Application of Genetic Algorithm methodologies in fuel bundle burnup optimization of Pressurized Heavy Water Reactor

Nuclear Engineering and Design 281 (2015) 58–71

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Application of Genetic Algorithm methodologies in fuel bundle burnup optimization of Pressurized Heavy Water Reactor M.L. Jayalal a,∗ , Suja Ramachandran a , S. Rathakrishnan b , S.A.V. Satya Murty a , M. Sai Baba c a b c

Electronics, Instrumentation and Radiological Safety Group (EIRSG), Indira Gandhi Centre for Atomic Research (IGCAR), Kalpakkam, Tamil Nadu, India Reactor Physics Section, Madras Atomic Power Station (MAPS), Kalpakkam, Tamil Nadu, India Resources Management Group (RMG), Indira Gandhi Centre for Atomic Research (IGCAR), Kalpakkam, Tamil Nadu, India

h i g h l i g h t s • We study and compare Genetic Algorithms (GA) in the fuel bundle burnup optimization of an Indian Pressurized Heavy Water Reactor (PHWR) of 220 MWe.

• Two Genetic Algorithm methodologies namely, Penalty Functions based GA and Multi Objective GA are considered. • For the selected problem, Multi Objective GA performs better than Penalty Functions based GA. • In the present study, Multi Objective GA outperforms Penalty Functions based GA in convergence speed and better diversity in solutions.

a r t i c l e

i n f o

Article history: Received 24 July 2014 Received in revised form 1 November 2014 Accepted 4 November 2014

a b s t r a c t The work carried out as a part of application and comparison of GA techniques in nuclear reactor environment is presented in the study. The nuclear fuel management optimization problem selected for the study aims at arriving appropriate reference discharge burnup values for the two burnup zones of 220 MWe Pressurized Heavy Water Reactor (PHWR) core. Two Genetic Algorithm methodologies namely, Penalty Functions based GA and Multi Objective GA are applied in this study. The study reveals, for the selected problem of PHWR fuel bundle burnup optimization, Multi Objective GA is more suitable than Penalty Functions based GA in the two aspects considered: by way of producing diverse feasible solutions and the convergence speed being better, i.e. it is capable of generating more number of feasible solutions, from earlier generations. It is observed that for the selected problem, the Multi Objective GA is 25.0% faster than Penalty Functions based GA with respect to CPU time, for generating 80% of the population with feasible solutions. When average computational time of fixed generations are considered, Penalty Functions based GA is 44.5% faster than Multi Objective GA. In the overall performance, the convergence speed of Multi Objective GA surpasses the computational time advantage of Penalty Functions based GA. The ability of Multi Objective GA in producing more diverse feasible solutions is a desired feature of the problem selected, that helps the reactor operator in getting more choices when deciding the appropriate discharge burnups of the core zones. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The Pressurized Heavy Water Reactors (PHWRs) play an important role in the Indian nuclear power program. PHWR is characterized by the use of natural uranium dioxide as fuel, heavy water as moderator and heavy water at high pressure/temperature in a separate circuit as coolant. The reactor consists of a low-pressure

∗ Corresponding author. Tel.: +91 44 27480210; fax: +91 44 27480210. E-mail address: [email protected] (M.L. Jayalal). http://dx.doi.org/10.1016/j.nucengdes.2014.11.013 0029-5493/© 2014 Elsevier B.V. All rights reserved.

horizontal reactor vessel (‘calandria’) containing the moderator at near ambient pressure and temperature. The fuel bundle burnup optimization of PHWR involves finding the optimum average discharge burnup of fuel bundles within the reactor core which gives the maximum fuel economy, while ensuring that operational and safety related constraints are always satisfied. In the present study, the reactor core considered is of two burnup zones. The aim of optimization is to find appropriate reference discharge burnup values for the two zones in order to obtain the optimum average discharge burnup for the total core. The zones reference discharge burnups obtained from the optimization can be used as the reference in

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selecting channels for refueling. The present study is based on 220 MWe Indian PHWR and that type forms the highest number of nuclear power plants operating in India. Nuclear Power Corporation of India Limited (NPCIL) reports that there are fourteen 220 MWe PHWR plants under operation in India (NPCIL, 2014). The aim of the present work is to apply and study the suitability of two flavors of the evolutionary optimization algorithm namely Genetic Algorithm (GA) in deriving burnup reference values for the two zones, which give maximum average discharge burnup of the total core without violating various operational and safety aspects of the reactor. Genetic Algorithm (GA) is an optimization tool based on Darwinian Theory of biological evolution. The method was developed by John Holland (Holland, 1975) and later popularized by one of his students, David Goldberg, who successfully applied to various practical engineering problems (Goldberg, 1989). GA has several advantages over the traditional optimization techniques. While calculus based optimization techniques depend on the derivative information of the objective functions, GA based techniques do not have this dependency. Furthermore, they are 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 advantages brought GA as a suitable and efficient tool in nuclear fuel management applications (Poon and Parks, 1993). The remaining part of the paper is organized as follows: a brief description about nuclear fuel management and the optimization techniques applied in the field is given in Section 2. The overall procedure of GA, as applied to nuclear fuel management problems is included in Section 3. The neutronics simulation code used in the present work is described in Section 4. The features of the optimization of fuel bundle for PHWR and the mathematical model formulation is given in Section 5. The two GA based methodologies applicable for the selected problem is formulated in Section 6. Section 7 gives GA related implementation details of the present work. Section 8 comprises the results followed by the conclusion of the study in Section 9.

2. Nuclear fuel management techniques and Genetic Algorithms The study about PHWR fuel bundle burnup optimization presented here comes under the in-core nuclear fuel management. In-core nuclear fuel management deals with the arrangement of fresh and partially burned fuel assemblies and reactivity control mechanisms within the core that optimizes the performance of the reactor over operating cycles, while ensuring operational and safety related constraints always being satisfied (Poon and Parks, 1993). In-core fuel management and optimization is a classical nuclear engineering problem, which has been studied for more than four decades and several techniques have been employed for solving them (Turinsky, 1999; Turinsky et al., 2005). Genetic Algorithm (GA) is one among the major global optimization techniques, used in the field of in-core fuel management and optimization. Apart from Genetic Algorithms, there are other global optimization techniques applied in the in-core nuclear fuel management. Some of them are listed as, Simulated Annealing (Kropaczek and Turinsky, 1991), Tabu Search (Lin et al., 1998; Castillo et al., 2004), Ant Colony Optimization (ACO) (Ortiz et al., 2007; Lin and Lin, 2012), Ant-Q Optimization (Machado and Schirru, 2002), Particle Swarm Optimization (PSO) (Meneses et al., 2009; Waintraub et al., 2009), Artificial Bee Colony Optimization (ABCO) (Oliveira and Schirru, 2011; Safarzadeh et al., 2011), Harmony Search Algorithm (HSA) (Poursalehi et al., 2013a) and Continuous Firefly Algorithm (CFA) (Poursalehi et al., 2013b). The above listed techniques come under the category of nature inspired intelligent algorithms. There are other types of optimization techniques also applied like Mixed

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Integer Programming (Kim and Kim, 1997), Estimation Distribution Algorithm (EDA) (Jiang et al., 2006; Mishra et al., 2009), and Particle Collision Algorithm (PCA) (Sacco et al., 2006). The survey carried out reveals that many of the GA applications in the field of in-core nuclear fuel management were applied for Light Water Reactors (i.e. for Pressurized Water Reactors and Boiling Water Reactors; Jayalal et al., 2014). For example, GA is applied for optimization of Pressurized Water Reactors core loading pattern and burnable poison (Alim et al., 2008; Chapot et al., 1999; DeChaine and Feltus, 1995, 1996; Haibach and Feltus, 1997; Hongchun, 2001; Khoshahval et al., 2011; Khoshahval and Fadaei, 2012; Norouzi et al., 2011; Pereira et al., 1999; Poon and Parks, 1993; Rafiei Karahroudi et al., 2013; Yamamoto, 1997; Yamamoto and Hashimoto, 2002; Yilmaz et al., 2006). Similarly, for Boiling Water Reactors, GA is applied for loading pattern optimization and control rod positioning (Kobayashi and Aiyoshi, 2001, 2002, 2003; Martin-del-Campo et al., 2001, 2009; Ortiz and Requena, 2004). There are a few studies related to Pressurized Heavy Water Reactor in-core fuel management. Quang Do et al. (2006) and Huo and Xie (2005) have applied GA for online refueling of PHWR. Mishra et al. (2009) considered special case of optimization of thorium loading in fresh core of PHWR and applied GA. It can be seen from the survey carried out, that there are two major methodologies in formulating nuclear fuel optimization model for GA (Jayalal et al., 2014). The two methodologies being: Penalty Functions based Genetic Algorithms (referred to as Penalty Functions based GA in the rest of the paper) and Multi Objective Genetic Algorithms (referred to as Multi Objective GA in the rest of the paper). These two approaches are further explained in Section 6. A brief description of the previous studies reported in the literature, regarding nuclear fuel management optimization problems using different methodologies of GA are given next. Penalty Functions based GA is applied in fuel management of Pressurized Water Reactors by DeChaine and Feltus (1995), Yamamoto (1997), Pereira et al. (1999), Yilmaz et al. (2006), Alim et al. (2008), Khoshahval et al. (2011), Norouzi et al. (2011), Khoshahval and Fadaei (2012) and Rafiei Karahroudi et al. (2013). Similarly, Penalty Functions based GA is applied for Boiling Water Reactors by Martin-del-Campo et al. (2001, 2009) and Ortiz and Requena (2004). Similar works for PHWRs are done by Huo and Xie (2005) and Mishra et al. (2009). The early applications of Multi Objective GA in the nuclear fuel management were for Pressurized Water Reactor by Parks (1996) and later by Pereira (2004). Fuel management optimization for Boiling Water Reactor’s by Multi Objective GA was carried out by Kobayashi and Aiyoshi (2001, 2002, 2003). Quang Do et al. (2006) applied the same concept for online refueling simulation of PHWR. There were some initiatives to apply Multi Objective GA in Fast Breeder Reactors also (Toshinsky et al., 1999, 2000). In the present study, we are applying the concepts of Penalty Functions based GA and Multi Objective GA separately, to find their suitability in the selected fuel bundle burnup optimization problem of PHWR. Before going into the details of the implementation of these GA based methodologies, a brief description about the overall GA procedure, applicable for both Penalty Functions based GA and Multi Objective GA methodologies is given in the next section.

3. Genetic Algorithm in nuclear fuel management: overall procedure The first step in applying GA to reactor fuel optimization is to determine the representation method which is suitable for the algorithm. As part of GA representation, a candidate solution (in the present study, burnup values of inner and outer zones) is encoded as a digital chromosome which has enough information to reproduce the original solution. While being executed, GA generate a

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module. Similarly the output values generated by the neutronics simulation module should be read back by the interface module and given back to the GA based optimization module for further calculations. This bidirectional data flow is shown by the two directional arrows in the flowchart. The fitness evaluation calculation using the neutronics simulation module is one of the important and computational time consuming step in the above procedure. The neutronics simulation module used for this study is called TAQUIL and is explained in detail in the next section. 4. Neutronics simulation module: TAQUIL

Fig. 1. Overall flowchart of GA when applied in nuclear fuel management.

collection of trial solutions i.e. a population of chromosomes, and the fitness values of each chromosome are evaluated. For example, in the case of fuel bundle burnup optimization, one set of inner and outer zone burnup values represents one chromosome. The fitness value for each such chromosome is calculated by running the neutronics simulation module. Similar to the natural selection process of biological evolution, chromosomes which have higher fitness values will have more chances of getting selected as ‘parents’ which participate in reproduction process (Michalewicz, 1994; Gen and Cheng, 1997). The ‘offspring’ solutions are produced from the parents using the genetic operations like crossover and mutation. The overall procedures of reactor fuel optimization using GA can be summarized as follows: (i) Generate the initial population with suitably encoded chromosomes. (ii) Calculating the fitness values using neutronics simulation model, for each individual or chromosome of the population (this is the objective function evaluation step of the optimization process). (iii) Select the parents according to the evaluated fitness, and perform the genetic operations such as crossover and mutation to produce next generation of offspring. (iv) Repeat steps (ii) and (iii) until the GA search process of finding the optimal solution is converged. The overall flowchart of GA when applied in nuclear fuel management optimization, which includes GA procedures, interface module and neutronics simulation module, is given in Fig. 1. As can be seen in the flowchart, there is an interface module present between the GA implementation part and the neutronics simulation module. Most of the neutronics simulation models used in the nuclear fuel management have been developed in FORTRAN programming language and are specific to the type of the reactor (such as the TAQUIL code used for this study, which will be described in Section 4). If the GA implementation part is developed in any other programming language (for example, C programming language is used for the present study), then the interface module should be able to generate the input files which satisfy the requirements of the FORTRAN programming language based neutronics simulation

The reactor physics based simulation code used for this study is called TAQUIL (Rastogi, 1989; Srinivasan, 1986) which is an in-house developed FORTRAN programming language based program, particularly suitable for PHWR equilibrium core simulation studies. In TAQUIL, the refueling scheme is simulated using time-average model of the selected PHWR core. The presence of on-power refueling in PHWR results in the absence of a constant core power distribution shape for the equilibrium core. In the time-average model, channel wise power distribution is averaged over a period of fuel residence time to get approximately constant core power distribution. In the present work, the number of burnup zones is fixed as two; a central high burnup inner zone and an outer low burnup zone. The discharge burnup values for the inner and outer zones act as input for TAQUIL code. The total core power and the refueling frequency are fixed; the code generates bundle powers, channel powers, effective reactivity multiplication factors and average value of discharge burnup for the total core. During every iteration of GA (irrespective of the GA methodology considered), inner and outer burnup values are assigned to each of the chromosomes of the GA population. The TAQUIL code is used to calculate corresponding neutronic parameter values like maximum bundle power (MBP), maximum channel power (MCP), effective reactivity multiplication factor (Keff ) and average discharge burnup of the total core (BUave ). The units of MBP, MCP, and BUave are kW, MW and MWd/t respectively. These output values from TAQUIL are used for the fitness evaluation stage in every iteration of GA. The passing of input burnup values from GA chromosomes to TAQUIL and parsing the required output values from the output file generated by TAQUIL to the GA are done by the interface module developed in the C programming language. The fuel bundle optimization problem is implemented with GA as the optimization tool and the neutronics simulation module (TAQUIL) as the objective function evaluation tool. The communication among these modules is smoothly achieved by the interface module. The mathematical model of the optimization problem is incorporated into the GA module. In the next section, we consider the mathematical model formulation of the selected problem. 5. Fuel bundle burnup optimization problem: model formulation The reactor core of 220 MWe PHWR contains 306 coolant channels. Each coolant channel contains 12 cylindrical fuel bundles made up of zircaloy, through which pressurized heavy water coolant circulates. Each fuel bundle holds 19 fuel elements and each fuel element consists of a stack of sintered cylindrical fuel pellets of natural uranium (Bajaj and Gore, 2006). In PHWR, the fueling is on-power (i.e. without reactor shutdown) and on a daily basis to maintain the reactor criticality and the reference core power distribution. The fuelling operations are carried out by two remotely controlled fuelling machines, operating at each end of a fuel channel. The 8-bundle shift scheme is followed in the daily refueling operation in which eight fresh fuel bundles are inserted into a

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followed to achieve the flux flattening requirement of the selected core. The maximization of Keff is also considered as an objective. This is because the more Keff of the core leads to more core excess reactivity that can be utilized for increasing the discharge burnup further during the reactor operation. If the optimization study results in two burnup patterns, the one with higher Keff , with all other parameters like BUave , MBP and MCP remains the same, then the pattern with higher Keff is the better option. The optimization problem given comes under the category of multi-objective optimization with four objectives and four constraints. A solution to the problem can be termed as feasible, only if it satisfies all the four constraints given. The optimization problem is defined with following objectives and constraints: Objectives:

Fig. 2. The two burnup zones for the reactor core used for the study.

channel from one end and eight burnt fuel bundles are taken out from the other end of the channel. The fuelling direction is opposite in adjacent channels. It helps in axial flux flattening (Mishra et al., 2010a,b). The division of the core into two burnup zones is important for the PHWR. It is seen that without the burnup zones division, the radial neutron flux shape is peaked at the central region of the core and the bundle power at that region will exceed the operating limit at full power operation (Mishra et al., 2010a,b). The radial flux flattening at the central region is essential in order to keep the maximum bundle power (MBP) below the operating bundle power limit at full power operation. In order to get radial flux flattening, the reactor core is divided into two burnup zones as shown in Fig. 2. The inner zone contains 78 fuel channels and the outer zone contains 228 fuel channels. The present study considers the discharge burnup of inner zone is in the range between 8500 and 11,000 MWd/t and that of outer zone is in the range between 4000 and 6500 MWd/t. The typical discharge burnups for a 220 MWe PHWR are around 10,200 MWd/t and 5500 MWd/t for inner and outer zones respectively. By keeping the discharge burnup of inner zone being higher, the fissile inventory is kept relatively lower at the inner zone as compared to the outer zone channels. This leads to flattening of the neutron flux at the inner region and with that the maximum bundle power (MBP) is within operating bundle power limit (Mishra et al., 2010a,b). The reference channel powers for the two burnup zones are arrived using the simulation code TAQUIL (refer Section 4). The channel coolant flows are fixed according to the reference channel powers in order to get uniform channel coolant outlet temperature. Since the channel flows are already fixed for the operating 220 MWe PHWR, the actual channel power cannot be kept more than its reference channel power. The aim of the optimization carried out in the present study is to find optimum reference discharge burnup values for the inner and outer zones, in order to obtain maximum average discharge burnup for the total core (BUave ). The discharge burnups arrived can be utilized in fixing the most suitable reference discharge burnups for the two zones. Once the fuel bundle burnup value reaches the corresponding zone’s reference discharge burnup value, then that fuel bundle can be replaced during the online refueling operation. The selection of appropriate reference discharge burnup values is important in maximizing the fuel utilization while satisfying the safety and operational constraints of the reactor. The present work explores the suitability of the two flavors of GA in deriving the optimum BUave which give maximum fuel economy without violating various constraints such as MBP and MCP. The primary objective for the study is the maximization of BUave . The minimization of MBP and MCP are considered as other two objectives. This approach is

to maximize, BUave (of the total core) to minimize, MBP to minimize, MCP to maximize, Keff Constraints: MBP should be less than 430 kW MCP should be less than 3.2 MW Keff should be greater than 1.0005 BUave should be greater than 6700 MWd/t The given objectives are a function of the burnups of inner and outer zones. The mathematical formulation of the given problem is given as: Max(BUave , Keff ) and Min(MBP, MCP) = f (inner zone burnup, outer zone burnup) Such that, MBP < 430 KW, MCP < 3.2 MW, Keff > 1.0005 and BUave > 6700 MWd/t where, Max represents the maximization, Min represents the minimization and f () represents “function of”. The problem has the two following boundary conditions for the input values, as follows: 8500 MWd/t ≤ inner zone burnup ≤ 11, 000 MWd/t 4000 MWd/t ≤ outer zone burnup ≤ 6500 MWd/t GA is an optimization technique well suited to solve such problems. The GA methodologies and the implementation details applied in the present work are covered in the next section. 6. Application of GA methodologies As mentioned earlier, GA is a general purpose versatile optimization tool well suited for complex optimization tasks like the nuclear fuel management problem that we consider for the study. There exist many flavors of GA for dealing such multi-objective optimization problems. The literature survey carried out reveals that, the major approaches in formulating nuclear fuel optimization model for GA are Penalty Functions based GA and Multi Objective GA (Jayalal et al., 2014). These two methodologies are further explained in Sections 6.1 and 6.2. 6.1. Penalty Functions based GA In the case of Penalty Functions based GA, the multi-objective problem of fuel management optimization is converted into single

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objective by adding penalty functions and constraints. The concept of penalty function is applied to convert the multi-objective optimization problem into the form that is suitable for GA to deal with. In general, this approach transforms a constrained optimization problem to an unconstrained optimization problem by defining a suitable penalty function. If the penalty coefficients and the constraints are properly selected, this approach will give feasible solutions. The selection of proper penalty coefficients and the constraints is a difficult task in the implementation of Penalty Functions based GA. For the nuclear fuel management problems, the selection is usually based on the expert opinion. Now we look into the model formulation of Penalty Functions based GA for the selected PHWR fuel bundle burnup optimization problem. Among the four objectives that were described in Section 5, the maximization of BUave is taken as the objective for Penalty Functions based GA and other three objectives are converted to penalty functions. The penalized objective function for the selected problem is formulated as follows: Fitness = BUave − P1 − P2 − P3 where, P1 = (MBP − MBP 0 ) × Abp P2 = (MCP − MCP 0 ) × Acp 0 P3 = (Keff − Keff ) × Akef

Fitness represents the penalized objective function used for the fitness evaluation in GA. P1 is the penalty function related to maximum bundle power constraint. P2 is the penalty function related to maximum channel power constraint. P3 is the penalty function related to effective multiplication factor constraint. MBP0 is the maximum permitted bundle power (=430). MCP0 is the maximum permitted channel power (=3.2). 0 Keff is the minimum required effective multiplication factor (=1.0005). The terms Abp, Acp and Akef are used to denote constant values selected to give proper weightage to penalty functions. These constants were estimated such that the penalty caused by each factor is of the order of BUave (the magnitude being in the order of thousands). For the present work, the values of Abp, Acp and Akef are fixed as 10, 1000 and 100,000 respectively. The penalty functions (P1, P2 and P3) are formulated in a modified way for the present problem, as compared with the commonly followed approach. In the commonly followed approach, penalty function is 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 (Goldberg and Deb, 1991; Michalewicz, 1994). In the present work, the penalty functions additionally help in reaching the three objectives of the problem, i.e. Max(Keff ) and Min(MBP, MCP). In order to achieve this, the penalty functions are formulated in such a way that, penalty function values (P1, P2 and P3) are becoming negative under the 0 . conditions when MBP < MBP0 or MCP < MCP0 or Keff > Keff 6.2. Multi Objective GA Multi Objective GA relies on the concepts of Pareto-optimality and dominance (Censor, 1997). Essentially, the main task of Multi

Objective GA is to find the Pareto-optimal solutions for the given problem with multiple conflicting objectives (Deb, 2008). The Pareto-optimal solution is the one in which an improvement in one of the objective requires a degradation of another. The set that consists of all the Pareto-optimal solutions for a given problem forms the Pareto-optimal front (or non-dominated front). In Paretooptimal front, one solution cannot be considered as better than the other. The method makes it possible to identify the trade-of-surface between competing objectives in a single optimization run. In the present work, implementation of Multi Objective GA followed is Elitist Non-dominated Sorting Genetic Algorithm (commonly referred as NSGA-II), which is proposed by Deb et al. (Deb et al., 2002; Deb, 2008). The Multi Objective GA implementation followed (NSGA-II) is having advantages like the ability to find better spread of solutions, better convergence and better performance with less degree of computational complexity. The diversity preserving feature is one of the advantages that make the NSGA-II superior than many other multi-objective implementations of GA (Deb, 2008). This feature is particularly valuable for the selected nuclear fuel management optimization problem. The overall procedure followed in the implementation of Multi Objective GA (i.e. NSGA-II) is summarized below (Deb et al., 2002, Deb, 2008). According to the concept of dominance, a solution x(1) is said to dominate another solution x(2) , if both of the following two conditions are satisfied: 1. The solution x(1) is not inferior than x(2) in all objectives. 2. The solution x(1) is better than x(2) in at least one objective. In the implementation, the initial parent population, Pt and the offspring population, Qt are combined to form a population Rt of size 2N, where N is the population size and is referred as: Rt = Pt ∪ Qt Subsequently, non-dominated sorting is applied to classify the entire population, Rt into different non-dominated sets (called as Pareto-optimal fronts). Solutions belonging to the best Paretooptimal front (termed as F1 ) are the best solutions in the combined population. If the size of F1 is smaller than N, all members of F1 are added to the new population, Pt+1 . The remaining members of Pt+1 are chosen from subsequent Pareto-optimal fronts in the order of their ranking. This procedure is continued until no further sets can be accommodated, as only N (population size) solutions can be chosen from 2N solutions in Rt . For selecting exactly N population members, solutions of the last front are sorted using the crowded comparison operator (normally denoted by dj . The crowded comparison operator guides the selection process at various stages of the algorithm toward a uniform spread of solutions along the best-known Pareto front. The main advantage of using crowded comparison operator is that a measure of population density around a solution is computed without requiring a

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user-defined niche size or the kth closest neighbor (Konak et al., 2006; Hedayat et al., 2009). The handling of constraint violations for the selected problem is explained next. As mentioned in Section 5, the fuel bundle burnup optimization problem that we consider is having four objectives and four constraints. The constraint violations are handled by an approach which is similar to the penalty handling mechanism of Penalty Functions based GA. The constraint functions are first normalized and then the violation for each constraint is calculated. For the four constraints of the selected problem, corresponding constraint violations are calculated as: C1 = =

MBP − MBP 0 MBPmax − MBP 0

if MBP >

0,

otherwise.

MBP 0

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Table 1 Genetic parameters and methods or values used. Parameter

Methods/values

Encoding Population size Crossover method Crossover probability Mutation method Mutation probability Maximum no. of generations

Floating point 50 Arithmetical 0.6 Non-uniform 0.05 100

to each of the objective function values to get the modified objective function values as: MBPmod = MBP + Bbp × Ctot MCPmod = MCP + Bcp × Ctot Keffmod = Keff − Bkef × Ctot

C2 = =

C3 = =

C4

= =

− MCP 0

MCP MCPmax − MCP 0

if MCP > MCP 0

0,

otherwise.

0 −K Keff eff 0 −K Keff effmin

0,

0 if Keff < Keff

otherwise.

0 − BU BUave ave 0 − BU BUave avemin

0,

0 if BUave > BUave

otherwise.

C1 is the constraint violation value related to maximum bundle power constraint. C2 is the constraint violation value related to maximum channel power constraint. C3 is the constraint violation value related to effective multiplication factor constraint. C4 is the constraint violation value related to average discharge burnup constraint. MBP0 is the maximum permitted bundle power (=430). MBPmax is the maximum possible value of bundle power. MCP0 is the maximum permitted channel power (=3.2). MCPmax is the maximum possible value of channel power. 0 Keff is the minimum required effective multiplication factor (=1.0005). Keffmin is the minimum possible value of effective multiplication factor. 0 is the minimum limit for average discharge burnup (=6700). BUave BUavemin is the minimum possible value of average discharge burnup. The individual constraint violations corresponding to the four constraints of the problem are calculated. Then the overall constraint violation (Ctot ) is calculated as: Ctot = C1 + C2 + C3 + C4 The next step is to modify each objective function value according to the overall constraint violation. The overall constraint violation is multiplied with suitable constant values and the product is added

BUavemod = BUave − Bbu × Ctot MBPmod is the modified value for maximum bundle power objective. Ctot is the overall constraint violation. MCPmod is the modified value for maximum channel power objective. Keffmod is the modified value for effective multiplication factor objective. BUavemod is the modified value for average discharge burnup objective. The terms Bbp, Bcp, Bkef and Bbu are used to denote constant values selected to make both terms on the right side of the above equations to have the same order of magnitude. For a feasible solution, Ctot is calculated as 0 and the modified values of the objective functions are same as that of actual objective function values. For an infeasible solution, a penalty is added to each of the objective functions corresponding to overall constraint violation. Once the modified objective functions are calculated, those values are used by the algorithm for Pareto-optimal fronts sorting. In the present work, the efficiency of the adopted Multi Objective GA (i.e. NSGA II) in approaching Pareto-optimal fronts is not analyzed, as it is normally done for the comparison of different Multi Objective GAs. 7. Implementation details The real-parameter GA (Michalewicz, 1994; Gen and Cheng, 1997, Jayalal et al., 2011) is selected for the implementation of both Penalty Functions based GA and Multi Objective GA and the C programming language is used for program development. The interface module for the communication between the GA modules and the neutronics simulation module (i.e. TAQUIL) is also developed in the C programming language. While comparing the performance of the two algorithms, it is important to keep the GA related parameters, corresponding values and methods as same. The information relating to GA parameters is given in Table 1. The methods and values given in Table 1 are common for Penalty Functions based GA and Multi Objective GA. The GA encoding scheme followed (for Penalty Functions based GA and Multi Objective GA) in the present work is floating point or real-coded representation (Michalewicz, 1994). The burnup values of inner and outer zones are of floating point values (rounded to two decimal places) and can be directly encoded as chromosome vector with two floating point values. In other words, from the genotype (i.e. the chromosomes) the phenotype (solution to the problem) can easily be extracted by taking the corresponding floating point

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value. Each element in the chromosome vector is assigned to be within the desired range at the time of generation of the initial population. The crossover and mutation operators selected for the GA implementations are also designed to meet this requirement. As an example, two typical chromosome vectors, x1 and x2 generated in the study presented can be represented as: x1 = (10, 281.35, 6034.42) x2 = (8941.24, 5281.71) The first elements in x1 and x2 denote the inner zone burnup values and the second elements denote the outer zone burnup values. Arithmetical crossover is the crossover mechanism selected for the implementation of both the algorithms. The arithmetical crossover is a method suitable for floating point chromosome encoding, which linearly combines two parent chromosome vectors to produce two new offspring (Michalewicz, 1994). If x1 and x2 are the chromosome vectors to be crossed, the resulting offspring are generated by the linear combination of the two vectors as: ax1 + (1 − a)x2 and ax2 + (1 − a)x1 where a is a random floating point value generated between 0 and 1.For the chromosomes vectors considered in the above example, i.e. x1 = (10,281.35, 6034.42) and x2 = (8941.24. 5281.71), the resulting offspring, o1 and o2 , after arithmetical crossover operation is represented as: o1 = (a × 10, 281.35 + (1 − a) × 8941.24, a × 6034.42 + (1 − a) × 5281.71)

where r is a random floating point value generated between the range 0 and 1, T is the maximal generation number (for the present study, T = 100) and b is a parameter determining the degree of dependency on the generation number (b = 5 is used for the present study). In continuation of the example considered, assume that one of the offspring (o1 ) derived from crossover operation is undergoing the non-uniform mutation at 40th generation (t = 40). Consider that the second element of o1 (that represents the outer zone burnup) is chosen for the mutation operation (i.e. vk = 5782.19). The UB and LB values, representing the upper and lower bounds for the outer zone burnup, are 6500.00 and 4000.00 respectively. If the random digit generated as part of non-uniform mutation operator is 0, then

vlk = 5782.19 + [40, (6500.00 − 5782.19)] Now, if r = 0.70, then (t, y) function in the above step will return 19.63 and the mutated element vlk will get 5801.82. In essence, for the considered example, the non-uniform mutation operation changes the outer burnup value from 5782.19 to 5801.82. Now we consider the methods of generating the population and selecting the new population followed in the present study. The initial population is generated by calling the random number generation function of the C programming language. Using the function, random numbers are generated within the range of input boundary conditions (see Section 5 for boundary conditions range). The selection used in Penalty Functions based GA is Roulette-Wheel method where the selection or survival probability for each chromosome is proportional to the fitness value (Goldberg, 1989; Michalewicz, 1994). The selection in Multi Objective GA is based on the ranks assigned to individual chromosomes by nondominated sorting of Pareto-optimal fronts, as discussed in Section 6.2. The concept of Elitism is also incorporated in the selection method of Penalty Functions based GA, where a fixed number of the chromosomes having higher fitness values are considered as elite chromosomes and are retained in the new generation.

o2 = (a × 8941.24 + (1 − a) × 10, 281.35, a × 5281.71 + (1 − a) × 6034.42) At a particular instance, if a = 0.3351, then the resultant offspring chromosomes are: o1 = (9390.31, 5533.94) o2 = (9832.28, 5782.19) The mutation method applied is non-uniform mutation (for both cases of algorithms). The special feature of this method is that the mutation probability is not uniform i.e. it is related to the generation number (Michalewicz, 1994). The mutation probability decreases as the generation number increases and approaches a value close to zero. This property causes the algorithm to search the space uniformly initially and very locally at later stages and allows to fine tune the results. If x1 = (v1 , . . ., vm ) is the chromosome vector, the element vk was selected for the mutationand the resultant chromosome vector is x1 = v1 , . . ., vlk , . . ., vm , then the non-uniform mutation operator is defined as:



vlk

=

vk + (t, UB − vk ) if a random digit is 0 vk − (t, vk − LB) if a random digit is 1

where UB and LB are upper and lower domain bounds of the variable vk and t is the generation number. The function (t, y) which returns a value in the range [0, y] is defined as:



t

(t, y) = y 1 − r (1− T )

b



,

8. Results Several trial runs were conducted with randomly generated initial GA population (for Penalty Functions based GA and Multi Objective GA). Each trial run was started with entirely different initial population ensuring different initial search space for different trial runs. Based on the results generated initially, GA parameters and penalty coefficients (refer Section 6.1) were fine tuned. Both GA methodologies were studied in detail to ascertain the suitability for the selected problem. During the comparison process: population size, maximum number of generations, encoding scheme of GA, crossover mutation methods and their probability values, are kept same for both the algorithms (see Table 1). The fine tuned versions of the algorithms are used to generate final results by conducting 20 trial runs for each of the algorithms. 8.1. Comparison of maximum and minimum values of an objective function The feasible solutions obtained in the final generation (set as 100th generation in the present study) of Penalty Functions based GA and Multi Objective GA are compared. The results generated by four trial runs selected randomly from the 20 trial runs, are presented in Tables 3 and 4. The objective function values corresponding to the maximum and minimum values for BUave from the final generation of Penalty Functions based GA and Multi Objective GA are presented in Tables 2 and 3, respectively. We have selected BUave as the reference for the comparison. Among the feasible solutions at the final generation, two solutions are extracted,

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Table 2 Maximum and minimum values obtained for BUave along with the corresponding other parameter values generated in the final generation (Penalty Functions based GA). Trial No. 1 2 3 4

Inner zone burnup

Outer zone burnup

MBP

MCP

Keff

BUave

10,417.09 10,416.32 10,399.88 10,397.19 10,415.50 10,414.07 10,413.69 10,412.48

5953.96 5953.03 5961.03 5960.42 5953.51 5953.00 5955.65 5954.95

423.23 423.24 423.92 423.99 423.27 423.27 423.37 423.38

3.0941 3.0943 3.0990 3.0995 3.0944 3.0945 3.0952 3.0952

1.000503 1.000508 1.000518 1.000527 1.000508 1.000514 1.000503 1.000510

6835.59 (Max) 6834.61 (Min) 6840.88 (Max) 6840.07 (Min) 6835.00 (Max) 6834.35 (Min) 6836.94 (Max) 6836.05 (Min)

one with maximum value for BUave and another with minimum value for BUave . The TAQUIL code (refer Section 4) can accept inputs and generate outputs with accuracy up to six decimal places. The results presented for burnups and MBP are rounded to two decimal places. In the case of MCP, the selected values were rounded off to four decimal places and that of Keff to six decimal places, to reflect the variation in the expected range. It can be seen from the results that, both Penalty based GA and Multi Objective GA are capable of arriving at around same converged area in the total search space of burnup values for inner and outer zones. It is also seen that the maximum and the minimum values obtained for BUave by the Penalty Functions based GA lies in narrower range as compared to Multi Objective GA. This behavior is consistent with the four trial runs of each algorithm. The major observations from the results are: • Both, Penalty based GA and Multi Objective GA, are able to produce feasible solutions at the final generation. • Always, the Multi Objective GA is capable of generating wide range of feasible solutions for the considered objective function i.e. BUave . The ability of Multi Objective GA to generate wide range of different feasible solutions is an important feature for the selected nuclear fuel management problem. That helps the reactor operator in getting more choices when deciding the appropriate discharge burnups of the core zones. 8.2. Aggregate diversity of objective functions: generation wise The term “diversity” represents the distribution of the selected parameter within the allowable range. The aim is to consider the generation wise evolution of Penalty Functions based GA and Multi Objective GA, by comparing the aggregate diversity of all the four objective functions. The diversity in the values of these objective functions, gives further insight about the range of generated final objective values. A common procedure is followed to find the aggregate diversity of four objective functions. The aggregate diversity of generated solutions, at successive stages of generations is calculated using the following procedure: • The arithmetic mean is calculated for each objective function in the population (the population size is 50 and each individual

• • •

• •

of the population represents a solution set with four objective function values). The distance from the arithmetic mean is calculated for each individual’s objective functions and the distances are squared. Summation of the distances (for every objective function separately) for all the individuals of the population. Dividing the total distance found for every objective function by the population size (corresponding to the four objective functions, four values are thus obtained). Normalize the four values obtained in the previous step, on a 0–10 scale. Calculate the average of the normalized values which represents the aggregate diversity of the four objective functions for that generation.

Using the above procedure, the normalized aggregate diversity is calculated for all successive generations for both of the algorithms i.e. Penalty Functions based GA and Multi Objective GA. Fig. 3 shows the complete diversity plots (i.e. for all generations from 1st generation to 100th generation) for a trial run of Penalty Functions based GA and Multi Objective GA. A wide range of objective functions values are produced in both of the algorithms’ initial generations. The reason for this is that both algorithms are started from the randomly generated initial solution space which has the maximum diversity. The range of diversity narrows down toward the end generations. The variation in diversity is approaching to a very small value (near zero in a normalized scale between 0 and 10) for Penalty Functions based GA but still better for Multi Objective GA (see Fig. 3). The diversity toward end generations can be represented in a wider scale, if a few initial generations are removed from the plots. The first nine generations are omitted (to represent the diversity at the end generations in a wider scale) and the resulting plots are given in Figs. 4 and 5. The results generated by four trial runs (randomly selected from the 20 trial runs) are presented here. The normalized aggregate diversity over the evolution from 10th generation up to 100th generation, for the Penalty Functions based GA is shown in Fig. 4. It can be seen that for the Penalty Functions based GA, diversity among the objective functions is drastically reducing and approaching to a very small value (near zero) toward the end generations. This clearly indicates that a high number of solutions are generated within a very small segment of the entire solution range by the Penalty Functions based GA. Similarly, Fig. 5

Table 3 Maximum and minimum values obtained for BUave along with the corresponding other parameter values generated in the final generation (Multi Objective GA). Trial No. 1 2 3 4

Inner zone burnup

Outer zone burnup

MBP

MCP

Keff

BUave

10,278.69 10,254.69 10,269.54 10,286.74 10,281.35 10,266.12 10,273.44 10,175.80

6037.28 5828.04 6034.63 5824.56 6034.42 5827.52 6039.73 5834.69

429.70 424.89 429.91 423.80 429.55 424.53 429.93 427.51

3.1403 3.1046 3.1418 3.0970 3.1368 3.1021 3.1419 3.1231

1.000514 1.001432 1.000550 1.001363 1.000520 1.001404 1.000518 1.001607

6903.36 (Max) 6700.70 (Min) 6899.97 (Max) 6700.00 (Min) 6900.87 (Max) 6701.15 (Min) 6905.17 (Max) 6700.59 (Min)

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Fig. 3. Normalized aggregate diversity of four objective functions for the Penalty Functions based GA and the Multi Objective GA (from 1st generation to 100th generation).

shows the normalized aggregate diversity over the evolution from 10th generation up to 100th generation for the Multi Objective GA. The result shows the ability of Multi Objective GA to retain the diversity among the objective functions, in a much better way as compared to Penalty Functions based GA. Thus Multi Objective GA produces wide range feasible solutions as final results. Following

observations can be made from the comparison of the normalized aggregate diversity plots given in Figs. 4 and 5: • Aggregate diversity among the four objective function values are shrinking to a narrow range (near zero) toward the end of generations for the Penalty based GA.

Fig. 4. Normalized aggregate diversity of four objective functions for the Penalty Functions based GA (from 10th generation to 100th generation).

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Fig. 5. Normalized aggregate diversity of four objective functions for the Multi Objective GA (from 10th generation to 100th generation).

• On the other hand, Multi Objective GA is capable of retaining the wider range of solutions with respect to all the four objectives of the selected problem.

8.3. Objective functions diversity: final generation The feasible solutions generated at the final generations of Penalty Functions based GA and Multi Objective GA are considered in the first comparison. The results generated by four trial runs are presented in Fig. 6. The distribution of generated objective function values in feasible solutions, at the final generation (i.e. 100th generation) is effectively represented by box plots (also known as box and whisker plots; Benjamini, 1988). The bottom and top of the box represent the first and third quartiles (Q1 and Q3). The vertical dotted lines at each end of the box are called whiskers. The bottom whisker goes from first quartile to the smallest non-outlier (outlier represents data point that diverges greatly from the overall pattern of data) in the data set; the top whisker goes from third quartile to the largest non-outlier. The outliers are represented by small circles (for our results, there is no outliers and hence no circles shown in Fig. 6). The inter-quartile range, which is a measure of variability, is represented by the vertical height of the box (i.e. Q3 minus Q1). The objective functions distribution for four trial runs of Penalty Functions based GA are plotted in left hand side and those for Multi Objective GA are plotted in the right hand side in the figure (Fig. 6). The two algorithms results are plotted under the same

scale, which enables a good comparison. The vertical height of the boxes represents the spread of feasible solutions for 50% of the data samples. The boxes generated for the Penalty Functions based GA are so narrow that they are represented as simple thick lines. The feasible solutions generated at the final generations of 20 trial runs for each of the algorithms are considered next. The maximum, minimum, average and standard deviation values for the four objective functions are given in Table 4. The maximum values (Max) are calculated by taking the average of maximum values produced (separately for each of the four objective functions) at the final generation in 20 trial runs. Similarly the minimum values (Min) are calculated by taking the average of minimum values generated. The average values (Ave) shown in the table are calculated by finding the average of respective objective function values for 20 trial runs and the standard deviation (SD) is also calculated. Following observations can be made from the comparison of the distribution of feasible solutions in final generation given in Fig. 6 and Table 4: • Multi Objective GA produces wide range of feasible solutions in the final generation with respect to the four objectives of the selected problem. • Penalty Functions based GA produces feasible solutions in much narrow range in the final generation. • The results produced are consistent (for the above observed behavior) with trial runs.

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Fig. 6. Box plots to represent the four objective functions distribution in generated feasible solutions in final generation (for four trial runs).

8.4. Solution diversity: initial population vs final population The inner and outer zone burnup values forms the solution candidates for the selected problem. The solutions candidates’ distribution in the initial generation population and final generation population in the overall search space are compared in Fig. 7(a and b). In Section 8.3, the feasible solutions and corresponding objective function values of four trial runs are considered for the comparison. The total population and corresponding solution candidates of a single trial run are being considered here. The floating point values (with the given accuracy of two decimal places) in the inner and outer zones burnup ranges form the total search space of the problem (refer Section 5 for the burnup ranges). Fig. 7(a) shows the initial and final populations’ solution diversity for Penalty Functions based GA and Fig. 7(b) shows that of Multi Objective GA. The solution values converge to a very narrow area (represented as single point in the diagram) in the case of Penalty Functions based GA,

but to a wider area for Multi Objective GA. This shows that as in the case of objective functions diversity, the solution diversity also better for Multi Objective GA.

8.5. Comparison of GA performance: generation wise The number of feasible or good solutions produced in successive generations can be treated as a measure of effectiveness of GA implementation (Michalewicz, 1994; Gen and Cheng, 1997). This indicates a measure of speed of convergence of the algorithm. For the selected problem of fuel bundle optimization, one generated solution is termed as feasible, if it satisfies all the four constraints. That is, a feasible solution satisfies the conditions such as, MBP should be less than 430 kW, MCP should be less than 3.2 MW, Keff should be greater than 1.0005 and BUave should be greater than 6700 MWd/t.

Table 4 Feasible solution values obtained in final generation considering 20 trial runs. Penalty Functions based GA

Max Min Ave SD

Multi Objective GA

MBP

MCP

Keff

BUave

MBP

MCP

Keff

BUave

423.33 423.29 423.30 0.1600

3.0948 3.0945 3.0946 0.0012

1.000515 1.000511 1.000512 1.5 × 10−05

6835.02 6834.43 6834.83 1.6100

429.90 423.25 426.12 0.2900

3.1413 3.0710 3.1112 0.0024

1.001569 1.000507 1.000897 5.1 × 10−05

6898.58 6702.79 6797.50 8.5200

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Fig. 7. (a) Solution diversity comparison of Penalty Functions based GA: initial population vs final population. (b) Solution diversity comparison of Multi Objective GA: initial population vs final population.

Fig. 8. Average number of feasible solutions (generation wise) for 20 trial runs.

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Fig. 8 shows the average number of feasible solutions produced in successive generations by 20 trial runs of Penalty Functions based GA (Fig. 8 (a)) and 20 trial runs of Multi Objective GA (Fig. 8 (b)). The conditions of feasibility are kept the same for both the algorithms. The Multi Objective GA produces feasible solutions in a faster rate at earlier generations and has a better convergence speed. A comparison is made on GA performance based on the average CPU time taken to produce equal number of feasible solutions. The comparison is done on a computer system with Dual Six Core 64 bit Intel Xeon @ 3.06 GHz processor and 48 GB RAM. The average times and generations taken for producing 80% of population with feasible solutions (i.e. 40 feasible solutions) are taken in the comparison. The Penalty Functions based GA took 32 generations and 163.10 s to produce 40 feasible solutions. The Multi Objective GA took 12 generations and 122.30 s to produce the same number of 40 feasible solutions; implying that the Multi Objective GA is 25.0% faster than Penalty Functions based GA, with respect to CPU time for generating 80% of the population with feasible solutions. The number of generations taken (for generating 40 feasible solutions) by Multi Objective GA is 62.5% less than Penalty Functions based GA. The behavior observed can be utilized in reducing the computational time by lowering the number of required generations of the algorithm. Another observation is that the computational time requirement for a fixed number of generations is greater for Multi Objective GA. This is due to the computational complexity involved in various stages (like sorting of Pareto-optimal fronts) of the algorithm. The average CPU time requirement (for 20 trial runs, each with 100 generations) for Penalty Functions based GA is 546.31 s whereas for Multi Objective GA it is 983.62 s. It is resulting in, when average computational time of fixed generations are considered, Penalty Functions based GA is 44.5% faster than Multi Objective GA. As seen earlier, the required number of generations to reach the given convergence criteria is 62.5% less in the case of Multi Objective GA. The observation from the comparison is that the convergence speed of Multi Objective GA surpasses the computational time advantage of Penalty Functions based GA. 9. Conclusion In this study, two different GA methodologies namely Penalty Functions based GA and Multi Objective GA are applied in the fuel bundle burnup optimization of PHWR. The result obtained from this study shows the suitability of Multi Objective GA over Penalty Functions based GA, in solving problems like the one selected for this study. Multi Objective GA is able to generate diverse optimal solutions with respect to all the four objectives considered in this study. The ability to find much better spread of solutions by Multi Objective GA is a key point with respect to the present study. It is resulting in getting more choices for the reactor operator while deciding the inner and outer zones’ fuel bundle discharge burnups of the reactor core. Another behavior shown by Multi Objective GA for the selected problem is the better convergence speed compared with Penalty Functions based GA. This implies that the total number of generations required for the problem convergence is less in the case of Multi Objective GA. A worth mentioning point here is that, the observations from the above comparison suits only for the present study and cannot be generalized. In the present study, even though there is computational time advantage for Penalty Functions based GA (when the fixed number of generations is considered) the convergence speed of Multi Objective GA surpasses it in the overall performance. Acknowledgements The authors thankfully acknowledge for the useful discussions with Smt. S. Rajeswari and other colleagues of Computer Division,

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