300W小型垂直轴风力发电机的设计【说明书+CAD】
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附录 固定风力发电机和风力集成园建模系统暂态稳定性的研究抽象程度越来越高的风力发电涡轮机，在现代电力系统中需要一项准确的风力发电系统暂态稳定模式. 因为许多风力发电机往往集合在一起，其中等价建模几个风力发电机尤为关键. 本文介绍的降阶动态固定风力发电机模型适合暂态稳定模拟. 该模型是使用一个模型还原技术所构建的高阶有限元模型. 然后， 用等价方式表明如何将几个风力发电机的风力合并成一个 单降阶模型. 用模拟个案来说明一些独特性能的动力系统，含风力发电机. 所以说，本文着重于介绍水平轴风力涡轮机用异步电机直接连到电网作为 系统的发电机. 用参数计算暂态稳定模拟系统，计算风力发电机组的建模，计算风力涡轮机造型. 一.最近，大家对风能的发展展现出了浓厚的兴趣. 伴随着使用风力发电机的热潮，现在需要对电力动态系统， 电力传输规划的设计评估. 本文的第一个目的是提出一个准确的低阶动态模型的风力发电机组，它是 符合现代机电暂态模拟计算机程式的. 本文中，开发的模式着重于水平轴的风力发电机， 或风力机直接连到同步网时采用异步发电机. 这其中还包含许多现代大型发电系统. 由于大型风力装置的构建是由许多个风力发电机组成的， 风力发电场的建模是一个迫切的需求. 因此， 本文的第二个目的是提供一种方法，它结合数个风力发电机连接到一个电网上，然后通过一个共同模式整合成一个单一的等效模型. 风力发电机主要分为定速或变速. 以最小单位，涡轮驱动的感应发电机为例，它是直接连接到电网上的. 涡轮转速变化很小，那是由于陡坡的发电机转矩和转速的特性所制； 因此， 它被称为定速系统. 还有变速装置，发电机连接到电网利用电力电子变换的技术使涡轮速度受到控制，以最大限度地表现出来(例如，电力的控制) . 这两种方法在风力工业均非常普遍. 在本文中， 我们将目光集中在建模定速装置和等效模拟几个固定转速风力发电集成园. 第一种典型的风力机械频率是在0至10赫兹范围； 这也是各种机电振荡的频率. 因此，这涉及到机械振动的风力互动学与机电动力学. 这方面的例子参见本文. 因此，为了构建一个精确的模型，风力发电机可用于暂态稳定的研究. 第一种涡轮机械动力学必须能准确的代表模型. 这里的风力发电机模型建出了导电模型，减少了一个详细的650阶有限元模型的一个典型的 横向轴. 气动力和机械动力的减少与非线性四阶双涡轮惯性模型相结合生成了一个标准发电机模型. 模拟计算表明了模型的精确性.几个风力发电机连接到传输系统上通过 一个单一的模型建模，因为每个涡轮暂态稳定系统都过于繁琐， 我们的目的是整和风力发电园成为相当于风力发电机模型的极小系统. 我们对等价建模的风园涉及到把所有涡轮以同样的机械固有频率整和成单一当量的涡轮机. 模拟结果表明，这种方法能够提供准确的结果. 二. 范例关于风力发电机建模的代表范例是关于暂态稳定系统的，它包括在2 - 10 . 模拟结果表明，固定频率的风力发电机组主要集中在以下两个主要方法. 第一种方式是把汽轮机和发电机转子作为一个单一的惯性体从而忽略系统的机械固有频率 2 - 5 . 第二种方式是把涡轮叶片和枢纽之一的惯性体接上发电机加上一个弹簧 6 9 . 在所有这些论文中，弹簧刚度的计算是从系统的主要部分中提取的. 我们的研究显示，较第一型机械频率来说第二型才是至关重要的一个精确的模型. 有限元分析表明，第一类动力的变化主要是因为灵活的涡轮叶片不够精确. 根据建模方法的算法，我们得知的主要事实是，小而灵活的机械部件是涡轮上的刀片. 结果7集中表明了几个风力发电机系统和降阶风园模型的类型和与类型相结合的方法. 但是， 作者不能解决水轮机和风力发电机相结合时采用这种方法保存的机械要求. 我们的研究结果表明:这关键在于有一个准确的风示范园. 10详细讨论了降阶变速涡轮机载的建模. 作者称涡轮的机械能所代表的类型是一个单一的个体， 从动态的机电动力学分析，那是因为机械的惯性使它的变速性能产生堵塞. 我们分析时不考虑变速情况.2 - 10的工作阐述着重于低阶水轮机模型，从而可以容易地实现大型暂态稳定代码的测量.相当多的研究集中在建模定额一个更深入的层次. 17是一个很好的概况和文献. 从高度详细的有限元模型角度，详细的阐述了建模方法，还较简单的叙述了六转五转，三转水轮机模型.这些模型中的大部分都采用动量理论来计算气动力. 三.我们对发展涡轮动力的一个降阶模型为出发点，把所有机械和气动涡轮机动态效果以高度详细的用机电射程的形式表示出来. 在这个还原过程中，是以消费者的角度来分析涡轮轴驱动发电机的. 目的是为了准确反映轴转速和扭矩特性与最小模型的秩序和复杂性. 数值调查表明，机械气动和机械效应的一个例子所展现的测试系统实现了有限元建模环境. 该系统是一种新兴的横向风轴机床，包括三个31.7米叶片，叶片的一套点俯仰角度为2.6 ， 一个82.5米的主轴，它们的额定功率为18.2 - RPM和1.5兆瓦，在15米/秒的风速条件下. 汽轮机是透过一个简单的异步发电机模型直接连接到60赫兹的机械. 它还利用ADAMS有限元软件(来自机械动力学 公司) ，加上毫微克(即由国家可再生能源实验室)软件进行模拟. 这两个软件一起被称为亚当斯. 所有参数测试系统的模型研制出一个现实的大型机器. 整个系统包含325个自由度，包括非常详细地模拟动力和外部作用力. 由于机械设计中的大多数水平轴风力涡轮机极为相似， 结果使该方法的适用面广. 研究者在用亚当斯/分数制进行了研究以后，还广泛接触了以一个制动脉冲对该系统的瞬态响应的研究方法.为了模仿长达0.1毫米的三相短路，发电机轴对电路的混乱反应进行了分析. 1 . 从图1 ，系统的反应是一个阻尼振荡的过程. 详细的拟态分析表明，系统的振荡是由于外层部分的叶片振动对两者的内在部位的叶片的作用.这样的结果是很典型的. 1)亚当斯仿真结果. 现代风力涡轮叶片非常大，有弹性， 而且往往颤动. 1表明，它主要包含4 Hz分量.这也是典型的大型涡轮机， 它通常有第一型机械自然频率在0至10赫兹范围内. 因为这个范围也是典型的机电振荡频率范围， 这还是风力涡轮机的关键频率范围.而研究者会倾向于研究机电振荡的频率. 模态的第一振荡模式会产生一系列的主导反应. 从图1起见，该模型的描图可以代表两标准单弹簧阻尼系统，这是基础的降阶模型和一个的外部分的叶片2 ) . 叶片尖端硬性连接描图. 2 )刀环 叶片的细片(忽略质量)作为一个单一的惯性体，其所有的瞬态干扰行为通过发电机轴的所有刀片.其他惯性力的代表如集聚效应的叶根，轮毂，涡轴，齿轮，轴发电机，发电机的惯性都很大.一个典型的系统，内部惯性主导地位取决于叶根和发电机的惯性量.许多研究者都推断整个涡轮机和发电机成为一个单一的惰性体从而忽略第一机械型动态系统的作用.别人都认同第一动态模式，但不认同模式叶片弹性模式.相反，这些作者都假设叶片是一个惯性体而把模型涡轮轴作为一个弹簧体. 但是，在一个典型的系统中，轴上的刀片相比其他元件来说灵活得多. 我们的研究表明，第一机械模式的叶片可以与竖轴作为一个刚体. 我们的研究还表明，正确建模是研究力学的关键，以获取准确的瞬态仿真结果. 四.单一风力发电机模型由两个基本部分组成: 降阶双涡轮惯性模型和驱使风力的力矩.在本文中， 我们假设发电机是一个标准的异步电机直接连接起来的网络，这也是最常见的配置方法. ( 1 )叶片数目:有效传动比=实际涡轮转速/额定涡轮转速； 电气频率基数； 每个叶尖惰性体:每个叶片根部惰性+惯性+惯性涡轮轴传动力/惯性力+发电机轴转子的惯性力； 叶片刚度，叶片阻尼，气动风力矩.发电机电气扭矩和叶尖角度通过齿轮传动反映出发电机轴向角.计算这个角需要有叶片断裂的惯性力和弹簧减振器的相关参数(见图2).如果叶片放置在不破裂的正确位置，然后得到的机械模态形状就会正确了. 研究的突破点主要在一个刀片力学性能上，可以从有限元分析或试验的叶片得到相应的数据，这个关键的数据似乎发生在第二个节点弯曲的叶片上.在研究实例个案上，降阶系统的灵敏度放置不当的突破点是很大的. 所幸的是， 最先进的叶片或制成品设施(如在国家可再生能源实验室的设施)有所需的资料用以确定叶片的断裂点.电力工程师只需要这一信息请求便可轻易计算出典型制造的数据.还可以计算出知识系统的第一型机械固有频率的使用刚度.（2）哪里第一模型机械研究技术领先，其机械的固有频率与系统连接到一起的几率就大. 例如，在上一节系统的系统情况就是这样.一般来说，制成品可以提供这样的频率范围.它可以很容易的用制动脉冲对水轮机进行计算和分析.在大多数情况下叶片阻尼很小，并假定为零.在旋转机中，衡量叶片的刚度是用弹簧刚度来计算的.主要衡量叶片的边缘刚度.可以看出，在( 3 )中 ，计算刚度是依靠俯仰的角度的. 这也仅限于从零度至10度的典型情况. （3）根据这一限制表明，差异很小的不同位置需要设置不同的点.这意味着，根据实验的支持，这是水轮机模型很小敏感性变异系统的准确的俯仰角. 假设一个理想的转盘来进行风力矩的计算.（4）在叶尖部分反映出的实际速度，加上空气密度的影响，通过清扫面积的叶片的磨合，计算出了机组的功率系数. 不幸的是，这不是一个常数. 然而，大多数涡轮制成品的特性反映出同一条曲线. 曲线表示，作为功能机组的叶尖速比. 叶尖速比的定义是自由风速度比涡轮叶片的冰山速度. ( 5 )叶片扫描半径单元叶尖速比. 3显示了一个典型的风力涡轮机曲线. 我们的研究已表明，可以假设固定情况下极高的风力条件下进行暂态稳定研究. 这是因为典型的变异叶尖速比下一个10秒的瞬态叶尖比小.假定风并没有显著的改变模拟时间， 实际上，涡轮轴的扭矩实际上是一个调制版. 调制是众所周知的，而且主要是考虑由于大楼遮蔽和力学失衡的作用，在专业人员和模式上才能出现典型的调制频率(注: 1人，是一种模式，每一个涡轮叶片).我们不把这些效应考虑在内，我们假定扭矩引起的暂时性故障比调制扭矩的多. 许多其他研究者已进行了这个假设.今后的研究将侧重于检验这一假设. 在一般情况下，双涡轮惯性模型在这里是一个相对稳健的模式，涵盖了许多汽轮机运行条件. 所有模型参数相对恒定，缺少敏感性的俯仰角度. 因为主要组成部分能量是短暂的，那是由于汽轮机的惯性能量的影响， 而且失速型风力涡轮机可准确模拟这种方式. 乙发电机模型中的标准做法是行之有效的建模发生器1.标准而详细的两轴感应机模型是用来代表异步发电机1的.由此方程( 6A )可知，凡是暂态开路的时间常数，滑移速度，都是同步的电抗，还是暂态电抗.而且并在D轴和q轴定子电压中， 并在D轴和Q轴的每单位定子电流中. 转矩的计算是从( 6B )及定子电流的计算中得到的，是通过( 6C )款的发电机模型参数 ( 6 )计算出(第562 ) ( 106 ) ( 7C )的相关参数.风园造型中的风园分为几个风力发电机连接到传输系统中整和为一个单一的系统.这需要建模，因为每个涡轮暂态稳定，可过于繁琐.我们的目标是整和风园成为一套最起码的等效模型.等价建模风园涉及到把所有涡轮以同样的机械固有频率成一个单一相当于涡轮机的系统. 每个这些等效的涡轮然后连接到异步发电机上.甲相当于水轮机模型的前提，我们的做法是: 因为轮机都离不开一个共同的系统，每个涡轮也受到了同样的干扰力矩. 因此，涡轮机的性能相似于震荡阶段.因此涡轮可合并为一个平行的机械组合.模态分析风力公园系统支持这个假说。例如，考虑要予以合并的涡轮相同的自然频率机械.，那么等于涡轮建模方程( 1 ) ( 7 )式中，弹簧和阻尼条件汽轮机分别是惯性体.涡轮得到的风力矩是利用( 4 ) ，并迫使水轮机具有相同输出功率为涡轮的总和，是机组的功率系数为涡轮机. 乙相当于发电机模型用异步发电机参数的纳加权平均法16来进行计算.用此方法，相当于机床参数和计算，以加权平均纳每一科的异步电机等效五 结论研究者已提交了降阶动态风力发电机模型适合于暂态稳定性的方案.该模型是汽轮机作为一个四阶非线性模型与风速作为输入参数得出的结论.涡轮方程符合标准发电机的用于暂态稳定的电气方程.一个等效办法还表明如何在几个风力发电机的情况下整和成风园，还可以组合成单一模式的风园. 模拟案例的提交证明这是正确的做法.今后的研究将侧重于测试效果用于调制力矩的建模方法.附录Fixed-Speed Wind-Generator and Wind-Park Modeling for Transient Stability StudiesIncreasing levels of wind-turbine generation in modern power systems is initiating a need for accurate wind-generation transient stability models. Because many wind generators are often grouped together in wind parks， equivalence modeling of several wind generators is especially critical. In this paper， reduced-order dynamic fixed-speed wind-generator model appropriate for transient stability simulation is presented. The models derived using a model reduction technique of a high-order finite-element model. Then， an equivalency approach is presented that demonstrates how several wind generators in a wind park can be combined into a single reduced-order model. Simulation cases are presented to demonstrate several unique properties of a powersystem containing wind generators. The results in these paper focuson horizontal-axis turbines using an induction machine directly connected to the grid as the generator.Index TermsTransient stability simulation， wind-generator modeling， wind-park modeling， wind-turbine modeling.I. INTRODUCTIONThis encompasses many modern large-scale systems. Because large wind installations consist of many wind generators， wind-park-modeling is a critical need. Consequently， the second goals to present a methodology for combining several wind generators connected to the grid through a common bus into a singleequivalent model.Wind generators are primarily classified as fixed speed or variable speed. With most fixed-speed units， the turbine drives an induction generator that is directly connected to the grid.The turbine speed varies very little due to the steep slope of the generators torque-speed characteristic； therefore， it is termed fixed-speed system. With a variable-speed unit， the generator is connected to the grid using power-electronic converter technology. This allows the turbine speed to be controlled to maximize performance (e.g.， power capture). Both approaches areManuscript received February 3， 2004. This work was supported in part bythe Western Area Power Administration. Paper no. TPWRS-00388-2003.The authors are with Montana Tech， University of Montana， Butte， MT59701USA (e-mail: ).Digital Object Identifier 10.1109/TPWRS.2004.836204 common in the wind industry. In this paper， we focus on modeling the fixed-speed unit and an equivalent model of severalA wind park consists of several wind generators connected toothed transmission system through a single bus. Because modeling each individual turbine for transient stability is overly cumbersome，our goal is to lump the wind park into a minimal setoff equivalent wind-generator models. Our approach for equivalence modeling of a wind park involves combining all turbines with the same mechanical natural frequency into a single equivalent turbine. Simulation results demonstrate this approach provides accurate results.A representative example of published results for modeling wind generators for transient stability is contained in 210.Results for modeling fixed-speed wind generators have focused on two primary approaches. The first approach represents the turbine and generator rotor as a single inertia thus ignoring the systems mechanical natural frequency 25. The second approach represents the turbine blades and hub as one inertia connectedto the generator inertia through a spring 69. In all of these papers， the spring stiffness is calculated from the systems shaft.Our research indicates that representing the first-mode mechanical frequency is critical to an accurate model. Finite-element analysis has shown that the first-mode dynamics are primarily a result of the flexibility of the turbine blades not the shaft as assumed by others 11. The modeling approach presented in this paper centers on the fact that the primary flexible mechanical component is the turbine blade. The results in 7 focus on reduced-order wind-park modeling. The authors use a standard induction generator equiva-0885-8950/04$20.00 2004 lancing method to combine several wind generator systems. But，the authors do not address the problem of combining the turbines in such a way to preserve the mechanical natural frequencies. Our research indicates this is critical to having an accurate wind park model. A thorough discussion of reduced-order modeling of variable-speed turbines is contained in 10. The authors argue the turbine mechanics can be represented as a single inertia because the variable-speed connection decouples the mechanical dynamics from the electromechanical dynamics. Our results do not consider the variable-speed case. The work described in 210 focuses on low-order turbine models that can be easily implemented in large-scale transient stability codes. Considerable research has focused on modeling at a more detailed level. An excellent overview and literature review is contained in 17. Detailed modeling approaches range from highly-detailed finite-element models to more simplified six-mass， five-mass， and three-mass turbine models. The majorityof these models use momentum theory 13 to calculate aerodynamic forces.III. TURBINE DYNAMICSOur approach for developing a reduced-order model consists of starting with a highly-detailed mechanical and aerodynamic turbine model and then removing all dynamic effects outside the electromechanical range. In this reduction process， all analysis is done from the perspective of the turbine shaft that drives the 325 cillation. Detailed modal analysis of the system shows that the oscillation is the result of the outer portions of the blades vibrating against both the inner portions of the blades and all other inertias on the shaft 11， 12. Such a result is typical， especially forlarge turbines. Modern wind-turbine blades are very large and flexible， and tend to vibrate at their first mode when excited from the hub. Pony analysis of the oscillation in Fig. 1 shows it primarily contains a 4-Hz component 12. This is also typical of large-scale turbines， which usually have a first-mode natural mechanical frequency in the 0- to 10-Hz range. Because this range is also typical for electromechanical oscillations， it is critical to represent the mechanical oscillations of the wind-turbine as they will tend to interact with the electromechanical oscillations. The mode shape of the first-mode oscillation that dominates the response in Fig. 1 dictates that the model can be represented by a two-inertia， single spring-damper system as depicted in Fig. 2. This is the basis for the reduced-order model that follows. One inertia represents the outer portion of the blades (the blade tips in Fig. 2). The blade tips are rigidly connected as depicted in Fig. 2 with a mass less “blade ring.” The blade tips act as a single inertia because all transient disturbances equally act on all blades through the generator shaft. The other inertia represents the combined effect of the blade roots， hub， turbine shaft， gearing， generator shaft， and generator inertia. For a typical system， the inner inertia is dominated by the blade roots and generator inertia. The reduced turbine model depicted in Fig. 2 is considerably different than what other researchers have proposed 29.Many have lumped the entire turbine and generator into a single inertia and ignored the mechanical first-mode dynamics 25.Others has considered first-mode dynamics， but do not model the blade flexibility 69. Instead， these authors have assumed the blades to be a single inertia and model the turbine shaft as a spring. But， in a typical system， the blades are much more flexible than the shaft. Our research indicates that the blades dominate the mechanical first mode and the shaft acts as a rigid body. Our research also indicates that correctly modeling the mechanics is critical to obtaining accurate transient simulation results. SINGLE WIND-GENERATOR MODEL The single wind-generator model consists of two primary components: the reduced-order two-inertia turbine model from the previous section driven by a wind torque； and a standard TRUDNOWSKI et al.: FIXED-SPEED WIND-GENERATOR AND WIND-PARK MODELING FOR TRANSIENT STABILITY STUDIESelectric generator. For this paper， we assume the generator to be a standard induction machine directly connected to the grid as this is the most common configuration. A. Turbine ModelThe two-inertia reduced-order turbine in Fig. 2 is the basis for the turbine model. The equations of motion for the system in Fig. 2 are(1)where number of blades；effective gear ratio = /rated-turbine-speed；electrical frequency base；inertia of each blade tip；inertia of each blade root+ inertia of + inertia of turbine shaft and gearing/+ inertia of generator shaft and rotor；blade stiffness；blade damping；aerodynamic wind torque；generator electrical torque；blade tip angle reflected through the gearing；generator shaft angle. Calculating the inertias and in (1) requires knowledge of the blade break point where the spring-damper is placed (see Fig. 2). If the blade is not broken at the correct position， then the mechanical mode shape will not be correct. The break point is primarily a function of the blade mechanics and can be determined from finite-element analysis or testing of the blade and seems to occur at the second bending node of the blade. In the example cases studied in 12， the reduced-order systems sensitivity to improper placement of the break point is significant. This is demonstrated in the example section. Fortunately， most modern blade manufactures or blade testing facilities (such as the facility at the National RenewableEnergy Laboratory in the United States) have the required information to determine the blade break point. The power engineer simply needs to request this information. Once one has the blade break point， the inertia parameters can easily be calculated from typical manufactures data. The stiffness in (1) can be calculated from knowledge of the systems first-mode mechanical natural frequency using(2)where is the first-mode mechanical lead-lag natural frequency with the system connected to infinite bus. For example，in the system in the previous section， .Typically， manufactures can provide this frequency. It can be easily calculated by applying a brake pulse on the turbine and analyzing its response (for example， Fourier analysis of the generators speed). In most cases the blade damping is very small and assumed to be zero. The spring stiffness is a measure of the blades stiffness in the rotational plane which is a combination of the blades edge stiffness and flat stiffness 12. Relating to the edge and flat results in(3)where is the edge stiffness， is the flat stiffness， and is the pitch angle. Both and are constant. As can be seen in(3)， is dependent on the pitch angle . Typically， is limitedto be between zero and ten degrees. Analysis of (3) under this restriction shows that varies very little for different pitch set points. This implies， and experiments support， that the accuracy of the turbine model has very small sensitivity to variations in the systems pitch angle 12.The wind torque is calculated assuming an ideal rotor disk from the equation 13(4)where is the velocity of the blade tip sections reflected through the gearing， is the air density， is the sweep area of the blades， is the free wind velocity， and is the turbines power coefficient. Unfortunately， is not a constant. However， the majority of turbine manufactures supply the owner with a curve. The curve expresses as a function caused primarily by tower shadowing and unbalanced mechanics. Typical modulation frequencies are at the 1P and 3Pmodes (note: 1P is once per revolution of a turbine blade) 6.We do not include these effects as we assume that the torque induced from the transient fault is much larger than the modulation torque. This assumption has been made by many other researchers (for example， 7). Future research will focus on testing this assumption. In general， the two-inertia turbine model proposed here is a relatively robust model that covers many turbine operating conditions. All model parameters are relatively constant with very little sensitivity to the pitch angle. Because the main component of energy in a transient is due to turbine inertial energy，stall-controlled turbines can be accurately modeled using this approachs. Generator Model Standard practices are well established for modeling the generator 1. A standard detailed two-axis induction machine model is used to represent the induction generator 1. The resulting equations are(6a) where is the transient open-circuit time constant， is the slip speed， is the synchronous reactance， is the transient reactance， and are the d-axis and q-axis stator voltages， and are the d-axis and q-axis per-unit stator currents. The torque is calculated from(6b)TRUDNOWSKI et al.: FIXED-SPEED WIND-GENERATOR AND WIND-PARK MODELING FOR TRANSIENT STABILITY STUDIES where is the sweep area， is the free wind velocity， and is the turbines power coefficient for turbine .B. Equivalent Generator Model The equivalence induction generator parameters are obtained using the weighted admittance averaging method in 16. With this method， the equivalent machine parameters ，and are calculated by taking the weighted average admittances of each branch of the induction machine equivalent circuit. The weighting for the averages are calculated using the rated power of the generators. I. SIMULATION RESULTS Many example test cases have been studied to evaluate the properties of the modeling approach； these are contained in 12，14， 15. A select few are presented in this section.For this example， we compare the response of the two-inertia reduced-order turbine in (1) to the response of the finite-element model and a detailed five-inertia model. Each model is connected to an infinite bus through an induction generator. The response of the finite-element model is shown in Fig. 1.Thefive-inertia model represents each blade with edge and flap spring-dampers； the slow-speed shaft spring stiffness is also represented； and the aerodynamics are modeled using Gluer vortex momentum theory 13. The five-inertia model also contains the centrifugal， gravity， and carioles effects. Derivation of the five-inertia model is contained in 11， 12. The turbine properties are described in Section III. It is directly connected to a 60-Hz infinite bus through the 1.68-MW induction generator. Turbine and induction-generator model parameters for the reduced-order model are provided in the Appendix. The simulation is compared to the ADAMS finite-element simulation which includes highly detailed aerodynamic and mechanical modeling. The two-inertia reduced-order model is a 6th-order model (1)， (4)， and (6) while the finite-element model is approximately 650th-order， and the five-inertia model is 18th order. Simulation results are shown in Fig. 4 and Fig. 5. As can be seen， the two-inertia reduced-order model closely matches the highly detailed finite-element and five-inertia models.In this example， we demonstrate the sensitivity of the two turbine model to the choice of the blade break point. The responses of three modeling cases are shown in Fig. 6. The 50% break-point places the blade spring at the center of the blade radius and is the same model used in example 1. This response is compared to a 43% break point and a 56% break point. The percentage indicates the location from the hub where the blade spring is placed along the blade radius. The differences between the responses are significant enough to merit careful selection of the blade break poiAll of the following simulations were implemented in a modified version of the Power System Toolbox (from Cherry Tree Scientific Software， Ontario，Canada). The Toolbox was modified to allow for simulation of wind generators. Fig. 8 shows the real power out of the wind generator for then two modeling cases for a disturbance consisting of a 5-cycle fault at bus 15 followed by a line opening at bus 15. Pony analysis of the two-inertia turbine response shows two modes in the oscillation: a 4.5-Hz mode and a 2.0-Hz mode. The 4.5-Hz mode is due to the mechanical mode of the turbine and the 2.0-Hz mode is the electromechanical mode. Similar analysis of the one-inertia response indicates only one mode at 2.4 Hz， which is an electromechanical mode. Because of the errors in the first-swing and oscillatory response of the single inertia system in Fig. 8， a power engineer would likely come to a different conclusion concerning the transient and small-signal stability properties of the system. The one-inertia response indicates a more stable system with a lower first-swing deviation and higher oscillatory damping. Other examples in 14 demonstrate cases
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