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Annals of Nuclear Energy 34 (2007) 207–212

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NUCLEAR ENERGY

Fuel loading and control rod patterns optimization in a BWRusing tabu search

Alejandro Castillo *, Juan Jose Ortiz, Jose Luis Montes 1, Raul Perusquıa

Instituto Nacional de Investigaciones Nucleares, Km. 36.5 Carretera Mexico-Toluca, Ocoyoacac 52750, Estado de Mexico, Mexico

Received 19 May 2006; received in revised form 11 December 2006; accepted 18 December 2006Available online 15 February 2007

Abstract

This paper presents the QuinalliBT system, a new approach to solve fuel loading and control rod patterns optimization problem in acoupled way. This system involves three different optimization stages; in the first one, a seed fuel loading using the Haling principle isdesigned. In the second stage, the corresponding control rod pattern for the previous fuel loading is obtained. Finally, in the last stage, anew fuel loading is created, starting from the previous fuel loading and using the corresponding set of optimized control rod patterns.For each stage, a different objective function is considered. In order to obtain the decision parameters used in those functions, the CM-PRESTO 3D steady-state reactor core simulator was used. Second and third stages are repeated until an appropriate fuel loading and itscontrol rod pattern are obtained, or a stop criterion is achieved. In all stages, the tabu search optimization technique was used. The Qui-nalliBT system was tested and applied to a real BWR operation cycle. It was found that the value for keff obtained by QuinalliBT was0.0024 Dk/k greater than that of the reference cycle.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

The operation cycle design for a boiling water reactor(BWR) has several optimization stages. One is the fuelloading (FL) design, which implies to place an inventoryof fuel assemblies into the core, in order to maximize thecycle length, while thermal limits, hot excess reactivityand cold shutdown margin are satisfied. Other optimiza-tion stage is related to the reactivity control during theoperation cycle. The goal in this part of the cycle designis to predict the full power control rod pattern (CRP). Inthis stage, the control rod axial locations during the wholeoperation cycle are considered. To obtain these CRPs it is

0306-4549/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.anucene.2006.12.006

* Corresponding author. Address: Instituto Nacional de InvestigacionesNucleares, Km. 36.5 Carretera Mexico-Toluca, Ocoyoacac 52045, Estadode Mexico, Mexico. Tel.: +52 55 53297200; fax: +52 55 53297301.

E-mail addresses: jacm@nuclear.inin.mx (A. Castillo), jjortiz@nuclear.inin.mx (J.J. Ortiz), jlmt@nuclear.inin.mx (J.L. Montes), mrpc@nuclear.inin.mx (R. Perusquıa).

1 Also PhD student at the Facultad de Ciencias of the UniversidadAutonoma del Estado de Mexico, Mexico.

necessary to take into account several constraints, includ-ing some of those mentioned above.

FL and CRP problems have been commonly solved inan independent way by using several optimization tech-niques, such as neural networks, ant colonies, genetic algo-rithms, tabu search and fuzzy logic (Ortiz and Requena,2004, 2006, 2004; Castillo et al., 2004, 2005; Francoiset al., 2004), among others. In some cases, Haling’s princi-ple (Haling, 1964) has been used to obtain FL design. TheHaling’s principle guarantees a safe operation of the reac-tor but it does not maximize cycle length. It is possible tomaximize cycle length using an adequate operation strat-egy. A good option is operating the reactor by using ‘‘Spec-tral Shift’’ strategy (Specker et al., 1978), which permits ‘‘tobreed’’ fissile material and allows a longer cycle length.Spectral Shift operation can be favored if control rod posi-tions (CRP design) are such, as to produce a peaked axialpower distribution at the bottom of the core. However, it isimportant to keep in mind that CRP design depends on FLdesign. In other words, if we do not have a good FL design,then we will not obtain a good CRP design. Therefore, an

208 A. Castillo et al. / Annals of Nuclear Energy 34 (2007) 207–212

approximation with greater scope is to consider the solu-tion of FL and CRP problems in a coupled manner.

QuinalliBT derives its name from ‘‘QUItze’’ and ‘‘toN-ALLI’’, two words of two ancient Mexican languages,and ‘‘Busqueda Tabu’’ (tabu search) in Spanish. The sys-tem solves both FL and CRP designs in a coupled way.In each stage a different multi-objective function is usedand then evaluated with CM-PRESTO (Scandpower,1993) 3D code. The tabu search (TS) heuristic techniqueis applied in the whole process. The QuinalliBT systemwas designed using FORTRAN-77 language in an Alphaworkstation with UNIX operation system.

2. Problem description

Fuel loading design problem may be described as follows:for a given inventory of fuel assemblies, an arrangement ofthem inside the reactor core has to be found, in such a waythat some restrictions will be satisfied during cycle opera-tion. Among these restrictions can be mentioned, for exam-ple, that some fuel bundle allocation rules have to besatisfied, as well as thermal limits, hot excess reactivity andcold shutdown margin; also energy extraction has to be max-imized. The Haling’s principle has been one of the most usedtechniques for designing FL. With this operation strategy asafe operation of the reactor can be achieved without prob-lems. However, the obtained energy is far from being themaximum energy of the cycle.

In CRP design, it is necessary to establish axial controlrod positions for each one of several burnup steps. Becausean adequate CRP design allows for axial power distribu-tion to be adjusted to a peak at the bottom of the core,CRP has an important contribution in achieving SpectralShift operation in the reactor. Another important consider-ation in this stage of the design is that reactor core has tobe critical, and also that thermal limits must be satisfiedthroughout the cycle. Although cycle length is stronglyaffected by CRP throughout the cycle, its maximizationat this stage is not a goal. This task will be made later bymeans of fuel loading pattern optimization.

As it was mentioned above, these problems have beensolved in an independent way, being one of the main rea-sons that the search space is too large if the problem is con-sidered in a coupled way in which case, the total number ofparameters included in the whole process is increased. Onthe other hand, the computers in the past did not havethe capacity to make the necessary calculations in a reason-able time. With present computers this is not a problemanymore. So, the coupled problem has begun to be investi-gated recently. An approach for solving this new problemwas proposed in a paper by Kobayashi and Aiyoshi(2002), by using a method based on Genetic Algorithms.In order to optimize the FL-CRP design, an ‘‘if-then’’ heu-ristic rule was implemented. As it was mentioned at thebeginning of this paper, the Haling’s principle has beenused by many authors, for example, Kobayashi et al. In

that paper the authors, pay special attention on the numberof fresh assemblies during the iteration process.

In a BWR reactor with a total of 444 fuel assemblies and109 control rods, we can make the following deliberationsabout the search space size. In the first case, if all fuelassemblies are different, then we have 444! possible solu-tions to FL design. In the control rods case, if we are con-sidering 25 axial nodes and 10 burnup steps, then we have((109)25)10 possible solutions to CRP design. We can seethat, both FL and CRP designs are combinatorial prob-lems. For this reason, a combinatorial technique can beused to solve these problems.

If some heuristic rules and simplifications are consid-ered, the number of possible solutions can be reduced con-siderably. Then, for the FL search space, it is possible toreduce it from 2.1 · 10984 to 7.3 · 1054 possible solutions,applying the following rules or simplifications:

1. Fuel load design with low leakage strategy.2. Eighth reactor core symmetry.3. Control cell core load strategy.

On the other hand, for the CRP search space, it is alsofeasible to reduce the total possible solutions, from 5.683 ·101523 to 7.8 · 1069 (with 10 burnup steps), applying the fol-lowing heuristic rules or simplifications:

1. Eighth reactor core symmetry.2. Control cell core load strategy.3. The intermediate positions of control rods are for-

bidden.

In this work, we take advantage of previously designedsystems, which were developed to individually optimizeboth problems (Castillo et al., 2004, 2005). By using them,it was possible to develop a new system to optimize thecoupled FL-CRP problem. In the following section, thenew system is described in detail.

3. QuinalliBT system

The QuinalliBT system was developed in order to opti-mize FL and CRP design in a coupled way, in a BWRoperating cycle. The system has three stages, each oneusing a different multi-objective function. In the first stage,an optimized FL is obtained using the Haling’s principle(Haling, 1964) while thermal limits, hot excess reactivityand cold shutdown margin are satisfied. When the Haling’sprinciple is applied in order to design an operation strat-egy, power peaks within reactor core are minimized. How-ever, the obtained FL is not an optimized design from thepoint of view of maximum energy extracted from the fuel,therefore it is used as a ‘‘seed’’ FL. This seed FL is used inthe second stage to obtain a set of optimized CRP using‘‘Spectral Shift’’ strategy (Specker et al., 1978). In the laststage, the CRP design attained in the second stage is usedto obtain a new FL design, instead of using the Haling’s

A. Castillo et al. / Annals of Nuclear Energy 34 (2007) 207–212 209

principle. Thus, second and third stages are repeated in aniterative process, until optimized FL and CRP designs areobtained, or a stop criterion is achieved. Fig. 1 shows theflowchart of the QuinalliBT system. In the following para-graphs, QuinalliBT system stages are described.

As mentioned above, the Haling’s principle is used infirst stage in order to obtain an initial FL design. The mainidea is to maximize keff value at the end of the cycle, whilethermal limits, hot excess reactivity and cold shutdownmargin are satisfied. The objective function to be maxi-mized is the following:

F obj ¼ keff � w1 þ DLim1 � w2 þ DLim2 � w3 þ DLim3 � w4

þ DLim4 � w5 þ DLim5 � w6 ð1Þ

wherekeff effective multiplication factor at end of the cycleDLim1 = MFLPDlim �MFLPDobtained

DLim2 = MPGRlim �MPGRobtained

DLim3 = MFLCPRlim �MFLCPRobtained

DLim4 = SDMobtained � SDMlim

DLim5 = HERlim � HERobtained

MFLPD maximum fraction of linear power densityMPGR maximum power generation ratioMFLCPR maximum fraction of limiting critical power

ratioSDM cold shutdown margin at the beginning of cycleHER hot excess reactivity at the beginning of cycle

and w1, w2, w3, w4, w5, w6 are non-negative weighting fac-tors obtained from a statistical analysis. When restrictionsLimi (i = 1, . . . , 5) are satisfied, their respective wi are equalto zero. It can be seen that if all constraints are satisfied,then Eq. (1) works only to maximize effective multiplica-tion factor keff.

Input Data stop

YES Fuel Loading with Haling’s stop principle NO criterion

was satisfied?

Seed Fuel Loading

Control Rod Fuel Loading Patterns using CRP

Fig. 1. QuinalliBT’s flowchart.

In the next stage, operation cycle is divided into ten bur-nup steps. In each one of these steps control rod positionsmust be fixed, taking into account that the reactor must becritical, the axial power shape must match a target axialpower profile and that thermal limits must be satisfied.The objective function to minimize is the following:

F ¼X25

i¼1

ðP iobj � P i

actÞ � w1 þ jkeff � kcritj � w2 þ DLim1 � w3

þ DLim2 � w4 þ DLim3 � w5 ð2ÞwherePobj target axial power profilePact obtained axial power profilekeff obtained effective multiplication factorkcrit objective effective multiplication factorLim1 = MFLPDlim �MFLPDobtained

Lim2 = MPGRlim �MPGRobtained

Lim3 = MFLCPRlim �MFLCPRobtained

and wi, i = 1, . . . , 5 are weighting factors. It is desirable tominimize the differences between the target and the obtainedaxial power profiles. When the constraints Limi (i = 1, . . . , 3)are satisfied, then its respective wi, i 2 {3, . . . , 5} is equal tozero. w2 is zero if jkeff � kcritj < 0.00010. It can be seen thatif all of these constraints are satisfied, Eq. (2) works onlyto minimize the difference between the axial power profiles.

In the last stage of the iterative process, the optimizedCRP design is used instead of Haling’s principle to obtaina new FL design. In this case, keff at the end of cycle is max-imized, while the thermal limits in each burnup step, thecold shutdown margin and the hot excess reactivity at thebeginning of cycle must be satisfied. The objective functionused in this stage is the following:

F ¼ kEOR � w1 �Xn�1

i¼1

jkieff � kcritj � w2 þ

Xn

i¼1

Limi1 � w3

þXn

i¼1

Limi2 � w4 þ

Xn

i¼1

Limi3 � w5 þ Lim4 � w6 þ Lim5 � w7

ð3Þ

wheren number of burnup stepskEOR obtained effective multiplication factor at the end

of cycleki

eff obtained effective multiplication factor in eachburnup step

kcrit objective effective multiplication factorLimi

1 = MFLPDi,lim �MFLPDi,obtained

Limi2 = MPGRi,lim �MPGRi,obtained

Limi3 = MFLCPRi,lim �MFLCPRi,obtained

Lim4 = SDMobtained � SDMlim

Lim5 = HERlim � HERobtained

i denotes one of the burnup steps in which cycle length isdivided. In the same way wi, i = 1, . . . , 7 are weighting factorswith the same considerations of those for Eqs. (1) and (2).

Table 1Results in the Haling’s calculation

Parameter Limit Obtained value

MFLPD <0.9 0.7378MPGR <0.9 0.6980MFLCPR <0.9 0.8944HER <1.03 1.024SDM >1.5 1.501keff 0.9986 1.006

210 A. Castillo et al. / Annals of Nuclear Energy 34 (2007) 207–212

As it was mentioned, the second and third stages per-form an iterative process until FL-CRP design is obtained,or a stop criterion is achieved.

4. Tabu search

In both FL and CRP design, the tabu search (TS) tech-nique was applied. Following is a brief description aboutthis technique. See reference Glover (1968) for more detailson these concepts.

The TS technique is an iterative process used to obtaina solution that maximizes (or minimizes) an objectivefunction, in a set X of feasible solutions. The TS techniquestarts from a randomly chosen initial feasible solution.Then, space X is explored by moving in a neighborhoodfrom one solution to another. In this process, each feasiblesolution x has an associated set of neighbors, called theneighborhood of x, N(x) 2 X. If N(x) is very large, it ispossible to choose, at each iteration of the process, a sub-set SN(x) � N(x) in which case, a search from the currentsolution x to the best one x* in SN(x), whether or notf(x*) is better than f(x), is carried out. In some problems,evaluation of the entire neighborhood becomes veryexpensive in terms of time and computing resources. Inthat case, it is possible to reduce the size of SN takingthe first move that improves the current solution. If thisis not possible, then it will be necessary to examine thewhole neighborhood.

During search process, the TS technique allows cyclingprevention using a short-term memory, an array with eitherfixed or variable length. The purpose of this memory is tostore certain prohibited movements (the reason why thisarray is called tabu list). In this sense, a movement remainstabu (forbidden) during n iterations (tabu tenure). It is verycommon also to apply a long-term memory to diversifyneighbors search, being the objective of this memory tomove toward unvisited regions.

It is important to mention that the TS technique mayforbid some interesting moves because of their tabu sta-tus. In such case, if a move produces the best solution ofthe whole process, but it is forbidden, it is possible tocancel its tabu status and take it into account (aspiration

criterion).In the present analysis a tabu list with variable length

from 7 to 15 is used. These values were obtained after a sta-tistical analysis in the development process of QuinalliBTsystem. Since function evaluation is a process that takesso much time and computing resources, the neighborhoodwas not revised in its entirety, neither for FL nor for CRPdesigns, but only 10% and 40%, respectively. During FLdesign, four CM-PRESTO simulator executions arerequired in order to evaluate the objective function. ForCRP design only one CM-PRESTO execution is required.

In order to penalize the high frequency movements thatdo not improve the objective function during the iterativeprocess, an aspiration criterion and a long-term memorywith a FVEC (Frequency VECtor) array were applied in

the present study. This array is equal to zero at the begin-ning of the process and updated in the following way:FVEC(i, j) = FVEC(i, j) + 2, if (i, j) move was made.

In order to finish the process, it is necessary to definesome stop criteria. The first one is to stop the algorithmwhen a maximum number of iterations is achieved; the sec-ond one is to stop the process after k iterations, if the objec-tive function does not improve.

5. Results

To test the QuinalliBT system, data of a real operationcycle were used. The chosen FL design (including itsCRP’s) corresponds to an 18-month cycle length, with108 fresh fuel assemblies, a cycle burnup of 11,020MWD/T and keff = 0.9986. This keff value was obtainedfrom the simulation of the reference cycle with CM-PRE-STO simulator. The QuinalliBT system was executed sev-eral times, with all executions having similar results. Bestresults obtained are shown however.

Table 1 shows the results obtained for the first stage,according to Haling calculation. It can be seen that thermallimits, hot excess reactivity and cold shutdown margin weresatisfied. The corresponding limit values for each parame-ter are shown in this table. Eighty-four iterations were car-ried out to obtain these results, and the objective functionwas evaluated 1996 times using CM-PRESTO simulator.In this stage the effective multiplication factor obtained ishigher than the target value for keff.

Once seed FL is obtained, QuinalliBT system performsan iterative process, until coupled FL-CRP is optimized.Fig. 2 presents the results obtained after four CRP-FL iter-ations. It can be seen that through the iterative process,effective multiplication factor decreases at the end of cycle,due to the fact that in the first iterations some thermal lim-its are violated. In the last iteration, the thermal limits aresatisfied and keff values in the intermediate burnup steps areadjusted to the target keff. At the end of cycle, keff value ishigher than target keff. In the last CRP optimization caseexecuted, 637 iterations were performed and the objectivefunction was evaluated 10,867 times. The number ofCRP iterations decreases when the CRP-FL optimizationprocess is almost finished. In Table 2, keff values, burnupsteps and thermal limits are shown.

Finally, Fig. 3 shows keff behavior through the CRP-FLiterative process, when FL is obtained. The optimized CRPis taken into account in this stage instead of Haling’s prin-

Table 2keff and thermal limits values

Burnup step keff MFLPD MPGR MFLCPR

0 1.0080 0.8967 0.7823 0.87951037 1.0090 0.8758 0.8774 0.83912093 1.0080 0.8449 0.8541 0.85623052 1.0070 0.7290 0.7350 0.86744081 1.0060 0.7651 0.7778 0.90105032 1.0050 0.8318 0.8742 0.87286065 1.0060 0.8341 0.8836 0.88587096 1.0040 0.7667 0.8204 0.87748157 1.0030 0.7711 0.8374 0.89919279 1.0020 0.7569 0.8432 0.8838

10,307 1.0010 0.7567 0.8751 0.896611,020 1.0000 0.7439 0.8681 0.8818

k eff behaviour

0.9980

1.0000

1.0020

1.0040

1.0060

1.0080

1.0100

0 2000 4000 6000 8000 10000burnup (MWd/T)

k eff

kcrit

keff first iteration

keff second iteration

keff third iteration

keff fourth iteration

Fig. 2. keff Behavior after four iterations.

Table 3Parameters obtained in the last iteration

Burnup step kcrit keff obtained

0 1.0076 1.0091037 1.0091 1.0092093 1.0085 1.0093052 1.0073 1.0074081 1.0061 1.0065032 1.0053 1.0056065 1.0045 1.0047096 1.0042 1.0048157 1.0033 1.0049279 1.0018 1.002

10,307 1.0011 1.00111,020 0.9986 1.001

HER 1.0024SDM 1.731

A. Castillo et al. / Annals of Nuclear Energy 34 (2007) 207–212 211

ciple. In the last FL optimization executed case, QuinalliBTsystem performed 170 iterations and evaluated the objec-tive function 8279 times. keff value decreases during theCRP-FL iterative process, due to some thermal limits vio-lated in the first iterations. Later on, thermal limits are sat-isfied and keff at the end of cycle is higher than target keff.

k eff behaviour

0.9960

0.9990

1.0020

1.0050

1.0080

0 2000 4000 6000

burnup (MWd/T)

k eff

Fig. 3. keff Behavior through

Table 3 shows the parameters obtained in this stage andit can be seen that all of these parameters are satisfied. Itis necessary to comment that, according to the stop criteriaof QuinalliBT system, the third stage of FL optimizationwas only performed three times.

8000 10000

kcrit

keff first iteration

keff second iteration

keff third iteration

out the iterative process.

212 A. Castillo et al. / Annals of Nuclear Energy 34 (2007) 207–212

6. Conclusions

A new system to optimize a FL-CRP design in a coupledway for BWRs was developed. Tests showed a convergentbehavior of the system throughout CRP-FL iterative pro-cess. Although at the beginning of the process some ther-mal limits are not satisfied, once iterative process isfinished all parameters are satisfied. The effective multipli-cation factor keff so obtained is better than the target valuefor keff used to test the QuinalliBT system.

According to Table 3, the keff obtained with QuinalliBTsystem is 0.0024 Dk/k greater than target keff. This valuerepresents around 10 full power days of reactor operation,with the corresponding extra energy generated. The suc-cessful executions of QuinalliBT system through differentcases suggest the robustness of the system.

During the performed tests, it could be seen that the firststage of the process can be eliminated, because seed FL isindependent of the last FL design. New proposals to obtainseed FL in a simpler way are under study, in order toreduce CPU time. Actually, the maximum number of iter-ations in the different executed tests was 10, which still rep-resents a considerable amount of CPU time.

Acknowledgement

This research has been partly supported through theresearch project SEP-2004-C01-46694 by CONACYT,Mexico.

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