Electrical Power and Energy Systems 44 (2013) 153–159

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Delay-dependent robust stabilization for steam valve opening of uncertaintime-delay multi-machine power system with sector saturating actuator

Sun Miao-ping ⇑, Nian Xiao-hong, Pan HuanSchool of Information Science and Engineering, Central South University, Changsha City, Hunan Province 410004, China

a r t i c l e i n f o

Article history:Received 2 May 2011Received in revised form 14 June 2012Accepted 5 July 2012Available online 10 August 2012

Keywords:Multi-machine power systemSteam valve openingSaturating actuatorDelay-dependentLinear matrix inequality (LMI)

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.07.004

⇑ Corresponding author.E-mail address: miaopingsun@mail.csu.edu.cn (S.

a b s t r a c t

In this paper, the time-dependent robust decentralized controller has been proposed for the steam valveopening of time-delay multi-machine power system with sector saturating actuator. Here, the nonlinearinterconnected function is expressed by sinusoidal functions, and furthermore, is enlarged into quadrat-ically bounded inequalities in the subsystem states via transformation of trigonometric functions. Twosufficient conditions of closed-loop multi-machine power system asymptomatic stability are derivedbased on Lyapunov stability theory and state transformation. Controller gain matrices are presented bysolving linear matrix inequalities (LMIs). A two-machine infinite bus system is considered as an applica-tion example, and the simulation results demonstrate the effectiveness of the proposed method.

� 2012 Elsevier Ltd. All rights reserved.

1. Induction

Multi-machine power system is known as a complex intercon-nected large-scale system that is composed of many electric de-vices and mechanical components, thus it is very common tohave parameter uncertainty and time-delay which are often thesource of instability. In the past years, scholars were accustomedto employ excitation control to enhance its stability [1–6], How-ever, application scope of this method is very limited for the con-straints on the maximum value and rising rate of excitationcurrent. Thus it is necessary to make further improvement on tran-sient stability level by regulating the steam valve or water valveopening of prime mover to control the injected mechanical power.In addition, the control of steam value opening is more effective byreason of the existence of water hammer effect which weakens theefficiency of its improvement on stability of system [7]. At thesame time, the physical limitations of the actuator are unavoidablein the operation of driving the actuator by signals emitted from thedesigned controllers, thus causing actuator saturation, which notonly deteriorates the control system performance, but also leadsto undesirable stability effects. Hence, it is significant to investigatetime-independent robust steam valve opening control of large-sized steam turbine with actuator saturation to enhance stabilitylevel of multi-machine power system.

In the past decade years, scholars paid close attention to thecontrol of steam valve opening and several contributions had been

ll rights reserved.

Miao-ping).

published, which mainly comprised two kinds: one was linearfeedback controller, for example, by virtual of solving Ricatti equa-tion [8], making use of LMI approach [9], based on state observer[10,11], etc., the other was nonlinear feedback controller, forexample, using backstepping method [12–14], availing itself ofHamiltonian control technique with dissipation and recursivemethod [15–17], etc. Among the above mentioned reports, only[14,17] took actuator saturation into consideration, meanwhile,they had only thought over that the actuator reached its extremelimit. So far as we known, report on the control of steam vale open-ing for time-delay multi-machine power system has not been cov-ered yet. Though there are many useful theoretical results aboutlinear uncertain time-delay systems with linear interconnectionfunction considering actuator saturation [18–20] and some goodresults about special linear time-delay systems with constrainednonlinear interconnection function [21,22], there are no construc-tive results related to linear uncertain time-delay system withnonlinear interconnection function.

Based on the above investigation, the chief idea in [19] isextended to large-scale system with nonlinear interconnectionfunction and utilized to deal with the design of robust delay-dependent controller for steam vale opening of uncertain time-delay multi-machine power systems with actuator saturation.The system parameter uncertainties are unknown but boundedand the delays are time-varying. Two sufficient conditions ofuncertain multi-machine power system asymptomatic stabilityhave been developed based on the Lyapunov stability theory com-bined with LMI technique, and the design algorithm of controllergain matrices and four corollaries are presented.

Nomenclature

pij constant of either 1 or 0 and pij = 0 means that the jthgenerator has no connection with ith generator

Hi inertia constant for ith generator, in secondsDi damping coefficient for ith generator, in puFIPi fraction of the turbine power generated by the interme-

diate pressure sectionTMi and KMi time constant and gain of ith machine’s turbineTEi and KEi time constant and gain of ith machine’s speed gover-

norRi regulation constant of ith machine in pu

Bij ith row and jth column element of nodal susceptancematrix at the internal nodes after eliminated all physicalbuses, in pu

PMi mechanical power for ith machine, in puXEi steam valve opening for ith machine, in puxi relative speed for ith machine, in radian/sdi rotor angle for ith machine, in radianx0 the synchronous machine speed�Eqi and �Eqj internal transient voltage for ith and jth machine, in

pu, which are assumed to be constantdi0, Pmi0 and XEi0 the initial values of di, Pmi and XEi, respectively

Fig. 1. Sector nonlinear saturation function.

154 S. Miao-ping et al. / Electrical Power and Energy Systems 44 (2013) 153–159

2. Model description

Considering delays, input constraints and the time-varyingstructured uncertainties, an N-machine power system with steamvalve control, which is composed of turbine generator with reheat-er, is described by the interconnection of N subsystems as follows[5,6]

_xiðtÞ ¼ ½Ai þ DAiðtÞ�xiðtÞ þ ½Bi þ DBiðtÞ�usiðtÞ

þXN

j¼1;j–i

pij½Gij þ DGijðtÞ�gijðxiðtÞ; xjðt � sijðtÞÞÞ

usiðtÞ ¼ satðuiðtÞÞxiðtÞ ¼ /iðtÞ; t 2 ½�s;0�

8>>>>>><>>>>>>:ð1Þ

where xiðtÞ ¼ DdiðtÞ xiðtÞ DPMiðtÞ DXEiðtÞ½ �T and Ddi(t) = -di(t) � di0, DPMi(t) = PMi(t) � PMi0, DXEi(t) = XEi(t) � XEi0.

ui(t), which is generated from the designed controller, is thecontrol input vector to the actuator and usi(t) is the control inputvector to the plant.

gij(xi(t), xj(t � sij(t))) is the nonlinear function vector character-izing the interconnection between the ith generator and jth gener-ator and

gijðxiðtÞ; xjðt � sijðtÞÞÞ ¼ sinðdiðtÞ � djðt � sijðtÞÞÞ � sinðdi0 � dj0Þ ð2Þ

sij(t) is the time-varying delay and satisfied

0 6 sijðtÞ 6 s 61; _sijðtÞ 6 s� < 1 ð3Þ

The nominal system matrices are represented as follows:

Ai ¼

0 1 0 00 � Di

2Hi

x02Hið1� FIPi

Þ x02Hi

FIPi

0 0 � 1TMi

KMiTMi

0 � KEiTEiRix0

0 � 1TEi

2666664

3777775; Bi ¼

000

1=TEi

2666437775;

Gij ¼ 0 �x0�Eqi

�EqjBij=2Hi 0 0h iT

:

The notion for the multi-machine power system model is givenin the Nomenclature.

During the practical operation, there are some changes in elec-tric power flow through each transmission line and parameter TMi,TEi, which result in the uncertainties of multi-machine power sys-tem. DAi(t), DBi(t) and DGij(t) are time varying and assumed to beof the following structure:

DAiðtÞ DBiðtÞ½ � ¼ LiFiðtÞ Mi Ni½ �; DGijðtÞ ¼ LijFijðtÞEij ð4Þ

with Fi(t) and Fij(t) being unknown matrix functions with Lebesguemeasurable elements and satisfying

FTi ðtÞFiðtÞ 6 Ii; FT

ijðtÞFijðtÞ 6 Iij ð5Þ

where Li, Mi, Ni, Lij and Eij are known real constant matrices withappropriate dimensions.

The nonlinear saturation function usi(t) is considered to be in-side the sector (ai, 1) and is shown in Fig. 1, where 0 6 ai 6 1.

From Fig. 1, it is obvious that

usiðtÞ � 0:5ð1þ aiÞuiðtÞ ¼ DtiuiðtÞ ) usiðtÞ ¼ wiuiðtÞ ð6Þ

where wi = 0.5(1 + ai) + Dti, Dti is a real number which varying be-tween �0:5ð1� �aiÞ and 0:5ð1þ �aiÞ, ai < �ai < 1.

3. Main results

For convenience, the nonlinear interconnection function gij(xi(t),xj(t � sij(t))) is converted into the following form via transforma-tion of triangle functions

gijðxiðtÞ; xjðt � sijðtÞÞÞ ¼ �2cosbiðdÞsinðDdiðtÞ � Ddjðt � sijÞÞ=2

¼ �2cosbiðdÞsinðWixiðtÞ �Wjxjðt � sijðtÞÞÞ=2

where biðdÞ¼ ½diðtÞ�djðt�sijðtÞÞþdi0�dj0�=2;Wi¼Wj¼ 1 0 0 0½ �. Itis immediately got that

gTijðxiðtÞ; xjðt � sijðtÞÞÞgijðxiðtÞ; xjðt � sijðtÞÞÞ

¼ 4cos2biðdÞsin2½WixiðtÞ �Wjxjðt � sijðtÞÞ�=2

6 ½WixiðtÞ �Wjxjðt � sijðtÞÞ�2

6 xTi ðtÞW

Ti WixiðtÞ þ xT

j ðt � sijðtÞÞWTj Wjxjðt � sijðtÞÞ ð7Þ

S. Miao-ping et al. / Electrical Power and Energy Systems 44 (2013) 153–159 155

Remark 1. The deduced enlarged result (7) of the nonlinearinterconnection function is smaller than the Assumption 3.1 in [6].

In the sequel, some useful lemmas, which are needed to solveour problem, are introduced.

Lemma 1. Let D, E and F be real matrices of appropriate dimensionswith FT(t)F(t) 6 I, then for any given e > 0, we have that

DFEþ ET FT DT6 e�1DDT þ eET E ð8Þ

Lemma 2. Consider the partitioned matrix F ¼ C DDT E

� �where

C = CT, E = ET with appropriate dimensions, then F is positive definiteif and only if either the following conditions hold

E > 0 and C � DE�1DT > 0; or C > 0 and E� DT C�1D > 0 ð9ÞNext, the state feedback control law is designed as

uiðtÞ ¼ KixiðtÞ ð10Þ

where Ki is control gain matrix.Thus, the resulting closed-loop system is written as

_xiðtÞ ¼ ½ACi þ DACiðtÞ�xiðtÞ þXN

j¼1;j–i

½Gij þ DGijðtÞ�gijðxiðtÞ; xjðt � sijðtÞÞÞ

ð11Þ

where ACi = Ai + wiBiKi, DACi(t) = D Ai(t) + wiDBi(t)Ki.In order to analyze the stability of (11), a new state variable is

defined as

ziðtÞ ¼ eatxiðtÞ ð12Þ

where a > 0 is stability degree.Following (12), it holds that

ziðt � sijðtÞÞ ¼ eaðt�sijðtÞÞxiðt � sijðtÞÞ ) eatxiðt � sijðtÞÞ¼ easijðtÞziðt � sijðtÞÞ 6 easziðt � sijðtÞÞ ð13Þ

and

_ziðtÞ ¼ aeatxiðtÞ þ eat _xiðtÞ

¼ ½aI þ ACi þ DACiðtÞ�ziðtÞ þXN

j¼1;j–i

½Gij

þ DGijðtÞ�eatgijðxiðtÞ; xjðt � sijðtÞÞÞ ð14Þ

The main results are given in the following theorems.

Theorem 1. The closed-loop system of (11) is robustly stable withfeedback gain matrix Ki ¼ YiX

�1i if there exist symmetric and positive-

definite matrix Xi, matrix Yi , positive scalars gi, mi such that thefollowing LMI holds:

Si XiMTi þ wYT

i NTi XiW

Ti

MiXi þ wiNiYi �giIi 0WiXi 0 �mi

264375 < 0 ð15Þ

where Si ¼ AiXi þ XiATi þ wiðBiYi þ YT

i BTi Þ þ 2aXi þ giLiL

Ti þ

PNj¼1;j–ipij

hiLijLTij þ #iGijG

Tij

� �, other variables are defined below.

Proof 1. Choose the Lyapunov–Krasovskii functional candidate inthe following form

VðziðtÞ;ziðt�sijðtÞÞ¼XN

i¼1

(zT

i ðtÞPiziðtÞ

þXN

j¼1;j–i

Z t

t�sijðtÞpijlije

2aszTj ðnÞW

Tj WjðnÞzjðnÞdn

)ð16Þ

where Pi is a positive definite symmetric matrix and lij is any givenpositive constant. h

Taking the time derivative of V(zi(t), zi(t � sij(t)) along thetrajectory of (14) is given by

_VðziðtÞ;ziðt�sijðtÞÞÞ¼XN

i¼1

2zTi ðtÞPi

"ðaIþACiþDACiðtÞÞziðtÞ

(

þXN

j¼1;j–i

pijðGijþDGijðtÞÞeatgijðxiðtÞ;xjðt�sijðtÞÞÞ#

þXN

j¼1;j–i

pijlije2as� zT

j ðtÞWTj WjzjðtÞ�ð1� _sijðtÞÞzT

j

h�ðt�sijðtÞÞWT

j Wjzj� t�sijðtÞÞ� io

ð17Þ

According to (4) and (5) and Lemma 1, it follows that

2zTi ðtÞPiDACiðtÞziðtÞ 6 zT

i ðtÞ½giPiLiLTi Pi þ g�1

i ðMi þ wNiKiÞTðMi

þ wNiKiÞ�ziðtÞ ð18Þ

and

2zTi ðtÞPi

XN

j¼1;j–i

pijDGijðtÞeatgijðxiðtÞ;xjðt�sijðtÞÞÞ

6 hizTi ðtÞPi

XN

j¼1;j–i

pijLijLTijPiziðtÞ

þh�1i

XN

j¼1;j–i

pijeatgijðxiðtÞ;xjðt�sijðtÞÞÞT ET

ijEijgijðxiðtÞ;xjðt�sijðtÞÞÞ

6pij hizTi ðtÞPi

XN

j¼1;j–i

LijLTijPiziðtÞ

(

þh�1i

XN

j¼1;j–i

kij zTi ðtÞW

Ti WiziðtÞþe2aszT

j ðt�sijðtÞÞWTj Wjzjðt�sijðtÞÞ

h i)ð19Þ

and

2zTi ðtÞPi

XN

j¼1;j–i

pijGijeatgijðxiðtÞ; xjðt � sijðtÞÞÞ

6 pij #izTi ðtÞPi

XN

j¼1;j–i

GijGTijPiziðtÞ

(

þ#�1i

XN

j¼1;j–i

zTi ðtÞW

Ti WiziðtÞ þ e2aszT

j ðt � sijðtÞÞWTj Wjzjðt � sijðtÞÞ

h i)ð20Þ

where gi, hi and #i are any given positive constants, kij are themaximum eigenvalues of ET

ijEij.Substituting (18)–(20) into (17), it is obtained that

_VðziðtÞ; ziðt � sijðtÞÞÞ 6XN

i¼1

(zT

i ðtÞNiziðtÞ

�XN

j¼1;j–i

pijbije2aszT

j ðt � sijðtÞÞWTj Wjzjðt � sijðtÞÞ

h i)ð21Þ

where Ni ¼ PiACi þ ATCiPi þ 2aPi þ PiHiPi þXi þ m�1

i WTi Wi, Hi ¼ giLi

LTi þ

PNj¼1;j–ipijðhiLijL

Tij þ #iGijG

TijÞ, Xi ¼ g�1

i ðMi þ wiNiKiÞTðMi þ wiNiKiÞ,m�1

i ¼PN

j¼1;j–ipijðljie2as þ h�1

i kij þ #�1i Þ, bij ¼ ð1� s�Þ � h�1

i kij � #�1i .

For hi and #i are any given positive constants, it is reasonable tolet bij P 0. Hence, if Ni < 0 holds, the system (14) is robust asymp-totically stable. Because Ni < 0 is a bilinear matrix inequality (BMI),it is necessary to find a way to transform the inequality to a form

156 S. Miao-ping et al. / Electrical Power and Energy Systems 44 (2013) 153–159

which is affine in the unknown variables. To achieve this, lettingthe variables Xi ¼ P�1

i and Yi = KiXi, pre-multiplying and post-mul-tiplying Ni by Xi, the following condition is obtained that

Ni < 0() ACiXi þ XiATCi þ 2aXi þHi þ XiXiXi þ m�1

i XiWTi WiXi < 0

ð22Þ

According to Lemma 2, it is obvious that LMI (15) is equivalent to(22) and (15) guarantees that the negativeness of _VðziðtÞ;ziðt � sijðtÞÞ whenever zi(t) is not zero, which immediately impliesthat the asymptotic stability of the nominal system of (11). Theproof of Theorem 1 is completed.

The other robust time-delay dependent LMI condition is inves-tigated via different choice of Lyapunov–Krasovskii functional can-didate in succession.

Theorem 2. The closed-loop system of (11) is robustly stable withfeedback gain matrix Ki ¼ YiX

�1i if there exist symmetric and positive-

definite matrix Xi and Qij, any matrix Yi, positive scalars gi and mi suchthat the following LMI holds:

Si XiMTi þ wYT

i NTi XiW

Ti Xi

MiXi þ wiNiYi �giIi 0 0WiXi 0 �mi 0

Xi 0 0 �Ci

2666437775 < 0 ð23Þ

where C�1i ¼

PNj¼1;j–ispijQ ji, the definition of other variables are the

same as the ones in Theorem 1 or defined below.

Proof 2. Choose another Lyapunov–Krasovskii functional candi-date in the following form

VðziðtÞ; ziðt � sijðtÞÞ ¼XN

i¼1

zTi ðtÞPiziðtÞ

�þXN

j¼1;j–i

pij

Z t

t�sijðtÞlije

2aszTj ðnÞW

Tj WjzjðnÞdn

"

þZ 0

�sijðtÞ

Z t

tþjzT

j ðnÞQ ijzjðnÞdndj

#)ð24Þ

Taking the time derivative of VðziðtÞ; ziðt � sijðtÞÞ along thetrajectory of (14) is given by

_VðziðtÞ;ziðt�sijðtÞÞÞ¼XN

i¼1

2zTi ðtÞPi

"ðaIþACiþDACiðtÞÞziðtÞ

(

þXN

j¼1;j–i

pijðGijþDGijðtÞÞeatgijðxiðtÞ;xjðt�sijðtÞÞÞ#

þXN

j¼1;j–i

pij lije2as zT

j ðtÞWTj WjzjðtÞ

hn� 1� _sijðtÞÞzT

j ðt�sijðtÞÞWTj Wjzjðt�sijðtÞÞ

� iþsijðtÞzT

j ðtÞQ ijzjðtÞ�ð1� _sijðtÞÞ

�Z t

t�sijðtÞzT

j ðjÞQijzjðjÞdjÞ�))

ð25Þ

Substituting (13), (14), (18)–(20) into (25), it is obtained that

_VðziðtÞ;ziðt�sijðtÞÞÞ6XN

i¼1

zTi ðtÞNiziðtÞ

��XN

j¼1;j–i

pij bije2aszT

j ðt�sijðtÞÞWTj Wjzjðt�sijðtÞÞ

hþð1�s�Þ�

Z t

t�sijðtÞzT

j ðjÞQ ijzjðjÞdjÞ#)

6

XN

i¼1

zTi ðtÞNiziðtÞ ð26Þ

where Ni¼PiACiþATCiPiþ2aPiþPiHiPiþXiþm�1

i WTi Wiþ

PNj¼1;j–ispijQ ji . h

If Ni < 0 holds, the system (14) is robust asymptotically stable.Using the same manipulation as the proof of Theorem 1, it is shownthat LMI (23) is equivalent to Ni < 0 and (23) guarantees that thenegativeness of _VðziðtÞ; ziðt � sijðtÞÞ whenever zi(t) is not zero,which immediately implies that the asymptotic stability of thesystem (11). The proof of Theorem 1 is achieved.

Remark 2. Both (15) and (23) are LMIs, the variables mi, which isclosely related to the maximum value of s, can be minimized, thesolution of Xi and Yi are obtained, and the corresponding controllergain matrix is derived as Ki ¼ YiX

�1i .

Remark 3. Compared with Theorem 2, there are less variables inTheorem 1. Consequently, LMI (15) is easily solved, so is thefeedback gain matrix Ki. However, from the proof process, it isseen that Theorem 2 has less conservation owing to morevariables.

Meanwhile, when some parameters in the Theorem 1 or Theo-rem 2 are taken some special values, the following results are eas-ily derived.

Corollary 1. When wi = 1, if LMI (15) or (23) holds, the close-loopsystem of (11) is time-dependent asymptotic stability without actu-ator saturation.

Corollary 2. When a = 0, if LMI (15) or (23) holds, the close-loop sys-tem of (11) is time-independent asymptotic stability with actuatorsaturation.

Corollary 3. When sij(t) = 0, wi = 1 and a = 0, designing Ki ¼ iiBTi Pi,

the following result is obtained that

Ni < 0orNi < 0() PiAi þ ATi Pi þ 2iiPiBiB

Ti Pi þ PiHiPi þXi

þ m�1i WT

i Wi þ Hi ¼ 0 ð27Þ

where Xi ¼ g�1i Mi þ iiNiB

Ti Pi

� �TMi þ iiNiB

Ti Pi

� �; �m�1

i ¼PN

j¼1;j–i2pij

ðhikij þ #iÞ and Hi is a positive definite matrix.It is obvious that Corollary 3 is the main result of [6], therefore

Theorem 1 has wider range of application.

Corollary 4. When sij = 0, wi = 1 and a = 0,in addition, withoutparameter uncertainty, the LMI (15) or (23) can be simplified asfollows" #

eSi XiW

Ti

WiXi �~miIi

< 0 ð28Þ

where eSi ¼ AiXi þ XiATi þ BiYi þ YT

i BTi þ

PNj¼1;j–ipijcijGijG

Tij; ~m�1

i ¼2PN

j¼1;j–icijpij.If letting AD, BD, KD and GD the same as that in [7] and

Gi ¼PN

j¼1;j–ipijGijgijðxiðtÞ; xjðtÞÞ, LMI (25) implies the main result in[7]. In addition, LMI (25) is simpler and more general than (20)in [7] in description form.

4. Simulation results

In this section, a two-machine infinite bus example systemwhich is shown in Fig. 2 is chosen to demonstrate the designprocedure and the effectiveness of the proposed decentralizedcontroller. Since generator 3# is an infinite bus, we have�Eq3 ¼ 1\0�.

From Fig. 2, it is easily seen that p12 = p21 = 1 andp13 = p23 = 1.

The system parameters used in the simulation are as follows[6]

Fig. 2. A two-machine infinite bus example power system.

S. Miao-ping et al. / Electrical Power and Energy Systems 44 (2013) 153–159 157

xd1 ¼ 1:863; �xd1 ¼ 0:257; xT1 ¼ 0:129; �Td01 ¼ 16:9s;H1 ¼ 4s;D1

¼ 5; kc1 ¼ 1; xd2 ¼ 2:36; �xd2 ¼ 0:319; xT2 ¼ 0:11; �Td02

¼ 7:96s;H2 ¼ 5:1s;D2 ¼ 3 kc2 ¼ 1 x0 ¼ 314:159 FIP1 ¼ FIP2

¼ 0:3;KM1 ¼ KE1 ¼ 1;KM1 ¼ KM2 ¼ 1rad=s;R1 ¼ R2 ¼ 0:05; TM1

¼ TM2 ¼ 0:35s; TE1 ¼ TE2 ¼ 0:1s; x12 ¼ 0:55; x13 ¼ 0:53; x23

¼ 0:6; xad1 ¼ xad2 ¼ 1:712:

The matrices Ai, Bi and Gij that describe the nominal systemmodel are as follows

A1 ¼

0 1 0 00 �0:625 27:48 11:7810 0 �2:85 2:8570 �0:637 0 �10

2666437775;

A2 ¼

0 1 0 00 �0:392 20:56 9:240 0 �2:857 2:8570 �0:637 0 �10

2666437775;

0 0.5 1 1.5 2−2

0

2

Δδ1(d

egre

e)

0 0.5 1 1.5 2−20

0

20

ω1(ra

d/s)

0 0.5 1 1.5 2−5

0

5

ΔPM

1(pu)

0 0.5 1 1.5 2−50

0

50

ΔXE1

(pu)

Time/sec

Fig. 3. The state trajectories of the 1# an

G12 ¼ G13 ¼ 0 �27:49 0 0½ �T ;

G21 ¼ G23 ¼ 0 �23:1 0 0½ �T ;

B1 ¼ 0 0 0 10½ �T ;

B2 ¼ 0 0 0 10½ �T :

For simplicity, only the parametric perturbation in TMi is consid-ered, which is used to emulate the time constant uncertainties inthe high-pressure and low-pressure sections. Letting ciðtÞ ¼1=TMi � 1=ðTMi � DTMiÞ ¼ 0:635~/iðtÞ with j~/iðtÞj 6 1, it follows that

DAiðtÞ ¼

0 0 0 00 0 0 00 0 �ciðtÞ ciðtÞ0 0 0 0

2666437775:

The structure of parametric uncertainty is expressed as follows:

Li ¼ 0 0 1:41jciðtÞjmax 0½ �T ; Mi ¼ diagf1;1;1;1g;

FiðtÞ ¼ 0 0 �0:707ciðtÞ 0:707ciðtÞ½ �=jciðtÞjmax:

Here constant delay is considered to explain the method simply.Assuming a1 = a2 = 0.3 and �a1 ¼ �a2 ¼ 0:5, so w1 = w2 varies between0.4 and 0.9. In addition, g1 = g2 = 2 and #1 = #2 = 0.1 are chosen.

According to Theorem 1 with a = 0.2 and wi = 0.4, solving (15)with Xi > 0, the solution of mi = 1.74, therefore, the upper bound ofdelay s = (2a)�1ln[(m�1 � 2#)/2#] = 2.5 � ln[(1.74�1 � 0.2)/0.2] =1.57 is obtained. Then the system (11) is robust stable for0 6 sij(t) 6 1.57, with the feedback controller whose gain matricesare

I :K1 ¼ Y1X�1

1 ¼ �283:09 �26:43 �23:65 �4:04½ �K2 ¼ Y2X�1

2 ¼ �266:42 �26:69 �20:53 �3:70½ �

(According to Theorem 2 with a = 0.2 and wi = 0.4, solving (23) withXi > 0 and Qij > 0, the solution of mi = 1.18, therefore, the upperbound of delay s = (2a)�1ln[(m�1 � 2#)/2#] = 2.5 � ln[(1.18�1 �0.2)/0.2] = 2.93 is obtained. Then the system (11) is robust stable

0 0.5 1 1.5 2−2

0

2

Δδ2(d

egre

e)

0 0.5 1 1.5 2−20

0

20

ω2(ra

d/s)

0 0.5 1 1.5 2−5

0

5

ΔPM

2(pu)

0 0.5 1 1.5 2−50

0

50

ΔXE2

(pu)

Time/sec

d 2# generator with the I controller.

Table 1The comparison of simulation results between Theorems 1 and 2.

kK1k kK2k s Stability time

Theorem 1 285.34 268.56 1.57 0.83Theorem 2 248.68 240.96 2.93 1.25

158 S. Miao-ping et al. / Electrical Power and Energy Systems 44 (2013) 153–159

for 0 6 sij(t) 6 2.93, with the feedback controller whose gain matri-ces are

II :K1 ¼ Y1X�1

1 ¼ �246:58 �22:54 �22:45 �5:18½ �K2 ¼ Y2X�1

2 ¼ �238:94 �23:62 �19:66 �4:77½ �

(

0 0.5 1 1.5 2−2

0

2

Δδ1(d

egre

e)

0 0.5 1 1.5 2−20

0

20

ω1(r

ad/s

)

0 0.5 1 1.5 2−2

0

2

ΔPM

1(pu)

0 0.5 1 1.5 2−50

0

50

ΔXE

1(pu)

Time/sec

Fig. 4. The state trajectories of the 1# an

0 1 2 366

67

681# generator

Δδ1

(deg

ree)

0 1 2 3300

320

340

ω1

(rad/

s)

0 1 2 3−5

0

5

ΔPM

1(p

u)

0 1 2 3−20

0

20

ΔXE1

(pu)

Time/sec

Fig. 5. The state trajectories of the 1# and

With time-delay s12 = s21 = 0.5s and parameter uncertaintyc1(t) = c2(t) = 0.625 and w1 = w2 = 0.6, the state trajectories of gener-ator 1# and 2# are shown in Fig. 3 with the I controller and Fig. 4with the II controller.

The comparisons of simulation results between Theorems 1 and2 with the same gi, #i, a and wi are shown in Table 1. Comparedwith Theorem 1, it is obvious that norms of controllers gain ob-tained from Theorem 2 are smaller and the upper bound of delayis larger. That is to say that Theorem 2 has a big advantage overTheorem 1 in stabilization level. However, stabilization time of(1) with the controller in Theorem 1 is shorter than with the con-troller in Theorem 2. In one word, stabilization criteria of Theorems1 and 2 have their own characteristics, and could be applied themin accordance with different needs.

0 0.5 1 1.5 2−2

0

2

Δδ2(d

egre

e)

0 0.5 1 1.5 2−20

0

20ω

2(rad

/s)

0 0.5 1 1.5 2−5

0

5

ΔPM

2(pu)

0 0.5 1 1.5 2−50

0

50

ΔXE

2(pu)

Time/sec

d 2# generator with the II controller.

0 1 2 366

67

682# generator

Δδ2

(deg

ree)

0 1 2 3300

310

320

ω2

(rad/

s)

0 1 2 3−5

0

5

ΔPM

2(p

u)

0 1 2 3−20

0

20

ΔXE2

(pu)

Time/sec

2# generator with the III controller.

S. Miao-ping et al. / Electrical Power and Energy Systems 44 (2013) 153–159 159

Taking the same constant values without considering uncer-tainty, saturating actuator and time-delay, by virtual of LMI tool-box, the control gain matrices are obtained as follows

III :K1 ¼ Y1X�1

1 ¼ �115:01 �10:60 �9:14 �1:45½ �K2 ¼ Y2X�1

2 ¼ �105:45 �10:38 �8:14 �1:31½ �

(With the same time-delay, parameter uncertainty and saturatingdegree, the state trajectories of generator 1# and generator 2# areshown in Fig. 5 with the III controller.

Compared Fig. 5 with Figs. 3 and 4, it can be easily seen that thenorms of controller gain obtained from the III controller is smaller,while the stabilization time of system (1) is longer (about 3 s), theovershoot of state trajectories is bigger, and number of oscillationsis more with the III controller than that with the proposed I and IIcontrollers. The uncertainty, time delay and saturating actuator arevery common in the practical multi-machine power system, it isvery necessary to consider their effective for the stabilization ofmulti-machine power system.

5. Conclusion

In this paper, a decentralized feedback control scheme has beenproposed to enhance the transient stability of uncertain and timedelay multi-machine power system with sector nonlinear saturat-ing actuator. Two sufficient conditions of closed-loop powersystem asymptomatic stability have been presented. The LMImethod has been used to compute the control gain matrices anddefine the upper bound of delay s. It has been shown from the sim-ulation results that the presented control schemes are efficient andpermits the rapid stability of the closed-loop system. The compar-ison with the III control without considering time-delay, uncertain-ties and saturating actuator shows that it is of great significance tothe study of multi-machine power system in this paper. What ismore, it is obvious that the results in [6,7] are special cases of The-orem 1 or Theorem 2 in our paper.

Acknowledgements

The authors express their sincere gratitude to the reviewer forhis constructive suggestions which help improve the presentationof this paper. This work is supported by the National Science Foun-dation of China under Grant 61075065,60774045, U1134108 andPh.D. Programs Foundation of Ministry of Education of China underGrant 20110162110041.

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