喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== 喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== 譯文 外文翻譯 題 目 純電動汽車動力傳動 系統(tǒng)匹配設計 專 業(yè) 班 級 學 生 指導教師 面向對象數(shù)學建模蓄電池的電動汽車仿真 Aden N. Seaman, Jone McPhee 摘要： 我們提出了一種在MapleSim軟件中基于數(shù)學模型設計出來的蓄電池電動汽車。這個模型有個優(yōu)點是：模型是在一種物理一致的方式下利用因果系統(tǒng)部件進行描述的。我們利用一個由Chen和Rincon-Mora建立的蓄電池模型來開發(fā)了一個基于數(shù)學模型的完整蓄電池組，并開發(fā)簡單控制器，電動機/發(fā)電機，地形模型，和驅動循環(huán)模型，以此在不同工況下測試電動車性能。由此產(chǎn)生的微分方程是被象征性地簡化的，并進行數(shù)值模擬來給出物理一致的結果，還有便是清楚地表明了蓄電池和縱向車輛動力學的緊密耦合。 1 簡介 車輛建模是一個復雜而又極具挑戰(zhàn)性的工作。汽車公司每年發(fā)布一些新的車型,所有的這些汽車都需要模擬和測試,然后才能進行車輛試制。 隨著推動清潔、高效汽車的發(fā)展，傳動系統(tǒng)正逐漸包含電機、發(fā)動機、無級變速器、類似電池的能量儲存裝置，以及傳統(tǒng)內燃機等。 在此，有一項技術能夠降低建立復雜車輛模型難度的便是非因果數(shù)學模型，該模型是利用控制方程組內組成部分動作的物理方程組來描述的。在最終被求出數(shù)值解以產(chǎn)生輸出數(shù)據(jù)之前，這些方程組特征地運行。這種方法使設計者們指定各部分動作，并約束各部分在一個更物理一致的語言環(huán)境中去描述各部分變得更容易。這使得交換或是修改各部分，甚至于簡化系統(tǒng)描述更為容易[1]。 Modelica[2]描述語言已被許多作者運用在建立混合動力汽車系統(tǒng)上了[3-7]，并且絕大多數(shù)運用Dymola[8]仿真環(huán)境。 我們選擇運用MapleSoft軟件中的MapleSim[9]仿真模塊作為我們的仿真環(huán)境，因為該模塊允許我們利用控制BEV系統(tǒng)仿真的基礎的數(shù)學方程組。 我們選用的這種方法產(chǎn)生一個簡化了的基于方程的可有效仿真的系統(tǒng)描述。方程組也可以運用在HIL實時仿真中，同時可以被運用于靈敏度分析和系統(tǒng)最優(yōu)化中[10,11]。 在本文中，我們提出一個蓄電池電動汽車 (BEV)，這是在軟件MapleSim中我們基于數(shù)學建模技術已經(jīng)建立的模型。如圖1中總體BEV系統(tǒng)框圖所示。這是一個更復雜的數(shù)學化的混合動力電動汽車整車模型建立的開始，我們旨在建立一個可運用的符號化數(shù)學模型。 圖1 總體BEV系統(tǒng)框圖 我們將一個Chen 和 Rincon-Mora[12]建立的鋰離子電路電池模型應用到BEV系統(tǒng)中。我們修改電池方程來模擬一個電池組，該電池組是由單個的電池單元通過串、并聯(lián)方式組合起來的。為了將電池組和驅動電機聯(lián)系起來，我們必須建立一個能量控制器模型作為系統(tǒng)集成的一部分。我們進一步結合一個簡單的在一個斜面驅動的一維動力學模型，一個地形模型控制傾斜度、一個驅動循環(huán)模型控制車輛所期望的速度。 通過改變驅動循環(huán)和地形模型，我們在不同的駕駛環(huán)境下檢測了所設計BEV純電動汽車的性能。 2 系統(tǒng)建模和仿真 我們決定使用的技術是利用MapleSim 數(shù)學化模型作為仿真環(huán)境，它有一個圖形界面互連系統(tǒng)部件。該系統(tǒng)模型通過Maple數(shù)學引擎進行運行，并且最后描述系統(tǒng)的微分方程(DAEs)被用于數(shù)值模擬以產(chǎn)生輸出數(shù)據(jù)。作為三維多體系統(tǒng)仿真，利用以線性圖論為基礎的DynaFlex-Pro引擎對系統(tǒng)進行仿真[1,11]。 2.1 蓄電池 無論BEV電動車還是HEV混合動力汽車，其中一個最重要組成部分是蓄電池。根據(jù)所需保真度和主要研究的電池參數(shù)，這里有很多種建立不同電池化學物質的方法。參考Rao所著論文[13]中總結的一些建模方法。一般來說,隨著計算設備精度的提高，模型的精度也必將隨著提高。 一些我們所回顧的電池建模技術有：Salameh建立的鉛酸蓄電池模型[14]；Rong 和Pedram建立的鋰離子電池數(shù)學模型[15]，其考慮了電池的SOH值和溫度效應;在3.1節(jié)PNGV電池測試手冊中的集總參數(shù)模型[16]；Piller發(fā)明的卡爾曼濾波技術[17]；Chen 和 Rin′con-Mora建立的電氣電路模型[12]；Nelson建立的阻抗模型[18]。這些不同的技術都有其優(yōu)點和缺點，也有其適用范圍。 在此，我們對電動汽車采用鋰離子電池具有極大的興趣，因為鋰離子電池質量輕并且具有高于鉛酸蓄電池和鎳基蓄電池的重量質量比和能量體積比。當司機加速和再生制動時，電池將受到持續(xù)高電流和反復充電的作用，因此，電動汽車對電池的性能要求很高。而且，隨著駕駛環(huán)境變化，電池溫度大范圍變化可能會嚴重影響電池的性能和壽命。 因此我們需要建立一個鋰離子電池化學模型，其具有較寬范圍SOC值，能承受較大范圍電流變化，適應較大范圍溫度變化。因此，最后我們更傾向于在HIL系統(tǒng)中建立這個電動汽車模型，并且我們需要的是一個成本不太昂貴，保真度也不十分高的模型。 這些要求把我們注意引向Chen 和 Rin′con-Mora提出的電氣電路蓄電池模型。我們在軟件MapleSim中執(zhí)行這些不同部分并且在充電狀態(tài)和電器元件之間（在他們論文中方程2至6）運用常用功能模塊代替非線性關系。見圖2 電池的框圖。 圖2 電池結構框圖 因為他們的模型是一個單一的單元,我們通過調整他們的方程用串、并聯(lián)的方式來模擬由若干單元組成的電池。Chen 和 Rin′con-Mora的電池可分為兩個線性電路以及兩個線性電路之間的非線性耦合關系。見圖2不同電路的標簽。一個電路是一種大型的電容器并聯(lián)電阻，這一電路是模擬電池充電狀態(tài)和電池自放電。這可以稱為“電容電路”。另一個電路是一個電壓源串聯(lián)一個電容電阻網(wǎng)絡，這一電路是模擬電池時域響應。這可以稱為“時域響應電路”。 調整單個單元模型來模擬整個電池組，令Nparallel是眾單元中的一個并聯(lián)單元，令Nseries 是許多并聯(lián)單元中的串聯(lián)單元，由此構成整個電池組。在時域響應電路中，開路電壓乘以Nseries 。當電流在電容電路中流動時，流經(jīng)電流在時域響應電路中為除以Nparallel 。在時域響應電路中，電阻為乘以Nseries Nparallel 并且電容為乘以Nparallel Nseries 。 電池模型的單個單元擁有的開路電壓為3.3 V，并且在從100%荷電狀態(tài)以1A的恒定電流放電情況下，其容量為837.5 mAh 。將每8個電池單元并聯(lián)起來組成一個并聯(lián)單元，再將74個這樣的并聯(lián)單元串聯(lián)起來組成一個最大電壓為244.2V和容量為6.7Ah的電池組。如此得到的電池組是可以和應用在2007款豐田凱美瑞混合動力汽車上的電池組相媲美的[19]。 Chen和Rin′con-Mora的電池模型在短時間內用于仿真是十分簡單的，然而，在以下提供的方式中是比較復雜的，如；開路電壓隨SOC值的變化；充電損耗和恢復的暫態(tài)效應；以及電量損耗和電量恢復對SOC值的依賴性；電池容量隨放電電流的變化等。此外，因為此模型是一個電氣電路模型，所以很容易并入BEV電動汽車模型的電氣系統(tǒng)，并且，這易于代替利用數(shù)學建模技術的方法。 該模型的一個負面因素是在沒有設置任何溫度影響的情況下建模，盡管Chen和Rin′con-Mora陳述了要包含一個溫度影響模塊并不是難事。對于電動汽車，其溫度會隨外部環(huán)境條件，電池內部耗散熱量和熱化學反應等變化。我們唯一遇到的明確包括溫度依賴性模塊的數(shù)學模型是Rong 和Pedram 所建立的[15]，但是他們的模型假定的是一個恒定的放電電流，因此，并不適合我們的BEV電動汽車系統(tǒng)。 Chen和Rincon-Mora的模型也能承受超過額定電流的充電電流，同時不用考慮電池內部增加的電阻值，因為其影響很小，即使有內阻，充電后的電量也接近完全充滿電的狀態(tài)。此外，電池的SOH值隨時間和充電循環(huán)次數(shù)的變化情況也未建立模型。這些負面因素是可接受的，考慮到在以后的模型中車輛控制系統(tǒng)將要限制電池的最大充電量，并且盡管本文沒有研究模型的溫度或者SOH值，但他們應該不至于太難編入。 2.2 能量控制器 接下來，純電動汽車的一個重要組成部分是能量轉化器。能量轉換器在蓄電池和傳動電機/發(fā)電機之間起著紐帶作用。在行駛過程模式下，能量轉換器控制大部分能量輸入電機；當在再生制動的模式下，大部分制動能量回流到電池。 通常，升壓或升壓去磁轉換器的使用取決于輸出電壓是高于還是低于輸入電壓[20]。通過改變高頻切換電路的工作周期，從而可以控制電機的輸出電壓、電流和功率。 圖3 能量控制器框圖 為避免在MapleSim中建立高頻電路模型，我們決定選用一個簡單的近似值，該值能作為能量從電池流向電機的升壓或是升壓去磁轉換器，反之亦然。如圖3所示是能量控制器框圖。盡管當前模型擁有一個100%效率的轉換器，但一種Hellgren[3]在其論文中所采用的效率更為現(xiàn)實的模型是可以被采用的。 在輸出循環(huán)中運用一種由信號驅動的電流源，據(jù)此可以測量輸出電壓和計算輸出功率。輸入電流是受PID控制器調整的，以致根據(jù)輸入功率匹配輸出功率。無論是對于決定功率流方向的正向電流還是反向電流，該電路都能很好地工作。當輸出電壓和輸出電流趨近于零時，這個模型解決了一個簡單代數(shù)功率轉換器“除以零”的問題，并且能適應變化的輸入輸出阻抗。但是其并未考慮該部件的物理限制，例如：電池的最大充放電率，電機、電線或是功率電子元件的電壓、電流限制等。 2.3 電機 本汽車模型中電機是選用的Modelica直流永磁電機，該電機包括內電阻，電感和轉子轉動慣量[21]。 電機的機械和電氣動作是通過方程1和2進行建模，在方程中Ja是電樞慣性，是點數(shù)轉角，Vnom, Inom和 fnom分別是電機公稱電壓、電流和旋轉頻率。是電機軸扭矩，La和Ra分別是電樞電感和電阻。最后，和分別是電機輸出端電壓和電流。 我們選擇由L.M.C公司[22]生產(chǎn)的型號為LEM-200的D127直流永磁電機模型。然而，我們需要修改電機的額定電壓和電流以適應我們所選電池電壓。這要求我們用不同的線束和改變電機自身磁體來得到重繞線圈電機。 電機所用到的參數(shù)已在表1中給出。我們可以注意到電機的電壓和功率均是各自額定值的兩倍。 2.4車輛動力學 我們所使用的車輛模型十分簡單。其物理參數(shù)基于2007款豐田凱美瑞混合動力汽車。因為我們只關心傳動部件的性能，我們不關心車輛自身的懸架系統(tǒng)或是轉向系統(tǒng)。我們運用了一個具有規(guī)定重量的位于斜面上的無阻力運輸車一維模型。驅動電機與運輸車變形車輪通過9:1的固定轉速比變速器進行彈性連接。車胎和凱美瑞汽車輪徑相同，型號為P215/60V R16.0。 方程3描述了電機旋轉和電機軸轉矩關系。是電機軸上轉矩，m是汽車的整車質量，R是驅動輪的半徑，是電機到車胎的傳動比，是電機主軸的轉動位移，g是重力加速度常數(shù)，且是傾斜角度。 表2列出了所用到的參數(shù)值。 在本模型中唯一的一種制動方式是再生制動，在再生制動的過程中，電機電流反向流動，利用車輛的動能給蓄電池充電。我們沒有將反復充電時電池的電流限制考慮在內。 對于這個車輛模型我們附加上了一個簡單的地形模型。根據(jù)時間查表控制地形的傾斜度，該地形是車輛的行駛環(huán)境。有了這樣的地形模型，我們可以仿真電動汽車在平原和丘陵地帶的性能。 駕駛循環(huán)系統(tǒng)是一個車輛理想速度隨時間的對照表。PID控制器將理想速度與實際速度進行對比，并驅動能量控制器輸入傳送動力到電機或是從電機獲得動力，直到車輛的實際速度和理想速度相匹配。 如圖1總體BEV框圖所示。 2.5數(shù)值仿真 在MapleSim軟件將車輛模型轉換成微分方程組過后，象征性地降低和減少了系統(tǒng)的方程組。然后用減少了的方程求出數(shù)值解以得到最終的輸出數(shù)據(jù)。 MapleSim 是利用自身的非剛性求解器來仿真我們建立的車輛系統(tǒng)，該非剛性求解器使用一個Fehlberg fourth-fifth命令四階插值Runge-Kutta 法。我們采用一種絕對誤差和相對誤差值均為1e-7的自適應時間步長，并打開MapleSim的使仿真程序運行更快的自身代碼生成能力。這個模型是在運用適合于Linux系統(tǒng)的MapleSim版本3的3兆英特爾Core2 Duo環(huán)境中運行的。它被設定在一個仿真超過30秒時間間隔，并且需10秒鐘實際時間才能完成。 3 仿真結果 圖4是單一電池單元脈沖放電在MapleSim仿真模型和實際電池單元中的對照。實際電池單元數(shù)據(jù)可以從Chen和Rin′con-Mora論文中圖5提取。類似在他們的論文中一樣，我們的模型也不考慮自放電電阻。最初98% SOC值和實驗結果很接近，直到電池容量耗盡之前都很貼近實際值。我們的模型要求一個放電循環(huán)而不僅僅是實際上看到的電池終端電壓快速下降。 運用我們的車輛模型進行了兩個簡單而直觀的測試。表3中列出了在驅動循環(huán)系統(tǒng)中應用到的參數(shù)。 3.1加速度 我們所做的第一個測試是在平坦地形上以硬和軟的加速度模擬車輛的駕駛狀況。由于內部損失，如果是軟加速而硬加速，那么蓄電池電動車和內燃機車的效率將更高。硬加速循環(huán)和軟加速循環(huán)的初始加速度是不同的，但是最大速度和減速度是相同的。見圖5是駕駛循環(huán)速度隨時間變化的硬和軟加速曲線圖 圖6為電池SOC值隨時間變化圖。曾描述該模型沒有滾動阻力。你可以看到硬加速驅動周期以一個低于軟加速循環(huán)的SOC值結束加速狀態(tài)。不相同的地方是由于電阻損失來自于電機繞組和電池內部化學損失 3.2山地 我們所做的第二個測試是測試汽車上坡和下坡的情況。當汽車上坡時，電池消耗能量并部分轉化為汽車重力勢能，然而，在下坡的時候，汽車減少的部分重力勢能轉化到電池當中。見圖5駕駛循環(huán)速度隨時間變化的山地循環(huán)曲線。地形循環(huán)非常簡單：在t=9.5s時，車輛遇到陡坡，并駛上陡坡，或是在t=20.5s之前從坡度為8度的斜坡上駛下，返回平地。 圖7為這個測試中電池SOC值隨時間變化曲線。在兩種情況下，電池消耗能量使車輛加速，將電池的能量部分轉化為車輛的動能。 在上坡的情況下，SOC值減小。駕駛控制器應用更多能量到電機以使車輛的速度和理想速度相匹配，并且電池能量轉化成了車輛的重力勢能。 在下坡的情況下，SOC值增加。駕駛控制器應用蓄熱式“制動”以使車輛保持速度恒定，并且車輛的重力勢能隨著轉化成電能回流到電池中。 最后，汽車運動到平緩的地點并利用再生制動實現(xiàn)剎車，同時將車輛動能轉化到電池中儲存起來。 3.3驗證 在基于能量守恒的原則下我們對在MapleSim中的仿真結果和近似計算結果做了一下對比。對硬和軟加速循環(huán)做了以下幾點對比：在車輛啟動之前和啟動后達到最大速度開始直至再生制動以前。因為車輛在平直道路上無滾動阻力地運動，僅僅包含車輛動能和電機、電池上必須考慮的阻力損失。 見表4，基于能量守恒的近似理論計算和MapleSim 軟件為硬和軟加速度循環(huán)做的仿真結果在以下參數(shù)上做的對比結果。J——轉化到車輛的能量；P——加速全程的平均功率；SOC——電機和電池上納入考慮的損失中電池的SOC值變化。詳見Appendix A在硬加速驅動循環(huán)計算中的步驟。 MapleSim仿真結果與近似理論結果比較吻合。考慮到近似理論公式的使用，出現(xiàn)較小的誤差并不奇怪。 4 總結 我們利用了運用MapleSim軟件的基于數(shù)學的方法模擬了一個簡單的蓄電池電動汽車。這項技術減少了汽車開發(fā)時間，并使系統(tǒng)更接近物理系統(tǒng)。 運用一個基于Chen和Rin′con-Mora的電池模型建立的完整電池組數(shù)學模型，一個簡單的功率控制器模型和一個標準Modelica直流電機模型，我們能夠組成一個BEV傳動系統(tǒng)并將其與一個簡單的車輛動力學模型聯(lián)系起來。 通過運用不同的地形條件和駕駛循環(huán)，對兩個不同的情景進行測試以比較我們汽車模型的性能和人們期望的實際汽車的性能。在兩種情況下，得到的測試結果和直覺想象以及近似理論計算都是想符合的。 基本的描述系統(tǒng)的數(shù)學方程能用到靈敏度分析、優(yōu)化或是實時HIL仿真等運用中。 后續(xù)工作將包括給系統(tǒng)增加內燃機作為一個增程器，增加功率控制器、電機模型的保真度，增加更復雜車輛模型、地形模型和駕駛循環(huán)模型 致謝 我們特別感謝豐田公司，MapleSoft公司以及加拿大自然科學與工程研究委員會的大力支助和支持！ - 11 - Proceedings of the ASME 2010 International Design Engineering Technical Conferences &Computers and Information in Engineering ConferenceIDETC/CIE 2010August 15-18, 2010, Montreal, Quebec, CanadaDETC2010-28814SYMBOLIC MATH-BASED BATTERY MODELINGFOR ELECTRIC VEHICLE SIMULATIONAden N. SeamanDepartment of Systems Design EngineeringUniversity of WaterlooWaterloo, Ontario, Canada. N2L 3G1Email: anseamanreal.uwaterloo.caJohn McPheeDepartment of Systems Design EngineeringUniversity of WaterlooWaterloo, Ontario, Canada. N2L 3G1Email: mcpheereal.uwaterloo.caABSTRACTWe present results of a math-based model of a battery elec-tric vehicle (BEV) designed in MapleSim1. This model has thebenefits of being described in a physically consistent way us-ing acausal system components. We used a battery model byChenandRin con-Moratodevelopamath-basedmodelof acom-plete battery pack, and developed simple power controller, mo-tor/generator, terrain, and drive-cycle models to test the vehicleunder various conditions. The resulting differential equationsare simplified symbolically and then simulated numerically togive results that are physically consistent and clearly show thetight coupling between the battery and longitudinal vehicle dy-namics.1IntroductionVehiclemodelingisacomplicatedandchallengingtask. Au-tomotive companies release several new vehicles each year, andall of these need to be simulated and tested before they are actu-ally manufactured.With the push towards cleaner and more energy-efficientvehicles, powertrains are incorporating motors, generators,continuously-variable transmissions, energy storage devicessuch as batteries and fuel-cells, and traditional internal combus-tion engines (ICEs).One of the techniques that can ease the growing complex-ity of vehiclemodelingis acausal math-basedmodelingin which1Maple and MapleSim are trademarks of MapleSoftthe system is described using the physics-based equations thatgovern the behaviour of its components. These mathematicalequations are processed symbolically before finally being solvednumerically to generate output data. This approach makes it eas-ier for designers to specify component behaviour, and constrainsthemtodescribecomponentsinamorephysically-consistentlan-guage. This makes it easier to swap or modify components andsimplifies the description of the system 1.The Modelica 2 description language has been used bymany authors 37 to model hybrid electric vehicle systemsacausally, mostly using the Dymola 8 simulation environment.We have chosen to use MapleSim 9 from MapleSoft asour simulation environment, as this allows us to access the un-derlying mathematical equations which govern the system beingsimulated.Thisapproachyieldsasimplifiedequation-baseddescriptionof the system which can be simulated efficiently. The equationscan also be used in real-time simulationfor hardware-in-the-loop(HIL) applications, and can be used in sensitivity analysis andsystem optimization 10,11.In this paper we present the results of a battery electric ve-hicle (BEV) we have modeled using math-based modeling tech-niquesinMapleSim. See Fig.1forablockdiagramoftheoverallBEV system. This is the beginning of a more complex math-based hybrid electric vehicle (HEV) model we aim to developusing symbolic mathematics.We have incorporated a lithium-ion electric-circuit batterymodel by Chen and Rin con-Mora 12 into the BEV system. We1Copyright c ? 2010 by ASMEProceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2010 August 15-18, 2010, Montreal, Quebec, Canada DETC2010-? Downloaded 25 Jun 2011 to 113.204.33.35. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmmodified the battery equations to simulate a battery pack com-posed ofseries and parallel combinationsof single cells. Inorderto connect the battery pack to a motor we had to develop a powercontroller model as part of the system integration. We furtherincorporateda simple one-dimensionalvehicle model that driveson an inclined plane, a terrain model that controls the incline,and a drive cycle model that controls the vehicles desired speed.By varying the drive cycle and terrain model, we tested theBEV under various driving conditions.FIGURE 1.BLOCK DIAGRAM OF OVERALL BEV MODEL2System Modeling and SimulationThe technique we decided to use was math-based model-ing using MapleSim as the simulation environment, which has agraphical interface for interconnecting system components. Thesystem model is then processed by the Maple mathematics en-gine, and finally the differential-algebraic equations (DAEs) de-scribing the system are simulated numerically to produce out-put data. For 3D multibody simulation it uses the DynaFlex-Pro engine, which uses linear graph-theory for system simula-tion 1,11.2.1BatteryOne of the most importantcomponentsof an electric vehicle either BEV or HEV is the battery. There are many ways ofmodeling different battery chemistries depending on the fidelityneeded and the battery parameters of interest. See the article byRao et al. 13 for an overview of some of the techniques. Gen-erally, with increasing model accuracy comes increased compu-tational requirements.Some modeling techniquesthat we reviewed were: the lead-acid model of Salameh et al. 14; the mathematical lithium-ionmodel of Rong and Pedram 15 that incorporates state-of-healthand temperature effects; the lumped-parameter model in section3.1 of the Partnership for a New Generation of Vehicles (PNGV)Battery Test Manual 16; the Kalman filtering techniques ofPiller et al. 17; the electrical circuit modelofChen and Rin con-Mora12; andthe impedancemodelof Nelson et al. 18. Thesedifferenttechniqueshavetheirstrengthsandweaknessesandlim-ited ranges of application.There is a great interest in using lithium-ion batteries inelectric vehicles, as they are light and have a higher power-to-weight and power-to-size ratio than Lead-Acid or Nickel-basedbatteries. Great demands are placed on vehicle batteries as thedriver accelerates and brakes regeneratively,putting the batteriesthrough periods of high current draw and recharge. Dependingon the driving environment,the batteries can also be subjected tolarge temperature variations, which can have a significant effecton the batterys performance and lifetime.Thus we needed to model a lithium-ion battery chemistryover a wide state-of-charge (SOC) range, under widely-varyingcurrents, for various temperatures. Since we would eventuallylike to model this vehicle in a hardware-in-the-loop (HIL) sys-tem, we neededa model that was not computationallyexpensive,and we did not require a high-fidelity model.These requirements led us to the electrical circuit model ofChen and Rin con-Mora. We implemented their components inMapleSimandusedacustomfunctionblocktorepresentthenon-linear relationship between the state of charge and the electricalcomponents (Equations 2 to 6 in their paper). See Fig. 2 for ablock diagram of the battery.FIGURE 2.BLOCK DIAGRAM OF SINGLE-CELL BATTERYMODELSince their model is of a single cell, we modified their equa-tions to simulate a battery of cells in parallel and series. TheChen and Rin con-Mora battery can be divided into two linearcircuits with a non-linear coupling between them. See Fig. 2 forlabels of these different circuits. One circuit is a large capaci-tor in parallel with a resistor that models the charge state of thebattery and self-discharge. This can be called the “capacity cir-cuit”. Anothercircuit is a voltage source in series with a resistor-2Copyright c ? 2010 by ASMEDownloaded 25 Jun 2011 to 113.204.33.35. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmcapacitor network that models the time response of the battery.This can be called the “time response circuit”.To adapt their single cell model to simulate an entire bat-tery pack, let Nparallelbe the number of cells in a parallel pack,and let Nseriesbe the number of parallel packs placed in seriesto make the whole battery. The open circuit voltage in the timeresponse circuit is multiplied by Nseries. The current flowing inthe time response circuit is divided by Nparallelwhen it flows inthe capacity circuit. The resistors in the time response circuit aremultipliedbyNseries/NparallelandthecapacitorsaremultipliedbyNparallel/Nseries.A singlecell ofthebatterymodelhas anopen-circuitvoltageof 3.3 V and a capacity of 837.5 mAh at a 1 A discharge ratestarting at 100% state of charge. By placing 8 cells in parallel,and 74 of these parallel packs in series, a 244.2 V, 6.7 Ah batterypack was created. This pack is comparable to that in a 2007Toyota Camry hybrid 19.The Chen and Rin con-Mora battery model is simple enoughto simulate in a short amount of time while being complexenough to provide the following: variations in the open circuitvoltage with SOC; transient effects of charge depletion and re-coveryandtheir dependenceonSOC; andthe variationin batterycapacity with discharge current. Furthermore, since it is an elec-tricalcircuitmodelitcaneasilybeincorporatedintotheelectricalsystem of the BEV model and is amenable to being representedusing math-based modeling techniques.Oneofthedownsidesofthis modelis thatnotemperatureef-fects of any kind are modeled, although Chen and Rin con-Morastate it would not be difficult to include them. In an electric ve-hicle the temperature will vary with external environmental con-ditions, with heating of the battery due to internal losses, andwith endo- and exothermic chemical reactions. The only modelwe encountered that explicitly included temperature dependencewas the mathematical model of Rong and Pedram 15, but theirmodel assumes a constant discharge current and thus is not suit-able for our BEV system.The Chen and Rin con-Mora model can also be overchargedand does not consider the increasing resistance of the battery asit nears a full charge. Furthermore the variations of the batterysstate-of-health (SOH) with time and charge cycles is not mod-eled. These downsides are acceptable given that in future mod-eling the vehicles control system will limit maximum batterycharge,andalthoughinthis paperwe are notinterested inmodel-ingtemperatureorstate-of-health,theyshouldnotbetoodifficultto incorporate.2.2Power ControllerThe next important component of an electric vehicle is apower converter that acts as an interface between the battery andthe drive motor/generator. This component controls the amountof power going to the motor during driving, and the amount ofpower going back into the battery during regenerative braking.Generally, boost or buck converters are used depending onwhether the output voltage is higher or lower, respectively, thanthe input voltage 20. By varying the duty cycle of a high-frequency switching circuit, the output voltage and thus currentand power can be controlled.Insteadof modelingthe highfrequencycircuit in MapleSim,we decided to use a simple approximation that can serve as botha boost or buck converter with power flowing from the batteryto the motor, or vice-versa. Figure 3 is a picture of the powercontrollerblock diagram. Althoughthe currentmodelhas a fixedconverter efficiency of 100%, a more realistic efficiency modelsuch as the one used by Hellgren 3 can be incorporated.FIGURE 3.BLOCK DIAGRAM OF POWER CONTROLLERMODELUsing a signal-driven current source in the output loop, theoutput voltage is measured and the output power is calculated.The input currentis adjusted by a PID controller so that the inputpower matches the output power. This circuit works both forpositive or negative current, which determines the direction ofpower flow. This model avoids the divide-by-zero problem of asimple algebraic power converter when the output voltage andcurrent goes to zero, and adapts to changing input and outputimpedance. However it does not take into consideration physicallimitations of componentssuch as the batterys maximumchargeor discharge rates, and voltage and current limits of the motor,wires, or power electronics.2.3Electrical MotorThe electrical motor used in the vehicle model is the Model-ica DC permanent magnet motor, which includes internal resis-tance, inductance, and rotor inertia 21.Its mechanical and electrical behaviour are modelled byEquations 1 and 2 where Jais the armature inertia,(t) is thearmaturerotationangle,Vnom, Inom, and fnomare the nominalmo-tor voltage, current, and rotational frequency, respectively.(t)is theshafttorque,andLaandRaare thearmatureinductanceand3Copyright c ? 2010 by ASMEDownloaded 25 Jun 2011 to 113.204.33.35. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmresistance, respectively. FinallyV(t), andI(t) arethe voltageandcurrent at the motor terminals, respectively.Ja(t)30(VnomRaInom)I(t)fnom(t) = 0(1)LaI(t)+RaI(t)V(t)+30(VnomRaInom)(t)fnom= 0(2)We chose to use the physical parameters of the LEM-200ModelD127DCpermanentmagnetmotorfromL.M.C.Ltd22.However we modified the rated current and voltage of the motorto be more compatible with our battery voltage. This would ef-fectively require re-winding the motor with different wire andchanging its magnets.The parameters used for the motor are presented in Table 1.Note that the peak current and power of the motor are twice therated value.TABLE 1.MOTOR MODEL PARAMETERSNameValueResistance0.0175 Inductance13HInertia0.0236 kgm2RPMrated3600 rpmVrated150 VIrated96 APrated12.56 kW2.4Vehicle DynamicsThe vehicle model we used was very simple.Its physi-cal parameters were based on the 2007 Toyota Camry hybrid.Since we were concerned only with the performance of the pow-ertrain components, we did not concern ourselves with vehiclesuspension or steering. We used a one-dimensional model of africtionless cart on an incline under the force of gravity. Thedrive motor is connected to one of the slipless wheels of the cartthrough a fixed transmission with a ratio of 9 motor revolutionsper wheel revolution. The wheels have the same diameter as theP215/60VR16.0 tires on the Camry.Equation 3 describes the relationship between the rotationand torqueof the motor shaft.(t) is the torqueseen at the motorshaft, m is the vehicles mass, R is the drive tire radius,is thegear ratio from the motor to the tire,(t) is the motor shaftsrotational displacement, g is the gravitational constant, and(t)is the terrain inclination angle.Table 2 lists the values used for these parameters.(t) =mR?Rd2dt2(t)+gsin(t)?(3)TABLE 2.VEHICLE MODEL PARAMETERSNameSymbolValuemassm1613 kgtire radiusR32.25 cmgear ratio9gravityg9.8 m/s2The only type of braking included in this model is regener-ative braking where the current to the motor is reversed and thebattery is charged with the kinetic energy of the vehicle. We didnot take into consideration recharge current limits of the battery.To this vehicle model we attached a simple terrain model.A time-dependent lookup table controlled the inclination of theterrainonwhich thevehicletraveled. This allowed us to simulatethe vehicles performance on flat and hilly terrain.Thedrivecycle is a time-dependentlookuptable of the vehi-cles desired speed. A PID controllercomparesthe desired speedto the actual speed and drives the input of the power controller totransfer power to the motor, or to extract power from the motoruntil the vehicles speed matches the desired speed.See Fig. 1 for the block diagram of the overall BEV model.2.5Numerical SimulationAfter MapleSim converts the vehicle model into differentialequations, it simplifies and reduces the system of equations sym-bolically. Then using this reduced equation set it solves themnumerically to produce the final output data.MapleSim simulated our system with its non-stiff solver,which uses a Fehlberg fourth-fifth order Runge-Kutta method4Copyright c ? 2010 by ASMEDownloaded 25 Jun 2011 to 113.204.33.35. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmwith degree four interpolant.We used an adaptive time-stepwith absolute and relative error tolerances of 1e-7, and turned onMapleSims native code generation ability which runs the sim-ulation faster. The model was simulated on a 3 GHz Intel Core2 Duo using MapleSim version 3 for Linux. It was set to simu-late over a 30 second time interval, and took 10 seconds of actualtime to complete.3Results0500010000150002000025000Time (s)33.23.43.63.844.2Voltage (V)ModeledActualFIGURE 4.MODELED-VS-ACTUAL 12 BATTERY UNDERPULSED CONSTANT-CURRENT DISCHARGEFigure 4 is a comparison between the MapleSim model andan actual cell for a pulsed current discharge of a single batterycell. The actual cell data was extracted from Fig. 5 of Chenand Rin con-Moras paper. Like the model in their paper, ourmodel does not include a self-discharge resistor. An initial SOCof 98% gives a close match to the experimental results, trackingthem very well until the battery capacity is almost exhausted.Our model requires one discharge cycle more than the actual tosee a rapid collapse in the battery terminal voltage.Using our vehicle model we performedtwo simple and intu-itive tests. Table 3 lists the parameters used in the drive cycles.3.1AccelerationsThe first test we did was to simulate the vehicle driving un-der hard and gentle accelerations on flat terrain. Battery and in-ternal combustion engine vehicles are more efficient if gentle ac-celeration is used compared to hard acceleration, due to internallosses. The initial accelerations of the hard and gentle cycles aredifferent,but the maximumspeed and rate of decelerationare thesame. See the hard and gentle curves of Fig. 5 for a plot of thedrive cycle speed with time.TABLE 3.DRIVE CYCLE AND TERRAIN MODEL PARAME-TERSNameValueVmax9 m/sahard1.607 m/s2agentle0.968 m/s2hill height8.67 mhill angle8Figure 6 plots the batterys state of charge versus time. Re-call that this model is without rollingresistance. One can see thatthe hard acceleration drive cycle ends up with a lower final stateof charge than the gentle cycle. This difference is due to ohmiclosses in the motor windings and chemical losses in the battery.3.2HillsThe second test we did was to drive the vehicle up and downa hill. The battery should lose energy going uphill as the vehi-cle gains gravitational potential energy, and gain energy goingdownhill as the vehicle loses potential energy. See the hill cyclecurve of Fig. 5 for a plot of the drive cycle speed with time. Theterrain cycle is very simple: at t=9.5 s the vehicle encounters thehill, then it drives up or down an 8incline before returning toflat terrain at t=20.5 s.Figure 7 plots the batterys state of charge versus time forthis test. In both cases the battery loses energy as it accelerates051015202530Time (s)0123456789Velocity (m/s)Hard & Hill cyclesGentle cycleFIGURE 5.DRIVE CYCLE: SPEED-VS-TIME FOR HARD, GEN-TLE, AND HILL CYCLES5Copyright c ? 2010 by ASMEDownloaded 25 Jun 2011 to 113.204.33.35. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmthe vehicle, transferring energy from the battery to the vehicleskinetic energy.In the uphill case the state of charge decreases. The drivecontrollerappliesmore powerto the motorto match the vehiclesspeed tothe desiredspeed, and thebatterys energyis put intothevehicles gravitational potential energy.In the downhill case the state of charge increases. The drivecontroller applies the regenerative “brakes” to keep the vehiclesspeed constant, and the vehicles gravitational potentialenergyistransferred to the battery.Finally the vehicle encounters a flat spot and uses regenera-tive braking to come to a halt, transferring the vehicles kineticenergy to the battery.051015202530Time (s)78.87979.279.479.679.880State of Charge (%)Gentle startHard startFIGURE 6.STATE OF CHARGE FOR HARD AND GENTLE AC-CELERATIONS ON FLAT TERRAIN051015202530Time (s)76777879808182State Of Charge (%)DownhillUphillFIGURE 7.S