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Time optimal spatial offset control in nuclear power reactors
Annals of Nuclear Science and Engineering, Vol. 1, pp. 529 to 536. Pergamon Press 1974. Printed in Northern Ireland
TIME OPTIMAL SPATIAL OFFSET CONTROL IN NUCLEAR POWER REACTORS A. A. EL-BAsSIONI and C. G. PONCELET Nuclear Science and Engineering Division, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A. (Received 25 March 1974)
Abstract--The problem of minimal time control of xenon spatial oscillations in nuclear power reactors is investigated and the effect of constraints on the control variable (rod position), spatial offset magnitude and rate of change, which is closely related to the rate of change of the local power density, is studied. The concept of spatial offset phase plane is extended to include direct reference to the measured spatial offset. The optimal constrained and unconstrained trajectories and switching curves are analyzed in the phase plane. Work done on the Carnegie-Mellon University digital reactor simulator confirmed the success of the derived theoretical results. Operational strategies that embody both the strength of the theory and the simplicity needed for practical application are recommended. INTRODUCTION
theoretical aspects of this problem with the spatial Xenon-induced spatial flux oscillations in large offset which is a reactor observable, generating the nuclear power reactors are of significant operational spatial offset phase plane switching curves and concern because of their relevance to fuel element trajectories. In this work, the problem of suppressing the performance, plant performance and operational capability, flexibility and reliability. These oscil- spatial xenon oscillations is pursued from a still lations are normally triggered by previous control more practical point of view. Constraints on the actions, load changes or other system perturbations. power tilt and the rate of change of local power Effective and versatile operational control procedures density are imposed. Such constraints are implied are essential to insure operational capability, by fuel integrity requirements. The Minimum including load follow capability of the power plant, Principle is applied and the phase plane behaviour is and to minimize stress cycles on fuel elements in the thoroughly analyzed. Exact timing for different facets of the optimal control policy is well established. reactor core. The problem of optimal control of spatial xenon These new features make the derived control oscillations in nuclear power reactors has been strategy superior to those of the work of E1-Bassioni studied by Wiberg (1965, 1967), Stacey (1968), and Poncelet (1974). E1-Bassioni (1969), Christie and Poncelet (1969), and THE REACTOR MODEL Bauer (1972). Optimal control strategies as well as Assuming that the reactor is subject to oneoperationally motivated control policies have been dimensional oscillation in the s direction, we shall developed. The former usually results in compli- define the spatial offset A by cated control procedures that cannot be easily implemented; while the latter, in spite of being A = 4~(s)ds ¢(s)ds ¢(s)ds (1) /o simple and motivated by realistic need, require exhaustive trials to improve the control policy, and where $(s) is the flux distribution in the s direction its conclusions usually lack the generality of ana- and H is the reactor core dimension in that direction. lytical optimal controls. In case of a PWR, s can be taken to be the axial Minimal time controls associated with this problem direction and H the core height. The spatial offset have been investigated by Christie and Poncelet is then referred to as the axial offset and it is con(1974) and E1-Bassioni and Poncelet (1972), using tinuously monitored by programming the output of the Pontryagin Minimum Principle. The control two neutron detectors located in the upper and was shown to be of the bang-bang type. Such policy lower halves of the core. can be implemented by instantaneous variation of We further assume that our system can be the control strength. In case of pressurized water satisfactorily described by the one group diffusion reactors (PWR) this can be achieved practically by equation with temperature and doppler feedbacks shifting full length control rods or part length control and the associated xenon and iodine equations. rods among preassigned core locations. Energy transients during xenon spatial redistribution A semi-operational procedure has been developed are negligible, such that a one energy group treatby E1-Bassioni and Poncelet (1974), coupling the ment is fully adequate. It has been shown by
{;/
529
fo
}/f?
A. A. EL-BAsSIONIand C. G. PONCELET
530
E1-Bassioni and Poncelet (1974) that A can be decomposed into two main components. The first is an oscillatory component which can be associated with the xenon-iodine behaviour. The real and imaginary parts of this component are found to satisfy zi1 = bA 1 - coA2 - u(t)
(2a)
z ~ = ~oA 1 + bA2 - a u ( t )
(2b)
THE CONTROL PROBLEM Due to fuel integrity and operational flexibility considerations, it is desirable to eliminate any spatial oscillation in the shortest possible time. In that sense we can state the control problem as follows. Given an initial perturbation Ato, A~0 we want to find the optimal control Uopt(t) which minimizes the functional J =
with the initial condition AI(0) = A10, A2(0) = A20
dt
(4)
subject to the following control and state constraints:
where b
is the stability index which is taken to be either positive or negative and close to zero to simulate a typical case of unstable or stable oscillations to be controlled; co is the angular frequency of the oscillations; u(t) is the control variable which depends on the control rod position and strength; and cr is a constant. The second component of the spatial offset is associated with the prompt flux response and it can be approximated by
-
M <_ u ( t ) < N
(5a)
I±R(t)l < c
(5b)
d AR(t )
~ K
(5c)
where tf is the final control time and where N, M, C and K are constants. The choice of the control bounds N and M of (5a) depend on the control rod strength and preassigned core location. The bounds C and K on the spatial offset and its time rate of change are chosen to satisfy requirements on the power density distribution and fuel integrity. Constraints (5b) and (5c), including the quantitative u(t) Aa(t ) ~--- - (2c) values of C and K, correspond to operational q restrictions imposed on plant operation by reguwhere q is a constant. All parameters b, ~o, or, and q latory agencies. The constraints (5a, b and c) can be expressed can be determined experimentally. The component Aa is not excited as long as u(t) is at its equilibrium more uniformly as: Wl = - ( N -- u ) ( M + u) < O (6a) value u(t) = 0 which corresponds Aa = 0. In terms of these two components the measured ~2 = -- (C -- AR)(C + AR) < 0 (6b) spatial offset can be shown to satisfy ~a = - (K - z~R)(K + AR) --< 0 (6C) b +crco AR(t) = Al(t) + Aa(t) bZ + e° 2 u(t) (3a) The application of the Pontryagin Minimum Principle includes the definition of the Hamiltonian associated with equations (2a, b and c) as
1 = Al(t) + - 7 u(t) q
H = 1 + 21(t)[bAl(t ) -- coA2(t) -- u(t)]
where
+ 2a(t)[coAl(t ) + bA2(t ) -- tru(t)] 1,
{1
b +_cro~t
--/q(t)~l --/~2(t)~2 --/~a~%
>+ 2j
With the measured spatial offset, associate an imaginary component Az(t ) given by Gb
Ai(t) = A2(t)
--
where ~t(tl) and 22(0 are called the costate variables. They are continuous and together with the spatial offset satisfy the Euler Lagrange equations OH
O)
b2 + °~2 u(t)
(3b)
(7)
~'i(t) =
-
-
0A---i.'
OH
Ai = ' - ~ i
i = 1,2 (8)
This new component when associated with AR(t ) which hold along the optimal trajectory ko~. will essentially provide the same amount of informa& ( t ) , j = 1-3 are Lagrange multipliers that are tion given by Al(t), A2(t), and Aa(t). piecewise continuous and are continuous at each
Time optimal spatial offset control in nuclear power reactors point of continuity of
u(t) and
531
satisfy
when
~j < 0
#~=0
and when
~oj >_ 0
Fj > 0
(9)
,
Z~MI o l5 I o~s
The optimal control policy Uo~,will be selected such that H[Aio,(t), ;tio,(t), Uov(t)]
~_ H[Aio~(t),
2~o~(t),u(t)] (10)
"
x
T
will hold subject to the condition, (6) namely V~ -<- 0
j = 1-3
THE CASE OF CONTROL CONSTRAINTS This is the case where only constraint (6a) is considered. Analysis of this problem has been completed by E1-Bassioni and Poncelet (1974), and the main conclusions of that work are reviewed here '-o o'i-o,o o. -" 'o,o I >o%o' ~._ Z~~,~--/~ R-~ for completeness. These are: b:OO2/hr J,,,~_N 4 ~ : 0 2 ra d/hr. (a) Singular control is not possible. q' (b) The optimal control is of the bang-bang type Fig. l. The A1 -- A2 phase plane. that requires instantaneous motion in and outside the core in case of full length control rod, or between A R - - A z trajectory behaves quite differently. At an upper and lower core position in case of part the moment of switching the state will experience a length rod. It was also shown that the residence jump from/°2 to P2 followed by the portion P2'ON" time of the control rod in any of the two assigned that corresponds to 1'20 of Fig. 1. Once the origin is locations must not exceed a period of ~r/~o. reached in the At - A2 plane or the point ON' in the (c) As shown in Fig. 1, the free and controlled AR -- Az plane, the control is switched back to its trajectories in the A~ -- 2x~ phase plane are spirals equilibrium position. This will cause a state jump with centers at the points 0, AM and AN for free, from O~v'to 0 as shown in Fig. 2. M type and N type controls respectively. Two In short the A1 -- A2 trajectory PtP~Ocorrespond portions of the controlled spirals A0 and B0 (Fig. 1) to P1P~P2'ON'Oin the AR -- A z phase plane. The identify the only trajectories leading to the origin of superiority of the latter is shown from the extra the phase plane and both will serve as switching curves. Figure 1 also shows the point P on the switching ~-.~ s'~ o~, curve AO. In case the control is of N type, Aa.p represents the oscillatory component of the spatial offset while ANO~v is the prompt mode (Az) contribution. Thus O yP represents the total spatial offset vector with A~ the measured spatial offset as its horizontal projection and Az its vertical component. Equations(3a andb) arerestatements oftheserelations. Since it is more practical to work directly with the \\ 4- a'~\ measured spatial offset, the 2x~ -- Az phase plane is introduced in Fig. 2. The points 0, AN', AM', and P ' in this figure correspond to the points 0, A~v, A ~ and P of Fig. 1 respectively. The trajectory P1P~in both Figs. 1 and 2 represent typical free oscillation trajectory. Once the switching O~ S t - - ~ , , , . \ \\ curve is crossed at P~ control switching to the N type is required. Once this is accomplished the trajectory will follow the P20 portion of the switching curve in the At -- A2 plane as shown in Fig. 1. In Fig. 2 the Fig. 2. The A~ -- Az phase plane. '
\\ - ' ~ -----~
',i\
\/
,
A. A. EL-BAsSlONIand C. G. PONCm.ET
532
details about the measured spatial offset response to any control action. We should also note the presence of AR jumps whenever the control is switched. These jumps are in violation of the constraint 6c. On the other hand, it is possible to violate constraint 6b in case of large size free oscillations or if the optimal control action is in such direction as to increase the measured spatial offset AR beyond the assigned upper or lower bound. Lines S1S 1' and $2S~' in Fig. 2 define the region in which AR -- Az state is allowed to vary. THE CASE OF CONTROL AND STATE CONSTRAINTS I n this section we intend to solve the control problem of Section 3 with constraints 6a, b and c. I n this case the costate variables satisfy
C 3 = -- q'K(b 2 + co~)/e C 4 = { - 2q'Kb + q'(b 2 + co2)AR¢ -- (2b + q')C3)/e A B =
•
aA R
a~iR
42 = coAz -- b22 -- 2/~2/~R 0A2
(lla)
(llb)
= Ee q~ + Fe t~ + C3(t f - t )
--C2
1
+ (b + q' --12)OI + q'K +12C 4 + C a } F=Os-E-C4
0t = N o r -- M
e =b 2+co ~+bq'+wqaand b + q
We have mentioned above that constraint 6c is not satisfied each time control is switched. Thus a constrained arc on which :xr~ = -4- K has to substitute for the undesirable state jump. Satisfaction of 6c implies that any controlled trajectory has to start and end with ziR = 4- K arcs. Also in case the optimal control policy as defined in Section 4 requires switching between N and M types control or vice versa. The transition between these two control levels has to be accomplished such as to satisfy 6c. From the operational point of view, this requirement means that the control rod has to be inserted or withdrawn gradually in a way that will result in a rate of change of spatial offset equal to its maximum or minimum allowable values. The control variation that will achieve a constant rate of change of measured spatial offset and the corresponding A2 variation given by:
u(t)
--A
Cl)
E - ll _ 12 { - bq'AR¢ + o~q'A2¢
OAR 21 = -- b2z - °~22 - 2p2AR(t) 0A1
-- 2/~3AR " ~ 1
1
tl - t~ { t l C 2 -
- (o)2 + bq'a)
The constraint 6b will not be satisfied if the A_~ - AI trajectory tends to cross the A• = --I-(7 lines $1S1,, S2Sv shown in Fig. 2. I n such a case control action has to be taken such as to keep AR on the proper bound. Fig. 3 shows AR = C curves in the A1 - A2 phase plane, with each curve starting at a different value of A2 and with the control u = - M. I n this case the spatial offset and control
It
+ C 4 (12)
A2~(t) = AR0 + K t
= A R f -- K(tf -- t) A2(t) = e q t + B e Z ~ * + C l ( t 1 _ t ) + 6 " 2
(13) (14) thr.
where AR0 and AR¢ are the initial and final values of the spatial offset and
C x = --q'K(ba - co)/e C2 = -(crKq' + 2b + q')/e
a,
Fig. 3. An = Cand An = Kcurves in the A1--A2 plane.
Time optimal spatial offset control in nuclear power reactors variations are described by
TA, (15)
AR = C A2 = A ' e q t + B'et2 ~ +
(16)
u : E ' e h t + F'et~ t +
(17)
where A' =
(b -- l.,)A2o +
¢DC +
12n
,I -020 B' =,520--A
533
-0
--
E ' = {(b ÷ q' -- 12)0 M + 12~ -- b q ' C + q'ooA2o}/(ll_l~) F ' = 0 M -- E ' -= q ' C ( b 2 + o~2)/e = q'C(ba - w)/e 0M =
-
-
M
T h e s w i t c h i n g curves
F r o m the above discussion we conclude that any optimal trajectory will consist of some or all of the following types of arcs:
Fig.
4.
Switching curves and An = K lines in the An -- AI plane.
curve is intersected a n d a [d]-type constrained arc will carry the state to the second switching curve A g o . Once the state is on this curve u = + N a n d it will be (b) Arcs corresponding to the N type control followed until the point g where a d-type arc will lead (u = N). the state to the origin. (C) A R = :£Carcs. I n Fig. 4 all the A R = C curves are m a p p e d on the S 1 S 1' line while the control variations are shown in d (d) ~tt AI: = ~ K arcs. Fig. 5 with the dotted line showing the control at the m o m e n t of switching to [(/]-type arcs. These d-type Thus the switching curve of Fig. 1 a n d 2 has to be arcs are the trajectories shown in Fig. 4. modified such that the origin will be reached through an arc of type [d]. This modified switching curve is N I u(~) shown in Fig. 3 as A g o consisting of [a]-type arc .4g 006~ linked to d-type arc go. I Also, since we c a n n o t switch instantaneously between arcs of type a and b, they have to be linked 0,04 by d-type arcs or if (6b) is n o t satisfied this link has ~2o" -,30 to be of type c arcs followed by type d. 0.02 i I n Fig. 3 the line B F represents the switching curve between [c]-type a n d [d]-type constrained arcs. This 4 5 6Time switching curve can be defined by substituting in _.oU 3 hrs equations (12), (13) a n d (14) with the proper con-002 stants and final states selected on the switching curve A g o . The points at which A• = C must be on -0.O4[ the BF curve. Figure 3 shows the fourth q u a d r a n t of the A 1 -- A o phase plane. I f the state lies somewhere in the lower left-hand region, then the proper control -0081" -M is M type. Once the trajectory intersects S1Sz', the control is changed such that a [c]-type arc will be followed. This will continue till the B F switching Fig. 5. Control variations for An = C,/Xn = K sequence. (a) Arcs corresponding to the M type control (u = -- 3,/).
~
V
A. A. EL-BASSIONIand C. G. PONCELET
534
DIGITAL SIMULATION Several control experiments were performed on the Carnegie-Mellon University reactor digital simulator to demonstrate the effectiveness of the suggested optimal controls. Slight deviations from optimal were introduced on purpose to simulate a possible reactor operator error. Two of these experiments were selected since they represent all the features of the suggested theoretical control policies. In the first of these experiments the physical variables, the A 1 - A 2 and the A R - - A I phase planes are shown in Figs. 6-8 respectively. As shown in Fig. 6, decision to interfere was made at the point P0. Control rod was inserted such as to
,08 :.=
52
P / L Rod Posilion"
0o
0.8
o
oo 00
I
i
I
I
I
I
1
T
I
r
1
o.18 o.22 o.z6 o,3o
I
~
I
-0.08 -0.12 P~
-0.16 -0.20
/
I
I/ I/
[]
O0
04
o
§ 0.2
;:on,
U 0 ~
o.ed ×.o°
Controlled Oscillations e.,..-- Fr ee Oscillotions
0.1 0
<:- 0.1 0 ~r,[
plane. It is distinguished from Fig. 7 by the details of variation of the measured axial offset while it lacks the presence of the BF switching curve. Figure 6 shows the xenon and iodine deviations as introduced by Bauer (1972). Physically the first part
Pr
't
0
o
Fig. 7. Control experiment 1A1 -- A2 trajectory.
Controlled Iodine
0.2C
o
~
- 0.04
-0'32 t
OOCl O
0
oOO
0.6
d
~o ~ - - - . ~ 1 4
-0.2BI
F r e e Xenon
Free Iodine/'~
F
0.04
-0.24
..•
J
O.OB
0.55~-. 0.45~
PO
PS
w~.~0~3 5 ~
-0.2 0 -z
I
I 0
I
] 4
I
] 8
I
2 1
I
6 1
I 2I 1 2I I I ] ~ 0 4 28 Z
T i m e in hrs
\t
Fig. 6. Control experiment 1. produce a d-type of constrained arc PoPz. At P1 u = - M, which means that the rod has reached its lower core position. The arc P1P2is an [a]-type arc and at P2A2 = C = 0 . 2 . P2Ps is a c-type arc with Pa almost on the switching curve BF (Fig. 7). At P3, control is varied to produce PaPa, a [d]-type arc, with P4 on the switching curve Ago (Fig. 7) and u = N. The last part P4PsP6is simply following the switching curve to the origin. As we see in Fig. 7, a wrong decision as to the exact moment of switching resulted in residual oscillation PeP7 of maximum amplitude 0.02 which is small when compared to the free oscillation amplitude of 0.225. Figure 8 shows the controlled trajectory in the A R - - A I phase
/
_~,I-O o5
/I
I F"t~ Pe i
I
Is~ ~%,
I
- 0 . 3' 0' ' -0,20
o.;o •
-
P3
~Po -0.I 5
\
-0.4~ _0.55 1
Fig. 8. Control experiment 1 An
0K==0O. 2. 3 / h r ,
--
Az trajectory.
535
Time optimal spatial offset control in nuclear power reactors of the control PoP1P2P3 is characterized by shifting the flux such as to enhance burning xenon and build iodine in the upper half; in the second part P4PsP6 the excess xenon is burned while keeping the declining trend of iodine. At the end of the control interval both of the xenon and iodine keep decreasing but the difference in phase keeps the small size oscillation going. PRACTICAL CONTROLS As shown above there are four types of optimal arcs. Any optimal trajectory will consist of a subset or all of these arcs. An automatic control system can be programmed to find the sequence and duration of these arcs required to eliminate the oscillation. However, in the case of a human operator a simple fail-safe policy could be adopted. Christie and Poncelet (1973) suggested a policy which
p~,
0.4£
ONv. . . . . . . . . . . . . .
. "Pz
\x
\\\
"%, ,I
,
•-oJ~
,
,
i
',,,,\ .NP,±
-oo~?~.~
-o.~5~o
,
oos~15",,
\
t: _!o.2o. :
,
../" !
-0.40 91
o
g. -~
108
~.
eo
-
Fig. 10. Experiment 2 An -- A1 phase plane.
4 P~ • 0.20
•
" Free O s c i l l o t i o n
•
°•~
0.10
8
I
p ••o
o Po
o
Controlle~
I' I ' - ' ~ - ' - ' ' ' ' ' ~ P e
O s c i l l ~ an'NN~
-OAO
P3 -0.20 I
I
2
t
i
4
I
i
6
i
I
S Time
I
I
10 in hrs.
I
I
'f
12
Fig. 9. Experiment 2 failsafe policy.
i
14
I
16
CONCLUSIONS The minimum time control of spatial xenon oscillations subject to constraints on the control, spatial offset and its rate of change is analyzed. Four types of constrained phase plane arcs were identified as the building blocks of any optimal trajectory. A measured spatial offset phase plane was introduced and its switching curves were defined. Digital simulation showed the effectiveness of the proposed optimal control policy. To suit the capacity of a human operator, a suboptimal fail-safe control strategy was proposed. This policy is not very sensitive to small errors in
consists of waiting until AR is close to its maximum switching times; it is clear and simple and can be or its minimum, then move the control rod in the flux implemented using full length or part length control peak. In terms of this work this is a suboptimal rods. policy and it has to be modified to include gradual motion of the control rod to produce a type [d] arc 5.5 while inserting the rod in the flux peak and finally when withdrawing it to its equilibrium position. 4.2 The moment of start of this fail-safe control action and the time duration of each arc of the hrs.~5 controlled trajectory can be determined from the 2.~ AR -- Az phase plane and the location of its switching curves which can be easily generated. 1.5 Figure 9 shows the second control experiment (fail-safe policy), while Fig. 10 shows its A• -- Ae o.5 I I I I I I I I I I I I I I I 02 0.06 0.10 014 018 0.22 0.26 0.30 trajectory. In Fig. 1 I, the time before reaching the A m mox, peak and control duration are plotted against the Fig. 11. Failsafe control timing curves. value of h• peak for free oscillations.
A. A. EL-BASSIONI and C. G. PONCELET
536
E1-Bassioni A. A. (1969) Ph.D. Thesis, University of Michigan. Bauer D. C. (1972) Ph.D. Thesis, Carnegie-Mellon E1-Bassioni A. A. and Poncelet C. G. (1972) Trans. Am. University. nucl. Soc. 15, 285. Bauer D. C., Poncelet C. G., Impink A. J. Jr., Jones K. A. EI-Bassioni A. A. and Poncelet C. G. (1974). NucL Sci. and Newton R. A. (1972), Trans. Am. nucL Soc. 15,887. Engng. 54, 2. Christie A. J. and Poncelet C. G. (1969) Trans. Am. nucL Stacey W. M. Jr. (1968) Nucl. Sci. Engng 33, 2. Soc. 12, 764. Wiberg D. M. (1965) Ph.D. Thesis, California Institute of Christie A. J. and Poncelet C. G. (1973) Nucl. Sci. Engng Technology. 51, 10. Wiberg D. M. (1967) Nucl. Sci. Engng 27, 3. REFERENCES
TIME OPTIMAL SPATIAL OFFSET CONTROL IN NUCLEAR POWER REACTORS A. A. EL-BAsSIONI and C. G. PONCELET Nuclear Science and Engineering Division, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A. (Received 25 March 1974)
Abstract--The problem of minimal time control of xenon spatial oscillations in nuclear power reactors is investigated and the effect of constraints on the control variable (rod position), spatial offset magnitude and rate of change, which is closely related to the rate of change of the local power density, is studied. The concept of spatial offset phase plane is extended to include direct reference to the measured spatial offset. The optimal constrained and unconstrained trajectories and switching curves are analyzed in the phase plane. Work done on the Carnegie-Mellon University digital reactor simulator confirmed the success of the derived theoretical results. Operational strategies that embody both the strength of the theory and the simplicity needed for practical application are recommended. INTRODUCTION
theoretical aspects of this problem with the spatial Xenon-induced spatial flux oscillations in large offset which is a reactor observable, generating the nuclear power reactors are of significant operational spatial offset phase plane switching curves and concern because of their relevance to fuel element trajectories. In this work, the problem of suppressing the performance, plant performance and operational capability, flexibility and reliability. These oscil- spatial xenon oscillations is pursued from a still lations are normally triggered by previous control more practical point of view. Constraints on the actions, load changes or other system perturbations. power tilt and the rate of change of local power Effective and versatile operational control procedures density are imposed. Such constraints are implied are essential to insure operational capability, by fuel integrity requirements. The Minimum including load follow capability of the power plant, Principle is applied and the phase plane behaviour is and to minimize stress cycles on fuel elements in the thoroughly analyzed. Exact timing for different facets of the optimal control policy is well established. reactor core. The problem of optimal control of spatial xenon These new features make the derived control oscillations in nuclear power reactors has been strategy superior to those of the work of E1-Bassioni studied by Wiberg (1965, 1967), Stacey (1968), and Poncelet (1974). E1-Bassioni (1969), Christie and Poncelet (1969), and THE REACTOR MODEL Bauer (1972). Optimal control strategies as well as Assuming that the reactor is subject to oneoperationally motivated control policies have been dimensional oscillation in the s direction, we shall developed. The former usually results in compli- define the spatial offset A by cated control procedures that cannot be easily implemented; while the latter, in spite of being A = 4~(s)ds ¢(s)ds ¢(s)ds (1) /o simple and motivated by realistic need, require exhaustive trials to improve the control policy, and where $(s) is the flux distribution in the s direction its conclusions usually lack the generality of ana- and H is the reactor core dimension in that direction. lytical optimal controls. In case of a PWR, s can be taken to be the axial Minimal time controls associated with this problem direction and H the core height. The spatial offset have been investigated by Christie and Poncelet is then referred to as the axial offset and it is con(1974) and E1-Bassioni and Poncelet (1972), using tinuously monitored by programming the output of the Pontryagin Minimum Principle. The control two neutron detectors located in the upper and was shown to be of the bang-bang type. Such policy lower halves of the core. can be implemented by instantaneous variation of We further assume that our system can be the control strength. In case of pressurized water satisfactorily described by the one group diffusion reactors (PWR) this can be achieved practically by equation with temperature and doppler feedbacks shifting full length control rods or part length control and the associated xenon and iodine equations. rods among preassigned core locations. Energy transients during xenon spatial redistribution A semi-operational procedure has been developed are negligible, such that a one energy group treatby E1-Bassioni and Poncelet (1974), coupling the ment is fully adequate. It has been shown by
{;/
529
fo
}/f?
A. A. EL-BAsSIONIand C. G. PONCELET
530
E1-Bassioni and Poncelet (1974) that A can be decomposed into two main components. The first is an oscillatory component which can be associated with the xenon-iodine behaviour. The real and imaginary parts of this component are found to satisfy zi1 = bA 1 - coA2 - u(t)
(2a)
z ~ = ~oA 1 + bA2 - a u ( t )
(2b)
THE CONTROL PROBLEM Due to fuel integrity and operational flexibility considerations, it is desirable to eliminate any spatial oscillation in the shortest possible time. In that sense we can state the control problem as follows. Given an initial perturbation Ato, A~0 we want to find the optimal control Uopt(t) which minimizes the functional J =
with the initial condition AI(0) = A10, A2(0) = A20
dt
(4)
subject to the following control and state constraints:
where b
is the stability index which is taken to be either positive or negative and close to zero to simulate a typical case of unstable or stable oscillations to be controlled; co is the angular frequency of the oscillations; u(t) is the control variable which depends on the control rod position and strength; and cr is a constant. The second component of the spatial offset is associated with the prompt flux response and it can be approximated by
-
M <_ u ( t ) < N
(5a)
I±R(t)l < c
(5b)
d AR(t )
~ K
(5c)
where tf is the final control time and where N, M, C and K are constants. The choice of the control bounds N and M of (5a) depend on the control rod strength and preassigned core location. The bounds C and K on the spatial offset and its time rate of change are chosen to satisfy requirements on the power density distribution and fuel integrity. Constraints (5b) and (5c), including the quantitative u(t) Aa(t ) ~--- - (2c) values of C and K, correspond to operational q restrictions imposed on plant operation by reguwhere q is a constant. All parameters b, ~o, or, and q latory agencies. The constraints (5a, b and c) can be expressed can be determined experimentally. The component Aa is not excited as long as u(t) is at its equilibrium more uniformly as: Wl = - ( N -- u ) ( M + u) < O (6a) value u(t) = 0 which corresponds Aa = 0. In terms of these two components the measured ~2 = -- (C -- AR)(C + AR) < 0 (6b) spatial offset can be shown to satisfy ~a = - (K - z~R)(K + AR) --< 0 (6C) b +crco AR(t) = Al(t) + Aa(t) bZ + e° 2 u(t) (3a) The application of the Pontryagin Minimum Principle includes the definition of the Hamiltonian associated with equations (2a, b and c) as
1 = Al(t) + - 7 u(t) q
H = 1 + 21(t)[bAl(t ) -- coA2(t) -- u(t)]
where
+ 2a(t)[coAl(t ) + bA2(t ) -- tru(t)] 1,
{1
b +_cro~t
--/q(t)~l --/~2(t)~2 --/~a~%
>+ 2j
With the measured spatial offset, associate an imaginary component Az(t ) given by Gb
Ai(t) = A2(t)
--
where ~t(tl) and 22(0 are called the costate variables. They are continuous and together with the spatial offset satisfy the Euler Lagrange equations OH
O)
b2 + °~2 u(t)
(3b)
(7)
~'i(t) =
-
-
0A---i.'
OH
Ai = ' - ~ i
i = 1,2 (8)
This new component when associated with AR(t ) which hold along the optimal trajectory ko~. will essentially provide the same amount of informa& ( t ) , j = 1-3 are Lagrange multipliers that are tion given by Al(t), A2(t), and Aa(t). piecewise continuous and are continuous at each
Time optimal spatial offset control in nuclear power reactors point of continuity of
u(t) and
531
satisfy
when
~j < 0
#~=0
and when
~oj >_ 0
Fj > 0
(9)
,
Z~MI o l5 I o~s
The optimal control policy Uo~,will be selected such that H[Aio,(t), ;tio,(t), Uov(t)]
~_ H[Aio~(t),
2~o~(t),u(t)] (10)
"
x
T
will hold subject to the condition, (6) namely V~ -<- 0
j = 1-3
THE CASE OF CONTROL CONSTRAINTS This is the case where only constraint (6a) is considered. Analysis of this problem has been completed by E1-Bassioni and Poncelet (1974), and the main conclusions of that work are reviewed here '-o o'i-o,o o. -" 'o,o I >o%o' ~._ Z~~,~--/~ R-~ for completeness. These are: b:OO2/hr J,,,~_N 4 ~ : 0 2 ra d/hr. (a) Singular control is not possible. q' (b) The optimal control is of the bang-bang type Fig. l. The A1 -- A2 phase plane. that requires instantaneous motion in and outside the core in case of full length control rod, or between A R - - A z trajectory behaves quite differently. At an upper and lower core position in case of part the moment of switching the state will experience a length rod. It was also shown that the residence jump from/°2 to P2 followed by the portion P2'ON" time of the control rod in any of the two assigned that corresponds to 1'20 of Fig. 1. Once the origin is locations must not exceed a period of ~r/~o. reached in the At - A2 plane or the point ON' in the (c) As shown in Fig. 1, the free and controlled AR -- Az plane, the control is switched back to its trajectories in the A~ -- 2x~ phase plane are spirals equilibrium position. This will cause a state jump with centers at the points 0, AM and AN for free, from O~v'to 0 as shown in Fig. 2. M type and N type controls respectively. Two In short the A1 -- A2 trajectory PtP~Ocorrespond portions of the controlled spirals A0 and B0 (Fig. 1) to P1P~P2'ON'Oin the AR -- A z phase plane. The identify the only trajectories leading to the origin of superiority of the latter is shown from the extra the phase plane and both will serve as switching curves. Figure 1 also shows the point P on the switching ~-.~ s'~ o~, curve AO. In case the control is of N type, Aa.p represents the oscillatory component of the spatial offset while ANO~v is the prompt mode (Az) contribution. Thus O yP represents the total spatial offset vector with A~ the measured spatial offset as its horizontal projection and Az its vertical component. Equations(3a andb) arerestatements oftheserelations. Since it is more practical to work directly with the \\ 4- a'~\ measured spatial offset, the 2x~ -- Az phase plane is introduced in Fig. 2. The points 0, AN', AM', and P ' in this figure correspond to the points 0, A~v, A ~ and P of Fig. 1 respectively. The trajectory P1P~in both Figs. 1 and 2 represent typical free oscillation trajectory. Once the switching O~ S t - - ~ , , , . \ \\ curve is crossed at P~ control switching to the N type is required. Once this is accomplished the trajectory will follow the P20 portion of the switching curve in the At -- A2 plane as shown in Fig. 1. In Fig. 2 the Fig. 2. The A~ -- Az phase plane. '
\\ - ' ~ -----~
',i\
\/
,
A. A. EL-BAsSlONIand C. G. PONCm.ET
532
details about the measured spatial offset response to any control action. We should also note the presence of AR jumps whenever the control is switched. These jumps are in violation of the constraint 6c. On the other hand, it is possible to violate constraint 6b in case of large size free oscillations or if the optimal control action is in such direction as to increase the measured spatial offset AR beyond the assigned upper or lower bound. Lines S1S 1' and $2S~' in Fig. 2 define the region in which AR -- Az state is allowed to vary. THE CASE OF CONTROL AND STATE CONSTRAINTS I n this section we intend to solve the control problem of Section 3 with constraints 6a, b and c. I n this case the costate variables satisfy
C 3 = -- q'K(b 2 + co~)/e C 4 = { - 2q'Kb + q'(b 2 + co2)AR¢ -- (2b + q')C3)/e A B =
•
aA R
a~iR
42 = coAz -- b22 -- 2/~2/~R 0A2
(lla)
(llb)
= Ee q~ + Fe t~ + C3(t f - t )
--C2
1
+ (b + q' --12)OI + q'K +12C 4 + C a } F=Os-E-C4
0t = N o r -- M
e =b 2+co ~+bq'+wqaand b + q
We have mentioned above that constraint 6c is not satisfied each time control is switched. Thus a constrained arc on which :xr~ = -4- K has to substitute for the undesirable state jump. Satisfaction of 6c implies that any controlled trajectory has to start and end with ziR = 4- K arcs. Also in case the optimal control policy as defined in Section 4 requires switching between N and M types control or vice versa. The transition between these two control levels has to be accomplished such as to satisfy 6c. From the operational point of view, this requirement means that the control rod has to be inserted or withdrawn gradually in a way that will result in a rate of change of spatial offset equal to its maximum or minimum allowable values. The control variation that will achieve a constant rate of change of measured spatial offset and the corresponding A2 variation given by:
u(t)
--A
Cl)
E - ll _ 12 { - bq'AR¢ + o~q'A2¢
OAR 21 = -- b2z - °~22 - 2p2AR(t) 0A1
-- 2/~3AR " ~ 1
1
tl - t~ { t l C 2 -
- (o)2 + bq'a)
The constraint 6b will not be satisfied if the A_~ - AI trajectory tends to cross the A• = --I-(7 lines $1S1,, S2Sv shown in Fig. 2. I n such a case control action has to be taken such as to keep AR on the proper bound. Fig. 3 shows AR = C curves in the A1 - A2 phase plane, with each curve starting at a different value of A2 and with the control u = - M. I n this case the spatial offset and control
It
+ C 4 (12)
A2~(t) = AR0 + K t
= A R f -- K(tf -- t) A2(t) = e q t + B e Z ~ * + C l ( t 1 _ t ) + 6 " 2
(13) (14) thr.
where AR0 and AR¢ are the initial and final values of the spatial offset and
C x = --q'K(ba - co)/e C2 = -(crKq' + 2b + q')/e
a,
Fig. 3. An = Cand An = Kcurves in the A1--A2 plane.
Time optimal spatial offset control in nuclear power reactors variations are described by
TA, (15)
AR = C A2 = A ' e q t + B'et2 ~ +
(16)
u : E ' e h t + F'et~ t +
(17)
where A' =
(b -- l.,)A2o +
¢DC +
12n
,I -020 B' =,520--A
533
-0
--
E ' = {(b ÷ q' -- 12)0 M + 12~ -- b q ' C + q'ooA2o}/(ll_l~) F ' = 0 M -- E ' -= q ' C ( b 2 + o~2)/e = q'C(ba - w)/e 0M =
-
-
M
T h e s w i t c h i n g curves
F r o m the above discussion we conclude that any optimal trajectory will consist of some or all of the following types of arcs:
Fig.
4.
Switching curves and An = K lines in the An -- AI plane.
curve is intersected a n d a [d]-type constrained arc will carry the state to the second switching curve A g o . Once the state is on this curve u = + N a n d it will be (b) Arcs corresponding to the N type control followed until the point g where a d-type arc will lead (u = N). the state to the origin. (C) A R = :£Carcs. I n Fig. 4 all the A R = C curves are m a p p e d on the S 1 S 1' line while the control variations are shown in d (d) ~tt AI: = ~ K arcs. Fig. 5 with the dotted line showing the control at the m o m e n t of switching to [(/]-type arcs. These d-type Thus the switching curve of Fig. 1 a n d 2 has to be arcs are the trajectories shown in Fig. 4. modified such that the origin will be reached through an arc of type [d]. This modified switching curve is N I u(~) shown in Fig. 3 as A g o consisting of [a]-type arc .4g 006~ linked to d-type arc go. I Also, since we c a n n o t switch instantaneously between arcs of type a and b, they have to be linked 0,04 by d-type arcs or if (6b) is n o t satisfied this link has ~2o" -,30 to be of type c arcs followed by type d. 0.02 i I n Fig. 3 the line B F represents the switching curve between [c]-type a n d [d]-type constrained arcs. This 4 5 6Time switching curve can be defined by substituting in _.oU 3 hrs equations (12), (13) a n d (14) with the proper con-002 stants and final states selected on the switching curve A g o . The points at which A• = C must be on -0.O4[ the BF curve. Figure 3 shows the fourth q u a d r a n t of the A 1 -- A o phase plane. I f the state lies somewhere in the lower left-hand region, then the proper control -0081" -M is M type. Once the trajectory intersects S1Sz', the control is changed such that a [c]-type arc will be followed. This will continue till the B F switching Fig. 5. Control variations for An = C,/Xn = K sequence. (a) Arcs corresponding to the M type control (u = -- 3,/).
~
V
A. A. EL-BASSIONIand C. G. PONCELET
534
DIGITAL SIMULATION Several control experiments were performed on the Carnegie-Mellon University reactor digital simulator to demonstrate the effectiveness of the suggested optimal controls. Slight deviations from optimal were introduced on purpose to simulate a possible reactor operator error. Two of these experiments were selected since they represent all the features of the suggested theoretical control policies. In the first of these experiments the physical variables, the A 1 - A 2 and the A R - - A I phase planes are shown in Figs. 6-8 respectively. As shown in Fig. 6, decision to interfere was made at the point P0. Control rod was inserted such as to
,08 :.=
52
P / L Rod Posilion"
0o
0.8
o
oo 00
I
i
I
I
I
I
1
T
I
r
1
o.18 o.22 o.z6 o,3o
I
~
I
-0.08 -0.12 P~
-0.16 -0.20
/
I
I/ I/
[]
O0
04
o
§ 0.2
;:on,
U 0 ~
o.ed ×.o°
Controlled Oscillations e.,..-- Fr ee Oscillotions
0.1 0
<:- 0.1 0 ~r,[
plane. It is distinguished from Fig. 7 by the details of variation of the measured axial offset while it lacks the presence of the BF switching curve. Figure 6 shows the xenon and iodine deviations as introduced by Bauer (1972). Physically the first part
Pr
't
0
o
Fig. 7. Control experiment 1A1 -- A2 trajectory.
Controlled Iodine
0.2C
o
~
- 0.04
-0'32 t
OOCl O
0
oOO
0.6
d
~o ~ - - - . ~ 1 4
-0.2BI
F r e e Xenon
Free Iodine/'~
F
0.04
-0.24
..•
J
O.OB
0.55~-. 0.45~
PO
PS
w~.~0~3 5 ~
-0.2 0 -z
I
I 0
I
] 4
I
] 8
I
2 1
I
6 1
I 2I 1 2I I I ] ~ 0 4 28 Z
T i m e in hrs
\t
Fig. 6. Control experiment 1. produce a d-type of constrained arc PoPz. At P1 u = - M, which means that the rod has reached its lower core position. The arc P1P2is an [a]-type arc and at P2A2 = C = 0 . 2 . P2Ps is a c-type arc with Pa almost on the switching curve BF (Fig. 7). At P3, control is varied to produce PaPa, a [d]-type arc, with P4 on the switching curve Ago (Fig. 7) and u = N. The last part P4PsP6is simply following the switching curve to the origin. As we see in Fig. 7, a wrong decision as to the exact moment of switching resulted in residual oscillation PeP7 of maximum amplitude 0.02 which is small when compared to the free oscillation amplitude of 0.225. Figure 8 shows the controlled trajectory in the A R - - A I phase
/
_~,I-O o5
/I
I F"t~ Pe i
I
Is~ ~%,
I
- 0 . 3' 0' ' -0,20
o.;o •
-
P3
~Po -0.I 5
\
-0.4~ _0.55 1
Fig. 8. Control experiment 1 An
0K==0O. 2. 3 / h r ,
--
Az trajectory.
535
Time optimal spatial offset control in nuclear power reactors of the control PoP1P2P3 is characterized by shifting the flux such as to enhance burning xenon and build iodine in the upper half; in the second part P4PsP6 the excess xenon is burned while keeping the declining trend of iodine. At the end of the control interval both of the xenon and iodine keep decreasing but the difference in phase keeps the small size oscillation going. PRACTICAL CONTROLS As shown above there are four types of optimal arcs. Any optimal trajectory will consist of a subset or all of these arcs. An automatic control system can be programmed to find the sequence and duration of these arcs required to eliminate the oscillation. However, in the case of a human operator a simple fail-safe policy could be adopted. Christie and Poncelet (1973) suggested a policy which
p~,
0.4£
ONv. . . . . . . . . . . . . .
. "Pz
\x
\\\
"%, ,I
,
•-oJ~
,
,
i
',,,,\ .NP,±
-oo~?~.~
-o.~5~o
,
oos~15",,
\
t: _!o.2o. :
,
../" !
-0.40 91
o
g. -~
108
~.
eo
-
Fig. 10. Experiment 2 An -- A1 phase plane.
4 P~ • 0.20
•
" Free O s c i l l o t i o n
•
°•~
0.10
8
I
p ••o
o Po
o
Controlle~
I' I ' - ' ~ - ' - ' ' ' ' ' ~ P e
O s c i l l ~ an'NN~
-OAO
P3 -0.20 I
I
2
t
i
4
I
i
6
i
I
S Time
I
I
10 in hrs.
I
I
'f
12
Fig. 9. Experiment 2 failsafe policy.
i
14
I
16
CONCLUSIONS The minimum time control of spatial xenon oscillations subject to constraints on the control, spatial offset and its rate of change is analyzed. Four types of constrained phase plane arcs were identified as the building blocks of any optimal trajectory. A measured spatial offset phase plane was introduced and its switching curves were defined. Digital simulation showed the effectiveness of the proposed optimal control policy. To suit the capacity of a human operator, a suboptimal fail-safe control strategy was proposed. This policy is not very sensitive to small errors in
consists of waiting until AR is close to its maximum switching times; it is clear and simple and can be or its minimum, then move the control rod in the flux implemented using full length or part length control peak. In terms of this work this is a suboptimal rods. policy and it has to be modified to include gradual motion of the control rod to produce a type [d] arc 5.5 while inserting the rod in the flux peak and finally when withdrawing it to its equilibrium position. 4.2 The moment of start of this fail-safe control action and the time duration of each arc of the hrs.~5 controlled trajectory can be determined from the 2.~ AR -- Az phase plane and the location of its switching curves which can be easily generated. 1.5 Figure 9 shows the second control experiment (fail-safe policy), while Fig. 10 shows its A• -- Ae o.5 I I I I I I I I I I I I I I I 02 0.06 0.10 014 018 0.22 0.26 0.30 trajectory. In Fig. 1 I, the time before reaching the A m mox, peak and control duration are plotted against the Fig. 11. Failsafe control timing curves. value of h• peak for free oscillations.
A. A. EL-BASSIONI and C. G. PONCELET
536
E1-Bassioni A. A. (1969) Ph.D. Thesis, University of Michigan. Bauer D. C. (1972) Ph.D. Thesis, Carnegie-Mellon E1-Bassioni A. A. and Poncelet C. G. (1972) Trans. Am. University. nucl. Soc. 15, 285. Bauer D. C., Poncelet C. G., Impink A. J. Jr., Jones K. A. EI-Bassioni A. A. and Poncelet C. G. (1974). NucL Sci. and Newton R. A. (1972), Trans. Am. nucL Soc. 15,887. Engng. 54, 2. Christie A. J. and Poncelet C. G. (1969) Trans. Am. nucL Stacey W. M. Jr. (1968) Nucl. Sci. Engng 33, 2. Soc. 12, 764. Wiberg D. M. (1965) Ph.D. Thesis, California Institute of Christie A. J. and Poncelet C. G. (1973) Nucl. Sci. Engng Technology. 51, 10. Wiberg D. M. (1967) Nucl. Sci. Engng 27, 3. REFERENCES