Nonlinear co-ordinated excitation and TCPS controller for multimachine power system transient stability enhancement

YWang, A.A.Hashmani and T.T.Lie

Abstract: A robust nonlinear co-ordinated generator excitation and thyristor-controlled phase shifter controller is proposed to enhance the transient stability of a multimachine power system. To eliminate the nonlinearities and interconnections of the multimachine power system, a direct feedback linearisation (DFL) compensator through the excitation loop is designed. Considering the effects of plant parametric uncertainties and remaining nonlinear interconnections, a robust decentralised generator excitation controller is proposed to ensure the stability of the DFL compensated system. Only the bounds of the generator parameters need to be known in the design of the proposed controller, while the transmission network parameters, system operating points or the fault locations do not need to be known. As the proposed controller can ensure the stability of a large-scale power system within the whole operating region for all admissible parameters, the transient stability of the overall system can be enhanced significantly. Digital simulation studies were conducted on a three- machine power system to show that the proposed control scheme can enhance the transient stability of the system significantly, regardless of the network parameters, operating points and fault locations.

List of symbols Qe,{t) = the reactive power of the ith generator, p.u.

the power angle of the ith generator, rad the power angle of the ith generator at the oper- ating point the relative speed of the ith generator, rad/s the mechanical input power which is constant, p.u. the active electrical power delivered by the ith generator, p.u. the synchronous machine speed of generators, in rad/s (q = 2M0) the per unit damping constant the per unit inertia constant, s the transient EMF in the quadrature axis of the ith generator, p.u. the EMF in the quadrature axis of the ith genera- tor, p.u. the equivalent EMF in the excitation coil, p.u. the direct axis transient short-circuit time constant, s the direct axis reactance of the ith generator, p.u the direct axis transient reactance of the ith gener- ator, p.u.

= the excitation current of the ith generator, p.u. = the quadrature axis current of the ith generator,

= the direct axis current of the ith generator, p.u. = the gain of the excitation amplifier of the ith

generator, p.u. = the input of the SCR amplifier of the ith genera-

tor, p.u. = the ith row and jth column element of nodal

transient admittance matrix at the internal nodes after eliminating all physical busbars, p.u.

= the mutual reactance between the excitation coil and the stator coil of the ith generator, p.u.

P.U

= a t > - q{t) = TCPS reactance = the phase shift angle of the TCPS = the time constant of the control system of the

= the phase shift angle of the TCPS at the operat-

= the gain of the control system of the TCPS = the input to the control system of the TCPS = transformation coefficient of voltage magnitude

TCPS

ing point

- - of TCPS

0 IEE, 2001 IEE Proceedings online no. 20010117 DUL 10.1M9/ip-gtd2~10117 Paper received 1st August 2000 The authors are with the School of Electrical & Electronic Engineering, Nan- yang TechnologiCal University, Singapore 639798

q ( t ) A

= the relative frequency at the TCPS busbar = the fraction of the faulted line to the left of the

fault (if A = 0, the fault is on the busbar of the generator; A = 0.5 puts the fault in the middle of the line, and so on)

IEE Proc.-Gener. Trnnsm. Disfuib.. Vol. 148. No. 2, Morch 2001 133

self-admittances of busbars a and b, respectively, with the TCPS installed admittance between busbars a and b with the TCPS installed admittance between busbars b and a with the TCPS installed self-admittances of busbars a and b, respectively, excluding the admittance of line a-b matrix composed of self- and mutual admittances identified with nodes to be eliminated (in this case nodes a and b between which TCPS is installed) matrix composed of self- and mutual admittances identified only with nodes to be retained (in this case n number of gen- erator nodes) matrix composed of only those mutual admittances common to a node to be retained and to one to be eliminated transpose of Y,

1 Introduction

Recently, with the development of modern power systems and increasing demand for power supply, the electric power industry has been facing a great challenge to meet the increased load demand with highest reliability and security by minimum transmission expenditure. Hence, power sys- tems have been called upon to operate transmission lines at high transmission levels, for some practical reasons such as environmental and cost considerations. The flexible AC transmission system (FACTS) devices have helped in increasing the transmission capacities of the transmission lines, and, thus, the transmission lines are operating at high transmission levels. Owing to these conditions, the stability margin of a power system has decreased significantly. Thus, new techniques in power system control which can improve the dynamic performance and transient stability of power systems present an even more formidable challenge. The control system should be able to suppress the potential instability and poorly damped power angle oscillations that can be dangerous for the system stability. The control sys- tem should have the ability to regulate the system under diverse operating conditions. Fast controllers are needed to improve the transient stability of power systems. While excitation controllers are helpful for the enhancement of stability of power systems, however the system stability may not be maintained when a large fault occurs close to the generator terminal [ 11.

A thyristor-controlled phase shifter (TCPS) is a device which injects a variable voltage to (essentially) shift the phase angle of the transmission-line phase voltage. This modifies the phase angle difference between the sending- and receiving-end voltages of the line. Through the fast control of the phase shift, a TCPS controls the active power flow through the transmission line [2] and provides additional damping to power systems. Thus, the TCPS is, essentially, a piece of equipment that is used to modulate active power transmission in power systems. A TCPS can be employed to boost voltages in a power system as well [3]. The high speed of the TCPS makes it attractive for it to be used to improve stability.

The generator excitation and TCPS controller is commonly designed by applying linear control theory. A linear controller may not be able to maintain sufficient

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stability when a large fault occurs. This is due to the fact that a power system is a highly nonlinear complex system. Moreover, when a large fault does occur, the behaviour of the power system changes. Nonlinear controllers are suita- ble for the control and improvement of the performance of power systems when operated under such conditions.

A robust nonlinear co-ordinated generator excitation and TCPS controller is proposed to enhance the transient sta- bility of a multimachine power system. The TCPS is located at the mid-point of the transmission line.

By using the direct feedback linearisation (DFL) tech- nique [4], a direct feedback linearisation compensator through the excitation loop is designed to eliminate the nonlinearities and interconnections of the multimachine power system. Although the DFL compensating law has the ability to alleviate the nonlinearities in the multima- chine power system, the DFL compensated model still contains uncertain parameters, nonlinearities and intercon- nections. To design a feedback controller to ensure the overall stability of the multimachine power system, many factors are taken into consideration. Among these factors are the variation of network parameters, the presence of parameter uncertainties and the necessity to get a co-ordi- nated control with the use of a TCPS. Thus, a robust nonlinear control technique [5] is suitable to be employed for the robust feedback controller design for the DFL compensated system. The controller is designed based on local measurements only. The procedure for the design of a robust nonlinear excitation controller for an n-machine power system requires solving n Riccati equations. The resulting nonlinear controller can ensure the overall stabil- ity of the large-scale power system considering the paramet- ric uncertainties. For the design of the robust nonlinear controller, only the bounds of generator parameters need to be known, but not the transmission network parameters, system operating points or the fault locations. As the proposed robust nonlinear controller can ensure the stabil- ity of the large-scale power system in the whole operating region for all admissible parameters, transient stability of the overall system can be enhanced significantly.

A three-machine example system, as shown in Fig. 2, is presented to illustrate the effectiveness of the proposed design method. A TCPS is located at the mid-point of the transmission line between generators 1 and 2. A first-order linear model of the TCPS is used. Generator 3 is assumed to be an infinite busbar. Digital simulation results show that the proposed nonlinear controller can enhance the transient stability of the power system, even when there are large operating point variations, such as when a three- phase short circuit fault occurs near to the generator. Power angle oscillations are also damped out rapidly regardless of operating point variations, fault locations and network parameters.

2 Multimachine power-system model

In this Section, a power system model has been considered that consists of n synchronous machines. The motion of the interconnected generators, under some standard assump- tions, can be described by a classical model with flux decay dynamics [6]. In this model, the generators are modelled as the voltages behind direct axis transient reactances; the angles of the voltages coincide with the mechanical angles relative to the synchronous rotating frame. The network is reduced to internal busbar representation. The dynamical model of the Ith (i = 1, 2, ...., n) machine in the lossless power system with generator excitation control can be

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written as follows [7]:

&(t ) = U i ( t )

Di WO

2Hi 2Hi Lji(t) = ---U&) + -[P,;o - P&)]

The TCPS dynamic model can be expressed as follows PI:

1 4(t) = --(-d@) + do + ~ P P P ( t ) ) (11)

T P It can be assumed that a TCPS is installed in the power system between nodes 1 and 2, as shown in Fig. 1 with K = kL@. The relevant elements in the network admittance matrix are [9]:

Yl1 = Yll + l / k2Z1z K2 = -e-Jd/kZ12 Y Z ~ = -eJ@/,tZ12 (12)

Yzz = YiZ + 1/z12

0 0

0 I yps 1 - 1 I 0

0

0 I I

awb k: 1

Fig. 1 Multimachine power system installed with c1 TCPS

The first step in forming the system admittance matrix Y, which is obtained by deleting all nodes except the n genera- tor internal nodes in the network admittance matrix of the power system, is to assume that the TCPS damping con- troller is initially not connected in the system. An initial system admittance matrix Y,, is first formed with the n generator internal nodes, and in addition the two nodes, nodes a and 6, between which the TCPS is installed:

IEE Proc.-Gener. Trcirzsm Distrih., Vol. 148. No. 2, March 2001

When the TCPS is connected to the power system, it can be seen from eqn. 12 that the admittance matrix YK can be expressed as follows:

By deleting nodes 1 and 2, the system admittance matrix can be obtained as follows:

Y = YM - YLTYE'YL (14) It is evident from eqns. 13 and 14 that, after the insertion of the phase shifter, the admittance matrix Y is no longer constant and it becomes the function of phase angle @ and k. In this paper, the authors only focus on controlling the phase angle @, thus assuming IC = 1 . Hence, when the TCPS is added to the power system, the power angle Sl{t) in power system model eqns. 1-10 will be replaced by 6i(t) +

Remark 1: The dimension of matrix YK depends on the number of phase shifters in the power system and can be given as (twice the number of phase shifters in the power system) x (twice the number of phase shifters in the power system).

The basic idea behind the phase shifter is to keep the transmitted power at a desired level independently of the power angle S,(t) in a predetermined operating range. In this way, the actual transmitted power can be increased significantly, even though the phase shifter does not increase the steady-state power transmission limit (but, it increases the transient and dynamic stabilities of the system) [lo]. To keep the transmitted power at a desired level, the effective power angle [ll] si(t) + @t) can be kept constant by varying the parameter $(t) [8]. For example, when the fault occurs, the electrical power output of the generator will decrease, while the input mechanical power is constant. As a result, the rotor speed will increase and hence the power angle S[{t) will also increase. Therefore, to maintain the effective power angle Sl{t) + @(t) constant, the value of gt) should be decreased.

s(0.

3 Nonlinear robust controller design

In this Section, a direct feedback linearisation (DFL) [8] compensator is designed to eliminate the nonlinearities and to minimise the interconnection effects between different generators. The DFL compensated system model, even after the DFL compensating law is employed, contains nonlinearities and interconnections. To design a feedback controller to ensure the overall stability of the multi- machine power system, irrespective of network parameters, a robust nonlinear control technique [5] has been extended for the design of a robust feedback control law, to ensure the stability of the DFL compensated system. The pro- posed robust nonlinear controller is able to stabilise the large scale power system taking into consideration the effects of parametric uncertainties, particularly caused by network parameters, fault locations and interconnections between generators.

3. I Feedback linearisation compensator design To eliminate the nonlinearities in the equations given in Section 2, Ebi(t) in the generator electrical dynamics is first

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1 1 aPez(t) = - 7 a P e z ( t ) + T ~ f z ( t )

TdOz TdOz n

+ E&(t)E;, (W’, s i n ( W ) )

- E;z(t)

,=1 n

E;J (W’, cos(&, (t))LJJ ( t ) .7=1

(18) where vfi(t) is the new input of the excitation loop of the ith generator.

It can be noticed that the mapping eqn. 15 from ufi(t) to vdt) is invertible, except for the point where IqL{t) = 0 (which is not in the normal working region for a genera- tor). From eqn. 15, the DFL compensating law can be obtained as

Remark 2: In power systems, Pci(t), Qe,(t) and Idt) are readily measurable variables. From eqns. 8 and 9 it is seen that Pci = E71(t)14L{t) and Qej = f14i(t)IdL{t) and using these available vanables Zdit) and Zql{t) can be calculated from

136 IEE Proc.-Gcner. Trrmsm. Dbtrib., Vol. 148, No. 2, March 2001

eqns. 4 and 10. Thus, compensating law eqn. 19 is practi- cally realisable by the use of only local measurements because m,(t) (i = 1, 2, ..., n) are also measurable variables, and the method for measuring the power angle s(t) can be found in [12]. The validity of the compensating law eqn. 19 is in the whole practical operating region, except when &(t) = 0. The compensating law eqn. 19 is a decentralised con- trol scheme because no remote signal transmission is needed in the controller. Remark 3: In eqn. 18, the terms&' .(t), q{t) and o,Ct), on the right hand-side, represent the effects of remote dynam- ics of the jth machine on the ith machine. The DFL com- pensator eqn. 19 does not cancel these remote dynamics. As a result, even after the authors employ the compensator eqn. 19, the DFL compensated system model (eqns. 16-18) contains the nonlinearities and interconnections. Therefore, a robust nonlinear control technique is proposed to be employed for the DFL compensated system eqns. 16-18, to ensure the overall stability of the multimachine power sys- tem irrespective of network parameters.

3.2 Robust feedback controller design When a major fault occurs on a transmission line in a multimachine power system, the effective impedance of the transmission line changes. The variations in the effective transmission line parameters are treated as parametric uncertainties. Taking the uncertainties and interconnections into consideration, the DFL compensated model (eqn. 1 6 - 18) can be generalised as follows:

ii(t) ( A , + AAi)zi(t) + (Bi + A B i ) ~ f i ( t ) n

+ { 4 1 i j [ Q i j + ~ u l i j ( t ) l u l i j ( ~ i , z j ) }

+ {42ij[UZij + nu,ij(t)Iulij(zi,zj)}

j=1

n

j=1

(20) where, for the ith (i = 1, 2, ..., n) subsystem, xi E Rni repre- sents the state; vfi E Rmi represents input; Ai, Bi, Ulij and U2ii are the known real constant matrices of appropriate dhensions that describe the nominal model; AAL{& AB,(t), AUl,{t) and AU,,{t) are the real time-varying parameter uncertainties; ul,{xi, xj) E R'U and u2,{xi, xi> E are unknown nonlinear vector functions that represents nonlin- earities in the ith subsystem and in the interactions with the other subsystems; 41d and 42ij are constant parameters hav- ing values either 1 or 0 (both having 0 value indicates that thejth subsystem has no connection with the ith subsys- tem).

In the DFL compensated model (eqns. 16-18), if the jth machine is an infinite busbar then 41ij = q2U = 0.

The uncertain matrices AA,{& AB,(t), AU,,{t) and AU2,{t) are assumed to be of the following structures:

[AAi( t ) , A&@)] = LiFi(t)[El,EZ] (21)

Auiij(2) = LiijFiij(t)Eiij (22)

Au2ij ( t ) = L2ijF2ij (t)Ezij (23) with Fi(t) E Riixji, Fl,{t) E Ri1 UiXjIUi and F 2& t 1 E RiZUiXJ2Ui

(for all i, 1) being unknown matrix functions with Lebesgue measurable elements and satisfying

f?,T(t)f?i(t) 512 Flij(t)F&(t) I Ilij F2 i j (t)F& ( t ) L: 12 i j (24)

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where El , E,, Ely, L,, L1, and L2, are known real con- stant matrices with appropriate dimensions.

Regarding the unknown nonlinear vector functions and the matrix E,,, the following assumptions have been made:

(i) The known constant matrices wll, W2,, Flrl and WZy exist such that, for all x, E Rni and xl E Rnj,

IlW2,x,(t)\l + 1lW2,xJ{t)ll for all i, j and for all t 2 0.

Remark 4: The parametric uncertainty in the parameter T',, has been considered as AT',, in the n-machine power system case. Therefore, taking the DFL compensated model (eqns. 16-18> into consideration, for i, j = 1, 2, ..., n and i # j , provides the following:

It!!&,, x,)ll g I I ~ l l ~ ~ ~ ~ ) l l + l l ~ i ~ $ ) l l ~ llu2yG% x,>ll 2

(ii> For all i = 1, 2, ..., n, R, = E2,TE21 > 0.

A = [ ! -$ 1 -.] 0

B = [ l ]

U123 = U223 = [ i] A A z = [ : 0" : ] A B z = [ 0 0

U123 = [ :: ] U223 = [ 0 ]

-- TA"? Tic,*

ulz3 = sin(&(t) - s3( t ) ) ~ 2 % ~ = w 3 ( t )

The uncertainty matrices are

0 P - I . L z ( t ) -cl$)

Y1v ( t ) Y2z3 ( t )

0

1 1 P$) = - -

TAOZ G o , 4- W O z

7 1 2 3 ( t ) = wY,; 7 2 % 3 ( t ) = -E;&)q , (t)Y,> cos(&, ( t ) )

The uncertainty for the ith generator has one possible decomposition, which can be expressed as follows

- w1,= [1 0 01 WIZj = [ 1 0 01 w2,= [ O 1 01 W2%% = [ O 0 01 -

and W223 = [ 0 1 01

Remark 5: It can be noticed that in the multimachine power system case, when the bounds of the parametric uncertainties related to the network transmission parame- ters and interconnections are estimated PJt ) = E',,(t) Cpl E',(t)Yi sin(8&t)). As the electrical power of each gen- erator and the electrical power flow through each transmis- sion line are all bounded, therefore Eb,(t)E',(t)Yk(i= 1, 2, ..., n and i # j ) are also bounded. Hence, 13/2,{t)l I E>!(t) E;i(t)Ybl I IPe,(t)lmdx . It can be noticed that tie excitation voltage Eht) may rise by 5 times that of the E,,(t) when

I37

there is no load in the system. Taking into consideration eqn. 3,

1 I E i 3 ( t ) l m a x IT/ I

do3 m a x

It follows that 4

IYlV (t)I I I pet ( t ) I m a x IT:oJ Imzn

It is clear that the bounds of x,{t) and y2&t) are only dependent on generator parameters T;O, and IPCL{t)l,,, For designing the robust nonlinear controller for the ith genera- tor, only the bounds of the generator parameters need to be known, while the exact information of the network parameters, system operating points and fault locations are not required.

The following algebraic Riccati equations are helpful for the solution to robust decentralised stabilisation of the DFL compensated system model (eqns. 16-18):

ATP, + P,A, + PtB,BTPt - u ~ ~ B ~ ~ RFIBpt n

n -T- + c q 2 i j ( W 2 i W 2 t + WTjiW2ji) + Qi = O

i=l

j=1

+ A;; L~~~ L;. 4 (26)

and the designer can choose Qi > 0. vi > 0, A,, > 0, > 0 (i, j = 1, 2, ..., n) are the scaling parameters to be chosen, with x&t) and y2,{t) satisfying Al$El,TEl, < Z and &ij2E2ijT E2, < I , Vi, j = 1, 2, ..., n.

An appropriate decentralised feedback linear controller is given as follows:

v&) = -Kizi(t) (27) As the feedback signals required implementing, the

robust control law eqn. 27 has measurable variables that are available locally and the control strategy is decentral- ised.

Thus, the DFL compensated power system model (eqns. 16-1 8) or the generalised uncertain interconnected system (eqn. 20) is stabilisable for all admissible uncertain- ties, satisfying eqns. 21-24, via the decentralised controller eqn. 27, if there exists stabilising solutions Pi 2 0 for the Riccati equations (eqn. 25). Remark 6 The multimachine power system (eqns. 1-10) can be compensated into the model (eqns. 16-18> or the generalised model (eqn. 20) by using the DFL compensat-

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ing law eqn. 19. Therefore, to design a robust nonlinear control law uf(t) to transiently stabilise the power system model (eqns. 1-10) is equivalent to design a robust linear control law vfj(t) to stabilise the DFL compensated system with parametric uncertainties (eqn. 20). Under a symmetri- cal three-phase short-circuit fault the multimachine power system model (eqns. 1-10) is transiently stable via the DFL compensating law:

+ Pmio + T & i Q e i ( t ) ~ ( t ) } (28) and

v f i ( t ) = -Ri l (BTPi + EZEl , ) z;( t ) (29)

if there exists stabilising solutions Pi > 0 for the Riccati eqn. 25 and Iqi(t) # 0.

From eqn. 29, the following holds:

U f i ( t ) = - - K g i ( & ( t ) - SiO) - KUiUi(t)

- I{PC% (Pe;(t) - Pmio) (30) where [Kg, K,, KPeJ = -R;'(B?Pi + E2irEli) Remark7: The robust nonlinear decentralised controller (eqns. 28 and 29) requires only local signals. It is obvious, from the design procedure of the controller, that the pro- posed control scheme can maintain the system stability regardless of the transmission network parameters, system operating points or the fault locations, as long as the math- ematical model used is valid.

For the solution of the Riccati eqn. 25, the algorithm proposed in [13] can be used. In the following Section, the effectiveness of the proposed nonlinear decentralised con- troller has been demonstrated via a three-machine power system exaniple.

4 Three-machine power system example

A three-machine power system example shown in Fig. 2 has been selected to illustrate the effectiveness of the proposed robust nonlinear decentralised controller. A TCPS is located at the midpoint of the transmission line between generators G1 and G2. Generator G3 is assumed to be an infinite busbar.

k:l x57

"4 /"" 3&3vs

Fig. 2 the trunsmission line

Three-machine power system with U TCPS located in the nzichle of

In the simulation the following example system parame- ters were used: xdl = 1.863p.u.; x:,, = 0.257p.u.; xT1 = 0.129p.u.; TZnl = 6.9s; X& = 2.36p.u.; I!& = 0.319p.u.; XE = 0.llp.u.; T h 2 = 7.96s; D1 = 5.Op.u.; HI = 4.0s; kcl = 1; D 2 = 3.Op.u.; H 2 = 5.1s; kc2 = 1; X46 = l/y,gj = 0.5p.u.; X57 = 11~57 = 0.5; ~ p s = 0.1; ~ 3 4 = 11~34 = 0.53p.u.; x3j = 1/y35 = 0.6p.u. and q = 314.159radJs. The excitation control input limitations are -3 I Efit) = k,,ufit) 5 5.2, i = 1, 2.

As the generator G3 is an infinite busbar in the example system, therefore is constant with a value of 1LO"p.u.

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and the generator G3 is used as reference. The parametric disturbances, for illustration purposes, are considered as AThi = 0.1 T:Mi, i = 1, 2.

The DFL compensated model for the generator G1, as discussed in remark 4, can be rewritten as

21 ( t ) = (Ai + nAi)~i ( t ) + (BI + ABi)ufi ( t ) + nu11~ sin(h(t) - 6 2 ( t ) ) + Au211w(t) + n u 2 1 2 w 2 ( t )

where

0 0 -0.1449

0 1 A1 = 0 -0.625 -39.27 [

A A 1 = [ 0 ° 0 0 : ] B l = [ : ~ A B l = . [ 0 0 ] 0 0 P l N

0.1449 -P1 ( t ) lPel(t)lmax = 1.4 has been chosen. In remarks 4 and 5 the structures and bounds of the parameter uncertainties have been explained. For the parameters given here the follow- ing values also hold: Ipl(t)l 5 0.0132, IT;02/min = 7.164s, Inld4l 5 0.7817, Iril$>l 5 1.4 and Iy212(t)l S 1.4.

In a similar way, the DFL compensated model for the generator G2 is given as

22(t) = (Az + A A z ) ~ 2 ( t ) + (Bz + a B z ) ~ f z ( t ) + nu121 sin(&(t) - &(t)) + AU221w(t) + AU222w2(t)

where

-0.1256 O I

1 A2 = 0 -0.2941 -30.8 [:: 0

A A 2 = [ 0 ° 0 0 : ] 0 0 P 2 ( t )

Bz = [ 0.&,,] = [ p2!t)] IPe2(t)lmax = 1.5 has been chosen. Therefore, lp2(t)l IO.0111,

2 1.5. For all i , j = 1, 2, let xii = = 0.99 and vi = 0.2. Ql = diag{500, 10, 2000) and Q2 = diag(500, 10, 3000) are chosen.

IThlvImin = 6*21s, In21(t>l 0.9662, Iri21(t)l 1.5 and IYU~(~>I

Solving algebraic Riccati eqn. 2 gives:

[Khl ,Kwl, Kpe,] = [ -40.19 -13.24 4.581 [ K , J ~ , Kw2, Kpe2] = [ -31.07 -12.71 7.891

Therefore, the robust stabilising controller for the generator G1 is found as

1 { U f l ( t ) - ( Z d l - 4 1 )Idi(t)Iql(t) u f l ( t ) = Kc&&)

+ pm10 + TAoiQe1 ( ~ ) u I ( t ) } and

~ f l ( t ) = 40.19(61(t) - 610) + 13.24wl(t) - 84.58(Pe1 ( t ) - Pm1o)

IEE Proc.-Gener. Trnnsnz. Distrib., Vol. 148, No. 2, March 2001

and for generator G2

5 TCPS controller

The control law for the TCPS is as follows:

U&) = - K S ( t ) (31) where K = [K, K2]

S ( t ) = [ &(t) W L ( ~ ) ] and A 4 = 4( t ) - 40 Using the parameters q, = 0.05s and kp = 1 for the

TCPS, the robust controller gain for the TCPS is found as K = [12 41. Thus, the control law for the TCPS can be expressed as:

U p @ ) = -12a4 - 4WL(t) The new co-ordinated generator excitation and TCPS con- troller consists of two subcontrollers, an ith generator exci- tation controller and TCPS controller. The excitation controller can be co-ordinated with the TCPS controller because, for the design of the excitation controller, only the bounded values were considered. The designs of the con- trollers for the generator and the TCPS are independent of each other because the feedback linearisation technique applied decouples the power system model. Only the local measurements are used for the controller design.

6 Simulation results

In this Section, the example system shown in Fig. 2 has been used to demonstrate the effectiveness of the proposed robust decentralised nonlinear co-ordinated generator exci- tation and TCPS controller under different fault locations.

The physical limit for the phase-shift angle is -10" S 4 I 10". A fault sequence given in the Appendix (Section 9) has been considered. The fault considered is a symmetrical three-phase short-circuit that occurs on one of the parallel transmission lines between busbars 4 and 6. The following operating points have been considered: SI, = 60.78", Pmlo = l.lp.u., V,, = l.Op.u., h0 = 60.64", Pm2, = l.0p.u. and V,, = l.0p.u.

The responses of the power angles for generators G1 and G2 for the fault location A = 0.063 are shown in Figs. 3a and b, respectively. For the fault location A = 0.062 see Figs. 3c and d, respectively, with the generator excitation controller but without TCPS controller. For the fault loca- tion A = 0.062 Figs. 3e and J I respectively. For the fault location A = 0.01 see Figs. 3g and h, respectively, with the proposed excitation and TCPS controller = 10"). Fig. 3a and b illustrate that the system is stable at the fault location of A = 0.063 with the excitation controller only. Figs. 3c and d illustrate that the system is unstable at the fault location of A = 0.062, because the physical limit of the excitation voltage has been considered. It is evident from these results that, when the fault occurs close to the genera- tor terminal, the system using only the excitation controller cannot maintain the synchronism. Figs. 3e and f illustrate that the system is stable at the same fault location (A = 0.062) when the TCPS controller is used in co-ordination with the excitation controller. The reason for this is that,

139

when the power angle s(t) increases, the phase-shift angle @((t) decreases (Fig. 4). As a result, the effective angle (qt) + s(t)) tends to remain constant. Figs. 3g and h illustrate that the system remains stable even when the fault occurs much closer to the generator busbar (at the fault location A = 0.0 1). Thus, by using the proposed nonlinear controller, the transient stability enhancement can be achieved. The capa- bility of the TCPS for the stability enhancement has been demonstrated through the digital simulation studies on a simple three-machine power system.

160 r I 150 140

8 130 ; 120 9 110 d 100 E 90

80 70 60

0 1 2 3 4 5 time, s

Responses ofS,(t) and &(t)for generators I and 2, respectively Fig.3 __generator 1 plots - - _ _ generator 2 plots n and 6 For generators 1 and 2, respectively, with 1 = 0.063 c and d For generators 1 and 2, respectively, for 1 = 0.062 with excitation controller only e andfFor generators 1 and 2, respectively, for 1 = 0.062 with the proposed excita- tion and TCPS controller g and h For generators 1 and 2, respectively, for d = 0.01 with the proposed excita- tion and TCPS controller

0 1 2 3 4 5 time, s

Fig. 4 controller

Response of $(t) ai A = 0.01 with the proposed excitation and TCPS

6 r

0 1 2 3 4 5

time, s Fig. 5 Responses of q ( t ) f o r enerators I (solid curve) and 2 (dotied curve) at the fault location A= 0.01 wit$ the proposed excitation and TCPS controller ($ma.Y = 10")

1.6 r

I I I

0 1 2 3 4 5 time, s

Fig. 6 Responses of Pei(t) for generators I (solid curve) and 2 (dotted curve) at the fault location A = 0.01 with the proposed excitation and TCPS controller = IO')

0 1 2 3 4 4

time, s Fig.7 curve) ai tlzejzuli locatiorl I = 0.01

Responses of Efi t ) for generators I (solid curve) and 2 (dotted

The responses of the relative speeds, the real powers and excitation control signals for the generators G1 and G2 at the fault location A = 0.01, when the proposed excitation and TCPS controller is used (for $,,,,,, = loo), are shown in Figs. 5-7, respectively. It is obvious from the simulation results presented in this Section that the proposed control- ler can enhance the system transient stability and dampen out the power angle oscillations.

7 Conclusion

The paper presents the development of a nonlinear co-ordi- nated generator excitation and TCPS controller design. The proposed controller is able to control two main parameters affecting AC power transmission: namely, excitation volt- age and phase angle in a co-ordinated manner. The TCPS is located at the mid-point of the transmission line. A non- linear feedback control law is proposed to linearise and decouple the power system. The main advantages of the proposed controller are as follows: (i) The two controllers in the system are co-ordinated, (ii) Only local measurements are required as feedback signals, (iii) The design of the resulting controller is independent of the operating point, and (iv) Controller design is done independent of the fault loca- tion and the controller can overcome the variation of the reactance of transmission line. The proposed controller consists of two subcontrollers: one for the generator excitation voltage and the other for the TCPS. The goal of the co-ordinated controller design is to allow the subcontrollers to co-operatively improve the transient performance of the power system. To simplify the analysis, a nonlinear feedback is derived to linearise and decouple the power system. The subcontrol- lers are designed separately based on local measurements only and the design of the proposed co-ordinated control- ler is independent of the operating point and fault loca- tion. Even if the information due to other controller actions is not available locally, the consideration of the bounds of the interactions between various interconnected subsystems in the controller design has resulted in a co- ordinated scheme. Thus, the proposed controller has good robustness against the generator parameter variations and the design result is irrespective of the network parameters and configuration.

Digital computer simulation results on the three-machine power system have shown that the proposed controller is robust against network parameter uncertainties. In addi- tion, it enhances the power system transient stability regardless of power transfer conditions and fault locations, it also improves the system damping.

IEE Proc-Genet Tiatism Distil6 Val 148, No 2 Murch 2001 140

References

WANG, Y., HILL, D.J., GAO, L., and MIDDLETON, R.H.: ‘Tran- sient stability enhancement and voltage regulation of power systems’, IEEE Trans. Power Syst., 1993, PWRS-8, pp. 62M27 XING, K., and KUSIC, G.: ‘Applications of thyristor-controlled phase shifters to minimize real power losses and augment stability of power systems’, IEEE Trans. Energy Convers., 1988, 3, pp. 792- 798 ARRILLAGA, J., BASSET, B., and VOVOS, N.A.: ‘Thyristor-con- trolled regulating transformer for variable voltage boosting’, IEE Proc. C, 1976, 123, (IO), pp. 1005-1009 GAO, L., CHEN, L., FAN, Y., and MA, H.: ‘DFL-nonlinear control design with applications in power systems’, Automatica, 1992, 28, pp. 975-979 WANG, Y., XIE, L., and DE SOUZA, C.E.: ‘Robust control of a class of uncertain nonlinear systems’, Syst. Control Lett., 1992, 19, pp.

ANDERSON, P.M., and FOUAD, A.A.: ‘Power system control sta- bility’ (Iowa State University Press, Ames, Iowa, 1977) WANG, Y., GUO, G., and HILL, D.J.: ‘Robust decentralised nonlin- ear control design for multimachine power systems’, Auiomaticu, 1997,

TAN, Y.L., and WANG, Y.: ‘Nonlinear excitation and phase shifter controller for transient stability enhancement of power systems using adaptive control law’, Int. J. Electr. Power Energy Syst., 1996, 18, pp. 397403

139-149

133, pp. 1725-1733

9

10

11

12

13

9

WANG, H.F., SWIFT, F.J., and LI, M.: ‘Analysis of thyristor-con- trolled phase shifter applied in damping power system oscillations’, Electr. Power Energy Syst., 1997, 19, (I), pp. 1-9 GYUGYI, L.: ‘Unified Power-flow control concept for flexible AC transmission systems’, IEE Proc. C, Gener. Transm. Distrib., 1992, 139, (4), pp, 323-331 O’KELLY, D., and MUSGRAVE, G.: ‘Improvement of power sys- tem transient stabilitv by phase shift insertion’. IEE Proc. C, 1973,

~ _ _ 120, pp. 247-252 DE MELLO, F.P.: ‘Measurement of synchronous machine rotor angle from analysis of zero sequence harmonic components of machine terminal voltage’, IEEE Trans. Power Deliv., 1994, PWRD-9, (9), pp. 177G1775 PETERSEN, I.R.: ‘A stabilisation algorithm for a class of uncertain linear system’, S y f . Control Lett., 1987, 22, pp. 39741 1

Appendix: Fault sequence

Stage I : The system is in prefault steady-state; Stage 2: A fault occurs at t = 0.1 s; Stage 3: The fault is removed by opening the breakers of the faulted line at t = 0.25s; Stage 4: The transmission lines are restored with the fault cleared at t = 1 .Os; Stage 5: The system is in postfault state.

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