水轮机非线性特性作用下的水电站调速系统稳定性研究

针对水轮机非线性特性,采用Hopf分岔理论研究了水电站调速系统稳定性。即建立了考虑水轮机非线性特性的水电站调速系统数学模型,基于Hopf分岔和临界稳定判据分析了系统的稳定性,通过对比分析揭示了水轮机非线性特性对调速系统稳定性的作用机理。结果表明,考虑水轮机非线性特性的水电站调速系统Hopf分岔是超临界的,系统的稳定域位于分岔线下方。水轮机非线性特性对调速系统稳定性和调节品质的影响由水头非线性和转速非线性两种因素引起,水头非线性起主要作用,转速非线性几乎无影响。减负荷工况下水头非线性对系统稳定性有利,系统具有较好的调节品质;增负荷工况则相反。     

Investigation of torsional instability, bifurcation, and chaos of agenerator set

In this paper, the nonlinear phenomenon known as Hopf bifurcation, chaos and asynchronous operation of a simple power system are explored. Firstly, taking into account the nonlinearity of the generator shaft and the interaction of mechanics and electrics in the generator sets, the authors obtain a transient model by combining Park equations and mechanics equations. Then the Hopf bifurcation, period-doubling bifurcation and chaos caused by too large a line resistance are investigated with nonlinear mode and Floquet theory. The bifurcation figure of the system is also given. Further study shows that the chaos attractor breaks up into asynchronous operation when the resistance becomes larger. This way of loss-of-stability is different from that caused by loss-of-excitation     

Hopf bifurcation and chaos in synchronous reluctance motor drives

This paper first presents the occurrence of Hopf bifurcation and chaos in a practical synchronous reluctance motor drive system. Based on the derived nonlinear system equation, the bifurcation analysis shows that the system loses stability via Hopf bifurcation when the d-axis component of its three-phase motor voltages loses its control. Moreover, the corresponding Lyapunov exponent calculation further proves the existence of chaos. Finally, computer simulations and experimental results are used to support the theoretical analysis.     

Bifurcation Behaviors Analysis on a Predator–Prey Model with Nonlinear Diffusion and Delay

In this paper, a class of predator–prey model with nonlinear diffusion and time delay is considered. The stability is investigated and Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, we derive the explicit formulas for determining the properties of the bifurcating periodic solutions. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are included.     

非线性刚性转子--轴承系统的分叉研究

以转子动力学和非线性动力学理论为基础,针对非线性转子——轴承系统的具体特点,用数值积分和庞加莱映射方法对采用长轴承模型的刚性Jeffcott转子轴承系统在较宽参数范围内进行稳定性研究。计算结果表明,系统存在Hopf分叉及低周期运动。用数值方法得到系统在某些参数域中的分叉图、响应曲线、频谱图、相图、轴心轨迹及庞加莱映射图,直视显示了系统在某些参数域中的运行状态;数值分析结果为该类转子─—轴承系统的设计和安全运行提供理论参考。     

Nonlinear analysis of tilted toroidal thermosyphon models

We analyze one-dimensional models for single-phase tilted toroidal thermosyphons for three different heating conditions: known heat flux, known wall temperature and mixed heating. For the first two the governing equations lend themselves to exact reduction to a set of three ordinary differential equations, while for the third the equations remain coupled as an infinite set. For all three cases, the tilt angle is stabilizing while the heat rate is a destabilizer. A nonlinear analysis is carried out using center manifold theory and normal form analysis. The known heat flux solutions lose stability through a supercritical Hopf bifurcation, while for the other two heating conditions the Hopf bifurcation is supercritical under some conditions and subcritical under others. Stable limit-cycle oscillations exist only for the supercritical cases, otherwise instability leads directly to chaos. Analysis also provides an estimate for the amplitude of oscillation for the supercritical conditions. Numerical experiments have confirmed the theoretical predictions qualitatively and quantitatively.     

碰摩转子-轴承系统周期运动稳定性分析

在同时考虑轴承油膜力和碰摩发生时转定子问的相对速度对非线性摩擦力的影响基础上,构造了具有碰摩故障转子一轴承系统的动力学模型,利用求解非线性非自治系统周期解的延拓打靶法和Floquet理论,研究了系统周期运动的稳定性。发现碰摩转子一轴承系统在不同的转速下会发生鞍结分岔、倍周期分岔和Hopf分岔等现象。研究结果为转子一轴承系统的故障诊断提供了一定的参考。     

变顶高尾水洞水电站水轮机调节系统的Hopf分岔及其动态仿真

基于刚性水击、非线性水轮机、三阶发电机、PID调速器等模型,建立了变顶高尾水洞水电站的水轮机调节系统的非线性模型,运用Hopf分岔理论分析了动力系统的分岔现象,以PID调节参数作为动力系统的分岔参数分析了接力器时间常数、尾水洞坡度和甩负荷对系统稳定域的影响,并采用RKF方法对系统进行了负荷变化的动态仿真。结果表明,系统的Hopf分岔为亚临界,接力器时间常数增大时系统稳定域减小,尾水洞坡度应取较小值,机组甩负荷时系统稳定域增大;在稳定域内系统是渐进稳定的,所建立的模型可用于大波动和小波动的动态仿真。     

Rayleigh–Benard convection in rotating fluids

Linear and weakly nonlinear properties of Rayleigh–Benard convection in rotating fluids are investigated. Linear stability analysis is studied to investigate analytically the effect of Coriolis force on gravity-driven convection for idealised stress-free boundary conditions. We have derived a nonlinear one-dimensional Landau–Ginzburg equation with real coefficients near the onset of stationary convection at the supercritical pitchfork bifurcation. A coupled Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection at the supercritical Hopf bifurcation are derived and discussed the stability regions of travelling and standing waves.     

An analysis of the hopf bifurcation in a hydroturbine governing system with saturation

A set of state equations of the polarization index governing system with saturation nonlinearity is established for a hydroturbine in small perturbation. The dynamic behavior of the proportional-integral governor (controller) governing system with saturation is analyzed and the conditions for the existence of a Hopf bifurcation are obtained by using a simple nonlinear model. The stability condition, which must be satisfied in the governing system, is simulated and supported by numerical calculations. The analysis and simulation show that a supercritical Hopf bifurcation may exist in hydraulic turbine systems. The results of this paper can be considered as an explanation for the sustained oscillations recorded in hydroelectric power stations.     

Analysis and study of two‐dimensional parameter bifurcation of wind power farms and composite loads

This study focuses on the stability of power system based on codimension‐two bifurcation theory. In this paper, we investigate the impact of load modeling on permissible wind power generation margins in distribution networks. The study considers codimension‐two bifurcations of equilibria and limit cycles in wind power systems depending on varying two parameters simultaneously. The principle parameter is the wind power generation, and the other parameter depends on the different types of loads. The types of loads are ZIP, exponential recovery, dynamic induction loads, and composite load models. To study the effects of the induction motor loads, the proportion of the static component in the motor load is changed and assessed with respect to their mechanical loads. Wind generation margin boundaries are traced, and saddle‐node, Hopf, and limit‐induced bifurcation branches are obtained, delimiting the stable and unstable operating regions in the parameter space. The analysis presented in this paper can pave the way for determining methods for improving and monitoring these margins with consideration to the system parameters and load composition.     

Nonlinear dynamic characteristics and bifurcation analysis of Ti–Zr–Ni quasicrystal as hydrogen storage material

In this article, the nonlinear dynamic characteristics and bifurcation of a Ti–Zr–Ni quasicrystal impacted by hydrogen atoms are studied. New nonlinear damping terms are proposed to express the delay characteristics of Ti–Zr–Ni quasicrystal, and the accurate natural frequency is obtained by the harmonic balance method. A new method based on the developed largest Lyapunov exponent is proposed to analyze the local stability of any point in the system, and the system's global stability is determined. Finally, a new way to realize the switch between hydrogen storage and release based on stochastic Hopf bifurcation is proposed. The results of theoretical analysis and numerical simulation show that the system's motion can be switched between a periodic orbit and a balanced point near the bifurcation boundary with little energy consumption, which is helpful for hydrogen storage and release.     

Stability analysis of the perturbed rest state and of the finite amplitude steady double-diffusive convection in a shallow porous enclosure

Transition from rest and steady convective states to oscillatory flows is investigated in a shallow porous enclosure subject to vertical thermal and solutal gradients. Various combinations of Dirichlet and Neumann thermal and solutal boundary conditions is considered. The unsteady form of Hazen-Darcy law with the Boussinesq approximation is used to model the convective flow through the porous medium. The governing and perturbation equations are solved numerically using finite element method. The threshold of transition, which characterizes the transition from steady to oscillatory finite amplitude flows, and the threshold of overstability (Hopf bifurcation), which characterizes the transition from the rest to oscillatory state, are obtained for a wide range of the governing parameters. The porosity and the acceleration parameter of the porous medium have a strong effect on the thresholds of transition and overstability. An increase in the acceleration parameter and the normalized porosity delays the onset of overstability and the transition to oscillatory finite amplitude flows. For Neumann boundary conditions type, the wavenumber is zero at the onset of overstabilities and finite at the transition threshold.     

Extended Hopf Bifurcation for Abstract Integral Equations at Resonant Eigenvalue

In this paper, we apply an extension of the abstract Hopf bifurcation theorem stated in Ja?imovi? (Non Anal TMA 73(8):2426–2432, 2010) to abstract integral equations (AIE) and retarded functional differential equations (RFDE). This yields sufficient conditions of what we refer to as extended Hopf bifurcation for AIE and RFDE, in which we have a relaxation of the non-resonance condition on the eigenvalues of the generator of corresponding semigroup. We illustrate our results with an explicit example of a system of two delay differential equations (DDE), undergoing extended Hopf bifurcation at the resonant eigenvalue.     

非线性水轮机调节系统的Hopf分岔分析

水力发电系统具有非线性特性,利用线性模型分析水轮机调节系统会导致仿真结果不准确。为此,在考虑水轮机调节系统各部分非线性情况下建立了水轮机调节系统的非线性模型。基于Hopf分岔理论对建立的非线性调节系统模型进行分析和仿真,比较线性模型与非线性模型动态特性。结果表明,非线性系统模型能更准确地反映系统大扰动情况下的本质特性,PID参数变化会引起非线性系统拓扑结构的变化。     

Study on thermal stability and bifurcation analysis of FCC carbon nanotube-based nanolattice as hydrogen storage materials

The thermal stability and bifurcation of a kind of carbon nanotube-based face-centered cubic (FCC) nanolattice as hydrogen storage materials are studied in this article. The dynamical model of a FCC nanolattice subjected to thermal disturbance is developed where its nonlinear stiffness is considered. The system's resonance frequency is given by the developed L-P method, and the effects of different terms on the resonance frequency are discussed. The system's abundant dynamic behaviours are revealed, which include Hopf bifurcation, limit cycle bifurcation and high-order limit cycle bifurcation in stochastic sense. Analysis results show that the FCC nanolattice's reliability and the first-passage time are almost unaffected by the outside disturbance, and the nanolattice's structures maybe play a main role in the stability and reliability.     

Linear Stability Analysis of Thermocapillary Flow in Rotating Shallow PoolsHeated from Inner Wall

Thermocapillary flow of silicon melt(Pr=0.011)in shallow annular pool heated from inner wall was simulated at the dimensionless rotation ratewranging from 0 to 7000.The effect of pool rotation on the stability of the thermocapillary flow was investigated.The steady axisymmetric basic state was solved by using the spectral element method;the critical stability parameters were determined by linear stability analysis;the mechanism of the flow instability was explored by the analysis of energy balance.A stability diagram,exhibiting the variation of the critical Marangoni number versus the dimensionless rotation ratewwas presented.The results reveal that only one Hopf bifurcation point appeared in the intervals ofω<3020 andω>3965,and the corresponding instability was caused by the shear energy,which was provided by the thermocapillary force and pool rotation,respectively.In addition,the competition between thermocapillary force and pool rotation leads to three Hopf bifurcation points in the range of 3020<ω<3965 with the increase of Marangoni number.     

一种用于电力系统电压稳定分析的雅可比矩阵关键特征值算法

分岔是一种常见的非线性现象.提出了一种追踪雅可比矩阵关键特征值的连续性算法,该方法首先利用特征值实部、特征值关于分岔参数的一阶和二阶灵敏度系数,构造一个判断关键特征值的指标,然后利用该指标来确定某几个特征值为待选的关键特征值,最后利用连续性方法对关键特征值进行连续追踪,直至霍普夫(Hopf)分岔点.     

INFLUENCE OF VIBRATION ON SORET-DRIVEN CONVECTION IN POROUS MEDIA

Two-dimensional thermosolutal natural convection with Soret effect under the simultaneous action of vibrational and gravitational accelerations is investigated. We consider a porous cavity saturated by a binary mixture and adopt the time-averaging formulation. For an infinite horizontal layer, a stability analysis is performed from which the threshold of stability is determined. Numerical simulations, using a pseudo-spectral Chebyshev collocation method, are performed to describe the convective motion. The problem is investigated for different aspect ratios with various directions of vibration. It is concluded that, for both the stationary and the Hopf bifurcation, the vertical vibration has a stabilizing effect while the horizontal vibration has a destabilizing effect on the onset of convection.     

Lorenz系统动力学行为的MATLAB仿真与分析

以Lorenz系统为例,采用相图和功率谱两种方法,借助MATLAB软件对之进行仿真研究,观察状态变量在时域和频域中的变化来了解系统的非线性特性。通过调整控制参数,观察Lorenz系统动力学行为的演变过程,得知Lorenz系统可通过Pomeau-Manneville途径走向混沌,间歇性与Hopf分岔和倍周期分岔有关。