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Optimum policy for reactor shut-down
Journal
of Nuclear
Energy Parts A/B, 1965, Vol. 19, pp. 645 to 651. Pergamon Press Ltd. Printed in Northern Ireland
OPTIMUM
POLICY FOR REACTOR
SHUT-DOWN*
V. N. ARTAMKIN,G. V. VASENKOVA, 1. V. OTROSHCHENKO and R. P. FEDORENKO (Received 2 March 1964)
Abstract-A discussion is given of the optimum control policy for shutting down a reactor which, by carefully controlling the reactor power over a specified period of time (of the order of several hours), will minimize the peak poisoning. The problem of finding the optimum control is solved by a method devised by FEDORENKO by which numerical solutions can be obtained for non-linear control problems. The method consists of a process of iteration whereby a certain initial (not optimum) control is systematically varied with the aim of reducing the value of the functional. This procedure leads quite rapidly to a control which is sufficiently close to the optimum. An analysis of a large number of numerical solutions allows some plausible assumptions to be made about the simple structure of the exact solution to the problem. INTRODUCTION A high-flux thermal reactor it is possible, in principle, to build in sufficient additional reactivity to compensate for iodine poisoning by increasing the reactor charge. This complicates the control system, however, and gives rise to a number of reactor safety problems. Such measures also have an adverse effect upon the economic characteristics of the plant. In fact, this approach to the problem of iodine poisoning is possible only with comparatively low neutron fluxes since, with large fluxes, the increase in charge needed to compensate for the xenon poisoning arising after the reactor has been shut down is several times the critical charge. ‘Topping up’ the core after shutdown therefore has serious drawbacks-not to mention the complications this procedure introduces into the design of the reactor. The xenon concentration in a reactor is very sensitive to any changes in the neutron flux. This encourages one to believe that if the reactor power were purposefully changed before shut-down according to a set pattern, it might be possible to effect a significant reduction of the xenon concentration The first attempts of this kind were made in 1959 by ASH et ~1.‘~’who examined the possibility of applying dynamic programming methods to the problem of finding the optimum reactor shut-down policy. However, the method developed by these authors required the services of a computer with such a large operative memory that apparently it was not possible at the time to put the method into practical use (so far, no reports have appeared which indicate that this work has actually been carried out). FRESDALL and BABB(~)have described some results of calculations performed on an analogue machine giving the change in xenon concentration for specified forms of reactor power variation. It was shown that for all the types of reactor power variation tested (linear, exponential and a curve with a continuously increasing slope) it was always possible to achieve some reduction in the peak xenon concentration accompanied by an increase in the time required for the xenon concentration to build up to the equilibrium value. Since no attempt was made in this work to determine and to IN
* Translated by D. L.
ALLAN
from Atomnuya Energiya 17, 189 (1964).
645
646
V. N. ARTAMKIN,G. V. VASENKOVA,I. V. OTROSHCHENKO and R. P. FEDORENKO
investigate the optimum shut-down policy for the reactor, the feasibility of the method remains an open question. In this paper, results are given of a solution to the problem of finding the optimum shut-down policy for a reactor using an approximate method* of solving non-linear optimum control problems numerically. Essentially, the method is a straightforward procedure for solving variational problems which leads quite rapidly to a control decision which is near enough to the optimum (after 20-30 million arithmetical operations) and which places no great demands on the operative memory of the computer. An analysis of a large number of such numerical solutions has enabled us to make some plausible assumptions about the simple structure of the exact solution to the problem. STATEMENT
OF
THE
PROBLEM
To describe the change in the concentration of xenon, use was made of a simplified model based on the assumption that the fission product concentration is determined by the neutron flux averaged over the volume of the fuel. On the basis of this assumption, and using the usual notationt3), one can write down equations describing the change in the concentration of iodine and xenon:
The following constraints are imposed on the thermal neutron flux 0: Q(t) = @II,
t < 0;
0 < Q(t) < @‘o,
O
O(t) = 0,
T<
t,
(2) I
where T is the specified control period. In the period of time O-T, Q(t) should be chosen so as to minimize the functional F(O) max X(t). OG< m
(3)
As will become apparent later, it is better to take O(t) =
d;,
as the unknown control rather than Q(t), and to take F(0) =
max X(r) 0
(3)
as the functional to be minimized. Although this is a more complicated way of setting up the problem than that adopted by ASH et al. (l), it turns out to be more convenient for the numerical method of solution to be applied here. A constraint on the time rate of change of the flux ought to have been added to the * Devised by R.
P. FEDORENKO.
Optimum policy for reactor shut-down
647
equations describing our problem. However, such a constraint does not noticeably affect the solution when the method described here is applied since the time step is, as a rule, appreciably smaller than the attainable asymptotic reactor periods. In addition, this constraint would lead to the appearance of an additional parameter which would complicate the analysis and limit the generality of the results. We will not, therefore, impose this constraint. One should regard the shut-down conditions obtained from the analysis as ideal and then try to realize them in practice by taking into account the special kinetic peculiarities of each reactor. Formally, the solution of our problem is a function of the four parameters: x:r, cp, CD,,T. In general, the decay constants J.r and J.Zremain unchanged and the iodine and xenon yields (yl and yZ respectively) change only when one changes over from one kind of fissile nucleus to another. It is not difficult to see that, on transforming to the dimensionless variables
where
are solutions of the set (1) with t G 0, the solution of the problem involves only the two parameters, T and 2 =a,@,/&. Thus, in terms of the notation we have introduced, the set of equations assumes the form
(7)
METHOD
OF
SOLUTION
The problem is solved numerically in accordance with the following scheme. Starting with some value of the control U, we solve set (7) numerically and find the points tti) (i = 1,2, . . . , 8); t(l) is the point corresponding to the localized maximum of the function x(t) with t > T; tc2) and tt3) are points on the section (0, T) at which the value of x(t) exceeds x(t’l)) or is close to it; tC4)and tC5)are points at which the condition Q, G 1 is most strongly violated; tt6) and t(‘) are analogous points, but for the condition CD> 1; t@) = T. The section (0, T) was divided into a number (usually 50) of equal intervals At and then the problem was analysed in terms of the lumped constants U. Let us consider the point 5 of the eight-dimensional space: i=
1,2,3;
i = 4, . . . ,8.
To investigate the variation of the control it is necessary to know how the point E
648
V. N. ARTAMKIN, G. V. VASENKOVA, I. V. OTROSHCHENKOand R. P. FEDORENKO
is displaced when, in one interval At, only, the control U, is replaced by Uj* = Uj + SUj. Information about this was given by the set of vectors h j (eight-dimensional). To compute the i-th components of the vectors hj ti) in all the intervals Atj, it is necessary to solve, from right to left, the set conjugate to the set (7) which was the subject of the variation process, having first specified the initial data at the point tci) intheformofavector(1,0,0)fori=1,2,3,and(0,0,1)fori=4 ,..., 8. Forthe intervals At located further to the right of t ti), the i-th component of h is zero. A variation of the control by the amount SUj shifts the point l to the new position In choosing the numbers SUj, one shomd be guided in such a way that max p(i) i=1,2,3
is minimized, with the conditions &?(i)< 1, i= F(i) > 0, i= -‘ (a) = 0. 5 Taking into account, in addition, the conditions
4, 5; 6, 7;
lSU,l < dj, associated with the fact that all the constructions are carried out within the framework of the theory of small perturbations, we find ourselves presented with a linear programming problem (here cYiis the parameter of smallness which tends to zero in a definite manner during the course of the iteration process). The problem was solved approximately using a relaxation procedure. Having obtained a new control, the set (7) was recalculated, new values of tci’ were determined, and so on. The transition from the control Q, to U = d@/dt was made in accordance with the following considerations : in the optimum solution, the function x(t) assumes a maximum value at the point of local maximum with t > t, and over the entire interval (t’, t”) which in some cases, contained 30-40 intervals Atj. Since only two points served to define this plateau in the function x(t), it was desirable to deal with a function x(t) which everywhere had a continuous derivative. This method has been described in detail elsewherec4). The above numerical method of solving the problem is not unique nor has its convergence been fully tested. However, as a result of solving a large number of problems, it has been demonstrated that the method yields solutions which are not only satisfactory in practice but require only quite a short computing time. Questions of convergence and uniqueness can be studied by solving a problem using different initial control functions U(t). This was done for certain values (Z, T). In all cases the iteration process yielded virtually the same value of the functional. Differences in U(t) were more marked but they too were not very large. The form of a typical numerical solution is given in Fig. I. DISCUSSION
From a study of the large number of variants calculated it was possible to divide the section (0, r) into three intervals each containing one simple control operation (see Fig. 1). At the instant of shut-down the reactor power should drop to zero at the maximum
Optimum policy for reactor shut-down
649
rate possible. Over the section (0, tr) the flux is held at zero level and the decay of iodine and the build-up of xenon proceeds as for a reactor which is shut down instantaneously. At time t,, the xenon concentration reaches a magnitude x(t,) = F(U) = x*-the maximum xenon concentration under such a control. Starting at time t,, the neutron flux in the reactor should be maintained at a level which will ensure a constant concentration of xenon. To find the appropriate law of variation of the neutron flux with time, use must be made of the second equation of the set (7) which, with dx/dt = 0 and x = F(U), enables one to find the relation between y and cp. 6.0
:/
t,
--
I,
i
‘I
hr
FIG. I.-Optimum reactor shut-down conditions (Z = 16, q = 0.56): change in xenon concentration for an instantaneous reactor shut-down; - - - - proposed optimum control pattern; O--calculation.
If this equation is used to eliminate y from the first equation proceed to solve it with respect to q?:
(1 +
z>--&
(
exp
Qi=
Yl (_
+
--CL
Zx” - (1 + Z)
mx*(Z + 1) -&
x* z(x*
-1)
-1
’
+
X* z(x*
Y%
x exp i
(-44)
of the set (7), one can
l)-ll4(t--tl) + z(x*
-
I)
I
‘I -
1) -1
I
t, & t < t,.
After some time, when t = t,, the flux rises to the original
value (9’ = 1) and is
650
V. N. ARTAMKIN,
G. V. VASENKOVA, I. V. OTROSHCHENKO
and R. P. FEDORENKO
maintained there up to time t = T, where an intense burn-up of xenon occurs accomAfter the final shut-down of the reactor, panied by a rise in the iodine concentration. the xenon concentrations begins to increase again reaching once more the magnitude F(U) at the local maximum (at time t3). The advantages of following the optimum control policy are illustrated in Fig. 2
0
0.2
0.4 qO6
@a
1.0
FIG. 2.-Dependence concentration
of reactor shut-down time on the ratio of the increase in xenon under optimum control to the increase in xenon concentration for an instantaneous shut-down.
where, along the abscissa, is plotted the ratio of the maximum increase in the xenon concentration during a controlled reactor shut-down to the maximum increase in the xenon concentration occurring in an instantaneous shut-down: F(U) -1 q=F(o)l and along the ordinate is plotted the control time T. Curves giving the dependence of q on T for different Z (2-64) fill the shaded region. We do not exclude the possibility that this spread amongst the curves arises from inaccuracies in the calculations. This is a question that needs to be looked into more closely. Figure 2 gives also the time T* which must elapse after an instantaneous shut-down before the xenon concentration falls to the same level q as that obtained under optimum control conditions (for Z equal to 4,8 and 16). The difference between the two ordinates, T*-T, is a measure of the time saved when the optimum control policy is followed. For each reactor the available reactivity margin determines the maximum xenon concentration x* for which the reactor will remain critical. From this maximum value of x*, the value of q is computed which can be realized with a given reactivity margin and, with the aid of Fig. 2, the minimum control time T is calculated. Once a reactor operating under optimum control has been fully shut down it may again be brought up to power at any time. Thus, for a given shut-down time, the optimum control of a reactor is determined
Optimum policy for reactor shut-down
651
by the two parameters, t, and t,. The calculations enable one to find these parameters for times T lying within the limits 415 hr, and for reactors with fluxes corresponding to values of Z lying within the limits 1-128. In conclusion, we would point out that once the optimum control pattern is known it is possible to find the optimum pattern quite simply for each individual case without having to go to the numerical solution for the full optimum control problem as formulated above. Indeed, let us suppose that x* for a given reactor is known. The problem then consists simply in finding the times t,, t, and T. To these we must add t,, since it is necessary to verify that at that particular instant of time the xenon concentration is a maximum and has reached the value x*. Having made use of the particular characteristics of each of the control integrals, namely $!3=0 06 t
t,
<
t
<
I
we can find an analytical solution of the first two equations of the set (7) in each interval. These solutions will depend parametrically on t,, t, and T. The conditions to be imposed on the functions x and y are
These conditions, along with the requirement that T must be a minimum, permit one to determine t,, t,, T and also t,, the times characterizing the optimum control. If x* has been chosen to allow for some margin and a small increase in x* still permits the reactor to be brought up to power, it is possible to use Fig. 2 to determine the approximate shut-down time T. The first of the equations in the set (10) gives t, directly. The only other quantity which one needs to find [using the two remaining equations in the set (lo)] is t,. It should be noted here that with the correct values of t, and T the uncertainty in the determination of t, slightly affects the magnitude of the local maximum at point t,. This is connected with the fact that the correct value of fz ensures a minimum value of the local maximum:
WcJ _
o
at,
’
i.e. the uncertainty in t, leads to deviations of the second order in I+ (naturally, these are always on the high side). REFERENCES I. ASH M., BELLMANR. and KALABA R. Nucl. Sri.
Engng. 6, 152 (1959). 2. FRESDALLJ. and BABB A. Trans. Amer. Nucl. Sot. 4, 316 (1961). 3. GLASSTONES. and EDLUND M. C. The Elements ‘of Nuclear Reactor Theory, Van Nostrand Inc., New York (1952). 4. FEDORENKOR. P. Zh. oichisl. matem. matem.jz. 4, 559 (1964).
of Nuclear
Energy Parts A/B, 1965, Vol. 19, pp. 645 to 651. Pergamon Press Ltd. Printed in Northern Ireland
OPTIMUM
POLICY FOR REACTOR
SHUT-DOWN*
V. N. ARTAMKIN,G. V. VASENKOVA, 1. V. OTROSHCHENKO and R. P. FEDORENKO (Received 2 March 1964)
Abstract-A discussion is given of the optimum control policy for shutting down a reactor which, by carefully controlling the reactor power over a specified period of time (of the order of several hours), will minimize the peak poisoning. The problem of finding the optimum control is solved by a method devised by FEDORENKO by which numerical solutions can be obtained for non-linear control problems. The method consists of a process of iteration whereby a certain initial (not optimum) control is systematically varied with the aim of reducing the value of the functional. This procedure leads quite rapidly to a control which is sufficiently close to the optimum. An analysis of a large number of numerical solutions allows some plausible assumptions to be made about the simple structure of the exact solution to the problem. INTRODUCTION A high-flux thermal reactor it is possible, in principle, to build in sufficient additional reactivity to compensate for iodine poisoning by increasing the reactor charge. This complicates the control system, however, and gives rise to a number of reactor safety problems. Such measures also have an adverse effect upon the economic characteristics of the plant. In fact, this approach to the problem of iodine poisoning is possible only with comparatively low neutron fluxes since, with large fluxes, the increase in charge needed to compensate for the xenon poisoning arising after the reactor has been shut down is several times the critical charge. ‘Topping up’ the core after shutdown therefore has serious drawbacks-not to mention the complications this procedure introduces into the design of the reactor. The xenon concentration in a reactor is very sensitive to any changes in the neutron flux. This encourages one to believe that if the reactor power were purposefully changed before shut-down according to a set pattern, it might be possible to effect a significant reduction of the xenon concentration The first attempts of this kind were made in 1959 by ASH et ~1.‘~’who examined the possibility of applying dynamic programming methods to the problem of finding the optimum reactor shut-down policy. However, the method developed by these authors required the services of a computer with such a large operative memory that apparently it was not possible at the time to put the method into practical use (so far, no reports have appeared which indicate that this work has actually been carried out). FRESDALL and BABB(~)have described some results of calculations performed on an analogue machine giving the change in xenon concentration for specified forms of reactor power variation. It was shown that for all the types of reactor power variation tested (linear, exponential and a curve with a continuously increasing slope) it was always possible to achieve some reduction in the peak xenon concentration accompanied by an increase in the time required for the xenon concentration to build up to the equilibrium value. Since no attempt was made in this work to determine and to IN
* Translated by D. L.
ALLAN
from Atomnuya Energiya 17, 189 (1964).
645
646
V. N. ARTAMKIN,G. V. VASENKOVA,I. V. OTROSHCHENKO and R. P. FEDORENKO
investigate the optimum shut-down policy for the reactor, the feasibility of the method remains an open question. In this paper, results are given of a solution to the problem of finding the optimum shut-down policy for a reactor using an approximate method* of solving non-linear optimum control problems numerically. Essentially, the method is a straightforward procedure for solving variational problems which leads quite rapidly to a control decision which is near enough to the optimum (after 20-30 million arithmetical operations) and which places no great demands on the operative memory of the computer. An analysis of a large number of such numerical solutions has enabled us to make some plausible assumptions about the simple structure of the exact solution to the problem. STATEMENT
OF
THE
PROBLEM
To describe the change in the concentration of xenon, use was made of a simplified model based on the assumption that the fission product concentration is determined by the neutron flux averaged over the volume of the fuel. On the basis of this assumption, and using the usual notationt3), one can write down equations describing the change in the concentration of iodine and xenon:
The following constraints are imposed on the thermal neutron flux 0: Q(t) = @II,
t < 0;
0 < Q(t) < @‘o,
O
O(t) = 0,
T<
t,
(2) I
where T is the specified control period. In the period of time O-T, Q(t) should be chosen so as to minimize the functional F(O) max X(t). OG< m
(3)
As will become apparent later, it is better to take O(t) =
d;,
as the unknown control rather than Q(t), and to take F(0) =
max X(r) 0
(3)
as the functional to be minimized. Although this is a more complicated way of setting up the problem than that adopted by ASH et al. (l), it turns out to be more convenient for the numerical method of solution to be applied here. A constraint on the time rate of change of the flux ought to have been added to the * Devised by R.
P. FEDORENKO.
Optimum policy for reactor shut-down
647
equations describing our problem. However, such a constraint does not noticeably affect the solution when the method described here is applied since the time step is, as a rule, appreciably smaller than the attainable asymptotic reactor periods. In addition, this constraint would lead to the appearance of an additional parameter which would complicate the analysis and limit the generality of the results. We will not, therefore, impose this constraint. One should regard the shut-down conditions obtained from the analysis as ideal and then try to realize them in practice by taking into account the special kinetic peculiarities of each reactor. Formally, the solution of our problem is a function of the four parameters: x:r, cp, CD,,T. In general, the decay constants J.r and J.Zremain unchanged and the iodine and xenon yields (yl and yZ respectively) change only when one changes over from one kind of fissile nucleus to another. It is not difficult to see that, on transforming to the dimensionless variables
where
are solutions of the set (1) with t G 0, the solution of the problem involves only the two parameters, T and 2 =a,@,/&. Thus, in terms of the notation we have introduced, the set of equations assumes the form
(7)
METHOD
OF
SOLUTION
The problem is solved numerically in accordance with the following scheme. Starting with some value of the control U, we solve set (7) numerically and find the points tti) (i = 1,2, . . . , 8); t(l) is the point corresponding to the localized maximum of the function x(t) with t > T; tc2) and tt3) are points on the section (0, T) at which the value of x(t) exceeds x(t’l)) or is close to it; tC4)and tC5)are points at which the condition Q, G 1 is most strongly violated; tt6) and t(‘) are analogous points, but for the condition CD> 1; t@) = T. The section (0, T) was divided into a number (usually 50) of equal intervals At and then the problem was analysed in terms of the lumped constants U. Let us consider the point 5 of the eight-dimensional space: i=
1,2,3;
i = 4, . . . ,8.
To investigate the variation of the control it is necessary to know how the point E
648
V. N. ARTAMKIN, G. V. VASENKOVA, I. V. OTROSHCHENKOand R. P. FEDORENKO
is displaced when, in one interval At, only, the control U, is replaced by Uj* = Uj + SUj. Information about this was given by the set of vectors h j (eight-dimensional). To compute the i-th components of the vectors hj ti) in all the intervals Atj, it is necessary to solve, from right to left, the set conjugate to the set (7) which was the subject of the variation process, having first specified the initial data at the point tci) intheformofavector(1,0,0)fori=1,2,3,and(0,0,1)fori=4 ,..., 8. Forthe intervals At located further to the right of t ti), the i-th component of h is zero. A variation of the control by the amount SUj shifts the point l to the new position In choosing the numbers SUj, one shomd be guided in such a way that max p(i) i=1,2,3
is minimized, with the conditions &?(i)< 1, i= F(i) > 0, i= -‘ (a) = 0. 5 Taking into account, in addition, the conditions
4, 5; 6, 7;
lSU,l < dj, associated with the fact that all the constructions are carried out within the framework of the theory of small perturbations, we find ourselves presented with a linear programming problem (here cYiis the parameter of smallness which tends to zero in a definite manner during the course of the iteration process). The problem was solved approximately using a relaxation procedure. Having obtained a new control, the set (7) was recalculated, new values of tci’ were determined, and so on. The transition from the control Q, to U = d@/dt was made in accordance with the following considerations : in the optimum solution, the function x(t) assumes a maximum value at the point of local maximum with t > t, and over the entire interval (t’, t”) which in some cases, contained 30-40 intervals Atj. Since only two points served to define this plateau in the function x(t), it was desirable to deal with a function x(t) which everywhere had a continuous derivative. This method has been described in detail elsewherec4). The above numerical method of solving the problem is not unique nor has its convergence been fully tested. However, as a result of solving a large number of problems, it has been demonstrated that the method yields solutions which are not only satisfactory in practice but require only quite a short computing time. Questions of convergence and uniqueness can be studied by solving a problem using different initial control functions U(t). This was done for certain values (Z, T). In all cases the iteration process yielded virtually the same value of the functional. Differences in U(t) were more marked but they too were not very large. The form of a typical numerical solution is given in Fig. I. DISCUSSION
From a study of the large number of variants calculated it was possible to divide the section (0, r) into three intervals each containing one simple control operation (see Fig. 1). At the instant of shut-down the reactor power should drop to zero at the maximum
Optimum policy for reactor shut-down
649
rate possible. Over the section (0, tr) the flux is held at zero level and the decay of iodine and the build-up of xenon proceeds as for a reactor which is shut down instantaneously. At time t,, the xenon concentration reaches a magnitude x(t,) = F(U) = x*-the maximum xenon concentration under such a control. Starting at time t,, the neutron flux in the reactor should be maintained at a level which will ensure a constant concentration of xenon. To find the appropriate law of variation of the neutron flux with time, use must be made of the second equation of the set (7) which, with dx/dt = 0 and x = F(U), enables one to find the relation between y and cp. 6.0
:/
t,
--
I,
i
‘I
hr
FIG. I.-Optimum reactor shut-down conditions (Z = 16, q = 0.56): change in xenon concentration for an instantaneous reactor shut-down; - - - - proposed optimum control pattern; O--calculation.
If this equation is used to eliminate y from the first equation proceed to solve it with respect to q?:
(1 +
z>--&
(
exp
Qi=
Yl (_
+
--CL
Zx” - (1 + Z)
mx*(Z + 1) -&
x* z(x*
-1)
-1
’
+
X* z(x*
Y%
x exp i
(-44)
of the set (7), one can
l)-ll4(t--tl) + z(x*
-
I)
I
‘I -
1) -1
I
t, & t < t,.
After some time, when t = t,, the flux rises to the original
value (9’ = 1) and is
650
V. N. ARTAMKIN,
G. V. VASENKOVA, I. V. OTROSHCHENKO
and R. P. FEDORENKO
maintained there up to time t = T, where an intense burn-up of xenon occurs accomAfter the final shut-down of the reactor, panied by a rise in the iodine concentration. the xenon concentrations begins to increase again reaching once more the magnitude F(U) at the local maximum (at time t3). The advantages of following the optimum control policy are illustrated in Fig. 2
0
0.2
0.4 qO6
@a
1.0
FIG. 2.-Dependence concentration
of reactor shut-down time on the ratio of the increase in xenon under optimum control to the increase in xenon concentration for an instantaneous shut-down.
where, along the abscissa, is plotted the ratio of the maximum increase in the xenon concentration during a controlled reactor shut-down to the maximum increase in the xenon concentration occurring in an instantaneous shut-down: F(U) -1 q=F(o)l and along the ordinate is plotted the control time T. Curves giving the dependence of q on T for different Z (2-64) fill the shaded region. We do not exclude the possibility that this spread amongst the curves arises from inaccuracies in the calculations. This is a question that needs to be looked into more closely. Figure 2 gives also the time T* which must elapse after an instantaneous shut-down before the xenon concentration falls to the same level q as that obtained under optimum control conditions (for Z equal to 4,8 and 16). The difference between the two ordinates, T*-T, is a measure of the time saved when the optimum control policy is followed. For each reactor the available reactivity margin determines the maximum xenon concentration x* for which the reactor will remain critical. From this maximum value of x*, the value of q is computed which can be realized with a given reactivity margin and, with the aid of Fig. 2, the minimum control time T is calculated. Once a reactor operating under optimum control has been fully shut down it may again be brought up to power at any time. Thus, for a given shut-down time, the optimum control of a reactor is determined
Optimum policy for reactor shut-down
651
by the two parameters, t, and t,. The calculations enable one to find these parameters for times T lying within the limits 415 hr, and for reactors with fluxes corresponding to values of Z lying within the limits 1-128. In conclusion, we would point out that once the optimum control pattern is known it is possible to find the optimum pattern quite simply for each individual case without having to go to the numerical solution for the full optimum control problem as formulated above. Indeed, let us suppose that x* for a given reactor is known. The problem then consists simply in finding the times t,, t, and T. To these we must add t,, since it is necessary to verify that at that particular instant of time the xenon concentration is a maximum and has reached the value x*. Having made use of the particular characteristics of each of the control integrals, namely $!3=0 06 t
t,
<
t
<
I
we can find an analytical solution of the first two equations of the set (7) in each interval. These solutions will depend parametrically on t,, t, and T. The conditions to be imposed on the functions x and y are
These conditions, along with the requirement that T must be a minimum, permit one to determine t,, t,, T and also t,, the times characterizing the optimum control. If x* has been chosen to allow for some margin and a small increase in x* still permits the reactor to be brought up to power, it is possible to use Fig. 2 to determine the approximate shut-down time T. The first of the equations in the set (10) gives t, directly. The only other quantity which one needs to find [using the two remaining equations in the set (lo)] is t,. It should be noted here that with the correct values of t, and T the uncertainty in the determination of t, slightly affects the magnitude of the local maximum at point t,. This is connected with the fact that the correct value of fz ensures a minimum value of the local maximum:
WcJ _
o
at,
’
i.e. the uncertainty in t, leads to deviations of the second order in I+ (naturally, these are always on the high side). REFERENCES I. ASH M., BELLMANR. and KALABA R. Nucl. Sri.
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