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where 0 is the angle hetween the voltage and the current vectors.
Clearly, only that component
of
the current vector which is in phase
w i t h t h e 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 c o n t r i b u t e s t o t h e
instantaneous power. The remaining current component could be
r e m o v e d w i t h o u t c h a n g i n g t h e p o w e r a n d t h i s c o m p o n e n t i s
therefore the instantaneous reactive current. The se observations can
be extended to the following definit ion of instantaneous reactive
power:
Q = f I ~ I
ilsin(0)
( 4 )
where the constant 312 is chosen
so
that the definition coincides with
t h e c l a s s i c a l p h a s o r d e f i n i t i o n u n d e r b a l a n c e d s t e a d y - s t a t e
conditions.
- _
(Vdsiqs vqsids)
Figure 4shows how further manipulation of the vector coordinate
frame leads to a
useful
separation of variables for power control
purposes. A new coordinate system is defined where the d-axis is
always coincident with the instantaneous voltage vector and the q-
axis is in quadrature with i t . The d-axis current component, id .
accounts for the instantaneous power and the q-axis current, i is the
instantaneous reactive current.
The
d,q axes
are
n o t s t a t i o n s i n h e
plane. They follow the trajectory
of
the voltage vector, and the d,q
coordinates within this synchronously-rotating reference frame are
given by the following tim e- vq ing transformation:
Isz
1
I C 1 1
= f
IC11,
and substituting in ( I ) we obtain
Under balanced steady state conditions the coordinates
of
the voltage
and current vectors in the synchronous reference frame are constant
quanti t ies . This feature is useful for analysis and for deco upled
control of the two current components.
Equivalent Circuit and Fqations
Figure
5
show s a simplified repre sentation of the ASVC, including a
dc-side capacitor, an inverter, and series inductance in the three lines
connecting to the transmission line. This inductance accounts for the
leakage of the actual power transformers. The circuit also includes
resis tance in shunt with the capacitor to represent the switching
losses in the inverter, and resistance in series with the ac-lines to
rep resen t the inver te r and t rans fo rmer conduct ion lo s ses. The
inverter block in the circuit is treated as an ideal , lossless powe r
transformer.
In terms of the instantaneous variables shown in Fig.
5,
the ac-side
circuit equations can be written as follows:
7 )
where p.
=
dldt, and a per-unit system has been adopted according to
the following definitions:
O L
L = b B
.
C ' = L
. R r = A
..=A
base b bas e base ase
w c z '
e
i =
;
v =
; e '= ; = base
base
ase
base i
base ase
i
( 8 )
Using the transformation of variables defined in
3,
quations (7)
can be transformed to the synchronously-rotating reference frame as
follows:
where
w =
dQ1dt. Figure
6
illustrates the ac-side circuit vectors in the
synch ronous f rame. When i i s pos i t ive , the ASVC i s d rawing
inductive vars from the line, an% for negative i' it is capacitive.
Types Of Voltage-Sourced Inverter
Neglecting the voltage harmonics produced by the inverter, we can
write a pair of equations for e i and
e '
q
e
= kvLCcos(m) ( 1 0 )
e
= k v; ic s in ( o) ( 1 1 )
where k is a factor for the inverter which relates the dc-side voltage
to the amplitude (peak) of the phase-to-neutral voltage at
the
inverter
ac-side terminals, and 01 is the angle by which the inverter voltage
vector leads the line voltage vector. It is important to distinguish
hetween two basic types of voltage-sourced inverter that can
be used
in ASVC systems.
Inverter Type I allows the instantaneous values of both 01 and k to be
vane d for control purposes. Provided that vi c is kept sufficiently
high,
e i
and
e
can be independently controlled. T his capability can
he achieved b4y various pulse-width-modulation (PW M) techniques
that invariably have a negative impact on the efficiency, harmonic
content,
or
utilization of the inverter. Type I inverters are presently
considered uneconomical for transmission-line applications and their
co nm l will only he briefly considered here.
Inverter T ype I1 is of primary interest for transmission line A SVCs.
In this case, k is a constant factor, and the only available control
input is the angle,
01, of
the inverter voltage vector. This case will be
discussed in greater detail.
Inverter Type I Control System
Inspection of
(9)
leads directly to a rule that will provide decoup led
control of ih and i i . The inverter voltage vector is controlled
as
follows:
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Substitution of (12) and (13) in (9) yields
Equation (14) shows that nd . espond to x1 and x2 respectively
through a simple first order trans & function. with no cross-coupling.
The control rule of
(12)
and
(13)
s thus completed by defining the
feedback loops an d proportional plus integral compensa tion as
follows:
x
- (k +
5 )
( i f * -
i
(15)
1 a p . d d )
The control is thus actually performed using feedback variables in
the synchronous reference frame. The reactive current reference,
v,
s supplied from the ASVC outer-loop voltage control system,
and the real power is regulated by varying iJ
in
response to error in
the dc-link voltage via a proportional plus integral compensation. A
block diagram of the control scheme is presented in Figure 7.
Further Model Develooment For Inverter T y ~ e Control
For Type II nverter control it is necessary to include the inverter and
dc-side circuit equation into the model. The instantaneous power at
the ac- nd dc-terminals of the inverter is equal, giving the following
power balance equation:
v&ci&c=
2
(e?
+
e ' i ' )q
(17)
and the &-side circuit equation is
Combining (9), (10). (11). (17) and (18). we obtain the following
state equations for the
ASVC:
p . [ l = [ A ]
li]-l i'i
- R ' W
ko
dc
o b c o s ( = )
b
L
L'
[ A I
=
Steady state solutions for (19) using typical system parameters
are
plotted in Fig. 8 as a function of a (subscript 0 denotes steady state
values.) Note that ';lovaries almost linearly with respect to
a@
and
t h e r a n g e o f
mo
fo r one per un i t swing in i ' o i s very smal l .
Neglecting losses (i.e.
R;
= 0, R; = m) the ste&y state solutions
would
be
as ollows:
= o b ; i = i
u o
o
; ih0 = o ; v -
4
qo
; I V ' I
=
b
ihOL 20)
dcO - E [
vb
Linearization
of
ASVC Fauations f or Small Perturbations
The AS VC state equations (19)
are
non-linear if a s regarded as an
input variable. We can, however, find useful solutions for small
deviations about a chosen steady state equil ibrium point . The
linearization process yields the following perturbation equations:
Standard frequency domain analysis can
be
used to obtain transfer
functions from (21). Numerical methods have been used to obtain
specific results, but it is useful to first consider some general results,
neglecting the system pow er losses (i.e. R;
=
0,
R; =
-.) For this
case, the block diagram of Fig.
9
shows how the control input,
Aa
influences the system states. The corresponding transfer function
relating
A';l
and Aa is
as
follows:
AI' 5 )
~ - * [ 3 ' L ~ - l v & ~ ~ +
~ c w
i
s I s 2 + o2 + L C 1
( 2 2 )
- b q o
_
Au(s)
ko 3kw
C'
L
=
.
C'.
=
L '
The undam ped poles of the system are hus at
t .
s
= 0
and
s = - ]ob 11 t 3klC
L '
(23)
The transfer function, (22), also has a pair of complex
zeroes
on the
imaginary axis. These move along the imaginary axis as a function
of ';lo.occuring at lower frequency than the poles only when
2 V '
(24)
cO i,
3kC
q O X
ho
A numerical computation of AI$s)/Aa(s) from (21), including the
losses, has been done for two operating points to i l lustrate the
movement of the complex zeroes. Figure 1Oapresents the result
for
each case in a plot
of
log gain and phase vs. frequency.
Case 1: Full Capacitive Load
;lo= -1.01 pa., a0 = -0.011 rad (0.63 )
9 -
')
2893 (3t8.7tj1330) (st8.7-jl330)
- -
A-
5 )
(st23.8) ~t15.4 +j1476) st15.4- 1476)
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Case 2: Full Inductive Load
;lo= 1.07
p a . , 0
=
0.01 rad (0.57 )
' () 2111(~+ 11.4+j 1557) s+11.4-j1557)
- - -
Ao(s) (s+23.8) (s+15.4+j1476) (s+15.4-j1476)
While Case 1 is amenable to feedback control , Case 2 clearly has
little phase margin near the system resonant frequency. The latter
si tuation is typical for the co ndit ions O>
;lox
(
= 0.44
p.u. in this
example.) A controller has been designed to overc ome this problem
by using non-linear state-variable feedback to improve the phase
margin when
' '0
> ox The non-linear feedback function, Aq', has
the following?ormb
qo - ibex 1.Av
25)
q - g.[ i
dc
q, = Ai
where
g
is a gain factor to be set by design. Figure 10b shows the
transfer function, AQ(s)/A a(s), for the same operating points a s Fig.
loa, with g
= 2.0.
The improved phase margin in Case 2 is clearly
seen. The control scheme block diagram is shown in Fig. 11,with
the additional integral compensation required to obtain zero steady
state error in This scheme has been implemented in the ASVC
s c a l e d m o d e w i t h a c l o s e d - l o o p c o n t r o l b a n d w i d t h s e t t o
approximately 20 0 rads. This makes i t possible to swing between
full inductive mode and full capacitive mode in slightly more than a
quarter of a cycle.
y
L I N E V O L T A G E U N B A L A N C E A N D H A R M O N I C
DISTORTION
With balanced sinusoidal line voltage and an inverter pulse-number
of
24 or
greater, the ASVC draws no low-order harmonic currents
from the l ine. However, harmonic currents of low order d o occur
when th e l ine vo l t age i s unba lanced
or
d is tor t ed . As migh t be
e x p e c t e d f r o m a n o n - li n e a r l o a d , t h e A S V C c u r r e n t s i n c l u d e
harmon ics no t p resen t in the l ine vo l t age . I t
is
impor tan t to
understand ASV C behaviour under these condit ions s ince i t can
influence equipment rating and component selection.
The ASV C harmonic currents can be calculated by postulating a set
of harmonic voltage sources in series with the ASVC tie l ines as
shown in Fig. 12. If we further neglect losses (Le. RI =
0,
R;1
=
m)
and as sume the s t eady s t a te cond i t ion , 01 =
0
and w = wb. then
equations (19) are modified as follows:
( 2 6 1
wher e vid . v i are the d ,q components of the harmonic voltage
vector. E qua tsns (26) are inear and can be solved using Laplace
transforms. Consider the effect of a single balanced harmonic set of
order , n, whe re negative values of n denote negative sequence. The
associated harmonic voltage vector has magnitude, v; and rotates
with angular velocity nob. In the synchronous reference frame i t
rotates with angular velocity (n-l)% as shown in
Fig. 13
and
vhq=
v;
sin((n-1)o
t)
2 7 )
These sinusoidal inputs on the d- and q-axes give rise to sine wave
responses ihd. i
,
and vidc of frequency (n-l)wb. Generally i i d
and ii do not form a balanced two -phase sinusoidal set. They can
be re A ve d into a positive sequence set and a negative seque nce set
using normal two-phase phasor symmetrical components. We thus
find
two dist inct current component vectors in response to the n-
order harmonic voltage vector. Within the synchronous reference
9
frame, these rotate with frequency (n-l)wb and (l-n)wb respectively.
The corresponding ASVC line currents have frequency nub and
(2-n)wb respectively. Note also that the inverter develops an
a l t e rna t ing vo l t age componen t o f f requency (n - l )o b a t i t s dc
terminals.
Equat ions (26 ) and (27 ) have been so lved to ob ta in a lgeb ra ic
expressions for the magnitudes of these harmonic currents in the
particular case where n = -1 (Le., fundamental negative sequence
voltage.) In this case the ordinals of the harmonic currents are 1 and
3 and the magnitudes
are
calculated from the following:
V'
lijl
=
4L
l
-
2LI
kzC
( 2 8 1
( 2 9 1
These expressions have be en evaluated using typical parameters with
v i
= 1
PA., and are plotted ag ainst per-unit capacitive reactance in
Fig. 14. Notice that for
C' =
2L'/k2 both
iL1
and
i j
become infinite.
This condition occurs if the second harmonic of the line frequency is
equal to the ASVC-resonant-pole frequency defined in equation
(23).
Also when C'= 8L'/k2, ill is
zero
and the ASVC draws no negative
sequence fundamental current from the line.
EXPERIMENTAL RESULTS FROM ASVC SCALED MODEL
It is beyond the scope
of
this paper to discuss the
EPN
ASVC scaled
model in detail. However, two Sets of measured waveforms from the
model are presented in Figs. 1 5 and 1 6 to i l lustrate th e system
behaviour under transient conditions. Figure
15
shows the dynamic
response of the instantaneous reactive current controller. In this case
a square-wave reference, ;1*, is injected, and the oscillogram shows
i i , a nd the ASVC l ine cu rren t s . F igu re 16 shows the ASVC
response to a simulated transient unbalanc ed fault. In this case the
full ASV C control system is functional and i '* come s from the
system voltage controller. Initially, the ASVCqis supplying 1 p.u.
capac i t ive vars to the line . A phase- to -neu t ra l fau l t , l as t ing
approximately 5 cycles, is simulated and the oscillogram shows the
associated ASV C currents. Note that the reactive current reference,
i y s l imited in magnitude
to
2 p.u.
CONCLUSION
There
is
every indication that ASVCs will be an important part of
powe r transmission system s in the future . A sound analytical basis
has now been established for studying their dynamic behaviour. The
mathemat ica l m odel der ived here can read i ly be ex tended to
represent the ASVC in broader system studies. The ASVC analysis
has also led to control system designs for both Typ e I and Type
I1
v o l t a g e - s o u r c e d i n v e r t e r s . T h e T y p e I 1 i n v e r t e r c o n t r o l i s
par t i cu la r ly s ign i f i can t because it makes i t poss ib le to ob ta in
excellent dynamic performance from the lowest cost inverter and
transformer combination.
ACKNOWLEDGEMENT
The ASVC scaled model was designed and built at the Westinghouse
Science and Technology Center through the combined efforts of
severa l ind iv idua l s . In par t i cu la r , the au tho rs wou ld l ike to
acknowledge the important contributions made by
Mr.
M.
Gernhardt
and Mr. M. Brennen.
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REFERENCES
1.
Gyugyi, L., et al., 1990, Advanced Static VAR Com pensator
Using Gate Turn-O ff Thyristors For Uti l i ty Applications,
CIGRE Paper No. 23-203.
2. Edwards, C.W., et al., 1988, Advanced Static VAR Generator
E m p l o y i n g G T O T h y r i s t o r s , I E E E P E S W i n t er P o w e r
m aper No. 38WM109-1
3.
Hingorani, N.G., 1988, High Power Electronics and Flexible
AC Transmission System, IEEE Power Engineering Review,
J
4.
Electric Power Research Institute, 1990, Development of an
Advanced Static VAR Compensator, Conuact
No.
RP3023-1.
Park , R.H., 192 8, Definition of an
Ideal
Synchronous Machine
and Formula for the Armature
Flux
Linkages, General Elecmc
Review, 31.
Lyon, W.V., 1954, Transient Analysis of Alternating Current
Machinery, John Wiley & Sons, New
Yo ,
USA.
5.
6.
\\ -ESE
+qs-AXIS
i
B- xis)
bt
''''\\\\\\\,,,,,&V *
'\
iqs
'\ v4s
_ _ _ _ _ _ _ _ _ _ _ _
ids Vds +ds- mS
Figure 3.
Definition of Orthogonal Coordinates. (A-axis)
+C-PHASE
AXIS
Vector Representation of Instantaneous
Three-phase Variables.
Figure
1.
Figu re 6.
ASVC Vectors in Synchronous Frame.
25
FIFTH HARMONIC
Example of Vector Trajectory.
J '
Figure
2.
+ds-axis
*-axis)
Figu re
4.
Definition of Rotating Reference Fra me
VOLTAGE
INVERTER
iC
Figure
5.
Equivalent Circuit Diagram for ASVC
+d- AXIS
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1
3.0
2.5-
2.0-
1.5-
p
E
R
0.5-
Figure
7 .
Block Diagram of Inverter Type I Control
I
I
I
I
I
I
I
I
I
v i c o
L
I
I
I
-3.0
I
I
R+
=
0.0
I R, = 10C
I = 1.P
un =
377
4
= 377 I
A
vdc
Figure
9.
Small Signal Block Diagram Showing
Dynamic Behaviour of
ASVC
System with
Type I1 Inverter.
40
10
B
-10
-20
' (1)
apacit ive
Rp
= lOO.O/k
D -90
E
G
E (2) nduct ive
s
-270 IQO
=
1.07
0
100 2
300 4
FREQUENCY
(HZ)
Figure loa. Transfe r Function AI' (s)/Aa(s).
I
Figure lob. Transfer Function AQ'(s)/An(s).
R e a c t i v e
CONTROL
c u r r e n t
I
l e
r e f e r e n c e
Iia
Iic
Figure 11.
Block Diagram for Inverter Type I1 Control
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Figure 12. ASVC Equivalent Circuit with Harmonic
Voltage Sources.
Figure
13.
Harmonic Vectors in the Synchronous
Reference Frame.
k
=
4/x
=
-
L =
0.15
p.u.
R,=
0
PER
UNIT DC-LINK
CAPACITANCE (C')
Figure 14.
ASVC Current Components with
Fundamental Negative Sequence Voltage
on
the Line.
I I I I I I I I ~ ~ ~
0 3 6.4 96
128
160 192
T i m e ( m s )
Figure 15.
Measured Transient Response of Reactive
Current Control System.
I , , , I , I
0
32 61 96
130
160 193
ASVC System Response to Line to Neutral
Fault.
Time ( ins )
Figure 16.