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

TRAN VIET PHUSTUDY ON FUEL LOADING PATTERN OPTIMIZATION FOR VVER-1000 NUCLEAR REACTOR Major: Nuclear and Atomic Physics Code: 9.44.01.06 SUMMARY OFDOCTORAL DISSERTATION OF PHYSICS Hanoi

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 OFDOCTORAL DISSERTATION OF PHYSICS

Hanoi - 2022

2.1 VVER-1000 MOX core benchmark 4

2.2 Data preparation for core calculations 4

2.3 Development of LPO-V code for core physics calculations 5

2.3.1 Steady-state multi-group diffusion equations 5

2.3.2 Finite difference method for spatial discretization 5

2.3.3 Boundary conditions 6

2.3.4 Successive over-relaxation method 6

2.3.5 Core modeling by LPO-V code 7

2.3.6 Verification of core calculations 7

2.4 Development of ESA method 8

2.4.1 SA and ASA methods 8

2.4.2 ESA method 9

2.5 Development of a discrete SHADE method 9

2.5.1 Classics Differential Evolution (DE) 9

2.5.2 SHADE operators 9

2.5.3 Success-history based adaptation 10

2.5.4 Discrete SHADE method 11

2.6 Fitness functions 12

2.7 Mann-Whitney U Test 13

i

3 Loading pattern optimization of VVER-1000 reactor 13

3.1 LP optimization of VVER-1000 reactor using ESA method 14

3.1.1 Selection of ESA method 14

3.1.2 Comparison among SA, ASA and ESA 14

3.1.3 LP optimization of the VVER-1000 MOX core using

ESA method 16

3.2 LP optimization of VVER-1000 reactor using SHADE method 17

3.2.1 Determination of control parameters 17

3.2.2 LP optimization of the VVER-1000 MOX core using

SHADE method 17

3.3 Optimal core loading pattern of SHADE and ESA 19

4 Conclusions and future work 20

4.1 Conclusions 20

4.2 Future works 21

List of Abbreviations

ASA Adaptive Simulated Annealing

BC Boundary Condition

DE Differential Evolution

ENDF Evaluated Nuclear Data File

ESA Evolutionary Simulated Annealing

FDM Finite Difference Method

GA Genetic Algorithms

ICFM In-core Fuel Management

kef f Effective Multiplication Factor

LP Loading Pattern

LPO-V Loading Pattern Optimization of VVER

MCNP Monte Carlo N-Particle

MOX Mixed Oxide Fuel

PPF Power Peaking Factor

RPI Relative Position Indexing

The issue of the In-core Fuel Management (ICFM) problem is that thesearch space is too large, so the developed optimization search processes cannot confirm that the final found solution is global optimal one Therefore,the current optimization studies focus on improving the convergence speed ofthe search process in order to find better and better solutions with the samenumber of trials This study will address the issue by considering, improvingand applying advanced optimization methods to enhance the convergencespeed for solving the LP optimization problem The two objectives of thisdissertation are as follows:

1 Investigation of advanced optimization methods to the problem ofloading pattern optimization of nuclear reactors

2 Development of a calculation tool for optimizing the loading patternfor the VVER reactor

In this dissertation, such following works have been concerned:

ˆ Chapter 1 describes briefly overview of the nuclear reactor gies, LP optimization problem and optimization methods, and theobjectives of the dissertation

technolo-ˆ Chapter 2 presents the methods and developments of advanced mization methods and a core physics code The VVER-1000 MOXcore is used to illustrate the application of the advanced optimiza-tion methods for solving the LP optimization of VVER reactors A

opti-core physics code, LPO-V, has been developed based on finite ence method for solving multi-group diffusion equations in triangularmeshes to calculate neutronic characteristics of VVER reactor, and toevaluate the fitness functions of the LP optimization problem Ver-ification calculations shows that the code has guaranteed accuracyand fast calculation speed, suitable for the requirements of the LPoptimization problem.

differ-Two advanced methods have been developed: Evolutionary SimulatedAnnealing (ESA) method and Success-History based Adaptive Differ-ential Evolution (SHADE) method ESA is an improvement of orig-inal SA method by using crossover operator similar to GA SHADEhas been proven as a high performance advanced method with manyoptimization problems In this dissertation, a discrete SHADE hasbeen developed and applied to the LP optimization problem Finnessfunctions based on power distribution, kef f and PPF have been used

to evaluate LPs to find out the optimal one A Mann-Whitney U testalso applied to compare the efficiency of the optimization methods

ˆ Chapter 3 presents the numerical calculations for VVER-1000 MOXcore applying ESA and SHADE methods The results of ESA methodwere compared with SA and ASA Statistical differences between thesemethods were also evaluated based on the Mann-Whitney U test Theresults show that the ESA method is advantageous over the SA andASA The results of discrete SHADE method is also comparable withthe ESA method The found optimal LP by the two method areidentical The results show that the kef f of the optimal LP is greaterthan that of the reference core by about 1580 pcm Whereas, theradial power peaking factor (P P F ) of the optimal LP is about 2.4%smaller than that of the reference core

ˆ Chapter 4 summarizes the results of this dissertation and providessome future plans

Chapter 2

Methods and development

In this research, numerical calculations for LP optimization have beenperformed based on a VVER-1000 benchmark core loaded with 30% MOXfuel The VVER-1000 benchmark core was proposed by OECD/NEA forstudying the neutronics performance of a mixed UO2-MOX core, and veri-fying computational codes and methods [1] The reference 1/6th core con-figuration consists of 28 fuel assemblies, including 19 UO2 fuel assembliesand 9 MOX fuel assemblies [1]

2.2 Data preparation for core calculations

The PIJ module of the SRAC code was used for lattice calculations offuel and non-fuel bundle using the collision probability method From thelattice calculations, a set of multi-group macroscopic cross-sections (groupconstants) of the fuel and non-fuel bundle is obtained and will be used forcore calculations using LPO-V code For the VVER-1000 assemblies, ahexagonal model with 60o rotational symmetry has been applied

Chapter 2: Methods and development

cal-culations

2.3.1 Steady-state multi-group diffusion equationsDiffusion equation is a combination of equation of continuity andFick’s law [2] In the case of multi-group in nuclear reactor, the diffusionequations are written for each g − group as follows:

diffu-in the FDM The form of FDM equation for a triangular mesh is derivedfrom the multi-group diffusion equation Eq (2.1) as follows:

or a matrix equation form:

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 neutronfission source term and neutron scattering from other groups to g − group

Figure 2.1: 2D triangular mesh (a) and mesh’s neighbours (b) in the FDM.The boundary conditions were applied to the boundary meshes toform 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

Three boundary condition (BC) was applied to solve the diffusionequation in nuclear reactor Free surface BC is usually applied to outersurface of the reactor core, while reflective BC and periodic BC are appliedwhen the reactor is simulated with considering reactor symmetry

The free surface BC assumes that the neutron flux is zero at a smalldistance beyond the surface

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:

2.3.4 Successive over-relaxation method

Iterative methods are used as a way of of solving the Eq (2.3) rectly when the matrix A becomes very large in the reactor simulation In

indi-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 thesystem of linear equations Eq (2.3) [3] The equation of the iterative SORmethod is written as follows:

The 1/6th core model is simulated using an in-house code for LoadingPattern Optimization for VVER reactors (LPO-V) The LPO-V code isdeveloped based on multi-group diffusion theory for hexagonal geometrysystems The code consists of a core physics module and a search modulefor LP optimization Fig 2.2displays the 1/6th core model of the VVER-

1000 MOX core with 24 triangular meshes per fuel assembly in the LPO-Vcode Four-group cross section set of the fuel assemblies was prepared fromlattice calculations using the PIJ code of the SRAC2006 code system andthe ENDF/B-VII.0 data library [4] The cross section set was then used inthe 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 beenconducted based on the reference configuration of the VVER-1000 MOX

fuel benchmark core [1] Table2.1shows the comparison of the kef f valuesobtained from the LPO-V code in comparison with that obtained from theMCNP4c calculations presented in the benchmark report The comparisonresults imply that the two codes have a good agreement A calculationTable 2.1: Comparison of the kef f values calculated with the LPO-V andMCNP4c codes for five states (S1–S5) of the VVER-1000 MOX benchmarkcore.

speed test was also performed by comparing between LPO-V and TION module of the SRAC2006 code The computational time of 2000core LPs using the LPO-V code is about 340 s, which is faster than theCITATION module of the SRAC code system (1040 s) running on the samesystem with the same core models by a factor of 3.0 Thus, the LPO-V code

CITA-is suitable for the problem of fuel loading optimization with a large number

of calculated LPs

Simulated Annealing (SA) is originally based on the phenomenon ofcrystal vibration in annealing metal, which has been soon applied to fuelloading optimization of nuclear reactors In the SA method, a new trial

LP is generated from the base LP by binary or ternary exchange The SAmethod has ability to escape local optima by an acceptance probability of

a worse trial LP as new base LP

The Adaptive SA (ASA) is a advanced version of SA method TheASA was implemented by applying simultaneously two strategies as follows:

ˆ Return to the best: If the current best LP does not change after a

Chapter 2: Methods and development

given number of successive updated base LPs, the current best LP isreused as a base LP

ˆ Restriction LP lists: The numbers of the nearest calculated trial LPsand base LPs are archived in restriction lists If the new trial LP isincluded in the lists, a new trial LP is regenerated

Evolutionary SA (ESA) method has proposed A new approach forgenerating trial LPs using crossover and mutation operators to replace thebinary/ternary exchange in the SA methods Two base LPs are used as theparents to generate an offspring by a crossover Then, a new trial LP isgenerated from the offspring by performing the mutation

Combining the two crossovers and the five approaches for selectingthe base LP and the mother, one obtains ten versions of the ESA method,and denoted as ESA–CiAj with i = {1, 2} and j = {1, , 5} Numericalinvestigation has been performed to compare and to select the most suitableone for the problem of LP optimization

2.5.1 Classics Differential Evolution (DE)

Starting from a random initial population, DE generates next lutionary population by applying mutation, crossover and selection opera-tors A population (P ) in the DE algorithm is a set of individual vectors

evo-xi = (x1,i, , xD,i) with i = 1, , N P Where, D is dimension of the tions, and N P is the population size In generation G, a set of N P trialvectors are generated from the current population via the mutation and thecrossover Then, a new generation G + 1 is updated by a selection operator

strategy with an external archive set is used for the mutation of the SHADEmethod The formula of the mutation is written as:

where vi,Gis the mutant vector in generation G; Fiis the mutant scale factorfor the individual xi, which is determined automatically by an adaptationmechanism; xpbest,Gis randomly selected from N P ×p (p ∈ (0, 1)) top mem-bers 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, thecrossover is performed to generate a trial vector In the crossover, a mutantvector vi,G is crossed with its parent xi,G to generate a trial vector ui,G Acommon binomial crossover is used as follows:

(

where, uj,i,Gwith j = {1, , D} is an element of a trial vector ui,G; rand[0, 1)

is a random number selected from 0 (inclusive) to 1 (exclusive); jrand is arandom 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 Ghave been generated, the selection operator is performed to determine thepopulation of next generation G + 1 In general, the selection compareseach xi,G with its corresponding trial ui,G to keep the better one for thenext generation G + 1 as:

(

ui,G if f (ui,G) ≥ f (xi,G)

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

pro-Chapter 2: Methods and development

cess, SHADE maintains two historical sets with H entries for the controlparameters F and CR, denoted as MF and MCR, respectively

In each generation, the Fi is determined according to Cauchy bution with a location parameter MF,r i and a scale parameter of 0.1 asfollows:

(meanW L(SF) if SF ̸= ∅

and

(meanW A(SCR) if SCR̸= ∅

In the problem of fuel LP optimization, the components of the vectorare integers Therefore, a relative position indexing (RPI) approach hasbeen implemented for converting the real variables into integer ones and

preserving the number of fuel types in the core configuration [6] In theRPI approach, the real components of a vector are sorted in ascendingorder Then, the integer components are generated by assigning the valuesequal to their orders in the real vector If two or more components receiveequal values, random numbers are generated and added to the component toreorder them In the Discrete SHADE method, the RPI is used to discretizethe mutant vector.

2.6 Fitness functions

In the present work, two problems of fuel LP optimization have beenconducted with the use of two OFs to evaluate the performance of the op-timization methods The first problem with the fitness function F 1 aims

at comparing the possibility of ESA with SA and ASA in reproducing areference LP, i.e., reproducing the power distribution of the reference LP[7] The F 1 is written as follows:

The second problem with the fitness function F 2 is to optimize the

LP of the VVER-1000 core by maximizing the kef f at the BOC, flatteningthe power distribution and ensuring the constraint of PPF The formula of

F 2 is written as follows:

F 2 = 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: