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Le CASL (Consortium for Advanced Simulation of Light Water Reactors) a ajouté des capacités de modélisation et de simulation directement applicables aux réacteurs à eau pressurisée dans les domaines du transport du rayonnement, de la thermo-hydraulique, de la thermomécanique et de la chimie de surface du combustible, rassemblées dans VERA (Virtual Environment for Reactor Applications). Étant donné le nombre de plateformes de calcul ciblées, allant de vastes technopoles à des ordinateurs haute performance à la pointe de la technologie, il s’est avéré nécessaire de disposer d’au moins deux modèles de simulation dans chaque domaine physique, certains étant dimensionnels, par exemple 2D ou 3D, et d’autres physiques, par exemple l’approche par sous-canal ou la CFD (computational fluid dynamics). À ce jour, on a mis au point un simulateur de coeur haute résolution, caractérisé par un transport du rayonnement basé sur la théorie multigroupe, un appauvrissement isotopique détaillé, la thermo-hydraulique de sous-canal, et, facultativement, la thermomécanique du combustible, chacun de ces composants physiques étant en cours de développement. Des capacités de transfert de solutions, des solveurs itératifs et des algorithmes de résolutions parallèles ont été mises en oeuvre pour venir étayer VERA et ses composants. Une simulation monophase avec le modèle CFD a également été intégrée séparément à une modélisation de la chimie de surface du combustible pour améliorer la représentation des performances thermiques et de dépôts de bore dans la couche de CRUD. Presque tous les modèles utilisent des bibliothèques tierces qui fournissent des solveurs linéaires et non linéaires permettant de tirer profit des avancées réalisées ailleurs.

In this work, a scalable algorithm for model calibration in nuclear engineering applications is presented and tested. The algorithm relies on the construction of surrogate models to replace the original model within the region of interest. These surrogate models can be constructed efficiently via reduced order modeling and subspace analysis. Once constructed, these surrogate models can be used to perform computationally expensive mathematical analyses. This work proposes a surrogate based model calibration algorithm. The proposed algorithm is used to calibrate various neutronics and thermal-hydraulics parameters. The virtual environment for reactor applications-core simulator (VERA-CS) is used to simulate a three-dimensional core depletion problem. The proposed algorithm is then used to construct a reduced order model (a surrogate) which is then used in a Bayesian approach to calibrate the neutronics and thermal-hydraulics parameters. The algorithm is tested and the benefits of data assimilation and calibration are highlighted in an uncertainty quantification study and requantification after the calibration process. Results showed that the proposed algorithm could help to reduce the uncertainty in key reactor attributes based on experimental and operational data.

AbstractThis paper discusses the utilization of an uncertainty quantification methodology for nuclear power plant thermal-hydraulic transient predictions, with a focus on small modular reactors characterized by the integral pressurized water reactor design, to determine the value of completing experiments in reducing uncertainty. To accomplish this via the improvement of the prediction of key system attributes, e.g., minimum departure from nucleate boiling ratio, a thermal-hydraulic simulator is used to complete data assimilation for input parameters to the simulator employing experimental data generated by the virtual reactor. The mathematical approach that is used to complete this analysis depends upon whether the system responses, i.e., sensor signals, and the system attributes are or are not linearly dependent upon the parameters. For a transient producing mildly nonlinear response sensitivities, a Bayesian-type approach was used to obtain the a posteriori distributions of the parameters assuming Gaussian distributions for the input parameters and responses. For a transient producing highly nonlinear response sensitivities, the Markov chain Monte Carlo method was utilized based upon Bayes’ theorem to estimate the a posteriori distributions of the parameters. To evaluate the value of completing experiments, an optimization problem was formulated and solved. The optimization addressed both the experiments to complete and the modifications to be made to the nuclear power plant made possible by using the increased margins resulting from data assimilation. The decision variables of the experiment optimization problem include the selection of sensor types and locations and experiment type imposing realistic constraints. The decision variables of the nuclear power plant modification optimization problem include various design specifications, e.g., power rating, steam generator size, and reactor coolant pump size, with the objective of minimizing cost as constrained by required margins to accommodate the uncertainty. Since the magnitude of the uncertainty is dependent upon the experiments via data assimilation, the nuclear power plant optimization problem is treated as a suboptimization problem within the experiment optimization problem. The experiment optimization problem objective is to maximize the net savings, defined as the savings in nuclear power plant cost due to the modified design specifications minus the cost of the experiments. Both the experiment and the nuclear power plant optimization problems were solved using the simulated annealing method.

The safe and economical design of new, innovative nuclear reactors will require uncertainty reduction in basic nuclear data that are input to simulations used during reactor design. These data uncertainties propagate to uncertainties in design responses, which in turn require the reactor designer to incorporate additional safety margins into the design, often increasing the cost of the reactor. Therefore, basic nuclear data need to be improved, and this is accomplished through experimentation, which is often done using cold critical experiments. Considering the high cost of nuclear experiments, it is desired to have an optimized experiment that will provide the experimental data needed for maximum uncertainty reduction in the design responses. However, the optimization of the experiment is coupled to the reactor design itself because with reduced uncertainty in the design responses the reactor design can be re-optimized. It is thus desired to find the experiment design that gives the most optimized reactor design. Solution of this nested optimization problem is made possible by the use of the simulated annealing algorithm. Cost values for experiment design specifications and reactor design specifications are estimated and used to compute a total savings by comparing the a posteriori reactor cost to the a priori cost accounting for the offsetting cost of the experiment. This was done for the Argonne National Laboratory-developed Advanced Burner Test Reactor design concept employing a modified Zero Power Physics Reactor as the experimental facility.

AbstractThis paper introduces the concepts and derives the mathematical theory of efficient subspace methods (ESMs) applied to the simulation of large-scale complex models, of which nuclear reactor simulation will serve as a test basis. ESMs are intended to advance the capabilities of predictive simulation to meet the functional requirements of future energy system simulation and overcome the inadequacies of current design methods. Some of the inadequacies addressed by ESM include lack of rigorous approach to perform comprehensive validation of the multitudes of models and input data used in the design calculations and lack of robust mathematical approaches to enhance fidelity of existing and advanced computational codes. To accomplish these tasks, the computational tools must be capable of performing the following three applications with both accuracy and efficiency: (a) sensitivity analysis of key system attributes with respect to various input data; (b) uncertainty quantification for key system attributes; and (c) adaptive simulation, also known as data assimilation, for adapting existing models based on the assimilated body of experimental information to achieve the best possible prediction accuracy. These three applications, involving large-scale computational models, are now considered computationally infeasible if both the input data and key system attributes or experimental information fields are large. This paper will develop the mathematical theory of ESM-based algorithms for these three applications. The treatment in this paper is based on linearized approximation of the associated computational models. Extension to higher-order approximations represents the focus of our ongoing research.

Use of adaptive simulation is intended to improve the fidelity and robustness of important core attribute predictions such as core power distribution, thermal margins, and core reactivity. Adaptive simulation utilizes a selected set of past and current reactor measurements of reactor observables, i.e., in-core instrumentation readings, to adapt the simulation in a meaningful way. The companion paper, “Adaptive Core Simulation Employing Discrete Inverse Theory - Part I: Theory,” describes in detail the theoretical background of the proposed adaptive techniques. This paper, Part II, demonstrates several computational experiments conducted to assess the fidelity and robustness of the proposed techniques. The intent is to check the ability of the adapted core simulator model to predict future core observables that are not included in the adaption or core observables that are recorded at core conditions that differ from those at which adaption is completed. Also, this paper demonstrates successful utilization of an efficient sensitivity analysis approach to calculate the sensitivity information required to perform the adaption for millions of input core parameters. Finally, this paper illustrates a useful application for adaptive simulation - reducing the inconsistencies between two different core simulator code systems, where the multitudes of input data to one code are adjusted to enhance the agreement between both codes for important core attributes, i.e., core reactivity and power distribution. Also demonstrated is the robustness of such an application.

The focus of this overview for this issue of Nuclear Technology, which contains papers presented at the American Nuclear Society Advances in Nuclear Fuel Management III (ANFM-III) 2004 topical meeting, is to introduce the subject of nuclear fuel management for light water reactors. A total of 23 papers was presented on this topic at ANFM-III. Nuclear fuel management involves making the so-called out-of-core and in-core decisions. Simply put, the out-of-core decisions address the attributes of the new (fresh) fuel that will be fabricated and the partially burnt (shuffled) fuel to reinsert into the core for additional energy production. The in-core decisions address where the fresh and burnt fuel along with burnable poisons should be located in the core. The above applies to batch refueling strategies, e.g., pressurized water reactors and boiling water reactors (BWRs). For BWRs, additional in-core decisions enter to address control rod pattern paired with core flow rate as a function of burnup. It is obvious that the out-of-core and in-core decisions are coupled.The objective of nuclear fuel management is to minimize the cost of electrical energy generation subject to operational and safety constraints. Since fuel resides in the core for several cycles, a multicycle assessment is required to make nuclear fuel management decisions. For nearly four decades there has been an effort to develop automated computational capability to assist the reload core nuclear design engineer in making nuclear fuel management decisions. This development has ranged from employment of heuristic rules to utilization of mathematical optimization approaches. This overview reviews the development of nuclear fuel management optimization capabilities by first defining the problem, then describing current capabilities, and finally projecting where future capabilities need to be developed to support the needs of reload core nuclear design engineers.

Use of adaptive simulation is intended to improve the fidelity and robustness of important core attribute predictions such as core power distribution, thermal margins, and core reactivity. Adaptive simulation utilizes a selected set of past and current reactor measurements of reactor observables, i.e., in-core instrumentation readings, to adapt the simulation in a meaningful way. A meaningful adaption will result in high-fidelity and robust adapted core simulator models. To perform adaption, we propose an inverse theory approach in which the multitudes of input data to core simulators, i.e., reactor physics and thermal-hydraulic data, are to be adjusted to improve agreement with measured observables while keeping core simulator models unadapted. At first glance, devising such adaption for typical core simulators with millions of input and observables data would spawn not only several prohibitive challenges but also numerous disparaging concerns. The challenges include the computational burdens of the sensitivity-type calculations required to construct Jacobian operators for the core simulator models. Also, the computational burdens of the uncertainty-type calculations required to estimate the uncertainty information of core simulator input data present a demanding challenge. The concerns however are mainly related to the reliability of the adjusted input data. The methodologies of adaptive simulation are well established in the literature of data adjustment. We adopt the same general framework for data adjustment; however, we refrain from solving the fundamental adjustment equations in a conventional manner. We demonstrate the use of our so-called Efficient Subspace Methods (ESMs) to overcome the computational and storage burdens associated with the core adaption problem. We illustrate the successful use of ESM-based adaptive techniques for a typical boiling water reactor core simulator adaption problem.

AbstractA Newton-Krylov iterative solver has been developed to reduce the CPU execution time of boiling water reactor (BWR) core simulators implemented in the core simulator part of the Fuel Optimization for Reloads Multiple Objectives by Simulated Annealing for BWR (FORMOSA-B) code, which is an in-core fuel management optimization code for BWRs. This new solver utilizes Newton’s method to explicitly treat strong nonlinearities in the problem, replacing the traditionally used nested iterative approach. Newton’s method provides the solver with a higher-than-linear convergence rate, assuming that good initial estimates of the unknowns are provided. Within each Newton iteration, an appropriately preconditioned Krylov solver is utilized for solving the linearized system of equations. Taking advantage of the higher convergence rate provided by Newton’s method and utilizing an efficient preconditioned Krylov solver, we have developed a Newton-Krylov solver to evaluate the three-dimensional, two-group neutron diffusion equations coupled with a two-phase flow model within a BWR core simulator. Numerical tests on the new solver have shown that speedups ranging from 1.6 to 2.1, with reference to the traditional approach of employing nested iterations to treat the nonlinear feedbacks, can be achieved. However, if a preconditioned Krylov solver is employed to complete the inner iterations of the traditional approach, negligible CPU time differences are noted between the Newton-Krylov and traditional (Krylov) approaches.

AbstractA Super-Nodal method is developed to improve computational efficiency of core simulations for three-dimensional (3-D) core neutronics models. Computational performance of the neutronics model is increased by reducing the number of spatial nodes used in the core modeling. The Super-Nodal method reduces the errors associated with the use of coarse nodes in the analyses by providing a new set of cross sections and discontinuity factors for the new nodalization. These so-called homogenization parameters are obtained by employing a consistent collapsing technique.During this research a new type of singularity, namely, “fundamental mode singularity,” is addressed in the analytical nodal method solution. The “coordinate shifting” approach is developed as a method to address this singularity. Also, the “buckling shifting” approach is developed as an alternative to address the “zero buckling singularity.” In the course of addressing the treatment of these singularities, an effort was made to provide better and more robust results from the Super-Nodal method by developing several new methods for determining the collapsed diffusion coefficient. A simple error analysis based on the relative residual in the 3-D few-group diffusion equation at the fine mesh level is also introduced in this work.

AbstractA methodology has been developed whereby a three-dimensional (3-D) geometry, nodal expansion method (NEM), pressurized water reactor (PWR) core simulator model is collapsed to form an equivalent two-dimensional (2-D) geometry model that preserves approximately, but with negligible loss of fidelity, the global quantities and axially integrated reaction rates and surface currents of the 3-D model. In comparison with typical licensed-quality 3-D models, the 2-D collapsed NEM model typically requires a factor of 50 less computational time and exhibits root-mean-square (rms) assembly relative power fraction errors, as compared with the original 3-D model, of 5 × 10–3over an entire fuel cycle, and average maximum errors over the fuel cycle of 1 × 10–2. The collapse methodology includes a pin reconstruction methodology, which exhibits assemblywise rms pin power errors of 5 × 10–3and average maximum assemblywise pin power errors of 1.2 × 10–2. When coupled with FORMOSA-P’s existing assembly power response generalized perturbation theory reactor core simulator, this permits loading-pattern evaluations at a speed approximately 100 to 150 times faster than full, 3-D models, providing the computational efficiency needed for efficient incore fuel management optimization using stochastic methods.

AbstractAs part of the continuing development of the boiling water reactor in-core fuel management optimization code FORMOSA-B, the cold shutdown margin (SDM) constraint evaluator has been improved. The SDM evaluator in FORMOSA-B had been a first-order accurate Rayleigh quotient variational technique. It was deemed unreliable for difficult perturbed loading patterns (LPs) and thus was replaced by a high-fidelity, robust, computationally efficient evaluator. The new model is based on the solution of the one-group diffusion equation using approximate albedo boundary conditions for a three-dimensional, variable axial node, 10 × 10 assembly subregion around the stuck rod location. The fidelity and robustness of the model are first demonstrated by performing calculations on difficult perturbed LPs and for different plant cores. It is shown that the SDM reactivity is estimated within 40 pcm for the highest worth rod and that the speedup factors are 50 to 100 for small cores (and even more for larger cores) in comparison to the full-core three-dimensional simulations. Next, the successful implementation of the model in imposing the SDM constraint for FORMOSA-B’s adaptive simulated annealing (SA)-based optimization strategy is presented. The results demonstrate SA’s ability to remove large SDM violations (>700 pcm) along with thermal margin and critical flow constraint violations. Finally, the importance of having the SDM constraint on during optimization is shown by comparing results with a simulation in which the constraint is off.

As part of the continuing development of the boiling water reactor in-core fuel management optimization code FORMOSA-B, the fidelity of the core simulator has been improved and a control rod pattern (CRP) sampling capability has been added. The robustness of the core simulator is first demonstrated by benchmarking against core load-follow depletion predictions of both SIMULATE-3 and MICROBURN-B2 codes. The CRP sampling capability, based on heuristic rules, is next successfully tested on a fixed fuel loading pattern (LP) to yield a feasible CRP that removes the thermal margin and critical flow constraint violations. Its performance in facilitating a spectral shift flow operation is also demonstrated, and then its significant influence on the cost of thermal margin is presented. Finally, the heuristic CRP sampling capability is coupled with the stochastic LP optimization capability in FORMOSA-B—based on simulated annealing (SA)—to solve the combined CRP-LP optimization problem. Effectiveness of the sampling in improving the efficiency of the SA adaptive algorithm is shown by comparing the results to those obtained with the sampling turned off (i.e., only LP optimization is carried out for the fixed reference CRP). The results presented clearly indicate the successful implementation of the CRP sampling algorithm and demonstrate FORMOSA-B’s enhanced optimization features, which facilitate the code’s usage for broader optimization studies.

AbstractThe computational capability to determine optimal core loading patterns (LPs) for boiling water reactors (BWRs) given a reference control rod program has been developed. The design and fidelity of the reference BWR core simulator are presented. The placement of feed and reload fuel is solved by an adaptive optimization by simulated annealing (OSA) objective algorithm. Objective functions available for BWR fuel management are maximization of end-of-cycle core reactivity, minimization of peak linear power density, maximization of critical power ratio, maximization of region average discharge burnup, and minimization of total reload cost. Constraints include thermal and fuel exposure related limits and cycle energy production, when appropriate. The results presented demonstrate the utility of OSA to improve LPs in this highly nonlinear and constrained search space.

AbstractBoiling water reactor (BWR) loading pattern assessment requires solving the two-group, nodal form of the neutron diffusion equation and drift-flux form of the fluid equations simultaneously because these equation sets are strongly coupled via nonlinear feedback. To reduce the computational burden associated with the calculation of the core attributes (that is, core eigenvalue and thermal margins) of a perturbed BWR loading pattern, the analytical and numerical aspects of a higher order generalized perturbation theory (GPT) method, which correctly addresses the strong nonlinear feedbacks of two-phase flow, have been established. Inclusion of Jacobian information in the definition of the generalized flux adjoints provides for a rapidly convergent iterative method for solution of the power distribution and eigenvalue of a loading pattern perturbed from a reference state. Results show that the computational speedup of GPT compared with conventional forward solution methods demanding consistent accuracy is highly dependent on the number of spatial nodes utilized by the core simulator, varying from superior to inferior performance as the number of nodes increases.

AbstractThe computational capability of automatically determining the optimal control strategies for pressurized water reactor core maneuvering, in terms of an operating strategy generator (OSG), has been developed. The OSG was developed for use with an on-line, three-dimensional core simulator and applies optimal control theory. To reduce computer run time, the optimization engine employs a one-dimensional axial core model. A method has been developed for generating a consistent one-dimensional axial core model from the three-dimensional on-line core simulator based on the consistent collapse methodology. From the one-dimensional, model-based, optimal control strategy, the associated axial offset versus time is obtained. These axial offsets are subsequently used in the three-dimensional simulator to determine with enhanced accuracy the associated control rod insertions and boration/dilution operations versus time.Various operational objectives are defined as the performance index to be minimized. The axial flux difference limit constraint and the maximum boration/dilution limit constraint are treated as penalty functions added to the performance index. The control rod insertion/withdraw limit constraint is treated as a hard constraint on the control variable. The optimality condition is obtained by applying Pontryagin’s maximum principle for constrained optimization. The resulting nonlinear, two-point boundary-value problem is solved via an iterative approach based on the first-order gradient method.Several sample OSG maneuvering problems have been studied to assess the robustness and efficiency of the optimization search and nonlinear iterations. The algorithm exhibited excellent control of the axial power distribution during maneuvering. For the cases of minimizing the boron system duty during maneuvering, the optimal strategies produced reduced volumes of primary water generated by dilution and boration operations of 12% for beginning-of-cycle cases and 10% for end-of-cycle cases over the volumes generated using heuristic rules.

AbstractAn error analysis is presented of the quartic polynomial nodal expansion method for solving the one-dimensional, neutron diffusion equation that originates from employing the transverse integration technique. Error bound expressions are determined for the L∞ error norms associated with the nodal surface flux and various moments of the nodal flux. Employing several test problems, these global error bounds were found to be conservative, but not excessively, in bounding the true errors Utilizing a functional form of the local error estimate for the node average flux, it is shown that a mesh-doubling technique can be effectively utilized to estimate the required cell size for uniform mesh refinement to achieve a specified global error fidelity. When employed in conjunction with a multigrid acceleration technique, this provides the foundations upon which to develop an adaptive spatial mesh algorithm.