SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models

Abstract

We propose an efficient sparsity-preserving reduced-order modelling approach for index-1 descriptor systems extracted from large-scale power system models through two-sided projection techniques. The projectors are configured by utilizing Gramian based singular value decomposition (SVD) and Krylov subspace-based reduced-order modelling. The left projector is attained from the observability Gramian of the system by the low-rank alternating direction implicit (LR-ADI) technique and the right projector is attained by the iterative rational Krylov algorithm (IRKA). The classical LR-ADI technique is not suitable for solving Riccati equations and it demands high computation time for convergence. Besides, in most of the cases, reduced-order models achieved by the basic IRKA are not stable and the Riccati equations connected to them have no finite solution. Moreover, the conventional LR-ADI and IRKA approach not preserves the sparse form of the index-1 descriptor systems, which is an essential requirement for feasible simulations. To overcome those drawbacks, the fitting of LR-ADI and IRKA based projectors from left and right sides, respectively, desired reduced-order systems attained. So that, finite solution of low-rank Riccati equations, and corresponding feedback matrix can be executed. Using the mechanism of inverse projection, the Riccati-based optimal feedback matrix can be computed to stabilize the unstable power system models. The proposed approach will maintain minimized ℌ2 -norm of the error system for reduced-order models of the target models.

Sparsity-Preserving Two-Sided Iterative Algorithm for Riccati-Based Boundary Feedback Stabilization of the Incompressible Navier–Stokes Flow

“…Assuming Z q is the low-rank Gramian factor of the Gramian Q that needs to be determined. In reference [50], a practically feasible technique is derived to overcome this situation. Ten, ‖G(s)‖ H 2 can be written as follows:…”

Memory-Efficient Interpolatory Projection Techniques for the Stabilization of Incompressible Navier-Stokes Flows

“…The optimal feedback matrix for the system (1) via X can be achieved in plenty of ways. When reduced-order matrices must be stored and used for subsequent manipulations in some of them, optimal feedback matrices can be approximated from the reduced-order feedback matrix using the inverse projection scheme or any other counter approach, such as the Singular-Value Decomposition (SVD)-based Balanced Truncation (BT) [15][16][17][18], the Krylov subspace-based Iterative Rational Krylov Algorithm (IRKA) [19][20][21][22], and a recently developed hybrid approach Iterative SVD-Krylov Algorithm [23][24][25][26]. In those methods, storing the reduced-order matrices claims redundant memory allocation and delays the convergence of the simulation.…”