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Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms
Annals of Nuclear Energy xxx (2015) xxx–xxx
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Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms 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 18 December 2014 Received in revised form 1 July 2015 Accepted 2 July 2015 Available online xxxx Keywords: Thorium Uranium Parametric Optimization Regression Predictive
a b s t r a c t An analysis and optimization of a set of neutronics parameters of a thorium-fueled pressurized heavy water reactor core fuel has been performed. The analysis covers a detailed pin-cell analysis of a seed-blanket configuration, where the seed is composed of natural uranium, and the blanket is composed of thorium. Genetic algorithms (GA) is used to optimize the input parameters to meet a specific set of objectives related to: infinite multiplication factor, initial breeding ratio, and specific nuclide’s effective microscopic cross-section. The core input parameters are the pitch-to-diameter ratio, and blanket material composition. Recursive partitioning of decision trees (rpart) multivariate regression model is used to perform a predictive analysis of the samples generated from the GA module. Reactor designs are usually complex and a simulation needs a significantly large amount time to execute, hence implementation of GA or any other global optimization techniques is not feasible, therefore we present a new method of using rpart in conjunction with GA. Due to using rpart, 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 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 rpart model is implemented as a library using R programming language. The results suggest that the initial breeding ratio tends to increase due to a harder neutron spectrum, however a softer neutron spectrum is desired to limit the parasitic absorption of Pa-233. The neutronics model, design and analysis have been done using Serpent 1.1.19 Monte Carlo code. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction 1.1. Objective
shown to pressurized heavy water moderated thermal reactors. Therefore, in a long term perspective of designing a heavy water moderated and cooled thermal breeder reactor, we analyzed the r
The research is based upon the following objectives. Firstly, identify, and analyze the behavior of core parameters in the reactor design of breed-and-burn type thorium fuel. Secondly, determine an optimized combination of the pitch-to-diameter ratio P ratio , and material composition to satisfy a set of objectives. D Finally, explore the feasibility and applicability of regression based predictive analysis models, and genetic algorithms (GA) based global optimization techniques in parametric analysis. In recent times, due to the successful implementation of CANDU (Torgerson et al., 2006), and PHWR (Bajaj and Gore, 2006; Balakrishnan and Kakodkar, 1994) type reactors, considerable interest has been
⇑ Corresponding author. E-mail addresses: [email protected] (A. Kumar), [email protected] (P.V. Tsvetkov).
r
, and rc;Th-232 in behavior of parameters such as k1 ; rc;Pa-233 ; rcc;Th-232 ;U-238 c;Th-232 a single fuel pin cell with a combination of natural uranium as the fuel, and enriched thorium as blanket. ‘‘Enriched’’ thorium is a mixture of thorium dioxide (ThO2 ) and uranium dioxide (UO2 ), with (UO2 ) having a certain enrichment of U-235. Current research is based on the determination of an optimum enrichment of U-235, and DP ratio to meet specific objectives related to criticality, breeding, and burn-up in a pressurized, heavy water moderated, and cooled, thermal breeder reactor fuel. The DP ratio, and material composition are used as the input parameters in this work. The model, design and analysis has been done using Serpent 1.1.19 Monte Carlo code (Leppnen). GA is used to perform optimization of the input variables to meet a set of desired objectives. In GA, a random sample of input variables are used to determine the parameters from the neutronics model. Based on the desired objectives, a fitness value, determined from the output, is assigned to every input
http://dx.doi.org/10.1016/j.anucene.2015.07.006 0306-4549/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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sample. Based on this fitness value, GA generates a new set of input parameters. It is a recursive method that results in an optimized set of input parameters. A detailed methodology used by GA is presented in Section 1.3.3. Neutron transport solver is very expensive in terms of computation time if higher accuracy of results is desired. And, in a global optimization technique, like GA, the neutronics model needs to be run for a significant number of input samples before a desired optimized solution is obtained. Hence, we are using a decision trees based multivariate regression fit to perform predictive analysis. The regression fit is obtained from a set of trial data. The trial set of data is obtained when Serpent is run on a predetermined sample of input set, and 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 used 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 model, rather it is obtained from the regression fit. The analysis of whole core, thermal hydraulics aspects, material stability and structural parameters are out of scope from this work. Review and a detailed theoretical analysis of rpart, and comparative analysis with other multivariate regression models is out of scope. 1.2. Previous work The concept of pressurized, heavy water moderated, and cooled, thermal reactor has shown very impressive applications due to the successful implementation and operation of CANada Deuterium Uranium (CANDU) and, Pressurized Heavy Water Reactor (PHWR) power reactors. Also, extensive research on the design and analysis of CANDU (Torgerson et al., 2006) and PHWR (Bajaj and Gore, 2006; Balakrishnan and Kakodkar, 1994) has been done since the last few decades. These references provide us useful information on the overall design of the reactors, fuel cycle and balance of plant. Analysis of the core parameters, also known as ‘‘parametric core analysis’’ has not been explored to a satisfactory level, however Perry and Weinberg (1972) and Taraknath and Ricci (2013) have done appreciable research in understanding the behavior of core parameters, and presenting their significance in reactor design. Perry and Weinberg (1972) performed the, reactor physics calculations, and analysis of behavior of reactor parameters in thermal breeder reactors, those include fluid fuel-thermal breeders such as aqueous-slurry systems and molten salt systems, and solid fuel-thermal breeders such as heavy water reactors, gas cooled reactors (Kumar et al., 2014), and light water reactors. Taraknath and Ricci (2013) focuses on PHWR core parameter analysis of PHWR reactors design using thorium based fuel. Taraknath and Ricci (2013) used two types of fuel, reprocessed ThO2 þ UO2 , and ðTh þ U233ÞO2 fuel. However, in the current work, we are focusing on a fuel combination of natural uranium as the fuel and enriched thorium (ThO2 þ UO2 ) as the blanket. Previous work related to optimization, in problems related to nuclear engineering using GA includes, core design (Pereira and Lapa, 2003; do Nascimento Abreu Pereira et al., 1999; Haibach and Feltus, 1997), plant design (Cantoni et al., 2000; Kumar and Tsvetkov, 2015), 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 predictive analysis and optimization using GA in parametric analysis has not been extensively explored in reactor design problems. We present a integration based approach to optimization and show that, if coupled with a multivariate regression model, it is an effective tool in reactor core design.
1.3. Current work 1.3.1. Parametric analysis In the design of a nuclear reactor, neutronics analysis of a single fuel pin cell using reflecting spatial conditions on all boundaries, plays an important role to understand the reactor behavior, safety, power and burn-up. It is important to perform an analysis of the parameters such as the, infinite neutron multiplication factor (k1 ), breeding ratio (BR), and capture and fission cross-sections of fissile and fertile material used in the fuel. We performed studies analyzing the above parameters, and design a optimization strategy for thermal breeder reactors using U–Th fuel. 1.3.2. Multivariate regression Predictive analysis is done using multivariate regression. Regression is a method used to predict a response variable from predictor variables. In this work, DP ratio, U-235 enrichment (U235En), and U weight fraction in the mixed fuel (UPct), represent r
r
, and rc;Th-232 reprethe predictor variables, and k1 ; rc;Pa-233 ; rcc;Th-232 ;U-238 c;Th-232 sent the response variables. A detailed description of all the parameters are presented in future sections. Predictive analysis involves the following steps: Firstly, fit a regression model to a set of trial data, and secondly, use the fit to predict the response for test input data in a desired confidence level. Trial data is obtained from the neutronics analysis using Serpent and test input data is provided by GA code. The regression model is built using a program called rpart (Therneau and Atkinson, 2000). 1.3.3. Genetic algorithms 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. 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, 1989; Davis, 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
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
A. Kumar, P.V. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx
‘‘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
3
stage using this new population, and the above steps are performed iteratively till the desired fitness is obtained. Fig. 1 presents a graphical view of GA (Kumar and Tsvetkov, 2015) that we have used in this research. Flow of the paper. The remainder of the paper is organized as follows. Firstly, we present the design and the approach for the analysis. This includes the Serpent model design, followed by GA input parameters and objectives. Then, we present the results with a discussion for each table and figure. Finally, we conclude with a summary and suggestions for future efforts. 1.4. Research flow The flow of the research and analysis is given in Fig. 2. A set of trial inputs (TRIALINP) is developed with a random combination of p ratio, UPct, and U235En. For each input set, output parameters D (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. 2. Design and approach 2.1. Neutronics model and analysis
Fig. 1. Genetic algorithms flow chart.
Parametric analysis has been done using an infinite pin-cell configuration of a hypothetical reactor core where, the criticality and burn-up calculations have been performed with a reflecting boundary condition. A reflecting boundary condition can be literally interpreted as an infinite number of pin-cells, that ensures that no neutron leaks out of the system. The infinite pin-cell configuration used in the analysis is given in Fig. 3. The pin-cell consists of an infinite configuration of fuel-blanket single fuel pins representing both the fuel assembly, and the blanket assembly regions at a
Fig. 2. Research flow.
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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A. Kumar, P.V. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx
Fig. 3. Pin-cell model used in the Th-fueled PHWR design.
coarser level within the whole core of a HWR. In the rest of the paper, a ‘‘fuel pin-cell’’ is composed of two fuel pins, and two blanket pins. In simpler words, our pin cell is a homogenized configuration of the assembly. For a steady state parametric analysis, with no depletion, and power deposition analysis, an assembly level spatial homogenized model is valid. In addition, we are not performing high fidelity assembly level and whole core analysis, hence a simplified homogeneous configuration would provide useful information about the parameters those affect breeding in PHWR fuel designs. The fuel pin (seed) consists of a cylindrical fuel rod element containing natural uranium oxide. This type of fuel pin is commonly used in the Indian PHWR design. The blanket pin contains Th-232 as the dominant fertile material. It is modeled under the same dimension of the seed with multiple variations of the fissile composition. The variation in the fissile composition is based on U235En, and UPct. Analysis has also been done with a variation of the DP ratio keeping the material composition constant. The core parameters analyzed in this work depend on the flux weighted effective one-group microscopic cross given by,
R
r /ðE; V ÞdEdV rx ¼ R x ; /ðE; V ÞdEdV
ð1Þ
where, x represent a specific material (U-235, U-238, Th-232, etc.), and an interaction (fission, capture, absorption, etc.) in the fuel or the blanket region, and /ðE; VÞ is the neutron energy spectrum in the evaluated region. These parameters are evaluated in the paper to give insights into the characteristics of the pin-cell configuration. For simplicity, and to suggest an area of improvement using existing design, we have assumed our seed region to have an identical composition with the fuel of an existing reactor design. The blanket has the same thickness and material composition for the gap and the cladding, as is the seed except that the material in the blanket is different. Enrichment in this blanket is even higher than the seed region. The blanket here gives us a room for improvement by adjusting the parameters DP , U235En and UPct.
3. Results and discussion 3.1. Neutronics This section presents the results from the simulation of the models, and provides a discussion with appropriate inherent
Fig. 4. Correlation between TRIALINP and OUTPUT.
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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assumptions. Values for the parameters obtained from the neutronics module is given in Fig. 4. The figure presents a correlogram (Kabacoff, 2011) of the correlations among the variables in the TRIALDATA. The lower triangle contains red smoothed best fit lines and, the blue ellipse depicts the confidence level. The upper triangle presents the same information using pie charts. A blue color represents a positive correlation, and a red color represents a negative correlation between the two variables that meet at the cell. The strength of the correlation is displayed by the size of the filled pie slice. Positive correlations fill the pie in a clockwise direction starting at 12 o’clock, and negative correlations fill the pie in a counter clockwise direction. Fig. 4 presents a macroscopic view of the effect on breeding based on the variations in DP ratio, UPct, r
and U235En. Except the ratio rf ;Th-232 , all other parameters are disc;Th-232 cussed in Figs. 5–8. With an increase in the DP ratio, due to the thermalization of neutrons there is more chances of capture of neutrons by Th-232 as compared to fission. Due to an increase in UPct there is an increasing trend towards fission by Th-232 as compared to capture. However, it is to be noted that the fission cross section for thorium is much smaller than the capture cross section. Fig. 5 presents the effect of kinf of the fuel pin-cell with respect to the variation in DP ratio, UPct, and U235En. In the figure we observe that, with an increase in DP ratio, kinf increases due to the thermalization of the neutrons. Due to an increase in the thermalization of neutrons there is an increase in the possibility of thermal fission in the core, because, fissile material in the fuel have a higher
Fig. 7. Growth of U-233 (Th-232 cross-section in the blanket, units: cm2).
Fig. 8. Ratio of
Fig. 5. Infinite neutron multiplication factor (kinf ).
rc;Th-232 and rc;U-238 .
fission cross-section in the thermal energy range. Within the scope of this research, the reactor model is in the under-moderated region. It is not reasonable to have a DP ratio greater that 3.5, as it is not economical in terms of the need for a higher pumping power of the coolant. Also, at a higher DP ratio the model operates in the under-moderated regime, which undermines the tendency of a nuclear reactor inherent safety feature, especially void moderator reactivity. No appreciable changes in kinf is observed with a variation in UPct. One of the important necessities of using Th-232 as a fertile material is the production of U-233. Th-232 transmutes to Th-233, which later produces U-233 through beta decays given by, 232
capture
b decay
b decay
Th þ 1 n ! 233 Th ! 233 Pa ! 233 U:
ð2Þ
Protactinium, Pa-233, the nuclide that separates Th-233 and U-233, has a longer half-life and a very high absorption cross-section that significantly increases the parasitic absorption, which is not desirable for a sustained criticality and breeding condition, hence not viable in an economic point of view. Therefore, it r is necessary to have a lower value for the ratio, rc;Pa-233 where, c;Th-232
rc;Pa-233 is the microscopic capture cross-section of Pa-233, and rc;Th-232 is the microscopic cross-section of Th-232. This ratio is
Fig. 6. Ratio of
rc;Pa-233 and rc;Th-232 .
used to evaluate the efficiency of Th-232 breeding in the presence of Pa-233. The effect of this ratio with respect to DP ratio, UPct, and U235En is presented in Fig. 6. We have observed that for any
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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Fig. 9. Neutron capture cross-section of Pa-233, units:- cm2.
combination of DP ratio, UPct, and U235En, Pa-233 has a superior absorption towards the fertile nuclide in the blanket. Analogous to the previous trend, harder spectrum seems to be the most benign environment to grow U-233. With an increase in enrichment, capture by Pa-233 is more prominent as compared to Th-232. In Fig. 7, we have plotted the effect of DP ratio, UPct, and U235En, on the rc;Th-232 in the blanket. Softening of the neutron energy spectrum adversely affects capture by Th-232. This is also seen from the cross-section tables. Lower U235En with higher UPct is more favorable for Th-232 capture as compared to higher enrichment and lower UPct. However, if compared with the production of Pu-239, U-233 is more likely to be produced in the thermal region. This can be inferred from Fig. 8. With softening of the neutron energy spectrum, capture by Th-232 is more favorable as compared to U-238. Higher U235En with lower UPct is more favorable for capture by U-238 as compared to lower U235En with higher UPct. Interestingly, from Fig. 9 we observe that, a higher U235En, and a lower UPct in the mixed fuel leads to less aggressive Pa-233 parasitic absorption, due to the flux depression from the presence of fissile material where Pa-233 has a high cross-section at this energy range. Neutron spectrum comparison depicted in Fig. 10 shows that both the configurations DP : 2:5; 3:5 have a strong thermal reactor
Fig. 11. Capture cross-section as a function of energy. Obtained by JANIS 4.0.
characteristics with a recognizable thermal peak. The difference occurs in the resonance region where a higher flux is expected with respect to a decrease in a lower DP ratio. Since the effective microscopic cross-section highly depends on the neutron spectrum, this could be the factor that increase parasitic absorption in Pa-233. At the early range of resonance energy, the value of relative microscopic capture cross-section of Pa-233-to-Th-232 is high as presented Fig. 11. Therefore, in this case, softer spectrum would likely give lower parasitic absorption of Pa-233. Hence, we desire to have a higher value for k1 to obtain more neutrons for breeding, minimize the r ratio rc;Pa-233 to ensure limited parasitic absorption by Pa-233, maxc;Th-232
r
, and imize rcc;Th-232 ;U-238
rc;Th-232 to breed higher amount of U-233.
3.2. Genetic algorithms 3.2.1. Input parameters The input parameters as defined in Section 2.1, are DP ratio, U235En, and UPct. We have used 16 bits for each of the variables. The conversion formula and the ranges for each of the variables are given in Table 1, 3.2.2. Optimization objectives We have identified the following objectives for optimization. Maximize k1 to obtain more neutrons for breeding, minimize rc;Pa-233 to ensure limited parasitic absorption by Pa-233, maximize r c;Th-232
rc;Th-232 rc;U-238 ,
and
rc;Th-232 to breed higher amount of U-233. In simpler
Table 1 Input value ranges fo the design parameters. Parameter
Range
p D
2:5 ! 3:5
ratio U235En UPct
Units
10 ! 20 40 ! 20
% %
Table 2 Optimization objectives.
Fig. 10. Neutron spectrum comparison of two different pin-cell configuration. Green is the pin-cell configuration with a DP ratio of 2.5, and red is the pin-cell configuration with a DP ratio of 3.5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Parameter
Objective
Objective Id (o)
Weight (wo )
k1
maximize minimize
OBJ1 OBJ2
0.25 0.25
maximize
OBJ3
0.25
maximize
OBJ4
0.25
rc;Pa-233 rc;Th-232 rc;Th-232 rc;U-238
rc;Th-232
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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A. Kumar, P.V. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx
Fig. 12. Input parameters generated by GA.
words, we determine the optimized combination of DP ratio, U235En, and UPct to meet the above mentioned objectives. The optimization objectives are given in Table 2. Total fitness of the system is the sum of the fitness contribution from all the objectives. Total fitness F ðsÞ is given by,
F ðsÞ ¼
O X f o ðsÞ wo ;
Table 4 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 )
20 16 10
ð3Þ
o¼1
where s is a chromosome, O is the total number of objectives, f o ðsÞ is the contribution of a specific objective, o to the total fitness of chromosome s, and wo is the weight of a specific objective o. It is to be P noted that wo is positive and Oo¼1 wo ¼ 1. 3.2.3. Results Fig. 12 presents the input parameters for Dp ratio, UPct, and U235En in five generations. The x-axis represents the input set in a specific generation. To present the effectiveness of GA we have presented only the first five generation, however for the optimized solution in our case we had ten generations. The y-axis represents the parameter value. It can be inferred from the figure that with generations, the input parameters become flat across multiple
Table 3 Optimized solution. Parameter p D
ratio U235En UPct
Value
Units
3.31 11.25 35.55
% %
chromosomes (input sets). This phenomenon is expected from GA, where every generation makes GA learn more about the system and, hence the inputs generated by GA are closer to the optimized input set. The optimized solution is given in Table 3. Table 4 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.
4. Conclusion and future work A parametric study of thorium-fueled heavy water moderated reactor fuel was performed. The analysis demonstrated the characteristic of thorium based reactor under HWR spectrum. Based on the results, within the range of the variation conducted in this paper, softer spectrum was needed to minimize the parasitic absorption in the blanket, but harder spectrum tends to give a higher breeding ratio. We obtained an optimized solution for Dp ratio, UPct, and U235En. We demonstrated the use of multivariate regression analysis and GA for optimization, which is one of the major objectives of this work. Future work involves detailed
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms 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
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Article history: Received 18 December 2014 Received in revised form 1 July 2015 Accepted 2 July 2015 Available online xxxx Keywords: Thorium Uranium Parametric Optimization Regression Predictive
a b s t r a c t An analysis and optimization of a set of neutronics parameters of a thorium-fueled pressurized heavy water reactor core fuel has been performed. The analysis covers a detailed pin-cell analysis of a seed-blanket configuration, where the seed is composed of natural uranium, and the blanket is composed of thorium. Genetic algorithms (GA) is used to optimize the input parameters to meet a specific set of objectives related to: infinite multiplication factor, initial breeding ratio, and specific nuclide’s effective microscopic cross-section. The core input parameters are the pitch-to-diameter ratio, and blanket material composition. Recursive partitioning of decision trees (rpart) multivariate regression model is used to perform a predictive analysis of the samples generated from the GA module. Reactor designs are usually complex and a simulation needs a significantly large amount time to execute, hence implementation of GA or any other global optimization techniques is not feasible, therefore we present a new method of using rpart in conjunction with GA. Due to using rpart, 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 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 rpart model is implemented as a library using R programming language. The results suggest that the initial breeding ratio tends to increase due to a harder neutron spectrum, however a softer neutron spectrum is desired to limit the parasitic absorption of Pa-233. The neutronics model, design and analysis have been done using Serpent 1.1.19 Monte Carlo code. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction 1.1. Objective
shown to pressurized heavy water moderated thermal reactors. Therefore, in a long term perspective of designing a heavy water moderated and cooled thermal breeder reactor, we analyzed the r
The research is based upon the following objectives. Firstly, identify, and analyze the behavior of core parameters in the reactor design of breed-and-burn type thorium fuel. Secondly, determine an optimized combination of the pitch-to-diameter ratio P ratio , and material composition to satisfy a set of objectives. D Finally, explore the feasibility and applicability of regression based predictive analysis models, and genetic algorithms (GA) based global optimization techniques in parametric analysis. In recent times, due to the successful implementation of CANDU (Torgerson et al., 2006), and PHWR (Bajaj and Gore, 2006; Balakrishnan and Kakodkar, 1994) type reactors, considerable interest has been
⇑ Corresponding author. E-mail addresses: [email protected] (A. Kumar), [email protected] (P.V. Tsvetkov).
r
, and rc;Th-232 in behavior of parameters such as k1 ; rc;Pa-233 ; rcc;Th-232 ;U-238 c;Th-232 a single fuel pin cell with a combination of natural uranium as the fuel, and enriched thorium as blanket. ‘‘Enriched’’ thorium is a mixture of thorium dioxide (ThO2 ) and uranium dioxide (UO2 ), with (UO2 ) having a certain enrichment of U-235. Current research is based on the determination of an optimum enrichment of U-235, and DP ratio to meet specific objectives related to criticality, breeding, and burn-up in a pressurized, heavy water moderated, and cooled, thermal breeder reactor fuel. The DP ratio, and material composition are used as the input parameters in this work. The model, design and analysis has been done using Serpent 1.1.19 Monte Carlo code (Leppnen). GA is used to perform optimization of the input variables to meet a set of desired objectives. In GA, a random sample of input variables are used to determine the parameters from the neutronics model. Based on the desired objectives, a fitness value, determined from the output, is assigned to every input
http://dx.doi.org/10.1016/j.anucene.2015.07.006 0306-4549/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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sample. Based on this fitness value, GA generates a new set of input parameters. It is a recursive method that results in an optimized set of input parameters. A detailed methodology used by GA is presented in Section 1.3.3. Neutron transport solver is very expensive in terms of computation time if higher accuracy of results is desired. And, in a global optimization technique, like GA, the neutronics model needs to be run for a significant number of input samples before a desired optimized solution is obtained. Hence, we are using a decision trees based multivariate regression fit to perform predictive analysis. The regression fit is obtained from a set of trial data. The trial set of data is obtained when Serpent is run on a predetermined sample of input set, and 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 used 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 model, rather it is obtained from the regression fit. The analysis of whole core, thermal hydraulics aspects, material stability and structural parameters are out of scope from this work. Review and a detailed theoretical analysis of rpart, and comparative analysis with other multivariate regression models is out of scope. 1.2. Previous work The concept of pressurized, heavy water moderated, and cooled, thermal reactor has shown very impressive applications due to the successful implementation and operation of CANada Deuterium Uranium (CANDU) and, Pressurized Heavy Water Reactor (PHWR) power reactors. Also, extensive research on the design and analysis of CANDU (Torgerson et al., 2006) and PHWR (Bajaj and Gore, 2006; Balakrishnan and Kakodkar, 1994) has been done since the last few decades. These references provide us useful information on the overall design of the reactors, fuel cycle and balance of plant. Analysis of the core parameters, also known as ‘‘parametric core analysis’’ has not been explored to a satisfactory level, however Perry and Weinberg (1972) and Taraknath and Ricci (2013) have done appreciable research in understanding the behavior of core parameters, and presenting their significance in reactor design. Perry and Weinberg (1972) performed the, reactor physics calculations, and analysis of behavior of reactor parameters in thermal breeder reactors, those include fluid fuel-thermal breeders such as aqueous-slurry systems and molten salt systems, and solid fuel-thermal breeders such as heavy water reactors, gas cooled reactors (Kumar et al., 2014), and light water reactors. Taraknath and Ricci (2013) focuses on PHWR core parameter analysis of PHWR reactors design using thorium based fuel. Taraknath and Ricci (2013) used two types of fuel, reprocessed ThO2 þ UO2 , and ðTh þ U233ÞO2 fuel. However, in the current work, we are focusing on a fuel combination of natural uranium as the fuel and enriched thorium (ThO2 þ UO2 ) as the blanket. Previous work related to optimization, in problems related to nuclear engineering using GA includes, core design (Pereira and Lapa, 2003; do Nascimento Abreu Pereira et al., 1999; Haibach and Feltus, 1997), plant design (Cantoni et al., 2000; Kumar and Tsvetkov, 2015), 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 predictive analysis and optimization using GA in parametric analysis has not been extensively explored in reactor design problems. We present a integration based approach to optimization and show that, if coupled with a multivariate regression model, it is an effective tool in reactor core design.
1.3. Current work 1.3.1. Parametric analysis In the design of a nuclear reactor, neutronics analysis of a single fuel pin cell using reflecting spatial conditions on all boundaries, plays an important role to understand the reactor behavior, safety, power and burn-up. It is important to perform an analysis of the parameters such as the, infinite neutron multiplication factor (k1 ), breeding ratio (BR), and capture and fission cross-sections of fissile and fertile material used in the fuel. We performed studies analyzing the above parameters, and design a optimization strategy for thermal breeder reactors using U–Th fuel. 1.3.2. Multivariate regression Predictive analysis is done using multivariate regression. Regression is a method used to predict a response variable from predictor variables. In this work, DP ratio, U-235 enrichment (U235En), and U weight fraction in the mixed fuel (UPct), represent r
r
, and rc;Th-232 reprethe predictor variables, and k1 ; rc;Pa-233 ; rcc;Th-232 ;U-238 c;Th-232 sent the response variables. A detailed description of all the parameters are presented in future sections. Predictive analysis involves the following steps: Firstly, fit a regression model to a set of trial data, and secondly, use the fit to predict the response for test input data in a desired confidence level. Trial data is obtained from the neutronics analysis using Serpent and test input data is provided by GA code. The regression model is built using a program called rpart (Therneau and Atkinson, 2000). 1.3.3. Genetic algorithms 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. 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, 1989; Davis, 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
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
A. Kumar, P.V. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx
‘‘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
3
stage using this new population, and the above steps are performed iteratively till the desired fitness is obtained. Fig. 1 presents a graphical view of GA (Kumar and Tsvetkov, 2015) that we have used in this research. Flow of the paper. The remainder of the paper is organized as follows. Firstly, we present the design and the approach for the analysis. This includes the Serpent model design, followed by GA input parameters and objectives. Then, we present the results with a discussion for each table and figure. Finally, we conclude with a summary and suggestions for future efforts. 1.4. Research flow The flow of the research and analysis is given in Fig. 2. A set of trial inputs (TRIALINP) is developed with a random combination of p ratio, UPct, and U235En. For each input set, output parameters D (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. 2. Design and approach 2.1. Neutronics model and analysis
Fig. 1. Genetic algorithms flow chart.
Parametric analysis has been done using an infinite pin-cell configuration of a hypothetical reactor core where, the criticality and burn-up calculations have been performed with a reflecting boundary condition. A reflecting boundary condition can be literally interpreted as an infinite number of pin-cells, that ensures that no neutron leaks out of the system. The infinite pin-cell configuration used in the analysis is given in Fig. 3. The pin-cell consists of an infinite configuration of fuel-blanket single fuel pins representing both the fuel assembly, and the blanket assembly regions at a
Fig. 2. Research flow.
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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Fig. 3. Pin-cell model used in the Th-fueled PHWR design.
coarser level within the whole core of a HWR. In the rest of the paper, a ‘‘fuel pin-cell’’ is composed of two fuel pins, and two blanket pins. In simpler words, our pin cell is a homogenized configuration of the assembly. For a steady state parametric analysis, with no depletion, and power deposition analysis, an assembly level spatial homogenized model is valid. In addition, we are not performing high fidelity assembly level and whole core analysis, hence a simplified homogeneous configuration would provide useful information about the parameters those affect breeding in PHWR fuel designs. The fuel pin (seed) consists of a cylindrical fuel rod element containing natural uranium oxide. This type of fuel pin is commonly used in the Indian PHWR design. The blanket pin contains Th-232 as the dominant fertile material. It is modeled under the same dimension of the seed with multiple variations of the fissile composition. The variation in the fissile composition is based on U235En, and UPct. Analysis has also been done with a variation of the DP ratio keeping the material composition constant. The core parameters analyzed in this work depend on the flux weighted effective one-group microscopic cross given by,
R
r /ðE; V ÞdEdV rx ¼ R x ; /ðE; V ÞdEdV
ð1Þ
where, x represent a specific material (U-235, U-238, Th-232, etc.), and an interaction (fission, capture, absorption, etc.) in the fuel or the blanket region, and /ðE; VÞ is the neutron energy spectrum in the evaluated region. These parameters are evaluated in the paper to give insights into the characteristics of the pin-cell configuration. For simplicity, and to suggest an area of improvement using existing design, we have assumed our seed region to have an identical composition with the fuel of an existing reactor design. The blanket has the same thickness and material composition for the gap and the cladding, as is the seed except that the material in the blanket is different. Enrichment in this blanket is even higher than the seed region. The blanket here gives us a room for improvement by adjusting the parameters DP , U235En and UPct.
3. Results and discussion 3.1. Neutronics This section presents the results from the simulation of the models, and provides a discussion with appropriate inherent
Fig. 4. Correlation between TRIALINP and OUTPUT.
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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A. Kumar, P.V. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx
assumptions. Values for the parameters obtained from the neutronics module is given in Fig. 4. The figure presents a correlogram (Kabacoff, 2011) of the correlations among the variables in the TRIALDATA. The lower triangle contains red smoothed best fit lines and, the blue ellipse depicts the confidence level. The upper triangle presents the same information using pie charts. A blue color represents a positive correlation, and a red color represents a negative correlation between the two variables that meet at the cell. The strength of the correlation is displayed by the size of the filled pie slice. Positive correlations fill the pie in a clockwise direction starting at 12 o’clock, and negative correlations fill the pie in a counter clockwise direction. Fig. 4 presents a macroscopic view of the effect on breeding based on the variations in DP ratio, UPct, r
and U235En. Except the ratio rf ;Th-232 , all other parameters are disc;Th-232 cussed in Figs. 5–8. With an increase in the DP ratio, due to the thermalization of neutrons there is more chances of capture of neutrons by Th-232 as compared to fission. Due to an increase in UPct there is an increasing trend towards fission by Th-232 as compared to capture. However, it is to be noted that the fission cross section for thorium is much smaller than the capture cross section. Fig. 5 presents the effect of kinf of the fuel pin-cell with respect to the variation in DP ratio, UPct, and U235En. In the figure we observe that, with an increase in DP ratio, kinf increases due to the thermalization of the neutrons. Due to an increase in the thermalization of neutrons there is an increase in the possibility of thermal fission in the core, because, fissile material in the fuel have a higher
Fig. 7. Growth of U-233 (Th-232 cross-section in the blanket, units: cm2).
Fig. 8. Ratio of
Fig. 5. Infinite neutron multiplication factor (kinf ).
rc;Th-232 and rc;U-238 .
fission cross-section in the thermal energy range. Within the scope of this research, the reactor model is in the under-moderated region. It is not reasonable to have a DP ratio greater that 3.5, as it is not economical in terms of the need for a higher pumping power of the coolant. Also, at a higher DP ratio the model operates in the under-moderated regime, which undermines the tendency of a nuclear reactor inherent safety feature, especially void moderator reactivity. No appreciable changes in kinf is observed with a variation in UPct. One of the important necessities of using Th-232 as a fertile material is the production of U-233. Th-232 transmutes to Th-233, which later produces U-233 through beta decays given by, 232
capture
b decay
b decay
Th þ 1 n ! 233 Th ! 233 Pa ! 233 U:
ð2Þ
Protactinium, Pa-233, the nuclide that separates Th-233 and U-233, has a longer half-life and a very high absorption cross-section that significantly increases the parasitic absorption, which is not desirable for a sustained criticality and breeding condition, hence not viable in an economic point of view. Therefore, it r is necessary to have a lower value for the ratio, rc;Pa-233 where, c;Th-232
rc;Pa-233 is the microscopic capture cross-section of Pa-233, and rc;Th-232 is the microscopic cross-section of Th-232. This ratio is
Fig. 6. Ratio of
rc;Pa-233 and rc;Th-232 .
used to evaluate the efficiency of Th-232 breeding in the presence of Pa-233. The effect of this ratio with respect to DP ratio, UPct, and U235En is presented in Fig. 6. We have observed that for any
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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Fig. 9. Neutron capture cross-section of Pa-233, units:- cm2.
combination of DP ratio, UPct, and U235En, Pa-233 has a superior absorption towards the fertile nuclide in the blanket. Analogous to the previous trend, harder spectrum seems to be the most benign environment to grow U-233. With an increase in enrichment, capture by Pa-233 is more prominent as compared to Th-232. In Fig. 7, we have plotted the effect of DP ratio, UPct, and U235En, on the rc;Th-232 in the blanket. Softening of the neutron energy spectrum adversely affects capture by Th-232. This is also seen from the cross-section tables. Lower U235En with higher UPct is more favorable for Th-232 capture as compared to higher enrichment and lower UPct. However, if compared with the production of Pu-239, U-233 is more likely to be produced in the thermal region. This can be inferred from Fig. 8. With softening of the neutron energy spectrum, capture by Th-232 is more favorable as compared to U-238. Higher U235En with lower UPct is more favorable for capture by U-238 as compared to lower U235En with higher UPct. Interestingly, from Fig. 9 we observe that, a higher U235En, and a lower UPct in the mixed fuel leads to less aggressive Pa-233 parasitic absorption, due to the flux depression from the presence of fissile material where Pa-233 has a high cross-section at this energy range. Neutron spectrum comparison depicted in Fig. 10 shows that both the configurations DP : 2:5; 3:5 have a strong thermal reactor
Fig. 11. Capture cross-section as a function of energy. Obtained by JANIS 4.0.
characteristics with a recognizable thermal peak. The difference occurs in the resonance region where a higher flux is expected with respect to a decrease in a lower DP ratio. Since the effective microscopic cross-section highly depends on the neutron spectrum, this could be the factor that increase parasitic absorption in Pa-233. At the early range of resonance energy, the value of relative microscopic capture cross-section of Pa-233-to-Th-232 is high as presented Fig. 11. Therefore, in this case, softer spectrum would likely give lower parasitic absorption of Pa-233. Hence, we desire to have a higher value for k1 to obtain more neutrons for breeding, minimize the r ratio rc;Pa-233 to ensure limited parasitic absorption by Pa-233, maxc;Th-232
r
, and imize rcc;Th-232 ;U-238
rc;Th-232 to breed higher amount of U-233.
3.2. Genetic algorithms 3.2.1. Input parameters The input parameters as defined in Section 2.1, are DP ratio, U235En, and UPct. We have used 16 bits for each of the variables. The conversion formula and the ranges for each of the variables are given in Table 1, 3.2.2. Optimization objectives We have identified the following objectives for optimization. Maximize k1 to obtain more neutrons for breeding, minimize rc;Pa-233 to ensure limited parasitic absorption by Pa-233, maximize r c;Th-232
rc;Th-232 rc;U-238 ,
and
rc;Th-232 to breed higher amount of U-233. In simpler
Table 1 Input value ranges fo the design parameters. Parameter
Range
p D
2:5 ! 3:5
ratio U235En UPct
Units
10 ! 20 40 ! 20
% %
Table 2 Optimization objectives.
Fig. 10. Neutron spectrum comparison of two different pin-cell configuration. Green is the pin-cell configuration with a DP ratio of 2.5, and red is the pin-cell configuration with a DP ratio of 3.5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Parameter
Objective
Objective Id (o)
Weight (wo )
k1
maximize minimize
OBJ1 OBJ2
0.25 0.25
maximize
OBJ3
0.25
maximize
OBJ4
0.25
rc;Pa-233 rc;Th-232 rc;Th-232 rc;U-238
rc;Th-232
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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A. Kumar, P.V. Tsvetkov / Annals of Nuclear Energy xxx (2015) xxx–xxx
Fig. 12. Input parameters generated by GA.
words, we determine the optimized combination of DP ratio, U235En, and UPct to meet the above mentioned objectives. The optimization objectives are given in Table 2. Total fitness of the system is the sum of the fitness contribution from all the objectives. Total fitness F ðsÞ is given by,
F ðsÞ ¼
O X f o ðsÞ wo ;
Table 4 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 )
20 16 10
ð3Þ
o¼1
where s is a chromosome, O is the total number of objectives, f o ðsÞ is the contribution of a specific objective, o to the total fitness of chromosome s, and wo is the weight of a specific objective o. It is to be P noted that wo is positive and Oo¼1 wo ¼ 1. 3.2.3. Results Fig. 12 presents the input parameters for Dp ratio, UPct, and U235En in five generations. The x-axis represents the input set in a specific generation. To present the effectiveness of GA we have presented only the first five generation, however for the optimized solution in our case we had ten generations. The y-axis represents the parameter value. It can be inferred from the figure that with generations, the input parameters become flat across multiple
Table 3 Optimized solution. Parameter p D
ratio U235En UPct
Value
Units
3.31 11.25 35.55
% %
chromosomes (input sets). This phenomenon is expected from GA, where every generation makes GA learn more about the system and, hence the inputs generated by GA are closer to the optimized input set. The optimized solution is given in Table 3. Table 4 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.
4. Conclusion and future work A parametric study of thorium-fueled heavy water moderated reactor fuel was performed. The analysis demonstrated the characteristic of thorium based reactor under HWR spectrum. Based on the results, within the range of the variation conducted in this paper, softer spectrum was needed to minimize the parasitic absorption in the blanket, but harder spectrum tends to give a higher breeding ratio. We obtained an optimized solution for Dp ratio, UPct, and U235En. We demonstrated the use of multivariate regression analysis and GA for optimization, which is one of the major objectives of this work. Future work involves detailed
Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006
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Please cite this article in press as: Kumar, A., Tsvetkov, P.V. Optimization of U–Th fuel in heavy water moderated thermal breeder reactors using multivariate regression analysis and genetic algorithms. Ann. Nucl. Energy (2015), http://dx.doi.org/10.1016/j.anucene.2015.07.006