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

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<ul><li><p>Received 12 February 2003; accepted 6 March 2003</p><p>Annals of Nuclear Energy 30 (2003) 12131229</p><p>www.elsevier.com/locate/anuceneAbstract</p><p>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</p><p>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</p><p>power peaking factor (PPF) lower than a limit value. Time-dependent-depletion (TDD) eectswere 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</p><p>reduce the computing time employed by the optimization was developed and is explained inthis paper. An example is presented for a 1010 fuel lattice with 10 dierent 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</p><p>strategy to economize computing time.# 2003 Elsevier Science Ltd. All rights reserved.A practical optimization procedure for radialBWR fuel lattice design using tabu search</p><p>with a multiobjective function</p><p>J.L. Francoisa,*, C. Martn-del-Campoa, R. Francoisb,L.B. Moralesc</p><p>aLaboratorio de Analisis en Ingeniera de Reactores, Nucleares, Universidad Nacional Autonoma de Mexico</p><p>(UNAM), Facultad de Ingeniera, Paseo Cuauhnahuac 8532, Jiutepec, Mor., 62550, MexicobCentro Universitario Anglo Mexicano, Luna 44, Cuernavaca, Mor., 62360, Mexico</p><p>cInstituto de Investigaciones en Matemat`cas Aplicadas y Sistemas, UNAM. Apdo. Postal 70-221, Mexico,</p><p>D.F., 04510, Mexico0306-4549/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0306-4549(03)00055-0</p><p>* Corresponding author.</p><p>E-mail addresses: jl@-b.unam.mx (J.L. Francois), cmcm@-b.unam.mx (C. Martn-del-Campo).</p></li><li><p>1. Introduction</p><p>A typical BWR fuel assembly (FA) consists of an 88, 99 or 1010 pin array,with a set of ve or six axial sections with dierent 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 nding 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 coecients, 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 signicantly 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 predened lattice geo-metry with xed fuel rod size, pitch and water zones. Only the pin enrichment andgadolinia distributions within the FA are investigated. The dierent pin enrichmentsto be utilized in the lattice are restricted by manufacturing requirements.Several studies related to the optimization of fuel rod enrichment distribution for</p><p>BWR 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 in</p><p>BWRs (Castillo et al., 2002) and for the axial fuel assembly optimization in BWRs(Martn-del-Campo et al., 2002a) obtaining good results. Other work uses a simplelinear perturbation method and a modied 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 fuel</p><p>1214 J.L. Francois et al. / Annals of Nuclear Energy 30 (2003) 12131229lattices, improving a previous work in this area (Martn-del-Campo et al., 2002b).</p></li><li><p>2. Methodology</p><p>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 determine</p><p>the 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 a</p><p>time-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 representation</p><p>of the solution is explained and the objective function is formulated, nally theimplementation of the TS to the specic problem using a strategy to economizecomputing time is described.</p><p>2.1. Tabu search method description</p><p>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. Briey, 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, asetM of simple modications must be dened. These modications 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 denitionof 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, except</p><p>J.L. Francois et al. / Annals of Nuclear Energy 30 (2003) 12131229 1215that the move from x may be to a worse solution x*, and thus may escape from local</p></li><li><p>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 dened, as a xed lower value f*, or a xed number of</p><p>solutions investigated, or a xed number of iterations.</p><p>2.2. Representation of a candidate solution</p><p>As it was mentioned, typical BWR fuel assemblies are 88, 99 or 1010 pinarrays with a set of approximately ten fuel compositions with dierent 235U enrich-ment and gadolinia (Gd2O3) concentrations. In order to investigate the performanceof dierent 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.</p><p>2.3. Mathematical model of the objective function</p><p>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 nd the solution x with the following four objectives:</p><p>1. the minimum average lattice enrichment;2. the best approximation to a prescribed innite-multiplication-factor (k-inf) as</p><p>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;</p><p>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.function via frequency counts, a tabu frequency list. A move is forbidden if its fre-mance is diversication, that is, to encourage the method to search unexploredregions. This mechanism can be achieved by implementing a long-term memorywhile the oldest is removed from T. Another mechanism that can improve perfor-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,</p><p>1216 J.L. Francois et al. / Annals of Nuclear Energy 30 (2003) 12131229However, in this study they were considered to be additional objectives.</p></li><li><p>seleczero(HE</p><p>obtalocaoptimThe 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 of(HELIOS code).</p><p>PPF0 x PPFmax04 0 5Gd0 x XNp</p><p>j0G j =Np 4</p><p>To satisfy objective 4, is not necessary to optimize any function, only Eq. (5) mustbe satised, 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-lator0</p><p>ined using Eq. (4), where G( j) is the gadolinia concentration of the fuel pinted in position j in the lattice with Np total fuel pin positions. The individualal solution of this function is also zero.</p><p>Minimize G x abs Gd0 x Gdtarget </p><p>=Gdtarget 3XN</p><p>i0kinf i x kinf target i 2 2</p><p>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 Gd (x) isted using a reference reactivity curve. Obviously the optimal value for S(x) is. The values for kinf_i(x) are obtained executing the neutronic simulatorLIOS code).</p><p>Minimize S x sum of the squared k-deviations x In order to accomplish the second objective function, S(x), the sum of the squaredk-deviations at dierent 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 beMinimize E x XNp</p><p>j0E j =Np 1The rst 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.</p><p>J.L. Francois et al. / Annals of Nuclear Energy 30 (2003) 12131229 1217the multiple objectives.</p></li><li><p>3. Application to the study case</p><p>The optimization process was developed for a 1010 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 array</p><p>indicating the fuel composition located at each pin position in the lattice.Minimize Z x wE E x wS S x wG G x wP</p><p> PPF0 x PPFmax 6</p><p>Here wk indicates the relative importance that the decision-maker attaches toobjective k and it mus...</p></li></ul>

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