Analysis and Design of Nonlinear Circuits With a Self-Consistent Carleman Linearization

Weber, Harry; Mathis, Wolfgang

doi:10.1109/tcsi.2018.2837677

Cited by 7 publications

(3 citation statements)

References 49 publications

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“…According to the Hartman-Grobman theorem, the nonlinear PS model ( 1) can be replaced at a steady operating point by the linearized model ( 2) only if the linearized model is a hyperbolic system i.e. all eigenvalues of the state matrix have non-zero real parts [4,11,12,23]. This assumption was made in the investigations presented in this paper.…”

Section: The Linearized Ps Modelmentioning

confidence: 99%

“…The power system (PS) is a large nonlinear dynamic system for the production, transmission and distribution of electrical energy. Its stability is associated with transient states and regulation processes occurring in PS [6,23]. Generally, the PS stability is understood as its ability to maintain a specific operating state after a disturbance.…”

Section: Introductionmentioning

confidence: 99%

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Archives of Control Sciences

Pruski

Paszek

2020

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In the paper, the results of investigations on the location of generating units most affecting the angular stability of a large power system (PS) are presented. For their location, the eigenvalues of the PS model state matrix associated with electromechanical phenomena (electromechanical eigenvalues) were used. The eigenvalues were calculated on the basis of the analysis of the disturbance waveforms of instantaneous power of the generating units operating in the PS. The used method of calculating eigenvalues consists in approximation of the disturbance waveforms of generating units by the waveforms being the superposition of modal components. The parameters of these components depend on the sought eigenvalues and their participation factors. The objective function was defined as the mean square error between the approximated and approximating waveforms. To minimize it, a hybrid algorithm, being a combination of genetic and gradient algorithms, was used. In the instantaneous power waveforms of generating units most affecting the PS angular stability, the least damped or undamped modal components dominate. They are related to eigenvalues with the largest values of real parts. The impact of individual modal components on the disturbance waveforms of subsequent generating units was determined with the use of participation factors and correlation coefficients of electromechanical eigenvalues.

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Section: The Linearized Ps Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Archives of Control Sciences

Pruski

Paszek

2020

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“…Therefore, an approximation of the infinite-dimensional linear system is necessary. In case of the Carleman linearization, a self-consistent technique is used to derive a suitable approximation for a given interval (Weber and Mathis, 2018; Weber et al , 2021). In the next section, it is described how the self-consistent technique is implemented in the context of the Hilbert space embedding.…”

Section: Hilbert Space Embedding and Carleman Linearizationmentioning

confidence: 99%

DC operating points of nonlinear circuits and generalized Carleman linearization

Weber

Mathis²

2023

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Purpose The purpose of this paper is to present a procedure for approximating DC operating points of nonlinear circuits. The presented approach can also be applied in case of multiple DC operating points. Design/methodology/approach A generalized Carleman linearization is used, which transforms an algebraic nonlinear equation into an equivalent infinite-dimensional linear system. In general, no close-form solution can be given for the infinite-dimensional linear system. Hence, the infinite-dimensional linear system is approximated by a finite one over a predefined interval using a self-consistent technique. The presented procedure allows to approximate all possible DC operating points within a predefined interval. To isolate all DC operating points, the initial interval is gradually divided into subintervals. Findings It is shown that the presented approach is not restricted to the polynomial case and allows to approximate all DC operating points. The presented approach can be applied in case of multiple DC operating points and does not depend on the domain of attraction of the DC operating points. Originality/value A new procedure for the approximation of DC operating points of nonlinear circuits based on a generalized Carleman linearization is presented. This approach can be applied in case of multiple DC operating points and is independent of the domain of attraction. Further, this generalized approach is not restricted to the polynomial case and can be applied to a variety of circuits.

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