Subspace Methods for Computing the Pseudospectral Abscissa and the Stability Radius

Kreßner, Daniel; Vandereycken, Bart

doi:10.1137/120869432

Cited by 38 publications

(42 citation statements)

References 24 publications

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“…"; A/ and r C .A/ can be accelerated. In [42], subspace acceleration techniques for both quantities were proposed for which, under certain mild conditions, locally superlinear convergence was proven. In practice, these subspace methods exhibit a quite robust convergence behavior, their local convergence is observed to be even locally quadratic, and they can be much faster than the methods from [24,31].…”

Section: Lemma 1 ([57])mentioning

confidence: 99%

Distance Problems for Linear Dynamical Systems

Kreßner

Voigt

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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This chapter is concerned with distance problems for linear timeinvariant differential and differential-algebraic equations. Such problems can be formulated as distance problems for matrices and pencils. In the first part, we discuss characterizations of the distance of a regular matrix pencil to the set of singular matrix pencils. The second part focuses on the distance of a stable matrix or pencil to the set of unstable matrices or pencils. We present a survey of numerical procedures to compute or estimate these distances by taking into account some of the historical developments as well as the state of the art. IntroductionConsider a linear time-invariant differential-algebraic equation (DAE)with coefficient matrices E; A 2 R n n , a sufficiently smooth inhomogeneity f W OE0; 1/ ! R n , and an initial state vector x 0 2 R n . When E is nonsingular, (20.1) is turned into a system of ordinary differential equations by simply multiplying with E 1 on both sides. The more interesting case of singular E arises, for example, when imposing algebraic constraints on the state vector x.t/ 2 R n . Equations of this type play an important role in a variety of applications, including electrical circuits [54,55] and multi-body systems [48]. The analysis and numerical solution of more general DAEs (which also take into account time-variant coefficients, nonlinearities and time delays) is a central part of Volker Mehrmann's work, as witnessed by the monograph [43] and by the several other chapters of this book

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Section: Lemma 1 ([57])mentioning

confidence: 99%

Distance Problems for Linear Dynamical Systems

Kreßner

Voigt

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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“…The dynamic models of (6) and (10) as well as the algebraic equations (4a) and (7) are computed using the Power System Toolbox (PST) [26]. α ε (A) is evaluated using the Matlab code provided with [27] with a tolerance of 1e−12. The gradients of α ε (A) are computed numerically and the Matlab Optimization Toolbox is used to compute the local optimum.…”

Section: Test Casesmentioning

confidence: 99%

Dynamics-aware optimal power flow

Mallada

Tang

2013

52nd IEEE Conference on Decision and Control

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Abstract-The development of open electricity markets has led to a decoupling between the market clearing procedure that defines the power dispatch and the security analysis that enforces predefined stability margins. This gap results in market inefficiencies introduced by corrections to the market solution to accommodate stability requirements. In this paper we present an optimal power flow formulation that aims to close this gap. First, we show that the pseudospectral abscissa can be used as a unifying stability measure to characterize both poorly damped oscillations and voltage stability margins. This leads to two novel optimization problems that can find operation points which minimize oscillations or maximize voltage stability margins, and make apparent the implicit tradeoff between these two stability requirements. Finally, we combine these optimization problems to generate a dynamics-aware optimal power flow formulation that provides voltage as well as small signal stability guarantees.

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“…As we will see later, in Section 4, both approaches often suffer from slow convergence and lack of means to quantify the obtained accuracy. Nevertheless, the projection‐based approaches have been successfully applied to computation of pseudospectral quantities, providing a significant improvement over the previous work …”

Section: Introductionmentioning

confidence: 99%

“…21,22 As we will see later, in Section 4, both approaches often suffer from slow convergence and lack of means to quantify the obtained accuracy. Nevertheless, the projection-based approaches have been successfully applied to computation of pseudospectral quantities, 23,24 providing a significant improvement over the previous work. 25,26 In addition to the mentioned computationally oriented approaches, the asymptotic behavior of -pseudospectra has been recently studied in the work of Gong et al, 27 while a priori estimates for pseudospectra using first-order approximations have been derived in the work of Hannukainen.…”

Section: Introductionmentioning

confidence: 99%

A reduced basis approach to large‐scale pseudospectra computation

Sirković

2018

Numerical Linear Algebra App

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Summary For studying spectral properties of a nonnormal matrix A∈Cn×n, information about its spectrum σ(A) alone is usually not enough. Effects of perturbations on σ(A) can be studied by computing ε‐pseudospectra, i.e. the level sets of the resolvent norm function sans-serifgfalse(zfalse)=false‖false(zI−Afalse)−1‖2. The computation of ε‐pseudospectra requires determining the smallest singular values σminfalse(zI−Afalse) for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation, which provides highly accurate estimates of pseudospectra in the region of interest, in particular, for pseudospectra estimates in isolated parts of the spectrum containing few eigenvalues of A. It incorporates the sampled singular vectors of zI − A for different values of z, and implicitly exploits their smoothness properties. It provides rigorous upper and lower bounds for the pseudospectra in the region of interest. In addition, we propose a domain splitting technique for tackling numerically more challenging examples. We present a comparison of our algorithms to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time.

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Subspace Methods for Computing the Pseudospectral Abscissa and the Stability Radius

Cited by 38 publications

References 24 publications

Distance Problems for Linear Dynamical Systems

Distance Problems for Linear Dynamical Systems

Dynamics-aware optimal power flow

A reduced basis approach to large‐scale pseudospectra computation

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