1

Abstract – This paper presents a systematic method for

the robust design of a supplementary controller for a TCSC device to damp inter-area oscillations in electric power systems. The method is based on a robust control technique structured in the form of linear matrix inequalities (LMIs). The polytopic model is used to guarantee the robustness of the controller with respect to the variations in the operating points of the system. The minimum damping ratio is used in the design stage as performance index for the closed loop system. The proposed controller is based on dynamic output feedback and uses only a local measurement of the active power flow as input signal for the controller, so communications links are avoided. The “feasp” solver, available in Matlab LMI Control Toolbox, is used to solve the set of LMIs formulated for the control problem. Performance analyses of the closed loop system were carried out by means of modal analysis and nonlinear simulations, showing that the controller is effective in improving the damping of inter-area oscillations of a two-area four-machine test system for several different operating points.

Index Terms— TCSC device, supplementaty damping

controller, inter-area oscillations, electric power systems, robust control.

I. INTRODUCTION

HE construction of extensive interconnection lines among different power systems areas has been considerably

increased due to the several advantages inherent to such arrangements [1,2]. The large interconnected systems are advantageous since they improve the reliability of power supply as well as make possible to minimize the total power generation capacity of the system. In other words, these systems can improve the use of the existing generation resources in addition to allowing the co-operation among the interconnected systems in case of faults and to guarantee the power supply in contingency periods.

On the other hand, the increased number of large interconnected power systems, associated with the higher loading of the transmission network, has been leading most of the power systems worldwide to different types of stability problems, such as the presence of poorly damped (or unstable) low-frequency inter-area oscillations [1,3,4]. If no adequate damping is available, the inter-area oscillations

This work was supported by CAPES, CNPq and FAPESP. R. Kuiava, R. A. Ramos, and N. G. Bretas are with Escola de Engenharia de

São Carlos/USP, São Carlos CEP 13566-590, Brazil (e-mails: kuiava@sel.eesc.usp.br; ramos@sel.eesc.usp.br, ngbretas@sel.eesc.usp.br).

(which can be observed, for example, as active power oscillations among interconnected systems) may cause operational limitations (due to the restrictions in the power transfers across the transmission lines) or, in certain cases, they may be sustained and grow, causing the interruption of the energy supply and, consequently, of the system separation.

Since the end of 1960’s, the Power System Stabilizer (PSS) has been widely used to damp oscillations in power systems by means of a stabilizing signal added to the Automatic Voltage Regulator (AVR) [5]. However, such controller may be less effective, in some cases, for inter-area oscillations with very low frequency [3]. In this context, the use of FACTS devices in transmission lines has allowed more effective damping of the inter-area oscillations with the addition of supplementary controllers [2,6].

The Thyristor Controlled Series Capacitor (TCSC) is a kind of FACTS device that has been successfully used for inter-area oscillation damping purposes. In 1999, a 1000 km 500 kV AC interconnection between North and South Brazilian systems was implemented. In such interconnection, two TCSC devices have been installed at both ends of the line to damp the inherent low-frequency inter-area oscillations (0.2 Hz) that occur between the systems [4]. Until the end of 2004, three TCSCs devices came into operation in Asia, being two in China and the other one in India, with the purpose of improving the stability of inter-area oscillations in such systems [7,8]. In addition to these commercial applications, an examination of the potential damping that could be achieved with the use of shunt and series FACTS devices in the European Power System (which has several inter-area oscillations with low damping ratio) was presented in [3], and [9] examines the use of TCSC devices in the radial power system of Argentina.

The inter-area oscillation damping can be achieved through a supplementary controller suitably designed for the TCSC device. Design methodologies based on frequency domain are presented, for example, in [10,11]. In addition, TCSC supplementary controller design methodologies based on robust control were also proposed (H∞ mixed-sensitivity formulation [12], combined H∞ control problem with regional pole placement constraints using LMIs [13] and also multi-objective genetic algorithms approach [14] are examples of these propositions).

In the presented context, this paper proposes a method for the robust design of a supplementary controller for a TCSC device by means of a methodology previously developed for

Robust Design of a TCSC Supplementary Controller to Damp Inter-Area Oscillations

Roman Kuiava, Rodrigo A. Ramos, Member, IEEE, and Newton G. Bretas, Senior Member, IEEE

T

1-4244-1298-6/07/$25.00 ©2007 IEEE.

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designing PSS-type damping controllers [15,16] and already applied for a simultaneous design of PSS and TCSC supplementary controllers [17]. The main innovation of this proposition is the design of a supplementary controller for a TCSC device only, which was made possible by combining this previously proposed methodology with a model order reduction technique.

The design procedure is based on robust control theory and structured in the form of LMIs. The minimum damping ratio is used as performance index for the system in closed loop form and a polytopic model is used to guarantee the robustness of the controller with respect to the variations in the operating points of the system. The proposed controller is based on dynamic output feedback and uses only local measurements of the active power flow as input signal (so communications links over large distances are not needed for the implementation of the proposed controller).

This paper is structured as follows: section II presents the system modelling, including the adopted TCSC model; section III presents the fundamentals of the control technique; the design procedure is summarized in section IV; section V shows the tests and the results obtained with the designed controller and section VI presents the conclusions and the final comments.

II. SYSTEM MODELLING

This section gives a brief description of some aspects involved in the power system model used in this work.

A. TCSC model

The TCSC is a device constituted by a series capacitor bank with fixed value (CTCSC) and a Thyristor-Controlled Reactor (TCR), as can be seen in Fig. 1 [2].

Fig. 1. Basic configuration of a TCSC device.

The TCSC device is installed directly in the transmission system and its equivalent reactance can be varied by adjusting the fire angle of the thyristors. A supplementary controller is required to guarantee that such device supplies additional damping to the inter-area oscillations of interest (by an adequate control of the power flow in the transmission network).

The TCSC is usually represented by a first order linear model in small signal stability studies [18] and this kind of model is also used in the studies carried out in this paper. The block diagram of the adopted TCSC with a supplementary

controller is given in Fig. 2.

Fig. 2. Small-signal dynamic model of TCSC with a supplementary damping controller.

In Fig. 2, TCSCX∆ is the deviation of the equivalent TCSC

reactance with respect to the nominal value, refX∆ is the

reference for the desired reactance deviation (from its nominal value) in steady state,

supX∆ is the stabilizing signal

from the proposed supplementary controller and TCSCT is the

device time constant.

B. Multimachine power system model

In general, multimachine models consider the various components that constitute the system, such as generators with their respective AVRs, transmission systems, loads and FACTS devices. In the adopted multimachine model, the synchronous generators are described by a sixth order model that considers the swing equations and the transient and subtransient effects in d and q axes [19,20]. A first order model of a static type AVR (thyristor controlled without transient gain reduction) is used. The set of nonlinear differential and algebraic equations used in this paper to describe the behavior of the multimachine power system is given by

i i i i i i= + ⋅ − ⋅d d q q a dV E'' X'' I R I (2)

i i i i i i= + ⋅ − ⋅q q d d a qV E'' X'' I R I (3)

,i s i sw w= −δ w& (4)

1[ ],i i i i i

i

= − − ⋅m ew P T D wΗ

& (5)

- - ( ) ( )1[ ],

- -i i i i i i i i

i i i ii i i i i i i

⋅= ⋅ − ⋅ + ⋅q q q l q q q l

d d d qqo q l q l q l

X X' X X X -X' X'' -XE' E'' E' I

τ' X' X X' X X' -X& (6)

- - ( - ) ([ - ]

- -i i i i i i i i

i i i i ii i i i i i i

⋅= + ⋅ ⋅ + ⋅d d d l d d d l

q fd q q ddo d l d l d l

X X' X X X X' X'' -X )1E' E E'' E' I ,

τ' X' X X' X X' -X& (7)

-[- ( - ) ]

-i i

i i i i i i ii i i

+ + ⋅ + ⋅q ld d d q q q d

qo q l

X'' X1E'' = E'' E' X' X'' I E' ,

τ'' X' X& & (8)

[- - ) ] i ii i i i i i i

i i i

+= + + ⋅ ⋅d lq q q d d d q

do d l

X'' -X1E'' E'' E' (X' X'' I E' ,

τ'' X' -X& & (9)

for 1, 2, ,i n= L, where n is the number of synchronous

generators. The state variables iδ and iw are, respectively, the

generator rotor angle and angular speed for the i-th synchronous machine and the state variables idE' , iqE' , idE''

and iqE'' are the direct and quadrature axis transient and

subtransient internal voltages for the i-th synchronous

3

machine. Detailed information regarding the equations of the presented model and their respective parameters can be obtained in [19,20].

III. DESIGN METHODOLOGY

An extension of the methodology proposed in [15] is used in this paper for the design of a TCSC supplementary controller. The control technique is described in this section.

A. System and linear controller model

Small signal stability studies consider that the operating condition of the system do not diverge significantly from its original equilibrium after a small disturbance in the system. In this case, the system analyses and controller design may be carried out by means of linear models. The multimachine system is represented in the design procedure by a set of linear equations, in the state space form, given by

( ) ( ) ( )t t t= ⋅ + ⋅x A x B u& , (10)

( ) ( )t t= ⋅y C x , (11)

where ( )∈ℜx nt is a vector composed of the deviations of the

system state variables (with respect to a nominal operating point), p(t)∈ℜu is a vector with the system control input

(which corresponds to the stabilizing signals to be added to the TCSC input), and q(t) ∈ ℜy is the vector with the system

outputs (in this case, chosen as the active power flow deviations across the transmission line in which the TCSC is installed).

In this paper, a control structure based on dynamic output feedback is adopted. Such control structure is used due to difficulties in obtaining measurements of all model state variables in the real system. The proposed controller based on dynamic output feedback is represented by

( ) ( ) ( )t t t= ⋅ + ⋅c c cx A x B y& , (12)

( ) ( )t t= ⋅c cu C x , (13)

where ( )∈ℜCx nt is a vector with the controller states. The

dynamic behavior of the controllers, as a function of the plant output (t)y , is described by (12). The control input for the

system (t)u is produced by (13) with the application of the

matrix gain CC to the states generated by the controller.

B. Fundamentals of the Control Technique

Robustness with respect to the variations in the operating points of the system is an important and desirable characteristic that may be provided to the closed loop system by the application of damping controllers. Having this in mind, the uncertainties of the electric power system models with respect to the variations of the operating points were treated in this paper by means of the polytopic modelling technique [21]. Such modelling technique is constituted by a set of L linear models structured from the connection of the model (10)-(11) (obtained from the linearization of the multimachine model in L different operating points) with the

controller model (represented by (12)-(13), whose matrices should be determined by the design procedure). The set of linear models is given by

i=x A x%%

& % , i ii

i

⎡ ⎤⎢ ⎥⎣ ⎦

c

c c

A B CA =

B C A% , (i = 1,...,L), (14)

where the matrices iA , iB and iC are obtained by the

linearization of the multimachine model with respect to the i-th operating point of the system, 2( )∈ℜx% nt is a vector with

the states of both the system and the controllers, and cA , cB

and cC are the matrix variables to be determined by the

design procedure. Now, consider the following system [22]:

0( ) , (0)= =x A x& %

% % %a x x , (15)

where ( )∈A Ω% a and Ω is a subset of a matrix space of

dimension 2nx2n, which is formed by the convex combination of the matrices

iA% , and (0)x% is the initial condition of the

closed loop system. Set Ω can be written as

1 1

( ) : ( ) ; 1; 0L L

i i i ii i

a a a a a= =

⎧ ⎫= = ≥⎨ ⎬⎩ ⎭

∑ ∑Ω = A A A% % % , (16)

It can be observed in (16) that set Ω is a polytope in the matrix space and matrices

iA% are the vertexes of this

polytope. A sufficient condition that proves the asymptotic stability of (15) is the determination of matrices cA , cB , cC

and 0= >P P% % such that

0Ti i+ <A P PA% % , para i = 1, ..., L. (17)

The use of the polytopic modelling technique in the design procedure, associated with the quadratic stability theory, guarantees the stabilization of the closed loop system, not only for the operating points used in the construction of the polytopic system, but also for all the operating points that can be generated from the convex combination of the L adopted operating points [21,22]. This feature ensures better robustness of the designed controllers with respect to variations in the operating conditions.

A performance index is included in the control problem to ensure an acceptable performance for the polytopic system in closed loop. The adopted design methodology uses a minimum damping ratio for all modes of the polytopic system in closed loop as a performance index. The adopted performance index is included in the control problem formulation by using the Regional Pole Placement (RPP) technique [23]. By means of this technique, a region in the complex plane is specified to guarantee a minimum damping ratio for the oscillation modes of the polytopic closed loop system. Such region is defined by ζ > ζ0, and it can be viewed in Fig. 3, where ζ0 is the minimum damping ratio to the poles of the polytopic system in closed loop.

4

Pole placement region

Re

(minimum damping ratio)0ζ=ζ

θ

Im

Fig. 3. Region for the pole placement.

The matrix inequalities related to the polytopic modelling, associated with the RPP technique, are given by

( )0

)~~~~

()~~~~

(cos

)~~~~

(cos~~~~

<⎥⎦

⎤⎢⎣

⎡

+⋅−⋅−⋅+⋅

iT

iiT

i

Tiii

Ti

sen

sen

APPAAPPA

PAAPAPPA

θθθθ , (18)

for i = 1, ..., L and 10cos ( )θ ξ−= .

The inequalities (18) are bilinear matrix inequalities (BMIs) on the problem matrix variables ( cA , cB , cC and

P~

). However, such inequalities are changed into LMIs by means of a different parameterization and variable changes applied to (18). The parameterization procedure is developed and detailedly presented in [15,24]. The design procedure is presented in the following section.

IV. DESIGN PROCEDURE

The design procedure is divided in three stages: (i) construction of the polytopic model; (ii) calculation of the state feedback gain matrix of the controller ( cC ) and; (iii)

calculation of the matrices that describe the controller dynamics (matrices cA and cB ). Each stage is described in

this section.

A. Construction of the polytopic modelling

The first step of the controller design procedure consists in choosing some typical operating points of the system to obtain

the matrices iA% (i = 1,..., L) that define the vertexes of the

polytopic model. The steps for carrying out the first stage are: (i) Build the multimachine model as a set of nonlinear

differential and algebraic equations in the form

f , ,x = ( )& x z u , (19)

0 , ,= ( )g x z u , (20)

, ,y = ( )h x z u , (21)

where f and g are, respectively, vectors of differential and algebraic equations and h is a vector of output equations [25];

(ii) Choose L typical operating points of the system through a power flow solution and obtain L

equilibrium points 0 0 0 , , ix z u for the system

equation (19)-(21); (iii) Linearize the system equation (19)-(21) around these

L equilibrium points and after the network reduction

(which is required to eliminate the vector of algebraic variables z), the linear systems are obtained in the space state form ( ) ( ) ( )i it t t= +x A x B u& and

( ) ( )it t=y C x for (i = 1,...,L) which will form the

vertexes of the polytope Ω defined by (16).

B. Calculation of the state feedback gain matrix of the controller ( cC )

After the first stage, the purpose of the two following stages is to design a supplementary controller based on dynamic output feedback structure, defined by matrices cA ,

cB and cC . This controller must stabilize the polytope Ω

while fulfilling a minimum damping ratio 0ξ for all

oscillation modes of the closed loop systems. In this design stage, the state feedback gain matrix of the controllers ( cC ) is

determined by means of the solution of an LMI set written for the polytope vertexes. These steps can be carried out in the following way:

(i) Define the minimum damping ratio 0ξ required and

calculate 10cos ( )θ ξ−= . Therefore, a region in the

complex plane defining the minmum acceptable performance index is specified for all the oscillation modes of the polytopic system;

(ii) Build the computational representation of the matrix variables DY and DL . The matrix DY must be

symmetric with dimension xn n and the matrix DL

must be rectangular with dimension xp n . The

following LMI set must be structured and solved for the variables DY and DL :

,DY > 0 (22)

( cos ( -

) - )

0

cos (- (

- ) )

T Ti i i i

T T T T

T Ti i i i

T T T T

sin

sin

θ θ

θ θ

⎡ ⎤⋅ + ⋅⎢ ⎥⎢ ⎥+ + +⎢ ⎥⎢ ⎥ <⎢ ⎥⎢ ⎥⋅ + ⋅ +⎢ ⎥⎢ ⎥+ + +⎢ ⎥⎣ ⎦

D D D D

D D D D

D D D D

D D D D

Y A A Y Y A A Y

L B BL L B BL

Y A A Y Y A A Y

L B BL L B BL

,(23)

for i = 1,...,L; (iii) After determining the variables DY and DL , by

solving LMIs (22)-(23), the state feedback gain matrix cC can be calculated by -1=c D DC L Y .

In the next stage, the matrices cA and cB are calculated

with the previously designed matrix cC .

C. Calculation of the matrices that describe the controller dynamics (matrices cA and cB ).

In the last stage, the matrices cA and cB are calculated

by means of the solution of a new LMI set established in the polytope vertexes. To do so, the following steps must be executed:

5

(i) Build the computational representation of the new matrix variables DP , DX , DF and DS , with

dimension xn n , xn n , xn q and xn n , respectively. The

following set of LMIs must be solved:

0⎡ ⎤

>⎢ ⎥⎢ ⎥⎣ ⎦

D D

D D

P P

P X, (24)

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

11 12 13 14

22 23 24

33 34

44

N N N N

* N N N< 0

* * N N

* * * N

, (25)

where

( ),Ti isinθ= ⋅ +11 D k k DN P A A P (26)

( ),T Ti isinθ= ⋅ + + +T

12 D k D D DN P A A X C F S (27)

cos ( - ),Ti iθ= ⋅13 k D D kN A P P A (28)

cos ( ),T T Ti iθ= ⋅ + + +14 D k D D DN -P A A X C F S (29)

( ),T T Ti isinθ= ⋅ + + +22 D D D DN X A A X F C C F (30)

cos (- - ),T T Ti iθ= ⋅ + +24 D D D DN X A A X F C C F (31)

,T=23 14N N =33 11N N , ,=34 12N N =44 22N N , (32)

i i= +k CA A BC , ( 1,..., )i L= . (33)

(ii) After finding DP , DX , DF and DS , the matrices cA

and cB can be calculated by:

-1 T=C D DA U M P , (34)

-1=C D DB U F , (35)

where -1=D DM P S and -=D D DU P X .

After concluding the three design stages, the matrices cA ,

cB and cC which define the structure of the TCSC

supplementary controller are obtained. Such controller can also be represented in the transfer function form obtained by

s s -1c c cH( ) = C ( I - A ) B . (36)

V. TESTS AND RESULTS

The tests to verify the performance of the proposed controllers were carried out in a well-known power system model, and the results obtained by means of the modal analysis and nonlinear simulations are presented in this section2. The adopted system is commonly used in small signal stability studies, and it is constituted by two areas interconnected by a tie-line (as shown in the diagram of Fig. 4). The complete data of this system can be obtained in [20,26].

A TCSC is included in the tie-line, since this branch is a weak connection which limits the power transfer between the two areas and therefore characterizes the inter-area oscillation presented in this system. The value of the TCSC reactance in

2 Modal analysis and nonlinear simulations were carried out, respectively, by means of PACDYN and ANATEM software’s, both developed by CEPEL (Brazilian Electrical Research Centre).

steady-state condition corresponds to a compensation of 40% in the tie-lie reactance.

L1G2

G1

2

L2 G4

4

G3

Area 11 5 6 7

Area 298 10 3

TCSC

Fig. 4. Single line diagram of two-area system with a TCSC.

The parameters of the AVRs used in all generators are Ke=200 and Te=0.01s. The time constant for the TCSC is 10 ms.

The system loads in the operating conditions for the base case are PL1=967 MW, QL1=100 MVAr, PL2=1767 MW and QL2=100 MVAr. The tie-line active power is 401 MW. The other operating points analyzed were obtained using ±10% variations of the system loads with respect to the base case, as shown the eigenvalue plot in Fig. 5. This figure shows that the open loop system presents adequately damped local modes (which correspond to a damping ratio greater than 5%) and a poorly damped inter-area mode for the base case and for vertexes 3 and 4. This inter-area mode is unstable for vertexes 2 and 5. Therefore, a supplementary controller for the TCSC was designed according to the procedure described in previous section aiming to enhance damping of the inter-area oscillation mode.

-0.5 -0.4 -0.3 -0.2 -0.1 00

0.5

1

1.5

2

2.5

3

real

imag

inár

y (r

ad/s

)

Local modes and inter-area mode for open loop system for five different operating conditions

Base case

Case 2: +10%L1, +10%L2

Case 3: -10%L1, -10%L2

Case 4: +10%L1, -10%L2

Case 5: -10%L1, +10%L2

Inter-area mode

Local mode - area 2Local mode - area 1

damping rat io 5%

Fig. 5. Local modes and inter-area mode for open loop system for five

different operating points.

The active power flow deviation in the tie-line (∆Ptie) was used as the input signal for the supplementary controller. A washout filter with a time constant equal to 4 s was added to the controller. Generator 3 was used as an infinite-bus (only for design purposes), supplying an angular reference to the model.

The five operating points used in open loop system analysis were also used for the construction of the polytopic model. A minimum damping ratio of 5% (ζ0=0.05) was defined as an acceptable performance index for the closed loop polytopic system. The “feasp” solver, available in Matlab LMI Control Toolbox [28] was used to solve LMIs related to

6

this control problem. The LMIs were solved in a computer with a Pentium IV 2.4 GHz processor and 512 MB of RAM, and the time taken to generate the controller was approximately 1 hour and 20 minutes.

The power system model has 23 states. One of the drawbacks of the proposed design methodology is its requirement that the order of the designed supplementary controller must be equal to the order of the power system model. In this presented application, since the power system model has 23 states, the resulting supplementary controller also had order 23.

High order controllers are difficult to implement in practice and, for this reason, it would be desirable that the controllers designed by this proposed methodology could have smaller dimensions. To achieve the goal of producing a low order controller, the proposed procedure was combined with a step of model order reduction. After a high order controller was designed, the balanced truncation method [25] was applied, reducing the dimension of the controller from order 23 to only order 4.

The inclusion of a model order reduction in the design procedure requires a posteriori verification of the controller effectiveness. The validation of the effectiveness of the reduced order controller was carried out by comparison of the frequency responses of the full and reduced order controllers, as shown in Fig. 6, and by quadratic stability analysis with regional pole placement constraints (using the formulation given by (18) and considering the same polytope adopted in design stage). It can be seen in Fig. 6 that the reduction of the controller order did not significantly affect the frequency response of the controller obtained from the design procedure in the frequency range of interest. The transfer function of the reduced TCSC supplementary controller with the washout filter is presented in the appendix.

The local modes and inter-area mode for the closed loop system in all five operating conditions used as the vertexes of the polytopic model are shown in Fig. 7. These results were obtained considering a full model of type (2)-(9) for generator 3, which was taken as the angular reference for the system model.

It was verified that the performance index obtained with the designed controller is better than the minimum acceptable performance specified in the design stage, as shown in Fig. 7. It can also be seen, by comparison of eigenvalues for the open loop system (Fig. 5) and closed loop system (Fig. 7), that the damping ratio for the local mode associated to area 2 increased with the addition of the supplementary controller. On the other hand, the damping ratio for the local mode associated to area 1 remained practically unchanged, which is probably associated to the fact that generator 3 was modeled as an infinite-bus in the design stage.

Fig. 6. Comparison of frequency response of full and reduced order controllers.

The results of the nonlinear simulation in the base case with the designed TCSC supplementary controller are shown in Figs 8, 9 and 10. These simulations were carried out to validate the results of the linear analyses. The perturbation used to stimulate the oscillation modes is a variation of 10% in both loads in t = 2s and after 0.5 s the system returns to the base case condition. The TCSC was constrained to operate in the capacitive region, in the range of 1% to 10% of the TCSC reactance.

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 00

0.5

1

1.5

2

2.5

3

real

imag

inar

y (r

ad/s

)

Local modes and inter-area mode for closed loop system for five different operating conditions

Base caseCase 2: +10%L1,+10%L2Case 3: -10%L1,-10%L2Case 4: +10%L1,-10%L2Case 5: -10%L1,+10%L2

Inter-area mode Local mode - area 1

Local mode- area 2

damping ratio 5 %

Fig. 7. Local modes and inter-area mode for the closed loop system in all

operating points considered in polytopic model.

The robust controller presented a satisfactory performance as can be seen in Fig. 8 and 9. The controller performance was also verified for other operating points used in the design stage and for intermediate operating points, where variations of ±2%, ±5% and ±7% were used. This satisfactory performance of the closed loop system in these intermediate points (which were not considered in the design stage) is a good indicative of the robustness of the designed controller with respect to variations in the operating conditions.

7

Fig. 8. Response of tie-line active power flow to a load disturbance.

Fig. 9. Response of rotor speed of generators 1 and 2 to a load disturbance.

Fig. 10. Response of equivalent TCSC reactance to a load disturbance.

VI. CONCLUSIONS

A robust design of TCSC supplementary controller for inter-area damping purposes was presented in this paper. The design procedure proposed in this paper is an extension of a methodology previously proposed by the authors. This extension includes a further step of model order reduction, addressing the problem of controller order, which was until now a drawback of the previous proposition.

The presented tests showed that the designed controller is robust with respect to variations in the operating conditions and guarantees that the response of the closed loop system to perturbations in these conditions will exhibit an acceptable performance.

The design of supplementary controllers to larger power systems and using other kinds of FACTS devices, as well as the treatment of aspects related to the construction of the polytopic model, are among the future directions of this research.

VII. APPENDIX

The transfer function of the reduced model of the designed supplementary controller (including the washout filter), is given by

6 3 2

5 2

s4 2.5556 10 1 8.3056 10 1 0.34557 0.02424( ) 0.003845

1+s4 0.6959 1 50 10 1 673.4 318100.0TCSC

s s s sF s

s s s s

− −⋅ + ⋅ + + += ⋅ ⋅ ⋅ ⋅+ ⋅ + + +

(A.1)

VIII. REFERENCES [1] P. Kundur, J. Paserba, V. Ajjarapu et al, “Definition and classification of

power system stability,” IEEE Transactions on Power Systems, vol. 19, pp. 1387-1401, May 2004.

[2] N. G. Hingorani, and L. Gyugyi, Understanding FACTS: Concepts and Technology of Flexible AC Transmission Systems. New York: IEEE Press, 2000.

[3] E. Handschin, N. Schnurr, and W.H. Wellssow, “Damping potential of FACTS devices in the European power system,” IEEE Power Engineering Society General Meeting, vol.4, July 2003.

[4] C. Gama, “Brazilian North-South Interconnection control-application and operating experience with a TCSC,” IEEE Power Engineering Society Winter Meeting, vol.2, pp.1103-1108, July 1999.

[5] F. P. DeMello, and C. Concordia, "Concepts of synchronous machine stability as affected by excitation control,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-88, pp. 316-329, April 1969.

[6] J.J. Paserba, “How FACTS controllers benefit AC transmission systems,” IEEE Power Engineering Society General Meeting, vol.2, pp. 1257 – 1262, June 2004.

[7] G. Jianbo, C. Gesong, L. Jiming et al, “Chengxian 220kV Thyristor Controlled Series Compensation: Parameters Design, Control & Overvoltage Protection,” IEEE Transmission and distribution conference and exhibition: Asia and Pacific, August 2005.

[8] F. Yue, and Q. Bailu, “Electrical Design Aspects of Pingguo TCSC Project,” IEEE Transmission and distribution conference and exhibition: Asia and Pacific, August 2005.

[9] A. Del Rosso, C. A. Cañizares, V. Quintana et al, “Stability improvement using TCSC in radial power systems,” NAPS-2000, Waterloo, ON, October 2000.

[10] N. Martins, H.J.C.P. Pinto, J.J. Paserba, “Using a TCSC for line power scheduling and system oscillation damping-small signal and transient stability studies,” IEEE Power Engineering Society Winter Meeting, vol. 2, pp. 1455-1461, January 2000.

[11] L. Fan, and A. Feliachi, “Robust TCSC control design for damping inter-area oscillations,” IEEE Power Engineering Society Winter Meeting, Vol. 2, pp. 784-789, July 2001.

[12] B. Chaudhuri, and B. Pal, “Robust damping of multiple swing modes employing global stabilizing signals with a TCSC,” IEEE Transactions on Power Systems, vol. 19, n. 1, pp. 499 – 506, February 2004.

[13] L. Qian, V. Vittal, and N. Elia, “LMI Pole Placement Based Robust Supplementary Damping Controller (SDC) for A Thyristor Controlled Series Capacitor (TCSC) Device,” IEEE Power Engineering Society General Meeting, pp. 1085-1090, June 2005.

[14] S. Panda, R.N. Patel, N.P. Padhy, “Power system stability improvement by TCSC controller employing a multi-objective genetic algorithm approach,” International journal of intelligent technology, vol 1, 2006.

[15] R. A. Ramos, L. F. C. Alberto, and N. G. Bretas, “A New Methodology for the Coordinated Design of Robust Decentralized Power System Damping Controllers,” IEEE Transactions on Power Systems, vol. 19, n.1, pp. 444-454, February 2004.

[16] R. A. Ramos, L. F. C. Alberto, and N. G. Bretas, “Decentralized Output Feedback Controller Design for the Damping of Electromechanical

8

Oscillations,” Int. J. Electrical Power & Energy Syst., vol. 26, n.3, pp. 207-219, March 2004.

[17] R. Kuiava, R.V. de Oliveira, R.A. Ramos et al, “Simultaneous coordinated design of PSS and TCSC damping controller for power systems,” IEEE Power Engineering Society General Meeting, June 2006.

[18] A. D. Del Rosso, C. A. Canizares, and V. M. Dona, “A study of TCSC controller design for power system stability improvement,” IEEE Transactions on Power Systems, vol, 18, n. 4, pp. 1487 – 1496, November 2003.

[19] P. M. Anderson, and A. A. Fouad, Power System Control and Stability. Piscataway, NJ: IEEE Press, 1994.

[20] P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1994.

[21] R. A. Ramos, N. G. Bretas, and L. F. C. Alberto, “Damping Controller Design for Power Systems with Polytopic Representation for Operating Conditions,” Proceedings of the IEEE Power Engineering Society Winter Meeting , 2002.

[22] S. Boyd, L. El Gahoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM 1994.

[23] M. Chiali, P. Gahinet, and P. Apkarian, “Robust Pole Placement in LMI Regions,” IEEE Transactions on Automatic Control, vol. 44, n.12, pp. 2257-2270, December 1999.

[24] M. C De Oliveira, J. C. Geromel, and J. Bernussou, “Design of Dynamic Output Feedback Decentralized Controllers via a Separation Procedure,” International Journal of Control, vol. 73, n.5, pp. 371-381, 2000.

[25] B. Pal, and B. Chaudhuri, Robust control in power systems. New York, NY: Springer Science+Business Media, 2005.

[26] G. Rogers, Power system oscillations. Norwell, MA: Kluwer, 2000. [27] K. Zhou, J. Doyle, and K. Glover, Robust and optimal control.,

Englewood Cliffs, NJ: Prentice-Hall, 1995. [28] P. Gahinet, A. Nemirovski, A. J. Laub et al, LMI Control Toolbox User´s

Guide. Natick, MA: The Mathworks Inc, 1995.