Nghiên cứu tối ưu thay đảo nhiên liệu lò phản ứng hạt nhân VVER 1000 TT TIENG ANH

MINISTRY OF EDUCATION AND TRAINING MINISTRY OF SCIENCE AND TECHNOLOGY VIETNAM ATOMIC ENERGY INSTITUTE TRAN VIET PHU STUDY ON FUEL LOADING PATTERN OPTIMIZATION FOR VVER-1000 NUCLEAR REACTOR Major: Nuclear and Atomic Physics Code: 9.44.01.06 SUMMARY OF DOCTORAL DISSERTATION OF PHYSICS Hanoi - 2022 Contents List of Abbreviations 1 Introduction 2 Methods and development 2.1 VVER-1000 MOX core benchmark 2.2 Data preparation for core calculations 2.3 Development of LPO-V code for core physics calculations 2.3.1 Steady-state multi-group diffusion equations 2.3.2 Finite difference method for spatial discretization 2.3.3 Boundary conditions 2.3.4 Successive over-relaxation method 2.3.5 Core modeling by LPO-V code 2.3.6 Verification of core calculations 2.4 Development of ESA method 2.4.1 SA and ASA methods 2.4.2 ESA method 2.5 Development of a discrete SHADE method 2.5.1 Classics Differential Evolution (DE) 2.5.2 SHADE operators 2.5.3 Success-history based adaptation 2.5.4 Discrete SHADE method 2.6 Fitness functions 2.7 Mann-Whitney U Test i 4 5 6 7 8 9 9 10 11 12 13 Author: Tran Viet Phu Loading pattern optimization of VVER-1000 reactor 3.1 LP optimization of VVER-1000 reactor using ESA method 3.1.1 Selection of ESA method 3.1.2 Comparison among SA, ASA and ESA 3.1.3 LP optimization of the VVER-1000 MOX core using ESA method 3.2 LP optimization of VVER-1000 reactor using SHADE method 3.2.1 Determination of control parameters 3.2.2 LP optimization of the VVER-1000 MOX core using SHADE method 3.3 Optimal core loading pattern of SHADE and ESA 13 14 14 14 Conclusions and future work 4.1 Conclusions 4.2 Future works 20 20 21 REFERENCES 22 ii 16 17 17 17 19 List of Abbreviations ASA BC DE ENDF ESA FDM GA ICFM kef f LP LPO-V MCNP MOX PPF RPI SA SHADE SOR VVER Adaptive Simulated Annealing Boundary Condition Differential Evolution Evaluated Nuclear Data File Evolutionary Simulated Annealing Finite Difference Method Genetic Algorithms In-core Fuel Management Effective Multiplication Factor Loading Pattern Loading Pattern Optimization of VVER Monte Carlo N-Particle Mixed Oxide Fuel Power Peaking Factor Relative Position Indexing Simulated Annealing Success-History based Adaptive Differential Evolution Successive Over-Relaxation Vodo-Vodyanoi Energetichesky Reaktor Chapter Introduction The issue of the In-core Fuel Management (ICFM) problem is that the search space is too large, so the developed optimization search processes can not confirm that the final found solution is global optimal one Therefore, the current optimization studies focus on improving the convergence speed of the search process in order to find better and better solutions with the same number of trials This study will address the issue by considering, improving and applying advanced optimization methods to enhance the convergence speed for solving the LP optimization problem The two objectives of this dissertation are as follows: Investigation of advanced optimization methods to the problem of loading pattern optimization of nuclear reactors Development of a calculation tool for optimizing the loading pattern for the VVER reactor In this dissertation, such following works have been concerned: ❼ Chapter describes briefly overview of the nuclear reactor technologies, LP optimization problem and optimization methods, and the objectives of the dissertation ❼ Chapter presents the methods and developments of advanced optimization methods and a core physics code The VVER-1000 MOX core is used to illustrate the application of the advanced optimization methods for solving the LP optimization of VVER reactors A core physics code, LPO-V, has been developed based on finite difference method for solving multi-group diffusion equations in triangular meshes to calculate neutronic characteristics of VVER reactor, and to evaluate the fitness functions of the LP optimization problem Verification calculations shows that the code has guaranteed accuracy and fast calculation speed, suitable for the requirements of the LP optimization problem Two advanced methods have been developed: Evolutionary Simulated Annealing (ESA) method and Success-History based Adaptive Differential Evolution (SHADE) method ESA is an improvement of original SA method by using crossover operator similar to GA SHADE has been proven as a high performance advanced method with many optimization problems In this dissertation, a discrete SHADE has been developed and applied to the LP optimization problem Finness functions based on power distribution, ke f f and PPF have been used to evaluate LPs to find out the optimal one A Mann-Whitney U test also applied to compare the efficiency of the optimization methods ❼ Chapter presents the numerical calculations for VVER-1000 MOX core applying ESA and SHADE methods The results of ESA method were compared with SA and ASA Statistical differences between these methods were also evaluated based on the Mann-Whitney U test The results show that the ESA method is advantageous over the SA and ASA The results of discrete SHADE method is also comparable with the ESA method The found optimal LP by the two method are identical The results show that the kef f of the optimal LP is greater than that of the reference core by about 1580 pcm Whereas, the radial power peaking factor (P P F ) of the optimal LP is about 2.4% smaller than that of the reference core ❼ Chapter summarizes the results of this dissertation and provides some future plans Chapter 2: Methods and development Chapter Methods and development 2.1 VVER-1000 MOX core benchmark In this research, numerical calculations for LP optimization have been performed based on a VVER-1000 benchmark core loaded with 30% MOX fuel The VVER-1000 benchmark core was proposed by OECD/NEA for studying the neutronics performance of a mixed UO2 -MOX core, and verifying computational codes and methods [1] The reference 1/6th core configuration consists of 28 fuel assemblies, including 19 UO2 fuel assemblies and MOX fuel assemblies [1] 2.2 Data preparation for core calculations The PIJ module of the SRAC code was used for lattice calculations of fuel and non-fuel bundle using the collision probability method From the lattice calculations, a set of multi-group macroscopic cross-sections (group constants) of the fuel and non-fuel bundle is obtained and will be used for core calculations using LPO-V code For the VVER-1000 assemblies, a hexagonal model with 60o rotational symmetry has been applied Chapter 2: Methods and development 2.3 Development of LPO-V code for core physics calculations 2.3.1 Steady-state multi-group diffusion equations Diffusion equation is a combination of equation of continuity and Fick’s law [2] In the case of multi-group in nuclear reactor, the diffusion equations are written for each g − group as follows: G −▽(Dg ▽ϕg ) + Σa,g ϕg + G Σg→h ϕg = h=1,h̸=g Σh→g ϕh h=1,h̸=g + χg kef f (2.1) G νΣf,h ϕh h=1 where D is diffusion coefficient; ϕ is neutron flux; Σa is macroscopic absorption cross section; Σg→h is scattering cross section from g − group to h − group groups; ν is an average number of neutrons released per fission; Σf is fission macroscopic cross section; kef1 f is a constant factor; χg is the probability that a fission neutron will be produced with an energy in g − group 2.3.2 Finite difference method for spatial discretization Finite difference method (FDM) is usually applied to solve the diffusion equations In the FDM, the core region is divided into meshes and the flux in each mesh point only directly related to the fluxes in its neighbours Fig 2.1 presents triangular mesh model and a sample of neighbour meshes in the FDM The form of FDM equation for a triangular mesh is derived from the multi-group diffusion equation Eq (2.1) as follows: ag,i,i−1 ϕg,i−1 + ag,i,i ϕg,i + ag,i,i+1 ϕg,i+1 + ag,i,k ϕg,k = Sg,i (2.2) or a matrix equation form: AΦ = S (2.3) where ag,i,k are factors derived from the group constants of meshes i, k; Sg,i is neutron source term for g −group in the mesh i, that includes the neutron fission source term and neutron scattering from other groups to g − group Chapter 2: Methods and development Figure 2.1: 2D triangular mesh (a) and mesh’s neighbours (b) in the FDM The boundary conditions were applied to the boundary meshes to form the final system of linear equations The results include eigen value kef f and neutron flux distribution ϕg,i One can obtain power distribution by multiplying ϕi and the fission cross-section Σf,g,i 2.3.3 Boundary conditions Three boundary condition (BC) was applied to solve the diffusion equation in nuclear reactor Free surface BC is usually applied to outer surface of the reactor core, while reflective BC and periodic BC are applied when the reactor is simulated with considering reactor symmetry The free surface BC assumes that the neutron flux is zero at a small distance beyond the surface ϕg (rB + dg ) = (2.4) where rB is boundary and dg is extrapolation distance of group g In most cases, the dg is given dg = 2.13Dg [2] The reflective BC is given by zero neutron current density in all energy groups at reflective surface (ref_sur): J(ref _sur) = (2.5) The periodic BC is given by the neutron flux and the neutron current density at two periodic surface (per_sur_1 and per_sur_2) are identical: ϕ(per_sur_1) = ϕ(per_sur_2) and J(per_sur_1) = J(per_sur_2) 2.3.4 (2.6) Successive over-relaxation method Iterative methods are used as a way of of solving the Eq (2.3) indirectly when the matrix A becomes very large in the reactor simulation In Chapter 2: Methods and development Figure 2.2: VVER-1000 core model with 24 triangular meshes per assembly in the LPO-V code this study, Successive over-relaxation (SOR) method was used to solve the system of linear equations Eq (2.3) [3] The equation of the iterative SOR method is written as follows: Φ(k) = Φ(k−1) + ω LΦ(k) + U Φ(k−1) + Q − Φ(k−1) 2.3.5 (2.7) Core modeling by LPO-V code The 1/6th core model is simulated using an in-house code for Loading Pattern Optimization for VVER reactors (LPO-V) The LPO-V code is developed based on multi-group diffusion theory for hexagonal geometry systems The code consists of a core physics module and a search module for LP optimization Fig 2.2 displays the 1/6th core model of the VVER1000 MOX core with 24 triangular meshes per fuel assembly in the LPO-V code Four-group cross section set of the fuel assemblies was prepared from lattice calculations using the PIJ code of the SRAC2006 code system and the ENDF/B-VII.0 data library [4] The cross section set was then used in the LPO-V code for core physics calculations and LP optimization 2.3.6 Verification of core calculations Verification of the core calculations using the LPO-V code has been conducted based on the reference configuration of the VVER-1000 MOX Chapter 2: Methods and development given number of successive updated base LPs, the current best LP is reused as a base LP ❼ Restriction LP lists: The numbers of the nearest calculated trial LPs and base LPs are archived in restriction lists If the new trial LP is included in the lists, a new trial LP is regenerated 2.4.2 ESA method Evolutionary SA (ESA) method has proposed A new approach for generating trial LPs using crossover and mutation operators to replace the binary/ternary exchange in the SA methods Two base LPs are used as the parents to generate an offspring by a crossover Then, a new trial LP is generated from the offspring by performing the mutation Combining the two crossovers and the five approaches for selecting the base LP and the mother, one obtains ten versions of the ESA method, and denoted as ESA–Ci Aj with i = {1, 2} and j = {1, , 5} Numerical investigation has been performed to compare and to select the most suitable one for the problem of LP optimization 2.5 2.5.1 Development of a discrete SHADE method Classics Differential Evolution (DE) Starting from a random initial population, DE generates next evolutionary population by applying mutation, crossover and selection operators A population (P ) in the DE algorithm is a set of individual vectors xi = (x1,i , , xD,i ) with i = 1, , N P Where, D is dimension of the solutions, and N P is the population size In generation G, a set of N P trial vectors are generated from the current population via the mutation and the crossover Then, a new generation G + is updated by a selection operator 2.5.2 SHADE operators Mutation SHADE operators are similar to that of the DE algorithm, but the mutant scale factor F and the crossover ratio CR are determined automatically by an adaption mechanism In the present work, DE/current−to−pbest/1 Chapter 2: Methods and development strategy with an external archive set is used for the mutation of the SHADE method The formula of the mutation is written as: vi,G = xi,G + Fi × (xpbest,G − xi,G ) + Fi × (xr1,G − xr2,G ), (2.8) where vi,G is the mutant vector in generation G; Fi is the mutant scale factor for the individual xi , which is determined automatically by an adaptation mechanism; xpbest,G is randomly selected from N P ×p (p ∈ (0, 1)) top members of the population PG of the generation G; xr1,G is randomly selected in the population PG in the generation G; xr2,G is randomly selected in PG ∪ A, in which A is the external archive set The selections of xpbest,G , xr1,G and xr2,G are required to satisfy the condition of r1 ̸= r2 ̸= pbest Crossover When a mutant vector is generated by the mutation operator, the crossover is performed to generate a trial vector In the crossover, a mutant vector vi,G is crossed with its parent xi,G to generate a trial vector ui,G A common binomial crossover is used as follows: uj,i,G = vj,i,G if rand[0, 1) ≤ CRi or j = jrand xj,i,G otherwise , (2.9) where, uj,i,G with j = {1, , D} is an element of a trial vector ui,G ; rand[0, 1) is a random number selected from (inclusive) to (exclusive); jrand is a random integer in the range of [1, D]; and CRi ∈ [0, 1] is the crossover ratio, which is determined by an adaptation mechanism Selection When all trial vectors ui,G with i = {1, , N P } in the generation G have been generated, the selection operator is performed to determine the population of next generation G + In general, the selection compares each xi,G with its corresponding trial ui,G to keep the better one for the next generation G + as: xi,G+1 = ui,G if f (ui,G ) ≥ f (xi,G ) xi,G otherwise , (2.10) where, f (ui,G ) is the fitness function value of vector ui,G 2.5.3 Success-history based adaptation In the SHADE method, the parameters F and CR are determined by a success-history based adaptation mechanism During the search pro10 Chapter 2: Methods and development cess, SHADE maintains two historical sets with H entries for the control parameters F and CR, denoted as MF and MCR , respectively In each generation, the Fi is determined according to Cauchy distribution with a location parameter MF,ri and a scale parameter of 0.1 as follows: Fi = randci (MF,ri , 0.1) (2.11) Similarly, the crossover ratio CRi is generated by a normal (Gaussian) distribution with mean MCR,ri and a standard deviation of 0.1 as follows: CRi = randni (MCR,ri , 0.1) (2.12) In a generation, the values of Fi and CRi , which are used to generate a succeeding trial vector ui,G better than the parent xi,G in Eq (2.10), are recorded to the two historical sets SF and SCR , respectively At the end of the generation, the contents of the historical sets are updated as follows: MF,j,G+1 = meanW L (SF ) if SF ̸= ∅ , MF,j,G otherwise (2.13) and MCR,j,G+1 = meanW A (SCR ) if SCR ̸= ∅ MCR,j,G otherwise (2.14) In Eqs (2.13) and (2.14), meanW A is the weighted mean, and meanW L is weighted Lehmer mean, which are calculated as [5]: meanW L = |SF | k=1 wk |SF | k=1 wk × SF,k × SF,k |SCR | wk × SCR,k , and meanW A = (2.15) k=1 where, wk = 2.5.4 | f (uk,G ) − f (xk,G ) | |SF | i=1 (2.16) | f (ui,G ) − f (xi,G ) | Discrete SHADE method In the problem of fuel LP optimization, the components of the vector are integers Therefore, a relative position indexing (RPI) approach has been implemented for converting the real variables into integer ones and 11 Chapter 2: Methods and development preserving the number of fuel types in the core configuration [6] In the RPI approach, the real components of a vector are sorted in ascending order Then, the integer components are generated by assigning the values equal to their orders in the real vector If two or more components receive equal values, random numbers are generated and added to the component to reorder them In the Discrete SHADE method, the RPI is used to discretize the mutant vector 2.6 Fitness functions In the present work, two problems of fuel LP optimization have been conducted with the use of two OFs to evaluate the performance of the optimization methods The first problem with the fitness function F aims at comparing the possibility of ESA with SA and ASA in reproducing a reference LP, i.e., reproducing the power distribution of the reference LP [7] The F is written as follows: N i i |Ptrial − Pref |), F = −w × ( (2.17) i=0 i i where, Ptrial and Pref are the relative power densities of the ith assembly in the trial and reference LPs, respectively w is a coefficient, which is chosen as w = 1/N = 0.0357 N = 28 is the total number of fuel assemblies in the 1/6th VVER-1000 core The second problem with the fitness function F is to optimize the LP of the VVER-1000 core by maximizing the kef f at the BOC, flattening the power distribution and ensuring the constraint of PPF The formula of F is written as follows: F = kef f − wp × max(0, P P F − P P F0 ) − wf × F latness, (2.18) The last term in Eq (2.18) represents the flatness of power distribution, which is written as: N i |Ptrial − 1| F latness = (2.19) i=0 where, P i is the relative power density of ith assembly P P F0 = 1.45 is the constraint value of P P F wp and wf are the weighting factors related to the P P F and F latness terms wp = 2.5 and wf = 0.006 were chosen for optimizing the LP of the VVER-1000 benchmark core 12 2.7 Mann-Whitney U Test Statistical test was also performed using the Mann-Whitney U Test to evaluate statistically significant difference between two independent samples [8] In the Mann-Whitney U Test, the null hypothesis states that there is no tendency for the optimal OF values of one method to be significantly different (better or worse) to that of the other The important result of the statistical test is the probability value, denoted as P -value If the null hypothesis is accepted, i.e P > threshold, the two methods are not significantly different In the case that the null hypothesis is rejected (P < threshold), the two methods are significantly different Then, the sum of ranks, R1 and R2 , of the two methods are compared to determine the better one corresponding to the larger value In this study, the Mann-Whitney U Test has been applied with two-tailed hypothesis and significance threshold of 0.05 Chapter Loading pattern optimization of VVER-1000 reactor Chapter presents LP optimization calculations for the VVER-1000 MOX core using ESA and discrete SHADE methods The calculations using each method were performed with 50 independent runs With the ESA method, 10 different versions were surveyed to select the best version for the LP optimization problem, that was compared with SA and ASA In discrete SHADE method, control parameters H and NP were surveyed to find out the appropriate values The discrete SHADE method was then applied to find out the optimal LP for the VVER-1000 MOX core and compared with the ESA and DE methods Comparison with reference core has also been presented 13 Chapter 3: Loading pattern optimization of VVER-1000 reactor 3.1 3.1.1 LP optimization of VVER-1000 reactor using ESA method Selection of ESA method A calculation survey has been conducted to compare the performance of the ten versions the ESA method using the two fitness functions F and F Table 3.1 summarizes the average values of F and F 2, their statistic standard deviations and the P -values obtained by comparing the ESA-C1A4 with other ESA approaches Combining all comparisons, ESA-C1A4 is selected as the best ESA method for further application and comparison with SA and ASA Table 3.1: Comparison of ten approaches of the ESA method using the two fitness functions Method ESA-C1A1 ESA-C1A2 ESA-C1A3 ESA-C1A4 ESA-C1A5 ESA-C2A1 ESA-C2A2 ESA-C2A3 ESA-C2A4 ESA-C2A5 3.1.2 fitness function Standard deviation F1 -0.0612 -0.0665 -0.0425 -0.0389 -0.0459 -0.0594 -0.0544 -0.0385 -0.0372 -0.0421 F1 0.0256 0.0274 0.0153 0.0152 0.0176 0.0296 0.0280 0.0183 0.0147 0.0163 F2 1.1086 1.1102 1.1106 1.1142 1.1112 1.1104 1.1101 1.1130 1.1133 1.1128 F2 0.0046 0.0043 0.0085 0.0023 0.0084 0.0036 0.0078 0.0029 0.0020 0.0028 P-value F1 0.000 0.000 0.195 0.015 0.000 0.000 0.499 0.295 0.243 F2 0.000 0.000 0.000 0.003 0.000 0.000 0.054 0.029 0.004 Comparison among SA, ASA and ESA To compare the performance of SA, ASA and ESA, the search processes using the three methods were firstly performed with the fitness function F for reproducing a reference LP The average results of the 50 independent runs are summarized in Table 3.2 This result indicates the better performance of ESA in reproducing a reference LP compared to SA and ASA The results of the Mann-Whitney U Test show that ESA is better than SA in all cases, and ESA is better than ASA in the cases of α = 0.90 and 0.95 (Table 3.4) 14 Chapter 3: Loading pattern optimization of VVER-1000 reactor Table 3.3 shows the results of the SA, ASA and ESA methods with the fitness function F One can see that the F values of ESA are greater than that of SA and ASA with smaller standard deviations, at the same time the number of calculated LPs in ESA are smaller than SA and ASA by about 5–10% in most cases The the statistical test in Table 3.4 show that ESA is significantly better than SA and ASA in most cases The results indicate that the new generation of trial LPs via the crossover and mutation could improve the performance of the ESA method in the LP optimization problem of VVER-1000 core Table 3.2: Comparison of the SA, ASA and ESA methods with the use of fitness function F in reproducing a reference LP Parameters F1 No of calculated LPs Standard deviation No of optimal findings α 0.85 0.9 0.95 0.85 0.9 0.95 0.85 0.9 0.95 0.85 0.9 0.95 SA -0.0535 -0.0476 -0.0745 4033 5860 9110 0.0168 0.0129 0.0438 0 ASA -0.0456 -0.0427 -0.0387 3949 5694 10575 0.0137 0.0154 0.0206 0 ESA -0.0424 -0.0349 -0.0281 3576 4999 9647 0.0167 0.0165 0.0198 1 12 Table 3.3: Comparison of the SA, ASA and ESA methods with the use of fitness function F α F2 SA ASA 0.85 1.1126 1.1118 0.9 1.1122 1.1137 0.95 1.1093 1.1136 *0.95 1.1140 1.1141 *: Additional criterion is set No of calculated LPs Standard deviation ESA SA ASA ESA SA ASA ESA 1.1139 5636 5510 5151 0.0022 0.0059 0.0022 1.1141 8299 8290 7703 0.0025 0.0021 0.0018 1.1153 12484 14266 14643 0.0065 0.0023 0.0016 1.1152 16694 15843 15108 0.0020 0.0019 0.0018 to stop the calculation loop if the best LP does not change after 10000 trial solutions 15 Chapter 3: Loading pattern optimization of VVER-1000 reactor Table 3.4: Comparison of the SA, ASA and ESA methods with fitness function F and F using the Mann-Whitney U Test P -value F1 α P -value F2 SA-ASA SA-ESA ASA-ESA SA-ASA SA-ESA ASA-ESA 0.85 0.719 0.006 0.011 0.005 0.001 0.638 0.9 0.003 0.000 0.267 0.156 0.000 0.016 0.95 0.000 0.000 0.000 0.000 0.000 0.003 0.95* 0.920 0.001 0.001 *: Additional criterion is set to stop the calculation loop if the best LP does not change after 10000 trial solutions 1 P P F 1 F la tn e s s k e ff 1 k e ff P P F 1 0 0 0 0 0 0 0 0 0 N u m b e r o f c a lc u la te d L P s (a) 0 0 0 0 0 0 0 0 0 N u m b e r o f c a lc u la te d L P s (b) Figure 3.1: Evolution of the kef f and P P F (a), and F latness (b) obtained by the ESA with the use of fitness function F 3.1.3 LP optimization of the VVER-1000 MOX core using ESA method The ESA method was applied to LP optimization of VVER-1000 reactor The numerical calculations were performed with values of α as 0.95 and the additional criterion as 10000 Fig 3.1 shows the evolution of the average objective parameters such as kef f , P P F and F latness of the VVER-1000 MOX core during the ESA search process At the end of the search, the P P F is smaller than the restriction and the kef f is better than that of the reference one [1] Besides, the F latness term in F is equivalent to that of the reference LP 16 Chapter 3: Loading pattern optimization of VVER-1000 reactor O p tim a l fitn e s s 1 1 1 1 1 H S e le c te d p o in t 1 2 3 4 5 N P Figure 3.2: A survey of control parameters N P and H The values of N P = 20 and H = were selected for further optimization process 3.2 3.2.1 LP optimization of VVER-1000 reactor using SHADE method Determination of control parameters A survey has been conducted to determine the control parameters N P and H , in which, N P was varied from 10 to 50 with a step of 5, and H was varied from to 10 Fig 3.2 shows the dependence of the average optimal fitness as a function of N P and H The selection of N P and H is primarily based on the maximization of the fitness function From Fig 3.2, the values of N P = 20 and H = have been selected for further optimization using the SHADE method 3.2.2 LP optimization of the VVER-1000 MOX core using SHADE method The discrete SHADE method was applied to LP optimization of VVER1000 reactor with the fitness function F Fig 3.3 (a) depicts the evolution of the average fitness function with the number of generations The results show that the advantage of the individuals selected in each generation compared to the trial solutions Also, the fitness function of the best solutions is usually better that that of the individuals Fig 3.3 (b) displays the variation of the mutant scale F and the crossover ratio CR determined based on the adaptation mechanism during 17 Chapter 3: Loading pattern optimization of VVER-1000 reactor Figure 3.3: Evolution of the fitness function (a) and Evolution of F and CR (b) with the number of generations The value is taken as the average of 50 independent runs the search process It can be seen that the value of F varies between 0.5 and 0.7 Whereas, CR varies between 0.25 and 0.5 The values can be used as a suggestion for other DE variants which use the mutant scale F and the crossover ratio CR as the predefined control parameters Table 3.5 shows the comparison of the best LPs and the average objective parameters obtained by the three methods It was found that the three methods could find the same optimal LP based on the same number of calculated LPs The P-values are all greater than 0.05, which indicate that the three methods exhibit comparable performance in the problem of LP optimization of the VVER-1000 MOX core The results imply that when the control parameters of F and CR are selected appropriately, the performance of DE is comparable to that of SHADE Table 3.5: Comparison of SHADE with DE and ESA Average fitness Average kef f Average PPF Average Flatness Average No of calculated LPs Fitness of the best LP P-value in comparison with SHADE 18 SHADE 1.11479 1.14945 1.41508 5.77763 15000 1.11727 - DE 1.11452 1.14801 1.41526 5.58076 15000 1.11727 0.34722 ESA 1.1152 1.15159 1.42268 6.06508 15108 1.11727 0.40654 Figure 3.4: Optimal core LP (a) and its relative radial power distribution (b) in comparison with the reference one 3.3 Optimal core loading pattern of SHADE and ESA Table 3.5 shows the optimal LPs found by the ESA and the SHADE are identical Fig 3.4 presents the optimal LP and its radial power distribution of the VVER-1000 MOX core Table 3.6 shows a comparison of the objective parameters such as the kef f , PPF and F latness between the optimal LP and the reference core A significant improvement of the kef f value of the optimal LP is obtained, which is greater than than the reference value by about 2087 pcm The P P F of the optimal LP is 1.443, which is smaller than the reference value (1.478) by about 2.4% Whereas, the flatness of the optimal LP is obtained as 6.870, slightly higher than the reference one (6.367) Table 3.6: Comparison of the optimal LP obtained from the search process with the reference one kef f PPF Flatness Reference 1.13762 1.478 6.367 19 Optimal LP 1.15849 1.443 6.870 Chapter 4: Conclusions and future work Chapter Conclusions and future work 4.1 Conclusions In this dissertation, the studies aim at developing advanced metaheuristics methods, i.e ESA and SHADE, and applying to the ICFM problem of the VVER-1000 reactor The following studies have been carried out: (1) Development of the LPO-V code, has been conducted to calculate the neutronic characteristics of VVER reactors with fast computational speed and acceptable accuracy This code is coupled with the newly developed optimization methods for solving the ICFM problem of VVER-1000 reactor (2) Development of advanced optimization methods, ESA and SHADE, has been conducted and applied successfully to the LP optimization problem of VVER-1000 reactor The ESA is an improved version of the original SA method, that was proposed by the PhD student Instead of using binary/ternary exchange operator to generate a new trial LP as in the SA, the ESA used a crossover of two base LPs for generating a new trial LP The SHADE method is an advanced version of the DE method with the use of success-history based adaptation to determine the control parameters F and CR automatically The RPI approach was deployed to convert real vectors to integer vectors in the discrete SHADE method This research is the first research to apply the SHADE method to the ICFM problem (3) Numerical calculations were performed for optimizing the LP of the reference VVER-1000 MOX core using the ESA and SHADE methods in comparison with other methods The comparison between ESA, SA and 20 Chapter 4: Conclusions and future work ASA show that the ESA is advantageous over SA and ASA in the problem of fuel LP optimization In comparison of ESA, DE and SHADE methods, the three methods have comparable performance However, the advantage of SHADE is that the adaptive mechanism simplifies significantly the determination of the control parameters compared to DE The optimal core LPs selected from the SHADE and ESA methods were similar This LP have a significant improvement of the kef f value, which is greater than that of the reference core by about 1580 pcm Whereas, the P P F is smaller than the reference value by about 2.4%, and the flatness values are approximate The results demonstrate that the ESA and the discrete SHADE methods have been successfully developed and applied to the fuel loading optimization of the VVER-1000 reactor 4.2 Future works (1) The LPO-V code is being upgraded to simulate 3D reactor with triangular and rectangular meshes and perform burn-up calculation (2) Further investigation of the ESA, SHADE methods and new advanced methods are being planned The extension of this study to multicycle optimization is also taken into account (3) Extension of the application of these methods to the LP optimization and core design for other reactor types is also being planned 21 References [1] E Gomin, M Kalugin, D Oleynik, VVER-1000 MOX Core Computational Benchmark, Specification and Results, Vol NEA/NSC/DOC(2005)17, 2006 [2] J R Lamarsh, A J Baratta, Introduction to Nuclear Engineering, third edition, Prentice-Hall, 2001 [3] J David M Young, Iterative methods for solving partial difference equations of elliptic type, Ph.D thesis, Havard University (1950) [4] K Okumura, T Kugo, K Kaneko, K Tsuchihashi, SRAC2006: A comprehensive neutronics calculation code system, Tech rep., JAEAData/Code 2007-004 Japan Atomic Energy Agency, Tokai, Japan (2007) [5] R Tanabe, A Fukunaga, Success-history based parameter adaptation for differential evolution, in: 2013 IEEE Congress on Evolutionary Computation, 2013, pp 71–78 [6] G C Onwubolu, D Davendra, Differential Evolution: A Handbook for Global Permutation-Based Combinatorial Optimization, Vol 175, Springer, Berlin, Heidelberg, 2009 doi:10.1007/978-3-540-92151-6 [7] T Smuc, D Pevec, B Petrovic, Annealing strategies for loading pattern optimization, Annals of Nuclear Energy 21 (6) (1994) 325–336 doi: 10.1016/0306-4549(94)90028-0 [8] G W Corder, D I Foreman, Nonparametric Statistics for NonStatisticians, John Wiley and Sons, Inc., 2009 22 Published during the dissertation Viet-Phu Tran, Giang T.T Phan, Van-Khanh Hoang, Haidang Phan, Nhat-Duc Hoang, Hoai-Nam Tran; Success-history based adaptive differential evolution method for optimizing fuel loading pattern of VVER1000 reactor; Nuclear Engineering and Design 377 (2021) 111125 Viet-Phu Tran, Giang T.T Phan, Van-Khanh Hoang, Pham Nhu Viet Ha, Akio Yamamoto, Hoai-Nam Tran; Evolutionary simulated annealing for fuel loading optimization of VVER-1000 reactor; Annals of Nuclear Energy 151 (2021) 107938 Viet-Phu Tran, Hoai-Nam Tran, Akio Yamamoto, Tomohiro Endo; Automated Generation of Burnup Chain for Reactor Analysis Applications; Kerntechnik, ISSN 0932-3902, 82 (2017 ) 196-205 Viet-Phu Tran, Hoai-Nam Tran, Van Khanh Hoang; Application of Evolutionary Simulated Annealing Method to Design a Small 200 MWt Reactor Core; Nuclear Science and Technology, ISSN 1810-5408, Vol 10, No (2020), pp 16-23 Nguyen Huu Tiep, Nguyen Thi Dung, Tran Viet Phu, Tran Vinh Thanh and Pham Nhu Viet Ha; Burnup calculation of the OECD VVER1000 LEU benchmark assembly using MCNP6 and SRAC2006; Nuclear Science and Technology, ISSN 1810-5408, Vol 8, No (2018), pp 1019 Tran Vinh Thanh, Tran Viet Phu, Nguyen Thi Dung; A study on the core loading pattern of the VVER-1200/V491; Nuclear Science and Technology, ISSN 1810-5408, Vol 7, No (2017), pp 21-27 Tran Viet Phu, Tran Hoai Nam; Discrete SHADE method for in-core fuel management of VVER-1000 reactor; 45th Vietnam Conference on Theoretical Physics (VCTP-45), 2020 (Poster) Viet-Phu Tran, Hoai-Nam Tran, Van Khanh Hoang; Application of Evolutionary Simulated Annealing Method to Design a Small 200 MWt Reactor Core; 6th Conference on Nuclear Science and Technology for young researcher, 08-09/10/2020 ... development 2.1 VVER- 1000 MOX core benchmark In this research, numerical calculations for LP optimization have been performed based on a VVER- 1000 benchmark core loaded with 30% MOX fuel The VVER- 1000 benchmark... 4 5 6 7 8 9 9 10 11 12 13 Author: Tran Viet Phu Loading pattern optimization of VVER- 1000 reactor 3.1 LP optimization of VVER- 1000 reactor using ESA method 3.1.1 Selection of ESA method ... assemblies in the 1/6th VVER- 1000 core The second problem with the fitness function F is to optimize the LP of the VVER- 1000 core by maximizing the kef f at the BOC, flattening the power distribution