Dieter Hennig

A nodal modal method for the neutron diffusion equation. Application to BWR instabilities analysis

Abstract Fast codes, capable of dealing with three-dimensional geometries, are needed to be able to simulate spatially complicated transients in a nuclear power reactor. In this paper, we propose a modal method to integrate the neutron diffusion equation in which the spatial part has been previously dicretized using a nodal collocation method. For the time integration of the resulting system of differential equations it is supposed that the solution can be expanded as a linear combination of the dominant Lambda modes associated with a static configuration of the reactor core and, using the eigenfunctions of the adjoint problem, a system of differential equations of lower dimension is obtained. This system is integrated using a variable time step implicit method. Furthermore, for realistic transients, it would be necessary to calculate a large amount of modes. To avoid this, the modal method has been implemented making use of an updating process for the modes at each certain time step. Five transients have been studied: a homogeneous reactor, a non-homogeneous reactor, the 3D Langenbuch reactor and two transients related with in-phase and out-of-phase oscillations of Leibstadt NPP. The obtained results have been compared with the ones provided by a method based on a one-step backward discretization formula.

On the regional oscillation phenomenon in BWR's.

Abstract In the last 20 years many papers have been devoted to the study of BWR stability phenomenon. While the physical mechanisms of global power oscillations are well-known, the regional power oscillation phenomenon is not understandable in all details. Our paper should be a contribution to the understanding of conditions under which regional power oscillations can be expected. With this aim we have analyzed, with the system code RAMONA3–12, some stability experiments which were conducted on the NPP Leibstadt. To test the ability of the monitoring system to cope with demanding operation situations, the power oscillations during the experiments were deliberately transformed from the in-phase into the out-of-phase mode, via changing some control rod positions. Hence, we have been able to study the “real world” of a BWR core in the regional oscillation mode. We focused our work on the analysis of the higher mode feedback reactivities (dynamical reactivities) and the calculation of some spatial indices. The feedback reactivities have been calculated with the code LAMBDA-REAC. From the results obtained we conclude that it is in the case of certain specific types of power distribution that a particular mode coupling mechanism can cause regional oscillations to occur.

A Transient Modal Analysis of a BWR Instability Event

To solve the time dependent neutron diffusion equation a modal method, based on the expansion of the neutronic flux in terms of the dominant Lambda modes of a static configuration of the reactor is presented. This method is used to analyse transients of a nuclear power reactor where an instability event can be developed. A simulation of a transient with the same conditions given for the case 9 of Ringhals stability benchmark has been analysed. It is shown that with these conditions an out of phase oscillation associated with the two first azimuthal modes can be developed. These results are corroborated using a power modal decomposition, using the local power distribution provided by RAMONA code. To complete the analysis, the modal feedback reactivities have been calculated to study the coupling mechanism among modes.

A NOVEL RESULT IN THE FIELD OF NONLINEAR STABILITY ANALYSIS OF BOILING WATER REACTORS

The nonlinear stability analysis of boiling water nuclear reactors (BWRs) is conducted with the aid of so-called advanced, well validated, system codes and an advanced reduced order model to build a detailed mathematical understanding of the BWR behavior in the practical relevant parameter space. In the last years, the existence of Hopf-bifurcation points was confirmed by some researchers. In the framework of this paper, a parameter region was analyzed in which the coexistence of different stability states is realized. As a novel result, we found a parameter region in which stable fixed points, unstable limit cycles and stable limit cycles coexist. This system behavior can be explained by a saddle-node bifurcation of cycles (turning point). The existence of this solution type in a BWR system indicates the possibility of large amplitude limit cycle oscillations in the linear stable region.

Remarks on boiling water reactor stability analysis – part 2: stability monitoring

Abstract In part 1 of this article we explained the partly relative complex solution manifold of the differential equations describing the stability behaviour of a BWR, in particular the coexistence of different types of solutions, such as the coexistence of unstable limit cycles and stable fixed points are of interest from the operational safety point of view. The part 2 is devoted to the surveillance of the stability behaviour. We summarize some stability monitoring methods and suggest to support stability tests by RAM-ROM analyses in order to reveal in advance the stability “landscape” of the BWR in a parameter region high sensitive for appearing of linear unstable states. The analysis of an especial stability test, performed at NPP Leibstadt (KKL), makes it clear that the measurement results can only be interpreted by application of bifurcation analysis.

Remarks on boiling water reactor stability analysis - part 1: theory and application of bifurcation analysis

Abstract Modern theoretical methods for analysing the stability behaviour of Boiling Water Reactors (BWRs) are relatively reliable. The analysis is performed by comprehensive validated system codes comprising 3D core models and one-dimensional thermal-hydraulic parallel channel models in the frequency (linearized models) or time domain. Nevertheless the spontaneous emergence of stable or unstable periodic orbits as solutions of the coupled nonlinear differential equations determining the stability properties of the coupled thermal-hydraulic and neutron kinetic (highly) nonlinear BWR system is a surprising phenomenon, and it is worth thinking about the mathematical background controlling such behaviour. In particular the coexistence of different types of solutions, such as the coexistence of unstable limit cycles and stable fixed points, are states of stability, not all nuclear engineers are familiar with. Hence the part I of this paper is devoted to the mathematical background of linear and nonlinear stability analysis and introduces a novel efficient approach to treat the nonlinear BWR stability behaviour with both system codes and so-called (advanced) reduced order models (ROMs). The efficiency of this approach, called the RAM-ROM method, will be demonstrated by some results of stability analyses for different power plants.

Analysis of a BWR Stability Behavior using Generalized Discrete Shannon Functions. Application to Ringhals and Forsmark Benchmarks Data

In this paper we have introduced a new methodology to on-line signal conditioning and monitoring to determine the stability parameters of the BWR NPP, that is, the determination of the effective Decay Ratio DR and the frequency of the main oscillation causing instability events. This method is based on the generalized discrete Shannon function convolution, which removes the noise and filters the signal in a specified frequency band. We have focused our attention in noise signals, first on analytic ones to check how the algorithm works, and then we have tested it with some real neutron signals. The algorithm works very well with dirty real signals providing good results, even in the case of short time series. Main attemption has been focused on decomposing signals to detect when a global and/or a regional oscillations are taking place in a BWR. This methodology can be implemented in on-line monitors to determine the stability parameters of the BWR reactors, that is, the determination of the effective decay ratio (DR) and the frequency of the main oscillation causing instability events.