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Annals of Nuclear Energy 38 (2011) 2488–2495

Contents lists available at SciVerse ScienceDirect

Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene

Huitzoctli: A system to design Control Rod Pattern for BWR’s using a hybrid method Alejandro Castillo a,⇑, Juan José Ortiz-Servin a, Raúl Perusquía a, Luis B. Morales b a b

Instituto Nacional de Investigaciones Nucleares, Carretera México-Toluca s/n, La Marquesa, Ocoyoacac, México, C.P. 52750, Mexico Universidad Nacional Autónoma de México, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Apartado Postal 70-221, México, D.F. 04510, Mexico

a r t i c l e

i n f o

Article history: Received 1 March 2011 Received in revised form 14 June 2011 Accepted 17 June 2011 Available online 24 August 2011 Keywords: BWR Control Rod Pattern Optimization Scatter Search technique

a b s t r a c t Huitzoctli system was developed to design Control Rod Patterns for Boiling Water Reactors (BWR). The main idea is to obtain a Control Rod Pattern under the following considerations: (a) the critical reactor core state is satisﬁed, (b) the axial power distribution must be adjusted to a target axial power distribution proposal, and (c) the maximum Fraction of Critical Power Ratio (MFLCPR), the maximum Fraction of Linear Power Density (FLPD) and the maximum Fraction of Average Planar Power Density (MPGR) must be fulﬁlled. Those parameters were obtained using the 3D CM-PRESTO code. In order to decrease the problem complexity, Control Cell Core load strategy was implemented; in the same way, intermediate axial positions and core eighth symmetry were took into account. In this work, the cycle length was divided in 12 burnup steps. The Fuel Loading Pattern is an input data and it remains without changes during the iterative process. The Huitzoctli system was developed to use the combinatorial heuristics techniques Scatter Search and Tabu Search. The ﬁrst one was used as a global search method and the second one as a local search method. The Control Rod Patterns obtained with the Huitzoctli system were compared to other Control Rod Patterns designs obtained with other optimization techniques, under the same operating conditions. The results show a good performance of the system. In all cases the thermal limits were satisﬁed, and the axial power distribution was adjusted to the target axial power distribution almost completely. It is very important to point out that, even though the cycle length improvement was not the main idea of this work, the effective multiplication factor (keff) at the end of the cycle was improved in all cases tested. The Huitzoctli system was programmed using Fortran 77 language in an Alpha Workstation with UNIX operating system. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The Control Rod Pattern design to BWR has been solved using different methodologies. For instance, Lin and Lin (1991) proposed the use of IF–THEN rules, obtaining an acceptable solution to the problem. Karve and Turinsky (1999a) used heuristic rules and the resulting implementation was added to the FORMOSA code (1999b). Montes et al. (2004) proposed a Genetic Algorithm, where the core coolant ﬂow remains without changes throughout of the cycle. Francois et al. (2004) applies Fuzzy Logic to optimize the Control Rod Patterns in a BWR taking into account the coolant changes into the core. Castillo et al. (2005) uses the Tabu Search technique and compares the results with other similar works. Recently, the Ant Colony System (Ortiz and Requena, 2006), has been proposed to achieve a good Control Rod Pattern. In all previous cases, the thermal limits and axial power proﬁle were considered to obtain the Control Rod Pattern design. ⇑ Corresponding author. Tel.: +52 55 53297200; fax: +52 55 53297301. E-mail addresses: [email protected] (A. Castillo), juanjose.ortiz@ inin.gob.mx (J.J. Ortiz-Servin), [email protected] (R. Perusquía), [email protected] (L.B. Morales). 0306-4549/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2011.06.017

In this work, a new system is proposed, where the Scatter Search (SS) and the Tabu Search (TS) methodologies are combined. The ﬁrst technique is used as a global search technique, whereas the second one is used to improve, in a local way, the solutions obtained with SS during the iterative process. The results obtained are compared with other similar works, with the idea of analyzing the advantages and disadvantages of our system. Huitzoctli is a word in an ancient Mexican language, whose meaning is metal rod.

2. Problem description The Control Rod Pattern (CRP) design consists on ﬁnding axial control rod positions in the different burnup steps, in which the operation cycle is divided, to compensate all reactivity changes into the core throughout the cycle. In this way, it is possible to extract the greatest energy from the fuel, satisfying the core’s safety throughout the operating cycle. In this work, the reactor core considered has 444 fuel channels and 109 control rods (Fig. 1). Axially, the control rods can be positioned in 25 different locations. The positions are labelled by even integer numbers as following (00, 02, 04, . . . , 44, 46, 48). The ﬁrst

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distribution can be determined by a core simulation with all control rods withdrawn of the core and without burnup step. In fact, the target axial power distributions throughout the cycle will change from a peaked form in the bottom at the beginning of the cycle, to a slightly peaked form at the top of the core towards the end of the cycle. This operation strategy, called spectral shift (Glasstone and Sesonske, 1994), is the best way to design the CRP, since more energy can be extracted from the fuel at the end of cycle. 3. Objective function

Fig. 1. Full BWR core.

The design of a Control Rod Pattern must consider in each burnup step: (a) maintaining critical the reactor core, (b) the adjustment of the axial power distribution to be as close as possible to a target axial power distribution, and (c) maintaining the parameters MFLCPR, FLPD and MPGR within their operational limits. Thus, the objective function to be minimized is the following:

F ¼ w1 jkeff ;o keff ;t j þ w2 10 positions, from 00 to 18, are named ‘‘deep positions’’; the positions 20 to 30 are called ‘‘intermediate positions’’; ﬁnally, the positions from the 32 to the 48 are named ‘‘shallow positions’’. The use of intermediate positions is forbidden due to the deformation of the axial power distribution shape (Almenas and Lee, 1992). Thus, there remain 19 axial different positions for each control rod. In the same sense, if the Control Cell Core (CCC) load strategy (Specker et al., 1978) is used, only a quarter part of the control rods are considered for the operating cycle. Moreover, if the 1/8 symmetry is applied then only ﬁve control rods are ﬁnally used to design CRPs. Likewise, in a typical case, there are 12 burnup steps throughout the operating cycle. Thereby, the total number of possibilities to design a CRP is ((19)5)12, which is a great number of possibilities. A scheme of the problem is shown in Fig. 2; CRi, i = 1, . . . , 5 represents each control rod in the core. CRP design must satisfy the following constraints in each burnup step: (a) Reactor core must be critical. (b) Thermal limits must be satisﬁed. (c) The differences between a target axial power distribution and the obtained axial power distribution must be minimized.

25 X

jP oi Pti j þ w3 jFLPDt FLPDo j

i¼1

þ w4 jMPGRt MPGRo j þ w5 jMFLCPRt MFLCPRo j

ð1Þ

where keff,o is the current effective multiplication factor, keff,t the target effective multiplication factor, Poi the axial power distribution for node i, Pti the target axial power distribution for node i, FLPDt the maximum fraction of linear lower density permitted, FLPDo the maximum fraction of linear power density calculated, MPGRt the maximum fraction of average planar power density permitted, MPGRo the maximum fraction of average planar power density calculated, MFLCPRo the maximum fraction of critical power ratio calculated, MFLCPRt the maximum fraction of critical power ratio permitted, and w1, . . . , w5 are the weighting factors obtained with an statistical analysis, wi > 0, i = 1, . . . , 5. During the optimization process, when the fractions of the thermal limits (FLPD, MPGR and MFLCPR) are lower than the limit values (Table 1), the weighting factors are equal to zero; otherwise a penalty is added to the objective function. Thus, only the differences between the target effective multiplication factor and the calculated multiplication factor and the target axial power distribution and the calculated axial target distribution will be minimized when the thermal limits are satisﬁed. 4. Optimization techniques

The thermal limits are: the maximum Fraction of Critical Power Ratio (MFLCPR), the maximum Fraction of Linear Power Density (FLPD) and the maximum Fraction of Average Planar Power Density (MPGR). These parameters must be lower than 0.94. Now, an interesting question is the following: how does Huitzoctli determine the target axial power distribution? Haling (1964) proposed a ﬂattened axial power distribution unchanged throughout the operating cycle. In this way, the reactor safety can be assured. On the other hand, when an axial power distribution peaked at the bottom is used, plutonium can be produced, in such way that the cycle length can be extended in comparison with the Haling axial power distribution. A peaked-bottom axial power

CR1

CR2

CR3

CR4

CR5

CR3

To design CRPs, combinatorial optimization techniques called Scatter Search (SS) and Tabu Search (TS) were used. SS has been used successfully to solve different problems in scientiﬁc and engineering areas. This technique was proposed in the 1980s by Glover (1989); the main idea is to use decision rules over the proposal problem. The strategy of this technique is to improve a set of solutions previously built, using the best attributes of their elements. On the other hand, given that SS is a global search method, it is desirable to modify the method to improve the performance in a local way. The main idea of the modiﬁed method is to ﬁnd the best solution in a neighbourhood based on a local search. To realize the

Table 1 Limit values for safety parameters.

CR2

Parameter

Limit value

CR1

FLPD MPGR MFLCPR

xi > > : xi

¼0

if xi < 0

¼ 18 if 18 < xi < 26

ð7Þ

¼ 32 if 26 6 xi < 32 ¼ 48 if xi > 48

For each iteration, the combination type is chosen in a randomized way. As already mentioned, the iterative process to solve the Control Rod Pattern design using the above combinatorial techniques consists of two stages. In the ﬁrst one, a global search is realized with Scatter Search, which was described above. Then, it is possible to improve the SS solution using a local search method (Tabu Search). In Fig. 3, the algorithm used is shown. It is necessary to say that the algorithm is applied to each burnup step.

Table 3 The best elements of the initial Reference Set’s obtained for example 1. RefSet1

RefSet2

BS

Objective function

keff

FLPD

MPGR

MFLCPR

Distance

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.6239 0.4427 0.2201 0.2868 0.3268 0.3940 0.2922 0.2876 0.2224 0.3517 0.2253 –

1.0040 1.0020 0.9981 0.9929 0.9915 0.9931 0.9949 0.9900 0.9873 0.9949 0.9914 –

0.8968 0.8794 0.8876 0.8945 0.8803 0.8897 0.8089 0.7808 0.8176 0.7920 0.8449 –

0.8511 0.8550 0.8760 0.8844 0.8719 0.8975 0.8322 0.8045 0.8179 0.8376 0.8945 –

0.7861 0.8265 0.7874 0.7457 0.8167 0.8168 0.7983 0.8161 0.8546 0.8861 0.8932 –

2848.0 3420.0 3412.0 3652.0 3040.0 3228.0 2952.0 3992.0 4416.0 3608.0 3756.0 –

0.9842 0.9919 0.9942 0.9881 0.9667 0.9882 0.9799 0.9924 0.9865 0.9751 0.9747 –

1.0780 0.9517 0.8208 0.9990 1.0010 0.8561 0.7915 0.7413 0.8329 0.9244 0.9383 –

1.0540 0.9482 0.8271 0.9840 0.9873 0.8750 0.8182 0.7714 0.8787 0.9755 0.9654 –

0.8896 0.9969 0.9166 1.0070 0.9434 0.9260 0.8866 0.9444 0.9864 1.0950 1.0090 –

Table 4 The best elements of the initial Reference Set’s obtained example 2. RefSet1

RefSet2

BS

Objective function

keff

FLPD

MPGR

MFLCPR

Distance

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.7458 0.4360 0.2409 0.2550 0.3680 0.3911 0.3143 0.2724 0.3138 0.2079 0.2893 –

1.0080 1.0020 0.9980 0.9975 0.9970 0.9948 0.9958 0.9940 0.9954 0.9924 0.9922 –

0.8869 0.8776 0.8868 0.8653 0.8519 0.8853 0.8219 0.7714 0.8346 0.8271 0.8948 –

0.8454 0.8424 0.8931 0.8539 0.8260 0.9081 0.8456 0.7879 0.8307 0.8231 0.9006 –

0.8243 0.8187 0.7767 0.7709 0.7760 0.7964 0.8093 0.8179 0.8645 0.8739 0.8907 –

2500.0 3272.0 2956.0 3624.0 3404.0 2940.0 3676.0 3372.0 2844.0 3608.0 3540.0 –

1.0060 0.9784 0.9784 0.9883 0.9846 0.9924 0.9931 0.9899 0.9829 0.9822 0.9779 –

0.9860 0.9373 0.8820 0.9302 0.9464 0.8969 0.8183 0.7647 0.7948 1.0080 0.9183 –

0.9449 0.8902 0.8537 0.9380 0.9686 0.9203 0.8421 0.8047 0.8387 1.0650 0.9700 –

0.8416 0.8960 0.8873 0.9939 0.8518 0.9231 0.9235 0.9819 0.9483 1.0340 1.0340 –

Table 5 The best elements of the initial Reference Set’s obtained example 3. RefSet1

RefSet2

BS

Objective function

keff

FLPD

MPGR

MFLCPR

Distance

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.7420 0.4103 0.2732 0.2736 0.2621 0.3624 0.3456 0.2641 0.3359 0.2088 0.1816 –

0.9997 1.0010 1.0010 0.9980 0.9977 0.9923 0.9962 0.9937 0.9877 0.9935 0.9919 –

0.8911 0.8966 0.8761 0.8656 0.8722 0.8613 0.8059 0.7624 0.7717 0.8327 0.8836 –

0.8394 0.8552 0.8652 0.8802 0.8706 0.8869 0.8280 0.7825 0.7935 0.8404 0.8945 –

0.8022 0.7744 0.8203 0.7792 0.7973 0.7722 0.8098 0.8203 0.8519 0.8803 0.8994 –

2652.0 3088.0 3892.0 3004.0 3048.0 2944.0 3836.0 3252.0 2888.0 3732.0 3240.0 –

0.9793 0.9832 0.9672 0.9678 0.9910 0.9943 0.9708 0.9913 0.9796 0.9827 0.9719 –

1.1830 0.8982 1.0350 1.0380 0.8098 0.8922 0.8703 0.7344 0.9261 0.9622 0.9582 –

1.0870 0.8362 0.9912 0.9600 0.8302 0.9157 0.8414 0.7620 0.9768 1.0150 0.9179 –

0.8148 0.9451 0.8604 0.8771 0.9898 0.9114 0.9666 0.9196 0.9835 1.0820 0.9345 –

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A. Castillo et al. / Annals of Nuclear Energy 38 (2011) 2488–2495

k eff behaviour

1.01 1.005 1 0.995 0.99 0.985 0.98

1

3

5

7

9

11

Burnup step keff target

Example 1

Example 2

Example 3

Fig. 4. Obtained results for effective multiplication factor.

Table 10 The best elements of the ﬁnal Reference Set’s obtained example 1 (RefSet1).

Table 6 The best Control Rod Pattern for the example 1. CRi

1 2 3 4 5

BSi 1

2

3

4

5

6

7

8

9

10

11

12

48 38 4 38 42

36 48 48 0 8

38 44 48 2 0

40 46 44 0 0

38 46 44 2 0

38 48 44 4 4

44 42 0 44 0

38 40 2 2 48

38 38 48 0 0

34 36 48 0 0

36 34 48 0 48

48 48 48 48 48

Table 7 The best Control Rod Pattern for the example 2. CRi

1 2 3 4 5

1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

11

12

42 40 36 0 46

36 46 46 0 12

38 48 44 2 6

40 44 48 0 0

38 46 46 2 0

38 48 2 42 6

42 42 44 2 0

40 40 0 2 38

38 38 48 0 0

38 36 48 0 0

46 34 36 0 34

48 48 48 48 48

BSi 1

2

3

4

5

6

7

8

9

10

11

12

42 38 6 38 4

36 38 48 0 12

40 48 40 0 0

40 44 48 0 0

38 46 44 2 0

38 48 4 42 0

42 44 0 42 2

40 40 2 0 48

40 38 0 46 0

36 36 48 0 0

36 34 48 0 32

48 48 48 48 48

Table 9 Global results. Example

1 2 3

Objective function

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.5403 0.3041 0.1310 0.1071 0.1060 0.1460 0.0847 0.1621 0.1400 0.1311 0.1603 0.4452

1.0040 1.0000 0.9978 0.9964 0.9952 0.9949 0.9939 0.9931 0.9930 0.9922 0.9911 0.9936

0.8993 0.8986 0.8979 0.8917 0.8830 0.8820 0.8556 0.8236 0.8349 0.8497 0.8932 0.8507

0.8596 0.8512 0.8737 0.8977 0.8953 0.8998 0.8807 0.8464 0.8311 0.8479 0.8949 0.9000

0.7756 0.7800 0.7738 0.7760 0.7808 0.7834 0.8196 0.8266 0.8559 0.8791 0.8874 0.8866

BSi

Table 8 The best Control Rod Pattern for the example 3. CRi

BS

Control rod movements

45 52 45

Movements between shallow and deep positions

10 11 13

Objective function evaluations Tabu Search

Scatter Search

9473 9199 9920

223 229 247

Table 11 The best elements of the ﬁnal Reference Set’s obtained example 2 (RefSet1). BS

Objective function

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.5429 0.2775 0.1516 0.0980 0.1176 0.1570 0.0918 0.1668 0.1375 0.1314 0.1499 0.5083

1.0400 1.0000 0.9980 0.9963 0.9952 0.9951 0.9936 0.9932 0.9929 0.9920 0.9911 0.9936

0.8987 0.8998 0.8957 0.8921 0.8868 0.8809 0.8651 0.8293 0.8389 0.8453 0.8996 0.8524

0.8639 0.8562 0.8687 0.8988 0.8962 0.8977 0.8904 0.8426 0.8301 0.8344 0.8928 0.8929

0.8367 0.7797 0.7698 0.7759 0.7793 0.8082 0.8063 0.8270 0.8549 0.8767 0.8867 0.8863

For each iteration, the global search with SS executes 10 inner iterations, whereas the local search (TS) is performed with 40 inner iterations. We can see that once the global search is applied, the major effort is used in the local search. The number of inner iterations was ﬁxed by a statistical analysis. 6. Results To test Huitzoctli system performance, an equilibrium cycle of Laguna Verde Nuclear Power Plant (LVNPP) was used. That equilibrium cycle used two different fuel batches both with 3.66w/o of U235 but different gadolinium concentration. In this case the keff value is equal to 0.9928 (according to a CM-PRESTO simulation)

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A. Castillo et al. / Annals of Nuclear Energy 38 (2011) 2488–2495

for a cycle length of 10,896 MWD/T. In our investigation the water core ﬂow was kept constant in all tests equals to 100%. The values above mentioned correspond to reactor full power operation condition. The cycle length was divided into 12 burnup steps. We performed different runs of the Huitzoctli system with the same fuel reload. The three best runs are shown; however, it is convenient to mention that the other ones have similar results. On the other hand, in the ﬁrst tests, Huitzoctli system was executed using Scatter Search technique only. We can verify that better results were obtained when a local search method (Tabu Search in this case) was implemented. In Table 2, the burnup steps used for this work are shown. In addition, we included the effective multiplication factor keff target for each one burnup step. The theoretical value for keff is not used as target given that the simulator models are not accurate. At the beginning, a Control Rod Pattern seed is built, which is used to generate a diverse set with 100 elements. Next, using the objective function and Eqs. (2) and (3), the Reference Set is generated. This set is divided in two parts; the ﬁrst one (RefSet1) has the best elements of the diverse set (eight elements), according the objective function evaluations; the second one (RefSet2) includes the elements farthest from RefSet1. The ‘‘far’’ concept is obtained taking into account the distance from Eq. (2). As it was mentioned, the Reference Set has 16 elements. The number of burnup steps is 12; however, since the last burnup step is performed with all control rods in the 48 position, this burnup step is not considered in the optimization process. Therefore, the number of elements is 11 16 = 176. On the other hand, each element has ﬁve parameters (keff, Axial Power Distribution, FLPD, MPGR and MFLCPR) and its respective control rod conﬁguration. It is very difﬁcult to show the whole information, for this reason, we will show the best results only. To show the results, the explanation will be divided in two parts, in the ﬁrst one the 3 best results are shown; in the second one, the results obtained are compared to other similar works found. As it was mentioned, in Tables 3–5, the best elements of the initial Reference Set generated for the three examples are included. An element of each one of these sets is a Control Rod Pattern Conﬁguration; however, the most important parameters for each one are their FLPD, MPGR, MFLCPR and effective multiplication factor (keff). These parameters are included in those tables and its respective Objective Function value. The next step is to generate the Control Rod Pattern for each one burnup step applying the iterative process using both SS and TS techniques. In this sense, in Fig. 4 effective multiplication factor (keff) for the best elements in each burnup step of the three examples are shown. In this ﬁgure, the target effective multiplication factor (keff) is included. A good manner to show the Control Rod Pattern for each burnup step is the following. In Fig. 2, a Control Rod Pattern for one burnup step is shown; this pattern can be shown in a vector form [04, 36, 38, 46, 06]. In the previous example, the ﬁrst entry (04) represents the axial position of the ﬁrst control rod; the second entry represents the second axial position of the second control rod and so on. Thereby, the Tables 6–8 include the best Control Rod Pattern for the three analyzed examples. To conclude this part, in Table 9, the number of control rod movements, the number of interchanges between shallow and deep positions and the number of objective function evaluations for each technique are included. In this case, the number of inner iterations remains without changes, 10 for global search (SS) and 40 for local search (TS) in each burnup step. Besides, in Tables 10–12 the best elements of the ﬁnal RefSet1 set of the three examples are shown. The RefSet2 is not included due to it remains without changes during the iterative process. In the Introduction Section, some papers about Control Rod Pattern design were mentioned. It is possible to do a comparison with

Table 12 The best elements of the ﬁnal Reference Set’s obtained example 3 (RefSet1). BS

Objective function

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.5117 0.2583 0.1524 0.0978 0.0974 0.1401 0.0744 0.1667 0.1507 0.1108 0.1424 0.4085

1.0030 1.0000 0.9978 0.9964 0.9953 0.9951 0.9939 0.9935 0.9933 0.9922 0.9913 0.9936

0.8996 0.8998 0.8975 0.8916 0.8840 0.8818 0.8524 0.8195 0.7891 0.8349 0.8891 0.8428

0.8648 0.8542 0.8849 0.8982 0.8960 0.8986 0.8771 0.8415 0.8106 0.8400 0.8985 0.8936

0.7854 0.7820 0.7720 0.7768 0.7805 0.8061 0.8163 0.8254 0.8672 0.8805 0.8988 0.8849

Table 13 Results comparison. Technique

Cycle length (MWD/TU)

Axial movements

Movements between deep and shallow positions

Tabu Search Ant Colony System Huitzoctli (SS + TS) example 1 Huitzoctli (SS + TS) example 2 Huitzoctli (SS + TS) example 3 Huitzoctli (SS alone) example 1 Huitzoctli (SS alone) example 2

11,005 11,096

48 47

10 13

10,956

45

10

10,956

52

11

10,956

45

13

10,916

52

14

10,916

54

15

some of them, taking into account they have similar conditions. In this sense, Ant Colony System was implemented by Ortiz and Requena (2006) and Castillo et al. (2005) used Tabu Search technique to solve the Control Rod Pattern design. Both cases taking into account the equilibrium cycle of LVNPP to apply their methodologies. In Table 13 the energy obtained with the different systems is shown, which includes both the number of axial movements and the interchanges number between shallow and deep positions. In this Table 13, the obtained results in the ﬁrst tests, where Scatter Search was used alone, are included. Thus, it is possible to verify how the Huitzoctli system (SS + TS) improves its performance by combining SS and TS.

7. Conclusions According to the obtained results with Huitzoctli system, it is viable to afﬁrm that it is a good tool to Control Rod Pattern design. It can be seen that the safety limits are satisﬁed. Besides, the differences between effective multiplication factor (keff,t) target and effective multiplication factor (keff,o) obtained during the iterative process are minimal. The main idea of this work is to obtain an adequate Control Rod Pattern design taking into account a fuel reload proposal. It was possible to obtain, effective multiplication factor’s (keff = 0.9936) greater than the reference effective multiplication factor (keff = 0.9928) target at the end of the cycle. This behavior can be seen in all real-

A. Castillo et al. / Annals of Nuclear Energy 38 (2011) 2488–2495

ized tests. Only three examples are shown, in fact, in the other results obtained, the keff value is equal to 0.9934 approximately. A great advantage of this system is the following, when a run has ﬁnished, it is possible to obtain eight Control Rod Pattern conﬁgurations, because this is the RefSet1 set length. In all tests, the RefSet1 obtained has similar results. Only the best element of RefSet1 has been shown. On the other hand, the obtained results with Huitzoctli system were similar to the other works (Table 13), taking into account the obtained energy at the end of cycle. In the same way, the number of realized axial movements by Huitzoctli system are similar than the others systems. Something like this happened with interchanges between shallow and deep positions, which produce a good performance. It is interesting to note that the hard work has been realized by local search technique (TS), while the global search technique (SS) was used to ﬁnd a neighbor to realize an intensive search during the iterative process. Huitzoctli system can be implemented with a fuel reload system to obtain a Control Rod Pattern and Fuel Reload in a coupled way. This is the main idea for future work.

Acknowledgement The authors acknowledge grateful to Departamento de Gestión de Combustible of the Comisión Federal de Electricidad of México.

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References Almenas, K., Lee, R., 1992. Nuclear Engineering: An Introduction. Springer Verlag, New York, USA. Castillo, A., Ortiz, J., Alonso, G., Morales, L., del Valle, E., 2005. BWR control rod design using tabu search. Annals of Nuclear Energy 32, 741–754. Francois, J., Martín del Campo, C., Tavares, A., 2004. Development of a BWR control rod pattern design based on fuzzy logic and knowledge. Annals of Nuclear Energy 31, 343–356. Glasstone, S., Sesonske, A., 1994. Nuclear Reactor Engineering, Reactor Systems Engineering. Chapman & Hall, Inc., USA. Glover, F., 1989. Tabu search Part I. ORSA, Journal of Computing 1, 190–206. Glover, F., 2001. Tabu Search. Kluwer Academic Publishers, Boston, USA. Haling, R., 1964. Operating Strategy for Maintain an Optimum Power Distribution Throughout Life. General Electric Company, Atomic Power Equipment Department, Atomic Energy Commission, pp. 205–210. Karve, A., Turinsky, P., 1999a. Effectiveness of BWR control rod pattern sampling capability in the incore fuel management code Formosa-B. In: Proc. Int. Conf. Mathematics and Computation, Reactor Physics and Environmental Analysis in Nuclear Applications, Madrid, Spain, vol. 2, September 27–30, pp. 1459–1473. Karve, A., Turinsky, P., 1999b. FORMOSA-B: a boiling water reactor in-core fuel management optimization package II. Nuclear Technology 131, 48–68. Laguna, M., Martí, R., 2003. Scatter Search, Methodology and Implementations in C. Kluwer Academic Publishers, Boston, USA. Lin, L., Lin, C., 1991. A rule-based expert system for automatic control rod pattern generation for boiling water reactors. Nuclear Technology 95, 1–12. Montes, J., Ortíz, J., Requena, I., Perusquía, R., 2004. Searching for full power control rod patterns in a boiling water reactor using genetic algorithms. Annals of Nuclear Energy 31, 1939–1954. Ortiz, J., Requena, I., 2006. Azcatl-CRP: an ant colony-based system for searching full power control rod patterns in BWRs. Annals of Nuclear Energy 33, 30–36. Scandpower, A.S., 1993. User Manual CM-PRESTO 9 Version CM914B, Rev. 6, July 16. Specker, S., Fennern, L., Brown, R., Stark, K., Crowther, R., 1978. The BWR control cell core improved design. Transactions of American Nuclear Society 30, 336–338.

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Huitzoctli: A system to design Control Rod Pattern for BWR’s using a hybrid method Alejandro Castillo a,⇑, Juan José Ortiz-Servin a, Raúl Perusquía a, Luis B. Morales b a b

Instituto Nacional de Investigaciones Nucleares, Carretera México-Toluca s/n, La Marquesa, Ocoyoacac, México, C.P. 52750, Mexico Universidad Nacional Autónoma de México, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Apartado Postal 70-221, México, D.F. 04510, Mexico

a r t i c l e

i n f o

Article history: Received 1 March 2011 Received in revised form 14 June 2011 Accepted 17 June 2011 Available online 24 August 2011 Keywords: BWR Control Rod Pattern Optimization Scatter Search technique

a b s t r a c t Huitzoctli system was developed to design Control Rod Patterns for Boiling Water Reactors (BWR). The main idea is to obtain a Control Rod Pattern under the following considerations: (a) the critical reactor core state is satisﬁed, (b) the axial power distribution must be adjusted to a target axial power distribution proposal, and (c) the maximum Fraction of Critical Power Ratio (MFLCPR), the maximum Fraction of Linear Power Density (FLPD) and the maximum Fraction of Average Planar Power Density (MPGR) must be fulﬁlled. Those parameters were obtained using the 3D CM-PRESTO code. In order to decrease the problem complexity, Control Cell Core load strategy was implemented; in the same way, intermediate axial positions and core eighth symmetry were took into account. In this work, the cycle length was divided in 12 burnup steps. The Fuel Loading Pattern is an input data and it remains without changes during the iterative process. The Huitzoctli system was developed to use the combinatorial heuristics techniques Scatter Search and Tabu Search. The ﬁrst one was used as a global search method and the second one as a local search method. The Control Rod Patterns obtained with the Huitzoctli system were compared to other Control Rod Patterns designs obtained with other optimization techniques, under the same operating conditions. The results show a good performance of the system. In all cases the thermal limits were satisﬁed, and the axial power distribution was adjusted to the target axial power distribution almost completely. It is very important to point out that, even though the cycle length improvement was not the main idea of this work, the effective multiplication factor (keff) at the end of the cycle was improved in all cases tested. The Huitzoctli system was programmed using Fortran 77 language in an Alpha Workstation with UNIX operating system. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The Control Rod Pattern design to BWR has been solved using different methodologies. For instance, Lin and Lin (1991) proposed the use of IF–THEN rules, obtaining an acceptable solution to the problem. Karve and Turinsky (1999a) used heuristic rules and the resulting implementation was added to the FORMOSA code (1999b). Montes et al. (2004) proposed a Genetic Algorithm, where the core coolant ﬂow remains without changes throughout of the cycle. Francois et al. (2004) applies Fuzzy Logic to optimize the Control Rod Patterns in a BWR taking into account the coolant changes into the core. Castillo et al. (2005) uses the Tabu Search technique and compares the results with other similar works. Recently, the Ant Colony System (Ortiz and Requena, 2006), has been proposed to achieve a good Control Rod Pattern. In all previous cases, the thermal limits and axial power proﬁle were considered to obtain the Control Rod Pattern design. ⇑ Corresponding author. Tel.: +52 55 53297200; fax: +52 55 53297301. E-mail addresses: [email protected] (A. Castillo), juanjose.ortiz@ inin.gob.mx (J.J. Ortiz-Servin), [email protected] (R. Perusquía), [email protected] (L.B. Morales). 0306-4549/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2011.06.017

In this work, a new system is proposed, where the Scatter Search (SS) and the Tabu Search (TS) methodologies are combined. The ﬁrst technique is used as a global search technique, whereas the second one is used to improve, in a local way, the solutions obtained with SS during the iterative process. The results obtained are compared with other similar works, with the idea of analyzing the advantages and disadvantages of our system. Huitzoctli is a word in an ancient Mexican language, whose meaning is metal rod.

2. Problem description The Control Rod Pattern (CRP) design consists on ﬁnding axial control rod positions in the different burnup steps, in which the operation cycle is divided, to compensate all reactivity changes into the core throughout the cycle. In this way, it is possible to extract the greatest energy from the fuel, satisfying the core’s safety throughout the operating cycle. In this work, the reactor core considered has 444 fuel channels and 109 control rods (Fig. 1). Axially, the control rods can be positioned in 25 different locations. The positions are labelled by even integer numbers as following (00, 02, 04, . . . , 44, 46, 48). The ﬁrst

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distribution can be determined by a core simulation with all control rods withdrawn of the core and without burnup step. In fact, the target axial power distributions throughout the cycle will change from a peaked form in the bottom at the beginning of the cycle, to a slightly peaked form at the top of the core towards the end of the cycle. This operation strategy, called spectral shift (Glasstone and Sesonske, 1994), is the best way to design the CRP, since more energy can be extracted from the fuel at the end of cycle. 3. Objective function

Fig. 1. Full BWR core.

The design of a Control Rod Pattern must consider in each burnup step: (a) maintaining critical the reactor core, (b) the adjustment of the axial power distribution to be as close as possible to a target axial power distribution, and (c) maintaining the parameters MFLCPR, FLPD and MPGR within their operational limits. Thus, the objective function to be minimized is the following:

F ¼ w1 jkeff ;o keff ;t j þ w2 10 positions, from 00 to 18, are named ‘‘deep positions’’; the positions 20 to 30 are called ‘‘intermediate positions’’; ﬁnally, the positions from the 32 to the 48 are named ‘‘shallow positions’’. The use of intermediate positions is forbidden due to the deformation of the axial power distribution shape (Almenas and Lee, 1992). Thus, there remain 19 axial different positions for each control rod. In the same sense, if the Control Cell Core (CCC) load strategy (Specker et al., 1978) is used, only a quarter part of the control rods are considered for the operating cycle. Moreover, if the 1/8 symmetry is applied then only ﬁve control rods are ﬁnally used to design CRPs. Likewise, in a typical case, there are 12 burnup steps throughout the operating cycle. Thereby, the total number of possibilities to design a CRP is ((19)5)12, which is a great number of possibilities. A scheme of the problem is shown in Fig. 2; CRi, i = 1, . . . , 5 represents each control rod in the core. CRP design must satisfy the following constraints in each burnup step: (a) Reactor core must be critical. (b) Thermal limits must be satisﬁed. (c) The differences between a target axial power distribution and the obtained axial power distribution must be minimized.

25 X

jP oi Pti j þ w3 jFLPDt FLPDo j

i¼1

þ w4 jMPGRt MPGRo j þ w5 jMFLCPRt MFLCPRo j

ð1Þ

where keff,o is the current effective multiplication factor, keff,t the target effective multiplication factor, Poi the axial power distribution for node i, Pti the target axial power distribution for node i, FLPDt the maximum fraction of linear lower density permitted, FLPDo the maximum fraction of linear power density calculated, MPGRt the maximum fraction of average planar power density permitted, MPGRo the maximum fraction of average planar power density calculated, MFLCPRo the maximum fraction of critical power ratio calculated, MFLCPRt the maximum fraction of critical power ratio permitted, and w1, . . . , w5 are the weighting factors obtained with an statistical analysis, wi > 0, i = 1, . . . , 5. During the optimization process, when the fractions of the thermal limits (FLPD, MPGR and MFLCPR) are lower than the limit values (Table 1), the weighting factors are equal to zero; otherwise a penalty is added to the objective function. Thus, only the differences between the target effective multiplication factor and the calculated multiplication factor and the target axial power distribution and the calculated axial target distribution will be minimized when the thermal limits are satisﬁed. 4. Optimization techniques

The thermal limits are: the maximum Fraction of Critical Power Ratio (MFLCPR), the maximum Fraction of Linear Power Density (FLPD) and the maximum Fraction of Average Planar Power Density (MPGR). These parameters must be lower than 0.94. Now, an interesting question is the following: how does Huitzoctli determine the target axial power distribution? Haling (1964) proposed a ﬂattened axial power distribution unchanged throughout the operating cycle. In this way, the reactor safety can be assured. On the other hand, when an axial power distribution peaked at the bottom is used, plutonium can be produced, in such way that the cycle length can be extended in comparison with the Haling axial power distribution. A peaked-bottom axial power

CR1

CR2

CR3

CR4

CR5

CR3

To design CRPs, combinatorial optimization techniques called Scatter Search (SS) and Tabu Search (TS) were used. SS has been used successfully to solve different problems in scientiﬁc and engineering areas. This technique was proposed in the 1980s by Glover (1989); the main idea is to use decision rules over the proposal problem. The strategy of this technique is to improve a set of solutions previously built, using the best attributes of their elements. On the other hand, given that SS is a global search method, it is desirable to modify the method to improve the performance in a local way. The main idea of the modiﬁed method is to ﬁnd the best solution in a neighbourhood based on a local search. To realize the

Table 1 Limit values for safety parameters.

CR2

Parameter

Limit value

CR1

FLPD MPGR MFLCPR

xi > > : xi

¼0

if xi < 0

¼ 18 if 18 < xi < 26

ð7Þ

¼ 32 if 26 6 xi < 32 ¼ 48 if xi > 48

For each iteration, the combination type is chosen in a randomized way. As already mentioned, the iterative process to solve the Control Rod Pattern design using the above combinatorial techniques consists of two stages. In the ﬁrst one, a global search is realized with Scatter Search, which was described above. Then, it is possible to improve the SS solution using a local search method (Tabu Search). In Fig. 3, the algorithm used is shown. It is necessary to say that the algorithm is applied to each burnup step.

Table 3 The best elements of the initial Reference Set’s obtained for example 1. RefSet1

RefSet2

BS

Objective function

keff

FLPD

MPGR

MFLCPR

Distance

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.6239 0.4427 0.2201 0.2868 0.3268 0.3940 0.2922 0.2876 0.2224 0.3517 0.2253 –

1.0040 1.0020 0.9981 0.9929 0.9915 0.9931 0.9949 0.9900 0.9873 0.9949 0.9914 –

0.8968 0.8794 0.8876 0.8945 0.8803 0.8897 0.8089 0.7808 0.8176 0.7920 0.8449 –

0.8511 0.8550 0.8760 0.8844 0.8719 0.8975 0.8322 0.8045 0.8179 0.8376 0.8945 –

0.7861 0.8265 0.7874 0.7457 0.8167 0.8168 0.7983 0.8161 0.8546 0.8861 0.8932 –

2848.0 3420.0 3412.0 3652.0 3040.0 3228.0 2952.0 3992.0 4416.0 3608.0 3756.0 –

0.9842 0.9919 0.9942 0.9881 0.9667 0.9882 0.9799 0.9924 0.9865 0.9751 0.9747 –

1.0780 0.9517 0.8208 0.9990 1.0010 0.8561 0.7915 0.7413 0.8329 0.9244 0.9383 –

1.0540 0.9482 0.8271 0.9840 0.9873 0.8750 0.8182 0.7714 0.8787 0.9755 0.9654 –

0.8896 0.9969 0.9166 1.0070 0.9434 0.9260 0.8866 0.9444 0.9864 1.0950 1.0090 –

Table 4 The best elements of the initial Reference Set’s obtained example 2. RefSet1

RefSet2

BS

Objective function

keff

FLPD

MPGR

MFLCPR

Distance

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.7458 0.4360 0.2409 0.2550 0.3680 0.3911 0.3143 0.2724 0.3138 0.2079 0.2893 –

1.0080 1.0020 0.9980 0.9975 0.9970 0.9948 0.9958 0.9940 0.9954 0.9924 0.9922 –

0.8869 0.8776 0.8868 0.8653 0.8519 0.8853 0.8219 0.7714 0.8346 0.8271 0.8948 –

0.8454 0.8424 0.8931 0.8539 0.8260 0.9081 0.8456 0.7879 0.8307 0.8231 0.9006 –

0.8243 0.8187 0.7767 0.7709 0.7760 0.7964 0.8093 0.8179 0.8645 0.8739 0.8907 –

2500.0 3272.0 2956.0 3624.0 3404.0 2940.0 3676.0 3372.0 2844.0 3608.0 3540.0 –

1.0060 0.9784 0.9784 0.9883 0.9846 0.9924 0.9931 0.9899 0.9829 0.9822 0.9779 –

0.9860 0.9373 0.8820 0.9302 0.9464 0.8969 0.8183 0.7647 0.7948 1.0080 0.9183 –

0.9449 0.8902 0.8537 0.9380 0.9686 0.9203 0.8421 0.8047 0.8387 1.0650 0.9700 –

0.8416 0.8960 0.8873 0.9939 0.8518 0.9231 0.9235 0.9819 0.9483 1.0340 1.0340 –

Table 5 The best elements of the initial Reference Set’s obtained example 3. RefSet1

RefSet2

BS

Objective function

keff

FLPD

MPGR

MFLCPR

Distance

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.7420 0.4103 0.2732 0.2736 0.2621 0.3624 0.3456 0.2641 0.3359 0.2088 0.1816 –

0.9997 1.0010 1.0010 0.9980 0.9977 0.9923 0.9962 0.9937 0.9877 0.9935 0.9919 –

0.8911 0.8966 0.8761 0.8656 0.8722 0.8613 0.8059 0.7624 0.7717 0.8327 0.8836 –

0.8394 0.8552 0.8652 0.8802 0.8706 0.8869 0.8280 0.7825 0.7935 0.8404 0.8945 –

0.8022 0.7744 0.8203 0.7792 0.7973 0.7722 0.8098 0.8203 0.8519 0.8803 0.8994 –

2652.0 3088.0 3892.0 3004.0 3048.0 2944.0 3836.0 3252.0 2888.0 3732.0 3240.0 –

0.9793 0.9832 0.9672 0.9678 0.9910 0.9943 0.9708 0.9913 0.9796 0.9827 0.9719 –

1.1830 0.8982 1.0350 1.0380 0.8098 0.8922 0.8703 0.7344 0.9261 0.9622 0.9582 –

1.0870 0.8362 0.9912 0.9600 0.8302 0.9157 0.8414 0.7620 0.9768 1.0150 0.9179 –

0.8148 0.9451 0.8604 0.8771 0.9898 0.9114 0.9666 0.9196 0.9835 1.0820 0.9345 –

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Effective Multiplication Factor (k eff )

A. Castillo et al. / Annals of Nuclear Energy 38 (2011) 2488–2495

k eff behaviour

1.01 1.005 1 0.995 0.99 0.985 0.98

1

3

5

7

9

11

Burnup step keff target

Example 1

Example 2

Example 3

Fig. 4. Obtained results for effective multiplication factor.

Table 10 The best elements of the ﬁnal Reference Set’s obtained example 1 (RefSet1).

Table 6 The best Control Rod Pattern for the example 1. CRi

1 2 3 4 5

BSi 1

2

3

4

5

6

7

8

9

10

11

12

48 38 4 38 42

36 48 48 0 8

38 44 48 2 0

40 46 44 0 0

38 46 44 2 0

38 48 44 4 4

44 42 0 44 0

38 40 2 2 48

38 38 48 0 0

34 36 48 0 0

36 34 48 0 48

48 48 48 48 48

Table 7 The best Control Rod Pattern for the example 2. CRi

1 2 3 4 5

1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

11

12

42 40 36 0 46

36 46 46 0 12

38 48 44 2 6

40 44 48 0 0

38 46 46 2 0

38 48 2 42 6

42 42 44 2 0

40 40 0 2 38

38 38 48 0 0

38 36 48 0 0

46 34 36 0 34

48 48 48 48 48

BSi 1

2

3

4

5

6

7

8

9

10

11

12

42 38 6 38 4

36 38 48 0 12

40 48 40 0 0

40 44 48 0 0

38 46 44 2 0

38 48 4 42 0

42 44 0 42 2

40 40 2 0 48

40 38 0 46 0

36 36 48 0 0

36 34 48 0 32

48 48 48 48 48

Table 9 Global results. Example

1 2 3

Objective function

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.5403 0.3041 0.1310 0.1071 0.1060 0.1460 0.0847 0.1621 0.1400 0.1311 0.1603 0.4452

1.0040 1.0000 0.9978 0.9964 0.9952 0.9949 0.9939 0.9931 0.9930 0.9922 0.9911 0.9936

0.8993 0.8986 0.8979 0.8917 0.8830 0.8820 0.8556 0.8236 0.8349 0.8497 0.8932 0.8507

0.8596 0.8512 0.8737 0.8977 0.8953 0.8998 0.8807 0.8464 0.8311 0.8479 0.8949 0.9000

0.7756 0.7800 0.7738 0.7760 0.7808 0.7834 0.8196 0.8266 0.8559 0.8791 0.8874 0.8866

BSi

Table 8 The best Control Rod Pattern for the example 3. CRi

BS

Control rod movements

45 52 45

Movements between shallow and deep positions

10 11 13

Objective function evaluations Tabu Search

Scatter Search

9473 9199 9920

223 229 247

Table 11 The best elements of the ﬁnal Reference Set’s obtained example 2 (RefSet1). BS

Objective function

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.5429 0.2775 0.1516 0.0980 0.1176 0.1570 0.0918 0.1668 0.1375 0.1314 0.1499 0.5083

1.0400 1.0000 0.9980 0.9963 0.9952 0.9951 0.9936 0.9932 0.9929 0.9920 0.9911 0.9936

0.8987 0.8998 0.8957 0.8921 0.8868 0.8809 0.8651 0.8293 0.8389 0.8453 0.8996 0.8524

0.8639 0.8562 0.8687 0.8988 0.8962 0.8977 0.8904 0.8426 0.8301 0.8344 0.8928 0.8929

0.8367 0.7797 0.7698 0.7759 0.7793 0.8082 0.8063 0.8270 0.8549 0.8767 0.8867 0.8863

For each iteration, the global search with SS executes 10 inner iterations, whereas the local search (TS) is performed with 40 inner iterations. We can see that once the global search is applied, the major effort is used in the local search. The number of inner iterations was ﬁxed by a statistical analysis. 6. Results To test Huitzoctli system performance, an equilibrium cycle of Laguna Verde Nuclear Power Plant (LVNPP) was used. That equilibrium cycle used two different fuel batches both with 3.66w/o of U235 but different gadolinium concentration. In this case the keff value is equal to 0.9928 (according to a CM-PRESTO simulation)

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A. Castillo et al. / Annals of Nuclear Energy 38 (2011) 2488–2495

for a cycle length of 10,896 MWD/T. In our investigation the water core ﬂow was kept constant in all tests equals to 100%. The values above mentioned correspond to reactor full power operation condition. The cycle length was divided into 12 burnup steps. We performed different runs of the Huitzoctli system with the same fuel reload. The three best runs are shown; however, it is convenient to mention that the other ones have similar results. On the other hand, in the ﬁrst tests, Huitzoctli system was executed using Scatter Search technique only. We can verify that better results were obtained when a local search method (Tabu Search in this case) was implemented. In Table 2, the burnup steps used for this work are shown. In addition, we included the effective multiplication factor keff target for each one burnup step. The theoretical value for keff is not used as target given that the simulator models are not accurate. At the beginning, a Control Rod Pattern seed is built, which is used to generate a diverse set with 100 elements. Next, using the objective function and Eqs. (2) and (3), the Reference Set is generated. This set is divided in two parts; the ﬁrst one (RefSet1) has the best elements of the diverse set (eight elements), according the objective function evaluations; the second one (RefSet2) includes the elements farthest from RefSet1. The ‘‘far’’ concept is obtained taking into account the distance from Eq. (2). As it was mentioned, the Reference Set has 16 elements. The number of burnup steps is 12; however, since the last burnup step is performed with all control rods in the 48 position, this burnup step is not considered in the optimization process. Therefore, the number of elements is 11 16 = 176. On the other hand, each element has ﬁve parameters (keff, Axial Power Distribution, FLPD, MPGR and MFLCPR) and its respective control rod conﬁguration. It is very difﬁcult to show the whole information, for this reason, we will show the best results only. To show the results, the explanation will be divided in two parts, in the ﬁrst one the 3 best results are shown; in the second one, the results obtained are compared to other similar works found. As it was mentioned, in Tables 3–5, the best elements of the initial Reference Set generated for the three examples are included. An element of each one of these sets is a Control Rod Pattern Conﬁguration; however, the most important parameters for each one are their FLPD, MPGR, MFLCPR and effective multiplication factor (keff). These parameters are included in those tables and its respective Objective Function value. The next step is to generate the Control Rod Pattern for each one burnup step applying the iterative process using both SS and TS techniques. In this sense, in Fig. 4 effective multiplication factor (keff) for the best elements in each burnup step of the three examples are shown. In this ﬁgure, the target effective multiplication factor (keff) is included. A good manner to show the Control Rod Pattern for each burnup step is the following. In Fig. 2, a Control Rod Pattern for one burnup step is shown; this pattern can be shown in a vector form [04, 36, 38, 46, 06]. In the previous example, the ﬁrst entry (04) represents the axial position of the ﬁrst control rod; the second entry represents the second axial position of the second control rod and so on. Thereby, the Tables 6–8 include the best Control Rod Pattern for the three analyzed examples. To conclude this part, in Table 9, the number of control rod movements, the number of interchanges between shallow and deep positions and the number of objective function evaluations for each technique are included. In this case, the number of inner iterations remains without changes, 10 for global search (SS) and 40 for local search (TS) in each burnup step. Besides, in Tables 10–12 the best elements of the ﬁnal RefSet1 set of the three examples are shown. The RefSet2 is not included due to it remains without changes during the iterative process. In the Introduction Section, some papers about Control Rod Pattern design were mentioned. It is possible to do a comparison with

Table 12 The best elements of the ﬁnal Reference Set’s obtained example 3 (RefSet1). BS

Objective function

keff

FLPD

MPGR

MFLCPR

1 2 3 4 5 6 7 8 9 10 11 12

0.5117 0.2583 0.1524 0.0978 0.0974 0.1401 0.0744 0.1667 0.1507 0.1108 0.1424 0.4085

1.0030 1.0000 0.9978 0.9964 0.9953 0.9951 0.9939 0.9935 0.9933 0.9922 0.9913 0.9936

0.8996 0.8998 0.8975 0.8916 0.8840 0.8818 0.8524 0.8195 0.7891 0.8349 0.8891 0.8428

0.8648 0.8542 0.8849 0.8982 0.8960 0.8986 0.8771 0.8415 0.8106 0.8400 0.8985 0.8936

0.7854 0.7820 0.7720 0.7768 0.7805 0.8061 0.8163 0.8254 0.8672 0.8805 0.8988 0.8849

Table 13 Results comparison. Technique

Cycle length (MWD/TU)

Axial movements

Movements between deep and shallow positions

Tabu Search Ant Colony System Huitzoctli (SS + TS) example 1 Huitzoctli (SS + TS) example 2 Huitzoctli (SS + TS) example 3 Huitzoctli (SS alone) example 1 Huitzoctli (SS alone) example 2

11,005 11,096

48 47

10 13

10,956

45

10

10,956

52

11

10,956

45

13

10,916

52

14

10,916

54

15

some of them, taking into account they have similar conditions. In this sense, Ant Colony System was implemented by Ortiz and Requena (2006) and Castillo et al. (2005) used Tabu Search technique to solve the Control Rod Pattern design. Both cases taking into account the equilibrium cycle of LVNPP to apply their methodologies. In Table 13 the energy obtained with the different systems is shown, which includes both the number of axial movements and the interchanges number between shallow and deep positions. In this Table 13, the obtained results in the ﬁrst tests, where Scatter Search was used alone, are included. Thus, it is possible to verify how the Huitzoctli system (SS + TS) improves its performance by combining SS and TS.

7. Conclusions According to the obtained results with Huitzoctli system, it is viable to afﬁrm that it is a good tool to Control Rod Pattern design. It can be seen that the safety limits are satisﬁed. Besides, the differences between effective multiplication factor (keff,t) target and effective multiplication factor (keff,o) obtained during the iterative process are minimal. The main idea of this work is to obtain an adequate Control Rod Pattern design taking into account a fuel reload proposal. It was possible to obtain, effective multiplication factor’s (keff = 0.9936) greater than the reference effective multiplication factor (keff = 0.9928) target at the end of the cycle. This behavior can be seen in all real-

A. Castillo et al. / Annals of Nuclear Energy 38 (2011) 2488–2495

ized tests. Only three examples are shown, in fact, in the other results obtained, the keff value is equal to 0.9934 approximately. A great advantage of this system is the following, when a run has ﬁnished, it is possible to obtain eight Control Rod Pattern conﬁgurations, because this is the RefSet1 set length. In all tests, the RefSet1 obtained has similar results. Only the best element of RefSet1 has been shown. On the other hand, the obtained results with Huitzoctli system were similar to the other works (Table 13), taking into account the obtained energy at the end of cycle. In the same way, the number of realized axial movements by Huitzoctli system are similar than the others systems. Something like this happened with interchanges between shallow and deep positions, which produce a good performance. It is interesting to note that the hard work has been realized by local search technique (TS), while the global search technique (SS) was used to ﬁnd a neighbor to realize an intensive search during the iterative process. Huitzoctli system can be implemented with a fuel reload system to obtain a Control Rod Pattern and Fuel Reload in a coupled way. This is the main idea for future work.

Acknowledgement The authors acknowledge grateful to Departamento de Gestión de Combustible of the Comisión Federal de Electricidad of México.

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