SSR results obtained with a dynamic phasor model of SVC using modal analysis

Electrical Power and Energy Systems 32 (2010) 571–582

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

SSR results obtained with a dynamic phasor model of SVC using modal analysis Fernando Cattan Jusan a,*, Sergio Gomes Jr. b, Glauco Nery Taranto c a

Furnas Centrais Elétricas S.A, Av. Real Grandeza, 219, Sala B-605, Rio de Janeiro, RJ, Brazil CEPEL, Av. Horácio Macedo, 354, 21941-911, Rio de Janeiro, RJ, Brazil c Federal University of Rio de Janeiro/COPPE, Centro de Tecnologia, Sala H-343, Ilha do Fundão, P.O. Box 68504, Rio de Janeiro, RJ, Brazil b

a r t i c l e

i n f o

Article history: Received 6 July 2008 Received in revised form 5 November 2009 Accepted 6 November 2009

Keywords: Subsynchronous resonance Torsional interaction SVC Dynamic phasors Control systems design Modal analysis

a b s t r a c t This paper presents the application of an improved dynamic phasor model of Static Var Compensator (SVC) in small-signal subsynchronous resonance (SSR) studies. The model is suitable for high frequency analysis (above 5 Hz) and takes into account the influence of Phase Locked Loop (PLL) circuit dynamics. A supplementary controller is designed for damping torsional modes due to SSR. The controller is designed using modal control theory to damp out critical modes in a wide range of series compensation and loading conditions. The study is conducted on the system-2 of the IEEE Second Benchmark Model. Excitation systems and power system stabilizers (PSS) are properly represented and incorporated into the system. Thus, the dynamic interactions among the several power system controllers and the network are considered in the supplementary controller design. The program PSCAD/EMTDC is used for the validation of the results obtained in the time domain simulations. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Subsynchronous resonance (SSR) is a phenomenon that occurs in thermal power plants, whose turbo-generator components (synchronous generator, rotating exciter and multi-stage steam turbines) are connected through a very long shaft. These turbogenerator components are modeled as lumped masses and have considerably high inertia constants. Shaft sections are modeled by torsional springs, whose stiffness varies with shaft section diameter. This spring-mass system has therefore several modes of oscillation, known as torsional modes [1,2]. SSR problems occur when there is adverse interaction between the electrical network subsynchronous modes and the poorlydamped torsional modes of the turbo-generators. Electrical torques at subsynchronous frequencies occur in generators associated with series-compensated transmission systems. Series-capacitor compensation has been used widely in AC transmission systems as an efficient and economic alternative of enhancing power transfer capability and improving transient stability. When the series-compensation ratio is increased to a certain level, an electrical resonance involving the generators, transformers, transmission lines and series capacitors will usually appear at the subsynchronous frequency range. When the electrical resonant frequency becomes complementary to one of the torsional frequencies of the turbine* Corresponding author. Tel.: +55 21 2528 5451; fax: +55 21 2528 5576. E-mail addresses: [email protected] (F.C. Jusan), [email protected] (S. Gomes), [email protected] (G.N. Taranto). 0142-0615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2009.11.013

generator mechanical system, some of these torsional modes may become undamped or unstable and can cause significant shaft damage [3]. Torsional modes can be also destabilized by fast excitation control systems of turbo-generators equipped with power system stabilizers derived from rotor speed, HVDC converter controls, FACTS devices and speed governors. In fact, any device that responds to changes in power and/or speed in the subsynchronous frequency range can adversely interact with torsional modes. Numerous countermeasures to SSR problem such as static blocking filters, supplementary excitation controllers, torsional relays, dynamic filters and many others have been reported in literature [4]. Among these, the application of a Static Var Compensator (SVC) for the damping of SSR is discussed in this paper. Static Var Compensators are conventionally used for bus voltage regulation. However, supplementary signals can be added to the main input of the SVC controller to damp out power system oscillations [5–10]. Conventionally, EMTP-type programs are used for SSR studies with systems containing FACTS devices. These simulation tools are accurate but they need numerous non-linear time domain simulations and experienced engineers to analyze the results and infer system characteristics. Besides, they are too time-consuming for optimization of controller parameters. On the other hand, linear analysis techniques provide a large set of structural information on the system in a direct and effective way, being complementary to non-linear time domain simulations [11]. Simplified models of FACTS devices assume that the Thyristor Controlled Reactor (TCR) is represented by a variable inductance.

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The TCR inductance is determined based on the relationship among fundamental frequency components of variables at steady-state. In these models, the delays introduced by the dynamics of the firing system are simulated using lag blocks and transport delays. For SSR studies, the use of simplified models may lead to erroneous results [12]. Some analytical models of FACTS devices suitable for modal analysis have been proposed [13–19], but simple and reliable high-frequency models were still needed. In [20], a new accurate and modular model of SVC in a form suitable for control theory applications and, following the model linearization, suitable for modal analysis was proposed. This objective was achieved by using the Dynamic Phasor approach [15–17,21], that allows transforming the non-linear time varying system of equations into an autonomous non-linear system. The Phase Locked Loop (PLL) module of the SVC was properly considered in the model. This paper presents a detailed SSR study using the SVC model described in [20]. A supplementary controller is designed for damping critical torsional modes due to SSR in a wide range of series compensation. The control concept is based on utilizing the front and rear speeds as feedback signals, similar to that described in [10]. Modal analysis and time domain simulations are performed on the system-2 of the IEEE Second Benchmark Model [22] in order to demonstrate the effectiveness of the proposed controller and the accuracy of the new model. The results obtained in the time domain simulations are validated through PSCAD/EMTDC [23] simulations.

The following set of linear algebraic and differential equations can be used to describe the linearized power system dynamics:

y¼cxþdu

ð1Þ

where x is the vector of state and algebraic deviation variables of the system, u and y are respectively the chosen input and output deviation variables (Single-Input–Single-Output system model). J is the jacobian matrix and T is a diagonal matrix, having elements equal to 1 in those lines associated with state variables and elements equal to zero in the lines associated with algebraic variables. This formulation is known as the descriptor system form. Modal analysis, which is traditionally applied to state space formulation, can be applied to this augmented equation system. The eigenvalues and eigenvectors are defined as the non trivial solution for:

Jv ¼ kTv

3. Dynamic phasor model of SVC The main component of the SVC is the TCR shown in Fig. 1, together with a fixed capacitor connected in parallel. The SVC is controlled by varying the phase delay of the thyristor firing pulses synchronized through a PLL to the measured voltage waveform. In this paper, time domain variables are denoted by lower case letters and their associated phasors are denoted by capital letters. The subscripts Re and Im refer respectively to the real and imaginary part of the phasors, while the subscript a is associated with phase a voltages and currents. Using the notation of Fig. 1, the following equations may be used to model the device:

dv tcr ¼ il  itcr dt ditcr ¼ qða; hpll ; rÞ  v icr Ltcr dt

system

ð3Þ ð4Þ

where q is the switching function [18], a is the firing angle, r is the conduction angle and hpll is the reference angle for the thyristor firing control, given by the PLL. In the case of SVC, hpll represents the angle of the filtered voltage of the TCR. The switching function is a function of the time, which may be 1 or 0, depending on the state of the thyristors (conducting or not) at that time, as illustrated in Fig. 2. The Eqs. (3) and (4) may be put in a form of time invariant system using dynamic phasors [20]. The angles a, hpll and r are included as variables of the problem.

Ctcr il

Ltcr itcr

ð2Þ

where the non-null vector v is the augmented eigenvector and k is the eigenvalue. Such as in the state space formulation, the matrix eigenvalues are the system poles of the dynamic system in (1). Next section will present the linearized dynamic phasor model of the SVC as a set of differential and algebraic equations using this descriptor system formulation. With the SVC and network equations [24], the modal analysis can be applied. Modal analysis yields a set of structural information on: – Identification causes and solutions for oscillations. – Tuning of control systems. – Identification of adverse interactions among components.

In the following items, some of these tools will be applied to a test system.

C tcr

2. Modal analysis

T  x_ ¼ J  x þ b  u

– Calculation of mode-shapes, controllability, observability and participation factors. – Identification of controllers most adequate for oscillation damping, using residues. – Root-Locus. – Frequency response (Bode and Nyquist diagrams) for controller loop design. – Time response using numerical integration for evaluation of system performance in the time domain.

vtcr Fig. 1. Thyristor Controlled Reactor connected in parallel with a fixed capacitor.

Switching Function 1

vtcr

itcr

0

θ pll

α

σ

The following tools are commonly used for the linear analysis: – Calculation of poles and zeros.

Fig. 2. Switching function.

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The switching function has cosine symmetry around the zero crossing of the voltage. If the angle b is defined by:

b ¼ a  hpll þ

r

ð5Þ

2

The Fourier series of the switching function can be written as:

q ¼ q0 þ

X k

X Q m ejmxt þ Q  ejmxt m cos½2kðxt  bÞ ¼ 2 m

r p

Q m ¼ Q 2k ¼ qk ejmb

ð7Þ mr 4 ¼ sin ejmb mp 2

ð8Þ

The fundamental phasor of the TCR voltage multiplied by several harmonics will produce harmonics on the TCR current. The harmonic phasors of the TCR current will in turn produce harmonics on the current injected in the power system, thus producing harmonic phasors on the TCR voltage. These harmonic phasors of the TCR voltage will interact with the existing harmonic phasors of the TCR current. The final equations for the harmonic phasors of the SVC are presented in (9)–(12):

dV tcrakRe  kxC tcr V tcrakIm ¼ IlakRe  ItcrakRe dt dV tcrakIm  kxC tcr V tcrakRe ¼ IlakIm  ItcrakIm C tcr dt C tcr

Ltcr

dItcrakRe  kxLtcr ItcrakIm dt   X Q mRe Q ¼ V tcrRe  mIm V tcranIm 2 2 mþn¼k  X Q mRe Q þ V tcrRe  mIm V tcranIm 2 2 mþn¼k   X Q mRe Q þ V tcrRe  mIm V tcranIm 2 2 mn¼k

dItcrakIm þ K xLtcr ItcrakRe Ltcr dt   X Q mIm Q ¼ V tcranRe þ mRe V tcranIm 2 2 mþn¼k  X Q mIm Q þ V tcranRe þ mRe V tcranIm 2 2 mþn¼k   X Q mIm Q þ V tcranRe  mRe V tcranIm 2 2 mn¼k

ð9Þ ð10Þ

ð11Þ

 tcr  V tcr Re ¼ Il Im  Itcr Im  tcr dV tcra Im þ x  C C a a a dt Ltcr

dItcra Re  0  V tcr Re þ Q  2  V tcr Re  x  Ltcr  V tcra Im ¼ Q a a Re dt  þ Q 2Im  V tcra Im

0 ¼ 3  Q ¼ 3  r Q 0

ð17Þ

 2 ¼ 3  Q 2 ¼ 3  sinðrÞ  ej2ðahpll þr2Þ Q 2 p

ð18Þ

p

The coefficients presented in (17) and (18) are obtained from the Fourier series of the switching function. The simple voltage regulator of a SVC can be done through a proportional integral block (PI), as shown in Fig. 3, where Vrmsf is the measured and filtered rms value of the controlled voltage, Vref is the voltage reference, Vs is the output signal of the supplementary controller and Btcr is the susceptance order. The susceptance order is then normalized and multiplied by xLtcr , resulting in Bn. The firing angle a will be given by the solution of the following equation [12], indicated in Fig. 3 as a(Bn):

Bn ¼ 2 

2a

p

þ

sinð2aÞ

ð19Þ

p

where a is between p/2 and p rad, when Bn is between 1 and 0. This equation is transcendental but may be easily solved for a by a Newton–Raphson procedure or through look-up tables. The PLL model used in PSCAD/EMTDC is referred to as d-q-z or Transvektor type [25]. Due to the low-pass frequency behavior of the PLL, a much simpler model could be used in the phasor model with practically coincident results, as shown in [20]. The PLL was defined by the unitary feedback of the filter given in (20), yielding the closed-loop transfer function in (21):

ð20Þ ð21Þ

where hV is the angle of the fundamental phasor of the applied voltage, given by:

hV ¼ arctg

ð12Þ

ð16Þ

 tcr ¼ 3C tcr and: where C

  KI 1 GðsÞ ¼ K Ppll þ pll  s s GðsÞ  hV hpll ¼ 1 þ GðsÞ

  V tcr1Im V tcr1Re

ð22Þ

The last variable to be calculated is the conduction angle. One approximation for this angle is to consider that the switching function multiplied by the voltage is always in phase with the fundamental phasor of the current. In this approach the following expression may be used for the calculation of the conduction angle:



r ¼ 2  p  a þ hpll  hV þ hVI

The Eqs. (13)–(16) may be used for the case where the system is balanced (positive sequence model) and only the fundamental frequency phasors are considered. The fundamental frequency model assumes that the harmonic contents of the voltage vtcr of the SVC is very low and may be neglected. This model was used to produce the results of this paper.

 tcr  V tcr Im ¼ Il Re  Itcr Re  tcr dV tcra Re  x  C C a a a dt

dItcra Im  0  V tcr Im þ Q  2  V tcr Re  x  Ltcr  V tcra Re ¼ Q a a Im dt   Q 2Re  V tcra Im

ð6Þ

where k = 1, 2, 3, . . . and m = 2k. The coefficients of the complex form of the switching function are given by:

Q 0 ¼ q0 ¼

Ltcr



ð23Þ

where hVI is the angular difference between voltage and current phasors, which may be calculated by the following equation:

hVI ¼ angleðV tcra  Itcra Þ  p=2

ð24Þ

ð13Þ

V rmsf ð14Þ

ð15Þ

V ref

Bmax

-

+ Σ + Vs

PI B min

1 Btcr

Bn

−ω Ltcr 0

Fig. 3. Simple voltage regulator for a SVC.

α ( Bn )

α

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Rt = 0, 0002 X t = j 0, 02

G1

R1 = 0, 0052 X 1 = j 0, 054 R0 = 0, 0120 X 0 = j 0,120

Rsys = 0, 0014

Xc

X sys = j 0, 03

Infinite Bus

Rt = 0, 0004 X t = j 0, 04

G2

X t = j 0, 056 SVC 13,8 kV -200/+200 MVAr

L

C

Control System

Fig. 4. IEEE Second Benchmark Model – system 2.

Vt VRMAX

Vref

+

∑

1 + sTc 1 + sTb

- VF

Ka 1 + sTa

+ VR

VRMIN

1 sTe

∑ - VFE

∑

+

E fd

Ke

Vx + E fd ⋅ Aex e

Bex E fd

sK f 1 + sT f Fig. 5. DC rotating exciter of G1.

Vt E fd MAX

Vref

+

∑ +

KA 1 + sTA

E fd E fdMIN

V pss Fig. 6. Static excitation system of G2.

This approach is a more precise procedure and yields a model defined by equations that are non-linear function of the Laplace complex frequency s, requiring an s-domain representation. This second model was not used in this paper since the simpler first model has already shown very good results.

4. SSR analysis without SVC The test system data utilized in this paper is described in Appendix. Prior to the development and analysis of a SVC damping controller, it is convenient to describe the dynamic behavior of the benchmark system in absence of any system controllers. This may be achieved by using linear analysis techniques to obtain some important structural information such as: torsional natural frequencies and their mode-shapes, critical levels of series compensation and eigenvalue sensitivities. The initial conditions for the linear and non-linear system simulations were calculated using a power flow program (Anarede). First, it was assumed a light load condition, with the following steady-state operating point:

PG1 ¼ 60 MW; P G2 ¼ 70 MW; V tG1 ¼ 1:01 pu; V tG2 ¼ 1:01 pu; V 1 ¼ 0:98 pu; V SVC ¼ 1:038 pu The calculated initial condition depends on the series compensation degree. For a compensation level of 30%, the reactive power of generators and SVC are given by:

Q G1 ¼ 5:1 Mvar; Q G2 ¼ 2:2 Mvar; Q SVC ¼ 50:1 Mvar

Fig. 7. Power system stabilizer.

This representation yields a linearized model comprising both algebraic and differential equations, which can be modeled by a descriptor system. The results presented in this paper were obtained using this approach. A second approach to calculate the conduction angle, proposed in [20], is by numerical integration of the voltage in time domain.

Figs. 8 and 9 show the frequency response of the transfer functions where the input variable is the mechanical power (pu) and the output variable is the generator (GEN) rotor speed (pu) for generators G1 and G2, respectively. The response was calculated for a series compensation level of 30% (XC = 0.0162 pu). These figures show the presence of dominant poles in the frequencies of the torsional modes of each generator. Besides, the plots present a double resonance peak near the frequency of the common mode (24.65 Hz), resultant of the electrical coupling between the generators [26,27]. This means that disturbances applied to one of the generators may excite the torsional mode of the other and vice versa. As the coupling is increased, the frequency difference increases. With a very low coupling, the two frequencies would be equal [27].

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Amplitude (pu/pu)

-5

Table 2 Mode-shapes of G2.

Local Mode Mode 2 (32.4 Hz)

-25 Intraplant Mode

Mode 3 (51.1 Hz)

-45 -65

-8 -12 -16 -20 -24 -28 -32 -36 -40

-85 -105 -125

Common Modes 24.7 Hz

24.4

-145 0.

5.

24.8 Hz

24.6

10.

24.8

15.

25.

25.2

20.

25.

30.

35.

40.

45.

50.

55.

Mass

Natural frequencies 0 Hz

24.65 Hz

45 Hz

HP LP GEN

1.0000 1.0000 1.0000

1.0000 0.5873 0.9779

1.0000 0.3749 0.0865

60.

Frequency (Hz) Table 3 System poles with 30% of series compensation.

Fig. 8. Frequency response of xGEN1/Pmec1.

Amplitude (pu/pu)

0

Local Mode

-20

Mode 2 (45.0 Hz)

Intraplant Mode

-40 -60

-4

-80

Common Modes

-12

-100

-20 -28

-120

24.7 Hz

24.8 Hz

-36 -44

-140

-52

24.2

24.4

24.6

24.8

25.

25.2

25.4

-160 0.

5.

10.

15.

20.

25.

30.

35.

40.

45.

50.

55.

60.

Frequency (Hz) Fig. 9. Frequency response of xGEN2/Pmec2.

The full eigensolution of the linearized system was obtained by using the QZ method. The participation factors were the index used to identify the nature of each mode, as indicated in the fourth column of Table 3. All poles have negative real parts and the system is stable. Fig. 10 shows the Root-Locus plot obtained when varying the series-compensation ratio XC/XL from 10% (XC = 0.0054 pu) to 90% (XC = 0.0486 pu) of the total line reactance. As the reactance XC is increased, the frequency of subsynchronous network mode is reduced, while the frequency of supersynchronous network mode is increased. Whenever the frequency of the network subsynchronous mode approaches the frequency of a torsional mode, they strongly interact. The net effect is that the subsynchronous pole shifts to the left while the torsional pole shifts to the right and the system becomes unstable. This adverse interaction is result of SSR from torsional interaction (TI) mechanism. The critical levels of series compensation, which cause the maximum shifts of the torsional modes to the right plane, are shown in Fig. 11 and Table 4. The graphical speed mode-shapes of these three critical modes for the complete system are illustrated in Fig. 12. They were calculated considering the interaction between the two turbine-generator sets through the power system. For the second torsional mode of G1 (32.4 Hz), very little interaction is observed at all, since the participation of G2 on the torsional oscillations of G1 is very low. This is clear from the figure, in which the mode-shape components related to G2 barely come out. However, for the first common modes (24.65 Hz), a strong interaction is verified and then the torsional oscillations of one unit will be felt by the other unit and vice versa.

Table 1 Mode-shapes of G1. Mass

Natural frequencies 0 Hz

24.65 Hz

32.4 Hz

51.1 Hz

HP LP GEN EXC

1.0000 1.0000 1.0000 1.0000

1.0000 0.2583 0.7313 0.9555

1.0000 0.2807 0.2089 0.3512

0.0005 0.0012 0.0097 1.0000

Pole

Frequency (Hz)

Damping (%)

Description

12.022 ± j518.95 6.3190 ± j376.89 0.0505 ± j321.27 0.02496 ± j282.9 11.3460 ± j234.8 0.04038 ± j203.6 0.0792 ± j155.82 0.0137 ± j155.22 2.2909 ± j11.536 0.6650 ± j6.356 29.172 27.462 27.216 18.989 1.8969 1.4345 0.8429 0.4442

82.6 60.0 51.1 45.0 37.4 32.4 24.8 24.7 1.84 1.01 – – – – – – – –

2.32 1.68 0.016 0.0088 4.82 0.020 0.051 0.0088 19.5 10.4 100 100 100 100 100 100 100 100

Supersynchronous Network Torsional mode 3 – Torsional mode 2 – Subsynchronous Torsional mode 2 – Torsional mode 1 – Torsional mode 1 – Intraplant mode Local mode Other modes

700.

G1 G2 G1 G1 G2

Supersynchronous Mode

600. 500.

Torsional Modes

400. 300.

Subsynchronous Mode

200. 100. 0. -15.

-10.

-5.

0.

5.

Real Part (1/s) Fig. 10. Root-Locus of XC – without SVC.

Fig. 13 shows the time domain simulation results for a 0.02 pu step disturbance applied to the synchronous machines mechanical power (DPmec) for a capacitive reactance of 0.0162 pu (XC/XL = 30%). The step disturbance is positive for G1 and negative for G2. The monitored variables are generator rotor speeds (DxGEN) and output power (DPt) deviations. Fig. 14 shows the system response to the same disturbance for a compensation level of 72.9% (XC = 0.0394 pu), which represents the most critical value of XC for one of the common modes. The system is unstable and unbounded torsional oscillations are observed. The results obtained with the electromagnetic transient program PSCAD/EMTDC are shown in the same figures. All plots are visually coincident. Subsynchronous resonance may limit considerably the degrees of series compensation that can be used in the system, once serious damages can be caused to turbine-generator’s rotor. If unstable SSR or excessive shaft fatigue conditions are found, torsional protection and mitigation schemes are required. The next section discusses the application of a SVC to mitigate SSR in the system-2 of the IEEE Second Benchmark Model. 5. SSR analysis with SVC The SVC will be connected at the generators high-side voltage bus through a step-up transformer, as illustrated in Fig. 4. The

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F.C. Jusan et al. / Electrical Power and Energy Systems 32 (2010) 571–582 203.65

156.

Xc=0.02132 pu

Xc=0.0052 pu

155.6

203.6 Xc=0.02391 pu

203.55

Xc=0.03398 pu 155.2

Xc=0.0394 pu

154.8

Xc=0.0468 pu Xc=0.02668 pu

203.5 -0.1

-0.05

0.

0.05

0.1

154.4 -0.2

0.

0.2

0.4

0.6

0.8

1.

1.2

Real Part (1/s)

Real Part (1/s) Fig. 11. Detail of Root-Locus for critical poles.

Table 4 Critical levels of series compensation. XC (pu)

XL (%)

Critical pole

Frequency (Hz)

n (%)

0.03940 0.03398 0.02391

72.9 62.9 44.2

+1.0657 ± j155.13 +0.1294 ± j155.59 +0.0381 ± j203.58

24.8 24.7 32.4

0.687 0.083 0.019

24.7 Hz HP

HP LP

LP

GEN

GEN

EXC

G1

power source that allows a better utilization of reactive power capacity of generators. The SVC is represented using the model described in Section 2, while the damping of torsional oscillations can be done by employing an auxiliary (supplementary) controller with appropriate phase and gain tuning. The output of the controller (Vs) is added to the input reference of the SVC (Vref). Consider first a supplementary controller based on the measured generator speed deviation (DxGEN), as shown in Fig. 15. The generator speed is chosen as the control signal because it includes components of all torsional modes that need to be damped. The primary objective of the controller is to stabilize the common first torsional modes (24.65 Hz). Thus, the compensation level of 72.9% is singled out for the controller design. The parameters Tn and Td of the lead-lag network are selected to give the correct phase compensation for this mode. The washout high-pass filter eliminates low frequency components from the generator speed signal (mode 0) and allows the controller to respond only to torsional modes. The parameters chosen for the controller are:

T wS ¼ 0:03 s; T n ¼ 0:1 s; T d ¼ 0:01 s; K s ¼ 1:1 pu=pu

G2

24.8 Hz HP GEN

LP HP GEN

LP

EXC

G1 G2

32.4 Hz HP GEN

EXC

HP

LP

GEN

LP

G1 G2

Fig. 12. Speed mode-shapes for the critical poles.

Table 5 shows the system poles with the supplementary controller in service for XC/XL = 72.9%. The critical torsional modes have been shifted to the left side of the complex plane, as desired. However, it had an opposite effect on the second torsional mode of G1 (32.4 Hz), which becomes unstable. This instability is not the result of a torsional interaction between the electrical network and the mechanical system; instead, it is due to an adverse interaction with the designed supplementary controller. A control approach to avoid this adverse interaction is to use the modal speed deviation as the feedback signal for the supplementary controller [10]. To achieve this, speed measurements of all turbo-generator sections are required. The relationship between the mass and modal speeds is given by the mode-shape matrix, Q. However, such a measurement system may be in most cases impractical or even impossible. To overcome this problem, the supplementary controller may be modified to eliminate the mode 2 component from the control signal [10]. This may be accomplished by measuring the front and rear standard speeds (HP and EXC sections), which usually are measurable quantities, and using the following control signal:

u1 ¼ DxEXC;G1 

q42 DxHP;G1 q12

ð25Þ

In terms of the modal components: location of the SVC was selected taking into account the damping controllability of the torsional modes. Tests were conducted placing the SVC at the bus after the series capacitor but the results shown that it was less effective for this task. It is worth to be noted that the voltage control of the SVC for this tutorial example is less important than its ability in increasing damping of torsional modes, since the voltage control could be performed by the two generators. Nevertheless, the SVC is an addition dynamic reactive

u1 ¼



   q q q40  42 q10 Dx0 þ q41  42 q11 Dx1 q12 q12   q þ q43  42 q13 Dx3 q12

ð26Þ

where qij refers to the element associated with section i and to mode j of the mode-shape matrix of generator G1, QG1, given in Table 1. Note that the mode 2 component is absent of the control signal. The sub-

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F.C. Jusan et al. / Electrical Power and Energy Systems 32 (2010) 571–582 6.0E-5

1.5E-5 PACDYN

4.5E-5

5.0E-6

PSCAD/EMTDC

3.0E-5

-5.0E-6

1.5E-5

-1.5E-5

0.0E+1

-2.5E-5

-1.5E-5

-3.5E-5

-3.0E-5 0.5

-4.5E-5 1.5

2.5

3.5

PACDYN PSCAD/EMTDC

0.5

4.5

1.5

2.5

3.5

4.5

Time (s)

Time (s)

3.5E-2

0.0E+1

3.0E-2

-5.0E-3

PACDYN PSCAD/EMTDC

2.5E-2

-1.0E-2

2.0E-2

-1.5E-2

1.5E-2 -2.0E-2 1.0E-2

PACDYN

5.0E-3

PSCAD/EMTDC

0.0E+1 0.5

1.5

2.5

3.5

-2.5E-2

4.5

-3.0E-2 0.5

1.5

2.5

3.5

4.5

Time (s)

Time (s)

Fig. 13. Time response for XC = 0.0162 pu – PACDYN  PSCAD/EMTDC.

2.5E-5

6.0E-5

1.5E-5

PACDYN

4.5E-5

5.0E-6

PSCAD/EMTDC

3.0E-5

-5.0E-6 1.5E-5

-1.5E-5

0.0E+1

-2.5E-5

PACDYN

-1.5E-5

-3.5E-5

PSCAD/EMTDC

-3.0E-5 0.5

-4.5E-5 0.5

1.

1.5

2.

2.5

3.

3.5

1.

1.5

3.5E-2

0.0E+1

3.0E-2

-5.0E-3

2.5E-2

2.

2.5

3.

3.5

Time (s)

Time (s)

PACDYN PSCAD/EMTDC

-1.0E-2

2.0E-2 -1.5E-2

1.5E-2 1.0E-2

PACDYN

5.0E-3

PSCAD/EMTDC

0.0E+1 0.5

1.

1.5

2.

2.5

3.

-2.0E-2 -2.5E-2 3.5

-3.0E-2 0.5

1.

1.5

Time (s)

2.

2.5

3.

3.5

Time (s)

Fig. 14. Time response for XC = 0.0394 pu – PACDYN  PSCAD/EMTDC.

Fig. 15. Supplementary controller using generator speed as stabilizing signal.

script 0 refers to the system mode (unison mode), where the rotor behaves as a rigid body. An additional notch filtering block is added

in the control path to prevent against undesirable interaction with the third torsional mode of G1 (51.1 Hz). Further simulation indicated that this mode could be destabilized by the supplementary controller in other operating conditions. The notch filter was designed with a sharp cut off to minimize interference with other modes. The transfer function of the filter is given below

NFðsÞ ¼

s2 þ 2nn xn s þ x2n s2 þ 2nd xn s þ x2n

ð27Þ

Table 5 System poles with supplementary controller based on DxGEN. Pole 0.0369 ± j321.27 -0.02491 ± j282.9 +0.1735 ± j203.7 0.5770 ± j160.9 0.0356 ± j155.5

Frequency (Hz)

Damping (%)

Description

51.1 45.0 32.4 25.6 24.7

0.012 0.0088 0.085 0.359 0.023

Torsional mode 3 – G1 Torsional mode 2 – G2 Torsional mode 2 – G1 Torsional mode 1 – G1 Torsional mode 1 – G2

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Besides, according to Table 4 the second torsional mode of G1 (32.4 Hz) is unstable for degrees of series-compensation ratio about 44.2% (XC = 0.02391 pu), which represents the most critical SSR condition for this mode. Thus, it is necessary to provide the controller with a second channel so as to suppress the mode 2 instability. To eliminate the mode 1 component from the feedback signal, the same method used before may be applied. The resulting control signal is given as follows:

u2 ¼ DxEXC;G1 

q41 DxHP;G1 q11

350. 280. 210. 140. 70. 0. -14.

-12.

-10.

-8.

-6.

-4.

-2.

0.

2.

ð28Þ 350.

which in terms of the modal components:

280.

    q q u2 ¼ q40  41 q10 Dx0 þ q42  41 q12 Dx2 q11 q11   q41 þ q43  q Dx3 q11 13

210. 140. 70.

ð29Þ

The second channel must also be equipped with a notch filter tuned to the third torsional mode of G1 (51.1 Hz) to produce a filtered signal without mode 3 component. The phase and gain compensation are tuned individually for each controller channel and the output of the supplementary controller (Vs) is obtained by summing the control signal derived from each channel (Vs1 and Vs2). The complete block diagram of the proposed controller is shown in Fig. 16, where NF(s) refers to the transfer function of the torsional filters. A Root-Locus analysis may be carried out to set the controller gains (Ks1 and Ks2), as illustrated in Fig. 17. Note that only the specific torsional mode for which the controller channels have been designed moves as the gain is increased. However, it is evident that the SVC damping controller has a tendency to deteriorate the damping associated with the electrical subsynchronous mode, which increases the risk of SSR from the induction-generator effect. The design of the supplementary damping controller was based on trial and error procedure through successive Root-Locus plots for each of the controller parameters, observing their influence on the torsional modes. It should be noted that even though this methodology does not lead to an optimal set of parameters, it is possible to tune the controller in order to allocate the torsional modes in an appropriate locus on the complex plane, providing stable and sufficiently damped response. Adverse interactions with other modes (e.g. other torsional modes, exciter modes, etc.) are also easily identified using this methodology. In this work, a small positive damping for all of the torsional modes in the whole range of operation and in the practical range of series compensation values was considered as the criteria for adequate dynamic performance, since these modes are naturally low damped and a larger damping value could not be achieved. The parameters chosen for the controller are:

0. -12.

-10.

-8.

-6.

-4.

-2.

0.

2.

Fig. 17. Root-Locus of supplementary controller gains: (a) Ks1; (b) Ks2.

204. 203.8 203.6 203.4 203.2 203. -1.2

-1.

-0.8

-0.6

-0.4

-0.2

0.

0.2

Real Part (1/s)

185. 175. 165. 155. 145. 135. 125. -12.

-11.

-10.

-9.

-8.

-7.

-6.

-5.

-4.

-3.

-2.

-1.

0.

Real Part (1/s)

Fig. 18. Root-Locus of XC with supplementary controller.

Table 6 System poles with and without SVC damping controller. XC (pu)

XL (%)

Critical polewithout controller

Critical polewith controller

0.03940 0.03398 0.02391

72.9 62.9 44.2

+1.0657 ± j155.13 +0.1294 ± j155.59 +0.03809 ± j203.58

0.0521 ± j155.49 0.0734 ± j155.51 0.7968 ± j203.21

q42 ¼ 0:3512; T w1 ¼ 0:03s;

q12 ¼ 1:0000;

T n1 ¼ 0:12s;

q41 ¼ 0:9555;

T d1 ¼ 0:002s;

q11 ¼ 1:0000

K s1 ¼ 0:2pu=pu

xn ¼ 320:82rad=s; nn ¼ 0:005; nd ¼ 1:000 Fig. 16. Supplementary controller using front and rear speeds as stabilizing signals.

T w2 ¼ 0:03s;

T n2 ¼ 0:12s;

T d2 ¼ 0:02s;

K s2 ¼ 1:2pu=pu

1.

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F.C. Jusan et al. / Electrical Power and Energy Systems 32 (2010) 571–582

Fig. 18 shows the Root-Locus plot obtained when varying the series-compensation ratio from 10% (XC = 0.0054 pu) to 90% (XC = 0.0486 pu) with the supplementary controller in service. The system is stable for the whole range of series compensation. For higher degrees of compensation, the subsynchronous electrical mode shifts to the right side of the complex plane and the system becomes self-excited. The system poles with and without the SVC damping controller for the critical levels of series compensation are presented in Table 6.

Aiming to demonstrate the accuracy of the new model of SVC, time domain simulations of 0.02 pu step disturbance in the voltage reference of the SVC voltage regulator were performed. The model show very good results when compared to the PSCAD/EMTDC, as shown in Fig. 19 for a series-compensation ratio of 30% (XC = 0.0162 pu). A more extensive procedure for the validation of the model can be found in [20]. A linear time domain simulation for a 0.02 pu step disturbance applied to the synchronous machines mechanical power

1.065

36.

1.06

35.

1.055

34.

1.05 33.

1.045

PACDYN

1.04

PACDYN

32.

PSCAD/EMTDC

1.035 0.9

PSCAD/EMTDC

31. 1.1

1.3

1.5

1.7

1.9

0.9

1.1

1.3

Time (s) 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.9

1.5

1.7

1.9

Time (s) 2.0E-4 1.0E-4 0.0E+1 PACDYN

-1.0E-4

PACDYN

PSCAD/EMTDC

1.1

1.3 1.5 Time (s)

1.7

PSCAD/EMTDC

1.9

-2.0E-4 0.9

1.1

1.3 1.5 Time (s)

1.7

1.9

Fig. 19. Response for a step of 0.02 pu in the voltage reference of the SVC.

5.5E-5 4.5E-5 3.5E-5 2.5E-5 1.5E-5 5.0E-6 -5.0E-6 -1.5E-5 -2.5E-5 -3.5E-5 0.5

1.5

2.5

3.5

4.5

5.5E-5 4.5E-5 3.5E-5 2.5E-5 1.5E-5 5.0E-6 -5.0E-6 -1.5E-5 -2.5E-5 -3.5E-5 0.5

1.5

Time (s) 5.5E-5 4.5E-5 3.5E-5 2.5E-5 1.5E-5 5.0E-6 -5.0E-6 -1.5E-5 -2.5E-5 -3.5E-5 0.5

1.5

2.5

3.5

4.5

5.5E-5 4.5E-5 3.5E-5 2.5E-5 1.5E-5 5.0E-6 -5.0E-6 -1.5E-5 -2.5E-5 -3.5E-5 0.5

1.5

Time (s) 3.5E-2 3.0E-2 2.5E-2 2.0E-2 1.5E-2 1.0E-2 5.0E-3 0.0E+1 0.5

1.5

2.5

3.5

4.5

3.5E-2 3.0E-2 2.5E-2 2.0E-2 1.5E-2 1.0E-2 5.0E-3 0.0E+1 0.5 1.5E-4

1.0E-4

1.0E-4

5.0E-5

5.0E-5

0.0E+1

0.0E+1

-5.0E-5

-5.0E-5

-1.0E-4

-1.0E-4 2.5

Time (s)

(a)

4.5

2.5

3.5

4.5

1.5

2.5

3.5

4.5

3.5

4.5

Time (s)

1.5E-4

1.5

3.5

Time (s)

Time (s)

-1.5E-4 0.5

2.5

Time (s)

3.5

4.5

-1.5E-4 0.5

1.5

2.5

Time (s)

(b)

Fig. 20. Dynamic response of G1 for Xc = 0.0394 pu – (a) without supplementary controller; (b) with supplementary controller.

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F.C. Jusan et al. / Electrical Power and Energy Systems 32 (2010) 571–582

(DPmec 1–DPmec2) is performed. Typical responses of the system with and without supplementary controller at light load and 72.9% compensation are shown in Figs. 20 and 21. The plotted variables are speed (generator and turbine sections), electrical power and voltage deviations. In order to verify the behavior of the controller in different operating points, Root-Locus plots of series capacitor reactance XC varying from 10% (0.0054 pu) to 90% (0.0486 pu) are carried out at various loading conditions. Fig. 22 shows one of these plots for a heavy loading condition. The operating point considered in this case is given by:

203.8 203.6 203.4 203.2 203. 202.8 202.6 -1.2

190.

V tG2 ¼ 1:050 pu; V 1 ¼ 1:05 pu; V SVC ¼ 1:020 pu

180.

Q G1 ¼ 228:8 Mvar; Q G2 ¼ 241:9 Mvar; Q SVC ¼ 63:9 Mvar

-0.8

-0.6

-0.4

-0.2

0.

0.2

Real Part (1/s)

PG1 ¼ 540 MW; PG2 ¼ 630 MW; V tG1 ¼ 1:025 pu;

The calculated reactive powers of generators and SVC for a 30% of series compensation level are given by:

-1.

170. 160. 150. 140. 130. 120. -10.

6. Non-linear simulations results To further examine the effectiveness of the designed SVC damping controller on stabilizing SSR of the studied system, time domain simulations based on a non-linear system model are performed on PSCAD/EMTDC software. For simulating system nonlinearities in detail, exciter ceilings, firing angles and control signal limits, etc. are all included. Saturation effects of generators and transformers are not represented. 3.5E-5 2.5E-5 1.5E-5 5.0E-6 -5.0E-6 -1.5E-5 -2.5E-5 -3.5E-5 -4.5E-5 0.5

1.5

2.5

3.5

4.5

-9.

-8.

1.5

2.5

3.5E-5 2.5E-5 1.5E-5 5.0E-6 -5.0E-6 -1.5E-5 -2.5E-5 -3.5E-5 -4.5E-5 0.5

3.5

4.5

3.5E-5 2.5E-5 1.5E-5 5.0E-6 -5.0E-6 -1.5E-5 -2.5E-5 -3.5E-5 -4.5E-5 0.5 0.0E+1 -5.0E-3

-1.0E-2

-1.0E-2

-1.5E-2

-1.5E-2

-2.0E-2

-2.0E-2

-2.5E-2

-2.5E-2

-3.0E-2

1.5

1.5

3.5

4.5

-3.0E-2 0.5

1.5

Time (s)

-2.

2.5

3.5

4.5

2.5

3.5

4.5

2.5

3.5

4.5

3.5

4.5

Time (s)

3.0E-4

3.0E-4

1.0E-4

1.0E-4

-1.0E-4

-1.0E-4

-3.0E-4

-3.0E-4

-5.0E-4

-5.0E-4

-7.0E-4 0.5

-3.

Time (s)

0.0E+1

2.5

-4.

Time (s)

-5.0E-3

1.5

-5.

1.5

2.5

3.5

4.5

-1.

0.

1.

Figs. 23 and 24 show the dynamic responses of the open-loop and closed-loop systems, respectively, subjected to a temporary

Time (s)

0.5

-6.

Fig. 22. Root-Locus of XC with supplementary controller for a heavy load operating condition.

Time (s) 3.5E-5 2.5E-5 1.5E-5 5.0E-6 -5.0E-6 -1.5E-5 -2.5E-5 -3.5E-5 -4.5E-5 0.5

-7.

-7.0E-4 0.5

1.5

2.5

Time (s)

Time (s)

(a)

(b)

Fig. 21. Dynamic response of G2 for Xc = 0.0394 pu – (a) without supplementary controller; (b) with supplementary controller.

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F.C. Jusan et al. / Electrical Power and Energy Systems 32 (2010) 571–582

single-phase short-circuit at the 500 kV transmission line near the series capacitor. The fault starts at t = 1.0s and lasts for 100 ms. The plots presented refer to the light load operating condition and 72.9% compensation. Fig. 23 shows that the system without the supplementary controller is unstable. On the other hand, the dynamic responses shown in Fig. 24 exhibit well damped oscillations and the unstable common torsional modes have been effectively stabilized when the SVC damping controller is in service. A static limit of ±0.3 pu was considered for the supplementary controller. If a lower limit was used, the controller would not be capable of damping the torsional oscillations and the system would be unstable for this fault duration.

due to SSR in the whole range of series compensation for various loading conditions. The effectiveness of the controller was confirmed through linear and non-linear time domain simulations. A conclusion from the simulations is that the controller could damp out the torsional modes as desired but shifted the subsynchronous electrical mode to the right on the s-plane. However, this does not represents a serious problem since the system only becomes unstable for very high levels of series compensation.

7. Conclusions

The test system considered in the study is the system-2 of the IEEE Second Benchmark Model [22], of which the one line diagram is shown in Fig. 4. The system consists of two nonidentical steam turbine-generator sets having a common torsional mode connected to an infinite bus via two step-up transformers and a single series-compensated transmission line. The network data are given in per unit on 100 MVA base. G1 is equipped with a DC rotating exciter (IEEE type DC1A) while G2 is equipped with a static excitation system. Their block diagrams are shown in Figs. 5 and 6, respectively. G2 is further provided with speed input power system stabilizer (PSS), whose block diagram is given in Fig. 7. The PSS parameters are selected to provide damping for both local and intraplant electromechanical

The paper presented a detailed SSR study using modal analysis methodology with an improved dynamic phasor model of SVC in the system-2 of the IEEE Second Benchmark System. A detailed system model, including excitation systems, power system stabilizers and torsional dynamics was used. The model is suitable for high frequency analysis (above 5 Hz) and is useful to study adverse interactions among power system controllers or the network. The results obtained with the analytical model closely matched those achieved through PSCAD/EMTDC simulations. A multiple-channel supplementary controller was designed for damping critical modes

Appendix A A.1. System data

0.15

0.1

0.075

0.05

0.0

0.0

-0.075

-0.05 -0.1

-0.15 0.5

1.5

2.5

3.5

0.5

1.5

1.2 1.1 1.05 1.0 0.95 1.5

3.5

90. 80. 70. 60. 50. 40. 30. 20. 10.

1.15

0.9 0.5

2.5

Time (s)

Time (s)

2.5

3.5

0.5

1.5

Time (s)

2.5

3.5

Time (s)

Fig. 23. Non-linear simulation results of the open-loop system.

0.02

0.01

0.01

0.005

0.0

0.0

-0.01

-0.005

-0.02 0.5

1.5

2.5

3.5

4.5

-0.01 0.5

1.5

Time (s)

2.5

3.5

4.5

3.5

4.5

Time (s)

1.2

0.3

1.15

0.2

1.1

0.1

1.05

0.0

1.0

-0.1

0.95

-0.2

0.9

-0.3 0.5

1.5

2.5

Time (s)

3.5

4.5

0.5

1.5

2.5

Time (s)

Fig. 24. Non-linear simulation results of the closed-loop system.

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F.C. Jusan et al. / Electrical Power and Energy Systems 32 (2010) 571–582

modes. To avoid torsional interaction, the PSS is further provided with a one stage torsional filter tuned to the frequency of the common torsional modes (24.65 Hz). A SVC comprising a fixed capacitor and a TCR is incorporated in the system. All system components were implemented in PacDyn [28] that allows small-signal stability analysis regarding electromechanical oscillations and subsynchronous resonance using descriptor systems [24] and s-domain [29,30], modeling of network dynamics. The IEEE(2.2) model is used to represent the generators electrical system. The rotor circuits are represented by one damper winding and one field winding on the d-axis and two damper windings on the q-axes. The turbine-generator mechanical systems are represented by lumped spring-mass models. Data of G1 and G2 electrical parameters and spring-mass systems are found in [22]. Tables 1 and 2 show the calculated torsional mode-shapes for the mechanical system of each turbine-generator set (TG) alone, i.e. without considering the interaction with the power system. These mode-shapes indicate, for each oscillation mode excited individually, which masses oscillate against the others and what are the relative amplitudes of these oscillations [2]. The zero frequency modes are consequence of the singular nature of the torsional stiffness and damping matrices which model the shaft systems. For these modes there is no relative motion among rotor sections and thus all inertias participate equally. However, interaction with electrical system results in a nonzero frequency, which is normally considered in stability studies. The control systems data are given as follows. Excitation system of G1:

K a ¼ 46 pu=pu; T a ¼ 0:06 s; T b ¼ T c ¼ 0 s; T e ¼ 0:46 s K f ¼ 0:1 pu=pu; T f ¼ 1:0 s; Aex ¼ 0:014;

Bex ¼ 1:55

T r ¼ 0:55 s; V RMIN ¼ 0:9 pu; V RMAX ¼ 1:0 pu Excitation system of G2:

K A ¼ 80 pu=pu; T A ¼ 0:05 s; EfdMIN ¼ 6:0 pu; EfdMAX ¼ 6:3 pu Power system stabilizer of G2:

xn ¼ 154:88 rad=s; fn ¼ 0:05; fd ¼ 1:00 T wG2 ¼ 3:0 s; T n ¼ 0:1961 s; T d ¼ 0:0785 s; K pss ¼ 8:0 pu=pu Static var compensator:

V nom ¼ 13:8 kV; BL ¼ 4:0 pu; Bc ¼ 2:0 pu K P ¼ 0:5 pu=pu; K I ¼ 500 s1 ;

T f ¼ 0:0075 s

K Ppll ¼ 50 pu=pu; K Ipll ¼ 500 s1 where Tf is the time constant of a double first-order filter associated with the voltage measurement system. References [1] Anderson PM, Agrawal BL, Van Ness JE. Subsynchronous resonance in power systems. New York: IEEE Press; 1988.

[2] Kundur P. Power system stability and control. New York: McGraw-Hill; 1994. [3] Neto OM, Macdonald DC. Analysis of subsynchronous resonance in a multimachine power system using series compensation. Int J Electr Power Energy Syst 2006;28(8):565–9. [4] IEEE SSR Working Group. Countermeasures to subsynchronous resonance problems. IEEE Trans Power Ap Syst 1980;99:1810–8. [5] Ramey DG, Kimmel DS, Dorney JW, Kroening FH. Dynamic stabilizer verification tests at the san juan station. IEEE Trans Power Ap Syst 1981;100(12):5011–9. [6] Larsen EV, Rostamkolai N, Fisher D, Poitras A. Design of a supplementary modulation control function for the chester SVC. IEEE Trans Power Deliver 1993;8(2):719–24. [7] Hammad AE, El-Sadek M. Application of a thyristor controlled var compensator for damping subsynchronous oscillations in power systems. IEEE Trans Power Ap Syst 1984;103(1):198–212. [8] Hsu YY, Wu CJ. Design of PID static var controller for the damping of subsynchronous oscillations. IEEE Trans Energy Convers 1988;3(2):210–6. [9] Wang L, Hsu YY. Damping of subsynchronous resonance using excitation controllers and Static Var Compensators: a comparative study. IEEE Trans Energy Convers 1988;3(1):6–13. [10] Wasynczuk O. Damping subsynchronous resonance using reactive power control. IEEE Trans Power Ap Syst 1981;100(3):1096–104. [11] Swift FJ, Wang HF. The connection between modal analysis and electrical torque analysis in studying the oscillation stability of multi-machine power systems. Int J Electr Power Energy Syst 1997;19(5):321–30. [12] Mathur RM, Varma RK. Thyristor-based FACTS controllers for electrical transmission systems. IEEE Press; 2002. [13] Ghosh A, Kumar SVJ, Sachchidanand S. Subsynchronous resonance analysis using a discrete time model of thyristor controlled series compensator. Int J Electr Power Energy Syst 1999;21(8):571–8. [14] Othman HA, Angquist L. Analytical modeling of thyristor-controlled series capacitors for SSR studies. IEEE Trans Power Syst 1996;11(1):119–27. [15] Mattavelli P, Stankovic AM, Verghese GC. SSR analysis with dynamic phasor model of thyristor-controlled series capacitor. IEEE Trans Power Syst 1999;14(1):200–8. [16] Stankovic AM, Mattavelli P, Caliskan V, Verghese GC. Modeling and analysis of FACTS devices with dynamic phasors. In: IEEE power engineering society winter meeting; 2000. [17] Zhijun E, Fang DZ, Chan KW, Yuan SQ. Hybrid simulation of power systems with SVC dynamic phasor model. Int J Elect Power Energy Syst 2009;31(5):175–80. [18] Pilotto LAS, Alves JER, Watanabe EH. High frequency eigenanalysis of HVDC and FACTS assisted power systems. In: IEEE PES summer meeting, vol. 2. 16– 20 July; 2000. p. 823–9. [19] Jovcic D, Pahalawaththa N, Zavahir M, Hassan HA. SVC dynamic analytical model. IEEE Trans Power Deliver 2003;18(4):1455–61. [20] Gomes Jr S, Martins N, Stankovic A. Improved controller design using new dynamic phasor models of SVC’s suitable for high frequency analysis. In: Transmission and distribution conference and exposition, Dallas; 2006. p. 21– 4. [21] Mattavelli P, Verghese GC, Stankovic AM. Phasor dynamics of thyristorcontrolled series capacitor. IEEE Trans Power Syst 1997;12(3):1259–67. [22] IEEE SSR Working Group. Second benchmark model for computer simulation of subsynchronous resonance. IEEE Trans Power Ap Syst 1985;104(May):1057–66. [23] . PSCAD/EMTDC User’s manual. Canada: Manitoba HVDC Research Center; 1994. [24] Lima LTG, Martins N, Carneiro Jr S. Augmented state-space modeling of large scale linear network. In: Proceedings of the IPST ’99 – international conference on power system transients, Budapest, Hungary; June 1999. [25] Gole AM, Sood VK, Mootoosamy L. Validation and analysis of a grid control system using d-q-z transformation for static compensator system. In: Canadian conference on electrical and computer engineering, Montreal; 17– 20 September 1989. [26] Walker DN, Bowler CEJ, Jackson RL, Hodges DA. Results of subsynchronous resonance test at mohave. IEEE Trans Power Ap Syst 1975;94(5):1878–89. [27] Alden RTH, Nolan PJ, Bayne JP. Shaft dynamics in closely coupled identical generators. IEEE Trans Power Ap Syst 1977;96(3):721–8. [28] PacDyn User’s manual, CEPEL, Brazil; 2007. [29] Semlyen A. s-Domain methodology for assessing the small signal stability of complex systems in non-sinusoidal steady state. IEEE Trans Power Syst 1999;6(1):132–7. [30] Gomes Jr S, Martins N. Portela C. Modal analysis applied to s-domain models of ac networks. In: IEEE power engineering society winter meeting, Columbus, Ohio; January 2000.