robust continuous speed governor control for small-signal and transient stability2

7/27/2019 Robust Continuous Speed Governor Control for Small-signal and Transient Stability2

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IEEE Transac t ions on Power Systems, Vol. 12, No. 1,February 1997 129ROBUST CONTINUOUS SPEED GOVERNOR CONTROL FOR SMALL-SIGNAL ANDTRANSIENT STABILITY

H. Bourlksf, Member, IEEE, F. Colledani$, M. P. Houryt, Member, IEEE?Electricit& de France, 1 avenue du GBn6ral de G aulle, 92141 Clamart, France

sSup6lec, Plateau d e Moulon, 91 192 Gif-sur-Yvette, FranceAbstruct- A robust continuous governor control for small-signal and transient stability is proposed. The turbine underconsideration is a steam turbine with two sections and is ofreheat type. The controller can be viewed as including thespeed droop governor and the fast-valving; tlie latter is notstandard, in that it generates a continuous control. Hence,the disadvantages of the (usual) bang-bang fast-valving areavoided. The control signal is generated by two controllers,with the first one for small disturbances and the second onefor large disturbances. Our approach is innovative in thatboth controllers have the same structure and only differ inthe tuning of their parameters. They generate a continuouscontrol and are switched with a bumpless transition toensure the continuity of tlie overall control. The mostimportant non-linearity of the turbine is taken into account,and a feedback linearization is made -but in a somewhatunusual manner to obtain a robust controller. A linearquadratic control is also used to ensure that the steady-stateobjectives are met and that good transients are obtained.Simulations with the EUROSTAG software show improvedefficiency with the proposed controller compared with thecurrent one, especially in the case of a short-circuit, anislanding or a temporary loss of synchronism of the machineconsidered; the current controller has a standard structure,including a bang-bang fast-valving.Keywords - overnor, Fast-Valving, Droop Characteristic,Robust Feedback Linearization.1.INTRODUCTION1.1. Many speed governing systems for steam turbinesinclude two types of controllers.- On e of them is the speed droop governo r [2]: its role isto ensure that, at steady-state, the speed and the power suppliedby the turbine are related hy the "droop characteristic" [4]; thiscontroller, which generates a continuou s control, also has tokeep the system stable under normal conditions, i .e . ,when smalldisturbances occur ("small-sign al stability").

- The other one is the fast-valving [21], which helpsmaintain system stability following a severe fault by rapidlyclosing (and then opening) steam valves, thus reducing theturbine mecha nical power. The fast-valving system provides abang-bang contro l and onl y applies to transient stability. Th econnection of two different kinds of controllers -continuousand discontinuous- has several disadv antages. The mos tserious one is that the fast-valving triggering thresholds cannotbe optimal for all disturbances.

96 WM 198-2 PWRS A paper recommended and approved by the IEEEPower System Engineering Committee of the IEEE Power EngineeringSociety for presentation at the 1996 IEEYPES Winter Meeting, January 21-25 , 1996, Baltimore, MD . Manuscript submitted July 21, 1995; madeavailable for printing January 15, 1996.

As a matter of fact, the fast-valving may negatively affect thesystem performance when forming island s [20]. Its suddenaction may lead to lightly dam ped oscillations [20], [16]. Seealso [SI and related references for the precautions to be takenwhen using a fast-valving.1.2. In this paper, we propose a robust continuousgovernor control for small-signal and transient stability. Theresulting controller can be viewed as including the speed droopgovernor and the fast-valving; the latter is not standard, in that itgenerates a continuo us control. Hence, the disadvantages of the(usual) bang-bang fast-valving are avoided. Th e most importantnon-linearity of the turbine is taken into account and the controlis based in part on feedback linearization [25]; this method isused in a somewhat unusual manner to obtain a robust controller

[113, [121. The control should not be too sensitive to "networknoise", i .e. , to small disturbances occuring far from the unit. Ifthis was not the case, the valves would be constantly agitatedand thus prematurely damaged. These small disturbancescannot be dissociated from the large ones based on theircharacteristics in the frequency domain, and cannot therefore befiltered in the usual way. For this reason, our control signal isgenerated by two controllers ---one fo r small disturbances andthe other one for la rge dis turbances . This approach isinnovative in that both controllers have the sa me structure andonly differ in the tuning of their parameters: the gains of the firstcontro ller are smaller than those of the second one. Bothcontrollers generate a continuous control and they are switchedwith a bumpless transition [ l ] to ensure the continuity of theoverall control. The first controller switch es to the second onewhen the disturbance is larger than a threshold; as opposed towhat happens with the current solution, the only change whenreachin g the threshold is in the "spee d" of the closed loopdynamics; therefore, the exact value of this threshold is notimportant.

1.3. The inputs of the controller are the pressures XI, x3and x2 in the high pressure (HP) stage of the turbine, the lowpressure (LP) stage and in the reheater respectively, the LP-valve position x4, the rotor speed w,, the active power P,supplied by the generator, and the set-points Po nd fo ; it soutputs are the commands '11an d u2 of the HP and the LP valves(see Fig. 1 and 2 to fix ideas). Hence, it is assumed that the HPand LP valves can be controlled separately. Th e use of P, forthe speed control is a kind of (very siniple) coordination withthe excitation control. However, we do not claim to proposehere a true coordinated excitation-and-speed governor control,as , e . $ . , in [22], [13], [lo ]. Results on such a coordinatedcontrol, based on a methodology similar to the one used here,have been obtained in [12] and will be reported in a forthcomingpaper. As is well known, a good filtering of the speed signal isimportant to efficiently eliminate the shaft torsionals 1141;hence, the filter recently d esigned in [16] is assumed to b e used.

0885-8950/97/$10.00 0 99 6 IEEE


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1301.4. As is shown in the sequel, the proposed con trollerhas a rather simple structure, and is therefore easy to implement;in addition, the tuning of its coefficents is easy to do. Thiscontrol le r is much more effec t ive than the current one ,esp cia lly in the case of short-circuits, islandings and temporaryloss of synchronism of a machine.1.5. The paper is organized as follows: the problem tobe solved is formulated in Section 2: the physical system ( ie. ,the turbine) is briefly described, its mathematical model is

established, and the control objectives are specified. Section 3is devoted to control synthesis: the controller switching is firstdescribed, then the robust feedback linearization is made, andthe section ends with the linear quadratic control of thelinearized system. Section 4 contains simulation results, andSection 5 includes concluding remarks.2. PROBLEM SEITING

2.1. System descrintioiiThe steam turbine under consideration here has twosections and is of reheat type (see, e.g., [21]). In addition, thecontrol valves are assumed to respond to servo-actuators. Th esteam coming from the boiler or the ste am generator first goesthrough the HP stage, producing about 30% of the mechanicaltorque T,. The steam is then reheated, before going through the

LP stage (produ cing there the remain der of the torque -that is70%). Finally, after condensation, the steam returns to theboiler or the steam generator. The two actuators are the HP andLP valves. For efficiency and reliability, the LP valve should befully open at steady-state. A simplified standard block-diagramof the turbine [17], [21] is given in Fig. 1.

Fig. 1: Block-diagram of the turbine2.2. MatheinaticalmodelThe mechanical anci electrical torques T, and T, rerelated to the mechanical power P , (produced by the boiler)and the electrical power P , (supplied to the network) by the

relations P, = #,T,, P, = w,T,. Set w = p LB,, where p is thenumber of pole pairs [ 2 ] ;w is very close to its set-point LBO = 2 7c

f o , with o = SO Hz in Europe. Henc e, the swing equation J w ,= T,,, - T, can be written (setting H = l / ( J $ w ))

w = H{P,-P,) (1)A good approximation of H s 1/{J p 2 L B O } .he meaning of thefour state variables x, , x2, x,, x, and of the two input variablesUI, x2 in Fig. 1 has beeu precised in 1.3. Another variable is z= ~ 2 x 4 which is the delivery at reheater output; a is thefraction of total mechanical power generated by the HP stage,Le., a = 0.3; TR T , ,TL an d T I respectively are the timeconstants of the reheater, the HP stage with its valving system,the LP stage, and the LP valving system. For a 1300 MW

turboalternator, typical values are: TR = 4 s , T , = 1.2 s, TL=The block-diagram in Fig. 1 leads to the following0.2 S, T I = 0.6 S.nonlinear state equations:

The total mechanicalP, = ax 1iL a )

I t is assumed that the voltage controller [15] acts so fastthat P , can be considered as g piecewise constant; hence,the model used for the electrical part of the turboaltemator is,for the control synthesis (but not, of course,in Section 4):P , = o

2.3. Control obiectivesAt steady-state, P ,equation should be satisfied,

with k = 25 in Europe ( L e . , theIn addition, again at st efully open, as is mentioned in 9(because all quantities are expressed in per uniu2 = 1

T h edescribed in control objectives during transients5 1.2and are not recalled here. have been

3. CONTROLSYNTHESIS3.1. Controller switchinoAs is mentioned in 3 1.2, the control signal is generatedby two controllers having the same structure (but differentgains); these controllers are switched with bumpless transitionwhen a large deviation of the electrical power is detected. The

quantity AP, efined helow is calculated:T n

where L(.) enotes the Lap1time constant", chosen about 10 s. This quantity A P ,compared with a threshold equal, e.g., to 20% of the nominvalue of the power. The "sm all signal controller" works whenA P , < 20%, and the "large signal controller" otherwise. Thetechnique of switchingp. 226) and is easy tostates of both controllers are thoses of thstates are denoted el an d e2 below), thus,control is obtained byswitching .Only the synthesis of one of these controllers isdescribed in the sequel. Note that the system is stable with any


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131of these controllers, hence undesirable phenomena like limitcycles cannot occur (due to bumpless transition).

3.2. Rob us t feedback l ineariza t ionOne of the difficulties of the problem is the nonlinearityof system (2). Se t Y = (XI, x2 , ~ 3 ) ~nd x = (XI, x2, x3, z ) ~herez = x2 x 4 ; hen, (2) splits as follows:Y = F X+ G UI (8 )

(9)=fix) + s(x) u2where F, G, f an d g can be easily deduced from (2); here, z ischosen as the fourth state of the turbine, instead of x 4 .

3.2.1. The most usual method for the control synthesisis to first calculate the linear approximation of (9) around thenominal operating point. Let (no,u o ) denote this point; as it isan equilibrium point, we have

(where ax denotes the derivative with respect to x ). The system(S), (11) is linear (apart from an additive constant), and itscontrol law can be determined using standard linear controltheory. The disadvantage of this method is that the controlobtained is fully efficient only when x is close to xo . If thelinear control is sufficiently robust, the closed loop system canremain stable for x far from xo [18], but the price to pay for therobustn ess is the lack of perform ance -due to the basiccompromise robustness-performance (see, e.g., [9]).

3.2.2. Another method is to exacly linearize (9) byfeedback (see, e.g., [26], [19] for other applications of feedbacklinearization to control of power systems) . Thi s approachusually consists in setting

f(n) + g(x) u2 = 1'2 (12)where v2 is an auxiliary control; note that u2 can be calculatedin function of v2 because g(x) = x2 cannot be zero. Let v = (ul,v#; obviously, the system with input v and state x is linear, sothat, again, its control law can be obtained using linear controltheory. The main difficulty is then that the dynamics of thissystem is completely different from that of (2), and has nophysical meaning (due to the equation z = v2) . Therefore, i t isvery difficult to suitably choose the linear feedback v , and thisapproach usually leads to non robust con trol laws.

3.2.3. This difficulty is avoided by choosing anauxiliary control w2 such that the system with input w = (u l ,~ 2 ) ~nd state x (i) is linear (up to an additive constant) and (ii)has the same dynamics than (2) around the nominal operatingpoint. From (1l) , his auxiliary control w2 is defined byf(x) + s(x) u2 =

4t(xd (x - o) + dxs(xo) (x- o) u2o + s(xo) (9 u2o) (13)

(Le., we have replaced u2 by w2 in the right-hand side of (11)).Again, note that u2 can be calculated in function of w2 , o that uand w are related by an equationu = h(x, w ) (14)

where the function h(x, .): + h(x, w ) is one-to-one for everyx in the whole admissible state domain.The equilibrium equations (10) and the relation (3)yield: ulo = XIO = q = ~ 3 0 P,o = 1 (Le., the nominal power)and ~ 4 0 u20 ; n addition, u2o= 1 by the steady-state condition(6). Th e equations of the linearized turbine a re found to be

T H X ~-X I +T R X ~XI - ZT L X ~ - ~ 3 ZT ~ z - - ( x ~ - z ) + x ~ - z + w ~ITR

From (14),one obtainsu - w = h(x, W ) -w

The memoryless nonlinear system (1 6) generating the deviationu - w from the state x and the auxiliary input w , is called thelinearizing controller in the sequel.

Remark: Robustness of the feedback linearization. By(13), the first two terms ( i ~ ,he constant and the linear ones) ofthe Taylor expansion of h(x, w ) - w around the nominaloperating point (XO, UO) are zero. Therefore, the "local gain" ofthe linearizing controller is zero at the nominal operating point[8]; in other words, this controller does not act around thenominal operating point. Th is property is basic for therobustness of the overall control law. More specifically, assum ethat, for the linear system with input w and output x , helinear controller generating w is robust with respect touncertainties located at the input of Z; this is true for the LQcontroller below). Th en, using the theory developed in [8], it isnot difficult to prove that these robustness properties still holdnear the nominal operating point', fo r the nonlinear controllergenerating u , with respect to uncertainties located at the input ofthe physical system I: with input u and output x (more detailsare given in [121).

3.3. Linear a ua dra t i c control3.3.1. In order to safisfy the steady-state objectives ( S ) ,

(6), the plant model (l), (4), (15) is augmented with two statese l , e2 defined byel = z - x 2

where Pef is P , filtered by a low-pass filter (with unit staticgain and time constant around 1 s) ; the role of this filter isexplained below. At steady-state, all derivatives in theaugmented system are zero. In particular, z = x2 , ence x4 = u 2= 1 , hu s (6) is satisfied. So is also the droop characteristicequation (5) .'Using the so-called "Q-analysis" [9], it has been proved in [12]that these robustness properties hold even in the wholeadmissible domain.


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9323.3.2. = (xI, 2,x.9, z, W, el ,4, nd let 6X an d

denote the d e v ~ a ~ ~ ~ n sf X and w with respect to theirsteady-state values (see [6 ] for another equivalent formulation,but in a more general context). The control w is obtained byminimizing a quadratic index6 X T Q 6 X + 6wTR 6 w ) d t (1%

where Q an d R are symmetric real matrices; Q and R are,respectively, n oni ~eg at~v eefinite and positive definite.In practice, Q and R are chosen diagonal for theirmeaning to be clear: then, they weight the components of 6X

g, the larger is a weight, the quicker isthe conesponding response. As a result, the closed loopdynamics is easy to tune v i a Q and R. In our case, the onlycomponents of 6X to be weighted are 6e I and 6 e 2 , with a verysmall weight on &I because the LP valve can be reopenedslowly after a disturbance.Another advantage of the obtained optimal control is itsrobustness properties, especially with respect to uncertaintieslocated at the system input [24], [7] (a.g., nonlinearities andneglected dynam ics of actuators).As is well known, this optimal control is linear, i.e.

w = - K X (19)(u p to an additive constant), where K is a gain matrix ofdimension 2x7: see, e.g. , [3]. An algebraic Riccati equation hasto be solved to calculate K ; algorithms for this calculation areincluded in most "control toolboxes", such as M ATLAB .Equation (19) defines the linear quadratic (LQ) controller.

3 3 . 3 . Each component w of the auxiliary controlpossesses an ntegral term - Ki7 e2 . Assume that before time to,the system is at steady-state, and letfs be the steady value of thefrequency (which is close, but not necessarily equal, to fo ; inaddition, assume that at t ime to, a disturbance arises, e.g., astep of the electrical power (due to a short-circuit or anislanding). During the first moments after to , Po - P,f- k c f, -,To) z 0 (because P e . remains approximatively constant, due tothe filtering), hence e2 2 - k 0, where qt) = i'f-f,) dt.Thus, the controller includes a feedback of the phase shift of theturboalternator with respect to the network . This point is veryimportant to increase cri clearing times and it helpsreestablish the synchronism e machine under consideration,after a temporary loss of it (see $ 2.4).

3.3.4. Finally, let us specify that, due to its integrators,th e LQ controller is equipped with anti-windup ([l],p. 224).This is necessary because the control u is subject to th econstraints 05 u; ( t )S I , i=l , 2.3.4.The block-diagram of the resulting nonlinear controlleris given in Fig. 2. Recall that- the series connection of the linearizing controller andof the turbine is linear ("linearized turbine");- the series connection of the LQ controller and of thelinearizing controller consists in the overall controller (which isno n l i ne ~ , e ca us e so is the linearizing con troller).

pe omPOLfO

Fig. 2: Result ing nonl ine ar ~ m ~ r o ~ ~ e r4.SIMLn TION RESVLTSA comparison between the current and proposedgovernor controls has been made using EUROSTAG, a timesimulation software for stability studies [23]: see Fig. 3-5 . Th ecurrent solution is representative of speed governor controllersincluding an acceleration-based bang-bang fast-valving.

1.02~ 1.013 1Pa

0.990 . 9 8 159 60 61 62 63 64 6S 66 61 68 69

t ime (8 )

0.259 60 61 62 63 64 65 66 61 68 69m e 9)

0.2 159 60 61 62 63 64 6.5 66 67 68 69tme (s)

Figure 3: 110 ins t h re e phase fault


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133

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1

010 15 20 25 30 35 40 45 50 10 12 14 16 18 20 22 24

time (8 )

1.80.610.40.2010 1.5 20 25 30 3s 40 4.5 sa

time(s)

time (s)

time(s)

t i I

time(s)

Figure 4: islandingIn all cases, the model used for the electrical partcorresponds to a generator connected to an infinite bu s -i.e.,the classical fifth order nonlinear model [2]. In addition, a localload corresponding to 30% of the nominal power is representedin the case of an islanding (Fig. 4). The excitation system isequipped with the voltage controller presented in [lS]. In thefigures below, the solid lines correspond to the proposedcontroller and the dash-dotted lines to the current one. The fistvertical dashed line indicates when the "large signal controller"is switched on; the second on e indicates when the "small signalcontroller" is switched on again. The valves are rate limited andreopening is slower than closing.Figure 3 shows the case of a 110ms three phase fault.Before the fault, the generator is connected to two lines inparallel, with an impedance of 55% an d 164% each. The faultoccurs on the latter, which is disconnected after the fault. Theoscillations are clearly much better damped and the valves

movem ents are reduced with the proposed controller. Notice

time(s)

Fig. 5 t emp o ra ry loss of synchronismthat this cas e is very constrained. With the current governorcontrol, stability is kept thanks to the first valves closing(although it is slightly delayed due to the processing of th eturbine acceleration) but the natural generator oscil lat ionsgenerate additional (and unnecessary) closings, thus a poordamping. This problem is avoided with the proposed solutionbecause the closed loop dynamics is well controlled and noacceleration processing (delaying and filtering) is needed.Figure 4 shows the case of an islanding after a threephase fault; the residual power is 30%. The speed overshoot ismuch reduced with the proposed controller, and this is due inpart to the phase shift feedback (during the two first secondsafter the fault) presented i n 6 3.3.3.In Figure 5 , a case is presented w here the synchronismof the considered machine, which was temporarily lost, isreestablished with the proposed controller, while this does nothappen with the current one (see 6 3.3.3).


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1345 . CONCLUDING REMARKSSimulat ion resul ts in Sect ion 4 clearly show theimproved effect iveness of the proposed controller comparedwith the current one -which is representat ive of speedgovernor controllers including a bang-bang fast-valving. Thebehavior of the controlled system is much better in the case of ashort-circuit, an islanding or a temporary loss of synchronism ofthe machine consid ered. The key points for this are: (i) thecontinuity of the control law, which enables to avoid thedisadvantages of the bang-bang fast-valving ( $ 3 1.2, 3.1); (ii)the robust feedback linearization of the turbine ($ 3.2 .3) ; (iii) thelinear quadratic control (19), which ensures that the steady-stateobjectives are met and that good transients are obtained (63.3.1); (iv) the fact that this control includes a phase shiftfeedback with "forgetting factor" ($ 3.3.3).

REFERENCES[I] Astrom K. J., Wittenmark B., Computer-Controlled Systems:Theory and Design, Second Edition, Prentice-Hall, 1990.[23 Anderson P. M., Fouad A. A., Power System Control andStability, Revised Printing, IEEE Press, 1994.131 Anderson B. D. O., Moore J. B., Optinid Control: LinearQuudratic Methods, Prentice-Hall, 1989.[4] Barret J., "L'adaptation autoinatiyue de la production 2 la

consommation ; e reglage d e la frequence", Revue GtWraledklectricite?, vol. 12, pp. 935-948, 1985.[SI Bhat t N . B.., "Field Experience with Momentary FastTurbine Valving and Other Special Stabi l i ty ControlsEmployed at AEP'S Rockport Plant", IEEE PES, WinterMeeting, Paper n"95 WM 186-7 PWRS , 1995.161 Bourlks H., Mercies 0. L., "La r6gulation de poursuiteoptimale quadratique des systbmes lindaires perturb&",RAIRO Autonzutiqi*e/SystelllsAnalysis and Control, vol. 12,pp. 457-465, 1982.[7] Bourlks H., "Stabilite des systkines linkaires perturb&.Robustesse", RAIRO Autonzatique/Systerns Analysis andControl, vol. 18, n"3, pp. 297-314, 1984.[XI Bourlbs H., Colledani, F., "W-Stability and Local Input-Output Stabi l i ty Results", IEEE Trans. on AutomaticControl, vol. AC-40, n"6, pp. 1102-1108, 1995.[9] Bourlks H., Aiioun F., "Ap proche H, et p-synthkse", in Larobustesse : rzulyse et synth2se de comandes robustes, A.Oustaloup Ed., Herinks, pp. 163-235, 1994.[lo] Cheetam R. G., Walker P. A. W., "Co-ordinated Self-Tun i ng Con t r o l f o r Tu r b i ne a nd Exc i t e r o f aTurboalternator", Electric Machines and Power Systems,[11] Colledani F., BourlBs H., Vanhersecke M. P., "RobustCon t r o l l e r w i t h Loc a l L i ne a r i z i ng Fe e dba c k f o rFrequencyIPower Control of Power Plants", Proc. 32ndIEEE Con$ on Decision and Control, San Antonio (Texas),December 13-17, pp. 3740-3741,1993.[12] Colledani F., '%&plation coordon nee tensionlvitesse desgroupes turboalternateurs", Ph. D. Thesis, Universi tdd'Orsay (France), 1995.[13] Ghandakly A., Idowu P., "Design of a Model Reference

Adaptive Stabilizer for the Exciter and Governor Loops ofPower Generators", IEEE Truns. on Power Systems, vol. 5,n03, pp. 887-893, 1990.[14] Hammons T. J., "Impact of Shaft Torsionals in SteamTurbine Control", IEEE Trans. on Energy Conversion, vol.EC-4, n02, pp. 143-151, 1989.[15] Heniche A., Bourlbs H., Houry M. P., "A DesensitizedController for Voltage Regulation of Power Systems", IEEETrans. ow Power Systems, vol. PWRS-10, pp. 1461-1466,1995.

V O ~ . 5, pp. 177-198, 1988.

[I61 Houry M. P., Bourlks H., "Rotation Speed Measurement ofa Turbogenerator Shaft : Torsions Fi l ter ing By UsingKalman Filter", IEEE PES, Summer Meeting, Paper n"95SM 440-8 PWRD, 1995.[I71 IEEE C ommittee Report, "Dynamic models for steam andhydro-turbines in power system studies", IEEE Trans. onPower Apparatus and Systems, vol. PAS-92, pp. 1904-1915,1973.[18] Jiang J., "Design of an Optimal Robust Governor forHydraulic Turbine Generating Unit", IEEE PES, SummerMeeting, Paper n094 SM 376 -4 EC, 1994.[19] King C. A ., Chapman, J . W., I l ic, M. D., "FeedbackLinearizing Excitat ion Control of a Full-Scale PowerSystem Model", IEEE/PES, Summer Meeting, Paper n"93

[20] Km dur P., Lee D. C., Bayne J. P., Dandeno P. L., "Impactof Turbine Genera tor Overspeed Cont rol s on Uni tPerformance under System Disturbance C onditions", IEEE

SM 478-8 PWRS , 1993.

Trans. on Power Apphratus and Systems, vol. PAS-104, pp.1262-1269. 1985.1211 Kundur P., Power System Stability and Control, McGraw-Hill, 1994.[22] Lu H., Hazell, P. A,, Daniels, A. R., To-o rdina ted single-va r i a b l e e xc i t a t i on c on t r o l a nd gove r n i ng o fturboalternators", IEE Proc., vol. 1 29, Pt. C, 116, pp. 278-284,1982.[23] Meyer B., Stubbe M., "EUROSTAG, a single tool forpower-system simulation", Transmissions & DistributionInternational, March 1992.[24] Safonov M. G., Athans M., "Gain and Phase Margin forMultiloop LQG Regulators", IEEE Trans. on AutomaticControl, vol. AC-22, pp. 173-179, 1977.[25] Slotine J.-J. E. , Li W. , Applied Nonlinear Control ,PrenticeHall, 1991.[26] Wang Y., Hill D. J., Middleton R. , "Transient StabilityEnhancement and Voltage Regulati of Power Systems",IEEE Trans. on Power Systems, vol. PWRS-18, n"2, pp.620-626, 1993.BIOGRAPHIES

Henri Bourlks received hiCentrale de Paris in 1977Polytechnique de Grenoble in 1982, and his 'IUniversitd d'Orsay in 1992.the Electricit6 d e France RResearch Engineer.Automatic Control at Ecole Normale Supkrieure desince 1992. His main fields of research are: robustnessystem theory and application of automatic control to powersystem regulation.FredCric Colledani received his EnginEcole Sup6rieure d'Electricitk in 1991Universit6 d'Orsay in 1995.Marie-Pierre Houry received h&ole Superieure d' lglectr iciElectr ici td de France Research Center in 199 1. Sh e is aResearch Engineer in the team working on power systemdynamics and control.

ye has been Associate Professor of


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135of the generator to the turbine model. The electric powerbecomes then a state component of the whole system. It iseasy to synthesize in this way a coordinated excitation-an&

governor controller. Let us briefly explain this point.The stateq of the whole system is q = [V Pe o TlT, whereV is the terminal voltage; the other variables are defined in thepaper. One should augment this state with the variablesel , e2defined by (17) and the variable e3 defined by deg/dt = V -V cwhere Vc is the terminal voltage set-point. Denote by E theresulting augmented state and set p = [wT V JT. TheController can now be synthesized by minimizing a quadraticindex of the form (18), with X replaced by E and w replacedby p. The resulting control law is linear, as (19).Such a coordinated controller was designed and tested inref. [12]. The iinprovement -in comparison with thcassociation of the speed controller designed in the paper andof the excitation controller designed in ref. [151-- was notsignificant. This cm be explained as follows. Consider forinstance the case of a three phase fault (seeFig. 3). Just afterthe fault, the only thing for the speed controller is to veryquickly close the valves, and for the voltage controller toincrease the excitation voltage. Hence, the coordination is notvery helpful during large transients. After them, theassumption dPe/dt= 0 is roughly satisfied.

Lamine Mili (Virginia Tech, Blacksburg, VA). The authors are tobe commended for a well written paper which describes a robustCOntrQl scheme for speed governor. The simulation results given inthe paper clearly demonstrate the superiority of the proposedapproach. The authors' comments on the following would be verymuch appreciated:1. The design of th e LQ controller is made under the assumption thatthe electric power is constant, namely that dPe / dt = 0 (see Eq.

(4)). How ever, during transient conditions, the electric power P eundergoes a great variation due to the large excursion of th einternal angle of the machine. Ho w does this assumption affect theperformance and the robustness of the controller ?2. The performance of the linearizing controller relies on theappropriate choice of the function h(x, w) given by (14). Couldthe authors give the analytical expression of the function h(x , )for this particular application and provide some hints on how it hasbeen derived?

Manuscript received February 15, 1996.

Closure by Henri Bourlks (Electricit6 de France, Directiondes Etudes et Recherches, 92141 Clamart Cedex, France:First of all, I would like to thank Professor Mili for his interestin our paper.1. The hypothesis dP$dt = 0 is made only for thecontroller synthesis -as is emphasized in the paper- and iscertainly not satisfied when the system is actually working.However, this assumption does significantly reduce theperforinance and robustness of the controller. It is useful in

that, through it, this controller is simple and easy toimplement.The very role of this assumption is to decouple excitationcontrol and speed control. It is indeed possible to adda model

2. The equation (14) is composed of two equations: u1 =hl(x, w) = wl and u2 = h2(x, w), where h2 is calculated from(13), namely

~ x ~ ( x 0 ) ( x - x 0 ) ~ 2 0~ ( X O ) ( W 2 - ~ 2 0 ) 1f(x) =- XI-z)- -- , g(x)=- 21 z 1 1TR x2 T1 T1

XO = [l 1 1 l ]T , u20= 1This yieldsT1 1 Z 1 1X 2 TR T1 x2u2 =- - (Z-Xl)(--- -1) +- x2-l)] +- 2

Manuscript received April 3, 1996.

robust continuous speed governor control for small-signal and transient stability2

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