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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Physics-Informed Neural Network Solution of Point Kinetics Equations for a Nuclear Reactor Digital Twin

A digital twin (DT) for nuclear reactor monitoring can be implemented using either a differential equations-based physics model or a data-driven machine learning model. The challenge of a physics-model-based DT consists of achieving sufficient model fidelity to represent a complex experimental system, whereas the challenge of a data-driven DT consists of extensive training requirements and a potential lack of predictive ability. We investigate the performance of a hybrid approach, which is based on physics-informed neural networks (PINNs) that encode fundamental physical laws into the loss function of the neural network. We develop a PINN model to solve the point kinetic equations (PKEs), which are time-dependent, stiff, nonlinear, ordinary differential equations that constitute a nuclear reactor reduced-order model under the approximation of ignoring spatial dependence of the neutron flux. The PINN model solution of PKEs is developed to monitor the start-up transient of Purdue University Reactor Number One (PUR-1) using experimental parameters for the reactivity feedback schedule and the neutron source. The results demonstrate strong agreement between the PINN solution and finite difference numerical solution of PKEs. We investigate PINNs performance in both data interpolation and extrapolation. For the test cases considered, the extrapolation errors are comparable to those of interpolation predictions. Extrapolation accuracy decreases with increasing time interval.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

High-order accurate solutions of the point kinetics equations with the spectral deferred correction method

Solving initial value problems with higher-order methods can improve the accuracy of the simulation results or the efficiency of the calculation. In this paper, we apply the spectral deferred correction (SDC) method to solve the initial value problem of the point kinetics equations (PKE). SDC is a stable, robust, and efficient high-order time-integration method capable of an arbitrary order of accuracy. For our implementation we show that it is A-stable for orders up to 8 and the order of accuracy is verified for PKE problems with a range of different reactivities. A 5.-order SDC method was then implemented to solve the exact PKE (EPKE) in the Transient Multilevel (TML) method of MPACT. The error from solutions of the EPKE is shown to be negligible. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

A User Guide to PARET/ANL

PARET was originally created in 1969 at what is now Idaho National Laboratory (INL), to analyze reactivity insertion events in research and test reactor cores cooled by light or heavy water, with fuel composed of either plates or pins. The use of PARET is also appropriate for fuel assemblies with curved fuel plates when their radii of curvatures are large with respect to the fuel plate thickness. The PARET/ANL version of the code has been developed at Argonne National Laboratory (ANL) under the sponsorship of the U.S. Department of Energy/NNSA since the inception of the Reactor Conversion Program. Since Reduced Enrichment for Research and Test Reactors (RERTR) began in 1978, PARET/ANL has been benchmarked to experimental data including SPERT testing, and used to determine the expected transient behavior of a large number of reactors regardless of enrichment both inside and outside the Reactor Conversion Program. This document provides the pertinent information for the use of PARET/ANL Version 7.6. PARET/ANL models the various fueled regions of a reactor core as channels. Each of these channels consists of a single flat fuel plate/pin (including cladding and, optionally, a gap) with water coolant on each side. In slab geometry the coolant channels for a given fuel plate are of identical dimensions (mirror symmetry), but they can be of different thickness in each channel. There can be many channels, but each channel is independent and coupled only through reactivity feedback effects to the whole core. The time-dependent differential equations that represent the system are replaced by an equivalent set of finite-difference equations in space and time, which are integrated numerically. PARET/ANL uses fundamentally the same numerical scheme as RELAP5 for the time-integration of the point-kinetics equations. The one-dimensional thermal-hydraulic model includes temperature-dependent thermal properties of the solid materials, such as heat capacity and thermal conductivity, as well as the transient heat production and heat transfer from the fuel meat to the coolant. Temperature- and pressure-dependent thermal properties of the coolant such as enthalpy, density, thermal conductivity, and viscosity are also used in determining parameters such as friction factors and heat transfer coefficients. The code first determines the steady-state solution for the initial state. Then the solution of the transient is obtained by integration in time and space. Multiple heat transfer, DNB and flow instability correlations are available. The code was originally developed to model reactors cooled by an open loop, which was adequate for rapid transients in pool-type cores. An external loop model appropriate for Miniature Neutron Source Reactors (MNSR’s) was also added to PARET/ANL to model natural circulation within the vessel, heat transfer from the vessel to pool and heat loss by evaporation from the pool. PARET/ANL also contains models for decay heat after shutdown, control rod reactivity versus time or position, time-dependent pump flow, and loss-of-flow event with flow reversal as well as logic for trips on period, power, and flow. Feedback reactivity effects from coolant density changes and temperature changes are represented by tables. Feedback reactivity from fuel heat-up (Doppler Effect) is represented by a four-term polynomial in powers of fuel temperature. Photo-neutrons produced in beryllium or in heavy water may be included in the point-kinetics equations by using additional delayed neutron groups.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2021

Lie group analysis of the point-reactor neutron kinetics equations for various reactivity models

This work applies Lie Group Theory to the nonlinear point-reactor neutron kinetics equations for several reactivity feedback and reactivity insertion models. The goal of this work is to connect known analytical solutions to their corresponding Lie groups as well as to obtain new analytical solutions and the circumstances from which they arise. The reactivity models we consider are the Nordheim-Fuchs Model, an arbitrary-order polynomial reactivity insertion model, and the Fuchs Ramp-Insertion Model. In all three models, we generalize the analysis when possible and obtain analytical solutions in every case. For the first two models, we succeed in finding the Lie groups corresponding to classical solutions and find additional solutions that were otherwise unknown. Finally, for the final model, we obtain a reduced-order nonlinear kinetics equation which is not analytically solvable, but by applying a second-order truncation, we find a novel closed-form solution and demonstrate its validity with a numerical example.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

(Doublon) Benchmarking of Different Inverse Point Kinetics Implementations for an Autocorrected Reactimeter Algorithm

In November 2017, the Transient Reactor Test Facility returned to operation. Since that time, many transient test series have been completed, such as the Transient Heatsink Overpower Response capsule (THOR), the Transient Water Irradiation System for TREAT (TWIST), and Sirius. Each has provided valuable data for materials performance and reactor safety that can be applied in future designs. During each experimental series, detector count rates provided important information on the core behavior during transients. However, a limitation of these data is that variations in the neutron distribution during experiments can cause errors when attempting to infer reactivity evolution from detector signals. Neutron physics codes can be used to compute the flux shape variations. However, this is a poor solution when the experimental data is used for code verification, validation and uncertainty quantification. Indeed, if the output of the code is used both as a reference and to correct what the reference is compared to, the circular dependency limits the quality of the verification, validation and uncertainty quantification approach. To overcome this problem, the autocorrected reactimeter algorithm (ACRA) has been developed. This approach infers a time-dependent reactivity evolution by testing different spatial corrections and selecting the one that minimizes reactivity variations when the core is in a frozen configuration (i.e., when there is no variation in parameters affecting reactivity). However, the scope of this method was limited to transients where there were negligible thermal feedback. Indeed, the core is never in a frozen configuration when the fuel temperature varies during the whole transient. This is our motivation for developing an improved version of the ACRA that does not require frozen configurations. To develop this new algorithm, we need a precise and unbiased implementation of the inverse point kinetic equations (IPKEs) as any error in the reactivity evaluation will be propagated into the choice of the optimal spatial correction. Indeed, the previous reactimeter algorithm would use approximations, such as a negligible flux amplitude derivative, to focus on rapidity. For the numerical validation of ACRA, we aim at absolute error under for reactivity derived from signals similar to the one of this study. In this summary, we test eight different IPKE implementations. Each will process a mockup signal built for this study, similar to those that the future ACRA will process. Each reactivity output will be compared to the reference reactivity that has been used to generate the mockup signal. The implementation minimizing the difference with the reference reactivity will be used in the development of a new ACRA formulation.

73 - NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Generation of localized reactor point kinetics parameters using coupled neutronic and thermal fluid models for pebble-bed reactor transient analysis

The systems analysis of anticipated operating occurrences and design basis accidents for pebble-bed reactor systems requires knowledge of neutron point kinetics equations (PKE) parameters. Typically, the generation of PKE parameters is performed in a global manner using standalone neutronics calculations, without the inclusion of thermal fluid distributions. We utilize Griffin and Pronghorn for generating global and local PKE parameters which includes the use of thermal fluid distributions to account for localized effects. This work establishes a methodology for calculating PKE parameters for a pebble bed reactor with a coupled neutronics/thermal fluids analysis. PKE parameters generated on a global and local basis are compared against a diffusion solve for a typical load-following transient. Locally-generated neutron kinetic parameters are able to reduce the maximum error in the transient power level from 5% to below 1.5%; along with this, bulk temperature errors were reduced from 12 K to 4 K.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2024

The Transient Multi-Level method for Monte Carlo reactor statics calculations

The Transient Multi-Level (TML) method is applied to a time-dependent Monte Carlo transport solver to offload some of the computational burden of the expensive Monte Carlo solve to lower-order Coarse Mesh Finite Difference (CMFD) and Exact Point Kinetics Equations (EPKE) solvers via factorization of the neutron flux at the transport and CMFD levels using the Predictor Corrector Quasi-Static Method (PCQM). The Monte Carlo transient is solved by a modified fission source iteration scheme that introduces a single transient source bank. The method is implemented in the production-level Monte Carlo code, Shift, and verified with prescribed reactivity ramps from the two-dimensional version of the C5G7-TD reactor benchmark. The results show that, as compared to other quasi-static methods, the TML reduces the stochastic noise inherent to the transient Monte Carlo solver by factors of ~2 to 6 for various norm comparisons of the reactor power amplitude. Finally, the TML additionally reduces the number of Monte Carlo evaluations needed to simulate the transient, leading to roughly an order of magnitude improvement in CPU time relative to the standard PCQM for the problems tested.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

Sensitivity study of parameters important to Molten Salt Reactor Safety

This paper presents a molten salt reactor (MSR) design parameter sensitivity study using a nodal dynamic modelling methodology with explicitly modified point kinetics equation and Mann’s model for heat transfer. Six parameters that can impact MSR safety are evaluated. A MATLAB-Simulink model inspired by Thorcon’s 550MW th MSR is used for parameter evaluations. A safety envelope was formed to encapsulate power, maximum and minimum temperature, and temperature-induced reactivity feedback. The parameters are perturbed by ±30%. The parameters were then ranked by their subsequent impact on the considered safety envelope, which ranks acceptable parameter uncertainty. The model is openly available on GitHub.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

Physics-informed neural network with transfer learning (TL-PINN) based on domain similarity measure for prediction of nuclear reactor transients

Nuclear reactor safety and efficiency can be enhanced through the development of accurate and fast methods for prediction of reactor transient (RT) states. Physics informed neural networks (PINNs) leverage deep learning methods to provide an alternative approach to RT modeling. Applications of PINNs in monitoring of RTs for operator support requires near real-time model performance. However, as with all machine learning models, development of a PINN involves time-consuming model training. Here, we show that a transfer learning (TL-PINN) approach achieves significant performance gain, as measured by reduction of the number of iterations for model training. Using point kinetic equations (PKEs) model with six neutron precursor groups, constructed with experimental parameters of the Purdue University Reactor One (PUR-1) research reactor, we generated different RTs with experimentally relevant range of variables. The RTs were characterized using Hausdorff and Fréchet distance. We have demonstrated that pre-training TL-PINN on one RT results in up to two orders of magnitude acceleration in prediction of a different RT. The mean error for conventional PINN and TL-PINN models prediction of neutron densities is smaller than 1%. We have developed a correlation between TL-PINN performance acceleration and similarity measure of RTs, which can be used as a guide for application of TL-PINNs.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2026

Regression Analysis with the Directed Infusion of Data

Integrating artificial intelligence and machine learning tools into industry necessitates large-scale collaborative efforts that ensure the robust and accurate execution of downstream analytics such as time series prediction, uncertainty quantification, grid optimization, and condition monitoring. However, concerns related to data privacy pervade the nuclear industry due to the proprietary nature of its data and the possibility of data leakage. Legacy techniques such as encryption often require the explicit transmission of data to trustworthy parties, thereby inviting data leakage concerns. The ideal collaboration scenario avoids the explicit dissemination of data/code while maintaining experimental fidelity, which is currently accomplished using various techniques such as trusted execution environments, homomorphic encryption, differential privacy, and multimatrix masking. These techniques, however, often necessitate a trade-off between trust, efficiency, and utility. This article extends a previously proposed technique called the directed infusion of data (DIOD) that ensures data privacy, allows for scalable obfuscation, and combats the risk of data leakage without compromising utility. The experiments discussed in this article examine a regression-type scenario using DIOD with the goal of preserving the inferential link between two variables. Using the point-kinetics equations, regression experiments compare the performance of a model trained using the original data to that of a model trained using the obfuscated data, which produced identical results. Our claim is further strengthened by an information theoretic proof and experiment, which showed that the inferential content between variables remains the same after obfuscation, thereby avoiding the required communication of the proprietary data.

47 - OTHER INSTRUMENTATION↗
DOE OSTI2022

Inferring small-sample-reactivity worth from oscillation experiment with auto-corrected local flux

To estimate reactivity changes as a function of time from the signal of detectors count rates, one usually uses point kinetic equations. However, when detectors are located inside the core, it is often necessary to compute a correction factor to account for the flux-shape deformation (i.e. the change of detector efficiency that may have been induced by neutron absorbers movements). Various simulation methods have been developed depending on transients observed. They range from the resolution of static equations up to full-time-dependent neutron transport in 3D. However, those methods require high computational-cost calculations of local reaction rates that rely on nuclear data evaluations. Therefore, a 'measured' reactivity is never determined through pure measurements. Taking advantage of the known oscillation movement of a small sample at the center of the MINERVE reactor, the innovative algorithm presented in this paper derives the correction factor needed to get no time drift on reactivity steps. Derived correction factors are in good agreement with those obtained independently with TRIPOLI-4 through sensitivity analysis. Low sensitivity to the sample composition and moderate sensitivity to the detectors position reinforce the reliability of those results. This application paves the way to almost pure nuclear data reactivity measurements. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2022

INITIAL EXPLORATION OF A NOVEL TRANSIENT ARREST SYSTEM INVOLVING FUEL HEATING

A preliminary analysis on a novel accident response system to diminish the severity of supercritical transients was conducted. The novel accident response system, called the instant shock arrest system, involves using electricity to heat the nuclear fuel at the onset of a large accidental reactivity insertion. This system is specifically designed for reactors with metallic fuel, such that the fuel is capable of conducting electricity, and being resistively heated. A reactor dynamics model of the advanced test reactor was created using the point kinetics equations and a linear reactivity feedback model to simulate how the system would effect the maximum fuel temperatures experienced during the transient. Transients with the instant shock arrest system were compare to those without it. It was found that the instant shock arrest system initially heated the fuel more than the unaffected transient but the negative reactivity inserted from such heating was enough to lower the maximum fuel temperature experienced during the transient. After simulating six different accident scenarios with reactivity insertions ranging from 0.5 \$ to 1.3 \$, it was found that the an optimal system response could reduce peak fuel temperatures during the transient by 3.5\% to 5\%. Furthermore, discussion was given on how the optimal system response could be obtained using relatively simple numerical optimization algorithms due to the smoothness of the optimization problem.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2024

Creation Synthetic Data to Train a Digital Twin to Predict Reactor Operations

Understanding techniques to strengthen the nuclear safeguards regime is crucial in preventing nuclear proliferation due to recent advancements in the nuclear energy industry such as Generation IV reactors and microreactors. Prior to the construction of a nuclear power plant, it is necessary to understand the proliferation potential of the plant's reactor. Digital twins serve as a unique solution to recognizing reactor behavior indicative of nuclear proliferation. A digital twin is defined as a virtual model that works in unison to represent a physical asset, with a transference of data between the virtual and physical assets [1]. This work serves as validation for training a digital twin on synthetic data fabricated via means of Serpent reactor physics and point kinetics equations simulations. In this case this work is based on parameters of Idaho State University's AGN-201 reactor. The synthetic data can then be utilized to train machine learning models in the future to further investigate the utility of these methods. The accuracy of the predicted data is measured against real operational data to verify the reliability of the synthetic data creation methods and decide whether these methods should be used in the future to inform inspectors of a reactor's proliferation potential.

98 NUCLEAR DISARMAMENT, SAFEGUARDS, AND PHYSICAL P↗
DOE OSTI2025

Nuclear Thermal Rocket Emulator for a Hardware-in-the-Loop Test Bed

To support NASA’s mission to use nuclear thermal rockets for future Mars missions, an instrumentation and control test bed has been built at Oak Ridge National Laboratory. The system is designed as a hardware-in-the-loop test bed for testing control elements and autonomous control algorithms for nuclear thermal propulsion rockets. The mock reactor system consists of a modular and scalable framework, using inexpensive components and open-source software. The hardware system consists of a two-phase flow loop and a mock reactor with six control drums. A single-board computer (NVIDIA Jetson) handles reactor core emulation and hosts a message queuing telemetry transport broker that allows user-deployed control algorithms to interact with the system hardware. The reactor emulator receives sensor data from the hardware and provides the simulated performance of the reactor under steady-state, transient, and fault conditions. The emulator uses a reactivity lookup table and the point kinetics equations to solve for the reactor dynamics in real time. Emulated reactor dynamics and sensor input inform the autonomous control algorithm’s decision-making in a closed-loop manner. The current system is capable of operating at 10 Hz, but faster cycle rates are an area of ongoing research. This test bed will enable NASA and other space vendors to rigorously test their autonomous control systems for NTP rockets under transient (reactor startup and shutdown), steady-state, and fault conditions to reduce development time and risk for autonomous control systems in future missions.

autonomous control↗
DOE OSTI2022

Stability margin analysis of the Holos-quad microreactor design

A simplified point model of the Holos-Quad microreactor is introduced. The model is based on the point kinetics equations coupled to three heat balance equations, representing the mean temperatures of the fuel particles, the graphite moderator and the helium coolant. The differential equations are converted to the frequency domain, enabling the construction of the closed-loop reactor transfer function. Using this function the stability margins of the core design is analyzed for various power levels. It is shown that the gain margins approaches infinity, demonstrating the stability of the reactor for all power levels. The phase margin at nominal power is about 60 degrees, however it shows a non-monotonous dependence on power, with minimal value obtained for about 10% of nominal power. As power level is further increased, the phase margin also increases, demonstrating the reactor becomes more stable. This behavior may be of high importance in load-follow scenarios, where the power level of the reactor changes with time. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗