BWR control rod design using tabu search

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  • Abstract

    * Corresponding author. Tel.: +52 55 53297233; fax: +52 55 53297340.

    E-mail addresses: (J.A. Castillo), (J.J. Ortiz), galonso (G. Alonso), (L.B. Morales), edmundo@nuclear.esfm.

    (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-

    tor cycle.

    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

    Fig. 1).

    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.

    3. Methodology

    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-

    set M.

    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-





    NO YES



























    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.

    Table 2

    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.

    Table 3

    Comparison of results between GA and TST for the rst problem using a Haling axial power prole

    Method Cycle length


    Control rod


    Movements between

    deep and shallow positions

    Genetic algorithms 10,157 44 20

    Tabu search technique 10,321 41 18

    Table 4

    Comparison of results between GA and TST for the rst problem using an spectral shift axial power


    Method Cycle length


    Control rod


    Movements between

    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


    Control rod


    Movements between deep

    and shallow positions

    Engineer expertise 9461.0 44 20

    Tabu search technique 9943.7 41 18

    Table 5

    Comparison of results between GA and TST for the equilibrium cycle using an spectral shift axial power


    Method Cycle length


    Control rod


    Movements between

    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