Multigroup Energy Research Articles

Optimization of multi-group energy structures for solving the neutron transport equation

We develop a framework to determine an optimal broad group structure of the energy variable when solving the neutron transport equation, starting from a fine structure. This broad group structure is required to perform the energy condensation method, where the fine group cross sections are collapsed into the broad group cross sections. In this optimization work, we study the objective function based on the error of the reaction rates and the k-effective as well. Moreover, together with the automatic coarse energy (ACE) algorithm proposed in [1] and the particle swarm optimization (PSO) usually applied to optimize the broad group structure, we introduce the Leave Group Out (LGO) and Add Group In (AGI) approaches. Numerical results for the core calculation are investigated to illustrate the performance of the proposed algorithms.

Improved treatment of multi-material cells in thermal radiation transport codes

High-energy-density physics simulations with non-conformal meshes of materials require a multi-material (MM) closure that affects the thermal radiation transport (TRT). We propose a set of novel closures that work for an arbitrary number of materials, both grey and multigroup energy discretizations, and any angular discretization (such as Sn, IMC, or diffusion). For each spatial cell, our closures let each species (ion and electron) of each material have its own temperature, density, and internal energy, but use a single radiation distribution that interacts with all materials within the cell. Our closures maintain energy conservation, do not incur increased computational cost in the TRT solve itself, are compatible with single-material TRT solvers, and do not make any temperature-equilibrium assumptions. We test our closures on a wide range of increasingly realistic problems and find them to be robust.

A new approach for uncertainty quantification in predictor-corrector quasi-static Monte Carlo transient simulation

Two different approaches are widely accepted for transient Monte Carlo (MC) simulation namely the Dynamic Monte Carlo (DMC) and the Predictor-Corrector Quasi-Static Monte Carlo (PCQS-MC). The MC transport code iMC developed at the Korea Advanced Institute of Science and Technology incorporates both approaches. In this paper, an original method for properly assessing the uncertainty of PCQS-MC is proposed and demonstrated using the iMC code. A detailed description of applying the PCQS method for transient MC calculation is presented with an emphasis on the origins of PCQS-MC uncertainties. The implementation of a quasi-static method, i.e., calculation of point-kinetics (PK) parameters, incurs an additional uncertainty for PCQS-MC calculation alongside the conventional stochastic MC uncertainty. Such quasi-static treatment-driven stochasticity cannot be recognized through the conventional MC sampling scheme, insinuating a significant underestimation of uncertainty without proper measures. To verify the findings, null-transient simulations have been performed for GODIVA and C5G7 benchmarks, where the former and latter problems require continuous and multi-group energy treatment respectively. For an improved uncertainty evaluation, a new sampling scheme referred to as the PK-sampling scheme is proposed where cycle-wise PK correction is made. Proper estimation of uncertainty can be made from the heap of cycle-wise corrected power through a unique screening process based on the null hypothesis testing. The proposed PK sampling scheme is tested for C5G7-TD transient benchmarks using iMC where the result plainly attests to the effectiveness of the new sampling approach.

On a Response Matrix Solver for Slab-Geometry Neutral Particle Transport Problems in the Discrete Ordinates and Energy Multigroup Formulations Considering Non-Uniform Interior Sources

We present in this work an extension of the Response Matrix (RM) method for the numerical solution of slab-geometry neutral particle transport equation in the discrete ordinates (S) and energy multigroup formulations considering non-uniform sources. By using the term non-uniform we mean that the particle source is not spatially uniform inside the regions that compose the domain. The extended RM method, differently from the conventional RM method, is based on the solution of “source-independent” auxiliary problems (Green’s functions). The solution of these auxiliary problems is used in conjunction with the given non-uniform source to generate the sweeping matrices for the extended RM method. Numerical results with respect to both uniform and non-uniform source problems are given to illustrate the efficiency of the offered extended RM method.

Preservation of kinetics parameters generated by Monte Carlo calculations in two-step deterministic calculations

The generation of accurate kinetic parameters such as mean generation time Λ and effective delayed neutron fraction βeff via Monte Carlo codes is established. Employing these in downstream deterministic codes warrants another step to ensure no additional error is introduced by the low-order transport operator when computing forward and adjoint fluxes for bilinear weighting of these parameters. Another complexity stems from applying superhomogenization (SPH) equivalence in non-fundamental mode approximations, where reference and low-order calculations rely on a 3D full core model. In these cases, SPH factors can optionally be computed for only part of the geometry while preserving reaction rates and K-effective, but the impact of such approximations on kinetics parameters has not been thoroughly studied. This paper aims at studying the preservation of bilinearly-weighted quantities in the Serpent–Griffin calculation procedure. Diffusion and transport evaluations of IPEN/MB-01, Godiva, and Flattop were carried out with the Griffin reactor physics code, testing available modeling options using Serpent-generated multigroup cross sections and equivalence data. Verifying Griffin against Serpent indicates sensitivities to multigroup energy grid selection and regional application of SPH equivalence, introducing significant errors; these were demonstrated to be reduced through the use of a transport method together with a finer energy grid.

Lp-CMFD acceleration schemes in multi-energy group 2D Monte Carlo transport

The linear prolongation flux update scheme is extended to both regular CMFD acceleration, as well as partial CMFD acceleration in 2D multi energy group Monte Carlo k-eigenvalue neutron transport problems. The acceleration performance of these CMFD variants were investigated in simple 2D slab geometries, first with a monoenergetic case and then with a three group problem on the same geometry based on the monoenergetic cross sections. Flux convergence was determined via an on-the-fly convergence diagnostics developed by Ueki and Brown. It is found that on top of providing better acceleration in general, the linear prolongation scheme is also able to correct for instabilities in the CMFD scheme. Overall, the lp-pCMFD scheme employing a maximum history length is found to have the best performance across the cases presented.

Hierarchical optimal configuration of multi-energy microgrids system considering energy management in electricity market environment

Aiming at the global environmental pressure and energy crisis, an optimal configuration method of a multi-energy microgrids system is proposed in the context of the electricity market. Firstly, the hierarchical collaborative optimization configuration framework of a multi-energy microgrid system is established. The upper-level regional energy supply is centrally coordinated and planned, and the lower-level multi-energy microgrids are independent and autonomous. At the same time, to reflect the electric energy interaction between various entities, a dynamic electricity price model between microgrids based on a multi-microgrid energy trading mechanism oriented by market supply and demand is established. To obtain the operation cost of the multi-energy microgrid, the microgrid operation strategy is analyzed under the cooperative operation of microgrids. Then, considering the interaction of the source-storage-load, a hierarchical collaborative optimal configuration model of the multi-energy microgrid group is established to minimize the economic cost. After that, the analytical target cascading method is adopted and a penalty function is introduced to satisfy the consistency constraints. Finally, a case study is conducted using the proposed approach. Experiments show that the optimal configuration based on the extreme scenarios and typical scenarios can reduce the total cost and maximize the interests of each subject. The economy of the system configuration is improved by 2 ∼ 3.5 %.

Decentralized P2P power trading mechanism for dynamic multi-energy microgrid groups based on priority matching

In order to promote the interaction among interconnected microgrids (MGs) with multiply distributed energy resources (DERs) to local energy utilization, a decentralized peer-to-peer (P2P) power trading mechanism based on priority matching is proposed. The priority indices are calculated based on scheduling results and quotations by an independent MG operator (IMO), which protect effectively the privacy and represent the bargaining willingness of MGs. According to the supply–demand ratio within a matching pair, each MG is permitted to update its quotation multi-times until the convergence or updated number limitation is reached. The clearing price of each pair is calculated by the mid-market rate (MMR) method. Because the imbalanced power is different with the autonomous scheduling in a MG at different dispatching time, the different power requirements of MGs results in the formation of the dynamic MG groups. The simulation results in an interconnected multi-microgrid system (IMMGS) with 4 MGs show the proposed decentralized P2P mechanism is reasonable and effective.

Reactor physics study of square and hexagonal lattice cell using MANTRA (Method of ChAracteristics based Neutron TRAnsport Code) coupled with resonance treated multi energy group cross section library

Since last few years we have been developing a computer code MANTRA (Method of ChAracteristics based Neutron TRAnsport Code) in phased manner to solve neutron transport equation using Method of Characteristics (MOC) for lattice (i.e. assembly) as well as full core with a flexibility of choosing boundary condition out of vacuum/periodic/reflective options appropriate for the simulation. In literature, we have already reported about the development and performance of the code in triangular mesh network for mainly lattice level calculation with reflective and periodic boundary conditions and to some extent core level calculation with vacuum boundary condition using the processed cross section of benchmark problem. Later, we have explored the advantage of linear source assumption over the commonly used flat source assumption within a mesh in some selected transport calculations (where the variation in neutron flux is substantial) using MANTRA. We have also demonstrated the application of Krylov subspace iteration in a matrix-free framework to accelerate the solution of MANTRA for solving complex problems in effective manner. In continuation of the work, MANTRA is now coupled with WIMSD multi energy group cross section library. For the purpose of coupling, a module is developed in the code to process the cross section data and apply resonance treatment to account for the self-shielding effect of the resonant nuclides. The resonance treatment requires estimation of Dancoff factor, which is carried out using MANTRA for various arrangement of fuel elements. Finally, a number of benchmark problems based on PHWR, BWR, VVER lattices are studied with resonance treated library to compare the performance of MANTRA with other internationally recognized codes like HELIOS, DRAGON, WIMSD, MCNP etc.

Parallelization of a coarse-mesh neutron transport method applied in the solution of one-dimensional neutron shielding problems

In this work, a coarse-mesh method used in the solution of neutron shielding problems in slab geometry is parallelized, using the memory shared environment OpenMP. The neutron transport problem is modelled using the linearized Boltzmann equation considering the energy multigroup theory and the discrete ordinate formulation ( ). The numerical method, named Modified Spectral Deterministic (MSD), uses the neutrons transport intranodal general solution alongside an iterative process to calculate the outgoing neutron angular fluxes on the nodal interfaces. Two model-problems are solved, comparing the iterative process execution time using different number of threads. The numerical results, which achieved around 30% and 50% better execution time, were generated using the programming language C++. 

On the calculation of neutron sources generating steady prescribed power distributions in subcritical systems using multigroup X,Y-geometry discrete ordinates models

In this paper a methodology is described to estimate multigroup neutron source distributions which must be added into a subcritical system to drive it to a steady state prescribed power distribution. This work has been motivated by the principle of operation of the ADS (Accelerator Driven System) reactors, which have subcritical cores stabilized by the action of external sources. We use the energy multigroup two-dimensional neutron transport equation in the discrete ordinates formulation (SN) and the equation which is adjoint to it, whose solution is interpreted here as a distribution measuring the importance of the angular flux of neutrons to a linear functional. These equations are correlated through a reciprocity relation, leading to a relationship between the interior sources of neutrons and the power produced by unit length of height of the domain. A coarse-mesh numerical method of the spectral nodal class, referred to as adjoint response matrix constant-nodal method, is applied to numerically solve the adjoint SN equations. Numerical experiments are performed to analyze the accuracy of the present methodology so as to illustrate its potential practical applications.

An efficient exponential representation for solving the two-energy group point telegraph kinetics model

The multi-energy group kinetics system configuration is the most widely used in the modern-day reactor for the analysis, operation, control, and design. The present work aims to develop and analyze the system of the multi-energy group point telegraph model in the presence of I groups of delayed neutrons. The spatial and time-dependent stiff coupled partial differential equations for a multi-energy group of telegraph model with multi-group of delayed neutrons are analytically derived. Moreover, an analytical solution of the two-energy group point telegraph model is presented based on Magnus expansion with an exponential representation of the matrix exponential function using eigenvalues and eigenvectors. The developed model of the point kinetics model (PKM) and point telegraph model (PTM) is applied to different benchmarks in the time domain using successive approximations of Magnus expansion in case of a step, ramp, and sinusoidal reactivity. The effect of the fast and thermal relaxation times for the kinetics two-energy group point telegraph model is obtained and analyzed for time-varying reactivity. The results indicate that the relaxation times increase, the solution response is relaxed behind that of the diffusion model, while as the same parameters tend to zero, the response of the neutron density tends to that of the diffusion case.

Generalized perturbation techniques for uncertainty quantification in lead-cooled fast reactors

The design of innovative nuclear fission systems requires a careful evaluation of the uncertainties affecting the basic input data. Among them, nuclear data are particularly relevant, due to their dramatic energy dependence. Because of this feature and of the strong spatial heterogeneity of nuclear reactors arrangement, full-core calculations are carried out using energy collapsed and spatially homogenised constants. Nowadays, collapsing is often performed with Monte Carlo codes, which allow a discretisation-free treatment of the neutron transport equation.The most popular method to propagate the uncertainty in the nuclear data libraries throughout the Monte Carlo transport calculation is the Generalised Perturbation Theory (GPT). However, due to its multi-group nature, GPT often blurs the continuous-energy feature of the Monte Carlo method. Therefore, in order to fully exploit its advantages, the XGPT method has been recently proposed. After discussing the main differences between these two approaches, the paper presents the application to an uncertainty quantification study on the lead-cooled fast reactor ALFRED design, performed with GPT and focused on the multi-group cross sections.Afterwards, the two nuclides that most contribute to the overall uncertainties, i.e. Pu-239 and U-238, are considered to compare the GPT results to some XGPT calculations carried out with different multi-group energy structures. This analysis suggests that XGPT is a consistent method for uncertainty quantification in the continuous-energy Monte Carlo framework. Moreover, it can be concluded that an adequate number of low-energy groups is necessary for an accurate uncertainty evaluation in the case of a fast system.

Optimization of multi-group energy structures for diffusion analyses of sodium-cooled fast reactors assisted by simulated annealing – Part II: Methodology application

Part I of this study introduced a novel methodology for the optimization of energy group structures to be used in diffusion calculations of Sodium cooled Fast Reactors. The goal of the optimization is to speed up calculations, particularly in transient analyses, while maintaining an acceptable accuracy of the results.The proposed methodology is based by either direct or simulated annealing search techniques. The capabilities of the method were preliminarily demonstrated on a single core state of the Superphénix reactor by using the DYN3D nodal diffusion code.The scope of Part II is the further demonstration of the methodology efficiency through its application to more challenging “real-life” cases. In this respect, a static Superphénix neutronic benchmark comprising 13 different core states and a transient test initiated by an increase of the core inlet temperature at the Phénix reactor are considered.For both Superphénix and Phénix reactor cores, 2- to 12-group optimal condensed energy group structures are identified using the 24-group structure as a starting point. The obtained optimal structures are thus employed in DYN3D and compared to the reference 24-group DYN3D solutions.The outcomes show that also for a broader range of core configurations and under transient conditions the optimal energy group structures allow for significant acceleration of the DYN3D performance with practically negligible degradation of the solution accuracy.

Coordinated Control and Energy Management System of Microgrid Group

As an important part of the smart grid, the operation control of the microgrid group is the key technology to maintain its safe and stable operation. The reasonable distribution of microgrid energy is closely related to the stability of system control. Multiple micro-grids in a certain area are uniformly regulated and controlled to form a micro-grid group operation mode. Aiming at the operation and management requirements of the multi-energy complementary microgrid group, the article proposes a design scheme for the energy management system of the microgrid group. The entire system has high operational reliability and economy, and has certain engineering value for improving the comprehensive benefits of the microgrid.

The Effect of the Flux Separability Approximation on Multigroup Neutron Transport

The angular dependence of flux-weighted multigroup cross sections is commonly neglected when generating multigroup libraries. The error of this flux separability approximation is typically not isolated from other error sources due to a lack of availability of library generation and corresponding solvers that cannot relax this approximation. These errors can now be isolated and quantified with the availability of a multigroup Monte Carlo transport and multigroup library-generation capability in the OpenMC Monte Carlo transport code. This work will discuss relevant details of the OpenMC implementation, provide an example case useful for detailing the type of errors one can expect from making the flux separability approximation, and end with more realistic problems which show the impact of the approximation and highlight how it can strongly arise from an energy-dependent resonance absorption effect. Since the angle-dependence is intrinsically linked to the energy group structure, these examples also show that relaxing the flux separability approximation with angle-dependent cross sections could be used to reduce either the fine-tuning required to set a multigroup energy structure for a specific reactor type or the number of energy groups required to obtain a desired level of accuracy for a given problem. This trade-off could increase the costs of generating multigroup cross sections, and has the potential to require more memory for storing the multigroup library during the transport calculations, but it can significantly reduce the computational time required since the runtime of a discrete ordinates or method of characteristics neutron transport solver scales roughly linearly with the number of groups.

Optimization of multi-group energy structures for diffusion analyses of sodium-cooled fast reactors assisted by simulated annealing – Part I: Methodology demonstration

This study presents an approach to the selection of optimal energy group structures for multi-group nodal diffusion analyses of Sodium-cooled Fast Reactor cores. The goal is to speed up calculations, particularly in transient calculations, while maintaining an acceptable accuracy of the results.In Part I of the paper, possible time-savings due to collapsing of energy groups are evaluated using 24-group energy structure as a reference. Afterwards, focusing on energy structures with a number of groups leading to significant calculation speedups, optimal grid configurations are identified. Depending on a number of possible energy grid configurations to explore, the optimization is conducted by either a direct search or applying the simulated annealing method. Speedup and optimization studies are performed on a selected case of the Superphénix static neutronic benchmark by using the nodal diffusion DYN3D code. The results demonstrate noticeable improvements in DYN3D performance with a marginal deterioration of the accuracy.

On the spectral Green’s function-constant nodal method for fixed-source SN problems in X,Y-geometry with arbitrary L’th-order anisotropic scattering

Presented here is an extension of the spectral Green’s function-constant nodal (SGF-CN) method for the numerical solution of energy multigroup, fixed-source, discrete ordinates (SN) problems in X, Y-geometry with arbitrary L’th-order of scattering anisotropy, provided L<N. This analytical coarse-mesh method uses the multigroup SGF method for numerically solving the one-dimensional transverse-integrated SN nodal equations with constant approximations for the transverse leakage terms. The only approximations in the present version of the SGF-CN method occur in these transverse leakage terms, as the energy-group transfer scattering source terms are treated analytically within the offered method. Numerical results to typical model problems are given to illustrate the method’s accuracy and to analyze the efficiency of the offered SGF-CN computer code for neutral particle transport calculations.

On the Generalization of the Response Matrix Spectral Nodal Method for Neutral Particle SN Source–Detector Problems in Slab Geometry

A generalization of the Response Matrix spectral nodal method is applied to fixed–source discrete ordinates (SN ) radiative transfer and neutron transport problems in slab geometry. This method is extended to forward and adjoint energy multigroup problems with anisotropic scattering including the upscattering events. We present the numerical methodology which generates SN solutions absolutely free from spatial truncation errors. To discuss the efficiency of the method, we offer a number of numerical results for two typical test problems.

INVESTIGATION ON DETERMINISTIC TRUNCATION TO CONTINUOUS ENERGY MONTE CARLO NEUTRON TRANSPORT CALCULATION

This paper presents the application and evaluation of a deterministic truncation of Monte Carlo (DTMC) solution method in a whole core reactor problem based on a continuous energy transport calculation. The DTMC method has been studied and developed as a systematic way to truncate the high-fidelity Monte Carlo (MC) solution to reduce the computational cost without compromising the essential reliability of the solution. Its fea-sibility and capability were preliminarily validated in several benchmark problems using a multi-group energy MC code. In this paper, further study has been conducted in the more practical application with the continuous-energy based MC calculation. The con-cept of the DTMC method is briefly described. Improvements to enhance the numerical stability and efficiency are specified in details. The DTMC method is applied to an SMR problem, in which reactor parameters are estimated to characterize the numerical per-formance and are compared to the standard MC method. Last, the computing time and corresponding figure-of-merit are evaluated.