Download - A practical optimization procedure for radial BWR fuel lattice design using tabu search with a multiobjective function

A practical optimization procedure for radialBWR fuel lattice design using tabu search

with a multiobjective function

J.L. Francoisa,*, C. Martın-del-Campoa, R. Francoisb,L.B. Moralesc

aLaboratorio de Analisis en Ingenierıa de Reactores, Nucleares, Universidad Nacional Autonoma de Mexico

(UNAM), Facultad de Ingenierıa, Paseo Cuauhnahuac 8532, Jiutepec, Mor., 62550, MexicobCentro Universitario Anglo Mexicano, Luna 44, Cuernavaca, Mor., 62360, Mexico

cInstituto de Investigaciones en Matematıcas Aplicadas y Sistemas, UNAM. Apdo. Postal 70-221, Mexico,

D.F., 04510, Mexico

Received 12 February 2003; accepted 6 March 2003

Abstract

An optimization procedure based on the tabu search (TS) method was developed for thedesign of radial enrichment and gadolinia distributions for boiling water reactor (BWR) fuellattices. The procedure was coded in a computing system in which the optimization code uses

the tabu search method to select potential solutions and the HELIOS code to evaluate them.The goal of the procedure is to search for an optimal fuel utilization, looking for a lattice withminimum average enrichment, with minimum deviation of reactivity targets and with a local

power peaking factor (PPF) lower than a limit value. Time-dependent-depletion (TDD) effectswere considered in the optimization process. The additive utility function method was used toconvert the multiobjective optimization problem into a single objective problem. A strategy to

reduce the computing time employed by the optimization was developed and is explained inthis paper. An example is presented for a 10�10 fuel lattice with 10 different fuel composi-tions. The main contribution of this study is the development of a practical TDD optimiza-tion procedure for BWR fuel lattice design, using TS with a multiobjective function, and a

strategy to economize computing time.# 2003 Elsevier Science Ltd. All rights reserved.

Annals of Nuclear Energy 30 (2003) 1213–1229

www.elsevier.com/locate/anucene

0306-4549/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0306-4549(03)00055-0

* Corresponding author.

E-mail addresses: [email protected] (J.L. Francois), [email protected] (C. Martın-del-Campo).

http://www.sciencedirect.com



http://www.elsevier.com/locate/anucene/a4.3d

mailto:[email protected]

mailto:[email protected]

1. Introduction

A typical BWR fuel assembly (FA) consists of an 8�8, 9�9 or 10�10 pin array,with a set of five or six axial sections with different distributions of approximatelyten fuel compositions of 235U enrichment and gadolinia (Gd2O3). An axial section ofFA, called radial fuel lattice or more frequently radial fuel assembly, is the object ofthis study. The radial fuel lattice design is focused on finding the optimal distribu-tion of fuel compositions to obtain the optimal fuel utilization. Radial fuel assemblydesign is one of the principal tasks in BWR fuel management. A variety of con-straints and objectives are imposed on the nuclear design of a BWR fuel assemblydue to requirements of operation, safety and economy. The requirements include thereactivity for operation, the void and Doppler reactivity coefficients, the control rodworth, etc. These reactivity characteristics are strongly dependent on the geometryand material compositions; e.g. fuel rod size and pitch, hydrogen-to-heavy-metalratio, average fuel density and enrichment, control poison material and geometry,and so on. Other requirements are the shutdown margin (SDM) and the thermalmargins (e.g. maximum linear heat generation rate, MLHGR). These parameters arenot determined from the radial fuel assembly design alone; radial and axial fuelassembly design and in-core load design significantly interact with each other. Thus,the detailed design of a BWR FA needs to be carried out simultaneously takingthese interactions into account. The method described in this paper focuses on theproblem of determining optimal radial fuel enrichment and gadolinia distributionsunder given constraints. The axial FA design and the in-core load pattern design areoutside the scope of this study. The present study considers predefined lattice geo-metry with fixed fuel rod size, pitch and water zones. Only the pin enrichment andgadolinia distributions within the FA are investigated. The different pin enrichmentsto be utilized in the lattice are restricted by manufacturing requirements.

Several studies related to the optimization of fuel rod enrichment distribution forBWR fuel assemblies have contributed to the understanding and the solution of thislarge combinatorial problem. The method of approximation programming (Hiranoet al., 1996) gives quite satisfactory results producing feasible candidate designscomparable to those elaborated by an expert engineer. A methodology that com-bines the response matrix method with non-linear programming techniques (Limand Leonard, 1977) was applied to search for an optimal pin enrichment distribu-tion that gives the best approximation to a prescribed power distribution in a two-dimensional FA. Another application to the optimization of MOX enrichment dis-tributions in typical light water reactor assemblies (Cuevas et al., 2002) used a sim-plex method-based algorithm.

In nuclear applications TS has been used for in-core reload pattern optimization inBWRs (Castillo et al., 2002) and for the axial fuel assembly optimization in BWRs(Martın-del-Campo et al., 2002a) obtaining good results. Other work uses a simplelinear perturbation method and a modified TS method to select potential optimizedBWR load patterns (Jagawa et al., 2001). In this study, the TS method is appliedto the optimization of radial enrichment and gadolinia distributions for BWR fuellattices, improving a previous work in this area (Martın-del-Campo et al., 2002b).

1214 J.L. Francois et al. / Annals of Nuclear Energy 30 (2003) 1213–1229

2. Methodology

The tabu search method is an iterative heuristic procedure for solving complexcombinatorial optimization problems. The TS is a constrained search proceduredesigned to overcome local optimality. The TS is based on the idea of moving stepby step, from an initial candidate solution towards a solution giving the minimumvalue of some objective function, with a special feature designed to avoid beingtrapped by local minima. TS combines a deterministic iterative improvement withthe possibility of accepting worst solutions to direct the search away from localminima (Glover and Laguna, 1997).

In the present study, the TS optimization method was implemented to determinethe ‘‘optimum’’ fuel enrichment and gadolinia distributions within a BWR fuel lat-tice. In order to evaluate the objective function, the TS implementation was linkedto the lattice code HELIOS (Studsvik Scandpower, 1998) and executed on a Com-paq Alpha Work Station (500 MHz and 128 MB of RAM).

It must be mentioned that generally a lattice nuclear data bank is obtained from atime-dependent-depletion evaluation with several exposure steps from 0 to 60,000MWd/T (megawatts day per ton of uranium fuel) that utilizes considerable com-puting time. For this reason, it is necessary to develop a strategy to avoid the eval-uation of very bad solutions in the optimization process.

In this section, a brief TS method description is presented, then the representationof the solution is explained and the objective function is formulated, finally theimplementation of the TS to the specific problem using a strategy to economizecomputing time is described.

2.1. Tabu search method description

Tabu search is one of the so-called meta-heuristic search methods (methods thathelp conduit directed ‘‘intelligent’’ search of the potentially very large solutionspace). These are techniques used for moving step by step towards the minimumvalue of a function. A tabu list of forbidden movements is updated during iterationsto avoid cycling and being trapped in local minima. Briefly, the TS method mini-mizes f(x), subject to x in X, where f is a cost function, and x is a set of candidatesolutions. It starts from an initial candidate solution and tries to reach a globalminimum by moving from one candidate solution to another. To accomplish this, aset M of simple modifications must be defined. These modifications are called moves,which can be applied to a given candidate solution to move to another. The notationx0=m(x), m in M, indicates that m transforms x into x0. This leads to the definitionof a neighborhood. For each candidate solution x, the neighborhood N(x) is the setof all candidate solutions directly reachable from x by a single move m in M. WhenN(x) is large, at each step of the iteration process, a subset V* of N(x) is generatedand the move is made from x to the best solution x* in V*, whether or not f(x*) isbetter than f(x).

Up to this point, the algorithm is close to a local improvement technique, exceptthat the move from x may be to a worse solution x*, and thus may escape from local

J.L. Francois et al. / Annals of Nuclear Energy 30 (2003) 1213–1229 1215

minima of f. To prevent cycling, any move that returns to any local minimumrecently visited is tabu (forbidden). This is accomplished in a short-term memoryfunction by storing the tabu move in a set T, called the tabu recency list, of length t.A move will remain tabu during t iterations, so the tabu list can be represented by aqueue. At each iteration the opposite move from x to x* is added at the end of T,while the oldest is removed from T. Another mechanism that can improve perfor-mance is diversification, that is, to encourage the method to search unexploredregions. This mechanism can be achieved by implementing a long-term memoryfunction via frequency counts, a tabu frequency list. A move is forbidden if its fre-quency exceeds a maximum frequency value, and this tabu frequency list and its limitvalue must be updated dynamically when iterations increase. These tabu lists couldforbid certain interesting moves, such as moves that lead to a better solution thanthe best one found so far. Considering this, the aspiration criterion is introduced tocancel the tabu status of a move when it is judged to be a useful move.

Stopping rules must also be defined, as a fixed lower value f*, or a fixed number ofsolutions investigated, or a fixed number of iterations.

2.2. Representation of a candidate solution

As it was mentioned, typical BWR fuel assemblies are 8�8, 9�9 or 10�10 pinarrays with a set of approximately ten fuel compositions with different 235U enrich-ment and gadolinia (Gd2O3) concentrations. In order to investigate the performanceof different fuel distributions, a candidate solution x could be represented by a bi-dimensional array indicating the fuel composition located at each pin position in thelattice.

2.3. Mathematical model of the objective function

The radial FA design is a complex combinatorial optimization problem; severalconstraints and objectives could be searched in order to optimize the fuel utilizationand safety operation. At the stage of this study, the objective function was for-mulated to find the solution x with the following four objectives:

1. the minimum average lattice enrichment;
2. the best approximation to a prescribed infinite-multiplication-factor (k-inf) as
a function of the exposure.3. an average gadolinia concentration target Gdtarget, at 0 MWd/T of exposure;
4. a PPF, at 0 MWd/T of exposure, lower than a limit value PPFmax;
Objective 1 is the main objective of the optimization process. Objective 2 is anadditional objective in order to obtain the expected reactivity for the lattice at dif-ferent burnup steps. Objective 3 is used to reduce fuel reactivity excess at the begin-ning of lattice exposure. Finally objective 4 is a constraint used to limit the radialpower generation. Objectives 3 and 4 can be seen as constraints or objectives.However, in this study they were considered to be additional objectives.


The first objective function is to minimize the average lattice enrichment E(x), thatis obtained considering the fuel composition located in each pin position using Eq.(1), where E( j) is the enrichment of the fuel pin located in position j in the latticewith Np total fuel pin positions. Obviously the minimal value for E(x) is not known.

Minimize E xð Þ ¼XNp

j¼0

E jð Þ=Np ð1Þ

In order to accomplish the second objective function, S(x), the sum of the squaredk-deviations at different predetermined exposure steps i, between the k-inf of thesolution x (kinf_i(x)) and the k-inf target (kinf_target_i) must be minimized. The termsum of the squared k-deviations(x) is expressed in Eq. (2). The exposure steps must beselected using a reference reactivity curve. Obviously the optimal value for S(x) iszero. The values for kinf_i(x) are obtained executing the neutronic simulator(HELIOS code).

Minimize S xð Þ ¼ sum of the squared k-deviations xð Þ

¼XN

i¼0

kinf i xð Þ � kinf target i

� �2ð2Þ

To get the third objective function, G(x) expressed in Eq. (3) must be minimized. Itcorresponds to the fractional deviation of the target value. The value for Gd0(x) isobtained using Eq. (4), where G( j) is the gadolinia concentration of the fuel pinlocated in position j in the lattice with Np total fuel pin positions. The individualoptimal solution of this function is also zero.

Minimize G xð Þ ¼ abs Gd0 xð Þ � Gdtarget� �

=Gdtarget ð3Þ

Gd0 xð Þ ¼XNp

j¼0

G jð Þ=Np ð4Þ

To satisfy objective 4, is not necessary to optimize any function, only Eq. (5) mustbe satisfied, where PPF0(x) is the local PPF at 0 MWd/t for solution x and PPFmax isthe local PPF limit at 0 MWd/t. PPF0(x) is obtained executing the neutronic simu-lator (HELIOS code).

PPF0 xð Þ � PPFmax04 0 ð5Þ

The global objective function was modeled using the additive utility functionmethod (Tabucanon, 1989) to convert the multiobjective optimization problem intoa single objective problem. This method directs the search to the best solution usingweighting factors to attach decision-maker preferences to each objective Then theobjective is to minimize the function Z(x) presented in Eq. (6) as a weighted sum ofthe multiple objectives.


Minimize Z xð Þ ¼ wE � E xð Þ þ wS � S xð Þ þ wG � G xð Þ þ wP

� PPF0 xð Þ � PPFmaxð Þ ð6Þ

Here wk indicates the relative importance that the decision-maker attaches toobjective k and it must be specified for each of the k objectives a priori. All the wkare fixed positive values to penalize the solutions when individual objectives are notsatisfied. Nevertheless the weighting factor wP is equal to zero when Eq. (5) isaccomplished in order to penalize only the solutions with a PPF that exceeds thelimit. These weighting factors must be determined evaluating hundreds of latticesusing Eq. (6), in which the constraints PPFmax, Gdtarget, and kinf_target_i for differentdepletion steps i have values according to the desired lattice performance char-acteristics.

3. Application to the study case

The optimization process was developed for a 10�10 fuel pin array with two waterzones and diagonal symmetry. The fuel used is uranium dioxide (UO2) and some fuelpins are mixed with gadolinia (Gd2O3) as burnable poison. The lattice performanceis evaluated using the objective function that was previously presented. To calculatethe objective function, the FA is simulated using the HELIOS code, in which the fuelpins, the water regions, the channel and the control rod are explicitly represented intwo-dimensions. In HELIOS this lattice can be represented using half diagonalsymmetry with 51 pin positions as is shown in Fig. 1. Then Np is equal to 51.

As was mentioned, a candidate solution x is represented by a bi-dimensional arrayindicating the fuel composition located at each pin position in the lattice.

Fig. 1. Schematic representation of the lattice.


3.1. Fuel compositions to accommodate in the lattice

The fuel is uranium dioxide, taking into account manufacturing requirements onlyten different pin enrichments and gadolinia concentrations were utilized in thisapplication (see Table 1).

3.2. Heuristic accommodation rules

In order to accelerate and simplify the search, some heuristic accommodationrules can be used in the implementation. In this study only three heuristic rules wereapplied:

first, the lowest enriched fuel (UO2-1) is only used at the corner positions:(0,0), (0,9) and (9,9)

second, fuels containing gadolinia cannot be placed in the lattice’s edge; third, water region position is fixed.

Taking into account the available fuel compositions and the heuristic accom-modation rules in the lattice (see Fig. 1), there are 16 pins that could have five dif-ferent fuel compositions, and there are 32 pins that could have nine different fuelcompositions. The resulting number of total combinations is calculated as:

516 þ 932 5:239 � 1041

consequently the search must be well conducted to save computing time without
andloss of accuracy.
3.3. Algorithm to reduce computing time

Generally, a lattice nuclear data bank is obtained from a time-dependent-deple-tion (TDD) evaluation with several exposure steps from 0 to 60,000 MWd/T.Nevertheless, in the design process, in order to reduce computing time, the TDDcalculations can be stopped at 20,000 MWd/T of exposure, when the gadolinia istotally depleted in the study lattice case. These HELIOS calculations consumeapproximately 1.6 CPU minutes on the Compaq Alpha Work Station. If each can-didate solution takes this time to be evaluated, it is very important to develop astrategy to avoid the evaluation of very bad solutions.

An algorithm was developed in order to use two kinds of evaluations in the opti-mization process:

Table 1

Fuel enrichment and gadolinia concentrations in fuel compositions

UO2-1
UO2-2 UO2-3 UO2-4 UO2-5 UO2-6 Gd-1 Gd-2 Gd-3 Gd-4
U235 %w
2.0 2.8 3.6 4.4 3.95 4.9 3-95 4.4 4.4 4.4
Gadolinia %w
0 0 0 0 0 0 5 5 4 2

A full evaluation which includes TDD calculations and uses Eq. (6) to eval-uate the solutions

A partial evaluation that is only a 0 MWd/T calculation and uses Eq. (7) toevaluate the solutions.

It is important to notice that for a typical studied lattice, a partial evaluation usesapproximately 0.21 CPU minutes compared with approximately 1.6 CPU minutesfor a full evaluation.

Minimize F xð Þ ¼ wE � E xð Þ þ wD � D xð Þ þ wG � G xð Þ þ wP

� PPF0 xð Þ � PPFmaxð Þ ð7Þ

F(x) is the objective function in the partial evaluation to be minimized. The indivi-dual objective function D(x) searches for minimizing the squared k-deviation(x) at 0MWd/t and is calculated using Eq. (8). Where kinf_0(x) is the kinf at 0 MWd/t forsolution x, kinf_target_0 is the target kinf at 0 MWd/t and wD is the user-definedweighting factor the squared k-deviation(x) at 0 MWd/t, all the other terms in theequation are previously defined in Section 2.3.

Minimize D xð Þ ¼ squared k-deviation xð Þ0¼ kinf 0 xð Þ � kinf target 0

� �2ð8Þ

Three decision parameters were defined and are used to order or not a full eval-uation, these parameters are kupper, klower, and PPFgood with values fixed by the user.Each candidate solution is firstly evaluated only at 0 MWd/T (partial evaluation)and if the conditions shown in Eqs. (9) and (10) are satisfied, then a full evaluation isexecuted. Once the first full evaluation is done, each other candidate solution thatdoes not satisfy Eqs. (9) and (10) is eliminated before the TDD evaluation, and thenanother solution is generated to replace it. Fig. 2 shows the TS technique adapted tothe BWR FA radial optimization problem in which this strategy has been incorpo-rated.

klower4 kinf 0 xð Þ4 kupper ð9Þ

and

PPF0 xð Þ < PPFgood ð10Þ

3.4. Specific tabu search characteristics

In this study, the first candidate solution x was constructed using the analyst’sexperience. This solution is considered the first current solution in the search pro-cedure.

The TS method is designed to select, at each iteration, the best available moveusing the current solution x and creating the solution x*. Two types of moves weredefined: a move type 1, which allows a change in the quantity of different fuels in thelattice; and a move type 2, that allows the modification of the pins’ distribution in the


FA and finds the best location of a certain fuel type in the lattice. A move type 1 isentirely defined by the random selection of one pin position and the random selec-tion of one fuel composition from the list of available fuel types. A move type 2is a pair wise exchange (or swap), it exchanges the location of two different fuel

Fig. 2. Process diagram implementation.


compositions in the lattice and is defined by the random selection of two positionshaving different fuel compositions.

In this application, a maximum number of moves in each iteration step is fixed,but the number of moves is variable since the process stops the moves when a solu-tion x* is better than the best solution in the past (aspiration criterion) or when asolution x* is better than the current solution x and the move x to x* is not tabu.

The process stops iterations when a given number of iterations is reached. Theanalyst can also stop the process manually when the objective function has con-verged, or when very bad values for the constraints are obtained.

Table 2 shows general data associated with the TS characteristics utilized in theoptimization process.

3.5. Specific data for the application case

A reference lattice, which is used in a Laguna Verde Nuclear Power Plant fuelbundle, was selected in order to fix the values for the limits and targets employed inEqs. (2), (3) and (5)–(10). All HELIOS calculations were done at 40% voids, a fueltemperature of 793 �K, and moderator temperature of 560 �K. Table 3 showsPPFmax and Gdtarget imposed as constraints at 0 MWd/T. The limits kupper, klower,and PPFgood used to activate the TDD evaluations are also showed in Table 3.Table 4 shows the k-infinite targets, as function of the exposure. The selection of

Table 2

Tabu search characteristics

Maximum number of moves in each iteration
30
Maximum number of iterations
200
Size of the tabu list
15
Table 3

Pre-established data at 0 MWd/T

PPFmax
1.438
Gdtarget%
0.815
kupper
1.0355
klower
1.0230
PPFgood
1.470
Table 4

k-Infinite targets as a function of exposure

Exposure step
1 2 3 4 5 6
MWd/T
0a 0 1000 12,000 15,000 20,000
k-infinite
1.02929 1.00870 1.01129 1.13167 1.14318 1.11704
a No Xenon.


these targets is specific for the desired lattice performance. The exposure steps areselected in order to have the shape of the reactivity curve.

The weighting factors were determined by the evaluation of hundreds of candidatesolutions and the selected values are presented in Table 5.

4. Results

Fig. 3 shows the evolution of the objective function Z(x), as a function of thesolution number, for all fully evaluated candidate solution. It shows the values forthe candidate solution, the best solution and the current solution.

Fig. 4 shows the k-infinite at 0 MWd/T as a function of the solution number, itincludes values for all partially evaluated solutions and for all fully evaluated solu-tions, for the current solution, and for the best solution. The limits kinf_target_0, kupper

and klower are also indicated in this fig.Fig. 5 presents the evolution of the PPF at 0 MWd/T as a function of the solution

number, it includes values for the current solution, for the best solution, for thePPFmax, for all the partially evaluated candidate solutions and for all the fully eval-uated candidate solutions. The PPFgood limit is also indicated in Fig. 5.

Fig. 6 shows the evolution of the sum of the squared k-deviation as a function of thesolution number for the best solution and for all fully evaluated candidate solutions.

Table 5

Weighting factors

wE
wS wG wP wD
1
2 24.5 20 320
Fig. 3. Evolution of the objective function (full evaluation).


In Fig. 7 the sum of the squared k-deviation was plotted as a function of the averagelattice enrichment for all the fully evaluated candidate solutions. The minimumvalue of 1.3E-06 corresponds to a candidate solution of 4.086% average latticeenrichment, nevertheless this solution has a high PPF equal to 1.451 that exceeds thePPFmax.

Fig. 8 shows the evolution of the average lattice enrichment as a function of thesolution number. This fig. includes values for each candidate solution, for the currentsolution and for the best solution. It can be observed that a lot of candidates wereinvestigated in the space solution, and that the lattice average enrichment is stronglyminimized. Remembering that the main objective of the optimization process is tominimize the enrichment, we can say that the process works successfully.

Fig. 4. k-I=Infinite at 0 MWd/T evolution.

Fig. 5. PPF (at 0 MWd/T) evolution.


Fig. 9 shows the evolution of the number of partial and full evaluations as afunction of the iteration number. It is observed that at the beginning of the processthere are only partial evaluations and that the first full evaluation corresponds tothe candidate solution 61 at the iteration 6. It is also observed, that at severaliterations there are a lot of solutions only partially evaluated, which helps to save

Fig. 6. Sum of the squared k-deviation evolution.

Fig. 7. Sum of the squared k-deviation as a function of the average enrichment.


Table 6

Results for the best solution

Lattice
kinf_0 PPF0 Gd0 (w%) S E (w%)
BOL Optimization
1.02912 1.419 0.81522 307.86E-06a 3.9435
TDD Optimization
1.02975 1.438 0.81522 6.53E-06 4.0261
Reference
1.02929 1.439 0.81522 0 4.1065
a This parameter was externally calculated to the optimization process.

Fig. 8. Evolution of the average enrichment.

Fig. 9. Number of full and partial evaluations executed as a function of the iteration number.


computer time. The reason is that all these solutions do not satisfy Eqs. (9) and(10).

A total of 75 iterations were done, that included 1501 evaluations, of which only877 of them were full evaluations. The first candidate solution fully evaluated wasthe 61st solution and the last one was the 1500th solution. The best solution, whichis the evaluation number 1087, was found at the iteration 59. In this study case, theuser manually stopped the process at cycle 75 since the best solution could not beimproved after cycle 59. The results obtained for the final best solution are presentedin Table 6 (case TDD optimization). The lattice description of the best solution ispresented in Fig. 10.

To highlight the importance of the incorporation of the TDD effects in theobjective function, the results were compared with a beginning-of-life (BOL) opti-mization case. The BOL case was obtained with a similar methodology in which thetime-dependent-depletion effects were not considered in the objective function.

Table 6 also shows the results obtained for the final best solution of the BOLoptimization process. Data for the reference lattice were also included in Table 6. Itis observed that the BOL optimization case gave a solution with a very low averageenrichment. Nevertheless, the sum of the squared k-deviations was calculated for thebest solution obtained with the BOL optimization, and a very high value (3.1E-4)

Table 7

k-Infinit for the best solution

1
2 3 4 5 6
MWd/T
0a 0 1000 12,000 15,000 20,000
BOL Optimization
1.02603b 1.00505b 1.00793b 1.13061b 1.12969b 1.10651b
TDD Optimization
1.02975 1.00869 1.01052 1.13203 1.14421 1.11491
Reference
1.02929 1.00870 1.01129 1.13167 1.14318 1.11704
a No Xenon.b These parameters were externally calculated to the optimization process.

Fig. 10. Lattice description for the best solution in the TDD optimization.


was found. The values of kinfðbestÞ as a function of the exposure at six different stepsare presented in Table 7.

Fig. 11 shows the k-infinites as a function of the lattice exposure for the bestsolution of TDD optimization, for the best solution of BOL optimization and forthe reference case. The values ktargets are also shown in Fig. 11. It is observed thatthe BOL optimization does not satisfy the requirements of the lattice reactivity asfunction of burnup.

Table 8 summarizes the characteristics of the best solution for the TDD and BOLoptimizations, and for the lattice reference case.

5. Conclusions

As general result, a practical optimization procedure for BWR fuel lattice designusing TS with multiobjective function, and a strategy to save computing time hasbeen developed. The objective function could incorporate additional parameters tomake the lattice evaluation performance more robust. For example, control rod

Table 8

Characteristics of the best and reference lattices

Lattice
UO2-1 UO2-2 UO2-3 UO2-4 UO2-5 UO2-6 Gd-1 Gd-2 Gd-3 Gd-4
BOL Opt.
4 12 14 10 22 14 10 1 5 0
TDD Opt.
4 8 5 30 18 11 12 1 2 1
Reference
4 6 6 32 16 12 4 9 2 1
Fig. 11. k-Infinite VS Exposure for the best solution.


insertion, void effects, temperature effects, power peaking factor penalty in thegadolinia rods, hot to cold reactivity swing etc. could be incorporated. On the otherhand, several strategies to reduce computing time can be implemented withoutpenalizing the accuracy of the results.

The system’s coding and architecture is so flexible that it is very easy to change theobjective function to minimize. For instance, the minimization of the power peakingfactor can be searched under a fixed lattice average enrichment.

Three conclusions can be highlighted: the first one is that the TS method isapplicable to optimize the fuel enrichment distribution in a BWR fuel lattice, mini-mizing the average enrichment subject to reactivity and power peaking factor asconstraints; the second one is that TDD effects must be included in the objectivefunction in order to assure a good reactivity performance during lattice exposure;and the last one is that the strategy to reduce computing time is very useful.

Acknowledgements

This work was performed under the auspices of the Mexican Science and Tech-nology National Council (CONACyT) under agreement No. 34657-U, and waspartly supported by the National University of Mexico (PAPIIT-UNAM) underproject No. IN109400.

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