Multicycle Fuel Loading Pattern Optimization by Multiobjective ...

NUCLEAR SCIENCE AND ENGINEERING: 176, 226–239 (2014)

Multicycle Fuel Loading Pattern Optimization by Multiobjective Simulated Annealing Employing Adaptively Constrained Discontinuous Penalty Function Tong Kyu Park,* Han Gyu Joo,y and Chang Hyo Kim Seoul National University 1 Gwanak-ro, Gwanak-gu, Seoul, 151-744, Korea Received May 30, 2012 Accepted April 11, 2013 http://dx.doi.org/10.13182/NSE12-41

Abstract – The multiobjective simulated annealing (MOSA)–based fuel assembly loading pattern (LP) optimization method, employing the discontinuous penalty function (DPF), is extended for multicycle applications by introducing an adaptively constrained discontinuous penalty function (ACDPF). A discontinuous point in the penalty function is adaptively shifted to a better direction during the course of MOSA such that the search can be more efficient. The advantages of the ACDPF-based MOSA algorithm over the original DPF-based algorithm are first examined with a real single-cycle LP optimization problem of an operating reactor, as well as with a simple LP optimization problem that has known solutions. A direct multicycle LP optimization method is then formulated with an application to the first four cycles of the Younggwang Nuclear Unit 4 (YGN4) core. The rearrangement method is devised as a fuel shuffling method that can avoid drastic changes in the LPs of the subsequent cycles of a seed cycle. It is demonstrated that the ACDPF-based MOSA combined with the rearrangement method produces quite effectively the optimum LP sets for the four cycles, which outperform the LPs generated by a series of cyclewise optimizations as well as the actual LPs of YGN4 that were already used in the plant. I. INTRODUCTION

ness in producing a set of optimum LPs meeting all the imposed constraints while satisfying mutually exclusive objectives with affordable computing resources and time was demonstrated with a single-cycle, dual-objective LP optimization problem for a cycle of the Younggwang Nuclear Unit 4 (YGN4) reactor, which is a pressurized water reactor (PWR) in operation in Korea.1 The DPF-based MOSA algorithm has a salient feature distinguished from the algorithms by others3–5 in that the penalty function for each individual constraint or individual objective takes the same form, regardless of the type of constraint or objective, and it becomes zero when a trial LP meets the target value for each individual constraint or objective. If the target value is not met, the penalty function jumps to a value greater than unity, with the penalty term proportional to the square of the difference between the actual value of the trial LP and the target value. The global penalty function is obtained by the sum of individual penalties formed either for a constraint or for an objective. It becomes zero whenever a

Determination of an optimized fuel assembly (FA) loading pattern (LP) for each cycle of a light water reactor (LWR) is one of the most important tasks that nuclear designers must complete periodically. It involves a multiobjective constrained optimization problem in that it aims at finding the best possible LP that meets all the safety-related constraints, as well as all the performancebased design criteria, such as fuel enrichment, cycle length, and the discharge burnup of the FAs not to be reused in subsequent cycles. As a method for resolving such a multiobjective constrained optimization problem efficiently, a multiobjective simulated annealing (MOSA) algorithm employing the discontinuous penalty function1,2 (DPF) was proposed by the authors. Its effective*Current address: FNC Corp., 705-5, Gongse-dong, Giheung-gu, Yongin, 446-902, Korea yE-mail: [email protected] 226

MULTICYCLE FUEL LOADING OPTIMIZATION

trial LP becomes a feasible one that satisfies all the constraints and objectives. The zero value of the global penalty function of all the feasible LPs poses, however, a practical difficulty in performing MOSA optimization because the conventional simulated annealing (SA) algorithm can neither determine the acceptance of a new feasible LP when the current LP is also feasible, nor can it recognize the superiority of one feasible LP over the other feasible LP. Thus, it is not possible to sort out the optimum LP among the feasible LPs. To get around this difficulty, the concept of dominance6,7 was introduced into the DPF-based MOSA algorithm in previous work1 as an additional acceptance criterion, and was used after both the current LP and the new LP enter the feasible region. It is observed, however, that the incorporation of the dominance concept–based acceptance criterion into the MOSA algorithm causes an undesirable stochastic effect in that the resulting optimum LPs depend strongly on the random number sequence, let alone the complexity of the acceptance logic and the efficiency reduction of the MOSA algorithm. The first purpose of this paper is to introduce an improved DPF method, named the adaptively constrained discontinuous penalty function (ACDPF) method, that can overcome this undesirable stochastic problem. The second purpose is to establish a direct multicycle fuel LP optimization scheme utilizing ACDPF. The ACDPF method is based on the idea of resetting the target value or the limit value of each individual objective by using the value obtained for a new feasible LP whenever a better feasible LP appears during the MOSA optimization run. On the contrary, the limits for the constraints remain fixed at the initial values. The resetting of the limits of the individual objectives has the effect of pushing the objective values toward the minimum target values that the final optimum LP would have. Furthermore, this makes it possible to overcome naturally the difficulty of the original DPF-based MOSA algorithm by accepting or rejecting a new LP solely by the acceptance probability determined by the ACDPF-based global penalty function without resorting to the concept of dominance. Consequently, the MOSA algorithm with the ACDPF method uses exactly the same acceptance or rejection logic as the conventional SA algorithm without causing the aforementioned problem of the DPF-based MOSA. The details of the ACDPF-based MOSA are presented in Sec. II, together with its superior performance in the aspect of the quality of the optimum LPs in a single-cycle optimization, as well as the computing time. The effectiveness of the ACDPF-based MOSA turned out to be stronger in multicycle fuel LP optimization, as detailed in Sec. III. The direct multicycle optimization, which is to find a set of optimum LPs for several cycles simultaneously, is an interesting fuel management problem from the standpoint of long-term planning of in-core and out-of-core fuel management schemes, NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

227

including the design of nonequilibrium or transient cycles or the equilibrium cycles. Nonetheless, only a few studies have been performed, as reported by Yamamoto and Kanda,8 Anderson et al.,9,10 and Kropaczek.11 As in the case of single-cycle optimization, the MOSA optimization formulation calls for defining both the global penalty function that fits to the given multiobjective multicycle problem and an FA shuffling scheme that is used to determine a new LP set from the current LP set for the multicycle problem. It will be shown in Sec. III that the principle of the ACDPF method enables specifying straightforwardly the required global penalty function for multicycle LP optimization problems. It is done first by representing not only all the individual constraints or the objectives imposed in each individual cycle, but also any additional objectives set for the multicycle optimization by the same form of the DPF and then by adding all the involved individual DPFs. As for the FA shuffling scheme, it was observed12,13 that unlike the case for single-cycle LP optimization problems, the efficiency of the MOSA algorithm for the multicycle LP optimization problems was strongly affected by how to generate a new LP set from a current LP set. The conventional shuffling scheme turns out to be quite ineffective, as will be shown later in Sec. III. In order to resolve this problem, the rearrangement method is devised as an effective shuffling scheme that can enhance the efficiency of the MOSA algorithm exclusively for multicycle LP optimization. The global penalty function formulated in the spirit of the ACDPF method for multicycle LP optimization and the rearrangement method are integrated into an ACDPFbased MOSA algorithm. A multicycle multiobjective LP optimization module, McFLOP (Multicycle Fuel Loading Optimization Program), was developed by incorporating the ASTRA code,14 the nuclear design code of the KEPCO Nuclear Fuel (KNF) company, as the threedimensional (3-D) core depletion solver. To examine the performance of the ACDPF-based MOSA algorithm, the McFLOP/ASTRA code is applied to solve a four-cycle dual-objective problem of YGN4, which is to determine the optimum LP set of its first four cycles that will maximize the total cycle length and minimize the average end-of-cycle (EOC) burnup of the FAs of cycle 4 to be reloaded into the subsequent cycles. The given conditions are (a) the constraints for safety requirements, (b) the cycle length of each cycle, (c) the types of new FAs, and (d) the size of the batch that is to be replaced at each cycle. The examination results are presented in Sec. IV to demonstrate that the ACDPF-based MOSA algorithm for the direct multicycle LP optimization performs quite well in terms of quality of the optimum LP set represented by the net cycle lengths of the optimum LP sets and the computing time. It is also demonstrated that direct multicycle optimization by the MOSA algorithm outperforms the corresponding successive single-cycle optimization.

PARK et al.

228

II. ADAPTIVELY CONSTRAINED DISCONTINUOUS PENALTY FUNCTION– BASED SIMULATED ANNEALING

Since the details of the DPF method are available in a previous paper,1 only its main aspects are presented briefly in the following. The ACDPF method is then described with its improved performance. II.A. Discontinuous Penalty Function Method The DPF-based MOSA algorithm makes use of the global penalty function defined as the sum of all the DPFs representing m constraints and n objectives as follows: J ð X Þ~

mzn X

Ji ð X Þ ,

ð1Þ

i~1

where X denotes an LP and where Ji ð X Þ is the i’th DPF, defined by Ji ðX Þ~ 1zci

di ð X Þ2 d2iN

! uðdi ð X ÞÞ ,

ð2Þ

where di ðX Þ~ai ð X Þ{alim i , d2iN ~

N 1X di ðXk Þ2 uðdi ðXk ÞÞ , N k~1

and  1, uð xÞ~ 0,

xw0 , xƒ0 :

ai ð X Þ is either one of the design constraints or one of the is either the multiobjectives to be minimized, and alim i limiting value for each constraint or the minimum value for each individual objective. The optimum LP should for all i’s. On the other hand, ci is a satisfy ai ð X Þƒalim i user-defined weighting factor and di ð X Þ is the i’th penalty of a given LP X. d2iN is the mean value of the square of the i’th penalty over N preset LPs. It acts as a normalization factor of d2i so that Ji ð X Þ becomes more or less similar in magnitude regardless of the types of penalties. uð xÞ is a step function and therefore uðdi ð X ÞÞ becomes zero when a trial LP X meets the condition and jumps over unity otherwise. ai ð X Þƒalim i The DPF-based global penalty function or Eq. (1) poses a difficulty when executing the conventional SA algorithm. Note that Ji ð X Þ50 if ai ð X Þƒalim i . This implies that J ð X Þ50 for any X that satisfies all the constraints and meets all the required minimum objectives. Therefore, the so-called

feasible LPs, i.e., those with J50, are qualified for candidates of an optimum LP. However, the conventional SA algorithm can neither determine the acceptance of a new feasible LP when the current LP is also feasible nor recognize the superiority of one feasible LP over the other feasible LP. Thus, it fails to sort out the optimum LP among the feasible LPs. As a way to get around this difficulty, the concept of dominance was incorporated into the SA algorithm in previous work1 at the expense of both complexity and efficiency. As a new improved way to get around this difficulty of the conventional SA acceptance criterion, we now devise the ACDPF method that fits the SA algorithm more naturally than the concept of dominance. II.B. Adaptively Constrained Discontinuous Penalty Function Method for the In the ACDPF method, each value of alim i given objectives is reset to that of the new feasible LP whenever a new feasible LP is found, while the alim i values for the constraints are fixed at the limit values set initially, during the course of the MOSA optimization calculation. Suppose that a feasible LP Xf is generated for the first time during the MOSA optimization calculation. lim set initially for each objective is reset Each value  of ai lim to ai Xf , while ai values set initially for constraints redefines the feasible remain intact. The resetting of alim i and nonfeasible LPs because the LPs that were feasible before the resetting are no longer the feasible ones after resetting. The MOSA optimization runs continue to proceed with the reset objectives without introducing any changes in the MOSA optimization algorithm such as the cooling schedule, the stage control in terms of the number of LPs searched, the number of LPs accepted per stage, termination criterion, etc., until a new feasible LP is is carried out again found. Note that resetting of alim i repeatedly whenever a new feasible LP is generated, and therefore it provides a strong driving force for trial LPs to move toward optimum LPs. Note also that even though 2 the resetting of alim i may be made repeatedly, diN need not be recalculated because of the characteristics of the DPF, explained in the following. Let d2iN represent the mean value of the squared i’th penalty when the reset is made for alim i . The i’th DPF Ji ð X Þ with the i’th objective reset to new values may then be expressed by ! 2 d ð X Þ i ð3Þ uðdi ð X ÞÞ : Ji ð X Þ~ 1zci d2iN  By introducing ci defined as ci ~ci d2iN = d2 iN , Ji ð X Þ can be rewritten as ! 2 d ð X Þ i Ji ðX Þ~ 1zci uðdi ðX ÞÞ : ð4Þ d2iN

By comparing Eq. (4) and Eq. (2), we note that Ji ð X Þ and NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

MULTICYCLE FUEL LOADING OPTIMIZATION

229

Ji ð X Þ differ only in the user-specified coefficient, namely ci and ci . Because the DPF-based MOSA optimization is relatively insensitive to the values of ci , as demonstrated in previous work1 and which will be shown again shortly in Sec. II.C, there is practically no need to recalculate d2iN in accordance with the resetting of alim i . II.C. Performance of ACDPF-Based MOSA Algorithm To assess the effectiveness of the proposed ACDPFbased MOSA algorithm, a simple fuel loading optimization problem15 consisting of a two-dimensional (2-D) and two-group calculation model with a three-batch FA shuffling is solved first, and the resulting solution is compared with the optimum solutions that are already known. Actually, it was demonstrated in Ref. 2 that the DPF-based MOSA algorithm performed very well in this problem. Figure 1 shows the reference core and the infinite multiplication factor k? of each batch. The problem is to find all the LPs having an effective multiplication factor keff greater than 1.136 and a peak FA power FFA lower than 1.5, under a constraint that the five peripheral FAs directly contacting the reflector and the center FA are not moved from their current positions. Excluding these six FAs, there exist a total of 195 776 possible LPs, including the reference LP shown in Fig. 1. All the LPs are evaluated by the dual reference perturbation16 method based on an improved hybrid harmonics and linearization perturbation15 theory.

Fig. 1. Reference LP and k? for a 2-D three-batch FA

shuffling problem. Figure 2 shows a plot of FFA versus keff for all 195 776 LPs. It is observed that 102 LPs marked by red circles out of the 195 776 LPs satisfy the given constraints (keff §1:136 and FFA ƒ1:5) and that six LPs marked by green triangles out of 102 LPs are mutually undominated LPs (color online). These six mutually undominated LPs are identified as the optimum LPs of this three-batch FA LP optimization problem. The words ‘‘undominated LPs’’ refer to any two LPs X1 and X2 satisfying the condition that X1 is superior to X2 in one objective but inferior to X2 in the other objective. For example, the two LPs X1 and X2 are undominated if keff ðX1 Þwkeff ðX2 Þ and FFA ðX1 ÞwFFA ðX2 Þ or vice versa. The six mutually undominated feasible LPs are shown in Fig. 3. The LPs in Fig. 3 are finally chosen as the true optimum LPs of the problem through a dominance check. Table I lists FFA and keff of the six true optimum LPs. The six true optimum LPs are very different from the reference

Fig. 2. Fuel assembly power peak versus keff for a 2-D three-batch FA shuffling problem.

NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

PARK et al.

230

Fig. 3. Six optimum LPs for the 2-D three-batch dual-objective LP shuffling problem.

LP. Let us take LP 6 in Fig. 3, for example. Comparison with the reference LP shows that the two LPs differ in eight FA positions in the octant core representation. This implies that it is almost impossible to produce the optimum LP by mere arbitrary shuffling of FAs from the reference LP. Because the true optimum LPs of the problem are known,

this problem provides a very good test to assess the capability of any LP optimization algorithm. The 2-D three-batch FA shuffling problem can be formulated as a dual-objective LP optimization problem to be solved by the ACDPF-based MOSA optimization algorithm by treating both keff and FFA as the dual

TABLE I Multiplication Factors and Assembly Power Peaking Factors of the Six Optimum LPs of the 2-D Benchmark Problem Number of FAs Whose Positions Were Changed from the Reference LP Index

keff

FFA

1/8 Symmetry Line

1/4 Symmetry Line

1 2 3 4 5 6

1.14123 1.14086 1.13921 1.13884 1.13837 1.13646

1.4895 1.4800 1.4730 1.4411 1.4282 1.3953

6 5 5 5 6 4

6 4 6 6 5 4

NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

MULTICYCLE FUEL LOADING OPTIMIZATION

231

objectives. The user-specified initial objectives imposed on the two parameters keff and FFA are keff §1:136 and FFA ƒ1:5

ð5Þ

{keff ƒ{1:136 and FFA ƒ1:5 :

ð6Þ

or

In the spirit of the ACDPF method, the global penalty function J is expressed by J ð X Þ~ 1zc1

d1 ð X Þ2

z 1zc2

!

d21N d2 ðX Þ2 d22N

uðd1 ðX ÞÞ ! uðd2 ð X ÞÞ ,

ð7Þ

lim lim , with keff ~1:136, and where d1 ð X Þ~{keff ð X Þzkeff lim lim d2 ð X Þ~FFA ð X Þ{FFA , with FFA ~ 1.5. The global penalty function [Eq. (7)] for the dualobjective LP optimization problem contains two userdefined weighting factors c1 and c2 . To investigate how these factors affect the results of the MOSA optimization, 25 different sets of c1 and c2 listed in Table II are tested. There are thus a total of 25 global objective functions defined differently. Each function is examined by five MOSA runs with a different random number sequence for each run so that a total of 125 MOSA optimization calculations are conducted. Each MOSA calculation is performed on a fixed annealing schedule involving 40 stages with 80 LPs per stage, or a total of 3200 LPs. Figure 4 shows the core parameters of the candidate LPs observed at various stages during the first run for set 13. The candidate optimum LPs refer to the feasible LPs that are mutually undominated. In this run, the first feasible LP shows up at the 15th stage and the MOSA optimization calculation terminates at the 40th stage with the six candidate optimum LPs, all of which turn out to be the true optimum LPs shown in Fig. 3. Figure 5 shows the number of candidate optimum LPs produced from all the 125 ACDPF-based MOSA

Fig. 4. Evolution of mutually undominated LPs with the SA stage number in the first run using the weighting factor set 13.

optimization runs. Specifically, it shows the average number of the true optimum LPs that are produced by five independent MOSA runs, each employing a different random number sequence as a function of the set number of the weighting factor combination designated in Table II. The black squares are from the ACDPF-based objective functions. For the sake of comparison, Fig. 5 also shows the MOSA optimization results that are obtained by minimizing the conventional continuous penalty function (CPF),   J ð X Þ~c1 1:136{keff ð X Þ zc2 ðFFA ð X Þ{1:5Þ ,

that does not involve any discontinuity. For the CPF case, however, only the nine sets are shown in Fig. 5. It is noted in Fig. 5 that the ACDPF-based MOSA algorithm indeed produces the true optimum LPs with more than five true LPs sorted out for all 25 cases. This demonstrates that the ACDPF-based MOSA is insensitive to the choice of weighting factors. In contrast, the CPFbased MOSA calculations reveal strong sensitivity to the weighting factors.

TABLE II Set Number Assigned to 25 Different Combinations of the Two Weighting Factors c1 and c2 c2 c1

0.01

0.1

1

10

100

0.01 0.1 1 10 100

1 6 11 16 21

2 7 12 17 22

3 8 13 18 23

4 9 14 19 24

5 10 15 20 25

NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

ð8Þ

232

PARK et al.

Fig. 5. The number of true optimum LPs produced from the ACDPF-based and CPF-based MOSA optimization calculations versus the weighting factor set number.

The outstanding performance of the ACDPF-based MOSA algorithm as noted above was also observed with the original DPF-based MOSA. For a further examination of the performance of the new algorithm against the original algorithm, the reload design problem1 of cycle 4 of YGN4 is taken as a more realistic example of multiobjective LP optimization problems than the 2-D three-batch FA shuffling problem. YGN4 is a former combustion engineering PWR that loads 177 FAs in the core. The reference LP of the cycle 4 core1 consists of 11 types of FAs that differ in the batch enrichment and the number of burnable poison rods (BPRs). It is a dualobjective LP optimization problem aiming at finding the best LP to have the longest cycle length (Lc ) and the lowest possible radial peaking factor (Fr ), given the fresh, once-burned, and twice-burned FAs. This problem was already solved by the DPF-based MOSA algorithm as reported in Ref. 1. The problem is solved again by the ACDPF-based MOSA algorithm using the same computational conditions in terms of design constraints and objectives, SA parameters, neutronics evaluation method, etc. As done in the DPF-based MOSA case, ten independent ACDPF-based MOSA runs are conducted to find the optimum LPs. The resulting mutually undominated feasible LPs are compared in Fig. 6a, which is the Fr versus Lc plot of the feasible LPs obtained by the two methods. The 54 red circles and the 52 black squares represent the results produced from ten independent runs of the DPF-based MOSA algorithm and the ACDPFbased counterpart, respectively. The 5 red circles and the 15 black squares in Fig. 6b are the finally chosen optimum LPs of the two MOSA methods through dominance checks. The results in Figs. 6a and 6b clearly indicate that the ACDPF-based MOSA algorithm outperforms the DPF-based counterpart in terms of the number

of optimum LPs finally chosen and the quality of the optimum LPs in terms of the two objectives. Namely, the ACDPF comes up with more optimum LPs than the DPF, and the former produces optimum LPs far better than the latter in terms of cycle lengths and the radial peaking factor. Table III shows a comparison of the efficiency of the two methods in terms of the ten-run average values of the number of LPs generated (Ngen ), the number of LPs subject to 3-D evaluation (N3-D ), the number of LPs accepted (Na ), the screening efficiency (gST ), the number of feasible LPs (Nfea ), the number of candidate LPs (Ncan ), the turnaround time (Thour), and the number of CPUs (NCPU ). The data in Table III indicate again that the ACPDF method makes the MOSA optimization calculations much more efficient than the DPF method. Note that the average turnaround time of the ACDPF is shorter by a factor of 2 than the turnaround time of the DPF as a result of the fact that Ngen , Na , and Nfea by the former are far smaller than those by the latter, even though the two methods come up with almost the same number of candidate LPs in each run. III. DIRECT MULTICYCLE OPTIMIZATION USING ACDPF

Multicycle optimization involves LPs of several cycles that constitute a set of LPs. By defining the global objective function as a function of the set of LPs, not as a function of a single LP, the ACDPF-based MOSA method can be applied to determine the optimum LP. The global penalty function should represent the constraints and the objectives of all the individual cycles and any additional objectives of multicycles. In addition to this definition of the global penalty function, an algorithm to optimize the shuffling schemes for a set of LPs should be developed. In the following, the problem statement of the multicycle FA LP optimization problem is given first, and the ACDPF-based global penalty function that incorporates the multicycle constraints and objectives is defined. Then, the rearrangement method devised for efficient LP shuffling is described. III.A. Global Penalty Function for Multicycle Optimization Problem The multicycle LP optimization problem to be solved by the ACDPF-based MOSA algorithm is stated as follows.   Determine the optimum LP sets X1op , X2op ,    , Xtop for the subsequent t cycles of a PWR such that all Xkop ’s satisfy all the design constraints imposed on (a) various power peaking, (b) moderator temperature coefficients (MTC) at hot zero power (HZP) and hot full power (HFP), (c) maximum pin discharge burnup, and (d) NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

MULTICYCLE FUEL LOADING OPTIMIZATION

233

Fig. 6. (a) Collection of feasible LPs from ten independent DPF- and ACDPF-based MOSA optimization runs. (b) Comparison of optimum LPs chosen finally from DPF- and ACDPF-based MOSA optimization runs.

TABLE III Average Values of Ten Independent MOSA Optimization Runs for the Reload FA LP Design of the Cycle 4 Core of YGN4 by the DPF and ACDPF Methods*

DPF ACDPF

Ngen

N3-D

Na

gST

Nfea

Ncan

Thour

NCPU

14400 8473

4823 2505

3266 2377

0.66 0.70

372 290

5.4 5.2

8.1 3.0

35 35

*Ngen 5 number of LPs generated, N3-D 5 number of LPs subject to 3-D evaluation, Na 5 number of LPs accepted, gST 5 screening efficiency, Nfea 5 number of feasible LPs, Ncan 5 number of optimum LPs, Thour 5 turnaround time, NCPU 5 number of CPUs.

NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

PARK et al.

234

the minimum cycle length required for each individual cycle while all the Xkop ’s meet the objectives of (a) maximizing the total cycle length of the t cycles and (b) minimizing the average EOC burnup of the FAs at cycle t that will be reloaded into the cycle t z 1, given the types of FAs and the batch sizes that are to be replaced at each cycle, where Xkop represents the optimum LP for the k’th cycle. Note that this is a dual-objective optimization problem because of the two objectives mentioned above. For this multicycle problem, the global penalty function defined for a single-cycle LP optimization needs to be modified to take into account not only the constraints of individual cycles but also the dual objectives, J t ðXÞ~

t X nz1 X

Jk,i ðXk ÞzJt,nz2 ðXt ÞzJ TCL ðXÞ ,

 P~

1, if J t ðXnew ÞvJ t ðXold Þ , expf{DJ=C g, otherwise ,

ð10Þ

where DJ ~ J t ðXnew Þ{J t ðXold Þ C 5 cooling temperature. If a new LP XKnew is generated from the current LP for cycle k (v t), the new LPs of the subsequent cycles X‘new ðkv‘ƒtÞ need be generated in accordance with the change of XKnew . As will be presented shortly, the efficiency of the MOSA algorithm is strongly influenced by the way in which they are generated. The rearrangement method presented below offers a way to generate a new LP set from the current LP set efficiently. Xkcur

ð9Þ

k~1 i~1

where Jk,i ðXk Þ 5 DPF for the i’th design constraints of cycle k for iƒn Jk,nz1 ðXk Þ 5 DPF for the constraint on the cycle length of cycle k specified in terms of the minimum required length Jt,nz2 ðXt Þ 5 DPF for the objective for the minimum average burnup of the FAs at the EOC of cycle t that will be reloaded into cycle tz1 J TCL ðXÞ 5 DPF for the total cycle length of the t cycles that is to be maximized. There are a few points to note in Eq. (9). First, the last two terms in Eq. (9) correspond to the penalty terms of the dual objectives to be attained by the t-cycle LP optimization problem. Second, the minimum required cycle length of each individual cycle is treated as an extra constraint like the design constraints imposed on each cycle rather than as an objective of the cycle. Third, the sum of the total cycle lengths of the t cycles is the adaptively constrained objective parameter that must be repeatedly reset every time a feasible LP set shows up during the course of the MOSA optimization runs. Once the global penalty function is specified as above, it is a straightforward matter to apply the ACDPFbased MOSA algorithm. Unlike the MOSA algorithm for single-cycle optimization problems, however, the multicycle optimization problem requires moving a set of t LPs toward the region of feasible LP sets simultaneously. For example, suppose that a new set of LPs Xnew ~ (X1new , X2new ,    , Xtnew ) is generated  from the current set of LPs Xcur ~ X1cur , X2cur ,    , Xtcur . According to the principle of SA, Xnew is accepted according to the probability P defined as

III.B. Rearrangement Method As a way to introduce the rearrangement method, let us consider a two-cycle LP optimization problem for a PWR. Suppose that one has the current LP set like that in Fig. 7 in the course of the MOSA optimization for the problem. Let us ask how to generate the new LP set from the current set. Note that batch A FAs in cycle 1 are no longer used in cycle 2, and the fresh batch E FAs are loaded in cycle 2. The two FAs C1 and B, with vertical stripes in the current LP of cycle 2, come from the FAs with horizontal stripes in cycle 1. Suppose that a new LP for the first cycle is generated by a binary exchange of the two FAs C1 and B. One way to generate the new LP for the second cycle in response to the exchange in the LP of the first cycle is to exchange the sites of C1 and B denoted by vertical stripes in the current LP of the second cycle. If one adopts this practice (referred to as the conventional method hereafter) of generating a new LP set from the current LP set, one may observe a drastic change in the characteristics of the new LPs in terms of the power peaking and average burnup of the FAs from those of the current LPs in cycle 2, depending upon the case of the first cycle LP shuffling, which may in turn adversely affect the efficiency of the MOSA algorithm. The rearrangement method is designed to avoid such an undesirable effect. The idea is based on (a) ranking the FA positions of each batch of the current LP of cycle 2 according to the beginning-of-cycle (BOC) FA burnup to generate the BOC2 ranking tables, (b) ranking the FAs of each batch loaded in the new LP of cycle 1 according to the EOC FA burnup to generate the EOC1 ranking tables, and (c) placing the EOC1 FAs of each batch into the BOC2 FA positions having the same ranking to generate the new LP of cycle 2. As an illustration, Fig. 8a displays the burnup (rank) of batches B, C, and D FAs in the current LP set shown in Fig. 7, while Fig. 8b shows those in the new LP set. The new LP of cycle 1 X1new in Fig. 8b is generated by a binary NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

MULTICYCLE FUEL LOADING OPTIMIZATION

235

Fig. 7. A current LP set for the two-cycle LP optimization problem.

Fig. 8. (a) A current LP set for the MOSA algorithm: (i) a current LP of cycle 1 and EOC burnup (rank) of batches B, C, and D and (ii) a current LP of cycle 2 and BOC burnup (rank) of batches B, C, and D. (b) A new LP set for the MOSA algorithm: (i) a new LP of cycle 1 and EOC burnup (rank) of batches B, C, and D and (ii) a new LP of cycle 2 and BOC burnup (rank) of batches B, C, and D.

exchange of two FAs, the third-ranked FA B (with 15 104 MWd/tonne U) and the first-ranked FA C1 (with 15 750 MWd/tonne U) from X1cur . Note also that the reload batches B, C, and D FAs are arranged in the new LP of cycle 2 X2new such that the BOC burnup rank of each batch of X2new coincides with that of X2cur by using the EOC1 burnup rank information of X1new . If the new cycle 2 LP is obtained only by the rearrangement method, there will not be much variation in cycle 2 LPs because the BOC2 burnup rank of the seed LP will always be retained. This problem can be avoided by choosing cycle 2 LP for binary exchange once in a while, not determining it after cycle 1 LP. The probability of choosing a certain cycle for binary exchange will be discussed in Sec. IV, which is for the actual application. To examine the effectiveness of the rearrangement method, the first two cycles of YGN4 are chosen for a two-cycle optimization problem. A seed LP set for the first two cycles is selected. The axially integrated pin power peaking factor (Fr ) and the cycle length (Lc ) of cycle 2 of the seed LP set are 1.4935 and 276.7 effective NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

full-power days (EFPDs), respectively. A total of 50 LP sets are randomly generated from the seed LP set using binary or ternary exchanges for cycle 1. Figure 9 shows a comparison of the design characteristics of 50 LPs of cycle 2 produced by the rearrangement method and by the conventional method in terms of Fr and Lc of cycle 2. It is interesting to note that the cycle length of cycle 2 of some new LPs determined from the conventional method is similar to that of the seed LP, but that their Fr values are very different. Some LPs show Fr w 40%. Such a radical change may cause a runaway of the MOSA optimization calculation from the optimum path and thus a slow convergence of the MOSA optimization run. On the other hand, the two core characteristics parameters of the new cycle 2 LPs obtained from the rearrangement method are quite similar to those of the seed LP. Thus, the rearrangement method can avoid the above-mentioned shortfall and make the direct multicycle LP optimization by the MOSA algorithm more efficient. Before closing this section, it is worth pointing out that the multicycle optimization by SA or MOSA

PARK et al.

236

Fig. 9. Lc versus Fr plots of 50 new LPs of cycle 2 generated from a seed LP by the conventional FA shuffling and rearrangement method.

algorithms calls for an efficient scheme to generate a set of new LPs for the involved multicycles, namely a new LP set from a current LP set for the multicycles. It is desirable to devise the scheme in such a way that the trial LP sets created from it cover a wide range of LP sets search space so that the SA or MOSA optimization will not be trapped in local optima. The rearrangement method offers one such scheme. There is another scheme called the move generation strategy11 adopted by COPERNICUS, a commercial multicycle optimization code.a IV. APPLICATION TO REAL-CORE FOUR-CYCLE LP OPTIMIZATION PROBLEM

The rearrangement method proposed as a way to generate a new trial LP set from the current LP set, as well as the DPF-based global penalty function formulated for multicycle optimization, is integrated into the ACDPFbased MOSA algorithm. A multicycle multiobjective LP optimization module, McFLOP, was developed by incorporating the ASTRA code, the nuclear design code of KNF, as the 3-D core depletion solver. The McFLOP/ ASTRA code is then used to solve a multicycle dualobjective problem of YGN4, which is to determine the optimum LP set of its first four cycles that will maximize the total cycle length and to minimize the average EOC burnup of the FAs of cycle 4 to be reloaded into the subsequent cycle 5, given the types of FAs and the batch sizes that are to be replaced. As the design constraints, the power peaking limits Fr ƒ1:55 and Fq ƒ2:2 are imposed, a

Studsvick, Sweden

the BOC MTC limits for HZP and HFP are set to 5.0 pcm/uF and 0.0 pcm/uF, respectively, and the maximum pin burnup is set to 60 000 MWd/tonne U. The cycle lengths of the four-cycle reference LPs are set as the minimum required cycle lengths. Recall that the cycle length of each individual cycle is treated as an additional constraint like the design constraints. The multicycle optimization runs by McFLOP/ ASTRA require inputting the user-specified SA parameters such as the initial acceptance ratio, the number of LP sets generated and accepted per stage, and the stopping criterion, which are set at 0.99, 1000, 100, and 0.01, respectively. In addition to these input parameters, the probability for picking up a seed cycle out of the four cycles to generate a new LP set from the current LP set employing the rearrangement method is needed. The seed cycle here refers to the cycle of the current LP, which is used to generate a new LP out of it by binary exchange or some other LP change action. The new LPs of the remaining cycles following the seed cycle are determined successively from the current LP of the corresponding cycle following the rearrangement method. The probabilities that are used here for selecting the seed cycle i are determined by 1=i ði~1, 2, 3, 4Þ: PCi ~ P m 1=i

ð11Þ

i~1

Accordingly, probabilities for selecting cycles 1 through 4 as the seed cycle are 0.48, 0.24, 0.16, and 0.12 in the order of the four cycles. This implies that when 100 new LP sets are to be generated from so many current LP sets, approximately 48 LPs are generated by choosing the first cycle as the seed cycle, while 12 LPs are generated for the fourth cycle as the seed. Once a seed cycle is chosen, the new LP of the seed cycle is generated from its current LP by employing the shuffling schemes commonly adopted in single-cycle MOSA optimization runs such as the binary or ternary exchanges of FAs, the addition or removal of BPRs, and the rotation of burned FAs. A total of 20 029 LP sets are generated and a 122-h computing time is spent to complete the ACDPF-based four-cycle MOSA optimization of YGN4 on a cluster of forty-eight 2.13-GHz Xeon E5506 central processing units (CPUs). A total of 439 LP sets out of them are found feasible that satisfy all the design constraints and the dual objectives, including the minimum required cycle lengths of the first four cycles of YGN4. The four mutually undominated LP sets that are viewed as the optimum LP sets are obtained through a dominance check of these 439 LP sets. Table IV shows a comparison of the four optimum LP sets resulting from the direct four-cycle optimization with the reference LP set in terms of the cycle lengths of the first four cycles of YGN4 in the unit of EFPD. The cycle lengths of the reference LPs are NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

MULTICYCLE FUEL LOADING OPTIMIZATION

237

TABLE IV Comparison of Four Optimum LP Sets from Direct Multicycle Optimization and One Optimum LP Set from Successive Single-Cycle Optimization with the Reference LP Set in Terms of the Cycle Lengths of the First Four Cycles of YGN4*

Reference Successive single-cycle optimization run Multicycle optimization run First set Second set Third set Fourth set

Cycle 1

Cycle 2

Cycle 3

Cycle 4

Total

358.0 373.4

278.9 296.0

363.1 388.2

390.5 421.1

1390.5 1478.7

365.2 372.0 368.2 363.8

315.7 301.1 309.2 319.2

393.0 406.9 395.6 401.1

440.6 432.8 434.2 436.2

1514.5 1521.8 1507.2 1520.3

*In EFPDs.

calculated by ASTRA. Figure 10 illustrates the fourth optimum LP set of YGN4. As noted clearly in Table IV, the four optimum LP sets are superior to the reference four-cycle LP set in that the total cycle lengths of all the four optimum LP sets are longer by over 120 days than those of the reference set. For a further comparison, the first four-cycle optimization problem of YGN4 is also solved by executing the ACDPF-based single-cycle MOSA optimization runs successively for cycles 1 through 4. The

maximization of both the cycle length and the average of the EOC burnup of the FAs that will no longer be used for the next cycle are taken as dual objectives of each successive single-cycle MOSA optimization run. Because the single-cycle MOSA optimization runs from cycles 1 through 4 come up with several mutually undominant optimum LPs, it is necessary to sample one of them randomly to produce an LP with which one can initiate the single-cycle MOSA optimization runs to find the optimum LPs for the succeeding cycle. As a result, the

Fig. 10. (a) The reference LP set of the first four cycles of YGN4 PWR. (b) One example of their optimum LP sets from the successive single-cycle LP optimization. (c) Another example of their optimum LP sets from the direct four-cycle LP optimization, by the ACDPF-based MOSA algorithm. NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

PARK et al.

238

successive single-cycle MOSA optimization runs produce a number of different optimum LP sets, depending on the combinations of the LPs sampled randomly from the optimum LPs found for every cycle. Table IV and Fig. 10b show one of the optimum LP sets that is regarded as the best in terms of the net cycle length of the four cycles. As noted in Table IV, the optimum LP of each cycle from the successive optimization runs is better than the reference LP of the corresponding cycle of YGN4, in that it has a longer cycle length and higher average discharge burnup while satisfying all the design constraints for all four cycles. The total of the four cycle lengths of the optimum LP set from the successive singlecycle optimization is longer by *80 days than that of the reference LP set of YGN4, yet it fell shorter by roughly 40 days than that of the optimum LP sets from the direct four-cycle MOSA optimization.

V. CONCLUSION

The ACDPF method inherits all the advantages that its predecessor the DPF method has brought into the MOSA algorithm designed for various reload core FA LP optimization calculations of LWRs and PWRs in particular. The two methods construct the global penalty function in the same way from individual DPFs of similar magnitude. Like its DPF-based counterpart, therefore, the ACDPF-based MOSA algorithm is free from shortcomings or inefficiencies that the earlier MOSA algorithms1 encountered due to the inherent drawbacks of global penalty functions that they formulated. Besides, the ACDPF method has an extra advantage over the original DPF method in that the MOSA algorithm of the former consists of the conventional acceptance logic of the SA algorithm determined solely by the global penalty function, while that of the latter cannot but incorporate the concept of dominance into the acceptance logic of its MOSA algorithm in addition to the conventional acceptance logic based on its global penalty function. Because of this, the MOSA algorithm of the ACDPF method outperforms that of the original DPF method, as demonstrated in terms of a real reload cycle core LP optimization problem, namely the cycle 4 LP design of YGN4. The direct multicycle LP optimization problems of commercial PWRs using the MOSA algorithm call for generating a new LP set from the current LP set for the involved multicycle cores. If a new LP for a seed cycle selected randomly among the multicycles is generated from the current LP of the seed cycle, the LPs of the succeeding cycles should be changed according to the changes introduced into the new LP of the seed cycle. As pointed out in Sec. III.A, the efficiency of the MOSA algorithm for the direct multicycle optimization is strongly affected by how to change them. The conventional way to

do so may cause slow convergence of, or runaway from, the optimum path of the LP sets in the MOSA optimization calculations. The rearrangement method devised in this paper exclusively for the direct multicycle FA LP optimization calculations offers an effective way to drive the trial LP set toward the optimum LP sets by the MOSA algorithm. Therefore, it must be useful for other multicycle LP optimization studies employing the MOSA algorithm. In summary, the ACDPF method has several outstanding advantages. It makes it easy and straightforward to formulate the global penalty function that incorporates various constraints and multiobjectives required for a given LP optimization problem, regardless of whether it is a single-cycle or multicycle problem. It makes the MOSA algorithm as simple as the conventional SA algorithm by allowing one to use the acceptance probability determined solely by the global penalty function. It makes the MOSA optimization calculations very efficient and conducted with affordable computer resources and turnaround time, even with the use of 3-D nuclear design codes for neutronics evaluation of LPs, as demonstrated in the dual-objective four-cycle LP optimization problem of YGN4. Because of these advantages, it is concluded that the ACDPF method offers a very useful and effective means to deal with a variety of fuel management optimization problems, including the singlecycle and multicycle FA LP optimization problems presented in this paper. ACKNOWLEDGMENT This work was supported by the project funded by the Ministry of Knowledge Economy of Korea to develop primary design codes for nuclear power plants.

REFERENCES 1. T. K. PARK et al., ‘‘Multiobjective Loading Pattern Optimization by Simulated Annealing Employing Discontinuous Penalty Function and Screening Technique,’’ Nucl. Sci. Eng., 162, 134 (2009). 2. T. K. PARK, H. G. JOO, and C. H. KIM, ‘‘Performance Analysis of Discontinuous Penalty Function Based Simulated Annealing,’’ Proc. Advances in Nuclear Fuel Management IV, Hilton Head Island, South Carolina, April 12–15, American Nuclear Society (2009) (CD-ROM). 3. P. ENGRAND, ‘‘A Multiobjective Optimization Approach Based on Simulated Annealing and Its Application to Nuclear Fuel Management,’’ Proc. 5th Int. Conf. Nuclear Engineering (ICONE5), Nice, France, May 26–30, 1997. 4. G. T. PARKS and A. SUPPAPITNARM, ‘‘Multiobjective Optimization of PWR Reload Core Designs Using Simulated Annealing,’’ Proc. Int. Conf. Mathematics and Computation, NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

MULTICYCLE FUEL LOADING OPTIMIZATION

Reactor Physics, and Environmental Analysis in Nuclear Application (M&C’99), Madrid, Spain, September 27–30, 1999. 5. P. M. KELLER, ‘‘FORMOSA-P Constrained Multiobjective Simulated Annealing Methodology,’’ Proc. Int. Mtg. Mathematical Methods for Nuclear Applications (M&C2001), Salt Lake City, Utah, September 9–13, American Nuclear Society (2001). 6. K. I. SMITH et al., ‘‘Dominance-Based Multi-Objective Simulated Annealing,’’ IEEE Trans. Evol. Comput., 12, 323 (2008); http://dx.doi.org/10.1109/TEVC.2007.904345. 7. G. T. PARKS, ‘‘Multiobjective Pressurized Water Reactor Reload by Nondominated Genetic Algorithm Search,’’ Nucl. Sci., Eng., 124, 178 (1996). 8. A. YAMAMOTO and K. KANDA, ‘‘Comparison Between Equilibrium Cycle and Successive Multicycle Optimization Methods for In-Core Fuel Management of Pressurized Water Reactors,’’ J. Nucl. Sci. Technol., 34, 882 (1997); http:// dx.doi.org/10.1080/18811248.1997.9733760. 9. K. A. ANDERSON, P. J. TURINSKY, and P. M. KELLER, ‘‘Improvement of OCEON-P Optimization Capabilities,’’ Trans. Am. Nucl. Soc., 96, 201 (2007). 10. K. A. ANDERSON et al., ‘‘OCEON-P Linkage with SIMULATE3,’’ Trans. Am. Nucl. Soc., 96, 203 (2007).

NUCLEAR SCIENCE AND ENGINEERING

VOL. 176

FEB. 2014

239

11. D. J. KROPACZEK, ‘‘COPERNICUS: A Multi-Cycle Nuclear Fuel Optimization Code Based on Coupled In-Core Constraints,’’ Proc. Advances Nuclear Fuel Management IV, Hilton Head Island, South Carolina, April 12–15, 2009, American Nuclear Society (2009) (CD-ROM). 12. T. K. PARK, H. G. JOO, and C. H. KIM, ‘‘Performance Analysis of Discontinuous Penalty Function Based MultiObjective Simulated Annealing for Multi-Cycle Problems,’’ Trans. Am. Nucl. Soc., 104, 904 (2011). 13. T. K. PARK et al., ‘‘The Rearrangement Method for Generating a New Loading Pattern Set in Multi-Objective Simulated Annealing for Multi-Cycle Optimization Problems,’’ Trans. Am. Nucl. Soc., 105, 831 (2011). 14. J. I. YOON et al., ‘‘Design and Implementation for Flux Distribution Solver Module of ASTRA Code,’’ S06NX08-F-2RD-18 Rev.00, KEPCO Nuclear Fuel (2008). 15. S. ZHANG et al., ‘‘Hybrid Harmonics and Linearization Perturbation Method for Fast Loading Pattern Evaluation,’’ Proc. Topl. Mtg. Reactor Physics (PHYSOR 2008), Interlaken, Switzerland, September 14–19, 2008, American Nuclear Society (2008). 16. T. K. PARK, H. G. JOO, and C. H. KIM, ‘‘Dual Reference Perturbation Method for Fast Loading Pattern Evaluation,’’ Trans. Korean Nucl. Soc. Autumn Mtg., Gyeongju, Korea, October 29–30, 2009, Korean Nuclear Society (2009) (CD-ROM).