1 ) , k = 1 , 2 , 3 . . . . ;

t h e + s ig n is f o r h a r m o n i c s o f o r d e r 6 k + 1 ,

t h e - - s ig n f o r h a r m o n i c s o f o r d e r ( 6 k - 1 ) ;

= 0 , 3" = a l , ~ = a 3 f o r l i n e c u r r e n t i a ; ~ =

2 ~ r /3 , 3 ' = a 2 , ~ = a l f o r l i n e c u r r e n t ib ; ¢ =

4 7 r /3 , 3' = a 3 , ~ = a 2 f o r l i n e c u r r e n t i c .F o r t r i p le h a r m o n i c s ( 3 r d , 9 t h , . . . )

s i n [ ( h + 1 ) 3 " / ] s i n [ ( h - - 1 ) 3 ']Gh ~

h + l h - - 1

2 s i n 7 c o s ( h 3 " ) s i n [ ( h + 1 )/ }]+

h h + l

s i n [ (h - - 1 )j3 ] 2 s in f~ cos(h i3)

h - - 1 h

H h = O



3 . COMPUTATIONALPROCEDURE

F o r a g i v e n u n b a l a n c e d r e a c t i v e p o w e r

l o a d d e m a n d [ Q L ] a be i t is p o s s i b l e to o b t a i n

m u l t i p le c o m b i n a t i o n s o f b a l a n c e d va l ue s

f o r [ Q C ] ~bc a n d [ Q S ] a b c, an d t h e u n b a l a n c e d

r e a c ti v e p o w e r [ Q R ] abe a b s o r p t io n s o f T C R .D i f f e r e n t c o m b i n a t i o n s o f f ir in g an g le s o f

t h e T C R a n d v o l t a g e c o n d i t i o n s g i v e d i f f e r e n t

s e t s o f p e r f o r m a n c e i n d i c e s T I F , I T a n d D .

T h e p a r t i c u l a r c o m b i n a t i o n w h i c h g i v e s

m i n i m u m T I F , I T a n d D is s e l e c t e d a s t h e

o p t i m u m . A c o m p u t e r p r o g r a m h as b e e n

d e v e l o p e d f o r t h e p u r p o s e . T h e p r o p o s e d

a l g o r i t h m is f l e x i b l e e n o u g h t o b e i m p l e -

m e n t e d u s i n g m i c r o p r o c e s s o r - b a s e d h a r d -

w a r e . T h e f o l l o w i n g is a s u m m a r y o f t h e

s t e p s i n v o lv e d i n t h e p r o g r a m .

S t e p 1 . R e a d t h e s y s t e m d a t a r e l a t e d to( a ) th e s o u r c e b u s ( b u s 1 ) v o lt a g e , ( b ) t h e

l in e i m p e d a n c e b e t w e e n t h e s o u rc e b u s a n d

l o a d b u s , ( c ) t h e u n b a l a n c e d l o a d d e m a n d

a t t h e l o a d b u s , ( d ) t h e r a t e d c a p a c i t y o f

t h e S V C , t h a t i s , ( i ) c a p a c i t i v e r a n g e , ( i i )

i n d u c t i v e r a n g e .

S t e p 2 . I n i ti a l l y a s s u m e t h e l o a d r e a c t i v e

p o w e r d e m a n d is m e t c o m p l e t e l y b y th e

c o m p e n s a t o r o n l y , t h a t is , u n i t p o w e r f a c t o r

l o a d i s s e e n b y t h e s o u r c e . S e t Q S ~ = Q S b =

Q S c = 0 . 0 .

S t e p 3 . S e t t h e F C - T S C v a l u e s c l o s e t o t h em a x i m u m o f ph a s e- w i se l o a d re a c ti v e p o w e r

d e m a n d s , t h a t i s, s e t Q C ~ = Q C b = Q C ~ =

c l o se t o m a x i m u m o f Q L a , Q L b a n d Q L c .

S t e p 4 . C o m p u t e t h e v o l t a g e s a t t h e l o a d

b u s ( b u s 2 ) a s s u m i n g t h e p h a s e - w i se l o a d

s e e n b y t h e s o u r c e a s P L a + j Q S a , P L b + j Q S b

a n d P L c + j Q S c .

S t e p 5 . C o m p u t e t h e p h a s e - w is e r e a c t iv e

p o w e r s Q R a , Q R b a n d Q R c t o b e a b s o r b e d

b y t h e T C R .

S t e p 6 . C o m p u t e t h e c o r r e s p o n d i n g v al u es

o f d e l t a - c o n n e c t e d r e a c t a n c e s X ab , X bc a n dXCa o f t h e T C R t o m e e t t h e p h a s e - w i s e v o l t -

a ge a n d r e a c ti v e p o w e r a b s o r p t i o n c o n d i t i o n s

a t t h e T C R .

S t e p 7 . C h e c k f o r t h e d e s i g n l im i t a t i o n s

o f t h e d e l t a - c o n n e c t e d r e a c t a n c es o f t h e

T C R . I f u n s a t i s f a c t o r y , g o to s t e p 1 1 .

S t e p 8 . C o m p u t e t h e u n s y m m e t r i c a l

f i r ing ang le s c~1 , a2 and c~3 o f t h e T C R c o r-

r e s p o n d i n g t o t h e r e a c t a n c e s .

S t e p 9. O b t a i n t h e v a l ue s f o r t h e p e r f o r-

m a n c e i n di c es T I F , I T a n d D b y p e r fo r m i n g

13 3

t h e h a r m o n i c a n a l y si s o f t h e t h r e e A C li n e

c u r r e n t s . P i c k t h e s e t h a v i ng a m a x i m u m

v a l u e f o r T I F , I T a n d D .

S t e p 1 0 . C h e c k w h e t h e r t h e p e r f o r m a n c e

i n d i c e s T I F , I T a n d D a r e w i t h i n t h e s a t is f a c -

t o r y l im i t s o r t h e l a t e s t c o m p u t e d v a l u e s f o r

T I F , I T a n d D a r e g r e a t e r t h a n t h e p r e v i o u s l yc o m p u t e d v a l ue s . I f t h e y a r e w i t h in t h e s at is -

f a c t o r y l i m i t s o r t h e y a r e i n c r e a s i n g c o m -

p a r e d w i t h t h e p r e v io u s l y c o m p u t e d v a lu e s,

g o t o s t e p 1 3 .

S t e p 1 1 . I n c r e a s e t h e v a l u e s o f Q C ~ =

Q C b = Q C c t o t h e n e x t h i g h e r s e t t in g p o s -

s i b l e , o r a s s u m e a n i n c r e a s e d b a l a n c e d r e a c -

t iv e p o w e r s u p p l y Q S a = Q S b = Q S o b y t h e

s o u r c e .

S t e p 1 2 . A d v a n c e t h e i t e r a t i o n a n d g o t o

s t e p 4 .

S t e p 1 3 . P r i n t t h e o p t i m u m s e t ti n g s o f t h ec o m p e n s a t o r a n d t h e c o r r e s p o n d i n g v al u es

o f th e p e r f o r m a n c e i n d ic e s T I F , I T a n d D .

4. TYPICAL SYSTEM STUDIES AND RESULTS

T h e s y s t e m u s e d i n t h e s t u d i e s i s s h o w n i n

F i g. 2 . T h e u n b a l a n c e d l o a d w a s c o n s i d e r e d

t o b e f e d f r o m a s u b s t a t i o n . T h e p a r a m e t e r s

o f t h e l i n e b e t w e e n t h e s o u r c e b u s a n d

t h e l o a d b u s w e r e t a k e n a s R ~ = R b = R c =

0 . 0 0 8 5 5 p . u . p e r p h a s e , a n d X a = X b = X c =0 . 0 8 0 3 0 p . u . p e r p h a s e .

4.1. Case 1

T h e p h a s e - w i s e lo a d c o n d i t i o n s c o n s i d e r e d

w e r e P L a = 3 0 . 0 , P L b = 2 9 . 0 , P L c = 2 8 . 0 M W ,

a nd Q L a = 2 0 . 0 , Q L b = 1 8 . 0 , Q L c = 1 6 . 0

M V A R . T h e r a t e d v a lu e s o f t h e S V C w e r e

2 0 .0 M V A R f o r t h e F C - T S C a n d 1 0 . 0 M V A R

f o r th e T C R .

T h e v a r io u s p o s s i b le c o m b i n a t i o n s o f S V C

t o m e e t th e u n b a l a n c e d l o a d d e m a n d s a n d

t h e c o r r e s p o n d i n g e f f e c t s o f h a r m o n i c s m e a -

s u r e d b y t h e p e r f o r m a n c e i n d i c es a r e g i ve n

i n T a b l e 1 . T h e o p t i m u m s e t t i n g s c o r r e -

s p o n d in g t o t h e m i n i m u m v a lu e s o f t h e p er -

f o r m a n c e i n d i ce s T I F , I T a n d D a r e Q S a =

Q S b = Q S c = 4 . 0 M V A R , a n d Q C a = Q C b =

Q C c = 2 0 .0 M V A R .

T h e p h a s e - w i s e u n b a l a n c e d r e a c t iv e p o w e r

a b s o r p t i o n a n d t h e c o r r e s p o n d i n g f i r i n g a n g l e s

o f t h e T C R a r e Q R a = 4 . 0 , Q R b = 6 . 0 , Q R c =

8 . 0 M V A R , an d cz = 5 8 . 3 , a2 = 3 3 . 3 , c~3 =

4 3 . 2 d e g .



1 3 4

T A B L E 1

S u m m a r y o f t h e r e s u lt s o f h a r m o n i c a n a ly s i s f o r c a se 1 ; p e r f o r m a n c e i n d i c es f o r v a r io u s S V C c o m b i n a t i o n s

R a t e d v a l u e o f F C - T S C = 2 0 .0 M V A R ; r a t e d v a l u e o f T C R = 1 0 .0 M V A R ; P L a = 3 0 .0 , P L b = 2 9 . 0, P L c

M W ; Q L a = 2 0 . 0 , Q L b = 1 8 . 0 , Q L c = 1 6 . 0 M V A R .

= 2 8 . 0

F C - T S C S o u r c e

s e t t i n g s s e t t i n g sQCa, QCB QS~, QSba n d Q C c a n d Q S c( M V A R ) ( M V A R )

P h a s e- w i se M V A R a b s o r p t i o n s o f T C R a n d

c o r r e s p o n d i n g f i r i n g an g l e s (d e g )

M a x i m u m o f th r e e - p h a se

p e r f o r m a n c e i n d i c e s

Q R a Q R b Q R c ~ 1 ~ : ~ 3 T I F I T D

2 0 . 0 0 . 0

2 0 . 0 1 . 0

2 0 . 0 2 . 0

2 0 . 0 3 . 0

2 0 . 0 4 . 0

2 0 . 0 5 . 0

2 0 . 0 6 . 0

0 . 0 2 . 0 4 . 0 - - - N o t f e a s i b l e

1 . 0 3 . 0 5 . 0 - - - - - - N o t f e a s i b l e

2 . 0 4 . 0 6 . 0 8 7 . 4 3 8 . 0 4 9 . 6 5 7 1 . 9 0 . 2 4 1 0 . 3 7 3

3 . 0 5 . 0 7 . 0 6 5 . 0 3 5 . 6 4 6 . 2 3 2 5 . 0 0 . 2 5 1 0 . 2 0 8

4 . 0 6 . 0 8 . 0 5 8 . 3 3 3 . 3 4 3 . 2 1 8 4 . 0 0 . 1 9 0 0 . 1 4 4

( O p t i m u m )

5 . 0 7 . 0 9 . 0 5 3 . 4 3 1 . 1 4 0 . 4 3 2 6 . 0 0 . 2 5 9 0 . 2 4 4

6 .0 8 .0 1 0 .0 . . . . N o t f e a s ib l e

T h e m i n i m u m v a l u es o f t h e p e r f o r m a n c e

i n d i c e s f o r t h i s c a s e a r e T I F = 1 8 4 . 0 , I T =

0 . 1 9 0 , D = 0 . 1 4 4 .

4 . 2 . Ca s e 2

T h e p h a s e - w i s e l o a d c o n d i t i o n s c o n s i d e r e d

i n th i s c a s e w e r e t h e s a m e a s in c a s e 1 . T h e

r a t e d v a l u e s o f t h e S V C w e r e 2 0 . 0 M V A R

f o r t h e F C - T S C a n d 1 5 . 0 M V A R f o r t h e

T C R .

T h e o p t i m u m s e t t i n g s c o r r e s p o n d i n g t o

t h e m i n i m u m v a l u e s o f t h e p e r f o r m a n c e

i n d i c e s a r e Q S a = Q S b = Q S c = 8 . 0 M V A R ,

a n d Q C a = Q C b = Q C c = 2 0 . 0 M V A R .

T h e p h a s e - w i s e u n b a l a n c e d r e a c t i v e p o w e r

a b s o r p t i o n s a n d t h e c o r r e s p o n d i n g f i r i n g

a n g l e s o f t h e T C R a r e Q R a = 8 . 0 , Q R b = 1 0 . 0 ,

Q R c = 1 2 . 0 M V A R , a n d ~ 1 = 4 9 . 4 , ( ~: = 3 4 . 6 ,

~ 3 = 4 1 . 2 d e g .

T h e m i n i m u m v a l u e s o f t h e p e r f o r m a n c e

i n d i c e s a r e T I F = 2 7 6 . 8 , I T = 0 . 3 4 2 , D =

0 . 2 2 4 .

4 . 3 . Ca s e 3

T h i s c a s e w a s s t u d i e d w i t h b a l a n c e d l o a d

c o n d i t i o n s f o r c o m p a r i s o n a n d t o c o n f i r m

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

c a s e s t u d i e s . T h e b a l a n c e d l o a d c o n d i t i o n s

c o n s i d e r e d w e r e P L a = P L b = P L c = 3 0 . 0 M W ,

a n d Q L a = Q L b = Q L e = 2 0 . 0 M V A R . T h e

r a t e d v a l u e s o f t h e S V C w e r e 2 0 . 0 M V A R

f o r t h e F C - T S C a n d 1 5 . 0 M V A R f o r t h e

T C R .

T h e v a r i o u s b a l a n c e d c o m b i n a t i o n s o f

S V C t o m e e t t h e b a l a n c e d l o a d r e a c t i v e p o w -

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

t h e p e r f o r m a n c e i n d i c e s a r e g i v e n i n T a b l e 2 .

T h e o p t i m u m s e t t i n g s c o r r e s p o n d i n g t o t h e

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

a r e Q S a = Q S b = Q S c = 3 . 0 M V A R , a n d Q C a =

Q C b = Q C c = 2 0 . 0 M V A R .

T h e p h a s e - w i s e b a l a n c e d r e a c t i v e p o w e r

a b s o r p t i o n s a n d t h e c o r r e s p o n d i n g f i r i ng

a n g l e s o f t h e T C R a r e Q R a = Q R b = Q R c =

8 . 0 M V A R , a n d a 1 = ( ~: = ~ 3 = 4 5 . 0 d e g .

T h e m i n i m u m v a l u es o f t h e p e r f o r m a n c e

i n d i c e s o b t a i n e d i n t h is c a s e a r e T I F = 4 7 . 2 ,

I T = 0 . 0 5 8 , D = 0 . 1 2 3 .

4 .4 . Case 4

T h i s c a s e s t u d y w a s c a r r i e d o u t t o f i n d

t h e e f f e c t o f S V C r a t i n g t o m e e t a c e r t a i n

l o a d d e m a n d . I n t h i s ca s e t h e l o a d c o n d i t i o n s

c o n s i d e r e d w e r e t h e s a m e a s i n c a s e 1. F o r

t h e d i f f e r e n t r a t e d S V C s , t h e o p t i m u m c o m -

b i n a t i o n s o f S V C c o r r e s p o n d i n g t o t h e m i n i -

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

g i v e n i n T a b l e 3 .

4 .5 . Case 5

T h e p u r p o s e o f t h i s c a s e s t u d y w a s t h e

s a m e a s f o r c a s e 4 , t h a t i s, t o f i n d t h e e f f e c t

o f S V C r a t i n g t o m e e t a c e r t a i n l o a d d e m a n d .

I n th i s c a s e th e l o a d c o n d i t i o n s c o n s i d e r e d

w e r e d i f f e r e n t f r o m t h e c a s e 4 c o n d i t i o n s .

T h e l o a d c o n d i t i o n s c o n s i d e r e d w e r e P L a =

6 0 . 0 , P L b = 5 8 . 0 , P L e = 5 6 . 0 M W , a n d Q L a =

4 0 . 0 , Q L b = 3 6 . 0 , Q L ¢ = 3 2 . 0 M V A R .

F o r t h e d i f f e r e n t r a t e d S V C s , t h e o p t i m u m

c o m b i n a t i o n s o f S V C c o r r e s p o n d i n g t o t h e

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

a r e g i v e n in T a b l e 4 .



1 3 5

T A B L E 2

S u m m a r y o f t h e r e s u l ts o f h a r m o n i c a n a ly s is f o r c as e 3 ; p e r f o r m a n c e i n d ic e s f o r v a ri o u s S V C c o m b i n a t i o n s

R a t e d v a l u e o f F C - T S C = 2 0 . 0 M V A R ; r a t e d v a l ue o f T C R = 1 5 . 0 M V A R ; P L a = P L b = P L c = 3 0 . 0 M W ; Q L a =

Q L b = Q L c = 2 0 . 0 M V A R .

F C - T S C S o ur ce

settings settingsQCa, QCb QSa, QSba n d Q C c a n d Q S c

( M V A R ) ( M V A R )

B a l an c e d M V A R a b s or p t io n s o f T C R

an d corr esponding firing ngles (deg)

P e r f o r m a n c e i n d i c e s

Q R a = Q R b = QRc ~ 1 = ~ 2 = ~ 3 T I F I T D

2 0 . 0 1 . 0 1 . 0 6 8 . 3 9 0 6 . 0 0 . 4 3 7 0 . 5 9 8

2 0 . 0 2 . 0 2 . 0 6 2 . 5 3 2 3 . 8 0 . 2 0 7 0 . 3 4 6

2 0 . 0 3 . 0 3 . 0 5 8 . 3 2 3 5 . 2 0 . 1 8 0 0 . 2 8 2

2 0 . 0 4 . 0 4 . 0 5 5 . 0 4 6 0 . 7 0 . 4 0 3 0 . 3 4 9

2 0 . 0 5 . 0 5 . 0 5 2 . 1 4 5 8 . 8 0 . 4 4 7 0 . 3 2 5

2 0 . 0 6 . 0 6 . 0 4 9 . 5 3 3 1 . 3 0 . 3 5 3 0 . 2 6 8

2 0 . 0 7 . 0 7 . 0 4 7 . 2 1 5 9 . 0 0 . 1 8 4 0 . 1 8 6

2 0 . 0 8 . 0 8 . 0 4 5 . 0 4 7 . 2 0 . 0 5 8 0 . 1 2 3

( O p t i m u m )

2 0 . 0 9 . 0 9 . 0 4 3 . 0 1 6 3 . 1 0 . 2 1 5 0 . 1 6 6

2 0 . 0 1 0 . 0 1 0 . 0 4 1 . 1 2 5 3 . 0 0 . 3 5 4 0 . 1 9 2

2 0 . 0 1 1 . 0 1 1 . 0 3 9 . 4 2 9 6 . 9 0 . 4 3 8 0 . 1 9 5

2 0 . 0 1 2 . 0 1 2 . 0 3 7 . 7 2 9 8 . 9 0 . 4 6 3 0 . 1 8 1

2 0 . 0 1 3 . 0 1 3 . 0 3 6 . 0 2 6 7 . 4 0 . 4 3 4 0 . 1 7 6

2 0 . 0 1 4 . 0 1 4 . 0 3 4 . 4 2 1 1 . 7 0 . 3 5 9 0 . 1 5 7

2 0 . 0 1 5 . 0 1 5 . 0 3 2 . 9 1 4 1 . 7 0 . 2 5 1 0 . 1 3 0

T A B L E 3

S u m m a r y o f t h e r e s u l t s o f h a r m o n i c a n a l y si s f o r c a se 4 ; m i n i m u m p e r f o r m a n c e i n d ic e s f o r v a r i o u s r a t i n g s o f S V C

F C - T S C r a t i n g --- 2 0 . 0 M V A R ; P L a = 3 0 . 0 , P L b = 2 9 . 0 , P L c = 2 8 . 0 M W ; Q L a = 2 0 . 0 , Q L b = 1 8 .0 , Q L c = 1 6 . 0

M V A R .

T C R P h a se - w i se O p t i m u m c o m b i n a t i o n o f M V A R a b s o r p t io n s o f M i n i m u m p e r f o r m a n c e

r a t i n g b a l a n c e d T C R a n d c o r r e s p o n d i n g f i ri n g a n g l e s ( d e g ) i n d i c e s

( M V A R ) M V A R f r o ms o u r c e Q R a Q R b Q R c ~ 1 0~2 C~3 T I F I T D

1 0 . 0 4 . 0 4 . 0 6 . 0 8 . 0 5 8 . 3 3 3 . 3 4 3 . 2 1 8 4 . 0 0 . 1 9 0 0 . 1 4 4

1 5 . 0 8 . 0 8 . 0 1 0 . 0 1 2 . 0 4 9 . 4 3 4 .6 4 1 . 2 2 7 6 . 8 0 . 3 4 2 0 . 2 2 4

2 0 . 0 1 6 . 0 1 6 . 0 1 8 . 0 2 0 . 0 4 0 . 0 3 0 . 7 3 5 . 2 2 5 8 . 4 0 . 5 3 8 0 . 1 7 2

2 5 . 0 2 0 . 0 2 0 . 0 2 2 . 0 2 4 . 0 3 9 . 4 3 1 . 8 3 5 . 5 2 6 6 . 7 0 . 7 1 4 0 . 1 7 8

3 0 . 0 2 6 . 0 2 6 . 0 2 8 . 0 3 0 . 0 3 7 . 2 3 1 . 0 3 4 . 0 2 6 6 . 2 0 . 9 1 0 0 . 1 8 3

4 0 . 0 1 9 . 0 1 9 . 0 2 1 . 0 2 3 . 0 4 8 . 3 4 2 . 0 4 5 . 0 2 2 4 . 0 0 . 7 7 1 0 . 2 0 1

5 0 . 0 2 4 . 0 2 4 . 0 2 6 . 0 2 8 . 0 4 7 . 6 4 2 . 6 4 5 . 0 1 9 2 . 5 0 . 8 1 6 0 . 1 8 4

6 0 . 0 2 9 . 0 2 9 . 0 3 1 . 0 3 3 . 0 4 7 . 2 4 2 . 9 4 5 . 0 1 7 2 . 1 0 . 8 6 8 0 . 1 7 6

4.6. D I S C U S S I O N O F T H E C A S E S T U D I E S

From the results of the above case studies,

the following observations may be made.

(i) The AC system line current harmonic

magnitudes vary depending upon the firing

angles of the TCR.

(ii) The effect of the harmonics is consid-

erable if the firing angles are either too low

or too high, that is, corresponding to either

higher or lower values of the delta reactances

of the TCR.

(iii) The effect of harmonics is a minimum

for a particular combination of reactive

power absorptions which is around half the

rated value of the TCR, that is, for firing

angle combina tions around 45 ° .

(iv) The combination of a low rated FC-

T S C a n d a h i g h e r r a t e d T C R g i ve s r i s e t o

h i g he r m a g n i t u d e s o f h a r m o n i c e f f e c t s. T o



1 3 6

T A B L E 4

S u m m a r y o f t h e r e s u l ts o f h a r m o n i c a n a ly s i s f o r c a se 5 ; m i n i m u m p e r f o r m a n c e i n d ic e s f o r v a r i o u s ra t i ng s o f S V C

F C - T S C r a t i n g = 4 0 . 0 M V A R ; P L a = 6 0 . 0 , P L b = 5 8 .0 , P L c = 5 6 .0 M W ; Q L a = 4 0 . 0 , Q L b = 3 6 .0 , Q L c = 3 2 .0

M V A R .

T C R P h as e -w i se O p t i m u m c o m b i n a t i o n o f M V A R a b s o r p t io n s o f M i n i m u m p e r f o r m a n c e

r a t i n g b a l a n c e d T C R a n d c o r r e s p o n d i n g f i r in g a n g le s ( d e g ) i n d ic e s( M V A R ) M V A R f r o m

s o u r c e Q R a Q R b Q R c C tl ~ 2 C/3 T I F I T D

1 5 . 0 4 . 0 4 . 0 8 . 0 1 2 . 0 8 6 . 2 3 1 . 7 4 5 . 1 2 8 7 . 0 0 . 3 1 1 0 . 1 9 5

2 0 . 0 8 . 0 8 . 0 1 2 . 0 1 6 . 0 5 8 . 1 3 3 . 0 4 3 . 0 1 9 6 . 9 0 . 3 4 7 0 . 1 9 7

2 5 . 0 1 0 . 0 1 0 . 0 1 4 . 0 1 8 . 0 5 6 . 0 3 5 . 6 4 4 . 1 2 8 8 . 5 0 . 7 0 2 0 . 1 9 6

3 0 . 0 2 2 . 0 2 2 . 0 2 6 . 0 3 0 . 0 4 2 . 5 2 9 . 6 3 5 . 6 2 8 2 . 1 0 . 8 3 3 0 . 1 8 9

4 0 . 0 3 1 . 0 3 1 . 0 3 4 . 0 3 8 . 0 4 0 . 1 3 0 . 5 3 5 . 1 2 5 5 . 8 1 . 0 5 4 0 . 1 7 1

5 0 . 0 3 9 . 0 3 9 . 0 4 3 . 0 4 7 . 0 3 9 . 2 3 1 . 4 3 5 . 2 2 6 2 . 9 1 . 4 2 5 0 . 1 7 9

6 0 . 0 2 7 . 0 2 7 . 0 3 1 . 0 3 5 . 0 4 9 . 5 4 0 . 9 4 5 . 0 2 8 5 . 5 1 . 4 2 8 0 . 2 2 7

m i n i m i z e t h e e f f e c t o f h a r m o n i c s , t h e T C Rh a s t o b e c o n t r o l l e d t o a b s o r b m o r e r e a c t iv e

p o w e r , w h i c h h a s t o b e d r a w n f r o m t h e A C

s y s t e m . H e n c e , t h e T C R h a s t o b e p r o p e r l y

r a te d s o a s t o d e m a n d m i n i m u m V A R s f r o m

t h e s o u r c e t h r o u g h o u t i ts c o n t r o l r a n g e. T h i s

i m p l i e s t h a t t h e s i z e o f S V C i s t o b e b a s e d

u p o n t h e l o a d r e q u i r e m e n t s .

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

A n a l g o r it h m f o r e v a lu a t in g t h e o p t i m u m

c o m b i n a t i o n o f p h a s e - w i se r e a c ti v e p o w e r

g e n e r a t i o n s / a b s o r p t i o n s f r o m s t a ti c V A R

c o m p e n s a t o r s t o m e e t u n b a l a n c e d re a c t iv e

p o w e r l o a d d e m a n d s b a s e d o n t h e p e r fo r -

m a n c e i n d ic e s T I F , I T a n d D h a s b e e n p r e -

s e n t e d . T h e m o d e l d e v e l o p e d p r o v i d e s t h e

c o m p e n s a t o r r e q u i re m e n t s in t er m s o f th e

m e a s u r a b l e q u a n t i t ie s o f l o a d r e a c ti v e p o w e r

d e m a n d s . T h e a p p r o a c h r e s u lt s in m i n im i z a -

t i o n o f t h e e f f e c t o f h a r m o n i c s i n j e c te d b y

t h e u n s y m m e t r ic a l o p e r a t i o n o f t h e c o m -

p e n s a t o r i n t o t h e A C s y s t e m a n d t h e r e b y

r e d u c e s t h e b u r d e n o n t h e e x t e r n a l h a r m o n i c

f il te r s. A c o m p u t e r p r o g r a m b a s e d o n t h e

p r o p o s e d a l g o r it h m h a s b e e n d e v e l o p e d a n d

i m p l e m e n t e d i n a f e w ty p i c a l d i s tr i b u t i o n

s y s t e m s . T h e r e su l ts o b t a i n e d f o r o n e t y p i c a l

s y s t e m f o r d i f f e r e n t l o a d c o n d i t i o n s a n d

S V C r a ti n g s h a v e b e e n p r e s e n t e d . T h e p r o -

p o s e d a l g o r i t h m h a s t h e a d v a n t a g e t h a t i t

c a n b e i m p l e m e n t e d u s i n g m i c r o p r o c e s s o r -

b a s e d h a r d w a r e .

N O M E N C L A T U R E

[ E L ] a b c = [ E L a, E L b , E L c ] t

= [ V a / S a , V b / S b , V c / _ ~ ] t , p h a s e -

w i s e c o m p l e x v o l t a g es a t l o a d b u s

[ E S ] a bc = [ E S a , E S b , E S c ] t , p h a s e - w i s e c o m -

p l e x v o l t a g e s a t s o u r c e b u s

[ Q S ] a bc = [ Q S ~ , Q S b , Q S c ] t , p h a s e - w i s e

r e a c t i v e p o w e r s u p p l i e d b y s o u r c e

[ Q C ] ab c = [ Q C a , Q C b , Q C c ] t , p h a s e - w i s e

r e a c ti v e p o w e r s u p p l i e d b y F C -

T S C

[ Q R ] a b e = [ Q R a , Q R b , Q R c ] t , p h a s e - w i s e

r e ac ti v e p o w e r a b s o r b e d b y T C R

[ Q L ] a b c = [ Q L a , Q L b , Q L c ] t , p h a s e - w i s e

r e ac t i v e p o w e r l o a d d e m a n d

[ Z S ] ab~ 3 × 3 m a t r i x r e p r e s e n t i n g s o u r c e

i m p e d a n c e

h h a r m o n i c o r d e r

I t R M S v a lu e s o f f u n d a m e n t a l l in e

c u r r e n t s

Ih R M S v a l u es o f h a r m o n i c l in e cu r -

r e n t s

L i n d u c t a n c e o f e a c h r e a c t o r , HV m m a x i m u m v a l u e o f l in e - to - l in e v o l t-

a g e

Wh w e i g h t in g f a c t o r f o r h a r m o n i c o f

o r d e r h

c~ f u n d a m e n t a l f r e q u e n c y , r a d s 1

R E F E R E N C E S

1 H . F r a n k a n d B . L a n d s t r o m , P o w e r f a c t o r c o r re c -

t i o n w i t h t h y r i s t o r c o n t r o l l e d c a p a c i t o r s , A S E A

J . , 4 4 ( 6 ) ( 1 9 7 1 ) 1 8 0 - 1 8 4 .



2 L . O . B a r t h o l d et al. ( C I G R E W o r k i n g G r o u p

3 1 - 0 1 ) , M o d e l l i n g o f s t a t ic s h u n t V A R s y s t e m s

( S V S ) f o r s y s t e m a n a ly s i s , Electra, 51 ( M a r . )

(1977) 45 - 74.

3 K. Engberg and S. Torseng, Reactors and capaci-

tors controlled by thyristors for optimum power

system VAR control, EPRI Seminar on Transmis-

sion Stat ic VA R Systems, Duluth, MN, Oct . 24 -25, 1978.

4 A. E. Hammed and R. M. Mathur, A new gen-

eralized concept for the design of thyristor

137

p h a s e - co n t r o l le d V A R c o m p e n s a t o r s . P a r t I :

S t e a d y s t a t e p e r f o r m a n c e , P a r t II : T r a n s i en t p e r -

f o r m a n c e , I E E E T r an s. , A S - 9 8 ( 1 9 7 9 ) 2 1 9 - 2 3 1 .

L . G y u g y i , R . A . O t t o a n d T . H . P u t m a n , P r i n c i -

p l e s a n d a p p l i ca t i o n s o f s t a ti c , t h y r i s to r c o n -

t r o ll e d s h u n t c o m p e n s a t o r s , I E E E T r an s. , P A S -

9 7 ( S e p t . /O c t . ) ( 1 9 7 8 ) 1 9 3 5 - 1 9 4 5 .

L . G y u g y i a n d E . R . T a y l o r , J r ,, C h a ra c t e ri s t i co f s t at i c, t h y r i st o r c o n t r o ll e d s h u n t c o m p e n s a -

t o r s f o r p o w e r t r a n s m is s i o n s y s t e m a p p l i c at i o ns ,

IEEE Trans., PAS-99 (1980 ) 1795 - 1804.

an algorithm for optimum control of static var compensators to meet phase-wise unbalanced reactive...

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