regional pole placement

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1

Abstract—This paper addresses a systematic controller

design for the TCSC device in power systems based on a Linear
Matrix

Inequality

(LMI)

pole

placement

technique.

Requirements of performance can be expressed in terms of
LMI’s. In our formulation we combine a

H

controller design

with some regional pole placement constraints using LMI’s.
The application of this controller, which is presented as an
example in this paper, is effective in improving the damping
ratio of a two-area four-machine test system. Furthermore,
good damping ratio can be achieved and maintained over a
larger range of operation with this LMI pole placement
controller than with a conventionally designed

H

controller.

Another benefit of this approach is that it results in a fixed
parameter controller.

I.

I

NTRODUCTION

OWER system operating conditions vary with system
configuration and load level in a complex manner.

Obtaining a robust controller to damp system oscillations is a
primary objective in the controller design in power systems.
The most commonly used controller is the power system
stabilizer (PSS), which is installed at the generator so that it
can damp both local oscillations and inter-area oscillations.
With the successful application of FACTS devices in larger
systems, a lot of interest has arisen in designing
supplementary damping controllers (SDC) for FACTS
devices in recent years. Several papers [1, 2] have shown the
significant impact of SDC in damping inter-area system
oscillations. There are several approaches which mainly use
a linear time invariant (LTI) controller to guarantee the
robust stability and robust performance after describing the
changes of operating condition as uncertainties [3].

Normally the problem is formulated as a weighted mixed

sensitivity design. References [4,5] dealt with the

H

controller design in order to guarantee the robust stability
and performance. The standard Riccati solution to this

The Authors are with Department of Electrical and Computer

Engineering, Iowa State University, Ames, IA, 50011 USA (corresponding
author Qian Liu e-mail: qliu@ iastate.edu).

problem will usually cause some pole-zero cancellation
problems and requires careful weighting function selection.
It is clearly known that the design objective of a conventional

H

controller is to minimize the infinity norm from some

output signals to some input signals. But the obtained

H

controller with the minimal infinity norm does not guarantee
the closed loop system with a largest damping ratio at some
critical modes because the

H

norm index is not directly

related to the damping issue. Though we can carefully select
some weighting functions to get the better controller, the
conventional

H

controller design still has its intrinsic limit

in achieving this specific goal.

Recently the Linear Matrix Inequality (LMI) approach

has been successfully applied in the control area [6]. The
stability problem can be formulated as LMI’s. In addition,
other specific objectives from the time domain performance
can be easily expressed in terms of some LMI’s. Previously
these would have been translated into some weighting
function selection. In this paper, a supplementary controller
for a TCSC device in power systems is designed with this
LMI pole placement technique.

The LMI based pole placement SDC is applied to a

two-area four-machine system. The simulation results show
that the designed controller can enhance the damping ratio
for the inter-area oscillations in a much wider operating
range than the conventionally designed

H

controller,

which doesn’t take the pole placement into account.

Section II of this paper introduces the LMI pole

placement controller design technique. The test system is
described in Section III. Details of the robust pole placement
output-feedback controller are addressed in Section IV.
Simulation results of its application in the test system are
shown in Section V as well as comparisons with a
conventionally designed

H

SDC. Conclusions are drawn

in Section VI.

LMI Pole Placement Based Robust Supplementary

Damping Controller (SDC) for A Thyristor

Controlled Series Capacitor (TCSC) Device

Qian Liu, Vijay Vittal, Fellow, IEEE, Nicola Elia

P

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2

II.

REGIONAL

POLE

PLACEMENT

VIA

LMI

OPTIMIZATION

The objective of the controller design is to place the

closed loop poles in some regions while still satisfying some
infinity norm constraints at the same time. It is clearly known
that the transient response of a linear system is related to the
location of its poles [7]. For example, the step response of a
second–order system with poles

d

n

j

ω

ζω

λ

±

=

is fully

characterized in terms of the undamped natural frequency

λ

ω =

n

, the damping ratio

ζ

, and the damped natural

frequency

d

ω

. Confining the closed-loop poles to some

region can ensure a minimum damping ratio

θ

ζ

cos

=

,

which is the objective of this damping problem in the power
systems.

A.

LMI regions

Definition 1: A LMI region is the any subset D of the

complex plane that can be defined as

{

}

0

:

<

+

+

=

T

M

z

zM

L

C

z

D

(1)

where

L

and

M

are real matrices such that

L

L

T

=

.

The matrix-valued function

T

M

z

zM

L

z

f

+

+

=

)

(

D

is

called the characteristic function of D. Some typical LMI
regions are shown in Fig. 1:

(1) Half-plane

α

<

)

Re(

z

:

0

2

)

(

<

+

+

=

α

z

z

z

f

D

(2) Disk centered at (-q,0) with radius r:

0

)

(

<

+

+

=

r

z

q

z

q

r

z

f

D

;

(3) Conic sector with apex at the origin and inner angle

θ

2 (

)

,

0

,

0

(

θ

S

):

0

)

(

sin

)

(

cos

)

(

cos

)

(

sin

)

(

<

+

+

=

z

z

z

z

z

z

z

z

z

f

θ

θ

θ

θ

D

LMI region is a subset of the complex plane that is

representable by an LMI in

z

and

z

.

B.

Quadratic

D

-stable

Definition 2: The system

Ax

x

=

&

is called D -stable if

all its poles lie in D.

Theorem 1: The matrix A is D-stable if and only if there

exists a symmetric matrix X such that

0

)

,

(

<

X

A

M

D

,

0

>

X

where

T

T

AX

M

AX

M

X

L

X

A

M

)

(

)

(

:

)

,

(

+

+

=

D

(2)

In the case of confining the poles in a conic sector of

)

,

0

,

0

(

θ

S

,

,

0

=

L

=

θ

θ

θ

θ

sin

cos

cos

sin

M

(3)

Based on Theorem 1, the system matrix has poles in

)

,

0

,

0

(

θ

S

if and only if

0

)

(

sin

)

(

cos

)

(

cos

)

(

sin

<



+

+

T

T

T

T

A

X

AX

AX

A

X

A

X

AX

A

X

AX

D

D

D

D

D

D

D

D

θ

θ

θ

θ

(4)

Note that when this D is the entire left–half plane, this

notion reduces to asymptotic stability, which is characterized
in LMI terms by the Lyapunov theorem [8].

C.

The

H

constraint

The classical

H

robust controller design in the mixed

sensitivity problem [9] is represented as

γ

KS

W

T

W

S

W

3

2

1

,

(5)

where

γ

is the upper bound on the

H

norm.

The

H

constraint is equivalent to the existence of a

solution

0

>

X

to the LMI

0

2

<

+

I

D

X

C

D

I

B

C

X

B

A

X

X

A

cl

cl

T

cl

T

cl

T

cl

cl

T

cl

cl

γ

(6)

where

cl

cl

cl

cl

D

C

B

A

,

,

,

are

D

C

B

A

,

,

,

matrices of the closed

loop system.

By combining the pole placement constraint in (4) with the

H

constraint in (6), we restrict our goal to the following

suboptimal formulation of

H

synthesis with pole

placement constraints.

The controller design objective is to find

0

>

X

and a

controller

)

(s

K

that satisfy (3) and (4) with

D

X

X

X

=

=

.

The difficulty is that there are non-linearities in both (3) and
(4). These can be solved by make appropriate changes of
controller variables. The details can be found in [10, 11].

θ

r

α


Fig. 1. LMI region

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3

III.

T

EST

S

YSTEM

A.

Test System

A two-area four-machine system [12] is used in this paper

for the simplicity of demonstration. This system has been
specially designed by Ontario Hydro for fundamental studies
of inter-area oscillations in power systems. The system has
the complexity to verify the efficiency of the proposed
procedure and is characterized by the presence of both
inter-area and local modes.

The test system consists of two identical areas, each

including two generators with the same power output and a
load. All generators are represented by the two-axis model
[13] equipped with (IEEE AC4A) ETMSP Type-30 [13]
Excitation system. Thus each generator with its exciter is
modeled by seven first-order nonlinear differential
equations. The loads are modeled as constant impedances.
A TCSC has been placed in series with the transmission line
to extend the power transfer capability. The diagram of this
system is shown in Fig. 2.

B.

TCSC model

Fig. 3 shows a block diagram for the TCSC model for

typical transient and oscillatory stability studies [14]. An
open loop auxiliary signal (

Auxiliary

X

) could be the output

of a power scheduling controller, which is used to extend the
power transfer limit and improve transient stability. A small
modulation input (

Modulation

X

) is produced by a

supplementary damping controller, which will be discussed
later in this paper. The desired compensation level is set by a

reference input (

Reference

X

).

The TCSC thyristor firing and other delays are usually

represented by a single lag of about 15ms. It will not be
modeled here for simplicity because it does not significantly
impact the electromechanical modes. In this paper, we
restrict ourselves to the design of the damping controller for
the inter-area mode.

C.

State Space Description of The System

The power system is represented by a differential

algebraic model that captures the differential dynamics of the
various components and the algebraic relationship that
governs the network.

)

,

(

0

)

,

(

.

Y

X

G

Y

X

F

X

=

=

(7)

The above equations can be linearized at a specific

operating point in the following manner:

u

B

w

B

Ax

x

2

1

+

+

=

&

u

D

w

D

x

C

z

12

11

1

+

+

=

(8)

u

D

w

D

x

C

y

22

2

2

+

+

=

We want to find an output-feedback controller in the state

space form as follows:

y

B

x

A

x

k

k

k

k

+

=

&

y

D

x

C

u

k

k

k

+

=

(9)

Usually a robust controller is designed based on a nominal

operating point (OP) with the plant uncertainty represented
as appropriate weighting functions [15]. The obtained
controller can improve the damping ratio at some operating
points. The damping performance, however, doesn’t relate
well to its achieved

H

norm. Without carefully selecting

weighting functions, the obtained controller can even worsen
the closed loop system damping. Therefore, it takes a lot of
efforts to select the appropriate weighting functions because
there is no clear and explicit connection between the
weighting functions and the damping ratio of the closed loop
system. With the introduction of the pole placement
constraints, the system damping issue can be directly
addressed and enhanced in a very wide operating range.

IV.

S

UPPLEMENTARY

D

AMPING

C

ONTROLLER FOR A

TCSC

DEVICE

We want to obtain a damping controller for the TCSC to

provide sufficient damping for the inter-area oscillatory
modes in the test system at all six operating points, which are
characterized by a 100-600MW range of power transfer from
area 1 to area 2 with a compensation level range of 0-50%. A
0% TCSC compensation level is used at P

tie

=100MW and

200MW while a 50% TCSC compensation level is used at
P

tie

=300MW to 600MW. The operating point at

P

tie

=400MW is chosen as a nominal case in the controller

synthesis. The absolute value of the tie-line current is used
as the output feedback control signal due to its largest


Fig. 2. The two-area four-machine test system

Fig. 3. The block diagram for the TCSC model for typical stability studies

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4

observability of the poorly damped modes. The setup of this
controller design problem is shown in Fig. 4, where P
represents the system model of the 2-area power system and
the primary part of the TCSC device. The K in this figure is
the controller we want to design. The output of this controller
is fed into the summing point of the TCSC device as the

Modulation

X

signal in Fig. 2. The major objective of the

controller design is to improve the damping instead of the
reference tracking or disturbance rejection. In the SDC
setup, because the regional pole placement constraint will be
imposed in terms of LMI’s, the weighting function is not a
critical issue so that we simply set them as

12

.

1

1

+

=

s

W

perf

,

1

=

u

W

,

1

.

0

=

noise

W

.

The original system obtained under the nominal OP is of

27

th

order. Then it is reduced to 4

th

order by using a Hankel

norm [16] reduction. To get a better damping,

o

63

=

θ

is

used to define the conic sector to ensure a minimum damping
ratio of 45% at the nominal OP while maintaining the
damping ratio good enough at all of the other five OPs.

LMI toolbox in MATLAB is used to form the three LMIs

(10) (11) (12), which are derived from (4), (6) with the
changes of some controller parameters in order to make the
problem convex and linear [14].

0

R

I

I

S

>

(10)

perf

W

noise

W

u

W

P

K

tie

I

d

noise

ref

TCSC

dV

_

Fig. 4. Supplementary damping controller (SDC) design setup

0

T

T

R

I

L

M

M

I

S

+

⊗ Φ +

⊗ Φ

<

(11)

11

21

21

22

0

T

Ψ

Ψ

<

Ψ

Ψ

(12)

The LMI of (10) sets the constraint of

0

>

X

; the second

LMI imposes the pole region constraint; and the last one
imposes

the

H

constraint.

The

details

of

22

21

11

,

,

,

Ψ

Ψ

Ψ

Φ

can be found in [16]. Therefore, a 5

th

order controller is obtained, which is of the same order as the
augmented open loop system.

The parameters of the obtained controller are shown in

the Appendix. A washout filter of

)

10

1

/(

10

s

s

+

is applied in

series with the controller to maintain the original steady state
gain of the open loop system.

V.

P

ERFORMANCE RESULTS

Apply this supplementary damping controller (SDC) to the

test system. A conventional

H

controller that is designed

based on the same nominal OP without pole placement
constraints is tested in the two-area system at the same time.

Small signal analysis is performed using EPRI’s MASS

[9] software package. Table 1 shows the damping ratio of the

system with the pole placement based

H

SDC, without

SDC and with the conventional

H

SDC.

The frequency jump of the poorly damped modes from

P

tie

=200MW to P

tie

=300MW is due to the compensation

level change of the TCSC device.

From Table 1, it is obvious that the closed loop system

with the LMI based pole placement SDC has the largest
damping ratio at all six OPs. The damping ratio at the
nominal case of P

tie

=400MW is 19.71%, which is smaller

than the expected value in the design phase. It is reasonable
because the system at the design phase is different from the
real system due to the linearization and model reduction.

Nonlinear time domain simulation is performed using

ETMSP [9] software. A three-phase short circuit fault is
applied at bus 5 for 100 ms; the tie-line real power flow is
monitored. The simulation results are shown in Fig. 5~10.
Comparisons are made between the system with the LMI

based pole placement SDC, with conventional

H

SDC and

without SDC.

From figures 6-10 it is very clear that the inter-area modes

are well damped from P

tie

=100MW to P

tie

=600MW. The

system takes a longer time to be settled down to its
equilibrium point at P

tie

=100MW than at P

tie

=600MW even

though the result from the small signal analysis shows the
damping ratio at P

tie

=100MW is greater than the damping

ratio at P

tie

=600MW. This inconsistency between the small

signal results and the transient results is due to the
non-linearity being taken into account in the transient
simulation. Another important point to mention is that the
transient response looks better with the increase in the level
of P

tie

. That is to say, the controller is more effective when

the power transfer level between areas is much higher. The
TCSC device is a series-connected device that is installed
between the tie-line, which makes it totally different from
other FACTS devices in the way that it affects the
controllability

in

this

problem.

Furthermore,

the

controllability increases with the increase of the power
transfer level between the two areas. Consequently, the
benefit of this controller is its increasing effect in an
increasing stressed power network. In other words, with the

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5

TABLE

I

D

AMPING RATIO COMPARISON BETWEEN CLOSED LOOP SYSTEM WITH POLE PLACEMENT BASED

SDC

AND CONVENTIONAL

H

SDC

Pole placement based

H

SDC

H

SDC

Without SDC

Ptie

(MW)

Frequency

(Hz)

Damping

ratio

(%)

Frequency

(Hz)

Damping ratio

(%)

Frequency

(Hz)

Damping ratio

(%)

100

0.4821

18.08

0.5225

8.04

0.4750

2.81

200

0.4774

34.64

0.5443

10.45

0.4549

2.68

300

0.8413

23.35

0.7318

9.69

0.6011

3.80

400

0.8518

19.71

0.7393

9.73

0.5829

3.61

500

0.8427

17.68

0.7346

9.53

0.5575

3.31

600

0.8167

16.29

0.7151

9.04

0.5193

2.74


Fig. 5. Three phase fault at Ptie=100MW

Fig. 6. Three phase fault at Ptie=200MW

Fig. 7. Three phase fault at Ptie=300MW

Fig. 8. Three phase fault at Ptie=400MW

Fig. 9. Three phase fault at Ptie=500MW

Fig. 10. Three phase fault at Ptie=600MW

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6

increase of the inter-area power transfer stress, the
supplementary damping controller of TCSC will be more
and more efficient.

VI. CONCLUSIONS

Instead of using an adaptive controller for TCSC, a

parameter-fixed controller is used to achieve better
damping ratio at all of the six operating points. This
controller is designed by imposing the regional pole
placement constraints. From its application in a two-area
four-machine system, it is obvious that the benefit of this
controller in enhancing the system damping is much greater
than the conventional

H

controller. This LMI based pole

placement SDC is easier to design compared with the
conventional

H

controller design in the way that the

performance requirement is explicitly formed in terms of
LMIs instead of being included in a complex weighting
function selection. The approach is practical and provides a
fixed structure controller. The designed controller is robust
and it can improve the system damping over a very wide
operating range. It is also shown that the controller
becomes more and more effective with the increase of
power transfer level between the two areas.

In future work this approach will be tested on a larger

test system to verify its efficacy.


A

PPENDIX

Zeros and poles of the controller:

+

=

+

+

=

9734

.

0

8880

.

7

7159

.

5

3030

.

3

7159

.

5

3030

.

3

2906

.

66

,

2054

.

1

4285

.

0

2054

.

1

4285

.

0

4665

.

4

3021

.

3

4665

.

4

3021

.

3

i

i

p

i

i

i

i

z

Controller matrices:

[

]

0

877

.

3

418

.

5

298

.

3

727

.

0

314

.

9

3812

.

7

0251

.

17

0121

.

0

19

.

14

0026

.

0

524

.

38

649

.

37

625

.

241

945

.

63

438

.

1322

570

.

83

040

.

83

975

.

547

087

.

143

348

.

3033

135

.

0

001

.

0

368

.

5

080

.

0

468

.

10

507

.

65

423

.

71

565

.

458

863

.

118

121

.

2535

062

.

0

108

.

0

568

.

2

069

.

0

043

.

2

=

=

=

=

k

k

k

k

D

C

B

A

R

EFERENCES

[1]

Bikash C. Pal and Alun H. Coonick. “A Linear Matrix Inequality
Approach to Robust Damping Control Design in Power Systems

with Superconducting Magnetic Energy Storage Device”, IEEE
Trans. on Power Systems, Vol.15, No.1, February 2000.

[2]

Jaw-Kuen Shiau and Glauco N. Taranto, “Power Swing Damping
Controller Design Using an Iterative Linear Matrix Inequality
Algorithm”, IEEE Trans. on Control Systems Technology, vol. 7,
No. 3, May 1999.

[3]

M.Klien, L.X. Le, G.J. Rogers, and S. Farrokhpay, “

H

Damping

Controller Design in Large Power Systems”, IEEE Trans. on Power
Systems, vol. 10, no. 1, Feburary 1995, pp158-166.

[4]

G. N. Taranto, J. K. Shiau, “Robust Decentralized Control Design
for Damping Power System Oscillations”, Proceeding of the 33rd
Conference on Decision and Control, Lake Buena Vista, FL,
December 1994.

[5]

Qihua Zhao, Jin Jiang, “Robust SVC Controller Design for
Improving Power System Damping”, IEEE Trans. on Power
Systems, Vol.10, No.4, November 1995.

[6]

P. Gahinet, and P. Apkarian, “A linear Matrix Inequality Approach

to

H

Control”, Int. J. Robust and Nonlinear Contr., 4 (1994), pp.

421-448.

[7]

J. Ackermann, Robust Control: Systems with Uncertain Physical
Parameters, London Springer_Verlag, 1993.

[8]

P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, The LMI
Control Toolbox, Natick, MA: The MathWorks, 1995.

[9]

S. Skogestad and I. Postlethwaite, “Multivariable Feedback Control
Analysis and Design”, John Wiley and Sons, 1996.

[10]

M. Chilaili and P. Gahinet, “

H

Design with Pole Placement

Constraints: an LMI approach,” IEEE Trans. On Automatic Control,
vol. 41, o. 3, pp. 358-367, Mar. 1996.

[11]

C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective
Output-feedback Control via LMI Optimization,” IEEE Trans. On
Automatic Control, vol. 42, no. 7, pp896-911, July 1997.

[12]

M. Klein, G. J. Rogers, and P. Kundur, “ A Fundamental Study of
Inter-area Oscillations in Power Systems,” IEEE Trans. on Power
systems, vol. PWRS-6, no. 3, pp. 914-921, Aug. 1991.

[13]

P. Kundur, G. J. Rogers, and D.Y. Wong, “Extended Transient
Midterm Stability Program Package: Version 2.0, user’s manuals”,
EPRI EL-6648, December 1989.

[14]

John J. Paserba (M) and Nicholas W. Miller (SM), “A Thyristor
Controlled Series Compensation Model for Power System Stability
Analysis”, IEEE Trans. on Power Delivery, Vol. 10, No. 3, July
1995.

[15]

Qihua Zhao and Jin Jiang, “A TCSC Damping Controller Design
Using Robust Control Theory”, Electrical & Energy System, Vol.
20, No. 1, pp.25-33, 1998.

[16]

D.J. Limebeer and M. Green, Linear Robust Control: Prentice-Hall,
1995.

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