Fuel lattice design in a boiling water reactor using a knowledge-based automation system

Nuclear Engineering and Design 293 (2015) 63–74

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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes

Fuel lattice design in a boiling water reactor using a knowledge-based automation system 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 automation system was developed for the fuel lattice radial design of BWRs. An enrichment group peaking equalizing method is applied to optimize the design. Several heuristic rules and restrictions are incorporated to facilitate the design. The CPU time for the system to design a 10x10 lattice was less than 1.2 h. The beginning-of-life LPF was improved from 1.319 to 1.272 for one of the cases.

a r t i c l e

i n f o

Article history: Received 31 October 2014 Received in revised form 15 May 2015 Accepted 19 July 2015

a b s t r a c t A knowledge-based fuel lattice design automation system for BWRs is developed and applied to the design of 10 × 10 fuel lattices. The knowledge implemented in this fuel lattice design automation system includes the determination of gadolinium fuel pin location, the determination of fuel pin enrichment and enrichment distribution. The optimization process starts by determining the gadolinium distribution based on the pin power distribution of a flat enrichment lattice and some heuristic rules. Next, a pin power distribution flattening and an enrichment grouping process are introduced to determine the enrichment of each fuel pin enrichment type and the initial enrichment distribution of a fuel lattice design. Finally, enrichment group peaking equalizing processes are performed to achieve lower lattice peaking. Several fuel lattice design constraints are also incorporated in the automation system such that the system can accomplish a design which meets the requirements of practical use. Depending on the axial position of the lattice, a different method is applied in the design of the fuel lattice. Two typical fuel lattices with U235 enrichment of 4.471% and 4.386% were taken as references. Application of the method demonstrates that improved lattice designs can be achieved through the enrichment grouping and the enrichment group peaking equalizing method. It takes about 11 min and 1 h 11 min of CPU time for the automation system to accomplish two design cases on an HP-8000 workstation, including the execution of CASMO-4 lattice code. The results obtained with the application of the implemented system show the potential of the proposed methodology in the fuel lattice design automation for BWRs. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The BWR (Boiling Water Reactor) fuel lattice design includes the determination of several parameters, such as the fuel pellet enrichment, the number of fuel pellet enrichment, the enrichment distribution, the gadolinium weight percent, the number of

∗ Corresponding author. Tel.: +886 3 4711400x6036; fax: +886 3 4711404. E-mail address: [email protected] (W.-H. Tung). http://dx.doi.org/10.1016/j.nucengdes.2015.07.035 0029-5493/© 2015 Elsevier B.V. All rights reserved.

gadolinium fuel pin, and the gadolinium distribution. The complexity of fuel lattice design not only comes from the many degrees of freedom but also derives from the coupling with the core design. As a fuel lattice will reside in a core for many cycles, the fuel lattice design will be even more complex if the multi-cycle performance is considered. In the design process, several iterations between the fuel lattice design and the core loading pattern design are performed, and the fuel lattice design may be adjusted according to the feedback of core parameters, such as MFLCPR (Maximum Fraction Limiting Critical Power Ratio), MFLPD (Maximum Fraction

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Limiting Power Density), etc. For a fixed core loading pattern and control rod pattern, it is possible to correlate the core design parameters to the fuel lattice design parameters. A fuel lattice with better lattice design parameters is expected to have better core design parameters. For example, a fresh fuel bundle with a lower local peaking factor (LPF) at its bottom portion is expected to have a better MFLPD performance in the core loading pattern design. Therefore, in this study, LPF is the optimization target for the fuel lattice located at the lower portion of a fuel bundle. On the other hand, F-eff, which is the weighted sum of the power fractions of a fuel pin and its neighboring fuel pins, is the optimization target for the fuel lattice located at the upper portion of the fuel bundle. In order to reduce the complexity, an existing reload core loading pattern is used to evaluate the performance of the fuel lattice design. In this study, the fuel pin power distribution and the burnup status of a fuel lattice are obtained using CASMO-4 (Rhodes and Smith, 2007); whereas, the core performance parameters, such as MFLPD and MFLCPR, are calculated using SIMULATE-3 (Covington, 2007). Several kinds of optimization techniques have been used to solve this complex space-composition combinatorial problem, including the tabu search (Francois et al., 2003), the tabu search method and fuzzy logic (Martín-Del-Campo et al., 2007a), the genetic algorithms and knowledge (Martín-Del-Campo et al., 2007b), the path re-linking (Castillo et al., 2011), the ant-colony (Montes et al., 2011), and the hybrid method of greedy search and neural network (Ortiz-Servin et al., 2010), etc. Most of them apply the heuristic rules to increase the efficiency of the optimization process, and two heuristic rules are commonly used. The first one is that the four corner fuel pins should have the lowest enrichment in the fuel lattice, and the other one is that the gadolinium fuel pin is not allowed to exist in the periphery of a fuel lattice. Cuevas et al. (2002) used a modified linear programming technique to optimize the enrichment distribution in a typical LWR (Light Water Reactor) assembly, and the optimization process is initiated from flattening the enrichment distribution until a target maximum local power peaking factor is achieved. The optimal locations and values for a reduced number of fuel rod enrichment groups are obtained for an inputted target maximum local power peaking factor by applying the sensitivity to change techniques. Lin and Lin (2012) developed an automatic procedure based on a particle swarm optimization (PSO) algorithm and a local search with some heuristic rules to simplify the search for the radial enrichment and gadolinium distribution of BWR fuel lattices. Lin and Lin consider a different object function for the different axial zone in their study. The minimization of LPF and F-eff has to be taken care of separately, because the minimization of LPF doesn’t mean the minimization of F-eff. The MFLPD of a core is sensitive to the design of fresh fuel lattices located at a lower portion of a fuel bundle, and therefore, the minimization of LPF should be considered for these fuel lattices. The MFLCPR of a core is mostly sensitive to the design of fuel lattices located at a near top portion of a fuel bundle, and hence, the minimization of F-eff should be considered for these fuel lattices. As burnable poison is commonly used in the BWR fuel lattice to suppress Kinf at the beginning of life (BOL), the distribution of gadolinium has to be addressed. In this study, the first step of the automation process is to determine the gadolinium distribution, using a specified gadolinium concentration and number of gadolinium fuel pin as a design constraint. Several heuristic rules are incorporated in the automation system to facilitate the determination of gadolinium distribution. The enrichment selection for each fuel pin enrichment type and the enrichment distribution are another issue to be solved by a nuclear engineer in designing a fuel lattice. In this study, the enrichment of each fuel pin enrichment type is a variable to be

Bottom

Zone1

Zone2

Zone3

Top

Fig. 1. Fuel assembly axial zoning.

determined in the design, not a value specified before carrying out the automation process. Usually, the engineer has to spend a lot of time to go through a trial and error process, to find the best fuel pin enrichment distribution, with the enrichment of each pin location selected from a specified set of enrichments. In this study, a different approach is presented to facilitate the lattice design tasks. The enrichment of each fuel pin enrichment type is determined by using a process that flatten the pin power or F-eff distribution to obtain an ideal enrichment distribution, and a process that divides the ideal enrichment distribution into several enrichment groups. This method eliminates the need to determine the enrichment for each fuel pin enrichment type prior to executing the fuel lattice design automation process, and also reduces the time spent in searching for the best fuel pin enrichment distribution. Following the determination of fuel pin enrichment, the enrichment group peaking equalizing process, which gradually reduces the difference between the highest and the lowest enrichment group peaking, is performed to reduce the lattice peaking factors such as LPF and Feff. A fuel lattice design with the improved lattice peaking can be obtained if the distribution of enrichment group peaking is equalized. Due to the voiding of moderator, a BWR fuel assembly is characterized by its several different axial lattice zones with varying enrichment from the bottom zone to the top zone. Currently, the 10 × 10 fuel assemblies used in the BWR4 and BWR6 nuclear power plant in Taiwan contain five zones in the axial direction, as shown in Fig. 1 To better utilize the neutron moderation, zone 1, which generates most of the energy during the cycle, is designed to have a higher enrichment than the rest of lattice zones. As for the upper portion of the fuel assembly, the neutron moderation is inferior and the fuel utilization is not so efficient because of a larger void fraction there. Therefore, the enrichments at zone 2 and zone 3 are lower than that at zone 1. Zone 2, playing an important role in the cold shutdown margin value, is usually designed with the highest gadolinium concentration. As the transition boiling often occurs near the top of a fuel assembly, zone 3 is designed to have a minimized F-eff, which affects the critical power of a fuel assembly.

2. Concepts and knowledge for the fuel lattice design automation The basic concept of designing a BWR fuel lattice is to emplace the enrichment based on the degree of neutron moderation effect at each fuel pin location. It is expected that a lower enrichment is emplaced at a fuel pin location with a better moderation effect and a higher enrichment is emplaced at a fuel pin location with a poorer moderation effect. The ideal design is that each fuel pin in a fuel lattice has its own unique enrichment depending on the degree of neutron moderation effect at the pin location. In such ideal condition, it is possible to design a fuel lattice with LPF equal to 1.0 if no gadolinium is used and the 5% enrichment upper limit is not considered. Due to the manufacturing complexity, there is a limit on the number of fuel pin enrichment type used in a fuel lattice design, and thus will result in the enrichment at each fuel pin location to deviate from its enrichment in the ideal design and increase LPF. It is important to properly determine the enrichment for each fuel pin enrichment type and select the enrichment type of each fuel pin location appropriately such that LPF can be minimized.

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I\J

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1

2.11 2.52 2.84 3.05 3.15 3.12 2.99 2.76 2.44 2.06

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2.52 3.22 3.77 4.10 4.18 4.10 3.89 3.56 3.05 2.42

3

2.84 3.77 4.49 4.74 4.58 4.37 4.23 3.98 3.45 2.68

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3.05 4.10 4.74 4.58 3.89 3.57 3.62 3.82 3.55 2.80

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3.15 4.18 4.58 3.89

3.30 3.44 2.82 Water Channel

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3.12 4.10 4.37 3.57

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2.99 3.89 4.23 3.62

3.06 3.31 2.77

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2.76 3.56 3.98 3.82 3.30 3.06 3.10 3.29 3.11 2.53

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2.44 3.05 3.45 3.55 3.44 3.31 3.23 3.11 2.81 2.30

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2.06 2.42 2.68 2.80 2.82 2.77 2.68 2.53 2.30 1.98

3.10 3.23 2.68

Fig. 2. Enrichment distribution of a 10 × 10 fuel lattice with LPF = 1.0.

Besides, the use of gadolinium and 5% enrichment upper limit make the fuel lattice design more complex. Considering a fuel lattice without gadolinium, if it is free to adjust the enrichment of each fuel pin in the fuel lattice, a flat pin power distribution in which the LPF is equal to 1.0, can be obtained. To generate a lattice with the flat pin power distribution, a lattice calculation is performed to obtain the pin power distribution for a lattice with a flat pin enrichment distribution. It is expected that the pin power distribution for such a flat enrichment distribution will not be flat – that is, the relative power fraction (RPF) varies from pin to pin. For fuel pins with RPFs larger than 1.0, the enrichments of these fuel pins have to be reduced to lower their RPFs, such that the RPFs can be close to 1.0. On the contrary, the pin enrichments have to be increased for those fuel pins with RPFs smaller than 1.0. The amount of enrichment adjustment for each fuel pin is determined based on two parameters: the difference between the pin RPF and 1.0, and the sensitivity of the pin power to the variation of the fuel pin enrichment change. With Pave set to 1.0, Eq. (2) in Section 3.2 will be used to estimate the amount of pin enrichment adjustment. This adjustment process continues for several iterations until the convergence criterion for a flat pin power distribution is reached. Fig. 2 shows the enrichment distribution of a 10 × 10 fuel lattice with LPF adjusted to 1.0. The enrichment of each fuel pin shown in Fig. 2 is taken as its ideal pin enrichment for the specified lattice average enrichment. In this example, no gadolinium is used and the fuel lattice design constraints are not considered. The depletion of the ideal lattice in Fig. 2 was performed and the Kinf , depleted at a void history of 0%, is shown in Fig. 3. The determination of the enrichment for each fuel pin enrichment type can be considered as a fuel pin enrichment grouping problem. Fig. 4 illustrates the idea of fuel pin enrichment grouping.

Fig. 3. Kinf vs. burnup of the ideal lattice in Fig. 2 with 0% void history.

Fig. 4. Illustration of fuel pin enrichment grouping.

By ordering the fuel pin enrichments, from lowest to highest, and setting the enrichment boundaries for each enrichment group, the fuel pin enrichments are divided into groups. Please note that each pin enrichment in Fig. 4 is linked to a fuel pin location. Therefore, by dividing the pin enrichments into groups, each fuel pin location is assigned an enrichment group. In Fig. 4, the enrichment distribution is separated into six enrichment groups, corresponding to six fuel pin enrichment types. The average of the enrichments of the fuel pins in the same group, defined as the group average enrichment, is then calculated. Then, the enrichment of each fuel pin is replaced with its group average enrichment such that only a limited number of pin enrichment types exist in the lattice. By replacing the fuel pin enrichment with its group average enrichment, the lattice average enrichment will keep unchanged. The number of enrichment group is flexible. It can be more or less, depending on how flatter the fuel pin power distribution the designer wants and how much manufacturing effort can be afforded. With the increase of enrichment group number, the enrichment distribution after grouping is more likely close to the ideal enrichment distribution and the pin power distribution is expected to be flatter. However, the more the enrichment group is, the more the fuel pin enrichment types are, and the more the manufacturing effort will be, so the maximum number of fuel pin enrichment types allowed in the design practice is often limited. To transfer a fuel lattice design as shown in Fig. 2 into a fuel lattice design with a limited number of fuel pin enrichment types will definitely result in the increase of LPF. After replacing the fuel pin enrichment with its group average enrichment, some fuel pins will have enrichment increased as compared to their ideal pin enrichments; whereas, other fuel pins will have enrichment decreased as compared to their ideal pin enrichments. As Fig. 2 is a fuel lattice with LPF equal to 1.0, those pins with enrichment increased will have RPFs larger than 1.0 and those pins with enrichment decreased will have RPFs lower than 1.0. The way of grouping an ideal fuel pin enrichment distribution like Fig. 2 into a prescribed number of enrichment groups affects some important fuel lattice design parameters, such as LPF and F-eff. Two approaches can be adopted in the fuel pin enrichment grouping. One approach is to divide the fuel pin enrichments into groups by arbitrarily setting the enrichment boundaries for each enrichment group and then to adjust the enrichment boundaries of each enrichment group until better fuel lattice design parameters are achieved. The other approach is to set enrichment boundaries for each enrichment group and then to move the fuel pin with maximum peaking factor from its enrichment group to an adjacent enrichment group with a lower enrichment until better fuel lattice design parameters are achieved. In this study, the second approach is adopted to improve the peaking factor, which is LPF for a lower zone lattice and maximum pin F-eff for an upper zone lattice. The details of this approach will be given in the next section. In addition to the basic concepts and knowledge for the fuel lattice design automation mentioned above, it is necessary to take

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The pin power distribution of a fuel lattice with a flat enrichment distribution is used to indicate the degree of neutron moderation effect of each fuel pin. Basically, the fuel pins with a higher pin power will be selected as the locations for emplacing gadolinium. In the actual fuel lattice design, there are some restrictions on the emplacement of gadolinium fuel pins, and these restrictions may be different for fuel bundles with a different mechanical design. For the lattice of the 10 × 10 fuel assembly used in Taiwan’s BWRs, the heuristic rules and restrictions applied are listed below. 1. Fuel pins at peripheral locations are not allowed to have gadolinium. 2. The gadolinium is only allowed to be emplaced at locations one row from periphery and at fuel pin locations next to a water channel. 3. Part-length fuel pins – which are shorter than the full length fuel pins and do not have any fuel composition, including the fuel pellet and cladding, in zones 2 and 3 – are not allowed for the placement of gadolinium. 4. Two gadolinium fuel pins adjacent to each other are not allowed. 5. If the fuel pin with a highest power is at the peripheral location, the nearest fuel pin that is not excluded by the rules and restrictions 1–4 is selected to emplace gadolinium. 3.2. Calculation of the pin power sensitivity matrix

Fig. 5. Calculation flow chart of the fuel lattice design automation system.

more design constraints into consideration, such as the utilization of gadolinium fuel pin and U235 enrichment upper limit, so as to generate a fuel lattice design which can fulfill the requirements of practical use. The way of incorporating gadolinium placement and of dealing with U235 enrichment upper limit in the fuel lattice design automation process will be discussed in next section. 3. Lattice design automation model Fig. 5 shows the calculation flow chart of this fuel lattice design automation system. The details of each calculation step in the flow chart are given in the following subsections. Sections 3.1–3.3 describe the common processes of the lower zone and upper zone lattice design, and Sections 3.4 and 3.5 provide the details of enrichment adjustment process for the lower zone and upper zone lattice design.

In this study, the lattice design automation starts from a fuel lattice with a flat pin enrichment distribution and the fuel pin enrichment of the lattice is gradually adjusted to the final design through several processes. The amount of enrichment adjustment of each fuel pin depends on the sensitivity of the pin power to the variation of pin enrichment. A pin power sensitivity matrix is obtained by perturbing the enrichment of a fuel pin, calculating the pin power variation of that fuel pin using CASMO-4, and recording the ratio of power variation to enrichment variation of that fuel pin. The interaction effects among the fuel pins are not considered. It means the variation of pin power due to the enrichment variation of other fuel pins is not accounted for in the calculation of pin power sensitivity matrix. Although the sensitivity matrix used here is only an approximation, it is still good enough in the pin enrichment adjustment process. The pin power sensitivity matrix, denoted as Si,j , is calculated as: Si,j =

Pi,j Ei,j

(1)

where Pi,j denotes the variation of pin power of fuel pin located at (i, j) due to the variation of its enrichment, which is denoted as Ei,j . The use of the pin power sensitivity matrix to facilitate the adjustment of fuel pin enrichment will be described in the following sections.

3.1. Determination of gadolinium fuel pin location 3.3. Determination of the fuel pin type enrichment Because gadolinium is a strong neutron absorber, it can reduce the RPF of a gadolinium pin to a fairly low value and substantially suppress the power of the pins next to the gadolinium pin. Also, the placement of a gadolinium fuel pin causes the RPFs of the pins which are far away from the gadolinium pin to increase, thus induces a higher LPF. In the design practice, gadolinium fuel pins are commonly emplaced at the pin locations having a good neutron moderation effect such that the adverse effect of using gadolinium can be mitigated. The number of gadolinium fuel pins and the weight percent of gadolinium are important fuel lattice parameters which affect the characteristics of the fuel lattice Kinf . In this study, the number of gadolinium fuel pins and gadolinium concentrations are specified values.

3.3.1. Flattening pin power and F-eff distribution To obtain a flat pin power or a flat F-eff distribution, the enrichment of each fuel pin, except for the gadolinium fuel pin, is adjusted until all the fuel pins have the same RPF or F-eff, depending on which axial zone the fuel lattice is in. In order to reduce the calculation time, this process is only performed at BOL for both lower zone and upper zone lattices. To flatten the pin power distribution of the lower zone lattice, the enrichment of each non-gadolinium fuel pin is adjusted according to the RPF of that fuel pin – that is, the enrichment of a non-gadolinium pin is decreased if its RPF is greater than an average value and increased on the contrary. The iteration

W.-H. Tung et al. / Nuclear Engineering and Design 293 (2015) 63–74

of pin enrichment adjustment continues until a flat pin power distribution is obtained. On the other hand, for the upper zone lattice, the enrichment of each non-gadolinium fuel pin is adjusted until a flat F-eff distribution is obtained. To accelerate the flattening process, the amount of enrichment adjustment for a fuel pin is given by the equation shown below. Ei,j =

Pi,j − Pave Si,j

(2)

where Ei,j denotes the enrichment adjustment of fuel pin at (i, j), Pi,j denotes the RPF of fuel pin at (i, j) and Pave denotes the average RPF of non-gadolinium fuel pins. After this flattening process, the enrichment of each fuel pin is normalized so that the average enrichment of the fuel lattice is equal to the design target. In this study, the lattice average enrichment design targets of each axial zone are assumed to be specified values which can fulfill the operational specifications such as cycle length, thermal margins, shutdown margins, and so forth. As the determination of the lattice average enrichment value for each axial zone is out of the scope of this study, the enrichment of the lattice in the corresponding axial zone in the reference assembly is taken as the design target. 3.3.2. Consideration of lattice design constraints Three lattice design constraints described below are considered in this study. 1. The enrichment of a fuel pin should be lower than 5.0%, and usually, 4.95% is used as the highest enrichment in the fuel lattice design. 2. Due to the manufacturing cost, the number of fuel pin enrichment type is limited. A fuel lattice design with different enrichment for each fuel pin is not acceptable. 3. In order to prevent the pin power of gadolinium fuel pins from becoming too high after the gadolinium burns out, an enrichment upper limit is set for the gadolinium fuel pins. A fuel pin enrichment distribution with flat pin power is likely to have the pin enrichment higher than 4.95% if the lattice average enrichment is high. To satisfy the pin enrichment upper limit, the enrichments of those pins with the enrichment higher than the upper limit are decreased to 4.95%. The enrichments of other fuel pins with enrichments lower than 4.95% are increased to keep the lattice average enrichment unchanged. The total amount of lattice enrichment reduction resulting from the adjustment of fuel pin enrichment to meet the enrichment upper limit, denoted as EUL , is calculated as: EUL =



(Ei,j − EUL ), (i, j)

i,j

∈ {fuel pins with enrichment larger than EUL }

(3)

where Ei,j denotes the fuel pin enrichment at (i, j) and EUL denotes the fuel pin enrichment upper limit which is 4.95% in this study. This enrichment redistribution process will result in an increase of LPF or F-eff. To mitigate the impact on LPF or F-eff, the amount of enrichment adjustment for each fuel pin is given by the following equation: Ei,j =

S −1

 i,j −1 × EUL , (i, j) S

i,j i,j

∈ {fuel roads with enrichment less than EUL }

67

to be used in the lattice design. To satisfy the constraint (3), the enrichment upper limit for gadolinium fuel pins can be specified, and 4.7% of the enrichment upper limit is set for the gadolinium fuel pins when the maximum fuel pin enrichment is 4.95%. 3.3.3. Enrichment grouping In order to meet the second lattice design constraint mentioned in the last section, the fuel pin enrichments are grouped into a specified number of enrichment groups, which is equal to the number of fuel pin enrichment type required in the lattice design. To begin with, the fuel pin enrichments are ordered and the difference between the lowest and the highest pin enrichment are evenly divided, such that the spacing between the lower and upper enrichment boundary of an enrichment group is the same for all the enrichment groups. For example, the fuel pins with enrichments between 2% and 2.5% belong to enrichment group 1, the fuel pins with enrichments between 2.5% and 3% belong to enrichment group 2 . . . and so on. If there are fuel pins with enrichments equal to 4.95%, these fuel pins are grouped together; whereas the enrichments of the fuel pins with enrichments less than 4.95% are sorted and evenly divided such that the spacing between the lower and the upper enrichment boundary of an enrichment group is the same for all the enrichment groups. Next, the group average enrichment is calculated by averaging the enrichments of the fuel pins that belong to the same group. Then, the fuel pins in the same group are assigned the same enrichment, which is the group average enrichment, in order to generate a lattice enrichment distribution that meets the constraint on the allowed number of fuel pin enrichment type. The group average enrichment is calculated as:



E¯ g =

E (i, j) i,j 0 Ng

, (i, j) ∈ {enrichment group g}

(5)

where Ng is the number of fuel pin in group g and E0 (i, j) is the pin enrichment distribution prior to the execution of enrichment grouping. After the enrichment grouping, the pin enrichments in a group are defined as: Eg (i, j) = E¯ g , (i, j) ∈ {enrichment group g}

(6)

After the enrichment grouping, the enrichment of each fuel pin enrichment type is determined and an initial enrichment distribution that meets the design constraint (2) mentioned in Section 3.3.2 is generated. Please note that the group average enrichments usually are not spaced evenly because of the uneven fuel pin enrichment distribution between the lower and upper enrichment boundary of an enrichment group, which results in the group average enrichment to be close to one of the two enrichment boundaries. For example, if most of the fuel pins have enrichments close to one of the enrichment boundaries, the group average enrichment will then close to that enrichment boundary. As a result, the group average enrichments are not evenly spaced even though the spacing between the lower and upper enrichment boundary of an enrichment group is the same for all the enrichment groups. The group average enrichments are spaced more unevenly after performing the enrichment group peaking equalizing process mentioned in the following. The fuel pin enrichment and enrichment groups determined by this way may not give the best results; therefore, two fuel pin enrichment adjustment processes are performed to improve LPF. The next section describes these two processes.

(4) 3.4. Enrichment group peaking equalizing for a lower zone lattice

As to the constraint (2), the number of fuel pin enrichment groups used in the enrichment grouping process, described in the next section, is set equal to the number of fuel pin enrichment types

Due to the bottom peaked axial power shape from the beginning of cycle (BOC) to the middle of cycle (MOC) and a large LPF near

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BOL for a fuel lattice, the cycle limiting MFLPD usually appears in the lower zone lattice of a fresh fuel bundle. For fresh fuel bundles, larger FLPD shows up in the lower zone lattices because some of the fuel pins have larger linear power density, which is a consequence of larger LPF and larger axial peaking factor. It is anticipated that during the lattice design stage, the LPF of the lower zone lattice is minimized such that MFLPD can be smaller in case that one of the fresh fuel bundles has the cycle limiting FLPD. In general, the optimization of lower zone lattice LPF can help to achieve lower MFLPD in the core loading pattern design and thus enhance the operation flexibility in terms of MFLPD. As to the optimization of axial peaking factor, which is dependent on several variables such as axial enrichment distributions, control rod patterns, etc, is out of the scope of this study. In order to achieve good MFLPD in a core loading pattern, it is expected that the fresh fuel lower zone lattices are designed with a good LPF at their early exposures, which are exposures less than 15 GWD/MTU taking an 18-month cycle as an example. As the LPF for a fuel lattice is the largest at BOL, the processes adopted to minimize the LPF for the lower zone lattices are performed at BOL. This strategy not only saves the computational effort but also improves the LPF at the early exposures less than 15 GWD/MTU as observed from the results. In this study, it is observed that the minimization of LPF can be fulfilled by equalizing the enrichment group peakings. Sections 3.4.1 and 3.4.2 describe the processes implemented to equalize the enrichment group peakings.

3.4.1. Peak pin enrichment group migration To facilitate the following descriptions, we define the largest RPF in an enrichment group as the group peaking, the fuel pin with the highest RPF in a lattice as the peak pin, and the enrichment group containing the peak pin as the peak pin group. After the initial enrichment grouping, it is observed that the group peakings are not even. The LPF, which is the largest of the group peakings, can be improved by lowering the enrichment of the peak pin, which is accomplished in this study by transferring the fuel pin to the adjacent group with a lower enrichment. This process is called the peak pin enrichment group migration. The group average enrichments in these two groups, which are involved in the peak pin enrichment group migration process, need to be recalculated because the fuel pins contained in either of these two groups are changed. The steps of the peak pin enrichment group migration process are described below. Step 1: The peak fuel pin, assuming that it is in enrichment group g and the group average enrichment in that group g is E¯ g , is identified. Step 2: The peak fuel pin is transferred to the adjacent group with a lower enrichment. This lower enrichment group, denoted as group g−1, has a group average enrichment denoted as E¯ g−1 . If the group peaking in group g−1 is the lowest among all the enrichment groups, in which the group average enrichments are below the enrichment upper limits specified in Section 3.3.2, then Step 3 is performed; otherwise, Step 4 will be performed. Step 3: E¯ g and E¯ g−1 are recalculated using Eq. (5). Because the enrichment of the peak pin is higher than those of all the fuel pins in group g−1, the migration of peak pin will result in the group average enrichment in group g−1 to increase, which may induce a worse LPF. This is an adverse effect on group peaking equalizing. Therefore, if the lowest group peaking is not in group g−1, Step 4, rather than Step 3, is performed to avoid the adverse effect. Step 4: The lattice enrichment decrement is calculated and the decrement is compensated for by increasing the group average enrichments in the groups with the group average enrichments below the enrichment upper limits specified in Section 3.3.2. The

lattice enrichment decrement Em resulting from the migration of peak pin is: Em = E0 (l, m) − E¯ g−1 ,

(7)

where (l, m) are the position indexes of the peak pin and E0 (i, j) is the fuel pin enrichment distribution defined in Section 3.3.3. After the calculation of Em , the peak pin enrichment in the enrichment distribution E0 (i, j) is set to the group average enrichment in group g−1. E0 (l, m) = E¯ g−1 .

(8)

Due to the migration of peak pin, the group average enrichment in group g is recalculated using Eq. (5). The group average enrichments in the groups, with their group average enrichments not reaching the enrichment upper limits mentioned in Section 3.3.2, are increased to compensate for the lattice enrichment decrement Em ; and thus the lattice average enrichment is kept unchanged. The enrichment increment in a group depends on the average value of Si,j in the group. For the group with a larger average value of Si,j , the amount of the group average enrichment increment is smaller. Step 5: The enrichment distribution Eg (i, j), defined in Eq. (6), is updated by using the new group average enrichments calculated in Step 3 or Step 4, to obtain an enrichment distribution Eg (i, j) that meets the design constraint (2) mentioned in Section 3.3.2. Step 6: A lattice calculation for the enrichment distribution Eg (i, j) is performed by using CASMO-4 to obtain a new pin power distribution. Step 7: If the termination criterion is not met, Step 1 is returned to; otherwise, the peak pin enrichment group migration process is terminated. The process stops when a same group average enrichment occurs in two adjacent groups. As there is no lower adjacent group for the lowest enrichment group, the peak pin will not migrate if it is in the lowest group. In addition, due to that the lowest group is very sensitive to the variation of enrichment and it contains a small number of fuel pin, the migration of peak pin from its upper adjacent group is forbidden. Therefore, if the peak pin is in one of the two lowest groups, the average enrichment in the peak pin group is reduced and the amount of lattice enrichment reduction is compensated for by using the same lattice enrichment compensation method as that mentioned in Step 4. Among all the lattice designs generated in the peak pin enrichment group migration process, the one with the lowest LPF is selected. For this selected design, if the difference between the highest and the lowest group peaking is not smaller than a specified value, the group average enrichment adjustment process described in the following section is used to further improve the LPF of the selected lattice design. 3.4.2. Group average enrichment adjustment The changes in group average enrichments resulting from the peak pin enrichment group migration are discrete; therefore, the variations in pin powers and group peakings resulting from the group average enrichments changes are discrete. Due to this discrete nature in the peak pin enrichment group migration process, the group peakings usually can only close to each other to a certain extent that the difference between the lowest and the highest group peaking is still too large in terms of group peaking equalizing. To enhance the equalization of the group peakings to further improve LPF, the group average enrichment in the group with the highest power peaking is decreased by a small amount, 0.01%, in each iteration. The group average enrichment in the lowest power peaking group is then increased to compensate for the lattice enrichment decrement resulting from the group average enrichment reduction in the highest power peaking group, such that the lattice average enrichment is kept unchanged. The lowest power peaking

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group is selected from the enrichment groups with group average enrichments below the enrichment upper limits prescribed in the constraints (1) and (3) mentioned in Section 3.3.2. The group average enrichment adjustment process is continued until the difference between the highest and lowest group peaking is less than a specified convergence value, or two adjacent groups reach the same group average enrichment during this process. 3.5. Enrichment group peaking equalizing for an upper zone lattice According to the operational experiences of BWRs in Taiwan, MFLCPR near the end of cycle (EOC) is the most limiting of the cycle. As the cycle is close to EOC, the top-skewed axial power shape results in the boiling transition to occur at the top portion of a high power bundle, which usually is a fresh fuel bundle. Therefore, it is important to optimize the F-eff peaking, defined to be the maximum F-eff for all the fuel pins in a lattice, of upper zone lattices at the lattice exposure at which the MFLCPR is the most limiting of the cycle. The lattice exposure where the F-eff peaking should be optimized is called the selected exposure. Usually, the selected exposure is close to the exposure at which the Kinf reach the maximum value. To improve F-eff peaking of an upper zone lattice in zone 2 or zone 3, a peak pin enrichment group migration process similar to the one mentioned in Section 3.4.1 is used. At the selected exposure, the peak F-eff pin, defined to be the pin with the maximum F-eff value in a lattice, is moved to the lower adjacent group. The compensation for the lattice enrichment reduction caused by executing the peak pin enrichment group migration process, as well as the termination criteria of peak pin enrichment group migration process are the same as those mentioned in Section 3.4.1. To improve the deteriorated BOL F-eff peaking resulting from the peak F-eff pin enrichment group migration process, two fuel lattice designs are selected. One is the fuel lattice with the lowest F-eff at the selected exposure and the other one is the fuel lattice with the lowest average F-eff. The average F-eff is defined to be the average of the F-eff peaking from BOL to 15 GWD/MTU. To improve the BOL F-eff peaking, the group average enrichment in the BOL peak F-eff pin group is reduced, and therefore reduces the lattice enrichment. The reduction of lattice enrichment is compensated for by increasing the group average enrichments in all the enrichment groups with group average enrichments below the enrichment upper limits prescribed in the constraints (1) and (3) mentioned in Section 3.3.2. The adjustment process will be terminated when a fuel lattice design with the F-eff peaking worse than that of a previous one is generated. After the adjustment of the two selected fuel lattice designs, the one with a lower average F-eff is selected as the final design.

Fig. 6. Group peaking of each enrichment group in the iteration of peak pin enrichment group migration process for design case 1.

designs with the same design parameters as that shown in Table 1 are used for comparison with the fuel lattice designs generated by the automation system. 4.1. Lattice design results of a lower zone lattice In Table 1, design case 1 is a lower zone lattice with the average enrichment of 4.471%, 6 types of UO2 fuel pin enrichments, 15 gadolinium fuel pins and gadolinium concentration of 6%. As this is a lower zone lattice, the lattice design automation process and CASMO-4 calculations were only performed at BOL to reduce the computational effort. Fig. 6 shows the group peaking, peak RPF in an enrichment group for a lower zone lattice, at BOL of each enrichment group after grouping, indicated as iteration 0, and during the peak pin enrichment group migration process. Although the group peakings of group 5 and group 6 are lower than that of the other four groups, the enrichments of these two groups were not able to be increased any further because of the constraints (1) and (3) mentioned in Section 3.3.2. The group peakings of enrichment groups 1–4 are gradually reduced and close to each other before the 10th iteration and become more diverse after the 10th iteration. The 5th iteration in the peak pin enrichment group migration process has the lowest LPF, which is 1.278; whereas, the 10th iteration, having the group peakings of group 1–4 closest to each other, has an LPF of 1.279. Though the 10th iteration does not give the lowest LPF, its LPF is almost the same as the lowest one. By using the peak pin enrichment group migration process, the LPF was improved from a value of 1.292 after grouping to a value of 1.278 at the 5th iteration. Fig. 7 shows the average enrichment in each group during the iteration process. The peak pin enrichment group migration process was terminated because group 4 and group 5 had reached the same group average enrichment.

4. Results and discussions The knowledge-based automation system was used to design the fuel lattices in a reload batch of a BWR4 core with 408 fuel assemblies. In the reload cycle, three different reload batches are used and the assembly average enrichments of these three batches are 4.026%, 4.036%, and 4.053% respectively. The reload pattern and control rod patterns are kept constant, whereas two lattices of the fuel batch with the assembly average enrichment of 4.026% are redesigned using the automation system. The fuel assembly used in the reload core has a 10 × 10 fuel pin array of and a square water channel occupying a 3 × 3 fuel pin array. The application includes the design of lower zone and upper zone lattices and the results are compared with the reference designs. Table 1 lists the enrichment and gadolinium data of the application cases. The reference

Fig. 7. The average enrichment in each group in the iteration of peak pin enrichment group migration process for design case 1.

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Table 1 Enrichment and gadolinium data of the application cases. Case

Zone

Lattice enrichment (%)

No. of UO2 enrichment type

No. of Gd pin

Gd concentration (%)

1 2

Zone 1 Zone 3

4.471 4.386

6 6

15 12

6.0 3.0

Table 2 Group peaking of each enrichment group in the group average enrichment adjustment process for design case 1. Iteration

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

0 1 2 3 4 5 6 7 8

1.275 1.276 1.272 1.272 1.272 1.273 1.273 1.270 1.270

1.275 1.275 1.276 1.273 1.273 1.274 1.271 1.271 1.272

1.278 1.276 1.276 1.276 1.274 1.272 1.272 1.273 1.271

1.260 1.262 1.263 1.264 1.266 1.267 1.269 1.269 1.271

1.185 1.185 1.185 1.185 1.186 1.186 1.186 1.187 1.187

1.202 1.202 1.202 1.202 1.202 1.202 1.202 1.202 1.202

Table 3 The average enrichment in each group in the group average enrichment adjustment process for design case 1. Iteration

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

0 1 2 3 4 5 6 7 8

2.61 2.61 2.60 2.60 2.60 2.60 2.60 2.59 2.59

3.34 3.34 3.34 3.33 3.33 3.33 3.32 3.32 3.32

4.05 4.04 4.04 4.04 4.03 4.02 4.02 4.02 4.01

4.39 4.40 4.40 4.41 4.42 4.42 4.43 4.43 4.44

4.70 4.70 4.70 4.70 4.70 4.70 4.70 4.70 4.70

4.95 4.95 4.95 4.95 4.95 4.95 4.95 4.95 4.95

As the 5th iteration in the peak pin enrichment group migration process has the lowest LPF, the fuel lattice design generated in this iteration was selected for the following group average enrichment adjustment. Tables 2 and 3 list the group peaking and group average enrichment in each group in the group average enrichment adjustment process. Not accounting for group 5 and group 6, the difference between the highest and the lowest group peaking is 0.018, indicating there is still some room for the improvement of LPF by using the group average enrichment adjustment to further equalize the group peakings. After the execution of group average enrichment adjustment, the LPF was reduced from 1.278 to 1.272 and the difference between the highest and the lowest group peaking was reduced from 0.018 to 0.002. The iteration process stopped at the 8th iteration where the difference between the highest and lowest group peaking of group 1–4 is less than 0.003, which is the termination criteria. After the fuel pin enrichment grouping, the LPF of the design case 1 at BOL is 1.292, lower than the LPF of the reference design at BOL, which is 1.319. The peak pin enrichment group migration process improved LPF further from 1.292 to 1.278. Although the enrichment grouping method is simple, it still provides a good initial LPF before taking any pin enrichment adjustment process. Fig. 8 shows the fuel lattice design generated by the automation system. Fig. 9 is the LPF of the design and the reference lattice. Although the fuel pin enrichment adjustment is only based on the fuel lattice calculation results at BOL, the improvement of LPF is not only seen at BOL but also seen at later exposures. Fig. 10 shows the Kinf ’s of the design and the reference lattice. The Kinf ’s of the design and the reference lattice are almost the same. To observe the effect of the improved LPF on the bundle FLPD, a core calculation was performed for a known reload core loading pattern with the reference lattice replaced by the lattice from the automation design. The core calculation results show that the maximum FLPD of the bundles using the design lattice was reduced from 0.693 to 0.675 at BOC.

Also shown in Fig. 8 are the gadolinium fuel pin locations, which were determined following the rules and restrictions described in Section 3.1. In order to suppress the power peaking, the location of the peak pin itself, not excluded by the rules and restrictions, is first selected for placing gadolinium. If the peak pin location is excluded from placing gadolinium by the rules and restrictions, the nearest neighboring rod to the peak pin, not excluded by the rules and restrictions, is selected for placing gadolinium. As there are 15 gadolinium fuel pins for the lattice and because the enrichment and gadolinium distributions are diagonally symmetry, one of the gadolinium fuel pin must be located on the symmetry line. This gadolinium fuel pin location is (4, 4), determined by experience.

I\J

1

2

3

4

5

6

7

8

9

10

1

2.59

3.32

4.01

4.01

4.44

4.44

4.44

4.01

3.32

2.59

2

3.32

4.70

4.70

4.95

4.95

4.70

4.95

4.70

4.70

3.32

3

4.01

4.70

4.95

4.95

4.95

4.95

4.95

4.95

4.70

4.01

4

4.01

4.95

4.95

4.70

4.95

4.95

4.95

4.95

4.95

4.44

5

4.44

4.95

4.95

4.95

4.95

4.70

4.44

6

4.44

4.70

4.95

4.95

4.95

4.95

4.44

7

4.44

4.95

4.95

4.95

4.70

4.95

4.44

8

4.01

4.70

4.95

4.95

4.95

4.95

4.70

4.95

4.70

4.01

9

3.32

4.70

4.70

4.95

4.70

4.95

4.95

4.70

4.95

3.32

10

2.59

3.32

4.01

4.44

4.44

4.44

4.44

4.01

3.32

2.59

Fuel pin with 6% Gd Fig. 8. Lattice enrichment distribution for design case 1.

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Fig. 9. LPF comparison for design case 1.

71

Fig. 11. Group peaking of each enrichment group in the iteration of peak pin enrichment group migration process for design case 2.

Fig. 10. Kinf comparison for design case 1.

After placing gadolinium at location (4, 4), the LPF occurred at (10, 10), which is a peripheral fuel pin and thus is not allowed for placing gadolinium. The nearest fuel pins to (10, 10) are (10, 9), (9, 10) and (9, 9). The first two locations are peripheral and the third one is the location of a part-length fuel pin; all of these locations are excluded from placing gadolinium by the rules and restrictions applied in this study. Therefore, the nearest neighboring fuel pin locations available for placing gadolinium, to suppress the power peaking at (10, 10), are (9, 8) and (8, 9), which are at diagonally symmetric locations. Following this logic, the gadolinium fuel pin locations were determined in turn. It took about 11 min of CPU time for the automation system to accomplish the design case 1 on an HP-8000 workstation, including the execution of CASMO-4. This shows that the automation method is reasonably practical in designing a fuel lattice. The efficiency of the automation design comes from that CASMO-4 is only executed at BOL and the automation process takes less times of iteration in the fuel pin enrichment adjustment. 4.2. Lattice design results of an upper zone lattice The design case 2 is an upper zone lattice with the average enrichment of 4.386%, 6 UO2 fuel pin enrichment types, 12 gadolinium fuel pins and gadolinium concentration of 3%. As this is an upper zone lattice, CASMO-4 calculations were performed from BOL to 15 GWD/MTU and the lattice design automation processes were performed at 10 GWD/MTU which is close to the exposure at which Kinf reaches the peak value. Fig. 11 shows the group peaking, peak F-eff in an enrichment group for an upper zone lattice, at 10 GWD/MTU of each enrichment group after grouping, indicated as iteration 0 in the figure, and during the peak F-eff pin enrichment group migration process. The difference between the highest and the lowest group peaking are gradually reduced and gradually close to each other, as can be seen in Fig. 11. The F-eff peaking,

Fig. 12. The average enrichment in each group in the iteration of peak pin enrichment group migration process for design case 2.

highest F-eff in a lattice, at 10 GWD/MTU is 0.9115 at the 19th iteration where the difference between the highest and the lowest group peaking is the smallest. Comparing with the lowest F-eff peaking value of 0.9108 at the 15th iteration, the 19th iteration gives the F-eff peaking that is almost the same as the lowest one. The enrichments of group 5 and 6 were not able to be increased any further because of the constraints (1) and (3) mentioned in Section 3.3.2. The peak F-eff pin enrichment group migration process was terminated because two adjacent enrichment groups had reached the same enrichment in the process as shown in Fig. 12. It is observed in Fig. 12 that the group average enrichments in group 1–4 are increased in most of the iterations. The reason is that the migration of peak pin caused the group average enrichments in the non-peak-pin groups to increase. As described in Section 3.4.1, if the lower enrichment group to which the peak pin migrates has the lowest group peaking, the recalculated group average enrichment in the lower enrichment group will be increased. For most of the iterations, the peak pin migrated to a lower enrichment group which does not have the lowest group peaking. Therefore, the lattice enrichment reduction, resulting from the peak pin enrichment group migration, was compensated for by increasing the group average enrichments in all the non-peak-pin groups, except for the groups with group average enrichments reaching the enrichment upper limits as specified in Section 3.3.2. Unlike LPF for the design case 1, during the peak pin enrichment group migration process of the design case 2, the group peakings of groups 5 and 6 were gradually close to that of the other four groups. In design case 1, the group peakings of group 5 and group 6 were lower than that of in the other four groups, but the group average enrichments in these two groups could not be raised to increase the group peakings because of the enrichment upper limits mentioned in Section 3.3.2. On the contrary, the group peakings

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Fig. 14. F-eff peaking comparison for design case 2.

Fig. 13. Lattice enrichment distribution for design case 2.

of group 5 and group 6 in design case 2 were higher than that of the other four groups in the first several iterations, and the peak pins in these two groups were transferred to the lower enrichment group to reduce the group peakings of these two groups. As a result, the group peakings of these two groups were gradually close to that of the other four groups. The group average enrichments in group 5 and group 6 did not change during the iteration because the group average enrichments in these two groups had reached their respective enrichment upper limits before the beginning of the iteration process and the increase of the group average enrichments during the iteration were restricted by the enrichment upper limits mentioned in Section 3.3.2. Table 4 lists the F-eff peaking in the peak pin enrichment group migration process and BOL F-eff peaking improvement process of the design case 2. The F-eff peaking at BOL is 0.9567. Comparing with the value of 0.9572 in the reference design, the F-eff distribution flattening and enrichment grouping processes give a good initial BOL F-eff peaking. However, after enrichment grouping, the lattice F-eff peaking at 10 GWD/MTU is 0.9632, which is higher than 0.9243 of the reference lattice at the same exposure. Because the Feff distribution flattening and enrichment grouping processes were performed only at BOL, it did not help generating a low F-eff peaking lattice at 10 GWD/MTU after the execution of the two processes. The major improvement of the F-eff at 10 GWD/MTU was accomplished through the execution of the peak F-eff pin enrichment group migration process. The F-eff at 10 GWD/MTU was reduced from 0.9632 at iteration 0 to 0.9120 at the 12th iteration and to 0.9108 at the 15th iteration, as shown in Table 4. Also as can be seen from Table 4, the fuel lattice design with the lowest F-eff peaking at 10 GWD/MTU was generated at the 15th iteration which has an F-eff peaking of 0.9108, and the fuel lattice design with the lowest 0–15 GWD/MTU averaged F-eff peaking was generated at the 12th iteration. These two fuel lattice designs were selected for the following enrichment adjustment process to improve F-eff peaking at BOL. The F-eff peaking at BOL was deteriorated in the peak pin enrichment group migration process as shown in Table 4. The F-eff peaking at BOL is 0.9567 at iteration 0 and becomes 0.9717 and 0.9763 at the 12th iteration and the 15th iteration, respectively. After the enrichment adjustment process at BOL, the F-eff peaking at BOL of these two lattice designs were reduced from 0.9717 to 0.9648 for the 12th iteration design, and from 0.9763 to 0.9681 for the 15th iteration design, respectively. As shown in Table 4, the average F-eff after the BOL F-eff peaking improvement

Fig. 15. Kinf comparison for design case 2.

process is 0.9292 and 0.9305 for the 12th iteration and the 15th iteration design, respectively. Since the 12th iteration design has the lower average F-eff after the BOL F-eff peaking improvement process, it is selected as the final design. Tables 5 and 6 list the group peaking and the average enrichment in each group during the BOL F-eff peaking improvement process for the 12th iteration design. The iteration process was terminated when the enrichment adjustment generated the F-eff peaking that was worse than that of a previous iteration. Fig. 13 shows the fuel lattice design generated by the automation system. Fig. 14 is the comparison of F-eff peaking of the design and the reference lattice. The automation design has the lowest F-eff peaking of 0.9115 at 11.0 GWD/MTU, lower than the lowest F-eff peaking of the reference design. Though the F-eff peaking of the automation design at BOL is larger than that of the reference design, the cycle limiting MFLCPR will not be impacted because it usually occurs at the cycle exposure near EOC. To observe the effect of the improved F-eff on the bundle FLCPR, a core calculation was performed for a known reload core loading pattern with the reference lattice replaced by the lattice from the automation design. The core calculation results show that the maximum FLCPR of the bundles using the design lattice was reduced by an amount of 0.012 at the cycle exposure at which MFCLPR is most limiting in the cycle. The Kinf ’s of the design and the reference lattice are almost the same as shown in Fig. 15. Also shown in Fig. 13 are the gadolinium fuel pin locations, which were determined following the rules and restrictions described in Section 3.1. Although this is an upper zone lattice, the method used to determine the gadolinium fuel pin locations is similar to that used for the lower zone lattice. To suppress the power peaking, the first choice for placing gadolinium is the peak pin itself. If the peak pin location is excluded from placing gadolinium by the rules and restrictions, the nearest neighboring rod to the peak pin that is not excluded by the rules and restrictions for gadolinium placement is selected instead. Before any gadolinium is placed to

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Table 4 F-eff peaking in the peak pin enrichment group migration and BOL F-eff improvement process for design case 2. Iteration

0 12 15

Peak pin enrichment group migration

BOL F-eff improvement

BOL

10 GWD/MTU

Average

BOL

10 GWD/MTU

Average

0.9567 0.9717 0.9763

0.9632 0.9120 0.9108

0.9583 0.9311 0.9328

– 0.9648 0.9681

– 0.9120 0.9112

– 0.9292 0.9305

Table 5 Group peaking of each enrichment group in the BOL F-eff improvement process for design case 2. (Bold value – the maximum peaking of the six groups). Iteration

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

0 1 2 3

0.972 0.969 0.964 0.966

0.959 0.959 0.959 0.961

0.965 0.965 0.965 0.962

0.961 0.961 0.962 0.963

0.938 0.938 0.938 0.938

0.921 0.921 0.921 0.921

Table 6 The average enrichment in each group in the BOL F-eff improvement process for design case 2. Iteration

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

0 1 2 3

2.804 2.776 2.748 2.765

3.522 3.525 3.527 3.547

3.978 3.981 3.984 3.944

4.445 4.448 4.451 4.474

4.700 4.700 4.700 4.700

4.950 4.950 4.950 4.950

the lattice, LPF occurred at (10, 10), which is a peripheral location and thus is not allowed for placing gadolinium. The nearest fuel pins to (10, 10) not excluded by the rules and restrictions are (9, 8) and (8, 9), which are at diagonally symmetric locations. The second gadolinium fuel pin location was determined in a way similar to the first one. The rest of the gadolinium fuel pin locations were determined following the same logic. It took 1 h and 11 min for the automation system to accomplish the design case 2 on an HP-8000 workstation, including the execution of CASMO-4. The time spent for the design case 2 is much longer than that for the design case 1 because the fuel lattice was depleted from 0 to 15 GWD/MTU for each iteration in the design case 2 and more iterations were performed in the design case 2 than that in the design case 1. However, less than 1.5 h of execution time for a fuel lattice design automation case is quite acceptable. 5. Conclusions and future works A knowledge-based fuel lattice design automation method has been developed for the radial fuel lattice design of BWRs with the method of designing the enrichment distribution by equalizing the group peakings of all the enrichment groups, in order to obtain a fuel lattice design with the reasonably good LPF or F-eff peaking. Several heuristic rules, restrictions and lattice design constraints, related to the gadolinium placement and fuel pin enrichment, are incorporated in the automation system to facilitate the design and enhance the design for practical use. Because the fuel pin enrichment upper limits for gadolinium and non-gadolinium pins are imposed on the lattice design as design constraints, the goal of equalizing the group peakings is unattainable when the adjustment of the group average enrichments is restricted by the enrichment upper limits for the enrichment groups that have lower group peakings. Though the lattice design is subject to the enrichment upper limits, the group peaking equalizing processes can still improve the LPF or F-eff peaking by equalizing the group peakings for the enrichment groups that have group average enrichments lower than the enrichment upper limits. To demonstrate the capability of the developed fuel lattice design automation method, the 10 × 10 fuel lattice designs are performed with the same lattice design parameters as that of the reference lattices, including the lattice average enrichment, number of fuel pin enrichment types,

number of gadolinium fuel pins and gadolinium concentrations. The beginning-of-life LPF is improved from 1.319 to 1.272 in the design case at the axial zone where LPF is important, and the Feff peaking at the exposure crucial to MFLCPR is improved from 0.9243 to 0.9120 for the design case at the axial zone where F-eff is important. It took about 11 min and 1 h 11 min of the CPU time for the automation system to accomplish the design case 1 and design case 2 on an HP-8000 workstation, including the execution of CASMO-4 lattice code. Though the methods used in this automation system are simple, the applications of this automation system demonstrate that the methods are effective in designing a BWR fuel lattice, no matter the lattice is in lower zone or upper zone. The two major methods – the enrichment grouping method and the peak pin enrichment group migration method – are shown effective in designing a fuel lattice with improved LPF or F-eff peaking, depending on which axial zone the lattice is in. As the enrichment grouping process is performed at BOL, a low F-eff peaking will not be obtained for upper zone lattices at the selected exposure after grouping. The improvement of the F-eff peaking is primarily through the execution of the peak F-eff pin enrichment group migration process. Although the fuel lattice designs and the CASMO-4 calculations are only performed at BOL, the improvement of LPF is not only seen at BOL but also observed at later exposures. As to the upper zone lattice, the designing process was performed at the selected exposure, in order to improve the most limiting MFLCPR in the cycle effectively by reducing the Feff peaking at the selected exposure for the lattice that belongs to the MFLCPR limiting bundle. As the most limiting MFLCPR usually occurs at the lattice exposure less than 15 GWD/MTU for an 18month cycle, the upper zone lattice design will not require a full range of depletion from 0 to 70 GWD/MTU; whereas, the lattice depletion from 0 to 15 GWD/MTU is sufficient. It is expected that the selected exposure will be different for the upper zone lattice design when the cycle length changes. Though the simple enrichment grouping method adopted in this study provides the initial enrichment grouping that results in reasonably good lattice designs, a different way of enrichment grouping may lead to better solutions and is worth of further exploration. As gadolinium distribution affects the result of enrichment grouping, a study of using various gadolinium distribution methods is also recommended to provide more insight on the coupling

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of the gadolinium distribution and the enrichment grouping. Plural kinds of gadolinium concentration or multiple kinds of enrichment for gadolinium fuel pin are often adopted in a design practice to fulfill the aims of the fuel lattice design; therefore, it is expected that the lattice design automation system has the capability to perform those kinds of design in the future. In the consideration of manufacturing cost or quality assurance, the use of some pre-defined pin enrichment type may be set as a constraint for the lattice design. In this case, the lattice design of this automation system can serve as a reference for the selection of fuel pin enrichments such that the adverse impact on LPF or F-eff peaking can be minimized. Accordingly, one of the future works is to automate the selection method when the use of pre-defined pin enrichment types is set as a design constraint. References Castillo, A., et al., 2011. Fuel lattice design with path relinking in BWR’s. Prog. Nucl. Energy 53, 368–374.

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