UK Nonlinear News, May 2004
Boiling water reactor (BWR) stability analysis is usually carried out using large system codes. However, because of the large computational efforts required, such codes cannot in practice be employed for the detailed investigation of the complete manifold of solutions of the nonlinear differential equations describing the BWR system. In this context, reduced order models, containing a minimum number of system equations describing the most important physical phenomena, become necessary to provide deeper insight into the physical mechanisms underlying the different instability phenomena observed in BWRs, e.g. in-phase and out-of-phase power oscillations.
A novel analytical, reduced order model has been developed to simulate the different types of instabilities encountered in heated channels and BWRs, viz. density wave oscillations (DWOs), as well as in-phase and out-of-phase oscillations in the reactor core. The complete model comprises three main parts: spatial lambda-mode neutron kinetics with the fundamental and first azimuthal modes, fuel heat conduction dynamics, and core thermal-hydraulics based on a drift flux model representation of the two-phase flow.
Stability and semi-analytical bifurcation analysis is carried out for a purely thermal- hydraulic system (heated channel), as well as for a complete BWR (represented via two-channel nuclear-coupled thermal-hydraulics), using the current reduced order model in conjunction with the bifurcation code BIFDD. The impact of the drift flux parameters on the stability boundary (SB) and nature of bifurcation has thereby been investigated. Results show that both sub- and supercritical Hopf bifurcations are encountered along the stability boundary. Using a drift flux model instead of a homogeneous equilibrium model for the two-phase flow is found to have significant effects on the SB, as well as on the nature of Hopf bifurcation. For independent confirmation of the results of the semi-analytical bifurcation analyses, as well as to evaluate the system behaviour in regions away from the stability boundary, numerical integration has been carried out of the set of ordinary differential equations (ODEs) involved in each case.
With each of the two channels of the currently developed BWR reduced order model representing half of the reactor core, it has been possible to apply it to the investigation of out-of-phase instability phenomena as well. First, the stability limits for in-phase and out-of-phase BWR oscillation modes for a generic case are determined in parameter space. An in-depth investigation is then performed of the properties of the elements of the eigenvectors associated with these two oscillation modes. Results show that analysing the properties of the eigenvectors can provide full information as regards the corresponding oscillation mode (in-phase or out-of-phase) without solving the set of system ODEs. In addition, such analysis conclusively shows that in-phase and out-of-phase oscillation modes in a BWR are whole-system mechanisms and not just limited to the excitation of the fundamental and first azimuthal modes of the neutron flux.
In parallel to the generic studies with the reduced order model, a detailed local bifurcation analysis has been performed at two representative operational points for the Leibstadt and Ringhals-1 BWR nuclear power plants using the complex system code RAMONA. The goal in this analysis is to demonstrate how the system solution (behaviour) can, in some situations, vary in a significant manner when a certain parameter, e.g. the mass flow rate, is changed by small amounts. First, a correspondence hypothesis is proposed, underlining the unique relationship for BWRs between a stable (unstable) limit cycle solution and the occurrence of a supercritical (subcritical) Hopf bifurcation. This RAMONA analysis carried out clearly shows that stability and bifurcation analysis expertise using reduced order models is indeed very important for the understanding and appropriate interpretation of certain complicated nonlinear phenomena that are sometimes observed in simulations using system codes. Thus, the present investigations have revealed, for the first time, the occurrence of a subcritical Hopf bifurcation during BWR stability analysis using a system code. Such a study is thereby shown to allow the determination and characterisation of local stability boundaries within the exclusion area of a BWR's power-flow map.
Finally, in order to assess the applicability (as well as limitations) of the currently developed reduced order in a more quantitative manner, it has been applied to the analysis of a specific Leibstadt operational point. Comparison of the results obtained with those of RAMONA show that, although the current reduced order model could adequately predict certain characteristics, it was not able to correctly predict some others because of the highly simplified reactor core geometry, the uncertainties in evaluating the design and operating parameters, as also the limitations of the feedback reactivity model employed. The main conclusion to be drawn in this context is that, although reduced order models do indeed allow an in-depth understanding of the complex processes determining BWR stability (through the possibility of conducting fast and detailed semi-analytical bifurcation analysis), they need to be considered as complementary tools to complex system codes, and not as alternatives.2