Spectral and Modal Methods for Studying Stability and Control of Electric Power Systems

Воропай, Н.И.; Golub, Irina; Efimov, Dmitry; Iskakov, Alexey B.; Yadykin, Igor B.

doi:10.1134/s000511792010001x

Cited by 7 publications

(10 citation statements)

References 68 publications

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“…Thus, the system of linear algebraic Equations ( 24)-( 28) implements an alternative method to determine elements of the Gramian control matrices in a simpler way than solving the linear algebraic equation in the form of Kronecker products and applying vectorization method (8) [16]. In addition, Formulas (10) and (11) give analytical solution of Lyapunov equation based on Gramian method. If a linear system is stable and the eigen numbers of its dynamics matrix are different, the solution of the Lyapunov equation is singular.…”

Section: Separable Spectral Expansion Of the Controllability Gramian ...mentioning

confidence: 99%

“…Let us discuss properties of the matrix and solution elements on the basis of analysis of solutions to the system of Equations ( 24)- (28) and Formulas (10) and (11).…”

Section: Separable Spectral Expansion Of the Controllability Gramian ...mentioning

confidence: 99%

“…Let's define the bilinear system controllability Gramian by means of the Volterra matrix series of the form [10], namely…”

Section: N(s ρ )mentioning

confidence: 99%

“…New mathematical modelling techniques, such as generation and load forecasting, optimal control of energy storage systems, and new methods for energy systems control will play a key role in 21st century power engineering and in realizing the Internet of Energy concept [9]. The development of new innovative technologies requires the involvement of a modern mathematical control theory, in which the study of the structural properties and computational methods of energy systems Gramians is an important task [10,11].…”

Section: Introductionmentioning

confidence: 99%

“…Consider the formation of the right part of the generalized Lyapunov equation at the first step for the case γ = 1. A separable spectral decomposition of this solution by a pairwise spectrum of the matrix, was obtained above (10) [32].…”

Section: Let Us Introduce the Notationsmentioning

confidence: 99%

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Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms

Yadykin

2022

Mathematics

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The application of transformations of the state equations of continuous linear and bilinear systems to the canonical form of controllability allows one to simplify the computation of Gramians of these systems. In this paper, we develop the method and obtain algorithms for computation of the controllability and observability Gramians of continuous linear and bilinear stationary systems with many inputs and one output, based on the method of spectral expansion of the Gramians and the iterative method for computing the bilinear systems Gramians. An important feature of the concept is the idea of separability of the Gramians expansion: separate computation of the scalar and matrix parts of the spectral Gramian expansion reduces the sub-Gramian matrices computation to calculation of numerical sequences of their elements. For the continuous linear systems with one output the method and the algorithm of the spectral decomposition of the controllability Gramian are developed in the form of Xiao matrices. Analytical expressions for the diagonal elements of the Gramian matrices are obtained, and by making use of which the rest of the elements can be calculated. For continuous linear systems with many outputs the spectral decompositions of the Gramians in the form of generalized Xiao matrices are obtained, which allows us to significantly reduce the number of calculations. The obtained results are generalized for continuous bilinear systems with one output. Iterative spectral algorithms for computation of elements of Gramians of these systems are proposed. Examples are given that illustrate theoretical results.

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Section: Separable Spectral Expansion Of the Controllability Gramian ...mentioning

confidence: 99%

“…Let us discuss properties of the matrix and solution elements on the basis of analysis of solutions to the system of Equations ( 24)- (28) and Formulas (10) and (11).…”

Section: Separable Spectral Expansion Of the Controllability Gramian ...mentioning

confidence: 99%

“…Let's define the bilinear system controllability Gramian by means of the Volterra matrix series of the form [10], namely…”

Section: N(s ρ )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Let Us Introduce the Notationsmentioning

confidence: 99%

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Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms

Yadykin

2022

Mathematics

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Coherence indicators of generators for express assessment of electric power system transient stability

Efimov

Stashkevich

2023

E3S Web of Conf.

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A brief overview of the generator coherence indicators is presented, the areas and limitations of their applicability are indicated. The application of the area method for the rapid assessment of the transient stability of complex multi-machine electric power system based on the features of the dynamics of the system behavior, determined by the heterogeneous structure of the electrical network, is considered. A distinctive feature of the proposed approach is the use of the characteristics of the heterogeneity of the power system structure, which determine the presence of weak links (or cut-sets) of the electrical network, which are critical from the point of view of a possible violation of the system stability.

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On the Solution of Generalized Lyapunov Equations for a Class of Continuous Bilinear Time-Varying Systems

Yadykin

Galyaev²,

Vershinin³

2022

Autom Remote Control

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Spectral and Modal Methods for Studying Stability and Control of Electric Power Systems

Cited by 7 publications

References 68 publications

Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms

Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms

Coherence indicators of generators for express assessment of electric power system transient stability

On the Solution of Generalized Lyapunov Equations for a Class of Continuous Bilinear Time-Varying Systems

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