Computationally Efficient Optimal Control for Unstable Power System Models

Abstract

In this article, the focus is mainly on gaining the optimal control for the unstable power system models and stabilizing them through the Riccati-based feedback stabilization process with sparsity-preserving techniques. We are to find the solution of the Continuous-time Algebraic Riccati Equations (CAREs) governed from the unstable power system models derived from the Brazilian Inter-Connected Power System (BIPS) models, which are large-scale sparse index-1 descriptor systems. We propose the projection-based Rational Krylov Subspace Method (RKSM) for the iterative computation of the solution of the CAREs. The novelties of RKSM are sparsity-preserving computations and the implementation of time-convenient adaptive shift parameters. We modify the Low-Rank Cholesky-Factor integrated Alternating Direction Implicit (LRCF-ADI) technique-based nested iterative Kleinman–Newton (KN) method to a sparse form and adjust this to solve the desired CAREs. We compare the results achieved by the Kleinman–Newton method with that of using the RKSM. The applicability and adaptability of the proposed techniques are justified numerically with MATLAB simulations. Transient behaviors of the target models are investigated for comparative analysis through the tabular and graphical approaches.

A Multi-Rate Simulation Strategy Based on the Modified Time-Domain Simulation Method and Multi-Area Data Exchange Method of Power Systems

“…To analyze the stability of the interpolation method, the local truncation error (LTE) and global error (GE) for states S 1 and S 2 can be considered [31]. For Systems 1 and 2, the LTE can be expressed as Equation ( 16):…”

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

“…C satisfying the Wilson conditions. (1) Construct the Sylvester equations defned in (25) and solve for X and Y, respectively.…”

“…For solving the Riccati equation without explicit estimation of the ROM, some techniques are available in practice, for example, low-rank alternative direction implicit (LR-ADI), integrated Newton-Kleinman (NK) method, and Krylov subspace associated rational Krylov subspace method (RKSM). Tose methods are derived and analyzed in detail in references [22][23][24][25] and references therein. Te Newton-Kleinman method is a very complex approach and at each Newton step, LR-ADI iterations need to be executed once, which is a very time-laborious task [26][27][28][29].…”

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

“…BT method requires highly time-demanding computation steps with gigantic memory allotment because of the simulation of large dimensional Lyapunov equations, whereas IRKA is comparatively cheaper in computation cost but the stability of the reduced-order matrices is uncertain and there exists no error bound. On the other hand, some more approaches are available where the reduced-order matrices are used implicitly without storing and any suitable matrix factorization techniques are applied to find approximated factored solution of the Riccati equation of the target models, such as, Low-Rank Alternative Direction Implicit (LR-ADI) incorporated Newton-Kleniman (NK) method [27][28][29][30] and system-specific priori-based Rational Krylov Subspace Method (RKSM) [31][32][33][34]. Newton-Kleinman method consists of the time-consuming multi-layer nested iterative approach but no rigid convergence criteria of this method are well-defined and definiteness of the solution matrix is unrestrained, whereas RKSM is more functional for matrix-vector operations and well-suited for symmetric systems but the essence of adjustable shift parameter and lack of proper convergence measure makes it incompetent for the simulations of the Navier-Stokes models.…”