HUCLEAn SCIENCE AhiD EhfGfiHEER2NG: 110, 425—444 ( 1992 )

Considerotions in the Design and Implementation of Control Lows for the Digital Operation

of Research Reactors

John A. Barnard and David D. fanning

M achusetts I ns titute o,f Technolo gy , Wualear Reactor Laborato rr

Received October 8, 1990 Accepted Auguci 29, J99I

kb ctrn ei — Factors refevsiif to the design and implementation of digital controllers for research reac- tors are disinwed with emp hasis on the rationale for incorporating a system model in roe control J‹tw. For this purpo se , proportional-integral-derivative ord period-generated control are compared. The let- ter is a model-b sfzf technique that achieves excellent trajectory tracki n g of nonlinear purer It does this by combining feedback: and feed l o r ' rd control ection in a manner that cancelc the effec:ts of the system's dynamics on the controller's performance. Model - t uis ed control is also mi p erior in thot iI per- mits replication of some of the functions thai hum a z r s perform whey exercising control. In particu- lar, models tan be used to predict expected plant response ond thereby facilitate diagnosis. The importance o/ validated signots, supervisory algorithms, property designed man-machine inlerfacec, and automated diagnostics are d i sci t s re d in relation to control low implementation. In addition, a sum- mary is provided of reactor dynamics es refered to control, and argumentc are presented in support of using the rate of chonge of reactivity as the actuator signal. Experimental results obtained from Mials of digital confrollers ozi both the 5-MW{thermal ) M ‹z sr chusetts Instinite of Technology Research Re- actor and the Annular More Research Reactor ther is operated by Sgndia N'ationot Laboratoriec are included.

1. INTRODUCTION

Research reactors are multipurpose facilities that provide sources of neutron and, in some cases, gamma radiation for use in the basic and applied sciences, ed- ucation, materials research, medicine, earth and plan- etary studies, neutron activation analysis, engineering, and many other disciplines. For example, major re- search projects currently in progress at the 5-MW ( ther- mal ) Massachusetts Institute of Technology Research Reactor ( MITR-II ) include the evaluation of the radio- isotope " 5 Dy for the nonsurgical treatment of rheu- matoid arthritis, the study of coolant chemistries with the objective of reducing radiation exposure from ac- tivated corrosion products, the use of neutron activa- tion analysis to identify sources of airborne pollutants ( acid deposition ) , the poaible identification of earth- quake-prone geologic formations by using track-etch

techniques to assess microcracks, the development of neutron capture therapies for the treatment of deep- seated brain tumors such as glioblastoma multiforme, and the radiation-hardness testing of electronic devices for use in space travel.'* 3 This wide range of activities necessitates frequent adjustments of a research reac- tor's power. Given this need for flexible operation and the present availability of sophisticated yet low-cost, real-time computing equipment, the operators of re- search reactors are increasingly considerin g conversion of their facilities to digital control.

Digital systems are very different from the anal og controllers with which most research reactors were equipped when they were built some 15 to 35 yr ago.. To fully appreciate this difference, it is necessary to have some understanding of how humans achieve con- trol. Four tasks are involved. These are planning, pre- diction, implementation, and assessment. •* Planning

BERNARD and LANDING

entails first noting the operational objectives, then de- termining the current plant state, and finally identify- ing the most efficient means for achieving the objectives given the confines of approved procedures. A specific sequence of control actions is then chosen based on the operator's experience and understanding of the plant. For example, a skilled operator would know whether or not withdrawal of a control device by a certain distance would result in an acceptable rate of rise of power. Adjustments to the chosen control signal are made only if it appears that the response of the plant. will not be as projected. Thus, an operator's ability to anticipate or predict plant behavior is of the utmost importance. Implementation of the selected control ac- tion may require the simultaneous application of sig- nals to several plant subsystems. This process is already often entirely automated via electromechanicai means. The last of the four taska is assessment, and it is the most com plex. Operator must ascertain that the re- quired control action is being implemented. If all pro- ceeds as anticipated, there is no difficulty. But if that is not the case, the operator must determine why. 1s his or/her understanding of the plant's behavior deficient? Or has some component failed, and, if so, which one? Assessment is generally only done well by experienced personnel.

Analog devices are basically an extension of a hu- man‘s capability to implement a control action. Such equipment may provide either steady-state or transient control. But, in each case, a licensed reactor operator is performing the planning, prediction, and assessment functions. For example, an operator might use an an- alog controller to conduct a transient by programming a decade box to move the reactor’s control devices. The responsibility of decision remains with the operator. The decade box is merely a tool, Digital technology differs in that the software can be designed to provide the equivalent of most of the control fimctions tiiat are normally performed by a human. For example, numer- ical models could be used for prediction and techniques such as signal validati on for assessment. The decision to utilize digital technology for process control there- fore brings with it a number of questions. Is it possi- ble to design a digital controller that will replicate each of the functions now performed by a human? If not, which iasks should be assigned to the machine, arid how can one be certain that licensed operators will un- derstand and recognize the machine's limitations? These and related questions are now being debated within the nuclear, chemical, and aerospace commu- nities. This paper addresses one part of this debate by enumerating issues concerning the design and imple- mentation of closed-loop laws for the trajectory con- trol of power in research reactors.

The specific objectives of this paper are to ( a ) re-

change of reactivity, ( c ) describe and assess both proportional-integral-derivative ( PID ) and period- generated co n trol laws, ( d} delineate the benefits that can be achieved by i ncorporating a system model in a control law, and ( e ) enumerate factors relevant to the implementation of a closed-loop control law including signal validation, supervisory control, automated diag- nosis, and the man-machine interface.

II. REACTOR DYNAMICS

Figure I illustrates the aspects of the fission pro- cess that bear on reactor control. Of significance is that there are three parallel but separate mechanisms for the production of neutrons. Prompt neutrons appear di- rectly following the fission event and have lifetimes that are quite shon, typically 100 ps. Delayed neutrons are produced following the decay by beta-particle emis- sion of fission products that are referred to as “pre- cursors.” The delay in the appearance of a delayed neutron relative to the fission event is the result of the precursor half-life. There are six recognized groups of p_recursors with half-lives ranging from 0.23 to 55 s. The average value is 12.2 s. The third mechanism for neutron production is the interaction of fission prod- uct gamma rays with certain moderating materials, most notably heavy water and beryllium. Called photo- neutrons, their appearance is delayed relative to the fis- sion event because of the time required for the fission products to undergo radioactive decay and emit the needed gamma rays. In the ensuing discussion, the term “delayed” includes both those from precursors and photoneutrons.

The process shown in Fig. 1 is described in quan-

titative terms by the space-independent kinetics equa- tions. Those equations constitute a suitable model for research reactors because the small, compact cores that power those reactors do not exhibit spatial depen- dencies under conditions of normal operation. The spaoe-independent kinetics equations can be combined through differentiation and substitution to obtain the dynamic period equation, which gives the instantaneous reactor period as a function of the rate of change of re- activity, the reactivity, and the rate of redistribution of the delayed neutron precursors.‘ This relation is par- ticularly useful in the design of tracking controllers for nuclear reactors because it explicitly represents each of the parameters that can affect the rate at which a re- actor's neutronic power will increase or decrease.

The instantaneous reactor period r ( r ) is defined as

i ( f ) = hut ( / ) , where

T t ) ui ( t ) T l t ) ( 1 )

view reactor dynamics with emphasis on factors that and F ( f ) denotes the amplitude function, which is a affect system control,Jb \ summarize the rationale for weighted integral of all neutrons present in the reâctor. formulating reactor control laws in terms of the rate of In the discussions on controller design that fol ) ow, ihe

NUCLEAR SCIENCE AND ENGINEERIN G VOL. l IO APR. 1992

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427

DELAYED NEUTRONS		DAUGHTER fitLI L DES

Fig. 1. Fission process showing prompt, delayed, and photoneutron production.

amplitude function is approximated as the reactor neu-

tronic power a ( I ) . Thus,

The dynamic period equation may be written in either a standard or alternate form. The two are mathemat- ically equivalent, but the alternate is the easier to pro- gram. It is as follows:

( 3 )

where the alternate, effective, multigroup decay pa-

rameter is donned as

@ ( f ) - X 2 C; ( I j/ L; C, ( t ) for i = I, N , ( 4 )

and where

= effective delayed neutron fraction p ( r ) = net reactivity

f’ = prompt neutron lifetime

iii ( r ) = rate of change of the inverse of the dy- namic reactor period

ui ( r ) = inverse of the dynamic reactor period

( / ) = rate of change of the net reactivity

i = effective fractional yield of the i'th group of delayed neutrons

X; = decay constant for the i'th precursor group

C, ( t ) = concentration of the i'th precursor group normalized to the initial power

N = numb#r of groups of delayed neutrons, including photoneutrons.

Adjustments of power in research reactors are achieved by withdrawing a control device so as to in- sert positive reactivity and thereby place the reactor on a period, with period defined as the power level divided by the rate of change of power. Having established a period, the power is allowed to rise. Once the power level approaches the desired value, the control device is gradually returned to its original position to reduce the reactivity to zero and to level the power without ov#rshoot. The crucial aspect of the control process is that the lengthening of the reactor period must be ini- tiated before attaining the specified power level. Such anticipatory actions are necessary because the rate at which reactivity can be removed is finite, particularly

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42s BBRNARD and LASHING

when rods are used at normal speeds. Hence, if changes in the reactor power are to be achieved both efficiently and without c hal l engetothesafe t y system, some method must be available by which the proper time for initia- tion of the reactivity removal process can be reliably predicted. Licensed operators are sufficiently experi- enced so that they can make the appropriate judgments. The implementation pf digital technology requires that the equivalent capability be developed in software. Doing so is not easy because of the nonlinear, time- delayed nature of reactor dynamics. In particular, al- lowaitce must be made for the following:

1. The instantaneous reactor period is a function of the rate of changg of reactivity, the reactivity, and the rate of redistribution of the delayed neutron pre- cursors. This means that the period observed at any given moment in a reactor will depend on both the dis- tance that a control device has been moved beyond the critical position and the rate at which that dcviv is be- ing moved.

2. The dynamic response of a reactor is determined by that of its prompt and delayed neuron populations. Prompt neutrons appear simultaneously with the fis- sion event and are therefore a function of the current power level. Delayed neutrons appear some time after the fission event and are therefore a function of the power history. This dependence on the power history means that delayed neutrons will not be in equilibrium

with the observed power during a transient. Hence, upon attaining a desired power level, the delayed neu- trons will continue to rise, and an ovefshoot will occur imless the controller is capable of reducing the prompt neutron population at a rate sufficient to offsei the still rising delayed neutron population.

3. The relation between power and reactivity is nonlinear. Also, nonlinear reactivity feedback effects reautt from the changes in the fuel and moderator tem- perature that occur during power adjustments.

4. The differentia ) reactivity worth of the control devices is a strong function of the axial flux profile. Hence, the rate at which a controller can insert and re- move reactivity varies nonlinearly with the position of the device. In general, this rate will be a maximum at the core midplane and be quite low in the upper por- tion of the core.

Figures 2 and 3 illustrate the difficulties that can arise if the foregoing factors are ignored. Shown are results from demonstrations conducted on the MITR- 11 in 1983 and 1984. The control law consisted of a di- rective to withdraw a control rod at constant speed until a desired power level was attained. In addition, 8 JTtinimum allowed period was specified. The period limit was selected so that the controller functioned quite well under normal operating conditions. In par- ticular, it would raise the reactor's neutronic power to

TIME ts )

Fig. 2. Failure of controller to recognize need to limit rtact›vity insertion results in power overshoot.

NUCLEAR SCIENCE AND ENGINEERING MOL. 110 APR. 1992

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UPPER TOLERANCE

429

DESIRED

" POWER LEVEL

4.0 -

BOUND

— 18

3.S -

3.0 -

m 2.5-

RODAT OUT-UMlT

ROD HEIGHT

- 16

- 14

— t2

POWER

1.5

- 8

EXPERIMENTAL DATA MITR-II

APRIL 4, 1983

0 50 100 150 200

250 3o0 350 400 450

TtMEp}

Fig. 3. Failure of controller to restrict rod withdrawal in region of low differential worth results in power overshoot.

the desired value and hold it at that value. However, this controller did not track a specified power profile. Rather, the resulting trajectory was a function of the initial position and the predetermined speed of the con- trol device. Tests also showed that this controller was incapable of performing properly under off-normal conditions. Figure 2 shows a power increase in which a period shorter than the one for which the controller had been designed was allowed. The controller inserted the necessary reactivity but, upon attaining the speci- fied power level, could not remove reactivity with sufficient speed to preclude an overshoot. Delayed neu- trons were rising at a rate faster than the prompt ones could be decreased. This experiment demonstrated the

inflexibility that results from the failure to incorporate

something like this did occur at Chernobyl. Model- based control can preclude such difficulties.^

III. SELECTION OF THE RATE OF CHANGE OF REACTIVITY AS THE ACTUATOR SIGNAL

The control laws described in the following mate- rial are formulated in terms of the rate of change of re- activity. This means that the signal sent to the actuator is the speed at which the control device should be moved. This choice contrasts with the traditional ap- proach to the design of controllers for nuclear reactors, which is to specify a control action in terms of the magnitude of the reactivity. Use of this latter practice

means that the signal sent to the actuator is the desired

knowledge of the system d n«m›c•. In this case, the

position of the control device. There are a number of

controller was “tuned” to a ip‹cinc set of conditions and gave unsatisfactory results when those conditions were altered. Figure 3 shows a power increase for which the normal period was required, but the control device

reasons for selecting the rate of change of reactivity as the actuator signal. First, specification of the appro- priate rate of reactivity change means that both the

was initially positioned where its differential reactivity

worth was low. The controller fully withdrew the rod in order to insert enough reactivity and then could not halt the transient because the rod was so far withdrawn as to have no strength. An overshoot resulted. Such a scenario could occur during a xenon transient, and

‘The foregoing experiments were carefully controlled tests. Limits on power and period for the MITR-II were never exce#ded. Only those established for the individual tests were violated, and these were set below those associ- ated with the reactor.

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directiOll dlld speed of the control device are uniquely determined. In contrast, if only the reactivity were specified, then the desired final Fosition of the control device would be known, but the speed at which the de- vice should be moved to attain that position would be undetermined. Second, as is evident from the dynamic period equation, the response of a reactor depends on both the magnitude and the rate of change of the re- activity. Failure to allow for the latter means that sud- den variations will occur in the rate at which power is being raised whenever rod motion is started or stopped. Third, a major requirement in the design of controllers for safety-constrained processes such as nuclear reac- tors is that it be possible to alter the control signal on demand and thereby have an immediate effect on the process in question. Reactivity does not fulfill this re- quirement because it is a function of the distance that

cise review of the MIT program is given in Ref. 7, and detailed results are given in Refs. 8 through 11. The di8cussion here focuses on the design of PID and period-generated laws for the trajectory control of re- search reactor power. The emphasis is on the treatment of nonlinear behavior. These two laws were chosen for comparison because they represent opposing philoso- phies concerning the design of closed-loop digital con- trollers. At issue is whether or not a system model should be incorporated in the control law.

IN.A. Pr oportional-Integral-Derivative 'ontrol

The PID mode of control is the most commonly used method for the automation of process systems. It is also the simplest because it considers the system dy-. Ild2RiM tO be a box." No use is made of numer-

a control rod has been moved beyond the critical po-

ical models.

step in designing a PID controller

sition, arid adjustments in a device's position can only is to define an error signal as the difference between

be made over a interval. In contrast, the rate of the demanded and measured outputs of the process.

change of reactivity can be immediately altered by merely initiating movement of a control device. More- over, a wide range of rates of change is achievable through the use of variable-speed stepper motors. Fourth, the rate of change of reactivity corresponds to the effect of a changing prompt neutron population while the reactivity itself reflects other effects includ- ing changing delayed neutron precursor populations and changing distributions of delayed neutron precur- sors within the defined groups. Precursor populations are a function of the power history and therefore can- not be altered on demand. In contrast, the prompt neutron population is essentially a function of only the current power level and is therefore immediately con- trollable. Heñce, if an immediate change is required in a reactor period, an adjustment should be made in the rate of change of reactivity rather- than in the reactiv- ity itself.

In summary, the reason for using the rate of change of reactivity as the signal to the actuator is that it is itself directly controllable and, upon being changed, it will have an immediate effect on the course of the transient. Additional information has been given previously.’‘'°

IV. CONTROL LAW DESIGN

The Massachusetts Institute of Technology ( MIT ) has btcn engaged since the late 1970s in a program to develop and demonstrate advanced concepts for the control of research, spaoe, and power reactors. A dis- tinguishing feature of this program has been its com- mitment to experiment. New control concepts are tested first by simulation and then under conditions of

Multiplication of that efror by a constant ( the system gain ) results in a proportional controller. Thus,

( 5 )

and

( 6 )

where

n q ( I 1 —— demanded ( or reference ) reactor power

level

n ( i ) = measured reactor power level e ( i ) = error signal

u ( i ) = signal to the actuator

kp — proportional gain constant.

Figure 4 depicts the process. Unless the gain can be made quite large ( in theory, it would have to be infi- nite ) , proportional controllers will exhibit offsets in that the observed power will not be fully driven to the reference value. This situation can be rectified through the addition of integral action. By doing so, any dif- ference between the demanded and measured outputs will eventually accumulate to the point where it forces the system response to the specified value. Unfortu- nately, integral action can induce oscillations as the system converges about the setpoint. These can be mit- igated by the addition of an anticipatory or derivative term that is a function of the rate of change of the error. The resulting controller is’ of the form

closed-loop didtal control on either the MITR-II or the where k, and k are the gain constants for the integral

Annular Core Research Reactor ( ACRR ) that is oper- and derivative terms.

ated by Sandia National Laboratories ( SNL ) . A con- Proportional-illtegral-derivative control is well

NUCLEA R SCIENCE AND ENGINEERING VOL. 110 APR. 1992

REFERENCE

OUTPUT ( ñ )

CONTRO L FOR RESEARCH REACTORS

_

OUTPUT {n )

t31

ERROR SIGNAL

PROPORTIONAL CONTROLLER

GAIN

CONTROL

SIGNAL

lu )

ACTUATO R PROCESS

MEASURED

OUTPUT

Fig. 4., Proportional control.

suited to maintaining a system parameter at some steady-state value and, as a result, is the control method employed by most analog devices. It can be modified to track a dimanded trajectory by specifying both a de- manded power and a demanded rate of rise of Power. In this case, it becomes

where p, ( i ) is the rate of change of reactivity that will be requéed from the actuator.' Figure 5 illustrates the approach. Control laws based on Eq. ( 8 ) have been ex- tensively evaluated for robot movement, and while the approach is widely used, certain limitations have been noted.' 2 '" These include the following:

1. The technique will cause the system to move to the desired end point, but it will not result in highly ac- curate tracking of a specified trajectory. Accomplish- ment of the latter requires incorporation of a system model in the control law.

2. There are physical limits to the speed at which an actuator can respond. This means that the gains cannot be made arbitrarily large as might be desired to rapidly overcome perturbations.

3. Even if achievable, the use of high gains to off- set poor performance may be unsafe because high gains will amplify modeling errors, inaccuracies in pa- rameter estimates, and noise. Doing so may also lead to instability.

Both PID and proportional-derivative control h,ave beefi evaluated experimeritaEy pn the MITR-II. To the foregoing list, one should add that the values of the controller gains cannot be treated as constants for nonlinear systems. A set of gains chosen to properly execute one type of transient can result in poor perfor-

‘The minus sign in Eg. ( 8 ) is appropriate because the purpose in adding the velocity term is to dampen the speed of response, thereby reducing the likelih oo d of an overshoot.

Fig. 5. Proportional-derivative control for trajectory tracking.

mance for another. Other drawbacks to this mode of control are that the selection of the gains is empirical ( and hence time-consuming ) and that there is no the- oretical basis for determining system stability.

Figure 6 shows the results of a test of proportional- derivative control on the MITR-II. The controller was tuned to raise the reactor's power from 1.0 to 1.5 MW, and except for a small overshoot, it did this properly. However, its performance was less satisfactory when used for other types of maneuvers. This raises two con- cerns. First, proportional-derivative control does per- form quite well when used to conduct the specific t r ansients for which its gains have been calibrated. Might this observation not cause a reactor operator, who has little knowledge of control theory, to place undue reliance on the controller and attempt its use for other types of transients? Second, the performance of proportional-derivative controllers can be enhanced by increasing the speed of the control device that serves as the actuator. Doing so enables the controller to offset both disturbances and nonlinear effects more quickly. Yet, is this practice desirable? Repeated movement of a control device at high speed may induce wear, and if failure were to occur during transient conditions, the consequences could have safety ramifications.

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1500

1200

1100

BERNARD and LANNING

200

150

TOO

50

10 20 30 40

TIME {s )

50 60 70

Fig. 6. Demonstration of proportional-der iv ative control.

IV. B. Period - C i ene r ated Control

Period-generated control is a model-based tech- nique that was developed at'the MIT Nuclear Reactor Laboratory in conjunction with SNL for the purpose of adjusting nuclear reactor power in a very rapid yet safe manner .’" 0 " 4 One possible application is the con- trol of the spacecraft reactors that will be used to pro-

desired trajectory. Hence, the first step in the imple- mentation of period-generated control is to define an error signal in terms of the demanded and observed power levels. The conventional approach would be to take the difference between those two quantities. How- ever, superior performance is achieved if the error sig- nal is expressed as

pel manned expeditions to Mars. The technique has since been extended to nonnuclear systems and is now being studied as a genera ) -purpose method for the tra- jectory control of systems for which a demanded rate

where

e ( i ) = in gi + iai za ii ) ,

( 10 )

is to be observed.' 5 The most well-known examples of period-generated control are the MIT-SNL Period- Generated Minimum Time Control Laws, which have been extensively demonstrated on both the MITR-II and the ACRR. Derivations of these laws were previ- ously given.” The intent here is to demonstrate the utility of the model-based, period-generated approach for the trajectory tracking of nonlinear systems.

It is desired that the reactor neutronic power con- form to a certain trajectory, Accordingly, some mea- sure of the rate of rise of the power is needed. A logical choice is the inverse of the reactor period, which is de- fined as

( 9 )

The objective of the control method is to determine a demanded inverse period, compute the rate of change

np ( / ) = demanded trajectory n ( i ) = observed trajectory

j - positive integer.

A Taylor series expansion of this logarithmic expres- sion reveals the rationale for selecting this particular arithmetic form for the error signal:

e ( I ) — 1n[ny ( I + yA/ ) ] — In Qty ( ) ]

+ In ( np ( f ) } —In[n ( i ) ]

d

di

of reactivity ( the actuator signal ) needed to generate that inverse period, and then apply the calculated rate of reactivity change to the actual system. Dping so should cause the reactor's power to rise or fall on the

d In I ^d ! Ill + In ( n u t ) /o ( i ) )

dt

NUCLEAR SCIENCE AND ENG I NEER l NG VOL. 110 APR. 1992

CONTROL FOR RBSEARCH REACTORS 433

where ‹r, ( f ) is the inverse period that corresponds to the power trajectory n d I ! - us, the error signal used in period-generated control is the sum of a feedforward action from the inverre period associated with the demanded trajectory, and a proportional action from the quotient of the demanded and observed system outputs. The former defines the system path. The lat- ter provides corrective action against deviations. it has been shown that the value of y should be at least 2 to ensure stability against oscillations. *

The second step in the application of period- generated control is to define a demanded period in terms of the error signal, Thus,

( 12 )

where F, and Pg are the integral and derivative times. Equation ( 12 ) is a conventional PID feedback expres- sion. The quantity oq ( I I is the inverse period that will either maintain the system on the demanded traJectory or, should a deviation exist, restore it to that trajec- tory. That is, «p ( r ) equals «, ( I ) when the observed power i5 o n the demanded trajectory. Otherwise, the two differ with og ( I ) driving the system to the de- manded trajectory.

The third step in the , application of period- generated control to a reactor is to obtain an appropri- ate model and thereby relate the demanded period to the actuator signa ) , which is the required rate of change of reactivity. The needed expression is readily obtained by rearranging terms in the dynamic period equation. Doing so yields the following:

where the symbol p denotes the rate of change of re- activity associated witb temperature-iaa»ccd reedback from the reactor's fuel and coolant. Equation ( l3 ) is a

system model, and the parameters contained therein

where

Ac = time step

6 = number of time steps over wbich it is desired that the system attaip the specified trajectory.

The quantity k should be chosen to be small because the objective of period-generated control is to cause the controlled parameter to begin rising ( or falling ) quicRiy at the demanded rate. For this to occur, the accelera- tion term must rapidly die out. However, as a practi- cal matter, there is a lower limit to the value of k. Should it be made too small, ‹itf ) will be quite large, and an excessive rate of change of reactivity will be needed for transient initiation.

The demanded power trajectory can now be real- ized by applying the quantity p, to the actual system. Thus, the observed inverse period is as follows:

tJ ) = [d — p ( ! ) ] '

- t•. ( ‹ ) + k›« ) + ›. ( t › + Zé: ›,—›:‹»i

- i-‹a ( › ) + i•‹' ) i2+ s ( '›•‹'H ) . tu›

The sequence of calculations is as follows. First, the demanded trajectory and the time allowed for its at- tainment are specified. This allows determination of n z ( t ) and k. Next, measurement of the actual power level allows calculation of the error si gnal and estima- tion of the demanded inverse period from Eqs. ( 10 ) and ( 12 ) . The reactivity and the effective multigroup decay parameter are then determined via either mea- surement or calculation or a combination thereof. Sub- stitution of ‹rp ( / ) into Eg. ( 13 ) and then substituting Sq. ( 14 j for ñ ( / ) gives the rate at which the reactivity should be changed during the next time step in order to cause the reactor's power to move to the desired tra- jectory. Once on that trajectory, the acceleration term will in theory become zero. As a practical matter, it will remain finite, acting as a source of feedback to correct for minor deviations in the tracking of the spec- ified path. Repetition of the sequence of calculations fields the rate of change of reactivity needed to move the system output along the desired trajectory. Once n ( r} attains a new desired steady-state value, the quan-

are estimates. This is indicated symbolically through

tity is set to zero, and the rate of change of the

use of the superscript caret.

The next issue is the treatment of the quantity u› ( I}, which represents the system acceleration. For most transients, it is acceptable to make the prompt- jump approximation and set the quantity 1‘, which is the prompt neutron lifetime, to zero. Under such con- ditions, acceleration effects can be neglected. However, if trajectories with periods of a few seconds or l ess are to be tracked, then allowance must be made for system acceleration. This is achieved using the foilowiiig relation:

reactivity needed to halt the transient is generated. The acceleration term will be no n z ero at this time.

Shown in Fig. 7 are the power and reactivity pro- files obtained during a trial of period-generated con- trol on the MITR-II. Also, the strip-chart recording of this traiuient is shown as an inset. The reactors power was increased from I to 2 MW under conditions of closed-loop digital control using a variable-speed step- ped motor to adjust the net reactivity. The specified pe- riod was 100 s. The transient was completed at the expected time of 69 s, and the shape of the power pro- file was exponential. Also of interest is the steep slope

ñ ( / ) = [uy{t ) - u ( t ) ] /yAf ,

( 14 )

of the reactivity profile during both transient initiation

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2.0

BERt'4ARD and LAHNIlsfCi

1.6

1.4

1.2

1.0

TIME ( s )

Fig. 7. Gosed-loop power increase from 1 to 2 MW usirtg MIT-SNL Period-Generated Minimum Time Control Law on a 100-s period.

and termination. This occurred because the reactor's period was in effect being stepped from infinity ( aeady state ) to 100 s and, upon transient completion, from 100 s back to infinity. Adjustments of the prompt neutron population, which are indicated by the rate of change of reactivity, were being used to drive the transient.

Figure 8 shows the power and reactivity profiles

obtained during a trial of period-generated control on the ACRR. The power was increased by three orders of magnitude, from 0.49 to 490 kW, on a demanded period of 3.0 s with the reactor initially subcritical by 800 mbeta ( 80 e ) .* Again, note the rapid rates of change of reactivity needed for both transient initiation and termination. The tracking performance achieved in both of the foregoing demonstrations was excellent. Of significance is that both the MITR-II and the ACRR tests were performed with essentially the same soft- ware. Yet, the two reactors are very different. The MITR-II uses fully enriched 2 3 ’ U fuel and forocd cir- culation. The ACRR uses 209c enriched BeO-U fuel and operates under adiabatic conditions. Use of the same software to conduct power transients on these

*Reactivity is a fraction and therefore is dimensionless. Nevertheless, several systems of “units” are in use. The one used here is io define a reactivity equal to a reactor's de- layed neutron fraction as I Beta. This unit is further divided into mbeta with 1000 mbeta equaling 1 Beta. Another widely used system o/ reactivity uxits is dollars and’ cents. I Beta equals I $ or 100 t. Hence, 1 0 equals 10 mbeta.

two very different reactors derñonstrates the generic nature of period-generated conWol.

V. RATIONALE FOR MODEL-BASED CONTROL LAWS

What accounts for the superior tracking capability of period-generated control as compared with the di- rect use of proportional-derivative action? The most significant difference between the two methods is that the period-generated approach incorporates a model of the system dynamics. While this does make the tech- nique's implementation more complex, it also results in substantial benefits. In particular, major advantages to period-generated control are that it is applicable to nonlinear systems and that, in the case of rate- constrained systems, the resulting control action ap- proaches time-optimal behavior. The basis of these attributes is discussed here.

V. A, Nonlinear Control

The period-generated approach achieves proper control of nonlinear systems through the use of a model of the process dynamics. Specifically, a feed- back signal ( the demanded inverse period ) is computed from a comparison of the demanded and observed val- ues of the system output. This signal is then input to an inverse dynamics model of the process that is being controlled. The solution is a form of feedforward con- trol in the sense that the output of the inverse dynam- ics calculation is the actuator signal, which, upon

NUCLEAR SCIENCE AND ENGINEERING YOL. 110 APR. 1992