BWR control rod design using tabu search
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* Corresponding author. Tel.: +52 55 53297233; fax: +52 55 53297340.
E-mail addresses: email@example.com (J.A. Castillo), firstname.lastname@example.org (J.J. Ortiz), galonso
@nuclear.inin.mx (G. Alonso), email@example.com (L.B. Morales), firstname.lastname@example.org. ipn.mx
(E. del Valle).1 Also a Ph.D. Student at Universidad Autonoma del Estado de Mexico.2 COFAA-IPN Fellow
Annals of Nuclear Energy 32 (2005) 741754
annals ofNUCLEAR ENERGY0306-4549/$ - see front matter 2005 Elsevier Ltd. All rights reserved.An optimization system to get control rod patterns (CRP) has been generated. This system
is based on the tabu search technique (TS) and the control cell core heuristic rules. The system
uses the 3-D simulator code CM-PRESTO and it has as objective function to get a specic
axial power prole while satisfying the operational and safety thermal limits. The CRP design
system is tested on a xed fuel loading pattern (LP) to yield a feasible CRP that removes the
thermal margin and satises the power constraints. Its performance in facilitating a power
operation for two dierent axial power proles is also demonstrated. Our CRP system is com-
bined with a previous LP optimization system also based on the TS to solve the combined LP-
CRP optimization problem. Eectiveness of the combined system is shown, by analyzing an
actual BWR operating cycle. The results presented clearly indicate the successful implementa-
tion of the combined LP-CRP system and it demonstrates its optimization features.
2005 Elsevier Ltd. All rights reserved.BWR control rod design using tabu search
Jose Alejandro Castillo a,1, Juan Jose Ortiz a,Gustavo Alonso a,c,*, Luis B. Morales b, Edmundo del Valle c,2
a Instituto Nacional de Investigaciones Nucleares, Km 36.5 Carretera Mexico-Toluca,
Ocoyoacac 52045, Edo. de Mexico, Mexicob Universidad Nacional Autonoma de Mexico, Instituto de Investigaciones en Matematicas Aplicadas y en
Sistemas, Apartado Postal 70-221, Mexico, D.F. 04510, Mexicoc Instituto Politecnico Nacional, Escuela Superior de Fsica y Matematicas, Unidad Profesional Adolfo
Lopez Mateos, ESFM, Edicio 9, C.P. 07738, D.F. Mexico
Received 18 August 2004; received in revised form 7 December 2004; accepted 7 December 2004
Available online 23 February 2005doi:10.1016/j.anucene.2004.12.004
specic power prole and it is a two edges calculation, at the beginning and at theend of the cycle. This array of assemblies is called the loading pattern (LP). The
742 J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754objective of the second stage is to obtain a control rod pattern (CRP) that provides
sucient thermal margin and a satisfactory axial power prole at any time during
the reactor cycle.
In a previous paper (Castillo et al., 2004), the rst stage was solved using the tabu
search technique (TS), the LP obtained generates more energy than those designed
by using engineer expertise. However, the optimized LP needs to be tested to know
if it is feasible to operate without violating the thermal and safety operational limitsat any time during the whole operating cycle. For each LP obtained by TS there ex-
ists the possibility that it cannot be controlled by any CRP during the whole cycle, in
this case it will not be considered as a feasible LP.
Then, in this work, an optimization system based also on TS is introduced to
design a CRP that satises the thermal and operational constraints. This system
can be applied independently of the way that the LP was obtained or it can be
combined with our LP system (Castillo et al., 2004). Our combined system can
be considered as a tool to tackle the combined LP-CRP optimization problem,which, for a BWR, is a tightly coupled problem as it was noted by Turinsky
and Parks (1999). We will assess an actual BWR operating cycle to show the eec-
tiveness of the combined system. The results will be compared to those obtained
from engineer expertise.
Historically, the design of CRP is usually based on trial and error techniques. Re-
cently, this problem has been automated using IF-THEN rules (Lin and Lin, 1991),
heuristic rules and common engineering practices (Karve and Turinsky, 1999), gene-
tic algorithms (GA) (Montes et al., 2004), and fuzzy logic and heuristics (Francoiset al., 2004).
To test our CRP independent system, we search for the CRP for two dierent
problems, the rst one is an operating cycle with a specic loading pattern, for this
problem two dierent axial power proles are considered. The results are compared
against those obtained from a GA search (Montes et al., 2004) and engineering
expertise. The second problem is a loading pattern for an equilibrium cycle (Montes
et al., 2001), the cycle length obtained using the optimized CRP is compared against
the results of GA given by Montes et al. (2004) for the same problem and thoseobtained by using engineer expertise.
2. BWR control rod pattern
The reactor control system must be capable of compensating for all reactivity
changes that take place throughout a reactor operating cycle, and to do this at1. Introduction
Optimization of BWR fuel reloads is a two stages task: The rst one comprises
the allocation of fuel assemblies in the core to get maximum cycle length under aa rate that roughly matches that of the reactivity changes. The movable control
J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754 743rods have a great eect on power distribution, and the interaction between control
rod arrangement and power distribution must be considered throughout the reac-
Design of a CRP involves the control rod allocation in the core. An optimal CRP
will compensate for excess reactivity during the entire cycle, respecting the thermaland operational constraints with a minimal reduction in cycle length.
In this study, a BWR core with 444 fuel assemblies is analyzed to get a CRP that
satises operational constraints. The core has 109 control rods, and each one of these
can be placed in 25 dierent axial positions. A typical analysis of an operating cycle
is divided into 10 burnup steps (Total_Burnup_Steps), yielding ((25)109)10 possible
control rod patterns. If one assumes eighth core symmetry, this number is reduced
to ((25)19)10. Moreover, exploiting the control cell core (CCC) technique commonly
Fig. 1. BWR control rod distribution and its 1/8 classication.used in BWR plants, only control rods that have no fresh fuel are used for reactorcontrol. This reduces the number of possible control rod patterns to ((25)5)10 (see
The control rod axial positions are labeled as [00,02,04,06, . . ., 44,46,48]. Posi-tions 0018 are considered deep positions, positions 2030 are considered inter-
mediate positions, and positions 3248 are considered shallow positions. The
intermediate positions are forbidden during normal operation because if they are
used the axial power distribution shape is deformed (Almenas and Lee, 1992). There-
fore, only 19 of the possible 25 positions are allowed. Thus, the total number of pos-sibilities to generate all control rod patterns is ((19)5)10 1064.
The minimal critical power ratio (MCPR) and linear heat generation rate
(LHGR) must be satised. Furthermore, the reactor core must be critical, in this
case, the eective multiplication factor must be adjusted to an eective multiplication
factor target for each burnup step through the whole cycle, and the axial power dis-
tribution must be adjusted to a target axial power distribution. Thus, the objective
function that we propose is a function of the eective multiplication factor (ke),
axial power prole (P), linear heat generation rate (LHGR), and minimal critical
MCPRt Minimal critical power ratio 1.45
744 J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754power ratio (MCPR). Therefore, the CRP design problem can be formulated as the
following optimization problem:
minimizeF x w1 keff ;x keff ;tj j w2X25
i1Px;i P t;ij j w3 LHGRx LHGRtj j
w4 MCPRx MCPRtj j;where ke,x is the eective multiplication factor of the control rod pattern x; ke,t, tar-
get eective multiplication factor; Px,i, axial power distribution for node i of the con-
trol rod pattern x; Pt,i, target axial power distribution for node i; LHGRx, linear heat
generation rate of the control rod pattern x; LHGRt, maximum linear heat genera-
tion rate value permitted; MCPRx, minimal critical power ratio of the control rod
pattern x; MCPRt, minimal critical power ratio value permitted; and w1, . . .,w4 arecalled weighting factors and wiP 0, for i = 1, . . ., 4.
If the thermal limits LHGR and MCPR are satised, the corresponding weighting
factors will be zero, otherwise they will be the ones given by the user penalizing the
objective function. Thus, the objective function will have only the contribution due
to the eective multiplication factor and the axial power prole, when the thermal
limits are satised. Table 1 shows the MCPRt, LHGRt and kt values. Two dierent
target axial power distribution will be assessed, one obtained from the Haling calcu-
lation and a second one using spectral shift (Montes et al., 2004). These axial powerdistributions depend on the fuel reload studied. Moreover, we impose the following
keff ;x keff;tj j 6 d; d > 0;
Px;i P t;ij j 6 eP t;i; e > 0; for i 1; . . . ; 25;where d and e are the convergence criteria for the multiplication factor and the powerTable 1
Target and limits parameters
Symbol Meaning Limit value
kt Eective multiplication factor 1.0 (target)
LHGRt Linear heat generation rate 439 w/cmprole, respectively.
TS is an iterative heuristic procedure for optimization. It has been designed to
overcome local optimality. It is distinguished from other methods because it incor-
porates a tabu list of length t of moves that forbids the reinstatement of certain attri-
butes of previously visited solutions, this tabu list is called short term memory,
J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754 745because it stores information on the t most recent moves. These forbidden moves are
called tabu. For a more detailed presentation of TS, see Glover (1989).
Let us now describe how we use TS to get CRPs. In our approach, a feasible solu-
tion will be an octant of a reactor core and it is represented by a vector
(c1,c2,c3,c4,c5), where ci is the axial position of the ith control rod, for i = 1, . . ., 5(see Fig. 1). The range of each ci, avoiding intermediate positions, is
2,4, . . ., 18,32,34, . . ., 48.In our problem, a move is a transition from one CRP to another that is deter-
mined by the change of only one axial position. Whenever the ith rod is moved
to a dierent axial position, the tabu list forbids any movement of this rod to an
axial position considered in t preceding iterations. Formally, the tabu list consists
of vectors (i,ci), where the ith rod could not be allocated in that axial position ci.
During the process the tabu list is updated circularly. For this study, the lengthof the tabu list was randomly selected in the range (6 6 t 6 16). The value of amove is the dierence between the objective function (dened in Section 2) value
before and after the move. At each iteration the best move is choose, even if it
does not improve the objective function. The number of possible moves in each
iteration is 5 18 = 90. Since it is too expensive from a computational point ofview to evaluate all the moves, only a percentage of such moves will be consid-
ered in this study. The set M of these moves will be randomly generated, and the
rst move that improves the objective function is done. However, if there is nomove that improves the objective function, then one must examine the whole sub-
In this study, the set M corresponds to 40% of the whole moves. This value was
chosen from previous experimental analysis where it was observed that for higher
values of moves there was no apparent improvement in the objective function value.
The range of moves analyzed was from 10% to 100%.
The long-term memory is a function that records moves taken in the past in order
to penalize those that are non-improving. The goal is to diversify the search by com-pelling regions to be visited that possibly were not explored before (Glover, 1989). In
our particular TS implementation, the long-term memory is an array, which will be
denoted F. The array has zeroes at the beginning of the procedure. When a control
rod is settled in an axial position at a given iteration, the array F changes as follows:
F i;ci F i;ci 2. The entry F i;ci is the frequency at which the axial position ci of the ithcontrol rod has been settled. The values of non-improving moves that switch the ax-
ial position ci of the ith control rod are then increased by F i;ci .The two last concepts to explain are the aspiration and the stopping criteria used.
The aspiration criteria cancels the status tabu of a move when it nds a feasible solu-
tion with a better function value than the best solution in the past. Our TS will be
stopped if the number of iterations used without improving the best solution is greater
than 40. This number was chosen from a statistical analysis as it was done with the
percentage of moves.
As it is set in Section 2, the objective function includes the thermal safety lim-
its, the axial power distribution and the eective multiplication factor. TS wasimplemented along with the 3-D reactor core simulator CM-PRESTO (Scand-
SAVE BURNUP NO IS IT THE LAST STEP VALUES BURNUP STEP?
IS IT THEBURNUP STEP 1?
YES NO END N-TH ITERATION CALCULATIONS
OR ITERATIONS? HALING
CALCULATION ASPIRATION FUNCTION EQUAL TO THE BEST
CONTROL ROD PATTERN YES ASPIRATIONRANDOMLY GENERATED, FUNCTION > THE BEST
ASPIRATION FUNCTION = M NEIGHBOR?
NEW CONTROL ROD AXIAL POSITION UPDATE TABU TIME ARRAY,
FREQUENCIES AND THE BESTRETURN THE CONTROL ROD PATTERN
CONTROL ROD TO ORIGINAL AXIAL YES
RULE VIOLATED OR YES NO END40 % OF THE NEIGHBORHOOD NEIGHBORHOOD
ANALYZED? SEARCH? OBJECTIVE FUNCTION
EQUAL TO THE BEST NO NEIGHBOR
CM-PRESTO OBJECTIVE CALCULATIONS YES FUNCTION PREVIOUS >
THE BEST NEIGHBORHOOD?
CALCULATIONS RETURN THE UPDATE THE BEST CONTROL ROD NEIGHBORHOOD, TO ORIGINAL AXIALTHE BEST SAFETY POSITION
LIMITS AND THE BESTAXIAL POWER
OBJECTIVE DISTRIBUTION YES FUNCTION NO
< IS IT ASPIRATION A TABU
FUNCTION? NO MOVEMENT?
YES OBJECTIVE FUNCTION
EQUAL TO OBJECTIVE IS IT FUNCTION PLUS AXIAL THE BESTCONTROL ROD POSITION NEIGBORHOOD?
Fig. 2. Flowchart of the CRP search.
746 J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754
does succeed for all the burnup steps then the process is stopped, obtaining as a re-
sult a feasible LP-CRP. A pseudo-code for our iterative LP-CRP combined system
J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754 747can be described as follows:
Phase 1: run the LP system
Phase 2:Input: the LP output from Phase 1
do burnup_step_number = 1 to Total_Burnup_Steps
run the CRP system
if (not succeed) then Phase 1
4. Problems analyzed and results
The CRP design system based on tabu search was applied to get the optimal CRP
for actual BWR operating cycles. To test the system the 3-D simulator CM-PRESTO
was used to analyze the corresponding BWR operating cycles.
In the rst approach, a CRP was obtained for an actual BWR fuel assembly LP.
In the second approach, the combined LP-CRP optimization problem was solved for
another BWR operating cycle.
The results of the rst stage are compared against the ones given by Montes et al.(2004) for the same problem. The objective is to get a CRP that satises the thermal
and operational limits for the whole throughout of the reactor cycle. In the rst ap-
proach, we use BWR fuel reloads previously obtained using engineer expertise for
the actual cycle. These loading patterns use 120 fresh fuel assemblies in two batches
one of 116 fuel assemblies of 3.52 w/o of U-235, and a second one of four fuel assem-
blies of 3.03 w/o of U-235. Two dierent axial power proles were considered, the
rst one is a Haling prole (Graves, 1979) and the second one is a spectral shift pro-power, 1995) to compose the Optimization Tabu Search System. This system is a
FORTRAN-77 based program implemented in an Alpha computer with UNIX
operating system. All Tabu Search runs were carried out with a random initial
Due to the way that the objective function is constructed, the system requires twodierent runs of CM-PRESTO; the rst one is to calculate the target axial power dis-
tribution in each burnup step, the other run is used to calculate the three parameters
and the axial power distribution. Fig. 2 shows the owchart of the CRP optimization
We now describe how to combine the CRP system with the LP system developed
by Castillo et al. (2004). Our combined system is a process consisting in two iterative
phases with two complimentary multi-objective functions. In the rst phase, we use
the LP system to get an optimized LP. In the second phase, this LP is used as an in-put to our CRP system, a TS technique designed to obtain CRPs in certain burnup
steps previously dened along the whole cycle. If the CRP system does not succeed in
a burnup step then the process goes to the rst phase again. On the contrary, if itle (Glasstone and Sesonske, 1994).
748 J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754Using the CRP design system based upon the TS, we obtained several control rodpatterns for this fuel reload. Less than 22,000 evaluations of the objective function
were required for each TS. This is a very small number compared with the
8.66 1063 permutations that are possible in one octant symmetry. In each burnup
Fig. 3. Objective function behavior for burnup steps 13 for the CRP using TS.step, we obtained a CRP where the objective function was minimized. The CRP that
provides the maximum cycle length for the Haling prole is reported through the fol-
lowing gures. Fig. 3 shows the objective function behavior to burnup steps 13, Fig.
4 shows the objective function behavior to burnup steps 46, and Fig. 5 shows the
objective function behavior to burnup steps 79. Table 2 shows the values obtainedfor ke, LHGR and MCPR in each burnup step. We ran our CRP TS system 10
times and the CRP with the longest cycle length is reported here.
The results for the GA were taken from Montes et al. (2004), where the problem
analyzed was the same that we use here. In this study, the cycle length given by the
CRP obtained by engineer expertise was 10023 MWd/TU. By using the optimized
control rod patterns obtained by TS and GA the cycle length are signicantly greater
compared to the one obtained by the engineer expertise.
Figs. 68 show the CRP and axial power proles for several burnup steps usingTS. Table 3 shows the cycle length, the number of control rod movements and the
number of rod movements for positions between deep and shallow.
For the same problem, using the spectral shift prole, the results for cycle length,
the number of control rod movements and the number of rod movements for posi-
tions between deep and shallow are shown in Table 4. Also the results obtained from
GA calculations for the same axial prole calculated by Montes et al. (2004) are
shown here, and again the cycle lengths obtained by using the tabu CRP system is
greater than the one obtained by using the GA system.
Fig. 4. Objective function behavior for burnup steps 46 for the CRP using TS.
Fig. 5. Objective function behavior for burnup steps 79 for the CRP using TS.
Multiplication factor, LHGR and MCPR values in each burnup step for the rst problem using a Haling
axial power prole
Burnup step number LHGR (w/cm) MCPR ke Number of iterations
1 390.16 1.51 1.00002 71
2 387.64 1.55 0.99996 73
3 374.67 1.58 0.99990 42
4 402.53 1.51 0.99995 89
5 398.49 1.58 0.99999 30
6 407.90 1.56 1.00127 23
7 401.86 1.57 1.00000 63
8 406.02 1.52 1.00014 26
9 390.43 1.53 1.00004 24
10 371.95 1.58 0.99935 1
J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754 749
Fig. 6. Axial power distribution at 1000 MWd/TU and its CRP using TS.
Fig. 7. Axial power distribution at 5000 MWd/TU and its CRP using TS.
750 J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754
J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754 751In this stage, the second problem considered is the loading pattern for an equilib-
rium cycle loading, which uses 104 fresh fuel assemblies of 3.70 w/o of U-235 and the
axial power prole is an spectral shift prole (Montes et al., 2004). Results for this
problem are shown in Table 5. As can be seen from here, the results obtained from
the tabu CRP system produce more energy than those produced using GA.
Fig. 8. Axial power distribution at 9000 MWd/TU and its CRP using TS.
Comparison of results between GA and TST for the rst problem using a Haling axial power prole
Method Cycle length
deep and shallow positions
Genetic algorithms 10,157 44 20
Tabu search technique 10,321 41 18
Comparison of results between GA and TST for the rst problem using an spectral shift axial power
Method Cycle length
deep and shallow positions
Genetic algorithms 10,305 25 12
Tabu search technique 10,564 37 17
752 J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754The second stage considers the combination of the CRP and LP (Castillo et al.,
2004) systems based on the TS. The cycle analyzed is dierent to the one used in
the rst stage. We ran our LP-CRP combined system using a loading pattern of
112 fresh fuel assemblies of 3.53 w/o of U-235 and a Haling axial power prole.
The idea under this process is to minimize the objective function. The combined
LP-CRP that produces the maximum cycle length, which is 9756.86 MWd/TU (Hal-
ing calculation), is reported in Table 6, where the cycle length, the number of control
rod movements and the number of rod movements for positions between deep andshallow are shown. In this case the combined system provides more energy than the
one given by the LP-CRP generated by using engineer expertise.Table 6
Cycle length comparison between the loading and control rod patterns given by engineer expertise and the
one generated by the TST coupled system
Method Cycle length
Movements between deep
and shallow positions
Engineer expertise 9461.0 44 20
Tabu search technique 9943.7 41 18
Comparison of results between GA and TST for the equilibrium cycle using an spectral shift axial power
Method Cycle length
deep and shallow positions
Genetic algorithms 10,896 38 13
Tabu search technique 11,005 48 105. Discussion and conclusions
A TS technique has been implemented successfully to the optimization of BWR
control rod patterns. The design system developed in this work generates controlrod patterns that satisfy the operational and safety constraints and keeps critical
the reactor through the whole cycle length. These results indicate that CRP designed
using TS yield fuel cycles that can produce more energy than those obtained using
engineer expertise or genetic algorithms.
In general, it is not possible to claim the superiority of one optimization method
over other because it depends on the specic problem analyzed and the implementa-
tion of the optimization method. In particular in this study, the results obtained sug-
gest that TS produce better results than those produced by the GA implemented byMontes et al. (2004) for the problem here considered.
On the other hand, an extensive comparative study for the job shop problem and
the ow shop problem has shown the signicant superiority of TS over other ap-
proaches including iterative improvement, genetic algorithm, simulated annealing,
side is solved by evaluating only a percentage of the whole neighborhood as it isdone in this study. Also, Carter (1997) mention that he is not aware of compar-
J.A. Castillo et al. / Annals of Nuclear Energy 32 (2005) 741754 753ative studies between GA and TS. Finally, for the CRP independent problem or
the combined LP-CRP problem will be necessary to perform a comparative study
to demonstrate the superiority of one method over the other but it is beyond the
goals of this paper.
Turinsky and Parks (1999) mention that for a BWR the LP and CRP optimiza-
tion must be considered together since the LP and CRP decisions are tightly coupled
problem, which is part of the purpose of this work. Turinsky (1999), also mentionthat use of Tabu Search for PWR LP produces results comparable to those from
GA but extensive testing must be performed before to give a rm conclusion.
Although some testing for the independent CRP problem has been done in this
work, and it shows better results for the TS than the GA, an extensive comparative
analysis must be performed to claim any superiority.
On the other hand, use of TS to generate CRP for an actual operating LP, in the
rst problem using a Haling axial power prole, shows an increase in the cycle
length. This energy is 2.97% greater than was generated in the actual cycle (10023MWd/TU). It also produces more energy with fewer control rod movements than
the one designed using the GA.
This extra energy represents 13 days more of full power operation. To assess
the economical impact of this optimized fuel reload it will be necessary to per-
form a multi-cycle analysis to know the actual advantages of this fuel assembly
Furthermore, for the others problems using the CRP system independently, TS
yield a cycle length with more energy than the one produced by the GA for both,the previous problem but now using the spectral shift prole and for the equilibrium
cycle problem. Although there were fewer control movements in the CRP designed
using GA than the one using TS, in both problems.
Finally, eectiveness of the TS combined system is shown by comparing the re-
sults for an actual BWR operating cycle with those obtained using engineer exper-
tise. The LP and CRP generated by our optimized combined system generate
more energy satisfying all the operational and safety constraints. The extra energy
produced represents 21 days more of full power operation. Thus, our system achievesits objective of getting more energy satisfying all the imposed constraints.
The authors acknowledge the support given by CONACyT through the research
Project 33806-U. The authors also acknowledge the signicant comments of the re-threshold accepting, constraint satisfaction, neural networks, and other local search
methods (see Vaessens et al., 1996).
Furthermore, Carter (1997) claims that TS produces very good results but it
has a downside because it needs to evaluate the whole neighborhood. This down-viewer which has contributed to improve our paper.
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BWR control rod design using tabu searchIntroductionBWR control rod patternMethodologyProblems analyzed and resultsDiscussion and conclusionsAcknowledgementsReferences