Stability Regions In Parameter Space Research Articles

Parameter Stability Region Analysis of Islanded Microgrid Based on Bifurcation Theory

This paper studies the parameter stability region of droop-controlled ac microgrid (MG) with static ZIP (constant impedance, constant current, and constant power) load and dynamic induction motor (IM) load using bifurcation theory. First, the dynamic model of the MG with ZIP and IM loads is developed. Next, bifurcation analysis is used to predict the bifurcation boundaries where MG becomes unstable. Saddle node and hopf bifurcation are detected in the studied system when parameters change. The stability region in parameters space is bounded by bifurcation boundaries. To improve the computing efficiency for predicting the stable region of parameters, the reduced-order models of MGs are developed based on singular perturbation method. Finally, numerical simulations and experiment are used to verify the analysis result and the effectiveness of the proposed strategy.

Dynamical Behaviors of Coupled Memristor-Based Oscillators with Identical and Different Nonlinearities

In this work, the interesting dynamics of coupled nonlinear memristor-based oscillators in ring configuration are explored. The mathematical models are derived to describe the possible cases of employing identical or different nonlinearities. Analytical and numerical techniques, involving perturbation methods, normal forms, phase portraits, and Lyapunov exponents are used to investigate various types of dynamical behaviors along with their stability regions in parameters space. The effects of time-delayed coupling on the proposed system are numerically studied. It is demonstrated that the coupled oscillators show rich dynamics including periodic orbits, quasiperiodicity, two-dimensional, and three-dimensional tori.

Linear energy stable methods for an epitaxial growth model with slope selection

We consider a phase field model for molecular beam epitaxial growth with slope selection with the goal of determining linear energy stable time integration methods for the dynamics. Stable methods for this model have been found via a concave-convex splitting of the dynamics, but this approach generally leads to a nonlinear update equation. We seek a linear energy stable method to allow for simple and efficient time marching with fast Fourier transforms. Our approach is to parametrize a class of semi-implicit methods and perform unconditional von Neumann stability analysis to identify the region of stability in parameter space. Since unconditional von Neumann stability does not ensure energy stability, we perform extensive numerical tests and find strong agreement between the predicted and observed stable regions of parameter space. This analysis elucidates a novel feature that the stability region in parameter space differs for a mono-domain system (single equilibrium slope) versus a many-domain system (coarsening facets from an initially flat surface). The utility of these steps is then demonstrated by a comparison of the coarsening dynamics for isotropic and anisotropic variants of the model.

PI Controller Based Load Frequency Control Approach for Single-Area Power System Having Communication Delay

The modern power systems are becoming complicated day by day because of the delays introduced by the communication networks. Due to this reason, the traditional load frequency control (LFC) design scheme depicts a destabilizing impact and an unacceptable performance. Therefore, this paper proposes an analytico-graphical approach for designing PI controller for a single-area LFC system having communication delay. The concept is based on extracting stability region in parameter space (kp,ki) with predefined gain and phase margins. Further, the values of optimal kp and ki are selected using integral error criterion. The proposed scheme gives faster disturbance rejection response as compared to the recently developed LFC scheme. The controller also works well when the system parameters are perturbed from their nominal values.

Mode interactions and resonances of an elastic fluid‐conveying tube

AbstractWe consider the non‐linear two‐dimensional oscillations of a fluid conveying tube using dynamical bifurcation theory. The tube is clamped at the upper end, and at its free lower end a point mass is fixed. The tube is assumed to be slender and flexurally elastic, and its transversal motion is constrained by two symmetrically arranged springs. The flow rate of the incompressible fluid is used as a distinguished parameter in the problem. By determining the stability regions in parameter space, it is investigated whether Hopf and/or steady‐state bifurcations may occur, as it was found for similar cases in previous works [1,3]. The non‐linear behaviour close to the bifurcation points is analyzed. Of specific interest are low‐order resonances. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hopf bifurcation control of the Rössler system based on Washout filter controller

The Washout filter is adopted to control Hopf bifurcation in Rössler system. The effect of its parameters on the position of bifurcation point, the bifurcation type, and the amplitude of periodic solution is discussed in detail. Based on Routh criterion, the stability region in parameter space is found out firstly. Then the influence of the filter’s time constant and the linear gain on the location of bifurcation point is analyzed. By using the Direct Normal Form method, the normal form of the controlled Rössler system is deduced and the coefficient of resonant term is expressed as a function of the nonlinear gain. It is revealed that, by changing the sign and the absolute value of the real part of the coefficient, the nonlinear gain can modify the periodic solution’s amplitude and Hopf bifurcation type. All the results obtained are verified by numerical simulation.

A conjecture on the stability of the periodic solutions of Ricker’s equation with periodic parameters

In this work, we propose a conjecture about the stability of the periodic solutions of the Ricker equation with periodic parameters, which goes beyond the existing theory, and for the special case of period-two parameters we analytically show the conjecture is true. For this case we show that the stability region in parameter space obtained from the conjecture is larger than a previously proposed stability region. The period-three case is investigated numerically and similar extensions are realized. This suggests that the current theory cited in this paper, while giving sufficient conditions for stability is far from optimal.

Modal stability of inclined cables subjected to vertical support excitation

In this paper the out-of-plane dynamic stability of inclined cables subjected to in-plane vertical support excitation is investigated. We compute stability boundaries for the out-of-plane modes using rescaling and averaging methods. Our study focuses on the 2:1 internal resonance phenomenon between modes that occurs when the excitation frequency is twice the first out-of-plane natural frequency of the cable. The second in-plane mode is excited directly, while the out-of-plane modes can be excited parametrically. An analytical model is developed in order to study the stability regions in parameter space. In this model we include nonlinear coupling effects with other modes, which have thus far been omitted from previous models of parametric excitation of inclined cables. Our study reflects the importance of such effects. Unstable parameter regions are defined for the selected cable configuration. The validity of the proposed stability model was tested experimentally using a small-scale cable actuator rig. A comparison between experimental and analytical results is presented in which very good agreement with model predictions was obtained.

Stability condition for vertical oscillation of 3-dim heavy spring elastic pendulum

Equations of motion for 3-dim heavy spring elastic pendulum are derived and rescaled to contain a single parameter. Condition for the stability of vertical large amplitude oscillations is derived analytically relating the parameter of the system and the amplitude of the vertical oscillation. Numerical continuation is used to find the border of the stability region in parameter space with high precision. The stability condition is approximated by a simple formula valid for a large range of the parameter and of the amplitude of oscillation. The bifurcation responsible for the loss of stability is identified.

Numerical Exploration of Kaldorian Macrodynamics: Enhanced Stability and Predominance of Period Doubling and Chaos with Flexible Exchange Rates

We explore a discrete Kaldorian macrodynamic model of an open economy with flexible exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods marketα, and the degree of capital mobilityβ. We determine by a numerical grid search method the stability region in parameter space and find that flexible rates cause enhanced stability of equilibrium with respect to variations of the parameters. We identify the Hopf-Neimark bifurcation curve and the flip bifurcation curve, and find that the period doubling cascades which leads to chaos is the dominant behavior of the system outside the stability region, persisting to large values ofβ. Cyclical behavior of noticeable presence is detected for some extreme values of a state parameter. Bifurcation and Lyapunov exponent diagrams are computed illustrating the complex dynamics involved. Examples of attractors and trajectories are presented. The effect of the speed of adaptation of the expected rate is also briefly discussed. Finally, we explore a special model variation incorporating the “wealth effect” which is found to behave similarly to the basic model, contrary to the model of fixed exchange rates in which incorporation of this effect causes an entirely different behavior.

AdS and stabilized extra dimensions in multi-dimensional gravitational models with nonlinear scalar curvature terms R−1 and R4

We study multi-dimensional gravitational models with scalar curvature nonlinearities of types R−1 and R4. It is assumed that the corresponding higher dimensional spacetime manifolds undergo a spontaneous compactification to manifolds with a warped product structure. Special attention has been paid to the stability of the extra-dimensional factor spaces. It is shown that for certain parameter regions the systems allow for a freezing stabilization of these spaces. In particular, we find for the R−1 model that configurations with stabilized extra dimensions do not provide a late-time acceleration (they are AdS), whereas the solution branch which allows for accelerated expansion (the dS branch) is incompatible with stabilized factor spaces. In the case of the R4 model, we obtain that the stability region in parameter space depends on the total dimension D = dim(M) of the higher dimensional spacetime M. For D > 8 the stability region consists of a single (absolutely stable) sector which is shielded from a conformal singularity (and an antigravity sector beyond it) by a potential barrier of infinite height and width. This sector is smoothly connected with the stability region of a curvature-linear model. For D < 8 an additional (metastable) sector exists which is separated from the conformal singularity by a potential barrier of finite height and width so that systems in this sector are prone to collapse into the conformal singularity. This second sector is not smoothly connected with the first (absolutely stable) one. Several limiting cases and the possibility of inflation are discussed for the R4 model.

Effect of multiple equilibria on power system stability

This paper illustrates that for the same system loading, a power system may become unstable either due to angle instability or due to voltage instability or due to both depending upon the disturbance. The studies on two sample power systems demonstrate that for certain range of system loading there are many controlling unstable equilibrium points and their unstable manifolds give rise to different kinds of the instability for the same loading. The stability region in parameter space has also been characterized considering saddle-node and Hopf bifurcation to illustrate that the Hopf bifurcation further reduces the stable range of the operating parameters.

DYNAMIC FEEDBACK AND THE DESIGN OF CLOSED-LOOP DRUG DELIVERY SYSTEMS

A closed-loop drug delivery system is constructed in which external negative feedback is used to regulate the dynamics of a time-delayed negative feedback mechanism which regulates hormone concentration. This results in a control system composed of two time-delayed negative feedback loops arranged in parallel. Stability regions in parameter space and the location of steady states are determined for the cases when the time delays are equal and when they are unequal. The advantage of this paradigm for drug delivery is that both the steady states and stability of the multiple loop feedback system can be influenced in a precisely controllable manner.

The stability of parametrically forced coupled oscillators in sum resonance

For a system of two damped parametrically forced oscillators in sum resonance the planar stability diagram of amplitude versus frequency of the forcing shows a discontinuity at damping zero. This is a well known phenomenon, for which we give a geometrical explanation. A linear stability analysis suffices. We show that a versal (i.e. a structurally stable) matrix unfolding for this problem needs four parameters, indicating that the stability diagram is actually four dimensional. The boundary of the stability region in parameter space is singular, this provides a geometric explanation of the discontinuity in the planar stability diagram.

Oscillatory reaction-diffusion equations on rings

We study the behavior of traveling waves in λ-ω systems on both homogeneous and inhomogeneous rings. The stability regions in parameter space of λ-ω waves were previously known [15, 19]; the results are extended here. We show the existence of Hopf bifurcations of traveling waves and the stability of the limit cycles born at the Hopf bifurcation for some parameter ranges. Using a Lindstedt-type perturbation scheme, we formally construct periodic solutions of the λ-ω system near a Hopf bifurcation and show that the periodic solutions superimposed on the original traveling wave have the effect of altering its overall frequency and amplitude. We also study the λ-ω system on an annulus ofvariable width, which does not possess reflection symmetry about any axis. We formally construct traveling waves on this variable-width annulus by a perturbation scheme, and find that perturbing the width of the annulus alters the amplitude and frequency of traveling waves on the domain by a small (order e2) amount. For typical parameter values, we find that the speed, frequency, and stability are unaffected by the direction of travel of the wave on the annulus, despite the rotationally asymmetric inhomogeneity. This indicates that the λ-ω system on a variable-width domain cannot account for directional preferences of traveling waves in biological systems.

Instability in a granular bed fluidized by a gas flow

Stability in fluidization is considered in an approach in which the granular bed is considered as a structureless element with a definite working response, and the boundary to the stability region in parameter space is examined.

Helimagnetism in MnP-type compounds: MnP, FeP, CrAs and CrAs 1− xSb x mixed crystals

The solid solution CrAs 1− x Sb x has been investigated by neutron diffraction at various temperatures. The phase boundary between mixed crystals with MnP-structure and those with NiAs-structure moves from 59 mol% CrAs at room temperature to 48 mol% CrAs at liquid helium temperature. The mixed crystals with NiAs-type structure at 4.2K have the collinear magnetic structure of CrSb. Samples having the MnP-type structure exhibit a “double spiral” spin structure like CrAs. The variation of the magnetic moment and the spiral structure have been studied. Using neutron diffraction data for the magnetic propagation vector and the phase difference of the two helices, the signs and the ratios of the exchange integrals are discussed in a Heisenberg model. The stability regions in parameter space are determined by a matrix method for the ferromagnetic mode observed in MnP above 50 K and for the helical modes observed in MnP below 50 K, and in FeP, CrAs and CrAs 1− x Sb x . We have also considered the possibility of antisymmetric and symmetric exchange between nearest neighbours only, using a group theoretical approach. The high ratios of antisymmetric to symmetric exchange would be necessary to explain the observed helical structures preclude however antisymmetric exchange as a dominant factor.