Nonlinear state-variable-feedback excitation- and

governor-control design using decoupling theory

S.N. Singh, Ph.D.

Indexing terms: Nonlinear systems, Controllers, Simulation

Abstract: Modern microprocessor capabilities permit the control designer to consider using relativelycomplicated nonlinear control algorithms, which would have been considered impractical in the past. Thepaper presents the results of a study of the potential usefulness of nonlinear decoupling algorithms for thedesign of excitation and governor controllers for a power system using state-variable feedback. A control lawfor decoupling rotor angle and field flux is derived. For the rejection of load disturbance, the design of aservocompensator consisting of strings of integrators in the outer loop around the decoupled inner loop isproposed. The closed-loop system is shown to be asymptotically stable. The system can be transferred to anew operating condition corresponding to any desired terminal voltage Vt and tie-line power /"tie- Thesimulation results using a first-order compensator show that system asymptotically tracks the desired Vt and/"tie ur>der unknown piecewise constant disturbances. Results show good transient and steady-state responsesin rotor angle, field flux and frequency. Steady-state errors in Vt and Pfte owing to parameter uncertaintyare reduced to zero by changing the command inputs using an external loop. The effect of stochastic load onthe response is small.

List of principal symbols 1 Introduction

5CO

v,vt

Efd

*fT T1 mi x urfxd>xq> xafi xt> xf

Hkd

ke

Tt, Tg, Te

ue,ug

Pg

P P5c> 0cx,u,ydiku,kns, kn^a(x,t),B,c(x)

idJqJf

vd,vQx'd

= rotor angle= rotor speed= synchronous speed= infinite-busbar and generator terminal

voltage= field voltage= field flux= shaft torque input, electrical torque= field resistance= reactances of the d-axis armature,

<7-axis armature, <i-a\is mutual, trans-mission line, field

= inertia constant= damping constant= exciter gain= governor, turbine, exciter time con-

stants= exciter, governor valve-actuating sig-

nals= governor output= tie-line power= load-demand variations= commanded values of8,\jjf= state, control, output vectors= decoupling parameter= feedback parameters and functions= system matrices= natural frequencies= stator currents in d-, q-axis circuits,

field current= stator flux linkages in d-, q-axis

circuits= stator voltages in d-, q-axis circuits= direct-axis transient reactance

Paper 690D, first received 29th October 1979 and in revised form3rd March 1980The author is with the Dept. de Engenharia Eletrica, CentroTecnolo'gico UFSC, Cx. P. 476, 88.000 - Floriano'polis -SC, Brazil

IEE PROCEEDINGS, Vol. 12 7, Pt. D, No. 3, MAY 1980

The control of power systems consisting of interconnectednetworks of transmission lines linking generators and loadsis an important problem. Ideally, the loads must be fed atconstant voltage and frequency at all times. In practicalterms, this means that both voltage and frequency must beheld within close tolerances. It is also necessary thatmachines do not lose synchronism following a systemfault. Random power impacts occur during the normaloperation of a power system; this added power must besupplied by the generators. Further, it is required to main-tain a scheduled power exchange over the tie-line in theinterconnected system. The controller must be designed toperform these functions.

Traditionally, the problem of design of power systemsis split into two separate problems, and the design ofexcitation-control systems and governor-control systems arecarried out independently, for simplicity. Excitation con-trollers are designed assuming constant mechanical torqueinput for the regulation of terminal voltage and improvinggenerators' stability limits, and the governor controlsystem is designed assuming constant flux linkage forpower-frequency regulation. Under transient conditions,the excitation controller helps voltage recovery after asystem fault and improves transient stability. On the otherhand, the governor control system helps in reducingdeviations in frequency and tie-line power flow under loadvariation. Of course, an integrated excitation and governorcontroller design based on the complete model is moreeffective in improving the performance of a power system.However, increased order and nonlinearity pose difficultiesin designing an integrated controller.

The problem of power-system-controller design hasattracted many researchers, and considerable work has beendone using linear dynamic system theory. These designs arebased on linear models resulting from the linearisation ofthe nonlinear system about the steady-state operatingpoint. For large perturbations of state variables, the linear-model representation is not adequate, and there is need ofnonlinear representation of synchronous machines incontrol-system design.

131

0143-7054/80/030131 + 11 $01-50/0

Recently, some attempts have been made to design con-trollers for the nonlinear models of power systems.13"20

Using a dynamic-programming approach, an excitation andgovernor controller was designed.13 A quasilinearisationtechnique was used to obtain a nonlinear excitation con-troller.14 Wilson et al.16 obtained a nonlinear output-feedback excitation controller based on nonlinear optimal-control theory and identification methods. A dynamicsensitivity approach was used to design a linear excitationand governor controller.17"18 Recently, using the Lyapunovfunction of the Lure type, nonlinear load-frequency controldesigns were presented.19'20 In this paper, nonlinear de-coupling theory is applied to the design of excitation andgovernor controllers using state-variable feedback.

Several authors have considered the problem of decoup-ling nonlinear systems.21"24 For an w-input/m-outputmultivariable system, application of decoupling theory givesrise to m single-input/single-output systems. For eachsingle-input/single-output system it is relatively easy tochoose feedback parameters to obtain desirable responses.In the closed-loop decoupled system, the j'th output isindependently controlled by the /th external input.

Our objective in this paper is to design a nonlinearintegrated excitation and governor control system so thatthe closed-loop system is stable in a large region in statespace, and asymptotically tracks the nominal terminalvoltage, frequency, and tie-line power flow under loadand parameter variations. Although nonlinear decouplingtheory is applicable to multimachine systems, forsimplicity the power system model considered in thispaper consists of a single synchronous machine feedinginto an infinite busbar. The usefulness of decoupling theoryis critically dependent on the choice of controlled,decoupled outputs. In this study, rotor angle 5 and fieldflux iif are chosen as the controlled outputs. Indeed, 5 and\pf are the basic variables which are directly influenced bythe mechanical input torque to the synchronous machineand the field voltage, respectively. In stability analysis, 5 isassumed to be an important variable. The controlled inputsare the governor-valve and exciter-actuating signals.

The basic approach taken in this paper is to decouple themechanical variable 6 and electrical signal yfyf. Applicationof decoupling theory gives rise to two single-input/single-output subsystems, such that 5 is the output of one sub-system and iif of the other. Further, responses 5 and \pf areindependently controllable by the command inputs of therespective subsystems. The decoupled control system hassome interesting features. It will be seen that, by choosingcertain feedback parameters, desirable responses in 5 and\pf are obtained easily. The response characteristics such asrise time, overshoot, and settling time of 6 and \frf can beindependently modified. Further, load fluctuations do nothave any effect on \pf. \pf therefore stays at its nominalvalue even when the load is changing.

Decoupling-control law is computed on the assumptionthat the load demand and parameters are exactly known.When load demand and parameters vary, the system doesnot remain perfectly decoupled. In such a case, steady-stateerrors in 5 and \pf are observed. Recently, for linearsystems, robust servomechanism theory has been used fordisturbance rejection in linear decoupled systems.25"26 Theservocompensator consists of strings of integrators drivenby error signals. Motivated by this approach, a servo-compensator in an outer loop around the inner decoupledsystem will be designed so that the system will track Vt and

P t i e under load variations. Even when the parameters vary,the servocompensator accomplishes tracking of nominal 5and \pf. However, this causes small steady-state errors in Vt

and Ptie, since these parameters affect the equationsrelating Vt and Ptie to 5 and \jjf. This difficulty could havebeen avoided by decoupling Vt and Ptie. However, thecontrol law, which decouples Vt and Ptie, is very complex,since Vt and Pt i e are complicated nonlinear functions of 5and \pf. Instead, in this paper we shall design an externalloop to change the command inputs to reduce the steady-state errors to zero when parameters vary.

The organisation of the paper is as follows. Section2 presents the mathematical model of the system. Adecoupling-control law is derived in Section 3. In Section 4,a servocompensator is designed. Section 5 presents theasymptotic behaviour of the closed-loop system and thedesign of an external loop to be used under parameter vari-ations. Section 6 presents the results of computer simu-lation.

2 Problem formulaton

The system model used as a basis for the development ofthe present paper is shown in Fig. 1. It comprises a singlesynchronous machine feeding into an infinite bus, a prime-mover representation and an excitation system. Theconventional control loops have not been retained, sincethey are not necessary for the present study. The generatoris represented by a nonlinear third-order model based onPark's equations, with 5, 5 and \pf as the three state vari-ables.7 However, this simplified model is in no wayessential, and is used only to make the example moretractable.

exciter

Fig. 1 Power-system configuration

The nonlinear equations representing the machine, trans-mission line and transformers in terms of Park's d-q axes aregiven in Appendix 10.1. In eqn. 27, Pv(t) and Pv{i) repre-sent known or predicted load-demand variation andunknown fluctuation in load demand, respectively. Usingeqns. 27 and 28, one obtains the variables if, id,iq, &d and\j/Q as functions of the state variables 5 and tyf, and theseare substituted in the differential equations for 6 and \pf toobtain the state equations16

5 = -p28-p3\l/fSin8+p4(Tm-Pv-pv)

\jjf = p5 cos 5 - p6 \jjf + p1Efd0)

where

-x'd)/(4H(xt + x'd)(xt +xq))Pi =

P2 =

p3 = co0 Vxaf/(2Hxf(xt + x'd))

132 IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980

Ps = UorfVxaf/(Xf(xt+x'd))

Pe = uorf(xt + xd)/(xf(xt + x'd))

Pi = l (2)

It is assumed that the turbine dynamics can be representedby a first-order system with time constant Tt, and thegovernor, similarly, with time constant Tg.

20 The equationsrepresenting the turbine and governor are

~ T(3)

The excitation system is assumed to be of first-orderform7'28

kPut (4)

The load Pv(t) changes continuously about its mean value.The control of rapidly varying load Pv(t) may involve largeexpenditure of control energy and severe stress on turbine-generator parts.8 As such, the decoupling-control law willbe derived for a simplified model of the power system byassuming the unknown fluctuation Pv(f) = 0.

The set of equations for the simplified model of thepower system obtained from eqns. 1, 3 and 4 by settingPv(t) = 0 can be written in vector form:

8

cb

hEfd

t

CO

Pi sin28'—p2u>—p3\Jjfsm8

ps cos 5 -

-Efd/Te

(-Tm+Pg)/Tt

-Pg/Tg

0

0

0

0

0

\\Tg

0

0

0

Ke/Te

0

0

(5a)

£a(x,t)+Bu(t)

where state vector x = (5, co, \j/f, Efd, Tm, Pg)T, and

control vector u = (ug, ue)T, and T denotes transpose.

Also, a(x, t) = (ax(x), a2(x, t),. . . , a6(x))T. The outputvector to be controlled is given by

y{t) = (5b)

We are interested in deriving a nonlinear state variablefeedback-decoupling-control law of the form

= f(x, t) , t)v(t) (6)

where v = (z>x, v7 )T is a 2-vector external input. In the

closed-loop decoupled system of eqns. 5 and 6, it is desiredthat external inputs vx and v2 independently control 5 and\pf, respectively. A summary of the important results fromdecoupling theory is given in Appendix 10.2.

The following features of the decoupling-control schemeare desirable:

(a) The closed-loop system should be asymptoticallystable for large perturbations of intial state.

(b) The transient response should be good.(c) The controller should be able to transfer the system

to any operating point corresponding to any desiredmachine terminal voltage and Pt i e .

(d) The frequency, Vt and-P^e deviations should tend tozero under unknown variations in load and parameters.

3 Decoupling-control synthesis

3.1 Existence of decoupling-control law

In this Section, we shall obtain conditions for the existenceof state-variable decoupling-control law to control 5 andi//f. The decoupling-control law is given by eqn. 33.

The expressions for La(cx),. . . , Z,2(cx), La(c2) andLl(c2) are given in Appendix 10.3. By the definition ofindex dt (see Appendix 10.2), we have that dx = 3 andd2 = 1. Using these values in eqn. 31, we get

B*(x) =, sin 8/Te

PikelTe

(7)

The matrix B*(x) is nonsingular for each xeR6, since itsdeterminant A, given by

A = P^P-,ke/(TtTgTe) (8)

is nonzero. Therefore, the necessary and sufficient con-dition for the existence of the decoupling-control law issatisfied. It will be useful to compute the inverse of B*(x)to construct the decoupling-control law, as follows:

TtTg/p4 P3TgTt sin 5/

0 Te/(Plke)(9)

3.2 Control law and closed-loop system

We shall now present a decoupling-control law using eqn.33 to control 5 and \pf. Let us select functions ht whichdetermine the properties of the output responses and thecontrol law, as shown in Appendix 10.2, as follows:

hx =

A:n6(5,5<1\.. . ,6(3 )) (10)

IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980 133

where 5 ( ; ) = d;(5)/df;. Then, the resulting expression forthe control law u(t), using eqns. 10 and 33, is

U(t) = -B*~l(x)Ll(c3)

A:135(2)

+ B*~1(x)Av(t)

kn6(8, 5 ( 1 ) , . . . , 5 ( 3 ) )

(11)

The control law eqn. 11 is an explicit function of variousparameters of the power systems. By the proper choice offeedback parameters ky and nonlinear feedback functionsin control law eqn. 11, desirable responses in 5 and \\jf areobtained. Shortly, we shall describe some methods forselecting these parameters and nonlinear functions. In viewof eqn. 32a we have that

/ = 1,2,3

= La{c2){x,t) (12)

To represent control law eqn. 11 as a function of statevariables, we substitute for the derivatives of 5 and \pf fromeqn. 12 into eqn. 11.

In view of eqn. 34, the derivatives of the outputvariables along the trajectory of the closed-loop systemeqns. 5 and 11 are given by

5 ( 4 ) A:135(2 ) kn8

Here, A in eqn. 11 has been taken as the identity matrix.It is obvious from eqn. 13 that command inputs vx and

v2 independently control 5 and 4/f, respectively. Eqn. 13represents two single-input/single-output systems. Theconstant parameters ktj in eqn. 13 are chosen to providelinear dynamic properties for the decoupled closed-loopresponses in 5 and \\jf. The capabilities of choosing thenonlinear feedback functions kn$ and kn^f does provideadded design flexibility. For each of the single-input/single-output systems in eqn. 13, the approach of Bass andWeber27 may be used to obtain the constant parametersand nonlinear feedback functions by minimising a perform-ance index of quadratic and higher-order terms of statevariables of each subsystem. As shown by Bass andWeber,27 such nonlinear feedback can be useful in satisfyinginequality constraints on the state variables of each sub-system in eqn. 13.

For simplicity, it is assumed that kn8 =kn^ = 0 incontrol law eqn. 11. Later, it is shown by simulation that,even without nonlinear feedback functions kn$ and kn^ ineqn. 11, desirable responses for 6 and \pf are obtained.

The linear decoupled transfer functions correspondingto eqn. 13 are of the form

S(s) =(s2

v2(s)

ul2)

(14)

where s denotes a Laplace transform variable. In view ofeqn. 13,

= ^

kl2 =

kn =

k22 —

k21 = (15)

There is little difficulty in choosing the parameters ineqn. 14 to obtain desirable responses for the two outputs.The closed-loop system is shown in Fig. 2.

4 Outer-loop design

The decoupling-control law has been derived for the simpli-fied model of the power system in which the unknown loadPv = 0. In practice, the system has unknown fluctuations inthe load demand Pv and parameter variations duringoperation of the power system. Under such conditions,eqn. 13 is no longer valid, and responses 6 and \jjf may havesmall coupling when the nominal controller (eqn. 13) isused for the actual model (eqn. 1). The load disturbance Pv

and parameter variations may cause the system to settledown with undesirable steady-state errors in 5 and \pf.

Rederiving L^ici) and Ll(c2) to be used in eqn. 32 forthe actual model (eqn. 1), it can be shown that nonzero^,and parameter variations contribute additive terms Zi (x, Pv)and z2(x) in eqns. 13a and b, respectively, when eqn. 11 issubstituted in eqn. 32. However, when there is no par-ameter variation, for nonzero Pv, zx =£0 and z2 = 0 ineqn. 13. If, for the admissible values of x(t) and Pv, thedisturbance terms zx and z2 can be modelled by a lineardynamic system, linear decoupled servomechanismtheory25"26 is directly applicable to design the servo-compensator in the outer loop for each single-input/single-output subsystem (eqn. 13). For example, if zi can begenerated by a pth-order differential equation, theservocompensator is a string of p integrators. In general,the modelling of zt by linear systems is not obvious. Insuch a case, first a lower-order compensator is designed and,if the response is not satisfactory for admissible Pv andparameter variations, then the order of the compensator isincreased.

We shall design a servocompensator of first order foreach of the decoupled subsystems (eqns. 13a and b). Theextension of this procedure to the design of servocompen-sators of higher-order for the rejection of disturbanceinputs which are polynomial functions of time is straight-forward. Later, it will be shown by digital simulation thatdesirable responses with the derived compensator of first

134 IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MAY 1980

order are obtained. Following the procedure of Reference25, we obtain a compensator of the form

z>i = klo(8c-8)

v2 = k20(\pc - \Jjf

(16)

where 5C and \pc are the desired values of 5 and \}jf, respect-ively. In view of eqn. 16, we note that the control signalsVi and v2 are generated by integrating the error signals<?6 = fcio(5c ~

5 ) a n d e* ~ ^ o O c - ^/)-The parameters ktj are chosen so that the system of

eqns. 13 and 16 is stable. The linear transfer functions arenow given by

8c(s)

(s 2J2con2s + co2j2)

(s + X2)(s2(17)

Comparing the characteristic- polynomials of the system ofeqns. 13 and 16 with that of eqn. 17 one obtains newvalues of feedback gains fcy(new) in control law eqn. 11.The new values of these parameters, as functions of the oldvalues given in eqn. 15, can be calculated as follows:

k20 =k10 = \ifcn

fry (new) =

k14 (new) = k14 (old) + Xx

k21 (new) = k2l (old) + X2A;22 (old)

2̂2 (new) = fc22(old) + X2

/ = 1,2,3

(18)

This method of design can be easily extended when theservocompensator consists of strings of integrators. We haveobtained a linear servocompensator (eqn. 16) for con-trolling 5 and \jjf. In general, to obtain different responsecharacteristics we may choose even nonlinear error

loop is active afterparameter changes

functions giving

v2 = (195)

where vir, i= 1,2, are reference inputs.All that we need is that the system of eqns. 13 and 19

should be asymptotically stable. If the tie-line power isavailable for feedback, we may even choose

tie

= — ai sin 25 + a2 \pf sin 5 (20)

since the response i//̂ is decoupled, after initial transienti//f->- i//c. The values of kti may be obtained by using a.Lure-type Lyapunov function19 for the asymptoticstability of eqns. 13a, 19a and 20 with \pf=\pc- Thus,decoupling control enables us to carry out the stabilityanalysis of response 5 independently of other variables. Theresponse of the closed-loop system will be different for eachchosen error function. In the case of eqn. 20,Pt i e -»• vXr ast-*°°, under varying loads.

Later, by digital simulation, it will be shown that theouter loop of the form of eqn. 16 does give desirableresponses and therefore nonlinear outer loops of the form ofeqn. 19 will not be treated in this paper.

5 Asymptotic response of closed-loop system

It is of interest to investigate the response of the closed-loop system of eqns 5, 11 and 16 when constant externalinputs 5C and i//c have been applied, and Pv{^)—Pvo> a

constant. We note that the parameters k{j have been chosenso that eqns. 13 and 16 are asymptotically stable, and8(t)-+8c and \}/f(i)-* i//c as t->°°. Using eqn. 5a andsetting the derivatives of 8 and \jjf to zero, one obtains fort -> 00

5* = 5,

co* = 0

1 m

P* =r8

( -P i sin 26C + p3 \pc sin 5C + p^

(-ps cos5c+p6i//c)/p7

(21)

outer loop

decoupled inner loop

V2G(x)

systemequations

I(P «. p

L.f(x)

decoupledoutput

.Jouter loop

Fig. 2 Closed-loop decoupled system including servocompensatorand external loop used under parameter variations

IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MAY 1980 135

where x* = (5*, u>*, E*d, T^,Pg) is the equilibrium state.Thus, the closed-loop system of eqns. 5, 11 and 16 isasymptotically stable, and x(t) -> JC* as t -> °°. Although theabove analysis assumes that Pv(t) = 0, by digital simulationit will be shown that the system remains stable even when

ftIt appears from the above analysis that the closed-loop

system is globally asymptotically stable. This will bepossible when there is no limit on the maximum turbineoutput and the field voltage. With constraints on themagnitudes of Tm and Efd, the closed-loop system has afinite domain of stability. This domain of stability isenlarged if the limits on Tm and Efd are raised.

The terminal voltage and the tie-line power are given by7

V, =Vx'd cos 5xt + x'd)xf

2\ 1/2

= ~Pi sin 25 + p2 \}/f sin 5 (22)

As 5 and \}/f converge asymptotically to 6C and i//c, Vt andP t i e

a^o attain constant values. For a given pair (V*,P*ie),eqn. 22 has a solution (5*, i//*). Therefore, we can transferthe system from any initial condition to the desiredequilibrium state corresponding to the pair (V*, P*ie) byapplying external inputs 5C = 5* and i//c = \jj}.

It should be noted that, when some of the parameters ofthe power system change, Vt and P t i e may have smallsteady-state errors, since command inputs 5C and \pc havenot changed from their nominal values, and 5 -> 5C and^f^-^c- Since 5 and \pf are decoupled and independentlycontrollable, the steady-state errors in Vt and Ptie due toparameter variations can be reduced to zero by giving smallchanges in the command inputs 5C and \pc. Since Vt andPtie are functions of iif and 5, using eqn. 22, the differen-tials of Vt and P t i e , when 5 and \pf are perturbed, can bewritten as

dVt = {(V2x2q sin 25/(2Vt(xt + xq)

2))

-(LVx'd sinS/F,)} d5 + (Lxafxt/(xfVt))d^f (23)

dPtie = {(Pi tf cos 5 - 2pi cos 25)A44} d5

+ (p3 sin 8/A4)d\pf

where

L = {Vx'd cos 5 + {xafx^flxf)}l{xt +x'df

and the coefficients of d5 and di// in eqn. 23 are evaluated at5C, \pc and the nominal values of the parameters.

Eqn. 23 can be written in the form

dVt= Ac

d5(24)

which can be solved for dS and d\pf to yield

d5= A:

dVt(25)

Let

(AVt)S8 = V?-(Vt)s

136

and

(APtie)ss = P*ie ~ (Aie)ss

where V* and P*ie are the desired values corresponding to5C and i//c and (vt)ss and (Ptie)ss denote the steady-statevalues of Vt and Pt i e when the parameters have changedfrom their nominal values. The desirable changes in thecommand inputs 6C and i//c may be obtained from thesolution of

AS, Xvt)ss

[Ptiel(26)

Suppose (Ptie)ss and (Vt)8S are measured at some instantsth i= \,2,. .. . At instants th incremental pulses aregenerated according to eqn. 26. Then, the steady-stateerrors in P t i e and Vt due to parameter variations can bereduced to zero by changing 5c and \pc at the instant tk byft ft

X A5c(r.) and 2 A(//C(f,), respectively. The dotted loop ini = l i = l

in Fig. 2 is used under parameter variations. Simulationresults indicate that, after application of two pulses, theerrors in P t i e and Vt become insignificant.

6 Simulation results

In this Section, we present the results of digital simulationto evaluate the performance of the designed controller. Theresponses are presented for the closed-loop system of eqns.1,11 and 16. For comparison, some of the responses of theinner decoupled closed-loop system of eqns. 1 and 11 arealso shown as dotted trajectories in the Figures. The simu-lation studies are presented for the following values of theparameters:

(a) System with the outer loop

cjnl = 9, con2 = 9-5, o;n3 = 5,

?i = fa = f3 = 0-7, Xt = 10, X2 = 10,

Pv = 0

(b) Decoupled inner loop

conl = 9, ojn2 = 9-5, wn3 = 4,

fi = ?2 = fa = 0-7, Pv = 0

These values are selected so that desirable fast responseswithout excessive requirements in Tm and£yd are obtained.

6.1 Response to perturbation in state

Fig. 3 shows the response of the closed-loop system toperturbation in the initial state. The command inputs are5C = 72° and i//c = 0-8228, giving the equilibrium conditionJC* =(72°, 0, 0-8228, 2-06, 0-90147, 0-90147)T, and thecorresponding Vt and P*ie are 0-9746 and 0-90147,respectively. Fig. 3 shows that the system returns rapidlyto the equilibrium state from the perturbed condition of5 (0) = 108°, \Jjf(O) = 0-4 and 5 (0) = 200° s"1. Themaximum variation in Tm and Efd are 1-12 and 5-7,respectively. Extensive simulation results for variousperturbed initial states showed that well-damped responsesin 5, \}jf and 5 are obtained. The response of the system canbe made faster by increasing the natural frequencies coni.However, this requires larger values of Tm and Efd.

IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980

6.2 Control under unknown piecewise constant loadvariations

Fig. 4 shows the response of the system under the action ofunknown piecewise constant load variation Pv(t). The valueof Pv(t) is 01 for 0 < t < 2 and 4 < t < 6, and is zero for2 < t < 4. The command inputs are 5C = 66° and \pc =0-8512, which correspond to the equilibrium conditionx* =(66°, 0, 0-8512, 1-8983, 0-8, 0-8)T and Vt =0-985and P^e — 0-8. At each instant of load variation, transientbegins and the system returns to the desired equilibriumstate x* in about a second. Fig. 4 shows that, for thesystem with outer-loop, errors in P t i e , Vt and 5 are nearlyzero in about a second. Only small variations in Tm

and Efd are required for keeping \pf decoupled from5, since, even when Pv =£0, z2 = 0 and eqn. 13b remains

10

0-9

0-8

0-7

. 0 - 6D

0A

03

0-2

0-1

0

130

120

110

r 100

90

$ 8 0

^ 6 0

50

A0

30

20

L 10

_

1-5

10

Q . -

0 5

2time, s

P.:.

2time, s

b

valid. \pf does not vary from its steady-state value of0-8512. The maximum deviation in terminal voltage wasless than 0-002. The Figure shows that the rotor angle 5of the closed-loop system without the servocompensatorattains a steady-state value of 55-5° for Pv = 0 1 . For thesystem without the compensator, a steady-state error of0-24 in .Ptie was observed. Thus, the system with theservocompensator performs better.

6.3 Response to stochastic load

Fig. 5 shows the response of the system to random fluctu-ations in Pv. Pv was generated by using a first-order filterexcited by Gaussian white noise. The command inputs andx(0) are as for Fig. 4. Maximum variations in variables 5and Vt were 13° s'1 and 0001, respectively. The Figureshows that Ptie varies about its desired value of 0-8. The\pf remains at its steady-state value.

6.4 Transfer of equilibrium state

Command inputs of 5C = 66° and i//c = 0-8512, corre-sponding to V\ = 0-985 and Pt*e = 0-8, were applied.Initially, the system was delivering i^g = 090147 atVt = 0-9746, corresponding to 5C = 72° and \pc = 0-8228.Well-damped responses with only small control require-ments were obtained. The results are not shown here.

6.5 Pa ram e ter sensitivity

The control law is an explicit function of various parametersof the power system. Thus, for any known changes in theseparameters, the exact decoupling-control law can beobtained by using the new values of the perturbed par-ameters. However, in practice, some of the parameters arenot known precisely. Simulation results were obtained foruncertainties of 10% in transmission-line reactance xt and5% in infinite-busbar voltage V. The system was initiallyoperating with command inputs Sc = 72° and \pc = 0-8228,corresponding to desired values of .Pt*ie = 0-9014 and V* =

200

-200

2time.

Fig. 3 Response to perturbations in initial states

a 8 and \jjfb Tm, Vt and P t i ec 5 and Efd

IEEPROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980 137

0-9746. Fig. 6 shows the response of the system to 10%perturbation in*,. Control law shows only little sensitivity.The steady-state errors in Vt and Ptie are —0-003 and— 0-04, respectively. Rotor angle and \}/f attain the com-manded values of 5C = 72° and \pc = 0-8228. For 5%

02

01

10 r

0-9

0-8

0 7

0-6

0-5

40

20

en 0<V

•io"

-20

-40

-50

'60

50

-

\

I\ y\Vi tit i\\/

/

ri

1

P,ie

6 _,

/ ' \/ ' \

• V ,; ^ \iii

ii

K

\ /\ /\ / /\l /\» /\l /\\ 1V

_ •tie

time, sa

0 1 3 4time, s

b

perturbation in the infinite-busbar voltage, steady-stateerrors of — 0-028and — 0047 were observedin.Ptie and Vt,respectively. These steady-state errors were reduced to zero

Li.

20

18

1-6

2]-2UJt-E10

0-8

0-6

0-4

0 2

V

--

VV

3time, s

Fig. 4 Control under unknown piecewise disturbance loadsa 6, Ptje and Pvb Frequency deviationc Efd and Tm

Fig. 5 Response to stochastic loads

a Pv, 8 and Ptie

b Frequency deviation

80

76

64

60L

10

0-8

d. 0-6

0-2

1time, s

Fig. 6 Response to 10% parameter uncertainty in Xf: Vt, Pt ie,and 6

138 IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980

by changing 6C and \pc using the dotted loop of Fig. 2.It was observed that the errors in Vt and Ptie were lessthan 0-003 in magnitude after the application of one pulse,using the dotted external loop.

The results of the simulation study can be summarisedas follows:

(a) The closed-loop system rapidly attains its equilibriumstate for large perturbations in state.

(b) The rotor angle and field flux are independently con-trolled. The system state can be transferred to a new statecorresponding to any desired terminal voltage and tie-linepower.

(c) The control magnitude requirements are moderate.(d) The system with first-order servocompensator

asymptotically tracks the specified terminal voltage and tie-line power flow under piecewise-constant disturbances.

(e) Control law has little sensitivity to parameter vari-ations, and steady-state errors in Vt and Ptie, caused byuncertainty in parameters, can be wiped out by giving smallchanges in command inputs.

(/) The effect of stochastic disturbances on the responseis small, and the response \}jf remains unaffected.

7 Concluding remarks

We have carried out an exploratory study of the applic-ability of nonlinear decoupling theory. The main advantageof the application of decoupling theory is that the compli-cated nonlinear w-input/ra-output system is decoupled intom single-input/single-output subsystems. Then, for eachdecoupled subsystem, there is little difficulty in choosingthe feedback parameters and designing the servocompensatorto get desirable responses. The controlled variables chosenfor this study were 5 and \pf, using governor and excitationcontrol signals. In the closed-loop system, 6 and \}/fwere governed by fourth- and second-order differentialequations, respectively. The closed-loop system was shownto be stable for large perturbations in the initial state.Desirable fast and well-damped responses in 5 and 1/7 wereobtained. It was also shown that the system asymptoticallytracks desired Vt and Ptie under piecewise constant dis-turbance loads. Control law was shown to have only littlesensitivity to uncertainty in infinite-busbar voltage and trans-mission-line reactance. The small steady-state errors in Vt

and P t i e were eliminated by changing the command inputsby a small amount using the external loop.

There are several important problems which need to begiven consideration. Controller synthesis using only outputfeedback is an important area of further research. It isdesirable that the state estimators use only locally measuredstate variables. Determination of the domain of stability ofthe closed-loop decoupled system is also useful.

8 Acknowledgments

The author is thankful to the authorities of the UniversidadeFederal de Santa Maria, RS, for their support in completingthis work.

9 References

1 ANDERSON, J.H.: 'The control of a synchronous machine usingoptimal control theory', Proc. Inst. Elect. Electronics Engrs.,1971, 59, pp. 25-35

2 YU, Y.N., VONGSURIYA, K., and WEDMAN, L.N.: 'Appli-cation of an optimal control theory to a power system', IEEETrans., 1970, PAS-89, pp. 55-62

3 FOSHA, C.E., and ELGERD, O.I.: 'The megawatt frequencycontrol problem: a new approach via optimal control theory",ibid., 1970, PAS-89, pp. 563-578

* 4 DAVISON, E.J., and RAU, N.S.: 'The optimal output feedbackcontrol of a synchronous machine', ibid., 1971, PAS-90, pp.2123-2134

5 MOUSSA, H.A.M., and YU, Y.M.: 'Optimal power systemstabilization through excitation and/or governor control', ibid.,1972, PAS-91,pp. 1166-1174

6 MINIESY, S.M., and BOHN, E.V.: 'Optimum load-frequencycontinuous control with unknown deterministic power demand',ibid., 1972, PAS-91, pp. 1910-1915

7 ANDERSON, J.H., and RAINA. V.M.: 'Power system excitationand governor design using optimal control theory', Int. J.Control, 1972, 12, pp. 289-308

8 CALVIC, M.: 'Linear regulator design for a load and frequencycontrol', IEEE Trans., 1972, PAS-91, pp. 2271-2281

9 KWATNY, H.G., KALNITSKY, K.C., and BHATT, A.: 'Anoptimal tracking approach to load-frequency control', ibid.,1975, PAS-94, pp. 1635-1643

10 RAINA, V.M., ANDERSON, J.H. WILSON, W.J. andQUINT ANA, V.H.: 'Optimal output feedback control of powersystems with high-speed excitation systems', ibid., 1976, PAS-95,pp. 677-689

11 VENKATESWARLU, K., and MAHALANABIS, A.K.: 'Designof decentralised load-frequency regulators', Proc. IEE., 1977,124, (9), pp. 817-820

12 DAVISON, E.J., and TRIPATHI, N.K.: 'The optimal decentra-lized control of a large power system: load and frequencycontrol', IEEE Trans., 1978, AC-23, pp. 312-325

13 IYER, S.N., and CORY, B.J.: 'Optimal control of a turbo-alternator including exciter and governor', ibid., 1971, PAS-90,pp. 2142-2148

14 MUKHOPADHYAY, B.K., and MALIK, O.P.: 'Optimal controlof synchronous-machine excitation by quasilinearisation tech-niques', Proc. IEE, 1972, 119, (1), pp. 91-98

15 ELMETWALLY, M.M., RAO, N.D., and MALIK, O.P.: 'Experi-mental results on the implementation of an optimal control forsynchronous machines', IEEE Trans., 1975, PAS-94, pp. 1192—1200

16 WILSON, W.J., RAINA, V.M., and ANDERSON, J.H.: 'Non-linear output feedback excitation controller design based onnonlinear optimal control and identification methods'. Presentedat the IEEE PES summer meeting, Portland, 1976.

17 DANIELS, A.R., LEE, Y.B., and PAL, M.K.: 'Nonlinear power-system optimisation using dynamic sensitivity analysis', Proc.IEE., 1976, 123, (4), pp. 365-370

18 DANIELS, A.R., LEE, Y.B., and PAL, M.K.: 'Combined sub-optimal excitation control and governing of a.c. turbogeneratorsusing dynamic sensitivity analysis', ibid., 1977, 124, (5), pp.473-478

19 DORAISWAMI, R., and GONDAR, U.C.M.: 'The design andstability of a multiarea load-frequency control under varyingloads'. Presented at the IEEE PES winter meeting, New York,1978

20 DORAISWAMI, R.: 'A nonlinear load-frequency control design',IEEE Trans., 1978, PAS-97, pp. 1278-1284

21 PORTER, W.A.: 'Diagonalization and inverses for nonlinearsystems', Int. J. Control, 1970, 11, pp. 67-70

22 SINGH, S.N., and RUGH, W.J.: 'Decoupling in a class of non-linear systems by state variable feedback', Trans. ASME Ser. G.J. Dyn. Syst. Meas. & Control, 1972, 94, pp. 323-329

23 MAJUMDAR, A.K., and CHOUDHURY, A.K.: 'On decouplingof non-linear systems', Int. J. Control, 1972, 16, pp. 705-718

24 TOKUMARU, H., and IWAI, Z.: 'Non-interacting control ofmultivariable systems', Int. J. Control, 1972, 16, pp. 945-958

25 DAVISON, E.J.: 'The output control of linear time-invariantmultivariable systems with unmeasurable arbitrary disturbances',IEEE Trans., 1972, AC-17, pp. 621-630

26 WANG, S.H., and DORATO, P.: 'Robust decoupled servo-mechanism theory for linear multivariable systems'. Presented at12th Asilomar Conference on Circuits, Systems and Computers1978

27 BASS, R.W., and WEBER, R.F.: 'Optimal nonlinear feedbackderived from quartic and higher order performance criteria',IEEE Trans., 1966, AC-11, pp. 448-454

28 MELLO, F.P. de, and CONCORDIA, C: 'Concepts of synchron-ous machine stability as affected by excitation control', ibid.,1969, PAS-88, pp. 316-329

IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980 139

10 Appendixes

10.1 Equations of nonlinear system model

10.1.1 The machine

5 = (uo/2H)(Tm ~TU-PV -Pv) -

jjf = u)0(vf-rfif)

v d = - 0 «

vQ = </><*

^d = -xdid +xafif

&f ~ ~~ xafid ~^~ xfif

V? = vd+vl

10.1.2 Transmission line and transformers

vd = —xtiq + Fsin 5

vQ = xtid + Fcos5

(27)

(28)

10.1.3 System data

H = 3-82 xd = 1-75 xQ = 1-68 xaf = 1-562

xf = 1-665 rf = 0-0012 xt = 0-3

V = 1 JC^ = 0-2846 / = 60

Te = 0-04 r g = 008 Tt = 0-3

70.2 Definitions and decoupling theory

Consider a plant of the form

x(t) = a(x,t)+B(x,t)u(t)

y(t) = c(x, t)

kd = 0006

(29)

where state vector x(t)GRn, input vector u(f)ERm,output vector y(t)eRm and (JC, r)G£) = Xx [r0) U] c

^ " x [ r o , f i ] .In decoupling control synthesis, one selects a control law

of the form

"(0 = /(*, 0 + G(x, t)v{t) (30)

such that, in the closed-loop system of eqns. 29 and 30 forany m-vector external input v(i), the ith output y((t) isindependent of Vj(t), j^i, i, / = 1, . . . , m in D. It isassumed that G is invertible on D.

The following definitions are basic to the decouplingproblem:22

_ d_a ' dt

Li = LJti-1)

a 1— (ct(xtt))\a(x,t)

(31)

where the arguments x and t of the operator La have beensuppressed in eqn. 31, and dt is the least nonnegativeinteger such that

—px B(x, for each (x, t) G D

Using the definition of La, it is easy to show that thederivatives of y((t) along the trajectory of the closed-loopsystem are given by

yV\t) = i = 0 , 1 , . . . , (32a)

= L^ic,) -Lad'(c.-)|i?(*,O[/(*,r)

+ G(x, t)v(t)] (32b)

where yjj\t) denotes the ;th derivative of >",(0- Thecontrol u has been replaced by eqn. 30 in eqn. 32ZJ.

One can show that eqn. 30 decouples eqn. 29 if andonly if the m x m matrix B*(x, t) is invertible for all(x, t)&D.21~2A Moreover, the decoupling-control law canbe chosen, of the form

,• •• , y(dm>)

+ B*-1(x,t)/\v(t) (33)

140

where ht is an arbitrary nonlinear function of yt and itsderivatives, and A is an m x m diagnonal matrix. Using eqn.33 in eqn. 29 gives

y\*t*i> =-hi(yi,.. . , # ' > + X ^ ( 0 , i=\,...,m (34)

where X,- is the nonzero element in the / th row of A.

10.3 Control law

By definition, L°(*) = (•)• In the following, using eqn. 31successively, we shall derive expressions for L^c^lx, i),La(c2)(x, t) etc., which are useful in computing the dt

parameters and also the control law (eqn. 11). We use thematrices a(x, t), B(x) and c(x) given in eqns. 5a and 5b forthe following derivation:

La{cx)(x, t) = ( l , 0 , 0 , 0 , 0 , 0 ) a ( x , 0 = w

L2a(cx){x,t) = a2(x,t)

Ll(cx){x, t) = ~P^Pv + (2pl cos 25 -p3\pf cos 6,

- Pi, ~ Pz sin 5, 0, p 4 , 0)a(x, t)

= —pnPv + (2pi cos 25 — p3i///-cos5)co

~P2(Pi sin 25

-p2(*3-p3\J/f sin 8 +P^Tm -p*Pv)

-p3 sin5 (p5 cosS-p6\ l / f+ pnEfd)

+ P4(-Tm+Pg)/Tt (35)

Ll{cx){x, t) = - p 4 ^ + P 2 P 4 P I , + co[(-4p1 sin 25

+ P3^f sin5)cj

- 2 p , p 2 cos 25 +(p2p3tf

IEEPROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980

+ p3p5 sin2 5] +a2(2pl cos 25

-p3\pfcos5+pl)

(^Ll(Cl)(x, 0| B = (p,/TtTg-p3Plke sin 5/Te)2 5 ] + ( 2 2 5 \°x

= (0 0)

sin 5 / . \ <36)

Z,a(c2)(x, /) = c3 Using eqns. 33 and 9 gives

£a(c2)(*,0 = ( -p s s in6 ,0 , -p 6 ,p 7 ,0 ,0 ) f l (x ,0 ug = {TtTg/p,){-Hi

Using these expressions, it can be easily verified that + (p3TgTt sin 5/p4)(— Z-a(c2) ~h2(\pf, i/4!))

(37)9 . \— L'O(CI)(JC,0 5 = (0 0), / = 0,1,2

ErratumDUNNETT, R.M., and WELLS, R.: 'Improvements in steamtemperature control on a modern oil-fired power-stationboiler', IEE Proc. D, Control Theor. & AppL, 1980, 127,(1), pp. 7-12:

On page 12, column 2, eqn. 15 should read:

hSAT = 2667 - 2-38 0 - 001189 02

ETC62 D

IEE PR OCEEDINGS, Vol. 127, Pt. D, No. 3, MAY 1980 141