P-moment stability of power system under small Gauss type random excitation

Lu, Zhanhui; Zhao, Weixiang; Xie, Di; Li, Gengyin

doi:10.1016/j.chaos.2015.08.020

Cited by 11 publications

(15 citation statements)

References 31 publications

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“…For p 2, the proof is obtained in literature [28], and here only prove the situation of 0 < p < 2. When > 0, applying the formula of integration by parts to the following equation, we have…”

Section: Proofmentioning

confidence: 87%

“…The proof is now completed.Remark The theorem provides a simple and effective way for the determination of stability, which can be commonly used to analyze the stability of power system models. The p ‐moment stability theorem in this paper deals with p > 0, which is a generalization and promotion of the previous literature with p ≥2. Besides, mean stability and mean square stability in are special circumstances with p = 1 and p = 2 of the p ‐moment stability theorem.…”

Section: P‐moment Stability Theoremmentioning

confidence: 88%

“…Based on the related previous lemmas, the p ‐moment stability theorem in system (), when p is larger than 0, is given as follows.Theorem Given the linear constant coefficients stochastic differential system () under Gauss type random excitation, the initial values x (0) = x 0 are bounded and independent of the Wiener process B ( t ). When p is greater than 0, if all the eigenvalue real parts of A are less than zero, then the stochastic differential system () is of p ‐moment stability.Proof For p ≥2, see literature , and here, we only illustrate the condition of 0 < p < 2. The explicit expression of solution process of system () is achieved as follows

x (t) = e^{A t} x_{0} + \int_{0}^{t} e^{A (t - s)} Q d B (s) .

For 0 < p < 2, applying the Cauchy‐Schwarz inequality and the Cr‐inequality leads to the following equation

\begin{matrix} {((), normalE {(||, x (t))}^{p})}^{2} \\ \leq normalE {(||, x (t))}^{2 p} \\ = normalE {(||, e^{A t} x_{0} + \int_{0}^{t} e^{A (t - s)} Q d B (s))}^{2 p} \\ \leq C_{p} ((), normalE {(||, e^{A t} x_{0})}^{2 p} + normalE {(||, \int_{0}^{t} e^{A (t - s)} Q d B (s))}^{2 p}) \\ = C_{p} ((), {(||, e^{A t} x_{0})}^{2 p} + normalE (||, \int_{0}^{t} e^{A (t - s)})) \end{matrix}

…”

Section: P‐moment Stability Theoremmentioning

confidence: 99%

“…For p 2, see literature [28], and here, we only illustrate the condition of 0 < p < 2. The explicit expression of solution process of system (2) is achieved as follows…”

Section: Proofmentioning

confidence: 99%

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P‐moment stability under small Gauss type random excitation of stochastic system

Hao

Zhao

et al. 2016

Math Methods in App Sciences

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In this paper, the stochastic stability under small Gauss type random excitation is investigated theoretically and numerically. When p is larger than 0, the p‐moment stability theorem of stochastic models is proved by Lyapunov method, Ito isometry formula, matrix theory and so on. Then the application of p‐moment such as k‐order moment of the origin and k‐order moment of the center is introduced and analyzed. Finally, p‐moment stability of the power system is verified through the simulation example of a one machine and infinite bus system. Copyright © 2016 John Wiley & Sons, Ltd.

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Section: Proofmentioning

confidence: 87%

Section: P‐moment Stability Theoremmentioning

confidence: 88%

x (t) = e^{A t} x_{0} + \int_{0}^{t} e^{A (t - s)} Q d B (s) .

For 0 < p < 2, applying the Cauchy‐Schwarz inequality and the Cr‐inequality leads to the following equation

\begin{matrix} {((), normalE {(||, x (t))}^{p})}^{2} \\ \leq normalE {(||, x (t))}^{2 p} \\ = normalE {(||, e^{A t} x_{0} + \int_{0}^{t} e^{A (t - s)} Q d B (s))}^{2 p} \\ \leq C_{p} ((), normalE {(||, e^{A t} x_{0})}^{2 p} + normalE {(||, \int_{0}^{t} e^{A (t - s)} Q d B (s))}^{2 p}) \\ = C_{p} ((), {(||, e^{A t} x_{0})}^{2 p} + normalE (||, \int_{0}^{t} e^{A (t - s)})) \end{matrix}

…”

Section: P‐moment Stability Theoremmentioning

confidence: 99%

“…For p 2, see literature [28], and here, we only illustrate the condition of 0 < p < 2. The explicit expression of solution process of system (2) is achieved as follows…”

Section: Proofmentioning

confidence: 99%

See 2 more Smart Citations

P‐moment stability under small Gauss type random excitation of stochastic system

Hao

Zhao

et al. 2016

Math Methods in App Sciences

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“…An electricity model under Gauss type random excitation has been structured in [20], theoretically proving the power system mean stable and mean-square stable under small Gauss type random excitation. In [21], -moment stability of power system under small Gauss type random excitation is verified when is greater than or equal to 2 using Lyapunov method, Ito isometry formula and matrix theory. In [22,23], -moment stability of power system is also discussed.…”

Section: Introductionmentioning

confidence: 99%

Stability of the Stochastic Model for Power Markets with Interval Parameters

Wang

et al. 2016

Mathematical Problems in Engineering

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Pertaining to the random nature of demand sides and the range of demand elasticity with suppliers and consumers, a stochastic model for power markets with interval parameters is described to illustrate uncertain external disturbances, which is a generalization of the Alvarado dynamic model, stochastic model, and interval model. The interval stochastic stability criteria of the provided model are investigated by the theory of economics, interval dynamical system, and the theory stability of stochastic differential equations. The conclusions indicate that the demand elasticity stable interval can be calculated and the random excitation intensity does not impact the system stability. Some numerical examples are given to show the applicability and validity of the obtained results from a statistical perspective.

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A Novel Small Disturbance Stochastic Stability Analysis Method Based on Entropy

Zhong

Sun

Meng

et al. 2023

IEEJ Transactions Elec Engng

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Power system stability has become an increasing concern in renewable power integration because the active power generated by renewable energy varies randomly. Thus, we focus on random outside excitations in this study. Since the power obeys Non‐Gaussian distribution in most case, a novel small disturbance stochastic stability analysis method based on entropy is presented. First, the dynamic model of power system with random outside excitation is established by stochastic differential equation. Then, the values of mean and mean square stability are efficiently derived by transforming the stochastic differential equation into a deterministic mean or variance equation. For non‐Gaussian disturbance, entropy stability is presented by this transformation. At last, the numerical simulations are presented on a WSCC three‐generator nine‐bus power system with a wind farm © 2023 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC.

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P-moment stability of power system under small Gauss type random excitation

Cited by 11 publications

References 31 publications

P‐moment stability under small Gauss type random excitation of stochastic system

P‐moment stability under small Gauss type random excitation of stochastic system

Stability of the Stochastic Model for Power Markets with Interval Parameters

A Novel Small Disturbance Stochastic Stability Analysis Method Based on Entropy

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