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Annals of Nuclear Energy 31 (2004) 151–161 www.elsevier.com/locate/anucene

BWR fuel reloads design using a Tabu search technique Alejandro Castilloa,1, Gustavo Alonsoa,*, Luis B. Moralesb, Cecilia Martı´n del Campob, J.L. Franc¸oisb, Edmundo del Vallec a

Instituto Nacional de Investigaciones Nucleares, Km 36.5 Carretera Me´xico-Toluca, Ocoyoacac 52045, Edo. de Me´xico, Mexico b Universidad Nacional Auto´noma de Me´xico, Apartado Postal 70-221, Me´xico, D.F. 04510, Mexico c Instituto Polite´cnico Nacional, Escuela Superior de Fı´sica y Matema´ticas, Unidad Profesional ‘‘Adolfo Lo´pez Mateos’’, ESFM, Me´xico, D. F., 07738, Mexico Received 29 May 2003; accepted 29 June 2003

Abstract We have developed a system to design optimized boiling water reactor fuel reloads. This system is based on the Tabu Search technique along with the heuristic rules of Control Cell Core and Low Leakage. These heuristic rules are a common practice in fuel management to maximize fuel assembly utilization and minimize core vessel damage, respectively. The system uses the 3-D simulator code CM-PRESTO and it has as objective function to maximize the cycle length while satisfying the operational thermal limits and cold shutdown constraints. In the system tabu search ideas such as random dynamic tabu tenure, and frequency-based memory are used. To test this system an optimized boiling water reactor cycle was designed and compared against an actual operating cycle. Numerical experiments show an improved energy cycle compared with the loading patterns generated by engineer expertise and genetic algorithms. # 2003 Elsevier Ltd. All rights reserved.

* Corresponding author. Tel.: +52-55-53297233; fax: +52-55-53297340. E-mail addresses: [email protected] (A. Castillo), [email protected] (G. Alonso), lbm@ servidor.unam.mx (L.B. Morales), [email protected] (E. del Valle). 1 Also Ph. D. student at Universidad Autonoma del Estado de Mexico. 0306-4549/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0306-4549(03)00214-7

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1. Introduction From an economical point of view for a Boiling Water Reactor (BWR) it is necessary to get as much fuel energy as we can to avoid under burnt fuel and waste energy. Thus, the fuel loading pattern plays a very important role to achieve that goal. Design of BWR fuel reloads is usually based in engineer expertise, which is a technique that uses human knowledge. This technique does not optimize the use of the fuel assemblies having in some cases under burnt fuel when it is discharged of the core. To avoid this waste of energy a better design of fuel reloads can be done through optimization techniques. BWR fuel assembly reloads design can be considered a combinatorial problem, which has been tackled using genetic algorithms (Franc¸ois and Lopez, 1999), simulated annealing (Moore et al., 1999) and recently tabu search (Jagawa et al., 2001). All of these techniques have as objective to maximize the cycle length while satisfying the operational thermal limits and cold shutdown constraints. For the last technique, Jagawa et al. (2001) designed an automatic system that uses a tabu search method along with a simple linear perturbation method to avoid the extensive use of the 3-D simulator. A BWR presents strong three-dimensional material heterogeneities such as fuel enrichment, burnable poison, coolant void and control rods, besides the number of fuel assemblies embedded in the core in comparison with a PWR. These characteristics makes the loading pattern optimization problem very complex and it appeals for the use of a licensed 3-D core simulator to achieve the goal proposed in comparison with the use of 2-D simulators used for the PWR optimization problem. We develop a system named optimization tabu search system (OTSS) based on the tabu search (TS) optimization technique, using the 3-D simulator code CM-PRESTO to evaluate the objective function. Our TS uses a random tenure and long-term memory whose purpose is to diversify the search of the optimal value making the process more eﬃcient leading to explore more scenarios in less time than the original tabu search. On the other hand, using this technique it is not necessary to give an initial loading pattern; the system generates a random loading pattern automatically in contrast with the TS proposed by Jagawa et al. (2001), which starts from a reference loading pattern. Furthermore, to follow the strategies used in many BWR plants, two heuristic rules will be applied along the TS technique. These are the Control Cell Core (CCC) and Low Leakage (LL) techniques. The ﬁrst one does not allow the use of fresh fuel in Control Rod (CR) positions and the former does not allow also the use of fresh fuel assemblies in the periphery to avoid damage to the core vessel.

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

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arrangement of 444 positions, which means 444! fuel assembly permutations. Thus, the ‘‘best’’ assembly distribution is that which provides as much energy of the cycle as can be possible without violating the operational and safety limits and have enough shutdown margin to not jeopardize the integrity of the core. As a ﬁrst sight this is a very complex problem which will require enormous computer resources. To reduce the complexity of the problem, one octant symmetry can be assumed, then there will be only 60 diﬀerent positions to allocate the fuel assemblies which represent 8.321081 permutations or possible movements. Furthermore, if we introduce the low leakage which means that only once and twice cycles burnt fuel assemblies can be used in the periphery (LL) and the control cell core rule which means that we can not use fresh fuel in control cell positions, the optimization problem is reduced to 7.3611054 diﬀerent permutations instead of the 444! from the original problem. Given the symmetry of the problem the fuel assemblies in the diagonal can be exchanged only among them as long as they do not violate any heuristic rule. The main goal in this work is to obtain a maximized energy without violate the thermal operational limits and the cold shutdown constraints. It can be achieved through the implementation of the tabu search technique using the following objective function in terms of maximum possible energy value in the cycle (Energy), Mean Ratio of Nominal Power (MRNP), Radial Power Peaking Factor (RPPF), Linear Heat Generation Rate (XLHGR), Maximum Power Generation Rate (XMPGR), Minimal Critical Power Ratio (XMCPR), and Shutdown Margin (SDM): f ¼ Energy þ w1 DMRNP þ w2 DRPPF þ w3 DXLHGR þ w4 DXMPGR þ w5 DXMCPR þ w6 DSDM where: Energy=cycle mean core burnup MRNP=MRNPmaxMRNPc RPPF=RPPFmaxRPPFc XLHGR=XLHGRmaxXLHGRc XMPGR=XMPGRmaxXMPGRc XMCPR=XMCPRcXMCPRmin SDM=SDMcSDMmin w1,. . .,w6 are called weighting factors and wi > 0, i=1,. . .,6 According to the ’s deﬁnition they will be negative if they are violating the safety limits imposed in such case the corresponding weighting factors will be the ones given by the user in other case they would be zero not penalizing the objective function. If all constraints are achieved then the objective function will be the energy produced by the core analyzed.

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3. Tabu search technique The tabu search method is an iterative heuristic method used for ﬁnding, in a set X of feasible solutions, the solution that minimizes an objective function f based on neighborhood search (NS). In a neighborhood search, each feasible solution x has an associated set of neighbors, NðxÞ 2 X, called the neighborhood of x. NS starts from an initial feasible solution chosen randomly and explores the space X by moving from one solution to another in its neighborhood. At each iteration of the process, a subset V of N(x) is generated and we move from the current solution x to the best one x* in V, whether or not f(x*) is better than f(x). If N(x) is not large, it is possible to take V as the entire neighborhood. The method of examining the entire neighborhood becomes very expensive as the problem size increases or its elements are expensive to evaluate. Thus, to reduce the sampling size of V one takes the ﬁrst move that improves the current solution; however, if there is no move that improves the current solution, then one has to examine all neighbors in V. Nevertheless, the main shortcoming of NS algorithm is a cycling problem. Stopping rules must also be deﬁned; in many cases a lower bound f * of the objective function is known in advance. As soon as we have reached this bound, we may interrupt the algorithm. In general, f * is not available with suﬃcient accuracy, as it is the case of study; thus, the stop criterion is met whenever a ﬁxed maximum number of iterations is reached, or if a given maximum number of iterations have been performed without improving the best solution obtained so far. The tabu search algorithm oﬀers another interesting possibility for overcoming the above-mentioned obstacle of the NS technique. To prevent cycling, any move that reinstates certain attributes of solutions recently visited is forbidden. This is accomplished in a short-term memory framework by storing the forbidden (tabu) move in a tabu list. A move remains tabu during a certain period (or tabu tenure) to help aggressive search for better solutions. The tabu tenure may be ﬁxed or variable. In many TS implementations the short-term memory is complemented with a longterm memory, whose purpose is to diversify the search and to move unvisited regions of the solution space; its function is usually based on the frequency criterion. Unfortunately, the tabu list may forbid certain interesting moves, such as those that lead to a better solution than the best one found so far. An aspiration criterion is introduced to cancel the tabu status of a move when this move is judged useful. Note that neighborhood search is a tabu search method without an aspiration function and where the length of the tabu list is zero. 4. Adaptation of Tabu search In order to apply a tabu search algorithm to a combinatorial optimization problem one has to deﬁne the following elements: (a) the representation of a feasible solution (b) the way to generate a starting solution

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(c) the moves (exchanges allowed without violating the restrictions imposed) (d) the form of the objective function and the method to calculate its values (e) the structure of the tabu list. As it was mentioned, TS operates on a space of feasible solution and thus for our problem a feasible solution will be an octant of a reactor core and it is represented by an array of 60 positions to allocate the fuel assemblies (see Fig. 1). Furthermore, to follow the strategies used in many BWR plants, two heuristic rules will be applied along the TS technique. These are the CCC and LL techniques. The ﬁrst one does not allow the use of fresh fuel in Control Rod (CR) positions and the former does not allow also the use of fresh fuel assemblies in the periphery to avoid damage to the core vessel. The process starts knowing the characteristics of the fuel assemblies that will compose the core. Then an initial loading pattern is randomly generated taking into account the low leakage and control cell core rules. In general, TS starts from the hypothesis that it is possible to build up a neighborhood along the iterative search process. In our problem, a neighbor of a feasible solution is obtained from this solution by exchanging (without infringe any heuristic rules) two diﬀerent fuel assemblies settled in a 1/8-symmetry reactor core. Given the symmetry of the problem the fuel assemblies in the diagonal can be exchanged only among them as long as they do not violate any heuristic rule. Thus a move exchanges two assemblies in the 1/8-symmetry reactor core and is determined by the two positions, p1 and p2 having assemblies a1 and a2, respectively. For each current solution x, considering one octant reactor core and taking into account the heuristic rules, there are 723 neighbors for each feasible solution. This number is calculated using a simple combinatorial counting. The set N(x) is considerably large and moreover, its elements are expensive to evaluate. Thus, only subset V of N(x) of size 0.1|N(x)| is randomly generated, and the move is made from x the ﬁrst solution in V

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

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that improves the objective function. However, if there no solution that improves x, then one must to examine all neighbors in V. As it is set in Section 2, the objective function includes the thermal safety limits and the shutdown margin. Tabu search technique was implemented along with the low leakage and control cell core rules in the 3-D reactor core simulator CM-PRESTO-B (Scandpower, 1995) to compose the Optimization Tabu Search System. This system is a FORTRAN-77 based program implemented in an Alpha computer with UNIX operating system. Due to the way that the objective function is constructed, the system requires four diﬀerent runs of CM-PRESTO-B; ﬁrst one (a Haling calculation) is to calculate the energy and the six parameters associated with the operational and safety limits, the other three runs are used to calculate the cold shutdown margin. Thus, those four runs compose a full evaluation of the objective function. It is important to point out that one Haling calculation takes around 1 tenth of the time that will take to do each one of the cold shutdown margin calculations. However for many solutions it is not necessary to have a full evaluation of the objective function. OTSS makes a partial evaluation of the objective function to assess the energy and safety thermal limits and if they are violated then the shutdown margin runs are not performed. This reduces the calculation time because the objective function will not be acceptable anyway. Our tabu list is implemented as a tabu time, which records the earliest iteration that a move is removed from the list. The number of iterations tabu_tenure that a move or exchange will keep its tabu status is randomly selected in the range from 6 to 14 (Glover, 1989). This random selection provides a more versatile search process. The tabu time is represented by the array tabu_time(p,a) where p is a position and a is the assembly type allocated in p. The attributes of a swap move are stored in the vector m=(p1,p2,a1,a2). Then, the tabu_time is updated as follows in order to impose a tabu on the move m for tabu_tenure iterations: tabu timeðp1 ; a2 Þ ¼ tabu timeðp2 ; a1 Þ ¼ iter þ tabu tenure where iter is the current iteration number. Thus, the swap move m=(p1,p2,a1,a2) is tabu if both tabu_time(p1,a2) and tabu_time(p2,a1) are greater or equal to the current iteration number. Note that the tabu_tenure value used in our case is a random number between 5 and 10, this range comes from a trial and error test. Our tabu time forbids, during tabu_tenure iterations, any replacement of a2 and a1 in the positions p1 and p2, respectively. The long term memory is a function that records moves taken in the past in order to penalize those which are non-improving. The goal is to diversify the search by compelling regions to be visited that possibly were not explored before (Glover, 1989). In our particular TS implementation, the long-term memory is a vector which will be denoted F. The vector has zeroes at the beginning of the procedure. When a pair of assemblies (a,b) are swapped at a given iteration, the vector F changes as follows: Fa=Fa+3 and Fb=Fb+3. The entry Fa is the frequency at which the

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

5. Test problem and results OTSS was applied to the design of a fuel reload, taking as a base of comparison an actual operating cycle. This one is an 14-month cycle with 9281 MWD/TU of energy produced, which used 112 fresh fuel assemblies of 3.53 w/o of U-235, this loading pattern was generated using engineer expertise. The safety operational limits at the end of the cycle (Haling calculation) imposed on this calculation are given in Table 1. On the other hand the cold shutdown margin will be assessed at the beginning of the cycle and it needs to be more than 1% k/k to not jeopardize the integrity of the reactor core. Due to the random nature of the process here considered we performed several times the search of an optimized loading pattern and the best results that we found are shown in Table 2, beside Table 3 shows the thermal limits obtained. The results from Table 2 show a maximum energy produced of 9970 MWD/TU and the Table 1 Operational and safety limits Mean ratio of nominal power Radial power peaking factor Linear heat generation rate Maximum power generation rate Minimal critical power ratio Shutdown margin

MRNP RPPF XLHGR XMPGR XMCPR SDM

1.83 1.51 370 0.85 1.5 1.0

Maximum value Maximum value Maximum value Maximum value Minimum value Minimum value

Table 2 Optimized fuel reload results Tabu search run

SDM

Energy

Evaluations of the objective function

Iterations

CPU (s)

1 2 3 4 5 Average Standard deviation

1.000698 1.042602 1.011673 1.013669 1.001696

9970.544 9916.886 9809.330 9899.255 9851.790 9889.561 61.770

4994 4734 4408 5495 4071 4740 546

193 199 186 207 177 192 12

26434.7 28540.2 27608.8 27894.5 29106.9 27917.0 1012.1

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Table 3 Thermal limits from optimized fuel reloads Tabu Search Run

MRNP

RPPF

XLHGR

XMPGR

XMCPR

1 2 3 4 5

1.826 1.829 1.819 1.822 1.829

1.509 1.509 1.508 1.499 1.505

365.911 366.443 364.173 365.043 366.468

0.794 0.795 0.787 0.791 0.793

1.599 1.603 1.601 1.613 1.604

operational and safety limits along with the cold shutdown margin are achieved, this energy is about a 7.4% of extra energy than the loading pattern generated by engineer expertise. Furthermore, in each TS there is a CM-PRESTO-B runs saving of about 30% because the cold shutdown margin is not evaluated when the energy produced by the new pattern is less than the maximum objective function of the neighborhood. To get an optimized loading pattern using OTSS, assuming octant symmetry, it takes less than 7700 evaluations of the objective function, which is a very small quantity, compared with the 7.3611054 permutations. Fig. 2 shows the ﬂowchart of this optimization process. Figs. 3 and 4 show the behavior of the objective function and energy, respectively for some of the several searches of the optimized loading patterns generated using OTSS. Objective function plot, from Fig. 3, shows energy values penalized due to the violation of shutdown margin and or safety and or operational limits. Energy plot, from Fig. 4, shows energy produced although exists in some cases violation of shutdown margin and or safety and or operational limits. At the end of the iteration process, the objective function has the same value to the energy produced due to that all the constraints are satisﬁed. Furthermore, it can be seen that the objective function plot is not smooth at the end. It happens due to the way that the TS works trying to escape from local minima making the process robust. On the other hand, Francois and Lopez (1999) analyzed the same actual cycle considered in this work using genetic algorithms, but their objective function does not take into account the cold shutdown margin as a constraint, which can lead to not realistic results. However, our best result shows a 9970 MWD/TU energy produced against his best result, which shows a 9892 MWD/TU energy produced.

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

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Fig. 2. OTSS ﬂow chart.

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limits and has a cold shutdown margin of 1.000698% k/k, which is greater than the 1% k/k (minimal) limit value. This energy is 7.04% greater than the one produced of 9281 MWD/TU by the actual operating cycle. From a practical point of view it can be seen from Table 2 results that it will be enough to perform ﬁve TS to assure that an optimized loading pattern is obtained. On the other hand it takes less than 7700 evaluations per TS of the objective function, which is a very small number, compared with the 7.3611054 permutations that can take place in one octant symmetry.

Fig. 3. Objective function behavior under a TS.

Fig. 4. Energy behavior under a TS.

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

Acknowledgements The authors acknowledge the support given by CONACyT through the research project 33806-U.

References 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, 1053–1063. Glover, F., 1989. Tabu Search part I. ORSA Journal of Computing 1, 190–206. 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. 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. Scandpower, 1995. User Manual CM-PRESTO-B/91.

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BWR fuel reloads design using a Tabu search technique Alejandro Castilloa,1, Gustavo Alonsoa,*, Luis B. Moralesb, Cecilia Martı´n del Campob, J.L. Franc¸oisb, Edmundo del Vallec a

Instituto Nacional de Investigaciones Nucleares, Km 36.5 Carretera Me´xico-Toluca, Ocoyoacac 52045, Edo. de Me´xico, Mexico b Universidad Nacional Auto´noma de Me´xico, Apartado Postal 70-221, Me´xico, D.F. 04510, Mexico c Instituto Polite´cnico Nacional, Escuela Superior de Fı´sica y Matema´ticas, Unidad Profesional ‘‘Adolfo Lo´pez Mateos’’, ESFM, Me´xico, D. F., 07738, Mexico Received 29 May 2003; accepted 29 June 2003

Abstract We have developed a system to design optimized boiling water reactor fuel reloads. This system is based on the Tabu Search technique along with the heuristic rules of Control Cell Core and Low Leakage. These heuristic rules are a common practice in fuel management to maximize fuel assembly utilization and minimize core vessel damage, respectively. The system uses the 3-D simulator code CM-PRESTO and it has as objective function to maximize the cycle length while satisfying the operational thermal limits and cold shutdown constraints. In the system tabu search ideas such as random dynamic tabu tenure, and frequency-based memory are used. To test this system an optimized boiling water reactor cycle was designed and compared against an actual operating cycle. Numerical experiments show an improved energy cycle compared with the loading patterns generated by engineer expertise and genetic algorithms. # 2003 Elsevier Ltd. All rights reserved.

* Corresponding author. Tel.: +52-55-53297233; fax: +52-55-53297340. E-mail addresses: [email protected] (A. Castillo), [email protected] (G. Alonso), lbm@ servidor.unam.mx (L.B. Morales), [email protected] (E. del Valle). 1 Also Ph. D. student at Universidad Autonoma del Estado de Mexico. 0306-4549/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0306-4549(03)00214-7

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1. Introduction From an economical point of view for a Boiling Water Reactor (BWR) it is necessary to get as much fuel energy as we can to avoid under burnt fuel and waste energy. Thus, the fuel loading pattern plays a very important role to achieve that goal. Design of BWR fuel reloads is usually based in engineer expertise, which is a technique that uses human knowledge. This technique does not optimize the use of the fuel assemblies having in some cases under burnt fuel when it is discharged of the core. To avoid this waste of energy a better design of fuel reloads can be done through optimization techniques. BWR fuel assembly reloads design can be considered a combinatorial problem, which has been tackled using genetic algorithms (Franc¸ois and Lopez, 1999), simulated annealing (Moore et al., 1999) and recently tabu search (Jagawa et al., 2001). All of these techniques have as objective to maximize the cycle length while satisfying the operational thermal limits and cold shutdown constraints. For the last technique, Jagawa et al. (2001) designed an automatic system that uses a tabu search method along with a simple linear perturbation method to avoid the extensive use of the 3-D simulator. A BWR presents strong three-dimensional material heterogeneities such as fuel enrichment, burnable poison, coolant void and control rods, besides the number of fuel assemblies embedded in the core in comparison with a PWR. These characteristics makes the loading pattern optimization problem very complex and it appeals for the use of a licensed 3-D core simulator to achieve the goal proposed in comparison with the use of 2-D simulators used for the PWR optimization problem. We develop a system named optimization tabu search system (OTSS) based on the tabu search (TS) optimization technique, using the 3-D simulator code CM-PRESTO to evaluate the objective function. Our TS uses a random tenure and long-term memory whose purpose is to diversify the search of the optimal value making the process more eﬃcient leading to explore more scenarios in less time than the original tabu search. On the other hand, using this technique it is not necessary to give an initial loading pattern; the system generates a random loading pattern automatically in contrast with the TS proposed by Jagawa et al. (2001), which starts from a reference loading pattern. Furthermore, to follow the strategies used in many BWR plants, two heuristic rules will be applied along the TS technique. These are the Control Cell Core (CCC) and Low Leakage (LL) techniques. The ﬁrst one does not allow the use of fresh fuel in Control Rod (CR) positions and the former does not allow also the use of fresh fuel assemblies in the periphery to avoid damage to the core vessel.

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

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arrangement of 444 positions, which means 444! fuel assembly permutations. Thus, the ‘‘best’’ assembly distribution is that which provides as much energy of the cycle as can be possible without violating the operational and safety limits and have enough shutdown margin to not jeopardize the integrity of the core. As a ﬁrst sight this is a very complex problem which will require enormous computer resources. To reduce the complexity of the problem, one octant symmetry can be assumed, then there will be only 60 diﬀerent positions to allocate the fuel assemblies which represent 8.321081 permutations or possible movements. Furthermore, if we introduce the low leakage which means that only once and twice cycles burnt fuel assemblies can be used in the periphery (LL) and the control cell core rule which means that we can not use fresh fuel in control cell positions, the optimization problem is reduced to 7.3611054 diﬀerent permutations instead of the 444! from the original problem. Given the symmetry of the problem the fuel assemblies in the diagonal can be exchanged only among them as long as they do not violate any heuristic rule. The main goal in this work is to obtain a maximized energy without violate the thermal operational limits and the cold shutdown constraints. It can be achieved through the implementation of the tabu search technique using the following objective function in terms of maximum possible energy value in the cycle (Energy), Mean Ratio of Nominal Power (MRNP), Radial Power Peaking Factor (RPPF), Linear Heat Generation Rate (XLHGR), Maximum Power Generation Rate (XMPGR), Minimal Critical Power Ratio (XMCPR), and Shutdown Margin (SDM): f ¼ Energy þ w1 DMRNP þ w2 DRPPF þ w3 DXLHGR þ w4 DXMPGR þ w5 DXMCPR þ w6 DSDM where: Energy=cycle mean core burnup MRNP=MRNPmaxMRNPc RPPF=RPPFmaxRPPFc XLHGR=XLHGRmaxXLHGRc XMPGR=XMPGRmaxXMPGRc XMCPR=XMCPRcXMCPRmin SDM=SDMcSDMmin w1,. . .,w6 are called weighting factors and wi > 0, i=1,. . .,6 According to the ’s deﬁnition they will be negative if they are violating the safety limits imposed in such case the corresponding weighting factors will be the ones given by the user in other case they would be zero not penalizing the objective function. If all constraints are achieved then the objective function will be the energy produced by the core analyzed.

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3. Tabu search technique The tabu search method is an iterative heuristic method used for ﬁnding, in a set X of feasible solutions, the solution that minimizes an objective function f based on neighborhood search (NS). In a neighborhood search, each feasible solution x has an associated set of neighbors, NðxÞ 2 X, called the neighborhood of x. NS starts from an initial feasible solution chosen randomly and explores the space X by moving from one solution to another in its neighborhood. At each iteration of the process, a subset V of N(x) is generated and we move from the current solution x to the best one x* in V, whether or not f(x*) is better than f(x). If N(x) is not large, it is possible to take V as the entire neighborhood. The method of examining the entire neighborhood becomes very expensive as the problem size increases or its elements are expensive to evaluate. Thus, to reduce the sampling size of V one takes the ﬁrst move that improves the current solution; however, if there is no move that improves the current solution, then one has to examine all neighbors in V. Nevertheless, the main shortcoming of NS algorithm is a cycling problem. Stopping rules must also be deﬁned; in many cases a lower bound f * of the objective function is known in advance. As soon as we have reached this bound, we may interrupt the algorithm. In general, f * is not available with suﬃcient accuracy, as it is the case of study; thus, the stop criterion is met whenever a ﬁxed maximum number of iterations is reached, or if a given maximum number of iterations have been performed without improving the best solution obtained so far. The tabu search algorithm oﬀers another interesting possibility for overcoming the above-mentioned obstacle of the NS technique. To prevent cycling, any move that reinstates certain attributes of solutions recently visited is forbidden. This is accomplished in a short-term memory framework by storing the forbidden (tabu) move in a tabu list. A move remains tabu during a certain period (or tabu tenure) to help aggressive search for better solutions. The tabu tenure may be ﬁxed or variable. In many TS implementations the short-term memory is complemented with a longterm memory, whose purpose is to diversify the search and to move unvisited regions of the solution space; its function is usually based on the frequency criterion. Unfortunately, the tabu list may forbid certain interesting moves, such as those that lead to a better solution than the best one found so far. An aspiration criterion is introduced to cancel the tabu status of a move when this move is judged useful. Note that neighborhood search is a tabu search method without an aspiration function and where the length of the tabu list is zero. 4. Adaptation of Tabu search In order to apply a tabu search algorithm to a combinatorial optimization problem one has to deﬁne the following elements: (a) the representation of a feasible solution (b) the way to generate a starting solution

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(c) the moves (exchanges allowed without violating the restrictions imposed) (d) the form of the objective function and the method to calculate its values (e) the structure of the tabu list. As it was mentioned, TS operates on a space of feasible solution and thus for our problem a feasible solution will be an octant of a reactor core and it is represented by an array of 60 positions to allocate the fuel assemblies (see Fig. 1). Furthermore, to follow the strategies used in many BWR plants, two heuristic rules will be applied along the TS technique. These are the CCC and LL techniques. The ﬁrst one does not allow the use of fresh fuel in Control Rod (CR) positions and the former does not allow also the use of fresh fuel assemblies in the periphery to avoid damage to the core vessel. The process starts knowing the characteristics of the fuel assemblies that will compose the core. Then an initial loading pattern is randomly generated taking into account the low leakage and control cell core rules. In general, TS starts from the hypothesis that it is possible to build up a neighborhood along the iterative search process. In our problem, a neighbor of a feasible solution is obtained from this solution by exchanging (without infringe any heuristic rules) two diﬀerent fuel assemblies settled in a 1/8-symmetry reactor core. Given the symmetry of the problem the fuel assemblies in the diagonal can be exchanged only among them as long as they do not violate any heuristic rule. Thus a move exchanges two assemblies in the 1/8-symmetry reactor core and is determined by the two positions, p1 and p2 having assemblies a1 and a2, respectively. For each current solution x, considering one octant reactor core and taking into account the heuristic rules, there are 723 neighbors for each feasible solution. This number is calculated using a simple combinatorial counting. The set N(x) is considerably large and moreover, its elements are expensive to evaluate. Thus, only subset V of N(x) of size 0.1|N(x)| is randomly generated, and the move is made from x the ﬁrst solution in V

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

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that improves the objective function. However, if there no solution that improves x, then one must to examine all neighbors in V. As it is set in Section 2, the objective function includes the thermal safety limits and the shutdown margin. Tabu search technique was implemented along with the low leakage and control cell core rules in the 3-D reactor core simulator CM-PRESTO-B (Scandpower, 1995) to compose the Optimization Tabu Search System. This system is a FORTRAN-77 based program implemented in an Alpha computer with UNIX operating system. Due to the way that the objective function is constructed, the system requires four diﬀerent runs of CM-PRESTO-B; ﬁrst one (a Haling calculation) is to calculate the energy and the six parameters associated with the operational and safety limits, the other three runs are used to calculate the cold shutdown margin. Thus, those four runs compose a full evaluation of the objective function. It is important to point out that one Haling calculation takes around 1 tenth of the time that will take to do each one of the cold shutdown margin calculations. However for many solutions it is not necessary to have a full evaluation of the objective function. OTSS makes a partial evaluation of the objective function to assess the energy and safety thermal limits and if they are violated then the shutdown margin runs are not performed. This reduces the calculation time because the objective function will not be acceptable anyway. Our tabu list is implemented as a tabu time, which records the earliest iteration that a move is removed from the list. The number of iterations tabu_tenure that a move or exchange will keep its tabu status is randomly selected in the range from 6 to 14 (Glover, 1989). This random selection provides a more versatile search process. The tabu time is represented by the array tabu_time(p,a) where p is a position and a is the assembly type allocated in p. The attributes of a swap move are stored in the vector m=(p1,p2,a1,a2). Then, the tabu_time is updated as follows in order to impose a tabu on the move m for tabu_tenure iterations: tabu timeðp1 ; a2 Þ ¼ tabu timeðp2 ; a1 Þ ¼ iter þ tabu tenure where iter is the current iteration number. Thus, the swap move m=(p1,p2,a1,a2) is tabu if both tabu_time(p1,a2) and tabu_time(p2,a1) are greater or equal to the current iteration number. Note that the tabu_tenure value used in our case is a random number between 5 and 10, this range comes from a trial and error test. Our tabu time forbids, during tabu_tenure iterations, any replacement of a2 and a1 in the positions p1 and p2, respectively. The long term memory is a function that records moves taken in the past in order to penalize those which are non-improving. The goal is to diversify the search by compelling regions to be visited that possibly were not explored before (Glover, 1989). In our particular TS implementation, the long-term memory is a vector which will be denoted F. The vector has zeroes at the beginning of the procedure. When a pair of assemblies (a,b) are swapped at a given iteration, the vector F changes as follows: Fa=Fa+3 and Fb=Fb+3. The entry Fa is the frequency at which the

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

5. Test problem and results OTSS was applied to the design of a fuel reload, taking as a base of comparison an actual operating cycle. This one is an 14-month cycle with 9281 MWD/TU of energy produced, which used 112 fresh fuel assemblies of 3.53 w/o of U-235, this loading pattern was generated using engineer expertise. The safety operational limits at the end of the cycle (Haling calculation) imposed on this calculation are given in Table 1. On the other hand the cold shutdown margin will be assessed at the beginning of the cycle and it needs to be more than 1% k/k to not jeopardize the integrity of the reactor core. Due to the random nature of the process here considered we performed several times the search of an optimized loading pattern and the best results that we found are shown in Table 2, beside Table 3 shows the thermal limits obtained. The results from Table 2 show a maximum energy produced of 9970 MWD/TU and the Table 1 Operational and safety limits Mean ratio of nominal power Radial power peaking factor Linear heat generation rate Maximum power generation rate Minimal critical power ratio Shutdown margin

MRNP RPPF XLHGR XMPGR XMCPR SDM

1.83 1.51 370 0.85 1.5 1.0

Maximum value Maximum value Maximum value Maximum value Minimum value Minimum value

Table 2 Optimized fuel reload results Tabu search run

SDM

Energy

Evaluations of the objective function

Iterations

CPU (s)

1 2 3 4 5 Average Standard deviation

1.000698 1.042602 1.011673 1.013669 1.001696

9970.544 9916.886 9809.330 9899.255 9851.790 9889.561 61.770

4994 4734 4408 5495 4071 4740 546

193 199 186 207 177 192 12

26434.7 28540.2 27608.8 27894.5 29106.9 27917.0 1012.1

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Table 3 Thermal limits from optimized fuel reloads Tabu Search Run

MRNP

RPPF

XLHGR

XMPGR

XMCPR

1 2 3 4 5

1.826 1.829 1.819 1.822 1.829

1.509 1.509 1.508 1.499 1.505

365.911 366.443 364.173 365.043 366.468

0.794 0.795 0.787 0.791 0.793

1.599 1.603 1.601 1.613 1.604

operational and safety limits along with the cold shutdown margin are achieved, this energy is about a 7.4% of extra energy than the loading pattern generated by engineer expertise. Furthermore, in each TS there is a CM-PRESTO-B runs saving of about 30% because the cold shutdown margin is not evaluated when the energy produced by the new pattern is less than the maximum objective function of the neighborhood. To get an optimized loading pattern using OTSS, assuming octant symmetry, it takes less than 7700 evaluations of the objective function, which is a very small quantity, compared with the 7.3611054 permutations. Fig. 2 shows the ﬂowchart of this optimization process. Figs. 3 and 4 show the behavior of the objective function and energy, respectively for some of the several searches of the optimized loading patterns generated using OTSS. Objective function plot, from Fig. 3, shows energy values penalized due to the violation of shutdown margin and or safety and or operational limits. Energy plot, from Fig. 4, shows energy produced although exists in some cases violation of shutdown margin and or safety and or operational limits. At the end of the iteration process, the objective function has the same value to the energy produced due to that all the constraints are satisﬁed. Furthermore, it can be seen that the objective function plot is not smooth at the end. It happens due to the way that the TS works trying to escape from local minima making the process robust. On the other hand, Francois and Lopez (1999) analyzed the same actual cycle considered in this work using genetic algorithms, but their objective function does not take into account the cold shutdown margin as a constraint, which can lead to not realistic results. However, our best result shows a 9970 MWD/TU energy produced against his best result, which shows a 9892 MWD/TU energy produced.

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

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Fig. 2. OTSS ﬂow chart.

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limits and has a cold shutdown margin of 1.000698% k/k, which is greater than the 1% k/k (minimal) limit value. This energy is 7.04% greater than the one produced of 9281 MWD/TU by the actual operating cycle. From a practical point of view it can be seen from Table 2 results that it will be enough to perform ﬁve TS to assure that an optimized loading pattern is obtained. On the other hand it takes less than 7700 evaluations per TS of the objective function, which is a very small number, compared with the 7.3611054 permutations that can take place in one octant symmetry.

Fig. 3. Objective function behavior under a TS.

Fig. 4. Energy behavior under a TS.

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

Acknowledgements The authors acknowledge the support given by CONACyT through the research project 33806-U.

References 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, 1053–1063. Glover, F., 1989. Tabu Search part I. ORSA Journal of Computing 1, 190–206. 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. 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. Scandpower, 1995. User Manual CM-PRESTO-B/91.

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