1
Abstract—This paper presents a linear state-space model of a
Static VAR Compensator. The model consists of three individual
subsystem models: an AC system, a SVC model and a controller
model, linked together through d-q transformation. The issue of
non-linear susceptance-voltage term and coupling with a static
frame of reference is resolved using an artificial rotating suscep-
tance and linearising its dependence on firing angle. The model is
implemented in MATLAB and verified against PSCAD/EMTDC
in the time and frequency domains. The verification demonstrates
very good system gain accuracy in a wide frequency range
f<150Hz, whereas the phase angle shows somewhat inferior
matching above 25Hz. It is concluded that the model is sufficiently
accurate for many control design applications and practical sta-
bility issues. The model’s use is demonstrated by analyzing the
dynamic influence of the PLL gains, where the eigenvalue move-
ment shows that reductions in gains deteriorate system stability.
Index Terms—Modeling, Power system dynamic stability,
State space methods, Static VAR Compensators, Thyristor con-
verters.
I. I
NTRODUCTION
tatic VAR Compensators are mostly analyzed using EMTP
type programs like PSCAD/EMTDC or RTDS. These si-
mulation tools are accurate but they employ trial and error type
studies only, implying a tedious blind search for the best solu-
tion in the case of complex analysis/design tasks. In order to
apply dynamic systems analysis techniques or modern control
design theories that would in the end shorten the design time,
optimize resources and offer new configurations, there is a
need for a suitable and accurate system dynamic model. In
particular, an eigenvalue and frequency domain analysis based
on an accurate state space system model would prove invalua-
ble for system designers and operators.
There have been a number of attempts to derive an accurate
analytical model of a Static VAR Compensator (SVC), or a
Thyristor Controlled Series Capacitor (TCSC), that can be
employed in system stability studies and controller design [1]-
[7].
The SVC model presented in [1] belongs to classical power
system modeling based on the fundamental frequency repre-
This work is supported by The Engineering and Physical Sciences Re-
search Council (EPSRC) UK, grant no GR/R11377/01.
D. Jovcic and H.Hassan are with Electrical and Mechanical Engineering,
University of Ulster, Newtownabbey, BT370QB, United Kingdom, (e-mail:
d.jovcic@ulst.ac.uk, ha.hassan@ulster.ac.uk).
N.Pahalawaththa and M,Zavahir are with Transpower NZ Ltd, Po Box
1021, Wellington, New Zealand, (Nalin.Pahalawaththa@transpower.co.nz,
Mohamed.Zavahir@transpower.co.nz ).
sentation. These models are used with power flow studies and
for stability analysis at very low frequencies (f<5Hz) only,
whereas they show very poor performance with more detailed
stability studies. The model presented in [2] uses a special
form of discretisation, applying Poincare mapping, for the par-
ticular Kayenta TCSC installation. The model derivation for a
different system will be similarly tedious and the final model
form is not convenient for the application of standard stability
studies and controller design theories. A similar final model
form is derived in [3], however the model derivation is im-
proved since direct discretisation of the linear system model is
used. The importance of having a state-space represented, li-
near continuous system model, is well recognized in [4]. The
model derivation in this case is based on a complex mathemat-
ical procedure encompassing averaging and integration, fol-
lowed by discrete representation and the subsequent model
conversion into linear continuous form. The model also does
not have a modular form for subsystem representation, which
would enable studies of internal system dynamics and subsys-
tem interactions. The approach used in [5] recognizes the ben-
efits of modular system representation, with d-q transformation
used for coupling with the external AC system. However, since
the open loop approach is used, the model does not address
issues of coupling with the static controller model and coupl-
ing with the Phase Locked Loop (PLL). The modeling prin-
ciple reported in [6] employs rotating vectors that are difficult
to use with stability studies, and only considers the open loop
configuration. The SVC model developed in [7] is in a conve-
nient final form, nevertheless it is oversimplified and the deri-
vation procedure for non-linear segments is cumbersome. Most
of the reported models are therefore concerned with a particu-
lar system, a specific operating problem or particular type of
study and many do not include control elements.
An ideal SVC system dynamic model would possess, beside
high accuracy, a convenient (linear state-space) model form,
and it would adequately represent most practical parameters
and variables. The model should be compatible with modern
control theories and preferably be readily implemented with
software tools like MATLAB.
This research adopts a systematic modeling approach by
segmenting the system into three subsystems and individually
modeling them with d-q transformation and matrix coupling
between them to achieve the above properties. It also seeks to
offer complete closed loop model verification in the time and
frequency domains. The modeling method resembles the one
used with HVDC-HVAC systems in [8] and [9].
SVC Dynamic Analytical Model
D.Jovcic, Member, IEEE, N.Pahalawaththa, Member IEEE, M.Zavahir, Member IEEE, H.Hassan
S
2
II. TEST
SYSTEM
The test system in use consists of a SVC connected to an
AC system that is represented by an equivalent impedance and
a local load, as shown in Figure 1. The SVC is very similar to
the one proposed in [10] and used as tutorial example in [11],
except that non-linear transformer effects (saturation and mag-
netizing current) are neglected. The AC system model is also
based on [11], with the introduction of an additional local load
and a variation in system impedance to represent different and
extreme system strengths. Two AC system configurations are
considered: System 1 with equivalent impedance z
1
=72Ω∠54
o
(200MVA), and System 2 having ten times increased strength
z
2
=7.2Ω∠85
o
(2000MVA) in order to fully validate the model
accuracy and flexibility.
The control system is structurally based on [11] but gains
are adjusted to reflect changes in the AC system. The test sys-
tem data are given in the Appendix.
AC
L
1
R
1
R
2
R
3
C
1
e
si
L
tcr
C
SVC
R
4
L
2
SVC
Local load
PLL
PI
V
ref
+
−
ϕ
α
+
−
controller
PLL - Phase Locked Loop
PI - Voltage PI controller
TCR - Thyristor Controlled
Reactor
static coordinate
frame
rotating coordinate
frame
φ
θ
i
L1
v
v
1
2
i
L2
1
V
φ
α
- Calculated firing angle
- PLL reference angle
- Actual firing angle
- Mag, phase angle of
θ
)cos(
11
ϕω +=
tVv
ϕ
,
1
V
1
v
Figure 1. Test system configuration.
III. ANALYTICAL
MODEL
A. Model structure
To avoid pitfalls with modeling complex systems, the sys-
tem model is here divided into three subsystems: an AC system
model, a SVC model and a controller model. Each subsystem
is developed as a standalone state-space model, linking with
the remaining two subsystems and with the outside signals.
With this structure, the subsystems can be analyzed indepen-
dently and their influence after the model connections can be
investigated, whilst enabling convenient coupling with more
complex, future configurations.
The state-space model for a subsystem unit “i” takes the
following generic form:
outioutout
j
ijijoutiiout
iout
outioutk
j
ijijkiik
ik
outi
j
ijijiii
uDuDxCy
uDuDxCy
uBuBxAx
++=
++=
++=
∑
∑
∑
o
(1)
where each of the indices i, j and k, take all values from the set
of three textual labels: “ac” –AC system, “tc” –SVC, “co” –
Controller, where the following cases are excluded: i=j and
i=k. The variables with subscript “out” are the outside inputs
and outputs. All matrices in the model (1) belong to the sub-
system denoted by the first index “i”. The input matrices, B
ij
,
take the second index “j” from the particular input-side con-
necting subsystem (i.e.: B
acco
is the AC model input matrix that
takes input signals from the controller), and the output matric-
es C
ik
have the second index “k” associated with the linking
subsystem that takes the particular output vector. With D
ijk
matrices the second and the third index label inputs and out-
puts, respectively.
B. AC System Model
The AC system model is linear, developed in the manner
described in [8] and [9], and only a derivation summary is
presented here.
A single-phase dynamic model is developed firstly, using
the instantaneous circuit variables as the states. The test system
uses a third order model with i
L1
, i
L2
and v
1
as the states. A
phase “a” model is given below (to increase clarity of presen-
tation we consider only one input link, one output link and
only one D matrix):
acacoacacocoacaacaco
acaco
acacoacacoacaacaaca
uDxCy
uBxAx
+=
+=
o
(2)
where the subscript ”aca” denotes phase a of the AC system.
Using the single-phase model and assuming ideal system sym-
metry, a complete three-phase model in the rotating coordinate
frame is readily created. To enable a wider frequency range
dynamic analysis and coupling with the static coordinate
frame, the above model is converted to the d-q static frame
using Park’s transformation [8],[12]. The AC model is
represented in the d-q frame as:
accoacacocoaccoacco
acco
accoaccoacacac
uDxCy
uBxAx
+=
+=
o
(3)
[
]
[ ]
[
]
[ ]
[ ]
[ ]
[ ]
[ ]
,,
,,
=
=
=
−
=
acacoco
rxm
rxm
acacoco
accoco
acaco
rxn
rxn
acaco
acco
acaco
nxm
nxm
acaco
acco
aca
nxn
o
nxn
oaca
ac
D
D
D
C
C
C
B
B
B
AI
IA
A
0
0
0
0
0
0
ω
ω
(4)
where
f
π
ω
2
0
=
, n- is the AC system order, m – the number
of inputs, and r – the number of outputs. The states, inputs and
outputs in the above model are the d-q components of the in-
stantaneous system variables:
=
=
=
acacoq
acacod
acco
acacoq
acacod
acco
acaq
acad
ac
y
y
y
u
u
u
x
x
x ,,
(5)
C. Static VAR Compensator Model
The static VAR compensator under consideration is a
3
twelve pulse system with two six pulse groups in ∆ connection
and coupled with the network through a single, three-winding
transformer with Y and ∆ secondaries [10],[11].
The SVC impedances are converted to Y configuration and
transferred to the primary transformer voltage. Each six-pulse
group consists of the transformer model, the thyristor con-
trolled reactor (TCR) and the capacitor unit in parallel with a
resistance. An equivalent, six-pulse group model is shown in
the singe phase diagram in Figure 2.
L (
φ)
L
tcr
t
C
R
SVC
cp
v
v
2
1
Figure 2. SVC electrical circuit model
The model can be represented in the state-space domain as
follows:
21
11
v
L
v
L
i
tt
t
−=
o
(6)
tcr
svc
t
svc
i
C
i
C
v
11
2
−=
o
(7)
2
)(
1
v
L
i
tcr
tcr
φ
=
o
(8)
Equation (8) is non-linear in view of the fact that the TCR
reactance is dependent upon the firing angle obtained from the
controller model. This equation cannot be directly linearised
since the SVC model is developed in the AC frame with oscil-
lating variables, (i.e.
)cos(
ϕ
ω
+
=
tVv
22
) whereas the firing
angle signal is derived as a signal in the controller reference
frame (i.e. a non-oscillating signal).
To link the SVC model with the controller model, the ap-
proach of artificial rotating susceptance is adopted. It is firstly
presumed that the AC terminal voltage has the following value
in the steady state:
)cos(
ooo
tVv
ϕω
+=
22
, where superscript
“o” denotes the steady-state variable, i.e., V
2
0
is a constant
magnitude,
ϕ
0
is a constant angle and
o
v
2
is a rotating vector of
a constant magnitude and angle. The susceptance value in the
steady-state is
o
tcr
L/1
.
Assuming small perturbations around the steady state we have:
)(
222
vvv
o
∆+=
(9)
)/(//
tcrtcrtcr
LLL 111
0
∆+=
. (10)
Small perturbations are justified assuming an effective voltage
control at the nominal value. Multiplying the terms in (9) and
(10) and substituting in (8) results in:
)/(/)/(
tcr
o
tcrtcr
o
tcr
LvLvLvi 11
222
∆∆+∆+∆=∆
o
(11)
The susceptance in (11) is further represented, using only the
fundamental component, as [13]:
)sin()(
φπφπ
π
−−−
=
22
tcrm
tcr
L
L
, (12)
where L
tcrm
corresponds to the maximum conduction period,
φ
=90
o
. Equation (12) can be linearised as:
φ
∆=∆
svctcr
KL )/(1
,
(
)
φ
∂∂= //
tcrsvc
LK 1
. (13)
The above linearisation is justified in practice since most mod-
ern SVC control systems will have a gain compensation
scheme (look-up table) that maintains a constant system gain
[13].
In view of (13), and neglecting the small terms, equation
(11) is written as:
o
tcrsvc
o
tcr
LvKvi /∆+∆=∆
φ
2
o
(14)
and it replaces (8) in the model. Equation (14) is in the AC
coordinate frame, and the following term:
)cos(
o
svc
o
svc
o
tKVKv
ϕωφφ
+∆=∆
22
(15)
is an artificial oscillating variable (susceptance) that has a va-
rying magnitude and a constant angle equal to the voltage no-
minal angle. In this way, the SVC model (6),(7),(14) has all
oscillating variables that are converted to d-q variables, as is
done with the AC system model in (3)-(5). Subsequently, using
the d-q components of the inputs and outputs, this model is
linked with the other model units. In order to link the d-q com-
ponents of the rotating susceptance (15) with the controller
module, these components are further converted to magnitude-
angle components using the x-y to polar co-ordinate transfor-
mation [8].
It should be noted that the transformer impedance (L
t
) must
be included in this model since the eigenvalue analysis proves
that this parameter has noticeable effects on system dynamics.
This conclusion is contrary to HVDC modeling principles,
since it has been demonstrated [8],[9] that transformer dynam-
ics can be excluded from system dynamic models.
D. Controller Model
The controller model consists of a second order feedback
filter, PI controller, Phase Locked Loop (PLL) model and
transport delay model, as shown in Figure 3. The PLL system
is of the d-q-z type and its functional diagram is given in [14]
and [10], whereas the state space linearised second-order mod-
el is developed in [8].
The delay filter does not have dynamic equivalent in the ac-
tual system. It is introduced to represent the effects of the dis-
crete nature of the signal transfer caused by thyristor firings at
discrete instants in the fundamental cycle. This simplified con-
tinuous-element modeling of a discrete phenomenon has li-
mited accuracy, but the model application value is much in-
creased with the continuous form and, as demonstrated in the
following sections, accuracy proves satisfactory for most ap-
plications. Researchers in [1] conclude that the delay filter
time constant has a value of 3-6ms and reference [13] suggests
2.77ms. During the proposed model verification, simulation
studies have suggested that the value of approximately
T
d
=2.85ms is used, which is in agreement with the above rec-
4
ommendations.
PLL
V
ref
1
V
+
−
ϕ
α
+
−
PI controller
Feedback filter
Delay filter
θ
s
k
k
i
p
+
22
2
2
fff
f
wsws
w
++ ς
1
1
+
s
T
d
φ
Figure 3. Controller model
E. Model Connections
The above three models are linked to form a single system
model in the state-space form. The final model has the follow-
ing structure:
outsss
out
outssss
uDxCy
uBxAx
+=
+=
o
(16)
where “s” labels the overall system and the model matrices
are:
=
actcacactccoacacco
actctcactcctcco
accocoactccocco
s
ACBCB
CBACB
CBCBA
A
**
**
**
cot
cot
(17)
[ ] [ ]
0==
=
sacouttcoutcoouts
acout
tcout
coout
s
DCCCC
B
B
B
B ,,
All the subsystems’ D matrices are assumed zero in (17) since
they are zero in the actual model and this noticeably simplifies
development.
The matrix A
s
has the subsystem matrices on the main di-
agonal, with the other sub-matrices representing interactions
between subsystems. The model in this form has advantages in
flexibility since, as an example, if the SVC is connected to a
more complex AC system only the A
ac
matrix and the corres-
ponding input and output matrices need modifications. Differ-
ent FACTS can be modeled using the TCR/SVC model unit or,
similarly, more advanced controllers can be developed using
modern control theory (H
∞
, MPC,..) and implemented directly
by replacing the A
co
matrix. The above structure enables the
model to be readily interfaced with the MATLAB HVDC
model or other FACTS elements or Power Systems Blockset,
for the purpose of investigating interactions and coordination.
IV. M
ODEL
V
ERIFICATION
A. Time domain
The model was implemented in MATLAB and tested
against the detailed, non-linear simulation PSCAD/EMTDC.
In the time domain, step responses were verified using the con-
troller reference as the input, given by V
ref
in Figure 3, and the
disturbance represented by the e
si
magnitude variation in Fig-
ure 1.
Figures 4 and 5 show the System 1 verification for the refer-
ence and disturbance inputs, respectively. Very good response
matching is evident for the voltage magnitude output signal;
similar matching was confirmed for all other model variables
that are not shown. To confirm the model robustness with dif-
ferent system parameters, System 2 was also tested and the
results are shown in Figures 6 and 7. A satisfactory response
matching is clear and the accuracy is further emphasized, since
the lightly damped oscillatory mode at 70Hz in the case of the
disturbance input (Figure 7) is very well represented. As seen
in Figure 7, however, MATLAB gives more noticeable error in
the phase angle, particularly at high frequencies.
0 0.05 0.1 0.15
0.2
0.25
0.3
120
120.5
121
121.5
122
122.5
123
123.5
Time (s)
PSCAD
MATLAB
V
o
l
t
a
g
e
(
k
V
)
Figure 4. System 1 response following a 3kV voltage reference step change.
0 0.05 0.1 0.15
0.2
0.25
0.3
118
118.5
119
119.5
120
120.5
Time (s)
PSCAD
MATLAB
V
o
l
t
a
g
e
(
k
V
)
Figure 5. System 1 response following a 2kV disturbance (remote source) step
change.
0 0.05 0.1 0.15
0.2
0.25
0.3
120
120.5
121
121.5
122
122.5
123
123.5
Time (s)
PSCAD
MATLAB
V
o
l
t
a
g
e
(
k
V
)
Figure 6. System 2 response following a 3kV voltage reference step change.
B. Frequency domain
The two test systems were also tested against PSCAD in the
frequency domain. PSCAD does not possess a frequency do-
main analysis capability, and the results were obtained “ma-
nually”, by injecting a single frequency component at a time.
The individual points were then linked in a single curve with
minimal filtering of the experimental data.
5
0 0.05 0.1 0.15
0.2
0.25
0.3
117.5
118
118.5
119
119.5
120
120.5
Time (s)
PSCAD
MATLAB
V
o
l
t
a
g
e
(
k
V
)
Figure 7. System 2 response following a 2kV disturbance (remote source) step
change.
Figure 8 shows the gain frequency response comparison in
the frequency range 1-150Hz, where the “error” is the differ-
ence between the two curves. Very good matching was found
across the entire frequency range, and at higher frequencies the
error is mostly within a 5dB envelope. Certainly, below 40Hz
very high accuracy is evident.
The phase angle frequency response is shown in Figure 9. In
this case the error increased, and particularly in the frequency
range 25-60Hz was pronounced. This result is a consequence
of poor representation of the discrete system: if the delay filter
in the controller model is omitted the error increases. Research
is currently under way to offer new modeling approaches to
eliminate this phase error.
In the majority of applications at higher frequencies, such as
the analysis of amplification of a particular oscillatory mode in
the system, the system gain is of primary importance, and in
this aspect the model represents the system correctly in a wide
frequency range.
Regarding the overall time and frequency domain responses,
and being aware of the phase response errors, it can be con-
cluded that the model has reasonably good accuracy when em-
ployed as a design and analysis tool for phenomena such as
subsynchronous resonance, or interactions with other fast
FACTS/HVDC controls.
10
1
10
2
10
3
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
frequency [rad/s]
PSCAD
MATLAB
ERROR
g
a
i
n
[
d
B
]
Figure 8. System 1 gain frequency response with the reference voltage input.
10
1
10
2
10
3
-250
-200
-150
-100
-50
0
frequency [rad/s]
a) phase angle frequency response
b) error (PSCAD-MATLAB) in phase angle frequency response
frequency [rad/s]
PSCAD
MATLAB
p
h
a
s
e
[
d
e
g
]
phase [deg]
10
1
10
2
10
3
-80
-40
0
40
80
Figure 9. System 1 phase frequency response with the reference voltage input.
V. S
TUDY OF INFLUENCE OF
PLL
GAINS
This section gives an example of the model use in the sys-
tem dynamic analysis. A PLL is typically used with thyristor
converters to provide the reference signal that follows the syn-
chronizing line voltage or current. As a dynamic element, a
PLL will also have influence on the system’s dynamic res-
ponses and stability, although this aspect not been analyzed in
the FACTS/HVDC references. Further, since a PLL has two
adjustable gains (k
p
and k
I
), these can be used as a convenient
means of adjusting system performance in respect of stability
issues or improving performance.
Figure 10 shows the dislocation of dominant eigenvalues af-
ter reduction in the PLL gains. As the gains are reduced, the
eigenvalues migrate from the original “x” to the location “o”,
representing ten times reduced gains. It is seen that the PLL
gains have significant influence on the system dynamics and
that the frequency of the dominant oscillatory mode reduces,
accompanied by a small reduction in mode damping (branch
“a”). The next dominant real mode is at the same time moved
away from the imaginary axis, as shown by the branch “b”.
-120 -100 -80
-60
-40
-20
0
-30
-20
-10
0
10
20
30
a
b
Im
Re
a
Figure 10. System 1. Influence of the PLL gains on the system eigenvalue
location. “x” – original eigenvalues, “o” – final location with 10 times re-
duced gains.