Automatica 46 (2010) 330–336

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Automatica

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Brief paper

A power system nonlinear adaptive decentralized controller designI

Rui Yan a, ZhaoYang Dong b,∗, T.K. Saha c, Rajat Majumder da The Institute for Infocomm Research, Agency for Science, Technology and Research, Singapore 138632, Singaporeb Department of Electrical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kongc School of Information Technology and Electrical Engineering, The University of Queensland, St. Lucia, QLD, 4072, Australiad ABB Corporate Research, Sweden

a r t i c l e i n f o

Article history:Received 3 April 2007Received in revised form22 July 2009Accepted 12 October 2009Available online 30 November 2009

Keywords:Adaptive controlNonlinear decentralized controlPower systemsTransient stabilityBackstepping design

a b s t r a c t

In this paper, a novel excitation control is designed for improvement of transient stability of powersystems. The control algorithm is based on the adaptive backstepping method in a recursive way withoutlinearizing the system model. Lyapunov function method is applied in designing the controller to ensurethe convergence of the power angle, relative speed of the generator and the active electrical powerdelivered by the generator when a large fault occurs. Compared with the existing nonlinear decentralizedcontrol approaches, the proposed controller has no requirement for the bounds of interconnections inthe power system. And the new approach does not need the existence of solution of a designed algebraicRiccati equation. Furthermore, the transient stability performance of power systems can also be improvedby the designed control approach. The efficacy of the designed controller has been demonstrated in amultimachine power system. Simulation results show transient stability enhancement of a power systemin the face of a large sudden fault.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Power systems aremodeled as large nonlinear systems. Simpli-fied linear models (deMello & Concordia, 1969) have been used fora long time to design the excitation controller for synchronous gen-erators. Linear control methodologies were well accepted in utili-ties due to their inherent simplicity in design and ease of real timeimplementation. Linear power system stabilizers (PSS) (Larsen &Swann, 1981) are often used to provide supplementary dampingthrough excitation control to improve the dynamic stability limit.Such linear control mechanisms generally provide asymptotic sta-bility in a small region of the equilibrium and is only appropriatefor the effect of small disturbances. In Fan, Ortmeyer, and Mukun-dan (1990), an adaptive control method was designed for the im-provement of transient stability of multimachine power systems.An approachwas proposed in Dong, Hill, and Guo (2005) for powersystem security assessment and enhancement based on the infor-mation provided from the pre-defined system parameter space.The proposed scheme opens up an efficient way for real time secu-rity assessment and enhancement in a competitive electricitymar-ket for single contingency cases.

I This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Xiaohong Guanunder the direction of Editor Toshiharu Sugie.∗ Corresponding author. Tel.: +852 27666148; fax: +852 23301544.E-mail addresses: ryan@i2r.a-star.edu.sg (R. Yan), eezydong@polyu.edu.hk

(Z. Dong).

0005-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.10.020

The transient stability in power systems studies involve the de-termination of whether or not synchronism is maintained afterthe machine has been subjected to severe disturbance. In orderto improve the transient stability of a power system with largedisturbances, the crucial control objectives are to rapidly increaseexcitation and to decrease themechanical input power at the sametime (Wang,Hill,Middleton, &Gao, 1993). In the past fewyears, theDirect Feedback Linearization (DFL) technique has been widely re-ported to design an excitation controller (Chapman & Ilic, 1993;Gao, Chen, Fan, & Ma, 1992; Guo, Wang, & Hill, 2000; Wang &Hill, 1996; Wang et al., 1993; Wang, Guo, & Hill, 1997). Further-more, in recent years a number of different advanced nonlinearcontrol techniques were used to enhance the transient stabilityof power systems in Cong, Wang, and Hill (2004), Cong, Wang, andHill (2005) and Wang, Zhang, and Hill (2004).Generally, decentralized controllers are used widely for mul-

timachine systems due to physical limitations on the systemstructure. Sometimes it might become infeasible to transfer infor-mation among subsystems. So far there have been numerous con-trol propositions for designing decentralized excitation control ofpower systems (Chapman, Ilic, King, Eng, & Kaufman, 1993; Jain,Khorrami, & Fardanesh, 1994; King, Chapman, & Ilic, 1994; Lu, Sun,Xu, &Mochizuli, 1996;Wang et al., 1997). DFLwas applied to trans-fer a nonlinear multimachine power system model to a linear oneand robust decentralized control was designed. The interconnec-tions were considered only with linear bounds in the above works.This may result in more conservative controllers for global behav-ior since the essential interconnections in the large-scale power

R. Yan et al. / Automatica 46 (2010) 330–336 331

systems are nonlinear. The nonlinear decentralized control schemewas developed to solve the problem of general nonlinear boundsof interconnections in Guo, Hill, and Wang (2000). Both excita-tion control and steam valve control were designed to enhance thetransient stability. Previous works are based on the known boundsof parameters. However, some of them are very difficult to knowin practice especially when serious disturbances occurs. Recently,a constructive methodology utilizing the backstepping techniquehas been developed for designing nonlinear adaptive control sys-tems (Kanellakopoulos, Kokotović, & Morse, 1991; Krstić, Kanel-lakopoulos, & Kokotović, 1995). This method can overcome therelative degree one restriction and can be applied to handle powersystems with uncertainties.In this paper, a novel nonlinear decentralized excitation con-

troller has been designed by applying adaptive backsteppingmethod to improve the transient stability performance of multi-machine power systems. Compared with the previous decentral-ized excitation control in Chapman and Ilic (1993), Guo and Hillet al. (2000), Jain et al. (1994), King et al. (1994), Lu et al. (1996) andWang et al. (1997), the requirement of the known bounds of inter-connection parameters has been relaxed in this method. Further-more, the new approach does not need the existence of solutionof a designed algebraic Riccati equation. Some comparison resultsbetween the proposed approach and existing methods in Guo andHill et al. (2000) and Wang et al. (1997) will be given in Section 4.This paper is organized as follows. In Section 2, the dynamic modelof a power system is described. In Section 3, the adaptive backstep-ping design is proposed. Illustrative examples and design consider-ations of applying the proposed controller to multimachine powersystems are provided in Section 4. The conclusion is drawn inSection 5.

2. Power system dynamical model

In this section, a power system consisting of n synchronousma-chines is considered. Based on some standard assumptions, themotion of the interconnected generators canbedescribedby a clas-sical model with flux decay dynamics (Anderson & Fouad, 1994;Bergen, 1986; Kundur, 1994; Pai, 1981). In themodel, the generatoris modeled as the voltage behind direct axis transient reactance;the angle of the voltage coincides with the mechanical angle rel-ative to the synchronously rotating reference frame. The networkhas been reduced to internal bus representation.The dynamicalmodel of the ithmachinewith excitation control

can be written as follows:Mechanical equations:δi(t) = ωi(t)ωi(t) = −

Di2Hi+ω0

2Hi(Pmi0 − Pei(t)),

(1)

where δi(t) is the power angle of the generator, in radian; ωi(t) isthe relative speed of the generator, in rad/s; Pmi0 is the mechanicalinput power, in p.u., which is a constant; Pei(t) is the activeelectrical power delivered by the generator, in p.u.; ω0 = 2π f0is the synchronous machine speed, in rad/s; Di is the per unitdamping constant and Hi is the inertia constant in seconds.Generator electrical dynamics:

E ′qi(t) =1T ′d0i[Efi(t)− Eqi(t)] (2)

where E ′qi(t) is the transient EMF in the quadrature axis of the ithgenerator; Eqi(t) is the EMF in the quadrature axis, in p.u.; Efi(t)is the equivalent EMF in the excitation coil, in p.u. and T ′d0i is thedirect axis transient short circuit time constant, in seconds.

Electrical equations:

Eqi(t) = E ′qi(t)− (xdi − x′

di)Idi(t),Efi(t) = kciufi(t),

Pei(t) =n∑j=1

E ′qi(t)E′

qj(t)Bij sin(δi − δj),

Qei(t) = −n∑j=1

E ′qi(t)E′

qj(t)Bij cos(δi − δj),

Idi(t) =n∑j=1

E ′qj(t)Bij cos(δi − δj),

Iqi(t) =n∑j=1

E ′qj(t)Bij sin(δi − δj),

Eqi(t) = xadiIfi(t),

Vti =√(E ′qi + x

′

diIdi)2 + (x′

diIqi)2,

(3)

where Qei(t) is the reactive power, in p.u.; Ifi(t) is the excitationcurrent, in p.u.; Iqi(t) is the quadrature axis current, in p.u.; kci isthe gain of the excitation amplifier, in p.u.; ufi(t) is the input ofthe Silicon Controlled Rectifier (SCR) amplifier of the generator, inp.u.; Bij is the ith row and jth column element of nodal susceptancematrix at the internal nodes after eliminating all physical buses, inp.u.; xadi is the mutual reactance between the excitation coil andthe stator coil of the ith generator, in p.u.; xdi is the direct axisreactance of the ith generator, in p.u.; x′di is the direct axis transientreactance of the ith generator, in p.u.; Vti is the terminal voltage ofthe ith generator.It is obvious that the multimachine power system is highly

nonlinear and interconnected by the transmission network.

3. Decentralized nonlinear controller design

Let δmi0, ωmi0 and Pmi0 be the desired values for the powerangle δi, the relative speed ωi and the active power Pei of the ithgenerator at the operating point. Denote δi(t) = δi(t) − δmi0,ωi(t) = ωi(t)−ωmi0 = ωi(t) for ωmi0=0 and Pei(t) = Pei(t)− Pmi0.Based on the calculation in Wang et al. (1997), the differentiationof Pei(t) is

˙Pei(t) =1T ′d0i[kciufi(t)+ (xdi − x′di)Idi(t)Iqi(t)

−(Pei(t)+ Pmi0)] − Qei(t)ωi(t)

+

n∑j=1

E ′qi(t)E′

qj(t)Bij sin(δi − δj + δmi0 − δmj0)

− E ′qi(t)n∑j=1

E ′qj(t)Bij cos(δi − δj + δmi0 − δmj0)ωj(t).

Denoteγ1ij(t) = E ′qi(t)E

′

qj(t)Bij, γ2ij(t) = −E′

qi(t)E′

qj(t)Bij, δij = δi− δj+δmi0 − δmj0.We further have

˙Pei =1T ′d0ikciIqiufi − Qei(t)ωi(t)

+1T ′d0i[−Pei − Pmi0 + (xdi − x′di)IqiIdi]

+

n∑j=1

γ1ij(t) sin(δij)+n∑j=1

γ2ij(t) cos(δij)ωj.

Therefore, themultimachine power systemdynamicmodel (1)–(3)can be compensated into:

332 R. Yan et al. / Automatica 46 (2010) 330–336

˙δi(t) = ωi(t),

ωi(t) = −Di2Hi

ωi(t)−ω0

2HiPei

˙Pei =1T ′d0ikciIqiufi − Qei(t)ωi(t)

+1T ′d0i[−Pei − Pmi0 + (xdi − x′di)IqiIdi]

+

n∑j=1

γ1ij(t) sin(δij)+n∑j=1

γ2ij(t) cos(δij)ωj.

Remark 1. From Remark 3.4. in Wang et al. (1997), we have|γ1ij(t)| ≤

4|Pei|max|T ′d0j|min

, |γ2ij(t)| ≤ |Pei(t)|max. It is clear that the

bounds of γ1ij(t) and γ2ij(t) depend on generator parameters T ′d0jand |Pei(t)|max. A robust decentralized nonlinear controller is de-signed based on the known T ′d0j and |Pei(t)|max in Guo and Hill et al.(2000) and Wang et al. (1997). However, the above robust non-linear controller design involves estimating nonlinearity boundswithin a certain operating region. When serious disturbances oc-cur which cause the system to operate in a wider range outside theestimated one, the designed system may not perform well. There-fore, in this work, we will relax the assumption that the bounds ofγ1ij(t) and γ2ij(t) are known and design the controller by the adap-tive robust control.

Remark 2. In the power systems, Pei(t), Qei(t) and Ifi(t) are readilymeasurable variables. From (3), we have Pei(t) = E ′qi(t)Iqi(t), andQei(t) = −E ′qi(t)Idi(t). Therefore, Idi(t) and Iqi(t) can be calculatedusing available variables. Furthermore, ωi(t) can be measurableand the power angle δi(t) can be found in Mello (1994).

A practically realizable controller will be designed using only lo-cal measurements, i.e., a decentralized controller without remotesignal transmissions. The control objective is to design a decentral-ized nonlinear feedback control law ufi, i = 1, 2, . . . , n for an ex-citation control loop such that the systems are transiently stablewhen a major fault occurs.

Let xi(t) = [x1i, x2i, x3i]T = [δi, ωi, Pei]T ,

a1i = −Di2Hi

, a2i = −ω0

2Hi, θi =

1T ′d0i

,

ξi = −x3i − Pmi0 + (xdi − x′di)IqiIdi,

γ i = [γ1i1, γ1i2, . . . , γ1in, γ2i1, γ2i2, . . . , γ2in]T ,

ηi =[sin(δi1), sin(δi2), . . . , sin(δin),

cos(δi1)x21, cos(δi2)x22, . . . , cos(δin)x2n]T,

bi =kciT ′d0i

, ui(t) = Iqiufi.

In these new defined parameters, a1i and a2i are known constants;θi and bi are unknown constants; γ1ij and γ2ij, (j = 1, . . . , n) areunknown time varying uncertainties and ξi, the elements of ηi areknown nonlinear functions.Based on the new denotations, (4) can be rewritten as follows:x1i = x2ix2i = a2ix3i + a1ix2ix3i = Qeix2i + θiξi + γTi ηi + biui.

(4)

The Direct Feedback Linearization (DFL) method always needs thepriori knowledge of the dynamic system. If there are some un-known parameters or some uncertainties in the dynamic system,the backstepping method will be more suitable. In the following,an adaptive controller can be designed analytically using the Lya-punov function method. The main feature of the design is that the

Lyapunov function method is applied to each subsystem to find aproper fictitious control as if it could be controlled independently.In order to deal with the time varying uncertainties γ i(t) in the

third equation of (4), we denote S(x) = ρ1arctan(ρ2x), for anyvariable x, where ρ1 > 0 and ρ2 > 0 are positive constants to bechosen by the designer. Note that if we choose the gains ρ1 and ρ2such that 1

ρ2tan 1

ρ1≤ δ, then

xS(x) = xρ1arctan(ρ2x) ≥{|x| |x| ≥ δx2/δ |x| < δ.

(5)

It is easy to verify that S(x) is continuous and differentiable. Wehave the following property in Xu and Yan (2006).

Property 1. |x| − S(x)x ≤ δ.

Now design the controller based on the adaptive backsteppingmethod in the following. The procedure is divided into three steps.Step 1. Design the fictitious control u1i according to the Lyapunovfunction V1i.Denote new coordinates z1i = x1i and z2i = x2i− u1i, where the

fictitious control u1i = −q1iz1i with q1i > 0. Define a Lyapunovfunction V1i = 1

2 z21i. The derivative of V1i is V1i = z1iz1i = z1i(z2i +

u1i) = z1iz2i − q1iz21i.Step 2. Design the fictitious control u2i according to the Lyapunovfunction V2i.Define z3i = x3i − u2i,where the fictitious control u2i is

u2i = −q2ia2iz2i −

1a2iz1i −

a1i + q1ia2i

x2i. (6)

Define a Lyapunov function V2i = V1i + 12 z22i. The derivative of V2i

should be

V2i = −q1iz21i − q2iz22i + a2iz2iz3i. (7)

Step 3. Design the actual control ui according to the Lyapunov func-tion Vi.Denote

gi =(a21i + q1ia1i + 1

a2i+ Qi(t)

)x2i + (a1i + q1i)x3i

−q2ia2iz1i −

q1iq2ia2iz2i + q2iz3i

θi = [b−1i θi b−1i ]T , φi = b

−1i γ i, ξTi = [ξi gi].

Let βi = [β1i, . . . , β2ni]T , where βji is the bound of φji(t). Design

the controller ui as follows:

ui = −a2iz2i − q3iz3i − θTi ξi − S(β

Ti ηiz3i)β

Ti ηi, (8)

where the parameter updating law is

˙θi = ξiz3i,

˙βi = |ηiz3i| − αiβi, (9)

with a damping coefficient αi > 0.From (4) and (6), we have

z3i = x3i − u2i= Qx2i + θiξi + γTi ηi + biui

−q2ia2iz2i −

1a2iz1i −

a1i + q1ia2i

x2i

= θiξi + γTi ηi + biui + gi

= bi(θTi ξi + φTi ηi + ui). (10)

Substituting the controller ui in (8) into (10), we obtain

z3i = bi[−a2iz2i − q3iz3i + (θi − θi)

T ξi

+ φTi ηi − S(βTi ηiz3i)β

Ti ηi

]. (11)

R. Yan et al. / Automatica 46 (2010) 330–336 333

Now define the following Lyapunov function:

Vi = V2i +12biz23i +

12(θi − θi)

T (θi − θi)

+12(βi − βi)

T (βi − βi). (12)

Then the derivative of Vi is

Vi = V2i +1biz3iz3i − (θi − θi)

T ˙θi − (βi − βi)T ˙βi. (13)

Substituting (7), (11) and the parametric updating law in (9) into(13), we have

Vi = −q1iz21i − q2iz22i − q3iz

23i − (θi − θi)

T (˙θi − ξiz3i)

+φTi ηiz3i − S(βTi ηiz3i)β

Ti ηiz3i − (βi − βi)

T ˙βi

= −q1iz21i − q2iz22i − q3iz

23i + βTi |ηiz3i| − β

Ti |ηiz3i|

+ βTi |ηiz3i| − S(β

Ti ηiz3i)β

Ti ηiz3i − (βi − βi)

T ˙βi

≤ −q1iz21i − q2iz22i − q3iz

23i + δ − αi(βi − βi)

T βi

≤ −q1iz21i − q2iz22i − q3iz

23i −

αi

2‖(βi − βi)‖

2

+ δ +αi

2‖βi‖

2. (14)

Vi is negative definite outside the compact setM = {(z1i, z2i, z3i) :K(z1i, z2i, z3i) ≤ δ +

αi2 ‖βi‖

2}, K(z1i, z2i, z3i) := q1iz21i + q2iz

22i +

q3iz23i+αi2 ‖(βi−βi)‖

2. Further define an ε-neighborhood ofM withε > 0Mε = {(z1i, z2i, z3i) : K(z1i, z2i, z3i) ≤ δ+

αi2 ‖βi‖

2+ ε}, then

Vi ≤ −ε. The state zji, j = 1, 2, 3 will enter the ε-neighborhood,Mε , in a finite time, which implies the asymptotic convergence to

the region |zji| ≤√2δ+αi‖βi‖2

2qji.

Therefore, we have the following theorem for multimachinesystems:

Theorem 1. For the multimachine systems (4), define z1i = x1i,z2i = x2i − u1i and z3i = x3i − u2i, where u1i = −q1iz1i, u2i =−q2ia2iz2i − 1

a2iz1i −

a1i+q1ia2ix2i. The controller ui = −a2iz2i − q3iz3i −

θTi ξi − S(β

Ti ηiz3i)β

Ti ηi with the parameter updating law

˙θi = ξiz3i,

˙βi = |ηiz3i| − αiβi will guarantee that |zji| ≤

√2δ+αi‖βi‖2

2qji, for

j = 1, 2, 3.

Remark 3. To construct the control lawui(t)based on the adaptivebackstepping method, the power angle δi(t) should be available.Since the power angle δi(t) is difficult to measure, it is necessaryto show how to construct the power angle δi(t) from the measure-ments available. It is a standard way to use a state observer to es-timate δi(t). A simple way was proposed by constructing a device,named a δi-detector δi(t) = ωi(t) in Wang et al. (1993).

Remark 4. From Theorem 1, it is clear that the bound of zji is de-cided by the design parameters q1i, q2i, q3i, δ and αi. Therefore, thetracking error can be made sufficiently small by choosing appro-priate values for the design parameters.Furthermore, from theoretical analysis, it is only required qji >

0, j = 1, 2, 3 to get convergence performance. Similar to PID con-trol, different values of qji will affect the convergence speed. If qjiis too small, the system needs a long time to reach the stable state.However, if qji is too large, the control signal may be too large torealize in practice. The reader can find a suitable value to get a sat-isfied performance by tuning values of qji. Wewill demonstrate theabove discussion in Section 4.

Remark 5. From the equation in (3), Vti is a nonlinear function ofδi, Pei and the systemstructure. Therefore, any change in the systemstructure will cause the voltage to reach a different post-fault statewhich is undesirable in practice although δi and Pei can return theirprefault steady values by using the above proposed controller.Voltage regulation is also an important issue particularly in the

post-transient period. Its basic objective is to regulate the voltageto reach its nominal value. Differentiating equation of Vti in (3)gives

˙V ti = θiξi(t)+ γTi (t)ηi(t)+ bifi(t)ufi

θi =1T ′d0i

, ξi =(E ′qi + x

′

diIdi)Eqi√(E ′qi + x

′

diIdi)2 + (x′

diIqi)2,

bi =kciT ′d0i

, fi(t) = −E ′qi + x

′

diIdi√(E ′qi + x

′

diIdi)2 + (x′

diIqi)2,

γ i = [γ1i1, . . . , γ1in, γ2i1, . . . , γ2in]T ,

ηi = [η1i1, . . . , η1in, η2i1, . . . , η2in]T ,

γ1ij = E ′qjBij, γ2ij = E ′qjBij,η1ij = sin δijg2 + cos δijg1,η2ij = (cos δijg2 − sin δijg1)(ωi − ωj),

g1 =(E ′qi + x

′

diIdi)x′

di√(E ′qi + x

′

diIdi)2 + (x′

diIqi)2,

g2 =x′2di Iqi√

(E ′qi + x′

diIdi)2 + (x′

diIqi)2. (15)

In the above equation, θi and bi are unknown constants, γ i(t) areunknown time varying vectors, ξi(t), ηi(t) and fi(t) are knownnonlinear functions. Since γ i(t) is dependent on the operatingconditions, they are bounded. So a new robust adaptive controlapproach can be easily developed for the new system. The voltageis introduced as a feedback variable in the proposed controlapproach. Thus the post-fault voltage is prevented from excessivevariation. It is unnecessary to keep the power angle regulated oncetransient stability is assured.By now it can be seen that the nonlinear controller for transient

stability and voltage controller achieve different control objectivein different regions of the states. In Guo, Hill, and Wang (2001), aglobal controller is designed to coordinate the transient stabilizerand voltage regulator. The designed controller is smooth androbust with respect to different transient faults. The global controlapproach takes the form: uf = µδu

(1)f + µru

(2)f , where u

(1)f is the

nonlinear controller proposed in Theorem 1 and u(2)f is the voltagecontroller which can be easily designed by Eq. (15). The definitionsofµδ andµr are given for details in equations (25) and (26) of Guoet al. (2001).

4. Illustrative examples

A three-generator system where generator #3 is infinite bus inFig. 1 is chosen to demonstrate the effectiveness of the proposedadaptive nonlinear decentralized controller. The generator and thetransmission line parameters are listed as follows. For generator#1, xd = 1.863 p.u., x′d = 0.257 p.u., xT = 0.129 p.u., xad =1.712 p.u., T ′d0 = 6.9 p.u., H = 4 s, D = 5 p.u. and kc = 1. Andfor generator #2, xd = 2.36 p.u., x′d = 0.319 p.u., xT = 0.11 p.u.,xad = 1.712 p.u., T ′d0 = 7.96 p.u., H = 5.1 s, D = 3 p.u. andkc = 1. Moreover, x12 = 0.55 p.u., x13 = 0.53 p.u., x23 = 0.6 p.u.and ω0 = 314.159 rad/s. In the example, the excitation controlinput limitations are−3 ≤ Efi(t) ≤ 6, i = 1, 2.

334 R. Yan et al. / Automatica 46 (2010) 330–336

Fig. 1. A two-machine infinite bus power system.

The fault we consider in the simulation is a symmetrical three-phase short circuit fault that occurs on one of the transmission linesbetween generator #1 and generator #2. λ is the fraction of theline to the left of the fault. If λ = 0, the fault is on the bus bar ofgenerator #1,λ = 0.5 puts the fault in themiddle of the generators#1 and #2. The fault sequence we considered is the following:Stage 1. The system is in a prefault steady state.Stage 2. A fault occurs at t = 0.1 s.Stage 3. The fault is removed by opening the circuit breakers of

the faulted line at t = 0.25 s.Stage 4. The transmission lines are restored with the fault

cleared at t = 1.0 s.Stage 5. The system is in a post-fault state.For the power system in Fig. 1, we first calculate the values of

parameters for the dynamic system of generator #1 in (4). a11 =−D12H1= −0.250, a21 = −

ω02H1= −39.2699 are known constants.

ξ1 = −Pe1+ Pm10+ (xd1− x′d1)Iq1Id1 = −Pe1+ Pm10+ 1.6060Iq1Id1and η1 = [sin(δ11), sin(δ12), cos(δ11)ω1, cos(δ12)ω2]T , δij =δi − δj + δmi0 − δmj0, are known functions. θ1 = 1

T ′d01=

0.1449, b1 =kc1T ′d01= 0.1449 are unknown uncertainties. γ1 =

[E ′q1E′

q1B11, E′

q1E′

q2B12,−E′2q1B11,−E

′

q1E′

q2B12]T is an unknown time

varying uncertainty.Similarly, we can get the values of parameters for generator #2.

a12 = −0.2941, a22 = −30.7999,ξ2 = −Pe2 + Pm20 + 2.0410Iq2Id2,

η2 = [sin(δ21), sin(δ22), cos(δ21)ω1, cos(δ22)ω2]T ,

θ2 = 0.1256, b2 = 0.1256,γ2 = [E

′

q2E′

q1B21, E′

q2E′

q2B22,−E′

q1E′

q2B21,−E′2q2B22]

T .

In the example system, since the Generator #3 is an infinite bus,E ′q3 = const. = 1 < 0

o. We demonstrate the performance of theproposed excitation controller in the following operating points:

δ10 = 30.5◦, Pm10 = 0.57 p.u., Vt1 = 1.01 p.u.,δ20 = 32.5◦, Pm20 = 0.56 p.u., Vt2 = 1.00 p.u.

and the fault location λ = 0.05. Furthermore, saturation ofsynchronous machines is also considered, so (2) becomes

E ′qi =1T ′d0i[Efi − Eqi − (1− kfi)E ′qi],

where kfi = 1+cnicdi(E ′qi)

(ni−1), with cn1 = 0.95, cd1 = 0.051, n1 =8.727, cn2 = 0.935, cd2 = 0.064, n2 = 10.878.The controller design is according to Theorem 1. Let γ = 0.001,

δ = 0.01 and ρ1 = 1, ρ2 = 156 such that 1ρ2 tan1ρ1≤ δ. Now

choose the different values of qij = 1, qij = 5, qij = 10 and qij = 15,i = 1, 2, 3 and j = 1, 2. The power angles of generator #1 and #2are given in Fig. 2. And the limited excitation control for generator#1 is also shown in Fig. 3. From Figs. 2 and 3, it can be seen that thesystem does not reach the stable states in a short time for a smallvalue of qij = 1. And for a large value of qij = 15, it needs a large

Fig. 2. Power angle response of generator #1 and #2 with different values of qij .

Fig. 3. Responses of Efi (t) for generator #1 and #2 with different values of qij .

control signal at the beginning stage. Due to the limit of excitationcontrol signal in practice, the performance based on this value isalso not good. While using qij = 5 and 10, we can get the satis-fied performance for this example and the performance based onqij = 10 is better than that of qij = 5. The comparison results alsoverify the explanation for different values qij in Remark 4.Furthermore, the responses of generator terminal voltages for

generator #1 and generator #2 using qij = 10 into the proposedcontroller are shown in Fig. 4. From the results shown above itis obvious that the proposed controller can enhance the systemtransient stability and dampen out the power angle oscillations.

Remark 6. The proposed control scheme is effective for changesin voltage setpoint and adjustment to transformer tap changerposition since they only affect the conditions of operating points.The proposed control approach is effective for different operatingpoints. It can be guaranteed from the theorem’s proof. Now wedemonstrate the performance of theproposed excitation controllerin the following operating points: δ10 = 59.9667◦, Pm10 =0.95 p.u., Vt1 = 1.05 p.u., δ20 = 62.9675◦, Pm20 = 0.95 p.u.,

R. Yan et al. / Automatica 46 (2010) 330–336 335

Fig. 4. Responses of Vti(t).

Fig. 5. Power angle response of generator #1 and #2.

Fig. 6. Responses of Vti(t).

Vt2 = 1.02 p.u. In this case, we consider the fault location λ = 0.5and use qij = 10 in the controller design. The performance resultsare shown in Figs. 5 and 6.Finally, we will give some comparison between our proposed

approach and existing methods in Guo and Hill et al. (2000)and Wang et al. (1997). As we pointed in Introduction, compar-ing with the existing control approaches, the major advantagesof the proposed approach are: firstly, it has no requirement forthe bounds of interconnections by the transmission networks; sec-ondly, it does not need that the solution of a designed algebraicRiccati equation exists; thirdly, it can improve transient stabilityperformance. The comparison results are given in Fig. 7. It is clearthat the proposed control approach achieves the better perfor-mance although the bounds of interconnections are unknown inthe controller design.

Fig. 7. Power angle response of generator #1 and #2 with existing controlapproaches.

5. Conclusion

In this paper, the adaptive backstepping method is used todesign a controller to enhance the transient stability for multi-machine power systems. Nonlinear excitation controllers are de-signed analytically using the Lyapunov function method. The mainfeature of the design is that the Lyapunov function method is ap-plied to each subsystem to find a proper fictitious control as if itcould be controlled independently. The proposed controller hasbeen applied to improve the transient stability in the face of a largesudden fault. Furthermore, the effectiveness of controllers havebeen demonstrated with a number of case studies.

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Rui Yan received the B.S. degree and the M.S. degree inthe Department of Mathematics from Si Chuan University,Chengdu, China, in 1998 and 2001, and received the Ph.D.degree from the Department of Electrical and ComputerEngineering from the National University of Singapore, in2006. She is currently a Research Fellow in the Institute forInfocomm Research, Agency for Science, Technology andResearch, Singapore.Her research interests include advanced nonlinear

control, power system stability analysis and control, intel-ligent control and social robotics.

ZhaoYang Dong obtained his Ph.D. from the University ofSydney, Australia in 1999. He is currently with Hong KongPolytechnic University. He previously held academic po-sitions with the University of Queensland, Australia andNational University of Singapore. He also held industrialpositions with Powerlink Queensland, and Transend Net-works, Tasmania, Australia. His research interests includepower system planning, power system stability and con-trol, load modeling, electricity market, and computationalintelligence and its application in power engineering.

T.K. Sahawas born in Bangladesh and immigrated to Aus-tralia in 1989. He is a Professor in the School of InformationTechnology and Electrical Engineering at the University ofQueensland Australia. He is a senior member of the IEEEand a Fellow of the Institution of Engineers, Australia. Hisresearch interests include power systems, power quality,and equipment condition monitoring.

Rajat Majumder has received his Ph.D. in Electrical PowerSystems from Imperial College London, UK in January2006. He is with ABB Corporate Research, Power Tech-nology Department, Sweden working with ABB’s futuredevelopment in FACTS & HVDC Technologies. He hasalso worked as a lecturer in Power & Energy Systemsat University of Queensland, Brisbane, Australia between2006–2007. His research interest is Power Systems Dy-namics & Control.