Dresden Scientific Workshop on Reactor Dynamics and Safety 2012 ROM's of ROM's make sense: semi-analytical approach to simplified reduced order models, inertial manifolds and global bifurcations in BWR dynamics (original) (raw)

One of the goals of nuclear power systems design and operation is to restrict the possible states of certain critical subsystems to remain inside a certain bounded set of admissible states and state variations. In the framework of an analytic or numerical modeling process of a BWR power plant, this could imply first to find a suitable approximation to the solution manifold of the system of nonlinear partial differential equations describing the stability behavior, and then a classification of the different solution types concerning their relation with the operational safety of the power plant. Inertial manifold theory gives a foundation for the construction and use of reduced order models (ROM's) of reactor dynamics to discover and characterize meaningful bifurcations that may pass unnoticed during digital simulations done with full scale computer codes of the nuclear power plant. The main aspects of approximate inertial manifolds and forms are briefly reviewed in the introduction of the paper.A complete numerical study of reactor dynamics using a realistic ROM currently involves the digital simulation of the behavior of approximately twenty state variables interrelated by a corresponding system of coupled nonlinear ordinary differential equations. The success of hybrid analytical-numerical bifurcation codes to detect interesting behavior, such as global bifurcations in BWR's, may be enhanced by studying suitable simplifications of ROM's, that is ROM's of ROM's. A previous generalization of the classical March-Leuba's model of BWR is briefly reviewed and a nonlinear integral-differential equation in the logarithmic power is derived. The asymptotic method developed by Krilov, Bogoliubov and Mitropolsky (KBM) is applied to obtain approximate equations of evolution for the amplitude and the phase of a manifold of oscillatory solutions jointly with a relation between an offset and the abovementioned amplitude. First, to exemplify the method working with a simpler problem, the KBM tentative solution (ansatz) is applied to construct approximate solutions of, and to study local bifurcations in, a van der Pol equation with continuous and discrete distribution of time delays. Then, the afore-mentioned ansatz is applied to the full nonlinear integral-differential equation of the BWR model. Analytical formulae are derived for the offset, the rate of change in the phase (the instantaneous frequency of oscillation) and the rate of change in the amplitude of oscillation, given as functions of the amplitude and the model parameters (steady state power and coolant flow, temperature and void reactivity coefficients, fuel to coolant heat transfer coefficient and other parameters from neutronics and thermal hydraulics). The obtained analytical formulae are applied to start a semi-analytical, mainly qualitative, approach to bifurcations and stability of the steady states located in different regions of parameters space. This includes a qualitative discussion of the possibility of both, super and subcritical Poincaré-Andronov-Hopf bifurcations, as well as a Bautin's bifurcation scenario. The preliminary qualitative results outlined in this study are consistent with results of recent digital simulations done with a full-scale reduced order model of BWR (PSI-TU Valencia-TU Dresden) and with the results obtained with the application of hybrid approaches to bifurcation theory done with the simplified March-Leuba's model of BWR.