Saddle-node Bifurcation Point Research Articles

A New Method for Monitoring Local Voltage Stability using the Saddle Node Bifurcation Set in Two Dimensional Power Parameter Space

This paper proposes a new method for monitoring local voltage stability using the saddle node bifurcation set or loadability boundary in two dimensional power parameter space. The method includes three main steps. First step is to determine the critical buses and the second step is building the static voltage stability boundary or the saddle node bifurcation set. Final step is monitoring the voltage stability through the distance from current operating point to the boundary. Critical buses are defined through the right eigenvector by direct method. The boundary of the static voltage stability region is a quadratic curve that can be obtained by the proposed method that is combining a variation of standard direct method and Thevenin equivalent model of electric power system. And finally the distance is computed through the Euclid norm of normal vector of the boundary at the closest saddle node bifurcation point. The advantage of the proposed method is that it gets the advantages of both methods, the accuracy of the direct method and simple of Thevenin Equivalent model. Thus, the proposed method holds some promises in terms of performing the real-time voltage stability monitoring of power system. Test results of New England 39 bus system are presented to show the effectiveness of the proposed method.

A Critical Eigenvalues Tracing Method for the Small Signal Stability Analysis of Power Systems

The continuation power flow method combined with the Jacobi-Davidson method is presented to trace the critical eigenvalues for power system small signal stability analysis. The continuation power flow based on a predictor- corrector technique is applied to evaluate a continuum of steady state power flow solutions as system parameters change; meanwhile, the critical eigenvalues are found by the Jacobi-Davidson method, and thereby the trajectories of the critical eigenvalues, Hopf bifurcation and saddle node bifurcation points can also be found by the proposed method. The numerical simulations are studied in the IEEE 30-bus test system.

Voltage Stability Bifurcation Analysis for AC/DC Systems with VSC-HVDC

A voltage stability bifurcation analysis approach for modeling AC/DC systems with VSC-HVDC is presented. The steady power model and control modes of VSC-HVDC are briefly presented firstly. Based on the steady model of VSC-HVDC, a new improved sequential iterative power flow algorithm is proposed. Then, by use of continuation power flow algorithm with the new sequential method, the voltage stability bifurcation of the system is discussed. The trace of the P-V curves and the computation of the saddle node bifurcation point of the system can be obtained. At last, the modified IEEE test systems are adopted to illustrate the effectiveness of the proposed method.

Saddle Node Bifurcation and Voltage Stability Analysis of Sri Lanka

In electrical power systems, saddle node bifurcation (SNB) is the voltage stability margin at which voltage collapse takes place. It plays an important role in power system planning and operation as economic and environmental pressures force to operate always closer to the limit of stability. Voltage collapse is associated with the reactive power demands not being met because of limitations in the generation and transmission of reactive power. Determination of the SNB point greatly assists planning and operation of power system. Among various voltage stability analysis (VSA) methods, this paper specifies the continuation power flow (CPF) method as it accurately identifies the SNB point and the critical or the weaker bus. In this paper, power system of Sri Lanka is assessed for diverse applications of VSA by concerning about single element outage or (N-1) criterion and the role of voltage compensation devices. Furthermore, the optimum at the SNB point is verified by eigenvalue analysis.

A New Method for Static Voltage Stability Assessment based on The Local Loadability Boundary

Abstract This paper proposes a new method for assessing static voltage stability based on the local loadability boundary or P- Q curve in two dimensional power parameter space. The proposed method includes three main steps. The first step is to determine the critical buses and the second step is building the local loadability boundary or the saddle node bifurcation set for those critical buses. The final step is assessing the static voltage stability through the distance from current operating point to the boundary. The critical buses are defined through the right eigenvector by direct method. The boundary obtained by the proposed method that is combining a variation of standard direct method and Thevenin equivalent model of electric power system is a quadratic curve. And finally the distance is computed through the Euclid norm of normal vector of the boundary at the closest saddle node bifurcation point. The advantage of the proposed method is that it keeps the advantages of both efficient methods, the accuracy of the direct method and simple of Thevenin Equivalent based method. Thus, the proposed method holds some promise in terms of performing the real time voltage stability assessment of power system. Test results of New England 39 bus system are presented to show the effectiveness of the proposed method.

The “resultant bifurcation diagram” method and its application to bifurcation behaviors of a symmetric railway bogie system

The concept of symmetric bifurcation for a symmetric wheel-rail system is defined. After that, the time response of the system can be achieved by the numerical integration method, and an unfixed and dynamic Poincare section and its symmetric section for the symmetric wheel-rail system are established. Then the ‘resultant bifurcation diagram’ method is constructed. The method is used to study the symmetric/asymmetric bifurcation behaviors and chaotic motions of a two-axle railway bogie running on an ideal straight and perfect track, and a variety of characteristics and dynamic processes can be obtained in the results. It is indicated that, for the possible sub-critical Hopf bifurcation in the railway bogie system, the stable stationary solutions and the stable periodic solutions coexist. When the speed is in the speed range of Hopf bifurcation point and saddle-node bifurcation point, the coexistence of multiple solutions can cause the oscillating amplitude change for different kinds of disturbance. Furthermore, it is found that there are symmetric motions for lower speeds, and then the system passes to the asymmetric ones for wide ranges of the speed, and returns again to the symmetric motions with narrow speed ranges. The rule of symmetry breaking in the system is through a blue sky catastrophe in the beginning.

Singular response of bistable systems driven by telegraph noise

We show that weak periodic driving can exponentially strongly change the rate of escape from a potential well of a system driven by telegraph noise. The analysis refers to an overdamped system, where escape requires that the noise amplitude θ exceed a critical value θ(c). For θ close to θ(c), the exponent of the escape rate displays a nonanalytic dependence on the amplitude of an additional low-frequency modulation. This leads to giant nonlinearity of the response of a bistable system to periodic modulation. Also studied is the linear response to periodic modulation far from θ(c). We analyze the scaling of the logarithm of the escape rate with the distance to the saddle-node and pitchfork bifurcation points. The analytical results are in excellent agreement with numerical simulations.

Tuning the Nonlinear Behaviour of Resonant MEMS Sensors Actuated Electrically

The objectives of this present work is to study the stability and bifurcation control of an idealized electrostatically actuated microcantilever MEMS device that can widely observe in the field MEMS application. Here, the cantilever based device has been modelled as a spring-mass-damper system considering both the linear and nonlinear spring and damper. Simultaneously, the cantilever based device is excited harmonically by applied voltages. The method of multiple scales is employed to obtain the reduced order equations in terms of amplitude and phase those are directly used to determine the approximate the solutions for different resonance conditions. The catastrophic failure of the system may occur due to the presence of saddle-node and pitchfork bifurcation points as it leads the jump phenomenon. Basins of attractions are plotted in order to find the initial condition for a specific solution in a region having more than one solution. The obtained results can successfully be used in designing the microcantilever based devices that depict typical realistic nonlinear characteristics in the field of MEMS application.

A Coordinating Procedure for Tracing Static Bifurcation Points of Voltage Instability

Based on static bifurcation theory, a systematic procedure for tracing the bifurcation points of power system voltage instability was presented. Two main kinds of bifurcations, a common saddle node bifurcation (SNB) and a kind of bifurcation—limit induced bifurcation (LIB), which can cause the voltage collapse, were studied simultaneity and all types of device limits were considered. Firstly, make sure the mathematical model of the system. Secondly, starting from the basic running point of power systems, a predictor scheme was presented to find all the limit points (LP) and identify the LIB points; an adaptive step size control method considering the curvature of the manifold was introduced for tracing the equilibrium manifold and the detection of the SNB points without the existing of LPs. The two schemes were combined to lead a more effective method for SNB, LP, and LIB detection.

COHERENCE RESONANCE–INDUCED STOCHASTIC NEURAL FIRING AT A SADDLE-NODE BIFURCATION

Coherence resonance at a saddle-node bifurcation point and the corresponding stochastic firing patterns are simulated in a theoretical neuronal model. The characteristics of noise-induced neural firing pattern, such as exponential decay in histogram of interspike interval (ISI) series, independence and stochasticity within ISI series are identified. Firing pattern similar to the simulated results was discovered in biological experiment on a neural pacemaker. The difference between this firing and integer multiple firing generated at a Hopf bifurcation point is also given. The results not only revealed the stochastic dynamics near a saddle-node bifurcation, but also gave practical approaches to identify the saddle-node bifurcation and to distinguish it from the Hopf bifurcation in neuronal system. In addition, many previously observed firing patterns can be attribute to stochastic firing pattern near such a saddle-node bifurcation.

Displaced orbits generated by solar sails for the hyperbolic and degenerated cases

Displaced non-Keplerian orbits above planetary bodies can be achieved by orientating the solar sail normal to the sun line. The dynamical systems techniques are employed to analyze the nonlinear dynamics of a displaced orbit and different topologies of equilibria are yielded from the basic configurations of Hill’s region, which have a saddle-node bifurcation point at the degenerated case. The solar sail near hyperbolic or degenerated equilibrium is quite unstable. Therefore, a controller preserving Hamiltonian structure is presented to stabilize the solar sail near hyperbolic or degenerated equilibrium, and to generate the stable Lissajous orbits that stay stable inside the stabilizing region of the controller. The main contribution of this paper is that the controller preserving Hamiltonian structure not only changes the instability of the equilibrium, but also makes the modified elliptic equilibrium become unique for the controlled system. The allocation law of the controller on the sail’s attitude and lightness number is obtained, which verifies that the controller is realizable.

A new technique for computation of closest saddle-node bifurcation point of power system using real coded genetic algorithm

Estimation of voltage stability margin is essential for operation of the system with an adequate security margin. In this paper, a new technique to determine the worst case loading margin, i.e., shortest distance to voltage instability is developed. The problem of determining the closest saddle-node bifurcation point (CSNBP) is formulated as an optimization problem and solved using Real Coded Genetic Algorithm (RCGA). The method is capable of handling various operational constraints and can determine the CSNBP accurately even if the transfer limit surface is not smooth. The proposed approach has been applied on a simple radial system, IEEE 14-bus and IEEE 57-bus systems. The developed method has been compared with the method based on Particle Swarm Optimization. Simulation results show the validity and feasibility of the proposed method.

Stability and Self-Sustained Oscillations in a Ventricular Cardiomyocyte Model

The Luo-Rudy I model, describing the electrophysiology of a ventricular cardiomyocyte, is associated with an 8-dimensional discontinuous dynamical system with logarithmic and exponential non-linearities depending on 15 parameters. By numerical approaches appropriate to bifurcation problems, sections in the static bifurcation diagram were determined. For different values of a steady depolarizing/hyperpolarizing current (Ist), the corresponding projection of the static bifurcation diagram in the (Ist, V) plane is complex, featuring three branches of stationary solutions delimited by two saddle-node bifurcation points. In addition, on the upper branch oscillations can occur within an Ist range [−4.45, −0.51 μA/cm2] where the Jacobian of the linearized system features two complex conjugated eigenvalues. Oscillations are either damped at a stable focus or amplified until the system trajectory is diverted to the lower branch of stable stationary solutions when reaching the unstable manifold of a homoclinic saddle. The middle branch of solutions is a series of unstable saddle points, while the lower one a series of stable nodes. For variable slow inward and K+ current maximal conductances (gsi and gK), in a range between 0 and 4-fold normal values, the dynamics is even more complex, and in certain instances self-sustained oscillations tending to a stable limit cycle appear. All these types of behavior were correctly predicted by linear stability analysis and bifurcation theory methods based on numerical continuation algorithms. Both unsustained oscillations, resulting in early after-depolarizations, and sustained oscillations may trigger dangerous ventricular arrhythmias by multiple mechanisms. In particular settings, e.g. for a normal gK and one-fifth-of-normal gsi, these two arrhythmia-threatening conditions may occur simultaneously.

Some Design and Simulation of Sliding Mode Variable Structure Control for Hopf Bifurcation in Power Systems

In power systems, the Hopf bifurcation (HB) can occur before the saddle-node bifurcation (SNB) and be-comes one of main reasons of voltage instability and collapse, so the bifurcation control for HB has impor-tant significance in improving power system voltage stability. In this paper, the numerical bifurcation analy-sis software MATCONT was used to study bifurcation behavior of a single-machine dynamic-load (SMDL) system with SVC, and the simulation analysis results show that a unstable Hopf bifurcation (UHB) point oc-curring before SNB point and engendering potential harm to voltage stability. To delay or eliminate the UHB phenomenon and enhance voltage stability of the SMDL system with SVC, we designed a sliding mode variable structure controller. The switching function and control variables of the controller are clearly de-scribed and the derivations are directly provided in detail in this paper. The MATLAB simulation results prove that the designed controller can eliminate the UHB point effectively and ensure safe and stable opera-tion of the system.

Two-mode combination resonances of an in-extensional rotating shaft with large amplitude

In this paper, two-mode combination resonances of a simply supported rotating shaft are investigated. The shaft is modeled as an in-extensional spinning beam with large amplitude. Rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The equations of motion are derived with the aid of the Hamilton principle and then transformed to the complex form. The method of harmonic balance is applied to obtain analytical solutions. Frequency-response curves are plotted for the combination resonances of the first and the second modes. The effects of eccentricity and external damping are investigated on the steady state response of the rotating shaft. The loci of saddle node bifurcation points are plotted as functions of external damping and eccentricity. The results are validated with numerical simulations.

Nonlinear Responses of Dual-Pendulum Dynamic Absorbers

The nonlinear responses of a single-degree-of-freedom system with two pendulum tuned mass dampers under horizontal sinusoidal excitation are investigated. In the theoretical analysis, van der Pol’s method is applied to determine the expressions for the frequency response curves. In the numerical results, the differences between the responses in single- and dual-pendulum systems are shown. A pitchfork bifurcation occurs followed by mode localization where both identical pendula vibrate at constant but different amplitudes. Hopf bifurcations occur, and then amplitude- and phase-modulated motions including chaotic vibrations appear in the identical dual-pendulum system. The Lyapunov exponents are calculated to prove the occurrence of chaotic vibrations. In a nonidentical dual-pendulum system, a perturbed pitchfork bifurcation occurs and saddle-node bifurcation points appear instead of pitchfork bifurcation points. Hopf bifurcations and amplitude- and phase-modulated motions also appear. The deviation of the tuning condition is also investigated by showing the frequency response curves and bifurcation sets. The numerical simulations are shown to be in good agreement with the theoretical results. In experiments, the imperfections of the two pendula were taken into consideration, and the validity of the theoretical analysis was confirmed.

Theoretical analysis for critical fluctuations of relaxation trajectory near a saddle-node bifurcation

A Langevin equation whose deterministic part undergoes a saddle-node bifurcation is investigated theoretically. It is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle-node bifurcation point in the weak-noise limit, while the final value of the deterministic solution changes discontinuously at the point. A systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method. In this formulation, the critical nature turns out to originate from the neutrality of exiting time from a saddle point. The theoretical calculation explains results of numerical simulations.

Stability analysis of an electrically actuated microbeam using the Melnikov theorem and Poincaré mapping

In this article, the stability of a microbeam under an electric actuation is studied. The electric actuation is induced by applying a voltage between the microbeam and an electrode plate that lies at the opposite side of the microbeam. In microswitches, the electric actuation is applied as a DC voltage, and in microresonators it is applied as a combination of AC—DC voltages. It is assumed that the midplane of the microbeam is stretched when it is deflected. It is also shown that by the altering DC electric actuation as a control parameter in a microswitch system, a stable and an unstable branches of equilibrium solution is observed, which meet each other at a saddle-node bifurcation point. The stability of a microresonator is studied using the phase plane diagram and Poincaré mapping. It is shown that depending on the value of damping factor, AC and DC electric voltages, and other parameters of the microresonator, a periodic solution, a quasi periodic, or a pull-in instability may be realized. The prediction of possible chaotic behaviour for microresonator is studied using the Melnikov theorem. It is shown that although for selected domain of system parameters the Melnikov function is satisfied for occurrence of chaotic behaviour, for theses parameter values the pull-in instability occurs before going into the chaotic behaviour. Briefly, the system does not realize any chaotic behaviour.

Primary resonances of a nonlinear in-extensional rotating shaft

In this paper, primary resonances of a simply supported in-extensional rotating shaft with large amplitudes are studied. The method of multiple scales is applied to the discretized equations as well as to the complex form of partial differential equations of motion. The effects of diametrical mass moment of inertia, eccentricity and external damping are investigated on the steady state response of the rotating shaft. The loci of saddle node bifurcation points are plotted as functions of damping coefficient and eccentricity.

A Simple Method for Tracking Turning Points in Parameter Space

We describe a simple method for tracking solutions of nonlinear equations f (u,α ) � 0 through turning points (also known as limit or saddle-node bifurcation points). Our implementation makes use of symbolic software such as Mathematica to derive an exact system of nonlinear ODE equations to follow the solution path, using a parameterization closely related to arc length. We illustrate our method with examples taken from the engineer- ing literature, including examples that involve nonlinear boundary value problems that have been discretized by finite difference methods. Since the code requirement to implement the method is modest, we believe the method is ideal for demonstrating continuation methods in the classroom.