Optimization of axial enrichment distribution for BWR fuels using scoping libraries and block coordinate descent method

Nuclear Engineering and Design 313 (2017) 84–95

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Optimization of axial enrichment distribution for BWR fuels using scoping libraries and block coordinate descent method Wu-Hsiung Tung ⇑, Tien-Tso Lee, Weng-Sheng Kuo, Shung-Jung Yaur Nuclear Engineering Division, Institute of Nuclear Energy Research, 1000, Wenhua Rd., Jiaan Village, Longtan Township, Taoyuan County 32546, Taiwan, ROC

h i g h l i g h t s
An optimization method for axial enrichment distribution in a BWR fuel was developed.
Block coordinate descent method is employed to search for optimal solution.
Scoping libraries are used to reduce computational effort.
Optimization search space consists of enrichment difference parameters.
Capability of the method to find optimal solution is demonstrated.

a r t i c l e

i n f o

Article history: Received 18 August 2016 Received in revised form 29 November 2016 Accepted 3 December 2016

Keywords: Axial enrichment distribution BWR Scoping libraries Block coordinate descent method Enrichment difference parameter

a b s t r a c t An optimization method has been developed to search for the optimal axial enrichment distribution in a fuel assembly for a boiling water reactor core. The optimization method features: (1) employing the block coordinate descent method to find the optimal solution in the space of enrichment difference parameters, (2) using scoping libraries to reduce the amount of CASMO-4 calculation, and (3) integrating a core critical constraint into the objective function that is used to quantify the quality of an axial enrichment design. The objective function consists of the weighted sum of core parameters such as shutdown margin and critical power ratio. The core parameters are evaluated by using SIMULATE-3, and the cross section data required for the SIMULATE-3 calculation are generated by using CASMO-4 and scoping libraries. The application of the method to a 4-segment fuel design (with the highest allowable segment enrichment relaxed to 5%) demonstrated that the method can obtain an axial enrichment design with improved thermal limit ratios and objective function value while satisfying the core design constraints and core critical requirement through the use of an objective function. The use of scoping libraries effectively reduced the number of CASMO-4 calculation, from 85 to 24, in the 4-segment optimization case. An exhausted search was performed to examine the capability of the method in finding the optimal solution for a 4-segment fuel design. The results show that the method found a solution very close to the optimum obtained by the exhausted search. The number of objective function evaluation (OFV) in the exhausted search is 8707, and the number of OFV in the search using the optimization method is 41. The reduction in OFV shows the efficiency of the optimization method. The optimization designs with more than 4 enriched axial segments were also performed to see if better objective function values can be achieved by having more axial segments. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction A BWR (Boiling Water Reactor) core is widely known for its axial void distribution which strongly affects the axial power shape during operation. To better utilize uranium and to flatten the axial power shape in the fuel, a BWR fuel assembly is designed to have

⇑ Corresponding author. E-mail address: [email protected] (W.-H. Tung). http://dx.doi.org/10.1016/j.nucengdes.2016.12.003 0029-5493/Ó 2016 Elsevier B.V. All rights reserved.

several different axial zones with different enrichments. The ‘axial zone’ is also called a ‘segment’ in this paper. For each reload cycle, the axial enrichment distribution for each reload batch is designed to meet the cycle energy requirement and the safety constraints. The safety constraints that have to be fulfilled in a reload core design are the cold shutdown margin (CSDM), the maximum fraction of limiting critical power ratio (MFLCPR), the maximum fraction of limiting power density (MFLPD) and the maximum of average planar power density ratio (MAPRAT).

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Hida and Yoshioka (1988, 1989 and 1992) formulated the axial enrichment optimization for BWR fuels in a core that is represented by a one-dimensional axial model as a nonlinear programming problem. This nonlinear programming problem was solved using a method of approximation programming that iteratively linearizes a nonlinear function and solves the resulting linear programming problem with a linear programming algorithm. Mochida et al. (1996) investigated the reactivity effects of an axially zoned enrichment core in three different types, based upon the Haling power-exposure iteration (1963) using a threedimensional BWR core simulator. Martin-del-Campo et al. (2001) developed a system that finds the axial locations of different fuel compositions based upon a genetic algorithm in order to minimize the mean enrichment needed to obtain the cycle length under safety constraints. Martin-del-Campo et al. (2002) also implemented a similar system based upon a tabu search method to optimize the axial enrichment design. Ortiz-Servin et al. (2010) proposed a hybrid system composed of a simple greedy search technique and a neural network to optimize the radial and axial design in a fuel assembly using a coupled method. In the objective function used in these works, many core design constraints such as CSDM and minimum critical power ratio were considered. The eigenvalue at the end of cycle (EOC) acquired by a Haling calculation also was considered in the objective function. A complete fuel cycle design is an iterative process between fuel assembly designs and core loading pattern designs. The coupling of these two kinds of design makes the optimization of a fuel cycle design a complex problem that is quite difficult to be solved. Therefore, the two kinds of design usually were optimized individually. For example, Martin-del-campo et al. (2002) optimized the axial enrichment design using a tabu search method and without changing the core loading pattern; whereas, Kobayashi and Aiyoshi (2002) optimized a BWR loading pattern using a twostage genetic algorithm and without altering the fuel assembly design. The optimization method developed in this work mainly focused on optimizing the axial enrichment distribution, meaning that the axial gadolinium design of the fuel assembly to be optimized is not changed in the optimization process. The assembly average enrichment and the axial gadolinium design of the fuel assembly to be optimized are the same as those of a reference fuel assembly selected from a reload core design. To reduce the complexity of the optimization problem, an existing reload core loading pattern and control rod patterns for a BWR reload cycle were utilized for evaluating the axial enrichment distribution through an objective function, which is the weighted sum of core parameters such as MFLPD, MFLCPR and CSDM. The whole cycle k-effs, evaluated with 3-D core simulator, were also integrated into the objective function to find an axial enrichment design that satisfies the core critical requirement based on a given control rod programming of the cycle. A 3-dimensional steady-state core simulator, SIMULATE-3 (Covington, 2007), is used to evaluate the core parameters; whereas, a fuel assembly burnup program, CASMO-4 (Rhodes and Smith, 2007), is used to generate the required cross section data for SIMULATE-3. To perform a CASMO-4 calculation, which generates the cross section data covering nominal and branch conditions, needs about 3.5 h in an HP-8000 workstation. As the CASMO-4 calculation is very time consuming and a large number of cross section data are required in the optimization process, the long computational time is an issue if all the required cross section data are generated by running CASMO-4. As a solution to the computational time issue, scoping libraries are utilized so that the segment enrichments are allowed to be adjusted in SIMULATE-3 without rerunning CASMO-4 explicitly. The scoping libraries

enable the cross section data interpolation functionality such that the cross section data to be used in SIMULATE-3 can be obtained through linear interpolations with the segment enrichment as the interpolation variable. In this work, the enrichment difference parameters are used to functionalize the objective function and form the space for optimization search. Then, the block coordinate descent method (Bertsekas, 1999) is employed to search in the space of enrichment difference parameters to optimize the axial enrichment distribution in a fuel assembly while simultaneously addressing the core design constraints. The axial enrichment optimization problem is addressed as a problem of nonlinear programming with an objective function that does not have readily computable derivatives for determining the search direction. The block coordinate descent method, which is easy to implement, is one of the numerous developed schemes for choosing a search direction without derivatives. Further descriptions about the enrichment difference parameters, the block coordinate descent method and the objective function are provided in the following sections. 2. Optimization methods In a typical BWR fuel assembly design, the axial blankets are implemented by placing natural uranium at both ends of the fuel. The fuel design in this study has the axial blanket at one bottom node and two top nodes (in a total of 25 nodes). The axial enrichment distribution for the fuel segments between the two axial blankets is optimized while keeping the assembly average enrichment unchanged. In general, there are three or four segments of different enrichments between the two axial blankets in the BWR fuel assembly design. In this study, the fuel zone between the two axial blankets is subdivided into 2, 4, 8 and 12 segments as shown in Fig. 1, in which each segment is numbered. For the 2-segment fuel design,

2 seg.

4 seg.

3 1 4

8 seg.

12 seg.

7 8 9 10

7 8 9 10 15 16 17 18 19 20 21 22

11 5 12 2 13 6 14

Top and bottom blanket Fig. 1. Segment numbering for fuels with various number of segments.

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the boundary between the two segments needs to be at the top of a part length rod because the change in fuel rod number is not allowed within a segment in terms of the cross section data generation. The 4-segment fuel is derived by subdividing each segment in the 2-segment fuel into two segments, respectively; and the 8segment fuel is derived similarly. The 12-segment fuel design is derived by subdividing each of the lower 4 segments in an 8segment fuel into two segments, respectively. 2.1. Enrichment difference parameters Instead of using the segment enrichments, the enrichment difference parameters and the average of segment enrichments are used to define the enrichment distribution in the fuel zone between the two axial blankets. As the average of segment enrichments takes one degree of freedom, n  1 enrichment difference parameters are required to define the enrichment distribution of a fuel assembly with n segments. The enrichment difference parameter is defined as the enrichment difference between the two smaller segments that are subdivided from one larger segment.

Di ¼ Eu  El ; Em ¼

i ¼ 1; 2; . . . ; n  1; n 2 Z; u ¼ 2i  1; l ¼ 2i:

Lu Nu Eu þ Ll N l El : Lu Nu þ Ll Nl

ð1Þ ð2Þ

2.2. Block coordinate descent method The block coordinate descent method solves the optimization problem by minimizing the objective function along each of the coordinate directions one by one orderly. In this study, each enrichment difference parameter represents a coordinate direction. Thus, the coordinate directions for this axial enrichment optimization problem are D1, D2,. . .,Dn1 for a fuel with n segments, and the objective function with n1 coordinate directions is denoted as f(D1,. . .,Dn1). Each coordinate direction Di is constrained in a feasible range: UB DLB i 6 Di 6 Di ;

In the equations above, Di is the enrichment difference between two segments; Eu is the enrichment of the upper segment, which is numbered as u; El is the enrichment of the lower segment, which is numbered as l; Em is the average of Eu and El; n is the number of segments between the top and bottom blanket; Lu and Ll are the length of the segments numbered as u and l; and Nu and Nl are the number of fuel rods in the segments numbered as u and l. By using Eqs. (1) and (2), Eu and El can be expressed as a function of Di and Em:

Ll N l Di Eu ¼ Em þ ; Lu Nu þ Ll Nl

ð3Þ

Lu Nu Di : Lu Nu þ Ll Nl

ð4Þ

El ¼ Em 

mentioned in the previous paragraph. Thus, both E3 and E4 are defined by E0, D1 and D2. Similarly, E5 and E6 are defined by E0, D1 and D3. As a result, the enrichments of the four segments in a 4-segment fuel are defined by E0, D1, D2 and D3. Following the same definition procedure, the enrichments of the n segments, in a fuel design with n segments, can be defined by E0, D1,. . .,Dn1. Assuming that the average of segment enrichments in the fuel design is a given value that is not subject to change during the optimization, the axial enrichment distribution optimization becomes a problem to find a set of (D1,. . .,Dn1) that minimizes the objective function to be introduced in Section 3.1.

Referring to Fig. 1, the two segments in a 2-segment fuel design are numbered as ‘‘1” and ‘‘2,” and the corresponding notation for the enrichments in these two segments are E1 and E2. The average of E1 and E2 is denoted as E0, and the difference between the two enrichments is denoted as D1 per equation (1). According to equations (3) and (4), the enrichments E1 and E2 can be defined by the average of and the difference between these two enrichments, which are E0 and D1. For a 4-segment fuel design, the four segments are numbered as ‘‘3”, ‘‘4”, ‘‘5” and ‘‘6,” and the corresponding notations for the enrichments in these four segments are E3, E4, E5 and E6. As the segment ‘‘1” is subdivided into the segments ‘‘3” and ‘‘4”, the average of E3 and E4 is equal to E1. Similarly, the average of E5 and E6 is equal to E2. According to Eq. (1), D2 is the enrichment difference between E3 and E4, and D3 is the enrichment difference between E5 and E6. Per equations (3) and (4), the enrichments E3 and E4 are defined by D2 and E1; whereas, E1 is defined by E0 and D1 as

8Di 2 R;

ð5Þ

UB where DLB i is the lower bound of Di and Di is the upper bound of Di. LB UB Di and Di are determined based upon the lowest allowable and the highest allowable enrichment of a segment. The upper bound of Di, which is a positive value, is determined by increasing Di until the enrichment of the upper segment reaches the highest allowable enrichment or the enrichment of the lower segment reaches the lowest allowable enrichment. Similarly, the lower bound of Di, which is a negative value, is determined by decreasing Di until the enrichment of the upper segment reaches the lowest allowable enrichment or the enrichment of the lower segment reaches the highest allowable enrichment. The optimization problem for the axial enrichment distribution can be described mathematically below.

Minimizing an objective function f(D), subject to D 2 D, UB D = {D | DLB i 6 Di 6 Di , Di 2 R, i = 1,. . .,n  1}, D = (D1,D2,. . .,Dn1). Assuming that for every D 2 D and every i = 1,. . .,n  1, the optimization problem minimizing f(D1,. . .,Di1,n,Di+1,. . .,Dn1), subject to n 2 Di, has at least one solution. Then, the block coordinate descent algorithm will generate the next iterate Dk+1 = (D1k+1,D2k+1,. . ., Dn1k+1), given that the current iterate is Dk = (D1k,D2k,. . .,Dn1k), according to the iteration scheme


 kþ1 k k Dkþ1 2 argminf Dkþ1 i 1 ;...;Di1 ;n;Diþ1 ;...;Dn1 ; i ¼ 1;...;n  1: n2Di

ð6Þ

In each iteration, the objective function is minimized with respect to each of the ‘‘block coordinate” vector Dki , which is taken in a cyclic order. When the differences in all Di’s between two successive iterations are less than 0.1, the iteration converges and the block coordinate descent search is terminated.

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3. Application to axial enrichment optimization 3.1. Objective function The objective function is the weighted sum of several core parameters so that the quality of the axial enrichment distribution in a fuel assembly is determined based upon the core calculation results. In calculating the core parameters, the core loading pattern and the operation strategy, such as control rod patterns and core flows, are fixed when the axial enrichment distribution in a fuel assembly changes. Eq. (7) is the mathematical representation of the objective function that provides the flexibility to enhance the improvements of some core parameters by giving different weighting factors for different core parameters.

f ðD1 ; . . . ; Dn1 Þ ¼ C 

6 X wi xi ;

ð7Þ

i¼1

where C = a constant; n = the number of segments between the top and bottom blanket; and wi = a weighting factor for the parameter xi. The six xi’s used in Eq. (7) are listed below: x1 = MFLPDlim  MFLPDcycle-max; x2 = MFLCPRlim  MFLCPRcycle-max; x3 = MAPRATlim  MAPRATcycle-max; x4 = CSDMcycle-min  CSDMlim; x5 = BOCHEX  BOCHEXlim; x6 = EOCHEX  EOCHEXlim; and x7 = DKlim  DKcycle-max wherein, MFLPDlim = core design limit of MFLPD; MFLPDcycle-max = maximum MFLPD in a cycle; MFLCPRlim = core design limit of MFLCPR; MFLCPRcycle-max = maximum MFLCPR in a cycle; MAPRATlim = core design limit of MAPRAT; MAPRATcycle-max = maximum MAPRAT in a cycle; CSDMlim = core design limit of CSDM; CSDMcycle-min = minimum CSDM in a cycle; BOCHEX = hot excess reactivity (HEX) at the beginning of cycle (BOC); BOCHEXlim = core design limit of the HEX at BOC; EOCHEX = HEX at EOC; EOCHEXlim = core design limit of the HEX at EOC; DKlim = k-effective deviation limit; DKcycle-max = the maximum of |k-eff  k-efftrg| in a cycle; and k-eff, k-efftrg = calculated k-effective and target k-effective. In addition, if xi > 0.0, then wi = wi,c; otherwise, wi = wi,p, wherein wi,c is a weighting factor used to credit the designs that meet the design limit, and wi,p is a weighting factor used to penalize the designs that violate the design limit 3.2. Generation of cross section data As the axial enrichment distribution changes, the enrichments of the fuel segments change correspondingly. Each time when the enrichment of a segment changes, the cross section data of that segment usually has to be generated again by performing CASMO4 calculations, so that a 3-D core simulation can be performed to

87

evaluate the core parameters as a function of cycle exposure, according to the changed axial enrichment distribution. The radial enrichment distribution of a segment was considered by normalizing the radial enrichment distribution of a reference fuel segment to a new segment enrichment corresponding to an axial enrichment distribution for optimization. The enrichments of the fuel rods in the reference fuel segment are adjusted proportionally to attain the required segment enrichment specified by the axial enrichment distribution. For example, if the enrichment of a reference segment is Eref and the segment enrichment corresponding to a new axial enrichment distribution is Enew, then the enrichment of each fuel rod in the reference segment is adjusted by multiplying the enrichment by the ratio of Enew/Eref. After the adjustment of fuel rod enrichment, CASMO-4 is used to generate the cross section data for a segment. Adjusting the fuel rod enrichment to attain the required segment enrichment will result in the change in the local peaking factor (LPF), which may affects calculation results of MFLPD and MFLCPR. This is an inevitable effect accompanying a change in the segment enrichment because it is quite difficult to vary the segment enrichment by adjusting the fuel rod enrichments without making any alteration to LPF. Although the effect of LPF variation is not separable, the optimization method was shown to evolve the design toward optimum in next section. This is due to that this LPF variation is considered a minor impact to the objective function compared to the impact to the objective function resulting from the axial peaking factor variation caused by the change in the axial enrichment distribution. As it is very time consuming for CASMO-4 to complete calculations covering all the nominal and branch conditions, scoping libraries are employed to reduce the computational efforts of CASMO-4. By generating the cross section data of a fuel segment at multiple enrichments and combining the cross section data of each segment into a file, a scoping library is created. With the use of a scoping library, SIMULATE-3 can obtain the required cross section data through a linear interpolation functionality using the segment enrichment as the interpolation variable. As a result, generating cross section data using CASMO-4 during the optimization is not required. 3.3. Flowchart Fig. 2 illustrates the procedure of the block coordinate descent method for optimizing the axial enrichment distribution. The search starts from a flat enrichment distribution in the axial direction, i.e., all the enrichment difference parameters are 0.0. Each coordinate direction Di is searched for in a cyclic order, as shown in Fig. 2. The optimization search progresses from D1 to Dn1 sequentially in an iteration, and is repeated until the convergence criterion is met. As the differences in all Di’s between two successive iterations are less than 0.1, the convergence criterion is met and the optimization search stops. As the objective function is not readily differentiable, the search for optimum in one coordinate direction has to be determined using some kind of method that can identify where the optimum is in that coordinate direction. In this study, it was found that the objective function, with respect to a coordinate direction Di, can be approximated with a fourth-order polynomial of Di to helps locate the Di value that minimizes the objective function in that coordinate direction. The characteristic of the objective function versus Di will be illustrated in next section. For a coordinate direction Di, the objective function is evaluated at 5 different Di values which are uniformly distributed in the feasible range of Di to give the required data for the fourth-order polynomial fitting. Upon having the coefficients from the fitting, the fourth-order polynomial is used to locate the value that minimizes

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Start k=0

i=1 Search

i=i+1

No

i≥n-1? Yes

k=k+1

No

k≥1? Yes

No

Converged? Yes

End Fig. 2. Flow chart of the block coordinate descent method.

the objective function in the direction Di (denoted as Di,opt in Fig. 2) by extensively calculating the polynomial value over the range of Di. As the objective function is more sensitive to D1, the search range of D1 is gradually reduced by discarding the range of D1 that does not contain the optimum in that direction. The narrowing down process of the search range of D1 continues until the range is less than 0.4, which is small enough to give a good polynomial fitting of the objective function, based upon the experiences. Then, the fourth-order polynomial fitting is performed and the polynomial is used to locate D1,opt by extensively calculating the polynomial value over the narrowed range of D1. 4. Results and discussions The axial enrichment optimization was applied to one of the two reload batches in a selected reload cycle of a BWR4 plant, which has 408 fuel assemblies in the core. In the selected reload cycle, the assembly average enrichments of the two fresh fuel batches are 4.058% and 4.064%, and the corresponding assembly numbers of these two batches are 76 and 36, respectively. The fresh fuel batch with more assemblies, i.e., the batch with the assembly average enrichment of 4.058%, was selected for the axial enrichment optimization. The selected fuel batch has four segments in the zone between bottom and top axial blanket. Fig. 3 is the core loading map that shows the channels where the fresh fuel assemblies are allocated.

The axial enrichment distribution in the selected fuel batch was adjusted according to the optimization method presented in the previous sections, while the axial gadolinium design in this fuel batch was kept unchanged during the optimization process; in other words, only the axial enrichment distribution in the selected fuel batch was varied. The corresponding SIMULATE-3 inputs were revised to reflect the change in the axial enrichment distribution in the selected fuel batch. The core loading pattern and the control rod patterns in the selected reload cycle remained unchanged in the evaluation of the core parameters. The evaluation results of the core parameters were then used to calculate the objective function value that represents the performance of the axial enrichment distribution in the given core. The lowest allowable and highest allowable segment enrichment were set to 3.0 % and 5.0%, respectively. The highest allowable segment enrichment in this optimization study was relaxed to 5% to provide a broader search range, in order to gain some insights from the results. Nevertheless, a fuel segment with the enrichment of 5% would be impractical in the lattice design (i.e., the radial design of a fuel segment) as it implies all the fuel rods in the segment will have the enrichment of 5%, which violates the design constraint that requires some of the fuel rods in the peripheral location to have enrichments less than 5%. The weighting factors for the calculation of the objective function are listed in Table 1. The weighting factors related to penalties are given with a higher weighting value compared to the weighting factors related to credits, in order to exclude the axial enrichment designs that fail to meet the design limits listed in Table 2. In addition, the k-efftrg for each cycle exposure is shown in Fig. 7. Among the credit related weighting factors, those for the thermal limit ratios (MFLPD, MFLCPR and MAPRAT) are given with a higher weighting value, such that an axial enrichment design with improved thermal limit ratios can be obtained. The credit related weighting factor for BOC HEX is set to 0.0 because the axial enrichment design with a higher BOC HEX is not particularly required in this study. For a 4-segment fuel design commonly used in a reload core, the characteristics of the objective function in the coordinate directions D1, D2 and D3 are shown in Figs. 4–6, respectively. These figures show that there is an optimal objective function value in each coordinate direction. Also shown in these figures is the sensitivity of objective function to each Di. It was found that the objective function is more sensitive to D1 than to D2 or D3. This is reasonable as D1 is the enrichment difference between the two segments that occupy 132 in. long; whereas D2 is the enrichment difference between the two segments that occupy 42 in. long and D3 is the enrichment difference between the two segments that occupy 90 in. long. As D1 affects the objective function values more than the other enrichment difference parameters, it is reasonable to search for the optimal solution in the coordinate direction D1 first, following by the optimal solution search in the coordinate direction D2 and D3. For a fuel assembly with n segments, it is expected that the sequential search order (D1, D2,. . .,Dn1) is appropriate as the enrichment difference parameters with a smaller index are expected to have more impact on the objective function values when those parameters vary. As shown in Table 3, the search for the optimized axial enrichment distribution for the selected 4-segment fuel batch stopped at the 2nd iteration because the differences in all Di’s between 1st and 2nd iteration are less than 0.1. The objective function value of 0.944 at the end of the 2nd iteration was reported as the optimal value. The three enrichment difference parameters listed in the 2nd iteration of Table 3, along with the assembly average enrichment of 4.058%, define the optimal axial enrichment distribution for the 4-segment fuel. Using the same optimal axial enrichment

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I/J

1

2

3

4

5

6

7

8

9

10

11

12

1

1

13

14

15

16

17

18

19

20

21

22

1 2 3

2

4

2

5

2

2

6

2 1

2

7

2

8

2

9

2

10

2

1

1

2 1

2 2

2 1

2 2

2

2

1

2

2 1

2

2

2 2 2

1

2

2

2

2

1

2

1

1

2 1

2

11

1

1

2

2

2

2

1

12

1

1

2

2

2

2

1

13

2

14

2

15

2 1

2

16

1 1

2

2

2

2 2

1

2 2

2

1 2

2

2

19

2

1 2

20

2

2 1

1

2 2

1 2 1

1

1

1

2 2

2

2

2

2

2

2 1

1 2

2

17 18

1

2 2

2 2

21 22 1

Fresh fuel batch with assembly enrichment of 4.064%

2

Fresh fuel batch for optimization with assembly enrichment of 4.058% Exposed fuel assembly Fig. 3. Core loading map.

Table 1 Weighting factors used for the calculation of objective function. Index, i

Parameter

Credit, wi,c

Penalty, wi,p

1 2 3 4 5 6 7

MFLPD MFLCPR MAPRAT CSDM BOC HEX EOC HEX DK

5 5 5 1 0 1 0

5 5 5 5 5 5 15

Table 2 Limit values of core design parameters. Parameter

Type of Limit

Limit Value

MFLPD MFLCPR MAPRAT CSDM BOC HEX EOC HEX DK

Upper Upper Upper Lower Lower Lower Upper

0.92 0.92 0.92 1.0%Dk 1.0%Dk 0.0%Dk 0.05%Dk

limit limit limit limit limit limit Limit

distribution, the core parameters were re-evaluated with the use of an exact cross section library, which gives an objective function value of 0.916. As the scoping library is an approximation to the exact cross section library, the difference in the objective function values between the two evaluations using the scoping library and the exact cross section library can be expected. In this study, the axial enrichment distributions are designed to meet with the core critical requirement along the cycle. Fig. 7 compares the cycle k-effs of the 4-segment optimization design with

the target k-effs, which was derived by performing a SIMULATE-3 cycle depletion of the selected reload cycle. In the objective function, the k-eff deviation from the target at every exposure point except the EOC of the cycle was considered. If the maximum of the k-eff deviations is larger than 0.5 mk, a penalty by weighting the maximum deviation with a factor of 15 is imposed on the objective function value. Large penalty weighting factor is necessary because the deviations, in the order of mk, are quite small. The results show that all the k-effs of the optimization case are within 0.5 mk of their corresponding target values. The core parameters and the objective function values for the 4segment fuel optimization design and the reference design are shown in Table 4. These two designs have the same axial zoning and the same axial gadolinium design. For the optimization design, the core parameters were evaluated using the scoping library and the exact cross section library, respectively; whereas the evaluation of the core parameters for the reference design used the exact cross section library only. The objective function value of the optimization design evaluated using the exact cross section library is 0.916, which is an improvement as compared to the objective function value of 0.968 of the reference design. By comparing the core parameters in the optimization design with those in the reference design, it was found that the optimization design achieves a better objective function value due to the improved MFLPD, MFLCPR and MAPRAT. The improved thermal limit ratios was the result of using higher weighting factors to credit the axial enrichment designs that have thermal limit ratios smaller than the design limits. According to Table 4, the differences in core evaluation results between the two cases in the optimization design are small. The differences exist because the cross section data from scoping libraries interpolation are an approximation to the exact cross

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Fig. 4. Objective function characteristic in the coordinate direction D1.

Fig. 5. Objective function characteristic in the coordinate direction D2.

section data. Due to the approximation, the search for the optimal axial enrichment distribution could be misled somewhat when using scoping libraries for core evaluations. However, it is expected that the optimal solution found by using scoping libraries would be close to the optimal solution found by using exact cross section libraries for core evaluations. Fig. 8 is the axial enrichment distribution of the 4-segment fuel optimization design for the selected fuel batch. The improvements of MFLPD and MAPRAT were the results of a lower enrichment at the lowest segment. The enrichment of the lowest segment in the optimization design is lower than that in the reference design, which reduced the axial peaking and improved MFLPD and

MAPRAT. The enrichment of the top segment in the optimization design is the highest among the four enriched segments of the fuel, and apparently higher than that in the reference design. However, MFLCPR is not deteriorated because this segment is only 12 in. long and the cycle limiting MFLCPR occurred at the axial location below this segment. In order to confirm whether the optimization method is capable of finding the optimal axial enrichment design, an exhausted search method that evaluated all the combinations of D1, D2, and D3 values was performed for the 4-segment fuel design case using the scoping library for core evaluations. All the Di values in the evaluation are within their feasible ranges. The step size of the

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Fig. 6. Objective function characteristic in the coordinate direction D3.

Table 3 Enrichment difference parameters in the iterations of a 4-segment fuel design. Iteration

D1

D2

D3

Objective Function

1 2

0.1959 0.1954

0.9310 0.9310

0.1954 0.1956

0.9441 0.9441

Fig. 7. Target k-eff and the k-effs of the optimization cases.

exhausted search for each Di value was 0.1 such that the search can be completed in a reasonable time while having an enough precision in locating the optimal solution. The exhausted search performed 8707 OFVs; whereas, the number of OFV performed by

the optimization search is 41. In comparison with the exhausted search, the optimization method has successfully reduced the number of OFV required to find the optimal solution, demonstrating the efficiency of the method.

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Table 4 Evaluation results of the 4-segment fuel optimization design and the reference design. Parameters

MFLPD MFLCPR MAPRAT CSDM (%) BOC HEX (%) EOC HEX (%) Objective Function

Optimization Design (scoping library)

Optimization Design (exact cross section library)

Reference Design (exact cross section library)

0.873 0.869 0.821 1.151 1.732 0.016 0.944

0.885 0.869 0.827 1.200 1.736 0.003 0.916

0.891 0.874 0.833 1.225 1.766 0.000 0.968

The optimal objective function value of the exhausted search is 0.921, and the corresponding D1, D2 and D3 values are 0.256, 0.990 and 0.238, respectively. The exhausted search method found an optimal solution with the objective function value being slightly superior yet very close to that of the optimization design, which is 0.944. Table 5 shows the core parameter evaluation results obtained by the exhausted search and the optimization search, and Fig. 9 compares the axial enrichment distributions of these two searches. These results demonstrate the capability of the optimization method to optimize the axial enrichment design in terms of using a scoping library for core evaluations. The axial enrichment optimizations for fuel designs with the number of axial segments more than four were also investigated in this study, to see if better objective function values can be achieved with the increased number of segments. The optimization results for 4-segment, 8-segment and 12-segment designs are shown in Table 6. The k-effs of these cases are all within 0.5 mk of the target values as shown in Fig. 7. The evaluations of the core parameters for the three kinds of axial enrichment designs were executed using the scoping library, and the results showed that all the core parameters have met the design limits listed in Table 2.

Table 5 Evaluation results of the optimization design and the design from the exhausted search in the 4-segment case. Parameters

Optimization Design (scoping library)

Design from Exhausted Search (scoping library)

MFLPD MFLCPR MAPRAT CSDM (%) BOC HEX (%) EOC HEX (%) Objective Function

0.873 0.869 0.821 1.151 1.732 0.016 0.944

0.876 0.867 0.823 1.135 1.739 0.005 0.921

As expected, better objective function values were achieved with the increased number of segments, which is reasonable because the fuel design with more segments has more flexibility in adjusting the axial enrichment distribution to improve the core parameters. The optimized axial enrichment distribution for the 8segment and 12 segments fuel designs are shown in Figs. 10 and 11. As the highest allowable segment enrichment in this optimization study was relaxed to 5%, Figs. 8, 10 and 11 show axial enrichment distributions that might not be feasible. To generate a feasible design, the highest allowable enrichment of a segment should be lower than 5% such that a practical lattice design can be achieved under the design constraints. These constraints could be maximum allowable fuel rod enrichment, maximum allowable corner rod enrichment, etc. Another restriction on the feasibility of the optimized axial enrichment distribution is the number of axial zone allowed in the fuel rod manufacture. Too many axial zones could cause the manufacturing cost to increase. In general, four enriched axial zones are acceptable. The optimization of the axial enrichment distribution for the designs with axial zone more than four could show the amount of benefit gained by increasing the number of axial zone. The optimization results will help in deciding whether increasing the number of axial zone is worthy.

Fig. 8. Optimal axial enrichment distribution for a 4-segment fuel design.

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Fig. 9. Optimal axial enrichment distributions for a 4-segment fuel design using two different search methods.

Table 6 Optimization results of the fuel designs with different segments. Parameters

4 segments

8 segments

12 segments

MFLPD MFLCPR MAPRAT CSDM (%) BOC HEX (%) EOC HEX (%) Objective Function

0.873 0.869 0.821 1.151 1.732 0.016 0.944

0.868 0.868 0.810 1.114 1.793 0.001 0.813

0.868 0.869 0.804 1.141 1.762 0.000 0.770

multiple solutions for a core design. The change of weighting factors will result in the change of the ELs needed in the optimization process. By using scoping libraries, rerunning CASMO-4 to generate cross section data for each EL is not required and computational time can be saved in an optimization process. Utilizing scoping libraries provide more flexibility for the optimization search because the weighting factors in the objective function can be adjusted freely to accomplish multiple designs without rerunning CASMO-4, which is a burden in terms of computational effort. 5. Conclusions

In the optimization process of the 4-segment case, the total required number of enrichment level (EL) for the four enriched segments is 85. Each EL needs a corresponding cross section data which could be generated either by performing a CASMO-4 calculation or by using scoping libraries with linear interpolation functionality. If CASMO-4 was used to generate the cross section data, the total number of CASMO-4 calculation required is 85 for the 4segment optimization case. To perform a CASMO-4 calculation, which generates the cross section data covering nominal and branch conditions, needs about 3.5 h in an HP-8000 workstation. Consequently, performing those 85 CASMO-4 calculations in an optimization process is almost not acceptable. In this study, scoping libraries were used to reduce the amount of CASMO-4 calculation to save the computational time in an optimization process. By using the scoping libraries to generate the cross section data for the segments in the 4-segment optimization case, the total number of CASMO-4 calculation required is 24, five CASMO-4 calculations for each segment to create a scoping library and one CASMO-4 calculation for each segment to evaluated core parameters using exact cross section data as the axial enrichment design is finalized. The reduction of CASMO-4 calculation number from 85 to 24 saves a lot of computational time. As the number of axial zone increased in the optimization case, the saving in the amount of CASMO-4 calculation would be more. In the optimization search, it is quite possible to use multiple sets of weighting factors for sensitivity study or to generate

An optimization method for the axial enrichment design of BWR fuel assemblies has been developed and demonstrated for its capability to find a solution close to the optimum. The optimization method features: (1) employing the block coordinate descent method to find the optimal solution in the space of enrichment difference parameters, (2) using scoping libraries to reduce the amount of CASMO-4 calculation, and (3) integrating a core critical constraint into the objective function that is used to quantify the quality of an axial enrichment design. The method effectively reduced the number of OFV and CASMO-4 calculation required in the optimization process of an axial enrichment design, saving a large amount of computational time. For the axial enrichment design of the 4-segment case, the exhausted search performed 8707 OFVs; whereas, the number of OFV performed by the optimization search is 41. In performing these 41 OFVs, the total EL of the four segments is 85, which is also the required number of CASMO-4 calculation if exact cross section libraries are to be used for those ELs. By using scoping libraries, the number of CASMO-4 calculation was reduced from 85 to 24, releasing the burden in computation for the optimization. In terms of using a scoping library for core evaluations, the exhausted search method found an optimal solution with an objective function value being slightly superior yet very close to that found by the optimization method in the 4-segment axial enrichment optimization case. The axial enrichment optimizations for

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Fig. 10. Optimal axial enrichment distribution for an 8-segment fuel design.

Fig. 11. Optimal axial enrichment distribution for a 12-segment fuel design.

fuel designs with the number of axial segments more than four were also investigated in this study. As expected, better objective function values were achieved with the increased number of axial segment because the fuel design with more segments has more flexibility in adjusting the axial enrichment distribution for optimization. Many core design constraints were considered in the objective function. These constraints are thermal limit ratios, cold shutdown margin and cycle energy. The k-effs, evaluated with 3-D core

simulator, were also integrated into the objective function to find an axial enrichment design that satisfy the core critical requirement based on a given control rod programming of the cycle. By using the optimization method and crediting the thermal limit ratios with larger weighting factors, axial enrichment designs that have improved MFLPD, MFLCPR and MAPRAT were found when the highest allowable segment enrichment was relaxed to 5%. It was demonstrated that the optimization method can obtain an axial enrichment design with improved thermal limit ratios and objec-

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tive function value while satisfying the core design constraints and the core critical requirement through the use of an objective function. The cross section data from scoping libraries interpolation are an approximation to the exact cross section data, and thus there are differences in core evaluation results between the case using the scoping library and the one using the exact cross section library for core evaluation. Due to these differences, the search for the optimal axial enrichment distribution could be misled somewhat, yet it is expected that the optimal solution found by using scoping libraries would be close to that found by using exact cross section libraries for core evaluation. Though the problem of being slightly misled in the optimization process could be avoided by using exact cross section libraries instead, one has to endure much longer computational time once the scoping libraries are not used. Besides, utilizing scoping libraries provide more flexibility for the optimization search because the weighting factors in the objective function can be adjusted freely to accomplish multiple designs without rerunning CASMO-4, which is a burden in terms of computational effort. A comprehensive axial design in a fuel assembly should include the axial enrichment design and the axial gadolinium design. In this study, these two kinds of design were dealt with separately by optimizing the axial enrichment design while fixing the axial gadolinium design, in order to simplify the optimization problem. In the future, (1) the axial gadolinium design should be incorporated into the optimization method such that the optimization could have practical applications in the reload core design practice, (2) a lattice design automation method, such as that developed by Tung et al. (2015) that optimize the radial enrichment distribution

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within a segment, can be integrated with an axial design optimization method to complete a fuel assembly design. References Covington, L.J., 2007. SIMULATE-3, advanced three-dimensional two-group reactor analysis code, Studsvik of America (SOA-95/15 Rev 4, proprietary). Bertsekas, Dimitri P., 1999. Nonlinear Programming. Athena Scientific, Belmont, Mass. Haling R.K., 1963. Operating strategy for maintaining an optimum power distribution throughout life, Topl. Mtg. Nuclear Performance of Power Reactor Cores, TID 7672, American Nuclear Society. Hida, K., Yoshioka, R., 1988. Optimal axial enrichment distribution of the boiling water reactor fuel under the Haling strategy. Nucl. Technol. 80, 423–430. Hida, K., Yoshioka, R., 1989. Optimization of axial enrichment and gadolinia distributions for BWR fuel under control rod programming. J. Nucl. Sci. Technol. 26, 492–500. Hida, K., Yoshioka, R., 1992. Optimization of axial enrichment and gadolinia distributions for BWR fuel under control rod programming, (II), optimization of reload fuel. J. Nucl. Sci. Technol. 29, 313–324. Kobayashi, Y., Aiyoshi, E., 2002. Optimization of boiling water reactor loading pattern using two-stage genetic algorithm. Nucl. Sci. Eng. 142, 119–139. Martin-del-Campo, C. et al., 2001. AXIAL: a system for boiling water reactor fuel assembly axial optimization using genetic algorithms. Ann. Nucl. Energy 28, 1667–1682. Martin-del-Campo, C. et al., 2002. Boiling water reactor fuel assembly axial design optimizations using tabu search. Nucl. Sci. Eng. 142, 107–115. Mochida, T. et al., 1996. Improvement in boiling water reactor uranium utilization and operating experience with burnup increase. Nucl. Technol. 116, 91–107. Ortiz-Servin, J.J. et al., 2010. GreeNN: A hybrid method for the coupled optimization of the axial and radial design of BWR fuel assemblies. Prog. Nucl. Energy 52, 249–256. Rhodes J., Smith K., 2007. CASMO-4, a fuel assembly burnup program, user’s manual. Studsvik of America (SSP-01/400 Rev 5, proprietary). Tung, W.H. et al., 2015. Fuel lattice design in a boiling water reactor using a knowledge-based automation system. Nucl. Eng. Des. 293, 63–74.