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-ControlledSeries Capacitor Interfaced With a

DFIG-Based Wind FarmHossein Ali Mohammadpour, Student Member, IEEE , and Enrico Santi, Senior Member, IEEE

Abstract—This paper presents application and control ofthe gate-controlled series capacitor (GCSC) for series com-pensation and subsynchronous resonance (SSR) dampingin doubly-fed induction generator (DFIG)-based wind farms.The GCSC is a new series FACTS device composed of afixed capacitor in parallel with a pair of antiparallel gate-commuted switches. The study considers a DFIG-basedwind farm, which is connected to a series-compensatedtransmission line whose parameters are derived from theIEEE first benchmark model for computer simulation of theSSR. The small-signal stability analysis of the system ispresented, and the eigenvalues of the system are obtained.Using both modal analysis and time-domain simulation, itis shown that the system is potentially unstable due to theSSR mode. Therefore, the wind farm is equipped with aGCSC to solve the instability of the wind farm resulting fromthe SSR mode, and an SSR damping controller (SSRDC)is designed for this device using residue-based analysisand 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 othermodes, and using root-locus analysis, the required gain forthe SSRDC is determined. MATLAB/Simulink is used as atool for modeling, design, and time-domain simulations.

Index Terms—Doubly fed induction generator (DFIG),flexible ac transmission systems (FACTS), gate-controlledseries capacitor (GCSC), root-locus diagram, subsyn-chronous resonance (SSR).

I. INTRODUCTION

DUE to the recent rapid penetration of wind power into thepower systems [1], some countries in central Europe, e.g.,

Germany, have run out of suitable sites for onshore wind powerprojects, due to the high population density in these countries.Moreover, it has been found that the offshore wind powerresources 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. Dateof publication August 12, 2014; date of current version January 7,2015. This work was supported by the National Science FoundationIndustry/University Cooperative Research Center for Grid-ConnectedAdvanced 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:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2014.2347007

scale sustainable electric energy resources [2]. Recently, thedoubly fed induction generator (DFIG) has gained significantattention from the electric power industry in offshore windfarms and renewable energy sources [1], [3].

However, in offshore wind farms, the distance between thewind turbines and the shore is much longer [4] than thatin onshore wind farms. Therefore, unlike the onshore windfarms—where the voltage level of the wind farm is usually thesame as the voltage level of the distribution system—highervoltage levels with reliable and efficient transmission lines arerequired for the offshore wind farms to minimize the powerlosses [2].

Currently, there are numerous large offshore wind farmsoperating throughout the world [2], [5]. Future projects inoffshore wind farms will be larger in size and further away fromthe shore [2]. This requires defining new concepts for the trans-mission system, including transmission lines from the offshorewind farm to the shore and network integration to the onshorepower system. The transmission system options to transmit thewind power to the shore are high-voltage ac (HVAC) [2] orhigh-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 powertransmission lines has been used for many years to increase thetransmittable power of transmission lines [7]. For the purposeof increasing the power transfer capability of a transmissionline, series compensation is preferred compared with shuntcompensation. One of the main reasons is that, unlike shuntcompensation, series compensation is less sensitive to systemload 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 toa 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 supportmassive integration of renewable energy resources into thepower networks [8]. The advantages of using FACTS devicessuch as static synchronous compensator (STATCOM) [7], staticVAR compensator (SVC) [9], synchronous series compensator(SSSC) [7], unified power flow controller (UPFC) [10], andthyristor-controlled series compensator (TCSC) [9] in powersystems are well known.

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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 FACTSdevice proposed for series compensation of transmission lines[7]. The application of this device to control the power flowhas been investigated [11]. For each phase, the GCSC consistsof a fixed capacitor in parallel with a pair of antiparallel gate-commuted switches. By controlling the turn-off angle of thegates in the GCSC, this device can provide a variable seriescapacitor for the transmission line [12], [13]. Compared withother FACTS devices such as SSSC, the GCSC has the advan-tages of being less complex and less expensive. Moreover, agood comparison of the TCSC and GCSC proves that the latteris a better device from cost and performance point of view [14].

This paper proposes application and control of the GCSCFACTS for series compensation and SSR mitigation in off-shore DFIG-based wind farms. This paper is organized asfollows. In Section II, the studied power system is brieflydescribed. In Section III, modeling of the system for small-signal stability analysis is presented. The model of the systemincludes wind turbine aerodynamics, an eighth-order modelof grid-side converter (GSC) and rotor-side converter (RSC)controllers, a first-order model for the dc link between theGSC and the RSC, a sixth-order model for DFIG, a third-order model for the shaft system, and a fourth-order modelfor series-compensated transmission line. In Section IV, theSSR phenomenon in fixed-series-compensated DFIG is studied.Here, first, the SSR phenomenon in a fixed-compensated windfarm is briefly explained. Then, the eigenvalues of the systemare obtained, and the participation factor is used to show howdifferent parts of the system are related to each eigenvalue.Here, time-domain simulation is also presented to verify theeigenvalue analysis. In Section V, the principles of operationof the GCSC, generated harmonics, and its control systemare presented. The control system of the GCSC is composedof power scheduling control (PSC) and SSR damping control(SSRDC) blocks. In Section VI, an SSRDC is designed usingresidue-based analysis and root-locus diagrams. Using residue-based analysis, the most effective and optimal input controlsignal (ICS) to the GCSC’s SSRDC is identified, which candamp the SSR mode without decreasing the damping ratio ofother system modes. Moreover, using root-locus diagrams, therequired SSRDC gain in order to obtain the desired dampingratio for the SSR mode is computed. In Section VII, the

Fig. 2. Wind power Pm (p.u.), wind turbine shaft speed ωm (p.u.), andwind speed Vω (m/s) relationship.

time-domain simulation of the GCSC-compensated DFIG ispresented to verify the design process. Finally, Section VIIIconcludes this paper.

II. POWER SYSTEM DESCRIPTION

The studied power system, shown in Fig. 1, is adaptedfrom the IEEE first benchmark model for SSR studies [15].In this system, a 100-MW DFIG-based offshore wind farm isconnected to the infinite bus via a 161-kV series-compensatedtransmission line [16]. The 100-MW wind farm is an aggre-gated model of 50 wind turbine units, where each unit has apower rating of 2 MW. In fact, a 2-MW wind turbine is scaledup to represent the 100-MW wind farm. This simplification issupported by several studies [6], [17]. The systems data aregiven in the Appendix.

III. MODELING OF DFIG-BASED WIND TURBINE

The overall power system model shown in Fig. 1 includesdynamic models of wind turbine aerodynamics, shaft system,induction generator (IG), RSC and GSC controllers, dc link,and series-compensated transmission line. These models arenow described.

A. Wind-Turbine Aerodynamics

The wind power can be calculated from the wind speed Vw

as follows [18]:

Tω =0.5ρπR2CPV

2w

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 windturbine (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

RCfλω (2)

where Cf is the wind turbine blade design constant, and θ is thewind 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. 2shows the wind power versus wind turbine shaft speed in perunit for various wind speeds with indication of the MPPT curve.To enforce operation on the MPPT curve, for a given windspeed Vω, the optimal reference power and optimal rotationalspeed are obtained. Note that due to power converters ratings, itmay not be practical to always work on the MPPT cure. In thiscase, for very low wind speeds, the DFIG operates at almostconstant rotational speed. On the other hand, when the windspeed increases so that it exceeds the turbine torque rating, theDFIG will work in maximum constant torque [19].

The aim of the GSC and the RSC is to enable the DFIG towork on the MPPT curve. Note that the converters are assumedto store no energy so that their losses can be neglected andoperate 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 forregulating the electric torque, i.e., Te, and stator reactive power,i.e., Qs. In a steady-state condition, neglecting power losses, thewind torque, i.e., Tω = Pω/ωm, is equal to electric torque, i.e.,Te. Therefore, the reference torque, i.e., T ∗

e , can be calculatedbased 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 reactivepower or unity power factor [17]. In this paper, the latter methodis chosen.

Moreover, the GSC is responsible for controlling the dc-linkvoltage, i.e., Vdc, and the induction generator’s terminal volt-age, i.e., Vs [17]. The GSC and RSC controllers add eight statevariables to the system, due to the eight proportional–integral(PI) controllers, and their state variables are defined as avector XRG.

C. Modeling of the DC Link

The dynamic of the capacitor in the dc-link between the GSCand the RSC can be expressed by a first-order model as fol-lows [19]:

−Cvdcdvdcdt

= 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 by0.5 (vqriqr + vdridr), and Pg is the GSC active power and isgiven by 0.5 (vqgiqg + vdgidg).

D. Modeling of the Induction Machine

The IG currents are selected as state variable, and the IG isrepresented 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 zerosequence current and voltage components (p.u.), respectively.

The AIG and BIG matrices are defined as follows. We firstdefine the matrices given in (8), shown at the bottom of thepage, and (9) as

G =

⎡⎢⎢⎢⎢⎢⎣

Xss 0 0 XM 0 00 Xss 0 0 XM 00 0 Xls 0 0 0

XM 0 0 Xrr 0 00 XM 0 0 Xrr 00 0 0 0 0 Xlr

⎤⎥⎥⎥⎥⎥⎦ . (9)

F =

⎡⎢⎢⎢⎢⎢⎢⎣

Rsωe

ωbXSS 0 0 ωe

ωbXM 0

−ωe

ωbXSS Rs 0 −ωe

ωbXM 0 0

0 0 Rs 0 0 00 (ωe−ωr)

ωbXM 0 Rr 0 (ωe−ωr)

ωbXrr

− (ωe−ωr)ωb

XM 0 0 − (ωe−ωr)ωb

Xrr Rr 00 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 magnetizingreactance (p.u.), Xss = Xls +XM (p.u.), Xrr is equal toXlr +XM (p.u.), Rr is the rotor resistance (p.u.), Rs is thestator resistance (p.u.), ωb is the base radian frequency (rad/s),ωr is the generator rotor speed (rad/s), and ωe is the rotatingsynchronous frame frequency (rad/s).

E. Modeling of the Shaft System

The shaft of the wind turbine system can be represented as atwo-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 adamper. 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

−12Ht

Dtg

2Hg

(−Dt−Dtg)2Hg

−12Hg

Ktgωb −Ktgωb 0

⎤⎥⎦ (15)

Bshaft =

⎡⎣ 1

2Ht0 0

0 12Hg

00 0 1

⎤⎦ . (16)

In the shaft equations, Te is the electric torque and is givenby 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 windtorque (p.u.), Dg and Dt are the damping coefficient of thegenerator and turbine (p.u.), Dtg is the damping coefficientbetween the two masses (p.u.), Ktg is the inertia constant of theturbine and generator (p.u./rad), and Hg and Ht are the inertiaconstants 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 inthe 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)

XL0 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

−1XL

0

ωe−RL

XL0 −1

XL

XC 0 −ωe 00 XC ωe 0

⎤⎥⎥⎦ (20)

BTline =

⎡⎢⎣ωb 0 0 00 ωb 0 00 0 1 00 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 thetransmission 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. 1given in this section, the entire DFIG system is a 22nd-ordermodel and can be expressed as

X = f(X,U, t) (22)

where

X =[XT

IG XTshaft X

TTline vdc X

TRG

]T. (23)

The nonlinear system equations developed in Section III arelinearized around an operating point to calculate the linearizedstate space matrices A, B, C, and D [18]. This can be performedusing MATLAB/Simulink. Equation (22) was assembled inMATLAB/Simulink, and eigenvalues were obtained using the“linmod” function. In the next section, the DFIG performanceis briefly analyzed using modal analysis and time-domainsimulation.

IV. SSR IN FIXED-SERIES-COMPENSATED DFIG

A. IGE in Wind Farms

A series-compensated power system with a compensationlevel defined as K = XC/Xe excites subsynchronous currentsat frequency given by [15]

fn = fs

√KXe∑

X(24)

where Xe is equal to XL +XT ,∑

X is the entire inductivereactance seen from infinite bus, fn is the natural frequency ofthe electric system, and fs is the fundamental frequency of thesystem.


TABLE ISYSTEM MODES AND PARTICIPATION FACTORS AT 75% SERIES COMPENSATION AND 7-m/s WIND SPEED (PART I)

TABLE IISYSTEM 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 negativesince the natural resonance frequency, i.e., fn, is less than theelectrical 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 magnitudeof the equivalent rotor resistance, i.e., Rr/S < 0, exceeds thesum of the resistances of the armature and the network, therewill be a negative resistance at the subsynchronous frequency,and the subsynchronous current would increase with time. Thisphenomenon 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 aneigenvalue, 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 modesor state variables, and Ψ and Φ are right and left eigenvectors,respectively.


TABLE IIIλ5,6 AT DIFFERENT WIND SPEEDS AND COMPENSATION LEVELS

Tables I and II show the eigenvalues and participation factorsof the system when the wind speed is 7 m/s and the compen-sation level is 75%. In these tables, larger participation factorsin each column are bolded. By looking at these tables, one canreadily find the participation of each state variable in systemmodes. For example, based on Table I and using participationfactors related to λ9,10, one can see that this mode is primarilyassociated to iqs, idr, and dc-link voltage, i.e., vdc. Moreover,using Table I, it can be observed that ωm and the rotor-sideconverter PI-D (see Fig. 3) have a high participation in modeλ11,12. In Table II, λ13 to λ22 are nonoscillatory and stablemodes, and one can easily find the participation of each statevariables on these modes by looking at this table. These modeswill not be further discussed.

C. Identification of System Modes

Here, the nature of modes λ1,2, λ3,4, λ5,6, and λ7,8 isidentified.

1) Identification of SSR and SupSR Modes: Table Ishows that modes λ1,2 and λ3,4 are primarily associated withiqs, ids, iqr, and idr. With the frequency of 20.9947 Hz (or131.913 rad/s) and λ3,4 with the frequency of 98.23 Hz (or617.197 rad/s) are the SSR and supersynchronous (SupSR)modes (Mode 1 and Mode 2), respectively. This can be verifiedusing (24), where fn is calculated to be around 39 Hz. Giventhe synchronously rotating reference frame, the complementarySSR 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% compensationand 7-m/s wind speed is unstable as the real part of this modeis positive, whereas the SupSR mode is stable.

2) Identification of Electromechanical Mode: In order toidentify the nature of this mode, Table III shows this modefor different wind speeds and series compensation levels. Inthis table, the optimum shaft turbine speed and correspondingfrequency related to each wind speed is also given using theMPPT plot shown in Fig. 2. It is seen that the frequencyof this mode is changed with the change in the wind speed,whereas changing the compensation level has a slight impacton this mode. It can be observed that the frequency of thismode is the complimentary of the frequency of shaft turbinespeed. For example, for the wind speed equal to 7 m/s andcompensation level equal to 75%, the frequency of this modeis 99.97 rad/s or 15.9 Hz, and its complementary is calculatedto be 44.1 Hz (60− 15.9 = 44.1 Hz). This frequency coincideswith the frequency of the shaft turbine, i.e., 45 Hz. This canalso be applied to other wind speeds; thus, this mode is relatedto wind speed change and, therefore, mechanical dynamics.Moreover, using Table I, it is observed that λ5,6 is mostlyassociated 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 calledelectromechanical 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 highestparticipation in λ7,8. Therefore, λ7,8 is related to the shaftmode (Mode 4). The shaft mode has low frequency, i.e., about0.954 Hz (or 5.999 rad/s), and this mode at the present operatingcondition is stable. This mode might be unstable if the seriescompensation level becomes too high, which will cause TI(torsional interactions) SSR-Type.

Fig. 6 shows the real part of Mode 1 through Mode 4 atvarious compensation levels and different wind speeds. Asthis figure shows, Mode 2 through Mode 4 are stable fordifferent operating points of the wind farm. Fig. 6(a) showsthat Mode 1, i.e., SSR mode, becomes unstable when the seriescompensation level increases and wind speed is 7 m/s. Fig. 6(b)shows that at higher wind speed, when the series compensationlevel increases, Mode 1 remains stable, even at a very highcompensation level. This can be explained as follows: Theslip 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 compensationlevel, e.g., 90%, the network’s oscillating frequency, i.e., fn, isconstant, and it is less than fr. When wind speed increases, frwill be also increased, and therefore, the absolute value of theslip will be increased. This decreases the value of Rr/S, whichconsequently provides less negative resistance to the system.Therefore, increasing the wind speed makes the SSR modemore stable.

D. Time-Domain Simulation in SimPowerSystems

In order to confirm the eigenvalue analysis provided inTables I and II, time-domain simulation is also presented. Fig. 7shows the system response when the series compensation is75% and the wind speed is 7 m/s. Note that in the givensimulation result, the system is first started with a low se-ries compensation level, i.e., 15%, and then at t = 1 s, thecompensation level is changed. As this figure shows, as weexpected from Table I and Fig. 6, the system is unstable, and


Fig. 7. (a) Electric torque, (b) generator rotor speed, and (c) DFIGterminal voltage at 75% compensation level and 7-m/s wind speed.

Fig. 8. (a) Electric torque, (b) generator rotor speed, and (c) DFIGterminal voltage at 90% compensation level and 9-m/s wind speed.

the oscillating frequency is about 21.27 Hz, which coincideswith what is calculated in Table I using modal analysis.

In order to study the SSR in a wind farm at higher windspeeds, Fig. 8 shows the wind farm response at 9-m/s windspeed and 90% series compensation. As we expected in Fig. 6,the system is stable at this operating condition, even for a veryhigh series compensation level, i.e., 90%.

V. GCSC: STRUCTURE AND CONTROL

FACTS are defined as a high-power electronic-based systemand other static equipment controlling one or several trans-mission systems to improve their controllability and powertransfer capability. Generally, high-power electronic devicesinclude a variety of diodes, transistors, silicon-controlled rec-tifiers (SCRs), and gate turn-off thyristors (GTOs). Unlike theconventional 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 usesGTO 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 composedof a fixed capacitor in parallel with a pair of GTOs. The switchin the GCSC is turned off at the angle β, measured from the

Fig. 9. Single line configuration of the GCSC. vcg = voltage across theGCSC, 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 switchingfunction of the GCSC. β = GCSC’s turn-off angle γ = the angle of theadvance(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 thisfigure, 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 changesfrom 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 analysisof the GCSC and some methods to reduce the harmonic levelshave already been studied in the literature [22], [23]. It has beenshown that the maximum total harmonic distortion (THD) ofthe GCSC voltage, when a single GCSC module is used, isabout 4.5%. However, in practice, multimodule GCSCs, whichuse smaller GCSC modules in series so that each module com-pensates part of the total required series compensation level, areused in order to obtain the required power rating for the GCSC.Using this configuration, the THD generated by the GCSC canbe reduced down to 1.5%. In this method, the voltage of eachGCSC module still contains all the harmonic components of thesingle GCSC module, but with lower magnitude [22].

Another method for reducing harmonic levels in the GCSCvoltage is using multipulse arrangements [22]. In this method,transformers are used to inject the GCSC voltage into thetransmission line, and the transformer windings are connectedin 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 theGCSC voltage could be reduced to less than 0.34%, which isan acceptable level of the THD level in high-voltage powersystems and FACTS applications [22]. More details of theharmonic 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 reactanceis implemented within a well-defined time frame, i.e., a delay.The delay can be modeled by a first-order lag as shown inFig. 11, which will add one more order to the system. In Fig. 11,XPSC is determined by the PSC. In [12] and [13], a powercontroller has been used for the GCSC to damp SSR and poweroscillation; however, as shown later, this power controller maynot be adequate to damp the SSR. Therefore, an auxiliarySSRDC, as shown in Fig. 11, should be added to the GCSCcontroller to enable it to damp the SSR.

1) PSC: The block diagram of the GCSC’s PSC control isshown in Fig. 12. In this figure, Tm is the time constant of afirst-order lowpass filter associated with the measurement ofthe line current. In this controller, the measured line current Imis compared with a reference current I∗L, and the error ΔI ispassed 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 powerfactor, are used to obtain I∗L. If the power losses are ignored, theoptimum input wind power Pω , which can be obtained using theMPPT curve for different wind speeds, is equal to the desireddelivered real power to the transmission line, i.e., P ∗

L (p.u.).Furthermore, depending on the chosen reactive power controlstrategy for the transmission line, i.e., fixed Var flow or fixedpower factor, the desired reactive power of the transmissionline, i.e., Q∗

L (p.u.), can be determined. Then, the transmissionline reference line current can be calculated as follows:

I∗L =

√P ∗L2 +Q∗

L2

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 windfarm is performed when the GCSC model with PSC is addedto the system. Fig. 13 compares the real part of Modes 1 and 2at different compensation levels and different wind speeds fortwo cases: 1) when the DFIG wind farm is compensated onlywith a series fixed capacitor; 2) when the DFIG wind farm iscompensated with a GCSC without SSRDC and only with aPSC. As seen in this figure, using only the PSC in GCSC notonly does not enable this device to stabilize Mode 1 but alsodecreases the damping of Mode 1. This shows that an auxiliarySSRDC 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 anappropriate ICS, as shown in Fig. 11. The question is how anappropriate ICS should be selected. This question is answeredin the following sections.

VI. ICS SELECTION AND SSRDC DESIGN

A. ICS Selection Using Residues

The residues corresponding to SSR and SupSR modes fordifferent ICSs are computed. If the state-space model andtransfer function of the single-input single-output are definedas [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 bedescribed as follows. Suppose that dynamics of all eigenvaluesare ignored, except one specific eigenvalue λa. This means thatthe 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 inthe 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 smallergain is needed for the feedback control system.

Figs. 14–16 show the residues of the SSR and SupSR modesat different operating conditions of the wind farm, when ωr,IL, and Vcg are used as ICS. Fig. 14 shows that when ωr isselected as ICS, the residue magnitude of the SSR mode issmall. Therefore, if this signal is being used as ICS, a largergain will be needed for the feedback control. In addition, asFig. 14 shows, the residues of the SSR and SupSR modes arein 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 indicatedgain, i.e., Kgc.

the controller design. The reason is that a simple proportionalcontroller chosen to increase damping of the SSR mode willdecrease the damping of the SupSR mode, verifying that ωr isnot an optimum choice for ICS. Therefore, this signal will notbe further considered.

Fig. 15 shows that when IL is selected as ICS, the residuemagnitude of the SSR mode is rather large, and therefore, asmaller feedback gain is needed to stabilize the SSR mode.However, since the residues of the SSR and SupSR modes inthis case point in opposite directions, stabilizing the SSR modevia a feedback gain will decrease the SupSR mode damping.This shows that the line current may not be an optimumparameter as ICS. This signal as ICS will be further analyzed inthe next section.

Finally, Fig. 16 shows the residue of the SSR and SupSRmodes, when Vcg is selected as an ICS. This figure exhibits twofacts: First, it shows that the SSR and SupSR modes are in thesame direction; second, the magnitude of the residues is largeenough. These properties will make the design of the feedbackcontrol simple so that a small gain will be enough to force boththe SSR and SupSR modes to move to the left and make thesystem stable. In the next sections, both IL and Vcg are studiedin 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, whenthe 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 indicatedgain, i.e., Kgc.

opposite directions, as we expected from residues analysis. Inaddition, the maximum damping ratio for SSR mode is obtainedat 3%, and the corresponding gain in this case is about 0.282,as indicated in Fig. 17. For this gain, the corresponding SupSRmode will move toward the right-hand side of the root-locusdiagram but will not pass the imaginary axis, and the system isstill stable.

Fig. 18 represents the root-locus diagram of the system whenVcg is an ICS. This figure shows that when the gain increases,both the SSR and SupSR modes move to the left-hand side ofthe root-locus plane. In this case, in order to have 5% dampingratio for SSR mode, the gain is computed as 0.598, as indicatedin Fig. 18. For this gain, the corresponding SupSR mode willmove toward the left-hand side of the root-locus diagram andbecome more stable.

In conclusion, the root-locus diagram results presented in thissection and residues analysis presented in Section VI-A showthat both IL and Vcg could be used as ICS; however, usingthe latter, a larger damping ratio can be obtained, and also, boththe SSR and SupSR modes can be simultaneously stabilizedby 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 ispresented to verify the analysis presented in Section VI. Thesystem is simulated for different scenarios, namely, the windfarm compensated by the GCSC with no SSRDC (open loop),the wind farm compensated by the GCSC and IL as ICS to theSSRDC (IL as ICS), and the wind farm compensated bythe GCSC and Vcg as ICS to the SSRDC (Vcg as ICS). Inthe simulation study, initially, the compensation level is reg-ulated at 50%, and then at t = 1 s, the compensation levelis changed to 75%. The dynamic responses of the wind farmincluding electric torque Te, terminal voltage Vs, and dc-linkvoltage Vdc are plotted in Figs. 19–21, respectively.

Fig. 19 shows the electric torque Te of the system forthree cases. As Fig. 19(a) shows, the wind farm is unstabledue to the SSR mode when the GCSC is not equipped withSSRDC. The wind farm equipped by the GCSC and SSRDCwith either IL or Vcg as ICSs can effectively damp out the

Fig. 19. Comparing dynamic response of the electric torque withoutSSRDC and with SSRDC (IL and Vcg as ICS). (a) Simulation time fromt = 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 withoutSSRDC and with SSRDC (IL and Vcg as ICS). (a) Simulation time fromt = 0 s to t = 20 s. (b) Simulation time from t = 0.9 s to t = 1.9 s.

Fig. 21. Comparing dynamic response of the dc-link voltage withoutSSRDC and with SSRDC (IL and Vcg as ICS). (a) Simulation time fromt = 0 s to t = 20 s. (b) Simulation time from t = 0.9 s to t = 1.9 s.

SSR mode and stabilize the system. Fig. 19(b) shows that whenVcg is used as ICS, both SSR and SupSR modes are mitigatedfaster compared with the case when IL is used as ICS. This


Fig. 22. Power factor of the DFIG wind farm. (a) Simulation time fromt = 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 asICS decreases the damping of the SupSR mode and that themaximum damping ratio for the SSR mode is limited to lessthan 3%. A similar behavior can be observed using Figs. 20and 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 thesystem when IL or Vcg is used as ICS. As seen in this figure,the control system is able to maintain the unity power factorfor the wind farm using both ISCs. Once again, using Vcg asICS provides better SSR and SupSR damping for the systemcompared 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 transmissionline compensation and SSR mitigation in DFIG-based windfarms 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-ordermodel of the system are obtained. Moreover, using participationfactors, the participation of each state to the each system modeis identified. The main modes of the DFIG-based wind farmincluding SSR, SupSR, electromechanical, and shaft modes areidentified. The results show that the fixed-series-compensatedDFIG-based wind farm is highly unstable due to the SSR mode.

Therefore, in order to stabilize the SSR mode, a seriesFACTS 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 voltageacross the GCSC Vcg are examined in order to find the optimalICS to the GCSC’s SSRDC.

The residue-based analysis shows that the rotor speed is notan optimum ICS for the SSRDC for two reasons: First, a verylarge gain is needed in this case, and second, it is not possible tosimultaneously increase the damping of both SSR and SupSRmodes. Moreover, although the residue-based analysis for theline current as ICS predicts that a smaller gain is needed to

TABLE IVPARAMETERS OF THE SINGLE 2-MW AND 100-MW AGGREGATED DFIG.

VALUES ARE IN (p.u.), UNLESS IT IS MENTIONED

TABLE VPARAMETERS OF THE NETWORK AND SHAFT SYSTEM. VALUES

ARE IN (p.u.), UNLESS IT IS MENTIONED

TABLE VIPARAMETERS OF THE CONTROLLERS

damp the SSR mode, the SupSR mode’s stability is decreased inthis 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 ofboth the SSR ans SupSR modes, simultaneously.

In addition, using root-locus diagrams, the required gain todamp the SSR mode is computed for both line current andvoltage across the GCSC as ICSs. The results show that, unlikethe line current as ICS, using voltage across the series capacitoras ICS can guarantee the damping of the SSR mode, withoutsacrificing the SupSR mode’s stability, verifying what wasexpected from the residue-based analysis. Moreover, the maxi-mum SSR damping ratio, when the voltage across the GCSC isused as ICS, is 67% more compared with that of the line current.Finally, time-domain simulation is used to verify the designprocess using residue-based analysis and root-locus diagrams.

The work presented in this paper was completely simulationbased. A thorough discussion of technology feasibility issuesfor a practical implementation of the proposed control schemeis left as future work.

APPENDIX

The parameters used are given in Tables IV–VI.

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Hossein Ali Mohammadpour (S’10) receivedthe 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 currentlyworking toward the Ph.D. degree in electricalengineering at the University of South Carolina,Columbia, SC, USA.

His primary research interests include powersystems stability, power electronics, renewableenergy, flexible ac transmission system tech-

nologies, and electric ship system modeling and analysis.

Enrico Santi (S’90–M’94–SM’02) received theDr. Ing. degree in electrical engineering from theUniversity of Padua, Padova, Italy, in 1988 andthe M.S. and Ph.D. degrees from the CaliforniaInstitute of Technology, Pasadena, CA, USA, in1989 and 1994, respectively.

From 1993 to 1998, he was a Senior DesignEngineer 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 AssociateProfessor in the Department of Electrical Engineering. He has publishedover 100 papers on power electronics and modeling and simulationin international journals and conference proceedings and holds twopatents. His research interests include switched-mode power convert-ers, advanced modeling and simulation of power systems, modelingand simulation of semiconductor power devices, and control of powerelectronics systems.

modeling and control of gate-controlled series capacitor interfaced with a dfig-based wind farm

Engineering

offshore wind farms

offshore wind powerresources

onshore wind power sources

control of gate

rootlocus analysis

fromthe ssr mode

dfigbasedwind farm

axed capacitor