A Unified SVPWM Realization for Minimizing Circulating Currents of Dual Three Phase Machines

Transcription

1 IEEE PEDS 17, Honolulu, USA 1 - December 17 A Unified SVPWM Realization for Minimizing Circulating Current of Dual Three Phae Machine Hiham Eldeeb, Chritoph Hackl Munich School of Engineering Technical Univerity of Munich (TUM) Munich, Germany hiham.eldeeb@tum.de Mohamed Abdelrahem Intitute of Electrical Drive ytem and Power Electronic, TUM Munich, Germany Ayman Samy Abdel-Khalik Electrical Engineering Department Faculty of Engineering, Alexandria Univerity Alexandria, Egypt. Abtract Space-vector PWM (SVPWM) gained attention, a a voltage ynthei technique for aymmetrical dual three phae machine (ADTM); due to it ability to minimize the harmonic content of the output and minimize the witching loe through proper election of the witching vector. It ha been tudied extenively for ADTM mainly for induction machine with iolated neutral (N), uch that the inverter i aumed to be the only ource capable of inducing harmonic within the tator current. Thu, by etting the xy ubpace voltage to zero, the harmonic content i minimized. However, for dual three phae-permanent magnet ynchronou machine (DT-PMSM), the preence of non-inuoidal rotor flux invoke ignificant harmonic, mandating non-zero reference for the xy plane to enure proper compenation. Thi paper invoke the employment of non-zero harmonic reference within the SVPWM calculation for the N connection, a well a laying out the fundamental for the ingle neutral (1N) connection, which i more reliable from fault-tolerance point of view. Alo, a imple digital realization i preented, uitable for the 1-, -ector SVPWM method, both continuou and dicontinuou modulation. The theoretical finding are corroborated with a. kw DT-PMSM, demontrating ignificant improvement in the total harmonic ditortion (THD) of the tator current. Index Term Dual three phae IPMSM, dual three phae inverter, SVPWM. NOMENCLATURE R, N i the et of real and natural number. x R i a real caler, while x R n (bold) i real valued vector with n N. x i the tranpoe of x. X R n m (capital bold) i a real valued matrix with n N row and m N column. I n R n n := diag(1,..., 1): identity matrix, while O n R n n := diag(,..., ): zero quare matrix. x a1 c R n i the tator pace vector expreed in the natural (a 1 b 1 c 1 -a b c ) reference frame and may repreent voltage u (in V), flux linkage ψ (in Wb) or current i (in A), i.e. x {u, ψ, i}. The ubcript donate referencing to the tator. The electromechanical and load torque (in Nm), the rotor electrical angular peed (in rad/), electrical angular poition (in rad) with repect to the reference flux axi of phae a, and inertia (in kgm ) are repreented by m m, m load, ω k, φ k, and Θ, repectively. The vicou friction i repreented by ν (in N m 1 ). The electrical reitance (in Ω) and inductance (in H) are indicated by R and L, repectively /17$1. c 17 IEEE I. INTRODUCTION The lat decade experienced an increaing interet in multiphae machine in general; owing to their inherent faulttolerance capability, enhanced efficiency, rating reduction of the employed voltage ource inverter (), and lower torque ripple compared to conventional three phae machine [1] [7]. Dual three phae machine (DTM) are of particular interet, ince the winding of conventional three phae machine could be replaced without, dramatic change, by ix phae winding. The employment of DTM i addreed from three main apect: machine deign [], [], neutral point connection [5], and proper ynthee of the demanded reference voltage [], [] []. Aymmetrical DTM (ADTM) (i.e. patial phae hift of π/ rad between the two three phae et a 1 b 1 c 1 - a b c ) intrinically attain the ability to provide an improved harmonic content of flux linkage, thu enhanced efficiency and lower pace harmonic. When combining uch machine with a pace-vector pule-width-modulation-baed (SVPWM), witching loe and witche commutation are minimized, along with enhanced tator current quality. Upon applying the vector pace decompoition technique [7] [9], the choice of the variouly dicued SVPWM cheme i dependant on: (i) patial phae hift between the two three phae et (in thi paper π/ rad), (ii) the effect of the choen witching vector on exciting (xy) harmonic and zero-equence ( + ) current, and (iii) challenge concerning practical digital implementation of the choen SVPWM cheme. It wa pointed out in [9] that the impedance of the equivalent circuit in the xy and + plane of a practical DTM could be a few percentage of the fundamental impedance. Therefore, a minor degree of unbalance of the upplied tator voltage, at uch low harmonic impedance, could eaily excite the harmonic plane with ignificant current that would only contribute to the tator ohmic loe and the emiconductor loe. Achieving minimum current in both xy and + plane i atteted by minimizing per witching period the ditance between the reultant voltage in xy and + plane and the origin (i.e. minimiing thoe current from control point of view); and/or maximizing their equivalent impedance (i.e. machine deign point of view), where the former i the main topic in thi paper. Iolating the two neutral point (N) of the dual three phae 95

2 et ha the conequence of completely eliminating the zeroequence current and higher utilization of the available DClink voltage. Accordingly, the N configuration wa endored in everal publication [] [], epecially for DT-induction machine (DT-IM), howing everal SVPWM manipulation and the correponding effect on the tator current harmonic content, fundamental component, and witching loe. For DT-permanent magnet ynchronou machine (DT-PMSM), the non-ideal non-inuoidal rotor flux would reult in a non-inuoidal back-electromotive force (emf ), which would contribute, beide the DT-inverter, to the harmonic current. Thu, etting the reference voltage to zero in the harmonic plane would only compenate for the inverter harmonic, leading to ditorted tator current with ignificant torque ripple [1]. In term of practical realization, although the 1-ector SVPWM method require le memory than the -ector method to tore the offline precomputed dwell time per ector, it face challenge to implement digitally and may require an additional hardware board to ait in ucceful implementation []. Moreover, the total number of tranition (i.e witching from ON OFF) i coniderably high to yntheize the deired reference voltage. Thu, the -ector PWM technique wa introduced in [7], [], [11], implifying the digital implementation and reducing the harmonic content. Thi paper illutrate the neceity to account for nonzero harmonic reference and encompa them within the precomputed dwell timing for the 1-, -ector SVPWM technique utilized for N configuration; making it uitable for DT-PMSM. Alo, a imple digital implementation trategy i preented, which i uitable for the aforementioned SVPWM technique (i.e. 1-, -ector and the modified -ector method) and accommodate both continuou and dicontinuou SVPWM method. Thi trategy could alo be ued a a plug-in that can be extended to include the implementation of SVPWM trategie within the over-modulation region [1]. In addition to thi, the fundamental for an adequate SVPWM for a 1N connection will be demontrated, ince the 1N connection i characterized by better fault-tolerance capability [5]. The preented theoretical finding are experimentally teted on a. kw ADT-interior PMSM (ADT-IPMSM). II. ASYMMETRICAL DUAL THREE PHASE DRIVE MODEL A. Dual three phae IPMSM Model Motivated by the implicity of modelling and controlling of early DC machine, where tationary orthogonal armature (i.e. tator) flux and field flux exited, the Clarke tranformation wa introduced for three-phae drive, decompoing the π/ patially ditributed three-phae fluxe into two-orthogonal αβ flux axe in the ame ubpace (i.e. electromechanical energy converion ubpace), and one remaining axi orthogonal to the αβ ubpace, which i alo known a the zero "" ubpace. Upon applying the rotating Park tranformation, the machine model attain imilaritie with earlier DC machine, where the d-axi component i directly correlated with the flux within the air-gap and the orthogonal q-axi component control the generated electromechanical torque m m. With the development of electrical machine, a in thi cae IPMSM, a well a the different optimiation technique, both the dq fluxe are optimied for a given m m to minimie the tatorcoper loe [1]. Similarly, for a DTM drive, a generalied Clarke tranformation, alo known a vector pace decompoition (VSD), T VSD = , (1) mapping the ix-phae to three orthogonal plane, namely the equivalent αβ ubpace, + in the "" ubpace repreenting the zero equence component from both threephae et, and the xy ubpace, which directly control the degree of unbalance between the three-phae et and derating factor during fault [1], [], [5]. Uing the generalied Park tranformation [1] T p (φ k ) 1 O O T p (φ k ) 1 = O T p ( φ k ) 1 O, () O O I [ where T p (φ k ) 1 co(φk ] ) in(φ k ) =, the ADTM-IPMSM in(φ k ) co(φ k ) dynamic model, neglecting aturation, i u k = R i k + ω k Jψ k + d dt ψk, u xy = R i xy ω k Jψ xy + d dt ψxy, u = R i + d dt ψ, Θ n p dω k dt = n p i k J ψ k }{{} :=m m (m load + ν n p ω k ), where [ J := ] T p (π/) 1, ψ k = L k i k + ψpm, k L k = Ll +L d, i k = (, i d, i) q the leakage and dq-axe L l +L q inductance are denoted by L l, L d, and L q, repectively, ψpm k = ( ψ pm, ), ψ xy = L l i xy, i xy = ( ), i x, i y ( ). ψ = L l i, i = i +, i B. Dual three-phae Inverter Model 1) 1N/N inverter model and voltage limit: Di-, connecting the tar point (i.e. N and 1N mode) through the deignated witch in Fig. 1 i crucial; ince it define the degree of freedom (i.e. number of independent current to be controlled), which are four for the N configuration and five for the 1N connection. Auming balanced phae voltage, the phae voltage u ph := ( u a1, u b1, u c1, u a, u b, u c () ) can 9

3 + u dc i dc a1 C dc a1 u a1 i a1 b1 b1 i b1 u b1 c1 c1 u c1 i c1 a a i a u a b b i b u b c a 1 b 1 c 1 a b c c i c u c 1N N Figure 1: ADT drive howing the poible 1N/N connection. be expreed for N a [] 1 1 u ph = u 1 1 dc , () and for 1N [] u ph = u dc , (5) where := ( a1, b1, c1, a, b, c i the witching vector, where each of it element repreent the witching tate per phae κ {, 1} and κ {a 1, b 1,..., c, }, where the value and 1 indicate that the upper witch i turned OFF or ON, repectively, while the oppoite applie to the lower witch. ) Space Vector repreentation: The inverter in Fig. 1 ha poible witching tate, which can be repreented in a binary form (i.e. in decimal form ) where the mot ignificant bit correpond to c and the leat one for a1 (in conitency with [] []). Each tate can be mapped to the αβ, xy, and + ubpace with different magnitude and orientation, obtained for N and 1N cae by multiplying equation () and (5) with T VSD ; leading to the pace vector indicated in Fig. (a), (b) and (c), repectively. Note that for the N cae, the path for the zero equence current i blocked and all witching vector at that cae in + ubpace are inhibited to the origin. It i oberved in Fig. (a) that the voltage vector with highet amplitude in correpond to the the minimum voltage vector in the xy ubpace, while the minimum voltage magnitude in the αβ ubpace correpond to the highet one ) in the xy ubpace, thu inducing the mot ignificant i xy. Accordingly, the minimum voltage magnitude in the αβ plane are excluded from the elected voltage vector when carrying out the SVPWM. III. SVPWM AND CIRCULATING CURRENT MINIMIZATION In literature, there are two claification of SVPWM applied for DTM with uch type of inverter: 1 ector and -ector. Each can operate in either the continuou mode (CSVPWM) (ee Fig. (a)-i, iii), where all inverter witche are toggled per witching period or the dicontinuou mode (DSVPWM), where at leat one witch i clamped to the poitive or negative DC-link terminal per witching period T w ) a hown in Fig. (a)-ii. The former mode i known for it implicity, but higher witching loe compared to the latter, which achieve better harmonic performance [] []. For a N configuration, four ditinct dwell time for the active vector t V and t V = T for the zero vector, correponding to the t Vn n=1 equivalent volt-econd covered by the reference u αβ,ref and u xy,ref at the ampling intant T = T w /. The dwell time are computed for the 1-, -ector method by [] [] t u α V1,1 u α, u α, u α 1, u α,ref t V t V = u β,1 u β, u β, u β, u β,ref u x,1 u x, u x, u x, u x t V u y,1 u y, u y, u y,ref T, (), u y,ref where each of the column element of the invere matrix i the reult of ubtituting the elected active voltage vector in (), with the exception that u xy,ref. For intance, auming the reference u αβ,ref i a hown in Fig. (a), the poible vector for the CSVPWM a in [7] within ector I are, 1, 9, 11,, where i divided equally among vector and ( ee Fig. (a)-i ) [7]. Since the ADT-voltage dodecagon i ymmetric, a number of unique dwell time can be computed offline, tored on the real-time DSP; reducing the execution time of the SVPWM. For intance, it i required to tore dwell time for the 1-ector method [], 1 dwell time for the -method [], and for the modified -ector method in [7], which i the -ector method adopted in thi paper. However, ince u xy,ref and by excerpting the elected witching vector in [] [], the memory requirement of the real-time platform to tore the offline-precomputed dwell time would double, to account for u xy,ref. On the other hand, for a 1N configuration, the triplet harmonic are no longer retrained to zero. Within the + ubpace, i + = i, which i obervable in Fig. (c). The additional degree of freedom add an additional degree dwell time t V5. It i worth to mention that for the 1N connection the zero vector for SVPWM are reduced to only two vector and, excluding the vector 5 and 7 which have highet magnitude in the + ubpace a highlighted with in Fig. (c). Since the aim i to minimize harmonic current excitation, either 5 and 7 would be excluded from the poible voltage vector (i.e. imilar to neglecting vector 97

4 β y u αβ,ref I α 5 I x (a) 7 (b) t 5 = t + T 11 T 5 t 11 = t 1 + T t = t + T 1 t 9 = t + T 9 t 1 = T 5 t 7 = t + T 5 t = t + T 9 t 1 = t 9 + T 1 t = t 1 + T t = t 7 + T 11 u (I-) 57 5 t = t 5 + T x a 1 b 1 c 1 a b c T w (c) (d) Figure : Voltage pace vector in the (a) fundamental αβ plane with reference voltage u ref αβ, (b) harmonic xy plane (voltage vector (, 7, 5, ) at their origin, (c) + plane howing reidual voltage u (I ) for ector I- per T and (d) the correponding timing diagram. highlighted in for the N cae) or ue both per halfcycle a in the CSVPWM cae (vector and ), where the computed t V5 would be ditributed evenly among them; leading to average zero voltage per T w. For intance for the ame witching equence ued for ector I in DSVPWM upon adding the proper zero vector, the witching equence would be ( ). It i alo worth to mention, that due to the reduced number of zero vector to only and, the DSVPWM would not exit in the 1N mode, leading to higher witching loe compared to the N cae. At the moment, different tet and witching equence are invetigated and will be reported oon in the near future. IV. DIGITAL SVPWM IMPLEMENTATION An important apect for any PWM technique i the degree of complexity involved for digital realization. One of the main motivation for adopting the -ector SVPWM method, compared to 1-ector SVPWM, wa the eaine of converting the computed timing into duty cycle (ratio between ON time and witching period T w ) paed to ix timer, each reponible 9

5 a1 b1 c1 a b c T t V t V CSVPWM DSVPWM CSVPWM1 t V T T T w t V t V t V x T a1 b1 c1 a b c T t V t V t V t V t V x t V T T w a1 b1 c1 a b c T T T x t T V t V t t t T V V V t V (a)-i (a)-ii (a)-iii T w T w T a1 T w a1 T w a1 T w b1 b1 b1 c1 c1 c1 (b)-i (b)-ii (b)-iii a T w a T w a T w b b b c c c (c)-i (c)-ii (c)-iii Figure : (a) Simulation and (b)-(c) experimental witching pattern for CSVPWM, DSVPWM, and CSVPWM1 for u αβ,ref in Fig. (a). for modulating the voltage per inverter leg []. In [7], the developed method enable the decoupling of the ix phae into two three phae modulated with two decoupled three-phae SVPWM. For intance, the witching tate in ector I for the -ector method (labeled I- in Fig. (a)) in the continuou mode i ( ), with the correponding timing diagram in Fig. (d). It clear that each three-phae et tart the witching period with the ame high/low logic per et, which i the cae for any other ector or SVPWM method. Upon identifying the tarting logic, and plotting the timing diagram for the remaining ector, the duty cycle per phae can be computed per ector by oberving it relation with the witching tate timing. For intance, the ON time for phae a i (T 1 + T 9 + T 11 + T ), while for phae u i T 5. Alternatively, in thi paper a imple propoed method i capable of realizing both method and alo uitable for available DSP, which developed rapidly throughout the lat decade. By identifying the tarting logic per et per ector, the ON and OFF time, each leg tate can be toggled upon completing the ON/OFF time. For intance, for ector I- in Fig. (a)), a1 tart with a "" logic, toggled to "1" after T 5 +T (half the OFF time), and witched back to zero upon completing the ON time. By aigning the repective timing per witching tate per ector (i.e. T 5, T, T 1...etc.) to the repective t V, one can implement any deired SVPWM techinque, including the 1-ector method. For ector 1 of the 1-ector SVPWM (labeled I-1 in (a)), the timing diagram i hown in Fig. (c). For phae a, the tarting logic i "1", and a1 toggle at (T / + + t V ), T /, (T / + t V + t V ). Uing of the ymmetry per T w /, the remaining toggling intant are alo identified. V. EVALUATION RESULTS The SVPWM trategy in term of implementation and harmonic inhibition ha been experimentally evaluated by mean of the tet bench hown in Fig. and with ADT- 99

6 I Figure : Tet bench: (I) ADT-IPMSM, (II) induction machine and (III) torque enor. % of fundamental III Harmonic order II Figure 5: Harmonic content in the back emf of the ADT-IPMSM. Table I: Parameter of the ADT-IPMSM under tudy Paramter Value Stator reitance R =. Ω Pole-pair n p = d-axi inductance L d = 5.5 mh q-axi inductance L q = 1.5 mh Permanent magnet flux linkage ψ pm = 17. mwb Leakage inductance L l =.9 mh Rated tator current i,rated =.1 A Rated torque m m,rated = 1. Nm Rated power P rated =. kw IPMSM parameter hown in Table I. The two three-phae two level are witched at a rate of khz with a dead-time of 1µec. The control algorithm i implemented on dspace DS17 by mean of Matlab/Simulink interface. The PWM ignal are ent to the via the DS511 PWM card, which i manually flahed and programmed to enure precie timing and ampling intant. The A/D DS board i triggered by the DS511 board to carry-out the meaurement in the middle of the witching period, enuring minimum witching noie interference. The field-oriented control ha been carried out a in [1], while the xy harmonic were compenated by mean of a proportional-reonant (PR) controller, dicretized by mean of impule-invariant method [1]. The effectivene of the propoed witching/timing algorithm and it uitability for different SVPWM ha been evaluated by etting the ame reference voltage vector u αβ,ref (ee Fig. (a)) in ector I- and I-1. Fig. how that the output of the imulated witching/ampling algorithm matche their equivalent experimental equivalent. Since the -ector technique i advantageou in term of harmonic content, it will be employed in the upcoming reult to evaluate the effect of excluding and including the xy ubpace regulation. The induction machine run a a prime mover rotating at a contant peed of 5 rpm, while the ADT-IPMSM operate in generator mode. The torque command for the ADT-IPMSM wa et to the rated torque, correponding to optimum current reference i k,ref = (.51, 5 ) A, which were precomputed uing the maximum-torque-per-current (MTPC) criteria, a explained in [1]. The harmonic content of the back emf i hown in Fig. 5. The preence of 5 th and 7 th harmonic in the back emf make it inufficient to control the i xy by jut etting u xy,ref =. A hown in Fig. (a), the preence of ixy lead to a highly ditorted tator current (ee Fig. (b)), even though the fundamental current are well regulated. Analyzing the tator current by mean of FFT how ubtantial 5 th and 7 th harmonic content, reulting in a THD = 1.%, coinciding with imilar practical finding a hown in [1]. Upon conidering the effect of compenating the xy current component a in [1], along with propoed SVPWM, ignificant improvement ha been een in term of harmonic compenation a evident in Fig. (d) and (e) a well a enhancing the quality of the harmonic content of the tator current with THD = 1.9%. VI. CONCLUSION SVPWM technique for ADTM are well known to yntheize the deired reference voltage, enuring a high quality output, optimum witching equence, and minimum witching loe. Applying the different SVPWM method for ADT- IPMSM a when applying them for ADT-IM would lead to a ignificant dicrepancy in term of the quality of the tator current; owing to the non inuoidal PM flux which compenate, practically, low order harmonic a well a dicarding the contribution of the harmonic reference voltage to the calculation of the dwell time. Upon taking thi into account, ignificant improvement wa overeen in the quality of the output current a well a nulling the xy current. Alo, the fundamental for 1N SVPWM wa laid which hould be reported in the near future. On the other hand, a imple SVPWM digital realization technique ha been propoed which i uitable for implementing the different SVPWM method, a covered in literature. The theoretical finding were corroborated with experimental meaurement on a. kw ADT-IPMSM for the ake of validation. ACKNOWLEDGMENT Thi work i upported by the project AWESCO (H- ITN-) funded by the European Union Horizon reearch and innovation programme under the Marie Sklodowka-Curie grant agreement No.. REFERENCES [1] I. Gonzalez-Prieto, M. J. Duran, H. S. Che, E. Levi, M. Bermúdez, and F. Barrero, Fault-tolerant operation of ix-phae energy converion ytem with parallel machine-ide converter, IEEE Tran. on Power Electron., vol. 1, no., pp. 79, Apr. 1. [] E. Levi, R. Bojoi, F. Profumo, H. A. Toliyat, and S. Williamon, Multiphae induction motor drive - a technology tatu review, IET Electr. Power Appl., vol. 1, no., pp. 9 51, Jul. 7. 9

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