基于降階動(dòng)態(tài)相量模型的電感耦合式勵(lì)磁系統(tǒng)間接勵(lì)磁電流估計(jì)

付興賀 夏宏偉 熊嘉鑫

基于降階動(dòng)態(tài)相量模型的電感耦合式勵(lì)磁系統(tǒng)間接勵(lì)磁電流估計(jì)

付興賀 夏宏偉 熊嘉鑫

（東南大學(xué)電氣工程學(xué)院 南京 210096）

電勵(lì)磁同步電機(jī)（EESM）具有稀土永磁材料依賴性低、勵(lì)磁磁場(chǎng)可控、調(diào)速范圍廣的優(yōu)點(diǎn)，在新能源汽車驅(qū)動(dòng)領(lǐng)域擁有良好的應(yīng)用前景。利用電感耦合式無(wú)刷勵(lì)磁技術(shù)可以有效降低傳統(tǒng)有刷EESM的摩擦損耗和維護(hù)成本。但是，無(wú)刷勵(lì)磁技術(shù)的引入導(dǎo)致勵(lì)磁電流無(wú)法直接測(cè)量，因此提出一種基于降階動(dòng)態(tài)相量模型的間接式勵(lì)磁電流估計(jì)方法。首先利用電感耦合關(guān)系將勵(lì)磁電流的直接估計(jì)轉(zhuǎn)換為間接估計(jì)，有效地提高了算法的魯棒性；然后推導(dǎo)出勵(lì)磁電流降階動(dòng)態(tài)相量估計(jì)模型，進(jìn)一步提高間接估計(jì)方法的精度；接著考慮電流諧波影響，提出變系數(shù)改進(jìn)方法；最后通過(guò)仿真和實(shí)驗(yàn)驗(yàn)證了估計(jì)方法的有效性。

電勵(lì)磁同步電機(jī) 電感耦合式無(wú)刷勵(lì)磁技術(shù) 電流估計(jì) 動(dòng)態(tài)相量模型

0 引言

電勵(lì)磁同步電機(jī)（Electrically Excited Synch-ronous Machine, EESM）具有稀土永磁材料依賴性低、勵(lì)磁調(diào)節(jié)靈活、調(diào)速范圍廣等優(yōu)點(diǎn)，在新能源汽車電驅(qū)動(dòng)領(lǐng)域展現(xiàn)出特有的技術(shù)和成本優(yōu)勢(shì)[1]。但傳統(tǒng)EESM的電刷集電環(huán)結(jié)構(gòu)會(huì)引起摩擦損耗、增加維護(hù)成本、降低電機(jī)工作可靠性[2]。因此，無(wú)刷勵(lì)磁技術(shù)已成為EESM應(yīng)用發(fā)展的迫切需求和亟待解決的關(guān)鍵問(wèn)題[3]。現(xiàn)有無(wú)刷勵(lì)磁方式主要包括勵(lì)磁機(jī)式、諧波勵(lì)磁及無(wú)線電能傳輸式[4-6]。無(wú)線電能傳輸勵(lì)磁方式又可以分為電感耦合式和電容耦合式。其中，電感耦合式勵(lì)磁方法結(jié)構(gòu)簡(jiǎn)單、傳輸效率高，具有廣闊的應(yīng)用前景[7]。

對(duì)于無(wú)刷勵(lì)磁系統(tǒng)，采用開(kāi)環(huán)方式控制勵(lì)磁電流難以滿足EESM高精度勵(lì)磁磁場(chǎng)調(diào)節(jié)和速度控制要求，必須依靠勵(lì)磁電流反饋構(gòu)建閉環(huán)控制系統(tǒng)才能提高勵(lì)磁電流的控制精度和抗干擾能力[8-9]。但無(wú)刷勵(lì)磁系統(tǒng)的勵(lì)磁繞組隨轉(zhuǎn)子一起旋轉(zhuǎn)，勵(lì)磁能量發(fā)送回路和接收回路之間無(wú)直接電氣連接，致使勵(lì)磁電流反饋的獲取面臨技術(shù)挑戰(zhàn)[10]。

無(wú)刷勵(lì)磁方案的差異和特點(diǎn)決定了勵(lì)磁電流獲取方法各有區(qū)別。在勵(lì)磁機(jī)方案中，可以根據(jù)勵(lì)磁機(jī)模型計(jì)算出勵(lì)磁機(jī)轉(zhuǎn)子的電壓、電流，再根據(jù)整流器模型計(jì)算勵(lì)磁電流估計(jì)值[11]。在諧波勵(lì)磁方案中，可以通過(guò)靜態(tài)實(shí)驗(yàn)建立離線表格，根據(jù)定子諧波繞組電流估計(jì)勵(lì)磁電流[12]。電容耦合式勵(lì)磁方案比較特殊，勵(lì)磁靜止側(cè)和旋轉(zhuǎn)側(cè)共用同一電流回路，經(jīng)過(guò)處理和折算便可直接獲得勵(lì)磁電流[13]。在電感耦合式勵(lì)磁系統(tǒng)中，獲取勵(lì)磁電流的方式包括：無(wú)線通信式、模型估計(jì)式。無(wú)線通信方式需要在二次回路增加采樣電路，利用無(wú)線通信裝置將采樣到的勵(lì)磁電流數(shù)據(jù)發(fā)送至一次回路[14]。該方法原理簡(jiǎn)單，但是需要額外增加電路裝置，且傳遞的反饋信號(hào)易受到電機(jī)內(nèi)磁場(chǎng)的干擾。模型估計(jì)方式包括基于電機(jī)繞組模型估計(jì)和基于線圈耦合電感模型估計(jì)。基于電機(jī)繞組模型的電流估計(jì)方式需要先建立電機(jī)繞組與勵(lì)磁繞組之間的磁鏈和電壓關(guān)系，再根據(jù)采集到的定子端電壓和電流計(jì)算出勵(lì)磁電流[15]；基于線圈耦合電感模型估計(jì)則需建立電感等效電路模型，利用一次電壓、電流信息計(jì)算出勵(lì)磁電流[16]。模型估計(jì)方式獲取勵(lì)磁電流對(duì)硬件要求較小，但是估計(jì)效果受數(shù)學(xué)模型精度及參數(shù)擾動(dòng)的影響。

反映勵(lì)磁系統(tǒng)電路中物理關(guān)系的數(shù)學(xué)模型主要包括穩(wěn)態(tài)模型和瞬態(tài)模型[17]。前者計(jì)算簡(jiǎn)單但是精度較低；后者精度高但計(jì)算復(fù)雜。除此之外，還有精度較高且計(jì)算相對(duì)簡(jiǎn)單的全階和降階動(dòng)態(tài)相量模型[18]。但目前上述模型多用于無(wú)線電能傳輸系統(tǒng)建模和參數(shù)優(yōu)化，鮮有研究將其應(yīng)用于無(wú)刷電勵(lì)磁電機(jī)的勵(lì)磁電流估計(jì)。

針對(duì)模型估計(jì)中存在的參數(shù)擾動(dòng)問(wèn)題，文獻(xiàn)[19]提出一種利用母線電流修正勵(lì)磁電流估計(jì)值的方法，在一臺(tái)最大勵(lì)磁電流為18 A的樣機(jī)上進(jìn)行驗(yàn)證，變負(fù)載工況下的估計(jì)誤差在2%以內(nèi)。該方法具有一定的估計(jì)精度，但是實(shí)現(xiàn)過(guò)程依賴大量實(shí)驗(yàn)結(jié)果，普適性低，且需要增加一個(gè)電流傳感器用于檢測(cè)母線電流。文獻(xiàn)[20]利用一次側(cè)LCL型諧振補(bǔ)償結(jié)構(gòu)的特點(diǎn)，提出一種適用于變負(fù)載工況的電流估計(jì)方法，并考慮了一次電流的諧波影響，在一臺(tái)額定勵(lì)磁電流為2 A的樣機(jī)上完成了實(shí)驗(yàn)驗(yàn)證，最終的電流估計(jì)誤差約為5.7%。該方法計(jì)算簡(jiǎn)單，但受限于一次側(cè)特定的補(bǔ)償形式，需要用到兩個(gè)電流傳感器，且并未考慮負(fù)載參數(shù)變化對(duì)電流估計(jì)的影響。

鑒于此，本文針對(duì)串聯(lián)-串聯(lián)補(bǔ)償型電感耦合式無(wú)刷勵(lì)磁系統(tǒng)，提出了一種基于系統(tǒng)降階動(dòng)態(tài)相量模型的間接勵(lì)磁電流估計(jì)方法，具有計(jì)算簡(jiǎn)單、負(fù)載適應(yīng)性強(qiáng)、硬件成本低等特點(diǎn)。本文首先設(shè)計(jì)了勵(lì)磁能量傳輸電路的拓?fù)浣Y(jié)構(gòu)，建立勵(lì)磁系統(tǒng)的等效電路模型；然后為避免負(fù)載參數(shù)擾動(dòng)的影響，利用電感耦合關(guān)系，選取二次側(cè)反射電壓作為中間變量，提出一種間接式電流估計(jì)方法，建立降階動(dòng)態(tài)相量估計(jì)模型，進(jìn)一步提高間接估計(jì)方法的估計(jì)精度；接著考慮二次電流的諧波影響，提出變波形系數(shù)改進(jìn)方法；最后通過(guò)仿真和實(shí)驗(yàn)驗(yàn)證了上述電流估計(jì)方法的有效性。

1 感應(yīng)耦合式勵(lì)磁系統(tǒng)等效電路模型

根據(jù)電勵(lì)磁電機(jī)勵(lì)磁傳輸功率的要求，建立圖1所示的串聯(lián)-串聯(lián)感應(yīng)耦合式無(wú)線電能傳輸系統(tǒng)的電路拓?fù)洹Ｔ摻Y(jié)構(gòu)高階非線性的特性不利于勵(lì)磁電流的在線估計(jì)。但無(wú)刷勵(lì)磁系統(tǒng)正常工作時(shí)處于諧振狀態(tài)，可以采用基波分析法對(duì)系統(tǒng)非線性環(huán)節(jié)進(jìn)行簡(jiǎn)化并建立等效電路模型[21]，在一定誤差范圍內(nèi)可以降低模型復(fù)雜度但又不失電流估計(jì)的有效性。

圖1 串聯(lián)-串聯(lián)補(bǔ)償型感應(yīng)耦合式無(wú)刷勵(lì)磁系統(tǒng)結(jié)構(gòu)

1.1 逆變器等效電路模型

當(dāng)母線電壓保持不變時(shí)，根據(jù)系統(tǒng)實(shí)際的移相角即可計(jì)算出逆變器輸出電壓的基波有效值。在后續(xù)分析中利用式（2）將非線性的逆變器環(huán)節(jié)等效為一個(gè)電壓源。

1.2 整流橋和負(fù)載等效電路模型

勵(lì)磁系統(tǒng)中作為負(fù)載的勵(lì)磁繞組端電壓滿足

圖2 整流橋和負(fù)載等效電路模型

1.3 系統(tǒng)一次側(cè)等效電路模型

圖3 無(wú)刷勵(lì)磁系統(tǒng)等效電路模型

圖4 一次側(cè)等效電路模型

Fig.4 Equivalent model of wireless power transfer systems

利用一次側(cè)等效模型開(kāi)展勵(lì)磁電流間接估計(jì)擺脫了對(duì)負(fù)載參數(shù)的依賴，提高了估計(jì)方法的魯棒性，同時(shí)降低了等效電路的儲(chǔ)能元件數(shù)量和模型階數(shù)。

2 降階動(dòng)態(tài)相量估計(jì)模型

對(duì)于含諧振環(huán)節(jié)的系統(tǒng)，可以采用動(dòng)態(tài)相量法進(jìn)行數(shù)學(xué)建模，計(jì)算復(fù)雜度低于瞬態(tài)模型并且精度高于穩(wěn)態(tài)模型，還可以進(jìn)行降階處理簡(jiǎn)化計(jì)算。本節(jié)將推導(dǎo)基于降階動(dòng)態(tài)相量模型的勵(lì)磁電流估計(jì)數(shù)學(xué)表達(dá)式。

2.1 一次側(cè)等效電路降階動(dòng)態(tài)相量模型

動(dòng)態(tài)相量模型的每個(gè)狀態(tài)量對(duì)應(yīng)兩個(gè)微分方程，分別為實(shí)部方程和虛部方程，因此系統(tǒng)一次側(cè)等效電路模型對(duì)應(yīng)的全階動(dòng)態(tài)相量模型為4階。為了方便計(jì)算，可以利用諧振網(wǎng)絡(luò)的特點(diǎn)對(duì)全階動(dòng)態(tài)相量方程進(jìn)行降階處理。根據(jù)式（7）寫(xiě)出原始的諧振網(wǎng)絡(luò)動(dòng)態(tài)相量模型在域下的表達(dá)式為

利用式（9）即可將諧振網(wǎng)絡(luò)的動(dòng)態(tài)相量模型的階數(shù)由二階降為一階，如圖5所示。

降階動(dòng)態(tài)相量模型精度高于穩(wěn)態(tài)模型，計(jì)算復(fù)雜度遠(yuǎn)低于瞬態(tài)時(shí)域模型。就模型本身精度而言，全階動(dòng)態(tài)相量模型略微高于降階動(dòng)態(tài)模型，但降階動(dòng)態(tài)模型方程階數(shù)更低，計(jì)算簡(jiǎn)單，更適用于在線計(jì)算[18]。因此本文基于降階動(dòng)態(tài)相量模型來(lái)描述勵(lì)磁系統(tǒng)一次側(cè)等效電路，并在此基礎(chǔ)上推導(dǎo)出勵(lì)磁電流估計(jì)表達(dá)式。

一次側(cè)等效電路模型對(duì)應(yīng)的微分方程為

將方程中的實(shí)部和虛部進(jìn)行分離，得

2.2 擾動(dòng)觀測(cè)器設(shè)計(jì)

式中

對(duì)式（16）兩邊進(jìn)行求導(dǎo)并化簡(jiǎn)可得

圖6 擾動(dòng)觀測(cè)器結(jié)構(gòu)框圖

2.3 勵(lì)磁電流估計(jì)值表達(dá)式

在動(dòng)態(tài)相量模型中，擾動(dòng)項(xiàng)二次側(cè)反射電壓可以表示為

整理可得

結(jié)合擾動(dòng)觀測(cè)器輸出結(jié)果及式（4），可以寫(xiě)出基于降階動(dòng)態(tài)相量估計(jì)模型的勵(lì)磁電流表達(dá)式為

3 對(duì)比分析及改進(jìn)

3.1 穩(wěn)態(tài)估計(jì)模型和降階動(dòng)態(tài)相量估計(jì)模型對(duì)比

為了更直觀地表現(xiàn)基于降階動(dòng)態(tài)相量間接勵(lì)磁電流估計(jì)方法的效果，給出基于穩(wěn)態(tài)模型的勵(lì)磁電流估計(jì)式用于對(duì)比分析。

結(jié)合一次側(cè)等效電路模型中各元件的穩(wěn)態(tài)模型和基爾霍夫電壓方程，推導(dǎo)出基于穩(wěn)態(tài)模型的勵(lì)磁電流估計(jì)表達(dá)式為

式中，1rms為一次電流有效值。

表1 無(wú)刷勵(lì)磁系統(tǒng)參數(shù)

Tab.1 Parameters of wireless excitation system

圖7 時(shí)兩種估計(jì)模型結(jié)果對(duì)比

在不同溫度下，勵(lì)磁繞組電阻可由式（23）計(jì)算。

上述結(jié)果表明，本文所提的間接式估計(jì)方法是可行的，且相較于穩(wěn)態(tài)估計(jì)模型，基于降階動(dòng)態(tài)相量模型的估計(jì)方法具有更高的估計(jì)精度和魯棒性。

圖8 時(shí)兩種估計(jì)模型結(jié)果對(duì)比

3.2 考慮諧波影響的變波形系數(shù)估計(jì)方法

圖9 Rf =15 Ω時(shí)二次電流有效值及勵(lì)磁電流估計(jì)結(jié)果

圖10 Rf =20 Ω時(shí)二次電流有效值及勵(lì)磁電流估計(jì)結(jié)果

式（4）中基于基波假設(shè)的始終為0.9。但實(shí)際上受二次電流諧波影響，波形系數(shù)是一個(gè)與和相關(guān)的時(shí)變系數(shù)。不同工況下仿真計(jì)算得到的隨和變化曲線如圖11所示。

圖12 變系數(shù)勵(lì)磁電流估計(jì)結(jié)果

Fig.12 Results of variable coefficient estimator

圖13 時(shí)變系數(shù)勵(lì)磁電流估計(jì)結(jié)果

Fig.13 Variable coefficient estimation results when

綜上所述，基于降階動(dòng)態(tài)相量模型的估計(jì)方法精度顯著高于穩(wěn)態(tài)模型估計(jì)方法，并且采用考慮諧波影響的變波形系數(shù)改進(jìn)估計(jì)方法可以進(jìn)一步提高電流估計(jì)的精度和魯棒性。

圖14 時(shí)變系數(shù)勵(lì)磁電流估計(jì)結(jié)果

4 實(shí)驗(yàn)驗(yàn)證

本文搭建的實(shí)驗(yàn)平臺(tái)如圖15所示。

圖15 實(shí)驗(yàn)平臺(tái)

為了獲取勵(lì)磁電流真實(shí)值來(lái)驗(yàn)證電流估計(jì)效果，對(duì)電機(jī)轉(zhuǎn)子進(jìn)行改造，利用電刷集電環(huán)將勵(lì)磁繞組引出，如圖16所示。

圖16 勵(lì)磁電流測(cè)量原理

圖17 時(shí)的實(shí)驗(yàn)波形

圖18 時(shí)的實(shí)驗(yàn)波形

圖19 的勵(lì)磁電流跟蹤曲線

圖20 的勵(lì)磁電流跟蹤曲線

為了使勵(lì)磁電流估計(jì)結(jié)果更加平穩(wěn)，增加了軟件濾波環(huán)節(jié)。圖19和圖20中曲線左側(cè)數(shù)值為勵(lì)磁電流估計(jì)值的平均值，右側(cè)數(shù)值為勵(lì)磁電流的平均值。結(jié)果表明，所提電流估計(jì)方法具有較高的精度和魯棒性，給定工況下最大相對(duì)估計(jì)誤差約為4.7%。

綜上所述，實(shí)驗(yàn)結(jié)果與仿真結(jié)果基本一致，變波形系數(shù)改進(jìn)方法的假設(shè)和應(yīng)用都是合理的，所提的電流估計(jì)方法具有較高的精度和魯棒性。

5 結(jié)論

針對(duì)串聯(lián)-串聯(lián)補(bǔ)償型電感耦合式無(wú)刷勵(lì)磁系統(tǒng)，本文提出了一種基于降階動(dòng)態(tài)相量模型的勵(lì)磁電流間接估計(jì)方法。該方法建立了勵(lì)磁系統(tǒng)的一次側(cè)等效電路模型，對(duì)勵(lì)磁電流進(jìn)行間接估計(jì)，提高了估計(jì)方法的魯棒性；建立降階動(dòng)態(tài)相量估計(jì)模型并考慮二次電流的諧波影響，提出變系數(shù)改進(jìn)方法，進(jìn)而獲得更高的電流估計(jì)精度。

本文所提的勵(lì)磁電流估計(jì)方法在串聯(lián)-串聯(lián)電感耦合勵(lì)磁系統(tǒng)中具有較好的效果，但對(duì)整流模型中的波形系數(shù)的細(xì)致分析和準(zhǔn)確獲取還有待進(jìn)一步的研究。此外，利用間接估計(jì)思想及動(dòng)態(tài)相量模型，可以實(shí)現(xiàn)對(duì)更多非串聯(lián)-串聯(lián)型拓?fù)涞碾姼旭詈鲜絼?lì)磁系統(tǒng)高精度、高魯棒性的電流估計(jì)，拓寬所提電流估計(jì)方法的應(yīng)用范圍。

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Indirect Field Current Estimation Algorithm for Inductively Coupled Excitation Systems Based on Reduced-Order Dynamic Phasor Model

Fu Xinghe Xia Hongwei Xiong Jiaxin

（School of Electrical Engineering Southeast University Nanjing 210096 China）

Electrically excited synchronous machines (EESM) has the advantages of low dependence on rare earth permanent magnet materials, controllable excitation field and wide speed regulation range, and has a good application prospect in electric vehicles. However, the traditional EESM's brush-slip ring structure caused friction loss, increased maintenance costs, and reduced reliability. Therefore, brushless excitation has become an urgent requirement and a critical issue to be solved for EESM applications. Inductively coupled brushless excitation technology can effectively reduce friction losses and maintenance costs. Currently, mainstream brushless excitation methods include exciter type, harmonic excitation type, and wireless power transfer type. Wireless power transfer excitation can be divided into inductive coupling and capacitive coupling types. Inductive coupling excitation has a simple structure and high transmission efficiency, making it promising for applications.

However, the usage of brushless excitation technology will bring a new challenges. The excitation winding of brushless excitation system rotates with the rotor, and there is no direct electrical connection between the transmitting circuit and the receiving circuit, resulting in the acquisition of excitation current value facing technical challenges. To estimate the field current in similar scenarios has been the scope of some previous studies. The existing current estimation methods can achieve good results in their respective application fields, but there are some limitations and shortcomings, which need to be further developed.

In view of this, an indirect excitation current estimation method based on reduced order dynamic phasor model is proposed for series-series compensation inductively coupled brushless excitation system, which has the characteristics of simple calculation, strong load adaptability and low hardware cost. The topology structure of excitation energy transmission circuit is designed. The equivalent circuit model of excitation system is established. In order to avoid the influence of load parameter disturbance, an indirect current estimation method is proposed by using the inductive coupling relation and the secondary side reflection voltage as the intermediate variable. A reduced order dynamic phasor estimation model is established to further improve the estimation accuracy of the indirect estimation method. Considering the harmonic effect of subside current, an improved method of variable waveform coefficient is proposed. Finally, the validity of the current estimation method is verified by simulation and experiment.

The proposed excitation current estimation method has a good effect in series-series inductively coupled excitation system. And only one current sensor is required, resulting in low hardware cost. However, the detailed analysis and accurate acquisition of waveform coefficients in the rectification model need further research. In addition, the indirect estimation idea and the dynamic phasor model can be used to estimate the current of more inductively coupled excitation systems with non-series-series topology with high accuracy and high robustness, and broaden the application range of the proposed current estimation method.

Electrically excited synchronous machines, inductively coupled brushless excitation technology, current estimation algorithm, dynamic phasor model

10.19595/j.cnki.1000-6753.tces.230946

TM341

國(guó)家自然科學(xué)基金資助項(xiàng)目（51977035）。

2023-06-23

2023-08-06

付興賀 男，1978年生，博士，副教授，研究方向?yàn)楦邷靥胤N電機(jī)及其控制、伺服系統(tǒng)多源異構(gòu)擾動(dòng)抑制。E-mail：fuxinghe@seu.edu.cn （通信作者）

夏宏偉 男，1998年生，碩士研究生，研究方向?yàn)殡姍C(jī)控制。E-mail：220213084@seu.edu.cn

（編輯 赫蕾）