UDC 6 2 1 . 3 . 0 1 6 . 2 5
E l e c t r i c a l Engineering i n Japan, Vol. 103, No. 4 , 1983
Translated from Denki Cakkai Ronbunshi, Vol. 103B, No. 7, July 1983, pp. 483-490
Generalized Theory of Instantaneous Reactive Power and Its
Application
HIROFUMI AKAGI, YOSHIHIRA KANAZAWA, KOETSU FUJITA and AKIRA NABAE
Technological U n i v e r s i t y of Nagaoka
1.
Introduction
r e a c t i v e power can b e compensated f u l l y by
t h e i n s t a n t a n e o u s r e a c t i v e power compensator,
which c o n s i s t s of s w i t c h i n g elements o n l y and
c o n t a i n s no energy s t o r a g e d e v i c e .
With t h e r e c e n t development of semiconducting reactive power compensation equipment [ 1, 21 , new t e r m i n o l o g i e s such as "ins t a n t a n e o u s r e a c t i v e power" t e n d t o be used
more f r e q u e n t l y [ 3 , 41. To c l a r i f y t h e conc e p t and t h e p h y s i c a l meaning of " i n s t a n t a n e ous r e a c t i v e power" i s very important from
p r a c t i c a l viewpoints. The c o n v e n t i o n a l conc e p t of " r e a c t i v e power" i s based on t h e t i m e
average concept.
2.
I n s t a n t a n e o u s Imaginary Power and
I n s t a n t a n e o u s R e a c t i v e Power
I n t h i s s e c t i o n , we u s e t h e i n s t a n t a n e o u s v o l t a g e and t h e i n s t a n t a n e o u s c u r r e n t
v e c t o r s t o d e f i n e t h e i n s t a n t a n e o u s imaginary
power o f a three-phase c i r c u i t n o t c o n t a i n ing t h c zero-phase-sequence c i r c u i t . Next,
w e d e f i n e t h e i n s t a n t a n e o u s r e a c t i v e power
and c l a r i f y i t s p h y s i c a l meaning.
I n t h i s p a p e r , w e i n t r o d u c e a new conc e p t c a l l e d ''instantaneous imaginary power"
t o t h e three-phase c i r c u i t . This concept
i s a b s o l u t e l y f r e e from t h e energy concept
but i t can be measured i n p r a c t i c e and i t
i s , i n some r e s p e c t , similar t o t h e i n s t a n taneous real power which can a l s o be measured i n p r a c t i c e .
2.1
D e f i n i t i o n of i n s t a n t a n e o u s
imaginary power
Decomposing an i n s t a n t a n e o u s t h r e e phase v o l t a g e ea, e b , ec and i n s t a n t a n e o u s
b o t h n o t conthree-phase c u r r e n t i a , i b , icy
t a i n i n g t h e zero-phase-sequence component,
i n t o a- and 8-components, we o b t a i n
Next, we d e f i n e t h e i n s t a n t a n e o u s rea c t i v e power of each phase u s i n g t h e i n s t a n taneous v o l t a g e and t h e i n s t a n t a n e o u s c u r r e n t . The i n s t a n t a n e o u s r e a c t i v e power can
be defined uniquely whether t h e v o l t a g e o r
t h e c u r r e n t v a r i e s p e r i o d i c a l l y , aperiodica l l y , randomly, o r t e m p o r a r i l y .
U n t i l now, i t h a s been thought t h a t a
s t a t i c v a r compensator n o t c o n t a i n i n g a n
energy s t o r a g e d e v i c e [ 5 - 81 i s a b l e t o
compensate only t h e fundamental component rea c t i v e power.
V o l t a g e s ea, eg and c u r r e n t s i,, i B can be
r e p r e s e n t e d by mutually p e r p e n d i c u l a r i n s t a n taneous v e c t o r s . Vectors B and i are r o t a t i n g i n s t a n t a n e o u s v e c t o r s and are expressed
I n t h i s paper, we demonstrate theor e t i c a l l y t h a t t h e i n s t a n t a n e o u s harmonic
c u r r e n t generated by a semiconducting power
c o n v e r t e r can be decomposed i n t o two components-one
depending upon t h e i n s t a n t a n e o u s
r e a l power, and t h e o t h e r depending upon t h e
i n s t a n t a n e o u s imaginary power. A l s o , w e
show e x p e r i m e n t a l l y t h a t t h e i n s t a n t a n e o u s
imaginary power produced by t h e harmonic
c u r r e n t s and t h e fundamental component
as
e=e,
i=i,
+eg
+ig
(3)
(4)
I n s t a n t a n e o u s power p of a three-phase c i r c u i t i s r e p r e s e n t e d g e n e r a l l y by
58
~SS~0424-7~60/8~/000~-0058$7.50/0
0 1984 S c r i p t a P u b l i s h i n g Co.
2.2
Fig. 1.
P h y s i c a l meaning of i n s t a n t a n e o u s
imaginary power and i n s t a n t a n e o u s
reactive power
Now l e t u s c o n s i d e r t h e p h y s i c a l meani n g of i n s t a n t a n e o u s imaginary power q.
Although i n s t a n t a n e o u s r e a l power p e x p r e s s e s
t h e flow rate of r e a l energy, q does n o t
r e p r e s e n t t h e flow r a t e of r e a l energy. For
t h i s r e a s o n , we c a l l q " i n s t a n t a n e o u s imagi n a r y power
."
I n s t a n t a n e o u s v o l t a g e v e c t o r s and
instantaneous c u r r e n t v e c t o r s .
Needless t o s a y , q cannot b e expressed
i n t h e c o n v e n t i o n a l u n i t s such as W , VA and
Var. Reexpressing Eq. ( l o ) , we o b t a i n
(5)
p = e n ia+ebis+ecic.
On t h e a-8 c o o r d i n a t e s , p i s e x p r e s s e d a s
p = e , . L + e @*
ip
where
(6)
= e mi , feg i p
phase-,
We d e f i n e t h e new concept of " i n s t a n t a n e o u s
imaginary power v e c t o r q" by t h e d i f f e r e n c e
between v e c t o r p r o d u c t s e, x ig and eg y ia
o r by
i n s t a n t a n e o u s real c u r r e n t :
ea
l
a
P
=
m
P
phase-a i n s t a n t a n e o u s r e a c t i v e c u r r e n t :
-
i a ea2+
q = e,2
LQ
q=e, x i p -ep x i ,
(7)
instantaneous real current:
phase-8
i n s t a n t a n e o u s reactive c u r r e n t :
ipp=& em2+esz P
The magnitude of i n s t a n t a n e o u s imaginary
power v e c t o r q i s g i v e n by
q=eaip-epia
phase-g
1
ea
i R s = l q
(8)
ea
i-es
The f i r s t t e r m of Eq. (12) e x p r e s s e s t h e i n s t a n t a n e o u s real power component and t h e
second t e r m e x p r e s s e s t h e i n s t a n t a n e o u s imagi n a r y power component. These i n s t a n t a n e o u s
c u r r e n t components can be determined uniquely
from p and q i f e, and eg are known.*
Equations (6) and (8) g i v e
(9)
A s shown i n Eq. (9), p i s d e f i n e d i n a mann e r q u i t e similar t o q. We c a l l p " i n s t a n taneous real power. t '
L e t t i n g pa and p @ d e n o t e t h e phase-,
and phase-@ i n s t a n t a n e o u s powers, respect i v e l y , we o b t a i n
To c o n s i d e r t h e s i g n i f i c a n c e of inst-antaneous imaginary power q , l e t u s n o t i c e
t h a t it i s impossible t o determine i, and i g
uniquely using e,, eg and p o n l y . However,
i f n o t only p b u t a l s o q i s known, t h e n i,
and i g can be determined u n i q u e l y as f o l l o w s :
(14)
F u r t h e r , t h e i n s t a n t a n e o u s r e a l power p of
t h e three-phase c i r c u i t is g i v e n by
F u r t h e r , Eq. ( 9 ) g i v e s
Equation (9) i s
c u i t comprising
and Eq. (10) i s
c u i t comprising
suited for a
a three-phase
suited for a
a three-phase
three-phase c i r v o l t a g e source,
three-phase c i r c u r r e n t source.
*If i, and i g are known o r i f t h e c u r r e n t s o u r c e i s g i v e n , e, and eg can b e determined uniquely from p and q .
59
2.3
It i s important t o n o t e h e r e t h a t t h e sum
of t h e t h i r d and f o u r t h terms of Eq. (15)
v a n i s h always. T h e r e f o r e w e o b t a i n
p = e a iap+eg i s p
2 hZP+P#P
O= eb i , g + eg is4
PPaq+
R e l a t i o n between i n s t a n t a n e o u s reactive power and c o n v e n t i o n a l reactive power
The r e a c t i v e power of a s i n g l e - p h a s e
c i r c u i t w i t h s i n u s o i d a l v o l t a g e and c u r r e n t
i s d e f i n e d by t h e a m p l i t u d e of i n s t a n t a n e o u s
power v a r y i n g a t frequency twice a s h i g h a s
t h e power s o u r c e frequency. The a v e r a g e of
t h e p u l s a t i n g component of i n s t a n t a n e o u s
power i s e q u a l t o z e r o . The r e a c t i v e power
of a three-phase c i r c u i t w i t h s i n u s o i d a l
v o l t a g e and c u r r e n t i s g i v e n by t h e a l g e b r a i c
sum of r e a c t i v e powers of r e s p e c t i v e p h a s e s .
Pgo
where
phase a i n s t a n t a n e o u s real power:
1
eaz
Pap=- emz+e 2
phase a i n s t a n t a n e o u s r e a c t i v e power:
p a q = x
The c o n v e n t i o n a l c o n c e p t of t h e react i v e power produced by n o n s i n u s o i d a l v o l t a g e
and c u r r e n t i s e s t a b l i s h e d by a p p l y i n g t h e
r e a c t i v e power c o n c e p t f o r s i n u s o i d a l waves
t o the F o u r i e r series of n o n s i n u s o i d a l waves.
T h e r e f o r e i t cannot be a p p l i e d t o a p e r i o d i c
waves o r randomly v a r y i n g waves.
e,2+egZq
phase $ i n s t a n t a n e o u s r e a l power:
b
Below, we d i s c u s s t h e r e l a t i o n between
i n s t a n t a n e o u s r e a c t i v e power and c o n v e n t i o n a l
r e a c t i v e power.
esz
Pep=phase $ i n s t a n t a n e o u s r e a c t i v e power:
According t o Eq. (15), i n s t a n t a n e o u s
real power p of a three-phase c i r c u i t is
g i v e n by
Each term of Eq. ( 2 7 ) r e p r e s e n t s t h e i n s t a n taneous power produced by t h e i n s t a n t a n e o u s
r e a c t i v e c u r r e n t of phase a o r phase B.
The f a c t t h a t t h e sum of t h e s e two terms
always v a n i s h e s s u g g e s t s t h a t t h e r e exists
an i n s t a n t a n e o u s power c i r c u l a t i n g through
phases a and 6. I n t h i s p a p e r , w e r e f e r t o
t h i s k i n d of c i r c u l a t i n g i n s t a n t a n e o u s power
as " i n s t a n t a n e o u s r e a c t i v e power" and t h e
i n s t a n t a n e o u s power produced by t h e i n s t a n taneous r e a l c u r r e n t a s " i n s t a n t a n e o u s r e a l
power." As shown i n Eq. (18), t h e s e powers
can b e determined u n i q u e l y from i n s t a n t a n e ous real power and i n s t a n t a n e o u s imaginary
power.
P= P.P+PBP+
paq+psq
(19)
Denoting t h e power s o u r c e p e r i o d by T , we
obtain
A s shown i n Eq. (17), t h e second term of Eq.
(20) v a n i s h e s always and, from symmetry of
t h e c i r c u i t , w e o b t a i n t h e f o l l o w i n g equat i o n s f o r t h e b a l a n c e d three-phase c i r c u i t :
I n p a r t i c u l a r , t h e i n s t a n t a n e o u s imagi n a r y power i s an important e l e c t r i c a l quant i t y determining t h e i n s t a n t a n e o u s power
c i r c u l a t i n g through a- and B-phases o r t h e
i n s t a n t a n e o u s r e a c t i v e power*.
*Both t h e i n s t a n t a n e o u s r e a l and react i v e powers r e p r e s e n t electrical q u a n t i t i e s
d e f i n e d f o r each phase of t h e three-phase
c i r c u i t , whereas b o t h t h e i n s t a n t a n e o u s r e a l
and imaginary powers r e p r e s e n t e l e c t r i c a l
q u a n t i t i e s d e f i n e d f o r t h e t h r e e phases c o l l e ct i v e l y
From t h e above e q u a t i o n s , we l e a r n t h a t
phase-a i n s t a n t a n e o u s real power pap and
phase-a i n s t a n t a n e o u s r e a c t i v e power paq
c o i n c i d e s w i t h c o n v e n t i o n a l r e a l power and
r e a c t i v e power f o r s i n u s o i d a l waves. More
.
60
precisely, the phase-a instantaneous reactive
power of the balanced three-phase circuit
represents the instantaneous power circulating through phases a and 8. Also, it represents a phase-a instantaneous power whose
one-cycle average vanishes.
I
Instantaneous real power p and instantaneous imaginary power q of the balanced
three-phase sinusoidal ac circuit become
constant. Further, p agrees with the conventional real power and q takes the same
value as the conventional reactive power.
However, it should be noticed that the instantaneous imaginary power has a completely
different physical meaning from the conventional reactive power.
2.4
I
i , P
Fig. 2.
-pl
The instantaneous power flow.
Instantaneous power flow of threephase circuit
Instantaneous power po produced by zerosequence components eo, 10, instantaneous
real power p and instantaneous imaginary
power q are defined by
Figure 2 represents the instantaneous
power flow. The circuit between power converter and load i s decomposed into a'- and
B'-circuits which are different from a- and
B-circuits on the power source side. Generally speaking, paq i s not equal to paq'.*
Similarly,
Qf4'
which gives
(23)
However, for a power converter which consists
of switching elements only and contains no
energy storage device, we obtain
P = P'
(24)
Summarizing the above discussion, we can
state the following:
"Instantaneous imaginary power q defined for a three-phase circuit makes it possible to define the instantaneous reactive
power clearly. Instantaneous imaginary power
q also represents an instantaneous power circulating through phases. "
where
0
The above-stated concept of instantaneous
imaginary power is not applicable to a
single-phase circuit because the instantaneous imaginary power cannot be defined
uniquely i n the single-phase circuit.
0
0 - e g e,
Transforming 10, i,, ig in Eq. (28) to threephase currents fa, ib, icy we obtain
3. Instantaneous Reactive Power of a ThreePhase Circuit with Zero-PhaseSequence Circuit
l/VT -1/2
l/VT -1/2
v3/2
-1/3/2
If zero-phase-sequence components
exist, then we obtain
*For instance, it i s possible to let
paq' of a polyphase forced commutation-type
cycloconverter be equal to zero independently
of Paq.
Instantaneous Instantaneous Instantaneous
zero-sequence ireal current reactive curcurrent
rent
61
(29)
where
-
i.o=im=i,o=io/V 3
ljV2 I/VZ l / V T
1
0
-1/2
-1/2
>'S/2 - v 3 / 2
1
Phase-a, phase-b and phase-c i n s t a n t a n e o u s
powers Pa, pb, pc are d e r i v e d from Eq. ( 2 9 )
a s follows:
Reactive pwr.
compensator
Pc
Fig. 3 .
Instancaneous Instantaneous Instancanems
zero-sequence real power reactive power
power
ed
,i
ie8
iLS
I ZC
The r e a c t i v e power compensator.
(30)
T h e r e f o r e t h e t o t a l power f a c t o r of t h e
power s o u r c e i s improved b u t t h e fundamental
component power f a c t o r cannot b e improved
because ec = 0.
From Eq. (30), w e o b t a i n
( i i ) I n s t a n t a n e o u s r e a c t i v e power
compensator
4.
This compensates t h e i n s t a n t a n e o u s reactive power produced by t h e i n s t a n t a n e o u s
, obtain
imaginary power. S i n c e p s = p ~ we
C l a s s i f i c a t i o n of R e a c t i v e Power
Compensator and I t s C o n t r o l
F i g u r e 3 shows t h e p r i n c i p l e of a rea c t i v e power compensator.* I n F i g . 3 , p s
and q s a r e i n s t a n t a n e o u s real power and i n s t a n t a n e o u s r e a c t i v e power, r e s p e c t i v e l y ,
on t h e power s o u r c e s i d e , and p~ and qL a r e
t h o s e on t h e load s i d e . Denoting t h e i n s t a n t a n e o u s real power and t h e i n s t a n t a n e o u s
imaginary power of t h e r e a c t i v e power comp e n s a t o r by pc and qc, r e s p e c t i v e l y , w e obtain
p c = 0 from Eq. ( 3 2 ) and t h e r e f o r e t h e r e a c t i v e power compensator does n o t r e q u i r e a n
energy s t o r a g e d e v i c e . T h e r e f o r e t h e comp e n s a t o r c u r r e n t i s given by
(34)
G e n e r a l l y s p e a k i n g , qs = 0 and t h e fundamental component power f a c t o r of power
s o u r c e becomes e q u a l t o u n i t y .
(32)
(iii) I n s t a n t a n e o u s r e a l and r e a c t i v e
power compensator
According t o t h e t h e o r y developed i n
preceding s e c t i o n s , t h e r e a c t i v e power comp e n s a t o r can be c l a s s i f i e d as f o l l o w s .
(i)
T h i s s u p p r e s s e s t h e v a r i a t i o n of r e a l
power and compensates t h e i n s t a n t a n e o u s rea c t i v e power produced by t h e i n s t a n t a n e o u s
imaginary power. The compensator r e q u i r e s
an e n e r g y s t o r a g e d e v i c e and, s i n c e pc = p s
- p~ and q c = q s qL, we o b t a i n
I n s t a n t a n e o u s real power compen-
sator
-
T h i s s u p p r e s s e s t h e v a r i a t i o n of i n s t a n t a n e o u s r e a l power and t h e compensator
r e q u i r e s a n energy s t o r a g e d e v i c e . It i s
c o n t r o l l e d s o t h a t p c = p s = p~ and q c = 0.
T h e r e f o r e compensating c u r r e n t s iccl
and i c B
a r e g i v e n by
[1c.]=[
icp
- eu
ep ~
e ,o ] - qs-qr.
'~s-PL]
(35)
When q s = 0 , t h e power s o u r c e c u r r e n t becomes a s i n u s o i d a l wave w i t h u n i t y power
factor,
Next, l e t u s a n a l y z e t h e i n s t a n t a n e o u s
c u r r e n t decomposing t h e i n s t a n t a n e o u s r e a l
and imaginary powers i n t o dc and a c components.
*"Reactive power" i n c l u d e s n o t o n l y
t h e fundamental component r e a c t i v e power,
b u t a l s o t h e harmonic, subharmonic and
zero-sequence r e a c t i v e powers.
62
Assuming t h a t t h e power s o u r c e v o l t a g e
i s a balanced three-phase s i n u s o i d a l v o l t age, l e t u s decompose i n s t a f f t a n e o u s r e a l
power PI, i n t o dc component p~ and ac compon e n t PL. S i m i l a r l y , we decompose i n s t a n taneous imaginary power qL t o dc component
and ac component &. Then i n s t a n t a n e o u s
phase-a load c u r r e n t iIa i s d e r i v e d from
Eq. (12) a s f o l l o w s :
4~
Fundamental
real current
U
Harmonic real
current
Instantaneous real current
produced by p~
(36)
d =O"
Fig, 4 .
Fundamental Harmonic
reactive cur- reactive
rent
current
Instantaneous reactive current
produced by qL
Equation ( 3 6 ) i n d i c a t e s t h a t i t i s p o s s i b l e
t o decompose t h e i n s t a n t a n e o u s harmonic curr e n t t o two ac components-one
produced by
p ~ and
,
t h e o t h e r produced by qL.
Simulation r e s u l t s .
Fundamental
The i n s t a n t a n e o u s real power compensat o r s t a t e d i n i t e m ( i ) m a i n t a i n s p s = PI, and
compensates t h e second term of Eq. (36).
I f p~ i s t h e average r e a l power over a v e r y
long t i m e p e r i o d , t h e n t h e i n s t a n t a n e o u s r e a l
power p s s u p p l i e d from t h e power s o u r c e i s
kept c o n s t a n t and i t is t r a n s m i t t e d i n a n
However, such a compensator
i d e a l way [ 9 ] .
would r e q u i r e a l a r g e - c a p a c i t y energy s t o r age d e v i c e .
r
Control angle
F i g . 5.
The i n s t a n t a n e o u s r e a c t i v e power comp e n s a t o r s t a t e d i n i t e m ( i i ) compensates
t h e t h i r d and t h e f o u r t h terms. I f qc = -qL,
then i t c o i n c i d e s w i t h t h e c o n v e n t i o n a l
fundamental component r e a c t i v e power compens a t o r . S p e c i a l a t t e n t i o n should be p a i d t o
t h e f a c t t h a t t h e compensator does n o t req u i r e an energy s t o r a g e d e v i c e i f t h e ins t a n t a n e o u s r e a c t i v e c u r r e n t which i s produced by t h e t h i r d and t h e f o u r t h terms,
i s compensated.
5.1
Harmonic a n a l y s i s of i S a .
Simulation r e s u l t s
Simulation r e s u l t s a r e i l l u s t r a t e d i n
F i g . 4 , where t h e i n s t a n t a n e o u s r e a c t i v e
power compensator i s assumed t o o p e r a t e i n
an i d e a l way and t h e dc o u t p u t c u r r e n t of
t h e r e c t i f i e r i s assumed t o b e c o n s t a n t .
¶
Since t h e second term of Eq. ( 3 6 ) i s
n o t compensated, t h e waveform of phase-a
power s o u r c e c u r r e n t i S a is n o t s i n u s o i d a l ,
b u t i t s fundamental component power f a c t o r
i s k e p t a t u n i t y independently of c o n t r o l
angle a.
The i n s t a n t a n e o u s real and r e a c t i v e
power compensator s t a t e d i n i t e m ( i i i ) comp e n s a t e s a l l t h e c u r r e n t components appeari n g i n t h e second t o f o u r t h terms of Eq.
(36). The i n t e r f e r e n c e c u r r e n t compensator
i n [ l o ] i s r e a l i z e d u s i n g a s e p a r a t e l y exc i t e d t h y r i s t o r c o n v e r t e r , a r e a c t o r and a
condenser.
5.
d =30"
F i g u r e 5 shows t h e c a l c u l a t e d harmonic
amplitude c h a r a c t e r i s t i c s of i s a . When M
= O o , t h e 5 t h and 7 t h harmonic components
have an amplitude e q u a l t o 2.86% of t h e
fundamental amplitude and t h e 11th and 13th
harmonic components have a n amplitude e q u a l
t o 0.7% of t h e fundamental amplitude. A s
shown i n F i g . 5, t h e compensation e f f e c t i s
Experimental R e s u l t s
E f f e c t i v e n e s s of t h e proposed compens a t o r s is examined by d i g i t a l s i m u l a t i o n and
experiments.
63
t h e g r e a t e s t when a = 0'.
T h i s is because
t h e second term of Eq. (36) d e c r e a s e s and
t h e f o u r t h term i n c r e a s e s as a d e c r e a s e s .
s w i t c h i n g . They can b e n e g l e c t e d i f t h e
s w i t c h i n g frequency is i n f i n i t e l y h i g h .
5.3
5.2
Experimental r e s u l t s
Control c i r c u i t
Response waves t o sudden l o a d change
are shown i n F i g . 8. Waveforms of ea and i S a
show t h a t l t h e fundamental component power
f a c t o r on t h e power s o u r c e s i d e is e q u a l t o
u n i t y even i n t h e t r a n s i e n t s t a t e . Harmonic
c u r r e n t s are a l s o w e l l compensated. Spikeshaped n o t c h e s of i S a are caused by t h e comp e n s a t i n g c u r r e n t which cannot change instantaneously following t h e r e f e r e n c e c u rre n t
The c o n t r o l c i r c u i t of the i n s t a n t a n e ous r e a c t i v e power compensator is shown i n
Fig. 6 . It c o n s i s t s of an analog c i r c u i t
comprising 8 m u l t i p l i e r s and d i v i d e r s , which
c a l c u l a t e s q~ i n Eq. (8) and icas la, ice
i n Eq. ( 3 4 ) . The proposed compensator is
d i f f e r e n t e s s e n t i a l l y from t h e c o n v e n t i o n a l
one i n t h a t t h e compensating c u r r e n t can b e
determined without time d e l a y from i L a , ILb,
i L c and ea, eb, ec.
The main c i r c u i t of t h e i n s t a n t a n e o u s
r e a c t i v e power compensator is shown i n Fig.
7. The l o a d c o n s i s t s a l s o of a three-phase
t h y r i s t o r r e c t i f i e r . The main c i r c u i t i s a
voltage-source t y p e PWM c o n v e r t e r which cons i s t s of high-speed t r a n s i s t o r s , c i r c u l a t i n g
d i o d e s , f i l t e r condensers C (0.5 uF), react o r L ( 2 . 4 mH), and smoothing condenser cd
( 5 PF).
Elements L, C and Cd are n o t used f o r
energy s t o r a g e b u t are used f o r t r a n s i s t o r
ZC0
'Cb
ZCC
Fig. 6 .
Control c i r c u i t of t h e i n s t a n t a n e o u s
reactive power compensator.
Power source
InstantaIleous
reactive power!
Compensator
i
I
Fig. 7.
Main c i r c u i t of t h e i n s t a n t a n e o u s r e a c t i v e
power compensator.
64
elements L, C, cd and subharmonic compensat i o n of c y c l o c o n v e r t e r w i l l b e d i s c u s s e d on
another occasion.
Acknowledgement
The a u t h o r s thank members of Research
Laboratory of Kansai E l e c t r i c Power Co. f o r
t h e i r c o o p e r a t i o n . The work w a s c a r r i e d o u t
under t h e f i n a n c i a l s u p p o r t of t h e M i n i s t r y
of Education.
APPENDIX
I n s t a n t a n e o u s Imaginary _>wer of
Single-Phase C i r c u i t
Ordinate : iA/div
Abscissa. 5 0 H q d l v
F i g . 9.
Denoting t h e i n s t a n t a n e o u s v o l t a g e and
t h e i n s t a n t a n e o u s c u r r e n t of a single-phase
c i r c u i t by e g and i g , r e s p e c t i v e l y , w e l e t
i, = 0 i n Eq. ( 9 ) t o o b t a i n
Harmonic spectrum of i S a and i L a .
due t o t h e e x i s t e n c e of f i l t e r inductance
L . The maximum s w i t c h i n g frequency of t h e
PWM c o n v e r t e r is a b o u t 30 kHz and t h e t o t a l
l o s s of t h e compensator i s about 20 W when
a = 0" and t h e o u t p u t c u r r e n t i s e q u a l t o
10 A.
p=eoip
Equations (Al) and (A2) may see? t o s u g g e s t
t h e p o s s i b i l i t y of d e f i n i n g p and q. How
e v e r , a l t h o u g h eg i s a real i n s t a n t a n e o u s
v o l t a g e , ea i s a h y p o t h e t i c a l i n s t a n t a n e o u s
v o l t a g e i n t h e single-phase c i r c u i t .
Figure 9 shows t h e measured harmonic
s p e c t r a of iSa and iLa i n s t e a d y s t a t e .
Since a = 0, i S a has t h e same fundamental
amplitude a s i L a b u t t h e 5 t h and 7 t h h a r monic amplitudes of i S a are much smaller
than t h o s e of i L a .
I n t h i s r e s p e c t , t h e single-phase circ u i t i s d i f f e r e n t e s s e n t i a l l y from t h e t h r e e phase c i r c u i t . T h e r e f o r e E q s . (Al) and (A2)
are a p p l i c a b l e i f t h e three-phase v o l t a g e
e x i s t s and t h e c u r r e n t of one phase v a n i s h e s .
However, i t i s i m p o s s i b l e t o d e f i n e t h e ins t a n t a n e o u s imaginary power uniquely i f only
t h e single-phase v o l t a g e i s p r e s e n t .
From t h e e x p e r i m e n t a l r e s u l t s shown
above, we o b t a i n t h e f o l l o w i n g conclusions:
"The i n s t a n t a n e o u s r e a c t i v e power compensat o r n o t comprising an energy s t o r a g e d e v i c e
i s a b l e t o compensate t h e fundamental react i v e c u r r e n t [ t h i r d term of Eq. (36)] and
t h e harmonic c u r r e n t ( f o u r t h term) n o t only
i n steady-state condition but also i n transi e n t condition.
REFERENCES
1.
6.
(Al)
Conclusions
2.
I n t h i s p a p e r , we have introduced t h e
concept of " i n s t a n t a n e o u s imaginary power q"
f o r t h e three-phase c i r c u i t . I t s p h y s i c a l
meaning h a s been c l a r i f i e d . The proposed
concept is v a l i d f o r any v o l t a g e and c u r r e n t
waveforms. The proposed concept is u t i l i z e d
f o r constructing an instantaneous r e a c t i v e
power compensator, which d e t e c t s t h e i n s t a n taneous r e a c t i v e power w i t h o u t t i m e d e l a y
and compensates i t .
3.
4.
5.
The e f f e c t i v e n e s s of t h e proposed comp e n s a t o r h a s been demonstrated e x p e r i m e n t a l l y
R e l a t i o n s between s w i t c h i n g frequency and
6.
65
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