垂直軸風力發(fā)電機的設計

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附錄 固定風力發(fā)電機和風力集成園建模系統暫態(tài)穩(wěn)定性的研究 抽象程度越來越高的風力發(fā)電渦輪機，在現代電力系統中需要一項準確的風力發(fā)電系統暫態(tài)穩(wěn)定模式. 因為許多風力發(fā)電機往往集合在一起，其中等價建模幾個風力發(fā)電機尤為關鍵. 本文介紹的降階動態(tài)固定風力發(fā)電機模型適合暫態(tài)穩(wěn)定模擬. 該模型是使用一個模型還原技術所構建的高階有限元模型. 然后， 用等價方式表明如何將幾個風力發(fā)電機的風力合并成一個 單降階模型. 用模擬個案來說明一些獨特性能的動力系統，含風力發(fā)電機. 所以說，本文著重于介紹水平軸風力渦輪機用異步電機直接連到電網作為 系統的發(fā)電機. 用參數計算暫態(tài)穩(wěn)定模擬系統，計算風力發(fā)電機組的建模，計算風力渦輪機造型. 一.最近，大家對風能的發(fā)展展現出了濃厚的興趣. 伴隨著使用風力發(fā)電機的熱潮，現在需要對電力動態(tài)系統， 電力傳輸規(guī)劃的設計評估. 本文的第一個目的是提出一個準確的低階動態(tài)模型的風力發(fā)電機組，它是 符合現代機電暫態(tài)模擬計算機程式的. 本文中，開發(fā)的模式著重于水平軸的風力發(fā)電機， 或風力機直接連到同步網時采用異步發(fā)電機. 這其中還包含許多現代大型發(fā)電系統. 由于大型風力裝置的構建是由許多個風力發(fā)電機組成的， 風力發(fā)電場的建模是一個迫切的需求. 因此， 本文的第二個目的是提供一種方法，它結合數個風力發(fā)電機連接到一個電網上，然后通過一個共同模式整合成一個單一的等效模型. 風力發(fā)電機主要分為定速或變速. 以最小單位，渦輪驅動的感應發(fā)電機為例，它是直接連接到電網上的. 渦輪轉速變化很小，那是由于陡坡的發(fā)電機轉矩和轉速的特性所制； 因此， 它被稱為定速系統. 還有變速裝置，發(fā)電機連接到電網利用電力電子變換的技術使渦輪速度受到控制，以最大限度地表現出來(例如，電力的控制) . 這兩種方法在風力工業(yè)均非常普遍. 在本文中， 我們將目光集中在建模定速裝置和等效模擬幾個固定轉速風力發(fā)電集成園. 第一種典型的風力機械頻率是在0至10赫茲范圍； 這也是各種機電振蕩的頻率. 因此，這涉及到機械振動的風力互動學與機電動力學. 這方面的例子參見本文. 因此，為了構建一個精確的模型，風力發(fā)電機可用于暫態(tài)穩(wěn)定的研究. 第一種渦輪機械動力學必須能準確的代表模型. 這里的風力發(fā)電機模型建出了導電模型，減少了一個詳細的650階有限元模型的一個典型的 橫向軸. 氣動力和機械動力的減少與非線性四階雙渦輪慣性模型相結合生成了一個標準發(fā)電機模型. 模擬計算表明了模型的精確性.幾個風力發(fā)電機連接到傳輸系統上通過 一個單一的模型建模，因為每個渦輪暫態(tài)穩(wěn)定系統都過于繁瑣， 我們的目的是整和風力發(fā)電園成為相當于風力發(fā)電機模型的極小系統. 我們對等價建模的風園涉及到把所有渦輪以同樣的機械固有頻率整和成單一當量的渦輪機. 模擬結果表明，這種方法能夠提供準確的結果. 二. 范例 關于風力發(fā)電機建模的代表范例是關于暫態(tài)穩(wěn)定系統的，它包括在[2] - [10] . 模擬結果表明，固定頻率的風力發(fā)電機組主要集中在以下兩個主要方法. 第一種方式是把汽輪機和發(fā)電機轉子作為一個單一的慣性體從而忽略系統的機械固有頻率 [2] - [5] . 第二種方式是把渦輪葉片和樞紐之一的慣性體接上發(fā)電機加上一個彈簧 [6] [9] . 在所有這些論文中，彈簧剛度的計算是從系統的主要部分中提取的. 我們的研究顯示，較第一型機械頻率來說第二型才是至關重要的一個精確的模型. 有限元分析表明，第一類動力的變化主要是因為靈活的渦輪葉片不夠精確. 根據建模方法的算法，我們得知的主要事實是，小而靈活的機械部件是渦輪上的刀片. 結果[7]集中表明了幾個風力發(fā)電機系統和降階風園模型的類型和與類型相結合的方法. 但是， 作者不能解決水輪機和風力發(fā)電機相結合時采用這種方法保存的機械要求. 我們的研究結果表明:這關鍵在于有一個準確的風示范園. [10]詳細討論了降階變速渦輪機載的建模. 作者稱渦輪的機械能所代表的類型是一個單一的個體， 從動態(tài)的機電動力學分析，那是因為機械的慣性使它的變速性能產生堵塞. 我們分析時不考慮變速情況.[2] - [10]的工作闡述著重于低階水輪機模型，從而可以容易地實現大型暫態(tài)穩(wěn)定代碼的測量.相當多的研究集中在建模定額一個更深入的層次. [17]是一個很好的概況和文獻. 從高度詳細的有限元模型角度，詳細的闡述了建模方法，還較簡單的敘述了六轉五轉，三轉水輪機模型.這些模型中的大部分都采用動量理論來計算氣動力. 三.我們對發(fā)展渦輪動力的一個降階模型為出發(fā)點，把所有機械和氣動渦輪機動態(tài)效果以高度詳細的用機電射程的形式表示出來. 在這個還原過程中，是以消費者的角度來分析渦輪軸驅動發(fā)電機的. 目的是為了準確反映軸轉速和扭矩特性與最小模型的秩序和復雜性. 數值調查表明，機械氣動和機械效應的一個例子所展現的測試系統實現了有限元建模環(huán)境. 該系統是一種新興的橫向風軸機床，包括三個31.7米葉片，葉片的一套點俯仰角度為2.6 ， 一個82.5米的主軸，它們的額定功率為18.2 - RPM和1.5兆瓦，在15米/秒的風速條件下. 汽輪機是透過一個簡單的異步發(fā)電機模型直接連接到60赫茲的機械. 它還利用ADAMS有限元軟件(來自機械動力學 公司) ，加上毫微克(即由國家可再生能源實驗室)軟件進行模擬. 這兩個軟件一起被稱為亞當斯. 所有參數測試系統的模型研制出一個現實的大型機 器. 整個系統包含325個自由度，包括非常詳細地模擬動力和外部作用力. 由于機械設計中的大多數水平軸風力渦輪機極為相似， 結果使該方法的適用面廣. 研究者在用亞當斯/分數制進行了研究以后，還廣泛接觸了以一個制動脈沖對該系統的瞬態(tài)響應的研究方法.為了模仿長達0.1毫米的三相短路，發(fā)電機軸對電路的混亂反應進行了分析. 1 . 從圖1 ，系統的反應是一個阻尼振蕩的過程. 詳細的擬態(tài)分析表明，系統的振蕩是由于外層部分的葉片振動對兩者的內在部位的葉片的作用.這樣的結果是很典型的. 1)亞當斯仿真結果. 現代風力渦輪葉片非常大，有彈性， 而且往往顫動. 1表明，它主要包含4 Hz分量.這也是典型的大型渦輪機， 它通常有第一型機械自然頻率在0至10赫茲范圍內. 因為這個范圍也是典型的機電振蕩頻率范圍， 這還是風力渦輪機的關鍵頻率范圍.而研究者會傾向于研究機電振蕩的頻率. 模態(tài)的第一振蕩模式會產生一系列的主導反應. 從圖1起見，該模型的描圖可以代表兩標準單彈簧阻尼系統，這是基礎的降階模型和一個的外部分的葉片2 ) . 葉片尖端硬性連接描圖. 2 )"刀環(huán)" 葉片的細片(忽略質量)作為一個單一的慣性體，其所有的瞬態(tài)干擾行為通過發(fā)電機軸的所有刀片.其他慣性力的代表如集聚效應的葉根，輪轂，渦軸，齒輪，軸發(fā)電機，發(fā)電機的慣性都很大.一個典型的系統，內部慣性主導地位取決于葉根和發(fā)電機的慣性量.許多研究者都推斷整個渦輪機和發(fā)電機成為一個單一的惰性體從而忽略第一機械型動態(tài)系統的作用.別人都認同第一動態(tài)模式，但不認同模式葉片彈性模式.相反，這些作者都假設葉片是一個慣性體而把模型渦輪軸作為一個彈簧體. 但是，在一個典型的系統中，軸上的刀片相比其他元件來說靈活得多. 我們的研究表明，第一機械模式的葉片可以與豎軸作為一個剛體. 我們的研究還表明，正確建模是研究力學的關鍵，以獲取準確的瞬態(tài)仿真結果. 四.單一風力發(fā)電機模型由兩個基本部分組成: 降階雙渦輪慣性模型和驅使風力的力矩.在本文中， 我們假設發(fā)電機是一個標準的異步電機直接連接起來的網絡，這也是最常見的配置方法. ( 1 )葉片數目:有效傳動比=實際渦輪轉速/額定渦輪轉速； 電氣頻率基數； 每個葉尖惰性體:每個葉片根部惰性+慣性+慣性渦輪軸傳動力/慣性力+發(fā)電機軸轉子的慣性力； 葉片剛度，葉片阻尼，氣動風力矩.發(fā)電機電氣扭矩和葉尖角度通過齒輪傳動反映出發(fā)電機軸向角.計算這個角需要有葉片斷裂的慣性力和彈簧減振器的相關參數(見圖2).如果葉片放置在不破裂的正確位置，然后得到的機械模態(tài)形狀就會正確了. 研究的突破點主要在一個刀片力學性能上，可以從有限元分析或試驗的葉片得到相應的數據，這個關鍵的數據似乎發(fā)生在第二個節(jié)點彎曲的葉片上.在研究實例個案上，降階系統的靈敏度放置不當的突破點是很大的. 所幸的是， 最先進的葉片或制成品設施(如在國家可再生能源實驗室的設施)有所需的資料用以確定葉片的斷裂點.電力工程師只需要這一信息請求便可輕易計算出典型制造的數據.還可以計算出知識系統的第一型機械固有頻率的使用剛度. （2）哪里第一模型機械研究技術領先，其機械的固有頻率與系統連接到一起的幾率就大. 例如，在上一節(jié)系統的系統情況就是這樣.一般來說，制成品可以提供這樣的頻率范圍.它可以很容易的用制動脈沖對水輪機進行計算和分析.在大多數情況下葉片阻尼很小，并假定為零.在旋轉機中，衡量葉片的剛度是用彈簧剛度來計算的.主要衡量葉片的邊緣剛度.可以看出，在( 3 )中 ，計算剛度是依靠俯仰的角度的. 這也僅限于從零度至10度的典型情況. （3）根據這一限制表明，差異很小的不同位置需要設置不同的點.這意味著，根據實驗的支持，這是水輪機模型很小敏感性變異系統的準確的俯仰角. 假設一個理想的轉盤來進行風力矩的計算. （4）在葉尖部分反映出的實際速度，加上空氣密度的影響，通過清掃面積的葉片的磨合，計算出了機組的功率系數. 不幸的是，這不是一個常數. 然而，大多數渦輪制成品的特性反映出同一條曲線. 曲線表示，作為功能機組的葉尖速比. 葉尖速比的定義是自由風速度比渦輪葉片的冰山速度. ( 5 )葉片掃描半徑單元葉尖速比. 3顯示了一個典型的風力渦輪機曲線. 我們的研究已表明，可以假設固定情況下極高的風力條件下進行暫態(tài)穩(wěn)定研究. 這是因為典型的變異葉尖速比下一個10秒的瞬態(tài)葉尖比小.假定風并沒有顯著的改變模擬時間， 實際上，渦輪軸的扭矩實際上是一個調制版. 調制是眾所周知的，而且主要是考慮由于大樓遮蔽和力學失衡的作用，在專業(yè)人員和模式上才能出現典型的調制頻率(注: 1人，是一種模式，每一個渦輪葉片).我們不把這些效應考慮在內，我們假定扭矩引起的暫時性故障比調制扭矩的多. 許多其他研究者已進行了這個假設.今后的研究將側重于檢驗這一假設. 在一般情況下，雙渦輪慣性模型在這里是一個相對穩(wěn)健的模式，涵蓋了許多汽輪機運行條件. 所有模型參數相對恒定，缺少敏感性的俯仰角度. 因為主要組成部分能量是短暫的，那是由于汽輪機的慣性能量的影響， 而且失速型風力渦輪機可準確模擬這種方式. 乙發(fā)電機模型中的標準做法是行之有效的建模發(fā)生器[1].標準而詳細的兩軸感應機模型是用來代表異步發(fā)電機[1]的.由此方程( 6A )可知，凡是暫態(tài)開路的時間常數，滑移速度，都是同步的電抗，還是暫態(tài)電抗.而且并在D軸和q軸定子電壓中， 并在D軸和Q軸的每單位定子電流中. 轉矩的計算是從( 6B )及定子電流的計算中得到的，是通過( 6C )款的發(fā)電機模型參數 ( 6 )計算出(第562 ) ( 106 ) ( 7C )的相關參數. 風園造型中的風園分為幾個風力發(fā)電機連接到傳輸系統中整和為一個單一的系統.這需要建模，因為每個渦輪暫態(tài)穩(wěn)定，可過于繁瑣.我們的目標是整和風園成為一套最起碼的等效模型.等價建模風園涉及到把所有渦輪以同樣的機械固有頻率成一個單一相當于渦輪機的系統. 每個這些等效的渦輪然后連接到異步發(fā)電機上.甲相當于水輪機模型的前提，我們的做法是: 因為輪機都離不開一個共同的系統，每個渦輪也受到了同樣的干擾力矩. 因此，渦輪機的性能相似于震蕩階段.因此渦輪可合并為一個平行的機械組合.模態(tài)分析風力公園系統支持這個假說。例如，考慮要予以合并的渦輪相同的自然頻率機械.，那么等于渦輪建模方程( 1 ) ( 7 )式中，彈簧和阻尼條件汽輪機分別是慣性體.渦輪得到的風力矩是利用( 4 ) ，并迫使水輪機具有相同輸出功率為渦輪的總和，是機組的功率系數為渦輪機. 乙相當于發(fā)電機模型用異步發(fā)電機參數的納加權平均法[16]來進行計算.用此方法，相當于機床參數和計算，以加權平均納每一科的異步電機等效 五 結論 研究者已提交了降階動態(tài)風力發(fā)電機模型適合于暫態(tài)穩(wěn)定性的方案.該模型是汽輪機作為一個四階非線性模型與風速作為輸入參數得出的結論.渦輪方程符合標準發(fā)電機的用于暫態(tài)穩(wěn)定的電氣方程.一個等效辦法還表明如何在幾個風力發(fā)電機的情況下整和成風園，還可以組合成單一模式的風園. 模擬案例的提交證明這是正確的做法.今后的研究將側重于測試效果用于調制力矩的建模方法. 附錄 Fixed-Speed Wind-Generator and Wind-Park Modeling for Transient Stability Studies Increasing 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 power system 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 Terms—Transient stability simulation， wind-generator modeling， wind-park modeling， wind-turbine modeling. I. INTRODUCTION This 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 single equivalent 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 generator’s 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 are Manuscript received February 3， 2004. This work was supported in part by the Western Area Power Administration. Paper no. TPWRS-00388-2003. The authors are with Montana Tech， University of Montana， Butte， MT59701 USA (e-mail: dtrudnowski@mtech.edu).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 several A 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 [2]–[10].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 system’s mechanical natural frequency [2]–[5]. The second approach represents the turbine blades and hub as one inertia connected to the generator inertia through a spring [6]–[9]. In all of these papers， the spring stiffness is calculated from the system’s 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 [2]–[10] 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 majority of these models use momentum theory [13] to calculate aerodynamic forces. III. TURBINE DYNAMICS Our 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 for large 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 [2]–[9].Many have lumped the entire turbine and generator into a single inertia and ignored the mechanical first-mode dynamics [2]–[5].Others has considered first-mode dynamics， but do not model the blade flexibility [6]–[9]. 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 STUDIES electric 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 Model The 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 system’s 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 Renewable Energy 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 manufacture’s data. The stiffness in (1) can be calculated from knowledge of the system’s 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 generator’s speed). In most cases the blade damping is very small and assumed to be zero. The spring stiffness is a measure of the blade’s stiffness in the rotational plane which is a combination of the blade’s 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 limited to 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 system’s 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 turbine’s 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 approach’s. 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 turbine’s 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 simulat