Model predictive control with constant switching frequency using a discrete space vector modulation with virtual state vectors

Model P edic i e Con ol wi h Cons an Swi ching F equency Using a Disc e e Space Vec o Modula ion wi h Vi ual S a e Vec o s Abs ac — Fini e S a es Model P edic i e Con ol (FS-MPC) appea s as a p omising con ol echnique o be applied o powe con e e s in he indus y. Howe e , he FS-MPC p esen s some d awbacks as non cons an swi ching equency and high sampling equency. This wo k p oposes a FS-MPC wi h cons an swi ching equency and low sampling equency applying a Disc e e Space Vec o Modula ion (DSVM) echnique. The eal s a e ec o s o he con e e a e used oge he wi h new i ual s a e ec o s o ming swi ching sequences each sampling pe iod. The ad an ages and disad an ages o he p oposed FS-MPC wi h he DSVM a e analyzed using a wo-le el h ee-phase in e e connec ed o he g id as se up o in oduce he p oposed echnique. Simula ion esul s a e p esen ed, showing ha using he p oposed echnique he swi ching equency is ixed and he sampling equency can be lowe ed wi hou educing he quali y o he con e e beha io . I. INTRODUCTION SINCE powe con e e s appea he de elopmen o di - e en applica ions and hei associa ed con ol s a egies ha e been a challenge in he powe elec onics ield. Se e al con ol echniques ha e been applied o he powe con e e s. Linea con olle s and non linea con olle s including adap i e con ol, epe i i e con ol, uzzy con ol and p edic i e con ol ha e been widely used in di e en applica ions as ec i ie s, in e e s, ac i e powe il e s, unin e up ible powe supplies and mo o s con ol [1]–[4]. Among he p edic i e echniques, deadbea con olle s and Model P edic i e Con ol (MPC) ha e s and ou due o hei good p ope ies [5]–[8]. The implemen a ion o MPC o powe con e e s can be di icul due o he la ge amoun o calcula ions needed o sol e online he op imiza ion p oblem. To o e come his, wo solu ions has been p oposed: o line calcula ion o he op imal solu ions [9], and calcula ion o he op imal solu ion by e alua ion o all possible swi ching s a es. This las solu ion has been called Fini e S a es Model P edic i e Con ol (FS-MPC), because i akes in o accoun only he ini e numbe o possible swi ching s a es, and can be easily implemen ed using s anda d con ol ha dwa e. In he pa icula case o FS-MPC, se e al applica ions as ec i ie [10], in e e [11], [12], unin e up ible powe supply [13] and mo o con ol [14], ha e shown ha his ype o con ol p esen s high pe o mance wi hou he necessi y o adjus con olle pa ame e s o ob ain an op imized beha io o he o e all sys em. In despi e o he good ea u es shown, he FS-MPC has wo main d awbacks ha makes e y di icul i s use in eal indus ial powe con e e s. The FS-MPC needs a high sampling equency o achie e high pe o mance, his en ails high compu a ional cos and he e o e expensi e DSP o FPGA ha dwa e. Besides FS-MPC applica ions show non cons an swi ching wi h a widesp ead spec um o he ou pu cu en s and ol ages. This ac leads o he necessi y o la ge induc o and capaci o s alues o he connec ion passi e il e s, inc easing he weigh , olume and cos o he powe con e e . These d awbacks a e p esen in o he disc e e con olle s as Di ec Powe Con ol (DPC) in ec i ie s applica ions and Di ec To que Con ol (DTC) in mo o con ol [15], [16]. Bo h con ol echniques a e based on choosing in a look up able (LUT) he bes ou pu s a e o minimize he e o s be ween he command e e ences and he ac ual alues o ins an aneous ac i e and eac i e powe (DPC), o o que and lux (DTC). To enhance he pe o mance o DTC he use o a Disc e e Space Vec o Modula ion (DSVM) has been p oposed [17]– [22], imp o ing he d i e pe o mance h ough he use o new s a es wi h a p e ixed ime in e als. In his pape his concep is ex ended o he FS-MPC and he use o DSVM o ca y ou he FS-MPC is p oposed. This new echnique is called MPC-DSVM. The p oposed me hod allows o achie e he con ol objec i es wi h low sampling equency and cons an swi ching equency. To assess he MPC-DSVM, simula ions esul s a e p esen ed using a con en ional wo- le el h ee-phase in e e connec ed o he g id. Finally he ad an ages and disad an ages o he p oposed FS-MPC wi h he DSVM a e analyzed. II. FS-MPC FS-MPC is based on he minimiza ion o a de ined cos unc ion s udying he p edic ed esponse o each disc e e ou pu s a e o a powe con e e . Each sampling ime (Ts), he FS-MPC e alua es all possible ou pu s a es and chooses he s a e ha minimizes a cos unc ion as he s a e o be applied o he con e e . FS-MPC echnique p esen s he ad an ages and disad an ages summa ized in Table I and Table II espec i ely. Among he mos aluable cha ac e is ics o FS-MPC i can be no iced ha high pe o mance con olle s can be implemen ed s aigh o wa d. Howe e his equi es high sam- pling equency, which de e mines he con ol ha dwa e o TABLE I ADVANTAGES OF FS-MPC CONTROLLERS Easy implemen a ion wi h high pe o mance No necessi y o cons an s con olle uning p ocedu es Easy o include any pa ame e op imiza ion c i e ia in he cos unc ion Takes in o accoun he disc e e na u e o he con e e TABLE II DRAWBACKS OF FS-MPC CONTROLLERS Needs high sampling equency s o ob ain high pe o mance P esen s non cons an swi ching equency sw , limi ed by s/2 Does no use he comple e con e e con ol egion be used o con ol he powe con e e . Ano he no iceable ea u es a e he no necessi y o cons an s con olle uning p ocedu es and ease o include any cons ain o pa ame e op imiza ion c i e ia in he cos unc ion. Finally, FS-MPC akes in o accoun he disc e e na u e o he con e e . As a consequence, FS-MPC does no use he comple e con e e con ol egion, only he disc e e eal s a es o he con e e (Fig. 1). This ac oge he wi h he possibili y o he cos unc ion o choose se e al consecu i e imes he same disc e e eal s a e as he op imum s a e, leads FS-MPC o p esen non cons an swi ching equency( sw). Besides sw can be as high as hal he sampling equency( s)due o he e is no es ic ion in he swi ching sequence unless i is included in he cos unc ion. FS-MPC con olle s a e also cha ac e ized by he way hey gene a e he powe swi ch i ing pulses. In compa ison wi h ex e nal modula o based con olle s, FS-MPC does no need a modula o o gene a e he i ing pulses. The powe swi ch ga e signals can be ob ained om a LUT, whe e he co espondence be ween he disc e e eal s a es o he con e e and he ga e signals a e s o ed. Al hough i is no necessa y, i is possible o gene a e he FS-MPC i ing pulses using an ex e nal modula o as i is shown in Fig. 2. The use o an ex e nal modula o does no complica e he ha dwa e used in he applica ion, bu opens new possibili ies o FS-MPC as he comple e con e e con ol egion is easily accessible. III. MPC-DSVM FS-MPC only conside s he con e e disc e e eal s a es o p edic he sys em esponse and o e alua e he FS-MPC cos unc ion. Al hough his leads o high dynamic pe o mance o he sys em, he comple e con ol egion o he con e e is no used and as a consequence some d awbacks appea as was shown in sec ion II. The idea o MPC-DSVM is o use no only he con e e disc e e eal s a es bu in addi ion o he poin s o he con ol egion called i ual ec o s [23]. These i ual ec o s can be loca ed in any posi ion o he con e e con ol egion and ha e o be syn hesized by a linea combina ion o he con e e disc e e eal s a es. Fig. 3 shows he con ol egion o he wo-le el h ee-phase con e e whe e he con e e disc e e eal s a e ec o s ( ound ma ks) and wel e addi ional i ual ec o s (squa e ma ks) ha e been ep esen ed. In gene al, any numbe o i ual ec o s can be plo ed in he con ol egion. The con e e i ual ec o s can be gene a ed as a linea combina ion o he disc e e eal s a es o he con e e by using a disc e e space ec o modula ion (DSVM). The DSVM echnique is a space ec o modula ion based on he use o a ini e numbe o i ual ec o s placed in he con ol egion o he powe con e e ins ead o using he whole con e e α β (100) V1 (000) V0 (111) V7 (110) V2 (010) V3 (011) V4 (001) V5(101) V6 Con e e con ol egion Con e e eal s a es Fig. 1. Con ol egion o a wo-le el h ee-phase con e e in he αβ ame. Sa,k+1 = 0 Sb,k+1 = 0 Sc,k+1 = 1 Sa,k = 1 Sb,k = 0 Sc,k = 1 α β (100) V1 (000) V0 (111) V7 (110) V2 (010) V3 (011) V4 Con e e con ol egion Con e e eal s a es (001) V5(101) V6 a b TsTs Sa,k Sa,k+1 PWM TsTs On O LUT Ga e Signal Ga e Signal Fig. 2. Gene a ion o he powe swi ch i ing pulses wi h FS-MPC. a) Th ough LUT. b) Th ough ex e nal PWM modula o . α β (100) V1 (000) V0 (111) V7 (110) V2 (010) V3 (011) V4 (001) V5(101) V6 Con e e con ol egion Con e e eal s a es Con e e i ual s a es (1;0.5;0) Vk Fig. 3. Con ol egion o a wo-le el h ee-phase con e e in he αβ ame including wel e i ual s a es. con ol egion. The concep o DSVM was in oduced in [17], [18], and has been used o educe he o que and cu en ipple in DTC applica ions [19]–[22]. Any i ual ec o i can be achie ed by a linea combi- na ion o h ee eal ec o s V eal, whe e each eal ec o is applied in a swi ching sequence a ce ain ime quan i y. i =X j=1,2,3 jV eal j 1+ 2+ 3=Tsw (1) V eal j∈ {V0, V1, . . . , V7} In (1), Tsw ep esen s he swi ching ime, i Tsw is ixed equal o all he i ual ec o s hen Tsw is ans o med in he swi ching pe iod and he swi ching equency o MPC-DSVM is ixed, sol ing in his way one o he mayo issues o con en ional FS-MPC. As in FS-MPC, he numbe o ec o s o be e alua ed is ini e so i is possible o use a LUT o s o e he swi ching ec o sequence and imes. Howe e when he numbe o i ual ec o s inc eases a be e solu ion is o use an ex e nal modula o o gene a e he powe swi ch ga e signals as i is shown in Fig. 4. IV. SIMULATIONS RESULTS MPC-DSVM echnique can be employed in any powe con- e e applica ion ha uses FS-MPC as con ol echnique. Fo he shake o simplici y, o p esen he concep o MPC-DSVM a con en ional wo-le el h ee-phase in e e connec ed o he g id is used. The h ee-phase wo-le el powe con e e is depic ed in Fig. 5, whe e he neu al poin is deno ed by n. The sys em is connec ed o he g id h ough smoo hing induc o s Land a cons an DC ol age sou ce Vdc is connec ed in he DC e minals o he con e e . The sys em pa ame e s and a iables a e desc ibed in Table III. The equa ions ha desc ibe he inpu cu en s dynamics can be de i ed om he sys em model in he s a iona y αβ ame. The a iables in he s a iona y ame a e calcula ed as {·}T αβ = A{·}T abc. αβ =Ldiαβ d +Vdc 2Sαβ (2) A=q2 3"1−1 2−1 2 0√3 2−√3 2#(3) The con ol objec i e o he h ee-phase powe in e e is he inpu cu en acking owa ds hei e e ences, i∗ αand i∗ β TABLE III SYSTEM VARIABLES AND PARAMETERS Va iable Desc ip ion ={ an bn cn}TPhase o neu al inpu ol age ec o i={iaibic}TPhase inpu cu en ec o S={SaSbSc}TCon ol inpu ec o ωG id equency LSmoo hing induc o Vdc DC ol age sou ce alue a b TsTs Sa,k Sa,k+1 PWM TsTs On On/O /OnLUT Ga e Signal Ga e Signal Sa,k+1 = 0.5 Sb,k+1 = 0 Sc,k+1 = 1 Sa,k = 1 Sb,k = 0 Sc,k = 1 α β (100) V1 (000) V0 (111) V7 (110) V2 (010) V3 (011) V4 Con e e con ol egion Con e e eal s a es Con e e i ual s a es (001) V5(101) V6 Fig. 4. Gene a ion o he powe swi ch i ing pulses wi h MPC-DSVM. a) Th ough LUT. b) Th ough ex e nal PWM modula o . a b c L L L s dc V a i c i b i an cn n bn Fig. 5. Th ee-phase wo le el powe in e e . espec i ely. These e e ences can be calcula ed in such a way ha ce ain alues o ac i e and eac i e powe a e injec ed o consumed om he g id, howe e his is ou o scope o his pape . iα→i∗ α(4) iβ→i∗ β(5) In [11] a FS-MPC cu en con olle o he h ee-phase wo-le el in e e is p oposed. This con ol algo i hm has been modi ied as i is shown in Fig. 6, whe e he numbe o disc e e ou pu s a es unde conce n is n , including he e alua ion o he eal and i ual s a e ec o s. The inal ha dwa e con igu a ion is p esen ed in Fig. 7 showing he powe con e e diag am block, whe e an ex e nal modula o has been used o gene a e he i ing pulses. Se e al simula ions ha e been pe o med o analyze he in luence on he powe con e e pe o mance when i ual ec o s a e used diminishing he sampling equency. In Table IV he pa ame e alues used in he simula ions a e sum- ma ized. Fou di e en si ua ions ha e been conside ed. The Fig. 6. Flow diag am o he MPC-DSVM con ol algo i hm. L PWM-SVM Ga e Signals Con e e n Fig. 7. Powe con e e diag am block using MPC-DSVM. i s one is a FS-MPC e alua ing only he eigh eal s a e ec o s using high sampling equency. Secondly he FS-MPC pe o mance is e alua ed when he sampling equency is i e imes slowe . The hi d expe imen uses he p oposed MPC-DSVM wi h he same sampling equency as he second si ua ion, bu e alua es hi y eigh ou pu ec o s a es (eigh eal and hi y i ual s a es). In his way he compu a ional e o has been kep cons an compa ed wi h expe imen 1. Finally MPC-DSVM wi h a highe numbe o i ual ec o s is conside ed bu he sampling equency is kep as in he second and hi d expe imen . In Table V a e shown he sampling equency and he TABLE IV SIMULATION PARAMETERS VALUES Pa ame e Value an, bn, cn 230V ms i∗ a,peak, i∗ b,peak, i∗ c,peak 20A ω50Hz L2mH Vdc 750V TABLE V GRID CURRENT THD VALUES sn cu en THD Expe imen 1 50 kHz 8 2.0% Expe imen 2 10 kHz 8 23.1% Expe imen 3 10 kHz 38 11.0% TABLE VI GRID CURRENT THD VALUES FOR HIGH n s:10 kHz n 62 332 1262 4922 cu en THD 7.2% 3.7% 2.0% 1.8% numbe o disc e e ou pu ec o s (n ) used in expe imen s 1, 2 and 3, oge he wi h he cu en o al ha monic dis o ion (THD) alue ob ained in each case. The cu en THD has been calcula ed up o ha monic 49 h. I can be no iced ha expe imen 1, co esponding wi h FS-MPC echnique, p esen s he lowes ha monic con en . Expe imen 2 also co esponds wi h FS-MPC, bu as he sampling equency has been educed hen he pe o mance o he con olle is highly de e io a ed. Finally, he p oposed MPC-DSVM is pe o med in expe imen 3. When compa ed wi h expe imen 2, he THD alue has been educed in 50%, howe e his alue is highe han THD ob ained wi h FS-MPC in expe imen 1. Fig. 8, 9 and 10 show he g id cu en wa e o m o phase a, i s e e ence and i s ha monic spec um o expe imen s 1, 2 and 3 espec i ely. I can be no iced in Fig. 8 ha cu en ipple o FS-MPC wi h high sampling equency is e y small. In addi ion, ha monic con en is small bu he spec um is widesp ead. In Fig. 9 he cu en ipple has inc eased as he sampling equency is lowe , besides he spec um is widesp ead as FS-MPC is used. When MPC-DSVM is em- ployed he cu en ipple achie es lowe alues ha FS-MPC wi h he same sampling equency as i is shown in Fig. 10. Besides, in he ha monic spec um can be no iced ha equency componen s a e less sp ead and a peak appea s in he swi ching equency. In expe imen 4 is s udied he pe o mance o MPC-DSVM when he numbe o addi ional i ual ec o s is inc eased. Table VI shows he cu en THD o di e en alues o n . I can be no iced ha THD diminishes as he numbe o addi ional i ual ec o inc eases, being possible o ob ain e en lowe THD han wi h FS-MPC. In Fig. 11 he esul s ob ained wi h n = 4922 a e shown, i can be obse ed ha cu en ipple is almos iden ical o he FS-MPC con olle wi h high sampling equency in expe imen 1, bu in con as ha ing a ixed swi ching equency. To s udy a high numbe o ou pu s a es p o ides high pe o mance o MPC-DSVM. Howe e , MPC-DSVM has o e alua e all he disc e e ou pu ec o s o minimize he cos unc ion. The con ol ha dwa e de e mines he maximum numbe o disc e e ou pu s a es ha can be s udied each sampling ime, limi ing in his way he pe o mance o he sys em. 0 0.01 0.02 0.03 0.04 ÿ35 0 35 Time(s) ia,i* a(A) 0 5000 10000 15000 0 3.5 7 F equency(Hz) THD(%) Fig. 8. Simula ion esul s o s= 50kHz and n = 8. F om op o bo om: a) Phase a cu en and e e ence. b) Phase a cu en ha monic spec um. 0 0.01 0.02 0.03 0.04 ÿ35 0 35 Time(s) ia,i* a(A) 0 5000 10000 15000 0 3.5 7 F equency(Hz) THD(%) Fig. 9. Simula ion esul s o s= 10kHz and n = 8. F om op o bo om: a) Phase a cu en and e e ence. b) Phase a cu en ha monic spec um. V. ANALYSIS OF MPC-DSVM Tables VII and VIII summa ize he ad an ages and disad- an ages o p oposed MPC-DSVM. This echnique has he same ad an ages o FS-MPC and includes wo new powe - ul ea u es, cons an swi ching equency and low sampling equency. Bo h cha ac e is ics a e consequence o a be e use o he con e e con ol egion. The cons an swi ching equency makes easie he design o passi e il e s and he low equency educes he con ol ha dwa e equi emen s. On he o he side, when MPC-DSVM is employed new conce ns ap- pea compa ed wi h FS-MPC. MPC-DSVM needs an ex e nal modula o and his o ces a mo e complex con ol ha dwa e. Mode n DSPs include build-in pe iphe als o do his ask o i can be buil easily in a FPGA. Besides a new challenge eme ges in low sampling equency MPC-DSVM. To ob ain simila pe o mance o ha wi h high sampling equency FS-MPC i is necessa y o e alua e a high numbe o disc e e ou pu s a es. This issue could be sol ed wi h a sma choosing o he i ual ec o s o be analyzed in each sampling ime. 0 0.01 0.02 0.03 0.04 ÿ35 0 35 Time(s) ia,i* a(A) 0 5000 10000 15000 0 3.5 7 F equency(Hz) THD(%) Fig. 10. Simula ion esul s o s= 10kHz and n = 38. F om op o bo om: a) Phase a cu en and e e ence. b) Phase a cu en ha monic spec um. 0 0.01 0.02 0.03 0.04 ÿ35 0 35 Time(s) ia,i* a(A) 0 5000 10000 15000 0 3.5 7 F equency(Hz) THD(%) Fig. 11. Simula ion esul s o s=10kHz and n =4922. F om op o bo om: a) Phase a cu en and e e ence. b) Phase a cu en ha monic spec um. TABLE VII ADVANTAGES OF MPC-DSVM CONTROLLERS Easy implemen a ion wi h high pe o mance No necessi y o cons an s con olle uning p ocedu es Easy o include any pa ame e op imiza ion c i e ia in he cos unc ion Takes in o accoun he disc e e na u e o he con e e P esen s cons an swi ching equency sw High pe o mance wi h low sampling equency s Takes be e ad an age o he con e e con ol egion TABLE VIII DRAWBACKS OF MPC-DSVM CONTROLLERS Needs a LUT o an ex e nal modula o o gene a e he i ing pulses Low sinc eases he compu a ional e o o achie e high pe o mance VI. CONCLUSIONS This pape p esen s a Fini e S a es Model P edic i e Con ol (FS-MPC) s a egy wi h cons an swi ching equency. To achie e his ea u e, a Disc e e Space Vec o Modula ion (DSVM) is added o he FS-MPC. The p oposed MPC-DSVM is based on he e alua ion o new ou pu ec o s in he con olle algo i hm. These ou pu ec o s, called i ual ec- o s, a e no eal con e e s a e ec o s and ha e o be syn hesized h ough a ex e nal modula o . I has been shown ha MPC-DSVM p esen s he same ad an ages o FS-MPC and addi ionally can wo k wi h low sampling equency and ixed swi ching equency. Besides, he in luence o he num- be o ou pu ec o s e alua ed ha e been analyzed. I has been shown ha o low sampling equency wi h highe pe o mance, MPC-DSVM equi es a la ge numbe o i ual ec o s. Howe e e en be e pe o mance han FS-MPC can be achie ed, so new challenges ha e been add essed. REFERENCES [1] S. Vazquez, J.A. Sanchez, J.M. Ca asco, J.I. Leon and E.Gal an “A model-based Di ec Powe Con ol o Th ee-Phase Powe Con e e s,” IEEE T ansac ions on Indus ial Elec onics, ol. 55, no. 4, pp. 1647– 1657, Ap il 2008. [2] X. del To o Ga cia, A. A ias, M.G. Jayne and P.A. Wi ing, “Di ec To que Con ol o Induc ion Mo o s U ilizing Th ee-Le el Vol age Sou ce In e e s,” IEEE T ansac ions on Indus ial Elec onics, ol. 55, no. 2, pp. 956–958, Feb ua y 2008. [3] C. LAscu, L. Asiminoaei, I. Boldea and F. Blaabje g, “High Pe o mance Cu en Con olle o Slec i e Ha monic Compensa ion in Ac i e Powe Fil e s,” IEEE T ansac ions on Powe Elec onics, ol. 22, no. 5, pp. 1826–1835, Sep embe 2007. [4] J.M. Gue e o, L. Huang and J. Uceda, “Con ol o Dis ibu ed Un- in e up ible Powe Supply Sys ems,” IEEE T ansac ions on Indus ial Elec onics, ol. 55, no. 8, pp. 2845–2859, Augus 2008. [5] Y.A.R.I. Mohamed and E.F. El-Saadany, “Robus High Bandwid h Disc e e-Time P edic i e Cu en Con ol wi h P edic i e In e nal Model-A Uni ied App oach o Vol age-Sou ce PWM Con e e s,” IEEE T ansac ions on Powe Elec onics, ol. 23, no. 1, pp. 126–136, Janua y 2008. [6] E.F. Camacho and C. Bo dons, Model P edic i e Con ol. New Yo k, N.Y.: Sp inge -Ve lag, 1999. [7] J.M. Maciejowski, P edic i e Con ol wi h Cons ains. Englewood Cli s, N.J.: P en ice-Hall, 2002 [8] S.J. Qin and T. A. Badgwell “A su ey o indus ial model p edic i e con ol echnology,” Con ol Enginee ing P ac ice, ol. 11, pp. 733–764, 2003 [9] A. Linde and R. Kennel, “Model P edic i e Con ol o Elec ical D i es,” IEEE Powe Elec onics Specialis Con e ence (PESC’05), pp. 1793–1799, Reci e, B azil, 12-16 May 2005. [10] P. An oniewicz and M. P. Kazmie kowski, “P edic i e di ec powe con ol o h ee-phase boos ec i ie ,” Bulle in o Polish Academy o Science, ol. 54, no. 3, pp. 287–292,2006. [11] J. Rod iguez, J. Pon , C.A. Sil a, P. Co ea, P. Lezana, P. Co es and U. Ammann, “P edic i e Cu en Con ol o a Vol age Sou ce In e e ,” IEEE T ansac ions on Indus ial Elec onics, ol. 54, no. 1, pp. 495–503, Feb ua y 2007. [12] P. Co es, J. Rod iguez, D.E. Que edo and C.A. Sil a, “P edic i e “P e- dic i e Cu en Con ol S a egy Wi h Imposed Load Cu en Spec um,” IEEE T ansac ions on Powe Elec onics, ol. 23, no. 2, pp. 612–618, Ma ch 2008. [13] P. Co es, J. Rod iguez, “Th ee-phase in e e wi h ou pu LC il e using p edic i e con ol o UPS applica ions,” in Eu opean Con e ence on Elec onics and Applica ions, Aalbo g, Denma k, 2-5 Sep embe 2007 [14] J. Rod iguez, J. Pon , C. Sil a, P. Co es, S. Rees, and U. Ammann, “P edic i e di ec o que con ol o an induc ion machine,” in 11 h In e - na ional Powe Elec onics and Mo ion Con ol Con e ence, EPEPEMC 2004, Riga, La ia, 2-4 Sep embe 2004. [15] T. Noguchi, H. Tomiki, S. Kondo and I. Takahashi. “Di ec Powe Con ol o PWM Con e e Wi hou Powe -Sou ce Vol age Senso s,” IEEE T ansac ions on Indus y Applica ions, ol. 34, no. 3, pp. 473– 379, May/June 1998. [16] I.Takahashi and T. Noguchi,“A New Quick-Response and High- E iciency Con ol S a egy o an Induc ion Mo o ,” IEEE T ansac ions on Indus y Applica ions, ol. 22, no. 5, pp. 820–827, Sep embe 1986. [17] D. Casadei, G. Se a and A. Tani, “Imp o emen o di ec o que con ol pe o mance by using a disc e e SVM echnique,” IEEE Powe Elec onics Specialis Con e ence (PESC’98), ol. 2, pp. 997–1003, May 1998. [18] D. Casadei, G. Se a and A. Tani, “Implemen a ion o a di ec con ol algo i hm o induc ion mo o s based on disc e e space ec o modula- ion,” IEEE T ans. Powe Elec on., ol. 15, no. 4, pp. 769–777, July 2000. [19] Jun eng Xu, Jianping Xu, Yinglei Xu and Fengyan Wang, “Di ec o que con ol o induc ion machines using disc e e space ec o modula ion applied o ac ion,” The Fi h In e na ional Con e ence on Powe Elec onics and D i e Sys ems (PEDS 2003), ol. 2, pp. 1200–1202, 17-20 No embe 2003. [20] H. R. Keyhani, M.R. Zolghad i and A. Homai a , “An ex ended and imp o ed disc e e space ec o modula ion di ec o que con ol o induc ion mo o s,” IEEE Powe Elec onics Specialis s Con e ence (PESC’04), ol. 5, pp. 3414–3420, 20-25 June 2004. [21] D. Ocen, L. Rome al, J. A. O ega, J. Cusido and A. Ga cia, “Disc e e Space Vec o Modula ion Applied on a PMSM Mo o ,” IEEE Powe Elec onics Specialis s Con e ence (PESC’06), pp. 1–5, 18-22 June 2004. [22] B. Singh and D. Goyal, “Imp o ed DSVM-DTC Based Cu en Sen- so less Pe manen Magne Synch onous Mo o D i e,” The Se en h In e na ional Con e ence on Powe Elec onics and D i e Sys ems (PEDS 2007), pp. 1354–1360, 27-30 No . 2007. [23] J. Res epo, J. Viola, J.M. Alle and A. Bueno, “A Simple Swi ch Selec ion S a e o SVM Di ec Powe Con ol,” IEEE In e na ional Sysmposium on Indus ial Elec onics ISIE’06, pp. 1112–1116, Mon- eal, Canada, 9-13 July 2006.