multiobjective fuel management optimization for self-fuel-providing lmfbr using genetic algorithms
TRANSCRIPT
Multiobjective fuel management optimization
for self-fuel-providing LMFBR using
genetic algorithms
Vladimir G. Toshinskya, Hiroshi Sekimotoa,*,Georgy I. Toshinskyb
aResearch Laboratory for Nuclear Reactors, Tokyo Institute of Technology,
2-12-1, O-Okayama, Meguro-ku, Tokyo 152, JapanbState Scienti®c Center, Institute of Physics and Power Engineering,
Bondarenko sq. 1, Obninsk, Kaluga Region, 249020 Russia
Received 3 August 1998; accepted 17 October 1998
Abstract
One of the conceptual options under consideration for the future of nuclear power is the long-
term development without fuel reprocessing. This concept is based on a reactor that requires no
plutonium reprocessing for itself, and provides high e�ciency of natural uranium utilization, so
called Self-Fuel-Providing LMFBR (SFPR). Several design considerations were previously given
to this reactor type which, however, su�er from some problems connected with insu�cient power
¯attening, large reactivity swings during burnup cycles, and peak fuel burnup being signi®cantly
higher than recent technology experience, which is about 18% for U-10wt%Zr metallic fuel to be
considered. Yet, the mentioned core parameters demonstrate high sensitivity to the fuel man-
agement strategy selected for the reactor. Therefore, the aim of this study is to develop a practical
tool for the improvement of the core characteristics by fuel management optimization, which is
based on advanced optimization techniques such as Genetic Algorithms (GA). The calculation
results obtained by a simpli®ed reactor model can serve as estimates of achievable values for
mentioned core parameters, which are necessary to make decisions at the preliminary optimiza-
tion stage.# 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction
The Self-Fuel-Providing LMFBR (SFPR) is designed to operate in such a way
that once it is initially loaded, only fertile materials are needed as core reload fuel for
Annals of Nuclear Energy 26 (1999) 783±802
0306-4549/99/$Ðsee front matter # 1999 Elsevier Science Ltd. All rights reserved
PII: S0306-4549(98)00092-9
*Corresponding author. Tel.: +81-3-5734-3066; fax: +81-3-5734-2959; e-mail: [email protected]
the rest of the reactors lifetime and, thus, it requires no plutonium reprocessing for
itself. High e�ciency of natural uranium utilization in the reactor makes it possible
to generate nuclear power during several hundreds of years, using available uranium
resources. The concept of long-term nuclear power generation without fuel repro-
cessing (Subbotin et al., 1994), based on SFPR, is considered as a temporary strat-
egy which may provide signi®cant time reserves for high-level development of
industrial reprocessing for closing of nuclear fuel cycle after the mentioned period or
earlier.
The SFPR is generally featured by the following:
. An internal conversion ratio is maintained at greater than unity so that the
positive burnup reactivity swing may compensate the reactivity loss due to
natural (or depleted) uranium fuel loading at the beginning of each cycle.
. A portion of the fresh fuel containing natural (or depleted) uranium is loaded
at the beginning of each cycle to control both the excess reactivity and the
power distribution.
. Portions of the fuel assemblies are radially shu�ed at the beginning of each
cycle to moderate the power distribution changes during burnup.
. Since natural (or depleted) uranium is used as a reload core fuel, plutonium
recycling by fuel reprocessing is not necessarily required.
. The reactor allows for long residence time for the fuel and thereby reduces the
fuel fabrication costs.
As it was pointed out, for the SFPR to work, the internal conversion ratio should
be high enough so that the reactor can enrich the reload fertile material to the point
that no ®ssile materials are required as reload fuel to maintain the reactor criticality
during operation. To this end, an increase of the fuel volume fraction, a reduction of
the neutron leakage from the core due to increased core volume and a utilization of
high-density fuels are required. Feasibility studies on this reactor type have been
performed in several research works (Bencheikh and Karam, 1985; Subbotin et al.,
1994; Gromov et al., 1997; Toshinsky, 1997) and for the present analysis we choose
the most recently proposed design (Toshinsky, 1997). The basic reactor design
parameters are given in Table 1.
The typical operation mode of SFPR can be described as following. The reactor
core consists of several radial zones, and each zone contains the same number of fuel
assemblies as it is schematically shown on the Fig. 1. At the beginning of the cycle,
the content of one of the zones is replaced by the fresh fuel containing natural (or
depleted) uranium. The displaced fuel is moved to another zone. This fuel replace-
ment is repeated in a speci®c way over all zones until the content of the last zone is
discharged from the reactor. The core reloading is performed in the same way in all
consequent cycles. During the cycle of the reactor operation, there is a reactivity
gain due to the internal conversion. On the other hand, there is a reactivity loss at
the refueling, due to the fresh fuel loading. Thus, the balance of these two processes
provides the operation of the reactor.
To reduce computation time, the reactor physics calculations are carried out in a
two-dimensional cylindrical geometry as shown on Fig. 2. The homogenized refueling
784 V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802
Table 1
Self-Fuel-Providing LMFBR design parameters
Reactor design parameter
Reactor power, MWt 1500 Axial re¯ector volume fraction
Core height, cm 120 Zirconium 0.60
Core e�ective radius, cm 191.3 Clad 0.14
Coolant 0.26
Axial re¯ector thickness, cm 50 Radial re¯ector volume fraction
Radial re¯ector thickness, cm 50 Clad 0.9
Core refueled fraction, % optional Coolant 0.1
Fuel assembly volume fraction
No. of core assemblies 312 Fuel 0.60
No. of re¯ector assemblies 216 Clad 0.14
Coolant 0.26
Reactor coolant outlet temp (�C) 470
Coolant temperature rise (�C) 140 Density of clad 7.82
Coolant material Pb (44%)±Bi(56%) Density of coolant 10.3
Clad material HT-9 Density of fuel
Fuel material Theoretical 15.9
Initial core loading U-10wt%Zr Smeared, %TD 75
Reload core U-10wt%Zr
Fig. 1. Operation mode of Self-Fuel-Providing LMFBR.
V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802 785
model is used to approximate the more realistic discrete refueling.While the latter treats
the fuel density of each assembly discretely, the former treats the fuel density homo-
geneously over a zone. Control rods are assumed to be fully withdrawn from the core.
The control reactivity requirements depend on the cycle length (CL), the refueling
fraction (RF) and the refueling scheme (RS). Fig. 3 shows the burnup reactivity gain
and reactivity loss as functions of RF and CL with ®xed RS (fuel is reloaded from
Fig. 2. R±Z geometric representation of Self-Fuel-Providing LMFBR.
Fig. 3. The burnup reactivity gain and reactivity loss at refueling as functions of the core refueling frac-
tion and cycle length.
786 V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802
the core center to periphery). It is seen that the reactivity gain slightly decreases as
RF increases, while the absolute value of the reactivity loss increases with greater rate.
Calculations show that the reactivity loss is almost independent of CL, whereas the
reactivity gain is in proportion to CL. A larger CL is required to balance the reactivities
as RF increases. As the result, the decrease of RF (or increase of total number of radial
zones) leads to almost linear reduction of reactivity swing during cycle.
Regarding to dependency on RS, the studies (Bencheikh and Karam, 1985;
Toshinsky, 1997) show that proper selection of the fuel management strategy plays a
great role in the reactor design. The reason is that the reactor zones have sig-
ni®cantly di�erent amount of accumulated ®ssile materials and, thus, such core
characteristics as power density, burnup, k-in®nity distributions as well as reactivity
swings during operation cycle are very sensitive to selected RS. For references, three
RSs (RF is ®xed at 7.14%) were chosen (see RSs 2,3,4 in Table 4) and calculation
results are shown on Fig. 4. It is seen from the top ®gure that the k-in®nity may
increase from cycle to cycle due to plutonium building up in rather di�erent ways for
various RSs. The variations of ¯ux distribution may also be signi®cant, as it is
shown on the bottom ®gure. The peaking factor varies from 1.67 to 3.47 and the
reactivity swing during cycle±from 3.57 to 8.10% �K=K for the considered RSs.
Taking into account this high sensitivity, RSs are chosen as decision variables for the
optimization, while RF is considered as a parameter that is ®xed to 7.14% (14 radial
zones) (Toshinsky, 1997) in the present research. For the further optimization, the pro-
cedure to be described later on can be applied to the problem with other values of RFs.
It should be noted that previous studies proposed some RSs providing reasonable
core characteristics, which, however, were derived without using optimization tech-
nique (Bencheikh and Karam, 1985; Toshinsky, 1997). For reference, the most recently
it has been reported (Toshinsky, 1997) that the following core parameters were
achieved for RF 7.14%: peaking factorÐ1.44, reactivity swing during cycle±5.3%
�K=K and peak fuel burnupÐ23%. It is seen that the value of reactivity swing is rather
large to be conventional value and the peak fuel burnup is higher than the technology
experience at Argonne National Laboratory, Argonne, Illinois, USA for metallic fuels.
However, since the optimization by previous researchers (Bencheikh and Karam, 1985;
Toshinsky, 1997) are mainly based on a trial-and-error approach, a practical limit exists
on their optimization capabilities, especially for the case when small value of RF (large
number of radial zone quantity) and several objectives are considered.
Thus, the major objective of present study is to develop a practical tool for the
improvement of the core characteristics by fuel management optimization, which is
based on a conventional optimization technique such as Genetic Algorithms (GA).
GA are chosen for the optimization, since it is considered to be quite suitable to
complicated combinatorial problems. The following optimization problem is con-
sidered: simultaneous minimization of power peaking factor and reactivity swing
during cycle with imposed constraint on peak fuel burnup that is taken 18% for U-
10wt%Zr. The reactor is assumed to operate in equilibrium cycle.
GA di�ers from the most optimization techniques by searching from one group of
solutions to another, rather than from one solution to another, and it is this fact that
makes them uniquely suited to multiobjective optimization. Calculations show that
V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802 787
Fig. 4. K-in®nity and ¯ux distribution at BOC for three di�erent refueling schemes (RF=7.14%) (a,b,c,
correspond to schemes 2,3,4 in Table 4).
788 V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802
there is competition between two studied objectives, and a recently proposed
approach (Parks, 1996) makes it possible to get trade-o� between them in a single
optimization run. Results obtained by simpli®ed reactor model can serve as esti-
mates for achievable relevant parameters for the considered reactor design.
Lately GA has been successfully applied to several PWR fuel management optimiza-
tion problems (DeChaine and Feltus, 1995, 1996; Parks, 1996). However, the range of
possible variation of Pressurized Water Reactor (PWR) core parameters (k-in®nities,
peaking factors, reactivity swings) is signi®cantly smaller than in the case of the con-
sidered design. Therefore, the present problem is a harder task for GA to tackle. It
should also be noted that GA are traditionally applied to optimization of a core loading
pattern (DeChaine and Feltus, 1995, 1996; Parks, 1996), where involved fuel types are
given as initial data. In the present study the reloading pattern is optimized and, there-
fore, the involved fuel types are in fact functions of the considered reloading pattern. For
their computation, the iterative burnup calculation is inevitable. Although the exact
solution of this problem is not now feasible in the case of applying a stochastic optimi-
zation technique like GA, an approximate solution can be obtained with acceptable
computation cost for optimization procedure by a method described in the next chapter.
2. Genetic algorithms
GA (Holland, 1975) is an optimization technique based on the evolution of life.
According to the Darwinian theory, the ``survival of the ®ttest'' principle and
crossover of chromosomes promote the life towards the adaptive direction. For the
reloading pattern optimization, a candidate of RS is treated as an individual, and
the performance of the individual, which is de®ned by core characteristics such as
peaking factor, reactivity swing during cycle and peak fuel burnup for the present
research, de®nes the probability of survival or mating.
GA requires the followings:
(a) a way of expressing the problem's decision variablesÐthe coding
(b) computationally e�cient methods of evaluating the problem functions for
trial solutions and thus the determining of their ®tness
(c) a way of implementing the Darwinian ``survival of the ®ttest'' principleÐthe
selection procedure
(d) a mechanism for combining information from two solutions to produce new
solutionsÐthe crossover operator
(e) a mechanism for making small changes to a solution±the mutation operator.
A ¯ow diagram of GA is shown in Fig. 5.
2.1. Coding
In the present problem the decision variables are RSs. For the core consisting of
radial zones, each RS can be represented by the string of integers, for instance (in
case of 10 radial zones):
V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802 789
3 8 7 9 4 2 5 1 6 10
where the position of an integer corresponds to zone position from the core center,
and its valueÐburnup cycle number. According to that example, the fresh fuel is
loaded into zone 8 and stay there during the 1st cycle. It then goes to zone 6 and stay
there during the 2nd cycle, and so on. During the last cycle the fuel stays in zone 10
and ®nally it is discharged from the core.
2.2. Fitness evaluation
During the optimization it is necessary to evaluate the performance indexes pro-
vided by the trial RSs. In the present research it is assumed that the reactor operates
in an equilibrium cycle. The equilibrium cycle is de®ned as a steady state loading
pattern with the ®xed number of fresh fuel, feed enrichment and fuel reloading
Fig. 5. Diagram of the Genetic Algorithm (GA).
790 V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802
between two consecutive cycles. Thus, the core characteristics such as peaking factor
at the beginning of cycle (BOC) (calculations show that it is the largest value during
cycle in the most cases), reactivity swing during cycle, peak burnup and the cycle
length do not change cycle by cycle; these characteristics are identical in every cycle.
The equilibrium cycle can be attained through the repetition of cycle burnup calcu-
lations assuming an identical fuel reloading throughout consecutive cycles. Equili-
brium state provided by a particular RS is attained as rapidly as possible (Chitkara
and Weisman, 1974) by using the same enrichment distribution for start-up loading as
it is determined in the equilibrium study for this RS. In this case equilibrium states are
closely approached in only three or four cycles, as shown on Fig. 6. Since the operation
in equilibrium is prevalent, the optimization of the parameters at equilibrium leads to
nearly optimum solution over the plant life.
As GA requires several thousands evaluations of RS during optimization proce-
dure, evaluation of one RS can take only a few seconds. Thus, performing iterative
burnup computations directly, which are inevitable for an exact evaluation of equi-
librium, is not feasible from computation cost point of view and approximate solu-
tion is obtained by the following method.
The ¯ux distribution at beginning of cycle (BOC) is determined by the following
set of one-group di�usion equations:
ÿD jr2���ja� �
1
Keff
v�jf� j � 1; . . . ;N �1�
where j denotes zone number (zones are ordered from the core center).
Fig. 6. Attaining equilibrium core parameters (in case of peaking factor at BOC) for three di�erent
refueling schemes.
V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802 791
The composition of zone j and, consequently, macroscopic constants in Eq. (1)
can be written in terms of ¯ux time (Fagan and Sesonske, 1969)±X. These constants
are computed in advance and, using polynomial ®tting, they are given by
~� X� � � ~A0 � ~A1X� . . .� ~AnXn �2�
where
~� � D;�a;�f; v�f;B�
;
X = �T;~Ai = empirical constants;
B = burnup.
The considered burnup chains contains 21 heavy nuclides and 4 groups of lumped
®ssion products, as shown on Fig. 7. In order to calculate ~�, burnup subroutine of
CITATION (Fowler et al., 1969) code was used to perform one-group point burnup
computations. The group constants for homogeneous cell were generated by
SLAROM (Nakagawa and Tsuchihashi, 1984) code. Calculations show that 4th
order polynomial expansion in Eq. (2) is accurate enough for macroscopic constants
representation. The expansion coe�cients are given in Table 2. The ¯ux time should
be taken in units of 1025 n/cm2, when applying data of the table.
Eqs. (1) and (2) are solved iteratively, as demonstrated on Fig. 8. First, the
refueling scheme and cycle length T are ®xed and an initial guess of zone average
¯ux distribution at BOC- ��j is chosen. Then, the macroscopic constants are readily
calculated by Eq. (2) taking into account the equilibrium condition:
Xi�1 � Xi � ��iT i � 1; . . . ;N �3�
where i denotes cycle number. It should be noted that order of sequence ��i is dif-
ferent from ��j and, therefore, sequence ��j is reordered, according to considered
refueling scheme, when applying Eq. (3). With calculated constants a new ¯ux dis-
tribution is calculated using Eq. (1). Before applying Eq. (2), the ¯ux is normalized
to the reactor nominal power using the macroscopic ®ssion cross-sections obtained
by the previous iteration. The iteration continues until all constants and ¯ux dis-
tribution becomes consistent. The parameter T (required cycle length) is adopted to
meet the criticality condition at BOC.
Di�usion calculations [Eq. (1)] were performed for two-dimensional R±Z geo-
metry with relatively large meshes (5±10 mesh points per zone in radial and 5 mesh
points in axial directions, respectively) to reduce computation cost. Avoiding direct
iterative burnup calculations as well as quick convergence of described iterations
make it possible to perform RS evaluation within a few seconds that is necessary for
GA applicability to the problem.
The described method uses several approximations which produce errors in pre-
diction of equilibrium core parameters. First, the equilibrium condition [Eq. (3)] is
792 V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802
applied under assumption that zone average ¯ux ��i does not change during burnup
cycle and it is approximated by its value at BOC. Calculations show that despite ¯ux
distribution change during cycle, the mentioned assumption does not produce sub-
stantial error in prediction of equilibrium parameters. An accuracy of the prediction
Fig. 7. Nuclide chains.
Table 2
Expansion coe�cients
Ao A1 A2 A3 A4
��f 1.67001E-3 1.54950E-1 ÿ1.38609 5.74731 ÿ9.63519
�f 6.03858E-4 5.25451E-2 ÿ4.71573E-1 1.95766 ÿ3.28526
�a 4.24003E-3 5.60416E-2 ÿ4.72079E-1 1.88597 ÿ3.12025
D 1.27191 ÿ1.31117E-1 ÿ1.73797 1.51761E+1 0
B, % h.a. 0 6.16441E+1 6.16711E+2 ÿ1.85810E+3 0
V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802 793
can still be improved by introduction to the method a simple empirical formula
(Chitkara and Weisman, 1974) for approximation of cycle average zone ¯ux-
< ��i >T using zone ¯ux at BOC- ��i and core average ¯ux-�avg at BOC:
< ��i >T� ��i�1�
1ÿ��i
�avg
C� �4�
where
h ��iiT �1
T
�T
��idt;
C=empirical constant.
The formula [Eq. (4)] re¯ects the following physical situation (Chitkara and
Weisman, 1974): as the cycle depletion proceeds, all zones tend to approach the core
average ¯ux. A zone with an initial ¯ux higher than the core average has its ®nal ¯ux
decreased, while the ®nal ¯ux of a zone with an initial ¯ux less than the core average
will tend to increase. It is numerically found that the ¯ux distribution, calculated by
the expression [Eq. (4)] using C � 5:51, approximates the exact cycle average ¯uxdistribution with better accuracy than BOC ¯ux distribution does. Therefore, in
subsequent calculations the ¯ux computed by [Eq. (4)] is used when applying the
equilibrium equation [Eq. (3)].
Second source of an error is that the method does not take into account spatial
variation of neutron spectrum, since one microscopic cross-section set is used to
represent the whole core. In order to estimate an inaccuracy caused by this
Fig. 8. Flow diagram of an equilibrium core parameter evaluation.
794 V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802
assumption, two types of calculations are performed using described above itera-
tions. At ®rst, the calculation is performed using one set of microscopic cross-sec-
tions, as it is previously suggested. This set is obtained by collapsing the group
constants into one group using:
�g �
�core
�g r!ÿ �
dr!:
In the second calculation type the microscopic cross-section set for the particular
zone i is obtained by collapsing the group constants using the neutron spectrum in
this zone:
�ig �
�zonei
�g r!ÿ �
dr!:
The results obtained by these two computations are shown on Fig. 9. It is seen that
the results are in good agreement except the outer region. Rough estimation of k-
in®nity using Eq. (5) and data of Table 3
kinf �v�f
�a
�
��9fN9
N8
� ��8f
�9aN9
N8
� �8a
; 0 �N9
N8
��8c�9a
; �5�
(the �8;9;N8;9 denote microscopic cross sections and number densities of U-238 and
Pu-239, respectively) shows that the discrepancy in k-in®nity caused by signi®cant
spectrum softening in outer zones (See Fig. 10) can be about 10±30% what is in
accord with the results shown in Fig. 9. From this analysis it is clear that the accu-
racy of the method can be improved by using two microscopic cross sections sets
and, consequently, two sets of expansion coe�cients used in expression [Eq. (2)]
namely, for inner and for outer core regions. In addition, it is necessary to take into
account which region fuel belongs to, when performing RS evaluation. However, in
the present study we neglect the discussed spectrum e�ect. The reason is that the
large inaccuracy takes place just in a few zones adjacent to re¯ector and the error in
evaluation of such core characteristics as peaking factor and reactivity swing during
cycle, which are of major interest in this study, is relatively small.
The last source of the error is that the macroscopic constants are assumed to be
functions of ¯ux time. This assumption results in inaccurate treatment of radioactive
decay of nuclides for which this process is comparable with other reactions (mainly
absorption) and, strictly speaking, the constants should be considered as functions
of two variables±¯ux and time. However, the amount of such nuclides does not
exceed 1% of fuel mass and the error is therefore negligible.
Table 4 shows the results of RS evaluation by the discussed technique and using
conventional method, which takes into account spatial variation of neutron spectrum
and performs iterative burnup computations. It is seen that the accuracy of the
developed technique is reasonable for the preliminary study.
V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802 795
2.3. Selection procedure
The selection procedure determines the probability of each trial solution in the
current population being chosen to parent new o�spring. The ``survival of the ®t-
test'' principle means that ``®tter'' solutions should be chosen more often.
The considered selection procedure is the same as a recently proposed one (Parks,
1996), that based on the concept of pareto optimality and dominance. Namely, a
Fig. 9. Comparison of calculation results using zone-wise and core-wise cross-section sets.
796 V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802
Fig. 10. Spectrum softening due to adjacent re¯ector.
Table 3
One group microscopic cross-sections collapsed using spectrum in the central and adjacent to re¯ector
zones
Central zone Zone adjacent to re¯ector
�9a 1.881E+0 2.212E+0
��9f 4.732E+0 5.165E+0
�8a 2.249E-1 2.608E-1
��8f 8.937E-2 6.979E-2
�8c 1.925E-1 2.354E-1
Table 4
Comparison of developed and conventional methods
Peaking factor Reactivity swing during cycle, �K=K%
Refueling schemesDeveloped
method
Conven.
method
Relative
error (%)
Developed
method
Conven.
method
Relative
error (%)
3 8 7 9 4 2 12 14 5 1 13 6 10 11 1.97 1.99 ÿ1.0 5.47 5.58 ÿ2.0
2 5 14 1 8 7 10 6 9 4 11 12 13 3 1.67 1.68 ÿ0.5 4.54 4.62 ÿ1.7
8 10 13 3 9 1 6 12 2 14 4 11 5 7 3.47 3.41 +1.7 7.90 8.10 ÿ2.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 2.14 2.06 +3.9 3.33 3.57 ÿ6.7
14 13 12 11 10 9 8 7 6 5 4 3 2 1 9.46 9.89 ÿ4.3 8.57 9.24 ÿ7.3
1 2 5 4 11 7 9 10 6 12 8 14 13 3 1.78 1.80 ÿ1.1 4.40 4.47 ÿ1.6
V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802 797
solution X is said to be dominated by solution Y if Y is better on all counts (objec-
tives), i.e. if
fi Y� � < fi X� � i � 1; . . . ;M �6�
where
fi = ith objective to be minimized;
M = number of objectives.
Using this de®nition, the entire population can be ranked by sorting through to
identify all nondominated solutions, then removing them from consideration, and
repeating the procedure until all solutions have been ranked, as illustrated by Fig. 11.
A selection probability can then be assigned to each solution based on its ranking
using Baker's single criterion ranking selection procedure (Baker, 1985), so that the
lower the rank the higher probability of selection. The constraint on peak fuel
burnup is treated as hard, meaning that any RS violating it ranked last in the
population and does not survive.
While the algorithm is running, an archive of nondominated solutions is main-
tained. After each trial solution has been evaluated, it is compared with existing
members of the archive. If it dominates any members of the archive, those are
removed, and the new solution is added. If the new solution is dominated by any
Fig. 11. An example of population ranking.
798 V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802
members of the archive, it is not archived. If it neither dominates nor is dominated
by any members of the archive, it is archived.
It is well established that GA perform better if an elitist selection procedure is
used, i.e. if the best solution found so far is always chosen as a parent, even if it is
not a member of the current population. In multiobjective optimization there is
probably more than one best (nondominated) solution, so a slightly more elaborate
procedure is used (Parks, 1996). Every solution in the archive (or P/4 of the solu-
tions archived, where P is the population size, if the archive contains more than P/4)
is chosen once as parent for each generation, thus introducing some multiobjective
elitism to the selection process.
2.4. Crossover operator
The role of the crossover operator is to combine information from parent solu-
tions to create o�spring solutions with, it is hoped, better objective function values.
The ``partially matched crossover (PMX)'' (Goldberg, 1989) was used for this pur-
pose. This crossover exchange the crossing parts of parents and then changes the
``outer'' part to satisfy fuel inventory constraint, as shown on Fig. 12. The crossover
is applied with probability 0.6.
2.5. Mutation operator
The mutation operator makes small changes on trial RSs and is necessary to
ensure that the entire search space is accessible. The considered mutation operator
performs shu�ing of randomly selected two zones, as demonstrated on Fig. 13. The
mutation probability is set 0.01.
3. Optimization results and discussion
Fig. 14 illustrates the performance of the optimization by plotting projection in
two-objective space of the initial and ®nal populations and the ®nal archive contents.
Fig. 12. An example of PMX operator.
Fig. 13. An example of mutation operator.
V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802 799
In this optimization run, a population of randomly chosen 100 RSs was evolved for
50 generations. It is seen that the ®nal population is on average much better than the
initial population with respect to both objectives and trade-o� surface between the
competing objectives is clearly indicated by the content of ®nal archive. Thus, a core
designer is supplied by family of solutions of the problem, from which those worthy
of further consideration can be chosen.
Table 5 shows a selection of the RSs (with peaking factor less than 1.5) in the ®nal
archive. As it is seen from the table, the solutions provide good power ¯attening and
Fig. 14. The optimization performance±reactivity swing during cycle versus peaking factor.
Table 5
Archived RSs (with peaking factor less than 1.5) and their parameters: peaking factors and reactivity
swings during cycle
Refueling scheme PF �K=K (%)
3 7 5 8 1 12 4 10 2 11 9 14 13 6 1.373 4.273
3 13 4 2 5 10 11 1 8 14 12 6 7 9 1.429 3.892
7 1 11 13 3 2 8 10 5 12 4 14 6 9 1.445 3.876
3 8 5 4 1 10 9 7 2 14 12 11 13 6 1.447 3.807
5 9 1 4 12 3 6 8 2 10 7 14 13 11 1.461 3.767
2 12 8 1 10 7 5 3 13 14 11 6 9 4 1.480 3.700
3 12 5 4 1 7 9 2 10 8 14 11 13 6 1.481 3.633
800 V.G. Toshinsky et al./Annals of Nuclear Energy 26 (1999) 783±802
all of them dominate the solution that was found by trial-and-error approach (See
Introduction) with respect to reactivity swing during cycle and peak fuel burnup.
However, the values of reactivity swing of archived RSs remain larger than con-
ventional ones. For the further investigation the described optimization procedure
can be applied to the problems with other values of RF to obtain corresponding
trade-o�s. Taking into account the results shown on Fig. 3, it may be expected that
reduction of RF will result in better performance with respect to reactivity swing
during cycle. On the other hand, it will cause more frequent reactor shutdown for
the reloading as well as signi®cant tangling of the fuel management. Thus, an opti-
mum value of RF may be found by the repeated implementation of the described
optimization procedure. Simpli®ed reactor model developed in the study can sig-
ni®cantly reduce computation cost of this preliminary analysis.
Once a value of RF is ®nally selected, a switch can be made to more precise core
calculation model with respect to geometry, control rods positioning and discrete
refueling. The precise calculation model together with two-dimensional genetic
operators (Parks, 1996; Yamamoto, 1998) can be readily incorporated into the pre-
sent optimization system.
Although RSs obtained by the preliminary analysis can not be directly transferred
to realistic core model, their parameters can serve as estimates of achievable per-
formance for the considered design to make a decision about optimum value of RF.
Moreover, they may help to incorporate some heuristics into optimization proce-
dure for reduction of the search space and, consequently, computation cost, that
would make it possible to perform ®nal optimization run using precise core model,
while applying GA, and re®ne the optimum design parameters.
4. Conclusion
The approach to improve design parameters of Self-Fuel-Providing LMFBR by
fuel management optimization has been described. It has been observed that GA is a
proper tool to tackle the considered multiobjective optimization problem and the
trade-o� surface between competing objectives has been identi®ed in a single opti-
mization run.
In order to apply such a stochastic optimization technique as GA, which require
several thousands of evaluations during one optimization run, the simpli®ed itera-
tive model for the prediction of equilibrium core parameters such as power peaking
factor, reactivity swing during burnup cycle and peak fuel burnup has been estab-
lished. The calculation results obtained by this model can serve as estimates of
achievable values for mentioned core parameters, which are necessary to make
decisions at the preliminary optimization stage.
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