Saddle-node Bifurcation of Power Systems Analysis in the Simplest Normal Form

2012 ◽  
Author(s):  
Yong Fang ◽  
Hong-geng Yang
Keyword(s):  
Normal Form ◽  
Power Systems ◽  
Systems Analysis ◽  
Saddle Node Bifurcation ◽  
Saddle Node

scholarly journals Voltage collapse in power systems: Dynamical studies from a static formulation

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
Luis Fernando Mello ◽  
Antonio Carlos Zambroni de Souza ◽  
Gerson Hiroshi Yoshinari ◽  
Camila Vasconcelos Schneider
Keyword(s):  
Power Systems ◽  
Power System ◽  
Algebraic Approach ◽  
Voltage Collapse ◽  
Static Behavior ◽  
Theoretical Justification ◽  
Saddle Node Bifurcation ◽  
Dynamical Approach ◽  
Saddle Node

This paper addresses the problem of voltage collapse in power systems. More precisely, we exhibit a voltage collapse in a power system with two buses. This study is carried out with the help of two approaches. The first is a dynamical approach where a saddle-node bifurcation is analyzed and the second is an algebraic approach. Both approaches deal with the static behavior of the power system, but some dynamic aspects may be observed. An equivalence between the algebraic and dynamical approaches is obtained. The need to use both models comes from the fact that they are usually exploited in the literature, but a deep theoretical justification is still pending. Such a justification is meant in this work.

Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation

2016 ◽  
Author(s):  
Jinki Kim ◽  
R. L. Harne ◽  
K. W. Wang
Keyword(s):  
Normal Form ◽  
Analog Circuit ◽  
Critical Parameters ◽  
Material Structure ◽  
Escape Time ◽  
Estimation Of Parameters ◽  
Saddle Node Bifurcation ◽  
Analytical Approximations ◽  
Piezoelectric Energy ◽  
Saddle Node

Accurately predicting the onset of large behavioral deviations associated with saddle-node bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and adaptive material/structure applications such as structural health monitoring and piezoelectric energy harvesting. In many such practices, noise and non-stationarity are unavoidable and ever-present influences. As a result, it is critical to simultaneously account for these two factors towards the estimation of parameters that may induce sudden bifurcations. Here, a new analytical formulation is presented to accurately determine the probable time at which a system undergoes an escape event as governing parameters are swept towards a saddle-node bifurcation point in the presence of noise. The double-well Duffing oscillator serves as the archetype system of interest since it possesses a dynamic saddle-node bifurcation. Using this archetype example, the stochastic normal form of the saddle-node bifurcation is derived from which expressions of the escape statistics are formulated. Non-stationarity is accounted for using a time dependent bifurcation parameter in the stochastic normal form. Then, the mean escape time is approximated from the probability density function to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verify that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.

scholarly journals Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Jinki Kim ◽  
R. L. Harne ◽  
K. W. Wang
Keyword(s):  
Normal Form ◽  
Analog Circuit ◽  
Duffing Oscillator ◽  
Critical Parameters ◽  
Escape Time ◽  
Estimation Of Parameters ◽  
Saddle Node Bifurcation ◽  
Analytical Approximations ◽  
Saddle Node ◽  
Two Factors

Accurately predicting the onset of large behavioral deviations associated with saddle-node bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and microscale mass sensing. In many such practices, noise and non-stationarity are unavoidable and ever-present influences. As a result, it is critical to simultaneously account for these two factors toward the estimation of parameters that may induce sudden bifurcations. Here, a new analytical formulation is presented to accurately determine the probable time at which a system undergoes an escape event as governing parameters are swept toward a saddle-node bifurcation point in the presence of noise. The double-well Duffing oscillator serves as the archetype system of interest since it possesses a dynamic saddle-node bifurcation. The stochastic normal form of the saddle-node bifurcation is derived from the governing equation of this oscillator to formulate the probability distribution of escape events. Non-stationarity is accounted for using a time-dependent bifurcation parameter in the stochastic normal form. Then, the mean escape time is approximated from the probability density function (PDF) to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verifies that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.

TRAVELLING WAVES WITH SPATIALLY RESONANT FORCING: BIFURCATIONS OF A MODIFIED LANDAU EQUATION

1993 ◽  
Vol 03 (06) ◽  
pp. 1447-1455 ◽  
Author(s):  
PAUL GLENDINNING ◽  
MICHAEL PROCTOR
Keyword(s):  
Normal Form ◽  
Landau Equation ◽  
Travelling Waves ◽  
Duffing Equation ◽  
Invariant Curve ◽  
Invariant System ◽  
Van Der Pol ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Resonant Forcing

Normal form equations are derived representing the effects of spatially resonant forcing on a bifurcation to travelling waves in a nonreflexionally invariant system. The equations also describe the effects of forcing on the van der Pol–Duffing equation. It is found that the forcing prevents the propagation of the wave if it is sufficiently strong: the transition to a nonuniformly propagating solution can occur either via an oscillatory bifurcation or due to a saddle node bifurcation on a closed invariant curve.

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