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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006
Frequency-Domain Analysis of the Static
Synchronous Series Compensator
Anil C. Pradhan and P. W. Lehn, Member, IEEE
Abstract—This paper presents an analytical formulation of the
frequency-domain characteristics of the static synchronous series
compensator (SSSC). This paper investigates the characteristics
of using two different types of SSSC controllers–one with quadrature voltage regulation ( -controlled SSSC) and another with
impedance regulation ( -controlled SSSC). The influence of
the controller parameters on the characteristics is investigated
and it is demonstrated that an SSSC-compensated transmission
line displays an impedance minimum at a subsynchronous frequency similar to a capacitively compensated transmission line.
The analytical results are validated with the results obtained by
time-domain simulations.
Fig. 1.
Simplified diagram of the SSSC.
Index Terms—Frequency-domain characteristics, static synchronous series compensator (SSSC),
-controlled SSSC,
-controlled SSSC.
Fig. 2. Equivalent representations of the SSSC. (a) AC side. (b) DC side.
I. INTRODUCTION
T
HE STATIC synchronous series compensator (SSSC) has
been developed as an alternative to the conventional capacitor-based series compensators [1]–[3]. Apart from having other
advantages, the SSSC was thought to be immune to subsynchronous resonance (SSR), which is inherent in capacitor-based
compensators [4]. Through simulation, [5] has shown that the
SSSC, contrary to initial expectations is, in fact, susceptible to
SSR. Though frequency-domain characteristics of the SSSC and
the capacitor-based series compensator are different and resonance occurs at different frequencies for the same amount of
series compensation, both types of compensators can potentially excite SSR and have influence on neighboring turbine
generators.
Reference [5] also shows that the controller strategy has
a significant effect on the frequency-domain characteristics
of the SSSC. Furthermore, the control parameters also affect
the impedance characteristic of the SSSC. While this study is
sufficient for the observation of the frequency-domain characteristics of the SSSC, it offers little insight into the resonance
phenomenon.
The frequency response of the unified power-flow controller
(UPFC) is investigated in [6]. It develops an analytical model of
the UPFC containing other frequencies in addition to the fundamental. However, the analytical model developed in [6] ignores the influence of the controller which, for the SSSC, has
Manuscript received October 6, 2004; revised January 11, 2005. Paper no.
TPWRD-00471-2004.
A. C. Pradhan is with Teshmont Consultants LP, Winnipeg, MB R3T 0P4,
Canada (e-mail: apradhan@teshmont.com).
P. W. Lehn is with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail:
lehn@ecf.utoronto.ca).
Digital Object Identifier 10.1109/TPWRD.2005.852311
a significant impact on the impedance characteristic. Therefore,
consideration of the controller’s effect must be included in the
analytical model to understand the frequency-response behavior
of the SSSC in totality.
This paper develops a frequency-domain model of the SSSC
including the influence of the controls. The impedance characteristic of the SSSC in the subsynchronous frequency range is
derived for both the - and -controlled SSSC. The developed
model is employed to demonstrate the influence of the control
strategy and controller gains on the impedance characteristics.
Finally, the model is validated by comparison with time-domain
simulation results obtained from PSCAD/EMTDC–electromagnetic (EM) transient simulation software.
II. MODELING OF THE SSSC
A. Fundamental Frequency Model
The fundamental frequency SSSC model is based on the simplified circuit diagram given in Fig. 1.
In the figure, the SSSC is represented by a synchronous sinusoidal voltage source with a variable amplitude and phase angle
on the ac side. The ac side of the SSSC also consists of a boost
and leakage inductransformer represented with resistance
tance . The dc side of the SSSC is represented by a current
source connected to the dc capacitor. The equivalent representation of the ac and dc sides of the SSSC is given in Fig. 2, where
represents losses in the converter.
For simplification of the system equations, the following has
been considered:
0885-8977/$20.00 © 2006 IEEE
(1)
(2)
(3)
PRADHAN AND LEHN: FREQUENCY-DOMAIN ANALYSIS OF THE SSSC
441
The mathematical model of the SSSC given in Fig. 1 in the
reference frame can be written as follows:
TABLE I
RESULTS OF STEADY-STATE CALCULATIONS
(4)
The dynamics of the dc side of the voltage-source converter
(VSC) are obtained from Fig. 2 to be
where
(5)
and
(6)
where is the capacitance of the capacitor and
is the
side current.
reference frame are transformed into a
Equations in the
rotating reference frame
time-invariant form, namely the
(7)
The converter terminal voltage and dc voltage are given by
(8)
Equation (11) is the final form of the SSSC equation in a
reference frame when operated in capacitive mode. Similarly,
the equation can be derived for the inductive mode by changing
the angle between the injected voltage and the line current from
to
in (10).
Since the main objective is to study behavior of the SSSC and
its interaction with the network while working in a capacitive
mode, (11) has been derived to reflect the capacitive mode of
the SSSC.
III. STEADY-STATE MODEL OF THE SSSC
For a fixed value of , the steady-state solution of (11) may
be found by solving a linear system of the form
(12)
(9)
where
inverter constant, which gives the relation between
dc side voltage and ac side peak, phase-to-neutral
voltage;
angle of
relative to ;
modulation index of the inverter;
transformation ratio of the boost transformer.
In order to make the SSSC work in the capacitive mode, the
.
line current must lead the injected compensating voltage by
of the
Therefore, the controller must determine the angle
injected voltage as follows:
where the subscript
denotes that excitation is at the synand
is the state of the
chronous angular frequency of
in the steady-state
system in steady state. The state variables
regime can be calculated using (13)
(13)
Five steady-state operating points of interest are calculated
and the results are presented in Table I. The operating points of
interest correspond to 0%, 25%, 35%, 45%, and 55% compensation of transmission-line impedance. The parameters of transmission line, boosting transformer, and VSC are given in the
Appendix.
IV. SSSC CONTROLLER
(10)
is the angle between line voltage and the line current
where
and is a small perturbation added to the inverter voltage angle
needed to charge or discharge the capacitor voltage (Fig. 4).
Finally, combining (7)–(10) gives
(11)
A. General
The primary function of the SSSC is to control the power flow
in the transmission line. This objective can be achieved either by
direct control of the line current or, alternatively, by the indirect
or the compocontrol of either the compensating reactance
nent of that is quadrature to the line current [2], [4]. Indirect
power-flow control through either reactance regulation ( control) or quadrature voltage regulation ( control) is exclusively
used in the literature.
and
This section gives only a brief description of the
controllers; more details may be found in [4].
442
Fig. 3.
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006
TABLE II
CONTROLLER DESIGN FOR V -CONTROLLED SSSC
(K = 1:11 E -5 AND K = 0:001)
SUMMARY
OF THE
SUMMARY
OF THE
Control block diagram of SSSC (X control).
TABLE III
CONTROLLER DESIGN FOR X -CONTROLLED SSSC
(K = 1:11 E -5 AND K = 0:001)
Fig. 5. Simplified diagram of the SSSC with subsynchronous voltage
excitation.
C.
Fig. 4. Phasor diagram of the controller.
B.
Control of the SSSC
The control circuit used in [4] is described. The control block
diagram is given in Fig. 3. A phasor diagram is given in Fig. 4,
which helps to understand the function of the SSSC controller.
A phase-locked loop (PLL) is used to determine the instanin Fig. 1. The
taneous angle of the three phase voltage
and of the three-phase line currents are calcomponents
culated in a synchronously rotating reference frame using .
and are then used to determine the amplitude of the current
and its angle relative to , called . The required ampliis determined by
tude of the SSSC’s compensating voltage
multiplying the current amplitude by the desired compensating
.
reactance
The required angle
of the ac compensating voltage space
vector is determined by first calculating the instantaneous phase
, and then eiof the line current space vector
ther adding
(inductive) or subtracting
(capacitive) accordingly. The SSSC controller, in this case, assumes a fixed
across the inverter, so the reference value
dc to ac gain
, in turn, determines the reference inverter
of ac voltage
dc voltage
. A proportional-integral (PI) controller reguto
by adding a small phase offset to angle
lates
forming the instantaneous phase
of the injected compensating voltage space vector. The phase offset results in a power
exchange with the dc link, as required for dc voltage regulation.
Control of the SSSC
control. In this
A similar control strategy can be used for
with
case, instead of multiplying the current magnitude
to obtain
,
is directly given as a reference. All
of the remaining control functions given in Fig. 3 remain the
same.
D. Controller Design
A simple PI controller has been designed for both the -con-controlled SSSC. Tables II and III
trolled SSSC and the
summarize the system’s small-signal performance about the operating points calculated in Table I.
Comparing Tables II and III, there are differences in values
of overshoots, settling times, and closed-loop poles between the
two types of controllers. Nonetheless, in both cases, the system
is stable and well damped.
All subsequent analysis is done based on the steady-state solutions and controller designs described in Sections III and IV.
V. FREQUENCY-DOMAIN CHARACTERISTICS OF THE SSSC
A. General
This section investigates the response of the SSSC when the
ac source contains an additional excitation voltage source at subsynchronous frequency.
Fig. 5 shows the SSSC connected to a transmission system
operating at a fundamental frequency . A positive-sequence
subsynchronous excitation voltage
is applied to the system.
PRADHAN AND LEHN: FREQUENCY-DOMAIN ANALYSIS OF THE SSSC
443
Reference [7] shows that the interaction of the VSC at fundawith an additional positive-sequence excimental frequency
tation voltage at angular frequency causes ac-side currents at
angular frequency and
and dc side voltage ripple at
. Reference [8] also shows similar
angular frequency
expressions for ac-side currents and the dc-side voltage of the
inverter when the line voltage is unbalanced or distorted.
frame equations are used to develop the frequencyThe
domain characteristics of the SSSC. The VSC equations in the
frame can be written as follows:
ac side
(14)
dc side
(15)
Terminal voltage of the VSC is
(16)
Positive-sequence excitation voltages at the fundamental frequency and subsynchronous frequency can be expressed as follows:
(17)
(18)
In steady state, ac-side current, including the fundamental
component, can be written as follows:
;
and
are current amplitudes and
. Simiphases at angular frequencies of , , and
is the amplitude of the dc voltage at steady state and
larly,
are amplitude and phase angle of the dc-side voltage
.
at angular frequency
B. Equations of the SSSC With
Control
Substituting (17)–(20) into (14)–(16) and applying harmonic
balance, the following equations in phasor form are obtained as
shown in (21)–(23) at the bottom of the page, where
and the underlined parameters are peak phasors. Refer to the
Appendix for detailed derivations of (22) and (23).
Phasors are defined as
at angular frequency ;
at angular frequency ;
at angular frequency
;
at angular frequency
;
at angular frequency
;
phasor of at frequency ;
phasor of at frequency
.
Since solutions for the fundamental frequency current and
voltage components are already obtained, they are neglected in
(21)–(23). Similarly, higher-order components of the equations,
and components at other angular frequencies are neglected because of their very small magnitudes.1 Only the ac-side currents
at angular frequencies and
and dc-side voltage
are considered.
ripple at angular frequency
control mode of the SSSC, dc voltage controller
In the
is given by
output,
(24)
(19)
(20)
Combining (21)–(24), the -controlled SSSC equation can
be written as (25), shown at the bottom of the next page.
Due to the conjugate terms in (25), an expansion in terms of
the real and imaginary parts of the phasors is necessary to yield
Note that
, and
are amplitudes and phases
and
and
of excitation voltages at angular frequencies
1It has been assumed that harmonic components resulting from the subsynchronous excitation are much smaller than the fundamental frequency
components.
Finally, the dc voltage may be expressed as follows:
(21)
(22)
(23)
444
the final form of the -controlled SSSC equation as shown in
(26) at the bottom of the page, where
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006
In (26), subscripts and are real and imaginary parts of the
currents and voltages.
Equation (26) is a linear system of equations that can be expressed in the generalized matrix form as
Assuming
is invertible, this system may be solved as
(27)
, ,
, ,
,
,
, and
may
Using (27),
be calculated for different subsynchronous excitation voltages.
C. Equations of the SSSC With
Control
The derivation of equations for the SSSC with control will
control mode except for the
be valid for the SSSC in the
(25)
(26)
PRADHAN AND LEHN: FREQUENCY-DOMAIN ANALYSIS OF THE SSSC
control portion. The equation for
can be written as follows:
in the
445
-controlled SSSC
(28)
represents ripple in the dc reference command
where
controller. It may be found from
voltage, resulting from the
the linearization of (29)
(29)
Neglecting higher order terms and the fundamental solution,
can be approximated as
(30)
, therefore only contains terms at frequency
and may be expressed by a phasor
at that
frequency according to
where
(31)
Fig. 6. Theoretical results of frequency-response characteristics of the
V -controlled SSSC.
from (27) and then the impedance
is calculated using (33).
(The magnitude of the excitation voltage at subsynchronous
is known.)
frequency
(33)
Substituting (31) in (28), a new matrix may be found, shown
in (32) at the bottom of the page, where
VI. RESULTS OF FREQUENCY-DOMAIN ANALYSIS
A. General
and all other elements are as defined in (26). This will form
control, including
the complete equation of the SSSC with
influence of the control parameters.
D. Subsynchronous Impedance of the
-Controlled SSSC
A single-line diagram of a simplified power system with an
SSSC is given in Fig. 5 and its parameters are given in the
Appendix. The SSSC is located halfway between sending and
receiving ends. The transmission-line voltage is 138 kV and the
phase-angle difference between sending and receiving ends is
30 . The fundamental frequency of the power system is 60 Hz.
In addition to the supply voltage at the fundamental frequency, a 2-kV excitation voltage at the subsynchronous
frequency is inserted at the sending end. The frequency of the
excitation voltage is varied between 60 to 60 Hz.
- and
In order to calculate the impedance at different subsynand
are calculated
chronous frequencies, first currents
B. Currents
and
and Voltage
Figs. 6 and 7 show the frequency response of the ac currents at
and the dc voltage ripple
angular frequencies and
(32)
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006
Fig. 9. Comparison of impedance characteristics of the
and
(operating point 4).
SSSC with different
K
K
V - and X -controlled
C. Impedance
Fig. 7. Theoretical results of frequency-response characteristics of the
-controlled SSSC.
X
The impedance characteristics of the - and
-controlled
SSSC are given in Fig. 8. Four curves represent the impedance
associated with each of the four operating points (from Op. pt. 1
to Op. pt. 4 in Table I).
These figures demonstrate that the SSSC has a frequency-domain impedance characteristic somewhat similar to that of a
conventional series compensator. This analytically obtained result is consistent with the simulation results observed in [5]. The
SSSC has resonant minima at subsynchronous frequencies for
both types of controllers, which means it can potentially excite
SSR and have influence on nearby turbine generators. Second,
the controller type can be seen to strongly influence the frequency-response behavior of the SSSC.
D. Effects of Controller Parameters (
Fig. 8. Theoretical impedance characteristics of the
SSSC.
V - and X -controlled
at angular frequency
for the - and
-controlled
SSSC, respectively.
The top plot of the figures shows the subsynchronous ac current at angular frequency . This plot may be used to determine the impedance of the SSSC as a function of the frequency.
The second plot shows the ac current that will flow at frequency
due to the applied subsynchronous voltage of frequency . The last plot shows the additional dc voltage compo.
nent at angular frequency
From the figures, it can be noticed that the frequency-reand
controls are
sponse characteristics of the SSSC with
quite different. Consider the current at angular frequency . In
control, the current peaks at all operating points are of the
same magnitude and peaks are spread between 7 and 20 Hz.
control, magnitudes of the current
However, in the case of
peaks differ with an operating point and they are spread between
also
7 and 45 Hz. The dc voltages at angular frequency
differ for the two types of SSSC controls. In the -controlled
is zero at all frequencies for
SSSC, the magnitude of
all operating points. However, in the case of the
-controlled
differ with frequency and operSSSC, magnitudes of
ating point. Similarly, there are significant differences between
.
currents at angular frequency
and
)
Fig. 9 shows the impedance characteristics of the - and
-controlled SSSC with different controller parameters.
It can be noticed from the figure that impedance of the
-controlled SSSC is insensitive to the control parameters
and the resonant frequency is solely a function of the oper-controlled SSSC,
ating point. However, in the case of the
impedance and the resonant frequencies vary significantly with
control parameters. Therefore, the impedance characteristic
-controlled SSSC is a function of both the operating
of the
point and the control parameters.
VII. VALIDATION OF RESULTS
The proposed model is validated against PSCAD/EMTDC
simulation results. The simulation includes a complete representation of the SSSC controller, the gating logic, and the
power semiconductors. For simplicity, the SSSC simulated
consisted of a single three-phase VSC that employed sinusoidal
pulsewidth modulation to eliminate low-order harmonics.2
A. Steady-State Operation of the SSSC
The steady-state results of simulation and the analysis are
compared in Table IV. As expected, a close agreement is found
between the simulation and the analytical steady-state operating
2Comparable results are expected for SSSCs using multipulse converter systems as the time averaged behavior of a system is independent of the specific
switching scheme. In contrast, SSSCs using converter structures that employ
multiple separate dc-side capacitors may display additional resonant modes due
to the increased order of the state model.
PRADHAN AND LEHN: FREQUENCY-DOMAIN ANALYSIS OF THE SSSC
447
TABLE IV
STEADY-STATE SOLUTIONS
TABLE V
COMPARISON OF ANALYTICAL AND SIMULATION RESULTS (V -CONTROLLED
SSSC), OP. PT. 2–V = 2 kV, f = 15 Hz
Fig. 10.
Comparison of impedance with V and X controls.
VIII. CONCLUSION
TABLE VI
COMPARISON OF ANALYTICAL AND SIMULATION RESULTS (X -CONTROLLED
SSSC), OP. PT. 2–V = 2 kV, f = 15 Hz
point calculations. According to the table, the maximum error
is about 1.4%, which is negligible.
B. Frequency-Domain Characteristics
and
and voltage
are comCurrents
pared numerically with simulation results in Tables V and VI.
Operating point 2 is considered for the comparison. A posi15 Hz is
tive-sequence voltage of 2 kV with a frequency of
injected. As indicated in the previous section, the current will
have components at 15 and 105 Hz and the dc voltage will have
a component at 45 Hz. Extraction of frequency components of
currents and voltage is obtained using a discrete Fourier transform (DFT). Tables V and VI show the comparison of the analytical and simulation results of the ac currents and the dc voltage
-controlled SSSC, respectively. According to the
for - and
tables, errors are small.
This paper presents an exhaustive mathematical formulation
of the frequency-domain characteristics of the SSSC. It includes
all of the parameters that have an influence on characteristics
including operating point and control parameters. The accuracy
of the analytical formulation has then been established through
comparison with simulation. Analytical and simulation results
have been found to closely match. A small discrepancy has been
found in the neighborhood of 60 Hz (negative sequence, fundamental frequency); however, it is small and not in the frequency range of interest for SSR studies. The following conclusions can be made from this study.
-con• Both the -controlled (voltage reference) and
trolled (impedance reference) SSSCs are found to have
frequency-domain characteristics somewhat similar to
that of a conventional series capacitor with a resonant
minimum at subsynchronous frequency.
-con• For the same amount of compensation, - and
trolled SSSCs display different frequency-domain
impedance characteristics. The resonant minima occur
at different frequencies and impedance minima are of
differing amplitudes.
-controlled
• Frequency-domain characteristics of the
SSSC are a function of the operating point as well as the
and .
PI controller parameters
-controlled
• Frequency-domain characteristics of the
SSSC are only a function of the operating point and are
invariant to changes in the controller parameters.
-controlled SSSC has more natural damping than
• The
the -controlled SSSC.
C. Impedance Characteristic
The time-domain simulation is used to validate the frequencydomain impedance characteristic of the SSSC. The frequency of
the subsynchronous voltage is varied from 60 to 60 Hz. Sim-controlled SSSC.
ulation is completed for both the - and
Operating points 1 and 3 are chosen for validation of the frequency domain impedance of the SSSC.
The analytical and simulated impedance characteristics of
-controlled SSSC are plotted in Fig. 10. As can
the - and
be seen, analytical and simulation results closely match. In the
neighborhood of 60 Hz, there are minor mismatches, however,
they are not significant.
The impedance characteristic of the SSSC found through
analysis is substantiated by the simulation. Therefore, validity
of the analytical model is proven.
APPENDIX A
DERIVATIONS OF ANGLES
A. Angle
Since the dc side of the VSC contains voltages at angular fre, , the output of the dc voltage
quencies zero and
controller also contains components at these frequencies. Therefore, can be expressed as
(34)
where angle is the amplitude of at steady state and angle
is the amplitude at angular frequency
. is the phase
of the oscillatory component.
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006
B. Angle
According to Fig. 4, angle
is the angle between voltage
space vector and the current space vector . Due to the presence of subsynchronous voltage, there is a small perturbation
around its steady-state operating point that can be
of angle
represented as
(35)
is the angle at steady-state operation and related to
where
the operating point.
is a small angle perturbation around
due to the presence of subsynchronous voltage.
the angle
Since steady-state operating points have already been calcuis known. The next step is to define
lated (refer to Table I),
the angle
.
The ac-side current can be expressed as a space vector as
where
and so on.
The above equation in the
reference frame is
The expression of angle
in terms of currents is
The results of this Appendix are used to derive equations for
the ac terminal voltage and the dc-side equation of the VSC
presented in Appendix B.
APPENDIX B
AND dc VOLTAGE
TERMINAL VOLTAGE
Substituting (39) and (20) into (16), and neglecting higher
may be
order terms, the following expression for
obtained
(36)
where
and
.
The last two terms of the denominator are small compared
and are neglected. Similarly, the numerator is very small
to
compared to the denominator. Therefore,
can be approximated as
In the phasor form, this can be expressed as three equations–one at angular frequency , another at , and the other
at
. Since the solution for the fundamental currents
and voltages have already been obtained and accounted for by
the operating points, they are not considered. At frequency
(40)
(37)
At frequency
Equation (37) shows that apart from a dc component, angle
contains a component at angular frequency
.
(41)
C. Angle
By definition
(38)
Substituting (34) and (37) in (38), the expression for
be obtained
can
(39)
These expressions for
and
are represented by (22) in
matrix form.
Similarly, substituting (39) and (19) into (15), the dc side of
the VSC equation is obtained
PRADHAN AND LEHN: FREQUENCY-DOMAIN ANALYSIS OF THE SSSC
449
MVA;
2) Boosting transformer:
;
;
MVA;
kV;
3) VSC:
F; switching frequency
kV;
H
kV;
Hz
REFERENCES
[1] N. G. Hingorani and L. Gyugyi, Understanding FACTS Concept and
Technology of Flexible AC Transmission Systems. Piscataway, NJ:
IEEE Press, 2000.
[2] L. Gyugyi, C. D. Schauder, and K. K. Sen, “Static synchronous series
compensator: A solid-state approach to the series compensation of transmission lines,” IEEE Trans. Power Del., vol. 12, no. 1, pp. 406–417, Jan.
1997.
[3] L. Gyugyi, “Dynamic compensation of AC transmission lines by solidstate synchronous voltage sources,” IEEE Trans. Power Del., vol. 9, no.
2, pp. 904–911, Apr. 1994.
[4] K. K. Sen, “Static synchronous series compensator: Theory modeling
and applications,” IEEE Trans. Power Del., vol. 13, no. 1, pp. 241–246,
Jan. 1998.
[5] B. S. Rigby and R. G. Harley, “Resonant characteristics of inverter based
transmission line series compensators,” in Proc. 30th Annu. IEEE Power
Electronics Specialist Conf., vol. 1, Aug. 1999, pp. 412–417.
[6] I. Papic and A. M. Gole, “Frequency response characteristics of the unified power flow controller,” IEEE Trans. Power Del., vol. 18, no. 4, pp.
1394–1402, Oct. 2003.
[7] M. Mohaddes, A. M. Gole, and E. Sladjana, “Steady state frequency
response of STATCOM,” IEEE Trans. Power Del., vol. 16, no. 1, pp.
18–23, Jan. 2001.
[8] C. Schauder and H. Mehta, “Vector analysis and control of advanced
static VAR compensators,” Proc. Inst. Elect. Eng. C, vol. 140, no. 4, pp.
299–306, Jul. 1993.
(42)
Neglecting fundamental and second-order terms and simplifying (42), the following expression for the dc side of the VSC
equation is obtained:
(43)
Substituting (20) into (43) and writing it in a phasor form
gives
(44)
APPENDIX C
PARAMETERS OF POWER SYSTEM
1) Parameters of transmission line:
kV;
;
;
H.
kV;
;
H;
Anil C. Pradhan received the Diploma in electrical
engineering from Belorussian Polytechnic Institute,
Minsk, Belarus, in 1987 and the M.A.Sc. degree in
electrical engineering from the University of Toronto,
Toronto, ON, Canada, in 2004.
From 1987 to 2001, he was a Transmission and
Distribution Engineer with the Nepal Electricity
Authority, Kathmandu. His research interests include
power system planning and studies.
Mr. Pradhan is a Chartered Electrical Engineer in
the U.K. since 1996.
P. W. Lehn (M’99) received the B.Sc. and M.Sc. degrees in electrical engineering from the University of
Manitoba, Winnipeg, MB, Canada, in 1990 and 1992,
respectively, and the Ph.D. degree from the University of Toronto, Toronto, ON, Canada, in 1999.
Currently, he is an Associate Professor at the
University of Toronto. From 1992 to 1994, he was
with the Network Planning Group of Siemens AG,
Erlangen, Germany.