BWR fuel reloads design using a Tabu searchtechnique

Alejandro Castilloa,1, Gustavo Alonsoa,*, Luis B. Moralesb,Cecilia Martın del Campob, J.L. Francoisb, Edmundo del Vallec

aInstituto Nacional de Investigaciones Nucleares, Km 36.5 Carretera Mexico-Toluca, Ocoyoacac 52045,

Edo. de Mexico, MexicobUniversidad Nacional Autonoma de Mexico, Apartado Postal 70-221, Mexico, D.F. 04510, MexicocInstituto Politecnico Nacional, Escuela Superior de Fısica y Matematicas, Unidad Profesional

‘‘Adolfo Lopez Mateos’’, ESFM, Mexico, D. F., 07738, Mexico

Received 29 May 2003; accepted 29 June 2003

Abstract

We have developed a system to design optimized boiling water reactor fuel reloads. This

system is based on the Tabu Search technique along with the heuristic rules of Control CellCore and Low Leakage. These heuristic rules are a common practice in fuel management tomaximize fuel assembly utilization and minimize core vessel damage, respectively. The system

uses the 3-D simulator code CM-PRESTO and it has as objective function to maximize thecycle length while satisfying the operational thermal limits and cold shutdown constraints. Inthe system tabu search ideas such as random dynamic tabu tenure, and frequency-basedmemory are used. To test this system an optimized boiling water reactor cycle was designed

and compared against an actual operating cycle. Numerical experiments show an improvedenergy cycle compared with the loading patterns generated by engineer expertise and geneticalgorithms.

# 2003 Elsevier Ltd. All rights reserved.

Annals of Nuclear Energy 31 (2004) 151–161

www.elsevier.com/locate/anucene

0306-4549/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0306-4549(03)00214-7

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

E-mail addresses: jacm@nuclear.inin.mx (A. Castillo), galonso@nuclear.inin.mx (G. Alonso), lbm@

servidor.unam.mx (L.B. Morales), edmundo@esfm.ipn.mx (E. del Valle).1 Also Ph. D. student at Universidad Autonoma del Estado de Mexico.

1. Introduction

From an economical point of view for a Boiling Water Reactor (BWR) it isnecessary to get as much fuel energy as we can to avoid under burnt fuel and wasteenergy. Thus, the fuel loading pattern plays a very important role to achieve thatgoal. Design of BWR fuel reloads is usually based in engineer expertise, which is atechnique that uses human knowledge. This technique does not optimize the use ofthe fuel assemblies having in some cases under burnt fuel when it is discharged of thecore. To avoid this waste of energy a better design of fuel reloads can be donethrough optimization techniques.BWR fuel assembly reloads design can be considered a combinatorial problem,which has been tackled using genetic algorithms (Francois and Lopez, 1999), simu-lated annealing (Moore et al., 1999) and recently tabu search (Jagawa et al., 2001).All of these techniques have as objective to maximize the cycle length while satisfy-ing the operational thermal limits and cold shutdown constraints. For the last tech-nique, Jagawa et al. (2001) designed an automatic system that uses a tabu searchmethod along with a simple linear perturbation method to avoid the extensive use ofthe 3-D simulator.A BWR presents strong three-dimensional material heterogeneities such as fuelenrichment, burnable poison, coolant void and control rods, besides the number offuel assemblies embedded in the core in comparison with a PWR. These character-istics makes the loading pattern optimization problem very complex and it appealsfor the use of a licensed 3-D core simulator to achieve the goal proposed in com-parison with the use of 2-D simulators used for the PWR optimization problem.We develop a system named optimization tabu search system (OTSS) based on thetabu search (TS) optimization technique, using the 3-D simulator code CM-PRE-STO to evaluate the objective function. Our TS uses a random tenure and long-termmemory whose purpose is to diversify the search of the optimal value making theprocess more efficient leading to explore more scenarios in less time than the originaltabu search. On the other hand, using this technique it is not necessary to give aninitial loading pattern; the system generates a random loading pattern automaticallyin contrast with the TS proposed by Jagawa et al. (2001), which starts from a refer-ence loading pattern.Furthermore, to follow the strategies used in many BWR plants, two heuristicrules will be applied along the TS technique. These are the Control Cell Core (CCC)and Low Leakage (LL) techniques. The first one does not allow the use of fresh fuelin Control Rod (CR) positions and the former does not allow also the use of freshfuel assemblies in the periphery to avoid damage to the core vessel.

2. BWR fuel reloads design problem

The problem to be solved is to get the ‘‘best’’ assembly distribution, makingshuffling (permutations) of the fuel assemblies in the core. For a BWR having 444fuel assemblies, as the Laguna Verde reactors in Mexico, this problem requires the

152 A. Castillo et al. / Annals of Nuclear Energy 31 (2004) 151–161

arrangement of 444 positions, which means 444! fuel assembly permutations. Thus,the ‘‘best’’ assembly distribution is that which provides as much energy of the cycleas can be possible without violating the operational and safety limits and haveenough shutdown margin to not jeopardize the integrity of the core. As a firstsight this is a very complex problem which will require enormous computerresources.To reduce the complexity of the problem, one octant symmetry can be assumed,then there will be only 60 different positions to allocate the fuel assemblies whichrepresent 8.32�1081 permutations or possible movements. Furthermore, if weintroduce the low leakage which means that only once and twice cycles burnt fuelassemblies can be used in the periphery (LL) and the control cell core rule whichmeans that we can not use fresh fuel in control cell positions, the optimizationproblem is reduced to 7.361�1054 different permutations instead of the 444! fromthe original problem. Given the symmetry of the problem the fuel assemblies in thediagonal can be exchanged only among them as long as they do not violate anyheuristic rule.The main goal in this work is to obtain a maximized energy without violate thethermal operational limits and the cold shutdown constraints. It can be achievedthrough the implementation of the tabu search technique using the followingobjective function in terms of maximum possible energy value in the cycle(Energy), Mean Ratio of Nominal Power (MRNP), Radial Power Peaking Factor(RPPF), Linear Heat Generation Rate (XLHGR), Maximum Power GenerationRate (XMPGR), Minimal Critical Power Ratio (XMCPR), and Shutdown Margin(SDM):

f ¼ Energy þ w1 � DMRNP þ w2 � DRPPF þ w3 � DXLHGR þ w4 � DXMPGR

þ w5 � DXMCPR þ w6 � DSDM

where:

Energy=cycle mean core burnup�MRNP=MRNPmax�MRNPc�RPPF=RPPFmax�RPPFc�XLHGR=XLHGRmax�XLHGRc�XMPGR=XMPGRmax�XMPGRc�XMCPR=XMCPRc�XMCPRmin�SDM=SDMc�SDMminw1,. . .,w6 are called weighting factors and wi>0, i=1,. . .,6

According to the �’s definition they will be negative if they are violating the safetylimits imposed in such case the corresponding weighting factors will be the onesgiven by the user in other case they would be zero not penalizing the objectivefunction. If all constraints are achieved then the objective function will be the energyproduced by the core analyzed.

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3. Tabu search technique

The tabu search method is an iterative heuristic method used for finding, in a set Xof feasible solutions, the solution that minimizes an objective function f based onneighborhood search (NS).In a neighborhood search, each feasible solution x has an associated set of neigh-bors, N xð Þ 2 X, called the neighborhood of x. NS starts from an initial feasible solu-tion chosen randomly and explores the space X by moving from one solution toanother in its neighborhood. At each iteration of the process, a subset V of N(x) isgenerated and we move from the current solution x to the best one x* in V, whetheror not f(x*) is better than f(x). If N(x) is not large, it is possible to take V as theentire neighborhood. The method of examining the entire neighborhood becomesvery expensive as the problem size increases or its elements are expensive to evalu-ate. Thus, to reduce the sampling size of V one takes the first move that improvesthe current solution; however, if there is no move that improves the current solution,then one has to examine all neighbors in V. Nevertheless, the main shortcoming ofNS algorithm is a cycling problem.Stopping rules must also be defined; in many cases a lower bound f * of theobjective function is known in advance. As soon as we have reached this bound, wemay interrupt the algorithm. In general, f * is not available with sufficient accuracy,as it is the case of study; thus, the stop criterion is met whenever a fixed maximumnumber of iterations is reached, or if a given maximum number of iterations havebeen performed without improving the best solution obtained so far.The tabu search algorithm offers another interesting possibility for overcomingthe above-mentioned obstacle of the NS technique. To prevent cycling, any movethat reinstates certain attributes of solutions recently visited is forbidden. This isaccomplished in a short-term memory framework by storing the forbidden (tabu)move in a tabu list. A move remains tabu during a certain period (or tabu tenure) tohelp aggressive search for better solutions. The tabu tenure may be fixed or variable.In many TS implementations the short-term memory is complemented with a long-term memory, whose purpose is to diversify the search and to move unvisited regionsof the solution space; its function is usually based on the frequency criterion.Unfortunately, the tabu list may forbid certain interesting moves, such as thosethat lead to a better solution than the best one found so far. An aspiration criterion isintroduced to cancel the tabu status of a move when this move is judged useful. Notethat neighborhood search is a tabu search method without an aspiration functionand where the length of the tabu list is zero.

4. Adaptation of Tabu search

In order to apply a tabu search algorithm to a combinatorial optimization prob-lem one has to define the following elements:

(a) the representation of a feasible solution

(b) the way to generate a starting solution

154 A. Castillo et al. / Annals of Nuclear Energy 31 (2004) 151–161

(c) the moves (exchanges allowed without violating the restrictions imposed)

(d) the form of the objective function and the method to calculate its values (e) the structure of the tabu list.

As it was mentioned, TS operates on a space of feasible solution and thus for ourproblem a feasible solution will be an octant of a reactor core and it is representedby an array of 60 positions to allocate the fuel assemblies (see Fig. 1). Furthermore,to follow the strategies used in many BWR plants, two heuristic rules will be appliedalong the TS technique. These are the CCC and LL techniques. The first one doesnot allow the use of fresh fuel in Control Rod (CR) positions and the former doesnot allow also the use of fresh fuel assemblies in the periphery to avoid damage tothe core vessel.The process starts knowing the characteristics of the fuel assemblies that willcompose the core. Then an initial loading pattern is randomly generated taking intoaccount the low leakage and control cell core rules.In general, TS starts from the hypothesis that it is possible to build up a neigh-borhood along the iterative search process. In our problem, a neighbor of a feasiblesolution is obtained from this solution by exchanging (without infringe any heuristicrules) two different fuel assemblies settled in a 1/8-symmetry reactor core. Given thesymmetry of the problem the fuel assemblies in the diagonal can be exchanged onlyamong them as long as they do not violate any heuristic rule. Thus a move exchan-ges two assemblies in the 1/8-symmetry reactor core and is determined by the twopositions, p1 and p2 having assemblies a1 and a2, respectively. For each currentsolution x, considering one octant reactor core and taking into account the heuristicrules, there are 723 neighbors for each feasible solution. This number is calculatedusing a simple combinatorial counting. The set N(x) is considerably large andmoreover, its elements are expensive to evaluate. Thus, only subset V of N(x) of size0.1|N(x)| is randomly generated, and the move is made from x the first solution in V

Fig. 1. Fuel reload design rules in one octant symmetry.

A. Castillo et al. / Annals of Nuclear Energy 31 (2004) 151–161 155

that improves the objective function. However, if there no solution that improves x,then one must to examine all neighbors in V.As it is set in Section 2, the objective function includes the thermal safety limitsand the shutdown margin. Tabu search technique was implemented along with thelow leakage and control cell core rules in the 3-D reactor core simulator CM-PRE-STO-B (Scandpower, 1995) to compose the Optimization Tabu Search System. Thissystem is a FORTRAN-77 based program implemented in an Alpha computer withUNIX operating system.Due to the way that the objective function is constructed, the system requires fourdifferent runs of CM-PRESTO-B; first one (a Haling calculation) is to calculate theenergy and the six parameters associated with the operational and safety limits, theother three runs are used to calculate the cold shutdown margin. Thus, those fourruns compose a full evaluation of the objective function. It is important to point outthat one Haling calculation takes around 1 tenth of the time that will take to do eachone of the cold shutdown margin calculations.However for many solutions it is not necessary to have a full evaluation of theobjective function. OTSS makes a partial evaluation of the objective function toassess the energy and safety thermal limits and if they are violated then the shut-down margin runs are not performed. This reduces the calculation time because theobjective function will not be acceptable anyway.Our tabu list is implemented as a tabu time, which records the earliest iterationthat a move is removed from the list. The number of iterations tabu_tenure that amove or exchange will keep its tabu status is randomly selected in the range from 6to 14 (Glover, 1989). This random selection provides a more versatile search pro-cess. The tabu time is represented by the array tabu_time(p,a) where p is a positionand a is the assembly type allocated in p. The attributes of a swap move are stored inthe vector m=(p1,p2,a1,a2). Then, the tabu_time is updated as follows in order toimpose a tabu on the move m for tabu_tenure iterations:

tabu time p1; a2ð Þ ¼ tabu time p2; a1ð Þ ¼ iter þ tabu tenure

where iter is the current iteration number. Thus, the swap move m=(p1,p2,a1,a2) istabu if both tabu_time(p1,a2) and tabu_time(p2,a1) are greater or equal to the cur-rent iteration number. Note that the tabu_tenure value used in our case is a randomnumber between 5 and 10, this range comes from a trial and error test. Our tabutime forbids, during tabu_tenure iterations, any replacement of a2 and a1 in thepositions p1 and p2, respectively.The long term memory is a function that records moves taken in the past in orderto penalize those which are non-improving. The goal is to diversify the search bycompelling regions to be visited that possibly were not explored before (Glover,1989). In our particular TS implementation, the long-term memory is a vector whichwill be denoted F. The vector has zeroes at the beginning of the procedure. When apair of assemblies (a,b) are swapped at a given iteration, the vector F changes asfollows: Fa=Fa+3 and Fb=Fb+3. The entry Fa is the frequency at which the

156 A. Castillo et al. / Annals of Nuclear Energy 31 (2004) 151–161

assembly a has been swapped. Then, the values of non-improving moves that switchthe assemblies a and b are decreased by Fa+ Fb.The two last concepts to explain are the aspiration and the stopping criterionsused. The aspiration criterion cancels the status tabu of a move when it finds a fea-sible solution with a better function value than the best solution in the past. Our TSwill be stopped if the number of iterations used without improving the best solutionis greater than 50.

5. Test problem and results

OTSS was applied to the design of a fuel reload, taking as a base of comparisonan actual operating cycle. This one is an 14-month cycle with 9281 MWD/TU ofenergy produced, which used 112 fresh fuel assemblies of 3.53 w/o of U-235, thisloading pattern was generated using engineer expertise.The safety operational limits at the end of the cycle (Haling calculation) imposedon this calculation are given in Table 1. On the other hand the cold shutdown mar-gin will be assessed at the beginning of the cycle and it needs to be more than 1%�k/k to not jeopardize the integrity of the reactor core.Due to the random nature of the process here considered we performed severaltimes the search of an optimized loading pattern and the best results that we foundare shown in Table 2, beside Table 3 shows the thermal limits obtained. The resultsfrom Table 2 show a maximum energy produced of 9970 MWD/TU and the

Table 1

Operational and safety limits

Mean ratio of nominal power

MRNP 1.83 Maximum value

Radial power peaking factor

RPPF 1.51 Maximum value

Linear heat generation rate

XLHGR 370 Maximum value

Maximum power generation rate

XMPGR 0.85 Maximum value

Minimal critical power ratio

XMCPR 1.5 Minimum value

Shutdown margin

SDM 1.0 Minimum value

Table 2

Optimized fuel reload results

Tabu search run

SDM Energy Evaluations of the

objective function

Iterations

CPU (s)

1

1.000698 9970.544 4994 193 26434.7

2

1.042602 9916.886 4734 199 28540.2

3

1.011673 9809.330 4408 186 27608.8

4

1.013669 9899.255 5495 207 27894.5

5

1.001696 9851.790 4071 177 29106.9

Average

9889.561 4740 192 27917.0

Standard deviation

61.770 546 12 1012.1

A. Castillo et al. / Annals of Nuclear Energy 31 (2004) 151–161 157

operational and safety limits along with the cold shutdown margin are achieved, thisenergy is about a 7.4% of extra energy than the loading pattern generated by engi-neer expertise.Furthermore, in each TS there is a CM-PRESTO-B runs saving of about 30%because the cold shutdown margin is not evaluated when the energy produced by thenew pattern is less than the maximum objective function of the neighborhood.To get an optimized loading pattern using OTSS, assuming octant symmetry, ittakes less than 7700 evaluations of the objective function, which is a very smallquantity, compared with the 7.361�1054 permutations. Fig. 2 shows the flowchart ofthis optimization process.Figs. 3 and 4 show the behavior of the objective function and energy, respectivelyfor some of the several searches of the optimized loading patterns generated usingOTSS. Objective function plot, from Fig. 3, shows energy values penalized due tothe violation of shutdown margin and or safety and or operational limits. Energyplot, from Fig. 4, shows energy produced although exists in some cases violation ofshutdown margin and or safety and or operational limits. At the end of the iterationprocess, the objective function has the same value to the energy produced due tothat all the constraints are satisfied. Furthermore, it can be seen that the objectivefunction plot is not smooth at the end. It happens due to the way that the TS workstrying to escape from local minima making the process robust.On the other hand, Francois and Lopez (1999) analyzed the same actual cycleconsidered in this work using genetic algorithms, but their objective function doesnot take into account the cold shutdown margin as a constraint, which can lead tonot realistic results. However, our best result shows a 9970 MWD/TU energy pro-duced against his best result, which shows a 9892 MWD/TU energy produced.

6. Conclusions

TS technique using tabu time has been implemented successfully to the optimiza-tion of BWR fuel reload patterns. The system developed in this work generatesoptimized fuel reloads which produce in general energies greater than the producedby engineer expertise and genetic algorithms.Use of OTSS for an actual operating 14-month cycle shows a maximum cyclelength of 9970 MWD/TU which does not violate the operational and safety thermal

Table 3

Thermal limits from optimized fuel reloads

Tabu Search Run

MRNP RPPF XLHGR XMPGR XMCPR

1

1.826 1.509 365.911 0.794 1.599

2

1.829 1.509 366.443 0.795 1.603

3

1.819 1.508 364.173 0.787 1.601

4

1.822 1.499 365.043 0.791 1.613

5

1.829 1.505 366.468 0.793 1.604

158 A. Castillo et al. / Annals of Nuclear Energy 31 (2004) 151–161

Fig. 2. OTSS flow chart.

A. Castillo et al. / Annals of Nuclear Energy 31 (2004) 151–161 159

limits and has a cold shutdown margin of 1.000698% �k/k, which is greater thanthe 1% �k/k (minimal) limit value. This energy is 7.04% greater than the one pro-duced of 9281 MWD/TU by the actual operating cycle.From a practical point of view it can be seen from Table 2 results that it will beenough to perform five TS to assure that an optimized loading pattern is obtained.On the other hand it takes less than 7700 evaluations per TS of the objective func-tion, which is a very small number, compared with the 7.361�1054 permutationsthat can take place in one octant symmetry.

Fig. 4. Energy behavior under a TS.

Fig. 3. Objective function behavior under a TS.

160 A. Castillo et al. / Annals of Nuclear Energy 31 (2004) 151–161

This extra energy represents 34 days more of full power operation. To assess theeconomical impact of this optimized fuel reload it will be necessary to perform amulti-cycle analysis to know the actual advantages of this fuel assembly utilization.Finally, to get a whole optimized reload design system it is necessary to implementthe search of optimized control rod pattern, which will be a future work.

Acknowledgements

The authors acknowledge the support given by CONACyT through the researchproject 33806-U.

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Scandpower, 1995. User Manual CM-PRESTO-B/91.

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