Fuel loading and control rod patterns optimization in a BWR using tabu search

annals of

NUCLEAR ENERGY Annals of Nuclear Energy 34 (2007) 207–212 www.elsevier.com/locate/anucene

Fuel loading and control rod patterns optimization in a BWR using tabu search Alejandro Castillo *, Juan Jose´ Ortiz, Jose´ Luis Montes 1, Rau´l Perusquı´a Instituto Nacional de Investigaciones Nucleares, Km. 36.5 Carretera Me´xico-Toluca, Ocoyoacac 52750, Estado de Me´xico, Mexico Received 19 May 2006; received in revised form 11 December 2006; accepted 18 December 2006 Available online 15 February 2007

Abstract This paper presents the QuinalliBT system, a new approach to solve fuel loading and control rod patterns optimization problem in a coupled way. This system involves three different optimization stages; in the first one, a seed fuel loading using the Haling principle is designed. In the second stage, the corresponding control rod pattern for the previous fuel loading is obtained. Finally, in the last stage, a new fuel loading is created, starting from the previous fuel loading and using the corresponding set of optimized control rod patterns. For each stage, a different objective function is considered. In order to obtain the decision parameters used in those functions, the CMPRESTO 3D steady-state reactor core simulator was used. Second and third stages are repeated until an appropriate fuel loading and its control rod pattern are obtained, or a stop criterion is achieved. In all stages, the tabu search optimization technique was used. The QuinalliBT system was tested and applied to a real BWR operation cycle. It was found that the value for keff obtained by QuinalliBT was 0.0024 Dk/k greater than that of the reference cycle.  2007 Elsevier Ltd. All rights reserved.

1. Introduction The operation cycle design for a boiling water reactor (BWR) has several optimization stages. One is the fuel loading (FL) design, which implies to place an inventory of fuel assemblies into the core, in order to maximize the cycle length, while thermal limits, hot excess reactivity and cold shutdown margin are satisfied. Other optimization stage is related to the reactivity control during the operation cycle. The goal in this part of the cycle design is to predict the full power control rod pattern (CRP). In this stage, the control rod axial locations during the whole operation cycle are considered. To obtain these CRPs it is * Corresponding author. Address: Instituto Nacional de Investigaciones Nucleares, Km. 36.5 Carretera Me´xico-Toluca, Ocoyoacac 52045, Estado de Me´xico, Mexico. Tel.: +52 55 53297200; fax: +52 55 53297301. E-mail addresses: [email protected] (A. Castillo), jjortiz@nuclear. inin.mx (J.J. Ortiz), [email protected] (J.L. Montes), mrpc@nuclear. inin.mx (R. Perusquı´a). 1 Also PhD student at the Facultad de Ciencias of the Universidad Auto´noma del Estado de Me´xico, Mexico.

0306-4549/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2006.12.006

necessary to take into account several constraints, including some of those mentioned above. FL and CRP problems have been commonly solved in an independent way by using several optimization techniques, such as neural networks, ant colonies, genetic algorithms, tabu search and fuzzy logic (Ortiz and Requena, 2004, 2006, 2004; Castillo et al., 2004, 2005; Franc¸ois et al., 2004), among others. In some cases, Haling’s principle (Haling, 1964) has been used to obtain FL design. The Haling’s principle guarantees a safe operation of the reactor but it does not maximize cycle length. It is possible to maximize cycle length using an adequate operation strategy. A good option is operating the reactor by using ‘‘Spectral Shift’’ strategy (Specker et al., 1978), which permits ‘‘to breed’’ fissile material and allows a longer cycle length. Spectral Shift operation can be favored if control rod positions (CRP design) are such, as to produce a peaked axial power distribution at the bottom of the core. However, it is important to keep in mind that CRP design depends on FL design. In other words, if we do not have a good FL design, then we will not obtain a good CRP design. Therefore, an

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approximation with greater scope is to consider the solution of FL and CRP problems in a coupled manner. QuinalliBT derives its name from ‘‘QUItze’’ and ‘‘toNALLI’’, two words of two ancient Mexican languages, and ‘‘Bu´squeda Tabu’’ (tabu search) in Spanish. The system solves both FL and CRP designs in a coupled way. In each stage a different multi-objective function is used and then evaluated with CM-PRESTO (Scandpower, 1993) 3D code. The tabu search (TS) heuristic technique is applied in the whole process. The QuinalliBT system was designed using FORTRAN-77 language in an Alpha workstation with UNIX operation system.

2. Problem description Fuel loading design problem may be described as follows: for a given inventory of fuel assemblies, an arrangement of them inside the reactor core has to be found, in such a way that some restrictions will be satisfied during cycle operation. Among these restrictions can be mentioned, for example, that some fuel bundle allocation rules have to be satisfied, as well as thermal limits, hot excess reactivity and cold shutdown margin; also energy extraction has to be maximized. The Haling’s principle has been one of the most used techniques for designing FL. With this operation strategy a safe operation of the reactor can be achieved without problems. However, the obtained energy is far from being the maximum energy of the cycle. In CRP design, it is necessary to establish axial control rod positions for each one of several burnup steps. Because an adequate CRP design allows for axial power distribution to be adjusted to a peak at the bottom of the core, CRP has an important contribution in achieving Spectral Shift operation in the reactor. Another important consideration in this stage of the design is that reactor core has to be critical, and also that thermal limits must be satisfied throughout the cycle. Although cycle length is strongly affected by CRP throughout the cycle, its maximization at this stage is not a goal. This task will be made later by means of fuel loading pattern optimization. As it was mentioned above, these problems have been solved in an independent way, being one of the main reasons that the search space is too large if the problem is considered in a coupled way in which case, the total number of parameters included in the whole process is increased. On the other hand, the computers in the past did not have the capacity to make the necessary calculations in a reasonable time. With present computers this is not a problem anymore. So, the coupled problem has begun to be investigated recently. An approach for solving this new problem was proposed in a paper by Kobayashi and Aiyoshi (2002), by using a method based on Genetic Algorithms. In order to optimize the FL-CRP design, an ‘‘if-then’’ heuristic rule was implemented. As it was mentioned at the beginning of this paper, the Haling’s principle has been used by many authors, for example, Kobayashi et al. In

that paper the authors, pay special attention on the number of fresh assemblies during the iteration process. In a BWR reactor with a total of 444 fuel assemblies and 109 control rods, we can make the following deliberations about the search space size. In the first case, if all fuel assemblies are different, then we have 444! possible solutions to FL design. In the control rods case, if we are considering 25 axial nodes and 10 burnup steps, then we have ((109)25)10 possible solutions to CRP design. We can see that, both FL and CRP designs are combinatorial problems. For this reason, a combinatorial technique can be used to solve these problems. If some heuristic rules and simplifications are considered, the number of possible solutions can be reduced considerably. Then, for the FL search space, it is possible to reduce it from 2.1 · 10984 to 7.3 · 1054 possible solutions, applying the following rules or simplifications: 1. Fuel load design with low leakage strategy. 2. Eighth reactor core symmetry. 3. Control cell core load strategy. On the other hand, for the CRP search space, it is also feasible to reduce the total possible solutions, from 5.683 · 101523 to 7.8 · 1069 (with 10 burnup steps), applying the following heuristic rules or simplifications: 1. Eighth reactor core symmetry. 2. Control cell core load strategy. 3. The intermediate positions of control rods are forbidden. In this work, we take advantage of previously designed systems, which were developed to individually optimize both problems (Castillo et al., 2004, 2005). By using them, it was possible to develop a new system to optimize the coupled FL-CRP problem. In the following section, the new system is described in detail. 3. QuinalliBT system The QuinalliBT system was developed in order to optimize FL and CRP design in a coupled way, in a BWR operating cycle. The system has three stages, each one using a different multi-objective function. In the first stage, an optimized FL is obtained using the Haling’s principle (Haling, 1964) while thermal limits, hot excess reactivity and cold shutdown margin are satisfied. When the Haling’s principle is applied in order to design an operation strategy, power peaks within reactor core are minimized. However, the obtained FL is not an optimized design from the point of view of maximum energy extracted from the fuel, therefore it is used as a ‘‘seed’’ FL. This seed FL is used in the second stage to obtain a set of optimized CRP using ‘‘Spectral Shift’’ strategy (Specker et al., 1978). In the last stage, the CRP design attained in the second stage is used to obtain a new FL design, instead of using the Haling’s

A. Castillo et al. / Annals of Nuclear Energy 34 (2007) 207–212

principle. Thus, second and third stages are repeated in an iterative process, until optimized FL and CRP designs are obtained, or a stop criterion is achieved. Fig. 1 shows the flowchart of the QuinalliBT system. In the following paragraphs, QuinalliBT system stages are described. As mentioned above, the Haling’s principle is used in first stage in order to obtain an initial FL design. The main idea is to maximize keff value at the end of the cycle, while thermal limits, hot excess reactivity and cold shutdown margin are satisfied. The objective function to be maximized is the following: F obj ¼ k eff  w1 þ DLim1  w2 þ DLim2  w3 þ DLim3  w4 þ DLim4  w5 þ DLim5  w6 ð1Þ where keff DLim1 DLim2 DLim3 DLim4 DLim5

effective multiplication factor at end of the cycle = MFLPDlim  MFLPDobtained = MPGRlim  MPGRobtained = MFLCPRlim  MFLCPRobtained = SDMobtained  SDMlim = HERlim  HERobtained

MFLPD maximum fraction of linear power density MPGR maximum power generation ratio MFLCPR maximum fraction of limiting critical power ratio SDM cold shutdown margin at the beginning of cycle HER hot excess reactivity at the beginning of cycle and w1, w2, w3, w4, w5, w6 are non-negative weighting factors obtained from a statistical analysis. When restrictions Limi (i = 1, . . . , 5) are satisfied, their respective wi are equal to zero. It can be seen that if all constraints are satisfied, then Eq. (1) works only to maximize effective multiplication factor keff.

In the next stage, operation cycle is divided into ten burnup steps. In each one of these steps control rod positions must be fixed, taking into account that the reactor must be critical, the axial power shape must match a target axial power profile and that thermal limits must be satisfied. The objective function to minimize is the following: F ¼

25 X

ðP iobj  P iact Þ  w1 þ jk eff  k crit j  w2 þ DLim1  w3

i¼1

þ DLim2  w4 þ DLim3  w5 where Pobj Pact keff kcrit Lim1 Lim2 Lim3

target axial power profile obtained axial power profile obtained effective multiplication factor objective effective multiplication factor = MFLPDlim  MFLPDobtained = MPGRlim  MPGRobtained = MFLCPRlim  MFLCPRobtained

F ¼ k EOR  w1 

NO

stop criterion was satisfied?

Seed Fuel Loading

Control Rod Patterns

Fuel Loading using CRP

Fig. 1. QuinalliBT’s flowchart.

jk ieff  k crit j  w2 þ

n X

Limi2  w4 þ

n X

Limi1  w3

i¼1 n X

Limi3  w5 þ Lim4  w6 þ Lim5  w7

i¼1

ð3Þ

stop

Fuel Loading with Haling’s principle

n1 X i¼1

i¼1

YES

ð2Þ

and wi, i = 1, . . . , 5 are weighting factors. It is desirable to minimize the differences between the target and the obtained axial power profiles. When the constraints Limi (i = 1, . . . , 3) are satisfied, then its respective wi, i 2 {3, . . . , 5} is equal to zero. w2 is zero if jkeff  kcritj < 0.00010. It can be seen that if all of these constraints are satisfied, Eq. (2) works only to minimize the difference between the axial power profiles. In the last stage of the iterative process, the optimized CRP design is used instead of Haling’s principle to obtain a new FL design. In this case, keff at the end of cycle is maximized, while the thermal limits in each burnup step, the cold shutdown margin and the hot excess reactivity at the beginning of cycle must be satisfied. The objective function used in this stage is the following:

þ Input Data

209

where n kEOR k ieff kcrit Limi1 Limi2 Limi3 Lim4 Lim5

number of burnup steps obtained effective multiplication factor at the end of cycle obtained effective multiplication factor in each burnup step objective effective multiplication factor = MFLPDi,lim  MFLPDi,obtained = MPGRi,lim  MPGRi,obtained = MFLCPRi,lim  MFLCPRi,obtained = SDMobtained  SDMlim = HERlim  HERobtained

i denotes one of the burnup steps in which cycle length is divided. In the same way wi, i = 1, . . . , 7 are weighting factors with the same considerations of those for Eqs. (1) and (2).

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As it was mentioned, the second and third stages perform an iterative process until FL-CRP design is obtained, or a stop criterion is achieved. 4. Tabu search In both FL and CRP design, the tabu search (TS) technique was applied. Following is a brief description about this technique. See reference Glover (1968) for more details on these concepts. The TS technique is an iterative process used to obtain a solution that maximizes (or minimizes) an objective function, in a set X of feasible solutions. The TS technique starts from a randomly chosen initial feasible solution. Then, space X is explored by moving in a neighborhood from one solution to another. In this process, each feasible solution x has an associated set of neighbors, called the neighborhood of x, N(x) 2 X. If N(x) is very large, it is possible to choose, at each iteration of the process, a subset SN(x)  N(x) in which case, a search from the current solution x to the best one x* in SN(x), whether or not f(x*) is better than f(x), is carried out. In some problems, evaluation of the entire neighborhood becomes very expensive in terms of time and computing resources. In that case, it is possible to reduce the size of SN taking the first move that improves the current solution. If this is not possible, then it will be necessary to examine the whole neighborhood. During search process, the TS technique allows cycling prevention using a short-term memory, an array with either fixed or variable length. The purpose of this memory is to store certain prohibited movements (the reason why this array is called tabu list). In this sense, a movement remains tabu (forbidden) during n iterations (tabu tenure). It is very common also to apply a long-term memory to diversify neighbors search, being the objective of this memory to move toward unvisited regions. It is important to mention that the TS technique may forbid some interesting moves because of their tabu status. In such case, if a move produces the best solution of the whole process, but it is forbidden, it is possible to cancel its tabu status and take it into account (aspiration criterion). In the present analysis a tabu list with variable length from 7 to 15 is used. These values were obtained after a statistical analysis in the development process of QuinalliBT system. Since function evaluation is a process that takes so much time and computing resources, the neighborhood was not revised in its entirety, neither for FL nor for CRP designs, but only 10% and 40%, respectively. During FL design, four CM-PRESTO simulator executions are required in order to evaluate the objective function. For CRP design only one CM-PRESTO execution is required. In order to penalize the high frequency movements that do not improve the objective function during the iterative process, an aspiration criterion and a long-term memory with a FVEC (Frequency VECtor) array were applied in

the present study. This array is equal to zero at the beginning of the process and updated in the following way: FVEC(i, j) = FVEC(i, j) + 2, if (i, j) move was made. In order to finish the process, it is necessary to define some stop criteria. The first one is to stop the algorithm when a maximum number of iterations is achieved; the second one is to stop the process after k iterations, if the objective function does not improve. 5. Results To test the QuinalliBT system, data of a real operation cycle were used. The chosen FL design (including its CRP’s) corresponds to an 18-month cycle length, with 108 fresh fuel assemblies, a cycle burnup of 11,020 MWD/T and keff = 0.9986. This keff value was obtained from the simulation of the reference cycle with CM-PRESTO simulator. The QuinalliBT system was executed several times, with all executions having similar results. Best results obtained are shown however. Table 1 shows the results obtained for the first stage, according to Haling calculation. It can be seen that thermal limits, hot excess reactivity and cold shutdown margin were satisfied. The corresponding limit values for each parameter are shown in this table. Eighty-four iterations were carried out to obtain these results, and the objective function was evaluated 1996 times using CM-PRESTO simulator. In this stage the effective multiplication factor obtained is higher than the target value for keff. Once seed FL is obtained, QuinalliBT system performs an iterative process, until coupled FL-CRP is optimized. Fig. 2 presents the results obtained after four CRP-FL iterations. It can be seen that through the iterative process, effective multiplication factor decreases at the end of cycle, due to the fact that in the first iterations some thermal limits are violated. In the last iteration, the thermal limits are satisfied and keff values in the intermediate burnup steps are adjusted to the target keff. At the end of cycle, keff value is higher than target keff. In the last CRP optimization case executed, 637 iterations were performed and the objective function was evaluated 10,867 times. The number of CRP iterations decreases when the CRP-FL optimization process is almost finished. In Table 2, keff values, burnup steps and thermal limits are shown. Finally, Fig. 3 shows keff behavior through the CRP-FL iterative process, when FL is obtained. The optimized CRP is taken into account in this stage instead of Haling’s prinTable 1 Results in the Haling’s calculation Parameter

Limit

Obtained value

MFLPD MPGR MFLCPR HER SDM keff

<0.9 <0.9 <0.9 <1.03 >1.5 0.9986

0.7378 0.6980 0.8944 1.024 1.501 1.006

A. Castillo et al. / Annals of Nuclear Energy 34 (2007) 207–212

211

k eff behaviour 1.0100 1.0080 kcrit

1.0060

keff

keff first iteration keff second iteration

1.0040

keff third iteration

1.0020

keff fourth iteration

1.0000 0.9980

0

2000

4000 6000 burnup (MWd/T)

8000

10000

Fig. 2. keff Behavior after four iterations.

Table 2 keff and thermal limits values

Table 3 Parameters obtained in the last iteration

Burnup step

keff

MFLPD

MPGR

MFLCPR

Burnup step

kcrit

keff obtained

0 1037 2093 3052 4081 5032 6065 7096 8157 9279 10,307 11,020

1.0080 1.0090 1.0080 1.0070 1.0060 1.0050 1.0060 1.0040 1.0030 1.0020 1.0010 1.0000

0.8967 0.8758 0.8449 0.7290 0.7651 0.8318 0.8341 0.7667 0.7711 0.7569 0.7567 0.7439

0.7823 0.8774 0.8541 0.7350 0.7778 0.8742 0.8836 0.8204 0.8374 0.8432 0.8751 0.8681

0.8795 0.8391 0.8562 0.8674 0.9010 0.8728 0.8858 0.8774 0.8991 0.8838 0.8966 0.8818

0 1037 2093 3052 4081 5032 6065 7096 8157 9279 10,307 11,020

1.0076 1.0091 1.0085 1.0073 1.0061 1.0053 1.0045 1.0042 1.0033 1.0018 1.0011 0.9986 HER SDM

1.009 1.009 1.009 1.007 1.006 1.005 1.004 1.004 1.004 1.002 1.001 1.001 1.0024 1.731

ciple. In the last FL optimization executed case, QuinalliBT system performed 170 iterations and evaluated the objective function 8279 times. keff value decreases during the CRP-FL iterative process, due to some thermal limits violated in the first iterations. Later on, thermal limits are satisfied and keff at the end of cycle is higher than target keff.

Table 3 shows the parameters obtained in this stage and it can be seen that all of these parameters are satisfied. It is necessary to comment that, according to the stop criteria of QuinalliBT system, the third stage of FL optimization was only performed three times.

k eff behaviour 1.0080 kcrit

k eff

1.0050

keff first iteration keff second iteration

1.0020

keff third iteration

0.9990 0.9960

0

2000

4000

6000

8000

10000

burnup (MWd/T) Fig. 3. keff Behavior throughout the iterative process.

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6. Conclusions

References

A new system to optimize a FL-CRP design in a coupled way for BWRs was developed. Tests showed a convergent behavior of the system throughout CRP-FL iterative process. Although at the beginning of the process some thermal limits are not satisfied, once iterative process is finished all parameters are satisfied. The effective multiplication factor keff so obtained is better than the target value for keff used to test the QuinalliBT system. According to Table 3, the keff obtained with QuinalliBT system is 0.0024 Dk/k greater than target keff. This value represents around 10 full power days of reactor operation, with the corresponding extra energy generated. The successful executions of QuinalliBT system through different cases suggest the robustness of the system. During the performed tests, it could be seen that the first stage of the process can be eliminated, because seed FL is independent of the last FL design. New proposals to obtain seed FL in a simpler way are under study, in order to reduce CPU time. Actually, the maximum number of iterations in the different executed tests was 10, which still represents a considerable amount of CPU time.

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Acknowledgement This research has been partly supported through the research project SEP-2004-C01-46694 by CONACYT, Mexico.