Monte Carlo Criticality Calculations Research Articles

Generalized extreme value analysis of efficient evaluation of extreme values in random media criticality calculations

The uncertainty of fuel debris criticality can be estimated by Monte Carlo criticality calculations repeated over independent replicas of some properly modelled random medium. Incomplete randomized Weierstrass function (IRWF) is a modelling framework for such calculations. Under these approaches at hand, the theme of this paper is how to efficiently analyse extreme realizations of neutron effective multiplication factor (keff) over IRWF random media replicas. To this end, a new bounded amplification (BA) technique is applied to IRWF. The numerical results clearly indicate that the BA-applied IRWF reduces a required number of random media replicas at least by an order of magnitude. To rigorously validate this efficiency gain, generalized extreme value (GEV) analysis is applied to a data set of keff values obtained without applying BA. It turns out that the extreme values of these keff values follow the Weibull distribution. Therefore, the theory of GEV guarantees the existence of an upper limit for these keff values, and the actually computed upper limit is indeed smaller than or equal to the top two keff values obtained from an order-of-magnitude reduced number of BA-applied IRWF random media replicas. This means that the efficiency gain brought by BA has been confirmed by the GEV methodology. In addition, a method of rejecting spurious estimation values due to statistical fluctuation is demonstrated concerning the extreme value index in the GEV analysis.

A new algorithm of critical control rod position search in MC code

Delayed Material Replacement algorithm is used in control rod movement simulation. Then, critical control rod position search is studied in Monte Carlo criticality calculation. Support Vector Regression algorithm is used in fitting neutron absorption as a function of rod positions. Fission iteration batches are reorganized as blocks in order to avoid fission source perturbation caused by control rod movement. The keff result in AP1000 reactor calculation shows discrepancy less than 20 pcm, though stable fluctuations are found in rod position values during fission iteration.

A hybrid neutronics method with novel fission diffusion synthetic acceleration for criticality calculations

A novel Fission Diffusion Synthetic Acceleration (FDSA) method is developed and implemented as a part of a hybrid neutronics method for source convergence acceleration and variance reduction in Monte Carlo (MC) criticality calculations. The acceleration of the MC calculation stems from constructing a synthetic operator and solving a low-order problem using information obtained from previous MC calculations. By applying the P1 approximation, two correction terms, one for the scalar flux and the other for the current, can be solved in the low-order problem and applied to the transport solution. A variety of one-dimensional (1-D) and two-dimensional (2-D) numerical tests are constructed to demonstrate the performance of FDSA in comparison with the standalone MC method and the coupled MC and Coarse Mesh Finite Difference (MC-CMFD) method on both intended purposes. The comparison results show that the acceleration by a factor of 3–10 can be expected for source convergence and the reduction in MC variance is comparable to CMFD in both slab and full core geometries, although the effectiveness of such hybrid methods is limited to systems with small dominance ratios.

Single-step Monte Carlo criticality algorithm

A single-step Monte Carlo criticality algorithm is proposed to improve the efficiency of Monte Carlo criticality calculation. This algorithm adopts the strategy of increasing neutron population cycle by cycle and achieves the highest computational efficiency through a weighted technique and mathematical optimization analysis. The inactive cycles are no longer needed, and all neutron histories contribute to the final results, so the fission source's convergence diagnosis is no longer needed. The algorithm's effectiveness is demonstrated with a single-rod model, an infinite medium problem, and the VERA benchmark problem #5 using a modified RMC code.

Weierstrass function methodology for uncertainty analysis of random media criticality with spectrum range control

Randomized Weierstrass function (RWF) has been under development for evaluating the uncertainty of random media criticality due to the material mixture in disorder. In this work, the modelling capability of RWF is refined so that the spectrum range can be controlled by specifying its lower and upper ends of the frequency domain variable. As a result, it becomes possible to make fair criticality comparison among replicas of random media under inverse power law power spectra. Technically, the infinite sum of trigonometric terms in RWF is formally extended to cover the arbitrarily low frequency domain and then truncated to finite terms for the sole purpose of spectrum range control. This means that the refined RWF is free of the issue of convergence towards a reference fractal characteristic of Weierstrass function and thus termed Incomplete Randomized Weierstrass function (IRWF). As a demonstration, a three-dimensional version of IRWF is applied to the mixture of three fuels with different burnups in a sufficiently water-moderated environment. Monte Carlo criticality calculations are carried out to evaluate the uncertainty of neutron effective multiplication factor (keff) due to the indeterminacy of mixture formation.

Source extrapolation scheme for Monte Carlo fission source convergence based on RMC code

The source extrapolation method had been used to accelerate the convergence of fission sources in the deterministic methods. However, it has not been effectively used for the Monte Carlo criticality calculation. We apply the source extrapolation method to the Monte Carlo criticality calculation to accelerate the fission source convergence. An explicit and straightforward formula for determining the extrapolation factor is given based on the mathematical derivation and numerical calculations, and a new source extrapolation scheme for the Monte Carlo criticality calculation is proposed. This source extrapolation scheme is easily implemented and makes the fission sources converge twice as fast as the standard Monte Carlo criticality calculation. The validity and optimality of the source extrapolation scheme are tested by a single fuel rod model and the OECD source convergence benchmark. The results show that the new scheme can accelerate the fission source convergence.

Optimal Batch Size Growth for Wielandt Method and Superhistory Method

In a high dominance ratio system, the problem of slow fission source convergence is faced during the Monte Carlo criticality calculation. The Wielandt method and the Superhistory method have been proven to reduce the dominance ratio. Still, the Wielandt method and the Superhistory method have also proven to be unable to accelerate the convergence of the fission source. With the estimation of errors in the cumulative fission source and the batch size optimization methodology, the optimal batch size growth for the Wielandt method and the Superhistory method is proposed. Compared with the direct simulation with the optimal batch size growth, the Wielandt simulation and the Superhistory simulation better use the optimal batch size growth. A single fuel rod model was tested, and the results show that the new method is helpful for the acceleration of fission source convergence.

Uniform variance method for accelerated Monte Carlo criticality calculation

The problem of the non-uniform distribution of relative errors is encountered during the Monte Carlo criticality calculation, which not only reduces the efficiency and the accuracy of some local tallies, but also affects the stability of the Monte Carlo multi-physics coupling calculation. To solve this problem, the uniform variance method is proposed. In this method, the global information is recorded and updated during the inactive cycles, and the source bias technique and the weight windows technique are combined to achieve the uniform distribution of relative errors. The modified C5G7 benchmark is tested by the uniform variance method, and it is found that the variance is reduced in almost all tallies compared with that of the direct simulation. So the uniform variance method is helpful for the global variance reduction, and can be used for the acceleration of the Monte Carlo criticality calculation.

A pseudo-material method for graphite with arbitrary porosities in Monte Carlo criticality calculations

ABSTRACT The latest ENDF/B-VIII library adapted new porosity-dependent cross-section data of graphite. However, the porosity of the actual graphite does not necessarily correspond to the porosity given in the data. We have proposed a method to perform neutronic calculations at the desired porosity on the basis of the pseudo-material method. We have also compared the k eff values calculated by the pseudo-material method with the experimental values for the VHTRC. In addition, we have investigated the temperature dependance of the calculation values obtained by this method. From these results, we have concluded that this method allows us to perform the neutronic calculations in which we can reflect detailed information on the porosity of graphite.

Forced propagation method for Monte Carlo fission source convergence acceleration in the RMC

In loosely coupled or large-scale problems with high dominance ratios, slow fission source convergence can take extremely long time, reducing Monte Carlo (MC) criticality calculation efficiency. Although various acceleration methods have been developed, some methods cannot reduce convergence times, whereas others have been limited to specific problem geometries. In this study, a new fission source convergence acceleration (FSCA) method, the forced propagation (FP) method, has been proposed, which forces the fission source to propagate and accelerate fission source convergence. Additionally, some stabilization techniques have been designed to render the method more practical. The resulting stabilized method was then successfully implemented in the MC transport code, and its feasibility and effectiveness were tested using the modified OECD/NEA, one-dimensional slab benchmark, and the Hoogenboom full-core problem. The comparison results showed that the FP method was able to achieve efficient FSCA.

Does neutron clustering affect tally errors in Monte Carlo criticality calculations?

Monte Carlo criticality calculations of large, loosely-coupled problems are long known to suffer from slow convergence of the tally errors due to cycle-to-cycle fission source correlations. In several recent studies, it was suggested that these correlations could be possibly attributed to the neutron clustering phenomenon that is visible in calculations with a small number of neutrons per iteration cycle (batch size). Nevertheless, other studies have also shown the error convergence rate in such loosely-coupled problems to be batch size-independent during active criticality cycles. Here, we aim to address this inconsistency by studying the error convergence in a large number of test calculations, varying the neutron batch size from small to large. In our tests, we have observed that the presence of visible neutron clusters does not increase the cycle-to-cycle fission source correlations and does not worsen the convergence rate of the tally errors.

APPLICATION OF RESPONSE MATRIX METHOD TO TRANSIENT SIMULATIONS OF NUCLEAR SYSTEMS

Until recently, reactor transient problems were exclusively solved by approximate deterministic methods. The increase in available computing power made it feasible to approach the transient analyses with time-dependent Monte Carlo methods. These methods offer the first-principle solution to the space-time evolution of reactor power by explicitly tracking prompt neutrons, precursors of delayed neutrons and delayed neutrons in time and space. Nevertheless, a very significant computing cost is associated with such methods. The general benefits of the Monte Carlo approach may be retained at a reduced computing cost by applying a hybrid stochastic-deterministic computing scheme. Among such schemes are those based on the fission matrix and the response matrix formalisms. These schemes aim at estimating a variant of the Greens function during a Monte Carlo transport calculation, which is later used to formulate a deterministic approach to solving a space-time dependent problem. In this contribution, we provide an overview of the time-dependent response matrix method, which describes a system by a set of response functions. We have recently suggested an approach where the functions are determined during a Monte Carlo criticality calculation and are then used to deterministically solve the space-time behaviour of the system. Here, we compare the time-dependent response matrix solution with the transient fission matrix and the time-dependent Monte Carlo solutions for a control rod movement problem in a mini-core reactor geometry. The response matrix formalism results in a set of loosely connected equations which offers favourable scaling properties compared to the methods based on the fission matrix formalism.

Monte Carlo Criticality Calculation of Random Media Formed by Multimaterials Mixture Under Extreme Disorder

A dynamical system under extreme physical disorder has the tendency to evolve toward the equilibrium state characterized by an inverse power law power spectrum. In this paper, a practical, implementable, three-dimensional model is proposed for the random media formed by a multimaterials mixture under such a power spectrum using a randomized form of the Weierstrass function, its extension covering the white noise, and partial volume pairings of constituent materials. The proposed model is implemented in the SOLOMON Monte Carlo solver with delta tracking. Two sets of numerical results are shown using the JENDL-4 nuclear data libraries. First, the uncertainty of the neutron effective multiplication factor (keff) due to the inherent uncertainty in the formation of random media is shown for a randomized version of the Bigten core in the International Criticality Safety Benchmark Evaluation Project (ICSBEP). Second, the influence of the exponent of the power spectrum on the uncertainty of keff is evaluated for a randomized version of the Topsy core in the ICSBEP.

Use of finite-difference diffusion Monte Carlo method to generate Shannon entropy of the fission distribution in a nuclear reactor core

Monte Carlo criticality calculation is an iterative process which requires a guess fission distribution to start the simulations. It is important to assess when the fission source has converged, so that a sufficient number of the initial batches are discarded prior to beginning of the Monte Carlo (MC) tallies. Shannon entropy has been highly effective in characterizing convergence of the fission distribution in a reactor core. To compute Shannon entropy, transport-based MC code usually requires additional binning of the geometry along with the use of algorithms to get the distribution of fission source in these bins. This binning is done by superimposing a grid kind of structure over and above the geometry defined for transport simulation. The entropy computation in transport-based MC code thus involves additional computational efforts. In this paper, we have used finite-difference (FD) difference based diffusion MC method to calculate Shannon entropy. In this method, problem is divided into regular meshes. Shannon entropy is estimated in these mesh structures without any additional computational efforts. Shannon entropy for 3D PHWR benchmark is estimated by this method and is compared with Transport MC Code PATMOC and 3D Space Time Kinetics Code KINFIN and the results are found to be in good agreement.

Stochastic-deterministic response matrix method for reactor transients

Presented is a stochastic-deterministic, response matrix method for transient analyses of nuclear systems. The method is based on the response matrix formalism, which describes a system by a set of response functions. We propose an approach in which these response functions are computed during a set of Monte Carlo criticality calculations and are later used to formulate a deterministic set of equations for solving a space-time dependent problem. Application of the response matrix formalism results in a set of loosely connected equations, which leads to a favourable linear scaling of the problem. The method offers a simplified approach compared to previously proposed response matrix methods by avoiding phase-space expansions in sets of basis functions. We describe the method starting with the fundamental neutron transport considerations, provide a demonstration on two absorber movement transients in a 3 × 3 assembly PWR mini-core geometry, and compare the solutions against time-dependent Monte Carlo simulations.

New strategy for global tallying in Monte Carlo criticality calculation

Traditionally, the Monte Carlo criticality calculation must set a maximum inactive step by experience to ensure that a fission source distribution has converged. The tallying process can only be invoked after this maximum inactive step to avoid the system error caused by the non-converged fission source distribution. In the same way, the uniform fission site algorithm for increasing the whole efficiency of global tallying should also be invoked after the fission source distribution has converged fully. The calculation must reach a maximum iteration step, then, this process can be stopped and the tallies can be printed. This old strategy has two defects. Firstly, the appointed maximum inactive step can only be set by experience, which will be insufficient in some cases; secondly, some iteration steps can be wasted because the precision of tallies has been enough and no one knows it. So, a new strategy is proposed in this article to overcome these defects. Based on an on-the-fly diagnostic method for the convergence of Shannon entropy sequence corresponding to the fission source distribution of each iteration step, the uniform fission site algorithm will be invoked after the iteration step whose serial number is the maximum of the first active step and the first converged step diagnosed by the above-mentioned rule. This rule will be helpful in ensuring that the uniform fission site algorithm can use enough accurate data to bias the secondary fission neutron number, thus avoiding the system error to some degree. Then, a global precision index will be calculated at each fixed step to judge whether the precision standard is reached. If so, the whole calculation is stopped. This process will be repeated until the pre-set maximum step number is reached. In this way, superfluous calculations can be skipped but the calculation precision can be guaranteed. In a word, this new strategy is beneficial to increasing the efficiency of global tallying in the Monte Carlo criticality calculation when appropriate parameters are adopted. This conclusion can be proved by the numerical result from the C5G7 benchmark model.

Optimal neutron population growth in accelerated Monte Carlo criticality calculations

We present a source convergence acceleration method for Monte Carlo criticality calculations. The method gradually increases the neutron population size over the successive inactive as well as active criticality cycles. This helps to iterate the fission source faster at the beginning of the simulation where the source may contain large errors coming from the initial cycle; and, as the neutron population size grows over the cycles, the bias in the source gets reduced. Unlike previously suggested acceleration methods that aim at optimisation of the neutron population size, the new method does not have any significant computing overhead, and moreover it can be easily implemented into existing Monte Carlo criticality codes. The effectiveness of the method is demonstrated on a number of PWR full-core criticality calculations using a modified SERPENT 2 code.

A power spectrum approach to tally convergence in Monte Carlo criticality calculation

ABSTRACTIn Monte Carlo criticality calculation, confidence interval estimation is based on the central limit theorem (CLT) for a series of tallies from generations in equilibrium. A fundamental assertion resulting from CLT is the convergence in distribution (CID) of the interpolated standardized time series (ISTS) of tallies. In this work, the spectral analysis of ISTS has been conducted in order to assess the convergence of tallies in terms of CID. Numerical results obtained indicate that the power spectrum of ISTS is equal to the theoretically predicted power spectrum of Brownian motion for tallies of effective neutron multiplication factor; on the other hand, the power spectrum of ISTS of a strongly correlated series of tallies from local powers fluctuates wildly while maintaining the spectral form of fractional Brownian motion. The latter result is the evidence of a case where a series of tallies are away from CID, while the spectral form supports normality assumption on the sample mean. It is also demonstrated that one can make the unbiased estimation of the standard deviation of sample mean well before CID occurs.

An efficient evaluation of depletion uncertainty for a GBC-32 dry storage cask with PLUS7 fuel assemblies using the Monte Carlo uncertainty sampling method

It is important to evaluate an accurate depletion uncertainty resulting from the limitation of a computer code on accuracy in nuclear depletion calculations from the viewpoint of burnup credit with axial burnup distributions. In this work, the bias and bias uncertainty in keff resulting from the uncertainty in the calculation of isotopic concentrations are assessed using the Monte Carlo sampling method for a generic 32 PWR-assembly burnup credit cask loaded with the PLUS7 assemblies discharged from Hanbit Nuclear Power Plant Unit 3 in South Korea. In addition, an efficient method with the two-way ANOVA is suggested to obtain reasonably accurate values of total depletion uncertainties resulting from the isotopic uncertainties with reduction of Monte Carlo criticality calculations. Finally, the reasonable total depletion uncertainties in keff for 10, 30, and 40 GWD/MTU were efficiently estimated to be 0.011866, 0.013175, and 0.014262, respectively, with 100 Monte Carlo criticality calculations, which have about 1.6, 2.1, and 2.4% discrepancies, respectively, from the ones obtained with 200 calculations. The one for 50 GWD/MTU was efficiently estimated to be 0.030811 with 120 Monte Carlo criticality calculations, which has about 3.6% discrepancy from the one obtained with 200 calculations. Thus, these uncertainties showed good agreements with the uncertainties obtained from 200 nuclear criticality calculations.

Monte Carlo criticality analysis under material distribution uncertainty

ABSTRACTAnalysis framework under material distribution uncertainty is not yet established for the Monte Carlo (MC) criticality calculation of continuously mixed media formed via molten core concrete interaction. To approach this issue, one group cross-sections have specifically been prepared for burned fuel, stainless steel and concrete, and the effect of spatially continuous variation of volume fractions is inspected. Trigonometric functions and randomized Weierstrass functions (RWFs) are utilized to represent the spatially continuous variation. Here, trigonometric functions enable the amount of each material to be exactly preserved, and RWF is a randomized trigonometric series that can be generated within specified upper and lower bounds. With these functions, one can assess the influence of indeterminate material distribution. MC particle transport is conducted using the delta-tracking. Numerical results indicate that the effective multiplication factor (keff) under RWF variation can depart significantly from the keff of a reference uniform medium. These numerical results also show that the deterministic modeling of spatial variation provides a bound measure for extreme results from random realizations.