Stability Region Of System Research Articles

Topological analysis of stability regions of power systems considering global orbits

In this paper, the stability region of power systems is studied from a geometrical point of view. A new method is proposed to obtain the global orbits (GOs) which have the significance for making the stability region be more complete. This method is based on the theory of invariant manifolds and focuses on dynamical behavior at infinity. The boundary of the stability region (BSR) with respect to the stable manifolds of dimension-1 and dimension-2 are given to describe the topological characteristics of the stability region of a multi-machine system. The effect of damping coefficient on the stability region is discussed through the variation of the singularities at infinity. The GOs of the power system when subjected to large disturbance are analyzed with the time domain simulation. The validity of the proposed approach is demonstrated on a 16-machine 68-bus system. The simulation results reveal the types and characteristics of the GOs, which play an important role in the prediction of stability/instability of the postfault system.

Stability regions of fractional systems in the space of perturbed orders

When dealing with fractional order systems, perturbations in differentiation orders arise frequently due to issues with floating point arithmetics, or due to imprecisions of various order estimation algorithms. This study establishes new results regarding stability/instability of fractional systems with perturbed differentiation orders, knowing the related properties of their unperturbed counterparts. First of all, starting from a point in the space of differentiation orders, sufficient stability/instability conditions of all systems with differentiation orders varying along a line segment with a prescribed direction are established. Then, a continuation procedure is developed allowing computation of the maximum perturbation (along some given direction) which guarantees that the number of zeros in the closed right-half plane of the characteristic function remain unchanged. Finally, sufficient conditions are established guaranteeing stability/instability of all systems having differentiation orders within a domain. The established results allow concluding on the stability of incommensurate fractional transfer functions. They are illustrated by a number of examples, including an experimental one.

Stability region of fractional differential systems with Prabhakar derivative

This paper analyzes the stability of fractional differential equations with Prabhakar derivative, which is a generalization of the fractional differential equation with Caputo and Riemann-Liouville derivative. As a result, the sufficient condition for asymptotic stability has been obtained by studying the eigenvalues of system related matrix and the position of these eigenvalues in the complex plane. To show the application of the results a two-dimensional predator–prey model has been studied with Prabhakar derivative and the dynamic behavior has been extracted. Then, a numerical method along with its error has been presented for solving differential equations with Prabhakar derivative. The predator–prey model simulation has been conducted by using this numerical method.

The Stability and Complexity Analysis of a Low-Carbon Supply Chain Considering Fairness Concern Behavior and Sales Service.

This paper studies a low-carbon dual-channel supply chain in which a manufacturer sells products through the direct channel and traditional channel, and a retailer sells products through the traditional channel. The manufacturer considers carbon emission reduction and has fairness concern behavior. The retailer provides sales service in the traditional channel and considers fairness concern behavior. The objective of this paper is to analyze the effects of different parameter values on the price stability and utility of the supply chain system emphatically using 2D bifurcation diagram, parameter plot basin, the basins of attraction, chaos attractor and sensitivity to the initial value, etc. The results find that the retailer’s fairness concern behavior shrinks the stability of the supply chain system more than that of the manufacturer’s fairness concern behavior. The system stability region decreases with the increase of carbon emission reduction level and the retailer’s fairness concern. The customers’ preference for the direct channel decreases the stable range of the direct channel, while it enlarges the stable range of the traditional channel. The supply chain system enters into chaos through flip bifurcation with the increase of price adjustment speed. In a stable state, the manufacture improving customer’s preference for the direct channel and the retailer choosing the appropriate fairness concern level can achieve the maximum utility separately. In a chaotic state, the average utilities of the manufacturer and retailer all decline, while that of the retailer declines even more. By selecting appropriate control parameter, the low-carbon dual-channel supply chain system can return to a stable state from chaos again. The research of this paper is of great significance to price decisions of participants and supply chain operation management.

Small Signal Modeling of Inverter-based Grid-Connected Microgrid to Determine the Zero-Pole Drift Control with Dynamic Power Sharing Controller

This paper presents a small signal state space modeling of three-phase inverter-based microgrid (MG) system with consideration of improved droop control. The complete system matrices for one distribution source-grid connects to the local load have been elaborated by applying high, medium and low-frequency clusters to the system without considering the switching action on the inverter during power-sharing. Moreover, the final matrices will be used to determine the location of the eigenvalues for the control parameters gains due to dynamic effect of the MG, by observing the root locus graph on cluster identification. Sensitivity analysis of all types of frequency cluster showed that power-sharing control parameters such as load current, source current, and inverter voltage are influencing system stability and must be considered when designing the proportional-integral (PI) control when different load scenarios have been applied from the zero-pole drifting. Those eigenvalues of the system model are indicating the frequency and damping oscillatory components when there is sudden changed at the inverter-grid connection. The matrices’ eigenvalues are being plotted using MATLAB/Simulink to identify system stability region and find the PI controller parameters.

Handling bounded and unbounded unsafe sets in Control Lyapunov-Barrier function-based model predictive control of nonlinear processes

Control Lyapunov-Barrier function (CLBF) has been used to design controllers for nonlinear systems subject to input constraints to ensure closed-loop stability and process operational safety simultaneously. In this work, we developed Control Lyapunov-Barrier functions for two types of unsafe regions (i.e., bounded and unbounded sets) to solve the problem of stabilization of nonlinear systems with guaranteed process operational safety. Specifically, in the presence of a bounded unsafe region embedded within the closed-loop system stability region, the CLBF-MPC is developed by incorporating CLBF-based constraints and discontinuous control actions at potential stationary points (except the origin) to guarantee the convergence to the origin (i.e., closed-loop stability) and the avoidance of unsafe region (i.e., process operational safety). In the case of unbounded unsafe sets, closed-loop stability with safety is readily guaranteed under the CLBF-MPC since the origin is the unique stationary point in state-space. The application of the proposed CLBF-MPC method is demonstrated through a chemical process example with a bounded and an unbounded unsafe region, respectively.

Examination on the parameter stability region of the VSC control system in VSC‐HVDCtransmission systems

Nowadays, the voltage source converter (VSC)-based high-voltage direct current (HVDC) transmission technology is widely adopted for the integration of renewable power generations. However, the grid-connected VSC systems may also cause power system instability issues. This study focuses on the parameter stability region of VSC control system in VSC-HVDC transmission systems. The small-signal linearised model of the whole power system is firstly established and on the basis of which the modal analysis method is applied to obtain the parameter stability region of VSC control system. The results show that the region is mainly affected by both the selection of the initial operating point of VSC and the dynamic interaction between the AC system and the VSC. From these two perspectives, the parameter stability regions of the VSC control system under different operating conditions are investigated. Meanwhile, it is found in the study that when modal coupling occurs between the oscillation modes corresponding to AC system and VSC, the parameter stability regions of the VSC control system obtained in the previous analysis will be deteriorated. Consequently, in terms of stability region, an evaluated method for VSC control system is proposed and validated through case study.

Optimum Design of Fractional Order Pid Controller Using Chaotic Firefly Algorithms for a Control CSTR System

AbstractA fractional‐order PID controller is a generalization of a standard PID controller using fractional calculus. Compared with the standard PID controller, two adjustable variables, “differential order” and “integral order”, are added to the PID controller. Fractional‐order PID is more flexible, has better responses, and the precise adjustment closed‐loop system stability region is larger than that of a classic PID controller. But the design and stability analysis is more complicated than for the PID controller. Therefore, the optimal setting of parameters is very important. A firefly algorithm in standard mode has only local optimization and accuracy is low. In order to fix this flaw an improved chaotic algorithm firefly is proposed for a design controller FOPID. To evaluate the performance of the proposed controller, it has been used in the control of a CSTR system with a variety of fitness functions. Simulations confirm the optimal performance of the proposed controller.

Stability Robustness for Secondary Voltage Control in Autonomous Microgrids With Consideration of Communication Delays

The future extensive usage of open communication networks for secondary voltage control in autonomous microgrids (MGs) inherently introduce time delays, which would degrade the system performance or even cause instability in the worst case. However, existing literatures have largely ignored these communication delays between the microgrid central controller and primary local controllers. To fill the knowledge gap, this paper proposes an analytical method to determine the stability robustness of secondary voltage control in MGs by taking into account the communication latency effect. Based on static output feedback, a closed loop small-signal model of an MG installed with proportional-integral secondary voltage controllers is first derived. Delay-dependent criteria are then formulated to detect the stability posture of MG system by tracing critical eigenvalues and cluster treatment of characteristic roots, in the forms of delay margin for a single delay and stability tableau over the delay domain for multiple delays, respectively. By a series of trial declarations, the qualitative impact of secondary controller gains on system stability regions against uncertain delays is investigated, from which it can be observed that generally the increase of integral gains would yield to a less delay-dependent stability of control system and the results can be utilized to guide the controller design for a compromise between the dynamic performance and stability robustness. The effectiveness of the proposed methodology is verified by a simulation study.

Security region-based small signal stability analysis of power systems with FSIG based wind farm

Based on the Security Region approach, the impact of fixed-speed induction generator based wind farm on the small signal stability of power systems is analyzed. Firstly, the key factors of wind farm on the small signal stability of power systems are analyzed and the parameter space for small signal stability region is formed. Secondly, the small signal stability region of power systems with wind power is established. Thirdly, the corresponding relation between the boundary of SSSR and the dominant oscillation mode is further studied. Results show that the integration of fixed-speed induction generator based wind farm will cause the low frequency oscillation stability of the power system deteriorate. When the output of wind power is high, the oscillation stability of the power system is mainly concerned with the inter-area oscillation mode caused by the integration of the wind farm. Both the active power output and the capacity of reactive power compensation of the wind farm have a significant influence on the SSSR. To improve the oscillation stability of power systems with wind power, it is suggested to reasonably set the reactive power compensation capacity for the wind farm through SSSR.

Closed-Form Discretization of Fractional-Order Differential and Integral Operators

This paper introduces a closed-form discretization of fractional-order differential and integral Laplacian operators. The proposed method depends on extracting the angle information from the phase diagram. The magnitude frequency response follows directly due to the symmetry of the poles and zeros of the finite ztransfer function. Unlike the continued fraction expansion technique, or the infinite impulse response of second-order IIR-type filters, the proposed technique generalizes the Tustin operator to derive a 1st, 2nd,3rd, and 4th-order discrete-time operators (DTO) that are stable and of minimum phase. The proposed method depends only on the order of the Laplacian operator. The resulting discrete-time operators enjoy a flat phase response over a wide range of the discrete-time frequency spectrum. The closed-form DTOs allow us to identify the stability regions of fractional-order discrete-time systems and to design discrete-time fractional-order PIλDµ controllers. The effectiveness of this work is demonstrated via several numerical simulation.

Queueing Stability and CSI Probing of a TDD Wireless Network With Interference Alignment

This paper characterizes the performance in terms of queueing stability of a network composed of multiple MIMO transmitter-receiver pairs taking into account the dynamic traffic pattern and the probing/feedback cost. We adopt a centralized scheduling scheme that selects a number of active pairs in each time-slot. We consider that the transmitters apply interference alignment (IA) technique if two or more pairs are active, whereas in the special case where one pair is active point-to-point MIMO singular value decomposition (SVD) is used. We consider a time-division duplex (TDD) system where transmitters acquire their channel state information (CSI) by decoding the pilot sequences sent by the receivers. Since global CSI knowledge is required for IA, the transmitters have also to exchange their estimated CSIs over a backhaul of limited capacity (i.e. imperfect case). Under this setting, we characterize in this paper the stability region of the system under both the imperfect and perfect (i.e. unlimited backhaul) cases, then we examine the gap between these two resulting regions. Further, under each case we provide a centralized probing policy that achieves the max stability region. These stability regions and scheduling policies are given for the symmetric system, where all the path loss coefficients are equal to each other, as well as for the general system. For the symmetric system, we provide the conditions under which IA yields a queueing stability gain compared to SVD. Under the general system, the adopted scheduling policy is of a high computational complexity for moderate numbers of pairs, consequently we propose an approximate policy that has a reduced complexity but that achieves only a fraction of the system stability region. A characterization of this fraction is provided.

АЛГОРИТМЫ ПОСТРОЕНИЯ ГРАНИЦ ОБЛАСТЕЙ УСТОЙЧИВОСТИ ЛИНЕЙНЫХ ГАМИЛЬТОНОВЫХ СИСТЕМ С ПОМОЩЬЮ ПАКЕТА MATLAB

В статье приводятся основные этапы алгоритма построения областей устойчивости динамических систем, описываемых линейной гамильтоновой системой вида . Алгоритм основан на методах теории нелинейных колебаний исследования устойчивости стационарных решений линейных дифференциальных уравнений с периодическими коэффициентами, зависящих от малого параметра. Алгоритм реализован с помощью математического пакета Matlab. В качестве приложения рассмотрена задача построения области устойчивости треугольных точек либрации плоской ограниченной эллиптической задачи трех тел The article presents the main stages of the algorithm for constructing the stability regions of dynamical systems described by a linear Hamiltonian system of the form . The algorithm is based on the methods of the theory of nonlinear oscillations of stability studies of stationary solutions of linear differential equations with periodic coefficients depending on a small parameter. The algorithm is implemented by using Matlab CAS. As an application, we have solved the problem of constructing the stability region of libration points of a flat elliptic restricted three-body problem is analyzed in detail.

Impacts of Retransmission Limit on Stability and Throughput Regions of S-ALOHA Systems

Limiting the number of retransmissions in practical random access systems such as S-ALOHA and IEEE 802.11 systems has been used, since it was often considered as a way of preventing the system from being congested, by dropping packets in retransmission or letting users know network congestion after the maximum of retransmissions. Therefore, it can be expected that the smaller the maximum of retransmissions is allowed, the smaller the backlog size gets with a higher packet dropping probability. This paper explores how such a retransmission limit alters the shape of the stability and throughput regions of S-ALOHA systems. To do this, we first consider an S-ALOHA system, where two users with infinite queue length have different packet arrival rate and retransmission probability, which is called asymmetric user. Then, we obtain those results of the systems with $N$ symmetric users.

Dynamic pricing strategy and coordination in a dual-channel supply chain considering service value

• This paper investigates two dynamic pricing strategies in a dual-channel supply chain with service. • The dynamic phenomena, such as period doubling bifurcation and wave shape chaos are analyzed through numerical simulation. • We discuss the influence of service value on the model. • We analyze the Bullwhip effect of the system. • In coordination mechanism, we calculate that the fixed fee changes as service value increases. In this paper, we investigate two dynamic pricing strategies in a dual-channel supply chain consisting of a manufacturer, a retailer and focus on the influence of service value on the decisions and the insight of complexity. We study and analyze the evolution of the system under this long-term price forecasting mechanism. We find that the higher adjustment parameters can make the system lose its stability, then appear period doubling bifurcation or wave shape chaos. In the decentralized decision, the system stability region first becomes smaller with service value increases, then becomes larger. In the centralized decision, the system stability region becomes smaller with service value increases. In addition, we calculate the bullwhip effect of the system, and analyze the effect of the adjustment parameters on the bullwhip effect. In order to achieve channel coordination, we put forward a two-part tariff contract. In the contract, we calculate the effect of service value on the fixed fee. The results show that the fixed fee first increases as the service value increases, then decreases.

Allowable delay sets for the stability analysis of linear time-varying delay systems using a delay-dependent reciprocally convex lemma

This paper addresses the stability analysis of linear systems subject to a time-varying delay. The contribution of this paper is twofolds. First, we aim at presenting a new matrix inequality, which can be seen as an improved version of the reciprocally convex combination, which provides a more accurate delay-dependent lower bound. When gathering this new inequality with the Wirtinger-based integral inequality, efficient stability conditions expressed in terms of LMI are designed and show a clear reduction of the conservatism with a reasonable associated computational cost. The second original contribution of this paper consists in noting that stability conditions issued from the Wirtinger-based integral inequality depends in an affine manner on the bounds of the delay function and also on its derivative. This allows to refine the definition of allowable delay set and to relax usual convex on the delay function. As a result of this new characterization, the LMI conditions allows obtaining stability regions for slow time-varying delay systems which are very closed to the constant delay case.

Calculation of absolute stability regions for time-varying electromechanical systems

The paper proposes a numerical method of constructing piecewise-linear Lyapunov functions for absolute stability of nonlinear time-varying systems researching. There are examples and results of comparison with classical methods.

Generalization of a stability domain estimation method for nonlinear discrete systems

In this paper, we present a generalization approach of an algebraic existing method to estimate stability regions for discrete nonlinear polynomial systems of degree 3. The existing method is based on the enlargement of a guaranteed stability region by applying various steps of a proposed algorithm. Its main limitation is that the initial result has only been subsequently developed in a particular case of a single iteration. The stability domain obtained is consequently not the widest one. Our main contribution in this paper is to develop generalized functions that allow the enlargement of the guaranteed stability region after k iterations, for any value of k. A required fundamental tool is developed and consists in a general formula allowing to give the result of the Kronecker power calculation of two matrices sum. The advantages of this generalization are to reach a larger region of asymptotic stability and to improve the existing methods results. Two application examples illustrate the proposed method.

A class of interference induced games: Asymptotic Nash equilibria and parameterized cooperative solutions

We consider a multi-agent system with linear stochastic individual dynamics, and individual linear quadratic ergodic cost functions. The agents partially observe their own states. Their cost functions and initial statistics are a priori independent but they are coupled through an interference term (the mean of all agent states), entering each of their individual measurement equations. While in general for a finite number of agents, the resulting optimal control law may be a non linear function of the available observations, we establish that for certain classes of cost and dynamic parameters, optimal separated control laws obtained by ignoring the interference coupling, are asymptotically optimal when the number of agents goes to infinity, thus forming for finite N, an ϵ-Nash equilibrium. More generally though, optimal separated control laws may not be asymptotically optimal, and can in fact result in unstable overall behavior. Thus we consider a class of parameterized decentralized control laws whereby the separated Kalman gain is treated as the arbitrary gain of a Luenberger like observer. System stability regions are characterized and the nature of optimal cooperative control policies within the considered class is explored. Numerical results and an application example for wireless communications are reported.

Stability and Performance Analysis of Time-Delayed Actuator Control Systems

Time delay is a common phenomenon in robotic systems due to computational requirements and communication properties between or within high-level and low-level controllers as well as the physical constraints of the actuator and sensor. It is widely believed that delays are harmful for robotic systems in terms of stability and performance; however, we propose a different view that the time delay of the system may in some cases benefit system stability and performance. Therefore, in this paper, we discuss the influences of the displacement-feedback delay (single delay) and both displacement and velocity feedback delays (double delays) on robotic actuator systems by using the cluster treatment of characteristic roots (CTCR) methodology. Hence, we can ascertain the exact stability interval for single-delay systems and the rigorous stability region for double-delay systems. The influences of controller gains and the filtering frequency on the stability of the system are discussed. Based on the stability information coupled with the dominant root distribution, we propose one nonconventional rule which suggests increasing time delay to certain time windows to obtain the optimal system performance. The computation results are also verified on an actuator testbed.