Analysis of nonlinear modal interaction in stressed power systems using POD-Galerkin characterization

Vázquez, Sergio; Barocio, E.

doi:10.1109/iceee.2009.5393447

Cited by 2 publications

(1 citation statement)

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“…Non‐linear POD/Galerkin ROMs have been constructed in a number of applications. An analytical technique based on the POD method and Galerkin projection was presented for the analysis and characterization of inter‐area oscillations in stressed power systems in . Dynamical models for bifurcation analysis and control of self‐sustained cavity oscillations, also based on the POD/Galerkin approach, were examined by Rowley et al in .…”

Section: Introductionmentioning

confidence: 99%

Efficient non‐linear proper orthogonal decomposition/Galerkin reduced order models with stable penalty enforcement of boundary conditions

Kalashnikova

Barone

2012

Numerical Meth Engineering

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An efficient, stability-preserving model reduction technique for non-linear initial boundary value problems whose solutions exhibit inherently non-linear dynamics such as metastability and periodic regimes (limit cycles) is developed. The approach is based on the 'continuous' Galerkin projection approach in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product. The reduced order model (ROM) basis is constructed via a proper orthogonal decomposition (POD). In general, POD basis modes will not satisfy the boundary conditions of the problem. A weak implementation of the boundary conditions in the ROM based on the penalty method is developed. Asymptotic stability of the ROM with penalty-enforced boundary conditions is examined using the energy method, following linearization and localization of the governing equations in the vicinity of a stable steady solution. This analysis, enabled by the fact that a continuous representation of the reduced basis is employed, leads to a model reduction method with an a priori stability guarantee. The approach is applied to two non-linear problems: the Allen-Cahn (or 'bistable') equation and a convection-diffusion-reaction system representing a tubular reactor. For each of these problems, bounds on the penalty parameters that ensure asymptotic stability of the ROM solutions are derived. The non-linear terms in the equations are handled efficiently using the 'best points' interpolation method proposed recently by Peraire, Nguyen et al. Numerical experiments reveal that the POD/Galerkin ROMs with stability-preserving penalty boundary treatment for the two problems considered, both without as well as with interpolation, remain stable in a way that is consistent with the solutions to the governing continuous equations and capture the correct non-linear dynamics exhibited by the exact solutions to these problems. Published 2012.

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Section: Introductionmentioning

confidence: 99%