Stability performance for primary frequency regulation of hydro-turbine governing system with surge tank

This paper aims to study the stability for primary frequency regulation of hydro-turbine governing system with surge tank. Firstly, a novel nonlinear mathematical model of hydro-turbine governing system considering the nonlinear characteristic of penstock head loss is introduced. The nonlinear state equations under opening control mode and power control mode are derived. Then, the nonlinear dynamic performance of nonlinear hydro-turbine governing system is investigated based on the stable domain for primary frequency regulation. New feature of the nonlinear hydro-turbine governing system caused by the nonlinear characteristic of penstock head loss is described by comparing with a linear model, and the effect mechanism of nonlinear characteristic of penstock head loss is revealed. Finally, the concept of critical stable sectional area of surge tank for primary frequency regulation is proposed and the analytical solution is derived. The combined tuning and optimization method of governor parameters and sectional area of surge tank is proposed. The results indicate that for the primary frequency regulation under opening control mode and power control mode, the nonlinear hydro-turbine governing system is absolutely stable and conditionally stable, respectively. The stability of the nonlinear hydro-turbine governing system and linear hydro-turbine governing system is the same under opening control model and different under power control model. The nonlinear characteristic of penstock head loss mainly affects the initial stage of dynamic response process of power output, and then changes the stability of the nonlinear system. The critical stable sectional area of surge tank makes the system reach critical stable state. The governor parameters and critical stable sectional area of surge tank jointly determine the distributions of stability states.     

Standard passivity-based control for multi-hydro-turbine governing systems with surge tank

This paper addresses the problem of control design for hydro-turbine governing systems with surge tanks from the perspective of standard passivity-based control. The dynamic model of a synchronous machine is considered in conjunction with a model of the hydro-turbine to generate an eleventh-order nonlinear set of differential equations. An Euler–Lagrange representati of the system and its open-loop dynamics is developed. Then, the standard passivity-based control is applied to design a global and asymptotically stable controller in closed-loop operation. The proposed control is decentralized to avoid challenges of communication between the hydro-turbine governing systems. The proposed standard passivity-based control approach is compared with two control approaches. First, a classical standard cascade proportional-integral-derivative controller is applied for the governing system, the automatic voltage regulator, and the excitation system. Second, a sliding mode control is also implemented in the governing system. Two test systems were used to validate the performance of the proposed controller. The first test system is a single machine connected to an infinite bus, and the second test system is the well-known Western System Coordinating Council’s multimachine system. Overall, simulation results show that the proposed controller exhibits a better dynamic response with shorter stabilization times and lower peaks during the transient periods.     

Passivity-based control and stability analysis for hydro-turbine governing systems

Low-frequency oscillations can occur in hydropower systems under in the new context of power system and the classical controller for hydro-turbine governing systems need to be enhanced with the purpose of improving its stability. We propose a controller based on passivity theory with the aim of damping oscillations in a power system. Passivity-based control arises as a natural choice for hydro-turbine governing system since its open-loop dynamic has a port-Hamiltonian structure, which allows designing a controller that preserves the passive structure in closed-loop via interconnection and damping reassignment. The proposed controller considers the complete non-linear model of the system and guarantees global asymptotic stability in the sense of Lyapunov. Time-domain simulations demonstrate the robustness and proper performance of the proposed methodology under different operative conditions when is compared with the classical controllers.     

Nonlinear dynamics of a novel fractional-order Francis hydro-turbine governing system with time delay

This paper focuses on the stability of a hydropower station. First, we established a novel nonlinear mathematical model of a Francis hydro-turbine governing system considering both fractional-order derivative and time delay. The fractional-order α, which is introduced into the penstock system, in the range from 0.82 to 1.00 is on the left side of the model in a incommensurate manner in increment of 0.03 to provide an adjustable degree of system memory. The time delay τ, which exists between the signal and response in the hydraulic servo system, in the range from 0 s to 0.26 s is inserted on the right side of the model in increment of 0.04 s. Utilizing the principle of statistical physics, we respectively explored the effects of the fractional-order α and the time delay τ on the stable region of the system. Furthermore, we exhaustively investigated the nonlinear dynamic behaviors of the system with different governor parameters by using bifurcation diagrams, time waveforms and power spectrums, finding that only under the condition of reasonable collocation of governor parameters the system can maintain stable operation. Finally, all of the above numerical experiments supply new methods for studying the stability of a hydropower station.     

Non-linear analysis of a traffic flow

A nonlinear mathematical model, which takes into account the dissipative mechanism, is used to describe the signal transmission in a traffic flow. It is shown that dissipative mechanisms, under certain conditions, may produce attenuation effects against the typical nonlinear steepening of waves. An asymptotic analysis is carried out to discuss wave features when the governing hyperbolic system of equations is objective to different kinds of approximations.     

On a thermomechanical milling model

This paper deals with a new mathematical model to characterize the interaction between machine and workpiece in a milling process. The model consists of a harmonic oscillator equation for the dynamics of the cutter and a linear thermoelastic workpiece model. The coupling through the cutting force adds delay terms and further nonlinear effects. After a short derivation of the governing equations it is shown that the complete system admits a unique weak solution. A numerical solution strategy is outlined and complemented by numerical simulations of stable and unstable cutting conditions.     

Linear and nonlinear mathematical models for noninvasive ventilation

Two mathematical models are proposed for passive, noninvasive ventilation. Both models take the form of coupled ordinary differential equations that describe the volume in a single compartment lung. One model is linear and the other nonlinear; both models are derived from basic pressure balances in the lung-ventilator system. These models are also compared to a physical model using a test lung. Both the physical and mathematical models exhibit instabilities that appear to have important clinical implications. The simulations from these models, and the forms of their governing equations, suggest that the presence of an airway leak proximal to the airway opening during pressure support noninvasive ventilation may render this mode of ventilation dynamically unstable. The mathematical models are extended to incorporate a special type of nonpassive ventilation where the total cycle times of the ventilator depend on the inspiratory phases of these cycles.     

A fractional differential equation for a MEMS viscometer used in the oil industry

A mathematical model is developed for a micro-electro-mechanical system (MEMS) instrument that has been designed primarily to measure the viscosity of fluids that are encountered during oil well exploration. It is shown that, in one mode of operation, the displacement of the device satisfies a fractional differential equation (FDE). The theory of FDEs is used to solve the governing equation in closed form and numerical solutions are also determined using a simple but efficient central difference scheme. It is shown how knowledge of the exact and numerical solutions enables the design of the device to be optimised. It is also shown that the numerical scheme may be extended to encompass the case of a nonlinear spring, where the resulting FDE is nonlinear.     

Study of the dust acoustic waves in a magnetized dusty plasma with many different dust grains

The nonlinear dust waves in a magnetized dusty plasma with many different dust grains are analytically investigated. New analytical solutions for the governing equation of this system have been obtained for the dust acoustic waves in a dusty plasma for the first time. We derive exact mathematical expressions for the general case of the nonlinear dust waves in magnetized dusty plasma which contains different dust grains.     

Theoretical results of the dust acoustic waves in a magnetized dusty plasma with different dust particles

In this paper, the nonlinear dust acoustic waves (DAW) in a magnetized dusty plasmas with different dust grains are analytically investigated. New analytical solutions of the governing equation for this system have been obtained for the first time. The exact mathematical expressions of the nonlinear dust waves have been canvassed for the general case in magnetized dusty plasma containing different dust particles.     

二阶耗散动力系统的降维对解长期行为的误差估计

基于非线性动力学理论,对一类高维二阶耗散自治动力系统的降维及其对解的长期行为的影响进行了理论分析．该分析将方程的解投影到控制方程的线性算子的特征向量所张成的完备空间中,并在相空间中引入一距离的概念,方便地解决了缩减后系统与原始系统解之间的误差或距离的描述．基于此距离定义,首先,分析了由于高阶模态的截取对解的长期行为的影响,并推导出了相应的误差估计,该估计表明由于降维对系统长期行为的影响不仅与系统的高阶子空间中的固有频率和阻尼比乘积的最小值有关,并且与高阶子空间中的某一最大固有频率有关．然后,将一般的模态截取视为对原系统的解的一个扰动,对一些文献中由于降维程度的不同而造成解的拓扑性质发生变化的现象进行了定性的解释．     

Theoretic study of dust acoustic waves in a dusty plasma with dust charge variation

The nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation is investigated by using the formally variable separation approach. New solutions for the governing equation of this system have been obtained for dust acoustic waves in a dust plasma firsthand. We derive exact mathematical expressions and numerical simulation studies for the general case of the nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation.     

纤维增强聚合物加固木柱的非线性稳定性分析

考虑加固层中纤维增强聚合物布（FRP布）拉伸与压缩时的不同弹性模量,基于梁大挠度变形假定,首先建立了FRP加固细长木梁大挠度弯曲的一般数学模型,给出了考虑梁弯曲二阶效应的非线性控制方程．其次,研究了FRP布加固细长简支木柱的非线性稳定性问题,得到了FRP加固简支木柱的临界载荷公式．理论证明了其过屈曲解的存在性,并利用摄动法,得到了临界载荷附近过屈曲状态的渐近解析解．进行了参数分析,结果表明:FRP加固层对临界载荷有显著的影响,而对其无量纲过屈曲状态影响较小．     

Nonlinear dynamic analysis of a V-shaped microcantilever of an atomic force microscope

This paper is devoted to investigate the nonlinear behaviors of a V-shaped microcantilever of an atomic force microscope (AFM) operating in its two major modes: amplitude modulation and frequency modulation. The nonlinear behavior of the AFM is due to the nonlinear nature of the AFM tip–sample interaction caused by the Van der Waals attraction/repulsion force. Considering the V-shaped microcantilever as a flexible continuous system, the resonant frequencies, mode shapes, governing nonlinear partial and ordinary differential equations (PDE and ODE) of motion, boundary conditions, frequency and time responses, potential function and phase-plane of the system are obtained analytically. The governing PDE is determined by employing the Hamilton principle. Subsequently, the Galerkin method is utilized to gain the governing nonlinear ODE. Afterward, the resulting ODE is analytically solved by means of some perturbation techniques including the method of multiple scales and the Lindsted–Poincare method. In addition, the effects of different parameters including geometrical one on the frequency response of the system are assessed.     

Distributed model of a self-developing market economy

Fairly rigorous mathematical laws are applied to construct a mathematical model of a self-developing market economy with movement of time-dependent capital in the technology space. The model is a system of nonlinear partial differential equations. We analyze the bifurcations of the spatially homogeneous solutions that describe the dynamics of macro-variables of the economic system. We also study some properties of the solutions of the partial differential equations that follow from diffusion of capital and consumer demand.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 243–262, 2002.     

Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.     

A mathematical analysis of tension shocks in long belt conveyors

This paper presents a mathematical study of the development and propagation of tension shock waves in the topside of long belt conveyor systems. Employing conservation laws for mass and momentum we construct a system of nonlinear hyperbolic equations governing the behavior of tension and momentum per unit length. An important feature of this construction is the nonlinear relationship between tension and overdensity due to the geometric sag of the belt between the idlers. Numerical profiles of the tension are computed using the moving finite element method which is especially suited for the numerical study of shocks. With our model we believe we have identified two of the sources of the large-scale tension shocks which sometimes severely damage such long belt conveyor systems.     

基于Fuzzy推理的时变系统建模

提出一种基于Fuzzy推理的时变系统建模方法,其基本思想是:对时间维度进行分割,在每个较短的时间间隔内用时不变模型代替时变模型,将这些时不变模型组合在一起,最终获得一个整体非线性时变的微分方程模型.分别研究了输入输出型时变系统和状态空间型时变系统的模型建立方法,除了从理论上保证了所获得的模型对系统的逼近性,还从仿真实验验证了用该方法建立的模型对非线性时变系统有很好的逼近效果.