robust decentralized nonlinear controller design for multimachine power systems

Pergamon

Robust Decentralized Nonlinear Controller Multimachine Power Systems*

PII: SOOOS-1098(97)OM91-5

Auromor~cu. Vol 33, No ‘I. pp 1725 1733, 1997 ,c, 1997 Elsewer Saence Ltd All II&S reserved

Prmted tn Great Bntam cw5.1098197 $1700 + on0

Brief Paper

Design for

YOUYI WANG,? GUOXIAO GUO$ and DAVID J. HILLQ:

Key Words-Power system control: linearization; nonlinear systems; robust control: decentralized control.

Abstract-In this paper, a robust decentralized excitation control scheme is proposed for multimachine power system transient stability enhancement. First, a direct feedback linearization (DFL) compensator through the excitation loop is designed to eliminate the nonlinearitles and interconnections of the multlmachine power system. Then, a robust decentralized controller IS proposed to guarantee the asymptotic stability of the DFL compensated system considering the effects of plant parametric uncertainties and remaining nonlinear interconnections. The design procedure for an n-machine power system involves in solving n Riccati equations. In the design of the robust nonlinear decentralized controller, only the bounds of generator parameters need to be known, but not the transmission network parameters, system operating points or the fault locations. Since the proposed robust nonlinear decentralized controller can guarantee the stability of the large scale power system within the whole operating region for all admissible parameters, transient stability of the overall system can be greatly enhanced. The design procedure is tested on a three-machine example power system. Simulation results show that the proposed control scheme can greatly enhance the transient stability of the system regardless of the network parameters, operating points and fault locations. (_. 1997 Elsevier Science Ltd.

I. Introduction Power systems are increasingly called upon to operate transmission lines at high transmission levels for economic or envlron- mental reasons. This requires the control system to have the ability to suppress potential instability and poorly damped power angle oscillations that might threaten the system stability as the load is expected to increase in the future. In a lot of cases, transient stability transfer limits are more constraining than the steady-state transfer limit under contingency. All these practical demands require the control system to have the ability to regu- late the system under diverse operating conditions. Unfortu- nately, power systems are large scale nonlinear systems; the characteristics of conventional controllers that are designed based on approximately linearized power system models, such as power system stabilizers, vary significantly with respect to the changes in operating conditions. Application of nonlinear feedback to cancel the inherent system nonlinearities have the potential to enhance power system transient stability (see Wang et al.,

*Received 13 October 1995; revised 29 May 1996; revised 5 February 1997: received in final form 22 April 1997. This paper was not presented at any IFAC meeting. This paper was recom- mended for publication in revised form by Associate Editor Hassan Khahl under the direction of Editor Tamer Baqar. Corresponding author Dr. Youyi Wang. Tel. + 657991423; Fax + + 65 7920415; E-mall [email protected].

tSchool of Electrical and Electronics Engineering, Nanyang Technological University, Singapore 639798, Singapore

:Data Storage Institute. The National University of Sm- gapore, Singapore 119260, Singapore

SDepartment of Electrical Engineering, Sydney University, NSW 2006, Australia

1992, 1993, 1995, 1996; Hill et al., 1993; King et al., 1994; Gao et al., 1989, 1992; Chapman et al., 1993; Lu and Sun, 1989; Mielczarski and Zajaczkowski, 1989.1990,1994. 1994a: Marino, 1984).

In this paper, we shall concentrate on the transient stability enhancement of multimachine power systems by means of robust decentralized nonlinear excitation control. The idea of transient stability enhancement via robust nonlinear controller in single-machine infinite-bus power system reported in Wang et al. (1992) has been extended to multlmachine case. By using the direct feedback linearization (DFL) technique (see Gao et al., 1989;Wang et al.. 1993). a decentralized feedback linearizing controller can be found. Although the DFL compensating law has the ability to alleviate the nonlinearities m the multimachine power system, the DFL compensated model still contains nonlinearities and interconnections. In order to design a feedback controller to guarantee the overall stability of the multlmachine power system irrespective of network parameters, robust nonhn- ear control technique (see Wang et al., ,1992a, 1993) can be employed to design the robust feedback controller for the DFL compensated system. The robust decentralized nonlinear excitation controller design procedure for an n-machine power system involves in solving n Riccati equations. The resulting decentralized nonlinear controller can guarantee the overall stability of the large scale power system considering the parameter uncertainties.

A three-machine example system IS presented to illustrate the effectiveness of the proposed design method. Simulation results show that the proposed nonlinear decentralized controller can effectively enhance the transient stability of the power system even in the presence of large operation point variations, such as when a three-phase short circuit fault occurs near the generator. Power angle oscillations are also damped out very rapidly regardless of operating point variations, fault locations and network parameters.

2. Dynamical model In this section, we consider a power system consisting of

II synchronous machines. Under some standard assumptions, the motion of the interconnected generators can be described by a classical model with flux decay dynamics (Bergen, 1986; Pai, 1981; Anderson and Fouad, 1994; Kundur. 1994). In this model, the generator is modeled as the voltage behind direct axls transient reactance; the angle of the voltage coincides with the mechanical angle relative to the synchronously rotating reference frame. The network has been reduced to internal bus representation. The dynamical model of the ith machine with excitation control can be written as follows:

Mechanical equatrons.

8,(t) = to,(t)

c&(t) = - g+(f) + $P,,” - P,,(O).

Generator electrical dynamics:

(1)

(2)

(3)

1725

1726 Brief Papers

Electricnl equations:

E& = EbXO - (Xd, - .uiAl,,(t), (4)

‘J%,(t) = kC,ML (5)

P,,(t) = f: EbWb, (t) sln(&, (U, (6) ,=1

Q.,(t) = - ,il E;,W;,(d coG%,(tN, (7)

r,,(d = i -q,@) cm (4, Oh (8) ,=1

&1(t) = %dJf,(O. (10)

The Notation for the multimachine power system model 1s given in the appendix. From the model discussed above we can see that the multimachine power system is highly nonlinear and interconnected by the transmission network.

3. Nonlinear robust controller design In this section, first, a direct feedback linearization (DFL) (see

Gao et al., 1989;Wang et al., 1993) compensator will be designed to cancel the nonlinearities and to reduce the interconnectlon effects among different generators. After employing the DFL compensating law, the DFL compensated system model still contains nonlinearities and interconnections. In order to design a feedback controller to guarantee the overall stability of the multimachine power system irrespective of network parameters, robust nonlinear control technique (see Wang et al., 1992a, 1993) will be extended to design a robust decentralized feedback control law to guarantee the stability of the DFL compensated system. The resulting robust nonlinear decentralized controller can stabilize the large scale power system considering the effects of parametric uncertainties, especially those caused by network parameters, fault locations and interconnections among generators.

3.1. Feedback linearization compensator design. In order to eliminate the nonlinearities in electrical equations given in Sec- tion 2, we first eliminate EL,(t) in the generator electrical dynamics by differentiating the active power Pei(t) in (6).

P,,(t) = i: &Mb, (t)Bi, sin@,, 0)) ,=1

+ i &, @)=&, (t)B,, sin (h,,(t)). ,=1

Using (3) and (7) we have

Z,, (0 i: ~&)B,,cos@&M+o) ,=1

= Ebt U) i: E&tW,, cos(6,, (O)w,(t) ,=1

-E;,(t) i K,, W,, co@&, (W,id ,=I

It follows that

_ -%Zt) i -‘& VP,, CW,, (t)ko,(t) ,=1

Considering (3) and (4) gives that

Pei(t) = j&I&c(r) - &b(t) + (-Y& - X’&Vd~(t)]lqr(t)

dch

- Q&h(d + i ~,d#‘;l,(t)B,, sln(d,,(t))

,=1

n

- QeL(Wt(~) + 1 E&(O&,(t)B,, sin(S,, (t))

_ ‘%(t) i E’,, W,,cos(G,, (t))w,{t) ,=I

Let Al’,&) = P,Xt) - P,,O. we have

f i E&(t)&,(t)B,,sin(d,,(r)) ,=1

_ E;,(t) i; E~,(t)B,,cos(g,,(b))w,(t) ,=1

If we let

we have

Q,,(t) = - +P.,(t) + + L’f,(f) d-31 d”L

+ i E14,(t)p4,(t)Bl, sin(&, (t)) ,=1

- E;,(t) i: El,, Wt, COG,, (W,tG. ,=1

Then the multimachme power system model (l)-(3) has been compensated into:

&t, = w,(t), (12)

Q 00 WI = - -w(t) + -_CP,,O - Pe,(t)],

2H, 2H, (13)

Brief Papers 1727

_ E;,(t) i E;, W%,cos(~,, Wb,W> (14) ,=1

where sit) is the new input of the excitation loop of the ith generator.

Note that the mapping (11) from udt) to urAt) is invertible, except for the point where I,,(t) = 0 (which is not in the normal working region for a generator). From (11) we get the DFL compensating law as

+ ~~arQ.Afh(~)). (15)

Remark 3.1. In power systems, Pe,(t), QJf) and 1(,(t) are readily measurable variables. From (6H9) we can obtain that

p,,(t) = E’,,(r)r,&) and Q.dt) = - E’,,Wdt)

and from (4) and (lo), we know that r,i(t) and r,,(t) can be calculated using these available variables. Since W,(t) (i = 1.2, , n) are also measurable variables and the method for measuring the power angle 6(t) can be found in de Mello (1994), so the compensating law (15) IS practically realizable using only local measurements.

The compensating law (15) 1s valid in the whole practical operating region, except when I&) = 0.

No remote signal transmission is needed in the controller, so (15) IS a decentralized control scheme.

Remurk 3.2. The terms i?,,{t), d#) and w,(t) on the right-hand side of (14) represent the effects of remote dynamics of the jth machine on the ith machine. This remote dynamics is not cancel- led using the DFL compensator (15) so that, after employing (15), the DFL compensated system model (12)+14) still contains nonlinearities and interconnections. In order to design a feedback controller to guarantee the overall stability of the multimachine power system irrespective of network parameters, a robust nonlinear control technique (see Wang et al., 1992a. 1993) can be employed to design the robust feedback controller for the DFL compensated system (12H14). This problem will be ad- dressed in the following section.

3.2. Robust feedback controller design. In a multimachine power system, when a major fault occurs on a transmission line. the effective impedance of the transmission line will change. The variations of the effective transmission line parameters will be treated as parametric uncertainties. Considering the uncertainties and interconnections, the DFL compensated model (12H14) can be generalized as

.%(t) = [A, + nAi(t,]xi(t) + (B, + AB,(t))u&) n

+ 1 IP~,CG~,, + AG1,,(t)lg1,,t.~,,x,)l ,=, n

+ c h,CG,, + A’%i~~)lsz,,h x,,:. (16) ,=*

where for the lth (i = 1,2, , n) subsystem we have that: x, E R”, is the state, u,,ER”, is the input, the matrices A,, B,, G1,, and Gll, are known real constant matrices of appropriate dimensions that describe the nominal model, AAI( .), ABi( .), AG,,l.) and AG2,b. ) are real time-varying parameter uncertainties, and g&,, x,) E R”J and gf&i, x,) E RlaJ are unknown nonlinear vector functions that represent nonlinearities in the ith subsystem and in the interactions with the other subsystems. The parameters PI,, and pzv are constants with values either 1 or 0 (if they are 0, it means that jth subsystem has no connection with the ith subsystem). Notice that in the DFL compensated model (12Hl4). if the jth machine is the infinite-bus then Ph, = PL, = 0.

The uncertain matrlces AA,(t), AB,(r)AG,,jt) and A(&, (t) are assumed to be of the following structures:

CAA,(tl ~B,(dl = LF,(OCJ5 Ed (17)

AG,&) = &F1,,(0&,,. (18)

AG,,,(r) = &$2,,(t)EZ,, (19)

with F,(t)ER’ZX’,, Fl~,(t)ER”“‘xJ16, and Fz,,(t)~R’2c~*X11c,, (for all i, j) being unknown matrix functions with Lebesgue measurable elements and satisfying

F:(t)F,(t) 51,: F,,~rFf&) 5 I,,,: FZ,,V)FT&) I Zh,. 0)

where El,, EZJ. E,,,, Ez,,, I.,, L,,, and L,,, are known real constant matrices with appropriate dimensions.

We make the following assumptions concerning the unknown nonlinear vector functions and the matrix E2 :

Assumption There exist known constant matrices IV,,, WZ,, WI,, and W,,, such that for all J, E R”* and X, E R”J

IlsIv~Xv~,)ll I II~,,x,(tN + IIW,,,\-Jwl.

Il92&,~-y,)Il s IIW,,.w,(t)ll + llWz,).~Jv)ll

for all i, j and for all t 2 0. Foralli=1,2 ,.... n,R,=ET,E2,>0

Remark 3.3. In the n-machine power system case, we consider the parametric uncertainties m the parameter 7’&, as AT&,. Then we have, considering the DFL compensated mode1 (12Hl4), for i.j = 1.2, __ ,n and i #I’

B, =

0

GI,, = Gz,, = o [I 0

0 .

0

1

T’ d0l _

g1t, = s111(GAt) - b#)). Yz,, = m,(t).

The uncertainty matrices are

AA, =[ t ,.E,,]. AB, =[ _;,(t,#

AG~~J=[Y,J;IIY AG~~J=[;gJ~

,‘l,,(d = ~q,mb,u)B 1,. ;‘a,@) = - Ei,, (W&P,, cos 6,, (0.

One possible decomposltlon of the uncertamtles for the ith generator can be expressed as

L = w 0 I~z(tWm,,IT. F,(t) = [ 0 0 & ,

I m.xl E,,=diag(l,l.l~. EZ, = [O 0 - 11’.

;‘k,ilr) L, = co 0 lilr~JlanaJ. Fkl, = I;lLl,(t)l,.,

i 1 1 4,, = 1.

W,,=[l 0 01. W,,,=[l 0 01,

Iv’,, = [O 1 01, wz,, = [O 0 01. Wzc, = [O 1 O]

The assumptions are satisfied.

1728 Brief Papers

Remark 3.4. In the multimachine power system case, when we estimate the bounds of the parametric uncertainties related to the network transmission parameters and interconnections, notice that

P&) = E’,,(t) f E’ti(t)B,jsin a,, (t). ,=1

Since the electrical power of each generator and the electrical power flow through each transmission line are all bounded, so E’,,(t)E’, (c)B,~ (i = 1.2, , n and i # j) are also bounded. Thus we have:

Ir2,,G)l I IE’&)E’,, (G&I I IP&)lmax.

Notice also that the excitation voltage Efi(t) may raise by 5 times of the E&t) when there is no load in the system. Considering (3), we have

I.%,(~)1 I C&,0) - E,,(td 1

T;o, max 2 4l.%,(&l,X m

It follows that

It is obvious that the bounds of Ylij(t) and Yzi,{t) only depend on generator parameters T& and IP&)l,., In this way, when designing the robust nonlinear controller for the ith generator, only the bounds of the generator parameters have to be known. Exact information of the network parameters, system operating points and fault locations are not needed.

The solution to robust decentralized stabilization of the DFL compensated system model (12H14) depends on the following algebraic Riccati equations

ATP, + P,A, + P,&B:P, - v;‘B;, R;,Bp, + v,ZE:,E,,

+ i: Pl,,ovl~i,, + cJ,W,,,) ,=,

(21)

where i = 1.2. . ,N, B, = BTP, + vfEz,E,,

B,F = v;*L,L,T + i p,,,[cli,q - I:,,E:~~E,,,)-,G:~,

j=l

+ ~2rfLZ~~L~ijl (22)

and QZ > 0 can be chosen by the designer. v, z 0, Al,, > 0. &$, > 0 (i,j = 1,2, , N) are scaling parameters to be chosen, with Al,, and ?b2rj satisfying AfijETi,E,,, < I and &jETijEz,, < I. Vi,j= 1,2, . . . . N.

Moreover, a suitable decentralized feedback linear controller is given as follows:

Of@) = - K,x,(t)

where K, = v; ‘R;,(B:P, + v;E;,E,,). Our main result is as follows:

(23)

Theorem 3.1. Consider the DFL compensated power system model (12Hl4) or the generalized uncertain interconnected system (16). This system is stabilizable for all admissible uncertainties, satisfying (17H20), via the decentralized controller (23) if there exist stabilizing solutions P, 2 0 for the Riccati equations (21).

Proof Combining equation (23) with equation (16) gives a closed loop system of the form

?I, = (A, + L,F,Ei)X,

+ C (GI,, + L,,,F1,,E1,,)S,,,~x,,xj) ,= 1”811,

+ C (Gz~ + L2,,Fz,,E2,,)g*i,(X,, xj) ,= l"P2ij

From (21) and the results in Khargonekar et al. (1990) and Xie et al. (1992), it follows that

A?Pc + P,A, + P,&B,TP, + i: p,,#v:Fv,, + w:,,w,,,) ,=,

where B, is as defined in (22) and Pi is the positive defimte solution to (21). In view of (22), we obtain that

ATP, + P,A, + v;2P,L,L:P,

+ f’z i: P,,CG,,,U - %E:,,E,,,)-,G:,, + &:L,,,L:,,l ,=1

+ i: P,#Gcii,i + W,iW,,,) ,=,

n + ,;, PZLJJK m’,i + K,, Wz,J + v:-v.% < 0.

Applying the result in Wang et al. (1992a) gives that

ATP + PA. + C2P L.L’P + v+.F.E II I1 I LItI &II

+ i: P,zAW w,, + w,, WI,,) ,=,

+ Pi 1

i GlijC:i, + i G,ijGz,j P, 4 0 ,=, ,=, 1

and it follows that

‘x’Pi + P,A, + E,TF’(t)L:P, + P,L,F&)E,

+ i Pl,,(n~,, + W,,W,,J ,=1

+ i: P*,,oG w’2, + G,, Wz,J ,=,

+p, i

i G,&,, + i: G$T,, >

P, < 0. ,=, j=,

Then we have

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It follows immediately that there exist positive definite matrices Qi such that

+ p, i: G,,G,, + i (

G,G, >

P, + Q, = 0. (26) ,=I ,= I

Now, in order to prove the asymptotic stability of the closed loop system (24), let the Lyapunov function candidate

V(X) = i; xfP,x,. ,=I

where I = [XT. XT, . XT]. Note that V(x) r 0 whenever x # 0. Then, by using (24). we have

It follows that

n n

n n

_ Pd~:~T~ w,,x, + “:Wb,w*,,X, - d&,1}

Since

” n

n n

+ P2vC~~K~2t-y, + x,Tw:,,w2,,x, - gT,,g2,,1}

then defining U, = [x:g:,, gZn g$,l g$,,]‘, we have

s, P,~l,l “.

e:,,p, - I

: ‘.

A,= CT,“P, 0

&,P, 0

&PC 0

Next, taking into account (26) the result in Kreindler (1972) and the facts that

n n

+ P2,,Ciw, W2J, + x:w:,, wz,,x, -- d&J 2 0

it follows immediately that

;y(,, < 0

whenever x # 0. Hence, V(x) is a Lyapunov function for for system (24) and thus this system is asymptotically stable

all admissible uncertainties. Therefore, the DFL compensated power system model (12)+14) or the generalized uncertain interconnected system (16) IS globally asymptotically stabilizable via the linear decentralized controller (23) for all admissible uncertainties

Remark 3.5. Using DFL compensating law (15), the multimachine power system (lHl0) can be compensated into (12H14) or the generalized model (16). So designing a robust nonlinear control law ur,(t) to transiently stabilize the power system (l)-(lO) is equivalent to designing a robust linear control law v&) to stabilize the DFL compensated system with parametric uncertainties, (16). The multimachine power system (1HlO) under a symmetrical three-phase short circuit fault is transiently stable via the DFL compensating law

1 n&) = -{rtAr) + Pm,0 - (Xd, - .uX&)r,,{t)

kJ&)

and + GorQez(f)w(f)j (27)

o,,(t) = - R; ‘(B,*P, + E~,TEJ~#) (28)

if there exist stabilizing solutions P, 2 0 for the Riccati equations (21) and I,,(t) # 0.

Remark 3.6. The robust nonlinear decentrahzed controller (27) and (28) only require local signals. From the design procedure of the controller (27) and (28) and the proof of Theorem 3.1, it can been seen that as long as the mathematical model used is valid, the proposed control scheme can maintain the system stabihty regardless of the transmission network parameters, system operating points or the fault locatrons.

Remark 3.7. The design procedure IS summarized as follows: Step 1: Apply the DFL compensator (15) to the multimachine

power system model to obtain the DFL compensated system model (12Hl4):

Step 2: Find the uncertain system model m the form of (16) using the method described in Remarks 3.3 and 3.4;

Step 3: Decompose the uncertamties and nonlinear intercon- nection terms as described in Remark 3.3. Formulate the respective Riccati equations for each machine;

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Step 4: For all the machmes, select the matrices Qc > 0 and weighting parameters V, > 0, 2’1v > 0, y2,, > 0 (i.j = 1,2, . n and i fj), solve Riccati equations (21) and find the positive definite solutions P,. Calculate the respective robust feedback gains as given in (23).

The algorithm proposed in Petersen (1987) can be used to solve the Riccati equations (21).

The effectiveness of the proposed nonlinear decentralized controller will be demonstrated via a three-machine power system in the next section

4. A three-machme example

A three-machine example system Fig. 1 is chosen to demon- strate the effectiveness of the proposed robust nonlinear decentralized controller.

The system parameters used in the simulation are as follows:

xdl =1.863 p.u., x&, = 0.257 p.u., xT1 =0.129 p.u., Tiol =6.9s

xd2 =2.36 p.u., x’~* = 0.319 p.u., .xT2 = 0.11 p.u.. T& = 7.96s

H, = 4s. D, = 5 p.u., k,, = 1: Hz = 5.1s, Dz = 3 p.u.. kc2 = 1,

and

_y12 = 0.55 p.u., xi3 = 0.53 p.u., xz3 = 0.6 p.u.,

w,, = 314 159rad/s, lad1 = .xad2 = 1.71 pa.

The excitation control input limitations are

- 3 $ E,,(t) = k,,u,,(t) I 6, i = 1.2.

In the example system, since the generator # 3 is an infinite bus, we have E& = const. = 1~0’ and use the generator # 3 as the reference.

For illustration purpose, we consider the parametric per- turbation AT &,, = 0.1 T,&,, i = 1,2. As discussed in Remark 3.3, the DFL compensated model for the generator #l can be rewritten as

Al = (A, + nA,)x,(t) + (BI + U3,)v,,(t)

+ AGIl sin(a,(t) - 6,(t))

+ ~G,,,w,(t) + ~Gz1sdf).

where

#l G Q XT1

Fig. 1. Three-machine example system

where

AZ =

L

B, =

We choose iPel(t)/,,, = 1.4. The structures and bounds of the parameter uncertainties have been explained in Remarks 3.3 and 3.4. For the parameters given above we have

IpI 5 0.0132, ITl&,,ln = 7.164 s

I;‘~& I 0.7817, l;‘z,1(t)l 2 1.4, l~zlz(t)l < 1.4.

Similarly. the DFL compensated model for the generator # 2 is

b 1 0

0 - 0.2941 - 30.8

0 0 - 0.1256

We choose IPez(t)imar = 1.5 and it follows that

I/cl(t)1 5 0.0111, I TA,~Ylmm = 6.21~.

l1..1zlW 5 0.9662, I;‘zzI(t)l I 15, IY2n(t)l 5 1.5.

For all 1.1 = 1,2. let Ebl,, = E.2u = 0.99 and Y, = 0.2. Choose Q1 = diag{ 500, 10.2000) and Qz = diag{ 500,10,3000} Solv- ing algebraic Riccati equations (21) gives that.

Ckd,o,pel ] = [ -40.19 - 13.24 4.581,

[kdZuZpeZ] = [ - 31.07 - 12.71 7.891.

Thus the robust stabilizing controllers for the generators # 1 and #2 are found to be

u,,(t) =&1(t) + Prn1” - (Xl1 - LJt)

X’dlvqlwLl(~)

+ Tdo~Qe,(thV)j and

rrl(t) =40.19(6,(t) - a,,) + 13.24w,(t) - 84.58(P,,(t) - P,,,).

%2(t) = &:unw + Pm20 - (Xd.2 - &J~qZ(t)L12(~) qz

+ ThzQezWzW; and

Cfl(t) = 31 07(6,(t) - Sz,) + 12.71w,(t) - 7739(P,,(t) - Pm,,).

The effectiveness of this controller will be tested in the next section.

5. Simulation results

In this section, the performance of the example system with the proposed robust nonlinear decentralized controller given in Section 4 will be tested. Dynamical performance under different operating points, different fault locations and different system parameters will be tested. The fault we consider in the slmula- tion 1s a symmetrical 3-phase short circuit fault that occurs on one of the transmission hnes between the generator # 1 and the generator # 2. i is the fraction of the line to the left of the fault. If 1. = 0, the fault is on the bus bar of the generator # 1, E. = 0.5

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puts the fault m the middle of the generators # 1 and #2. The fault sequence we considered is the following:

Stage 1: The system is in prefault steady-state; Stage 2: A fault occurs at to = 0.1 s; Stage 3, The fault is removed by opening the breakers of the

faulted line at t, = 0.25 s; Stage 4: The transmtsston lines are restored wtth the fault

cleared at tz = 1.0 s, Stage 5: The system 1s in postfault-state.

In the simulatton. saturation of synchronous machines (see Arrillaga and Arnold, 1990) is also considered, so (3) becomes

-G*(r) = +dt) - E&) - (1 - k,,)E’&)], d01

where

kr, = 1 + $E.,.(r))‘“*- l’

and the parameters are

llr = 0.95. b, = 0.051, n, = 8.121.

a, = 0.935, bZ = 0.064, n2 = 10.878.

Simulation cases 1-3 are used to test the system dynamical responses under different operating points and fault locations.

Cuse 1: The operating points are as follows:

a,,, = 60.78’. I’,,,,, = I.lg.u., V,t = l.Op.u.,

6 z0 = 60 64’. P,,, = l.Op.u., v,z = 1.op.u.

The responses of power angles, relative speeds, real powers, terminal voltages and excitation control signals of the generators # 1 and #2 with i = 0.2 are shown in Figs. 2-6 respectively.

The responses of power angles without any controller are given m Fig. 7 for compartson.

From the results shown above it is obvious that the proposed controller can enhance the system transient stability and dam- pen out the power angle oscillattons.

95 1 I

Time (s)

Fig. 2. Responses of 6,(t).

-2 I 0 1 2 3 4 5

Time (s)

Fig. 3. Responses of o,(r).

0 1 2 3 4 5 Time (s)

Fig. 4. Responses of P,,(r).

3 0.90’

3 0.85.

>- 0.80.

0.75

0.70

‘_-,#2 -\ ,_----=-=----

1 \/ \ ,’ r \ /

#l

0 1 2 3 4 5 Time (s)

Fig. 5. Responses of V,,(r).

-41 0 1 2 4 5

Time (s)~

Fig. 6. Responses of E,#)

20 I 0 5 10 15

Time (s)

Fig. 7 Responses of s,(t). No controller.

Next we will test the effectiveness of the proposed controller at different operating points,

Case 2: The operatmg points are

6t, = 18.51’, P,,, = 0.3 p.u.. v,, = 0.95 p.u.,

6,, = 23.68’, P,,,a,, = 0.4 p.u.. v,, = 0.95 p.u.

The power angles and terminal voltages of the generators # 1 and #2 with I = 0.05 are shown in Figs. 8 and 9.

1732 Brief Papers

0 1 2 3 4 5 Time (s)

Fig. 8. Responses of &(t).

3 0.9

9 a 0.8

5 0.7

0.6

0.5 0.41 J

0 1 2 3 4

110

100 =‘ s 90 C e 80

2 s 70

a 60

Time (s)

Fig. 9. Responses of V,,(t).

1 2 3 4 Time (s)

Fig. 10. Responses of s,(t).

1

-I 50 I

0 1 2 3 4 5 Time (s)

Fig. 11. Responses of S,(t).

Case 3: The operatmg pomts are

c?~,, = 56.09’. Pmlo = 1.0 p.u., v,, = 1.0 p.u.,

C&O = 59.040. Pm20 = 1.0 p.u.. VQ = 1.0 p.u.

Figures 10 and 11 compare the power angles of the generators # 1 and # 2 with different fault locations (A = 0.05,0.5 and 0.95).

From the simulation results shown above, it can be seen that despite the different fault locations and the operating point

/

70 - ,I

60 0 1 2 3 4

Time (s)

Fig. 12. Responses of s,(t)

-2

-4 v 1 z

Time (s; 4 5

Fig. 13. Responses of w,(t).

variations, the system remains transiently stable. Power angle oscillations are also damped out rapidly in all cases.

In Section 3, we have stated that our controller design result does not depend on the network parameters. Next we will check the control effect under different transmission line parameters. The operating points and line parameters are

aI0 = 64.08*, Pmlo = 0.95p.u.. v,, = 1.op.u..

& = 65.33’. Pm20 = 0.95p.u., v,z = 1.op.u..

and x12 = xl3 = xZ3 = 0.7.

The power angle and excitation control signal responses of the generators # 1 and #2 when the fault occurs at i = 0.1 are shown in Figs. 12 and 13, respectively. Note here that the controllers and their parameters remain unchanged even when network parameters have changed a lot and the power angle oscillations are still damped quite well under the large fault.

6. Conclusions In this paper, the idea of transient stability enhancement via

robust nonlinear excitation control in single-machine infinite- bus power system reported in Wang et aI. (1992) has been extended to the multimachme case. The direct feedback linearization (DFL) technique has been extended to multimachme power systems and the DFL compensated model has been found. Then the robust nonlinear control technique in Wang et al. (1992a) 1s extended to the interconnected uncertain DFL compensated system to design the robust feedback controller. With the proposed control scheme, only the bounds of the generator parameters need to be known but not the network parameters, system operating points or fault locations. The robust nonlinear decentralized excitation controller design problem for an n-machine power system involves solving of n Riccati equations. The resulting decentralized feedback controller can guarantee the overall stability of the large scale power system considering all admissible network parameter uncertainties. Thus, this controller has good robustness against the generator parameter variations and the design result is irrespective of the network parameters and configuration.

For illustration purpose, the proposed robust nonlinear decentralized control scheme is demonstrated on a three-machine power system. Simulation results on this example system have

Brief Papers 1733

shown that the proposed controller can greatly enhance power system transient stability regardless of power transfer conditions, fault locations and network parameters as well as improve oscillation dampings.

Ackno~ledyements~Thls work was supported by the Nanyang Technological University Research Fund and the Australian Research Council (Ref: A49231875)

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Appendix A. Not&on 6,(t): the power angle of the ith generator, m radian; w,(t): the

relative speed of the ith generator, in rad/s: P,,o. the mechanical mput power, in p.u., which is constant; P,,(t): the electrical power. in p.u.; wO: the synchronous machme speed, m rad/s; co0 = 211f0; D,: the per unit damping constant; H,: the inertia _ constant in seconds; Eb, (t): the transient EMF in the quadrature axis of the lth generator, in p.u.; E,,(t): the EMF in the quadrature axis. m p.u.: E,,(t): the equivalent EMF in the excitation coil, in p.u.: T’,,,; the direct axis transient short circuit time constant, in second; x,,#: the direct axis reactance of the ith generator, m p.u.; xL,: the direct axis transient reactance of the ith generator. in p.u.; B,,’ the ith row and jth column element of nodal suscep- tance matrix at the internal nodes after eliminating all physical buses, in p.u.; s,,(t) = &(t) - 6,(t); Q_(t): the reactive power, in p.u.; I,,(t): the excitation current, in p.u.: Id,(t): the direct axis current, m p.u.; IsAt): the quadrature axis current, in p,u., k,,: the gain of the excitation amplifier, in p.u.; udt): the input of the SCR amplifier of the ith generator, in p.u.; I_,,: the mutual reactance between the excltatlon cod and the stator coil of the ith generator, m p.u.; xTI: the transformer reactance; x,,: the transmission line reactance between the lth generator and the jth generator.

robust decentralized nonlinear controller design for multimachine power systems

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