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非线性模型预测控制的若干问题研究一、概述随着现代工业技术的快速发展，非线性模型预测控制（Nonlinear Model Predictive Control，NMPC）已成为控制领域的研究热点。
非线性系统广泛存在于实际工业过程中，其特性复杂、行为多样，且具有不确定性，这使得传统的线性控制策略在面对非线性系统时往往难以取得理想的效果。
研究非线性模型预测控制策略，对于提高控制系统的性能、稳定性和鲁棒性具有重要意义。
非线性模型预测控制是一种基于非线性模型的闭环优化控制策略，其核心思想是在每个采样周期，以系统当前状态为起点，在线求解有限时域开环最优问题，得到一个最优控制序列，并将该序列的第一个控制量作用于被控系统。
这种滚动优化的策略使得非线性模型预测控制能够实时地根据系统的状态变化调整控制策略，从而实现对非线性系统的有效控制。
非线性模型预测控制的研究也面临着诸多挑战。
由于非线性系统的复杂性，其预测模型的建立往往较为困难，且模型的准确性对控制效果的影响较大。
非线性模型预测控制需要在线求解优化问题，这对计算资源的需求较高，限制了其在实时性要求较高的系统中的应用。
非线性模型预测控制的稳定性和鲁棒性也是研究的重点问题。
本文旨在深入研究非线性模型预测控制的若干关键问题，包括非线性模型的建立、优化算法的设计、稳定性和鲁棒性的分析等。
通过对这些问题的研究，旨在提出一种高效、稳定、鲁棒的非线性模型预测控制策略，为实际工业过程的控制提供理论支持和实践指导。
1. 非线性模型预测控制（NMPC）概述非线性模型预测控制（Nonlinear Model Predictive Control，简称NMPC）是一种先进的控制策略，广泛应用于各种动态系统的优化控制问题中。
NMPC的核心思想是在每个控制周期内，利用系统的非线性模型预测未来的动态行为，并通过求解一个优化问题来得到最优控制序列。
这种方法能够显式地处理系统的不确定性和约束，因此非常适合于处理那些对控制性能要求较高、环境复杂多变的实际系统。
数学建模中对非线性动力系统模型的认识和体会真实动力系统几乎总是含有各种各样的非线性因素，诸如机械系统中的间隙、干摩擦，结构系统中的材料弹塑性和黏弹性、构件大变形，控制系统中的元器件饱和特性、控制策略非线性等等。
非线性动力学理论的研究和发展已经经历了一个多世纪，在新世纪之初，为了使非线性动力学理论得到更好的发展，非常有必要回顾一下非线性动力学研究和发展的历史。
非线性动力学理论的发展大致经历了三个阶段。
第一个阶段是从1881年到1920年前后，第二阶段从20世纪20年代到70年代，第三阶段从20世纪70年代至今。
人们对于非线性系统的动力学问题的研究可以追溯到1673年Huygens对单摆大幅摆动非等时性的观察。
第一阶段的主要进展是动力系统的定性理论，其标志性成果是法国科学家Poincare从1881年到1886年期间发表的系列论文“微分方程定义的积分曲线”，俄罗斯科学家Liapunov 从1882年到1892年期间完成的博士论文“运动稳定性通论”，以及美国科学家Birkhoff在1927年出版的著作“动力系统"。
第二阶段的主要进展是提出了一系列求解非线性振动问题的定量方法，代表人物有俄罗斯科学家Krylov、Bogliubov，乌克兰科学家Mitropolsky，美国科学家Nayfeh等等。
他们系统地发展了各种摄动方法和渐近方法，解决了力学和程科学中的许多问题。
在这个阶段中抽象提炼出了若干著名的数学模型，如Duffing方程、vander Pol方程、Mathieu方程等，至今仍被人们用以研究非线性系统动力学现象的本质特征。
从20世纪60~70年代开始，原来独立发展的分岔理论汇入非线性动力学研究的主流当中，混沌现象的发现更为非线性动力学的研究注入了活力，分岔、混沌的研究成为非线性动力学理论新的研究热点。
俄罗斯科学家Arnold和美国科学家Smale等数学家和力学家相继对非线性系统的分岔理论和混沌动力学进行了奠基性和深入的研究，Lorenz和Ueda等物理学家则在实验和数值模拟中获得了重要发现。
非线性动力学系统的建模与分析深入探究非线性动力学系统的建模与分析在科学研究中，许多系统都具有非线性特征，只有对这些系统进行深入的研究和建模，才能更好地了解其规律和特性。
非线性动力学系统的建模与分析，便是其中重要的一个方面。
一、非线性动力学系统的基本概念非线性动力学系统是由一个或多个非线性微分方程组成的系统，其特点在于其响应不随着输入信号呈线性变化。
这种系统一般存在着混沌现象、周期现象或者其他的非线性现象，因此其建模和分析具有很大的挑战性。
二、非线性动力学系统的建模方法1. 全局建模法全局建模法是一种直接把原系统转化为通用数学形式的建模方法，其核心是准确地描述系统的动力学状态，并且建立一个合适的数学模型以描述其动态行为。
2. 基于神经网络的建模法基于神经网络的建模法通过构建一种可以学习的算法，来从实验数据中获取非线性系统的内在结构和动态特征。
3. 非线性滤波法非线性滤波法是以基本的线性和非线性滤波器为基础来建立非线性动力学系统模型的方法。
三、非线性动力学系统的分析方法1. 稳态分析法稳态分析法主要是通过计算系统的稳定点、特征值和特征向量等指标来研究非线性系统的稳定性和性态。
2. 线性化分析法线性化分析法是将非线性系统模型线性化后，研究其内在特征，例如特征值和特征向量。
3. 数值分析法数值分析法是通过计算机模拟和数值解析方法，来研究非线性系统的动态特性和性态。
其中最为常用的方法包括Euler法和Runge-Kutta法等。
四、实例分析以一个简单的非线性动力学系统为例，假设其状态方程如下：$$\begin{cases} \dot{x}=y \\ \dot{y}=-\sin{x}-\cos{y}\end{cases}$$应用数值分析法，我们可以通过Euler法进行模拟仿真。
在t=10时，得出系统的稳定点位于(x,y)=(nπ,nπ/2)，n为整数。
此外，我们还可以通过计算特征值和特征向量等指标，来研究该系统的特性。
非线性振动系统的动力学分析与控制方法研究引言非线性振动系统是指频率、振幅或相位等不随时间线性变化的振动系统。
由于非线性振动系统具有复杂的动力学特性，其分析与控制方法一直是科学家们关注的热点。
本文将针对非线性振动系统的动力学分析与控制方法进行深入研究，并探讨其在实际应用中的意义和前景。
第一节动力学分析1.1 非线性振动系统的基本特性非线性振动系统的基本特性包括振动模态的非线性现象、周期倍跳现象和混沌现象等。
其中，振动模态的非线性现象是指在系统振动过程中出现非线性响应，如频率变化、非谐波分量增强等。
周期倍跳现象是指系统在特定参数条件下，由一种周期运动突然跳变为另一种周期运动。
混沌现象是指系统长时间的不可预测行为，表现为无规律的、高度复杂的振动模式。
1.2 非线性振动系统的数学模型为了对非线性振动系统进行分析与研究，科学家们建立了一系列数学模型来描述系统的运动行为。
常用的数学模型包括Van der Pol振子模型、Duffing振子模型以及非线性函数模型等。
这些模型能够反映系统的非线性特性，为动力学分析提供了基础。
1.3 非线性振动系统的动力学特性基于数学模型，可以通过分析非线性振动系统的动力学特性来了解系统的运动规律和振动行为。
其中，周期解的存在性与稳定性是非线性振动系统分析的核心内容之一。
通过线性稳定性分析、周期解的Hopf分支、分岔理论等方法，可以得到非线性振动系统周期解的存在性与稳定性条件。
第二节控制方法研究2.1 基于线性控制方法的研究线性控制方法是最常见的控制方法之一，它基于线性系统理论进行研究。
在非线性振动系统的控制中，可以通过线性化处理，将非线性振动系统转化为线性系统进行分析与控制。
典型的线性控制方法包括PID控制、H∞控制和模型预测控制等。
这些方法能够在一定程度上降低系统的非线性特性，提高系统的稳定性与控制性能。
2.2 基于非线性控制方法的研究除了线性控制方法外，非线性控制方法也被广泛应用于非线性振动系统的控制中。
非线性动力系统的建模与分析非线性动力系统是指其运动方程包含非线性项的动力系统。
与线性动力系统不同，非线性动力系统具有更加复杂的行为和特性。
因此，建模和分析非线性动力系统是理解和预测实际系统行为的重要一环。
本文将介绍非线性动力系统的建模方法以及各种分析工具和技术。
一、非线性动力系统建模方法：1. 分析系统的特性：了解系统的背景和工作原理，找出系统的主要组成部分和相互作用关系。
这样可以更好地理解系统行为和特性，为后续的建模提供基础。
2. 选择适当的数学模型：非线性动力系统可以用多种数学模型进行描述，如微分方程、差分方程、动力学方程等。
根据系统的特性和需求，选择适合的数学模型是非常重要的。
3. 确定系统的状态变量：状态变量是描述系统状态的变量，可以是位置、速度、温度等。
根据系统的特性和需要，确定适当的状态变量是非线性动力系统建模的关键一步。
4. 构建系统的运动方程：根据数学模型和状态变量，建立非线性动力系统的运动方程。
这些方程描述了系统的演化规律和相互关系，是进一步分析系统行为的基础。
5. 校验和验证模型：将模型与实际数据进行比较和验证，确保模型能够准确描述系统的行为和特性。
如果有必要，可以对模型进行调整和改进，以提高模型的准确性和可靠性。
二、非线性动力系统分析工具和技术：1. 稳态分析：稳态分析是研究系统在长时间尺度下的行为稳定性和平衡点的性质。
通过稳态分析，可以判断系统的稳定性和吸引子的性质，进一步预测系统的长期行为。
2. 线性化分析：将非线性动力系统线性化为一组近似的线性方程，以便在局部范围内对系统进行分析。
线性化分析可以简化非线性系统的复杂性，从而更好地理解系统的行为和特性。
3. 相平面分析：相平面分析是用相图表示系统状态的演化和相互关系。
通过分析相图的特征，可以得到系统的稳定性和周期解等信息，为进一步研究系统的行为提供参考。
4. 分岔分析：分岔分析是研究系统参数变化时系统行为的变化和性质的分析方法。
非线性系统的动力学分析及控制研究随着科学技术的快速发展，对于动力学分析和控制研究的需求和重视也逐渐增加。
其中一种非常重要的研究对象就是非线性系统。
1.非线性系统概述非线性系统，简单来说就是不能被描述为线性关系的系统。
由于其比线性系统更复杂，因此难以进行精确的分析和控制，但非线性系统却可以描述许多自然界中的现象以及工程技术实践中的问题。
我们知道，线性系统的特性是“比例性”和“叠加性”，其输入和输出之间存在着数量上的线性关系。
但是，非线性系统在不同的输入下会产生系统响应的非线性变化。
其系统行为可能表现出变化多样、复杂、不可预知等特征。
这些性质决定了非线性系统的动力学不规则和不稳定性，对动力学的分析和控制构成了巨大的困难。
2.非线性系统的控制在非线性系统的控制领域中，最基本的方法就是通过反馈控制的方式，尽量减少系统的误差和稳态误差。
但对于非线性系统来说，它需要一些更为高级和复杂的控制策略，如模糊控制、神经网络控制、自适应控制等。
以自适应控制为例。
自适应控制方法是通过不断对过程进行监控，并改变控制器或控制算法的参数来实现快速、准确和自适应的控制。
这种方法的基本思想是根据系统的现实状况，进行实时修正和调整，使系统能更加灵活和稳定地运行。
但是，由于非线性系统的动力学特性，自适应控制系统设计也会面临很大的挑战。
这主要包括控制算法的设计、系统模型的定位和优化等一系列困难。
3.非线性系统的动力学分析非线性系统的动力学分析是非线性控制领域研究的核心问题之一。
涉及到非线性系统的稳定性、运动轨迹、系统响应等多个方面。
这里简单介绍一些非线性动力学分析方法。
首先是Lyapunov方法。
Lyapunov方法是通过构造Lyapunov函数，来判断非线性系统的稳定性。
主要思想就是找到一个函数，使得对于给定的初值，系统的状态必定会趋近于稳定。
通过求出Lyapunov函数的导数，然后判断其正负性，就能得出系统的稳定性。
另外还有基于相平面分析的方法。
非线性动力学系统的分析与控制随着科学技术的不断发展，人们对复杂系统的研究日益深入。
非线性系统时常出现在自然界和工程技术中，例如气象系统、化学反应、电路、生物系统、机械系统等等。
非线性系统具有极其丰富的动态行为，不同的系统之间存在着很大的差异性。
面对这些复杂多样的非线性系统，如何进行分析与控制是非常重要的。
一、非线性动力学系统的定义及特点非线性动力学系统是指在时间和空间上均发生动态行为的系统，其系统关系不是线性关系。
由于非线性因素的存在导致了系统的复杂性和不可预测性，系统可能表现出各种奇异的动态行为。
这些动态行为包括周期性运动、混沌、周期倍增等等。
一个非线性系统通常由多个部分组成，每个部分之间有相互作用，这种相互作用可以是线性的，也可以是非线性的。
与线性系统不同的是，非线性系统的各种状态和运动是非简单叠加的，微小的扰动可能会导致系统出现完全不同的行为，所以非线性系统的行为很难被准确地预测和控制。
二、非线性动力学系统的分析方法1. 数值方法数值方法是研究非线性系统的基本工具之一。
数值方法的核心是计算机程序，基本思路就是用计算机模拟系统的行为，通过计算机的演算，得出系统的动态变化。
在数值模拟中，巨大的数据量和模拟误差可能导致计算结果的不确定性。
为了解决这个问题，可以采用随机性和模糊性来描述不确定性，将非确定性的信息融入到模型和模拟中。
2. 动力学分析动力学分析是利用动力学知识进行对非线性系统的分析和研究。
通过对系统的本质特性进行分析，了解系统的发展趋势和行为特征。
动力学分析主要通过相空间画图、稳定性分析、流形理论等方法对非线性系统进行分析。
其中，相空间画图是研究非线性系统最常用的方法之一。
它可以将非线性系统的状态表示为相空间中的一点，通过画出系统在相空间中的运动轨迹，了解系统在不同初态下的动态行为。
3. 控制方法控制方法是为了改变非线性系统的行为，使其达到预期目标或保持稳定状态。
非线性系统的控制可以分为开环控制和反馈控制。
非线性动力学系统深度研究深度研究非线性动力学系统引言：非线性动力学系统是一类常见的复杂系统，广泛应用于物理、化学、生物学等领域。
与线性系统相比，非线性系统具有更为复杂的行为和动力学特性。
本文将对非线性动力学系统进行深度研究，探讨其定义、模型、稳定性和混沌等关键概念。
一、非线性动力学系统的定义和基本概念非线性动力学系统是指系统中的状态变量和控制参数之间的关系是非线性的系统。
其基本概念主要包括状态变量、动力学方程和相空间等。
1. 状态变量：状态变量是系统的内部变量，它们描述了系统在不同时间的状态。
通常采用向量形式表示，例如(x1, x2, ..., xn)。
2. 动力学方程：动力学方程是描述系统演化规律的数学方程。
对于非线性动力学系统，动力学方程通常是一组非线性微分方程或差分方程。
3. 相空间：相空间描述了非线性动力学系统的所有可能状态的集合。
在相空间中，每个状态被表示为一个点，而系统的演化则对应于在相空间中的运动轨迹。
二、非线性动力学系统的模型与常见例子非线性动力学系统的模型通常采用一组微分方程或差分方程来描述。
下面给出两个常见的非线性动力学系统模型。
1. Lorenz系统：Lorenz系统是一个三维非线性动力学系统，由爱德华·洛伦兹发展而来，主要用于描述大气环流的运动。
Lorenz系统的动力学方程如下：dx/dt = σ(y - x)dy/dt = ρx - y - xzdz/dt = xy - βz其中，x、y、z分别表示系统的三个状态变量，σ、ρ、β分别为控制参数。
2. Van der Pol振荡器：Van der Pol振荡器是一个二阶非线性动力学系统，广泛应用于电子工程和生物学中。
其动力学方程如下：d²x/dt² - μ(1 - x²)dx/dt + x = 0其中，x表示系统的状态变量，μ为控制参数。
三、非线性动力学系统的稳定性分析在研究非线性动力学系统时，稳定性是一个关键问题。
非线性动力学模型与应用研究随着人类科学技术的发展，数学和物理学科也在不断推陈出新。
其中，非线性动力学模型在近年来的研究中备受关注。
本文将对非线性动力学模型做一简要介绍，并重点关注其应用研究，以及目前常见的几类非线性动力学模型。
一、非线性动力学模型简介动力学研究是物理科学的一个重要分支。
在物理学领域中，动力学研究的主要目的是描述物体运动的规律，这种规律与物体运动的初始条件、外加力的大小和方向等关系紧密。
近年来，随着人们对物理世界深入的认识和理解，非线性动力学模型逐渐成为研究的热点。
非线性动力学模型一般采用微分方程来描述，这类微分方程通常比较复杂，解析的解不容易得到。
因此，非线性动力学模型的研究一般通过数值模拟的方式进行。
在模拟中，可以使用计算机进行计算，通过编写程序对微分方程进行离散化，从而得到数值解。
在非线性动力学模型的研究中，常见的问题包括混沌现象、相图、分叉现象等。
在研究这些问题时，非线性动力学模型可以提供一个有力的数学工具。
二、非线性动力学模型的应用研究1. 混沌现象的模拟混沌现象是非线性动力学模型中的重要问题之一，它指的是具有确定初态，但在运动过程中产生随机性的现象。
在混沌现象的研究中，非线性动力学模型可以提供精确的数学描述。
通过对非线性动力学模型的数值模拟，可以得到对混沌现象的深入了解。
2. 生物医学领域非线性动力学模型在生物医学领域中也有广泛的应用。
比如，在神经元的研究中，可以利用非线性动力学模型模拟神经元的活动，并且通过模拟得到神经信号的时序间隔、频率等信息。
此外，在心血管方面，非线性动力学模型也可以描述人体的生理功能状态和心脏病变的动力学机制。
3. 社会学研究非线性动力学模型也逐渐被应用于社会学领域的研究中。
比如，通过对非线性动力学模型数值仿真，可以确定人类群体行为、物种竞争、市场变化等诸多社会现象的演化规律。
此外，对于经济领域，非线性动力学模型能够描述经济周期、金融市场的波动和崩溃等现象。
动力学非线性系统建模与预测控制在现代科技的飞速发展下，高科技产业的生产要求越来越高，要求对各种机电系统进行合理的建模和控制。
其中，动力学非线性系统的建模和预测控制是一个十分重要的问题。
动力学非线性系统是指其运动状态、系统输出和控制输入之间存在非线性关系，通俗的讲就是不存在一个通用的数学函数可以描述系统的行为。
这种系统在日常生活中很常见：例如，弹簧振动、地震、车辆运动轨迹等等。
由于其极其复杂的性质，能够对其进行建模和预测控制对于人类解决很多实际问题具有重要的意义。
在这方面，我们先来谈谈建模的问题。
对于非线性系统的建模，主要有时间域和频域两种方法。
时间域方法是指通过差分方程或微分方程来描述系统的状态变化，而频域方法则是通过系统的传递函数或频率响应来描述系统的输入和输出关系，即不考虑系统的状态变化。
相对来说，频域方法建模简单易懂，广泛应用也是其中的原因之一。
但是，当非线性系统的系统建模前提不能满足输出具有平稳性时，频域方法就不能使用，这时需要使用更为复杂的时间域方法。
在开始进行动力学非线性系统的建模之前，需先了解系统的基础性质，如系统是否相对稳定等，而这些性质确定了之后，才可进行相应的状态方程和输出方程的推导。
举个例子，我们来看看质量悬挂在弹簧上进行简谐振动的建模过程。
对于这个系统，可以通过牛顿第二定律F=ma得到其状态方程为（m为质量，k为弹簧系数，x为质量相对平衡点的位移）：m(d2x/dt2)+kx=0此外，可以通过观察到系统的位移x与时间t的关系，得到其输出方程为：x=Asinωt其中A表示振幅，ω表示角频率。
将其代入状态方程，可以解得系统的频率为：ω=√(k/m)通过上述推导过程，我们就成功地建立了弹簧振动的动力学非线性系统模型。
除了建立系统模型，预测控制也是非常重要的一个环节。
在许多应用中，经常需要预测未来的状态，进而为控制提供依据。
例如，对于自主驾驶汽车来说，需要对未来的交通情况进行预测，以便进行合理的驾驶。
Study on Nonlinear Dynamical Model and Control Strategy of Transient Process in Hydropower Stationwith Francis turbineHaiyan Bao , Jiandong Yang, Liang FuState Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University No.8 Donghu South Road, Wuchang District, Wuhan 430072, ChinaHaiyan_8931@Abstract —The transient process in conduits of hydropowerstations is a very complicated dynamic procedure coupled withfluid, machines, electricity. In this paper, a whole nonlinear dynamical model of transient process in hydropower station with Francis turbine has been developed, and the control strategies of each transient process are studied. The nonlinear characteristics of hydraulic turbine and the elastic water hammer effect of pressure water supply conduit are considered in the model. The developed model is accurate enough to represent and simulate each transient process of the plant and may enable a plant operator to carry out economical, convenient study for the static stability and transient stability of the hydropower station under a wide range of transient processes. In addition, the literature takes a hydropower station as engineering case to simulate the transient processes of hydro-generator units ’ start-up, load variation, full load rejection from the grid and emergency stop. And the results of simulation are very satisfied.Keywords-hydraulic transients; nonlinear mathematical model; numerical simulation; control strategyI.I NTRODUCTIONH ydropower is an important and vital renewable energy resource, which converts energy in flowing water into electricity. Generally, a hydro-generator unit has many different operating conditions, and any operating condition changes will result in different hydraulic transients. The calculation of hydraulic transient is a key link for the safety and reliability of units and hydraulic installations. Traditionally, the objective of hydraulic transient calculation is to predict three primary regulation guaranteed parameters including the maximum dynamic pressure in the spiral case, the maximum rising ratio of rotating speed and the draft tube minimum pressure, consequently to ensure safety operation of hydropower station. H owever, with the development of hydroelectric construction and technology in China, the content of hydraulic transient calculation is continuously beingenriched, it already not only include calculation of regulationguaranteed parameters, but also include calculation and research of stabilization and dynamic quality [1]. In conventional hydropower stations, there are a series of hydraulic transient processes, such as start-up, load variation, full load rejection from the grid, and emergency stop, wherepower and frequency regulations may always be needed [2]. In order to design suitable control law, stabilize the nonlinear systems, solve many existing control problems, reduce operating costs and energy losses, and improve guarantee security and safety of equipments and plants, it is necessary to develop a whole nonlinear dynamical model that is accurate enough to represent and simulate each transient process of the plant. The developed model may enable a control system designer or a plant operator to carry out accurate, economical, convenient study for the static stability and transient stability of the hydropower station under a wide range of operational modes and nonlinear process conditions, and to design the suitable control strategy, so as to improve stability of hydro-generator units.The literature review carried out in this work finds some published research works. In [3], a new kind of start-up rule is proposed, by using this rule the contradiction between fast start-up and smooth start-up is eliminated; In [4], it analyses the adjusting mode of power adjustment in digital electric-hydraulic governor, and how to realize power adjustment; In [5], the transient performance index of hydro-generator unit in a full load rejection are studied. owever, in the aforementioned published research works, the effect of hydraulic turbine characteristics and the elasticity of conduit walls on the transient process are neglected . In addition, a whole nonlinear dynamical model that can simulate each transient process of the plant isn’t developed in predecessors’ research works. In china, some large-scale hydropower stations often use the complex arrangement nowadays, moreover, the hydraulic conduits are getting longer, and its nonlinearity is very obvious. Therefore, it is very important and necessary to develop a whole nonlinear dynamical model for the complex hydropower system.II.M ATHEMATICAL M ODELS For developing the whole nonlinear mathematical model, the hydropower plant system is decomposed into decoupled dynamical modules as illustrated in Fig. 1, and a mathematical model for each module is developed. 978-1-4244-2487-0/09/$25.00 ©2009 IEEEFigure 1. Hydropower plant dynamical systemA.Modelling of Hydraulic ConduitIn hydraulic transient calculation of hydropower stations, one-dimensional continuity equation and momentum equation will be applied to the unsteady flow in conduit [6]. These equations constitute a system of two equations of partial derivatives – of the hyperbolic type. An exact integration of the equations is however very complicated, analytical solutions are rare. Nevertheless, there exist different numerical techniques making solutions possible. The method of characteristics is often and efficiently used to obtain the solutions of the equations. The method is reasonably convenient to deal with complicated boundary conditions. In the system of hydropower station, there are some boundary conditions (such as turbine) that are nonlinear themselves. Therefore, adopting the method of characteristics, a system of complex nonlinear equations will be avoided, so as to make the solution process simplify greatly. By using the method of characteristics, put the system of the equations of partial derivatives into a system of equations of ordinary derivatives. And under the condition of fixed time step, the equations of ordinary derivatives will be put into two linear algebraic equations with two unknown functions, discharge P Q and water head P H , along the curves of the characteristics, the positive characteristic +C and the negative characteristic −C . The simple forms of the two equations can be written as [7]:+C :P QP CP P H C Q Q .−=, (1)−C :P QM CM P H C Q Q .+=. (2)where CP Q ,QP C ,CM Q ,QM C are known coefficients that are related to the velocity at the forward time step, the water headat the forward time step and the geometrical sizes of conduits. B.Modelling of Hydraulic TurbineThe type of hydraulic turbine is various, in this paper, the mathematical model of Francis turbine will be developed. It is well known that the characteristics of hydraulic turbine are usually represented by the synthetic characteristic curves. In order to develop the mathematical model of hydraulic turbine, it is necessary to carry on data processing for the synthetic characteristic curves. Using the special-purpose software, translate the synthetic characteristic curves into discharge characteristic curves and torque characteristic curvesthat respectively take the specific discharge 1Q ′, the specific torque 1M ′as the ordinate, take the specific speed 1n ′as the abscissa, and take the wicket gate servomotor stroke y as parameter. During the calculation of hydraulic transient process, by using the wicket gate servomotor stroke and water head at the time level, the operating point will be found at discharge characteristic curves and torque characteristic curves. Therefore, the mathematical model of the characteristic of hydraulic turbine may be written as:()y n f Q ,11′=′, (3) ()y n f M ,11′=′. (4) owever, there are still no detailed and precisemathematical models that can describe 1Q ′ and 1M ′; therefore, 1Q ′ and 1M ′are usually obtained by interpolation method. The discharge equation of every type hydraulic turbine may be written as follows:H D Q Q P 211′=. (5) where 1D is the turbine-specific diameter, H is the availablehydraulic head of turbine, the specific discharge 1Q ′ can be obtained using (3).And the specific speed 1n ′in (3) can be shown by following equationH D n n /11=′. (6) where nis the rotational speed of hydro-generator unit, and the computational method of nwill be discussed in next segment.According to the theory of hydraulic turbine, the mechanical torque of hydraulic turbine t M can be written as follows:H D M M t 311′=. (7)where the specific torque 1M ′ can be obtained using (4). The mechanical torque t Min terms of the hydraulic turbine mechanical power P may be written as follows:n P M t ʌ/30=. (8)The mathematical model of hydraulic turbine should be coupled with the characteristic curves, +C and −C (as shown in(1) and (2)), of the upper and downstream conduits linked with the hydraulic turbine. The schematic diagram of turbine boundary is shown in Fig. 2.Figure 2. Schematic diagram of turbine boundaryC.Modelling of Synchronous GeneratorThe synchronous generator converts the mechanical power of the hydraulic turbine to electrical power at a specific voltage and frequency. The electric dynamics have very short time constants compared to hydrodynamics and can be ignored [8]. The mechanical equations of a rotating machine are based on the swing equation of the rotating inertia. Constant shaft speed for a given machine is maintained when there is equilibrium between the mechanical shaft and braking electrical torques. Any imbalance between the torques will cause the acceleration or deceleration of the machine according to the laws of motion of a rotating body. The swing equation may be written as follows:g t M M dtdn−=30Jπ. (9)where J is inertia torque of the machine,g M is braking electrical torque.D.Modelling of Turbine GovernorThe governor is important assistant equipment of the hydroelectric generator, it cooperates with computer monitoring system to complete hydro-generator units’ start-up, load variation, full load rejection from the grid, emergency stop and so on.As for control mode of modern turbine governor, there are commonly speed regulation, power control mode and opening control mode. And now, the parallel connection PID controller has been widely used in hydropower process control. The schematic diagram of PID controller is expressed as Fig. 3 andFig.4.f cfcY cP gP Figure 3.Schematic diagram of PID controller (a)gff cY cP gP Figure 4. Schematic diagram of PID controller (b)Contrasting the two Figs, power control mode and opening control mode in each Fig are different. In Fig. 3, the power control mode and opening control mode adopt PI control law, whereas that adopts “Integral + feedforward loop” control law in Fig. 4.When the turbine governor carry out speed regulation, frequency deviation between the frequency set point and the present frequency come into parallel proportional (P) function, integral (I) function and differential (D) function , then get across mechanical servo system. The derivative equation of speed regulation can be written as follows:¸¸¹·¨¨©§++−=+++++++x dt dx T dt x d T T yb dtdyT b T b b T dtyd T T b T T b T T b dt y d T T T b d n d p y p d p t d y d p y d t n d p y n d p 22233)()(. (9)where x is the speed relative deviation given by rcn n n x −=;p b ,t b ,d T ,n T are governor parameters.Usually, turbine governor is in power control mode tocontrol load variation of operating units. H owever, opening control mode is only a supplement control mode of power control mode. Therefore, the literature mainly analyses power control mode, the derivative equation of the control mode in Fig. 3 and Fig. 4 can be written as follows respectively:()()g c p g c d p d t y d t P P b dt P P d T b dt dyT b dt y d T T b −+−=+22, (10) ()d t P P T b b P t y dt dy T t g c d t p c y ³−⋅⋅+Δ=+01)(. (11) where c P is power set point; g P is present power. Except the aforementioned mathematical models, the mathematical models of other boundary conditions (such as surge tank, bifurcated pipe) in hydropower station system are also developed. As limited by the scope of this paper, the expressions of the mathematical models are omitted. The omitted models and the aforementioned models form the whole nonlinear dynamical model which can be solved by timedomain method that is a kind of numerical methods. On thebasis of the nonlinear dynamical model, Section III will study the different control strategies of turbine governor, in order to carry out optimum control for every hydraulic transient. III.H YDROPOWER P ROCESS C ONTROL S TRATEGY According to the regulation and control of turbine governorfor hydroelectric generator, there are three steady states of hydroelectric generator: normal operation, on-load operation,shut-down waiting state. The schematic diagram of conversionconnection of the three steady states is shown in Fig. 5.Figure 5. Schematic diagram of conversion connectionThe literature takes a hydropower station as engineering case to simulate hydro-generator units’ start-up, load variation, full load rejection from the grid and emergency stop transient processes. Conduit parameters and essence data of the hydropower station are shown as follows: nominal power600=r P MW, nominal speed 7.166=r n rad/min, nominal discharge 5.232=r Q m 3/s, nominal head 0.288=r H m 3/s, the length of the conduit 166601=L m, the length of the penstock 83.2332=L m, the area of upstream throttled surge tank 3.349=A m, the water inertia time constant 40.1=w T s, the unit inertia time constant 361.9=a T s.A.Start-up Control StrategyFast start-up and synchronization depends directly upon a good start-up algorithm. Moreover, the start-up procedure is closely linked with the network reliability, dynamic performance and economic operation. Open loop start-up rule,closed loop start-up rule and “open loop + closing loop” start-up rule are the most common start-up rules. The lecture mainlysimulates “open loop + closed loop” start-up procedure. Theprinciple of the start-up rule is shown as follows: after a start-up command is issued to the turbine governor under shut-down waiting state, the gate starts to open at maximum speed until a start-up gate opening, the gate opening stays constant until thedifference between the present frequency and the frequency setpoint 40HZ attains certain values. Then PID regulation routineis put in to regulate the gate opening and make the frequency follow the frequency set point, the frequency will reposefully attain 50H Z. When the frequency deviations, as well as the phase and voltage deviation between the unit and the network reach certain values, the generating unit can be synchronized to the network. During the start-up process, the generating unit isn’t connecting with the network, so braking electrical torque is zero, i.e.0=g M . At the phase of open loop start-up, the opening rule of the wicket gate servomotor stroke y is known.And at the phase of closed loop start-up, the value of y can beobtained using the equation of speed regulation, as shown in(9). The turbine governor parameters under no-load condition can be estimated by the value of w T and a T [9], let t b =0.3,p b =0, d T =7s, n T =0.6s. And let start-up gate 0y =0.1, wicket opening time k T =50s, to simulate start-up transient process. The simulation results are shown in Fig.6.Figure 6. Simulation results of start-up transient process The results of simulation have shown that start-up process of the hydropower station is fast and smooth. The unit speed follows its own set point very good, and the unit speed has good dynamic response, short response time, small overshoot, and fast attenuation. Therefore, the “open loop + closed loop” start-up rule is a perfect and simple start-up control mode.B.Load Control StrategyThe generating unit is connected with the network during load variation. There are two main control mode, speed regulation and power control mode, to control load variation. 1)Speed Regulation RoutineWhen the generating unit is connected with small or isolated network, or connected with large network and operating under speed regulation routine, the speed regulation routine is put in to control load variation. Let p b=0.01, tb=0.3, d T=7s, n T=0.6s, to simulate the transient process of -20% step change in load. The simulation results are shown inFig.7.Figure 7. Simulation of -20% step change in load under speed regulation The results of simulation have shown: after -20% step change in load is detected, the unit speed rises quickly with the speed overshoot only once, then the unit speed stabilizes at normal speed newly; while the power decrease quickly, finally stabilizes at power set point with a very small overshoot. The response curves in Fig.7 have small oscillation that is caused by the fluctuation in upstream surge tank after step change in load.2)Power Control ModeWhen the generating unit is connected with large network, and is controlled by automatic generation control system (AGC), the power control mode is put in to control load variation. Because the generating unit is connected with large network, we can think that the speed is normal speed and stays constant during load variation. Let p b=0.04, t b=0.3, d T=7s, to simulate the transient process of -20% step change in load.The simulation result is shown in Fig.8.Figure 8. Simulation of -20% step change in load under power control modeAs shown in Fig.8, the two parallel center lines are ±5% [9] power set point. The solid is the simulation result under “Integral + feedforward loop” control law, and the dashed is the simulation result under PI control law. Under “Integral + feedforward loop” control law, the power decrease quickly, and stabilizes in bandwidth of ±5% power set point with a very small reverse overshoot at t=18.4s. However, in contrast to the previous control law, the variation velocity of the power is slower under PI control law. Although there is no overshoot, the system response under PI control law is not very satisfactory; because the regulating time which is 112s is too long. Therefore, the “Integral + feedforward loop” control law is superior to the PI control law.C.Control Strategy of Emergency Stop and Full LoadRejection1)Emergency StopWhile the unit is on normal operation, the unit suddenly rejects full load that is caused by the fatal accident of the unit itself, then wicket gate closes rapidly, finally brings on emergency stop. During emergency stop, the generating unit has broken away the network, so braking electrical torque is zero, the wicket gate closes at maximum speed. The simulation results of emergency stop are shown in Fig.9.Figure 9. Simulation results of emergency stop transient processThe gate opening and speed curves in Fig.9 show: at the time of emergency stop, the wicket gate closes at maximum speed; at the same time, the speed rapidly rises firstly, and then the speed descends slowly.2)Full Load Rejection from the NetworkIf the unit suddenly rejects full load that is caused by other accidents, such as oil switch tripping, the wicket gate will close until no-load gate opening, and braking electrical torque is zero. Let p b=0.01ˈt b=0.3ǃd T=7sǃn T=0.6s, to simulate the transient process of full load rejection. The simulation resultsare shown in Fig.10.Figure 10. Simulation results of full load rejection from networkAfter full load rejection from the network, the wicket gate closes at maximum speed, and the speed rises rapidly, then the turbine governor traces and regulates the speed, in order to make the speed stabilizes at normal speed, finally the unit is on no-load operation. The simulations of the full load rejection have shown that both the dynamic response of speed with overshoot being≥3% only once and the regulating time satisfy the desire of specification [10].IV.C ONCLUSIONIn this paper, a whole nonlinear dynamical model of transient process in hydropower station with Francis turbine has been developed. The models vary in complexity and are meant to be used for the study of hydro turbine governing system problems of different types. And these nonlinear characteristics of hydraulic turbine and the elastic water hammer effect of pressure water supply conduit are considered in the modelling.The developed model is used to simulate different types of transient processes, such as start-up, load variation, full load rejection from the grid, and emergency stop, and to study the different control strategies of turbine governor. It could be found that the hydro-generator unit has good stability and dynamic performances under each transient process with optimal control strategy. The results of simulation and control strategies are very satisfied, it shows that the developed model is accurate enough to enable a control system designer or a plant operator to carry out accurate, economical, convenient study for the static stability and transient stability of the hydropower station, and the model is valuable.R EFERENCES[1]L. Q. Ye, “H ydropower Process Control – Theory, Application &Development,” Huazhong University of Science and Technology Press, China, 2002.[2]J. S. Chang, “Transients of H ydraulic Machine Installations,” H igherEducation Press, China, 2005.[3]H. B. Zhang, J. C. Xie, S. B. Jiao, “Study on optimum start_up rule forhydroelectric generating units,” Journal of H ydraulic Engineering, vol.35, no. 3, pp. 53-59, 2004.[4]S. P. Wei, Y. J. Wang, P. Luo, “Power adjustment of digital electrichydraulic governor,” Hydropower Automation and Dam Monitoring, vol.27, no. 4, pp. 20-22, 2003.[5]H. B. Zhang, “Improvement of the governor,s control performance in afull load rejection,” Journal of Hydraulic Engineering, vol. 34, no. 3, pp.98-103, 2003.[6]W. Li, X. P. Xu, “H ydraulics,” Wuhan University of H ydraulic andElectric Engineering Press, China, 2000.[7]R. Q. Wu, J. Z. Chen, “Transient Process in Hydropower Station,” ChinaWaterPower Press, China, 1997.[8] F. deMello, C. Concordia, “Concepts of Synchronous Machine Stabilityas Affected by Excitation Control,” IEEE Trans. Power Apparatus and Systems, vol. PAS-88, pp. 316–329, 1969.[9]S. P. Wei, “Modern Technology of Turbine Regulation,” H uazhongUniversity of Science and Technology Press, China, 2002.[10]IEC 61362: Guide to specification of hydraulic turbine control system.1998.。
