HEURISTIC RULES EMBEDDED GENETIC ALGORITHM TO SOLVE
VVER LOADING PATTERN OPTIMIZATION PROBLEM
F. A lim , K. Iv a n o v
Department of Mechanical and NuclearEngineering Pennsylvania State University Department of Technology Turkish Atomic Energy Authority
ABSTRACT
Loading Pattern (LP) optimization is one of the most important aspects of the operation of nuclear reactors. A genetic algorithm (GA) code GARCO (Genetic Algorithm Reactor Optimization Code) has been developed with embedded heuristic techniques to perform optimization calculations for in-core fuel management tasks. GARCO is a practical tool that includes a unique methodology applicable for all types of Pressurized Water Reactor (PWR) cores having different geometries with an unlimited number of FA types in the inventoiy. GARCO was developed by modifying the classical representation of the genotype. Both the genotype representation and the basic algorithm have been modified to incorporate the in-core fuel management heuristics rules so as to obtain the best results in a shorter time. GARCO has three modes. Mode 1 optimizes the locations of the fuel assemblies (FAs) in the nuclear reactor core, Mode 2 optimizes the placement of the burnable poisons (BPs) in a selected LP, and Mode 3 optimizes simultaneously both the LP and the BP placement in the core. This study describes the basic algorithm for Mode 1. The GARCO code is applied to the W E R -1000 reactor hexagonal geometry core in this study. The “Moby-Dick” is used as reactor physics code to deplete FAs in the core. It was developed to analyze the W E R reactors by SKODA Inc. To use these rules for creating the initial population with GA operators, the worth definition application is developed. Each FA has a worth value for each location. This worth is between 0 and 1. If worth of any FA for a location is larger than 0.5, this FA in this location is a good choice. When creating the initial population of LPs, a subroutine provides a percent of individuals, which have genes with higher than the 0.5 worth. The percentage of the population to be created without using worth definition is defined in the GARCO input. And also age concept has been developed to accelerate the GA calculation process in reaching the optimum solutions. The computing time is divided into ages. It can be stated that the classical GA has only one age. It is assumed that in each age the operators work with a group of genes instead of with all of the genes. These groups are created according to in-core fuel management heuristic rules.
1. INTRODUCTION
In-core fuel management optimization is one of the most important aspects of the operation of nuclear reactors. It involves the arrangement of approximately 150 to 200 fuel assemblies in the pressurized water reactor (PWR). A typical 1/8 core sector of symmetry can have 1026 and more possible loading patterns. Loading pattern (LP) includes used fuel assemblies coming from previous cycles and fresh fuel assemblies which replace the discharged fuel assemblies at the end of the cycle (EOC). All fuel assemblies are reshuffled to a configuration that is optimal with respect to some performance criterion and which meets the safety constraints. Usually this requires finding an optimal fuel arrangement with maximum cycle length in the reactor core while satisfying the safety constraints.
Fuel assemblies with fixed properties must be placed in specific regions. It is a discrete problem and mathematical derivative information is not easily obtained to optimize the fuel arrangement in the nuclear reactor core. Evaluating a fuel arrangement requires determining the core lifetime and normalized power distribution using a reactor physics code, which performs complex iterative calculations. Because of these iterative calculations, it is very difficult to find mathematical derivative information that shows how the result changes when an input parameter is changed. Because of providing an opportunity to work well with discrete functions and without any derivative information Genetic Algorithms (GA) have been successfully applied to wide range of engineering problems, including core reload design problems in Nuclear Engineering.
The developed GA code, GARCO (Genetic Algorithm Reactor Core Optimization), is applied to the W E R - 1000 reactor core. W E R is a Russian-type pressurized water reactor. It means a water-moderated and water- cooled power reactor. Although the design concept is the same as for PWRs, it has some differences as compared to a PWR. The major difference between W E R and PWR is the fuel assembly and core geometry. W E R s use hexagonal fuel assemblies. Fuel rods are also arranged in hexagonal geometry.
The core physics code used in this study is Moby-Dick, which was developed to analyze the W E R reactors by SKODA Inc. Moby-Dick is based on the finite difference approximation to the few-group ( 2 - 1 0 energy
2. GENOTYPE REPRESENTATIN OF THE NUCLEAR REACTOR CORE
To demonstrate genotype representation of the nuclear reactor core, a sample core is used. Location numbers and symmetry definitions for each location are shown in the core layout (Figure 1). There are 15 locations. While the locations in the core axes have Vz symmetry, other locations have lA symmetry. If a location has 1/n symmetry, its symmetiy number is n. It is assumed that there are 20 different fuel assembly types (fl,f2 ,.. .,f20) with different numbers in the inventory. Using these fuel assembly types a loading pattern is created with assigning fuel assembly types to the locations randomly. For genotype representation the symmetry condition of the core can be used. Representation of LP in Figure 1 is shown in Figure 2.
In genotype representation these rules are defined as: • Columns represent fuel assembly type
• Rows represent location number
• Small squares show which fuel assembly type is in which location (or which gene is in which location). • Number in small squares represents symmetry for the location number. This number also represents how
many fuel types x (x is between 1 and 20) are replaced in location y (y is between 1 and 15) in the full core.
• # in the inventoiy shows how many fuel types x are in the inventory. • # in the core shows how many fuel types are used in the core.
• # in the inventory must be larger than or equal to # in the core for each fuel type.
• Each row must have only one square. (Only one fuel type should be defined for each fuel location) • A column may have more than one square as long as the # in the core is not larger than the # in the
inventory.
Location Number Sym m etry Number
FA Type L5 2 F20 L4 2 F5 L9 4 F17 L3 2 F3 L8 4 F18 L12 4 F20 L2 2 F7 L7 4 F16 L11 4 F17 L14 4 F14 L1 1 F1 L6 2 F12 L10 2 F16 L13 2 F9 L15 2 F12
Figure 2. Genotype Representation of Sample Core in Figure 1
3. USING IN-CORE FUEL M ANAGEM ENT HEURITI RULES
Nuclear reactors have been operated for the last 40 years. Many heuristic rules have been learned within these years to arrange the fuel assembly types in the nuclear reactor core. To use these rules for creating initial population and utilizing these rules with GA operators, worth definition application is developed. Each gene defines a worth value for each location in the genotype. This worth is between 0 and 1. If fuel management heuristic rules state that:
• FA type A should not be in the location x, then
0.0 <WorthAx< 0.5
• FA type A could be in the location x, then
0.5 <WorthA< 1.0
• There is no expression for locating the FA type A in location x, then
WorlhAx = 0.5
For creation of initial population a subroutine provides a percent of individuals, which have genes with higher than 0.5 worth. This number can be defined by the user. To provide diversity in the population some individuals’ worth values can be lower than 0.5. Generation to generation frequency of genes in locations of genotype with higher fitness is calculated. According to this frequency this worth increases or decreases.
4. ALGORITHM
A simple GA flow diagram is shown in Figure 3. The GA operators are modified to work with the new genotype representation
Figure 3. GA Algorithm
For the selection operator the tournament selection method is used. Two individuals are chosen randomly from population, their fitness values are compared, and the individual with higher fitness is selected for processing of other operators. Every individual has an equal chance of being chosen for creating breeding pool.
The crossover operator process is used to switch the genes between the genotype representations to generate offspring. Crossover is not applied to relative parents. A relativeness degree is calculated for the randomly selected parents. Fuel assembly type for each location of the first parent is compared to the fuel assembly type for the each location of the second parent. Number of similarity equals how many locations of the first parent have the same fuel assembly type as the second parent for the same locations. Relativeness degree for the randomly selected parents equals to;
If RD equals or lowers than 0.75, crossover is applied to these parents.
For this problem the heuristic knowledge demonstrated that some fuel assembly types should be placed at some specific locations to obtain longer cycle length. Due to that, crossover operator works as cycle crossover. After using crossover operator, the location of the fuel assembly types in the offspring is the same as of the parents.
To provide diversity, two different mutation operators are used.
Location Based Mutation: Two locations with the same symmetry condition are selected randomly and fuel
types in these locations are switched. The fuel management heuristics knowledge is utilized in the mutation operator. If the worth value of one of the switched genes is smaller than 0.5 in its new location in genotype new individual is not accepted with a certain probability. This probability is defined as 0.8 to provide diversity.
Fuel Assembly Type Mutation: Randomly a fuel assembly type is selected in the genotype and this fuel type
is switched with a different available fuel assembly type in the inventory.
Multi-Mutation: Due to the characteristics of the crossover operator, which provides less diversity for the
next population, a new operator called multi-mutation operator is developed to provide more diversity in the population. A gene is chosen by this operator and location of this gene is switched with locations of randomly selected genes from at least two times to maximum five times. To apply the fuel heuristic knowledge to this operator the same method is used as with the mutation operator.
5. APPLICATION to VVER-1000 CORE RE LOAD DESIGN
For the W E R -1000 reactor, the symmetrical structure of the core is used to generate genotypes. The W E R - 1000 reactor core can have the following symmetry options; 300, 600, and 1200 sectors of symmetry. As it can be seen in Figure 4 the reactor core can have a 600-sector of rotational symmetry. Location 1 shows the center of the core and symmetry number for this location is one while the symmetry numbers of the other locations are 6. In the center location the fuel assembly type is fixed for all LPs produced in the code. Fuel assembly type 1 (fl) is used for the center location. Locations 7, 13, 18, 22, 25, 27, and 28 are in the peripheiy of the core. In this problem 22 different fuel types are used. Table 1 shows some characteristics of the fuel assembly types in the inventory. As it can be seen in Table 1, f21 and f21 are fresh fuels.
The klnf characterizes the multiplication properties of the fuel assembly type in the reactor. Neutrons are absorbed by the material in the fuel assembly. Absorbing neutron by fissile or fissionable nuclei induce fission and the birth of new fission neutrons, which cause new fissions. It is called chain reaction. Multiplication factor is the ratio of the number of neutrons in one generation and the number of neutrons in preceding generation. In the inventory there are two fuel assembly types. They are burned fuel assembly coming from previous cycles and fresh fuel assemblies. Fresh fuels can induce more fissions and their kinf values is higher than burned fuel assembly.
L o c a t io n Number
4. Location Numbers of W E R -1000 Core (1/6
Figure layout)
Table 1. Characteristics of Fuel Assembly Types in the Inventory
Name fl f2 f3 f4 f5 f6 BO C kinf 0.962 1.037 1.046 0.965 1.119 1.094 # in the inventory 1 6 6 6 6 6 Name f? f8 f9 flO f l l f 12 BO C kinf 1.129 1.134 1.14 1.006 1.06 0.988 # in the inventory 6 6 6 6 6 6 Name f 13 f 14 f l 5 f 16 f 17 f 18 BO C kinf 1.042 1.039 1.119 1.094 1.129 1.139 # in the inventory 6 6 6 6 6 6 Name f 19 f20 f 21 f22 BO C kinf 1.006 0.989 1.175 1.273 # in the inventory 6 6 24 24
The main idea of this optimization problem is to maximize the core lifetime without exceeding the Normalized Power (NP) peaking limit that is 1.3 for this problem.
Fitness is calculated with using reactor physics code, which is Moby-Dick. Fuel assembly arrangement for each individual is sent as an input to Moby-Dick, which calculates NPs for each fuel assembly and EOC boron concentration. Due to safety constraints, maximum NP must be lower than 1.3. Optimal core should have larger boron concentration at the end of cycle.
Fitness= -Max NP for Max NP ^ 1.3
Fitness= EOC Boron Concentration for MAX NP < 1.3
As shown in Figure 4 location 7, 13, 18, 22, 25, 27, and 28 are at the periphery. The worth value of the fuel assembly type with the lower kinf in these locations is assigned to be higher than 0.5. On the other hand, the worth value of the fuel assembly type with higher kinf (especially fresh fuel) for these locations is lower than 0.5. It is assumed that the fresh fuel should be closer to the periphery and location 2 should have a fuel assembly type with lower kinf to provide maximum NP constraint.
In Figure 5, the comparison of GA with worth-definition with GA with non worth-definition is shown. For non-worth definition initial population was created randomly. Result shows that GA with worth definition gives better result in a shorter time.
To use heuristic rules effectively and decrease the problem size, age process is developed. Operation time is divided to ages as the ages in the real world life. (For example ice age). It is assumed that in each age operators works with a group of genes instead with all of the genes. Groups are created according to the in core fuel management heuristic rules and obtained frequency values in the first run.
• Fuel assembly types with lower kinf are placed at the periphery. In this way the neutrons are used efficiently in the core with providing less leakage.
• Fuel assembly type 22 has the highest kinf in the inventory. These fuel types are placed in location 12, 17, 24 and 26 with 90% frequency. These locations are very close to the periphery. Therefore, the maximum normalized power is kept below 1.3.
These points provide the heuristic in-core fuel management knowledge. We can use these rules with applying the aging process. In this process, these points are fulfilled;
• To create initial population, worth definition is used. In this population and for next generation fuel assembly type 1 is fixed in location 1.
• 250 generations are divided to three ages.
1. Age; Fuel assembly type 22’s locations are fixed in location 12, 17, 24 and 26 for the first 10
generations.
2. Age; After the end of the 10th generation, fuel types at the periphery locations (7, 13, 18, 22, 25, 27, and
28) are fixed for generations from 11th to 170th.
3. Age; After the end of the 170th generation all the fuel assembly types except fuel assembly types in the
periphery are fixed for generations from 170th to 250th.
The change of fitness value is shown in Figure 5. The better result is found with a smaller amount of population generation.
Figure 5. Comparison of the best fitness variations with and without specific worth definition and with age process for W E R - 1000
6 .CONCLUSION
GA was used to optimize LP elsewhere. In this research, The GA concept for LP problem optimization is adjusted further by using some novel developments. These developments are summarized below;
• The previous in-core fuel management codes are written for specific core structures with using different techniques such as deterministic and stochastic methods. This research is aimed to developing a general code, which is suitable for every core structure. GA is suitable for this goal. A new genotype representation is developed to define the different core structures by specifying in the input deck easily. Then, GA operators are modified to adapt the new representation and are applied to generate new populations. One of the advantages of a GA code is that this code is independent from the reactor physics code. This is defined as “black box” approach. This approach and the new representation provide the independency for GA code.
• In the last 50 years, the experience of operation of nuclear reactors provided in-core fuel management heuristic rules for loading pattern optimization. Using these rules in the optimization code will decrease the time to obtain optimal results. To use these rules, the worth definition concept is developed and combined with the GA code. Population is created with using worth definition and some restrictions are added to the operators to use worth definitions to create next generations. When using this concept the running time of the code is decreased.
• The in-core re load problem is very large problem. It includes approximately 1026 different combinations. If the burnable poison optimization is added to the problem, the size of problem will be larger than 1026. The size of the problem is decreased during the operation of the GA code with using the age definition, which is combined with the fuel management heuristic rules.
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