Boundary Of Stability Region Research Articles

Estimating the boundary of the asymptotic stability region of Lotka–Volterra system by using the trajectory reversing method

This paper proposes a topological approach for plotting the boundary of the region of asymptotic stability (RAS) of Lotka–Volterra predator-prey system. First, stability analysis was used to determine the specific saddle point that has eigenvalues with one positive and two negative real parts in a linearized Jacobian matrix. A set of initial states located around the saddle point on the specific eigenplane spanned by the two stable eigenvectors was then selected. Finally, the trajectory reversing method was used and the trajectories that had initial states on the eigenplane delineated the boundary of the asymptotic stability region. The trajectories of the initial states that started from the opposite sides of the RAS exhibited different dynamic behaviour. The numerical simulation are presented to demonstrate the effectiveness of the proposed approach.

Обобщенные критерии устойчивости особых линейных систем автоматического управления с запаздыванием

Matters concerned with stability of special linear automatic control systems with a time delay are considered. The determining feature of such systems is that the control loop contains a link of pure (or transport) delay by time T (or for the path S) of the signal at its output with respect to the signal at its input. Apart from time delay links, these systems also contain other linear elements. Such systems include both longstudied systems for transporting various materials and relatively new selflearning repetitive control systems for accurately reproducing cyclically repeated motions or other signals with a period T (or S). The stability of special linear systems can be studied with fully resting on the Nyquist frequency criterion. According to the Nyquist frequency criterion, the stability of a closedloop system is judged by analyzing the frequency transfer function (FTF) loci of the openloop system that includes the controller and the controlled object with respect to the critical point (1, j0) on the complex plane. However, in the majority of cases, the presence of transcendental links results in that the FTF loci of the openloop system have the shape of a star with infinitely long rays, which makes it difficult to interpret the stability of systems with a time delay according to the Nyquist criterion. In such cases, it is more convenient to use a group of graphic criteria, the application of which makes it possible to establish sufficient stability conditions without examining the entire openloop system (the controller plus the object) by analyzing the relative position of the controlled object frequency response loci (obtained without taking into account the controller properties) and a certain stability region boundary, which is determined by the controller properties (obtained without taking into account the controlled object properties). To date, stability region boundaries have already been found for many types of specific structures of systems with a time delay. The problem is that an individually shaped stability region boundary has to be found for each specific type of controller. It is shown, based on a structural transposition of systems with a time delay, that their stability can be studied by using almost any stability boundaries that were previously proposed for some specific types of transcendental controllers. By generalizing the criteria it will be possible not only to estimate for the first time the stability of systems with a time delay in combination with a linear part of an arbitrary kind, but also to expand the variety of applied types of transcendental controllers.

Power System Transient Stability Study by Involving Higher Order Corrections in Improved Quadratic Lyapunov Function for Singular Perturbed Model of Synchronous Generator

The assessment of transient stability of multi-machine power system, in a post-fault scenario, is a very challenging task due to highly non-linear nature of the system equations. In this work, the physical concept of maximum relative acceleration of a machine at a fault scenario is related with the boundary of stability region. This approach is extended to develop a quadratic Lyapunov function using singular perturbed modelling. Unlike the conventional energy function methods, this Lyapunov function candidate is a more general energy function and has a decaying nature around an equilibrium point. Moreover, with higher order corrections, an improved Lyapunov function candidate is presented with simulation results of power system examples. The decaying nature of Lyapunov function candidate remains similar even without higher order corrections. However, the relevance of inclusion of higher order corrections is brought out in the cases of faults occurring at other than generator buses. Also, the upper bound of singular perturbation parameter in the model is within allowable limits and ensures the accuracy of results.

A Homotopy-Based Method for Robust Computation of Controlling Unstable Equilibrium Points

This paper proposes a homotopy-based method for efficiently computing and precisely determining controlling unstable equilibrium points (CUEPs) in transient stability analysis and screening using direct methods. The proposed method is intended to overcome potential numerical convergence problems associated with computing CUEPs. These convergence problems are mostly due to irregularities in the region of convergence of CUEPs which make it difficult to determine suitable initial points. The most commonly used initial points in computing CUEPs are minimum gradient points (MGPs) on the stability boundary which in turn rely on the determination of exit points (EPs-points at which sustained-fault trajectories exit system stability boundaries). However, determination of MGPs and EPs are computationally involved and sophisticated numerical methods have been introduced such as stability-boundary-following procedure and shadowing method to correct the solution trajectory. The proposed method does not depend on MGPs and also does not require accurate EPs. The proposed method maps the solution from EPs to the CUEPs. The proposed method is applied on several known systems including the NE 39 bus system and the reduced WECC system and the results are provided. The results are compared with the traditional methods such as Boundary of stability region based CUEP method (BCU method).

Computation of Robust PI-Based Pitch Controller Parameters for Large Wind Turbines

This paper deals with the computation of all proportional-integral (PI)-based pitch controllers which achieve the desired frequency-domain specifications, namely, gain and phase margins (GPMs) of a large wind turbine (LWT) with communication delays. An efficient graphical method based on extracting the boundaries of stability regions in PI controller parameter space having user-defined GPMs has been employed to determine GPM-based stability regions for a wide range of time delays. The theoretical region boundaries are validated by using a powerful numerical method known as the quasi-polynomial mapping-based root finder (QPmR) and time-domain simulations. Results indicate that the proposed scheme gives an improved dynamic response compared to the recently developed scheme based on stability only for the pitch control of LWTs with communication delays.

Improved Generalized Procedure for Determining Critical State of a Thin-Walled Beam Under Combined Symmetric Loads

This paper presents an improved generalized procedure for dealing with the stability of thin-walled beams under combined symmetric loads based on the energy method. The differential equations for the case of complex loading conditions were developed using an axis transformation matrix. The work caused by external loads was related to the work of internal forces to simplify the computational procedure. The thin-walled beam subjected to axial force [Formula: see text], bending moment [Formula: see text] at both ends, and concentrated load [Formula: see text] at midspan was studied. The case of a concentrated load [Formula: see text] replaced by a distributed load [Formula: see text] over partial beam length was also examined. The stability region boundary of the beam was derived by two approaches: one was to estimate an approximate angle of twist prior to determination of the deflection and the other was to do it in the reverse way. Numerical results reveal that the first approach yields less error than the second; however, the outcome obtained by the former was more cumbersome than the latter. Above all, both approaches provided feasible results and are useful for further applications dealing with the stability analysis of thin-walled beams.

Basic Bifurcation Scenarios in Neighborhoods of Boundaries of Stability Regions of Libration Points in the Three-Body Problem

In this paper, we construct stability regions (in the linear approximation) of triangular libration points for the planar, restricted, elliptic three-body problem and examine bifurcations that occur when parameters of the system pass through the boundaries of these regions. A new scheme for the construction of stability regions is presented, which leads to approximation formulas describing these boundaries. We prove that on one part of the boundary, the main scenario of bifurcation is the appearance of nonstationary 4π-periodic solutions that are close to a triangular libration point, whereas on the other part, the main scenario is the appearance of quasiperiodic solutions.

Robust non‐fragile approach to resilient design of PID‐based blade pitch control for wind energy conversion system

AbstractThe design of a blade pitch controller (BPC) for wind energy conversion system (WECS) applications is load‐dependent and has to be adjusted for each operating condition. Thus, BPC robustness is important for coping with the endless variations in operating conditions. The boundaries of a robust stability region are determined in regards to the controller parameters plane using their relevant set of polynomial inequalities via Referential Integrity between Routh‐Hurwitz criterion and Root‐Locus (RI‐RH/RL) approach. Constrained and unconstrained stability regions respectively are defined through a novel hybrid control technique based on the combination of both RI‐RH/RL and Kharitonov (Kh) theorem. The hybrid RI‐Kh method is used for globally analyzing all vertex plants to ensure the proposed controller robustness, non‐fragility, and resilience by selecting its parameters at the center of the robust stability region. The optimal BPC‐PID parameters estimated using different optimization techniques are always located within the specified stability region. Thus, the capability of the RI‐Kh approach in determining the most robust, non‐fragile and resilient controller is verified. Through simulation results, the effectiveness of the proposed approach and its applicability to WECS' global stabilization are validated.

On Orbital Stability of Pendulum-like Satellite Rotations at the Boundaries of Stability Regions

On Orbital Stability of Pendulum-like Satellite Rotations at the Boundaries of Stability Regions

基于并集及特殊点求解的稳定域边界分析方法

基于并集及特殊点求解的稳定域边界分析方法

Stability Prediction Based Effect Analysis of Tool Orientation on Machining Efficiency for Five-Axis Bull-Nose End Milling

The traditional stability analysis by only considering cutting depth-spindle speed lobe diagram is appropriate for parameters optimization and efficiency improvement of the five-axis ball-end milling. However due to the complicated cutter-workpiece engagement (CWE) of bull-nose end cutter in five-axis milling, the maximal cutting depth may not produce the maximal material removal rate (MRR). Thus, the traditional stability analysis is not suitable for the five-axis bull-nose end milling in parameters optimization, and this paper presents a new stability analysis method to analyze the effect of tool orientation on machining efficiency for five-axis bull-nose end milling. In the establishing of stability prediction model, coordinate transformation and vector projection method are adopted to identify the CWE and dynamic cutting thickness, and the geometrical relationship of frequency response function (FRF) coordinate system and cutting force coordinate system with variable tool orientation is derived to establish the conversion of FRF and cutting force in stability equation. Based on the CWE sweeping, the cutting area along the feed direction is calculated to realize the critical MRR analysis in the stability model. Based on the established stability prediction model, the effects of tool orientation on critical cutting depth and MRR considering the chatter constraint are analyzed and validated by the cutting experiments, respectively. The lead-tilt diagram, which not only gives the boundary of stability region but also describes the contour line for MRR, is proposed for the further tool orientation optimization.

Global Approximation of Static Voltage Stability Region Boundaries Considering Generator Reactive Power Limits

Power system static voltage stability region boundary (SVSRB) is a complicated hypersurface in parameter space. Due to inequality constraints such as generator reactive power limits, different critical points on the SVSRB may be related to different groups of active constraints and can be associated with different types of loadability limit, which makes it difficult to approximate the whole SVSRB at once or assemble many local approximations of the SVSRB. To overcome the difficulty, this paper proposes an approach for globally approximating the whole SVSRB considering generator reactive power limits at one go. First, a parametric optimization model incorporating all inequality constraints is formulated to describe the SVSRB, where the optimal solutions are the critical points on the SVSRB. The optimality condition, i.e., parametric Karush–Kuhn–Tucker conditions (KKT conditions), is also derived. Then, by applying the Galerkin method to the parametric KKT conditions in an interior-point-method-like manner, the whole SVSRB is approximated and represented using a single polynomial expression. Case studies in a 10-bus system and IEEE 300-bus system verify the effectiveness and accuracy of the proposed method.

Computation of PI Controllers Ensuring Desired Gain and Phase Margins for Two-Area Load Frequency Control System with Communication Time Delays

This article deals with computation of all proportional-integral (PI) controllers which achieve the desired frequency-domain specifications namely, gain and phase margins (GPMs) of a traditional two-area load frequency control (LFC) system with constant communication delays. An efficient graphical method based on extracting the boundaries of stability regions in PI controller parameter space having user defined GPMs is proposed to obtain GPMs based stability regions. Regions are determined for a wide range of GPMs and time delay. The theoretical region boundaries are validated using a powerful numerical method known as the quasi-polynomial mapping-based root finder (QPmR) and time-domain simulations. Results indicate that the proposed scheme gives faster disturbance rejection and desired frequency response as compared to the recently developed scheme for LFC systems with communication delays.

Dynamics of unstable sound waves in a non-equilibrium medium at the nonlinear stage

A new dispersion equation is obtained for a non-equilibrium medium with an exponential relaxation model of a vibrationally excited gas. We have researched the dependencies of the pump source and the heat removal on the medium thermodynamic parameters. The boundaries of sound waves stability regions in a non-equilibrium gas have been determined. The nonlinear stage of sound waves instability development in a vibrationally excited gas has been investigated within CSPH-TVD and MUSCL numerical schemes using parallel technologies OpenMP-CUDA. We have obtained a good agreement of numerical simulation results with the linear perturbations dynamics at the initial stage of the sound waves growth caused by instability. At the nonlinear stage, the sound waves amplitude reaches the maximum value that leads to the formation of shock waves system.

Multi-Time Scale Model Order Reduction and Stability Consistency Certification of Inverter-Interfaced DG System in AC Microgrid

AC microgrid mainly comprise inverter-interfaced distributed generators (IIDGs), which are nonlinear complex systems with multiple time scales, including frequency control, time delay measurements, and electromagnetic transients. The droop control-based IIDG in an AC microgrid is selected as the research object in this study, which comprises power droop controller, voltage- and current-loop controllers, and filter and line. The multi-time scale characteristics of the detailed IIDG model are divided based on singular perturbation theory. In addition, the IIDG model order is reduced by neglecting the system fast dynamics. The static and transient stability consistency of the IIDG model order reduction are demonstrated by extracting features of the IIDG small signal model and using the quadratic approximation method of the stability region boundary, respectively. The dynamic response consistencies of the IIDG model order reduction are evaluated using the frequency, damping and amplitude features extracted by the Prony transformation. Results are applicable to provide a simplified model for the dynamic characteristic analysis of IIDG systems in AC microgrid. The accuracy of the proposed method is verified by using the eigenvalue comparison, the transient stability index comparison and the dynamic time-domain simulation.

On Parametrically Excited Vibration and Stability of Beams with Varying Rotating Speed

In this paper, principal parametric resonance of rotating Euler–Bernoulli beams with varying rotating speed is investigated. The model contains the geometric nonlinearity due to the von Karman strain–displacement relationship and the centrifugal forces due to the rotation. The rotating speed of the beam is considered as a mean value which is perturbed by a small harmonic variation. In this case, when the frequency of the periodically perturbed value is twice the one of the axial mode frequencies, the principal parametric resonance occurs. The direct method of multiple scales is implemented to study on the dynamic instability produced by the principal parametric resonance phenomenon. A closed-form relation which determines the stability region boundary under the condition of the principal parametric resonance is derived. Numerical simulation based on the fourth-order Runge–Kutta method is established to validate the results obtained by the method of multiple scales. The numerical analysis is applied on the discretized equations of motion obtained by the Galerkin approach. After validation of the results, a comprehensive study is adjusted for demonstrating the damping coefficient and the mode number influences on the critical parametric excitation amplitude and the parametric stability region boundary. A discussion is also provided to illustrate the advantages and disadvantages of the fourth-order Runge–Kutta method in comparison with the method of multiple scales.

A study of the boundaries of stability regions in two-parameter dynamical systems

We consider dynamical systems defined by autonomous and periodic differential equations that depend on two scalar parameters. We study the problems of constructing boundaries of stability regions for equilibrium points in the plane of parameters. We identify conditions under which a point on the boundary of a stability region has one or more smooth boundary curves coming through it. We show schemes to find the basic scenarios of bifurcations when parameters transition over the boundaries of stability regions. We distinguish types of boundaries (dangerous or safe). The main formulas have been obtained in the terms of original equations and do not require to pass to normal forms and using theorems on a central manifold.

Nonlinear equations of the ion vibration envelope in quadrupole mass filters with cylindrical rods

This paper develops the method of envelope equations to describe the oscillatory motion of ions in quadrupole mass analysers with round rods. Nonlinear equations for the description of the envelope dynamics are obtained taking into account the dodecapole (twelve-pole), the icosapole (twenty-pole) and octopole field distortions. It is shown that the effective potential which determines the ion vibration envelope has a quadratic part, with a magnitude and sign that depend on the location of the ion working point. It also contains a nonlinear part, which does not depend on the ion working point location near the tip of the first stability zone. The motion of ions in quadrupole mass filters with cylindrical rods has a nonlinear character at a theoretical mass resolving power over 1000. In the linear approach, nonlinear effects of the ion motion occur in a narrow zone near the boundaries of the first stability region. This explains low sensitivity of mass analysis within stability islands, which is a direct result of the quadrupole excitations at a frequency different from the main excitation frequency. Thus, the boundaries do not depend on nonlinear field distortions. The results of this theoretical approach are illustrated using the transmission peak shape modelling for mass analysers with different values of the rod radius r relative to the inscribed radius r0.It is shown, that the 6th and 10th order field distortions never compensate each other. However, the octopole field suppresses them completely in the range 1.115<r/r0<1.135, which is considered as an optimum. It is also shown that very substantial improvements in a peak shape can be achieved by adding minor (less than 1%) octopole distortions.

Towards Robust Non-Fragile Control in Wind Energy Engineering

&lt;p&gt;Blade Pitch Controller (BPC) that can cope system uncertainties is one of the most interesting topics in wind energy engineering. Therefore, this paper presents a step towards the design of robust non-fragile BPC for wind energy conversion system. The proposed approach presents all boundaries of stability region that can guarantee robust stability (RS) over a wide range of operating conditions. The proposed technique results from the complementarity of both Root-Locus and Routh-Hurwitz (RL/RH) approach. Continuous variation in the operating conditions is tackled through a new hybrid control technique based on the referential integrity of both RL/RH and Kharitonov (Kh) theorem. Simulation results confirm the effectiveness of the proposed designing approach in computing the most resilient and robust controller.&lt;/p&gt;

Global Parametric Polynomial Approximation of Static Voltage Stability Region Boundaries

A novel method for globally approximating the static voltage stability region boundaries (SVSRBs) of power systems is proposed by applying parametric polynomial approximation to the criterion equation which defines the SVSRB. Known as the Galerkin method, the implicit function portraying the SVSRB is described as a linear combination of basis polynomial functions, and the coefficients of the basis functions are obtained by projecting the SVSRB criterion onto each basis function. The approximation guarantees high precision globally in the whole domain of the parameter of interest rather than only in the neighborhood of a point, and the error can be controlled by the degree of polynomial basis functions. In the meantime, analytical expression of the left or right eigenvector of the system's Jacobian matrix corresponding to the zero eigenvalue is obtained in the form of polynomial, which provides valuable information for online voltage stability control or monitoring. Case studies in a 10-bus test system and IEEE 118-bus test system verifies the validity, accuracy, and flexibility of the proposed method.