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Progress in Nuclear Energy 52 (2010) 249–256

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Progress in Nuclear Energy

journal homepage: www.elsevier .com/locate/pnucene

GreeNN: A hybrid method for the coupled optimization of the axial and radialdesign of BWR fuel assemblies

Juan Jose Ortiz-Servin a,*, Jose Alejandro Castillo a, David Alejandro Pelta b

a Instituto Nacional de Investigaciones Nucleares, Carretera Mexico Toluca S/N, La Marquesa Ocoyoacac, Estado de Mexico, CP 52750, Mexicob ETS Ingenierıa Informatica y Telecomunicaciones, Universidad de Granada, C/Daniel Saucedo Aranda, s/n 18071, Granada, Spain

Keywords:BWRFuel Lattice DesignNeural NetworksGreedy Search

* Corresponding author.E-mail addresses: juanjose.ortiz@inin.gob.mx (

castillo@inin.gob.mx (J.A. Castillo), dpelta@decsai.ugr.

0149-1970/$ – see front matter � 2009 Elsevier Ltd.doi:10.1016/j.pnucene.2009.06.016

a b s t r a c t

Radial and axial optimization of the fuel assembly in a boiling water reactor are usually solved asindependent problems, despite they are highly related. In this work we propose GreeNN, a hybrid systemcomposed by a simple greedy search technique and a neural network that allows approaching thesolution of both problems in a coupled way. Firstly, GreeNN performs the radial optimization of the fuelassembly (minimizing the Local Power Peaking Factor according to a 2D simulation) and then, theobtained fuel lattice is added to a fuel lattices inventory. This inventory is used to solve the axial opti-mization of the fuel assembly where a 3D core simulator is used to make a Haling calculation at the endof the cycle and to estimate the generated energy. The method proceeds iteratively, with the aim ofdecreasing the uranium enrichment of the designed fuel lattices in the radial stage while keeping theenergy requirements.

GreeNN system was applied to design the fuel lattices for an equilibrium cycle of 18 months. The fuelassembly’s performance proposed by GreeNN system was better than the reference case, withoutjeopardizing the reactor safety.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Radial optimization and axial optimization of the fuel assemblyin a boiling water reactor can be understood as combinatorialoptimization problems where the global aim is to maximize theextracted energy from the uranium while strict safety constraintsare satisfied.

Despite both problems are highly related, they are usuallysolved independently. For example, regarding the radial optimiza-tion, (Zheng et al., 2001) used a linear superposition method toestimate the fuel lattice parameters in a boiling water reactor; thisimplementation was added to FORMOSA-B code (Karve andTurinsky, 1999). On the other hand, Cuevas et al. (2002) workedwith both, MOX fuel and the simplex method for the local PowerPeaking Factor optimization. Tests were performed in boiling waterreactors and in pressure water reactors, obtaining good results.Castillo et al. (2006) applied a Path Relinking technique (Glover,1999) for optimizing the fuel lattice design, minimizing the LocalPower Peaking Factor and keeping the Infinite Multiplication Factorinto a proposed reactivity interval. Ortiz et al. (2006) successfully

J.J. Ortiz-Servin), alejandro.es (D.A. Pelta).

All rights reserved.

applied a Neural Network to optimize fuel lattices in boiling waterreactors.

Regarding the axial design problem, it has not been widelystudied, but the available works represents important milestones inthe fuel management area. For example, Mochida et al. (1996)studied boiling water reactors to increase the burnup discharge;they worked both the spectral shift strategy and the axial distri-bution of uranium enrichment (U235). Recently Tohjoh et al. (2006)used the Linear Heat Generation Rate (LHGR) thermal limitaccording to both radial and axial Power Peaking Factors of the fuelassembly nodes for the axial optimization of the fuel assemblyusing the Monte Carlo method. Finally, Ortiz et al. (2007) developeda fuel assembly axial optimization system using a multi staterecurrent neural network.

In this contribution we recognize the strong relation betweenboth problems, and as a consequence, we propose an optimizationtool called GreeNN to carry out the design of a fuel assembly takinginto account simultaneously both the radial and the axial designs.

In order to present GreeNN system, its main features and vali-dation performed, we structure the contribution as follows: in thesecond section we describe the problem to be solved. In Section 3the GreeNN system is described while in Section 4 computationalexperiments and results for an equilibrium cycle are shown. Finally,the last section is devoted to the conclusions and future works.

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2. Problem description

In this section, we define both the radial and the axial design ofa fuel assembly. The reason to optimize them in a coupled way willbe indicated.

2.1. Radial design optimization

A typical fuel lattice section is a 10�10 array of rods as it isshown in Fig. 1. The small white circles represent rods havinguranium only, the black circles include both uranium and gadoli-nium rods, and the big circles are water channels. The fuel assemblyradial design problem is defined in the following way: given aninventory of uranium rods with different enrichments and gadoli-nium concentrations, the goal is to find an allocation of the rodsinto the small circles of Fig. 1, such as the Local Power PeakingFactor (LPPF) is minimized while the neutron Multiplication Factorin an infinite medium (k-inf) is kept within certain margins. Thesereactivity margins are established according to a statistical study offuel lattice k-inf versus the cycle energy requirements. Besides inthe radial-axial optimization, the energy requirements are checkedand only fuel lattices with appropriate reactivity levels are kept.The allocation is constrained by ’’rules of thumb’’ that must besatisfied:

1. Only 2% enrichment rods can be allocated in the fuel latticecorners.

2. Can not be allocated gadolinium rods in the fuel latticeperiphery.

The complexity of the problem can be reduced if we considerthat the upper triangular part and the lower triangular one of thelattice section (see Fig. 1) should be symmetric. Once an allocationis given (the uranium and gadolinium rods have been distributedinto the fuel lattice), a transport code is executed to solve theNeutron Transport Equation in 2 dimensions with several energygroups. In this study the CASMO-4 code (Rhodes and Edenius,

Fig. 1. Fuel Lattice example.

2004) is used to calculate the LPPF and k-inf parameters at thebeginning of the fuel lattice life and 40% of void fraction.

2.2. Axial design optimization

An axial fuel assembly can be understood as a quadrangularprism that is build piling up 25 fuel lattices. The way that thoselattices are selected and how they are distributed into the prism,informally define the axial design problem.

Axially the fuel assembly can have different sections, which canbe formed by different fuel lattices, depending on the uraniumcontent and uranium and/or gadolinium distribution in each rod.Fig. 2 shows a fuel assembly axial distribution with several fuellattices sections. Each square represents a fuel lattice and similarfuel lattices segments are indicated by the same colour.

Additionally, the fuel assembly axial distribution changesaccording to the manufacturer and his own designs. In this studya fuel assembly with 6 axial zones, which are labelled from A to Fupwards, will be considered. Some fuel assemblies’ new designsinclude empty and/or vanished rods in zones from C to F. Zones A, Eand F have natural uranium. Besides, the B zone can be divided intothree sub zones and the D zone can be divided in two sub zones ofvariable length. Some constraints should be taken into account: theB zone must not have either empty or vanished rods while the Dzone should contain fuel lattices with either empty or vanishedrods.

Given a fuel lattices inventory, the axial fuel assembly designaims to find a combination of them, in order to extract themaximum amount of energy while both, the thermal limits andcold shutdown margin are satisfied. There is one rule to constrainthe fuel lattices allocation, the natural uranium fuel lattices must beallocated in both upper and bottom extremes of the fuel assembly.Once the fuel assembly is designed, a number of them are loadedinto the reactor core and an operation cycle simulation is carriedout using Haling’s principle (Haling, 1964). Then, the extractedenergy, the thermal limits and cold shutdown margin are verified.In this work the verification is done with the SIMULATE-3 code(2005).

3. GreeNN system

GreeNN is a short name for a hybrid method composed bya greedy search and a multi state recurrent neural network that areused to solve the axial and radial design optimization problemsrespectively.

The greedy search (Rayward-Smith et al., 1996) is a multi-startmethod that consists of two stages: 1) construction (solutions’space exploration) and 2) improvement (exploitation). In the firstone, an initial solution is obtained by means of a random orheuristic procedure. Then, in the improvement stage, a local searchheuristic is applied. An important characteristic of this searchstrategy is that it takes fast decisions with the available informa-tion, without considering future consequences.

The Multi State Recurrent Neural Network (Merida et al., 2001)is a Hopfield’s generalized neural network type. While in Hopfield’sdiscrete neural network the neural states takes binary values, in themulti state neural network the neural states can be integer values inthe [0,.,N] interval. As in the Hopfield’s model, this neural networkhas an energy function associated to its behaviour. This neuralnetwork has been applied to diverse combinatorial optimizationproblems with good results (Ortiz and Requena, 2004; Mejıa andOrtiz, 2005).

In the next subsections, we describe how the optimizationtechniques are adapted and integrated to simultaneously solveboth design problems.

Fig. 2. Fuel assembly axial distribution example.

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3.1. Radial stage – greedy search

In order to apply a greedy search to an optimization problem, weshould define how the initial solution is constructed and how theiterative improvement is done.

As it was mentioned before, a solution to this problem isa particular allocation of certain rods to circles in the fuel lattice.The quality of an allocation (or fuel rod configuration) is measuredin terms of an objective function.

One way to transform a configuration to a different one is todefine a ‘‘move’’ operator that exchanges the position of twoselected rods. After a change is made, the new configuration shouldbe evaluated through the execution of the transport code whichconsumes several CPU seconds. As a consequence, and in order tomake the search more efficient, we first performed a sensitive studyon the variation of both the LPPF and neutron multiplication factor(k-inf), when interchanges of two rods of the fuel assembly ina proposal fuel lattice are done.

The study was made with a fuel lattice of 10�10 rods and twowater central channels. A random initial uranium and gadoliniumdistribution was assumed. This fuel lattice has a 3.84% averageenrichment and 14 gadolinium rods at 4%.

From the initial fuel lattice, 10,000 pair’s exchanges (moves)were performed. The selection of the rods to exchange was random.For each move, a record of the initial position of the two rods (i1, i2,j1, j2 are the rods coordinates of the fuel lattice according to theFig. 1), its uranium content (U %), its gadolinium (Gd2O3) content, itsk-inf value and its LPPF value (the last two values were obtainedusing CASMO-4) was recorded. The CASMO-4 calculi were made for40% void fraction at the beginning of the fuel lattice life.

With these data, we wanted to study the relation between thedifference of uranium enrichment DU% of the exchanged rods andthe variation of both LPPF and k-inf. Fig. 3 shows the LPPF behaviouraccording to above difference (DU%) when both rods (with andwithout gadolinium) are interchanged. Fig. 4 shows the k-infbehaviour versus DU% for rods with and without gadolinium.

The results led to the following observation: when the rods donot have gadolinium, a clear relation arises: small changes in theuranium content produce small changes in the variables. Bigchanges in the uranium content produce both lower and bigchanges in those variables. However, when the exchanged rods

Fig. 3. LPPF behaviour accord

have gadolinium, the LPPF or k-inf changes can be big or small,neither a clear tendency nor any pattern can be observed.

As a consequence, we can obtain a move that leads to a smallperturbation of the cost of the solution if we perform an exchangeof rods without gadolinium whose difference in uranium content islow. This move is called basicMove.

When a rod with gadolinium is involved in the exchange, thena greater perturbation is obtained, so we call this move perturba-tion. Given these moves, the greedy method operates as follows.

In order to construct the initial solution, a random gadoliniumand uranium enrichment distribution is generated. Then, 50perturbation moves are performed (one rod will have gadolinium,while the other one is randomly selected). The only restriction isthat any fuel rod with gadolinium must not be allocated in the fuellattice periphery. The fuel lattice obtained is very different,according to its objective function value, to the initial fuel latticeand it is taken as the final result of the construction stage. This fuellattice is named as Lattice 0.

Then, an iterative improvement stage is done. Starting fromLattice 0, a number of basicMoves are applied, making exchangesbetween randomly selected pairs of rods whose uranium enrich-ment difference is lower than 1%. This uranium enrichment limit isempirically set after the results shown in Fig. 3. A greater limit willproduce high LPPF variations that are undesirable at this stage. Thelowest uranium enrichment must be allocated in the fuel latticecorners. If the basicMove decreases the LPPF value then, theexchange is accepted and the process is repeated from that newconfiguration. If no improvement was made, then a new basicMoveis applied. This iterative process is made 200 times, at which 200different fuel lattices are evaluated. According to previous experi-ments (Ortiz et al., 2008), 200 interchanges are enough to obtain anacceptable result.

3.2. Axial stage – multi state recurrent neural network

The axial fuel assembly design is a combinatorial optimizationproblem, which is easier to solve than the fuel lattice design;however is still a complex problem. For this stage, a Multi StateRecurrent Neural Network with 25 neurons was used. Each neuronrepresents a square in Fig. 2. The neural states are integer numbersrepresenting fuel lattices. Every time a new fuel lattice is made at

ing to DU% differences.

Fig. 4. k-inf behaviour according to DU% differences.

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radial stage, that fuel lattice is added to fuel lattice list with anascending sequential number, which is labelled like a neural state.

Table 1 shows an example with the available fuel lattices to fuelassembly design, including their associated neural states. It can beseen that the available fuel lattices are grouped according to theaxial zone where they can be allocated. The neural states indicatethe axial zone that can be assigned: 1xx for the B zone; 2xx for the Czone; and 3xx for the D zone. The ‘‘% U/Gd’’ column indicates thefuel lattice average uranium enrichment and the ‘‘xGy.0’’ symbolindicates the number of gadolinium rods (x) and its concentrations(y%). Fuel rods with different concentrations can appear simulta-neously. Fig. 5 shows a recurrent neural network example initial-ized with neural states included in Table 1. The neural states A1, E1and F1 remain without changes for the natural uranium fuel latticesin A, E and F zones respectively.

The operation mechanism of this neural network type involveschanging the neural states so that the neural network decreases itsenergy level. Thereby, the neural network energy function mustmaximize the keff at the end of the cycle, while thermal limits andcold shutdown margin are satisfied. The proposed function is thefollowing:

F ¼ wijkeffðENÞ � keffTj þw2ðFLPDðENÞ � FPLDLimÞþw3ðMPGRðENÞ �MPGRLimÞ þw4ðMFLCPRðENÞ�MFLCPRLimÞ þw5

�SDMBOC;Lim � SDMBOCðENÞ

�

where wi are weighting factors obtained in an experimental way(It is important to say that they are not the connection weightingsbetween the neurons), keff neutron effective multiplication factorobtained by a Haling calculation, keffT objective neutron effectivemultiplication factor, FLPD density linear power fraction obtainedby a Haling calculation, FLPDLim limit density power linear

Table 1Neural status corresponding with the fuel lattices according to the axial zone.

Zone B Zone C

Neural State %U/Gd Neural State

B01 3.75 5G6.0,6G5.0 C01B02 3.76 9G6.0,2G5.0 C02B03 3.76 11G5.0 C03B04 3.71 11G5.0 C04B05 3.71 4G5.0, 10G4.0, 2G2.0 C05B06 3.70 12G4.0, 2G2.0 C06B07 4.06 8G5.0 C07B08 4.06 10G5.0

fraction, MPGR maximum power generation rate obtained Halingcalculation, MPGRLim maximum power generation rate limit,MFLCPR fraction of Limiting Critical Power Ratio obtained bya Haling calculation, MFLCPRLim fraction of Limiting Critical PowerRatio limit, SDMBOC,Lim cold shutdown margin limit at thebeginning of the cycle, SDMBOC cold shutdown margin at thebeginning of the cycle, EN is a vector with the 25 neural networkneural states.

The way the neurons change their neural states is called StatesTransition Rule. This rule can be better understood when the neuralnetwork operation algorithm is explained. This algorithm is thefollowing:

1. The neurons are initialized with neural states according toTable 1.

2. An axial zone is chosen in a random way (B, C or D zones).3. If the B or D zones are chosen then a sub zone number is

randomly chosen, according to the proposed constrains.4. For each axial sub zone do:

a) The neurons of the sub zone take all the possible neuralstates according to the Table 1.

b) The energy function for each neural state is evaluated.c) The new neural state of this sub zone is that with lowest

energy function value.d) Go to the step 6

5. If the C zone was chosen then:a) The two neurons take all possible neural states of the

Table 1.b) The energy function for each neural state is evaluated.c) The new neural state of this sub zone is that with lowest

energy function value.6. Repeat from step 2 to step 5 until the stop criteria are achieved.

Zone D

%U/Gd Neural State %U/Gd

3.75 13G6.0 D01 3.75 13G6.0,2G3.03.76 9G6.0,2G5.0 D02 3.75 13G6.03.71 11G5.0 D03 3.76 9G6.0,2G5.0,3G3.03.71 4G5.0, 10G4.0 D04 3.76 9G6.0,2G5.03.70 12G4.0, 2G2.0 D05 3.71 11G5.04.06 6G6.0, 3G5.0 D06 3.71 11G5.04.06 12G5.0 D07 3.71 4G5.0, 10G4.0,4G2.0

D08 3.71 4G5.0, 10G4.0, 2G2.0D09 3.70 12G4.0, 2G2.0D10 3.97 6G6.0, 3G5.0D11 3.97 12G5.0

Fig. 5. Neural network initialization example.

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In this algorithm, in the point a) of the steps 4 and 5, the StatesTransition Rule is included. There are two stopping criteria, either50 iterations are done or the energy function value remainswithout changes during 10 iterations. These limits were imposedbased on a statistical analysis realized during the tests made.

3.3. Radial – axial coupling

The radial stage builts an initial fuel lattice with a specifiedaverage uranium enrichment that is fed into the GreeNN system.The GreeNN objective is to design a fuel assembly of a fuel reloadbatch for a proposed length operation cycle. For this, a known fuelreload is available, in which the fuel batch obtained with GreeNNsystem is introduced. The fuel reload for that cycle is evaluated bySIMULATE-3 code according to the Haling principle, and theneutron multiplication factor (keff) value is obtained. If the safetyparameters are not satisfied, the fuel lattice is discarded. Otherwise,the fuel lattice quality is measured through the keff value.

The GreeNN system flowchart is shown in Fig. 6. The systeminput data are the following:

1. Average uranium enrichment of both the fuel lattice and thefuel assembly (U%FL and U%FA)

2. Enrichment inventory of both uranium and gadolinium3. Gadolinium rods number and their concentration

Fig. 6. GreeNN syst

4. Fuel lattice inventory5. Fuel reload6. Fuel reload batch length

As it was mentioned, both the empty and the vanished rodspositions remain fixed in the fuel lattice. In this way, once the fuellattice design was made, the fuel lattices corresponding to the C andD zones are built equal to B zone, but taking into account bothempty and vanished rods of those zones. The fuel lattices areevaluated using CASMO-4 code to different burnup steps and voidsfractions, and then, the fuel lattices inventory that can be used forthe multi state recurrent neural network is incremented.

Once the axial stage is finished, if the keff value obtained at theend of cycle is greater than the reference value, then the averageuranium enrichment value is reduced in 0.005% and a new fuellattice is designed. This process is repeated until the obtained keffvalue is lower or equals to the reference value.

4. Results

4.1. Uranium enrichment reduction deactivated

In order to test the GreeNN system, a reload fuel batch with3.66% of average uranium enrichment was designed. Consideringthe natural uranium fuel lattices included in the fuel assembly

em Flowchart.

Table 2Valid limit values in the safety parameters.

Parameter Limit Value

LPPF maximum 1.40k-inf range From 1.14 to 1.16FLPD maximum 0.80MPGR maximum 0.80MFLCPR maximum 0.80SDM % minimum 1.5keffTarget (10075 MWD/T) 0.9972

Table 4Reactor parameters for 10 GreeNN system runs.

NS LPPF k-inf keff FLPD MPGR MFLCPR SDM %

Ref. 1.240 1.1437 0.9972 0.683 0.706 0.791 l.784B20 1.347 1.1533 0.9980 0.689 0.707 0.789 1.514B21 1.364 1.1541 0.9980 0.688 0.704 0.789 1.728B22 1.379 1.1558 0.9981 0.690 0.704 0.789 1.594B23 1.338 1.1523 0.9980 0.689 0.707 0.789 1.528B24 1.370 1.1547 0.9983 0.688 0.706 0.789 1.548B25 1.378 1.1573 0.9982 0.689 0.706 0.789 1.510B26 1.370 1.1570 0.9984 0.689 0.704 0.788 1.554B27 1.364 1.1532 0.9981 0.688 0.706 0.789 1.501B28 1.365 1.1516 0.9980 0.689 0.707 0.789 1.601B29 1.323 1.1529 0.9981 0.687 0.705 0.789 1.596

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extremes, the fuel lattices in B and C zones must have 4.06% ofuranium enrichment. D zone must have 3.97% of uranium enrich-ment. In these fuel lattices 8 gadolinium rods at 5% are included.The designed fuel batch will substitute another fuel batch in anequilibrium cycle of 18 months. Using the original fuel batch fora burnup of 10,075 MWd/T according to a Haling calculation, thekeff value is equal to 0.9972. This is the reference case.

Table 2 shows the limit values of the variables involved in theneural network energy function and the limit values of the fuellattice parameters. Thermal limit fractions are calculated accordingto Haling Principle. It is an ideal reactor operation scheme in whichthermal limits are easily fulfilled. Maximum thermal limit valuesfixed to 0.8 guarantee us (according to our experience) that will bepossible to design control rod patterns with thermal limit fractionslower than 0.95, that is the limit commonly used in this stage of theoperation cycle optimization. In the same way, the cold shutdownmargin (SDM) is fixed to 1.5% at the beginning of the cycle becauseit guarantees a SDM equal or greater than 1.0% thorough the cycle.The reader should note that these safety limits help us to reduce theenergy function calculations complexity and the CPU time required.In order to deal with a more realistic operation scheme we wouldresort to pre-optimized control rod patterns or we would coupleour method to a control rod pattern optimization system. Thesetasks are out of the scope of this contribution and are left as futurework.

The uranium and gadolinium rods inventories available for thefuel lattices are presented in Table 3. To build the fresh fuelassembly, the fuel lattices of Table 1 were used.

The most important parameters of the reactor simulation areshown in the Table 4, using both the fuel lattices and fuel assem-blies designed. The LPPF and k-inf columns correspond to the fuellattice parameters at the beginning of its life. The keff, FLPD, MPGRand MFLCPR columns correspond to the end of the reactor opera-tion cycle according to the Haling calculation. The SDM column

Table 3Both uranium enrichment and gadolinium concentrations inventory.

%U235 %Gd %U235 %Gd

2.00 0 3.60 22.40 0 3.60 32.80 0 3.60 43.20 0 3.60 53.60 0 3.60 63.80 0 3.95 23.95 0 3.95 34.20 0 3.95 44.40 0 3.95 54.80 0 3.95 64.90 0 4.40 23.20 2 4.40 33.20 3 4.40 43.20 4 4.40 53.20 5 4.40 6

corresponds to the beginning of operation cycle. The column NSshows the neural state number associated to each fuel lattices inorder to be used in the neural network. Also, a reference fuel latticeis shown. For this experiment, the iterative process to reduce theuranium enrichment was deactivated.

As it can be seen from the Table, the three fractions of thethermal limits are lower than 0.8, while the cold shutdown marginis greater or equal than 1.5%. Likewise, the fuel lattice LPPF value inall cases is lower than 1.40. Thereby, the designed fuel lattices andfuel assemblies satisfied the core safety constrains. Then, the nextimportant requirement is to achieve the required energy. As it wasindicated in the Table 2, the keff value must be greater than 0.9972to guarantee that operation cycle energy was satisfied. In the 10cases this requirement is satisfied. On average, the keff value is0.00111 greater than the mentioned value. The above indicates thatoperation cycle could be extended for 1 day. Although it does notseem a great improvement, it is necessary to remember that oneextra day full power reactor operation is equivalent to generateextra profits by 1 million euros approximately.

Fig. 7 shows the axial composition of the 10 fuel assembliesobtained by GreeNN system. The labels show the fuel lattice designaccording to neural states shown in Table 1. Neural states from B20to B29 are the GreeNN fuel lattices of the Table 4. Neural states fromC20 to C29 are the fuel lattices of the C zone and neural states fromD20 to D29 are the fuel lattices of the D zone. In Fig. 8 the radialdistribution of the neural state B29 and B22 shown. Both have thelowest and the highest LPPF values of Table 4.

4.2. Uranium enrichment reduction activated

The previous results showed that it is possible to automaticallydesign fuel assemblies for a fuel reload. Besides, using a Halingcalculation it was possible to determine a reactivity excess at theend of the cycle. In the next computational experiment we willshow results for the uranium enrichment reduction using GreeNNsystem.

The above mentioned cycle was again used here. The initialuranium enrichment for the lattice fuel (U%FL) in B zone is 4.06%.Each time that a fuel reload, with keff greater than the referencevalue is found, the U%FL is reduced by 0.005% and a new optimi-zation process is made. This is repeated until the keff value is loweror equals to the reference value 10. GreeNN system executions withthis option were made and we only show typical tendencies.

Fig. 9 shows keff behaviour at the end of the cycle as a functionof the uranium enrichment reduction. The decreasing tendency isclear. Fig. 10 shows the LPPF fuel lattice behaviour as a function ofthe uranium enrichment reduction. No clear tendency can beobserved, although if we focus on uranium enrichments lower than3.95% a stabilization of the LPPF values occurred.

F1 F1 F1 F1 F1 F1 F1 F1 F1 F1E1 E1 E1 E1 E1 E1 E1 E1 E1 E1

D20 D21 D22 D23 D24 D25 D26 D27 D28 D10D20 D21 D11 D23 D24 D25 D26 D27 D28 D10D20 D21 D11 D11 D24 D11 D11 D27 D11 D10D20 D21 D11 D11 D11 D11 D11 D27 D11 D10D20 D21 D11 D11 D11 D25 D11 D11 D11 D10D20 D21 D11 D11 D11 D25 D11 D11 D28 D11C07 C21 C22 C23 C24 C25 C26 C27 C07 C29C07 C21 C22 C23 C24 C25 C26 C27 C07 C29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29B20 B08 B08 B23 B24 B25 B26 B27 B28 B29B20 B08 B08 B23 B24 B25 B26 B27 B28 B29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29B20 B21 B22 B23 B24 B25 B07 B27 B28 B29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29B20 B21 B22 B23 B24 B25 B26 B27 B28 B29A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

Fig. 7. Fuel assemblies’ axial composition obtained by GreeNN system.

2.00 %U= 4.054

3.20 4.90

3.20 3.95 3.20

4.90 3.95 4.40 4.40

3.60 3.20 3.95 4.40 3.60

4.90 4.90 3.95 3.95

4.90 3.20 4.90 4.90 3.60

4.90 3.95 4.90 3.60 3.60 3.60 4.40 4.90

3.95 4.40 3.95 4.90 3.60 4.90 3.95 3.95 4.40

2.00 3.95 4.90 3.60 3.20 4.90 4.90 3.20 3.20 2.00

Fuel lattice with LPPF= 1.323,NS = B29.

2.00 %U= 4.054

4.40 4.90

4.90 3.95 4.90

4.40 3.20 3.60 4.40

3.60 3.60 3.95 3.95 3.20

4.90 3.60 3.20 3.6

4.40 3.6 4.90 3.20 3.95

4.90 3.95 4.90 3.95 3.20 4.90 3.95 4.90

3.60 4.90 3.95 4.40 3.95 3.60 3.60 3.95 4.90

2.00 4.40 4.40 4.40 4.90 4.40 4.90 3.60 4.40 2.00

Fuel lattice with LPPF= 1.379, NS B22

Fig. 8. Fuel lattices obtained by GreeNN system.

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Fig. 9. Keff-EOC according to Haling Calculation as a function of uraniun enrichment.

Fig. 10. LPPF fuel lattices behaviour as a function of uraniun enrichment.

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5. Conclusions

In this work the GreeNN system used for the coupled optimi-zation of both fuel lattices and fuel assemblies for boiling waterreactors was described. The radial optimization is done by a greedysearch strategy taking into account the possible fuel rod combi-nations of both uranium enrichment and gadolinium concentra-tions for a 10�10 pins fuel lattice. The axial optimization problemis managed with a Multi State Recurrent Neural Network takinginto account the possible known fuel lattices combinations in a 25nodes array.

According to the experimental tests without uranium enrich-ment, the GreeNN system performance was satisfactory. The fuellattices have LPPF values lower than 1.4. The fuel assemblies’behaviour into the core shows both thermal limits and cold shut-down margin fulfilled. In fact, the energy generation is greater thanthe original reported energy. A typical optimization requiresaround 8hs of CPU time in a 2.6 GHz processor with 1 Gb of RAM.Right now, time is not a problem. If time reduction is needed,simple parallelization techniques can be applied.

For the uranium enrichment reduction case, all experimentsshowed a similar decreasing tendency for keff at the end of thecycle according to Haling calculation. When we described the radialstage, we said that 200 interchanges are enough to obtain goodresults. However, for fuel lattices with uranium enrichment lower

than 3.923%, more greedy initializations (one initialization is a cycleof 200 interchanges) to reach LPFF values lower than the limit wereneeded. With this GreeNN execution modality, it was determinedthat it is possible to further decrease the fuel lattice uraniumenrichment to values around of 3.91 to 3.93% and to satisfy theenergy requirements according to Haling Principle.

We should remark that this methodology can be extended toother fuel assembly designs in an easy way by firstly, changing theneural network size for the axial stage and secondly, by modifyingthe range of the fuel rods coordinates i1, j1, i2 and j2 according tothe fuel lattice geometry.

Acknowledgments

The authors acknowledge the support given by CONACyT,through the research project SEP-2004-C01-46694 and the ININthrough the research project CA-910. Dr. Pelta acknowledges thesupport given by the Spanish Ministry of Science and Innovationthrough project TIN2008-01948, and Junta de Andalucia, throughproject P07-TIC02970

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