Sensitivity and uncertainty analysis of a reduced-order model of nonlinear BWR dynamics: II adjoint sensitivity analysis

Abstract

This work applies the first order adjoint sensitivity analysis methodology to a reduced-order model of BWR dynamics to compute the exact first order sensitivities of the model’s state-functions with respect to the model’s initial conditions and parameters. These sensitivities are computed using exact forward and adjoint sensitivity functions, which are shown to yield identical results, within machine accuracy, in all of the phase-space regions characterizing this BWR-model, which comprise: (i) the “stable” region, in which the state-functions converge asymptotically to time-independent values; (ii) the “limit-cycle” regions, in which the state-functions oscillate periodically in time, and (iii) the “chaotic” region, in which the state-functions oscillate among infinitely many unstable attractors aperiodically in time. The exact results using the adjoint sensitivity analysis methodology are contrasted with the unreliable results produced by “brute-force” methods using finite-differences, which are often used in the literature to compute approximate response sensitivities to parameters. The finite-difference methods are shown to produce reasonable approximations when used inside the stable region but completely fail when used in the oscillatory regions.The sensitivity analysis results in the “stable” Region 1 indicate that, after an early oscillatory phase characterized by high-amplitude oscillations caused by the initial perturbation, the sensitivities of the state-variable reach the exact time-independent values predicted theoretically. In the “limit-cycle” Regions, the sensitivities of the state functions oscillate with increasing amplitudes among the respective unstable equilibrium points. In the chaotic region, the sensitivities oscillate with amplitudes that increase exponentially in time, reaching very large values (1023) very rapidly after the onset of the initial perturbation, thus confirming the model’s extreme sensitivity to parameter perturbations in this region.The major novelty resulting from this work is the first-ever demonstration that the First-Level Forward Sensitivity System (1st-LFSS) and the First-Level Adjoint Sensitivity System (1st-LASS) can be used reliably to compute the exact 1st-order sensitivities of state functions with respect to the model parameters not only in the stable region in phase-space, but also in the “limit-cycle” and “chaotic” regions, in contradistinction with the failure of methods based on finite-differences. Subsequent work will use the sensitivities presented in this work to compute the uncertainties induced in the reduced-order BWR-model’s state functions by uncertainties in this BWR-model’s parameters and initial conditions.