Independent Control of two PM motors using a single inverter: Application to Elevator Doors.

Transcription

1 Independent Contol of two PM motos using a single invete: Application to Elevato Doos. John Chiasson, Danbing Seto, Fanping Sun, Alex Stankovic and Scott Botoff Abstact This wok consides the contol of two PM synchonous motos using a single invete. The quadatue cuent of each moto is contolled and the diect cuent is uncontollable. The methodology is applied to a the contol of elevato doos. Keywods Elevato doos, PM synchonous motos, I. Intoduction This wok consides the contol of two PM synchonous motos using a single invete. The standad appoach to the contol of a PM synchonous moto is to use a single invete which povides independent contol of the diect and quadatue voltages (and theefoe of the diect and quadatue cuents) of the moto. The quadatue cuent is popotional to the moto toque and the diect cuent is used fo field weakening. Hee, an appoach is pesented that povides independent toque contol of two PM synchonous motos using a single invete. In this appoach, the quadatue cuent of each moto is contolled while the diect cuent is uncontollable. Such an appoach was suggested in the wok [] and hee the intent is to give a moe detailed exposition of this appoach. This appoach was motivated by the contol of elevato doos. A conventional elevato doo system has the two doos mechanically connected to a single cable. Consequently, the two doos must open and close togethe as they ae mechanically coupled. Using position senso feedback fom the wall, the position of the doos is then contolled by a moto/invete system that pushes/pulls on the cable. The objective hee was to conside a diffeent system whee the cable system is eliminated. Instead, each of the two doos of the elevato ae actuated using a linea synchonous moto. The two motos must eliably open and close the two doos of the elevato while maintaining astiffness in the diffeential diection of motion on the ode of,n/m to have the feel of the conventional cable diven doos. Fo example, in a conventional elevato doo system if one doo is held, the othe doo must stop at the same position since the doos ae attached to a single cable whose stiffness is,n/m. The outline is as follows: Section biefly descibes the modeling of PM synchonous motos, Section develops a linea PM moto model fom the otay model, gives the doo model and summaizes the standad PM synchonous moto con- J. Chiasson is with the ECE Dept, Univesity of Tennessee, Knoxville TN, 799 chiasson@utk.edu Danbing Seto, Fanping Sun and S. Botoff aewithunitedtechnologies Reseach Cente, Mail Stop 5, Silve Lane,East Hatfod, CT, botofsa@utc.utc.com Alex Stankovic is with the Depatment of Electical Engineeing, Notheasten Univesity, Boston, MA 5, astankov@cdsp.neu.edu tol algoithm, Section consides the contol of two linea PM motos using a single invete fo both the paallel and seies connection. Finally, sections 5 offes some conclusions. II. Modeling & Contol of PM Synchonous Motos The linea moto is modeled by consideing an equivalent -phase pemanent magnet (PM) synchonous moto. By an appopiate choice of the stato cuents, a otating magnetic field is setup in the aigap of the machine esulting in the pemanent magnet oto following it by magnetic attaction. A model of a thee-phase synchonous moto is [] (θ x/ eq, ω v/ eq ) L S di S M di S M di S M di S + L S di S M di S M di S M di S + L S di S v S R S i S + K m ω sin(n p θ) v S R S i S + K m ω sin(n p θ π ) v S R S i S + K m ω sin(n p θ π ) K m i S sin(n p θ) K m i S sin(n p θ π ) K m i S sin(n p θ π ) τ L ω () Hee L S is the self-inductance of a stato winding, M is the coefficient of mutual inductance between the phases, K m is the toque/back-emf constant, R S the esistance of a stato winding, J eqm the moment of inetia of the oto, τ L the load toque, θ is the oto angula position, ω the oto speed and n p is the numbe of pole pais (o the numbe of oto teeth fo a steppe moto). If the phases wee pefectly coupled, one would have M L S. The thee-phase to two-phase tansfomations fo cuents and voltages ae defined by i sa i sb, i / / / / / / / i S i S i S 9

2 v sa v sb, v / / / / / / / v S v S v S which tansfoms the oiginal model into the equivalent model (L S + M) di sa v sa R S i sa + K mω sin(n p θ) (L S + M) di sb v sb R S i sb K mω cos(n p θ) (L S M) di v R S i K mi sa sin(n p θ)+ K mi sb cos(n p θ) τ L ω Fo a balanced thee-phase system in which v (v S + v S + v S ) /,i (i S + i S + i S ) /, one obtains the two-phase equivalent model given by L di sa L di sb R S i sa + K eq ω sin(n p θ)+v sa R S i sb K eq ω cos(n p θ)+v sb K eq i sa sin(n p θ)+k eq i sb cos(n p θ) τ L ω q K m, i a Hee L L S + M (appoximately L S), K eq and i b ae the equivalent cuents in phases a and b, espectively. Letting V bus denote the bus voltage into a phase invete, then the maximum voltage out of the invete is obtained when it is un in six step mode. The peak of the fundamental of the six step wavefom is v max π V bus which we take as the maximum limit of the phase voltage. Finally, with i max,v max denoting the limits of the phase cuents and voltages of the -phase moto, the coesponding limits I max,v max fo the equivalent -phase moto ae then I max V max i max v max π V bus The diect-quadatue o dq tansfomation is given by id cos(np θ) sin(n p θ) isa i q sin(n p θ) cos(n p θ) i sb vd cos(np θ) sin(n p θ) vsa sin(n p θ) cos(n p θ) v q whee i d, i q and v d,v q ae the tansfomed cuents and voltages, espectively in the dq (fo diect and quadatue) v sb efeence fame. The definition of the dq efeence system assumes that the d axis is aligned with the oto s magnetic axis when θ. Note that when θ,thed axis is aligned the i a axiswhichintunisthesameasthe i S axis. The state-space model in the dq coodinates is L di d v d R S i d + n p ωli q () L di q v q R S i q n p ωli d K eq ω () K eq i q τ L () ω. (5) This model assumes that the oto is smooth (non-salient) and that the magnetics ae linea. III. Moto Specifications The moto paametes ae specified fo a linea moto and must be conveted to an equivalent otay moto. The linea moto paametes ae stato inductance L S.mH, stato esistance R S 9.5Ohms, coefficient of mutual inductance M.5L S. mh, moto mass m.7 kg, foce constant K M N/A, back emf constant K v. m/s/v, distance between poles d p.7/ m, n p (no. of pimay pole-pais). The maximum dc bus voltage to the invete is V max V esulting in a peak fundamental wavefom to the moto of v max π V max V. The phase cuents ae limited to I max Amps (peak) and the maximum (linea) foce put out by the moto is N. The adius of an equivalent otay moto satisfies π eq n p d p eq.7. The toque constant of an equivalent -phase otay moto is found fom the linea foce constant by setting K m eq K M (.7)().7 (Nm/Amp) and moment of inetia is J eqm.the paametes L S,M, R S,n p ae the same as fo the linea moto. If x, v denote the linea position and speed of the linea moto, espectively, then θ x/ eq, ω v/ eq ae the angle and angula velocity, espectively, of the equivalent otay moto. Hee x coeponds to the magnetic axis of oto phase a being lined up with the magnetic axis of stato phase a and similaly fo the otay moto. The coesponding equivalent two-phase paametes ae q then L L S + M 9.mH, R S 9.5Ohms, q K eq i max K m.9 N-m/Amps, I max (continuous). Amps, V max foce put out by this moto is then F K eq i q / eq K mi q / eq. q v max 9.5 Volts. The linea A. Doo Model The doo model is fom the technical epot of He [] and is given by dx/ Ax + bu y cx 9

3 whee A R,b R,c R. The values of the tiple {A, b, c}ae given in []. Hee x is the doo position and x is the doo speed and the input u tothedooisthelineafocef K eq i q / eq put out by the moto. The state vaiables x,x ae the two measued/computed state vaiables so that the output matix is c The mass of the doo is denoted by M c so that the total mass of the doo/moto combination is M c + m. The obsevability matix c ca ca ca ca ca 5 ca ca 7 T has ank while the contollability matix b Ab A b A b A b A 5 b A b A 7 b has ank 5. Howeve, A is stable. The contol appoach is to simply feed back x, v (θ x/ eq, ω v/ eq ) teating the tansfe function fom u F to x as a double integato. The esolution of the linea position feedback fom the wall to the doo contol system is.5 mm. The maximum doo speed is v max m/sec, the maximum acceleation is α max. m/sec, and the jek ate is limited to j max. m/sec. The total distance taveled by each doo is 555 mm. B. Contolle A staightfowad way to do sevo contol of this moto when thee is one invete fo each moto is to choose the linea foce as u M c (α ef + K (v ef v)+k (x ef x)+ K (x ef x) i qef ( eq /K eq )u i def () v q K p (i qef i q )+K I (i qef i q ) v d K p (i def i d )+K I (i def i q ). IV. Two Motos and One Invete One appoach to contolling two PM synchonous motos using one invete would be to just contol the two motos identically. Specifically, as they nominally follow identical tajectoies, just set v d v d,v q v q so that θ θ, ω ω,i d i d,i q i q. This is a standad appoach fo toque/speed contol of induction moto populsion systems (light ail vehicles, subway cas, etc.) Howeve, even fo toque contol, induction motos only equie the speed of oto (to estimate the oto fluxes). In this case, the aveage speed of the two motos ae used in the flux estimato and fo speed contol. Howeve, in the case of synchonous motos, the position of the oto is equied fo contol as (7)() show. The extenal distubances τ L, τ l ae not necessaily always equal and so the otos will misalign, i.e., θ θ. In this situation, the contol scheme would not be able to ecove, that is, to ealign θ θ. Theobjectiveheeistouseoneinvetetocontoltwo motos. Fo a system that has an invete fo each moto, the moto contolle is simply the feedback contolle system given in (). When thee is only one invete, the two motos can be connected to the invete eithe in seies o in paallel. A. Paallel Connection Fist, conside the motos connected in paallel. so that the applied voltage to each phase of the motos ae the same. The model of the two motos in the dq coodinate system ae then and L di d L di q L di d L di q R S i d n p ω Li q + v d R S i q n p ω Li d ω K eq + v q K eq i q τ L ω R S i d n p ω Li q + v d R S i q n p ω Li d ω K eq + v q K eq i q τ L ω. To un these two motos off of one invete, we must take into account how the voltages ae commanded to the moto. The same thee voltages v S,v S,v S,oequivalently,the same two phase voltages v a, ae commanded to both motos. That is, the dq voltages fo the two motos ae given by vd cos(np θ ) sin(n p θ ) va sin(n p θ ) cos(n p θ ) (7) v q vd v q cos(np θ ) sin(n p θ ) sin(n p θ ) cos(n p θ ) va whee θ, θ ae the angula position of moto and moto, espectively. As the contolle () indicates, this can be done by specifing the quadatue voltage v q of each moto which is then θ is assumed to coespond with the magnetic axis of phase a of moto and similaly fo moto. () 95

4 id a n d id D o o P o s itio n in m e te s S p e e d in m / s e c used to contol the toque poducing cuent i q. To do so, this equies choosing v a, such that vq sin(np θ ) cos(n p θ ) va sin(n p θ ) cos(n p θ ) o v q va sin(n p (θ θ )) cos(np θ ) cos(n p θ ) sin(n p θ ) sin(n p θ ) vq Clealy thee is a singulaity in the invese at v q. (9) Speed and Speed vs Time Fig.. (n p (θ θ )) mod π. At points sufficiently fa fom this singulaity, a contol scheme fo the two motos would be.7..5 Doo and Doo position i qef Jα ef + K ω (ω ef ω )+K θ (θ ef θ ) K eq v q K p (i qef i q )+K I (i qef i q ) i qef Jα ef + K ω (ω ef ω )+K θ (θ ef θ ) K eq v q K p (i qef i q )+K I (i qef i q ). The diect voltages ae detemined by the quadatue voltages given by (9). Specifically, substitute (9) into vd cos(np θ ) sin(n p θ ) va cos(n p θ ) sin(n p θ ) to get v d vd v d sin(n p (θ θ )) cos(np (θ θ )) cos(n p (θ θ )) vq so that when θ θ, v d,v d and consequently, the cuents i d,i d can be lage nea the singulaity. Thesimplestuseoftheabovecontolschemeistophysically offset the angula position of the two motos by (π/)/n p. Then, as both nominally tack the same tajectoy (except fo the position offset of (π/)/n p ), a tight tajectoy tacking contol loop would keep n p (θ θ ) close to π/ and thus keep the system away fom the singulaity. A. Simulations A Simulink simulation was un using the one invete two moto contolle. The speed pofile fo the two motos is shown in Figue and the position pofiles ae shown in Figue. These plots show that the doos open in. seconds. The diffeence in position shown in Figue is π due to the fact that x () and x () eq n p.5 m. v q Fig.. The uncontolled cuents i d,i d ae shown in Figue. The diffeence in the tajectoies of the two cuents can be explained by the fact the initial angula position of the two motos ae diffeent by (π/)/n p to avoid the singulaity in the contol. The consequence of not being able to contol the diect cuent to zeo esults in the wasted powe R S i d because in a one-invete/one-moto configuation, this cuent would be zeo D iect C uent Fig.. The quadatue cuents i q,i q ae contolled and ae basically on top of each othe as shown in Figue and epesent the foce ( K eq i q / eq ) ecquied to make the move. Finally, the phase voltage v S had a maximum voltage of just unde Volts. 9

5 iq a n d iq dq Cuents With τ, τ the toque efeences fo the two motos espectively, then the following two equations τ K eq u sa sin(n p θ )+K eq u sb cos(n p θ )K eq i q () τ K eq u sa sin(n p θ )+K eq u sb cos(n p θ )K eq i q Fig.. B. Seies Connection In the seies connection, the cuent in each moto is the same in thei coesponding phases. To analyze this situation, conside the two-phase equivalent model of the two synchonous motos: L di sa L di sb L di sa R S i sa + K eq ω sin(n p θ )+v sa R S i sb K eq ω cos(n p θ )+v sb () K eq i sa sin(n p θ )+K eq i sb cos(n p θ ) τ L ω R S i sa + K eq ω sin(n p θ )+v sa L di sb R S i sb K eq ω cos(n p θ )+v sb () K eq i sa sin(n p θ )+K eq i sb cos(n p θ ) τ L ω. The contol input is v sa v sa + v sa v sb v sb + v sb. Using high-gain cuent feedback, one may conside v sa K p (i sa_ef i sa ) v sb K p (i sb_ef i sb ) u sa i sa_ef u sb i sb_ef as new inputs. Note that the high-gain feedback does not have an integato. This is due to the fact that the cuents ae sinusoids and will be of high fequency at high speeds. Consequently, the integato can have touble tacking such a fast vaying signal. ae solved to detemine the inputs u sa,u sb to achieve these toques. usa u sb sin(n p (θ θ )) cos(np θ ) cos(n p θ ) τ sin(n p θ ) sin(n p θ ) τ () As in the paallel connection case, this has a singulaity when (n p (θ θ )) mod π. At points sufficiently fa fom this singulaity, a contol scheme fo the two motos would be τ Jα ef + K ω (ω ef ω )+K θ (θ ef θ ) τ Jα ef + K ω (ω ef ω )+K θ (θ ef θ ). The diect cuents ae detemined by the quadatue cuents given by (9). Specifically, substitute () into (u Sa i Sa,u Sb i Sb ) id cos(np θ ) sin(n p θ ) isa cos(n p θ ) sin(n p θ ) i d to get id i d sin(n p (θ θ )) cos(np (θ θ )) cos(n p (θ θ )) i sb iq When θ θ ± πk, i d,i d and consequently, the cuents i d,i d can be lage nea the singulaity. The simplest use of the above contol scheme is to offset the angula position of the two motos by (/)π/n p. Then as both nominally tack the same tajectoy, the contolle would nominally keep n p (θ θ )π/ and thus the system away fom the singulaity. B. Simulations To simulate the seies connected system, the fist and second equation of () ae added to the fist and second equation of (), espectively so that with v sa v sa + v sa,v sb v sb + v sb, along with the two speed and two position equations, the oveall dynamic model fo simulation is i q. 97

6 iq a n d iq D o o P o s itio n in m e t e s D i e c t C u e n t L di sa L di sb R S i sa + K eq ω sin(n p θ )+K eq ω sin(n p θ ) +v sa R S i sb K eq ω cos(n p θ ) K eq ω cos(n p θ ) +v sb K eq i sa sin(n p θ )+K eq i sb cos(n p θ ) τ L ω K eq i sa sin(n p θ )+K eq i sb cos(n p θ ) τ L ω. epesents wasted powe R S i d becauseinaoneinveteone moto seneio, this cuent would be zeo. The diect cuents below show a slight diffeence between the paallel and seies cases.finally, it tuned out that the phase id and id dx eq Hee θ x / eq, ω dx eq, θ x / eq, ω. The speed pofile was the same as the paallel connected case. The diffeence in position shown in Figue 5 is π due to the fact that x () and x () eq n p.5 m Doo and Doo position Fig. 5. The quadatue cuents shown below in Figue ae identical to the paallel case as it must since the toque equied fo eithe tajectoy is the same. 7 5 iq C u e n ts Fig.. The uncontolled cuents i d,i d ae shown in Figue 7. The diffeence in the tajectoies of the two cuents can be explained by the fact the initial angula position of the two motos was diffeent. As in the paallel case, this Fig. 7. voltage v S tuned out to be about Volts. C. Conclusions and Summay The diffeence between the paallel and seies connection is not eally significant and the choice could be made based on eliability consideations. Fo example, if a phase fails in the paallel case, then one of the doos could still be opeational (one would have to detect the failue and then contol the emaining moto in the nomal fashion). The ating of the invete could also detemine the choice of the connection. The seies connected moto system uses twice the voltage of the paallel connnected moto system, but half the cuent. This is simply a esult of consevation of enegy. The issue of stiffness still needs to be woked out. That is, when one doo is being blocked and theefoe commanded to eopen, then this same command (tajectoy efeence) must be sent to the othe doo so that they maintain thei sepeation of eq π/ to avoid the singulaity in the contolle. The stiffness is dependent on how fast the blocking of the doo can be detected and then a command given to the othe doo to open them while maintaining the sepeation eq π/. Refeences [] Bodson, M., Chiasson, J.N., Novotnak, R.T., & Rekowski R.B., High-Pefomance Nonlinea Feedback Contol of a Pemanent Magnet Steppe Moto, IEEE Tans. on Contol Systems Technology, vol., no.., 99, pp [] Seto, D, F. Sun, J. Jieas and N. Hootsmans, Single Electonic Dive Synchonizing Two Motos Via Modified Vecto Contol, to appea in the Applied Powe Electics Conf, Dallas, Texas, Mach -,. [] Leonhad, W. Contol of Electical Dives, Spinge Velag, Belin, 99. [] Blauch, A., Bodson, M., & Chiasson, J., High-Speed Paamete Estimation of Steppe Motos, IEEE Tans. on Contol Systems Technology, vol., no., 99, pp [5] Bodson, M., Chiasson J., & Novotnak, R.T., A Systematic Appoach to Selecting Optimal Flux Refeences in Induction Motos, IEEE Tans. Contol Systems Technology, vol., no., 995, pp [] He, Thomas, Computeized Simulation fo LIM Doo Mechanism and Contol System (LIM Doo), Technical Repot Otis- R/D Cente, July 99. 9