Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems
In this paper, we discuss stochastic differential-algebraic equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then...
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MDPI AG
2020-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/9/1393 |
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| author | Andrés González-Zumba Pedro Fernández-de-Córdoba Juan-Carlos Cortés Volker Mehrmann |
| author_facet | Andrés González-Zumba Pedro Fernández-de-Córdoba Juan-Carlos Cortés Volker Mehrmann |
| author_sort | Andrés González-Zumba |
| collection | DOAJ |
| description | In this paper, we discuss stochastic differential-algebraic equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous <inline-formula><math display="inline"><semantics><mrow><mi>Q</mi><mi>R</mi></mrow></semantics></math></inline-formula> decomposition-based numerical methods are implemented to compute the fundamental solution matrix and use it in the computation of the LEs. Important computational features of both methods are illustrated via numerical tests. Finally, the methods are applied to two applications from power systems engineering, including the single-machine infinite-bus (SMIB) power system model. |
| first_indexed | 2024-03-10T17:07:53Z |
| format | Article |
| id | doaj.art-6e20f3016acf49468bad08c7915859b1 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T17:07:53Z |
| publishDate | 2020-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-6e20f3016acf49468bad08c7915859b12023-11-20T10:45:05ZengMDPI AGMathematics2227-73902020-08-0189139310.3390/math8091393Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power SystemsAndrés González-Zumba0Pedro Fernández-de-Córdoba1Juan-Carlos Cortés2Volker Mehrmann3Departamento de Matemática Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, SpainInstituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, SpainInstituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, SpainInstitut für Mathematik MA 4-5, Technische Universität Berlin, Str. des 17. Juni 136, D-10623 Berlin, GermanyIn this paper, we discuss stochastic differential-algebraic equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous <inline-formula><math display="inline"><semantics><mrow><mi>Q</mi><mi>R</mi></mrow></semantics></math></inline-formula> decomposition-based numerical methods are implemented to compute the fundamental solution matrix and use it in the computation of the LEs. Important computational features of both methods are illustrated via numerical tests. Finally, the methods are applied to two applications from power systems engineering, including the single-machine infinite-bus (SMIB) power system model.https://www.mdpi.com/2227-7390/8/9/1393stochastic differential-algebraic equationslyapunov exponentpower system stabilityspectral analysisstochastic systemsnumerical methods |
| spellingShingle | Andrés González-Zumba Pedro Fernández-de-Córdoba Juan-Carlos Cortés Volker Mehrmann Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems Mathematics stochastic differential-algebraic equations lyapunov exponent power system stability spectral analysis stochastic systems numerical methods |
| title | Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems |
| title_full | Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems |
| title_fullStr | Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems |
| title_full_unstemmed | Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems |
| title_short | Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems |
| title_sort | stability assessment of stochastic differential algebraic systems via lyapunov exponents with an application to power systems |
| topic | stochastic differential-algebraic equations lyapunov exponent power system stability spectral analysis stochastic systems numerical methods |
| url | https://www.mdpi.com/2227-7390/8/9/1393 |
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