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