Boundary Of Stability Region Research Articles

Bifurcation Analysis for an OSN Model with Two Delays

In this research, we introduce and analyze a mathematical model for online social networks, incorporating two distinct delays. These delays represent the time it takes for active users within the network to begin disengaging, either with or without contacting non-users of online social platforms. We focus particularly on the user prevailing equilibrium (UPE), denoted as P*, and explore the role of delays as parameters in triggering Hopf bifurcations. In doing so, we find the conditions under which Hopf bifurcations occur, then establish stable regions based on the two delays. Furthermore, we delineate the boundaries of stability regions wherein bifurcations transpire as the delays cross these thresholds. We present numerical simulations to illustrate and validate our theoretical findings. Through this interdisciplinary approach, we aim to deepen our understanding of the dynamics inherent in online social networks.

Defining the Cortical Purchase Zone in New Minimally Invasive Bunion Surgery. A Retrospective Study of 638 Cases.

Introduction The stability of the screws in the cortical purchase zone (CPZ) with new minimally invasive bunion surgery (MIBS) is of utmost importance in creating a stable construct for maintaining position and providing a scaffold for bone healing. The distance between the osteotomy and where the screw(s) exit the lateral cortex (a distance we herein coin the “Cortical Runway”) within the CPZ has yet to be studied, and is the focus of this retrospective study. Methods A retrospective review was preformed of a single surgeons first and consecutive cases of patients who had undergone MIBS from January 2018 to November 2022. The CPZ measurement was obtained on early postoperative radiographs. The CPZ was divided into five stability regions using the statistical empirical rule to define the boundaries of each of the regions. Results We included 427 patients and 638 feet in this study. Radiographic measurements were performed at an average 20.7±24.2 days (95% CI, 15.3–15.8 days) from surgery. The mean age of the patients was 46.5±15.0 years (95% CI, 45.3–47.6 years). There were 330 (51.7%) right feet and 308 (48.3%) left feet. Of the 427 patients, 211 (49.4%) had surgery on both feet, with 97 (46%) having both feet operated on the same day. A 1-screw construct was performed in 526 feet (82.4%) and a 2-screw construct in 112 feet (17.6%) of the total 638 feet. The mean anchor screw (primary screw) distance was 10.4±3.7mm (95% CI, 10.1-10.7mm). The mean collateral screw (secondary screw) distance was 1.2±2.0mm (95% CI, 0.8-1.5) and the mean cortical bridge (distance between the two screws) was 10.3±3.2mm (95% CI, 9.7-10.9), in 112 feet. We were able to define the CPZ stability region boundaries, which are: danger (< 2.9mm), vulnerable (3.0 - 6.6mm), standard (6.6 - 14.1mm), safety (14.2 - 17.8mm) and the security (> 17.9mm). Conclusion These CPZ boundaries and newly delineated stability regions will serve not only as an intraoperative visual tool for surgeons to perform MIBS successfully, but will also provide a much more stable and safer construct, thus minimizing risks and complications.

Stability of floating vessels with cross sections in the form of elliptic and hyperbolic segments

The paper considers two problems on the stability of the trivial equilibrium position of floating vessels with cross sections in the form of elliptic and hyperbolic segments. A review of examples on the stability of floating bodies is given and the key principles of its study by methods of analytical statics are outlined. For each of the presented problems, by means of rather serious mathematical constructions, an exact expression for the potential energy is obtained within the framework of the accepted configuration, and its quadratic approximation is calculated near the equilibrium state under study. On its basis, stability conditions of the equilibrium position are established in terms of three dimensionless parameters, and limit cases are also analyzed. A comparison of intermediate expressions and final results obtained in the process of discussing each of the problems is carried out, and their common features and distinctive features are revealed. The found solutions are illustrated as families of boundaries of stability regions on the plane of two dimensionless parameters for different values of the third parameter. These results are of fundamental theoretical importance and may be useful for practical applications.

Dynamic perturbation analysis of fractional order differential quasiperiodic Mathieu equation

The paper investigates the influence of parameters on the stability of fractional order differential quasiperiodic Mathieu equations. First, we use the perturbation method to obtain approximate expressions (i.e., transition curves) for the stability and unstable region boundaries of the equation. After obtaining the approximate expression of the transition curve, we use Lyapunov's first method to analyze the stability of the fractional order differential quasiperiodic Mathieu system, thereby obtaining the conditions for the stability of the fractional order differential quasiperiodic Mathieu equation system. Second, by comparing the approximate expressions of the transition curve of the steady-state periodic solution of the quasiperiodic Mathieu oscillator under different parameter conditions, we obtained the conclusion that the fractional order differential term exists in the form of equivalent stiffness and equivalent damping in the fractional order differential quasiperiodic Mathieu system. By comparison, we have summarized the general forms of equivalent linear damping and equivalent stiffness of the system. Through this general form, we can define an approximate expression for the thickness of unstable regions to better study the characteristics of fractional order differential quasiperiodic Mathieu systems. Finally, the influence of the parameters of the fractional order differential quasiperiodic Mathieu equation on the transition curve of the equation was intuitively analyzed through numerical simulation, to analyze the stability changes in the equation.

Approaching the Transient Stability Boundary of a Power System: Theory and Applications

Estimating the stability boundary is a fundamental and challenging problem in transient stability studies. It is known that a proper level set of a Lyapunov function or an energy function can provide an inner approximation of the stability boundary, and the estimation can be expanded by trajectory reversing methods. In this paper, we streamline the theoretical foundation of the expansion methodology, and generalize it by relaxing the request that the initial guess should be a subset of the stability region. We investigate topological characteristics of the expanded boundary, showing how an initial guess can approach the exact stability boundary locally or globally. We apply the theory to power system transient stability assessment, and propose expansion algorithms to improve the well-known Potential Energy Boundary Surface (PEBS) and Boundary of stability region based Controlling Unstable equilibrium point (BCU) methods. Case studies on the IEEE 39-bus system well verify our results and demonstrate that estimations of the stability boundary and the critical clearing time can be significantly improved with modest computational cost. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper was motivated by the problem of determining whether a power system can retain transient stability subject to large disturbances such as short-circuit fault and generator tripping. The most well-known approaches to such a problem are the direct methods that can avoid time-costly simulations and “directly” determine the transient stability by estimating the stability boundary. Nevertheless, such estimations are generally too conservative. This paper proposes a new expansion method to improve the direct methods, which allows the initial estimation to lie partially outside the real boundary and hence enables applications to both global and local direct methods. Our method can expand the initial estimation towards the real stability boundary, thus reducing the conservativeness and providing a more accurate stability assessment such as the critical clearing time (CCT) of a fault. In this paper, we first mathematically establish the expansion theory and then propose numerical algorithms for power system applications. Simulations on the IEEE 39-bus benchmark verify our approach and show that the conservativeness can be significantly reduced with modest computational cost. Our approach can be further extended to scenarios where a valid Lyapunov (energy) function is unavailable.

Online assessment of multi-parameter stability region and stability margin of wind power plants

The wind power plant (WPP) operates in a wide range of operating points, resulting in diverse small-signal stability performances in a high-dimensional parameter space. The mapping between the stability performance and the operating point is implicit, which makes an obstacle for the stability region and margin assessment. Furthermore, the implicit mapping cannot guide the system operation in real-time applications. In this paper, a data-driven method is developed for the stability assessment and enhancement. Regarding the stability assessment, a regression-based analytical model is proposed to assess the stability region and margin as functions of multiple parameters, such as the active power, the reactive power, and the wind speed. The stability assessment makes use of the impedance data of the WPP. Based on the impedance data at diverse operating points, the generalized Nyquist criterion is analytically reformulated with the supportive vector regression method. Minimum characteristic locus is defined as a single-parameter index of the relative stability, and the stability region boundary is developed. Besides, the dominated oscillation frequency in stable and unstable regions is analytically formulated. The analytical solutions yield an explicit mapping between the stability performance and the operating points. Moving forwards to the stability enhancement, with the help of the explicit mapping achieved in the digital twin (DT) system, the real-time stability margin of the physical WPP could be enhanced by optimizing the operating point of the WPP. The proposed data-driven method for stability assessment and enhancement is implemented in a DT system of WPP. The DT system provides the impedance data of WPP and the effective interaction between the physical and virtual systems, offering opportunities for online applications. The validation is performed on a single wind turbine generator (WTG) and a WPP, respectively. With numerical simulations and experiments, the stability region is proven to be accurately and efficiently observed, and the stability margin is enhanced by adjusting the operating point.

Устойчивость стержня с упругим шарниром при нагружении распределенной неконсервативной нагрузкой

Numerous features of non-conservative mechanical systems identified in studying their stability determine significant theoretical and practical interest to analyze various options of the calculation schemes. The paper considers stability of a rectilinear rod connected at one end to a joint and loaded with follower and potential loads uniformly distributed along the rod length. The joint was rigid with respect to rotation of the rod end. In order to apply the method of expanding solution to the perturbed motion equation into a series in terms of eigenmodes, the problem of determining the system eigenfrequencies and modes was solved. The cases of separate and combined action of the follower and potential loads were considered. A study was made on the influence of rigidity of the rod fastening and damping in the system on the loads critical values and on the position of the stability region boundaries on the loading parameters plane.

Fractional-Order Active Disturbance Rejection Controller Design for Large Radio Telescope Antenna

large radio telescope antennas are required in current astronomical research and deep space communication applications. The wind disturbance becomes a serious problem with an increase in the size of the antennas. Based on the typical antenna model, this work aims to design a fractional-order active disturbance rejection controller (FOADRC) for improving the accuracy of the antenna pointing mechanism in wind disturbance. First, the fractional-order active disturbance rejection control scheme is chosen to estimate and compensate both the wind disturbance and other uncertain mechanical dynamics. Afterwards, the relationship between the parameters of fractional-order extended state observer and the performance of disturbance suppression and noise immunity is revealed. The stability of fractional-order extended state observer is proved under similar assumptions as an integer order extended state observer. In addition, a stability region boundary calculation method is presented to simplify the parameter setting procedure. Finally, we perform simulations to verify the superiority of FOADRC in the antenna control system under wind disturbance. The results demonstrate that FOADRC has a good capability of resisting wind disturbance; at the same time, it limits the response overshoot and converges faster compared to other industrial controllers.

Fasting the stabilization response for prevention of tractor rollover using active steering: Controller parameter optimization and real-vehicle dynamic tests

The rollover problem is prominent for a small tractor under complicated road conditions. The application of active steering (AS) technology is an effective approach to solve this problem. A lateral stability index based on the critical position is proposed and improved. Sliding mode control (SMC) technology based on an exponential terminal sliding mode surface is studied by considering the parameter changes and external disturbances in the rollover process of the tractor, which further accelerates the system’s response speed. A real vehicle test platform was built taking into account the limitations of the few real vehicle dynamic tests in current studies. Real vehicle dynamic tests with varying speeds, slopes, and obstacles were conducted for various road pavements and typical off-road pavements. For road pavements, the results demonstrated that the maximum roll angle of the tractor was 15.94 % less than with that under uncontrolled. For typical off-road pavements, AS control effectively restrained rollover accidents under non-extreme conditions, and SMC reduced the time the tractor was in the critical rollover region by 35.98 % on average compared with PID control. The stability region boundary of a small tractor was defined, and the stable driving region of a tractor on various slopes is obtained. This study provides a specific theoretical basis and implementation plan for guiding the research of tractor active safety technology.

Adaptive Frequency Control of Microgrid Based on Fractional Order Control and a Data-Driven Control With Stability Analysis

In this paper, a new adaptive fractional frequency control method for microgrids(MG) and low inertia networks is presented. The newly proposed grid frequency control is based on the droop and inertia controller for doubly-fed induction generator (DFIG) wind turbine (WT) using a data-driven algorithm. Also, in this paper, to ensure the grid frequency stability, the stability region boundary has been calculated using the parameter space method for above and below synchronous speeds. In the first step in the proposed method, a data-driven control (DDC) is used to update the frequency control coefficients. In the second step, in order to deployment more frequency data and to use both frequency error concavity and slope, the utilization of fractional gradient descent (FGD) and fractional inertia control is proposed. In the third step to maintain frequency stability, the updated frequency controller coefficients are compared with the stability region boundary. If the updated frequency controller coefficients are in the authorized band, they are applied. To test the proposed idea is a simulated real grid in Ontario, Canada. The simulation results indicate the proper operation of the control system in transient and steady-state and under different conditions.

Impedance Adaptive Dual-Mode Control of Grid-Connected Inverters With Large Fluctuation of SCR and Its Stability Analysis Based on D-Partition Method

The stable operation of grid-connected inverters (GCIs) with traditional current source mode (CSM) control is affected by the large fluctuations of short-circuit ratio (SCR) under weak grids. Improved CSM control enhances the stability of GCIs under weak grids but remains unstable for very weak grids, while the stability of GCIs with voltage source mode (VSM) is just the opposite. Therefore, previous research works have proposed the impedance adaptive dual-mode control strategy: when the SCR is large, the GCI is controlled by CSM, and when the SCR is small, it is controlled by VSM. However, the existing literature still lacks the analysis of the parameter stability region and mode switching boundary of GCI in different modes. Therefore, based on the D-partition method, this article derives the parameter stability region of GCI in different modes under the constraints of multiple performance indexes such as phase and gain margin, current-loop bandwidth and phase-locked loop bandwidth, and the switching boundaries of CSM and VSM are also proposed. Finally, based on the grid impedance identification algorithm, the impedance adaptive dual-mode control is realized, which effectively improves the stability of GCI when SCR fluctuates greatly, and the experimental results verify the correctness of the above analysis.

Polynomial Approximation of Transient Stability Region Boundary in Parameter Space Based on Collocation Method

Polynomial approximation based on collocation method is an emerging technique to establish the relationship between changeable parameters and interested transient trajectory in a given parameter space. If transient stability region boundary (TSRB) exists in the parameter space, the approximation may have low accuracy due to the high sensitivity of the real trajectory with respect to parameters on TSRB. Utilizing the approximated trajectory and the high sensitivity characteristic, this letter proposes a complementary method to obtain TSRB by calculating the local maxima of a sensitivity-based index. The effectiveness of the method is verified in IEEE 39-bus test system.

Study on mining area of roof surrounding rock stability of mining fissure development

The “double end sealing device of drilling drilling water injection leakage” section, to the different depth of sealing hole drilling water injection test, observation in different directions and different drilling water Kong Shen missed development in mining fracture analysis at different depth of stratum loss, combined with different directions borehole sections of fissure, preliminary judgment the test site of mining roof surrounding rock fracture stability region boundary of mining; use of field test data, analysis of Yu Yang coal mine N21110 mining fissure development scope and the degree of fracture development in mining area of roof stability.

Область стабилизации нелинейной модели синхронного генератора с ПИДД2-регулятором возбуждения

We take as the research object the well-known system of nonlinear differential and functional equations that describes a synchronous generator. PIDD2-control was carried out through the excitation block. Effective values of the steady state in relative units and the preliminary settings of suboptimal controllers were obtained earlier using a linearized model. Disturbing surges (spikes and slump) in bus voltages were modeled as line impedance changes. The system stabilization was studied at various surges values and controller parameters by means series of numerical experiments; the calculations were carried out by the methods of Runge-Kutta and Dorman-Prince. The main attention was paid to various transient types, both stabilizing and diverging, as well as a configuration of the stability region boundary in the controller parameters of and a surge magnitude; the paper presents the results for the proportional control parameter are as the most informative. We've identified five types of transients and several important features of the model's behavior near the stabilization area boundary. Thus, the nonlinear model turned out to be unstable to small and stable to bigger perturbations in the unstable area of its linearization. The steady-state oscillations at the boundary have a lower amplitude with increasing of an initial perturbation, etc.

Parametric Problems in Power System Analysis: Recent Applications of Polynomial Approximation Based on Galerkin Method

In power systems, there are many uncertainty factors such as power outputs of distributed generations and fluctuations of loads. It is very beneficial to power system analysis to acquire an explicit function describing the relationship between these factors (namely parameters) and power system states (or performances). This problem, termed as parametric problem (PP) in this paper, can be solved by Galerkin method, which is a powerful and mathematically rigorous method aiming to seek an accurate explicit approximate function by projection techniques. This paper provides a review of the applications of polynomial approximation based on Galerkin method in power system PPs as well as stochastic problems. First, the fundamentals of polynomial approximation and Galerkin method are introduced. Then, the process of solving three types of typical PPs by polynomial approximation based on Galerkin method is elaborated. Finally, some application examples as well as several potential applications of power system PPs solved by Galerkin method are presented, namely the probabilistic power flow, approximation of static voltage stability region boundary, and parametric time-domain dynamic simulation.

Математические модели для определения предельных режимов в электрических сетях с установками распределенной генерации

Decentralization of electricity generation based on distributed generation plants is an important segment of the new technology platform for the power industry. On the basis of this approach, significant positive effects can be obtained, which consist in reducing financial costs of energy supply, increasing the uninterrupted power supply, improving the quality of electricity and stimulating the use of renewable energy sources. Effective use of distributed generation in electric power systems requires the development of methods and tools that provide coordinated management of normal, emergency and post-emergency modes. Of particular relevance is the problem of determining the limit operating modes of networks, at the nodal points of which relatively low power generators are connected. In some situations, for example, when using small hydraulic stations, groups of such generators can be located at significant distances for 6-10-20 kV distribution networks from consumption centers. In this case there will be a noticeable limitation of the regions of static aperiodic stability. The article presents the results of developments aimed at implementing methods for determining the limit operating modes by static aperiodic stability in networks with distributed generation plants. The proposed approach is based on the limit modes equations which provide the formation of effective algorithms for the operational finding of points belonging to the boundaries of stability regions. The results of the construction of the indicated areas for a 6 kV electric network with distributed generation plants based on low-power hydraulic stations are presented. Additionally, the transient processes in the studied electric power system were simulated in the Matlab system for various space points of the controlled mode parameters.

Vehicle Lateral Stability Control Based on Shiftable Stability Regions and Dynamic Margins

Vehicle lateral stability control is one of the most critical aspects of vehicle safety control. In this paper, aiming at keeping the vehicle states (lateral velocity and yaw rate) always in the vehicle stability regions, a novel lateral stability control method, which consists of shiftable region-based stability analyses and a corresponding stability controller, is proposed. In the stability analyses, two features of the stability regions are introduced. First, a shifting vector is formulated to explicitly describe the shifting feature of lateral stability regions, so that the shiftable regions are not necessarily re-estimated with respect to a varying steering angle input. Second, dynamic margins of the stability regions are formulated and applied to avoid the penetration of vehicle state trajectory with respect to the stability region boundaries. With these two features, the shiftable stability regions are feasible for an accurate stability analysis, where a projection method is adopted to determine the relative location/distance between the vehicle state point and the closest boundary of the stability region. Based on the analyses, a dynamic sliding mode controller, where the sliding surfaces are selected as the stability region boundaries with the proposed dynamic margins, is designed to keep the vehicle states always in the shiftable stability region. To validate the effectiveness of the proposed control design and analyses, two maneuvers through the co-simulation between Matlab/Simulink and CarSim, namely a high-speed constant cornering and a double lane change maneuvers, are presented and discussed.

Modeling and Networked Control of Two-rigid link Robot Arm

A networked control system (NCS) is one in which controller(s), actuator(s),and sensor(s)exchange command signals and data through a limited-bandwidth communication network that may be used by other applications, devices, and control systems. Compared to classical wired controlled systems, NCSs possess many advantages. In this paper, we propose the modeling and networked control of two-rigid link robot arm. To deal with the time delays that may occur during communication between the components of the system through the network, a model of the system was first determined, and second, PID controllers were designed based on the obtained model and using the stability region boundary locus technique. To demonstrate the validity of the proposed approach, numerical simulations were conducted using TrueTime, Simscape, SimMechanics, and Simulink with the MATLAB environment

Relaxed decoupled direct calculation of voltage collapse points and its application in static voltage stability region boundary formation

Static voltage stability is critical for the operation of power grids. With the penetration of renewable energies, region-based method become a choice to present the stability of a power grid. Static voltage stability region is hence a useful and intuitive tool to determine the safety margin of a power grid. The formation of static voltage stability region is however time consuming. To overcome such issue, this work proposes a relaxed decoupled direct calculation (RDDC) method to find voltage collapse points (VCPs). The direct calculation of VCPs is a Newton-Raphson type method and faces the sensitivity issue of the initial values. Boundary tracking method provides a framework to utilize the information from neighbor VCPs and hence is integrated with the proposed RDDC method. The results show that the RDDC method is able to calculating VCPs as precise as continuation power flow with less computational time. When RDDC is integrated with boundary tracking, results show that the static voltage stability region boundary can be generated with high precision, and computational performance improvement can be observed for all the test systems from nine buses to ten thousand of buses. This demonstrates the applicability and scalability of the proposed method to power systems with different scales.

Entire stability region estimation using the invariant manifolds and singularities at infinity

This study provides a new perspective to study the formation and wholeness of the stability region of power systems. The main idea of the approach proposed in this study is to project the system into a closed area and then investigate the behaviours of the field in a neighbourhood of infinity on a sphere that is topologically equivalent to the original system. The invariant manifolds and singularities at infinity are described in detail to indicate the boundary of the entire stability region, based on the concept of the basin of attraction. The proposed method is applied to the single-machine model, two-machine model and multi-machine system to illustrate the complex geometry structure of the transient stability region. A singularity variation index is presented to estimate the size of the entire region and indicate the transient stability degree of the power system. Simulation results verify the correctness and effectiveness of the theoretical analysis.