A new approach to nuclear reactor design optimization using genetic algorithms and regression analysis

Annals of Nuclear Energy 85 (2015) 27–35

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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene

A new approach to nuclear reactor design optimization using genetic algorithms and regression analysis q Akansha Kumar ⇑, Pavel V. Tsvetkov Department of Nuclear Engineering, Texas A&M University 3133, TAMU, College Station, TX 77843, United States

a r t i c l e

i n f o

Article history: Received 29 January 2015 Received in revised form 20 April 2015 Accepted 23 April 2015

Keywords: Genetic Algorithm Reactor Design Modular Optimization

a b s t r a c t A module based optimization method using genetic algorithms (GA), and multivariate regression analysis has been developed to optimize a set of parameters in the design of a nuclear reactor. GA simulates natural evolution to perform optimization, and is widely used in recent times by the scientific community. The GA fits a population of random solutions to the optimized solution of a specific problem. In this work, we have developed a genetic algorithm to determine the values for a set of nuclear reactor parameters to design a gas cooled fast breeder reactor core including a basis thermal–hydraulics analysis, and energy transfer. Multivariate regression is implemented using regression splines (RS). Reactor designs are usually complex and a simulation needs a significantly large amount of time to execute, hence the implementation of GA or any other global optimization techniques is not feasible, therefore we present a new method of using RS in conjunction with GA. Due to using RS, we do not necessarily need to run the neutronics simulation for all the inputs generated from the GA module rather, run the simulations for a predefined set of inputs, build a multivariate regression fit to the input and the output parameters, and then use this fit to predict the output parameters for the inputs generated by GA. The reactor parameters are given by the, radius of a fuel pin cell, isotopic enrichment of the fissile material in the fuel, mass flow rate of the coolant, and temperature of the coolant at the core inlet. And, the optimization objectives for the reactor core are, high breeding of U-233 and Pu-239 in desired power peaking limits, desired effective and infinite neutron multiplication factors, high fast fission factor, high thermal efficiency in the conversion from thermal energy to electrical energy using the Brayton cycle, and high fuel burn-up. It is to be noted that we have kept the total mass of the fuel as constant. In this work, we present a module based (modular) approach to perform the optimization wherein, we have defined the following modules: single fuel pin cell, whole core, thermal–hydraulics, and energy conversion. In each of the modules we have defined a specific set of parameters and optimization objectives. The GA system (GAS), and RS together, play the role of optimizing each of the individual modules, and integrating the modules to determine the final nuclear reactor core. However, implementation of GA could lead to a local minimum or a non-unique set of parameters, those meet the specific optimization objectives. The GA code is built using Java, neutronic analysis using MCNP6, thermal–hydraulics calculations using Java, and regression analysis using R. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Genetic algorithms (GA) based optimization method has been successfully implemented in a wide range of engineering

q An abridged version of this paper has been submitted as a summary in the conference proceedings: Akansha Kumar, Pavel V. Tsvetkov An Optimization Methodology in Nuclear Reactor Design using Genetic Algorithms and Regression Analysis, 2015 ANS Annual Meeting. ⇑ Corresponding author. E-mail addresses: [email protected] (A. Kumar), [email protected] (P. V. Tsvetkov).

http://dx.doi.org/10.1016/j.anucene.2015.04.028 0306-4549/Ó 2015 Elsevier Ltd. All rights reserved.

problems. However, very limited implementation of this method has been observed in nuclear engineering, specifically, nuclear reactor design. In this work, we have demonstrated a module based optimization method using GA, and implement this method in the design of a conceptual gas cooled fast breeder reactor. A module herein is a functionality or a physics based phenomenon that solves a specific problem. A module can be understood as a set of equations representing a physical phenomenon i.e. heat conduction, thermal hydraulics of a coolant across fuel pin cell through a reactor core, and neutron transport. In this work, we present a very generic solution approach where, different modules interact among each other and finally correspond with the GA

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structure (GAS) to optimize certain defined parameters and meet specific objectives. A neutron transport and depletion code like MCNP, SERPENT takes a significantly long execution time if higher accuracy of results is desired. And, in a global optimization technique like GA, the code needs to be run for a significant number of input samples before a desired solution is obtained. Hence, we are using a spline based multivariate regression fit to perform predictive analysis. The regression fit is obtained from a set of trial data. Trial data is obtained when MCNP is run on a predetermined sample of input set, and the corresponding trial output parameters are obtained. The combination of the trial input set, and trial output parameters form the trial data. The trial data is obtained to build the regression fit. The regression fit is applied on the test data obtained from GA to determine the desired parameters in a satisfactory confidence level. It is to be noted that the output parameters for the test input is not obtained from the neutronics code, rather it is obtained from the regression fit. A GAS is the central module that interacts with a specified set of the defined modules and performs optimization. The GAS is a combination of the GA optimization module and the regression model. The objective is to design a gas cooled fast breeder reactor core that can yield a, high neutron fast fission fraction (T f ;f ), high fuel burn-up (BU), high breeding ration (BR), and high thermal efficiency (g) using the Brayton cycle for energy conversion. It is to be noted that, material structural feasibility, transient analysis, safety analysis, probabilistic risk analysis, and economics are out of scope from this work. We have defined the following modules those interact with the GAS for optimization. In the first module we determine the

optimum r F , and enrichment of U-233 in ðU  ThÞO2 fuel to obtain the desired BU, infinite neutron multiplication factor (K 1 ), and T f ;f . In the second module we determine the BR, radial power peaking factor (F PF;rad ), axial power peaking factor (F PF;ax ), and effective neutron multiplication factor (K eff ). In the third module we perform a hot channel analysis to analyze the heat transfer across the fuel pin cell i.e. the flow of heat from fuel pin to the coolant wherein, we determine the optimum core inlet temperature, T in , and, flow rate of the coolant (W) to determines the desired core outlet temperature, T out , and temperature peaking factors. Finally, we determine the optimum T in , and T out to obtain a high thermal efficiency. A detailed description of the parameters and the optimization objectives of each of the mentioned modules is presented in a future section. 1.1. Previous work Previous work related to optimization, in problems related to nuclear engineering using GA includes, core design (Pereira and Lapa, 2003; Pereira et al., 1999; Haibach and Feltus, 1997), plant design (Cantoni et al., 2000), nuclear system availability and maintenance scheduling (Lapa et al., 2000; Marseguerra and Zio, 2000), fuel management (Chapota et al., 1999; Dechaine and Feltus, 1995), and spent fuel management (Omori et al., 1997). However, coupled neutronics-thermal hydraulics problems have not been explored and the effectiveness of GA in solving the coupled problems has not been evaluated. In all the above stated work, researchers have explored GA in solving single physics problems in nuclear engineering. However, in the current work, we present a module based approach to optimization in conjunction with multivariate regression and show that it is an effective tool in reactor core design in a multiple physics domain, where each physics domain is defined as a module. In this work we have ignored the, safety aspects of a nuclear reactor, material and structural details, control elements including the control rods, fuel cycle, and economics of the nuclear reactor core design. A comparative analysis of different optimization methods is out of scope from this work. We have also ignored the analysis of different multivariate regression methods. The remainder of this paper is organized as follows. First we introduce the underlying concept of genetic algorithms based optimization method, then we present our implementation method. Then, we present the motivation, and implementation details of the various modules used in optimization. Finally, we present the results and then conclude with a summary. 2. Genetic algorithms

Fig. 1. Genetic algorithms flow chart.

GA is a search heuristic machine learning model which is derived from the process of natural selection based on the theory of species evolution (Darwin, 1859). It involves the processes, such as inheritance, reproduction, crossover, mutation, and others used for selection. Before we explain the implementation of GA in detail, it is very important to get the biological background of the concept. Every living organism consists of animal cells with, every cell consisting of a nucleus that has the genetic information of an organism. The DNA molecule in the nucleus consists of thread-like structures called chromosomes. The chromosome is a constitution of genes with, a gene located at specific locations called locus. A gene is the basic physical and functional unit consisting of instructions that define an organism i.e. how the organism survives, how it appears, and how it behaves in its environment. These characteristics determine the adaptability of the organism in the environment, also called as, the fitness of the organism. Basically, a gene encodes a trait of the organism, e.g. color of the skin.

A. Kumar, P.V. Tsvetkov / Annals of Nuclear Energy 85 (2015) 27–35

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Fig. 2. Research flow.

In GA, a population of individuals i.e. a set of chromosomes to an optimization problem is manipulated using the above mentioned processes to evolve towards a new generation of population with stronger individuals. The chromosome consists of a set of genes that carry intrinsic characteristics of a symbolic individual. The adaptation capability also known as, the fitness of an individual in the environment depends on these intrinsic characteristics. In GA (Goldberg and Edward, 1989; Davis and Lawrence, 1991), the selection and evolution process is defined in such a way that only the stronger individuals i.e. the individuals having a higher fitness level, in a generation, pass their characteristics to their off-springs, hence making them stronger. Therefore, the population in a newer generation is more fit as compared to the population in its previous generation. The GA flow starts with a random set of individuals selected from a set of possible configurations i.e. a set of possible values for the input parameters, called as a ‘‘population’’ in a ‘‘generation’’, with the first set of individuals called as the ‘‘initial’’ population or the ‘‘first’’ generation. Each individual is then evaluated, and a ‘‘fitness’’ value for that individual is calculated. This is the stage where the modules are executed and for the input parameters given by the ‘‘individual’’, a solution is obtained. The fitness value is calculated by how well the solution fits to our objectives. This stage is called as the ‘‘evaluation’’ stage. Next, we select a set of fit individuals from the population to obtain a ‘‘new’’ population for the next generation. Selection is made such that ‘‘bad’’ designs (individuals with low fitness value) are discarded and ‘‘good’’

designs are carried forward to the next generation. The selected individuals are called as ‘‘parents’’. This stage is called as the ‘‘selection’’ stage, and the set of selected individuals form a ‘‘mating pool’’. Then, crossover is performed by creating crosses of the parents i.e. the individuals in the ‘‘mating pool’’ to create a set of even ‘‘fitter’’ individuals. The idea is that, the individuals of the new population inherits the best characteristics of its parents. This stage is called as the ‘‘crossover’’ stage. Then, we perform the evaluation stage using this new population, and the above steps are performed iteratively till the desired fitness is obtained. Following figure (Fig. 1), presents a graphical view of GA. Following illustration presents the implementation details of GA, and discuss about the operations, crossover, mutation, and fitness calculation. Let us assume two types of input parameters given by G1 , and G2 . Each of these input parameters are a gene, and a combination of these form a chromosome. For the genes with a size, four bits, the size of the chromosome is eight bits. Let C 11 , and C 12 be two chromosomes selected randomly from the mating pool after the selection stage:

C 11 ¼ 01101100

ð1Þ

C 12 ¼ 11100010

ð2Þ

Suppose the crossover point is defined as the fifth bit, then the bits after the fifth bit in the parent-chromosomes, C 11 , and C 12 are swapped to create two children-chromosomes, C 21 , and C 22 , given by,

C 21 ¼ 01101010

ð3Þ

C 22 ¼ 11100100

ð4Þ

The chromosomes, C 21 , and C 22 retain some characteristics of their parent-chromosomes, C 11 , and C 12 , and those will explore the solution space not explored by the parents, C 11 , and C 12 . This operation is called as the ‘‘crossover’’ operation. The mutation operation is performed, when a specific bit in the chromosome is changed. Suppose a chromosome C 30 is defined as,

C 30 ¼ 01101010

ð5Þ

has the mutation operation on its fourth bit, results in Fig. 3. Single fuel pin cell with reflective boundary conditions.

C 33 ¼ 01111010

ð6Þ

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Fig. 4. Axial and radial view of the whole core.

where, the bit 0 at the fourth position in C 30 changes to 1 in C 33 . Mutation helps in exploring different regions of the search space and prevents from stagnation. Mutation plays an important role in species diversity. Mutation gives the children-chromosome characteristics that may be much unlike those of its parents-chromosomes, increasing the overall diversity of the population, and therefore enhancing exploration of the search space. To quantify how close a solution is to the specified objectives, a value is calculated and assigned to each chromosome after it is evaluated and a solution is obtained. This value is called as a ‘‘fitness’’ value. In other words, the fitness of a chromosome is a function of the variables those form the objective. These variables can have the objective of, maximization, minimization or proximity to limits. Let us define two variables, O1 , and O2 . Suppose, the objectives are to, maximize O1 , and minimize O2 . Let DO1 , and DO2 define the distance of the variables O1 , and O2 , from the constraint limits. The fitness function is defined as,

f ¼

1 1 þ DO1 DO2

ð7Þ

of neutrons produced per neutron absorbed in fuel (g), k1 , and BR. It is to be noted that the parameters g; m, and a are related as, g ¼ 1þm a. BR is the ratio of the total mass of fissile material at the beginning of the time step to the total mass at the end of the time step. To obtain a controlled nuclear fission chain reaction with breeding capability the following objectives need to be met: BR > 1, maximize T f ;f , and k1 > 1. To meet the objectives it is inevitable to have m > 2, a low value for the parameter, a, and a lower neutron absorption in structural materials. The parameter, T f ;f is the fraction of the fission events caused by the neutrons above the fast neutron energy of 100 keV. The parameters we optimize in a fuel pin cell are, the radius of the fuel pin in the fuel element (rf ), and the enrichment of U-233 in Th-U fuel (Kumar et al., 2014). It is to be noted that, T f ;f depends on the Dp ratio. In our case we have used a constant value for the pitch, and have varied the diameter. Following figure (Fig. 3) shows a graphical view of the single fuel pin cell used for the analysis. We have assumed the density of materials to be constant, even though in reality, density depends on the temperature, and we have assumed a constant value for the thickness of the gap (t G ), and the thickness of the cladding (tC ).

3. Research flow The flow of the research and analysis is given by the figure (Fig. 2). TODO A set of trial inputs (TRIALINP) is developed with a random combination of Dp ratio, UPct, and U235En. For each input set, output parameters (OUTPUT) are obtained using the neutronics simulation module in SERPENT. TRIALINP and OUTPUT together form the trail data (TRIALDATA). TRIALDATA is fed into the multivariate regression model (REG) to determine a regression fit (FIT) to be used in the predictive analysis. A random test input (TESTINP) is built in the GA model (GAM), and a predicted output (PREDOUT) with a confidence level is obtained from the FIT. This PREDOUT is used by GAM to obtain a new set of optimized TESTINP. PREDOUT is again determined for this new set of TESTINP. This cycle continues till we obtain an optimized solution. 4. Optimization modules In this section, we present the motivation, assumptions and implementation of each of the modules introduced in Section 1. 4.1. Fuel pin cell A single fuel pin cell analysis with reflective conditions on all external boundaries is a very efficient method for understanding the behavior of parameters such as the, number of neutrons produced per fission (m), ratio of the microscopic neutron capture cross-section and the microscopic fission cross-section (a), number

4.2. Whole core For the whole core analysis, we are using the configuration of the core of an existing gas cooled fast breeder reactor design (Kumar et al., 2014). However, for simplicity, we have ignored, the control rods, and the axial blankets. A detailed description of other components are presented in the paper (Kumar et al., 2014). The core consists of an array of hexagonal assemblies wherein, an assembly consists of an array of fuel elements in a hexagonal lattice. The assemblies with the fuel elements having fissile material are called as the ‘‘fuel’’ assemblies, and those with fertile material are called as the ‘‘blanket’’ assemblies. The fuel assemblies have fuel elements having a mixture of Th-232 and U-233. The blanket assemblies have light water reactor used fuel. The core consists of internal and external blanket assemblies. Our objectives for optimization in this module is to obtain a, desired keff , lower F PF;rad , lower F PF;ax , and higher BU. The total mass Table 1 System parameters for GA. Parameter

Value

Number of chromosomes in every generation (N ch;gen ) Number of bits per gene (b) Number of generations (N gen ) Crossover probability (P c ) Mutation probability (P m ) Number of elite elements preserved (N elite )

40 16 100 0.4 0.05 2

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of the fuel (fissile + fertile) material is kept constant. Therefore, when we vary the radius of the fuel element, the height of the core (LFC ) is affected, and due to having a constant density, the total mass of the fuel remains constant. The burn-up and criticality calculations of the whole core is performed using MCNP6 with a power of 100 MWth to determine the BU. For a safe operation in terms of preventing meltdown of the fuel rod, peaking factors play an important role. The peaking factor is the ratio between maximum local energy depositions to the average energy deposition in the reactor core. It has been assumed that the external blanket is not part of the reactor core while calculating the radial peaking factors. Axial peaking factors are also calculated to obtain the hottest region of the core. This helps in determining the hottest region in the core from a safety perspective. In the whole core analysis we have assumed a constant value for the, power, and positioning of blankets. Following figure (Fig. 4) presents a radial and an axial view of the whole core. 4.3. Thermal hydraulics and heat transfer A standard thermodynamic analysis of the transfer of heat across the fuel gap, cladding and bulk coolant has been performed to the core design. Since, the primary focus is the implementation of GA in the multiple module domain coupled with the regression analysis, a rigorous thermal hydraulics and energy transfer analysis has not been performed. However, for completeness, we implemented a simplified model. In a single phase coolant heat transfer domain, the pressure drop across the length of the active core is the sum of the pressure drop due to friction, form and elevation, given by,

DP ¼ DPfriction þ DPform þ DPelev ation :

ð8Þ

For simplicity we assume that the total pressure drop is only due to friction hence, DP form ¼ 0, and DPelev ation ¼ 0. Therefore, the primary loop pressure drop (DP) is given by a simplified equation,

qFC  V 2FC

DP ¼

2

!   LFC  f Darcy—Weisbac  ; DFC

ð9Þ

where

V FC ¼

W ; NFC  qFC  AFC

ð10Þ

where qFC is the density of the coolant, V FC is the velocity of the coolant, f Darcy—Weisbac is the Darcy–Weisbach constant with a value of 0:016; DFC is the diameter of the fuel element, N FC is the number of fuel elements, and AFC is the flow area of the coolant. The temperature of the coolant at core outlet (T out ) is given by,

T out ¼ T in þ

Q ; W  Cp

ð11Þ

Table 3 Input value ranges for the design parameters. Parameter

Range

Units

rF En T in W

0:2 ! 0:33 14 ! 20 50 ! 200 20 ! 100

cm wt.%  C

where Q is the total thermal power of the reactor core, and C p is the specific heat capacity of the coolant. The pumping power of coolant (Ppump ) is given by,

Ppump ¼

Module Fuel pin cell

Whole core

Thermal–hydraulics Energy conversion

Label

Parameter

Objective

Weight

OBJ1 OBJ2 OBJ3 OBJ4 OBJ5 OBJ6 OBJ7

k1 T f ;f keff F PF;rad F PF;ax BR Burn-up

Maximize Maximize 1:0—1:1 Minimize Minimize Maximize Maximize

0.1 0.1 0.1 0.1 0.1 0.1 0.1

OBJ8 OBJ9 OBJ10

T out DP

<600 <100 Maximize

0.1 0.1 0.1

geff

Units

DP  AFC  V FC

gpump

;

ð12Þ

where gpump is the pump efficiency. For a safe operation, and to ensure fuel material structural integrity, it is important to compute the maximum radial, and axial fuel temperature. There is a temperature variation radially on the fuel element due to the presence of heterogeneous components: fuel, gap, and clad. The governing equations with assumptions for simplicity are given by,

q0peak

DT b ¼

; 2  p  LFC  ðrF þ tG þ t C Þ   q0peak rF þ tG þ tC ; DT C ¼  ln 2  pkC rF þ tG   q0peak rF þ tG ; DT G ¼  ln 2  pkG rF q0peak DT F ¼ ; 2  pkF

ð13aÞ ð13bÞ ð13cÞ ð13dÞ

where DT F is the temperature drop across the fuel, DT G is the temperature drop across the gap, DT C is the temperature drop across the cladding, DT b is the temperature drop across the bulk coolant, q0peak is the peak linear heat generation rate i.e. linear heat generation rate multiplied by the radial and axial peaking factors, kC is the thermal conductivity coefficient of the cladding, kG is the thermal conductivity coefficient of the gap, and kF is the thermal conductivity coefficient of the fuel. The objectives are to maintain a peak fuel temperature in structural integrity limits, and DP in limit. The parameters we used to optimize are T in , and W. 4.4. Energy conversion Brayton cycle is used to analyze energy conversion of Helium. Efficiency of energy conversion from heat to electricity is important in the economics point of view. A higher efficiency is always desired. For simplicity, we implemented a simple variation of the Brayton cycle with no regeneration, and reheating. The optimum pressure ratio (r p;opt ) is given by,

rp;opt ¼ Table 2 Optimization objective.

kgs sec

c  c1 T3 ; T1

ð14Þ

where, T 1 is the temperature of coolant at core inlet, T 3 is the temperature of coolant at core outlet, and c is the heat capacity ratio. The amount of work done by the turbine per unit mass flow rate _ T ) is given by, of the coolant (W

2

GWd MTHM 

C Pa

3 1 _ T ¼ g  C p  T 3  41  5 W c1 ; T rpc

ð15Þ

where gT is the turbine efficiency, and C p is the specific heat capacity of the coolant. The amount of work done by the compressor per _ CP ) is given by, unit mass flow rate of the coolant (W

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A. Kumar, P.V. Tsvetkov / Annals of Nuclear Energy 85 (2015) 27–35

Fig. 5. The optimized solution for the design parameters vs generation.

 c1  _ CP ¼ C p  T 1 rpc  1 ; W

gCP

ð16Þ

where gCP is the compressor efficiency. Therefore, the maximum _ max ) is given by, amount of work done (W

  c1 _ max ¼ C p  T 1 T 3  r pc : W T1

ð17Þ

Thermal efficiency (geff ) is given by,

geff ¼

_ T W _ CP W : _ max W

5.1. Fitness function As mentioned in a previous section, the fitness function plays a major role in determining if a solution is an optimum solution or not. Therefore, it is important to have a fitness function that should meet all our objectives. The fitness of the system covering all the objectives, called as the total fitness (TF) is given by,

TF ¼

O X ZiW i ;

ð19Þ

i

ð18Þ

The objective is to obtain a higher geff , and the parameters we used for optimization are, T in , and T out . 5. Implementation of the genetic algorithms Table 1 presents the system parameters used to run the GA module. For a more precise optimization it is advisable to use a higher value for N ch;gen ; b, and N gen for a more precise optimized solution. However, this would lead to a slower convergence. The objectives are given in the table (Table 2), The input parameters are given in the table (Table 3),

with, O X W i ¼ 1;

ð20Þ

i

where Z i is the fitness of objective i; W i is the weight of the objective i, and O is the total number of objectives. The weight, W i determines the contribution of this objective to the overall fitness of the system. 5.1.1. Maximize fitness function For the sake of completeness we are presenting the method we used to determine the fitness of a maximize objective. Let the function is given by f ðxÞ, where x is our design variable. The fitness of f ðxÞ is given by,

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Fig. 6. Correlogram of reactor parameters.

Table 4 Optimized solution. Parameter

Value

Units

rF En T in W

0.217 18.476 63.476 35.937

cm wt.%  C kgs sec

Fig. 8.

av erage total fitness maximum total fitness

versus generation of the system.

6. Results and discussion

Fig. 7. Contribution of individual objectives to the overall fitness of the system.

Z¼

f ðxÞ W i; f max

ð21Þ

where, f max is the maximum value of f ðxÞ, and W i is the weight of the fitness function. We have used similar methods for other objective functions.

The figure (Fig. 5) presents the progress of the design variables to an optimized solution with respect to generation. We observe that with generation, the values of the design parameters become flatter and finally converge to a specific solution. The figure (Fig. 6) presents a correlogram of the reactor parameters. Correlogram can be an useful way to examine large numbers of bivariate relationships among quantitative. In our case we have four design parameters and ten objective parameters, hence correlogram presents a concise way of presenting bivariate relationships. The figure (Fig. 6) contains several smoothed best fit lines and confidence

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A. Kumar, P.V. Tsvetkov / Annals of Nuclear Energy 85 (2015) 27–35

Fig. 9. Fitness of the fittest chromosome in a generation versus generation. Fig. 11. Average fitness of all chromosomes versus crossover probability.

ellipses. The red smoothed fit line in each eclipse presents the bivariate relationship between the parameters given by a specific row and column. The optimized solution is given by the table (Table 4). The figure (Fig. 7) presents the contribution of each objective function to the overall fitness of the system. The values are determined by calculating the ratio between individual fitness (W i ), and the total fitness (TF). This shows that OBJ1 has more contribution as compared to OBJ9 to the overall fitness of the system. The figure (Fig. 8) presents the ratio between the average total fitness and the maximum total fitness with respect to generation. The ratio is very high in the proximity of 0.8, which is atypical indication that a close to optimal solution is plausible. This is a typical behavior of heuristic optimization methods like GA, where with generation we approach towards an optimal solution. Figure (Fig. 9) presents the fitness value of the fittest chromosome in a generation versus generation. The mean value is 0.7686 with a standard deviation of 0.0357. It is also interesting to know how the average fitness of all chromosomes in generation is distributed. The average fitness of all chromosomes in generation has a mean value of 0.6252, and the standard deviation is 0.0154. A lower value for the standard deviation provides an impression that the solution obtained, is near to optimal. Table 1 presents the mean and the standard deviation of the values of the fittest chromosome in each generation for the design parameters. The optimized solution given in Table 4 is in close proximity with the mean and the standard deviation values given in Table 1. That means, the values in the

Table 5 Mean and standard deviation of the design input parameters of the fittest chromosome across all generations. Parameter

Mean value

Standard deviation

Units

rF En T in W

0.2272 18.1317 90.6501 44.1429

0.0267 1.1442 34.9030 8.83

cm wt.%  C kgs sec

optimized solution lie inside one standard deviation from the mean. This shows that the obtained solution, Table 4 is near to optimal. It is interesting to understand the effect of P c , and Pm on the optimization algorithm. Most of the work done use an hit and trial method to determine the values of these parameters. A detailed analysis of these parameters are out of scope of this work, however Figures (Figs. 10, and 11) present the effect Pc , and P m have on the average fitness of the chromosomes. In figure (Fig. 10) Pm shows a visible trend, i.e. the average fitness is decreasing with an increase in P m , but the effect of P c as shown in Fig. 11 is completely random. In theory, if the P c is small, individual chromosomes would dominate the selection process and this would tend towards an undesired local maximum. Hence Pc significantly affects the diversity of the search. Pm determines the divergence probability and is intended to occasionally break one or more chromosomes of a population out of a local minimum/maximum space and potentially explore a better minimum/maximum space. Table 5 presents the mean and standard deviation of the design input parameters of the fittest chromosome across all generations. This table gives an idea about the search space of all the design parameters.

7. Summary and future work

Fig. 10. Average fitness of all chromosomes versus mutation probability.

A new approach towards optimization using GA and RS has been demonstrated. This approach has been implemented to the design of Gas cooled Fast Breeder reactor. We have also presented a module based approach where we have analyzed the following major components as modules for optimization, fuel pin cell, whole core, thermal–hydraulics, and energy transfer. We demonstrated the use of multivariate regression analysis and GA for optimization. Future work involves detailed analysis of the GAS, and exploration of other regression methods. It is also interesting to

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