Modeling and Control of Gate-Controlled Series Capacitor Interfaced With a DFIG-Based Wind Farm

1022 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 2, FEBRUARY 2015
Modeling and Control of Gate-Controlled
Series Capacitor Interfaced With a
DFIG-Based Wind Farm
Hossein Ali Mohammadpour, Student Member, IEEE, and Enrico Santi, Senior Member, IEEE
Abstract—This paper presents application and control of
the gate-controlled series capacitor (GCSC) for series com-
pensation and subsynchronous resonance (SSR) damping
in doubly-fed induction generator (DFIG)-based wind farms.
The GCSC is a new series FACTS device composed of a
fixed capacitor in parallel with a pair of antiparallel gate-
commuted switches. The study considers a DFIG-based
wind farm, which is connected to a series-compensated
transmission line whose parameters are derived from the
IEEE first benchmark model for computer simulation of the
SSR. The small-signal stability analysis of the system is
presented, and the eigenvalues of the system are obtained.
Using both modal analysis and time-domain simulation, it
is shown that the system is potentially unstable due to the
SSR mode. Therefore, the wind farm is equipped with a
GCSC to solve the instability of the wind farm resulting from
the SSR mode, and an SSR damping controller (SSRDC)
is designed for this device using residue-based analysis
and root locus diagrams. Using residue-based analysis,
the optimal input control signal to the SSRDC is identified,
which can damp the SSR mode without destabilizing other
modes, and using root-locus analysis, the required gain for
the SSRDC is determined. MATLAB/Simulink is used as a
tool for modeling, design, and time-domain simulations.
Index Terms—Doubly fed induction generator (DFIG),
flexible ac transmission systems (FACTS), gate-controlled
series capacitor (GCSC), root-locus diagram, subsyn-
chronous resonance (SSR).
I. INTRODUCTION
DUE to the recent rapid penetration of wind power into the
power systems [1], some countries in central Europe, e.g.,
Germany, have run out of suitable sites for onshore wind power
projects, due to the high population density in these countries.
Moreover, it has been found that the offshore wind power
resources are much larger than onshore wind power sources [2].
Therefore, offshore wind farms have a great potential as large-
Manuscript received September 13, 2013; revised December 29,
2013, May 23, 2014, and July 10, 2014; accepted July 29, 2014. Date
of publication August 12, 2014; date of current version January 7,
2015. This work was supported by the National Science Foundation
Industry/University Cooperative Research Center for Grid-Connected
Advanced Power Electronic Systems under Grant 0934378.
The authors are with the Department of Electrical Engineering,
University of South Carolina, Columbia, SC 29208 USA (e-mail:
Mohammah@email.sc.edu; Santi@cec.sc.edu).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2014.2347007
scale sustainable electric energy resources [2]. Recently, the
doubly fed induction generator (DFIG) has gained significant
attention from the electric power industry in offshore wind
farms and renewable energy sources [1], [3].
However, in offshore wind farms, the distance between the
wind turbines and the shore is much longer [4] than that
in onshore wind farms. Therefore, unlike the onshore wind
farms—where the voltage level of the wind farm is usually the
same as the voltage level of the distribution system—higher
voltage levels with reliable and efficient transmission lines are
required for the offshore wind farms to minimize the power
losses [2].
Currently, there are numerous large offshore wind farms
operating throughout the world [2], [5]. Future projects in
offshore wind farms will be larger in size and further away from
the shore [2]. This requires defining new concepts for the trans-
mission system, including transmission lines from the offshore
wind farm to the shore and network integration to the onshore
power system. The transmission system options to transmit the
wind power to the shore are high-voltage ac (HVAC) [2] or
high-voltage dc (HVDC) [5], [6]. The HVAC solutions are vi-
able for distances up to 250 km, and with series compensation,
they may be viable for distances longer than 250 km [2].
Reactive power injection, either shunt or series, into power
transmission lines has been used for many years to increase the
transmittable power of transmission lines [7]. For the purpose
of increasing the power transfer capability of a transmission
line, series compensation is preferred compared with shunt
compensation. One of the main reasons is that, unlike shunt
compensation, series compensation is less sensitive to system
load characteristics and equipment location along a transmis-
sion line. However, it was found at an early date that using se-
ries compensation can cause instability in power systems due to
a phenomenon known as subsynchronous resonance (SSR) [7].
Properly designed flexible ac transmission systems (FACTS)
could be used to take advantage of series compensation ben-
efits without causing the SSR problem in power systems [7].
Nowadays, FACTS devices are required in order to support
massive integration of renewable energy resources into the
power networks [8]. The advantages of using FACTS devices
such as static synchronous compensator (STATCOM) [7], static
VAR compensator (SVC) [9], synchronous series compensator
(SSSC) [7], unified power flow controller (UPFC) [10], and
thyristor-controlled series compensator (TCSC) [9] in power
systems are well known.
0278-0046 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
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MOHAMMADPOUR AND SANTI: GCSC INTERFACED WITH A DFIG-BASED WIND FARM 1023
Fig. 1. One line diagram of the studied power system. RL =
transmissionline resistance, XL = transmission line reactance, XT =
transformer reac-tance, Xsys = system impedance, XC = fixed-
series capacitor, Xtg = trans-former reactance in GSC, Vs =gen-
erator’s terminal voltage, iL =line current, ig = GSC current, is =
stator current, ir = rotor current [15].
The gate-controlled series capacitor (GCSC) is a FACTS
device proposed for series compensation of transmission lines
[7]. The application of this device to control the power flow
has been investigated [11]. For each phase, the GCSC consists
of a fixed capacitor in parallel with a pair of antiparallel gate-
commuted switches. By controlling the turn-off angle of the
gates in the GCSC, this device can provide a variable series
capacitor for the transmission line [12], [13]. Compared with
other FACTS devices such as SSSC, the GCSC has the advan-
tages of being less complex and less expensive. Moreover, a
good comparison of the TCSC and GCSC proves that the latter
is a better device from cost and performance point of view [14].
This paper proposes application and control of the GCSC
FACTS for series compensation and SSR mitigation in off-
shore DFIG-based wind farms. This paper is organized as
follows. In Section II, the studied power system is briefly
described. In Section III, modeling of the system for small-
signal stability analysis is presented. The model of the system
includes wind turbine aerodynamics, an eighth-order model
of grid-side converter (GSC) and rotor-side converter (RSC)
controllers, a first-order model for the dc link between the
GSC and the RSC, a sixth-order model for DFIG, a third-
order model for the shaft system, and a fourth-order model
for series-compensated transmission line. In Section IV, the
SSR phenomenon in fixed-series-compensated DFIG is studied.
Here, first, the SSR phenomenon in a fixed-compensated wind
farm is briefly explained. Then, the eigenvalues of the system
are obtained, and the participation factor is used to show how
different parts of the system are related to each eigenvalue.
Here, time-domain simulation is also presented to verify the
eigenvalue analysis. In Section V, the principles of operation
of the GCSC, generated harmonics, and its control system
are presented. The control system of the GCSC is composed
of power scheduling control (PSC) and SSR damping control
(SSRDC) blocks. In Section VI, an SSRDC is designed using
residue-based analysis and root-locus diagrams. Using residue-
based analysis, the most effective and optimal input control
signal (ICS) to the GCSC’s SSRDC is identified, which can
damp the SSR mode without decreasing the damping ratio of
other system modes. Moreover, using root-locus diagrams, the
required SSRDC gain in order to obtain the desired damping
ratio for the SSR mode is computed. In Section VII, the
Fig. 2. Wind power ¯Pm (p.u.), wind turbine shaft speed ¯ωm (p.u.), and
wind speed Vω (m/s) relationship.
time-domain simulation of the GCSC-compensated DFIG is
presented to verify the design process. Finally, Section VIII
concludes this paper.
II. POWER SYSTEM DESCRIPTION
The studied power system, shown in Fig. 1, is adapted
from the IEEE first benchmark model for SSR studies [15].
In this system, a 100-MW DFIG-based offshore wind farm is
connected to the infinite bus via a 161-kV series-compensated
transmission line [16]. The 100-MW wind farm is an aggre-
gated model of 50 wind turbine units, where each unit has a
power rating of 2 MW. In fact, a 2-MW wind turbine is scaled
up to represent the 100-MW wind farm. This simplification is
supported by several studies [6], [17]. The systems data are
given in the Appendix.
III. MODELING OF DFIG-BASED WIND TURBINE
The overall power system model shown in Fig. 1 includes
dynamic models of wind turbine aerodynamics, shaft system,
induction generator (IG), RSC and GSC controllers, dc link,
and series-compensated transmission line. These models are
now described.
A. Wind-Turbine Aerodynamics
The wind power can be calculated from the wind speed Vw
as follows [18]:
Tω =
0.5ρπR2
CP V 2
w
ωm
(1)
where Tω is the wind power (N.m), Vω is the wind speed (m/s),
ρ is the air density (kgm−3
), R is the rotor radius of the wind
turbine (m), and ωm is the wind turbine shaft speed (rad/s).
Moreover, CP is the power coefficient of the blade given by
CP = 0.5
RCf
λw
− 0.022θ − 2 e−0.225
RCf
λω (2)
where Cf is the wind turbine blade design constant, and θ is the
wind speed pitch angle (rad).
Furthermore, λω is the wind speed tip-speed ratio defined by
λω =
ωmR
Vω
. (3)

Fig. 3. RSC controllers.
Fig. 4. GSC controllers.
B. Modeling of the DFIG Converter Controllers
In this paper, both RSC and GSC controllers are modeled.
In order to achieve high efficiency in the DFIG wind farm,
maximum power point tracking (MPPT) is used [1], [17]. Fig. 2
shows the wind power versus wind turbine shaft speed in per
unit for various wind speeds with indication of the MPPT curve.
To enforce operation on the MPPT curve, for a given wind
speed Vω, the optimal reference power and optimal rotational
speed are obtained. Note that due to power converters ratings, it
may not be practical to always work on the MPPT cure. In this
case, for very low wind speeds, the DFIG operates at almost
constant rotational speed. On the other hand, when the wind
speed increases so that it exceeds the turbine torque rating, the
DFIG will work in maximum constant torque [19].
The aim of the GSC and the RSC is to enable the DFIG to
work on the MPPT curve. Note that the converters are assumed
to store no energy so that their losses can be neglected and
operate fast enough so that their dynamics can be neglected.
Figs. 3 and 4 show the block diagrams of the two controllers,
respectively. In this paper, the RSC controller is responsible for
regulating the electric torque, i.e., Te, and stator reactive power,
i.e., Qs. In a steady-state condition, neglecting power losses, the
wind torque, i.e., ¯Tω = ¯Pω/¯ωm, is equal to electric torque, i.e.,
Te. Therefore, the reference torque, i.e., T∗
e , can be calculated
based on the value of ¯T∗
ω determined by the MPPT shown in
Fig. 2 [17]. The value of Q∗
s depends on the chosen reactive
power control method, which could be either fixed reactive
power or unity power factor [17]. In this paper, the latter method
is chosen.
Moreover, the GSC is responsible for controlling the dc-link
voltage, i.e., Vdc, and the induction generator’s terminal volt-
age, i.e., Vs [17]. The GSC and RSC controllers add eight state
variables to the system, due to the eight proportional–integral
(PI) controllers, and their state variables are defined as a
vector XRG.
C. Modeling of the DC Link
The dynamic of the capacitor in the dc-link between the GSC
and the RSC can be expressed by a first-order model as fol-
lows [19]:
−Cvdc
dvdc
dt
= Pr + Pg (4)
where vdc is the dc-link voltage, and C is the dc-link capacitor.
In this equation, Pr is the rotor active power and is given by
0.5 (vqriqr + vdridr), and Pg is the GSC active power and is
given by 0.5 (vqgiqg + vdgidg).
D. Modeling of the Induction Machine
The IG currents are selected as state variable, and the IG is
represented by a sixth-order dynamic model as follows [19]:
˙XIG = AIGXIG + BIGUIG (5)
where
XIG = [iqs ids i0s iqr idr i0r]T
(6)
UIG = [vqs vds v0s vqr vdr v0r]T
(7)
where iqs, ids, iqr, idr are the stator and rotor qd-axis currents
(p.u.), vqs, vds, vqr, vdr are the stator and rotor qd-axis volt-
ages (p.u.), and i0s, i0r, v0s, v0r are the stator and rotor zero
sequence current and voltage components (p.u.), respectively.
The AIG and BIG matrices are defined as follows. We first
define the matrices given in (8), shown at the bottom of the
page, and (9) as
G =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Xss 0 0 XM 0 0
0 Xss 0 0 XM 0
0 0 Xls 0 0 0
XM 0 0 Xrr 0 0
0 XM 0 0 Xrr 0
0 0 0 0 0 Xlr
⎤
⎥
⎥
⎥
⎥
⎥
⎦
. (9)
F =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Rs
ωe
ωb
XSS 0 0 ωe
ωb
XM 0
−ωe
ωb
XSS Rs 0 −ωe
ωb
XM 0 0
0 0 Rs 0 0 0
0 (ωe−ωr)
ωb
XM 0 Rr 0 (ωe−ωr)
ωb
Xrr
−(ωe−ωr)
ωb
XM 0 0 −(ωe−ωr)
ωb
Xrr Rr 0
0 0 0 0 0 Rr
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(8)

Then,
AIG = −ωb · G−1
· F (10)
BIG = ωb · G−1
. (11)
In (8) and (9): Xlr is the rotor leakage reactance (p.u.), Xls
is the stator leakage reactance (p.u.), XM is the magnetizing
reactance (p.u.), Xss = Xls + XM (p.u.), Xrr is equal to
Xlr + XM (p.u.), Rr is the rotor resistance (p.u.), Rs is the
stator resistance (p.u.), ωb is the base radian frequency (rad/s),
ωr is the generator rotor speed (rad/s), and ωe is the rotating
synchronous frame frequency (rad/s).
E. Modeling of the Shaft System
The shaft of the wind turbine system can be represented as a
two-mass system. The first mass represents the low-speed tur-
bine, and the second mass represents the high-speed generator,
and the two mass connections are modeled as a spring and a
damper. The motion equations then can be expressed as a third-
order linear system in per unit as follows [17]:
˙Xshaft = AshaftXshaft + BshaftUshaft (12)
where
Xshaft = [¯ωm ¯ωr Ttg]T
(13)
Ushaft = [ ¯Tω Te 0]T
. (14)
The Ashaft and Bshaft matrices are defined as follows:
Ashaft =
⎡
⎢
⎣
(−Dt−Dtg)
2Ht
Dtg
2Ht
−1
2Ht
Dtg
2Hg
(−Dt−Dtg)
2Hg
−1
2Hg
Ktgωb −Ktgωb 0
⎤
⎥
⎦ (15)
Bshaft =
⎡
⎣
1
2Ht
0 0
0 1
2Hg
0
0 0 1
⎤
⎦ . (16)
In the shaft equations, Te is the electric torque and is given
by 0.5XM (iqsidr − idsiqr) (p.u.), ¯ωm is the turbine shaft speed
(p.u.), ¯ωr is the generator rotor speed (p.u.), ¯Tω is the wind
torque (p.u.), Dg and Dt are the damping coefficient of the
generator and turbine (p.u.), Dtg is the damping coefficient
between the two masses (p.u.), Ktg is the inertia constant of the
turbine and generator (p.u./rad), and Hg and Ht are the inertia
constants of the generator and turbine (s).
F. Modeling of the Transmission Line
Considering the line current and the voltage across the ca-
pacitor as the state variables, the transmission line equations in
the qd-frame can be expressed in matrix form as follows [20]:
˙XTline = ATlineXTline + BTlineUTline (17)
where
XTline = [iql idl vqc vdc]T
(18)
UTline =
(vqs − EBq)
XL
(vds − EBd)
XL
0 0
T
. (19)
Fig. 5. Equivalent circuit of the system under subsynchronous fre-
quency [16].
The ATline and BTline matrices are defined as follows:
ATline =
⎡
⎢
⎢
⎣
−RL
XL
−¯ωe
−1
XL
0
¯ωe
−RL
XL
0 −1
XL
XC 0 −¯ωe 0
0 XC ¯ωe 0
⎤
⎥
⎥
⎦ (20)
BTline =
⎡
⎢
⎣
ωb 0 0 0
0 ωb 0 0
0 0 1 0
0 0 0 1
⎤
⎥
⎦ (21)
where iql and idl are the transmission line qd-axis currents
(p.u.), vqc and vdc are the series capacitor’s qd-axis voltages
(p.u.), RL is the transmission line resistance (p.u.), XL is the
transmission line reactance (p.u.), XC is the fixed-series capac-
itor (p.u.), EBq and EBd are the infinite bus qd-axis voltages
(p.u.), and ¯ωe is the rotating synchronous frame frequency (p.u.)
Considering the modeling of the system shown in Fig. 1
given in this section, the entire DFIG system is a 22nd-order
model and can be expressed as
˙X = f(X, U, t) (22)
where
X = XT
IG XT
shaft XT
Tline vdc XT
RG
T
. (23)
The nonlinear system equations developed in Section III are
linearized around an operating point to calculate the linearized
state space matrices A, B, C, and D [18]. This can be performed
using MATLAB/Simulink. Equation (22) was assembled in
MATLAB/Simulink, and eigenvalues were obtained using the
“linmod” function. In the next section, the DFIG performance
is briefly analyzed using modal analysis and time-domain
simulation.
IV. SSR IN FIXED-SERIES-COMPENSATED DFIG
A. IGE in Wind Farms
A series-compensated power system with a compensation
level defined as K = XC/Xe excites subsynchronous currents
at frequency given by [15]
fn = fs
KXe
X
(24)
where Xe is equal to XL + XT , X is the entire inductive
reactance seen from infinite bus, fn is the natural frequency of
the electric system, and fs is the fundamental frequency of the
system.

TABLE I
SYSTEM MODES AND PARTICIPATION FACTORS AT 75% SERIES COMPENSATION AND 7-m/s WIND SPEED (PART I)
TABLE II
SYSTEM MODES AND PARTICIPATION FACTORS AT 75% SERIES COMPENSATION AND 7-m/s WIND SPEED (PART II)
At this frequency, the slip given by (25) becomes negative
since the natural resonance frequency, i.e., fn, is less than the
electrical frequency corresponding to the generator rotor speed,
i.e., fr. Thus,
S =
fn − fr
fn
. (25)
The steady-state equivalent circuit of the system under sub-
synchronous frequency is shown in Fig. 5. If the magnitude
of the equivalent rotor resistance, i.e., Rr/S < 0, exceeds the
sum of the resistances of the armature and the network, there
will be a negative resistance at the subsynchronous frequency,
and the subsynchronous current would increase with time. This
phenomenon is called the induction generator effect (IGE).
B. System Modes and Participation Factors
The participation factor is a measure of the relative partici-
pation of the jth state variable in the ith mode of the system.
The magnitude of the normalized participation factors for an
eigenvalue, i.e., λi, is deﬁned as [21]
Pji =
|Ψji||Φij|
n
i=1
|Ψji||Φij|
(26)
where Pji is the participation factor, n is the number of modes
or state variables, and Ψ and Φ are right and left eigenvectors,
respectively.

TABLE III
λ5,6 AT DIFFERENT WIND SPEEDS AND COMPENSATION LEVELS
Tables I and II show the eigenvalues and participation factors
of the system when the wind speed is 7 m/s and the compen-
sation level is 75%. In these tables, larger participation factors
in each column are bolded. By looking at these tables, one can
readily find the participation of each state variable in system
modes. For example, based on Table I and using participation
factors related to λ9,10, one can see that this mode is primarily
associated to iqs, idr, and dc-link voltage, i.e., vdc. Moreover,
using Table I, it can be observed that ¯ωm and the rotor-side
converter PI-D (see Fig. 3) have a high participation in mode
λ11,12. In Table II, λ13 to λ22 are nonoscillatory and stable
modes, and one can easily find the participation of each state
variables on these modes by looking at this table. These modes
will not be further discussed.
C. Identification of System Modes
Here, the nature of modes λ1,2, λ3,4, λ5,6, and λ7,8 is
identified.
1) Identification of SSR and SupSR Modes: Table I
shows that modes λ1,2 and λ3,4 are primarily associated with
iqs, ids, iqr, and idr. With the frequency of 20.9947 Hz (or
131.913 rad/s) and λ3,4 with the frequency of 98.23 Hz (or
617.197 rad/s) are the SSR and supersynchronous (SupSR)
modes (Mode 1 and Mode 2), respectively. This can be verified
using (24), where fn is calculated to be around 39 Hz. Given
the synchronously rotating reference frame, the complementary
SSR and SupSR frequencies are fs − fn = 21 Hz and fs +
fn = 99 Hz, which matches the frequency of λ1,2 and λ3,4.
Table I also shows that the SSR mode at 75% compensation
and 7-m/s wind speed is unstable as the real part of this mode
is positive, whereas the SupSR mode is stable.
2) Identification of Electromechanical Mode: In order to
identify the nature of this mode, Table III shows this mode
for different wind speeds and series compensation levels. In
this table, the optimum shaft turbine speed and corresponding
frequency related to each wind speed is also given using the
MPPT plot shown in Fig. 2. It is seen that the frequency
of this mode is changed with the change in the wind speed,
whereas changing the compensation level has a slight impact
on this mode. It can be observed that the frequency of this
mode is the complimentary of the frequency of shaft turbine
speed. For example, for the wind speed equal to 7 m/s and
compensation level equal to 75%, the frequency of this mode
is 99.97 rad/s or 15.9 Hz, and its complementary is calculated
to be 44.1 Hz (60 − 15.9 = 44.1 Hz). This frequency coincides
with the frequency of the shaft turbine, i.e., 45 Hz. This can
also be applied to other wind speeds; thus, this mode is related
to wind speed change and, therefore, mechanical dynamics.
Moreover, using Table I, it is observed that λ5,6 is mostly
associated with iqs and ids, iqr, and idr. Therefore, this mode is
Fig. 6. Real part of eigenvalues when wind speed is (a) 7 m/s or
(b) 9 m/s.
related to both mechanical and electrical dynamics and is called
electromechanical mode (Mode 3).
3) Identification of Shaft Mode: From Table I, it is ob-
served that the generator rotor speed ¯ωr and the mechani-
cal torque between two masses, i.e., Ttg, have the highest
participation in λ7,8. Therefore, λ7,8 is related to the shaft
mode (Mode 4). The shaft mode has low frequency, i.e., about
0.954 Hz (or 5.999 rad/s), and this mode at the present operating
condition is stable. This mode might be unstable if the series
compensation level becomes too high, which will cause TI
(torsional interactions) SSR-Type.
Fig. 6 shows the real part of Mode 1 through Mode 4 at
various compensation levels and different wind speeds. As
this figure shows, Mode 2 through Mode 4 are stable for
different operating points of the wind farm. Fig. 6(a) shows
that Mode 1, i.e., SSR mode, becomes unstable when the series
compensation level increases and wind speed is 7 m/s. Fig. 6(b)
shows that at higher wind speed, when the series compensation
level increases, Mode 1 remains stable, even at a very high
compensation level. This can be explained as follows: The
slip of the system, i.e., S, depends on the wind speed and,
consequently, on electric frequency corresponding to the gen-
erator rotor speed, i.e., fr. At the constant series compensation
level, e.g., 90%, the network’s oscillating frequency, i.e., fn, is
constant, and it is less than fr. When wind speed increases, fr
will be also increased, and therefore, the absolute value of the
slip will be increased. This decreases the value of Rr/S, which
consequently provides less negative resistance to the system.
Therefore, increasing the wind speed makes the SSR mode
more stable.
D. Time-Domain Simulation in SimPowerSystems
In order to confirm the eigenvalue analysis provided in
Tables I and II, time-domain simulation is also presented. Fig. 7
shows the system response when the series compensation is
75% and the wind speed is 7 m/s. Note that in the given
simulation result, the system is first started with a low se-
ries compensation level, i.e., 15%, and then at t = 1 s, the
compensation level is changed. As this figure shows, as we
expected from Table I and Fig. 6, the system is unstable, and

Fig. 7. (a) Electric torque, (b) generator rotor speed, and (c) DFIG
terminal voltage at 75% compensation level and 7-m/s wind speed.
Fig. 8. (a) Electric torque, (b) generator rotor speed, and (c) DFIG
terminal voltage at 90% compensation level and 9-m/s wind speed.
the oscillating frequency is about 21.27 Hz, which coincides
with what is calculated in Table I using modal analysis.
In order to study the SSR in a wind farm at higher wind
speeds, Fig. 8 shows the wind farm response at 9-m/s wind
speed and 90% series compensation. As we expected in Fig. 6,
the system is stable at this operating condition, even for a very
high series compensation level, i.e., 90%.
V. GCSC: STRUCTURE AND CONTROL
FACTS are defined as a high-power electronic-based system
and other static equipment controlling one or several trans-
mission systems to improve their controllability and power
transfer capability. Generally, high-power electronic devices
include a variety of diodes, transistors, silicon-controlled rec-
tifiers (SCRs), and gate turn-off thyristors (GTOs). Unlike the
conventional thyristors or SCRs, GTOs are fully controllable,
and they can be turned on and off by their gate. Nowadays,
SCRs and high-power GTOs are widely used for FACTS con-
trollers. GCSC is a family of series FACTS devices that uses
GTO switches that can be turned on and off by its gate [7].
A. Principle of Operation and Generated Harmonics
A GCSC (one per phase), as shown in Fig. 9, is composed
of a fixed capacitor in parallel with a pair of GTOs. The switch
in the GCSC is turned off at the angle β, measured from the
Fig. 9. Single line configuration of the GCSC. vcg = voltage across the
GCSC, iL =transmission line’s current, icg =GCSC capacitor current,
Xcg = fixed capacitance of the GCSC.
Fig. 10. Line current iL(t), capacitor voltage vcg(t), and switching
function of the GCSC. β = GCSC’s turn-off angle γ = the angle of the
advance(π/2 − β), δ = hold-off angle (π − 2β = 2γ).
peak value of the line current. Fig. 10 shows the line current,
capacitor voltage, and the GTO pulse waveform. As seen in this
figure, the GTO switch is closed, when vcg(t) is equal to zero.
The effective capacitance of the GCSC is given by [13]
XG =
Xcg
π
(2γ − sin 2γ) =
Xcg
π
(δ − sin δ) (27)
where γ is the the angle of the advance, δ is the hold-off angle,
and XCfg is the fixed capacitance of the GCSC. As δ changes
from 0◦
to 180◦
, XG varies from 0 to Xcg.
The voltage across the GCSC contains odd harmonics, in ad-
dition to the fundamental components. The harmonic analysis
of the GCSC and some methods to reduce the harmonic levels
have already been studied in the literature [22], [23]. It has been
shown that the maximum total harmonic distortion (THD) of
the GCSC voltage, when a single GCSC module is used, is
about 4.5%. However, in practice, multimodule GCSCs, which
use smaller GCSC modules in series so that each module com-
pensates part of the total required series compensation level, are
used in order to obtain the required power rating for the GCSC.
Using this configuration, the THD generated by the GCSC can
be reduced down to 1.5%. In this method, the voltage of each
GCSC module still contains all the harmonic components of the
single GCSC module, but with lower magnitude [22].
Another method for reducing harmonic levels in the GCSC
voltage is using multipulse arrangements [22]. In this method,
transformers are used to inject the GCSC voltage into the
transmission line, and the transformer windings are connected
in such a way that some lower order harmonics of the GCSC

Fig. 11. Block diagram of the GCSC controller.
Fig. 12. Block diagram of the GCSC PSC.
voltage are canceled out. Using this method, the THD of the
GCSC voltage could be reduced to less than 0.34%, which is
an acceptable level of the THD level in high-voltage power
systems and FACTS applications [22]. More details of the
harmonic analysis of the GCSC can be found in [22] and [23].
B. GCSC Modeling and Control
The operation of the GCSC is modeled as a variable capaci-
tor. It is assumed that the desired value of the GCSC reactance
is implemented within a well-defined time frame, i.e., a delay.
The delay can be modeled by a first-order lag as shown in
Fig. 11, which will add one more order to the system. In Fig. 11,
XPSC is determined by the PSC. In [12] and [13], a power
controller has been used for the GCSC to damp SSR and power
oscillation; however, as shown later, this power controller may
not be adequate to damp the SSR. Therefore, an auxiliary
SSRDC, as shown in Fig. 11, should be added to the GCSC
controller to enable it to damp the SSR.
1) PSC: The block diagram of the GCSC’s PSC control is
shown in Fig. 12. In this figure, Tm is the time constant of a
first-order lowpass filter associated with the measurement of
the line current. In this controller, the measured line current Im
is compared with a reference current I∗
L, and the error ΔI is
passed through a lead controller and a PI regulator.
The MPPT curve and the chosen reactive power control strat-
egy for the transmission line, i.e., fixed Var flow or fixed power
factor, are used to obtain I∗
L. If the power losses are ignored, the
optimum input wind power ¯Pω, which can be obtained using the
MPPT curve for different wind speeds, is equal to the desired
delivered real power to the transmission line, i.e., P∗
L (p.u.).
Furthermore, depending on the chosen reactive power control
strategy for the transmission line, i.e., fixed Var flow or fixed
power factor, the desired reactive power of the transmission
line, i.e., Q∗
L (p.u.), can be determined. Then, the transmission
line reference line current can be calculated as follows:
I∗
L =
P∗
L
2
+ Q∗
L
2
V ∗
s
. (28)
Fig. 13. Real part of Modes 1 and 2 when wind speed is (a) 7 m/s or
(b) 9 m/s with GCSC and fixed capacitor in line.
A modal analysis at different operating points of the wind
farm is performed when the GCSC model with PSC is added
to the system. Fig. 13 compares the real part of Modes 1 and 2
at different compensation levels and different wind speeds for
two cases: 1) when the DFIG wind farm is compensated only
with a series fixed capacitor; 2) when the DFIG wind farm is
compensated with a GCSC without SSRDC and only with a
PSC. As seen in this figure, using only the PSC in GCSC not
only does not enable this device to stabilize Mode 1 but also
decreases the damping of Mode 1. This shows that an auxiliary
SSRDC is needed to enable the GCSC to damp the SSR.
2) SSRDC: In order to enhance the SSR damping, an aux-
iliary controller is added to the GCSC control system with an
appropriate ICS, as shown in Fig. 11. The question is how an
appropriate ICS should be selected. This question is answered
in the following sections.
VI. ICS SELECTION AND SSRDC DESIGN
A. ICS Selection Using Residues
The residues corresponding to SSR and SupSR modes for
different ICSs are computed. If the state-space model and
transfer function of the single-input single-output are defined
as [21]
˙X = AX + BU (29)
Y = CX (30)
G(s) =
Y (s)
U(s)
=
n
i=1
Ri
s − λi
(31)
then for a complex root λi, the residue Ri is a complex number,
which can be considered as a vector having a certain direction,
and can be expressed as [21]
Ri = CΨiΦiB. (32)
In a root-locus diagram, Ri is a representation of the direc-
tion and speed of the closed-loop eigenvalue λci, which leaves

Fig. 14. Residues of the SSR mode with ¯ωr as ICS.
Fig. 15. Residues of the SSR mode with IL as ICS.
the pole λi. The effect of the residues in selecting ICS can be
described as follows. Suppose that dynamics of all eigenvalues
are ignored, except one specific eigenvalue λa. This means that
the open-loop transfer function of the system has only one pole,
which can be represented as
Ga(s) =
Ra
s − λa
. (33)
Using (33), the closed-loop system with a gain controller, i.e.,
Kgc, is represented as follows:
Gca(s) =
Ra
s − λa + KgcRa
. (34)
Finally, using (34), the root of the closed loop and the shift in
the eigenvalues, i.e., Δλsh, can be represented using (35) and
(36), as follows:
λca = λa − KgcRa (35)
Δλsh = −KgcRa. (36)
Equation (36) shows that the residue influences the closed-
loop system root, by determining the direction and speed of it.
If the magnitude of the residue is large enough, then a smaller
gain is needed for the feedback control system.
Figs. 14–16 show the residues of the SSR and SupSR modes
at different operating conditions of the wind farm, when ¯ωr,
IL, and Vcg are used as ICS. Fig. 14 shows that when ¯ωr is
selected as ICS, the residue magnitude of the SSR mode is
small. Therefore, if this signal is being used as ICS, a larger
gain will be needed for the feedback control. In addition, as
Fig. 14 shows, the residues of the SSR and SupSR modes are
in an opposite direction, which will increase the difficulty of
Fig. 16. Residues of the SSR mode with Vcg as ICS.
Fig. 17. Root-locus diagram of the SSR mode with IL as ICS. The
+ sign indicates the locations of the roots corresponding to the indicated
gain, i.e., Kgc.
the controller design. The reason is that a simple proportional
controller chosen to increase damping of the SSR mode will
decrease the damping of the SupSR mode, verifying that ¯ωr is
not an optimum choice for ICS. Therefore, this signal will not
be further considered.
Fig. 15 shows that when IL is selected as ICS, the residue
magnitude of the SSR mode is rather large, and therefore, a
smaller feedback gain is needed to stabilize the SSR mode.
However, since the residues of the SSR and SupSR modes in
this case point in opposite directions, stabilizing the SSR mode
via a feedback gain will decrease the SupSR mode damping.
This shows that the line current may not be an optimum
parameter as ICS. This signal as ICS will be further analyzed in
the next section.
Finally, Fig. 16 shows the residue of the SSR and SupSR
modes, when Vcg is selected as an ICS. This figure exhibits two
facts: First, it shows that the SSR and SupSR modes are in the
same direction; second, the magnitude of the residues is large
enough. These properties will make the design of the feedback
control simple so that a small gain will be enough to force both
the SSR and SupSR modes to move to the left and make the
system stable. In the next sections, both IL and Vcg are studied
in more detail as two potential ICSs.
B. Root-Locus Analysis
The analysis presented in Section VI-A is verified using root-
locus analysis. As represented in Fig. 17 for IL as ICS, when
the gain increases, the SSR and SupSR modes will move in

Fig. 18. Root-locus diagram of the SSR mode with Vcg as ICS. The
+ sign indicates the locations of the roots corresponding to the indicated
gain, i.e., Kgc.
opposite directions, as we expected from residues analysis. In
addition, the maximum damping ratio for SSR mode is obtained
at 3%, and the corresponding gain in this case is about 0.282,
as indicated in Fig. 17. For this gain, the corresponding SupSR
mode will move toward the right-hand side of the root-locus
diagram but will not pass the imaginary axis, and the system is
still stable.
Fig. 18 represents the root-locus diagram of the system when
Vcg is an ICS. This ﬁgure shows that when the gain increases,
both the SSR and SupSR modes move to the left-hand side of
the root-locus plane. In this case, in order to have 5% damping
ratio for SSR mode, the gain is computed as 0.598, as indicated
in Fig. 18. For this gain, the corresponding SupSR mode will
move toward the left-hand side of the root-locus diagram and
become more stable.
In conclusion, the root-locus diagram results presented in this
section and residues analysis presented in Section VI-A show
that both IL and Vcg could be used as ICS; however, using
the latter, a larger damping ratio can be obtained, and also, both
the SSR and SupSR modes can be simultaneously stabilized
by the use of the proposed procedure.
VII. TIME-DOMAIN SIMULATION OF
GCSC-COMPENSATED DFIG IN SIMPOWERSYSTEMS
Here, the time-domain simulation of the DFIG wind farm is
presented to verify the analysis presented in Section VI. The
system is simulated for different scenarios, namely, the wind
farm compensated by the GCSC with no SSRDC (open loop),
the wind farm compensated by the GCSC and IL as ICS to the
SSRDC (IL as ICS), and the wind farm compensated by
the GCSC and Vcg as ICS to the SSRDC (Vcg as ICS). In
the simulation study, initially, the compensation level is reg-
ulated at 50%, and then at t = 1 s, the compensation level
is changed to 75%. The dynamic responses of the wind farm
including electric torque Te, terminal voltage Vs, and dc-link
voltage Vdc are plotted in Figs. 19–21, respectively.
Fig. 19 shows the electric torque Te of the system for
three cases. As Fig. 19(a) shows, the wind farm is unstable
due to the SSR mode when the GCSC is not equipped with
SSRDC. The wind farm equipped by the GCSC and SSRDC
with either IL or Vcg as ICSs can effectively damp out the
Fig. 19. Comparing dynamic response of the electric torque without
SSRDC and with SSRDC (IL and Vcg as ICS). (a) Simulation time from
t = 0 s to t = 4 s. (b) Simulation time from t = 0.9 s to t = 1.9 s.
Fig. 20. Comparing dynamic response of the terminal voltage without
Fig. 21. Comparing dynamic response of the dc-link voltage without
SSR mode and stabilize the system. Fig. 19(b) shows that when
Vcg is used as ICS, both SSR and SupSR modes are mitigated
faster compared with the case when IL is used as ICS. This

Fig. 22. Power factor of the DFIG wind farm. (a) Simulation time from
t = 1 s to t = 2 s. (b) Simulation time from t = 1 s to t = 25 s.
confirms the analysis presented in Section VI that using IL as
ICS decreases the damping of the SupSR mode and that the
maximum damping ratio for the SSR mode is limited to less
than 3%. A similar behavior can be observed using Figs. 20
and 21, where the terminal voltage Vs and dc-link voltage Vdc
are plotted, respectively.
Finally, in order to show that the control system guarantees
the unity power factor, Fig. 22 compares the power factor of the
system when IL or Vcg is used as ICS. As seen in this figure,
the control system is able to maintain the unity power factor
for the wind farm using both ISCs. Once again, using Vcg as
ICS provides better SSR and SupSR damping for the system
compared with when IL is used as ICS.
VIII. CONCLUSION
This paper has proposed application, modeling, and con-
trol of the GCSC, a series FACTS device, for transmission
line compensation and SSR mitigation in DFIG-based wind
farms using modal analysis. First, the studied fixed-series-
compensated DFIG-based wind farm is modeled for small-
signal stability analysis. The eigenvalues of the 22nd-order
model of the system are obtained. Moreover, using participation
factors, the participation of each state to the each system mode
is identified. The main modes of the DFIG-based wind farm
including SSR, SupSR, electromechanical, and shaft modes are
identified. The results show that the fixed-series-compensated
DFIG-based wind farm is highly unstable due to the SSR mode.
Therefore, in order to stabilize the SSR mode, a series
FACTS device, i.e., GCSC, replaces with the fixed-series ca-
pacitor. Using residue-based analysis, three different signals,
namely, generator rotor speed ¯ωr, line current IL, and voltage
across the GCSC Vcg are examined in order to find the optimal
ICS to the GCSC’s SSRDC.
The residue-based analysis shows that the rotor speed is not
an optimum ICS for the SSRDC for two reasons: First, a very
large gain is needed in this case, and second, it is not possible to
simultaneously increase the damping of both SSR and SupSR
modes. Moreover, although the residue-based analysis for the
line current as ICS predicts that a smaller gain is needed to
TABLE IV
PARAMETERS OF THE SINGLE 2-MW AND 100-MW AGGREGATED DFIG.
VALUES ARE IN (p.u.), UNLESS IT IS MENTIONED
TABLE V
PARAMETERS OF THE NETWORK AND SHAFT SYSTEM. VALUES
ARE IN (p.u.), UNLESS IT IS MENTIONED
TABLE VI
PARAMETERS OF THE CONTROLLERS
damp the SSR mode, the SupSR mode’s stability is decreased in
this case, indicating that this signal may not be an optimum ICS.
The residue-based analysis for the voltage across the GCSC,
however, predicts that this signal can increase the stability of
both the SSR ans SupSR modes, simultaneously.
In addition, using root-locus diagrams, the required gain to
damp the SSR mode is computed for both line current and
voltage across the GCSC as ICSs. The results show that, unlike
the line current as ICS, using voltage across the series capacitor
as ICS can guarantee the damping of the SSR mode, without
sacrificing the SupSR mode’s stability, verifying what was
expected from the residue-based analysis. Moreover, the maxi-
mum SSR damping ratio, when the voltage across the GCSC is
used as ICS, is 67% more compared with that of the line current.
Finally, time-domain simulation is used to verify the design
process using residue-based analysis and root-locus diagrams.
The work presented in this paper was completely simulation
based. A thorough discussion of technology feasibility issues
for a practical implementation of the proposed control scheme
is left as future work.
APPENDIX
The parameters used are given in Tables IV–VI.
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Hossein Ali Mohammadpour (S’10) received
the B.Sc. and M.Sc. degrees in electrical engi-
neering and power systems from Iran Univer-
sity of Science and Technology, Tehran, Iran,
in 2006 and 2009, respectively. He is currently
working toward the Ph.D. degree in electrical
engineering at the University of South Carolina,
Columbia, SC, USA.
His primary research interests include power
systems stability, power electronics, renewable
energy, ﬂexible ac transmission system tech-
nologies, and electric ship system modeling and analysis.
Enrico Santi (S’90–M’94–SM’02) received the
Dr. Ing. degree in electrical engineering from the
University of Padua, Padova, Italy, in 1988 and
the M.S. and Ph.D. degrees from the California
Institute of Technology, Pasadena, CA, USA, in
1989 and 1994, respectively.
From 1993 to 1998, he was a Senior Design
Engineer with TESLAco, where he was respon-
sible for the development of various switch-
ing power supplies for commercial applications.
Since 1998, he has been with the University of
South Carolina, Columbia, SC, USA, where he is currently an Associate
Professor in the Department of Electrical Engineering. He has published
over 100 papers on power electronics and modeling and simulation
in international journals and conference proceedings and holds two
patents. His research interests include switched-mode power convert-
ers, advanced modeling and simulation of power systems, modeling
and simulation of semiconductor power devices, and control of power
electronics systems.

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