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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 3, NO. 2, JUNE 1995 23 1

Nonlinear Control Methods for Power Systems: A ComparisonV. Rajkumar and R. R. Mohler

Abstract-The design and performance of two nodinear con-trollers for power systems is explored in this paper. The designsare based on nonlinear discrete-time predictive control and feed-back linearizing control, consideringa simplifiednonlinear single-machine, infinite bus (SMIB) power system model equippedwith a series-capacitor controller. Simulations are presented fornonideal conditions such as bounded control, the uncertainties inthe chosen model parameters, and the presence of inter-area-typepersistent disturbances. The results indicate that superior perfor-mance can b e obtained using the nonlinear predictive controllerwhen compared with the feedback Linearizing controller and astandard linear quadratic regulator.

I. INTRODUCTIONARGE faults on the transmission system cause the elec-L rical power to collapse to a small value during the fault.This leads to an acceleration of the rotors of the synchronous

generators, due to unequal mechanical and electrical torquesacting on the shafts. Rapid detection of the fault and fast-acting circuit-breakers are normally used to isolate the faultedsections of the transmission system. In addition to the rapidisolation of faulted sections of the transmission equipment, itis generally recognized that auxiliary control action would benecessary to enhance the post-fault region of stability of thepower system.Synchronous generators can be analyzed conveniently usingParks transformation [ I ] , which results in a time-invariantnonlinear system. In the event that the post-fault systemexperiences small oscillations, simplified linear power sys-tem models such as [2] can be used with advantage foranalysis and linear controller design. For example, the widelypopular linear power-system stabilizers (PSS) [3] are usedto provide supplementary damping through excitation control,thus enhancing the dynamic stability limit. Such linear controlon the nonlinear power system generally provides asymptoticstability in a small region about the equilibrium and is thusappropriate for the dynamic stability problem, where theprimary concem is of providing damping following smalldisturbances.Nonlinear controllers are suitable for the transient stabilityproblem, by virtue of being designed to stabilize the systemfor large initial conditions (within the physical limits im-posed by the controller). We note that when the impedanceof the transmission lines are controlled, for example by aflexible ac transmission systems (FACTS) [4] device suchas a thyristor-controlled series-capacitor (TCSC), the control

Manuscript received August 4, 1993; revised August 7, 1994. Recom-mended by Associate Editor, R. Ravi. This work was supported by N SF GrantECS 9301 168, EPRI Conract RP 3573-05, and BPA Contract 94BI16785.The authors are with the Department of Electrical and Computer Engineer-ing, Oregon State University, Corvallis, OR 9733 I USA.IEEE Log Number 9410593.

action is multiplicative (product of the state and control) andthus necessitates the consideration of nonlinear models indesign for the most effective, economical use of the availablecontrol resources. In this context, nonlinear variable-structurecontrol [5] and nonlinear model-based self-tuning control[6] have been proposed for a class of transient stabilityproblems.In this paper, we illustrate the design of a nonlinear-time-series, model-based, generalized predictive controller for asimplified power system, using rotor angle as the measuredoutput and a TCSC controller. Generalized predictive con-trollers offer the advantages of being easy to implement inreal-time and allowing systematic methods of handling inputconstraints.Recently, a number of papers have addressed a new classof controllers for the power system, based on the conceptof cancellation of the nonlinear dynamics through feedback[7]-[14], using excitation or govemor control. Subject to theavailability of an accurate reference model and measurementsof the power system, the dynamics of the power systemmay be cancelled and replaced with some desired linearbehavior.This paper also examines a very simple design of a series-capacitor based feedback linearizing controller and subjectsit to three types of uncertainties: 1) bounded control, 2)uncertainties in model parameters, and 3) the presence ofunmodeled inter-area-type disturbances. The performance ofthe feedback linearizing controller is evaluated and comparedwith that of the nonlinear generalized predictive controller andan ordinary linear quadratic regulator (LQR), which are alsosubjected to similar conditions.

11. P O W E R S Y S T E M MODELS OR CONTROLLER DESIGNThe dynamics of the post-fault power system can be de-scribed by a set of nonlinear differential equations. Typically,the states can be associated with each machines armaturecurrents and rotor currents resolved through Parks equa-tions, rotor dynamics, automatic voltage regulator (AVR),and turbine-governor dynamics [151. Many controller designmethods utilize simplified models of the synchronous gener-ator with states composed of the rotor dynamics, change ofinternal voltage in the quadrature axis, and at least one stateto represent the highly nonlinear closed-loop AVR. With thelast consideration, we note that the field voltage is typically

a state variable and not an independent agent for control,except for a short while during field forcing in the eventthat the terminal voltage feedback collapses. We address twoissues regarding the simplification of power system modelsfor controller design.10634536/95$04.00 0 99 5 IEEE

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A. Considerations Regarding Voltage RegulatorsMost modern generators are equipped with AVRs. Thesedevices maintain the terminal voltage of the generator at aspecified value and in the process modulate the field voltageand hence the field current, thus supplying the required reactivepower to the load. It is well known that the AVR can injectnegative damping into the system at high power loadings,

leading power factors, and large tie-line reactance [ 2 ] . Inlinearized-analysis terms, this can result in unstable rotor modeeigenvalues at some power loadings. This phenomenon cannotbe simulated in models that neglect the AVR dynamics. An ad-hoc simulation of unstable rotor modes may be done, however,by making the mechanical damping constant negative in theswing equation.The neglect of the AVR in power system models used forfeedback control design purposes and the use of field voltageas an independent control variable as proposed in the feedbacklinearizing controllers of [7]-[ 141 cannot guarantee terminalvoltage regulation, except by adding nonstandard AVRs bythe incorporation of additional control loops in the feedbacklinearizers. But in this case, the nonlinear dynamics are notcancelled completely, since the feedback of terminal voltageresults in the injection of significant nonlinearities into thesystem.B. Considerations Regarding Infinite Bu s

The assumption of a single-machine, infinite bus (SMIB)may be valid for faults around low-inertia systems whichare connected to high-inertia external systems by long tie-lines. There is concern, however, that the SMIB-designedcontrollers may not perform well in the presence of inter-areamodes. It will thus be necessary to evaluate the SMIB-designednonlinear controllers on more realistic multimachine modelswith different oscillation frequencies. For an ad-hoc appraisalof controller performance, a simulation test may be devisedon SMIB systems, where the infinite bus voltage is modulatedin magnitude by an inter-area-type frequency, an approachtermed single-machine, quasi-infinite bus in [161.There are additional considerations with respect to neglectedturbine-governor dynamics, reactive power demand, and loadmodeling, which are not addressed in this paper for brevity.In the following sections, we design a candidate nonlinearpredictive controller and a candidate feedback linearizingcontroller for power system transient stability. The designsare based on series-capacitor control, and for simplicity ofillustration of the concepts involved and in the interest ofbrevity, use a second-order model of the power system.The candidate controllers are subjected to the testing criteriasuch as those listed above, and for comparison purposes areevaluated alongside an ordinary LQR which is also subjectedto the same test conditions.111. DISCRETE-TIMEONLINEARREDICTIVEONTROLLER

Consider a simplified power system model of the post-faultsingle machine connected to an infinite bus, represented withthe rotor speed and rotor angle, with equilibrium translated to

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 3, NO . 2. JUNE 1995

the origin, and controllable series susceptanced6-t = wbw

whereT,,, s the prime mover torque,E is the internal bus voltage of the generator,V is the infinite bus voltage,B is the series susceptance,M is the machine inertia,D is the mechanical damping factor,6 is the rotor angle in radians,w is the rotor speed in per-unit,

Wb = base speed, 377 radslsec,U is the control input (additional series susceptance),and suffix e represents equilibrium value.An input-output model can be obtained for the system

(1)-(2), by choosingY ( t ) = 6 ( t ) . (3)

Differentiating (3) twice and eliminating wd2 6~ + - - wbVEsin(6, + S)(B,+ u ) /Mdt2 (g )2- wbTm/M = 0 . (4)

Some properties of linear controllers with the linearizedinput-output model of (4) can now be illustrated, which willprovide insight into local behavior of the controllers to bedesigned.Consider a linear model of (4 ) asd 2 A 6 dA6dtz+ U- + bA 6 = C A U ( 5 )d t

where a, b , and c are appropriate linearization constants.( 5 ) by choosingA simple proportional output feedback may be designed for

AU= -KA6 (6)where K is some feedback gain constant.istic equation are obtained asSubstituting (6) in (3 , he roots of the resulting character-

Xl ,2 = ( - U f du2- 4( b + c K ) ) / 2 . (7)For normal operation of the power system, the rotor modesexhibit oscillatory damping, indicating complex conjugateroots. Clearly, the real parts of (7) are unaffected by theselection of K. Thus controller (6) cannot stabilize system

(4 ) locally, as long as u2 - 4(b + cK) < 0. Second, the rootsof ( 5 ) cannot be placed at will in the state-space, owing to thedependency of the roots on the single tunable parameter K.Now consider the controller as a PD-a proportional +derivative controller withdASd tAU z= -KIA6 - K2-

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By substituting (8) into (9 , e again check the roots of theJ(a + C K Z ) ~ - 4( b + c ~ 1 ) ) / 2 . 9)

Thus by introducing feedback of the derivative of the output,the real part and the imaginary parts of (9) can be adjustedindependently, by an appropriate choice of the gains.

closed-loop characteristic equation to yield~ 1 , 2 ( - ( a + C K ~ )

A. Discrete-Time Nonlinear Input-Output ModelSystem (4) can be discretized locally by using Eulersbackward shift approximations for the differentials, usingsufficiently small step-size h , i.e.,

Substituting (lo)-( 11) in (4) nd rearranging the time-shiftedterms, we obtain the discrete, nonlinear input-output equationfor the power system as6 k + i = a i S k + aaSk-1 + b i sin(6, + S k )

+ CI sin(& + 6 k ) u k + d (12)where

a1 = -a i /ab ; a2 = -.;/ab; bl = -6\/abc1 = -c i /ab ; d = -d /ab

andU ; = l / h 2+ D/ ( hM) ;a; = l / h 2 ; : = w b V E B , / M ;

U: = - ( 2 / h 2 + D/ ( hM ) ) ;e: = w b V E / M ;

d = -wbTm/h/l.B. Generalized P redictive Control ( GP C) Criterion

The generalized predictive controller is now based on themodel (12), by minimizing the following generalized predic-tive cost function

N NJ = (112)E + ]+ ( P l / 2 ) E ( S k + j - &+j-1)2

J = 1 3=1N+ ( ~ 2 1 2 ) u E + j - l . (13)3=1

We note that cost function (13) involves quadratic terms inthe output, the discrete rate-of-change of the output, and theinput. The rate-of-change of the output terms are included tohave derivative effect as discussed earlier for the linearizedmodel. N is the prediction horizon of the GPC.C. Predictions of the Nonlinear Discrete-Time Reference Model

The predictions of the plant output as demanded by (13)over the horizon N are highly nonlinear in the control andmay be obtained by running the time index j in (14) fromone to N . Note that when j = 1, the initial conditions forthe predictions are available as the current and immediate

past output measurements, b k , and b k - 1 . The control valuesu ~ + ~ - Ieeded for the predictions are obtained iteratively byminimizing (13). Consequently

6 k + j = alSk+j-l+ a zSk+ j - a + b l sin(8, + S k + j - l )+ ci sin(6, + 6k+j-i)uk+j-i + d . (14)D. Minimization of the Criterion Function

Criterion function (13) can now be minimized with respectto the controllers U k + j - 1 over j = 1,2 , . . ,N , to obtain Nnonlinear equations in the future controls. Consider the ithsuch equation obtained by minimizing (13) with respect touk+i-l

x [Ae+j- Ae+j-lm]) +~2ur~+i-1 0 (15)i.e., fi(u)= 0, for i = 1 , 2 , . . . ,N , where U = [Uk ,uk+l,. . ,

We define the derivative of the jth prediction of the outputwith respect to the ith control recursively as followsF o r j = 1 ,2 , .. . , N; = j , j + l , . . . , N ; S e t A i - A i p l = 0

uk+N-l] T .

when j = i ,Ai+j = CI sin(6, + 6k+j-1)when j < i ,Ai+j = 0when j > i , use (16) (causality)

whereA,+j-1 =[a1 + bi cos(Se + 6k+j-1)+ ci C O S ( & + 6 k + j - i ) u k + j - i ] . (17)

Nonlinear equations (15) can be solved numerically [17].When the controller saturates, additional considerations arenecessary to ensure that the controller remains a minimizingcontroller [18]. The control loop iS closed using the control U k ,and the whole procedure is repeated at the next time step. Wenote that this method provides a powerful method of nonlinearoutput feedback control. In general, large prediction horizonswill be necessary to assure stability. The framework developedallows the implementation of receding-horizon controllers. Inthis paper, we have considered N = 2 and have not madeany special consideration for controller saturation, to keep theprocedure simple.

IV. FEEDBACKINEARIZINGONTROLLERESIGNNonlinear systems and control have received renewed at-tention in recent years. For certain kinds of nonlinear systemswhere the control variable appears linearly, it has been shownthat subject to certain conditions being satisfied by the model,it would be possible to devise a transformation to design afeedback control law, which cancels all the dynamics of thesystem, and replace it with a desired linear dynamical system

[191, PO I .

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System (1)-(2) can be represented asd6d tdwd t controller.

pair of (25). When this controller is used on system (1)-(2),asymptotic stability is guaranteed in a small region about theorigin. While this is not suitable for the transient stabilityproblem, it would serve our purpose of comparison withthe nonlinear predictive controller and feedback linearizing= wbw= a ( w ,6) + b(6 )U

(18)-and is in a form amenable to direct feedback cancellation.

Assuming that b ( 6 ) # 0, a direct feedback linearizingVI. SIMULATION DATA, ESULTS, AN D DISCUSSIONcontrol law takes the form For easy reproduction of these results, a detailed accountof the simulation experiments and some numerical results areprovided for validation. The following simulation parametersare chosen in this study: Be = l/(X; + XI + X, - X,,),where X i = 2.0 p.u. is the direct axis transient reactance,

Xl = 0.5 p.u. is the equivalent transformer reactance, X , = 1value of the series capacitor reactance, corresponding to 50%compensation of X,. Let the capacitor reactance have a rangeof &loo% percent compensation of X,. Thus the maximumand minimum allowable incremental series susceptance is

U = [w- a ( w , 6 ) ] / b ( 6 ) (19)i.e.,U = [w-T,+VEsin(6 ,+6)B,+Dw]/VEsin(6 ,+6) . (20)

Substituting (20) n (18) esults in the equivalent system p,u- is the line reactance, and X,, = 0.5 p.u., is the equilibriumdSdwd t

- (21)(22)

dt = wbw- v /M.Now choosing the linear control law as

w = -k66 - ,w (23)system (21)-(22) can be provided with some desired linearresponse. The linear component of the control can be de-signed by solving the well-known algebraic Ricatti equation,considering the feedback-linearized plant (21)-(22) for the(A,)-pair.Remark 4.1: Controller (19) fails whenever

S = fn7r - 6,; n = 0 , 1 , 2 , . . (24)which may also be found to be the values at which thecolumns of the nonlinear controllability matrix [20] ail thelinear independence test.Remark 4.2: Controller (19) has two components-a feed-back linearizing component and a linear feedback component.The control must be restricted to the admissible control set.In the event that U ( X ) is restricted to the boundary of theadmissible control set, (19)no longer guarantees feedbacklinearization. In the transient stability problem, the initial post-fault states are typically large, and thus the control demandedby (19) to linearize the system can be very large and cancause controller saturation. In this event, the residual nonlineardynamics will not behave as the original nonlinear powersystem, but with different properties.

V. LINEARQUADRATICEGULATORA linear quadratic regulator may be designed on the nominallinearized system of (1) nd (2) round the origin. In this case

d x- A z + b ud tand the controller is chosen asU = -Kx (26)

where K is the vector of feedback gains, obtained by solvingthe algebraic Ricatti matrix equation, considering the (A ,b)

U,, = 0.06667 p.u. and umin= -0.04762 p.u. The machineinertia M = 3.25 seconds, internal bus voltage E = 1 p.u.infinite bus voltage V = 0.99 p.u. and synchronous speedWb = 377 radskec. The damping factor D is varied in eachsimulation and is listed below appropriately.All simulations to follow are done at the nominal operatingcondition, 6, = 50 degrees, w, = 0 and prime mover torqueT, = 0.252795. At a sampling (and numerical integration)time step of h = 0.01 second, and damping factorD = 0.5, thecoefficients of the model (12) re obtained as a1 = 1.99846,0.00293.a2 = -0.99846, b l = -0.00382, ~1 = -0.01147, d =

A . Effect of Bounded ControlAs mentioned earlier, the control demanded by (19) can beexcessively large and, in practical systems, would be restricted

to the control saturation values. For the above operatingconditions choosing D = 0.5, LQR state weighting matrixQ = diag(1,l) and control weight R = 1, the closed-loop feedback linearized system (21), (22)has the followingeigenvalues: Xl , 2 = -7.76173 f 7.6142, correspondingto the controller gains K = [1,49.5126]. For the sameLQR weights, the linearized closed-loop system (25)-(26)has the following eigenvalues: X l , z = -5.7785 f 7.613,corresponding to controller gains K = [.758679,48.8674].The GPC uses a weight of p1 = 100 on the rate-of-changeof output and p2 = 0 on the control. The fault is a short-circuit simulated by setting E = 0 P.u., cleared after 10 ACcycles, simulated by resetting E = 1.0p.u. Without control,the power system (1)-(2) is unstable.Fig. 1 shows the application of the feedback linearizingcontroller (19) (FBLC), the nonlinear generalized predictivecontroller (13) GPC), and the LQR (26).The nonlinear GPCyields the best damping among the controllers, in the shortestpossible time, using the least amount of control. The FBLCdemands more control (needs three positive peaks of control)to do the same task less efficiently than the LQR (needstwo positive peaks of control), as shown by the less damped

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1001 1 0.021 I'10 10Time (seconds) Time (seconds)-0BoldLine- NllPredictiveCOntKllBmkenUne- LQR contrdDoliedUne- UneadzlngControl

I .

10Time (seconds)Fig. 1.LQR . Response to 1 0 cycleshort-circuit.Effect of bounded control on nonlinear GPC, eedback linearizer, and

1 I

I I5 5 100 -0.021Time (SeCMlds) Time (seconds)0.1

a 11b+ , ~ot ted~ne-~ lnear i z ingcont ro ll ,$. _ - - - - - -I , >

g o3 I I'10-O050 'Time (seconds)

Fig. 2. Response to 10 cycleshort-circuit with nonlinear GPC,FBLC andLQR, based on imperfect reference model-20% error in line susceptance B.

swings of FBLC. The LQR neglects all nonlinearities in thesystem, and as a consequence, displays degraded performancewhen compared with the nonlinear GPC. This study brings outthe following features-i) the control input demanded by thefeedback linearizing controller (19) is greater than the controlinput demanded by LQR (26), since (19) spends considerableeffort in linearizing the system, and ii) LQR's exhibit degradedperformance on the nonlinear power system when comparedwith the nonlinear GPC.B. Imperject Reference Models

The nominal susceptance term Be is seen to incorporatethe line reactance X , , the measurement of which involvesconsiderable uncertainty. With this consideration, the plant(1)-(2) uses the Be as outlined earlier, while FBLC (19) andnonlinear GPC (13) uses a value of Be which is 20% in error,i.e., B' = 0.8Be. For appropriate comparison, the linearizedmodel (25) also uses B' = 0.8Be to design the LQR (26).

1001 I 0.021 I

I I5 5 100 -0.02'1001 Time (seconds) Time (seconds)

Bold Line- NIIPtedtcbve ContmlBroken Line - LOR ControlDonedUne - UneadzingContml!o\I 1 1 )1 ;I I 1 1 ' 11 1 1 1 5 10-0.05 Time (seconds)

Fig. 3. Response to nine cycleshort-circuit with nonlinear GPC, BL C an dLQR. Based on imperfect reference model-10 percent error in damping factorD, for unstable operating condition.

With D = 0.5, Q = diag(1, l ) ,R = 1 for the linear part of(19) and for (26), and p1 = 100,p2 = 0 for (13), the result ofa 10-cycle short-circuit is shown in Fig. 2.While all controllers stabilize the system, the offset seen inthe rotor angle and the control input, caused by the FBLC andnonlinear GPC controllers, bring out the next important featureof nonlinear controllers. If the reference model is inaccurate,the feedback linearizing controller would feedback-linearizethe system around a wrong post-fault equilibrium. Similarly,the nonlinear GPC would conduct the post-fault system to awrong equilibrium.In contrast to this, the LQR which was also designed on asimilar imperfect reference model is seen to bring the powersystem to the proper (only) post-fault equilibrium at the origin.It may be noted that the LQR exhibits poor damping comparedwith the nonlinear GPC. Similar observations can be alsomade with regard to uncertainties in the infinite-bus voltage,a fictitious far-end quantity.

C . Effect of Uncertain Parameters fo r a DynamicallyUnstable SituationBy virtue of its simplicity, model (1)-(2) can never representthe dynamic instability seen in AVR-equipped power systems.We may artificially endeavor to make (1)-(2) unstable, how-ever, by choosing the damping factor as a negative quantity. Itmay be noted that the damping factor D s a fictitious quantity,lumping the effects of mechanical and electrical damping in(1)-(2). Thus it can be determined only approximately fora practical power system. Let D = -1.91 in the model

(lt(2). Let D' = l . lD, representing a 10% error in theestimation of the damping factor, be the value of D used inthe feedback linearizer (19), the nonlinear GPC (13), and thelinear reference model (25). For Q = diag(1,I) ,R = 1,p1 =100,p2 = 0, and a nine cycle short-circuit, the response withthe controllers (13), (19), and (26) on the system (1)-(2) isshown in Fig. 3.

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100, 0.02 Ii I

5 10-100 -0.0210Time (seconds) Time (seconds)

-E7 0.05-dB o

5 10-0.05

Bold Line- NnPredictive COnIrolBroken Line - LOR ControlDotted Line- LinearizingControl

Time (seconds)Fig. 4.LQR, in the presence of persistent inter-area-type disturbance.Response to nine cycleshort-circuit with nonlinear GPC, FBLC and

It may be observed that nonlinear GPC (13) performsbest, and rapidly damps the rotor oscillations, using the leastcontrol input. The feedback linearizing controller fails instabilizing the system and results in an unstable transient.Under identical conditions, the LQR controller is seen tostabilize the system to the correct equilibrium, however, usingtwice the control demanded by the nonlinear GPC. This resultindicates the potential danger that exists by improper feedbackcancellation of unstable dynamics, a situation similar to theinexact cancellation of a nonminimum phase zero by anunstable pole in linear systems.We emphasize at this point that for values of D morepositive than -1.91 and for shorter-duration faults, the feed-back linearizer may stabilize the system, but is inferior inperformance when compared with the nonlinear GPC and theLQR controllers.D. ffect of Inter-Area-Oscillation-Type Disturbance

In the context of interconnected operation of power systems,the presence of various inter-area and local modes of oscilla-tion is inevitable. This situation is simulated approximately, bymodulating the infinite-bus voltage in magnitude by a smallfrequency, representative of an inter-area mode.Accordingly for the simulation of the plant (1)-(2), wechoose V ( t )= VO- 0.125cos(3.77t), corresponding to aninter-area fundamental frequency of 0.6 Hz. Further assumethat nonlinear GPC (13), controller (19), and linear referencemodel (25) are provided with the constant value of VOonly.Choosing D = 0.5;Q = diag(1,l ) , = 1forboth controllers(19) and (26), and p1 = 100 and pz = 0 in (13), the responseto a nine-cycle short-circuit is shown in Fig. 4. Again, thefeedback linearizer can be observed to fail in stabilizing thesystem, while the LQR stabilizes the system under identicalconditions. The nonlinear GPC performs best, and nearlyrejects the effect of the persistent disturbance.It may be noted that for smaller faults and lower inter-areafrequencies, the feedback linearizer may stabilize the system.Under identical circumstances, it may be noted that the LQR

stabilizes the system, and the nonlinear GPC offers the greatesteffectiveness for stabilization and damping.Similar observations can be made with respect to unmodeledprime-mover dynamics when comparing controller perfor-mance (e.g., consider T,(t) = T,, + 0 .05c os(w r t ) in thesimulations, and use only T,, in the control design, wherew, is a frequency close enough to resonate with the generatorshaft mode, say w, = 4.54 radshec).

VII. CONCLUSIONSThis paper investigates the performance of a nonlineargeneralized predictive controller, feedback linearizing con-troller and an ordinary LQR for a simplified power systemunder nonideal conditions. Owing to physical limitations onthe magnitude of the control, transient stability controllersinitially operate in a bang-bang mode. Nonlinear GPC isseen to provide a powerful, effective option for the nonlinearoutput feedback control of the power system. The nonlinear

GPC is seen to possess excellent robustness properties andrequires the least control when compared with the LQR andthe feedback linearizing controllers. A major advantage ofthe nonlinear GPC over the other two controllers is thatit allows a systematic way of handling control constraintsalbeit numerically intensive, which may restrict the look-aheadhorizon for real-time implementation.Feedback linearizing controllers demand excessive controleffort, compared with LQRs, due to the additional controleffort expended in feedback linearizing the nonlinear dynam-ics. When the control saturates, the nonlinear dynamics arenot cancelled entirely, and the system is left with residualnonlinear dynamics with properties that are different fromthe original nonlinear power system. Feedback linearizingcontrollers need a perfect reference model and measurementsto provide exact cancellation. Uncertainties in the referencemodel can lead to deteriorated robustness of the controller.There exists the danger of destabilization, by imperfectlycancelling power system dynamics which possess an unstableequilibrium, e.g., dynamically unstable situations. The pres-ence of dynamic uncertainties, such as time-varying infinite-bus voltage, can lead to nonrobust performance of the feedbacklinearizing controller.

LQ regulators perform worse than the nonlinear GPC, butbetter than the feedback linearizers, for identical conditions.Although the performance of LQR on nonlinear systems issuboptimal, it is seen to have good robustness properties.ACKNOWLEDGMENT

The authors wish to thank W. A. Mittelstadt and D.Maratukulam for their contributions.REFERENCES

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