High Dynamic Performance Programmed PWM Control of a Multilevel Inverter with Capacitor DC Sources

Transcription

1 High Dynamic Pefomance Pogammed PWM Contol of a Multileel Inete with Capacito DC Souces John N. Chiasson, hong Du, Buak Özpineci, and Leon M. olbet Abstact A cascade multileel inete consisting of a standad 3-leg inete supplied by a DC souce and thee full H-bidges each supplied by a capacito is consideed fo use as a moto die. he capacito H-bidges can only supply eactie oltage to the moto while the standad thee leg inete can supply both eactie and actie oltage. A switching contol algoithm is pesented that shows this inete topology can be used as an AC die achieing consideable pefomance adantages (e.g., highe moto speed) compaed to using a standad 3-leg inete while at the same time egulating the capacito oltages. he conete contolle is a fundamental fequency switching contolle based on pogammed PWM to achiee highe efficiency (less powe losses in the switches) compaed to high-fequency PWM appoaches. As is well known, the pogammed PWM switching times ae computed assuming the die is in sinusoidal steady-state, that is, the deied switching angles achiee the fundamental while ejecting specified hamonics if the oltage waefoms ae in sinusoidal steady-state. Hee it shown that the switching commands to the conete can be implemented in a smooth fashion fo oltage waefom commands whose fequency and amplitudes ae continuously aying. Index ems Cascade Multileel Inete, Capacito Souces, Reactie Voltage, Dynamic Pogammed PWM I. INRODUCION A cascade multileel inete (CMLI) is a powe electonic deice built to synthesize a desied AC oltage fom seeal leels of DC oltages. Such inetes hae been the subject of eseach in the last seeal yeas [1][][3][4], whee the DC leels wee consideed to be identical in that all of them wee eithe batteies, sola cells, etc. In [5], a multileel conete was pesented in which the two sepaate DC souces wee the secondaies of two tansfomes coupled to the utility AC powe. Cozine et al [6] hae poposed using a single DC powe souce and capacitos fo the othe DC souces. A method was gien to tansfe powe fom the DC powe souce to the capacito in ode to egulate the capacito oltage. A simila appoach was late (but independently) J. Chiasson is with the ECE Depatment, Boise State Uniesity, Boise ID 8375, johnchiasson@boisestated.edu.. Du is with the Pake Hannifin Copoation, 85 Hacks Coss Rd, Olie Banch, MS 38654, zhong.du@pake.com Buak Ozpineci is with he Oak Ridge National Laboatoy, Knoxille, N USA, buak@ieee.og. Leon M. olbet is with the ECE Depatment, he Uniesity of ennessee, Knoxille, N USA 37996, tolbet@utk.edu Pepaed by the Oak Ridge National Laboatoy, Oak Ridge, ennessee 37831, managed by U-Battelle fo the U.S. Depatment of Enegy unde contact DE-AC5-OR75. he submitted manuscipt has been authoed by a contacto of the U.S. Goenment unde Contact No. DE-AC5-OR75. Accodingly, the U.S. Goenment etains a nonexclusie, oyalty-fee license to publish fom the contibution, o allow othes to do so, fo U.S. Goenment puposes. poposed by Du et al [7]. Explicit conditions when the capacito oltage can be egulated ae gien in tems of the modulation index and powe facto angle in [8]. Othe simila methods hae also been poposed by Veensta and Rufe [9][1][11]. Hee a cascade multileel inete consisting of a standad 3-leg inete supplied by a DC souce and thee full H- bidges each supplied by a capacito is consideed fo use as an AC moto die. he basic idea is that the capacito H-bidges supply the eactie oltage equied of the moto and the standad thee leg inete supplies the actie oltage. A contol algoithm is pesented that shows this inete topology can poide this capability while simultaneously keeping the capacito oltage egulated. In addition, a fundamental fequency switching contolle based on pogammed PWM is used to achiee highe efficiency (lowe losses in the switches) compaed to highfequency (sine-tiangle) PWM appoaches. Howee, as is well known, the pogammed PWM switching times/angles ae computed assuming the die is in sinusoidal steady-state, that is, the deied switching angles achiee the fundamental while ejecting specified hamonics assuming the oltage waefoms ae in sinusoidal steady-state. We popose a method to dynamically implement the pogammed PWM switching times/angles fo smooth tansitions unde a aying (amplitude and fequency) oltage command. II. CMLI ARCHIECURE Figue 1 shows a DC souce connected to a single leg of a standad 3-leg inete and second DC souce consisting of a capacito. V dc i c V dc DC Souce C S S 1 S S S 5 S 6 + i + - Fig. 1. One leg of a 3-leg inete connected to a full H-bidge with a capacito DC souce.

2 he output oltage 1 of this leg (with espect to the gound) is eithe +V dc / (S 5 closed) o V dc / (S 6 closed). his leg is connected in seies with a full H-bidge which is supplied by a capacito. If the capacito is kept chaged to V dc /, then the output oltage of the H-bidge can take on the alues +V dc / (S 1 &S 4 closed), (S 1 &S closed o S 3 &S 4 closed), o V dc / (S &S 3 closed). Due to this seies connection in the topology of Fig. 1, we will use the cuent as the efeence fo actie/eactie quantities athe than the oltage. he oltage out of the multileel inete that is in phase with the cuent will be efeed to as the actie oltage while the oltage 9 out of phase with the cuent will be efeed to as the eactie oltage. he appoach hee is to hae the H-bidge poide the eactie pat of the oltage (the component 9 out of phase with the load cuent) so that the capacito is not dischaged. he standad thee leg inete then poides the actie pat of the oltage (the component in phase with the load cuent). As a consequence, this inete poides a oltage boost to account fo the eactie oltage equied by the load. A. Reactie Voltage Contol We now pesent a method fo poiding the eactie oltage while egulating the capacito oltage as poposed by one of the authos (B. Özpineci). Let i S1,i S, and i S3 be the stato cuents and define the space ecto fom [1] of the stato cuents as i S = i Sa + ji Sb, i S1 + e jπ/3 i S + e j4π/3 i S3. 3 (1) Similaly, let S1, S, and S3 be the stato phase oltages and define the space ecto fom of the stato oltages as S = Sa + j Sb, S1 + e jπ/3 S + e j4π/3 S3. 3 () We then decompose the oltage into its actie and eactie components with espect to the stato cuent as S_actie, i S S i S S_eactie, S ji S i S whee denotes the ecto dot poduct, i.e., S i S, ( Sa + j Sb ) (i Sa + ji Sb ), Sa i Sa + Sb i Sb. (5) Note that we ae making the identification Sa + j Sb [ Sa, Sb ] and i Sa + ji Sb [i Sa,i Sb ] so that isa ecto opeation and not complex multiplication. Futhe, we define ji S, j (i Sa + ji Sb )= i Sb + ji Sa [ i Sb,i Sa ]. With these definitions, we hae (i S ) (ji S ), (i Sa + ji Sb ) j (i Sa + ji Sb ) = (i Sa + ji Sb ) ( i Sb + ji Sa ) = i S = p q i S i S = i Sa + i Sb. (3) (4) heideaheeisthati S / i S and ji S / i S ae two othogonal ectos and ae used as a new set of basis ectos (the oiginal basis being [1, ] and [, 1] ). In (3) and (4) we pojected the oltage ecto S onto these two ectos to obtain the components of the oltage ecto with espect to this new basis, that is, a component in phase with the cuent and a component 9 degees out of phase with the cuent. We can then ewite the oltage ecto S with espect to this new basis as hen i S_actie, S S_actie (6) i S ji S_eactie, S S_eactie i S. (7) S = S_actie + S_eactie whee S_actie is a ecto paallel to i S and S_eactie is a ecto pependicula to i S. As a consequence, oe a cycle of peiod we hae S i S dt = S_actie i S dt + S_eactie i S dt = S_actie i S dt (8) as S_eactie i S =. Using the two-phase to theephase tansfomation, the command oltages to the thee leg inete (with the DC souce) ae then S1actie = 3 Re S_actie jπ/3 Sactie = 3 Re S_actie e j4π/3 S3actie = 3 Re S_actie e. while the command oltages to the H-bidge (with the capacito souces) ae S1eactie = 3 Re S_eactie jπ/3 Seactie = 3 Re S_eactie e j4π/3 S3eactie = 3 Re S_eactie e. If thee wee no losses, the capacitos would keep thei oltage leel as they ae only supplying eactie oltage. Howee, thee ae inaiably going to be losses and so a contol loop is equied to egulate the capacito oltages. o do so, we modify the command oltages to the H-bidges. Let V c be the desied alue of the capacito oltage and c be its measued alue. he command oltage to the fist H- bidge is modified to be S1eactie = 3 Re S_eactie K cap (V c c1 ) S1actie. (9)

3 Hee K cap > is the contolle gain fo egulating the capacito oltage leel. Note that oe one peiod of a cycle the enegy out of the capacito is gien by S1eactie i S1 dt = Re( 3 S_eactie )i S1 dt K cap (V c c1 ) S1actie i S1 dt = K cap (V c c1 ) S1actie i S1 dt (1) whee it was assumed that V c c1 does not ay significantly oe a single peiod. If c1 <V c, then the capacito needs chaging and enegy must be supplied to the capacito, i.e., the enegy out of the capacito must be negatie. Equation (1) shows that the contolle does indeed do this because S1actie and i S1 hae the same sign (they ae in phase) so that R S1actiei S1 dt > and theefoe S1eactie i S1 dt = K cap (V c c1 ) S1actie i S1 dt <. (11) hat is, enegy is being supplied to the capacito of phase 1 to echage it. Similaly, if c1 >V c, the contolle will dischage the capacito. In summay, the oltage commands to the H-bidges ae S1eactie = 3 Re S_eactie K cap (V c c1 ) S1actie jπ/3 Seactie = 3 Re S_eactie e K cap (V c c ) Sactie j4π/3 S3eactie = 3 Re S_eactie e K cap (V c c3 ) S3actie. B. Simulations he contolle is set up so that the H-bidge with capacito souces supply eactie oltage and each leg of the thee leg inete supplies only actie oltage. A PM synchonous machine was chosen as they often equie significant eactie oltage. he DC link oltage V dc was set to 5 V yet the oeall oltage used is less than V asshowninfig.3. Consequently, the supply fo the thee leg standad inete is 1 V and the capacitos fo the H-bidge ae chaged and egulated to V dc / = 15 V. With espect to the eactie oltage, the CMLI is thus poiding a boost of.5 times that of a standad thee leg inete. he moto s paamete alues ae an inetia of J =.1 kg-m, n p =4pole-pais, stato esistance is R S =.65 Ohms, stato inductance is L S =3mH, toque/back-emf constant K = K b =.37 Nm/A (V/ad/s) and a load toque of τ L =19Nm. he H-bidge capacito alue is C =.1 F. Fo compaison puposes, simulations wee pefomed using both the multileel inete of Fig. 1 and a standad 3-leg inete (i.e., only the bottom half of Fig. 1). hough the multileel inete can supply up to ± (V dc + V dc /)) = ±5 V, it can do this and maintain egulation of the capacito oltages only if 15 V of the 5 V is eactie oltage. he PM moto was un to achiee the highest possible speed unde the gien load and aailable oltage. his is shown in Fig.. he standad 3-leg inete could only achiee a maximum speed of 1 ad/s while the poposed multileel inete could get the moto up to 75 ad/s using the same DC souce oltage. Radians/Second Roto Speed (Max = 1 ad/sec) Radians/Second Roto Speed (Max = 75 ad/sec) Fig.. Roto speeds achieable using a standad 3-leg inete (left) and the poposed multileel inete (ight). he coesponding oltages fo the speed tajectoies of Fig. ae shown in Fig. 3. he peak oltage of the standad 3-leg inete is nealy a six step wae fom with V dc / = 1 V coesponding to a fundamental oltage of (4/π)V dc / = 17 V peak while the multileel inete is supplying up to 165 V peak. Volts 1-1 Stato Voltage Phase Volts Stato Voltage Phase Fig. 3. Left: Voltage using a standad 3-leg inete. Right: Voltage obtained using the poposed multileel inete. he coesponding toques fo the aboe tajectoies ae shown in Fig. 4. he chatteing shown in the toque esponse of the standad 3-leg inete is due to the fact its link oltage is limited to 1 V and the contolle is asking fo moe than 1 V on the output (see the left-side of Fig. 3). he toque poduced is geate in the case of the multileel inete as the moto is being acceleated to a highe speed in the same amount of time.

4 Newton-metes Moto oque Newton-metes Moto oque s1_actie i s Fig. 4. Left: Moto toque using standad 3-leg inete. Right: Moto toque using poposed multileel inete. he stato cuents ae shown in Fig. 5. Stato Cuent Phase 1 Stato Cuent Phase Ampees 4 - Ampees Fig. 5. Left: i S1 fo a standad 3-leg inete. Right: i S1 fo the poposed multileel inete Fig. 7. Stato cuent i S1 (A) and the actie component S1_actie (V) s. time (s). he capacito oltage as a function of time is plotted in Fig. 8 showing that it is kept within about olts of the desied alue he stato cuent and the eactie component of the stato oltage (the output of the H-bidge) ae plotted in Fig. 6 showing that they ae 9 out of phase. (Both signals wee put though a low pass filte with a cutoff of 5 khz.). Fig. 8. Capacito oltage (V) esus time (s). he electical powe S1 i S1 + S i S + S3 i S3 and the mechanical powe τω ae shown in Fig Electical Powe 15 s1_eactie i s1 4 Mechanical Powe Fig Electical and mechanical powe (W) s. time (s). Fig. 6. Stato cuent i S1 (A) and the eactie component S1_eactie (V) s. time (s). Fig. 7 is a plot of the stato cuent and the actie component of the oltage (output of the thee leg inete) showing that they ae in phase. (Both signals wee put though a low pass filte with a cutoff of 5 khz.) he powe facto angle φ is defined by µ φ, cos 1 S1 i S1 + S i S + S3 i S3 3V S I S V S = S I S = i S with i S, S gien by equations (1) and (), espectiely. φ is plotted in Fig. 1. Note that the machine opeates with

5 a athe lage powe facto angle so that the die equies significant eactie oltage making the choice of a capacito fed H-bidge CMLI attactie. 8 Powe Facto Angle V dc 1 V dc 1 V dc V dc θ 1 θ θ π π Degees Fig. 11. Output waefom of a fie-leel cascade multileel inete. As in the peious section, S = Sa + j Sb, p /3 S1 + e jπ/3 S + e j4π/3 S3. We ewite this space ecto in pola coodinate fom as S = S e jθ whee S, p Sa + Sb and θ, tan 1 ( Sb / Sa ) (see Fig. 1). Fig. 1. Powe facto angle φ s time. III. DYNAMIC PROGRAMMED PWM Pogammed PWM techniques hae been poposed fo oe 3 yeas fo choosing the switching angles in an inete to poide a specified fundamental oltage and eliminate aious highe-ode hamonics. If the switching losses in an inete ae not a concen (i.e., switching on the ode of a few khz is acceptable), then the sine-tiangle PWM method and its aiants ae ey effectie fo contolling the inete [13]. On the othe hand, fo systems whee high switching efficiency (low powe loss) is of impotance, it is desiable to keep the switching fequency much lowe. In this case, anothe appoach is to choose a finite numbe of switching times t i pe peiod (with a coesponding finite numbe of switching angles θ i =(t i / )π) such that a desied fundamental output is geneated and specifically chosen hamonics of the fundamental ae suppessed [13][14][15][16][17]. his is efeed to as hamonic elimination o pogammed PWM as the switching angles θ i ae chosen (pogammed) to eliminate specific hamonics. Howee, these methods assume the die is in sinusoidal steady-state, that is, the deied switching angles achiee the fundamental while ejecting specified hamonics if the oltage waefoms ae in sinusoidal steady-state. In an application such as a moto die, it is typically the case that the fequency and amplitude of the oltage waefom is changing in esponse to the changing speed of the moto, that is, dynamic pefomance is equied. As pointed out by Holtz and Beye [18][19], applications of pogammed PWM techniques ae few because of thei low dynamic pefomance. A new appoach which achiees high dynamic pefomance pogammed PWM has been poposed by one of the authos (. Du) and is now discussed. o fix ideas, conside again the fie-leel inete shown in Fig. 1. A fundamental switching scheme fo the output waefom is shown in Fig. 11 whee θ 1 and θ ae the switching angles elatie the fundamental peiod. Fig. 1. sb s θ s s e jθ = s sa = sa + j sb Space ecto fom of the commanded stato oltages. he switching angles θ 1 and θ of Fig. 11 ae detemined solely by the modulation index m, 4 S π s (V dc /) whee V dc is the link oltage and s =is the numbe of DC souces []. hat is, the output waefom is completely detemined by the modulation index and, in paticula, does not depend on the specific alue of the peiod of the fundamental waefom. he (instantaneous) angle θ of the commanded stato oltage is computed modulo π detemining its position in the inteal [, π] and thus accoding to Fig. 11 the coesponding switch positions ae detemined. We emak that fo a significant ange of the modulation index, the angles θ 1 and θ of the waefom of Fig. 11 can be chosen to achiee a gien fundamental oltage and no fifth hamonic [1][] (see the Appendix). Dynamic Pogammed PWM Algoithm 1) Compute the oltage command S to the inete (e.g., a field-oiented contolle based on measued stato cuents, speed and/o position). ) Compute the magnitude S, p Sa + Sb and phase θ, tan 1 ( Sb / Sa ) of the command oltage. hese will typically be noisy because, fo example, a fieldoiented contolle uses noisy measuements of the cuents. 3) Put the oltage magnitude though a low pass filte to eliminate some of the noise. 4) Put the filteed magnitude though a lookup table containing the pepogammed PWM switching pattens to

6 obtain the switching angles. (It typically will not help to filte the phase because it changes too apidly and any filte that emoes the noise will inaiably cause toomuchofadelayinthephase). 5) Check the alue of the computed phase angle θ against the switching angles to detemine the alue of the inete output oltage (and theefoe switch positions). A. Simulations with Dynamic Pogammed PWM As an illustation of the appoach, command oltages to the inete wee amped up in magnitude and fequency. Specifically, the phase 1 oltage is gien by S1 (t) = V S sin(ω S t) with ω S (t) =π(1)t and V S (t) =1t and is plotted in Fig Fig. 15. Bipola switching with V S =35Vandω S =77ad/s Fig. 16. Fundamental switching with V S =55Vandω S =155ad/s. Fig. 13. S1 (t) =V S sin(ω S t), ω S (t) =π(1)t and V S (t) =1t. 1 8 he inete simulated is a fie leel multileel inete with a DC link of V dc = 1 V. Fo V S 38 (modulation index of.484) the capacito DC souces ae not needed so a bipola scheme was used with the 3-leg inete while fo highe oltage amplitudes the fundamental switching scheme of Fig. 11 was used. Figues 14, 15, 16, 17, and 18 ae enlaged iews of Fig. 13 along with the inete output oltage computed fom the commanded oltage at t =.,.35,.5,.75 and 1., espectiely. hese sequence of figues demonstate the smooth tansition of CMLI output waefom as the fequency is inceased Fig Fundamental switching with V S =75Vandω S =35ad/s Fig. 14. Bipola scheme whee V S =Vandω S =5ad/s. Fig. 18. Fundamental switching with V S =1Vandω S =635ad/s.

7 B. Field-Oiented Contol with Dynamic Pogammed PWM We now use a dynamic pogammed PWM algoithm to die an induction moto. he induction moto paametes ae M =.11 H, R R = 3.9 Ohms, R S = 1.7 Ohms, L R =.14 H, L S =.14 H, J =.11 kgm,n p =3 pole-pais and V dc = 1 V. he moto was bought up fom zeo speed to 838 ad/s (4 Hz) as shown in Fig. 19. he coesponding cuent in stato phase 1 is shown in Fig.. A field-oiented contolle was used to hae the moto poide the tacking shown in Fig. 19. he output of the field oiented contolle is the stato oltage and its magnitude S fo the aboe tajectoy is shown in Fig. 1 (the coesponding modulation index is easily calculated). Figue is a plot of the command oltage fom the contolle and the inete output oltage with a oltage magnitude S = 5 V and a fequency ω S = 945 ad/s (15 Hz), and is implemented using a fie-leel multileel switching scheme. Figue 3 shows a plot of the command oltage fom the contolle and the inete output oltage with oltage magnitude S =67Vand fequency ω S = 513 ad/s (4 Hz), and is implemented using a fie-leel multileel switching scheme Fig. 1. S (V) esus time (sec) Fig.. Commanded and inete output oltage (V) with V S =5V and ω S =945ad/s (15 Hz) Fig. 19. ω and ω ef (ad/s) esus time (s) Fig. 3. Commanded and inete output oltage (V) with V S =67V and ω S =513ad/s (4 Hz) Fig.. Stato cuent i S1 (A) esus time (s). IV. CONCLUSIONS A fie-leel cascade multileel inete topology has been poposed that equies only a single standad 3-leg inete and capacitos as the powe souces. An algoithm was pesented that could simultaneously egulate the (capacito souced) oltage of the H-bidges and poduce the equisite eactie oltage equied by the AC machine. Simulations showed that the multileel PM die can attain significantly highe speeds compaed to the standad thee leg inete

8 simply because the capacito souced H-bidges poide the eactie oltage equied by the load. Additionally, a method to dynamically implement pogammed PWM switching pattens was pesented and was shown to smoothly set the switching angles while the oltage command aied in fequency. V. APPENDIX he Fouie seies expansion of the (stepped) output oltage waefom of the multileel inete as shown in Figue 11 is X 4V dc V (ωt) = cos(nθ 1 )+ cos(nθ ) sin(nωt) nπ n=1,3,5,... (1) Gien a desied fundamental oltage V 1, the goal hee is to detemine the switching angles θ 1,θ so as to make the fist hamonic equal to the desied fundamental oltage V 1 and the 5 th ode hamonic zeo. (As the application of inteest hee is a thee-phase moto die, the tiplen hamonics in each phase need not be canceled as they automatically cancel in the line-to-line oltages.) he mathematical statement of these conditions is 4V dc cos(θ 1 )+ cos(θ ) = V 1 π (13) cos(5θ 1 )+cos(5θ ) =. his is a system of two tanscendental equations in the two unknowns θ 1,θ. o sole these equations, let x 1 = cos(θ 1 ) and x =cos(θ ) and use the tigonometic identity cos(5θ) = 5cos(θ) cos 3 (θ) +16cos 5 (θ) so that the conditions (13) become p 1 (x 1,x ), x 1 + x m = p (x 1,x ), 5x 1 x x x x 3 +16x 5 = whee x =(x 1,x ) and m, V 1 / (4V dc /π). hisisaset of two equations in the two unknowns x 1,x. Futhe, the solutions must satisfy x <x 1 1. osolethisset one simply soles p (x 1,m x 1 )= fo all oots x 1i satisfying <x 1i 1. Fo each such oot x 1i, the coesponding oot x i = m x 1i keeping only those solution pais (x 1i,x i ) that also satisfy x i < x 1i 1. he coesponding angles ae θ 1 =cos 1 (x 1 ),θ = cos 1 (x ) whee the cos 1 is esticted to the angle in the fist quadant. (his appoach to soling the hamonic elimination equations is genealized in [][1].) REFERENCES [1] M. Klabunde, Y. hao, and. A. Lipo, Cuent contol of a 3 leel ectifie/inete die system, in Confeence Recod 1994 IEEE IAS Annual Meeting, pp , [] W. Menzies, P. Steime, and J. K. Steinke, Fie-leel GO inetes fo lage induction moto dies, IEEE ansactions on Industy Applications, ol. 3, pp , July [3] J. K. Steinke, Contol stategy fo a thee phase AC taction die with thee leel GO PWM inete, in IEEE Powe Electonic Specialist Confeence (PESC), pp , [4] J. hang, High pefomance contol of a thee leel IGB inete fed AC die, in Conf. Rec. IEEE IAS Annual Meeting, pp. 8, [5] M. Manjeka, P. K. Steime, and. 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