A Survey of Genetic Algorithm Applications in Nuclear ...

Journal of Nuclear Engineering & Technology ISSN: 2277–6184 Volume 4, Issue 1 www.stmjournals.com

A Survey of Genetic Algorithm Applications in Nuclear Fuel Management M.L. Jayalal*, S.A.V. Satya Murty, M. Sai Baba Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu, India Abstract The prime aim of nuclear fuel management is to achieve higher fuel utilization without compromising safety during reactor operation. Nuclear fuel management optimization is a classical nuclear engineering problem, which has been studied for more than four decades and several techniques have been used for its solution. Genetic algorithm is one among the major global optimization techniques, used in the field of nuclear fuel management. Aim of this paper is to survey genetic algorithm related developments happened during the last few decades in nuclear fuel management optimization. The objectives of this survey are: (i) to summarize how genetic algorithm is applied to the field of nuclear fuel management (ii) to compare different genetic algorithm techniques and operators used in solving the nuclear fuel management optimization problems (iii) to bring out current trends and future directions in this field. The survey will help the researchers in the field to get an overview about genetic algorithm techniques available for nuclear fuel management optimization and how to use them.

Keywords: Genetic Algorithm, nuclear fuel management, multi objective genetic algorithm, constrained optimization, parallel genetic algorithm, crossover, mutation *Author for Correspondence E-mail: [email protected]

INTRODUCTION Nuclear fuel management entails making decisions that influence how a nuclear reactor core’s reactivity, flux, power and burnup spatial distribution vary in order to extract electrical energy in an optimal way. The fuel management related decisions include the following: design of the fresh fuel lattices and fuel assemblies, identification of the number of fresh fuel assemblies and partially burned fuel assemblies inserted in to the reactor core, determination of core loading pattern and management of core reactivity via control materials. The objective of nuclear fuel management is to minimize the cost of electrical energy generation, considering the operational and safety constraints. The cost reduction can be achieved through various means; maximization of the cycle length, maximization of the burnup and minimization of fuel and reactivity control material inventory. The constraints can be imposed on the power peaking, core excess reactivity, linear heat rating of fuel assemblies and region

averaged burnups. The principle characteristics of nuclear fuel management related problems are high-dimensionality, large number of constraints, large number of feasible solutions, disconnected feasible regions in search space, lack of derivative information and the high computational cost of evaluation function [1]. Nuclear fuel management optimization is a Non-deterministic Polynomial-time hard (N-P hard) combinational problem which has been studied for more than four decades and several techniques have been used for its solution [2]. For example, early applications of mathematical programming methods in this area were made with Dynamic Programming and with Linear and Quadratic Programming during 60s [3, 4]. Subsequently global optimization techniques like Genetic Algorithm (GA) and Simulated Annealing were applied in nuclear fuel management field.

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Nuclear Fuel Management Jayalal et al. __________________________________________________________________________________________

Genetic Algorithm (GA) is an optimization tool based on Darwinian Theory of biological evolution. The method was developed by John Holland [5] and later popularized by one of his students, David Goldberg, who successfully applied to various practical engineering problems [6]. During the last three decades GA found its increasing applications in various fields like data mining [7], image processing [8] , pattern recognition [9] , signal processing [10] etc. GA can do an intelligent exploitation of a random search within a defined search space to solve optimization and search problems. This algorithm surpasses the more traditional methods of search and optimization in the quest for robustness. This is because GA does not get trapped in local optima and do not depend upon the existence of derivatives like calculus-based methods. Furthermore, they are much more efficient than enumerative schemes and random search algorithms as they do not require evaluation of a very large number of points in the search space. These vivid advantages brought GA as a suitable and efficient tool in nuclear fuel management applications. The aim of this paper is to survey GA related developments happened during the last few decades in nuclear fuel management optimization. This survey presents “when and how” GA has been used during the period from 1990 to 2013, in the field of nuclear fuel management. The main objectives of this survey can be formulated as: (i) to summarize how GA is applied to the field of nuclear fuel management (ii) to compare different GA techniques and GA operators used in solving the nuclear fuel management optimization problems (iii) to bring out current trends and future directions in this field. The survey will help the researchers in the field to get an overview about GA techniques available for nuclear fuel management optimization and how to use them. This survey primarily concerned with GA techniques and usage, rather than the reactor physics related part of fuel management problems. For more details about theory and practices of neutronics simulations in nuclear fuel management, the readers are referred to specialized literature on the topic [4, 11–13].

The survey is structured as follows. A brief history about optimization attempts happened in nuclear fuel management is given in Section 2. A general description about the nuclear fuel management problem and its types are given in Section 3. Section 4 gives a brief description about GA’s suitability in nuclear fuel management and also gives an idea about the overall procedures involved in GA. The basic types of GA used in this field and the standard approaches followed in those implementations are covered in Section 5. The implementation details of GA, particularly in nuclear fuel management is given in Section 6. Finally conclusions and future directions are given in Section 7.

NUCLEAR FUEL MANAGEMENT OPTIMIZATION: BRIEF HISTORY Optimal nuclear fuel management is a classical optimization problem in the field of nuclear engineering which was initially solved manually by experts that used their knowledge and experience. Finding out the optimal reactor core configuration is very crucial for initial fuel assembly loading as well as periodic fuel assembly reloading, irrespective of the type of nuclear reactor. Hence the need of suitable optimization technique in this field was evident from the very early stages of nuclear reactor design and implementation. All the early optimization techniques applied in nuclear fuel management were using gradient search or hill climbing like methods for finding the optimal solutions [14].These methods were having the possibility of getting trapped in local optima and hence the requirement of a global optimization technique for nuclear fuel management was there from early stages of nuclear fuel management. During the period of seventies and eighties, inventions of global optimization techniques like GA by John Holland [5] and simulated annealing by Kirkpatrick et al. [15] were happened. Nuclear fuel management problems found an immediate suitability of application of these global optimization techniques. There are many examples of early application of GA in nuclear fuel management [16–19]. Similarly simulated annealing also applied for the same purpose in early nineties [20]. The first application of GA in nuclear fuel management was presented by Poon and Parks [16]. A


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Journal of Nuclear Engineering & Technology Volume 4, Issue 1, ISSN: 2277–6184 __________________________________________________________________________________________

detailed comparison between the GA and simulated annealing were presented in their work and observed the advantages of GA in the global search and GA’s suitability of implementation for parallel computers. The early applications of stochastic intelligent algorithms in this field started with GA and Simulated Annealing in 90s but eventually GA found more suitable because of its inherent advantages. Subsequently GA became a well proven and robust tool in the field of nuclear fuel management, which got application in many types of reactors [21]. During the past two decades, several other computational intelligence techniques have been used in nuclear fuel management field by many researchers. Some examples are Artificial Neural Networks (ANNs) [22–24], Tabu Search (TS) [25–27], Ant Colony Optimization (ACO) [28–30], Particle Swarm Optimization (PSO) [31–33] Artificial Bee Colony Optimization (ABCO) [34,35] , Harmony Search Algorithm (HSA) [36] and Continuous Firefly Algorithm (CFA) [37], which have presented good results. Over the last few years many quantum inspired evolutionary algorithms such as Quantum Ant Colony Optimization (QACO) [38], Quantum Population-Based Incremental Learning (QPBIL) [39] and Quantum Evolutionary Algorithm (QEA) [40] are applied in nuclear fuel management and shown their advantage in potential reduction in computational time. Even though the developments in other computational intelligence techniques are happening in parallel, GA still stands as a well proven and widely accepted optimization technique in the field of nuclear fuel management. At present, many new variants of GA are being applied in nuclear fuel management of different types of reactors [41– 43].

NUCLEAR FUEL MANAGEMENT: THE PROBLEM AND TYPES Traditionally nuclear fuel management has been divided in to two categories, out-of-core and in-core fuel management [44, 45]. Out-ofcore fuel management focuses on answering the questions “What to manufacture?” and “What to insert?” in the context of multi-cycle operations of the reactor. It includes the

decision on number and composition of fresh assembles and which burned fuel assemblies to be inserted in the core. The decisions associated with out-of-core fuel management are lattice design (like pin wise fuel enrichment and type and configuration of reactivity control materials), axial placement of lattice designs, number of assemblies of each type, the selection of spent fuel assemblies to reinsert and the determination of appropriate cycle lengths. The two major deciding parameters of out-of-core fuel management are:  determination of the optimal lattice design of fuel assemblies and control materials  identification of optimal number and types of fuel assemblies to be loaded in the core The important constraints for out-of-core nuclear fuel management are; the cycle energy production requirements, the fuel enrichment requirements, the control materials worth requirements and various discharge burnup requirements. Hence the out-of-core fuel management decisions are specific to the type of reactor. The reactor core design related fuel management problems are typical examples of out-of-core fuel management. In-core nuclear fuel management focuses on answering the question “Where to position?” the fuel assembles or reactivity control materials in the core. The in-core fuel management entails the arrangement of fresh and partially burned fuel assemblies and reactivity control mechanisms within the core that optimizes the performance of the reactor over the next operating cycle, while ensuring that operational constraints are always satisfied [17]. The two major decisions to be taken under in-core fuel management are:  optimal core loading pattern determination where the location and orientation of different subassemblies are decided  optimal loading scheme of reactivity control materials in the core Majority of the nuclear fuel management problems in which intelligent optimization techniques like GA are applied, are coming under in-core fuel management. So we are looking in to more details about how it is applied in various types of reactors. In-core fuel management of Pressurized Water Reactor (PWR) and Boiling Water Reactor


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(BWR) are having many common features along with certain differences among them. In both types of reactors, the refueling is offpower i.e., refueling is done after shutting down the reactor. The refueling scheme is also similar in both the types of reactor where shuffling of burned fuel assemblies is done. The use of burnable poisons as a reactivity control material is also common in PWR and BWR. Normally burnable poisons are shaped as non-removable or integral part of the fuel assembly once it is manufactured. In PWR, separate burnable poison rods which are similar to fuel assembles but manufactured with neutron absorbing materials, may also be present. Apart from the burnable poison, soluble poison like boric acid (called as chemical shim) is used for excess reactivity control in PWR. Since chemical shim is used to ensure reactor criticality at the desired core flow rate (known as criticality constraint), control rod positions and insertions are not coming as a decision variable in PWR in-core fuel management optimization problems. But in the case of BWR, where chemical shim is not used, the criticality constraint is achieved by control rods positioning. The positioning of control rods as a function of cycle burnup is called control rod program. In BWR, the incore fuel management should consider optimum control rods program also, for each potential loading pattern of fuel assemblies which result in added complexity for the optimization problem [44]. The reactor physics neutronics simulation codes used in BWR are usually of 3-dimensional because of the presence of more numbers of fuel assemblies in the core, presence of strong axial heterogeneities and coolant voidity. This results in more computational time for in-core fuel management optimization problems of BWR [46, 47]. The in-core fuel management scheme of Pressurized Heavy Water Reactor (PHWR, sometimes referred as CANDU reactors) is different from that of PWR and BWR. In the case of PWR and BWR, the fuel assemblies are loaded at the beginning of cycle and there is no fuel assembly movement until the reactor is shut down for refueling operations. There the fuel management scheme aims to determine the best fuel arrangement throughout a fuel cycle [48, 49]. In PHWR, the

refueling is on-power (i.e., without reactor shutdown) and in daily basis to maintain the reactor criticality and the reference power distribution. Because of the on-power refueling scheme, the equilibrium PHWR core is not uniquely defined. Typically the timeaverage core calculation is used to search for an optimum power distribution by changing the number of fuel bundles loaded per refueling operation and adjusting the discharge burnups of the inner and outer cores in PHWR. The in-core fuel management scheme of fast breeder reactor (FBR) is off-power which is similar to that of PWR and BWR. One major difference in FBR is that the reshuffling of burned fuel assemblies is not usually done during refueling operation. This is due to the bowing of subassemblies in response to fast neutron flux and temperature gradients [50]. The thermal operating conditions are much tighter and the core neutronics related parameters (like k-infinity, peaking factors, reactivity swings) are significantly higher, which result in the requirement of more precise and sophisticated neutronics simulation models in handling fuel management problems of FBR [50-52]. Another noteworthy point is that in FBR, neutron moderators are absent and hence neutron poison mechanisms like burnable poisons or chemical shims are also absent. There are some special cases of in-core fuel management optimization problems where instead of regular fuel reloading, some special cases of fuel loading patterns are considered as the objective. Examples of such special case fuel management problems are optimization of thorium loading in fresh core [53] and optimization of depleted uranium bundle loading in fresh core [54,55] of PHWR. The in-core fuel management problems of research reactors or test reactors can also be treated as a special case where the absence of automatic refueling machine results in an additional constraint on the limited number of fuel shuffles [56–58]. It can be observed from the literature that, unless otherwise specified, the term “nuclear fuel management” represents “in-core fuel management” only. This survey also follows that convention. The “core design” related


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problems discussed in this survey are coming under out-of-core fuel management and all others are coming under in-core fuel management. The out-of-core fuel management decisions are taken at the time of core design of the reactor and the decisions taken are almost fixed. On the other hand, the in-core fuel management decisions are to be taken at initial core loading time as well as regular refueling times; hence are more frequent in nature. The applications of optimization tools like GA are more common in in-core fuel management. The following section describes about the suitability and methods of application of GA in nuclear fuel management. The strategies related to GA described here are common to both in-core and out-of-core fuel management. GA IN NUCLEAR FUEL MANAGEMENT As mentioned earlier, nuclear fuel management’s aim is to get an optimized initial fueling or subsequent refueling scheme by considering relevant reactor physics related parameters, safety and economics. When we consider all these objectives together, some of them will conflict with the other. That means any of the final solutions must represent some sort of compromise in which no further improvement in a given performance index can be obtained without a degradation in at least one of the other performance index. Hence nuclear fuel management optimization is considered commonly as a multi-objective optimization problem where the goal is to identify the solution vector which gives rise to best compromise among various objective functions [52]. It has been well proven that GA is an efficient and versatile tool for dealing such a complex multi-objective optimization problems [6]. The ease of parallelization of GA is an added advantage when we consider the computational cost involved in fuel management problems. As part of GA representation, a candidate solution is encoded as a digital chromosome which has enough information to reproduce the original solution. While being executed, GA generate a collection of trial solutions i.e., a population of chromosomes, and the fitness values of each chromosome is evaluated. Similar to the natural selection process, chromosomes which have higher fitness values will have more

chances of getting selected as ‘parents’ which participate in reproduction process. The ‘offspring’ solutions are produced from the parents using the genetic operations like crossover and mutation [59, 60]. In GA, there are many flexible and efficient encoding schemes and operators which are suitable in handling fuel management optimizations. The overall procedures of nuclear fuel management optimization using GA can be summarized as follows: (1) Generate the initial population with suitably encoded chromosomes which depend on the type of NFMO under consideration. For example, if the problem is of out-of-core fuel management, then each chromosome represents the lattice design parameter values. Similarly if the problem is of in-core fuel management, then each chromosome represents the loading pattern of fuel assemblies and/or reactivity control materials. (2) For each individual or chromosome of the population, calculate the fitness values using reactor physics simulation codes. This is the objective function evaluation step of the optimization process. (3) Select the parents according to the evaluated fitness, and perform the genetic operations such as crossover and mutation to produce next generation of offspring. (These GA specific operators are further discussed in Section 6.) (4) Repeat (2) and (3) until the GA search process of finding the optimal solution is converged. The fitness evaluation calculation using the reactor physics simulation codes is one of the important and computational time consuming step in the above procedures. Even though the GA strategy and procedures are common, the rector physics codes are specific to the type of the reactor and to the optimization problem being considered. The overall flowchart of GA when applied in nuclear fuel management optimization; which includes GA procedures, interface module and reactor physics simulation code is given in Fig. 1. As shown in the flowchart, there is an interface module normally present between the GA implementation part (steps shown in left side of the flowchart) and the reactor physics


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simulation code. Most of the reactor physics codes used in nuclear fuel management arena is developed in FROTRAN language. If the GA implementation part is developed in any other language (such as C), then the interface module should able to generate the input files which satisfies the FORTRAN based reactor physics simulation code’s file read format.

Similarly the output values generated by the reactor physics simulation module should be read back by the interface module and given back to the GA implementation module for further calculations. This bidirectional data flow is shown by the two directional arrows in the flowchart.

Fig.1: Flowchart of GA When Applied in Nuclear Fuel Management Optimization. Conceptually, the GA implementation part share many common features among different types of reactors fuel management optimization. From the literature, it can be observed that, there are some initiatives to develop GA part of as a flexible and adaptable module which suites for multiple types of fuel management optimization problems. The major advantage of this method is that with minimum change, the GA module can be used for different types of fuel management problems. Some typical examples are GARCO-Genetic Algorithm Reactor Code Optimization- developed by Alim et al. [61– 63] and CIGARO-Code Independent Genetic

Algorithm Reactor Optimization- developed by DeChaine et al. [18, 19, 64].

TYPES OF GA IN NUCLEAR FUEL MANAGEMENT As mentioned earlier, GA is a general purpose versatile optimization tool well suited for combinational problems like nuclear fuel management. There exist many flavors of GA for dealing such multi-objective optimization problems. It can be observed from the literature that, there are two major approaches in formulating nuclear fuel management optimization model for GA; they are Constrained optimization with penalty function and Multi Objective Genetic


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Algorithms (Multi Objective GA). In first case of the Constrained optimization with penalty function, the actual multi-objective problem of fuel management optimization is artificially converted to single objective by adding penalty functions and constraints. In the second case i.e., in Multi Objective GA, the algorithm’s power of handling multiple objectives together is exploited in an efficient way. The parallel program based GA can be considered as yet another category, where the execution part of GA is divided to parallel executable units. These three widely used approaches, when GA is applied in the nuclear fuel management filed, are discussed below. These three widely used GA categories in nuclear fuel management optimization are discussed below.

CONSTRAINED OPTIMIZATION WITH PENALTY FUNCTION The concept of penalty function is applied here to convert the constrained optimization problem in to the form that is suitable for GA to deal with. The penalty function is incorporated in to the objective function so that, a “penalized objective function” will be used by the GA. The penalty function is needed to be formulated in such a way that, it should not affect the actual objective function if constraints are not violated. On the other hand, in the case of constraint violation, the penalty function will put a high value in the opposite direction of the objective function;

f1 =

keff - 1, if keff

i.e., the penalty function will penalize the actual objective function, sharply for constraint violations. GA’s flexibility gives an option to incorporate other constraints handling mechanisms like “death penalty method” to the penalized objective function [65]. These concepts are further explained in the fuel assembly loading pattern optimization example explained below. The Constrained optimization with penalty function method has been used extensively in GA based nuclear fuel management. In order to explain the method and its application in the nuclear fuel management field, a typical example of fuel assembly loading pattern optimization is discussed here. In the loading pattern optimization of fuel management, where objective of the problem aims at finding the optimal loading pattern that maximize endof-cycle reactivity ( keff ) and minimize the power peaking factor (PPF) [57]. The penalized objective function can be formulated according to the Constrained optimization with penalty function method as follows: Maximize F = a1 f1 + a2 f2 ,

(1)

Where a1 and a2 are penalty factors and f1 and f2 are penalty functions. The penalty factors a1 and a2 are positive coefficients and are considered as a measure of the keff and PPF significance, respectively. The penalty functions are further defined as:

kmin 0, otherwise (2)

where kmin is the lower limit of keff , f2 =

PPF0 - PPF, if PPF ≤ PPFmax 0, otherwise (3)

where PPF0 is an input factor that is chosen so that the PPF is always lower than it and PPFmax is the maximum limit for power peaking factor. By writing the objective function in the form of Eq. (1), the problem of minimizing the PPF is transformed into a

maximization problem. For the given problem, the other constraints like discharge burnup (BU) can be incorporated to the model as: BU ≤ BUmax


(4)

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where BUmax is maximum allowable discharge burnup. The burnup related constraint can be added to the penalized objective function of Eqn.(1) as death penalty method where burnup values higher than BUmax values are considered as infeasible solutions and are rejected. GA based on Constrained optimization with penalty function is easier to model and implement; hence is widely used in nuclear fuel management optimization. If the penalty coefficients (e.g. a1 and a2 of Eqn. (1)) and the constraints are properly selected, this approach will give feasible solutions. One disadvantage here is that the selection of penalty coefficient and constraints are based on designer’s experience and preconceptions, which are prone to errors. Another problem of the approach is that the multi-objectives are artificially converted to single objective, which leads to the identification of just one solution in the trade-off surface. Moreover, it is not easy to aggregate these multiple objectives into a practicable single performance index until their relative importance is well appreciated.

MULTI OBJECTIVE GENETIC ALGORITHM (MULTI OBJECTIVE GA) Parks [66] suggested a novel method named Multi Objective GA in nuclear fuel management, which explores the unique suitability of GA in solving such true multiobjective optimization problems. This method addresses the major issues of Constrained optimization with penalty function method, like requirement of expertise in parameter value’s selection and single point convergence. The method makes it possible to identify the trade-of-surface between competing objectives in a single optimization run. The concepts of Pareto optimality and dominance [6, 67, 68] are used in Multi Objective GA, to rank the population of solutions without imposing any preconceptions about the relative importance of individual objective. According to the concept of Pareto optimality and dominance, a solution X is said to be dominated by solution Y if Y is better in all objectives, i.e., if fi (Y) < fi (X)

i=1,N

(5)

where fi are the N objectives to be minimized. If a solution is not dominated by any other solution, it is said to be non-dominated solution. Using the above definition the entire population can be sorted and ranked to identify all non-dominated solutions. Ranking the population appropriately, then removing them from consideration and repeating the procedure until all solutions have been ranked, are the major steps involved in Multi Objective GA. In Multi Objective GA, constraints can be treated as additional objectives to be optimized. A detailed implementation details covering various aspects of application of Multi Objective GA in PWR fuel management is given by Parks [66] and that of FBR is given by Toshinsky and team [51]. A detailed description about a modified Multi Objective GA in fuel management where concepts like fitness sharing and niches induction is given by Toshinsky et al. [52]. The major advantage of applying Multi Objective GA in nuclear fuel management is that, it solves the multi-objective optimization problem as true multi-objective optimization problem rather than artificially converting it in to single objective problem, as in the case of Constrained optimization with penalty function method. The Multi Objective GA will also generate a wide range of feasible solutions with multiple objectives. This feature is particularly valuable in nuclear fuel management optimization where the designer can use this additional information for the better informed decision making.

PARALLEL GENETIC ALGORITHM (PARALLEL GA) In general, GA is suitable for parallel computation because it is inherently data parallel in nature. In GA, the decision variables are encoded as chromosomes and a fixed number of such chromosome based candidate solution constitute the population. Each candidate solution in a population can be evaluated independently with respect to other individuals in the population, hence suitable for parallel computation. This inherent data parallelism is not available in many other intelligent algorithms like Simulated Annealing, since candidate solution


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generation and evaluation must be carried out sequentially there [60]. The computational overheads due to complex neutronic calculations involved in nuclear fuel management problems; hence the parallel computing concepts of Parallel GA are much promising there. One can implement the Parallel GA in solving nuclear fuel management problems by dividing the GA execution part in to different parallel executable units, normally called “islands”. The total population in conventional GA is divided in to predefined number of islands which has its own independent evolution process. The communication between the islands is achieved by the process called migration where information about different regions of the search space is exchanged between islands, providing more diversity in the search. This island model is the major class of Parallel GA when applied in nuclear fuel management problems, even though other approaches like master-slave and cellular parallelization can also be applied [69,70]. It is important to note here that the parallelization concept of Parallel GA is for dividing and distributing the computational burden among multiple processors. But for handling the constraints of fuel management problems in Parallel GA, one has to follow the Constrained optimization with penalty function or Multi Objective GA approach. That means Parallel GA can further divide in to two categories; one is Parallel GA with “constrained optimization with penalty function” approach and another is Parallel GA with “multiobjective genetic algorithm” approach. The first concept of Parallel GA with “constrained optimization with penalty function” is followed in the recent work by Norouzi and team [41]. It is observed from the literature that the application of Parallel GA with “multi-objective genetic algorithm” approach is a less explored research area.

SUMMARY OF TYPES OF GA IN NUCLEAR FUEL MANAGEMENT The three major categories of GA applied in nuclear fuel management optimization are Constrained optimization with penalty function, Multi Objective GA and Parallel GA.

The Table given below (Table 1) shows “where and how” different categories of GA are applied, in nuclear fuel management field. As mentioned earlier, the term “Core Design” in the Table is used to refer the out-of-core fuel management problems. It is evident from the Table that, the most common and frequent GA method in nuclear fuel management is with Constrained optimization with penalty function strategy. In fact, the four articles listed in the Table under Parallel GA are also in turn follow the Constrained optimization with penalty function strategy for handling the fuel management constraints. Hence, out of the forty one GA related nuclear fuel management articles reviewed here, thirty two (including four articles coming under Parallel GA) are of Constrained optimization with penalty function strategy. The reason for this vast application is the simplicity of this approach in GA model formulation, in which multiple constraints are converted to a single objective function. The potential of Multi Objective GA which is more suitable in nuclear fuel management is not explored enough. Similarly, it can be observed that the application of Parallel GA in nuclear fuel management is also very less. The design and application of parallel Multi Objective GA in this field is almost untouched and has a lot of scope in improving the performance of GA, when applied in nuclear fuel management.

GA IN NUCLEAR FUEL MANAGEMENT: IMPLEMENTATION DETAILS The GA implementation details will vary according to the fuel management problem that selected for optimization. In general, the chromosome encoding scheme, the GA specific operators used, the convergence criteria selected, these all will depend on the selected type of fuel management problem. Commonly used GA specific implementation details, available in nuclear fuel management related articles, are covered in this survey. The aim here is to give the developer, an idea about what GA specific parameters and operators those can be selected, for the nuclear fuel management problem under his consideration.


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Table 1: Summary of Different Types of GA in Nuclear Fuel Management Applications. Type of GA

Reactor Type Pressurized water

Constrained optimization with penalty function

Boiling water Pressurized heavy water Advanced gas cooled Research reactor Pressurized water

Multi Objective GA

Parallel GA

Boiling water Pressurized heavy water Fast breeder Research reactor Pressurized water

Type of Problem Loading pattern Burnable poisons Loading pattern and Burnable poisons Core Design Loading pattern Online Refueling

References 18, 21, 43, 59, 76, 85. 81, 90, 91. 2, 17, 19, 61, 62, 63,64, 79. 1, 88. 83,84,86. 89

Thorium Loading

53

Loading pattern

80, 92.

Loading pattern Loading pattern and Burnable poisons Core Design Loading pattern and CRP

56, 57.

87 46, 47, 82.

Online Refueling

49

Loading pattern Loading pattern Loading pattern Loading pattern and Burnable poisons Core Design

51,52. 58 41

GA ENCODING SCHEME Every search and optimization algorithm requires a suitable encoding scheme to represent the probable solutions. GA encoding scheme define a chromosome representation for each individual in the population. Choosing the right method of encoding chromosome is a crucial task and largely affects the efficiency of optimization problem solving. For example, in the loading pattern optimization, the values to be encoded as chromosomes are the position of fuel sub-assemblies or burnable poisons or control rods. The most commonly used encoding schemes of GA in nuclear fuel management related problems are binary encoding and integer encoding schemes. In the case of binary encoding, the decision variables are represented by binary string chromosomes. For integer encoding, the decision variables are directly represented as integer numbers in chromosomes. Nuclear fuel management optimization is a field where the decision variables in most of the cases are naturally of integer type like loading positions of fuel subassemblies. Therefore, the most common GA encoding scheme for such problems are of integer type. A special type of chromosomal encoding scheme named Random Keys

66

60 69, 70.

representation is used by some of the authors in nuclear fuel management field, to address the crossover difficulty of generating infeasible solutions [2, 71].

GA OPERATORS The working principle of GA is based on its three genetic operators which are responsible for evolving better solutions; those are selection, crossover, and mutation operators [6]. The selection operator is responsible for selecting the solutions to make up the breeding pool in a "survival of the fittest" competition. Individual solutions are selected with a probability based on their "fitness" (fitness here represents a figure of merit) or based on value of the objective function associated with each solution. The crossover operator, pairs solutions from the breeding pool and mixes their traits to create two new and different solutions. This is analogous to biological mating, where the children inherit a unique mixture of their parent’s genes. Finally, the mutation operator makes small random changes in the solutions. Mutation maintains the diversity of the population and allows the search to cover the entire search space [19]. Another concept applied in most of the fuel


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management implementations of GA is “elitism”. The detailed discussion about various types of GA operators and their working principles are given in GA related literatures [6, 67, 72-74].The subsequent sections give an overview about various types of GA operators used in nuclear fuel management problems.

SELECTION OPERATORS The selection operator works by selecting solutions from the population of chromosomes for the breeding pool, based on their fitness values. The commonly used GA selection operators in nuclear fuel management optimization field are roulette wheel selection (also known as fitness proportional selection), tournament selection and ranking selection. In roulette wheel selection method, one can imagine the roulette wheel divided into partitions, one partition corresponding to each solution. The size of the partition is proportional to the fitness of that particular solution. The wheel is then spun and is allowed randomly to point some partition of a solution, which will be selected for breeding through crossover. Thus greater is the fitness, the larger is the partition and greater is the probability of the solution to be selected for breeding as a parent. In tournament selection method, two solutions from the population are randomly selected and their finesses are compared. The one with the higher fitness is selected for breeding. Thus bad solutions with lesser values of fitness will lose out to a better solution; thus will have a very low probability of being selected for crossover. Next commonly used selection method in nuclear fuel management is ranking selection. The idea here is straightforward; sort the population from best to worst, assign the number of copies that each individual should receive according to a non-increasing assignment function, and then perform proportionate selection according to that assignment.

ELITISM Elitism can be considered as an augmentation to selection operators and is independent of the type of selection operator. The better few individuals namely “elite members” in any population is guaranteed to pass unaltered in to

the next generation of population by this strategy. It is well established that GA performs better and converge faster if the elitism strategy is incorporated in the selection procedure.

CROSSOVER OPERATORS This part of the survey focuses on commonly used crossover operators in GA based nuclear fuel management problems and some special types of crossover operators those have special relevance to the field. One-point and two-point crossover operators are widely used in nuclear fuel management optimization. In one-point crossover, two strings are picked from the mating pool at random. Then the crossover point is selected randomly which is common to both strings. The “parts of the strings” are exchanged between that crossover points. Two-point crossover is similar to the one-point crossover, except that two crossover pints or “sites” are chosen and the bits between the sites are exchanged. The classic crossover operators like one-point and two-point crossover operators are having some inherent limitations when used in nuclear fuel management optimization. The offspring resulted by these crossover operators may not represent a valid configuration for the selected problem. To overcome this crossover difficulty several specialized crossover operators have been developed to ensure the feasibility of generated solutions. Two of such specialized crossover operators applied in nuclear fuel management optimization are Heuristic tiebreaking crossover (HTBX) and Partially matched crossover (PMX). Heuristic tiebreaking crossover operator is a special type of customized crossover mechanism for nuclear fuel management optimization developed by Poon and Parks [17]. The Heuristic tiebreaking crossover maps the parent population in to ranked arrays based on the selected objective function values. It then combines randomly selected complementary parts of these arrays through a "cut and paste" operation and uses a simple tie-breaking algorithm to produce valid offspring [75]. In Partially matched crossover, two strings are aligned, and two crossover points are selected uniformly at random along the length of the


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strings. The parts of the strings between the crossover points are exchanged; up to this step, it is similar to two-point crossover. Then the remaining string portions or the “outer” parts of the two strings are changed or “adjusted” to satisfy the fuel management related constraints [6, 52].

MUTATION OPERATORS The mutation operator makes some small random changes in the solutions maintaining the diversity of the population to prevent a premature convergence to a local optimal solution. Mutation occurs during evolution according to a user definable mutation probability. Uniform mutation and nonuniform mutation are two types of mutations applied to most of the GA based nuclear fuel management problems [76, 35]. The mutation probability remains the same throughout the GA evolution process in the case of uniform mutation. The uniform mutation when applied to bit string chromosome representation performs bit flips at random positions. When applied to integer coded or real valued chromosome representation, the uniform mutation operator replaces the value of the chosen gene with a uniform random value selected between the user-specified upper and lower bounds for that gene. On the other hand, the mutation probability will vary (normally decreases with increase of generation numbers of GA) in the case of non-uniform mutation which tunes the search in later stages of evolution [42]. Uniform mutation is applied in the majority of the fuel management problems surveyed in this paper. This mutation mechanism is very effective with both binary encoding and integer encoding schemes of nuclear fuel management problems.

SUMMARY ON GA IMPLEMENTATION IN NUCLEAR FUEL MANAGEMENT The binary and integer encoding schemes are widely applied in nuclear fuel management GA implementations, because the majority of the problems addressed are of finding the most suitable “positions” of fuel assemblies or burnable poisons or control rods. These positions can perfectly be represented in

binary or integer encoding. As mentioned earlier (Section Elitism), the use of “elitism” in selection procedure of GA is always recommended for better performance. The Table given below (Table 2) shows different categories of GA operators and the specific type of operators from each category used in GA based nuclear fuel management. As given in the Table, roulette wheel and tournament selection mechanisms are implemented in nuclear fuel management, where the Constraint optimization with penalty function strategy is implemented. The reason is that, in the case of Constraint optimization with penalty function, there is only one objective function; there the probability based approaches like roulette wheel and tournament selection mechanisms will perform well. The ranking selection method is used in all Multi Objective GA based fuel management optimization problems. Ranking selection is suitable there because the sorting of the population from best to worst is essential when multiple objectives are considered together [67]. One-point and two-point crossover operators are widely used in GA based nuclear fuel management. They have well proven in both constraint optimization with penalty function and Multi Objective GA strategies. Hedayat et al. [58] compared the performance of the two crossover operators in nuclear fuel management optimization and concluded that the two-pint crossover operator is better for such applications. As mentioned earlier, Heuristic tiebreaking crossover and Partially matched crossover are crossover operators specially tuned for addressing the offspring feasibility of nuclear fuel management optimization. Among these two operators, Partially matched crossover is more suitable and widely used for Multi Objective GA based fuel management problems. Among mutation operators, uniform mutation is applied in the majority of problems. The reason is that the uniform mutation is easy to implement; simply flipping the selected chromosome portion with uniform probability will give the effect.


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Table 2: Summary of GA Operators in Nuclear Fuel Management Applications. GA Operator Category Selection

GA Operator (Specific Type) Roulette wheel

Tournament Crossover

Ranking One-point

Two point

Heuristic tiebreaking Partially matched

Mutation

Uniform

Type of GA Constrained optimization with penalty function Constrained optimization with penalty function Multi Objective GA Constrained optimization with penalty function Multi Objective GA Constrained optimization with penalty function Multi Objective GA Constrained optimization with penalty function Constrained optimization with penalty function Multi Objective GA Constrained optimization with penalty function

Multi Objective GA Non-uniform

Constrained optimization with penalty function

CONCLUSIONS Genetic algorithm (GA) has attracted considerable attention in nuclear fuel management. In this paper, a detailed survey has been presented about “when, where and how” GA has been applied in the field of nuclear fuel management. This will help the researchers in the field to get a clear picture about the GA related techniques available so that they can select the suitable one for their application. From the various types of GA surveyed here, it is observed that the full potential of Multi Objective Genetic Algorithm (Multi Objective GA) and Parallel Genetic Algorithm (Parallel GA) are not explored well in this field and hence a potential research scope is there. The open challenge in the field is that the design of a novel GA model for nuclear fuel management, in which the concepts of Multi Objective GA is embedded in to the Parallel GA. This results in an efficient algorithm which is suitable for running in parallel computers.

References 1, 2, 17, 18, 19, 21,41,43,57,64,69, 70,76,78, 79, 81,83,85, 86, 88. 53, 56, 58, 61, 62, 63, 80, 84, 92, 90, 91. 46, 47, 51, 52, 66, 82, 87. 49,53,56,57,58,61,62, 63, 76,81,83,90, 91. 49,58. 1, 18, 19, 21, 58, 64, 69, 70 79, 80, 83, 84, 86, 87, 88, 92. 58,87. 17, 21, 66. 43,76. 46, 47, 51, 52, 82. 1,2,17,18,19,21,43,53,56,57, 59,61,62,63, 64, 69, 70, 79, 80, 83, 84, 86, 88, 90, 91, 92. 46, 47, 49, 51, 52, 58, 66, 82, 87. 41, 76, 81, 85.

GA is proved to be useful in satisfying many of the requirements of nuclear fuel management problems such as highdimensionality, large number of feasible solutions and disconnected feasible regions in search space. The main short coming of all stochastic optimization methods including GA, in this field is that they normally need a large number of function evaluations, which is impractical because running reactor simulation code is extremely expensive in computational time [21]. A solution for this is to develop hybrid intelligent algorithms in which GA or such advanced optimization algorithm can be used as an efficient search tool and fast approximation models based on Artificial Neural Networks can be used as fast fitness evaluator to mimic the reactor simulation code. This survey is conducted as a part of research work, in which the application and evaluation of intelligent evolutionary algorithms including GA, in the field of nuclear fuel management is involved. The survey throws


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light in to the scope and future research directions in the field of nuclear fuel management optimization, where GA plays a major role. The findings of this survey are also useful in the evaluation of other modern intelligent evolutionary algorithms already used in nuclear fuel management; like Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Artificial Bee Colony Optimization (ABCO), Continuous Firefly Algorithm (CFA) etc.

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