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附錄
固定風(fēng)力發(fā)電機(jī)和風(fēng)力集成園建模系統(tǒng)暫態(tài)穩(wěn)定性的研究
抽象程度越來(lái)越高的風(fēng)力發(fā)電渦輪機(jī)，在現(xiàn)代電力系統(tǒng)中需要一項(xiàng)準(zhǔn)確的風(fēng)力發(fā)電系統(tǒng)暫態(tài)穩(wěn)定模式. 因?yàn)樵S多風(fēng)力發(fā)電機(jī)往往集合在一起，其中等價(jià)建模幾個(gè)風(fēng)力發(fā)電機(jī)尤為關(guān)鍵. 本文介紹的降階動(dòng)態(tài)固定風(fēng)力發(fā)電機(jī)模型適合暫態(tài)穩(wěn)定模擬.
該模型是使用一個(gè)模型還原技術(shù)所構(gòu)建的高階有限元模型. 然后， 用等價(jià)方式表明如何將幾個(gè)風(fēng)力發(fā)電機(jī)的風(fēng)力合并成一個(gè) 單降階模型. 用模擬個(gè)案來(lái)說(shuō)明一些獨(dú)特性能的動(dòng)力系統(tǒng)，含風(fēng)力發(fā)電機(jī). 所以說(shuō)，本文著重于介紹水平軸風(fēng)力渦輪機(jī)用異步電機(jī)直接連到電網(wǎng)作為 系統(tǒng)的發(fā)電機(jī). 用參數(shù)計(jì)算暫態(tài)穩(wěn)定模擬系統(tǒng)，計(jì)算風(fēng)力發(fā)電機(jī)組的建模，計(jì)算風(fēng)力渦輪機(jī)造型.
一.最近，大家對(duì)風(fēng)能的發(fā)展展現(xiàn)出了濃厚的興趣. 伴隨著使用風(fēng)力發(fā)電機(jī)的熱潮，現(xiàn)在需要對(duì)電力動(dòng)態(tài)系統(tǒng)， 電力傳輸規(guī)劃的設(shè)計(jì)評(píng)估. 本文的第一個(gè)目的是提出一個(gè)準(zhǔn)確的低階動(dòng)態(tài)模型的風(fēng)力發(fā)電機(jī)組，它是 符合現(xiàn)代機(jī)電暫態(tài)模擬計(jì)算機(jī)程式的. 本文中，開(kāi)發(fā)的模式著重于水平軸的風(fēng)力發(fā)電機(jī)， 或風(fēng)力機(jī)直接連到同步網(wǎng)時(shí)采用異步發(fā)電機(jī). 這其中還包含許多現(xiàn)代大型發(fā)電系統(tǒng). 由于大型風(fēng)力裝置的構(gòu)建是由許多個(gè)風(fēng)力發(fā)電機(jī)組成的， 風(fēng)力發(fā)電場(chǎng)的建模是一個(gè)迫切的需求. 因此， 本文的第二個(gè)目的是提供一種方法，它結(jié)合數(shù)個(gè)風(fēng)力發(fā)電機(jī)連接到一個(gè)電網(wǎng)上，然后通過(guò)一個(gè)共同模式整合成一個(gè)單一的等效模型. 風(fēng)力發(fā)電機(jī)主要分為定速或變速. 以最小單位，渦輪驅(qū)動(dòng)的感應(yīng)發(fā)電機(jī)為例，它是直接連接到電網(wǎng)上的. 渦輪轉(zhuǎn)速變化很小，那是由于陡坡的發(fā)電機(jī)轉(zhuǎn)矩和轉(zhuǎn)速的特性所制； 因此， 它被稱為定速系統(tǒng). 還有變速裝置，發(fā)電機(jī)連接到電網(wǎng)利用電力電子變換的技術(shù)使渦輪速度受到控制，以最大限度地表現(xiàn)出來(lái)(例如，電力的控制) . 這兩種方法在風(fēng)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(gè)固定轉(zhuǎn)速風(fēng)力發(fā)電集成園.
第一種典型的風(fēng)力機(jī)械頻率是在0至10赫茲范圍； 這也是各種機(jī)電振蕩的頻率. 因此，這涉及到機(jī)械振動(dòng)的風(fēng)力互動(dòng)學(xué)與機(jī)電動(dòng)力學(xué). 這方面的例子參見(jiàn)本文. 因此，為了構(gòu)建一個(gè)精確的模型，風(fēng)力發(fā)電機(jī)可用于暫態(tài)穩(wěn)定的研究. 第一種渦輪機(jī)械動(dòng)力學(xué)必須能準(zhǔn)確的代表模型. 這里的風(fēng)力發(fā)電機(jī)模型建出了導(dǎo)電模型，減少了一個(gè)詳細(xì)的650階有限元模型的一個(gè)典型的 橫向軸. 氣動(dòng)力和機(jī)械動(dòng)力的減少與非線性四階雙渦輪慣性模型相結(jié)合生成了一個(gè)標(biāo)準(zhǔn)發(fā)電機(jī)模型. 模擬計(jì)算表明了模型的精確性.幾個(gè)風(fēng)力發(fā)電機(jī)連接到傳輸系統(tǒng)上通過(guò)
一個(gè)單一的模型建模，因?yàn)槊總€(gè)渦輪暫態(tài)穩(wěn)定系統(tǒng)都過(guò)于繁瑣， 我們的目的是整和風(fēng)力發(fā)電園成為相當(dāng)于風(fēng)力發(fā)電機(jī)模型的極小系統(tǒng). 我們對(duì)等價(jià)建模的風(fēng)園涉及到把所有渦輪以同樣的機(jī)械固有頻率整和成單一當(dāng)量的渦輪機(jī). 模擬結(jié)果表明，這種方法能夠提供準(zhǔn)確的結(jié)果.
二. 范例
關(guān)于風(fēng)力發(fā)電機(jī)建模的代表范例是關(guān)于暫態(tài)穩(wěn)定系統(tǒng)的，它包括在[2] - [10] . 模擬結(jié)果表明，固定頻率的風(fēng)力發(fā)電機(jī)組主要集中在以下兩個(gè)主要方法. 第一種方式是把汽輪機(jī)和發(fā)電機(jī)轉(zhuǎn)子作為一個(gè)單一的慣性體從而忽略系統(tǒng)的機(jī)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(jī)加上一個(gè)彈簧 [6] [9] . 在所有這些論文中，彈簧剛度的計(jì)算是從系統(tǒng)的主要部分中提取的. 我們的研究顯示，較第一型機(jī)械頻率來(lái)說(shuō)第二型才是至關(guān)重要的一個(gè)精確的模型. 有限元分析表明，第一類動(dòng)力的變化主要是因?yàn)殪`活的渦輪葉片不夠精確. 根據(jù)建模方法的算法，我們得知的主要事實(shí)是，小而靈活的機(jī)械部件是渦輪上的刀片. 結(jié)果[7]集中表明了幾個(gè)風(fēng)力發(fā)電機(jī)系統(tǒng)和降階風(fēng)園模型的類型和與類型相結(jié)合的方法. 但是， 作者不能解決水輪機(jī)和風(fēng)力發(fā)電機(jī)相結(jié)合時(shí)采用這種方法保存的機(jī)械要求. 我們的研究結(jié)果表明:這關(guān)鍵在于有一個(gè)準(zhǔn)確的風(fēng)示范園. [10]詳細(xì)討論了降階變速渦輪機(jī)載的建模. 作者稱渦輪的機(jī)械能所代表的類型是一個(gè)單一的個(gè)體， 從動(dòng)態(tài)的機(jī)電動(dòng)力學(xué)分析，那是因?yàn)闄C(jī)械的慣性使它的變速性能產(chǎn)生堵塞. 我們分析時(shí)不考慮變速情況.[2] - [10]的工作闡述著重于低階水輪機(jī)模型，從而可以容易地實(shí)現(xiàn)大型暫態(tài)穩(wěn)定代碼的測(cè)量.相當(dāng)多的研究集中在建模定額一個(gè)更深入的層次. [17]是一個(gè)很好的概況和文獻(xiàn). 從高度詳細(xì)的有限元模型角度，詳細(xì)的闡述了建模方法，還較簡(jiǎn)單的敘述了六轉(zhuǎn)五轉(zhuǎn)，三轉(zhuǎn)水輪機(jī)模型.這些模型中的大部分都采用動(dòng)量理論來(lái)計(jì)算氣動(dòng)力.
三.我們對(duì)發(fā)展渦輪動(dòng)力的一個(gè)降階模型為出發(fā)點(diǎn)，把所有機(jī)械和氣動(dòng)渦輪機(jī)動(dòng)態(tài)效果以高度詳細(xì)的用機(jī)電射程的形式表示出來(lái). 在這個(gè)還原過(guò)程中，是以消費(fèi)者的角度來(lái)分析渦輪軸驅(qū)動(dòng)發(fā)電機(jī)的. 目的是為了準(zhǔn)確反映軸轉(zhuǎn)速和扭矩特性與最小模型的秩序和復(fù)雜性. 數(shù)值調(diào)查表明，機(jī)械氣動(dòng)和機(jī)械效應(yīng)的一個(gè)例子所展現(xiàn)的測(cè)試系統(tǒng)實(shí)現(xiàn)了有限元建模環(huán)境. 該系統(tǒng)是一種新興的橫向風(fēng)軸機(jī)床，包括三個(gè)31.7米葉片，葉片的一套點(diǎn)俯仰角度為2.6 ， 一個(gè)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(fēng)速條件下. 汽輪機(jī)是透過(guò)一個(gè)簡(jiǎn)單的異步發(fā)電機(jī)模型直接連接到60赫茲的機(jī)械. 它還利用ADAMS有限元軟件(來(lái)自機(jī)械動(dòng)力學(xué) 公司) ，加上毫微克(即由國(guó)家可再生能源實(shí)驗(yàn)室)軟件進(jìn)行模擬. 這兩個(gè)軟件一起被稱為亞當(dāng)斯. 所有參數(shù)測(cè)試系統(tǒng)的模型研制出一個(gè)現(xiàn)實(shí)的大型機(jī)
器. 整個(gè)系統(tǒng)包含325個(gè)自由度，包括非常詳細(xì)地模擬動(dòng)力和外部作用力. 由于機(jī)械設(shè)計(jì)中的大多數(shù)水平軸風(fēng)力渦輪機(jī)極為相似， 結(jié)果使該方法的適用面廣. 研究者在用亞當(dāng)斯/分?jǐn)?shù)制進(jìn)行了研究以后，還廣泛接觸了以一個(gè)制動(dòng)脈沖對(duì)該系統(tǒng)的瞬態(tài)響應(yīng)的研究方法.為了模仿長(zhǎng)達(dá)0.1毫米的三相短路，發(fā)電機(jī)軸對(duì)電路的混亂反應(yīng)進(jìn)行了分析.
1 . 從圖1 ，系統(tǒng)的反應(yīng)是一個(gè)阻尼振蕩的過(guò)程. 詳細(xì)的擬態(tài)分析表明，系統(tǒng)的振蕩是由于外層部分的葉片振動(dòng)對(duì)兩者的內(nèi)在部位的葉片的作用.這樣的結(jié)果是很典型的.
1)亞當(dāng)斯仿真結(jié)果. 現(xiàn)代風(fēng)力渦輪葉片非常大，有彈性， 而且往往顫動(dòng). 1表明，它主要包含4 Hz分量.這也是典型的大型渦輪機(jī)， 它通常有第一型機(jī)械自然頻率在0至10赫茲范圍內(nèi). 因?yàn)檫@個(gè)范圍也是典型的機(jī)電振蕩頻率范圍， 這還是風(fēng)力渦輪機(jī)的關(guān)鍵頻率范圍.而研究者會(huì)傾向于研究機(jī)電振蕩的頻率. 模態(tài)的第一振蕩模式會(huì)產(chǎn)生一系列的主導(dǎo)反應(yīng). 從圖1起見(jiàn)，該模型的描圖可以代表兩標(biāo)準(zhǔn)單彈簧阻尼系統(tǒng)，這是基礎(chǔ)的降階模型和一個(gè)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(xì)片(忽略質(zhì)量)作為一個(gè)單一的慣性體，其所有的瞬態(tài)干擾行為通過(guò)發(fā)電機(jī)軸的所有刀片.其他慣性力的代表如集聚效應(yīng)的葉根，輪轂，渦軸，齒輪，軸發(fā)電機(jī)，發(fā)電機(jī)的慣性都很大.一個(gè)典型的系統(tǒng)，內(nèi)部慣性主導(dǎo)地位取決于葉根和發(fā)電機(jī)的慣性量.許多研究者都推斷整個(gè)渦輪機(jī)和發(fā)電機(jī)成為一個(gè)單一的惰性體從而忽略第一機(jī)械型動(dòng)態(tài)系統(tǒng)的作用.別人都認(rèn)同第一動(dòng)態(tài)模式，但不認(rèn)同模式葉片彈性模式.相反，這些作者都假設(shè)葉片是一個(gè)慣性體而把模型渦輪軸作為一個(gè)彈簧體. 但是，在一個(gè)典型的系統(tǒng)中，軸上的刀片相比其他元件來(lái)說(shuō)靈活得多. 我們的研究表明，第一機(jī)械模式的葉片可以與豎軸作為一個(gè)剛體. 我們的研究還表明，正確建模是研究力學(xué)的關(guān)鍵，以獲取準(zhǔn)確的瞬態(tài)仿真結(jié)果.
四.單一風(fēng)力發(fā)電機(jī)模型由兩個(gè)基本部分組成: 降階雙渦輪慣性模型和驅(qū)使風(fēng)力的力矩.在本文中， 我們假設(shè)發(fā)電機(jī)是一個(gè)標(biāo)準(zhǔn)的異步電機(jī)直接連接起來(lái)的網(wǎng)絡(luò)，這也是最常見(jiàn)的配置方法.
( 1 )葉片數(shù)目:有效傳動(dòng)比=實(shí)際渦輪轉(zhuǎn)速/額定渦輪轉(zhuǎn)速； 電氣頻率基數(shù)； 每個(gè)葉尖惰性體:每個(gè)葉片根部惰性+慣性+慣性渦輪軸傳動(dòng)力/慣性力+發(fā)電機(jī)軸轉(zhuǎn)子的慣性力； 葉片剛度，葉片阻尼，氣動(dòng)風(fēng)力矩.發(fā)電機(jī)電氣扭矩和葉尖角度通過(guò)齒輪傳動(dòng)反映出發(fā)電機(jī)軸向角.計(jì)算這個(gè)角需要有葉片斷裂的慣性力和彈簧減振器的相關(guān)參數(shù)(見(jiàn)圖2).如果葉片放置在不破裂的正確位置，然后得到的機(jī)械模態(tài)形狀就會(huì)正確了. 研究的突破點(diǎn)主要在一個(gè)刀片力學(xué)性能上，可以從有限元分析或試驗(yàn)的葉片得到相應(yīng)的數(shù)據(jù)，這個(gè)關(guān)鍵的數(shù)據(jù)似乎發(fā)生在第二個(gè)節(jié)點(diǎn)彎曲的葉片上.在研究實(shí)例個(gè)案上，降階系統(tǒng)的靈敏度放置不當(dāng)?shù)耐黄泣c(diǎn)是很大的. 所幸的是， 最先進(jìn)的葉片或制成品設(shè)施(如在國(guó)家可再生能源實(shí)驗(yàn)室的設(shè)施)有所需的資料用以確定葉片的斷裂點(diǎn).電力工程師只需要這一信息請(qǐng)求便可輕易計(jì)算出典型制造的數(shù)據(jù).還可以計(jì)算出知識(shí)系統(tǒng)的第一型機(jī)械固有頻率的使用剛度.
（2）哪里第一模型機(jī)械研究技術(shù)領(lǐng)先，其機(jī)械的固有頻率與系統(tǒng)連接到一起的幾率就大. 例如，在上一節(jié)系統(tǒng)的系統(tǒng)情況就是這樣.一般來(lái)說(shuō)，制成品可以提供這樣的頻率范圍.它可以很容易的用制動(dòng)脈沖對(duì)水輪機(jī)進(jìn)行計(jì)算和分析.在大多數(shù)情況下葉片阻尼很小，并假定為零.在旋轉(zhuǎn)機(jī)中，衡量葉片的剛度是用彈簧剛度來(lái)計(jì)算的.主要衡量葉片的邊緣剛度.可以看出，在( 3 )中 ，計(jì)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(jù)這一限制表明，差異很小的不同位置需要設(shè)置不同的點(diǎn).這意味著，根據(jù)實(shí)驗(yàn)的支持，這是水輪機(jī)模型很小敏感性變異系統(tǒng)的準(zhǔn)確的俯仰角. 假設(shè)一個(gè)理想的轉(zhuǎn)盤(pán)來(lái)進(jìn)行風(fēng)力矩的計(jì)算.
（4）在葉尖部分反映出的實(shí)際速度，加上空氣密度的影響，通過(guò)清掃面積的葉片的磨合，計(jì)算出了機(jī)組的功率系數(shù). 不幸的是，這不是一個(gè)常數(shù). 然而，大多數(shù)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(jī)組的葉尖速比. 葉尖速比的定義是自由風(fēng)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(gè)典型的風(fēng)力渦輪機(jī)曲線. 我們的研究已表明，可以假設(shè)固定情況下極高的風(fēng)力條件下進(jìn)行暫態(tài)穩(wěn)定研究. 這是因?yàn)榈湫偷淖儺惾~尖速比下一個(gè)10秒的瞬態(tài)葉尖比小.假定風(fēng)并沒(méi)有顯著的改變模擬時(shí)間， 實(shí)際上，渦輪軸的扭矩實(shí)際上是一個(gè)調(diào)制版. 調(diào)制是眾所周知的，而且主要是考慮由于大樓遮蔽和力學(xué)失衡的作用，在專業(yè)人員和模式上才能出現(xiàn)典型的調(diào)制頻率(注: 1人，是一種模式，每一個(gè)渦輪葉片).我們不把這些效應(yīng)考慮在內(nèi)，我們假定扭矩引起的暫時(shí)性故障比調(diào)制扭矩的多. 許多其他研究者已進(jìn)行了這個(gè)假設(shè).今后的研究將側(cè)重于檢驗(yàn)這一假設(shè). 在一般情況下，雙渦輪慣性模型在這里是一個(gè)相對(duì)穩(wěn)健的模式，涵蓋了許多汽輪機(jī)運(yùn)行條件. 所有模型參數(shù)相對(duì)恒定，缺少敏感性的俯仰角度.
因?yàn)橹饕M成部分能量是短暫的，那是由于汽輪機(jī)的慣性能量的影響， 而且失速型風(fēng)力渦輪機(jī)可準(zhǔn)確模擬這種方式. 乙發(fā)電機(jī)模型中的標(biāo)準(zhǔn)做法是行之有效的建模發(fā)生器[1].標(biāo)準(zhǔn)而詳細(xì)的兩軸感應(yīng)機(jī)模型是用來(lái)代表異步發(fā)電機(jī)[1]的.由此方程( 6A )可知，凡是暫態(tài)開(kāi)路的時(shí)間常數(shù)，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(zhuǎn)矩的計(jì)算是從( 6B )及定子電流的計(jì)算中得到的，是通過(guò)( 6C )款的發(fā)電機(jī)模型參數(shù) ( 6 )計(jì)算出(第562 ) ( 106 ) ( 7C )的相關(guān)參數(shù).
風(fēng)園造型中的風(fēng)園分為幾個(gè)風(fēng)力發(fā)電機(jī)連接到傳輸系統(tǒng)中整和為一個(gè)單一的系統(tǒng).這需要建模，因?yàn)槊總€(gè)渦輪暫態(tài)穩(wěn)定，可過(guò)于繁瑣.我們的目標(biāo)是整和風(fēng)園成為一套最起碼的等效模型.等價(jià)建模風(fēng)園涉及到把所有渦輪以同樣的機(jī)械固有頻率成一個(gè)單一相當(dāng)于渦輪機(jī)的系統(tǒng). 每個(gè)這些等效的渦輪然后連接到異步發(fā)電機(jī)上.甲相當(dāng)于水輪機(jī)模型的前提，我們的做法是: 因?yàn)檩啓C(jī)都離不開(kāi)一個(gè)共同的系統(tǒng)，每個(gè)渦輪也受到了同樣的干擾力矩. 因此，渦輪機(jī)的性能相似于震蕩階段.因此渦輪可合并為一個(gè)平行的機(jī)械組合.模態(tài)分析風(fēng)力公園系統(tǒng)支持這個(gè)假說(shuō)。例如，考慮要予以合并的渦輪相同的自然頻率機(jī)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(jī)分別是慣性體.渦輪得到的風(fēng)力矩是利用( 4 ) ，并迫使水輪機(jī)具有相同輸出功率為渦輪的總和，是機(jī)組的功率系數(shù)為渦輪機(jī). 乙相當(dāng)于發(fā)電機(jī)模型用異步發(fā)電機(jī)參數(shù)的納加權(quán)平均法[16]來(lái)進(jìn)行計(jì)算.用此方法，相當(dāng)于機(jī)床參數(shù)和計(jì)算，以加權(quán)平均納每一科的異步電機(jī)等效
五 結(jié)論
研究者已提交了降階動(dòng)態(tài)風(fēng)力發(fā)電機(jī)模型適合于暫態(tài)穩(wěn)定性的方案.該模型是汽輪機(jī)作為一個(gè)四階非線性模型與風(fēng)速作為輸入?yún)?shù)得出的結(jié)論.渦輪方程符合標(biāo)準(zhǔn)發(fā)電機(jī)的用于暫態(tài)穩(wěn)定的電氣方程.一個(gè)等效辦法還表明如何在幾個(gè)風(fēng)力發(fā)電機(jī)的情況下整和成風(fēng)園，還可以組合成單一模式的風(fēng)園. 模擬案例的提交證明這是正確的做法.今后的研究將側(cè)重于測(cè)試效果用于調(diào)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