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通過集成磁軸承輔助有限元分析的一種新型飛輪儲能存儲系統(tǒng)的設計與建模 C.張，學生會員，IEEE，吳平，學生會員，IEEE和K. J. Tseng，高級會員，IEEE 新加坡共和國, 新加坡639798，南陽大道BLK S2，南洋理工大學，先進電力電子研究中心 摘要——本文提出的是緊湊和高效的飛輪存儲系統(tǒng)。該系統(tǒng)是由綜合力學性能和磁軸承輔助，飛輪作為轉子的驅(qū)動系統(tǒng)，并且該系統(tǒng)通過被夾在兩個磁盤式定子之間而節(jié)省空間。通過主動磁軸承，轉子飛輪旋轉和保持在垂直方向的磁懸浮機械軸承和軸向磁通永磁同步電動機的助攻結合使用，而限制在徑向方向的其他四個自由度的機械。所提出的系統(tǒng)的數(shù)學模型被推導出來。三維有限元方法是應用于通過研究和驗證數(shù)學模型系統(tǒng)分析結果而支持系統(tǒng)可行性。 一 正文 在現(xiàn)代化電力行業(yè)，具有強度高，重量輕的先進復合材料，控制技術和電子電力，飛輪能量存儲系統(tǒng)（FESS）正在成為一個傳統(tǒng)的化學電池系統(tǒng)的可行性替代。其優(yōu)點為儲能密度高，充電放電風險較低，放電深度容易檢測，能在較寬溫度范圍內(nèi)操作，壽命更長，有利于環(huán)境。所以FESS被認為是對于現(xiàn)在許多應用的一個有前景的技術，包括航空航天，交通運輸，電力工業(yè)，軍事，建筑服務。 一般來說，一個飛輪儲能系統(tǒng)是由一個磁性的或機械的軸承支撐的由電機帶動的飛輪，一個將機械能和電能內(nèi)部轉化系統(tǒng)的飛輪，控制增強電子的器件和觸地軸承組成的。這個單獨的除磁性軸承驅(qū)動電機使轉子長，容易產(chǎn)生彎曲振動。且大電機軸承系統(tǒng)使得小型化【5】困難。為了克服這些問題，自軸承永磁電機被引進。電機結合磁軸承和汽車功能為單一的磁性制動器。這樣的設計由于不需要機械軸承可以降低整體的一種電機長。因此能夠提高功率密度，減輕重量，降低轉子的動態(tài)振動【6】的敏感性。 如圖1所示，沿x，y，z在飛輪軸有三個方向，使每一個軸的位移和旋轉受機械或磁性的幫助來控制六個自由度。機械軸承具有結構簡單，操作方便的優(yōu)點，但由于摩擦損耗，應考慮潤滑油的使用。特別是發(fā)生在軸承，沿重力方向上即圖1沿z軸方向的摩擦要比其他方向上的摩擦大得多。由于這個原因，軸承使用機械軸承是不現(xiàn)實的，而其他的軸是可以承受的。主動磁軸承相對于傳統(tǒng)軸承是可以承受的。主動磁軸承相對于傳統(tǒng)軸承有許多優(yōu)點，這些優(yōu)點包括更高的能量效率，降低磨損，延長壽命，不需要潤滑機械維修和較寬的操作溫度。關于磁軸承有許多研究，但大多數(shù)人對待至少有五個自由度的對象是控制。由于控制每個自由度需要一個傳感器，執(zhí)行器和控制器，整個系統(tǒng)在機械/電氣部分和控制系統(tǒng)設計變得復雜。鑒于此，本文提出了一個新概念磁性軸承。其中軸只有兩個自由度受主動控制，即分別沿平移和旋轉方向。其他方向的運動方向由機械軸承完全限制。主動磁軸承和機械軸承的結合使用可以減少控制的復雜性，使系統(tǒng)運動更加穩(wěn)定，可行和具有成本效益。 圖1 飛輪的三個運動方向 目前，軸向磁通永磁電機（AFPM）在許多應用中 已成為一個有吸引力的研究場【8】【9】。它們有幾個獨特的功能，如效率高，高能，高扭矩密度，低轉子損耗和小磁厚度。然而缺點是該分布式繞組具有與線圈導體的有效部分相比的顯著長度的端繞組。這顯然會導致機器性能差。作為本機顯著成分（即總在大多數(shù)機器設計的50%以上）被產(chǎn)生熱量，但沒有轉矩。集中繞組可以解決這個問題。此外，他們有簡單的設計，更容易安排及更高效率。 有限元分析法（FEM）已被證明是特別靈活，可靠。有效的分析方法是工頻電磁場和機電裝置的合成。有限元法可以分析任何形狀和材料的PM電路，有限元分析與其他永磁電機的分析方法相比的一個顯著優(yōu)點是其準確計算電樞反應，電磁力和力矩的固有能力。 本文中，一種集成磁軸承輔助新型飛輪儲能系統(tǒng)被介紹。用電動機和發(fā)電機相結合并且使飛輪功能作為機器，以節(jié)省空間的轉子。機械軸承是用來限制沿徑向方向得位移和旋轉，位移和旋轉沿軸向方向由主動磁軸承控制。利用數(shù)學模型所提出的系統(tǒng)的結構和電磁設計被呈現(xiàn)。三維有限元分析的實現(xiàn)，驗證了數(shù)學模型和支持體系的可行性。本文中介紹的分析結果已經(jīng)獲得。 二 建設與計算所提出的系統(tǒng) （1）整個系統(tǒng)的配置 圖2所提出的系統(tǒng)的橫截面圖 圖2示出了所提出的的飛輪儲能系統(tǒng)的橫截面圖。它的組分列于表I項目1和8是固定在該裝置的殼體，其目的是從任何轉子碎片消散徑向動能，并確保在發(fā)生機械故障的情況下安全的上部和下部固定件。軸向磁通永磁同步電動機的實施來驅(qū)動其也用作轉子的飛輪。 機械旋轉球軸承安裝在轉子上，以限制其徑向運動和輔助飛輪/轉子的旋轉的外緣。這種安排使結構不使用軸非常緊湊。但是機械軸承的孔的最大直徑限制了最大速度。用油膜軸承DN值（孔直徑mm*轉速rpm）可以達到3，000,000【13】。這意味著最高車速小于2000轉時該孔的直徑為150毫米。在更高的速度飛輪系統(tǒng)上，兩個機械軸承可以安裝在被固定在所述轉子的中間軸的兩端。用這種結構，速度可以高達60000轉以上。 軸向運動可實現(xiàn)對旋轉球軸承的輪輞正交安裝的4個滑動球軸承的援助。當轉子旋轉時（圖中的項目＃2和＃102），非接觸式渦流位移傳感器和光電傳感器在兩個定子的中空的中心設置用以檢測沿z軸的位移和角位置。起動操作時或在磁懸浮軸承故障的情況下，需要著陸軸承。著陸軸承應安裝在對著轉子的外緣。在正常操作期間，存在所有的轉子表面和觸下軸承之間的小于0.5mm的空氣間隙，從而實現(xiàn)了機械接觸式的環(huán)境。 （2） 建議系統(tǒng)的基本特征 圖3顯示了所提出的系統(tǒng)的基本特征。電動機及發(fā)電機用盤式幾何組合成一個單一的電動機，如圖3所示（a）所示。轉子兼作飛輪和被夾持兩個圓盤型定子之間。此設計使盤式轉子的轉矩產(chǎn)生區(qū)。 如圖所示在圖3（b）中，每個上部和下部定子承載的一組三相繞組的銅與正弦電流供給;集中繞組被實現(xiàn)，以減少功率損耗。如果分布式繞組，繞組-末端將跨越轉子的半個圓周。線圈導體的有效部分比端部是更長，從而繞組的銅損會更大。在這個特定的設計中，有6個線圈，其中每個線圈都圍繞定子齒。三相和三相電流的方向在特定的實例中的分布，如圖4顯示。除了提高效率，結構簡單，安裝方便定子繞組也可實現(xiàn)這種設計。 永久磁鐵被安裝在轉子的兩個表面上，如圖3（c）所示。這些PM的與磁通流過電動機的結構被描繪在圖4中。預防性維護都定居在相反的方向上和下轉子面，所以他們會相互吸引，增加磁路的總光通量。 圖3所提出的飛輪系統(tǒng)的基本組成部分 （a）定子轉子組件 （b）定子的繞組（c）轉子（d）非磁性的護環(huán) 采用高強度非磁性材料制成的護圈是用來協(xié)助PM在抵抗離心力的作用，如圖3（d）所示。 圖4電機發(fā)展結構和二維通量模式 磁懸浮軸承可以用吸引力來實現(xiàn)。定子和轉子場之間的相互作用產(chǎn)生的軸向力，使得在轉子和定子相互吸引。每個定子的電流可以獨立調(diào)節(jié)，以控制轉子上的凈力，并保持它在兩個定子的中間。沿著軸向軸的凈力可求得 F = F2 ? F1 (1) 其中，F(xiàn)1是較低的定子和轉子之間的力; F2是上定子和轉子之間的作用力。 電動機 - 發(fā)電機相當于兩個電動機，總轉矩T可以寫為 T = T1 +T2 (2) 其中，T1和T2分別由上部和下部分別電機產(chǎn)生的轉矩 （3） 電機尺寸 軸向磁通電機的尺寸可通過下式被轉換到一個等效徑向尺寸的機器得到 D=Do+ Di/2 (3) L=Do- Di/2 (4) 其中DO和DI是軸向磁通盤式馬達，D和L的外徑和內(nèi)徑都內(nèi)徑的徑向當量機和長度。當KR = Do / Di = 3 .最大扭矩產(chǎn)生 從電機的輸出方程，我們可以得到D2L=QCons （5） 然后，我們就可以得到 其中C0是輸出系數(shù)，Q為機器的千伏安的評級，NS是額定轉速在RPS 其中Bgav代表的平均磁通密度超過氣隙的機器，也被稱為磁載荷，A為電負荷;千瓦是繞組系數(shù); PN，ηN和cosφN分別表示額定功率，效率和功率因數(shù); KE是感應電動勢和電壓之間的比率。在本設計中，KE =0.905。 空氣間隙的最小長度是由機械約束集并且不大可能小于0.3毫米。磁鐵'的深度一般應減少到最低值，以盡量減少磁體的成本。制造業(yè)的限制，很難有磁鐵大于2.0mm更薄。在此設計中，GL被選擇為0.5mm時，與毫升設定為2.5毫米。 根據(jù)在表II中示出的設計要求的數(shù)據(jù)時，電機設計的結果可以得到如在表III中。這只是一個測試設計驗證系統(tǒng)結構的可行性和數(shù)學模型的正確性。所以在額定轉速時只選擇為1500轉每分鐘。 三.數(shù)學模型 如圖3所示，在定子的三相繞組分別記為a，b和c具有相同的匝數(shù)。永久磁鐵被安裝在所述盤型轉子的表面上，一個非凸轉子最后獲得。只有當勵磁繞組被永久磁鐵所取代時，電機可以被視為一個常規(guī)同步電機， PM電機可以通過假設這里所述轉子的永磁體已被替換成等效的轉子電流，如果與卷繞數(shù)N F是容易分析。由定子相繞組與等效轉子電流產(chǎn)生，如果可以被認為是第φ和rφ，相同的繞組[14] [15]的分布的正弦函數(shù)的粗略近似的磁動勢的波形。其中sφ和rφ是從一個三相定子繞組軸與旋轉直軸，分別測得的角度。假設極對數(shù)為P，其功能如下 其中N s是相當于匝正弦分布繞組的定子的各相的數(shù)量。 對于被描繪為圖3的繞組分布（b）所示，音調(diào)因數(shù)KP= 1，分配系數(shù)KD= COS（π/ 6）= 3/2，所以繞組系數(shù)千瓦= KP×KD= 3/2。然后Ns個可以計算為 其中NPH是圈串聯(lián)每相的實際數(shù)目。 由PMs，MMFM，所產(chǎn)生的等效的MMF的最大值被計算為 其中，LM和HM表示磁體長度和當磁鐵由導磁的鐵短路的磁場強度。然后N個f如果該值，可以實現(xiàn)如 B是用于的PM的殘留磁通密度，R，μ為相對磁導率，μ0為空氣與4π×10-7的值的磁導率。 定子和轉子的表面之間的有效氣隙長度被定義為g時，磁通密度B與磁通如下圖所示： 作為一個例子，讓我們判斷，由于電流只在一個繞組漏感在這里忽略繞組的總磁鏈。 其中Ro和Ri是，定子的外表面和內(nèi)半徑。同樣地，我們可以得到 在a和f繞組之間的互感是通過確定 在與上述相同的方式，LAF，LBF，LCF可寫為 因此，其他的互感可求得 然后 其中L是電機的電感矩陣，該電感是由（18）（19）確定，（21） - （24）。 （31）的電感表達式可以當它們被表達的dq0變量方面被簡化 存儲的磁能可以被計算為 因此，可以得到的有吸引力的力Fs 從弗萊明左手法則，旋轉扭矩Ts可表示為 這里，定子和的PM在平衡點的表面之間的空氣間隙被定義為LG，所以在定子和轉子在平衡點之間的有效氣隙可求得 Kc為卡特的系數(shù)，它是約等于1。然后F1和T1可以通過代克= G0+ Z，ID = ID1和IQ = IQ1入（28）（29）進行計算，而F2和T2可以通過替換來計算G = G0 - Z，ID = ID 2和 IQ = IQ2到相同的等式，其中z是在垂直方向上的轉子的位移?？偟牧土厥怯桑?）和（2）得到的。 在轉子的徑向運動由機械球軸承的限制。因此，轉子的軸向運動是獨立的徑向運動。轉子的軸向運動的動力學方程為 其中FZ是在z軸方向上的外力，而重力被考慮在內(nèi)。 總轉矩的方程可以改寫為 和 其中J是轉動慣量，θ是轉子角，π是轉速。 電壓方程可寫為 四. 有限元分析和模型驗證 （1）理論 在永磁電機的磁場總是與瞬態(tài)激勵和非線性磁性材料相關。以下三個麥克斯韋方程有關的瞬態(tài)的應用程序。 其中，H表示磁場密度，J是電流密度，σ是介質(zhì)的電導率，和E是電場強度 從（36）和（37），可以得到 力和力矩可以計算為存儲磁共能W'相對于小排量的導數(shù)。助能量可以寫成 然后瞬時力Fs中的偏移量s的方向上的分量是 以小角度旋轉位移θ的瞬時轉矩T由下式表示 （2）有限元分析 使用時步三維有限元模擬[16]在第二節(jié)中描述的提出的系統(tǒng)進行了分析。分析模型的網(wǎng)格形狀被示為圖5。只有一個定子和轉子被實現(xiàn)在有限元分析中，為了節(jié)省計算時間，但它是有效的描述整個系統(tǒng)的性能。 圖5分析模型的網(wǎng)格形狀 圖6有限元模擬的結果時，定子伴隨50Hz正弦電流（a）磁鏈（b）引起的電壓（c）轉速 在開環(huán)的條件下，50赫茲的正弦波電流，并且給定的1500轉每分的初始速度，無論是交鏈磁通量和感應電壓是準正弦，而且速度穩(wěn)定到同步速度最終。有限元法 析結果如圖6示。事實證明，電機可以作為一個正弦波電機進行分析，數(shù)學分析是站得住腳的。 圖7（a）和（b）示出了磁通密度在定子和轉子。很明顯，有分別具有定子和轉子的表面磁通密度較高的區(qū)域4。它代表4極電機。永久磁鐵NdFe35 與留磁通密度Br=1.23 T的安裝在轉子的表面上，所以在根據(jù)的PM的區(qū)域中，磁通密度是肯定比在其他地方更高。 圖7定子和轉子的磁通分布（a）定子的磁通密度(b)轉子的磁通密度 （3）. 數(shù)學模型的驗證 三個相電流可被分解為直軸電流，如下所示 其中，θ是轉子的電角度。 使得IQ =0，我們得到平均為零的扭矩如圖8所示，（a）所示。顯而易見的是，該扭矩沒有關系的id。 分配ID = 0和Q =1時，力，轉矩和轉速也可以如圖8（b）中所示獲得的 。（d）所示上的力和力矩是大致恒定的，并且速度線性增加。事實證明，扭矩是成正比的iq 通過分配IQ或id到零，然后改變ID或IQ相應的值，我們可以得到的軸向磁力和 矩曲線在起點處，如圖9所示。實線代表從（28）的計算結果和（29），以及星標記都是在有限元分析的結果時，它被分配了。當他們比最大額定電流是1.85×2=2.62 A在這樣的設計更高的力和力矩偏離的計算曲線向下。這是由當高電流被輸入的磁飽和引起的。 圖8當qi=0，di=0的有限元分析結果（a）qi=0時扭矩（b）di=0時的力（c）di=0時的（d）di=0時的速度 在圖9（a），用小于電流有限元模型和數(shù)學模型之間的誤差仍然存在。這是因為當ID =0，22N f如果在占主導地位（28）所示的力值， N個之間f小錯誤，如果用（14）和有限元分析軟件計算出的值會導致對力值差異較大。圖9（e）及（f）是扭矩和動力的變化時，不同氣隙長度分配。結果通過這兩種方法獲得的幾乎是相同的。 圖9有限元模擬結果與解析計算結果的比較（a）id=0，iq是變量時的力（b）id=0，iq是變量時的力矩（c力（d）），id是變量時的力矩（e）id=0 ，iq=1，氣隙長度變化時的力（f）id=0 ，iq=1，氣隙長度變化時的力矩 為了進一步驗證的數(shù)學模型，Matl ab / Simulink環(huán)境可以采用來模擬電動機的性能的準確性，所導出的模擬結果隨后可被用于與有限元分析的數(shù)據(jù)進行比較。 當ID = 0和Q =1的電流被分配到定子繞組，通過Simulink中的仿真結果示于圖10。電機在模擬的參數(shù)在表IV中所示。通過比較從Simulink仿真和有限元分析，這示于圖8（BC）和圖中得到的力和扭矩曲線。10分別當相同的電流分配，可以看出，它們的平均值是非常相似的，盡管有在有限元分析結果有一定的波動。 圖10通過圖8指定相同的電流的仿真結果圖(a)軸向磁力MATLAB仿真（b）通過MATLAB仿真扭矩 同樣，我們也可以輸入相同的電壓，這是電角向電動機模型在上述兩種方法的功能，其結果，得到與圖1所示。 11。力的相應的曲線，扭矩是在形狀和價值觀相似。其結果是，支持該數(shù)學模型的正確性的證明。 從有限元分析結果和有限元法和模擬結果之間的比較，很顯然，所提出的系統(tǒng)是可行的，并且衍生數(shù)學模型是準確的，并且可以被用來設計的驅(qū)動系統(tǒng)。 圖11通過指定相同的電壓的MATLB仿真和有限元分析結果的比較（a）軸向磁力的MATLB仿真（b）軸向磁場力的有限元分析（c）通過MATLB仿真扭矩（d）轉矩的有限元分析 五.結論 一種新穎的飛輪儲能系統(tǒng)與局部自支承飛輪轉子已經(jīng)提出了這樣的紙張。系統(tǒng)的結構及設計方法的細節(jié) 進行了描述。數(shù)學模型是來自和三維有限元分析已經(jīng)進行，以驗證所提出的設計和數(shù)學模型。支持所有的分析結果所提出的系統(tǒng)的可行性，并證明了數(shù)學模型的正確性。該系統(tǒng)的原型目前正在開發(fā)中。 參考文獻 [1]R.Beach和D.A.Christopher，“航空航天應用的“飛輪儲能技術的開發(fā)，IEEE aerosp。電子系統(tǒng)雜志 卷13，頁9-14， 1998年6月。 [ 2 ] j.g.bitterly，“飛輪技術：過去，現(xiàn)在，和第二十一世紀的預測，”IEEE aerosp。電子系統(tǒng)雜志，13卷，第13-16頁， 1998年8月。 [3]J.Beno, R.Hebner and A.Walls, “飛輪電池再次到來，”IEEE譜，卷39，頁46-51。2002年4月。 [4] S.Ginter, G.Gisler, J.Hanks, D.Havenhill, W.Robinson 和L.Spina，“宇宙飛船能量存儲系統(tǒng)，IEEE aerosp”。電子系統(tǒng)雜志，13卷，第27-32， 1998。 [5] H. Kanebako 和 Y. Okada, “小的混合型自軸承馬達的新設計，高速主軸，IEEE / ASME跨。mechatron。，卷8，頁111-119， 2003年3月。 [6] M.A.Casemore 和 L.S Stephens; “驅(qū)動器的收益為toothless永磁軸承電機的自我”，IEEE跨。magn.，第35卷，第4482 -4489，1999年11月。 [7] I.da Silva 和 O. Horikawa; “吸引型磁軸承控制在一個單一的方向”，IEEE跨。工業(yè)應用，卷36，頁1138-1142， 20007 年/ 8月。 [8] A. Cavagnino, M. Lazzar和 F. Profumo, “軸向磁通內(nèi)部永磁同步電機：參數(shù)辨識和穩(wěn)態(tài)性能的測量，”IEEE跨。工業(yè)應用。，卷36，第1581-1588，，2000年十一月/十二月。 [9] A. Cavagnino, M. Lazzari 和 F. Profumo, “一個軸向磁場和徑向磁通結構之間的永磁同步電機相比，IEEE反式”。工業(yè)應用。，卷38，6，第1517-1524，2002年十一月/十二月。 [10] K. Sitapati 和 R. Krishnan, “性能比較徑向和軸向場中，永磁無刷電機，”，IEEE跨。公司applicat.，第37卷，第1219 - 1226， 2001年九月/十月 [11] S.J. Salon, 電機的有限元分析，Kluwer學術出版社，1995。 [12] J. F. Gieras 和 M. Wing, 永磁電機技術，Marcel德克爾公司，2002。 [13] M.M. Khonsari and E.R. Booser, 應用摩擦學：軸承的設計、潤滑。約翰威利父子有限公司，2001。 [14] A.E.Fitzgerald, C. Kingsley Jr., 和 S.D.Uman, 電動機械，第五版。紐約：麥格勞山，1992。 [15] P. C. Krouse, O. Wasynczuk 和S. D. Sudhoff分析電機械傳動系統(tǒng)，第二版。IEEE出版社。 [ 16 ]麥斯威爾三維參數(shù)化參考指南，Ansoft公司，2002。 Abstract—A compact and efficient flywheel energy storage system is proposed in this paper. The system is assisted by integrated mechanical and magnetic bearings, the flywheel acts as the rotor of the drive system and is sandwiched between two disk type stators to save space. The combined use of active magnetic bearings, mechanical bearings and axial flux PM synchronous machine assists the rotor-flywheel to spin and remain in magnetic levitation in the vertical orientation, while constrains the other four degrees of freedom in radial directions mechanically. The mathematical model of the proposed system has been derived. Three-dimensional finite element method is applied for studying the performances and verifying the mathematical model of the system. The analysis results support the feasibility of the system. I. INTRODUCTION N modern power industries, with the advances of high strength and light weight composite material, control technology and power electronics, the Flywheel Energy Storage System (FESS) is becoming a viable alternative to traditional chemical battery systems, with its advantages such as higher energy storage density, lower risk of overcharge and over-discharge, easier detection of the depth of discharge, operation over a wider temperature range, longer lifespan and environmental friendliness [1]-[4]. As a result, FESS is now considered a promising technology for many applications including spaceflight, transportation, power industry, military, and building services. In general, a flywheel energy storage system is composed of a flywheel, magnetic or mechanical bearings that support the flywheel, a motor-generator to drive the flywheel and inter-convert the mechanical energy and electrical energy, control and power electronic devices, and touchdown bearings. This separate driving motor-generator in addition to magnetic bearings makes the rotor long and apt to produce bending vibrations. And the large motor-bearing system makes it difficult for miniaturization [5]. To overcome these problems, a self-bearing permanent magnet motor is introduced. The motor combines magnetic bearing and motoring functionality into a single magnetic actuator. Such designs can reduce the overall length of a motor because less mechanical bearings are required, thus increasing power density, reducing weight, and lowering susceptibility to rotor dynamic vibrations [6]. As shown in Fig. 1, there are three directions along x , y and z axes within the flywheel, such that six degrees-of-freedom (DOF) which are the displacement and rotation of every axis should be controlled with the help of mechanical or magnetic bearings. Mechanical bearings have the advantages of simple structure and easy operation, but the frictional loss and thereby, the use of lubrication should always be taken into consideration. Especially, the friction occurring on the bearing which is along the direction of the gravity, i.e., the direction along z axis in Fig. 1, is much greater than those in the other directions. For this reason, it is not practical to use mechanical bearings along this axis, while for the other axes, they can still be tolerated. Active magnetic bearings have many advantages over the conventional bearings. Such benefits include higher energy efficiency, lower wear, longer lifespan, absence of need of lubrication and mechanical maintenance, and wider range of operating temperatures. There are many studies concerning magnetic bearings, but most of them treat the bearing in which at least five DOF of the object are controlled. Since the control of each DOF requires a sensor, an actuator and a controller, the entire system becomes complex in terms of the design of its mechanical/ electrical part and the control system [7]. Considering this, this paper presents a new concept of magnetic bearing, in which only 2 DOF of an axis, namely, the translation and rotation along and about axial directions respectively, are actively controlled. The motions in other directions are entirely restricted by mechanical bearings. The combined use of active magnetic bearings and mechanical bearings can cut down the complexity of control and make the system more stable, viable and cost-effective. Currently, axial flux permanent magnet motors (AFPM) used in many applications have become an appealing research field [8] [9]. They have several unique features such as high efficiency, high power and torque densities, low rotor losses and small magnetic thickness. However, the disadvantage is that the distributed windings have end-windings of significant length compared to the active part of the coil conductors. This FEM Analyses for the Design and Modeling of a Novel Flywheel Energy Storage System Assisted by Integrated Magnetic Bearing C. Zhang, Student Member, IEEE, P. Wu, Student Member, IEEE and K. J. Tseng, Senior Member, IEEE Centre for Advanced Power Electronics, Nanyang Technological University Blk S2, Nanyang Avenue, Singapore 639798, Republic of Singapore I Fig. 1. Three motion directions of flywheel. 0-7803-8987-5/05/$20.00 ?2005 IEEE. 1157 obviously results in poor machine performance, as a significant part of the machine copper (i.e., more than 50% of the total in most machine designs) is producing heat but no torque [10]. Concentrated windings can solve this problem. Furthermore, they have simpler design, easier arrangement and higher efficiency. The finite element method (FEM) has proved to be particularly flexible, reliable and effective in the analysis and synthesis of power-frequency electromagnetic and electromechanical devices [11] [12]. The FEM can analyze PM circuits of any shape and material. A remarkable advantage of FEM analysis over other approaches to analysis of PM motor is the inherent ability to calculate accurately armature reaction effects, electromagnetic force and torque. In this paper, a novel flywheel energy storage system assisted by integrated magnetic bearing is proposed. The motor and generator are combined to be a single machine and the flywheel functions as the rotor in order to save space. Mechanical bearings are used to restrict the displacement and rotation along radial directions, and the displacement and rotation along axial direction are controlled by active magnetic bearings. The structure and electromagnetic design of the proposed system is presented along with the mathematical model. 3D FEM analyses are implemented to verify the mathematical model and support the feasibility of the system. Analysis results have been obtained and are presented in this paper. II. CONSTRUCTION AND GEOMETRY OF THE PROPOSED SYSTEM A. Configuration of the Entire System Fig.2 shows the cross-sectional diagram of the proposed flywheel energy storage system. Its components are listed in Table I. Items 1 and 8 are the upper and lower stators fixed on the system housing which is designed to dissipate radial kinetic energy from any rotor debris and ensure safety in the event of mechanical failure. Axial flux permanent magnet synchronous motor is implemented to drive the flywheel which is also functioning as a rotor. Mechanical rotational ball bearings are mounted on the outer rim of the rotor to constrain its radial motion and assist the rotation of the flywheel/rotor. This arrangement makes the structure very compact without using the shaft. But the large diameter of the bore of the mechanical bearing limits the maximum speed. By using fluid-film bearings, the DN value (bore diameter mm× speed rpm) can reach 3,000,000 [13]. That means the maximum speed is less than 20,000 rpm when the bore diameter is 150 mm. In higher speed flywheel system, two mechanical bearings can be mounted at the ends of the shaft which is fixed in the middle of the rotor. With this arrangement, the speed may reach up to 60,000 rpm and above. The axial motion can be realized with the aid of 4 sliding ball bearings installed orthogonally on the rim of the rotational ball bearing. Non-contact eddy current displacement sensor and photo electrical sensor are set in the hollow center of the two stators to detect the displacement and angular position along z-axis when the rotor spins, (items #2 and #10 in Fig. 2). Touchdown bearings are necessary during starting operation or in the event of magnetic bearings failure. The touchdown bearings shall be mounted against the outer rim of the rotor. During normal operation, there is a less than 0.5 mm air-gap between all rotor surfaces and the touchdown bearings, thus achieving a mechanically contact-less environment. B. Basic Features of the Proposed System Fig.3 shows the basic features of the proposed system. The motor and generator with disk-type geometry are combined into a single electric machine as shown in Fig.3 (a). The rotor doubles as the flywheel and is sandwiched between two disk-type stators. This design maximizes the torque production area of the disk-type rotor. As shown in Fig.3 (b), each of the upper and lower stators carries a set of three-phase copper windings to be fed with sinusoidal currents; concentrated windings are implemented to reduce the power loss. If distributed windings are used, the winding-ends will span half the circumference of the rotor. The ends are much longer compared to the effective parts of coil conductors, and the copper loss of the windings will thus be larger. In this particular design, there are 6 coils, each of which surrounds a stator tooth. The distribution of three phases and directions of the three-phase currents at a particular instance are TABLE I COMPONENTS OF THE PROPOSED SYSTEM Item Number Item Name 1 Upper stator 2 Position sensor 3 Stator windings 4 Touchdown bearings 5 Rotational ball bearing 6 System housing 7 Rotor permanent magnets 8 Lower stator 9 Non-magnetic material guard ring 10 Rotation sensor 11 Flywheel-rotor 12 Fasteners 13 Sliding ball bearing Fig. 2. Cross-sectional diagram of the proposed system. 1158 shown in Fig. 4. Besides improved efficiency, simple structure and easy installation of the stator winding can also be realized in this design. Permanent magnets are mounted on both surfaces of the rotor, as shown in Fig.3 (c). The arrangement of these PMs and the magnetic flux flowing in the motor are depicted in Fig.4. PMs are settled in opposite directions in upper and lower rotor faces, so that they would attract each other and increase the total flux in the magnetic circuits. A guard ring made of high strength non-magnetic material is used to assist the PMs in resisting the centrifugal force, as shown in Fig.3 (d). Magnetic bearings can be realized by using attractive forces. The interaction between the stator and rotor fields produces an axial force that makes the rotor and stator attract each other. The currents of each stator can be independently adjusted to control the net forces on the rotor and keep it in the middle of the two stators. The net force along the axial axis can be obtained as 21 FF F= ? (1) Where 1 F is the force between the lower stator and rotor; 2 F is the force between the upper stator and rotor. The motor-generator is equivalent to two motors, the total torque T can be written as 12 TTT= + (2) Where 1 T and 2 T are torques generated by the upper and lower motor respectively. C. Dimensions of the Motor The size of axial flux motor can be transformed to that of an equivalent sized radial machine by the following formulas 2 oi DD D + = (3) 2 oi DD L ? = (4) where o D and i D are the outer and inner diameters of the axial flux disk-type motor, D and L are the inner diameter and length of the equivalent radial machine. Maximum torque is produced when /3 Roi KDD==. From output equation of the motor, we can get 2 0 s Q DL Cn = (5) and then, we can obtain 3 3 2 0 8 (1)(1) R o RRs QK D KKCn = +? (6) where 0 C is the output coefficient, Q is the rating of machine in kVA, s n is the rated speed in r.p.s. 3 0 11 10 gav w CBAK ? =×, cos EN N N KP Q η ? = (7) where gav B represents the average flux density over the air-gap of the machine, also known as magnetic loading; A is the electric loading; w K is the winding factor; N P , N η and cos N ? represent rated power, efficiency and power factor respectively; E K is the ratio between induced EMF and the voltage. In this design, 0.905 E K = . Minimum length of air-gap is set by mechanical constraints and is unlikely to be less than 0.3 mm. Magnets’ depth should generally be reduced to a minimal value so as to minimize the cost of the magnets. Manufacturing restrictions make it difficult to have magnets thinner than 2.0mm. In this design, g l is selected as 0.5 mm, and m l is set to be 2.5 mm. According to the design requirement data shown in Table II, the motor design results can be obtained as in Table III. This is just a test design to verify the feasibility of the system structure and the correctness of the mathematical model. So the rated Fig. 4. Motor development structure and 2D flux pattern. (a) (b) (c) (d) Fig. 3. Basic parts of the proposed flywheel system. (a) Stator-rotor Assembly. (b) Stator Windings. (c) Rotor-flywheel. (d) Non-magnetic guard ring. 1159 speed is only selected to be 1500 rpm. III. MATHEMATICAL MODEL As shown in Fig.3, the three-phase windings of the stator are denoted as a, b and c with the same winding number. Permanent magnets are mounted on the surface of the disk type rotor, a non-salient rotor is obtained as a result. The motor can be treated as a conventional synchronous motor, only if the field windings are replaced by permanent magnets. The PM motor can be readily analyzed by assuming that the permanent magnets of the rotor here have been replaced by an equivalent rotor current f i with the winding number f N . The waveform of the MMFs produced by the stator phase windings and the equivalent rotor current f i may be considered as coarse approximation of sinusoidal functions of s φ and r φ , the same as the distribution of the windings [14] [15]. Where s φ and r φ are the angles measured from the a phase stator winding axis and rotational d axis, respectively. Assuming the number of pole pairs is P , their functions are as follows cos 2 sin 2 s as as s s as s MMF N iP P N NP φ φ =? = (8) ( ) () 2 cos 3 2 2 sin 3 2 s bs bs s s bs s MMF N iP P N NP φ π φπ =? ? =? (9) ( ) () 2 cos 3 2 2 sin 3 2 s cs cs s s as s MMF N iP P N NP φ π φπ =? + =+ (10) cos 2 sin 2 f f fr f rf r MMF N iP P N NP φ φ =? = (11) where s N is the number of turns of equivalent sinusoidally distributed winding in each phase of the stator. For the winding distribution depicted as Fig.3 (b), the pitch factor 1 p k = , the distribution factor d k = cos( / 6)π = 3/2, so the winding factor 3/2 wpd kkk=×= . Then s N can be calculated as 4 s wph NkN π = (12) where ph N is the actual number of turns in series per phase. The maximum value of the equivalent MMF produced by PMs , m MMF , is calculated to be 2 ff mmm Ni MMF H l P == (13) where, m l and m H denote the magnet length and the magnetic field intensity when the magnet is shorted by permeable iron. Then the value of f f N i can be achieved as 0 22 rm ff mm r B l Ni PHl P μ μ == (14) r B is the remanent flux density for the PMs, r μ is the relative permeability, and 0 μ is the magnetic permeability of the air with the value 7 410π ? × . The effective air gap length between the surfaces of stator and rotor is defined as g , the magnetic flux density B and magnetic flux are shown as below: 0 s MMF B Bds g μφ== ∫ (15) As an example, let us determine the total flux linkages of the winding due to current flowing only in a winding, leakage inductances are ignored here. / 0 22 / 0 () () sin 2 () .[ ]cos 4 s s P s as as s as s s s P so i as s N NdPP NR R iPdd Pg π φπ φ λ φφ φ φ φ μ ξ ξφ + == ? ? ∫∫ ∫ (16) 222 0 2 () 8 as o i s as s as R RN LL i Pg λμπ? === (17) where o R and i R are the outer and inner radius of the stator. Similarly, we can get as bs cs s L LLL= == (18) TABLE III DESIGN GEOMETRICAL DATA No. of pole pairs 2 No. of slots 6 Outer diameter of stator 130 mm Inner diameter of stator 76 mm Permanent magnets length 2.5 mm Air gap length 0.5 mm Slot width 28 mm Slot depth 22 mm Stator yoke thickness 18 mm Rotor core thickness 60 mm Air gap flux density 0.805 T No. of turns per phase 416 TABLE II DESIGN REQUIREMENT DATA Rated power 1 kVA Phase current, rms 1.85 A Power factor 0.9 Efficiency 0.9 Rated speed 1500 rpm Frequency 50 Hz Slot fill factor 0.4 Remanent flux density 1.23 T Magnet recoil permeability 1.05 Carter’s factor 1.05 1160 222 0 2 () 8 f oif ff f as RRN LL i Pg λμπ? === (19) The mutual inductance between the a and f windings is determined by / 0 22 / 0 ()() sin 2 () .[ ]cos() 4 s s P S asf as s f r s s P fo i f s N NdPP NR R iP dd Pg π φπ φ λφφφ φ μ ξ θξφ + == ? ?? ∫∫ ∫ (20) In the same way as above, af L , bf L , cf L can be written as 22 0 2 () cos cos 8 oisf af m RRNN L PL P Pg μπ θ θ ? == (21) 22 0 2 () cos( 2 / 3 ) 8 oisf bf RRNN LP Pg μπ θ π ? =? (22) 22 0 2 () cos( 2 / 3 ) 8 oisf cf RRNN Pg μπ θ π ? =+ (23) Therefore, the other mutual inductances can be obtained as 1/2 ab ba ac ca bc cb s L LLLLL L======? (24) Then, () = ff af bf cf f T af as ab ac a fabc bf bs ac b ba cb cs ccf ca L L L Li L L L Li L L Li L L LiLL λλλλ ?? ?? ?? ?? ?? == ?? ?? ?? ?? ?? ?? λ Li (25) where L is the inductance matrix of the motor, the inductances are determined by(18)(19) and (21)-(24). The inductance expression of (31) can be simplified when they are expressed in terms of 0dq variables 3/2 0 3/2 0 003/2 f fm f dm s d sqq L Li L L i λ λ λ ?? ???? ???? ?? = ?? ?? ?? ?? ???? ?? (26) The magnetic energy stored may be calculated as ()( ) 1 i 2 T fdq f d q Wii λλλ= (27) The attractive force s F can thus be obtained () 22 0 22 22 2 2 2 () 16 53 . 22 oi s ff sffd s d q RRW F g Pg Ni NNii N i i μπ ?? =? = ? ?? +++ ?? ?? (28) From the Fleming’s left-hand rule, the rotating torque s T can be expressed as 22 0 3( ) 3 () 216 oisf s dq qd f q RRNN TPii i Pg μπ λλ ? =× ? = (29) Here, the air gap between the surfaces of the stator and PMs at the equilibrium point is defined as g l , so the effective air gap between the stator and rotor at the equilibrium point can be obtained as 0 (/) cg m r gKllμ= + (30) where c K is the Carter’s coefficient, which is approximately equal to 1. Then 1 F and 1 T can be calculated by substituting 0 g gz= + , 1dd ii= and 1qq ii= into (28)(29) whereas 2 F and 2 T can be calculated by substituting 0 g gz=?, 2dd ii= and 2qq ii= into the same equations, where z is the displacement of the rotor in the vertical direction. The total force and torque are obtained by (1) and (2). The radial motions of the rotor are restricted by mechanical ball bearings. Therefore, the axial motion of the rotor is independent of radial motion. The dynamic equation of the axial motion of the rotor is '' z mz F f= + (31) where z f is the external force in the direction of z-axis, and the gravity is taken into consideration. The equation of total torque can be rewritten as '' Tq TJ Kiθ== (32) And '' , Tq J Kiθ??== (33) where J is the moment of inertia, θ is the rotor angle and ? is the rotational speed. The voltage equations can be written as 1 () q a ddd rq ddd L Rd ivi pi dt L L L ?=?+ (34) 1 () () ad m qqd rd r qqq q RLd ivi pi p dt L L L L λ ? ?=?? ? (35) IV. FEM ANALYSIS AND MODEL VERIFICATION A. Theory Magnetic fields in PM motors are always associated with transient excitations and nonlinear