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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
Analysis of SSR Performance of TCSC Control
Schemes Using a Modular High Bandwidth
Discrete-Time Dynamic Model
S. R. Joshi and A. M. Kulkarni, Member, IEEE
Timing signal from PLL with reference to current
signal (rad).
TCSC voltage zero crossing instant with reference
to synchronizing signal.
Measuring instant of line current and TCSC voltage
with reference to synchronizing signal.
Reactance defined as:
.
SVR controller filter time constant.
Reactance of external fixed series capacitor.
TCSC capacitive reactance in blocked mode.
TCSC capacitive reactance in vernier mode.
Ratio of
to .
Discrete-time system eigenvalue.
Equivalent continuous-time system eigenvalue.
Vector of individual speed deviations of rotating
masses (rad/s).
Modal speed deviation of torsional mode (rad/s).
Washout time constant.
Rotational speed of generator rotor mass (rad/s).
Abstract—Subsynchronous resonance (SSR) performance of
various control schemes of a thyristor controlled series compensator (TCSC) has been studied in the previous literature
using digital simulation based analyses. With the availability of
a modular, high bandwidth (six-sample per cycle), discrete-time
model of a TCSC, accurate eigen-analysis of a TCSC compensated
system has become feasible. Therefore, we revisit the issue of SSR
performance in order to gain additional insight provided by this
analytical model. We use the model to assess fixed firing angle
delay (FFAD) and synchronous voltage reversal (SVR) under
different situations. The performance of synchronization schemes
and an output feedback subsynchronous damping controller
(SSDC) is also evaluated. The derivation of the model is outlined
and validated by comparing the results of eigen-analysis with
those obtained by detailed digital simulation.
Index Terms—Discrete time systems, dynamic modeling, subsynchronous resonance (SSR), synchronous voltage reversal (SVR),
thyristor controlled series compensator (TCSC).
NOMENCLATURE
System frequency (rad/s).
Inductance of TCR branch of TCSC.
Capacitance of TCSC capacitor.
Natural frequency,
(rad/s).
Instantaneous value of TCSC voltage.
I. INTRODUCTION
Instantaneous value of line current.
Current in TCR branch of TCSC.
Turn-on instant of thyristor.
Turn-off instant of thyristor.
Half of the conduction angle of thyristors (rad).
Instant of th sample.
Instant of
th sample.
Sampling interval.
Firing angle delay of thyristor with reference to
synchronizing signal (rad).
Quiescent firing angle delay of thyristor with
reference to synchronizing signal (rad).
Manuscript received October 16, 2008; revised December 11, 2008. First
published March 27, 2009; current version published April 22, 2009. Paper no.
TPWRS-00804-2008.
The authors are with the Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India (e-mail: anil@ee.
iitb.ac.in).
Digital Object Identifier 10.1109/TPWRS.2009.2016530
ERIES compensation using capacitors is an economical
way to enhance the power transfer capability of a transmission system. However, the possibility of adverse interaction
between the generator-turbine mechanical system and an electrical network compensated with fixed series capacitors—subsynchronous resonance (SSR) [1]—is a major concern. Variable
series compensation in the form of thyristor controlled series
compensator (TCSC)—see Fig. 1—can be used for control of
power flow and damping of power swings [2]. A TCSC can be
controlled so that its dynamic behavior is significantly different
than that of a fixed series capacitor. Therefore adverse torsional
interaction can be avoided. However, a TCSC is much costlier
than fixed series capacitors. Therefore for economic reasons, it
is used only as a top-up in a series compensation scheme. In
such a situation, SSR performance of the overall scheme, with
due representation of TCSC controls, need to be carefully evaluated.
While the nonlinear equations of a power system and time
variant equations of a TCSC can be easily handled by a simulation program, techniques like eigen-analysis are insightful and
useful, especially for design and parameter sensitivity studies.
S
0885-8950/$25.00 © 2009 IEEE
JOSHI AND KULKARNI: ANALYSIS OF SSR PERFORMANCE OF TCSC CONTROL SCHEMES
841
Fig. 1. TCSC.
Therefore, there has been an effort in the past to obtain linearized state space models of a TCSC. Accurate linearized discrete time dynamic models were obtained for a TCSC, with
little or few approximations in [3]–[6]. The model developed
by Othman and Angquist [5] is modular, i.e., the TCSC model
is derived independently and then interfaced with the rest of the
system. To do this, it is assumed that the line current in a rotating
frame of reference is constant between samples. This model is
easier to derive and incorporate in existing computer programs.
Kabiri et al. [6] refined the method of [4] to obtain a discrete
model with higher sampling rate (six-samples per cycle). However, this model is not modular and is sample variant. The six
sample per cycle model can be made sample-invariant using a
time-variant transformation of zero sequence variables [7]. The
higher sampling rate model is also amenable for modularization
because the assumption of line current in a rotating frame of
reference being constant during a sampling interval, is a better
approximation at a higher sampling rate.
A modular higher sampling rate model has adequate bandwidth to accurately predict behavior around SSR frequencies
[8]. While both modular and nonmodular TCSC models have
been used for the study of SSR [5], [9], [10], the modular higher
bandwidth model has not been used for the evaluation of specific
control schemes. However, a comprehensive comparative evaluation of TCSC control strategies was carried out using digital
simulation studies in [11].
It has been conjectured or shown in specific case studies,
that some TCSC control schemes can avoid adverse torsional
interactions. Eigen-analysis using an accurate analytical model
could clearly establish this or otherwise. Therefore in this paper,
we revisit the issue of performance of TCSC control schemes
in the context of SSR. We present case studies to assess fixed
firing angle delay (FFAD) and synchronous voltage reversal
(SVR) scheme. The validity of the model is demonstrated by
a quantitative comparison of torsional mode damping obtained
by eigen-analysis and detailed simulation. The use of the TCSC
analytical model to evaluate a synchronization scheme and an
SSR damping controller (SSDC) is also presented. Therefore
this paper presents a consolidation of the previous work on discrete-time TCSC models [3]–[11], and follows up on the application and validation of the modular higher bandwidth model.
The organization of this paper is as follows. Section II outlines the derivation of the six sample per cycle, modular TCSC
discrete time model. The modeling of TCSC control scheme is
discussed in Section III. Section IV describes the discretization
of the rest of the system and its interfacing with modular discrete-time TCSC model. The TCSC control schemes are analyzed using the developed models by presenting case studies in
Section V.
Fig. 2. Thyristor triggering-timing diagram.
II. TCSC MODEL
The model derivation closely follows the previous work in [5]
and [7], but includes some refinements in the method of interfacing the TCSC model with the rest of the system. Therefore for
the sake of completeness and clarity, we outline the derivation
of this model. The performance of the model has been validated
in [8].
The linearized sampled analytical model of TCSC is described in this section. The following assumptions are made.
1) TCSC is operated in capacitive mode only, and the conduc, is limited to 60 electrical
tion angle of thyristors,
degrees. TCSC usually operate in this range.
2) The thyristors are assumed to be ideal. For simplicity, the
reactor is assumed to be lossless.
3) The sampling instants (six in a cycle) are fixed in time and
are chosen such that the conduction period of thyristors lie
within the sampling intervals as shown in Fig. 2.
4) The line current in DQ0 reference frame can be assumed
to be constant as in [5]. Alternatively, we can assume that
it is varying linearly during the sampling interval as is
done here. This refinement is useful when fast dynamics
is present in the system [8]. Therefore, in the th interval,
the expression for instantaneous current which is used for
the TCSC model derivation is
(1)
.
where
The following transformation is used to convert phase variables
to DQ0 variables
:
(2)
where
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
The TCSC circuit of a phase is shown in Fig. 1. When any
thyristor of phase is conducting, the dynamic equations are
given by
(3)
When both thyristors of phase are not conducting, the TCSC
is modeled as series capacitor as in the following equation:
Remarks:
1) As is recognized in [6], the state space matrices are not
sample invariant but are periodic, i.e.,
, and
for any .
2) It is shown in [12] that
for all n.
3) An additional observation is that the sign of all coupling
terms between zero sequence variables and DQ variables
, and
alternate at consecutive samin the matrices
pling instants (e.g.,
and so on).
4) The sign of the last element of vector also alternates at
consecutive sampling instants.
Hence, to make the TCSC model sample invariant, we need to
transform the zero sequence variables as given in the following
[7]:
(4)
(7)
Similar equations can be obtained for the other phases.
Following the methodology given in [5], we can obtain a small
signal relationship in the th interval as follows:
If we apply this transformation of zero sequence variables, the
resulting sample invariant set of equations are
(8)
(5)
and
are the change in thyristor turn-on instant
and the change in thyristor turn-off instant respectively.
The time varying transformation matrix is evaluated at instants and
and applied to (5) to transform
the equation into DQ0 domain. The transformed matrices are
and
In a similar fashion, the equations for other intervals can be obtained. To avoid notational clutter, we drop the time argument in
current and voltage and replace it with the sample number, e.g.,
is replaced by
, etc.
For any sample , we can write following set of equations:
(6)
The sample invariance obtained by using a transformation is
elegant and straightforward as compared to the lifting of alternate samples described in [6]. For eigen-analysis, it obviates the
need to choose between 1) neglecting effect of zero sequence
variables, 2) using either
or
, and 3)
as in [6].
A. Computation of Equilibrium Conditions
The linearized discrete model requires equilibrium values of
the line current and TCSC voltages at the thyristor turn on instants. Considerable simplicity is achieved if we assume line
current to be sinusoidal and free of zero sequence components.
The following steps can be used to determine the equilibrium
conditions if firing angle delay (or conduction angle) is given.
1) The fundamental frequency reactance can be obtained
using the TCSC reactance formula as a function of the
delay angle [13].
2) Phasor analysis (a power flow) of the entire system can be
done to determine the fundamental frequency line current
and TCSC voltages.
3) The dominant harmonics of the TCSC voltage can then be
obtained from the TCSC fundamental frequency voltage
using the relationships given in [13].
4) The approximate TCSC instantaneous voltage can be obtained analytically from step 3, from which one can determine the TCSC voltages at turn on instants of the thyristors.
III. TCSC CONTROL
A typical TCSC control system (see Fig. 3) consists of the
firing angle generation and synchronizing block at the lowest
JOSHI AND KULKARNI: ANALYSIS OF SSR PERFORMANCE OF TCSC CONTROL SCHEMES
843
Fig. 4. Firing angle delay based reactance controller.
Fig. 3. TCSC controller.
level, a reactance controller which could be a closed loop or
open loop scheme, an “outer loop” power flow/current controller
or reactance scheduler, and an auxiliary power swing or SSR
damping controller (SSDC). The power or current controller is
likely to be a relatively slow controller (scheduling function),
although fast current and power flow controllers have been considered in the literature [11]. The reactance controller and firing
generation/synchronizing scheme, however, are fast acting and
can potentially affect SSR.
Fig. 5. Block diagram of SVR control scheme.
A. Modeling of Synchronizing Scheme
Although one may use the TCSC capacitor voltage for synchronization, line current is relatively free of harmonics and is
generally preferred. Also, phase locked loop (PLL) based synchronization schemes are preferred over zero crossing detection
based schemes due to their relative immunity to harmonics and
fast transients.
A PLL can generate a common reference signal from the three
phase currents, which can be used by the firing pulse generators of all three phases. Alternatively, three separate single phase
PLLs can be used, with individual phase currents as the respective inputs.
The common three phase PLL used in this paper is similar to
the one in [14], except that it uses currents instead of voltages as
inputs. In steady state, the reference timing signal for the firing
angle controller, , is locked in quadrature with the current, i.e.,
Considering the states corresponding to the controlled oscillator and PI controller of the PLL, the dynamic equations can
be expressed in state-space form as follows:
(9)
(10)
B. Fixed Firing Angle Delay Scheme
Fixed firing angle delay (FFAD) scheme is a rudimentary
open loop scheme in which the firing angle delay, , is obtained
directly from the reactance order,
, using the steady state
nonlinear algebraic relationship between them. The schematic
of the same is shown in Fig. 4.
The firing pulses are generated using the delay angle obtained
from reactance order and timing signal generated by the synchronizing scheme. If
is constant, then
and hence
the change in firing instant
can be written as follows:
(11)
C. Modeling of SVR Control Scheme
SVR scheme is a closed loop scheme of reactance control (see
Fig. 5). TCSC reactance during transients (and steady state) is
defined by the following expression:1
(12)
The reactance controller does not directly give the firing
angle delay, but determines the time delay
with reference to
the synchronizing signal, after which the TCSC voltage zero
crossing should occur. In steady state, this time delay corresponds to an angle of 180 . However, in transient conditions, or
when
is changed, the time delay is varied by the reactance
controller.
A detailed description of the scheme and motivation for its
use is given in [7]. One of the advantages of the scheme is that
the plant transfer function
is practically independent of the operating point, and has an integral
characteristic. Although the output of the reactance controller
is the desired , this has to be implemented by changing the
thyristor firing instant. In other words, one has to obtain from
1An
alternative
and
(v i 0 v i )=(i + i )
distinct
definition
is
X
=
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
where
Fig. 6. Schematic diagram of system with TCSC.
denotes the states of the network.
the order, by solving the algebraic equations given in the following [7]:
(13)
(14)
(15)
B. Discrete-Time Model of Rest of the System
We can combine (9) and (16) if SVR control scheme is used
and (19) to write the continuous time state space equations of
rest of the system in following form:
(21)
For the discretization of (21) we assume that
varies linearly within a sampling interval. For example, in the th interval
where
and
are instantaneous values of line current and
TCSC voltage, respectively, obtained at a delay of
with reference to the synchronizing signal. In this paper we assume that
in steady state, this instant coincides with the most recent sampling instant in a sampling interval.
The TCSC reactance,
, is regulated by a PI controller.
The general form of SVR controller equations are obtained as
follows:
(16)
(17)
(22)
Since the DQ0 variables are not constant in steady state due to
harmonics, the system which is linearized about the equilibrium
trajectory will not be time-invariant. However, the variables in
which feature in the nonlinearities (e.g., generator fluxes
in the electrical torque expression, line current in a PLL) are
relatively free of harmonics. Therefore for simplicity, it is reasonable to assume that
is constant. The discretized model for
the rest of the system can then be written as follows:
(18)
(23)
IV. DISCRETIZATION OF REST OF THE SYSTEM
AND INTERFACING
A schematic of the interconnection of various subsystems is
shown in Fig. 6. In this section we describe the models of the
subsystems of the rest of the system and interface them with the
TCSC model given in the previous section.
(24)
The equations of the transmission network and turbine-generators can be expressed in the DQ0 frame of reference. Phase
symmetry is assumed. Therefore, the equations involving zero
sequence variables are decoupled from the DQ variables.
The equations for small deviations from the equilibrium trajectory are written as follows:
It may be noted that instead of “exact” discretization used
above, one may discretize the equations by trapezoidal method
or use other approximations for
.
One would need to apply the transformation (7) to the zero
sequence variables in
and
in order to facilitate
interfacing with TCSC equations. The application of the sample
variant transformation to (24) does not result in sample variance
if 1) there is complete decoupling between the zero sequence
equations and the equations of the other variables, and 2)
is
time independent. In such a case, we may write the transformed
equations as follows:
(19)
(20)
(25)
A. Network and Turbine-Generator Equations
JOSHI AND KULKARNI: ANALYSIS OF SSR PERFORMANCE OF TCSC CONTROL SCHEMES
Fig. 7. IEEE FBM System with TCSC.
From (11) or (18), depending on the control scheme used, and
(10), we also obtain
845
, where is any positive or negative integer,
also satisfies the same. This is essentially a manifestation
of aliasing. However, subsynchronous torsional modes
can easily be identified because their frequencies do not
significantly change after interfacing with the network and
TCSC.
2) From the digital simulation studies, we plot the modal
speed deviations of the turbine-generator system in order
to clearly and separately observe the damping of various
corresponding to
modes. The modal speed deviation
a torsional mode is obtained from
(26)
The overall system model is obtained by combining (8), (25),
and the algebraic relationship (26). By gathering all the terms
th sample on the left and those
corresponding to the
corresponding to the th sample on the right-hand side of the
equation, we obtain an equation of the form
(27)
.
or equivalently,
V. CASE STUDIES
IEEE First Benchmark Model (FBM) for SSR [15] is adapted
to include a TCSC as shown in Fig. 7, and is used for the case
studies presented here. This system has several subsynchronous
torsional modes and is ideally suited for evaluating any modeling and control scheme. Note that in this study
1) mechanical damping is not considered in the turbine-generator model;
2) the series compensation of the system is made up of a fixed
pu) and a TCSC. Also,
capacitor (
, where is chosen to be 2.8;
3) a simple model of an AVR and exciter is considered, which
has a single time constant of 0.025 s and a gain of 200
p.u./p.u.
The case studies presented in this section utilize the discrete
time state space model developed in the previous sections.
Eigen-analysis of the discrete time system is compared
with the results obtained from digital simulation of the detailed and nonlinear system model which includes thyristor
switching and control. Digital simulation is carried out on
MATLAB/SIMULINK software [16] and it uses a relatively
small time step of 10 s with the trapezoidal rule of integration.
Note the following.
1) For ease of correlation with simulation results, the discretetime system eigen-values
of the system are converted
which are
to equivalent continuous-time eigen-values
related by
(28)
Some care must be taken while interpreting the
eigen-values since there is no unique value of
for
a given , i.e., if
satisfies the above equation, then
where
is the vector consisting of left eigenvector components corresponding to the speed of individual rotating
masses for the turbine-generator mechanical system.
is the vector of individual speed deviations.
A. Fixed Firing Angle Delay Control
Eigen-analysis of IEEE FBM system augmented by TCSC,
which uses fixed firing angle delay control (FFAD), is presented
here. The eigen-analysis is carried out for the case without
TCSC, and with TCSC for two values of quiescent firing angle
delay .
The following observations can be made.
1) Torsional mode 5 appears to be marginally stable; actually
the eigen-values have a very small negative real part [17].
This torsional mode is practically uncontrollable by electrical torque and is therefore unaffected by a TCSC.
2) Some torsional modes have very small nonnegative real
parts. However note that mechanical damping has been neglected, therefore the results shown are pessimistic; some
of these modes would actually be stable in a real life situation. Estimates of mechanical damping are difficult to obtain, but worst case estimates have been indicated in [18].
The other modes of the system are stable, although they are
not shown here.
3) The choice of fixed compensation is such that the subsynchronous resonant frequency of the network mode is higher
than the frequency of Torsional Mode 4. This is shown in
the first column of Table I. However, marginal negative
damping of Torsional Mode 4 is seen due its proximity with
the network mode.
4) Interestingly, with a TCSC operated with fixed delay angle
(and therefore fixed reactance), the frequency of the network mode reduces. This causes unfavorable torsional interaction. This fact has also been reported in a simulation
study [11]. The extent of negative damping is affected by
the firing angle.
We verify these results by detailed digital simulation. A small
pulse disturbance is given to the torque input at the LPA turbine
at 0.5 s and lasting for 0.05 s and the TCSC is de-blocked at
0.05 s. The real part of the eigen-value corresponding to the critical torsional mode, , obtained from eigen-analysis (Table I) is
matched with the calculated from the simulated response (see
Fig. 8). Clearly, there is a very good match.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
TABLE I
EIGENVALUES WITH FFAD CONTROL SCHEME
Fig. 9. Comparison of FFAD and SVR control scheme for Torsional Mode 4
(203.2 rad/s) at two values of quiescent firing angle.
Fig. 8. Modal speed deviations in rad/s of Torsional Mode 4 (202.8 rad/s) for
a small pulse disturbance in mechanical torque with FFAD firing scheme.
Fig. 10. Comparison of FFAD and SVR control scheme for Torsional Mode 3
(160.0 rad/s) at two values of quiescent firing angle.
B. Parametric Study of FFAD and SVR Control Schemes
The objective of this case study is to compare the performance
of FFAD and SVR control scheme. The real part of Torsional
Mode 3 and Torsional Mode 4, obtained from the eigen-analysis is plotted as a function of ratio of effective TCSC reacto the fixed external series capacitor reactance
tance
. The quiescent operating conditions are identical for both
schemes. The bandwidth of the SVR reactance controller is kept
varies slowly during transients.
low, therefore
Figs. 9 and 10 show the variation of the real part of eigenvalue of Torsional Mode 4 and Torsional Mode 3, respectively,
is varied from 0.02 p.u. to 0.1 p.u. For larger values of
as
and quiescent firing angle, the subsynchronous network
frequency approaches the frequency of Torsional Mode 3, which
explains the rise in the real part of the eigen-value corresponding
to this mode.
The variation of real part of eigen-value of Torsional Mode 4
for different values of
while varying is shown in Fig. 11.
Fig. 12 compares SVR reactance controllers with low and high
bandwidth (see controller data in the Appendix).
The results obtained from this study show that the SVR
scheme in most cases results in a marginal improvement in
SSR damping.
Fig. 11. Comparison of FFAD and SVR control scheme for Torsional Mode 4
.
(203.2 rad/s) at different values of
X
C. Effect of PLL Bandwidth
There is a marginal reduction in negative damping if the PLL
bandwidth is reduced (see PLL data in the Appendix). This is
reflected in the results given in Tables II and III.
D. Subsynchronous Damping Controller
The cases presented so far suggest that it is better to have
a specific SSR damping controller (SSDC) rather than rely on
the inherent properties of reactance control and synchronization
JOSHI AND KULKARNI: ANALYSIS OF SSR PERFORMANCE OF TCSC CONTROL SCHEMES
847
TABLE III
EFFECT OF PLL BANDWIDTH (USED WITH SVR)
Fig. 12. Comparison of SVR control scheme with high and low bandwidth
reactance controllers.
Fig. 13. SSDC structure.
TABLE II
EFFECT OF PLL BANDWIDTH (USED WITH FFAD)
TABLE IV
POLES AND PROXIMATE ZERO LOCATIONS: LINE CURRENT MAGNITUDE SIGNAL
schemes. The analytical TCSC model makes it feasible to design
a damping controller. The general form of an output feedback
SSDC is shown in Fig. 13.
, which is a local
Line current magnitude
signal, is considered as an input signal to SSDC first. The poles
and proximate zero locations of the equivalent continuous
domain plant transfer function
for two
is assumed
different operating points are given in Table IV.
to be constant. It can be observed that the zero locations are
very close to the pole locations for all torsional modes. On the
other hand, there is more network mode movement because
there is no zero close by. We now consider the use of generator
slip as an input signal. Note that generator slip is defined as
being the rotational speed of the generator rotor mass. Generator slip as an input to the SSDC results
in better pole-zero separation for all torsional modes as seen in
Table V, while the network mode observability in this signal
is lower. Therefore it will be much easier to attain robust SSR
damping with the remote generator slip.
While the design of an SSDC is beyond the scope of this
paper, it can be observed that a simple compensator transfer
function
with generator slip signal as an input and with
s can
achieve significant damping for all torsional modes without
destabilizing network modes—Table VI. The results are verified by simulation of the system with the SSDC for
(see Fig. 14).
TABLE V
POLES AND PROXIMATE ZERO LOCATIONS: GENERATOR SLIP SIGNAL
Fig. 14. Modal speed deviations in rad/s of Torsional Mode 4 (202.8 rad/s) for
a small pulse disturbance in mechanical torque with SSDC.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
TABLE VI
EIGEN-VALUE MOVEMENT WITH SSDC USING GENERATOR SLIP SIGNAL
TABLE VII
DATA OF SVR CONTROLLER AND PLL
VI. CONCLUSION
The modular sample-invariant, six sample per cycle model of
TCSC described in this paper has adequate bandwidth to accurately predict SSR performance of TCSC control schemes. This
has been verified by a quantitative comparison of torsional mode
damping obtained by eigen-analysis and detailed simulation. It
is found that the SVR control scheme is marginally better than
the FFAD control scheme. The damping of torsional modes with
SVR control scheme is not significantly affected by the bandwidth of the reactance controller. Reduction in PLL bandwidth
marginally improves the damping of torsional modes.
While the SVR control scheme can be used to marginally improve SSR performance, the availability of this analytical model
allows for direct design of an SSR damping controller. Although
a local signal may be used for this purpose, a remote signal like
generator slip appears to be more amenable for attaining robust
damping for all torsional modes.
APPENDIX
DATA OF SVR CONTROLLER AND PLL
[3] A. Ghosh and G. Ledwich, “Modelling and control of thyristor controlled switched compensators,” Proc. Inst. Elect. Eng., Proc. Gen.,
Transm., Distrib., vol. 142, no. 3, pp. 297–304, May 1995.
[4] S. G. Jalali, R. H. Lasseter, and I. Dobson, “Dynamic response of a
thyristor controlled switched capacitor,” IEEE Trans. Power Del., vol.
9, no. 3, pp. 1609–1615, Jul. 1994.
[5] H. A. Othman and L. Angquist, “Analytical modelling of thyristorcontrolled series capacitor for SSR study,” IEEE Trans. Power Syst.,
vol. 11, no. 1, pp. 119–127, Feb. 1996.
[6] K. Kabiri, S. Henschel, J. R. Martí, and H. W. Dommel, “A discrete
state-space model for SSR stabilizing controller design for TCSC compensated system,” IEEE Trans. Power Del., vol. 20, no. 1, pp. 466–473,
Jan. 2005.
[7] L. Angquist, “Synchronous voltage reversal control of thyristor controlled capacitor,” Ph.D. dissertation, Royal Inst. Technol., Stockholm,
Sweden, 2002.
[8] S. R. Joshi and A. M. Kulkarni, “Comparative evaluation of discretetime dynamic models of TCSC,” in Proc. 16th Power Systems Computation Conf., Glasgow, U.K., Jul. 14–18, 2008, accepted for publication.
[9] R. Rajaraman, I. Dobson, R. H. Lasseter, and Y. Shern, “Computing
the damping of subsynchronous oscillations due to thyristor controlled
switched capacitor,” IEEE Trans. Power Del., vol. 11, no. 2, pp.
1120–1127, Apr. 1996.
[10] B. K. Perkins and M. Iravani, “Dynamic modeling of a TCSC with
application to SSR analysis,” IEEE Trans. Power Syst., vol. 12, no. 4,
pp. 1619–1625, Nov. 1997.
[11] L. A. S. Pilotto, A. Bianco, W. F. Long, and A.-A. Edris, “Impact of
TCSC control methodologies on subsynchronous oscillations,” IEEE
Trans. Power Del., vol. 18, no. 1, pp. 243–252, Jan. 2003.
[12] I. Dobson, “Stability of ideal thyristor and diode switching circuits,”
IEEE Trans. Circuits Syst. I., vol. 42, no. 9, pp. 517–529, Sep. 1995.
[13] K. R. Padiyar, FACTS Controllers in Power Transmission and Distribution. New Delhi, India: New Age International, 2007.
[14] A. M. Gole, V. K. Sood, and L. Mootoosamy, “Validation and analysis
of a grid control system using D-Q-Z transformation for static compensator systems,” in Proc. Can. Conf. Electrical and Computer Engineering, Montreal, QC, Canada, Sep. 1989, pp. 745–748.
[15] IEEE Subsynchronous Resonance Task Force, “First benchmark model
for computer simulation of subsynchronous resonance,” IEEE Trans.
Power App. Syst., vol. PAS-96, no. 5, pp. 1565–1572, Sep./Oct. 1977.
[16] The Mathworks Incorporated, MATLAB and SIMULINK, 1994–2007.
[Online]. Available: http://www.mathworks.com.
[17] P. M. Anderson, B. L. Agarwal, and J. E. V. Ness, Subsynchronous
Resonance in Power Systems. New York: IEEE Press, 1990.
[18] IEEE SSR Working Group Sponsored Panel Discussion, “Comparison
of SSR calculations and test results,” IEEE Trans. Power Syst., vol. 4,
no. 1, pp. 336–344, Feb. 1989.
S. R. Joshi received the B.E. degree in electrical engineering from Sardar Vallabhbhai Regional College of Engineering, Surat, India, in 1992 and the M.E.
in electrical engineering from Gujarat University, Ahmedabad, India, in 2000.
He is currently pursuing the Ph.D. degree at the Indian Institute of Technology,
Bombay, India.
He is currently an Assistant Professor at Government Engineering College,
Surat. His research interests include power system dynamics and FACTS.
See Table VII for the data.
REFERENCES
[1] K. R. Padiyar, Analysis of Subsynchronous Resonance in Power Systems. Norwell, MA: Kluwer, 1999.
[2] N. G. Hingorani and L. Gyugyi, Understanding FACTS. New York:
IEEE Press, 2000.
A. M. Kulkarni (M’07) received the B.E. degree in electrical engineering from
the University of Roorkee, Roorkee, India, in 1992 and the M.E. degree in electrical engineering and Ph.D. degree from the Indian Institute of Science, Bangalore, India, in 1994 and 1998, respectively.
He is currently an Associate Professor at Indian Institute of Technology,
Bombay, India. His research interests include power system dynamics and
FACTS.