The concept of dynamic equilibrium of control in conjunction with a proportional corrector constitutes the structure of the new control technique. The controller utilizes direct measurements and model- based state estimations in the feedback loop. The method includes recovering unanticipated parametric variations or partially unknown dynamics. An application of this approach to the axial xenon oscillation problem of PWRs was considered. A two-point xenon oscillation model was used in designing the controller and testing it through simulations. 6 refs., 8 figs

Berkan, R. C.

Kisner, R. A.

Perez, R. B.

Upadhyaya, B. R.

UNT Digital Library

CONF-830555—16DE89 012502A NEW NONLINEAR "RECONSTRUCTIVE" CONTROL APPROACH APPLIEDTO -THE AXIAL XENON OSCILLATION PROBLEM IN PHRs*R. C. Berkan, B. R. Upadhyaya, R. 3. Perez**The University of TennesseeDepartment of Nuclear EngineeringKnoxville, Tennessee 37996-2300"Tha sutxnttad manuscript has bean r • • - • iauthored by a conuacior of tha US, and ' • '• ;-. * ' ' 0 "1 '• ' ' •  '< 5 '; \ \Govarnmant undar contract No. DE- " T. 1 i _ _j .. '' • - '•AC05-MOR21400. Accofdioglv. tha U.S. . :Govarnmant rttami • nonaxckoiv*, OaO a V i c n o T ~ «royalty-fraa tc«iM lo publish or raproduca ^ * rt# fti9IIBl kl&V 2 4lha pubtahad form of this contribution, or fnr\' * Jalow othars Jo do so. (or U.S. Govammant . _ . » , . j _ . 1 _ » j _Oak Ridge National Laboratory .Instrumentation and Controls DivisionOak Ridge, Tennessee 37831-6010*^ . ̂ABSTRACTThe concept of dynamic equilibrium of control inconjuction with a proportional corrector constitutes thestructure of the new control technique. The controllerutilizes direct measurements and model-based state estimationsin the feedback loop. The method includes recoveringunanticipated parametric variations or partially unknowndynamics. An application of this approach to the axial xenonoscillation problem of PWRs was considered. A two-point xenonoscillation model was used in designing the controller andtesting it through simulations.1. INTRODDCTIONThis paper presents a nonlinear control technique fortrajectory following processes including unknown parametricdeviations or partially unknown dynamics. Recent studiesentitled "inverse problems in dynamics" [1] address solutiontechniques in which the controls are calculated for assignedtrajectories. The embedded predictive models [2] use inversemodels and calculate controls numerically.Roearch ipomofcd by tbe Advanced Coatrok Profrua of the Office of R«*ctor lechnolopesDevelopment of U» VS. Deptnmeat of EDerp under contract No. DE-AC05-S4OR21400 with MminMarietta En«|y System*, lac" Re«ud» perfortd «ider S»bcootn>ct 41B-99732C X39 with The Ua**nity of Teo&ewee under MirtinMarietta E&erfjr Syuetat, b e , contract DE-AC05-MOR21400 with the U.S. Deptnmeot of Enerp.-A?DISTRIBUTION OF THIS DOCUMENT IS UNLIMITEaIn this study, we introduce the use of dynamic equilibrium cfcontrol which exerts a dissipative control force against theopen-loop dynamics. These forces appear when a system leavesits (minimum-energy) stable-equilibrium point due to soir.eexcess energy. The control is completed by reconstructing thedissipated dynamics for assigned trajectories. Compared withrelated methods in the literature, the dynamic equilibrium ofcontrol corresponds to a partial inverse dynamics of thecomplete trajectory following dynamics and recognizes theminimum effort to achieve continuous equilibria.It is a well-known phenomenon that pressurized waterreactors (PKH) exhibit xenon induced power oscillations.Analytical studies [3] showed that the nature ofnonlinearities are quite complicated and may go beyond thecontrol capability of reactor operators. During plantoperations, the axial power oscillations are avoided bylowering the power to some economically undesired levels. Inthis paper, it is not intended to propose a new completecontrol strategy for the xenon oscillation problem of PWRs.This application introduces the concept of dynamic equilibriumfor the compensation of complicated nonlinear behavior likethe xenon oscillation using a simple model. However, thesuperior results indicate that the methodology may find a realapplication in this field.2. CONTROL DESIGNThe method includes two design phases: control andadaptation, both utilizing a set of process measurements andmodel-based knowledge. The final form of controller is acombination of the two, determined by the availability ofsensory information and the importance of the unknown part ofprocess dynamics. Partially unknown dynamics is defined hereas mismatches between actual process behavior and predictionsobtained by analytical methods. Such mismatches may be due totime-dependent unknown parametric deviations, which areassumed to be constants in dynamic models, or by the lack ofmathematical representation of the actual physical behavior.The control system is shown in Figure 1, including the flow ofimportant system variables. The following describes the basicsof these two design steps.2.1 Control PhaseConsider a dynamic system bounded by a set of predefinedtrajectories. In state space representation, a nonlinearsystem is stated asPLRNT[x\ 7MUMODELPLRNTCONTROLu TRflCKINGYVVMODELCONTROLMFigure 1. Block-structure of ••Reconstructive"nonlinear control system.DISCLAIMERThis report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither the United States Government nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability or responsi-bility for the accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Government or any agency thereof. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of theUnited States Government or any agency thereof.X = f(X,U) (1)where X and ~U are state and control vectors, respectively. Wethen concentrate on the i-th state equation assuming that afixed trajectory was assigned to this state variable.L i ^ i , j=l,2,..,N (2)whereX^ = ith state variable,Xj = coupling state variables,Uj_ = ith control variable,N = number of state equations.Next we define the dynamic equilibria of U^ and of aselected coupling state Xj in case U^ does not exist providedUj (any other control) exists which i£ an implicit function ofthe selected coupling state Xj (controllable path).Establishing dynamic equilibrium of a selected coupling statecan also be used to overcome multiple-equilibrium problem(same control appearing in more than one dynamic equation).The dynamic equilibria are,U i E = f(Xi,XifUiJf. (3)lxi=°(Xj ) F = f'fXjL^j) . (4)whereXj = selected coupling state variable,E = subscript indicating equilibrium.Evaluating the functions in Eqs (3) and (4) for x^-0 andsolving for the control and for the selected state variablewill yield time-variant equilibria that are not functions ofU^ and Xj , respectively. We then introduce the desiredtrajectories to be followed by the process and associate itwith control through the following definition of a totalweighted errors,. i i i E i i R i (5)andE*- [Qi(Xj*-Xj*E) +R i(X i-XR i)] (6)vhereXRi= desired trajectory for i-th state variable,Ri»Qi = weights.Note that the state variables and controls arenormalized, nondimensional variables. Equation (5) does notevariaoxe or by simply using Uu when UJ " does not exist. Thecontrols are solved from Eqs. (5) and (6) by forcing E or E*to go to zero.UL = for(7)i*> (8)*Xj" = f»(Xi,Xj) (9)whereU-: = any other control (j=l,2,...,N ;Ki ~ tRi/Qi) adjustable gain.The final form of the control includes a dynamic equilibriumterm which works towards dissipating the excess energy of adynamic system between equilibrium and nonequilibrium pointsand a proportional term which forces the system to "rebuild"its dynamics in a desired form. It can be shown analyticallythat control (7) guarantees stability for linear systems.The control design stated by Eqs.(7), (8) and (9)requires that the state variables that appear in theseequations are directly measured or accurately estimated. Ifall the states are accessible through a measurement set, thenthe unanticipated transients (like state disturbance) will berecovered automatically by the dynamic equilibrium of control.If some of the state variables are estimated usingmathematical models then the parametric deviations may affectthe controller performance severely. This problem holds forevery model-based control scheme and can be overcome usingseveral adaptive techniques available in the literature [4].We proceed with the second design phase, including adaptivefeatures to predict the unknowns of the process.2.2 Adaptive PhaseWe introduce an adaptive philosophy for which the trackedparameters are used to update the control but not the model.In ether words, the model provides a reference state similarto the model-reference adaptive control technique.Consider process dynamics and its correspondingmathematical model in which the model assur.es constantparameters that are time-variant in the actual process,X ' — f f X ' X ' U ' Y * ) ( 1 0 )a n d«M ^ ^ (luf Vf XT V ^ f T 1 \whereYi = time-variant parameter,Mi = i-th state variable of the model,Mj = coupling state variables of the model,Vi = control in the model,Ym = constant parameter used in the model.i/ote that the control Vi of the model would exactly match^ if Yi did not change in time. Considering the first casewhere Ui exists, we present the following error criteriaB - CQi(Yi-Ym) + RifUi-Vi)], (12)and a solution for Y^*= Ym - Ky[(UiE-ViE) + Kim<Mi-XRi) - Kix(Xi-XRi) ], (13)where Ky, K±m and Kj,x are adjustable constants. The solution(13) includes dynamic equilibria which may include unknown Y^,ang will be updated by Yi . The control Ui is a function ofYi •The second case where Ui does not exist, the time-variantYt can be considered as partially unknown dynamics providedthe corresponding state measurement is available. A solutioncan be obtained by treating Yi as a control variableassociated with a state trajectory, which is continuouslyprovided by the model. An error expression is written in thefollowing fashion.A solution for Yj_* is obtained,Yi " YiE KyCXi-Mi) (15)whereYi E* (16)The Eqs. (15) and (16) are then used in UJ if it is afunction of Yi .The design procedure stated above may require assigningadditional trajectories not included in the frame of controltasks. This will be necessary when the estimated statesinclude time-variant parameters and there is no trajectoryfollowing assignment to those states. In this case, theadditional trajectory assignment can be selected intuitivelyprovided it is close to its steady-state value. Note that thenumber of trajectory assignments can not exceed the number ofdegrees of freedom of the system.3. AN APPLICATION TO THE XENON OSCILLATION PROBLEMA two-point xenon oscillation model of a PWR [5] is usedto demonstrate the method of "reconstructive" control. Themodel employs the nonlinear xenon and iodine balance equationsand one group, one-dimensional, neutron diffusion equationhaving nonlinear power reactivity feedback. A two-termspatial, harmonic-series solution was assumed for the flux,xenon and iodine distributions. The system was made as closeto critical as possible with assumed distributions using avariational principle. The xenon and iodine concentrationswere then obtained from their governing differentialequations. The input/output nature of the model makes it idealfor simulation of xenon-induced reactor transients. In themodel, the spatial average of the normalized flux for thelower half and for the upper half are expressed by thefollowing equations.^ - 0.6366 [1 - A(t)] (17)F2(t) « 0.6366 [1 + A(t)] (18)whereF,,F2 = upper and lower half average fluxes,A(t) -= amplitude of the first harmonic.Similarly, the average xenon and iodine oscillations inthe upper and lower part of the core are expressed asXx(t) = 0.6366 [1 - B(t)] (19)X2(t) = 0.6366 [1 + B(t)] (20)Yx(t) = 0.6366 [1 - C(t)J (21)Y2(t) » 0.6366 [1 + C(t)] (22)whereX^ = average xenon concentration in lower-half,X2 = average xenon concentration in upper-half,Yj = average iodine concentration in lower-half,Y, = average iodine concentration in upper-half,B(t)= xenon amplitude function,C(t) = iodine amplitude function.The amplitude functions described in the literature [5]satisfy the following equations.-Z2(t) A2(t) + 2(Z1-Z3) A(t) + Z2(t) - 0 (23)= ax A(t) - a2 C(t) (24)= [a3 - a4 B(t)] A(t) - a5 B(t) + a6 C(t) (25)Z2(t) = a7[R2(t) - + a8 B(t) (26)andZ^Z^a^ « neutronic and geometric constants,R2(t) » reactivity input with upper partial-lengthcontrol rod,R1(t) - reactivity input with lower partial-lengthcontrol rod.The control task is determined such that the amplitudefunctions -B(t) and C(t) rercain constant (close to a fixedtrajectory) which guarantees nonoscillatory flux response. Thesolution for [R^Rj^ (relative reactivity input with respectto upper and lower halves of the core) is obtained using themethod described by Eqs. (7) and (9) . In this example, two"coupling" states are chosen due to the fact that there aretwo differential state equations both without any direct entryof controls R^ and R2. First we select C (t) in Eqn. (25) andestablish dynamic equilibrium forB(t).a5 B(t)/a6 - [a3 - a4 B(t)] A(t)/a6 (27)C*(t) = C*EQ(t) - K-L [B(t)-Bd(t)] (28)where Bd(t) is a desired trajectory, close to its steady-statevalue. Then we establish a dynamic equilibrium in Eqn.(24) forC(t) by selecting A*(t).A*EQ(t) = a2 C*(t)/a3 (29)A*(t) = A*EQ - K2CC(t)-Cd(t)J (30)The final step is to use A*(t) in Eqs. (23) and (26) tosolve for2(Z, - Z3) A* a8 B(t)[R2(t) - Rx(t)] - — - - . (31)a7[l - (A*)2] a7Note that in Eqn. (31) , A* should not be equal to 1 atany tine. It can easily be seen from Eqn. (23) that A =1 doesnot satisfy Eqn. (23) due to Z^Zj. In applications wheresuch condition can not be established easily, the controlsignal can be protected by numerical methods or anotherstrategy of achieving equilibrium can be used. The gains Kjand K2 were arbitrarily chosen both equal to 100 in thisapplication.4. SIMULATION RESULTSTwo-point axial xenon oscillation model is firstsimulated without control. Figure 2 shows the normalized fluxamplitude in the lower half of the core for the Oconee plant.i 1 1—r, 1 1.7510 15 80 25 30 55 40 45 50 55 $0 66 70 75 80 85TIME (HOURS AFTER PERTURBATION)Figure 2. Time plot of model response compared withobserved axial oscillation at Oconee NuclearStation. [5].Results of Ref[5] and our simulation are compared. The dottedcurve shows the'time response of the Oconee reactor duringxenon oscillations. The reactor had been operating at 75% offull power with steady-state xenon prior to the perturbation(which lasted 2.5 hr.) by means of control rods. Figure 3shows the xenon, iodine and flux oscillations in the upperhalf of the core for those conditions in which the dynamicsexhibits a limit cycle behavior.The controller is turned on 30 minutes after the 2.5-hr.perturbation. As shown in Fig.4, flux, xenon and iodineoscillations are compensated rapidly. Figure 5 shows thereactivity insertion using an upper partial-length controlrod. The 2.5-hr, perturbation (about 1% worth of reactivity)is also performed using partial-length rods and appears as theinitial pulse in the same plot. Comparing this "perturbation"pulse with the rest of the curve indicates that the amount ofrod motion is reasonably small.In order to demonstrate the effect of "dynamicequilibrium of control", the controller is turned off for 17hours following the perturbation and then turned on. Figure-6indicates that the controller suddenly perceives the dynamicequilibrium point and acts accordingly. As shown in Fig. 7,the required rod motion is much larger compared to theprevious case which agrees with intuitive reasoning. Figure 8shows xenon-iodine phase-plane graphs of uncontrolled andcontrolled (after 17 hrs.) reactor. The limit-cycle behaviorof the uncontrolled plant is exhibited.5. CONCLUSIONSThe application of the "reconstructive" control techniqueto an axial xenon oscillation problem clearly emphasizes theeffectiveness of using dynamic equilibrium of control. The rodposition (or reactivity input) changes in small amounts over alarge interval of time, mapping the equilibrium trajectory ofthe overall dynamics. Obviously, such a trajectory is noteasily perceivable by the reactor operator. Due to the uniquestructure of the xenon oscillation problem, adaptive featuresare not addressed. The controller uses calculated values forxenon and iodine concentrations or their correspondingamplitudes. There are fast algorithms for such calculationsavailable in the literature [6]. In addition, the transient isquite slow enabling the system engineers to update theircalculations. Unanticipated transients are automaticallyreflected in the dynamic equilibrium part of the controller,thus there is no need for an adaptive design for this purpose.However, the design can easily be extended using aore<SJ'sor\ /NA / P A Iv M™71/0.20 0.-K3 0.60*3O2 (HOURS) 0.6D 1.00Figure 3. Limit-cycle behavior in the upper half ofthe core. Arrow indicates the perturbation.o'1OaCOoD 3aa"ao'-r0.00 0.20 0.*0T *3O2n.sn(HOURS)0.80 1 VFigure 4. Flux, xenon and iodine behavior in thecontrolled plant (upper core region)Controller is turned on 30 minutes after theperturbation.toenITT).5302> • "<ri11U1i1 ii0.00 0.20 0.40 0.60T MO* (HOURS)0.80COUSG5COI<rCM1.00Figure 5. Reactivity input in the upper core region.Controller is turned on 30 minutes after theperturbation.oOC3o11oCNJCO?°"o•'"8oool.COoGJXQCOx°oX"o "8oso.oo 0.20 0.40 0.50T *3O2 (HOURS) 0.6D 1.00Figure 6. Flux, xenon and iodine behavior in thecontrolled plant (upper core region).Controller is turned on 17 hours after theperturbation.(TVoCJdo.as<x.K0.00 0.20/ I0.40 0.60(HOURS)0.80CM1/5mCMCDCOinca.<xj1.00Figure 7. Reactivity input in the upper core region.Controller is turned on 17 hours after theperturbation.§IFI!0.00 °-i0 n'-ftooml1.20) 1.60CDenoex.vT!M2.00IB)oocOXx°o8inCMCMCDCOIVCO0 00 0.40 1.60 2.00Figure 8. Xenon-iodine phase-plane graphs for(a) uncontrolled plant, (b) controlleractivated IV hours after the perturbationsophisticated models and including the boron effect on thedynamic equilibrium.REFERENCES[1] P. D. Krutko, N. A. Lakota and E. P. Popov,"The Conceptof Inverse Problems in Dynamics and Synthesis ofAlgorithms to Control the Motion of Manipulator Robots,"IFAC 9th Triennial World Congress Trans., Budapest,Hungary, p. 457 (1984).[2] S. S. Godbole et al.,"Plant Control Using EmbeddedPredictive Models," Proc. EPRI Conference on PowerPlant Simulations and Modeling., Charlotte, NorthCarolina (June 1988).[3] W. Stacey,"Xenon-induced Spatial Power Oscillations,"Reactor Technology, Vol. 13, No.3, p.252, (1970).[4] R. B. Asher, I. I. Andrisani and P. Dorato,"Bibliography"on Adaptive Control Systems," Proc. IEEE, Vol.64, No.8,p. 1226-1240 (1981).[5] R. J. Onega and R. A. Kisner,"An Axial Xenon OscillationModel," Annals of Nuclear Energy, Vol.5, p.13-19,Pergaroon Press (1978).[6] H. Tochihara,"Simple Functional Method for CalculatingAxial Power Distribution of PWR Core," Nucl. Sci.Technol., Vol 19, p. 449-459 (1982).

A new nonlinear ''reconstructive'' control approach applied to the axial xenon oscillation problem in PWRs

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A new nonlinear ''reconstructive'' control approach applied to the axial xenon oscillation problem in PWRs

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