SLIDING-MODE DIRECT FIELD-ORIENTED CONTROL OF SIX-PHASE INDUCTION MOTOR

Transcription

1 TECHNICAL TRANSACTIONS ELECTRICAL ENGINEERING CZASOPISMO TECHNICZNE ELEKTROTECHNIKA DOI: / XCT E/2016 JACEK LISTWAN, KRZYSZTOF PIEŃKOWSKI SLIDING-MODE DIRECT FIELD-ORIENTED CONTROL OF SIX-PHASE INDUCTION MOTOR ŚLIZGOWE BEZPOŚREDNIE STEROWANIE POLOWO-ZORIENTOWANE SZEŚCIOFAZOWYM SILNIKIEM INDUKCYJNYM Abstact The pape pesents the combine contol of a six-phase squiel-cage inuction moto with the application of Sliing-Moe Contol an Diect Fiel-Oiente Contol (SMC-DFOC). The mathematical moel of the six-phase squiel-cage inuction moto has been pesente. The esign poceue of the contol stuctue of SMC-DFOC has been escibe. Simulation stuies on the consiee contol stem have been caie out an the esults of the simulation stuies have been pesente an iscusse. Keywos: six-phase inuction moto, Diect Fiel-Oiente Contol, Sliing-Moe Contol, simulation stuies Steszczenie W atykule pzestawiono metoę steowania sześciofazowym silnikiem inukcyjnym klatkowym z zastosowaniem steowania ślizgowego i bezpośeniego steowania polowo-zoientowanego. Omówiono moel matematyczny sześciofazowego silnika inukcyjnego. Opisano poces pojektowania ukłau steowania ślizgowego i bezpośeniego steowania polowo-zoientowanego. Pzepowazono baania mulacyjne ozpatywanej stuktuy steowania, pzestawiono i omówiono wybane wyniki baań mulacyjnych. Słowa kluczowe: sześciofazowy silnik inukcyjny, bezpośenie steowanie polowo-zoientowane, steowanie ślizgowe, baania mulacyjne M.Sc. Eng. Jacek Listwan, D.Sc. Ph.D. Eng. Kzysztof Pieńkowski, Depatment of Electical Machines, Dives an Measuements, Faculty of Electical Engineeing, Woclaw Univesity of Science an Technology.

2 96 1. Intouction In ecent yeas, thee has been a geat inteest in using multi-phase squiel-cage inuction motos (motos with the numbe of stato phases geate than thee). It is cause by the evelopment of powe electonics fequency convetes that can be opeate as multi-phase stems of enegy, powee by the inustial thee-phase gi. The main avantages of multiphase inuction motos ae: the euction of toque pulsations an the euction of losses cause by highe hamonics of moto cuents [5, 6, 8, 9]. Multi-phase motos have geat eliability because they can be conitionally opeate at failue of one o moe phases of the stato wining. The ate stato phase cuents of multi-phase inuction motos ae smalle than the ate stato phase cuents of thee-phase inuction motos of the same powe [5, 6, 8, 9]. In the liteatue, thee ae some papes about the constuctions an contol methos of the multi-phase inuction motos [1 6, 8, 9]. Most stuies ae base on the consieations of Fiel-Oiente Contol o Diect Toque Contol [5, 9]. Howeve, a small numbe of publications constitutes the poblems of using the Sliing-Moe Contol fo the multi-phase inuction motos. In this pape, the novel combine contol metho of a six-phase squiel-cage inuction moto with application of Sliing-Moe Contol an Diect Fiel-Oiente Contol has been pesente. It is expecte that the consieation of the sliing-moe contol technique fo the contol stems of a multi-phase inuction moto allows to achieve obust contol of the moto state vaiables, fast ynamic esponses an ea hawae implementation [1, 4, 6, 7, 10, 11]. It is known that the chatteing of contolle vaiables is the main isavantage of sliing-moe contol. In the liteatue, thee ae some aticles about the methos of elimination of these negative effects [1, 7, 10, 11]. The pape is ivie into six sections. Section 1 povies an intouction. Section 2 is eicate to the mathematical moel of the six-phase squiel-cage inuction moto. In Section 3, the intouction to the Sliing-Moe Contol is pesente. Section 4 eals with the esign metho of Sliing-Moe Diect Fiel-Oiente Contol. Simulation stuies ae pesente an iscusse in Section 5. Section 6 is eicate to the concluing emaks. 2. Mathematical moel of six-phase squiel-cage inuction moto A mathematical moel of six-phase squiel-cage inuction moto has been fomulate on the basis of commonly use simplifying assumptions [5, 6, 8, 9]: stato an oto phase winings ae consiee as concentate; the mmety of the phase moto winings an the lineaity of moto magnetic cicuits ae assume; the ey cuents an ion losses ae neglecte; the electomagnetic vaiables an paametes of oto wining have been tansfome to the stato wining sie. In this pape, the six-phase squiel-cage inuction moto with the axes of stato winings shifte electically by sixty egees is consiee. The sta connection of six-phase stato wining with single isolate neutal point an shotcicuite cicuits of six-phase oto wining ae assume. The initial mathematical moel of the six-phase squiel-cage inuction moto is fomulate fo moto state vaiables consiee in the stato an oto phase cooinate

3 97 stem. This moel has the fom of a set of iffeential equations with vaiable coefficients, which ae changing as a function of the oto angle position. By the use of appopiate tansfomation matices, which ae pesente in [5, 9], the stato an oto state vaiables ae tansfome to the state vaiables pesente in the common ectangula cooinate stem an the state vaiables pesente in some othe aitional cooinate stems. This way, the simple fom of mathematical moel of six-phase inuction moto, escibe by the iffeential equations with constant coefficients, is obtaine. The final equations of the tansfome mathematical moel of the six-phase inuction moto take the following fom [5, 9]: the stato an oto voltage equations, expesse in the geneal ectangula cooinate stem x-y, which otates elative to the stato at abitay angula electical spee ω k : u R i ω +, u Rsi + ω k sx + (1) t t sx s sx k sx 0 Ri x ( ωk ωe) y + x, 0 Ri y + ( ωk ωe) x + y (2) t t the stato voltage equations expesse in the aitional cooinate stem z1-z2: u R i +, u R i t, sz2 s sz2 + sz2 (3) t sz1 s sz1 sz1 the equations fo stato an oto flux linkages: +, Li s + Li m y (4) sx Li s sx Li m x +, y Li y + Li m (5) x Li x Li m sx the equation of the moto electomagnetic toque: the mechanical motion equation: whee: T n 6 p ( i i ) p ( i i ) (6) 2 2 e b y sx x b y sx x T e Jm Tm ω e (7) p t b u sx, u, i sx, i, i x, i y components of the stato voltage vecto, stato cuent vecto an oto cuent vecto in the x-y cooinate stem, espectively, u sz1, u, i, i components of the stato voltage vecto an stato cuent sz2 sz1 sz2 vecto efine in the aitional z1-z2 cooinate stem, espectively,

4 98 sx,, x, y components of the stato an oto flux linkage vecto in the x-y cooinate stem, sz1, components of the stato flux linkage vecto in the aitional z1-z2 sz2 cooinate stem, ω k abitay angula spee of the cooinate stem elative to the stato, ω e the electical angula spee of the moto, T e the moto electomagnetic toque, T m the loa toque, J m the inetia of the ive stem, R s, R stato an oto phase esistance, espectively, L ls, L l the leakage inuctance of stato an oto, espectively, L m the moto magnetizing inuctance, L s L ls + L m, L L l + L m the total inuctance of stato an oto, espectively, p b the numbe of moto pole pais, n the numbe of stato phases. In the mathematical moel of the six-phase inuction moto pesente above, the equations fo stato an oto zeo components an the equations fo oto components consiee in the aitional cooinate stem have been omitte, because une the afoementione assumptions, these state vaiables ae always equal to zeo. On the basis of the toque equation (6), it can be state that, in the multi-phase inuction moto, only the state vaiables etemine in the geneal cooinate stem x-y ae involve in the geneation of the electomagnetic toque. The othe state vaiables epesente in the aitional stems (labele with the inex z) o not contibute to the toque pouction. Howeve, these vaiables shoul be consiee in the exact analysis because the values of these vaiables may be iffeent fom zeo an can influence upon the esultant stato phase cuents an losses. 3. Intouction to the Sliing-Moe Contol Sliing moe contol is a nonlinea contol technique, featuing emakable popeties of accuacy, obustness, an ea tuning an implementation [10]. Nowaays, this type of contol is wiely use fo the contol of electomechanical stems an powe electonic convetes [7, 10, 11]. On this basis, the fist stuies have been taken on the use of the sliing moe contol to contol the multi-phase inuction motos [1, 4, 6]. In this pape, the impove sliing moe contol with the application of equivalent contol has been stuie an iscusse. Sliing moe contol stems ae esigne to ive the stem states onto a paticula suface name the sliing suface. Once the sliing suface is eache, sliing moe contol keeps the states on the sliing suface. Hence, the sliing moe contol is a two-pat contolle esign. The fist pat involves the esign of a sliing suface; the secon one is concene with the selection of a contol law that will make the switching suface attactive to the stem state [10].

5 99 In the aticle, the following fom of switching functions s(t) has been use [1, 6, 7, 10]: s () t x () t x (), t (8) i i i whee: x i the efeence value of contolle vaiable, x i the eal value of contolle vaiable, I 1,, m the numbe of the contolle vaiables. The contol laws applie in the contolles must meet the equiements fo the existence an eachability of the sliing moe. In the sliing-moe contol, the following conition has been inclue [1, 7, 10, 11]: st () t st () <0 (9) In the applie metho of the equivalent sliing-moe contol, the contol signal u is consiee as the sum of two components: continuous component u eq an iscontinuous component u n [1, 7, 11]: u u + u (10) The continuous component u eq is consiee as the signal of the equivalent sliing-moe contol. The continuous component u eq is etemine on the basis of the mathematical moel of the contol stem assuming that all uncetainties ae equal to zeo. The nonlinea contol stem of six-phase inuction moto can be escibe by the following geneal equation stem [1, 11]: x t eq n f() t + gt () ut () (11) whee: x the vecto, which epesents the state vaiables of the stem, f(t), g(t) vecto functions, u(t) the vecto of contol signals. The continuous component u eq of the contol signal of the sliing-moe contolle can be etemine as the solution of the following equation [1, 11]: An hence: s [ f() t + gt () ut ()] 0 (12) t s u t gt s t f t eq () () (13) In oe to account fo eal nonzeo uncetainties, the iscontinuous contol signal is intouce. The iscontinuous pat u n of the signal of sliing-moe contol has been esigne as the tem efine by the following equation [1, 10]: 1

6 100 u k sgn st ( ) (14) n n whee k n > 0 is the element of vecto of the gain coefficients of the iscontinuous pat of the sliing-moe contolle [1, 10]. 4. Sliing-Moe Diect Fiel-Oiente Contol In this pape, the metho of Diect Fiel-Oiente Contol (DFOC) with equivalent sliing-moe contolles has been applie fo the contol of the six-phase inuction moto. The main goal of a fiel-oiente contol is obtaining ecuple contol of the electomagnetic toque an oto flux of the inuction moto. The oto fiel-oiente contol is base on the mathematical moel of the six-phase inuction moto fomulate in the ectangula cooinate stem x-y, which otates at angula spee ω equal to the angula spee of the oto flux vecto: ω k Li m L ω + ωe, τ (15) τ R Fom the conitions fo the oto fiel-oiente contol, it is equie that the component x of the oto flux vecto is equal to the magnitue of the oto flux vecto an the component y of the oto flux vecto is equal to zeo [1, 8]: x, y 0 (16) In the conventional Diect Fiel-Oiente Contol (DFOC) methos, the PI contolles have been geneally use. In the impove contol stuctue pesente in this pape, the PI contolles have been eplace by the sliing-moe contolles. The evelope scheme of DFOC with sliing-moe contolles is pesente in Fig. 1. The estimato of the instantaneous magnitue of the oto flux vecto an the instantaneous angle of the oto flux vecto must be use in this metho. The oto-flux moel base on the measue stato cuents an moto angula spee is use in the estimation block. In the Sliing-Moe DFOC stem, two oute contol loops with Sliing-Moe Contolles (SMC) have been applie: the contol loop fo the moto angula spee ω e an the contol loop fo the magnitue of the oto flux vecto. The Sliing-Moe Contolle of the moto angula spee etemines the efeence component i of stato cuent vecto, which is esponsible fo the contol of moto electomagnetic toque. The Sliing-Moe Contolle of the magnitue of the oto flux vecto etemines the efeence component i sx of the stato cuent vecto, which is esponsible fo the oto flux contol. In the pesente contol stem, two inne contol loops have been also applie: the contol loop fo the component i sx of the stato cuent vecto an the contol loop fo the component i of the stato cuent vecto. The output signals fom the Sliing-Moe Contolles of the components i sx an i of the stato cuent vecto ae the efeence values of the components u sx an u of the stato voltage vecto. These efeence

7 101 values ae then tansfome to the stato phase cooinate stem an ae sent to PWM moulato, which etemines the switching states of the six-phase Voltage Souce Invete. Detaile esciptions of the esign an nthesis of the SMC contolles use in the DFOC stuctue ae pesente in the next sections Equivalent sliing-moe contolle of moto angula spee The equation of the moto electomagnetic toque (6) afte taking into account the conition fo the oto flux oientation [1, 8] can be epesente in the following fom: T 3p i (17) e b The mechanical motion equation (7) can be witten in the following fom: ω t ( ) p T T J e b e m m ( ) p 3p i T b b m J m (18) The switching function an its time eivative fo SMC contolle of moto angula spee can be epesente in the following foms: sω ωe ωe t s ω ωe ω t t e (19) (20) The equation (20) afte substitution of the equation (18) takes the following fom: t s 3pi 2 b pt b m ω ωe (21) t J On the basis of the equation (21), the contol signal i can be etemine as the sum of two appopiate components: continuous component i eq an iscontinuous component i n which ae establishe by the sepaate contol blocks. The contol law of sliing-moe contolle of moto angula spee can be pesente in the fom: i Jm ωe + pt b m t 3p 2 b ieq m + kn sgn ( sω ) whee: k n > 0 the gain of the iscontinuous pat of the sliing-moe spee contolle. The sliing-moe contolle of moto angula spee has been inclue in Fig. 1 as the block make as Spee contolle. in (22)

8 Equivalent sliing-moe contolle of the magnitue of the oto flux vecto Afte appopiate manipulations of the moto moel equations, the magnitue of the oto flux vecto can be pesente in the fom: The component i x of the oto cuent vecto is etemine as: t Ri (23) x i x Li The equation (23) afte substitution of the equation (24) takes the fom: L m sx (24) t Li m sx τ (25) The switching function an its time eivative fo sliing-moe contolle of the magnitue of the oto flux vecto can be epesente in the following foms: s (26) t s t t The equation (27) afte substitution of the equation (25) has the fom: (27) t s + t Li τ m sx (28) On the basis of the equation (28), the contol signal i sx can be etemine as the sum of two components: continuous component i sxeq an iscontinuous component i sxn which ae establishe by the sepaate contol blocks. Afte some manipulations, the following fom is obtaine: i τ + + ksxn sgn ( s ) Lm t Lm (29) sx i sxeq whee: k sxn >0 the gain of the iscontinuous pat of the contolle of magnitue of the oto flux vecto. isxn

9 103 The sliing-moe contolle of the magnitue of the oto flux vecto is inclue in the Fig. 1 as the block make as Flux contolle Equivalent sliing-moe contolle of component i sx of stato cuent vecto With taking into account that fo the oto fiel-oiente contol the y 0, the i y component of the oto cuent vecto can be expesse in the fom: i y Li m L (30) The equations of the components of stato flux vecto afte some manipulations can be pesente in the following foms: Li + (31) L Li + ls sx ls sx ( ) ω ω. (32) R l e The switching function an its time eivative fo the contolle of the component i sx of stato cuent vecto ae efine as: s i i ix sx sx (33) t s t i t i (34) ix sx sx The equation (1) of the component u sx of the stato voltage vecto afte substitution of the equations (31) an (32) has the fom: Ll ( e) u R i L i L R t i sx s sx ω ω ω ls + ls sx + (35) Equation (35) afte some algebaic opeations can be pesente in the fom: t i sx L usx Rsisx + ω Llsi + L ls ( ω ω ) R l e (36) Equation (34) afte substitution of the equation (36) takes the following fom: t s ix ( ) Ll ω ωe usx Rsisx + ω Llsi + R t i sx (37) L ls

10 104 The output signal fom the sliing-moe contolle of the component i sx of stato cuent vecto can be consiee as the sum of two components: continuous component u sxeq an iscontinuous component u sxn, which ae etemine by the sepaate contol blocks: u L t i R i L i Ll ω ωe sx ls sx + s sx ω ls + R u sx eq ( ) + ( ) ksuxn sgn six usxn (38) whee: k suxn > 0 the gain of the iscontinuous pat of the contolle of component i sx of stato cuent vecto. The contol pat esponsible fo equivalent sliing-moe contol of the i sx component of the stato cuent vecto is inclue in the Figue 1 as i sx contolle Equivalent sliing-moe contolle of i component of stato cuent vecto The switching function an its time eivative fo contolle of the component i of stato cuent vecto ae efine as: s i i iy (39) t s t i t i iy The equation of the component u of stato voltage vecto (1) afte substitution of the equations (31) an (32) an with using of the conitions equie fo the fiel-oiente contol has the fom: u R i L i t Li s + ω ( ls sx + )+ ls (41) Equation (41) afte some algebaic opeations can be epesente in the following fom: (40) t i u R i ω L i + L s ls sx ls ( ) (42) Equation (40) afte substitution of the equation (42) has the following fom: t s iy t i u R i ω L i + L s ls sx ls ( ) The output signal fom the sliing-moe contolle of the component i of stato cuent vecto can be consiee as the sum of two components: continuous component u eq an iscontinuous component u n which ae etemine by the sepaate contol blocks: (43)

11 105 u L t i R i L i ls + s + ω ( ls sx + )+ ksuyn sgn siy (44) u eq u n ( ) whee: k suyn > 0 the gain of the iscontinuous pat of contolle of i component of stato cuent vecto. The pat esponsible fo equivalent sliing-moe contol of the i component of the stato cuent vecto is inclue in the Fig. 1 as i contolle The scheme of the Sliing-Moe Diect Fiel-Oiente Contol The stem of the Sliing-Moe Diect Fiel-Oiente Contol of six-phase squielcage inuction moto with application of the nthesis poceues pesente above has been shown in Fig. 1. Fig. 1. Sliing-Moe DFOC of six-phase squiel-cage inuction moto The appopiate tansfomation blocks of the moto stato cuents have been applie in the Sliing Moe Diect Fiel-Oiente Contol stem. The values of the stato cuents tansfome fom the phase cooinate stem to the x-y an α-β cooinate stems ae use in the contol stem an in the estimato of the oto flux vecto. Six-phase inuction moto is supplie by the six-phase two level Voltage Souce Invete.

12 Simulation esults Simulation stuies have been caie out in oe to veify the pesente metho of Sliing-Moe Contol. The simulation moel of the Sliing-Moe Diect Fiel-Oiente Contol of six-phase inuction moto has been implemente in Matlab-Simulink Softwae. The simulation stuies wee caie out fo six-phase squiel-cage inuction moto with the paametes: P N 3 kw, U fn 230 V, f N 50 Hz, n N 2950 pm, p b 2, R s 1.9 Ω, R 2.1 Ω, L ls L l H, L m 0.6 H. The nominal values wee make by the inex N. Figue 2 pesents the wavefoms of efeence an calculate moto spee. It is assume that, at fist, the efeence spee is ising to the half of the nominal spee, then fo a cetain peio of time is constant, an then is ising again to the nominal spee. Duing the final tajectoy, the efeence spee is eceasing with the evesion of the nominal spee. Duing the peio of opeation with the nominal spee, the step change of the loa toque has been foce. It can be state that the calculate spee follows the efeence spee with geat accuacy. Fig. 2. The wavefoms of spees of six-phase inuction moto fo Sliing-Moe DFOC Figue 3 pesents the wavefoms of the electomagnetic toque of the six-phase inuction moto an the loa toque. The obtaine simulation esults confim the contol of the electomagnetic toque with geat accuacy an quick eaction of the Sliing-Moe DFOC fo the changes of the loa toque. Fig. 3. The wavefoms of the electomagnetic toque of six-phase inuction moto an the loa toque fo Sliing-Moe DFOC Figue 4 pesents the stato phase cuent of the six-phase inuction moto fo the Sliing- Moe DFOC. The values of the stato cuent amplitue ae geat in ynamic states an ae fixe at the steay-state opeation. The simulation stuies showe that fo the consiee

13 107 contol of the components of the stato cuent in the aitional cooinate stem z1-z2 have small amplitues an they have no negative impact on the value of the phase cuents. Figue 5a pesents the output signal i fom Sliing-Moe Contolle of moto angula spee an Fig. 5b pesents the output signal i sx fom Sliing-Moe Contolle of magnitue of the oto flux vecto. In the pesente examples of contol signals, the continuous components an iscontinuous contol components ae visible. Fig. 4. The stato phase cuent of six-phase inuction moto fo Sliing-Moe DFOC a) b) Fig. 5. Simulation esults: a) the output signal i fom Sliing-Moe Contolle of moto angula spee an b) the output signal i sx fom Sliing-Moe Contolle of magnitue of the oto flux vecto Figue 6 pesents the tajectoy of the oto flux vecto of the six-phase inuction moto fo Sliing-Moe DFOC. It can be state that the magnitue of the oto flux vecto is egulate at the efeence value without oveshoot an have ate value at all opeational times. Fig. 6. The tajectoy of the oto flux vecto of six-phase inuction moto fo Sliing-Moe DFOC

14 Conclusions In this pape, the analysis an the esign poceue of the novel stuctue of Sliing- Moe Diect Fiel-Oiente Contol of six-phase inuction moto have been pesente. The conucte simulation stuies of the ive stem with six-phase inuction moto allow to aw a conclusion that the consiee SMC-DFOC allows to obtain the contol of the moto electomagnetic vaiables with geat accuacy. Base on the analysis of the simulation esults, it can be state that the eal moto spee follows the efeence moto spee with geat accuacy. The values of moto electomagnetic toque an the amplitues of the stato cuents epen on the conition of the ive stem. The magnitue of the oto flux vecto is egulate at the efeence nominal value. Duing all teste states of the ive stem, the oveshoots an oscillations o not occu in the wavefoms of the contolle electomagnetic vaiables. Refeences [1] Amimeu H., Abesseme R., Aouzellag D., Meabet E., Hamoui F., A Sliing Moe Contol associate to the Fiel-Oiente contol of Dual-Stato Inuction Moto Dives, Jounal of Electical Engineeing, Vol. 10, No. 3, 2010, [2] Benatt J., AC motos an geneatos with six-phase stato winings, Acta Technica, Vol. 55, No. 2, 2010, [3] Dozowski P., Multiphase cage inuction motos fo contolle ives, Zeszyty Poblemowe Maszyny Elektyczne, No. 93, 2011, [4] Jafaifa M., Kianinezha R., Seifossaat S. Gh., Sliing Moe Sensoless Contol Of Symmetical Six-Phase Inuction Machines, Int. Conf. ELECO, Busa, Tukey 2009, [5] Levi E., Bojoi R., Pofumo F., Toliyat H.A., Williamson S., Multiphase inuction moto ives a technology status eview, IET Elect. Powe Appl., Vol. 1, Iss. 4, 2007, [6] Listwan J., Pienkowski K., Analysis of sliing-moe contol of multi-phase inuction moto, Zeszyty Poblemowe Maszyny Elektyczne (Komel), No. 4, 2015, , (in Polish). [7] Olowska-Kowalska T., Tachala G., Unifie appoach to the sliing-moe contol an state estimation application to the inuction moto ive, Bulletin of the Polish Acaemy of Sciences, Technical Sciences, Vol. 61, No. 4, 2013, [8] Pienkowski K., Analysis an contol of ual stato wining inuction moto, Achives of Electical Engineeing, Vol. 61, Iss. 3, 2012, [9] Pienkowski K., Analysis an Contol of Multi-Phase Squiel-Cage Inuction Moto, Pace Naukowe Instytutu Maszyn, Napęów i Pomiaów Elektycznych Politechniki Wocławskiej, No. 65, 2011, , (in Polish). [10] Utkin V.I., Gulne J., Shi J.X., Sliing moe contol in electomechanical stems, Taylo an Fancis, New Yok, [11] Venkataamanan R., Sliing Moe Contol of Powe Convetes, PhD Thesis, Califonia Institute of Technology, Pasaena, Califonia 1986.