development of a bwr loading pattern design system based on modified genetic algorithms and...

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Annals of Nuclear Energy 31 (2004) 1901–1911

Development of a BWR loading patterndesign system based on modifiedgenetic algorithms and knowledge

Cecilia Mart�ın-del-Campo a,*, Juan Luis Franc�ois a,Linda Avenda~no b, Mario Gonz�alez b

a Laboratorio de An�alisis en Ingenier�ıa de Reactores Nucleares, Facultad de Ingenier�ıa, Universidad

Nacional Aut�onoma de M�exico, Paseo Cuauhn�ahuac 8532, Jiutepec, Mor. 62550, Mexicob Facultad de Ingenier�ıa, Universidad Nacional Aut�onoma de M�exico, Circuito Exterior S/N, Ciudad

Universitaria, DF 04510, Mexico

Received 30 January 2004; accepted 8 March 2004

Available online 12 August 2004

Abstract

An optimization system based on Genetic Algorithms (GAs), in combination with expert

knowledge coded in heuristics rules, was developed for the design of optimized boiling water

reactor (BWR) fuel loading patterns. The system was coded in a computer program named

Loading Pattern Optimization System based on Genetic Algorithms, in which the optimiza-

tion code uses GAs to select candidate solutions, and the core simulator code CM-PRESTO to

evaluate them. A multi-objective function was built to maximize the cycle energy length while

satisfying power and reactivity constraints used as BWR design parameters. Heuristic rules

were applied to satisfy standard fuel management recommendations as the Control Cell Core

and Low Leakage loading strategies, and octant symmetry. To test the system performance, an

optimized cycle was designed and compared against an actual operating cycle of Laguna

Verde Nuclear Power Plant, Unit I.

� 2004 Elsevier Ltd. All rights reserved.

*Corresponding author.

E-mail addresses: [email protected] (C. Mart�ın-del-Campo), [email protected] (J.L. Franc�ois).

0306-4549/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.anucene.2004.03.015

mail to: [email protected]

1902 C. Mart�ın-del-Campo et al. / Annals of Nuclear Energy 31 (2004) 1901–1911

1. Introduction

The main goal of boiling water reactor (BWR) in-core fuel management is focused

on finding the optimal fuel utilization assuring the constraints imposed in the nuclear

BWR design due to requirements of operation, safety and economy. Radial and axial

fuel assembly designs, in-core fuel load design, control rod pattern design, andmulticycle plan design significantly interact with each other. However, the system

described in this paper focuses only on the problem of determining optimal fuel

loading patterns under given constraints.

For a typical BWR the reload designer’s task is to find out the best arrangement

of 444 fresh and partially burnt fuel assemblies (FAs) from a pool of about 520 FAs

which differ in various properties like enrichment, reactivity and burnup. The BWRs

present strong three-dimensional material heterogeneities such as fuel enrichment,

burnable poison, control rods and coolant voids. Furthermore, a BWR has a higherquantity of FAs in the core compared with a pressurized water reactor (PWR). These

characteristics make the loading pattern design problem a very complex one and it

appeals for the use of an appropriate optimization method and a licensed 3-D core

simulator to evaluate the reactor cycle operation during the optimization process.

Design of BWR fuel reloads is usually based on engineer expertise, which is a

technique that uses human knowledge (Franc�ois et al., 1999). This technique does

not optimize the fuel assemblies’ location, a better design of fuel assembly reloads

can be achieved through optimization techniques.Fuel assembly reload design for BWRs is a huge combinatorial problem and some

solutions have been proposed in the past: using Genetic Algorithms (GAs) (Franc�oisand L�opez, 1999) with a very simple objective function and two-dimensional cal-

culations, applying simulated annealing (Moore et al., 1999) with a better objective

function and using an automatic system based on the tabu search method, along

with a simple linear perturbation method, to avoid the extensive use of a 3-D sim-

ulator (Jagawa et al., 2001). Recently Castillo et al. (2004) used a tabu search

technique with some heuristic rules to optimize BWR fuel reloads.Genetic algorithms has already been used for nuclear fuel management related

problems, particularly for the reload pattern design optimization for PWRs (Poon

and Parks, 1993: De Chaine and Feltus, 1995; Yamamoto, 1996; B€ack et al., 1996).

All these works have contributed to the understanding of the application of GAs to

the optimization problem.

In the present work, GAs method is used to determine the optimal radial location

of a fixed set of different FAs in the core positions. The optimization algorithm

searches to maximize the cycle energy for one reactor cycle operation under thesafety constraints.

2. Genetic algorithms description

Genetic algorithms method (Goldberg, 1989) combines mathematical analysis

with random search to build artificial systems having properties similar to natural

C. Mart�ın-del-Campo et al. / Annals of Nuclear Energy 31 (2004) 1901–1911 1903

systems according to the Darwinian mechanisms of evolution. GAs is a highly

parallel mathematical algorithm that transforms a set (population) of individual

mathematical objects (chains of chromosomes); each of them associated to a fitness

criterion, into a new population (next generation) using genetic operations. These

operations are modeled based on the Darwin theory of reproduction and survival of

the fittest, as a result of the execution of a series of them. The principle of GAs issimple and it can be summarized as follows:

1. Encoding of the problem in a mathematical solution.

2. Random generation of an initial population. This one includes a genetic pool rep-

resenting a group of candidate solutions.

3. Classifying of the solutions in terms of their fitness. Reckoning of a fitness value

for each individual.

4. Selection of individuals that will mate according to their share in the population

global fitness.5. Genomes crossover and mutations that modify the composition of the descen-

dants.

6. If requirements are met, then stop. Otherwise go back to point 3.

The GAs has very good characteristics for solving complex combinatorial prob-

lems, they do not require any functional derivative information, they cover the

search space in a relatively fast way, and it works well with reduced search spaces.

This method may be very efficient, but there is no proof that the optimum has been

found.

3. BWR loading pattern design problem

The problem to be solved is to get the ‘‘best’’ distribution of FAs in the core. A

BWR, as the Laguna Verde reactors in Mexico, has 444 FAs and 109 control rods.

Some of these FAs are fresh and others have been once, twice or more cycle’s op-

eration. To reduce the complexity of the loading pattern (LP) design and the ma-neuvering of control rods during the cycle operation, a recommended heuristic rule

in BWR fuel management is to assign quarter core symmetry, and then there will be

only 111 different positions to allocate the FAs (see Fig. 1). It is important to

mention that all the FAs, which have been burnt during previews cycles must be

considered with its particular burnup.

For every nuclear power plant, the objective is to operate in such a way that the

fuel burnup, expressed in MWD/TU (mega-watt day per metric tone of uranium),

and the average linear power density, expressed in kW/cm (kilo-watt per centimeter)are as high as possible. However, this objective – of maximizing the two quantities

just mentioned – is also a constraint imposed to the operator of the power plant,

because both, burnup and linear heat generation rate may lead to fuel failure (release

of fission products into the primary coolant) if they exceed their design values. In the

LP design process is very important to define an objective function which takes into

account the design values for the most important quantities affecting the operation of

the power plant. In this work, a similar multiobjective function, like the proposed by

Fig. 1. Quarter core symmetry representation.


Mart�ın-del-Campo et al. (2001) in the optimization of the fuel assembly axial design,

is used.

3.1. Objective function

The main goal is to obtain a loading pattern r, with the maximum cycle energy,

without violation of the thermal power limits, the hot reactivity and the minimum

cold shutdown margin limits. This can be achieved through the implementation of

the optimization method using a mutiobjective function in terms of the quantities

defined in Table 1. The limit values for the design constraints depend on the main

characteristic of the fuel assemblies charged in the core, and also they depend on thetype of simulation employed to evaluate all the candidate loading patterns.

In the present work, in order to avoid the control rod pattern design and to save

computing time, all the power design constraints and the cycle energy are evaluated

at the end of cycle using the Haling strategy (Haling, 1965). The power constraints

are satisfied when:

PPF(r)<PPFmax,

MLHGR(r)<MLHGRmax,

XMPGR(r)<XMPGRmax,MRNP(r)<MRNPmax,

MCPR(r)>MCPRmin,

MFAB(r)<MFABmax.

Table 1

Quantities included in the objective function

Energy Cycle energy¼ cycle mean core burnup

PPF Power Peaking Factor

MLHGR Maximum Linear Heat Generation Rate

XMPGR Fraction of the Limiting Average Planar Heat

Generation Rate (APLHGR)

MRNP Maximum Relative Nodal Power

MCPR Minimum Critical Power Ratio

MFAB Maximum Fuel Assembly Burnup

SDM Shutdown Margin

HER Hot Excess Reactivity


The Cold Shutdown Margin and the Hot Excess Reactivity are evaluated at the

beginning of cycle (BOC) but conservative limiting values are used. They are satisfied

when:

SDM(r)> SDMmin,

HERmin <HER(r)<HERmax.

The objective function is to maximize F(r) in Eq. (1)

F ðrÞ ¼ EnergyðrÞ � w1 � DPPFðrÞ � w2 � DMLHGRkðrÞ � w3

� DXMPGRkðrÞ � w4 � DMRNPkðrÞ � w5 � DMCPRkðrÞ � w6

� DMFABðrÞ � w7 � DSDMðrÞ � w8 �HERðrÞ � w9 þ B ð1Þ

where

DPPFðrÞ ¼ PPFðrÞ � PPFmax

DMLHGRðrÞ ¼ MLHGRkðrÞ �MLHGRmax

DXMPGRðrÞ ¼ XMPGRkðrÞ �XMPGRmax

DMRNPðrÞ ¼ MRNPkðrÞ �MRNPmax

DMCPRðrÞ ¼ MCPRmin �MCPRkðrÞDMFABðrÞ ¼ MFABðrÞ �MFABmax

DSDMðrÞ ¼ SDMmin � SDMðrÞHERðrÞ ¼ ðKeffðrÞ �KeffcritÞ=Keffcritw1 to w9 are called the weighting factors

B is a constant with a high positive value.

In order to give the appropriate weight to each parameter in the objective functionthe wi0s are determined by the user observing the values of F for a large quantity of

different candidate loading patterns. According to the D’s definition, the weighting

factors used in Eq. (1), will be positive if their associated safety limits are violated, in

this case the weighting factors will be the ones given by the user, in other case wi0s will

be zero, in order to do not penalize the objective function.

The inclusion of B in the objective function has a double function:(a) To assure always a positive value of F(r), this is necessary in order to apply the

proportional roulette method for the selection of parent-individuals of the

crossover operation.


(b) To reduce differences in the fitness of very good and very bad solutions, espe-

cially when a ‘‘super-individual’’ being too often selected, the whole population

tends to converge towards his genome (FAs distribution). This is called the linear

transformation to each fitness, F 0 ¼ AF þ B.

3.2. Heuristic recommendations

To solve the LP design problem some common recommendations in BWR fuel

management were applied as heuristic rules:

• The low leakage (LL) loading strategy, which means that the set of the more burnt

FAs must be located in the periphery for the new LP. This helps to avoid damage

to the reactor vessel.

• FAs placed on periphery positions in precedent cycles only can be used in periph-

ery locations in the new LP.• The control cell core (CCC) strategy rule which means that fresh FAs do not can

be placed in the active CCC positions for the new LP. The active CCC control

rods are those of the typical BWR A2 sequence (see Fig. 1).

• FAs placed in positions of the active CCC locations during all previews cycles can

not be placed in the predetermined CCC positions for the new LP.

• To look for octant symmetry.

3.3. System description

The main components of the Loading Pattern Optimization System based on

Genetic Algorithms (LPOS-GAs) are the optimization program, coded in the C

language, and the simulator CM-PRESTO (Scandpower, 1993) used to obtain all the

core parameters considered in the objective function to evaluate the fitness of each

candidate solution.

CM-PRESTO is a three-dimensional (3D) steady state BWR core simulator with

coupled neutronic and thermal-hydraulic models. The neutronic model is based onan approximation of two groups (1.5 groups) diffusion theory with a special coarse

mesh treatment of the space variable. The thermal-hydraulic model calculates the

average void content in each volume associated with a neutronic node to account for

the void feedback. The void distribution is obtained given the nodal power distri-

bution, total core mass flow and core inlet subcooling. The coolant flow is described

individually in each fuel bundle together with one common bypass channel. CM-

PRESTO has been validated for Laguna Verde Nuclear Power Plant applications

(Franc�ois et al., 2001). CM-PRESTO can be used to analyze several reactor con-ditions. It is used to simulate operating cycles under the Haling strategy (Haling,

1965) and to predict cold shutdown margin, only to mention some of the code ca-

pabilities.

A loading pattern in CM-PRESTO with quarter core symmetry is represented by

an array of 111 elements. These elements are alpha-numeric data which identify the

FAs placed in each of the 111 positions (see Fig. 1).

To accomplish heuristic recommendations, the following rules were implemented:


• The core was divided into three regions:

1. periphery: 17 positions,

2. CCC: 24 positions,

3. in-core: 70 positions.

• The positions are coupled in octant geometry (see Fig. 2). FAs placed in octant

symmetry positions are always moved to symmetrical positions.• The set of 111 FAs to be placed in the core is predetermined. The process starts

knowing the characteristics of the FAs that will compose the core. The set is sub-

divided into three subsets:

(a) The subset of 17 partially burnt FAs, with the highest burnup but including all

the FAs which have been in periphery in previous cycles, will be placed in the

periphery region.

(b) The subset of 24 partially burnt FAs, which have not been in the CCC region

in previous cycles, will be placed in the CCC region in the new reload.(c) The subset of 70 fresh and partially burnt FAs includes: the partially burnt

FAs that have never been placed in periphery positions; all the FAs that have

been in CCC positions; all the fresh assemblies and any remaining FAs. This

subset of FAs will be placed in the in-core region.

• Each subset of FAs is ordered taking into account the fuel type, the batch and the

burnup. Then, pairs of FAs are selected in order to be manipulated as couples.

FAs without couples will be placed in diagonal symmetry positions.

Fig. 2. Identification of position pairs in octant symmetry.

Fig. 3. Crossover operation of CCC region: the subsets of 24 FAs are exchanged between Parent 1 and

Parent 2.


• The crossover operation is implemented in order to exchange the regions’ subsets

of FAs between two parents. This action forces the accomplishment of heuristicrecommendations and a duplication of FA identifiers will never be generated.

• The mutation operation exchanges the position between two FA pairs in the oc-

tant geometry without changing the region, FAs in the diagonal symmetry are

only exchanged with other FAs in the diagonal symmetry. Fig. 3 shows the case

of CCC region crossover.

• A very large size population is used in order to have a high diversity of individu-

als. The initial population of loading patterns (individuals) is randomly generated

taking into account the heuristic rules.

4. Test problem and results

LPOS-GA was validated using an actual BWR operating cycle. This is the fifth

cycle of Laguna Verde Nuclear Power Plant, Unit I, which loading pattern was

previously generated using engineering expertise. The actual loading pattern, de-

signed with engineering expertise, achieved a cycle length of 9281 MWD/TU usingHaling calculations with CM-PRESTO.

This loading pattern has 112 fresh fuel assemblies of 3.53 w/o U-235 enrichment.

The safety operational limits at the end of the cycle (Haling calculation) imposed on

this calculation are given in Table 2. On the other hand to avoid calculations of

SDM during the cycle, this parameter is evaluated only at the beginning of the cycle

and it needs to be higher than 1.5% Dk=k. The hot excess reactivity at the beginning

of cycle is recommended to have a value between 1.5% and 2.0% Dk=k.Due to the random nature of the process, here considered, we performed the

search of an optimized loading pattern several times and the best results are shown in

Table 3. The LP number 1 (LP1) was obtained using a population of 100 individuals

and 35 generations, the LP number 2 (LP2) was found using 100 individuals and 30

generations. Both cases were obtained with the same objective function. The

weighting factors and constants of Eq. (1) are shown in Table 4. The crossover and

mutation probabilities are 50% and 25%, respectively.

Table 3

Results for the best loading pattern

Quantity Units Optimization 1 Optimization 2

Cycle energy MWD/TU 9341 9340

PPF 1.52 1.53

MLHGR W/cm 365.2 365.3

XMPGR 0.79 0.79

MRNP 1.82 1.83

MCPR 1.59 1.58

MFAB MWD/TU 36,770 36,770

SDMBOC %Dk=k 1.86 1.86

HEX %Dk=k 1.59 1.59

Evaluations 3500 3000

Table 2

Limit values for all the design constraints in the test case

Quantity Units Limit value

PPFmax 1.53

MLHGRmax W/cm 374

XMPGRmax 0.85

MRNPmax 2.0

MCPRmin 1.5

MFABmax MWD/TUa 44,000

SDMmin %Dk=k 1.0

HERmin %Dk=k 1.5

HERmax % Dk=k 2.0

aTU is metric tone of uranium.

Table 4

Weighting factors and escalation factor B

w1 w2 w3 w4 w5 w6 w7 w8 w9 B

5 1000 100 10,000 1000 100 10 7000 1000 10,000


Results in Table 3 show a maximum energy produced of 9341 MWD/TU under

the operational and safety limits. It is important to mention that in each optimiza-

tion execution, several loading patterns were found that satisfy the constraints and

generate cycle energies fairly close to the highest energy. This can be very useful

because it allows Reactor Engineers to choose between several options considering

the history of each fuel assembly in the previous cycles.

Fig. 4 shows the average qualification evolution of the loading patterns’ popu-

lation, for cases LP1 and LP2.

Fig. 4. Objective function, average qualification evolution.


5. Conclusions

Genetic algorithms optimization method has been implemented successfully to the

optimization of BWR fuel reload patterns. The system developed in this work

generates loading patterns which produce more energy than the loading pattern

designed by engineer expertise.

Results for the best loading pattern show a maximum cycle length of 9341 MWD/

TU which does not violate the operational and safety thermal limits, it has a cold

shutdown margin of 1.86% Dk=k at BOC which is greater than the 1.5% Dk=k(minimal) limit value and it has a hot excess reactivity of 1.83% Dk=k at BOC, whichis a good value to operate the reactor during all the cycle.

The cycle energy is only 0.6% of extra energy than the cycle energy of the loading

pattern generated by engineer expertise of 9281 MWD/TU. This improvement is not

significant, however the goal of the system was to have a tool to design good LPs and

this goal was achieved.

To get an optimized loading pattern using LPOS-GA, assuming the heuristic

rules, it takes less than 3500 evaluations of the objective function, which is a very

small quantity, compared with the 7.361� l054 possible permutations in the full core.In order to improve the results, it could be possible to use less tight restrictions to

improve the cycle energy. Other option is to test several sets of fuel assemblies in the

Periphery, CCC and in-core regions searching to improve the cycle energy.

Acknowledgements

This work was performed under the auspices of the Mexican Science and Tech-

nology National Council (CONACyT) under agreement No. 34657-U, and was


partly supported by the National University of Mexico (PAPIIT-UNAM) under

project No. IN109400.

References

B€ack, T., Heistermann, J., Kappler, C., Zamparelli, M., 1996. Evolutionary algorithms support refuelling

of pressurized water reactors. In: Proceedings of the Third IEEE Conference on Evolutionary

Computation, Piscataway, NJ, p. 104.

Castillo, A., Alonso, G., Morales, L., Martin del Campo, C., Francois, J.L., del Valle, E., 2004. BWR fuel

reloads design using a Tabu search technique. Annals of Nuclear Energy 31 (2), 151–161.

De Chaine, M.D., Feltus, M.A., 1995. Nuclear fuel management optimization using Genetic Algorithms.

Nuclear Technology 111, 109.

Franc�ois, J.L., Esquivel, J.L., Cort�es, C., Esquivias, J., Mart�ın del Campo, C., 2001. Validation of a

Methodology for Fuel Management Analysis of Laguna Verde Nuclear Power Plant. Annals of

Nuclear Energy 28 (5), 489–501.

Francois, J.L., Martin del Campo, C., Cortes, C.C., Ramirez, E., Arellano, J., 1999. Development of an

automated system for fuel reload patterns design. Nuclear Engineering and Design 193, 13–21.

Franc�ois, J.L., Lopez, H.A., 1999. SOPRAG: a system for boiling water reactors reload pattern

optimization using Genetic Algorithms. Annals of Nuclear Energy 26 (12), 1053–1063.

Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-

Wesley Publishing Company, Addison-Wesley, Reading, MA.

Haling, R.K., 1965. Operating strategy for maintaining an optimal power distribution throughout life.

Nucleonics 23 (5).

Jagawa, S., Yoshii, T., Fukao, A., 2001. Boiling water reactor loading pattern optimization using simple

linear perturbation and modified tabu search method. Nuclear Science and Engineering 138, 67–77.

Mart�ın-del-Campo, C., Franc�ois, J.L., Lopez, H.A., 2001. AXIAL: a system for boiling water reactor fuel

assembly axial optimization using Genetic Algorithms. Annals of Nuclear Energy 28/16, 1667–1682.

Moore, B.R., Turinsky, P.J., Karve, A.A., 1999. FORMOSA-B A boiling water reactor in-core fuel

management optimization package. Nuclear Technology 126, 153–169.

Poon, P.W., Parks, G.T., 1993. Application of Genetic Algorithms to in-core fuel management

optimization. In: Proceedings of the Topical Mtg Math Methods and Supercomputing in Nuclear

Applications, vol. 2, Karlsruhe, Germany, p. 777.

Scandpower, 1993. User Manual CM-PRESTO-91.

Yamamoto, 1996. Loading pattern optimization using Genetic Algorithms. In: Proceedings of the

International Conference on Physics of Reactors PHYSOR 96, vol. 3, Mito, Japan, pp. 1–48.

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