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262 Feng Liu, Shengwei Mei, Qiang Lu, and Masuo Goto

Recursive Design of Nonlinear Disturbance Attenuation Control for

STATCOM

Feng Liu, Shengwei Mei, Qiang Lu, and Masuo Goto

Abstract: In this paper, a nonlinear robust control approach is applied to design a controller forthe Static Synchronous Compensator (STATCOM). A robust control dynamic model ofSTATCOM in a one-machine, infinite-bus system is established with consideration of the

torque disturbance acting on the rotating shaft of the generator set and the disturbance to theoutput voltage of STATCOM. A novel recursive approach is utilized to construct the energy

storage function of the system such that the solution to the disturbance attenuation controlproblem is acquired, which avoids the difficulty involved in solving the Hamilton-Jacobi-Issacs

(HJI) inequality. Sequentially, the nonlinear disturbance attenuation control strategy ofSTATCOM is obtained. Simulation results demonstrate that STATCOM with the proposedcontroller can more effectively improve the voltage stability, damp the oscillation, and enhance

the transient stability of power systems compared to the conventional PI+PSS controller.

Keywords: Disturbance attenuation, nonlinear robust control, recursive design, STATCOM.

1. INTRODUCTION

Power system control mainly includes three parts:generator control, transmission control and load

control. Before FACTS (Flexible AC TransmissionSystems) technology was brought forward, power

system control focused on the generator part and theload part, as the controllability in electric transmissionnetworks was very weak. The situation has been

changed since the development of FACTS technology,

which can be utilized to flexibly control the powerflow in the electric transmission network. STATCOM

is one of the most important devices of FACTS. Theability of STATCOM to support the voltage on theconnection point as well as improve system voltage

stability has been well documented [1-7]. In fact, withthe proper control strategies, STATCOM has

numerous benefits on systems, such as enhancing the

transmission ability of lines, improving the transientstability, and damping the oscillation.

Many large-capability STATCOM equipments have

been developed and put into operation in the U.S. and

Japan [1,5,7]. Conventional PID [8] and PSSsupplementary control [3] are employed in the

majority of cases. Although they have simplestructures and are easy to design, their dynamicresponses are comparatively slow, and their

performance is not very satisfying. Mitsubishi,Hitachi and Toshiba adopted the VQ control method,

which is based on instantaneous active and reactivepower [2]. Since the time constant of the DC voltageloop and that of the reactive power control loop have

the same order of magnitude, and because the system

is strongly nonlinearthis decoupling method maystray from what is desired. Additionally, most controlmethods have not considered the external disturbances

to systems. As complex and large-scale nonlinearsystems, power systems always suffer various types of

external disturbances. So it is important to takeaccount of these disturbances and try to reduce theirill effects on the system. For the sake of endowing the

system strong robustness under large disturbances, a

robust control model of STATCOM has beenestablished based on the 3rd-order dynamic equations

of STATCOM with the consideration of thegenerators dynamics and external disturbances.

In this paper, we manage to apply the nonlinear

robust control in the sense of L2-gain [9] toSTATCOM control. Usually, approaches to solve

nonlinear robust control depend on the resolution of

the Hamilton-Jacobi-Isaacs (HJI) inequality. Since HJI

__________

Manuscript received September 10, 2004; acceptedDecember 1, 2005. Recommended by Guest Editor Youyi

Wang. This work is supported partly by the Chinese NationalNatural Science Foundation (Grant 50377018), the Chinese

National Key Basic Research Fund (Grant 1998020309) andthe NSFC-JSPS Scientific Cooperation Program, as well as by

the New Energy and Industrial Technology DevelopmentOrganization of Japan.

Feng Liu is with the Institute of System Science, Beijing,

100080, China (e-mail: [email protected]).Shengwei Mei and Qiang Lu are with the Department of

Electrical Engineering, Tsinghua University, Beijing, 100084,China (e-mails: [email protected]; [email protected]).

Masuo Goto is with Power System Division, Hitachi Ltd.,

Japan (e-mail: [email protected]).

International Journal of Control, Automation, and Systems, vol. 3, no. 2 (special edition), pp. 262-269, June 2005

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Recursive Design of Nonlinear Disturbance Attenuation Control for STATCOM 263

inequality is too difficult to solve analytically, it

seems unfeasible to obtain nonlinear robust controlstrategies of STATOM in this way. In this paper, arecursive approach [11] is applied to construct the

energy storage function of the system so that thesolution of the control problem is acquired, which

avoids the difficulty caused by solving HJI inequality.This paper is organized as follows. The problem

description of nonlinear disturbance attenuationcontrol and a brief review of the recursive designmethod for constructing Lyapunov function are given

in Section 2. A robust control model of STATCOM

considering the generators dynamics has beenestablished in Sections 3 and 4. In Section 5, arecursive approach is applied to construct the energy

storage function of the system so that the nonlineardisturbance attenuation control strategy of STATCOM

is obtained. In Section 6, in order to ensure the effectsof the proposed control strategy, a simulation has been

carried out in a one-machine, infinite-bus system witha set of STATCOM located on the mid-point of thelong distance transmission line. Finally, some

conclusions are given.

2. NONLINEAR DISTURBANCE

ATTENUATION CONTROL

2.1. The problem of nonlinear disturbance attenuationcontrol

In this section, we briefly state the essential conceptof the nonlinear disturbance attenuation problem.

Consider a nonlinear affine system with externaldisturbances in the form of

1 2( ) ( ) ( )

( )

x f x g x w g x u

y h x

= + +

=

(1)

where xRn is the state vector, uRm is the control

input, wRr is the exogenous disturbance, and

y R is the regulation output. The functions fg1g2 are smooth vector fields with corresponding

dimensions, andf(0)=0h(0)=0

Problem Statement: For system (1), the nonlineardisturbance attenuation problem (NDAP) is toconstruct C

1state feedback controlleru=u(x) such that

for a given positive constant , the correspondingclosed-loop system can satisfy the following L2 gain

dissipative inequality

22 2

0 0

2

( (0))

(0, ), 0

T Ty w V x

w L T T

+

>

(2)

where

2 (0, ) { | :[0, ) ,n kL T w w T R = T 2

0 w }dt< + ;

( )V is a nonnegative storage function to beconstructed, x(0) is the initial state, and w is the

exogenous disturbance. denotes the usualEuclidean norm for vectors.

The dissipation inequality states that the input-

output maps have L2-gain for every initial

condition x(0)=x0. The L2-gain of a system reflectsessentially the amplification between the amplitude ofthe input-output of the system. We can try to enhancerobustness and dynamic performance of the system

under various disturbances by design of the so-calleddisturbance attenuation controller. The ability of

disturbance attenuation can be valued by thepredestined positive number . The smaller is, thestronger the effect of disturbance attenuation will be.

If reaches its minimum, then we can acquire the so-

called optimal robust control. Obviously, if the valueof prescribed is too small, which would be less than

its existing minimum value, the robust control issuehas no solution. To design a control strategy in thesense of disturbance attenuation, we often set a proper

positive firstly, and then solve the dissipativeinequality. From the point of view of engineering design,

we do not always seek for the optimal robust controllaw, but seek for an appropriate or satisfactory one.

As well known, the nonlinear disturbance problem

can be transformed into that of finding a nonnegativesolution of HJI inequality, which is a nonlinear partial

differential one. Because it is very difficult to acquire

an analytic nonnegative solution of dissipativeinequality in a large area from the view of modernmathematics, to seek a new treatment to avoid the

obstacle of solving HJI inequality is imperative fordesign disturbance attenuation controllers of power

systems. In reference [11], a novel recursive designapproach is proposed to directly obtain the solution ofthe attenuation control problem by constructing the

storage function of the dynamic system instead ofsolving the HJI inequality, which is suitable to a vast

class of engineering control systems.

2.2. A brief review of the recursive design methodThis section introduces the basic approach of

recursive design in control design [11].Consider the nonlinear cascade system

1 1 1 2

2

( ) ( , )x f x x x

x u

= +

=

(3)

where x1,x2R are state variables, uRm

is control

input, f( ) and ( , ) are C1 mappings with f(0)=0,

(0,0) 0 = . A basic process of recursive designtechnique for the control Lyapunov function isdescribed as follows:

Step 1: Choose a positive-definite function W(x1)

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such that

1 11

( ) 0 0W

f x xx

<

(4)

holds.

Step 2: Select functions (x1,x2) that satisfy

1 2 1 2 2( , ) ( , ) .x x x x x = (5)

Step 3: Construct the following positive-definitefunction

1 2 2

1( ) ( ) ,

2

TV x W x x x= + (6)

where 1 2[ , ]Tx x x= .

Consider that

1 2 2

1 1 2 2

1 2 1 2

( )

( ) ( , )

( ) { ( , ) }.

T

T

TT

W

V x x x xx

W Wf x x x x u

x x

W Wf x x x x u

x x

= +

= + +

= + +

(7)

Let

1 2 21

( , ) - ,T

T Wu x x xx

=

(8)

where is a positive number. Then there is

21 2

1

( ) ( ) 0.W

V x f x xx

= 0 is a given positive number and is a

smooth function to be designed. If we find

1 2 3( , , )v x x x= and 1 2 3( , , )x x x such that thefollowing inequality

2 22

00 0( ), 0

T Tz dt dt V x T + > (20)

holds, namely the L2-gain is less than or equal to

and the robust controller is solvable. Here, ( )V isthe storage function to be constructed, x0 the initial

state of the system, and a prescribed positive number.

From the coordinate transformation (9), we have

1x =

1 2 ,kx x +

2 1 2 2 2 1 1 1

3 1 2 31 2 3

( ) ,

,

ex kx x k a x a P a

x x x xx x x

= + = + + +

= + +

(21)

where e mo eP P P= .At first, we let

1 1( ,V x 2 )x2

= 21x +

22

1 ,

2x (22)

where 0 > is a positive number to be selected.Now, let us introduce a function denoted by H1

defined as

2 221 1 (14) 1

1{ },

2H V z = + (23)

where 1 (21)V denotes the differential of function

( )211 , xxV with respect to time t along the state

equation of system (21). Let

2 2 21 2

2 21 1 22

221

2 2 22

1 1,

2 2

1 1,

2

1,

2

e k q q k

ka kq

aa k q

=

= +

= + + +

(24)

and

3 1 2 3 1 1 2 2 3 ( , , ) .x x x x x x x = = + (25)

Then we know that

21 1H ex=

21 1 2

1 2( )

4a x

221

4

2x+ 1 3 3( )ea P x x+ + (26)

21ex + 2x 3x

2

4

21

2 1 3 ( )ex a P x+ + .

Considering (22) we construct the followingnonnegative function

2V 1( ,x 2 ,x 3 )x2

= 21x +

1

2

22x +

1

2

23x (27)

and define

2 222 2 (21)

2 22 2 21 1 2(21)

1 ( )2

1 1{ ( )}

2 2

H V z

V z

= +

= + +

3x 3 ,x

where)21(2V

denotes the differential of function

( )3212 ,, xxxV with respect to time t along the state

equation of system (21). Then we have

})2

(

)1({)(

)

(2

1)

(

4

1

12232

1222

3

222113312

2332

21231

2112

vPaaax

xakxxxPax

xaaxxeH

e

e

+++

++++++

+

According to inequality (29), if there is

1 1 2 2 2 2 1

22 23 3 22 1 1 32

3

(1 )

( ) ( )

2

e

e

v kx a x a P

x a xa a P x

x

= + + + +

+ + + +(30)

then the following equation will hold2

2 1 1 .H e x (31)

In view of (24)let

2 2 21 2

1( ).

2q q k

k + (32)

Then there is

1 0e .(32) and (31) yield that

{ }2 22 22 1 1(21)1

0.2

V z e x + (33)

That is:2 2

22 0 20 0

2

2 2 00

2 ( ) ( )

2 ( ).

T T

T

T

z dt dt V x V x

dt V x

+

+

(34)

. (29)

(28)

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Setting

2( ) 2 ( ),V x V x= we have

2 22

00 0( ).

T Tz dt dt V x +

Now, we have successfully solved the problem ofnonlinear disturbance attenuation control in the sense

ofL2-gain for system (21), and the control strategy is

2 2

1 1 2 2 2 2 1

22 23 3 22 1 1 32

3

arcsin( )

sin2

(1 )

( ) ( )

2

svg

s

e

e

Cxv

K V

v kx a x a P

x a xa a P x

x

= + = + + + + + + + +

.

(35)

Note that the regulating area of the firing

angle vg is limited by svg . From the

coordinate transformation (19), we have

,

,

,

svg

+

= +

(36)

where

2 2arcsin( )

sin2

Cxv

K V

= .

Finally, the control strategy can be written in the

form of

1 2 2 2 1 00

21 2 2 20 3

2 12

01 0

1 20

(1 ) ( )

( )2

( ( ) ).

t

m e

t

svg

t

m e svg t

svg

v k d a a P P

d V aa

k da P P V

d V

= + + + +

+ + +

+ + +

+

(37)

Remark 1: In this control strategy, 0 =

0vg svg svg V V V = . Note that the rotor speed is a

variable of the generator and cannot directly beacquired locally by STATCOM. Fortunately, thedevelopment of PMU and WAMS technology make it

feasible in practical operation. The power angel

deviation can be calculated by0

td =

or directly acquired from PMU. The parameters

1 2 1 2 3, , , , , ,k a a a are determined by equalities (15),

(19) and (24).

6. SIMULATION RESULTS

1) Description of a one-machine infinite-bus system

with STATCOM (see Fig. 4)2) Parameters of the Generator:

'0

'

1 2 3 4

12.922, 1.0, 6.55,

0.8285, 0.1045, 0.0292,

0.18

s d

d d T

L L L L

H V T

x x x

x x x x

= = =

= = =

= = = =

3) Parameters of STATCOM:

15000 , 0.9101846 , 0.1177 ,

2 6 / , 120 , 0.1 ( . .)

L

T

C uF x r

K x p u

= = =

= = =D

4) Parameters of theL2-gain Disturbances Attenua-tion Controller:

1 0.04, 2 0.06, 4, 80, 0.5q q k = = = = =

5) The control strategy of conventional controller is

shown in Fig. 56) Fault Setting:To analyze the performance of a power system

under large disturbances, we assume that a balancedthree-phase short-circuit fault occurs at the position

near to the connection point of STATCOM at t=0.1s,and is cleared at t=0.3s.

7) Analysis of Simulation Results:

Considering the prescribed system and fault, westudy the different responses of several indices between

conventional PID+PSS control as well as the nonlineardisturbance attenuation. The rotor angle, rotor speed,

AC voltage of STATCOM, voltage on the connectionpoint, and transmission active and reactive power on

the lines are monitored. The following figures indicatethe different response curves of those indices.

Figs. 6 and 7 show the responses of rotor angle

and rotor speed , which are the targets of the distur-

Fig. 4. A one-machine infinite bus system with

STATCOM.

Fig. 5. Block of the conventional controller.

PIDV

P

svgPSS

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bance attenuation control. With the proposed control

method, the system resumes its stability faster than the

conventional one. Except for rotor angle and rotor

speed , voltage is another index of the dynamic

performance. So we inspect the responses of the ACvoltage of STATCOM and that on the connection

point. Figs. 8 and 9 reflect that the two control methodshave almost identical voltage responses. The last twofigures demonstrate the responses of the active power

0 2 4 6 8 10

0.990

0.995

1.000

1.005

1.010

Disturbance attenuation control

PID control

RotorSpeed

Time (s)

Fig. 6. Rotor response.

0 2 4 6 8 10

30

40

50

60

70

80


PID control

GeneratorAngle

Time (s)

Fig. 7. Angle response.

0 2 4 6 8 10

0.90

0.95

1.00

1.05

1.10


PID control

TheVoltage(p.u.)

Times

Fig. 8. AC voltage response on the ASVG.

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

1.2


PID controlVoltage

Time (s)

Fig. 9. Voltage response on the point where the faultoccurs.

0 2 4 6 8 10

0.0

0.5

1.0

1.5

2.0

2.5 Disturbance attenuation control

PID control

ActivePower

Time (s)

Fig. 10. Active power response of generator.

0 2 4 6 8 10

0.5

1.0

1.5

2.0

2.5


PID control

ReactivePower

Time (s)

Fig. 11. Reactive power response of generator.

and reactive power. According to the simulation results,it is obvious that, compared to the conventional

control method, nonlinear disturbance attenuationcontrol is more effective in damping the oscillation,

maintaining the speedy recovery of the system voltage,and limiting the fluctuation of transmission power ofthe power system under large disturbances.

7. CONCLUSIONS

Numerous approaches of nonlinear robust control

have been developed in recent years. However, it isseldom applied to the control of FACTS devices. We

introduce the theory of nonlinear disturbance

attenuation control in the sense ofL2-gain into thedesign of the STATCOM controller. A recursiveapproach is applied to construct the energy storage

function of the system such that the solution of thecontrol problem is acquired, which avoids the

difficulty from solving the HJI inequality. Sequentially,

the nonlinear disturbances attenuation control in senseof L2-gain for STATCOM is obtained. Since the

impact of the extend disturbance has been consideredand attenuated by control to keep theL2-gain less than

or equal to a given constant the proposed control

strategy has a strong robustness. Simulation resultsdemonstrate the excellent damping and dynamic

performance of the OMIB system with the proposedcontroller.

REFERENCES[1] L. Gyugri, N. G. Hingorani, P. R. Nanney et al.

Advanced Static Var Compensator Using Gate-Turn-Off Thyristor for Utility Application,

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CIGRE Paris, August 1990.

[2] C. Jianye, W. Zhonghong, et al. The presentresearch situation of STATCOM, Journal ofTsinghua University (Science and Tech.), vol. 37,

no. 7, pp. 7-12, May 1997.[3] K. V. Patil, J. Senthil, J. Jiang, and R. M. Mathur,

Application of STATCOM for dampingtorsional oscillations in series compensated AC

systems, IEEE Trans. on Energy Conversion,

vol. 13, no. 3, pp. 237-243, September 1998.[4] M. Kristic, I. Kanellakopoulos, and P. Kokotovic.

Nonlinear and Adaptive Control Design, New

Jersey, Prentice Hall, 1995.[5] S. Mori, K. Matsuno, T. Hasegawa, S. Ohnishi,

M. Takeda, M. Seto, S. Murakami, and F.

Ishiguro, Development of a large static VArgenerator using self-commutated inverter for

improving power system stability, IEEE Trans.on Power Systems, vol. 8, no. 1, pp. 371-377,

1993.[6] S. Chen and S. Yuanzhang, The effects of

applying nonlinear control of STATCOM to

improve power systems damping ability,Power System Automation, vol. 21, no. 5, 1997.

[7] L. H. Walker, 10MW GTO converter for batterypeaking service, IEEE Trans. on Industry

Applications, vol. 26, no. 1, pp. 63-72, 1990.[8] P. W. Lehn and M. R. Iravani, Experimental

evaluation of STATCOM closed loopdynamics,IEEE Trans. on Power Delivery, vol.13, no. 4, pp. 1378-1384, October 1998.

[9] A. van der Schaft, L2Gain and PassivityTechniques in Nonlinear Control, Springer Press,

London, 1996.[10] C. Byrnes, A. Isidori, and J. Willems, Passivity,

feedback equivalence, and the globalstabilization of minimum phase nonlinearsystems, IEEE Trans. on Automatic Control,

vol. 36, no. 11, pp. 1228-1240, 1991.

[11] L. Qiang and M. Shengwei, Recursive design

of nonlinear H excitation controller, Science inChina (Series E), vol. 43, no. 1, pp. 23-31, 2000.

[12] Q. Lu, S. Mei, W. Hu, and Y. H. Song,

Decentralized nonlinearH control based onregulation linearization, IEE Proc. Generation,Transmission and Distribution, vol. 147, no. 4,pp. 245-251, 2000.

[13] Q. Lu, Y Sun, and S. Mei, Nonlinear Control

Systems and Power System Dynamics, KluwerPublisher, USA, 2001.

Feng Liu was born in Yun Nan, P. R.

China, in 1977. He received the B.S.and Ph.D. Degrees in electrical

engineering from Tsinghua University,

Beijing, P.R. China, in 1999 and 2004respectively. He is currently a post-

doctor of the Institute of System

Science of Chinese Academy ofScience, P. R. China.

Shengwei Mei was born in Xinjiang,China, on Sept. 20, 1964. He received

his B.S degree in mathematics from

Xinjiang University, M.S. degree in

operations research from Tsinghua

Universty, and Ph.D degree in auto-matic control from Chinese Academy

of Sciences, Beijing, in 1984, 1989,

and 1996 respectively. He is now aProfessor of Tsinghua University. His current interest is in

control theory applications in power systems.

Qiang Luwas born in Anhui, China,

on April 19, 1937. He graduated from

the Graduate Student School ofTsinghua University, Beijing, in 1963

and joined the faculty of the same

University. From 1984 to 1986 he was

a visiting scholar and a visiting

professor in Washington University, St.Louis and Colorado State University,

Ft. Collins, respectively. From 1993 to 1995 he was a

visiting professor of Kyushu Institute of Technology (KIT),Japan. He is now a Professor of Dept. of ElectricalEngineering, Tsinghua University, Beijing. Since 1991 he

has been elected to be an academician of Chinese Academyof Sciences. Currently, he is responsible for nonlinear

control of power system studies in Tsinghua University,

China. He is an IEEE Fellow.

Masuo Goto was born in Takamatsu,Japan, on May 9, 1942. He received

his B.E. and Dr. E. degrees from

University of Osaka prefecture, Japanin l965 and 1979, respectively. He

received Professional Engineer degree

from the Institution Professional

Engineers, Japan in 2001. He joinedHitachi Research Laboratory of

Hitachi, Ltd. in 1965. From l965 to 2000 he was engaged in

research and development in the field of power system

analysis, control and protection. In 2001 he moved to

Nagoya University. He is currently a Guest Professor ofNagoya University. He received the Advanced Technology

Award of IEE of Japan for the development of an advancedpower system simulator in 1991. Dr. Goto is a member of

IEEJ and an IEEE Fellow.

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