vector weighting approach and vector space decoupling
TRANSCRIPT
Research ArticleVector Weighting Approach and Vector SpaceDecoupling Transform in a Novel SVPWM Algorithm forSix-Phase Voltage Source Inverter
PengWu1 Lei Yuan 2 Zhen Zuo1 and JunyuWei3
1College of Artificial Intelligence National University of Defense Technology Changsha 410073 China2Institute of Communications Engineering The Army Engineering University of PLA Nanjing 210007 China3College of Electric and Information Engineering Hunan University of Technology Zhuzhou 412007 China
Correspondence should be addressed to Lei Yuan leiyuanvqqcom
Received 12 February 2019 Revised 13 April 2019 Accepted 6 May 2019 Published 13 June 2019
Academic Editor Zhixiang Zou
Copyright copy 2019 PengWu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
For six-phase permanent-magnet synchronous motor (PMSM) which has two sets of Y-connected three-phase windings spatiallyphase shifted by 30 electrical degrees to increase the utilization ratio of the DC bus voltage a novel space vector pulse widthmodulation (SVPWM) algorithm in full modulation range capability based on vector weighted method is proposed in this paperThe basic vector action time of SVPWM method is derived in detail employing vector space decomposition transformationapproach Compared with the previous algorithm this strategy is able to overcome the inherent shortcomings of the four-vector SVPWM and it achieves smooth transitions from linear to overmodulation region Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposed strategy
1 Introduction
Compared with the pulse width modulation (PWM) controlalgorithm for conventional three-phase motors the PWMmethod for six-phase motors is more flexible [1ndash6] Atpresent carrier-type PWM algorithm permanent-magnetsynchronousmotor SVPWMalgorithm and current hystere-sis PWM algorithm are three of the PWM algorithms thatare commonly used for multiphase two-level voltage sourceinverter (VSI) [7 8] However the current hysteresis PWMalgorithm is unsuitable for high power algorithms In recentyears carrier-type PWMalgorithms and SVPWMalgorithmsthat can be applied to multiphase motor systems have beenextensively studied
The PWM algorithm for multiphase inverters aims toeliminate low-order harmonic components as much as pos-sible which is different from that for the three-phase motorsystem [9ndash11] As these low-order harmonic componentscould result in the loss of the motor by leading to a largeamount of stator harmonic current components in the x-ysubspace In order to increase the utilization ratio of the DC
bus voltage multiphase carrier-type PWM algorithm basedon zero-sequence signal injection is applied for multiphaseinverter system which is composed of several three-phasewindings In particular the reference voltage vector can bedecomposed into two three-phase inverters sharing a com-mon DC bus voltage when used in practice After that thecarrier-type PWM algorithm based on dual zero-sequenceinjection can be used [12 13] Although this method is easyfor digital realization it is not an optimal PWM algorithmformultiphase motors as there are fifth and seventh harmoniccomponents in the stator current
As the multiphase AC motors have been widely usedthe conventional inverter control algorithm is expected to beoptimized for them At present many researches are devotedto the implementation of the SVPWM algorithm One of themultiphase SVPWM algorithms is based on the maximumadjacent two vectors in which the voltage of the subspace issinus without considering the influence of the x-y subspacevoltage [14] In this connection there are more harmonicvoltages in the output and a larger fundamental voltagemodulation factor can be gotten in this way [15] While for
HindawiMathematical Problems in EngineeringVolume 2019 Article ID 8729125 12 pageshttpsdoiorg10115520198729125
2 Mathematical Problems in Engineering
M
Vdc
A UB VC W
30∘
Figure 1 Six-phase converter drive system
multiphase AC motors the SVPWM algorithm will becomeincredibly complex And with the increase of the numberof motor phases the complexity of this direct calculationSVPWM algorithm will be greatly increased sector judg-ment vector duration calculation vector action sequencearrangement vector duration conversion into switchingaction time and other complex issues and the difficulty willbe greatly increased and the performance requirements ofthe controller will undoubtedly be greatly improved
In order to further improve the DC bus utilization rate atwo-vector and improved four-vector algorithm can be usedto increase the modulation factor by injecting the 5th and7th harmonics of the vector in the x-y subspaceThis methodonly increases the system loss and does not generate torqueripple But it only reached the maximum linear modulationrange When it is necessary to continue to improve the busvoltage utilization rate it can be said that the commonly usedthree-phase overmodulation technology [16 17] is extendedto the six-phase system and the 11th and 13th harmonicsare injected in the 120572-120573 subspace so that the maximum busvoltage utilization can be obtained However the torqueripple will be poor
In order to increase the utilization ratio of the DC busvoltage numerous studies have been done on improving thePWM algorithm for three-phase motor when overmodula-tion occurs As for the PWM overmodulation strategies formultiphase motors the fundamental amplitude of the outputreference voltage will rise by using vector space decom-position (VSD) in [10] transform matrix decomposition toget fundamental component of the x-y subspace [18] whilethe harmonic content of the reference voltage in the linearmodulation region also increases in that way
To solve above problems and to increase the utilizationratio of the DC bus voltage a novel SVPWM algorithmfor six-phase VSI with wide modulation range capabilitybased on vector weighted method is proposed in this paperThe concept of mid-vector presented in [18] is employed toobtain the expression for the time of algorithm of the activespace voltage vectors and using vector weighted method theovermodulation region is divided into three sections and theproposed SVPWM algorithm can also allow smooth transi-tion from a linear modulation region to an overmodulationregion The control strategy is verified through simulationanalysis
2 Dwell Time Calculation for Six-Phase VSI
The six-phase motor system fed by six-phase VSI is shownin Figure 1 There are two sets of Y-coupled three-phasestator windings which are spatially separated by 30∘ electricaldegrees Since there are two switch states for each bridge armthe six-phase inverter has 26 switch states According to theVSD transformation approach all of the space vectors can bemapped to the 120572-120573 x-y and o1-o2 three mutually orthogonalsubspaces The fundamental component and harmonics withthe order 12nplusmn1 (n=1 2 3 ) are mapped into the 120572-120573 subspace The harmonics with the order 6nplusmn1 (n=1 35 ) and the harmonics with the order 6nplusmn3 (n=1 3 5 )are mapped into the x-y subspace and the o1-o2 subspacerespectively The subspaces are divided into 12 sectors
To suppress the harmonic current component of statorcurrent a good four-vector SVPWM method should satisfythat the amplitude of the reference voltage vector in the 120572-120573subspace is maximum and in the x-y subspace is minimum
Mathematical Problems in Engineering 3
2
4
3b
r
a2
1
Figure 2 SVPWMmethod based mid-vector
In the 120572-120573 subspace any three adjacent voltage vectors canbe synthesized into a new vector called mid-vector and themid-vector has a same direction as the vector in the middleposition As shown in Figure 2 three adjacent basic vectorsv1 v2 and v3 are employed to synthesize into a mid-vector vaSupposing the dwell time of vector v2 is aTs the vectors v1 andv3 will be 05(1-a)TsThe amplitude of va in 120572-120573 subspace andx-y subspace respectively are as follows
10038161003816100381610038161003816Vmax101584010038161003816100381610038161003816 =
radic2 (radic3 + 1)6 119881119889119888 (119886 + radic32 (1 minus 119886)) (1)
10038161003816100381610038161003816Vmin101584010038161003816100381610038161003816 =
radic2 (radic3 minus 1)6 119881119889119888 (119886 minus radic32 (1 minus 119886)) (2)
where a is a positive constantSimilarly the three adjacent basic vectors v2 v3 and v4
are employed to synthesize into a mid-vector vb In this waywe can synthesize 12 voltage vectors which have the samedistribution as the traditional two vector SVPWMalgorithmbut the magnitude is different So the reference voltage vectorvr in 120572-120573 subspace can be synthesized by using 12 mid-vectors and the basic principle is the same as the traditionaltwo vector SVPWM algorithm the dwell time of mid-vectorsTa and Tb can be calculated according to (5) and |Vmax|mustbe replaced by the variable |Vmax
1015840| and the dwell time ofvectors v1 v2 v3 and v4 can be obtained by
1199051 = 05 (1 minus 119886) 1198791198861199052 = 119886119879119886 + 05 (1 minus 119886) 1198791198871199053 = 05 (1 minus 119886) 119879119886 + 1198861198791198871199054 = 05 (1 minus 119886) 119879119887
(3)
Moreover the relationship between 120579 and 120575 is obtained by120579 = 120575 + 12058712 minus 119896 minus 16 120587 (4)
where k is the sector with k=12 3 12According to the traditional three-phase SVPWM
method the dwell time of mid-vector can be used by
119879119886 =1003816100381610038161003816V11990310038161003816100381610038161003816100381610038161003816Vmax1003816100381610038161003816 sin (1205876) sin(
2119896 minus 112 120587 minus 120575)119879119904
linear
over-modulation I
over-modulation IIover-modulation III
Figure 3 Linear and overmodulation range
119879119887 =1003816100381610038161003816V11990310038161003816100381610038161003816100381610038161003816Vmax1003816100381610038161003816 sin (1205876) sin(120575 minus
2119896 minus 312 120587)119879119904(5)
where Ts is the switch period and |V119903| is the amplitude of thereference voltage vector
In the linear modulation range the control objectiveof four-vector SVPWM based on mid-vector method is toreduce the current harmonic component ie the amplitudeof reference is maximum in the 120572-120573 subspace and minimumin the x-y subspace For this purpose (1) is only to be satisfiedwith |Vmin
1015840| = 0 ie 119886 = 2radic3 minus 3 Equation (2) will changeinto |Vmax
1015840| = (3radic2minusradic6)3sdot119881119889119888The dwell time ofmid-vectorcan be obtained by replacing |Vmax| with |Vmax
1015840| and then theexpression of dwell time can be achieved by (3)
3 A Novel SVPWM Technology withWide Modulation Range
31 Overmodulation Region Division To facilitate the analy-sis the modulating index is defined by
119898 = 120587 1003816100381610038161003816V1199031003816100381610038161003816(2119881119889119888) (6)
when the reference voltage vector is more than this limitingvalue of |V119903| = 2119881119889119888radic3 and the inverter moves into theovermodulation region for 2119881119889119888radic3 lt |V119903| lt 2119881119889119888120587 Whenthe voltage vector is |V119903| = 2119881119889119888120587 the inverter operates in thetwelve-step mode as shown in Figure 3
The overmodulation region is further divided into threesubmodes the overmodulation region I (0907 lt 119898 le 0977)the overmodulation region II (0977 lt 119898 le 0988) and theovermodulation region III (0988 lt 119898 le 1)32 SVPWM Based on Vector Weighted Method When thesystem is running in different overmodulation region as
4 Mathematical Problems in Engineering
shown in Figure 3 the basic idea of vector weighting methodis to obtain a new integrated reference voltage vector byweighting the voltage vectors in different overmodulationregions with using the concept ofmodulation coefficient anduse the reference voltage vector to calculate the duty cycle ofPWM In order to facilitate calculation four voltage vectorsare defined in this paper such as 119881sin1 119881sin2 119881119889119900119889 and 119881119905119908119890119881sin1 is the voltage vector of the maximum linear modulationratio when the four vector SVPWM algorithm is used andthe trajectory is a circle with a radius of 119881119889119888radic3 119881sin2 isthe voltage vector corresponding to the two-vector SVPWMalgorithm and the trajectory is a dodecagon inscribed circlewith a radius of radic2(radic3 + 1)6 sdot 119881119889119888 sdot cos(12058712) 119881119889119900119889 isthe voltage vector that rotates on the contour line of regulardodecagon119881119905119908119890 is the voltage vector of twelve-step waveTheexpression of the four voltage vectorswill be shown as follows
119881sin1 = 119881119889119888radic3119890119895120575 asymp 0577119881119889119888 sdot 119890119895120575 (7)
119881sin2 = radic2 (radic3 + 1)1198811198891198886 cos 12058712119890119895120575 asymp 0622119881119889119888 sdot 119890119895120575 (8)
119881119889119900119889 = radic2 (radic3 + 1)1198811198891198886 cos (((2 (119896 minus 1) + 1) 12) 120587 minus 120579) cos 12058712119890119895120575
asymp 0622119881119889119888cos (((2 (119896 minus 1) + 1) 12) 120587 minus 120579)119890119895120575
(9)
119881119905119908119890 = radic2 (radic3 + 1)1198811198891198886 119890119895119896(12058712)asymp 0644119881119889119888 sdot 119890119895119896(12058712) 119896 minus 16 120587 le 120575 le 1198966120587
(10)
where 120575 is the angle between reference voltage vector Vr and120572 axis the relationship between 119881sin1 119881sin2 119881119889119900119889 and Vr isphase equality and the amplitude of four-vector is 0577Vdc0622Vdc 0629Vdc and 06366Vdc
321 Overmodulation Region I (0907 lt m le 0977) It canbe seen from (1) and (2) that the amplitude |Vmax
1015840| of funda-mental voltage vector will gradually become larger with anincrease of 119886 but the amplitude of harmonic subspace voltagevector will also gradually increase The modulation methodespecially will become two-vector SVPWM algorithm Toobtain the reference voltage vector Vr the amplitude need tobe satisfied with |V119903| = |Vmax
1015840| cos(12058712) it can be obtained asfollows
119886 = 12 1003816100381610038161003816Vlowast1003816100381610038161003816119881119889119888 minus 2radic3 minus 3 (11)
The unitary expression of dwell time can be achievedwhen (11) is substituted into (1) with considering function(3) When the reference voltage vector Vr is in the modulationregion I the amplitude of Vr will gradually increase to0622Vdc and the value of 119886 will also gradually increasefrom 2radic3 minus 3 to 1 Thus the smooth transition from linearmodulation region to overmodulation region I is realized
o
ok1VMCH2
B
lowast
(1 minus k1)VMCH1
A
B
Figure 4 The principle diagram reference voltage vector in over-modulation section
According to the principle of vector weighting methodthe overmodulation coefficient of overmodulation I is definedas
1198961 = 119898 minus 11989811198982 minus 1198981 (0 le 1198961 le 1) (12)
where m1=0907 and m2=0977 The value 1198961 will be set to 0when the vector Vr is in linear modulation region and thevalue 1198961 is set to 1 when Vr rotates along the inner-circle ofthe regular dodecagon
Like overmodulation I region to illustrate the processof adjusting the reference voltage vector sector 2 shown inFigure 4 will be taken as an example and 119881sin1 and 119881sin2 willbe used to reconstruct the reference voltage vector Vlowast ie
Vlowast = (1 minus 1198961)119881sin1 + 1198961119881sin2 (13)
According to (7) and (8) the expression of Vlowast in sector 2can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = 119881119889119888radic3 (1 minus 1198961)
+ 2 (radic3 + 1)2 11988111988911988824 (119886 + radic32 (1 minus 119886)) sdot 1198961
(14)
The function (14) is substituted into (5) with considering(3) so the dell time of nonzero vector of each sector canbe calculated and then get the PWM waveform of theovermodulation I
Figure 5 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0907 to0977 As can be seen from the figure with the gradualincrease of modulation ratio m the output voltage of the Aphase voltage has a certain distortion When m is close to0977 the reference voltage vector rotates along the inner-circle of the regular dodecagon
322 Overmodulation Region II (0977 lt m le 0988)When the vector Vr is in overmodulation II the outside ofthe regular dodecagon cannot be synthesized by the switch
Mathematical Problems in Engineering 5
Out
put p
hase
volta
ge (V
)
08
06
04
02
0
minus02
minus04
minus06
minus08
Modulation ratio (M)09709609509409309209109
Figure 5 The relationship between fundamental amplitude of output phase voltage and m in mode I
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)
0988098609840982098097809760974
Figure 6 The relationship between fundamental amplitude of output phase voltage and m in mode II
vector the amplitude or phase of Vr must be adjusted so thatthe average value of output voltage for the entire period isequal to the reference voltageTheovermodulation coefficientof over modulation II is defined as
1198962 = 119898 minus 11989821198983 minus 1198982 (15)
wherem3=0988 In addition the value of k2 is set to 0 whenvoltage vector is located in overmodulation I and the valueof value of k2 is set to 1 when voltage vector rotates along theregular dodecagon
According to (8) and (9) the expression of Vlowast in sector 2can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = 2 (radic3 + 1)2 11988111988911988824 (1 minus 1198962) + 2 (radic3 + 1)
2 11988111988911988824 cos (1205876 minus 120593)1198962 (16)
In addition when the reference voltage vector is locatedoutside the regular dodecagon an unreasonable situation thatthe dwell time of the zero vector is less than 0 will happen So(3) is modified by
119905119894 = 1199051198941199051 + 1199052 + 1199053 + 1199054 119894 = 1 2 3 41199050 = 0
(17)
Figure 6 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0974 to0988 As can be seen from the diagram with the gradualincrease of the modulation ratio m the distortion of theoutput phase voltage is more and more obvious when m isclose to 0988 the output voltage vector will rotate along thetrack of the regular dodecagon
323 Overmodulation Region III (0988 lt m le 1) Theovermodulation coefficient of overmodulation III is definedas
1198963 = 119898 minus 11989831 minus 1198983 (18)
where the value of k3 is set to 0 when voltage vector is locatedin overmodulation II and the value of k3 is set to 1 whenvoltage vector rotates along the twelve-step wave
Like overmodulation II region to illustrate the processof adjusting the reference voltage vector sector 2 shown inFigure 7 will be taken as an example and 119881119889119900119889 and 119881119905119908119890 willbe used to reconstruct the reference voltage vector Vlowast ie
Vlowast = 119881119889119900119889 (1 minus 1198963) + 1198811199051199081198901198963 (19)
Although the introduced 119881119905119908119890 vector makes the phaseof reconstructed voltage vector different from the reference
6 Mathematical Problems in Engineering
o
o
B
lowast
A
B
k3Vtwe
(1 minus k3)Vdod
r
Figure 7 The principle diagram reference voltage vector in overmodulation section III
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)109980996099409920990988
Figure 8 The relationship between fundamental amplitude of output phase voltage and m in mode III
voltage vector it increases the output amplitude of voltagevector According to (9) and (10) the magnitude of thereconstructed voltage vector on the 120572-120573 axis can be obtainedas
1003816100381610038161003816Vlowast1205721003816100381610038161003816 = 2 (radic3 + 1)2cos120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) cos (1205874) sdot 1198811198891198886 1198963(20)
10038161003816100381610038161003816Vlowast12057310038161003816100381610038161003816 = 2 (radic3 + 1)2 sin 120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) sin (1205874) sdot 1198811198891198886 1198963(21)
The amplitude and phase of the reconstructed voltagevector can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = radic1003816100381610038161003816Vlowast12057210038161003816100381610038162 + 10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038162
120574 = arctan(10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038161003816100381610038161003816Vlowast1205721003816100381610038161003816)
(22)
Figure 8 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0987 to 1As can be seen from the diagram with the gradual increaseof the modulation ratio m the distortion of the output phasevoltage is more and more obvious when m is close to 1 theoutput voltage vector will rotate along the track of the regulardodecagon vertex
4 Simulation Results Analysis
To verify the validity of the proposed SVPWM algorithm ofsix-phase VSI with wide modulation range the simulationmodel is built in MatlabSimulink environment Figure 9shows the phase voltage and phase voltage harmonic spectrain different modulation regions The voltage is per unitvalue When m=0906 the harmonic voltages vx and vyare zero in x-y subspaces and the phase voltage is sinewave THD=0 As the modulation ratio increases the THDof phase voltage gradually increases and the low-orderharmonic components are mainly from 120572-120573 subspace and x-ysubspace
Figure 10 shows the voltage vector of the different mod-ulation ratios in the 120572-120573 and x-y subspaces Among themthe left side of Figures 9(a) 9(b) and 9(c) is 120572-120573 subspaceand the right side is x-y subspace It can be seen from
Mathematical Problems in Engineering 7
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 0577 THD= 000
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
1 2 3 4 5 6 7 8 9 10 110 1312Harmonic order
(a) m=0906
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 06048 THD= 1050
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
Harmonic order43210 98765 13121110
5
(b) m=095
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u) Fundamental (5Hz) = 06288 THD= 1640
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(c) m=098
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u)
Fundamental (5Hz) = 0637 THD= 3138
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(d) m=1
Figure 9 Phase voltage and phase voltage harmonic spectra in different modulation regions
Figure 9 with the increase of the modulation ratio m thefundamental amplitude of voltage vector on 120572-120573 subspaceincreases gradually and the modulation ratio ism=0988 theoutput voltage vector tracks along the regular dodecagon andthe output voltage vector trajectory rotates along the trackof the regular dodecagon vertex with the modulation ratioof m=1 which verify the correctness of the above theoryanalysis
In addition although the harmonic component in the x-y subspace is gradually increasing the utilization ratio of DCvoltage is increased to a certain extent and the distortion rateand harmonic content of phase voltage increase graduallywith the increase of modulation ratio According to theabove analysis when the modulation ratio is higher thanthe linearmodulation region (mgt0907)modulation strategycompletely degenerates into two-vector SVPWM algorithmthe distortion rate of harmonic distribution is applicable to
all the two-largest-vector synthesis and only the harmonicamplitude is different
Figure 11 shows the modulation curve of SVPWMmethod with different modulation ratio it can be seen thatwhen the modulation ratio is m=1 the modulation wavebecomes a square wave signal
To verify the validity of the proposed SVPWM algorithmachieves the smooth transitions from linear to overmodula-tion region Figure 12 shows the curve of phase voltage whenthe modulation ratio is m increased from 0907 to 1 It canbe seen that as the modulation ratio increases stepwise thephase voltage waveform achieves a smooth transition fromlinear modulation region to overmodulation region
In order to illustrate the superiority of the control algo-rithmproposed in this paper Figure 13 shows the comparisonwith the traditional two-vector SVPWM algorithm and thesimulation results As can be seen from Figure 13 the THD
8 Mathematical Problems in Engineering
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(a) Overmodulation region I (m=0974)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(b) Overmodulation region II(m=0988)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(c) Overmodulation region III(m=1)
Figure 10 The voltage vector of 120572-120573 and x-y subspaces in different modulation regions
Mathematical Problems in Engineering 9
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus1
Times (s)
SVPW
M w
avef
orm
02018016014012010080060040020
Figure 11 The modulated wave with different modulation ratio
08
06
04
02
0
minus02
minus04
minus06
minus08
Times (s)08070605040302010
Phas
e vol
tage
(pu
)
Figure 12 The modulated wave in different modulation regions
091 092 093 094 095 096 097 098 099 1modulation ratio m
0
10
20
30
40
50
THD
The method in the paperTwo-vector methodThe method in Ref[1]
Figure 13The comparisonwith the traditional two-vector SVPWMalgorithm
content of the proposed control strategy is relatively smallThus it is verified that the proposed algorithm has bettercontrol performance
5 Experimental Results Analysis
To demonstrate the effectiveness of the proposed a novelSVPWM algorithm in full modulation region for six-phase
3MWConverter system
2MW IPMSM system
Figure 14 The overall experiment setup
voltage source inverter the drive operation is examined atexperiment platform shown in [19] Figures 14 and 15 illus-trate a 2MW IPMSMdriving system and detailed experimentsetup a high-power back-to-back converter system is used tofeed the IPMSM system and the parameters of IPMSM arepresent in Table 1 All three-phase currents and voltages aremeasured using magnetic current and voltage transducers
The system control algorithms are developed in Mat-labSimulink followed by implementation on an OPAL RT-Lab (Real-time Digital Simulator) controller board In thispaper the method of changing the amplitude of a given signalis used to change the modulation ratio m value The motor iscontrolled byVF and the dead time is 10usThe phase voltagewaveforms under different modulation degrees are observed
10 Mathematical Problems in Engineering
High-power inverter
Communication Circuit board
RT-Lab system
Figure 15 The detailed experiment setup
500V
div
Times (50msdiv)
(a) m=09
500V
div
Times (50msdiv)
(b) m=098
500V
div
Times (50msdiv)
(c) m=1
Figure 16 Phase A voltage experimental waveforms in different modulation index
Table 1 Parameters of IPMSM prototype
Rate power [kW] 2180Rate speed [rmin] 17Rate frequency [Hz] 85Rotor inertia [kgsdotm2] 16000Rate phase to phase voltage [V(rms)] 690Rate current [A] 1960Number of pole pairs 30Stator resistance per phase(R) [Ω] 00192d-axis inductance (Ld) [mH] 4q-axis inductance (Lq) [mH] 5Permanent magnet flux (Ψ119891) [Wb] 10228
by an oscilloscope as shown in Figure 16 It can be seen thatwith different modulation ratios along with the increasing
modulation the modulated wave in the PWM also graduallychanges from a sine wave to a square wave signal This isthe same as the simulation result shown in Figure 9 andthe phase voltage also transitions from the linear modulationregion to the overmodulation region119880A and 119880U phase voltage reconstruction waveforms areshown in Figure 17 It can be seen that the experimentalwaveform is consistent with the simulation waveform shownin Figure 10 Phase voltage UU lags UA 30 electrical anglesAs the modulation index m increases the voltage utilizationrate increases gradually and the phase voltage graduallytransitions from a sine wave to a twelve-step wave
6 Conclusion
To improve the utilization ratio ofDCbus voltage and restrainthe stator current harmonics and torque ripple a novel
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
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2 Mathematical Problems in Engineering
M
Vdc
A UB VC W
30∘
Figure 1 Six-phase converter drive system
multiphase AC motors the SVPWM algorithm will becomeincredibly complex And with the increase of the numberof motor phases the complexity of this direct calculationSVPWM algorithm will be greatly increased sector judg-ment vector duration calculation vector action sequencearrangement vector duration conversion into switchingaction time and other complex issues and the difficulty willbe greatly increased and the performance requirements ofthe controller will undoubtedly be greatly improved
In order to further improve the DC bus utilization rate atwo-vector and improved four-vector algorithm can be usedto increase the modulation factor by injecting the 5th and7th harmonics of the vector in the x-y subspaceThis methodonly increases the system loss and does not generate torqueripple But it only reached the maximum linear modulationrange When it is necessary to continue to improve the busvoltage utilization rate it can be said that the commonly usedthree-phase overmodulation technology [16 17] is extendedto the six-phase system and the 11th and 13th harmonicsare injected in the 120572-120573 subspace so that the maximum busvoltage utilization can be obtained However the torqueripple will be poor
In order to increase the utilization ratio of the DC busvoltage numerous studies have been done on improving thePWM algorithm for three-phase motor when overmodula-tion occurs As for the PWM overmodulation strategies formultiphase motors the fundamental amplitude of the outputreference voltage will rise by using vector space decom-position (VSD) in [10] transform matrix decomposition toget fundamental component of the x-y subspace [18] whilethe harmonic content of the reference voltage in the linearmodulation region also increases in that way
To solve above problems and to increase the utilizationratio of the DC bus voltage a novel SVPWM algorithmfor six-phase VSI with wide modulation range capabilitybased on vector weighted method is proposed in this paperThe concept of mid-vector presented in [18] is employed toobtain the expression for the time of algorithm of the activespace voltage vectors and using vector weighted method theovermodulation region is divided into three sections and theproposed SVPWM algorithm can also allow smooth transi-tion from a linear modulation region to an overmodulationregion The control strategy is verified through simulationanalysis
2 Dwell Time Calculation for Six-Phase VSI
The six-phase motor system fed by six-phase VSI is shownin Figure 1 There are two sets of Y-coupled three-phasestator windings which are spatially separated by 30∘ electricaldegrees Since there are two switch states for each bridge armthe six-phase inverter has 26 switch states According to theVSD transformation approach all of the space vectors can bemapped to the 120572-120573 x-y and o1-o2 three mutually orthogonalsubspaces The fundamental component and harmonics withthe order 12nplusmn1 (n=1 2 3 ) are mapped into the 120572-120573 subspace The harmonics with the order 6nplusmn1 (n=1 35 ) and the harmonics with the order 6nplusmn3 (n=1 3 5 )are mapped into the x-y subspace and the o1-o2 subspacerespectively The subspaces are divided into 12 sectors
To suppress the harmonic current component of statorcurrent a good four-vector SVPWM method should satisfythat the amplitude of the reference voltage vector in the 120572-120573subspace is maximum and in the x-y subspace is minimum
Mathematical Problems in Engineering 3
2
4
3b
r
a2
1
Figure 2 SVPWMmethod based mid-vector
In the 120572-120573 subspace any three adjacent voltage vectors canbe synthesized into a new vector called mid-vector and themid-vector has a same direction as the vector in the middleposition As shown in Figure 2 three adjacent basic vectorsv1 v2 and v3 are employed to synthesize into a mid-vector vaSupposing the dwell time of vector v2 is aTs the vectors v1 andv3 will be 05(1-a)TsThe amplitude of va in 120572-120573 subspace andx-y subspace respectively are as follows
10038161003816100381610038161003816Vmax101584010038161003816100381610038161003816 =
radic2 (radic3 + 1)6 119881119889119888 (119886 + radic32 (1 minus 119886)) (1)
10038161003816100381610038161003816Vmin101584010038161003816100381610038161003816 =
radic2 (radic3 minus 1)6 119881119889119888 (119886 minus radic32 (1 minus 119886)) (2)
where a is a positive constantSimilarly the three adjacent basic vectors v2 v3 and v4
are employed to synthesize into a mid-vector vb In this waywe can synthesize 12 voltage vectors which have the samedistribution as the traditional two vector SVPWMalgorithmbut the magnitude is different So the reference voltage vectorvr in 120572-120573 subspace can be synthesized by using 12 mid-vectors and the basic principle is the same as the traditionaltwo vector SVPWM algorithm the dwell time of mid-vectorsTa and Tb can be calculated according to (5) and |Vmax|mustbe replaced by the variable |Vmax
1015840| and the dwell time ofvectors v1 v2 v3 and v4 can be obtained by
1199051 = 05 (1 minus 119886) 1198791198861199052 = 119886119879119886 + 05 (1 minus 119886) 1198791198871199053 = 05 (1 minus 119886) 119879119886 + 1198861198791198871199054 = 05 (1 minus 119886) 119879119887
(3)
Moreover the relationship between 120579 and 120575 is obtained by120579 = 120575 + 12058712 minus 119896 minus 16 120587 (4)
where k is the sector with k=12 3 12According to the traditional three-phase SVPWM
method the dwell time of mid-vector can be used by
119879119886 =1003816100381610038161003816V11990310038161003816100381610038161003816100381610038161003816Vmax1003816100381610038161003816 sin (1205876) sin(
2119896 minus 112 120587 minus 120575)119879119904
linear
over-modulation I
over-modulation IIover-modulation III
Figure 3 Linear and overmodulation range
119879119887 =1003816100381610038161003816V11990310038161003816100381610038161003816100381610038161003816Vmax1003816100381610038161003816 sin (1205876) sin(120575 minus
2119896 minus 312 120587)119879119904(5)
where Ts is the switch period and |V119903| is the amplitude of thereference voltage vector
In the linear modulation range the control objectiveof four-vector SVPWM based on mid-vector method is toreduce the current harmonic component ie the amplitudeof reference is maximum in the 120572-120573 subspace and minimumin the x-y subspace For this purpose (1) is only to be satisfiedwith |Vmin
1015840| = 0 ie 119886 = 2radic3 minus 3 Equation (2) will changeinto |Vmax
1015840| = (3radic2minusradic6)3sdot119881119889119888The dwell time ofmid-vectorcan be obtained by replacing |Vmax| with |Vmax
1015840| and then theexpression of dwell time can be achieved by (3)
3 A Novel SVPWM Technology withWide Modulation Range
31 Overmodulation Region Division To facilitate the analy-sis the modulating index is defined by
119898 = 120587 1003816100381610038161003816V1199031003816100381610038161003816(2119881119889119888) (6)
when the reference voltage vector is more than this limitingvalue of |V119903| = 2119881119889119888radic3 and the inverter moves into theovermodulation region for 2119881119889119888radic3 lt |V119903| lt 2119881119889119888120587 Whenthe voltage vector is |V119903| = 2119881119889119888120587 the inverter operates in thetwelve-step mode as shown in Figure 3
The overmodulation region is further divided into threesubmodes the overmodulation region I (0907 lt 119898 le 0977)the overmodulation region II (0977 lt 119898 le 0988) and theovermodulation region III (0988 lt 119898 le 1)32 SVPWM Based on Vector Weighted Method When thesystem is running in different overmodulation region as
4 Mathematical Problems in Engineering
shown in Figure 3 the basic idea of vector weighting methodis to obtain a new integrated reference voltage vector byweighting the voltage vectors in different overmodulationregions with using the concept ofmodulation coefficient anduse the reference voltage vector to calculate the duty cycle ofPWM In order to facilitate calculation four voltage vectorsare defined in this paper such as 119881sin1 119881sin2 119881119889119900119889 and 119881119905119908119890119881sin1 is the voltage vector of the maximum linear modulationratio when the four vector SVPWM algorithm is used andthe trajectory is a circle with a radius of 119881119889119888radic3 119881sin2 isthe voltage vector corresponding to the two-vector SVPWMalgorithm and the trajectory is a dodecagon inscribed circlewith a radius of radic2(radic3 + 1)6 sdot 119881119889119888 sdot cos(12058712) 119881119889119900119889 isthe voltage vector that rotates on the contour line of regulardodecagon119881119905119908119890 is the voltage vector of twelve-step waveTheexpression of the four voltage vectorswill be shown as follows
119881sin1 = 119881119889119888radic3119890119895120575 asymp 0577119881119889119888 sdot 119890119895120575 (7)
119881sin2 = radic2 (radic3 + 1)1198811198891198886 cos 12058712119890119895120575 asymp 0622119881119889119888 sdot 119890119895120575 (8)
119881119889119900119889 = radic2 (radic3 + 1)1198811198891198886 cos (((2 (119896 minus 1) + 1) 12) 120587 minus 120579) cos 12058712119890119895120575
asymp 0622119881119889119888cos (((2 (119896 minus 1) + 1) 12) 120587 minus 120579)119890119895120575
(9)
119881119905119908119890 = radic2 (radic3 + 1)1198811198891198886 119890119895119896(12058712)asymp 0644119881119889119888 sdot 119890119895119896(12058712) 119896 minus 16 120587 le 120575 le 1198966120587
(10)
where 120575 is the angle between reference voltage vector Vr and120572 axis the relationship between 119881sin1 119881sin2 119881119889119900119889 and Vr isphase equality and the amplitude of four-vector is 0577Vdc0622Vdc 0629Vdc and 06366Vdc
321 Overmodulation Region I (0907 lt m le 0977) It canbe seen from (1) and (2) that the amplitude |Vmax
1015840| of funda-mental voltage vector will gradually become larger with anincrease of 119886 but the amplitude of harmonic subspace voltagevector will also gradually increase The modulation methodespecially will become two-vector SVPWM algorithm Toobtain the reference voltage vector Vr the amplitude need tobe satisfied with |V119903| = |Vmax
1015840| cos(12058712) it can be obtained asfollows
119886 = 12 1003816100381610038161003816Vlowast1003816100381610038161003816119881119889119888 minus 2radic3 minus 3 (11)
The unitary expression of dwell time can be achievedwhen (11) is substituted into (1) with considering function(3) When the reference voltage vector Vr is in the modulationregion I the amplitude of Vr will gradually increase to0622Vdc and the value of 119886 will also gradually increasefrom 2radic3 minus 3 to 1 Thus the smooth transition from linearmodulation region to overmodulation region I is realized
o
ok1VMCH2
B
lowast
(1 minus k1)VMCH1
A
B
Figure 4 The principle diagram reference voltage vector in over-modulation section
According to the principle of vector weighting methodthe overmodulation coefficient of overmodulation I is definedas
1198961 = 119898 minus 11989811198982 minus 1198981 (0 le 1198961 le 1) (12)
where m1=0907 and m2=0977 The value 1198961 will be set to 0when the vector Vr is in linear modulation region and thevalue 1198961 is set to 1 when Vr rotates along the inner-circle ofthe regular dodecagon
Like overmodulation I region to illustrate the processof adjusting the reference voltage vector sector 2 shown inFigure 4 will be taken as an example and 119881sin1 and 119881sin2 willbe used to reconstruct the reference voltage vector Vlowast ie
Vlowast = (1 minus 1198961)119881sin1 + 1198961119881sin2 (13)
According to (7) and (8) the expression of Vlowast in sector 2can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = 119881119889119888radic3 (1 minus 1198961)
+ 2 (radic3 + 1)2 11988111988911988824 (119886 + radic32 (1 minus 119886)) sdot 1198961
(14)
The function (14) is substituted into (5) with considering(3) so the dell time of nonzero vector of each sector canbe calculated and then get the PWM waveform of theovermodulation I
Figure 5 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0907 to0977 As can be seen from the figure with the gradualincrease of modulation ratio m the output voltage of the Aphase voltage has a certain distortion When m is close to0977 the reference voltage vector rotates along the inner-circle of the regular dodecagon
322 Overmodulation Region II (0977 lt m le 0988)When the vector Vr is in overmodulation II the outside ofthe regular dodecagon cannot be synthesized by the switch
Mathematical Problems in Engineering 5
Out
put p
hase
volta
ge (V
)
08
06
04
02
0
minus02
minus04
minus06
minus08
Modulation ratio (M)09709609509409309209109
Figure 5 The relationship between fundamental amplitude of output phase voltage and m in mode I
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)
0988098609840982098097809760974
Figure 6 The relationship between fundamental amplitude of output phase voltage and m in mode II
vector the amplitude or phase of Vr must be adjusted so thatthe average value of output voltage for the entire period isequal to the reference voltageTheovermodulation coefficientof over modulation II is defined as
1198962 = 119898 minus 11989821198983 minus 1198982 (15)
wherem3=0988 In addition the value of k2 is set to 0 whenvoltage vector is located in overmodulation I and the valueof value of k2 is set to 1 when voltage vector rotates along theregular dodecagon
According to (8) and (9) the expression of Vlowast in sector 2can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = 2 (radic3 + 1)2 11988111988911988824 (1 minus 1198962) + 2 (radic3 + 1)
2 11988111988911988824 cos (1205876 minus 120593)1198962 (16)
In addition when the reference voltage vector is locatedoutside the regular dodecagon an unreasonable situation thatthe dwell time of the zero vector is less than 0 will happen So(3) is modified by
119905119894 = 1199051198941199051 + 1199052 + 1199053 + 1199054 119894 = 1 2 3 41199050 = 0
(17)
Figure 6 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0974 to0988 As can be seen from the diagram with the gradualincrease of the modulation ratio m the distortion of theoutput phase voltage is more and more obvious when m isclose to 0988 the output voltage vector will rotate along thetrack of the regular dodecagon
323 Overmodulation Region III (0988 lt m le 1) Theovermodulation coefficient of overmodulation III is definedas
1198963 = 119898 minus 11989831 minus 1198983 (18)
where the value of k3 is set to 0 when voltage vector is locatedin overmodulation II and the value of k3 is set to 1 whenvoltage vector rotates along the twelve-step wave
Like overmodulation II region to illustrate the processof adjusting the reference voltage vector sector 2 shown inFigure 7 will be taken as an example and 119881119889119900119889 and 119881119905119908119890 willbe used to reconstruct the reference voltage vector Vlowast ie
Vlowast = 119881119889119900119889 (1 minus 1198963) + 1198811199051199081198901198963 (19)
Although the introduced 119881119905119908119890 vector makes the phaseof reconstructed voltage vector different from the reference
6 Mathematical Problems in Engineering
o
o
B
lowast
A
B
k3Vtwe
(1 minus k3)Vdod
r
Figure 7 The principle diagram reference voltage vector in overmodulation section III
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)109980996099409920990988
Figure 8 The relationship between fundamental amplitude of output phase voltage and m in mode III
voltage vector it increases the output amplitude of voltagevector According to (9) and (10) the magnitude of thereconstructed voltage vector on the 120572-120573 axis can be obtainedas
1003816100381610038161003816Vlowast1205721003816100381610038161003816 = 2 (radic3 + 1)2cos120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) cos (1205874) sdot 1198811198891198886 1198963(20)
10038161003816100381610038161003816Vlowast12057310038161003816100381610038161003816 = 2 (radic3 + 1)2 sin 120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) sin (1205874) sdot 1198811198891198886 1198963(21)
The amplitude and phase of the reconstructed voltagevector can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = radic1003816100381610038161003816Vlowast12057210038161003816100381610038162 + 10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038162
120574 = arctan(10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038161003816100381610038161003816Vlowast1205721003816100381610038161003816)
(22)
Figure 8 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0987 to 1As can be seen from the diagram with the gradual increaseof the modulation ratio m the distortion of the output phasevoltage is more and more obvious when m is close to 1 theoutput voltage vector will rotate along the track of the regulardodecagon vertex
4 Simulation Results Analysis
To verify the validity of the proposed SVPWM algorithm ofsix-phase VSI with wide modulation range the simulationmodel is built in MatlabSimulink environment Figure 9shows the phase voltage and phase voltage harmonic spectrain different modulation regions The voltage is per unitvalue When m=0906 the harmonic voltages vx and vyare zero in x-y subspaces and the phase voltage is sinewave THD=0 As the modulation ratio increases the THDof phase voltage gradually increases and the low-orderharmonic components are mainly from 120572-120573 subspace and x-ysubspace
Figure 10 shows the voltage vector of the different mod-ulation ratios in the 120572-120573 and x-y subspaces Among themthe left side of Figures 9(a) 9(b) and 9(c) is 120572-120573 subspaceand the right side is x-y subspace It can be seen from
Mathematical Problems in Engineering 7
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 0577 THD= 000
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
1 2 3 4 5 6 7 8 9 10 110 1312Harmonic order
(a) m=0906
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 06048 THD= 1050
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
Harmonic order43210 98765 13121110
5
(b) m=095
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u) Fundamental (5Hz) = 06288 THD= 1640
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(c) m=098
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u)
Fundamental (5Hz) = 0637 THD= 3138
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(d) m=1
Figure 9 Phase voltage and phase voltage harmonic spectra in different modulation regions
Figure 9 with the increase of the modulation ratio m thefundamental amplitude of voltage vector on 120572-120573 subspaceincreases gradually and the modulation ratio ism=0988 theoutput voltage vector tracks along the regular dodecagon andthe output voltage vector trajectory rotates along the trackof the regular dodecagon vertex with the modulation ratioof m=1 which verify the correctness of the above theoryanalysis
In addition although the harmonic component in the x-y subspace is gradually increasing the utilization ratio of DCvoltage is increased to a certain extent and the distortion rateand harmonic content of phase voltage increase graduallywith the increase of modulation ratio According to theabove analysis when the modulation ratio is higher thanthe linearmodulation region (mgt0907)modulation strategycompletely degenerates into two-vector SVPWM algorithmthe distortion rate of harmonic distribution is applicable to
all the two-largest-vector synthesis and only the harmonicamplitude is different
Figure 11 shows the modulation curve of SVPWMmethod with different modulation ratio it can be seen thatwhen the modulation ratio is m=1 the modulation wavebecomes a square wave signal
To verify the validity of the proposed SVPWM algorithmachieves the smooth transitions from linear to overmodula-tion region Figure 12 shows the curve of phase voltage whenthe modulation ratio is m increased from 0907 to 1 It canbe seen that as the modulation ratio increases stepwise thephase voltage waveform achieves a smooth transition fromlinear modulation region to overmodulation region
In order to illustrate the superiority of the control algo-rithmproposed in this paper Figure 13 shows the comparisonwith the traditional two-vector SVPWM algorithm and thesimulation results As can be seen from Figure 13 the THD
8 Mathematical Problems in Engineering
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(a) Overmodulation region I (m=0974)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(b) Overmodulation region II(m=0988)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(c) Overmodulation region III(m=1)
Figure 10 The voltage vector of 120572-120573 and x-y subspaces in different modulation regions
Mathematical Problems in Engineering 9
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus1
Times (s)
SVPW
M w
avef
orm
02018016014012010080060040020
Figure 11 The modulated wave with different modulation ratio
08
06
04
02
0
minus02
minus04
minus06
minus08
Times (s)08070605040302010
Phas
e vol
tage
(pu
)
Figure 12 The modulated wave in different modulation regions
091 092 093 094 095 096 097 098 099 1modulation ratio m
0
10
20
30
40
50
THD
The method in the paperTwo-vector methodThe method in Ref[1]
Figure 13The comparisonwith the traditional two-vector SVPWMalgorithm
content of the proposed control strategy is relatively smallThus it is verified that the proposed algorithm has bettercontrol performance
5 Experimental Results Analysis
To demonstrate the effectiveness of the proposed a novelSVPWM algorithm in full modulation region for six-phase
3MWConverter system
2MW IPMSM system
Figure 14 The overall experiment setup
voltage source inverter the drive operation is examined atexperiment platform shown in [19] Figures 14 and 15 illus-trate a 2MW IPMSMdriving system and detailed experimentsetup a high-power back-to-back converter system is used tofeed the IPMSM system and the parameters of IPMSM arepresent in Table 1 All three-phase currents and voltages aremeasured using magnetic current and voltage transducers
The system control algorithms are developed in Mat-labSimulink followed by implementation on an OPAL RT-Lab (Real-time Digital Simulator) controller board In thispaper the method of changing the amplitude of a given signalis used to change the modulation ratio m value The motor iscontrolled byVF and the dead time is 10usThe phase voltagewaveforms under different modulation degrees are observed
10 Mathematical Problems in Engineering
High-power inverter
Communication Circuit board
RT-Lab system
Figure 15 The detailed experiment setup
500V
div
Times (50msdiv)
(a) m=09
500V
div
Times (50msdiv)
(b) m=098
500V
div
Times (50msdiv)
(c) m=1
Figure 16 Phase A voltage experimental waveforms in different modulation index
Table 1 Parameters of IPMSM prototype
Rate power [kW] 2180Rate speed [rmin] 17Rate frequency [Hz] 85Rotor inertia [kgsdotm2] 16000Rate phase to phase voltage [V(rms)] 690Rate current [A] 1960Number of pole pairs 30Stator resistance per phase(R) [Ω] 00192d-axis inductance (Ld) [mH] 4q-axis inductance (Lq) [mH] 5Permanent magnet flux (Ψ119891) [Wb] 10228
by an oscilloscope as shown in Figure 16 It can be seen thatwith different modulation ratios along with the increasing
modulation the modulated wave in the PWM also graduallychanges from a sine wave to a square wave signal This isthe same as the simulation result shown in Figure 9 andthe phase voltage also transitions from the linear modulationregion to the overmodulation region119880A and 119880U phase voltage reconstruction waveforms areshown in Figure 17 It can be seen that the experimentalwaveform is consistent with the simulation waveform shownin Figure 10 Phase voltage UU lags UA 30 electrical anglesAs the modulation index m increases the voltage utilizationrate increases gradually and the phase voltage graduallytransitions from a sine wave to a twelve-step wave
6 Conclusion
To improve the utilization ratio ofDCbus voltage and restrainthe stator current harmonics and torque ripple a novel
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
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Mathematical Problems in Engineering 3
2
4
3b
r
a2
1
Figure 2 SVPWMmethod based mid-vector
In the 120572-120573 subspace any three adjacent voltage vectors canbe synthesized into a new vector called mid-vector and themid-vector has a same direction as the vector in the middleposition As shown in Figure 2 three adjacent basic vectorsv1 v2 and v3 are employed to synthesize into a mid-vector vaSupposing the dwell time of vector v2 is aTs the vectors v1 andv3 will be 05(1-a)TsThe amplitude of va in 120572-120573 subspace andx-y subspace respectively are as follows
10038161003816100381610038161003816Vmax101584010038161003816100381610038161003816 =
radic2 (radic3 + 1)6 119881119889119888 (119886 + radic32 (1 minus 119886)) (1)
10038161003816100381610038161003816Vmin101584010038161003816100381610038161003816 =
radic2 (radic3 minus 1)6 119881119889119888 (119886 minus radic32 (1 minus 119886)) (2)
where a is a positive constantSimilarly the three adjacent basic vectors v2 v3 and v4
are employed to synthesize into a mid-vector vb In this waywe can synthesize 12 voltage vectors which have the samedistribution as the traditional two vector SVPWMalgorithmbut the magnitude is different So the reference voltage vectorvr in 120572-120573 subspace can be synthesized by using 12 mid-vectors and the basic principle is the same as the traditionaltwo vector SVPWM algorithm the dwell time of mid-vectorsTa and Tb can be calculated according to (5) and |Vmax|mustbe replaced by the variable |Vmax
1015840| and the dwell time ofvectors v1 v2 v3 and v4 can be obtained by
1199051 = 05 (1 minus 119886) 1198791198861199052 = 119886119879119886 + 05 (1 minus 119886) 1198791198871199053 = 05 (1 minus 119886) 119879119886 + 1198861198791198871199054 = 05 (1 minus 119886) 119879119887
(3)
Moreover the relationship between 120579 and 120575 is obtained by120579 = 120575 + 12058712 minus 119896 minus 16 120587 (4)
where k is the sector with k=12 3 12According to the traditional three-phase SVPWM
method the dwell time of mid-vector can be used by
119879119886 =1003816100381610038161003816V11990310038161003816100381610038161003816100381610038161003816Vmax1003816100381610038161003816 sin (1205876) sin(
2119896 minus 112 120587 minus 120575)119879119904
linear
over-modulation I
over-modulation IIover-modulation III
Figure 3 Linear and overmodulation range
119879119887 =1003816100381610038161003816V11990310038161003816100381610038161003816100381610038161003816Vmax1003816100381610038161003816 sin (1205876) sin(120575 minus
2119896 minus 312 120587)119879119904(5)
where Ts is the switch period and |V119903| is the amplitude of thereference voltage vector
In the linear modulation range the control objectiveof four-vector SVPWM based on mid-vector method is toreduce the current harmonic component ie the amplitudeof reference is maximum in the 120572-120573 subspace and minimumin the x-y subspace For this purpose (1) is only to be satisfiedwith |Vmin
1015840| = 0 ie 119886 = 2radic3 minus 3 Equation (2) will changeinto |Vmax
1015840| = (3radic2minusradic6)3sdot119881119889119888The dwell time ofmid-vectorcan be obtained by replacing |Vmax| with |Vmax
1015840| and then theexpression of dwell time can be achieved by (3)
3 A Novel SVPWM Technology withWide Modulation Range
31 Overmodulation Region Division To facilitate the analy-sis the modulating index is defined by
119898 = 120587 1003816100381610038161003816V1199031003816100381610038161003816(2119881119889119888) (6)
when the reference voltage vector is more than this limitingvalue of |V119903| = 2119881119889119888radic3 and the inverter moves into theovermodulation region for 2119881119889119888radic3 lt |V119903| lt 2119881119889119888120587 Whenthe voltage vector is |V119903| = 2119881119889119888120587 the inverter operates in thetwelve-step mode as shown in Figure 3
The overmodulation region is further divided into threesubmodes the overmodulation region I (0907 lt 119898 le 0977)the overmodulation region II (0977 lt 119898 le 0988) and theovermodulation region III (0988 lt 119898 le 1)32 SVPWM Based on Vector Weighted Method When thesystem is running in different overmodulation region as
4 Mathematical Problems in Engineering
shown in Figure 3 the basic idea of vector weighting methodis to obtain a new integrated reference voltage vector byweighting the voltage vectors in different overmodulationregions with using the concept ofmodulation coefficient anduse the reference voltage vector to calculate the duty cycle ofPWM In order to facilitate calculation four voltage vectorsare defined in this paper such as 119881sin1 119881sin2 119881119889119900119889 and 119881119905119908119890119881sin1 is the voltage vector of the maximum linear modulationratio when the four vector SVPWM algorithm is used andthe trajectory is a circle with a radius of 119881119889119888radic3 119881sin2 isthe voltage vector corresponding to the two-vector SVPWMalgorithm and the trajectory is a dodecagon inscribed circlewith a radius of radic2(radic3 + 1)6 sdot 119881119889119888 sdot cos(12058712) 119881119889119900119889 isthe voltage vector that rotates on the contour line of regulardodecagon119881119905119908119890 is the voltage vector of twelve-step waveTheexpression of the four voltage vectorswill be shown as follows
119881sin1 = 119881119889119888radic3119890119895120575 asymp 0577119881119889119888 sdot 119890119895120575 (7)
119881sin2 = radic2 (radic3 + 1)1198811198891198886 cos 12058712119890119895120575 asymp 0622119881119889119888 sdot 119890119895120575 (8)
119881119889119900119889 = radic2 (radic3 + 1)1198811198891198886 cos (((2 (119896 minus 1) + 1) 12) 120587 minus 120579) cos 12058712119890119895120575
asymp 0622119881119889119888cos (((2 (119896 minus 1) + 1) 12) 120587 minus 120579)119890119895120575
(9)
119881119905119908119890 = radic2 (radic3 + 1)1198811198891198886 119890119895119896(12058712)asymp 0644119881119889119888 sdot 119890119895119896(12058712) 119896 minus 16 120587 le 120575 le 1198966120587
(10)
where 120575 is the angle between reference voltage vector Vr and120572 axis the relationship between 119881sin1 119881sin2 119881119889119900119889 and Vr isphase equality and the amplitude of four-vector is 0577Vdc0622Vdc 0629Vdc and 06366Vdc
321 Overmodulation Region I (0907 lt m le 0977) It canbe seen from (1) and (2) that the amplitude |Vmax
1015840| of funda-mental voltage vector will gradually become larger with anincrease of 119886 but the amplitude of harmonic subspace voltagevector will also gradually increase The modulation methodespecially will become two-vector SVPWM algorithm Toobtain the reference voltage vector Vr the amplitude need tobe satisfied with |V119903| = |Vmax
1015840| cos(12058712) it can be obtained asfollows
119886 = 12 1003816100381610038161003816Vlowast1003816100381610038161003816119881119889119888 minus 2radic3 minus 3 (11)
The unitary expression of dwell time can be achievedwhen (11) is substituted into (1) with considering function(3) When the reference voltage vector Vr is in the modulationregion I the amplitude of Vr will gradually increase to0622Vdc and the value of 119886 will also gradually increasefrom 2radic3 minus 3 to 1 Thus the smooth transition from linearmodulation region to overmodulation region I is realized
o
ok1VMCH2
B
lowast
(1 minus k1)VMCH1
A
B
Figure 4 The principle diagram reference voltage vector in over-modulation section
According to the principle of vector weighting methodthe overmodulation coefficient of overmodulation I is definedas
1198961 = 119898 minus 11989811198982 minus 1198981 (0 le 1198961 le 1) (12)
where m1=0907 and m2=0977 The value 1198961 will be set to 0when the vector Vr is in linear modulation region and thevalue 1198961 is set to 1 when Vr rotates along the inner-circle ofthe regular dodecagon
Like overmodulation I region to illustrate the processof adjusting the reference voltage vector sector 2 shown inFigure 4 will be taken as an example and 119881sin1 and 119881sin2 willbe used to reconstruct the reference voltage vector Vlowast ie
Vlowast = (1 minus 1198961)119881sin1 + 1198961119881sin2 (13)
According to (7) and (8) the expression of Vlowast in sector 2can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = 119881119889119888radic3 (1 minus 1198961)
+ 2 (radic3 + 1)2 11988111988911988824 (119886 + radic32 (1 minus 119886)) sdot 1198961
(14)
The function (14) is substituted into (5) with considering(3) so the dell time of nonzero vector of each sector canbe calculated and then get the PWM waveform of theovermodulation I
Figure 5 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0907 to0977 As can be seen from the figure with the gradualincrease of modulation ratio m the output voltage of the Aphase voltage has a certain distortion When m is close to0977 the reference voltage vector rotates along the inner-circle of the regular dodecagon
322 Overmodulation Region II (0977 lt m le 0988)When the vector Vr is in overmodulation II the outside ofthe regular dodecagon cannot be synthesized by the switch
Mathematical Problems in Engineering 5
Out
put p
hase
volta
ge (V
)
08
06
04
02
0
minus02
minus04
minus06
minus08
Modulation ratio (M)09709609509409309209109
Figure 5 The relationship between fundamental amplitude of output phase voltage and m in mode I
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)
0988098609840982098097809760974
Figure 6 The relationship between fundamental amplitude of output phase voltage and m in mode II
vector the amplitude or phase of Vr must be adjusted so thatthe average value of output voltage for the entire period isequal to the reference voltageTheovermodulation coefficientof over modulation II is defined as
1198962 = 119898 minus 11989821198983 minus 1198982 (15)
wherem3=0988 In addition the value of k2 is set to 0 whenvoltage vector is located in overmodulation I and the valueof value of k2 is set to 1 when voltage vector rotates along theregular dodecagon
According to (8) and (9) the expression of Vlowast in sector 2can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = 2 (radic3 + 1)2 11988111988911988824 (1 minus 1198962) + 2 (radic3 + 1)
2 11988111988911988824 cos (1205876 minus 120593)1198962 (16)
In addition when the reference voltage vector is locatedoutside the regular dodecagon an unreasonable situation thatthe dwell time of the zero vector is less than 0 will happen So(3) is modified by
119905119894 = 1199051198941199051 + 1199052 + 1199053 + 1199054 119894 = 1 2 3 41199050 = 0
(17)
Figure 6 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0974 to0988 As can be seen from the diagram with the gradualincrease of the modulation ratio m the distortion of theoutput phase voltage is more and more obvious when m isclose to 0988 the output voltage vector will rotate along thetrack of the regular dodecagon
323 Overmodulation Region III (0988 lt m le 1) Theovermodulation coefficient of overmodulation III is definedas
1198963 = 119898 minus 11989831 minus 1198983 (18)
where the value of k3 is set to 0 when voltage vector is locatedin overmodulation II and the value of k3 is set to 1 whenvoltage vector rotates along the twelve-step wave
Like overmodulation II region to illustrate the processof adjusting the reference voltage vector sector 2 shown inFigure 7 will be taken as an example and 119881119889119900119889 and 119881119905119908119890 willbe used to reconstruct the reference voltage vector Vlowast ie
Vlowast = 119881119889119900119889 (1 minus 1198963) + 1198811199051199081198901198963 (19)
Although the introduced 119881119905119908119890 vector makes the phaseof reconstructed voltage vector different from the reference
6 Mathematical Problems in Engineering
o
o
B
lowast
A
B
k3Vtwe
(1 minus k3)Vdod
r
Figure 7 The principle diagram reference voltage vector in overmodulation section III
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)109980996099409920990988
Figure 8 The relationship between fundamental amplitude of output phase voltage and m in mode III
voltage vector it increases the output amplitude of voltagevector According to (9) and (10) the magnitude of thereconstructed voltage vector on the 120572-120573 axis can be obtainedas
1003816100381610038161003816Vlowast1205721003816100381610038161003816 = 2 (radic3 + 1)2cos120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) cos (1205874) sdot 1198811198891198886 1198963(20)
10038161003816100381610038161003816Vlowast12057310038161003816100381610038161003816 = 2 (radic3 + 1)2 sin 120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) sin (1205874) sdot 1198811198891198886 1198963(21)
The amplitude and phase of the reconstructed voltagevector can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = radic1003816100381610038161003816Vlowast12057210038161003816100381610038162 + 10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038162
120574 = arctan(10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038161003816100381610038161003816Vlowast1205721003816100381610038161003816)
(22)
Figure 8 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0987 to 1As can be seen from the diagram with the gradual increaseof the modulation ratio m the distortion of the output phasevoltage is more and more obvious when m is close to 1 theoutput voltage vector will rotate along the track of the regulardodecagon vertex
4 Simulation Results Analysis
To verify the validity of the proposed SVPWM algorithm ofsix-phase VSI with wide modulation range the simulationmodel is built in MatlabSimulink environment Figure 9shows the phase voltage and phase voltage harmonic spectrain different modulation regions The voltage is per unitvalue When m=0906 the harmonic voltages vx and vyare zero in x-y subspaces and the phase voltage is sinewave THD=0 As the modulation ratio increases the THDof phase voltage gradually increases and the low-orderharmonic components are mainly from 120572-120573 subspace and x-ysubspace
Figure 10 shows the voltage vector of the different mod-ulation ratios in the 120572-120573 and x-y subspaces Among themthe left side of Figures 9(a) 9(b) and 9(c) is 120572-120573 subspaceand the right side is x-y subspace It can be seen from
Mathematical Problems in Engineering 7
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 0577 THD= 000
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
1 2 3 4 5 6 7 8 9 10 110 1312Harmonic order
(a) m=0906
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 06048 THD= 1050
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
Harmonic order43210 98765 13121110
5
(b) m=095
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u) Fundamental (5Hz) = 06288 THD= 1640
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(c) m=098
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u)
Fundamental (5Hz) = 0637 THD= 3138
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(d) m=1
Figure 9 Phase voltage and phase voltage harmonic spectra in different modulation regions
Figure 9 with the increase of the modulation ratio m thefundamental amplitude of voltage vector on 120572-120573 subspaceincreases gradually and the modulation ratio ism=0988 theoutput voltage vector tracks along the regular dodecagon andthe output voltage vector trajectory rotates along the trackof the regular dodecagon vertex with the modulation ratioof m=1 which verify the correctness of the above theoryanalysis
In addition although the harmonic component in the x-y subspace is gradually increasing the utilization ratio of DCvoltage is increased to a certain extent and the distortion rateand harmonic content of phase voltage increase graduallywith the increase of modulation ratio According to theabove analysis when the modulation ratio is higher thanthe linearmodulation region (mgt0907)modulation strategycompletely degenerates into two-vector SVPWM algorithmthe distortion rate of harmonic distribution is applicable to
all the two-largest-vector synthesis and only the harmonicamplitude is different
Figure 11 shows the modulation curve of SVPWMmethod with different modulation ratio it can be seen thatwhen the modulation ratio is m=1 the modulation wavebecomes a square wave signal
To verify the validity of the proposed SVPWM algorithmachieves the smooth transitions from linear to overmodula-tion region Figure 12 shows the curve of phase voltage whenthe modulation ratio is m increased from 0907 to 1 It canbe seen that as the modulation ratio increases stepwise thephase voltage waveform achieves a smooth transition fromlinear modulation region to overmodulation region
In order to illustrate the superiority of the control algo-rithmproposed in this paper Figure 13 shows the comparisonwith the traditional two-vector SVPWM algorithm and thesimulation results As can be seen from Figure 13 the THD
8 Mathematical Problems in Engineering
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(a) Overmodulation region I (m=0974)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(b) Overmodulation region II(m=0988)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(c) Overmodulation region III(m=1)
Figure 10 The voltage vector of 120572-120573 and x-y subspaces in different modulation regions
Mathematical Problems in Engineering 9
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus1
Times (s)
SVPW
M w
avef
orm
02018016014012010080060040020
Figure 11 The modulated wave with different modulation ratio
08
06
04
02
0
minus02
minus04
minus06
minus08
Times (s)08070605040302010
Phas
e vol
tage
(pu
)
Figure 12 The modulated wave in different modulation regions
091 092 093 094 095 096 097 098 099 1modulation ratio m
0
10
20
30
40
50
THD
The method in the paperTwo-vector methodThe method in Ref[1]
Figure 13The comparisonwith the traditional two-vector SVPWMalgorithm
content of the proposed control strategy is relatively smallThus it is verified that the proposed algorithm has bettercontrol performance
5 Experimental Results Analysis
To demonstrate the effectiveness of the proposed a novelSVPWM algorithm in full modulation region for six-phase
3MWConverter system
2MW IPMSM system
Figure 14 The overall experiment setup
voltage source inverter the drive operation is examined atexperiment platform shown in [19] Figures 14 and 15 illus-trate a 2MW IPMSMdriving system and detailed experimentsetup a high-power back-to-back converter system is used tofeed the IPMSM system and the parameters of IPMSM arepresent in Table 1 All three-phase currents and voltages aremeasured using magnetic current and voltage transducers
The system control algorithms are developed in Mat-labSimulink followed by implementation on an OPAL RT-Lab (Real-time Digital Simulator) controller board In thispaper the method of changing the amplitude of a given signalis used to change the modulation ratio m value The motor iscontrolled byVF and the dead time is 10usThe phase voltagewaveforms under different modulation degrees are observed
10 Mathematical Problems in Engineering
High-power inverter
Communication Circuit board
RT-Lab system
Figure 15 The detailed experiment setup
500V
div
Times (50msdiv)
(a) m=09
500V
div
Times (50msdiv)
(b) m=098
500V
div
Times (50msdiv)
(c) m=1
Figure 16 Phase A voltage experimental waveforms in different modulation index
Table 1 Parameters of IPMSM prototype
Rate power [kW] 2180Rate speed [rmin] 17Rate frequency [Hz] 85Rotor inertia [kgsdotm2] 16000Rate phase to phase voltage [V(rms)] 690Rate current [A] 1960Number of pole pairs 30Stator resistance per phase(R) [Ω] 00192d-axis inductance (Ld) [mH] 4q-axis inductance (Lq) [mH] 5Permanent magnet flux (Ψ119891) [Wb] 10228
by an oscilloscope as shown in Figure 16 It can be seen thatwith different modulation ratios along with the increasing
modulation the modulated wave in the PWM also graduallychanges from a sine wave to a square wave signal This isthe same as the simulation result shown in Figure 9 andthe phase voltage also transitions from the linear modulationregion to the overmodulation region119880A and 119880U phase voltage reconstruction waveforms areshown in Figure 17 It can be seen that the experimentalwaveform is consistent with the simulation waveform shownin Figure 10 Phase voltage UU lags UA 30 electrical anglesAs the modulation index m increases the voltage utilizationrate increases gradually and the phase voltage graduallytransitions from a sine wave to a twelve-step wave
6 Conclusion
To improve the utilization ratio ofDCbus voltage and restrainthe stator current harmonics and torque ripple a novel
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
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4 Mathematical Problems in Engineering
shown in Figure 3 the basic idea of vector weighting methodis to obtain a new integrated reference voltage vector byweighting the voltage vectors in different overmodulationregions with using the concept ofmodulation coefficient anduse the reference voltage vector to calculate the duty cycle ofPWM In order to facilitate calculation four voltage vectorsare defined in this paper such as 119881sin1 119881sin2 119881119889119900119889 and 119881119905119908119890119881sin1 is the voltage vector of the maximum linear modulationratio when the four vector SVPWM algorithm is used andthe trajectory is a circle with a radius of 119881119889119888radic3 119881sin2 isthe voltage vector corresponding to the two-vector SVPWMalgorithm and the trajectory is a dodecagon inscribed circlewith a radius of radic2(radic3 + 1)6 sdot 119881119889119888 sdot cos(12058712) 119881119889119900119889 isthe voltage vector that rotates on the contour line of regulardodecagon119881119905119908119890 is the voltage vector of twelve-step waveTheexpression of the four voltage vectorswill be shown as follows
119881sin1 = 119881119889119888radic3119890119895120575 asymp 0577119881119889119888 sdot 119890119895120575 (7)
119881sin2 = radic2 (radic3 + 1)1198811198891198886 cos 12058712119890119895120575 asymp 0622119881119889119888 sdot 119890119895120575 (8)
119881119889119900119889 = radic2 (radic3 + 1)1198811198891198886 cos (((2 (119896 minus 1) + 1) 12) 120587 minus 120579) cos 12058712119890119895120575
asymp 0622119881119889119888cos (((2 (119896 minus 1) + 1) 12) 120587 minus 120579)119890119895120575
(9)
119881119905119908119890 = radic2 (radic3 + 1)1198811198891198886 119890119895119896(12058712)asymp 0644119881119889119888 sdot 119890119895119896(12058712) 119896 minus 16 120587 le 120575 le 1198966120587
(10)
where 120575 is the angle between reference voltage vector Vr and120572 axis the relationship between 119881sin1 119881sin2 119881119889119900119889 and Vr isphase equality and the amplitude of four-vector is 0577Vdc0622Vdc 0629Vdc and 06366Vdc
321 Overmodulation Region I (0907 lt m le 0977) It canbe seen from (1) and (2) that the amplitude |Vmax
1015840| of funda-mental voltage vector will gradually become larger with anincrease of 119886 but the amplitude of harmonic subspace voltagevector will also gradually increase The modulation methodespecially will become two-vector SVPWM algorithm Toobtain the reference voltage vector Vr the amplitude need tobe satisfied with |V119903| = |Vmax
1015840| cos(12058712) it can be obtained asfollows
119886 = 12 1003816100381610038161003816Vlowast1003816100381610038161003816119881119889119888 minus 2radic3 minus 3 (11)
The unitary expression of dwell time can be achievedwhen (11) is substituted into (1) with considering function(3) When the reference voltage vector Vr is in the modulationregion I the amplitude of Vr will gradually increase to0622Vdc and the value of 119886 will also gradually increasefrom 2radic3 minus 3 to 1 Thus the smooth transition from linearmodulation region to overmodulation region I is realized
o
ok1VMCH2
B
lowast
(1 minus k1)VMCH1
A
B
Figure 4 The principle diagram reference voltage vector in over-modulation section
According to the principle of vector weighting methodthe overmodulation coefficient of overmodulation I is definedas
1198961 = 119898 minus 11989811198982 minus 1198981 (0 le 1198961 le 1) (12)
where m1=0907 and m2=0977 The value 1198961 will be set to 0when the vector Vr is in linear modulation region and thevalue 1198961 is set to 1 when Vr rotates along the inner-circle ofthe regular dodecagon
Like overmodulation I region to illustrate the processof adjusting the reference voltage vector sector 2 shown inFigure 4 will be taken as an example and 119881sin1 and 119881sin2 willbe used to reconstruct the reference voltage vector Vlowast ie
Vlowast = (1 minus 1198961)119881sin1 + 1198961119881sin2 (13)
According to (7) and (8) the expression of Vlowast in sector 2can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = 119881119889119888radic3 (1 minus 1198961)
+ 2 (radic3 + 1)2 11988111988911988824 (119886 + radic32 (1 minus 119886)) sdot 1198961
(14)
The function (14) is substituted into (5) with considering(3) so the dell time of nonzero vector of each sector canbe calculated and then get the PWM waveform of theovermodulation I
Figure 5 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0907 to0977 As can be seen from the figure with the gradualincrease of modulation ratio m the output voltage of the Aphase voltage has a certain distortion When m is close to0977 the reference voltage vector rotates along the inner-circle of the regular dodecagon
322 Overmodulation Region II (0977 lt m le 0988)When the vector Vr is in overmodulation II the outside ofthe regular dodecagon cannot be synthesized by the switch
Mathematical Problems in Engineering 5
Out
put p
hase
volta
ge (V
)
08
06
04
02
0
minus02
minus04
minus06
minus08
Modulation ratio (M)09709609509409309209109
Figure 5 The relationship between fundamental amplitude of output phase voltage and m in mode I
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)
0988098609840982098097809760974
Figure 6 The relationship between fundamental amplitude of output phase voltage and m in mode II
vector the amplitude or phase of Vr must be adjusted so thatthe average value of output voltage for the entire period isequal to the reference voltageTheovermodulation coefficientof over modulation II is defined as
1198962 = 119898 minus 11989821198983 minus 1198982 (15)
wherem3=0988 In addition the value of k2 is set to 0 whenvoltage vector is located in overmodulation I and the valueof value of k2 is set to 1 when voltage vector rotates along theregular dodecagon
According to (8) and (9) the expression of Vlowast in sector 2can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = 2 (radic3 + 1)2 11988111988911988824 (1 minus 1198962) + 2 (radic3 + 1)
2 11988111988911988824 cos (1205876 minus 120593)1198962 (16)
In addition when the reference voltage vector is locatedoutside the regular dodecagon an unreasonable situation thatthe dwell time of the zero vector is less than 0 will happen So(3) is modified by
119905119894 = 1199051198941199051 + 1199052 + 1199053 + 1199054 119894 = 1 2 3 41199050 = 0
(17)
Figure 6 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0974 to0988 As can be seen from the diagram with the gradualincrease of the modulation ratio m the distortion of theoutput phase voltage is more and more obvious when m isclose to 0988 the output voltage vector will rotate along thetrack of the regular dodecagon
323 Overmodulation Region III (0988 lt m le 1) Theovermodulation coefficient of overmodulation III is definedas
1198963 = 119898 minus 11989831 minus 1198983 (18)
where the value of k3 is set to 0 when voltage vector is locatedin overmodulation II and the value of k3 is set to 1 whenvoltage vector rotates along the twelve-step wave
Like overmodulation II region to illustrate the processof adjusting the reference voltage vector sector 2 shown inFigure 7 will be taken as an example and 119881119889119900119889 and 119881119905119908119890 willbe used to reconstruct the reference voltage vector Vlowast ie
Vlowast = 119881119889119900119889 (1 minus 1198963) + 1198811199051199081198901198963 (19)
Although the introduced 119881119905119908119890 vector makes the phaseof reconstructed voltage vector different from the reference
6 Mathematical Problems in Engineering
o
o
B
lowast
A
B
k3Vtwe
(1 minus k3)Vdod
r
Figure 7 The principle diagram reference voltage vector in overmodulation section III
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)109980996099409920990988
Figure 8 The relationship between fundamental amplitude of output phase voltage and m in mode III
voltage vector it increases the output amplitude of voltagevector According to (9) and (10) the magnitude of thereconstructed voltage vector on the 120572-120573 axis can be obtainedas
1003816100381610038161003816Vlowast1205721003816100381610038161003816 = 2 (radic3 + 1)2cos120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) cos (1205874) sdot 1198811198891198886 1198963(20)
10038161003816100381610038161003816Vlowast12057310038161003816100381610038161003816 = 2 (radic3 + 1)2 sin 120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) sin (1205874) sdot 1198811198891198886 1198963(21)
The amplitude and phase of the reconstructed voltagevector can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = radic1003816100381610038161003816Vlowast12057210038161003816100381610038162 + 10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038162
120574 = arctan(10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038161003816100381610038161003816Vlowast1205721003816100381610038161003816)
(22)
Figure 8 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0987 to 1As can be seen from the diagram with the gradual increaseof the modulation ratio m the distortion of the output phasevoltage is more and more obvious when m is close to 1 theoutput voltage vector will rotate along the track of the regulardodecagon vertex
4 Simulation Results Analysis
To verify the validity of the proposed SVPWM algorithm ofsix-phase VSI with wide modulation range the simulationmodel is built in MatlabSimulink environment Figure 9shows the phase voltage and phase voltage harmonic spectrain different modulation regions The voltage is per unitvalue When m=0906 the harmonic voltages vx and vyare zero in x-y subspaces and the phase voltage is sinewave THD=0 As the modulation ratio increases the THDof phase voltage gradually increases and the low-orderharmonic components are mainly from 120572-120573 subspace and x-ysubspace
Figure 10 shows the voltage vector of the different mod-ulation ratios in the 120572-120573 and x-y subspaces Among themthe left side of Figures 9(a) 9(b) and 9(c) is 120572-120573 subspaceand the right side is x-y subspace It can be seen from
Mathematical Problems in Engineering 7
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 0577 THD= 000
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
1 2 3 4 5 6 7 8 9 10 110 1312Harmonic order
(a) m=0906
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 06048 THD= 1050
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
Harmonic order43210 98765 13121110
5
(b) m=095
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u) Fundamental (5Hz) = 06288 THD= 1640
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(c) m=098
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u)
Fundamental (5Hz) = 0637 THD= 3138
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(d) m=1
Figure 9 Phase voltage and phase voltage harmonic spectra in different modulation regions
Figure 9 with the increase of the modulation ratio m thefundamental amplitude of voltage vector on 120572-120573 subspaceincreases gradually and the modulation ratio ism=0988 theoutput voltage vector tracks along the regular dodecagon andthe output voltage vector trajectory rotates along the trackof the regular dodecagon vertex with the modulation ratioof m=1 which verify the correctness of the above theoryanalysis
In addition although the harmonic component in the x-y subspace is gradually increasing the utilization ratio of DCvoltage is increased to a certain extent and the distortion rateand harmonic content of phase voltage increase graduallywith the increase of modulation ratio According to theabove analysis when the modulation ratio is higher thanthe linearmodulation region (mgt0907)modulation strategycompletely degenerates into two-vector SVPWM algorithmthe distortion rate of harmonic distribution is applicable to
all the two-largest-vector synthesis and only the harmonicamplitude is different
Figure 11 shows the modulation curve of SVPWMmethod with different modulation ratio it can be seen thatwhen the modulation ratio is m=1 the modulation wavebecomes a square wave signal
To verify the validity of the proposed SVPWM algorithmachieves the smooth transitions from linear to overmodula-tion region Figure 12 shows the curve of phase voltage whenthe modulation ratio is m increased from 0907 to 1 It canbe seen that as the modulation ratio increases stepwise thephase voltage waveform achieves a smooth transition fromlinear modulation region to overmodulation region
In order to illustrate the superiority of the control algo-rithmproposed in this paper Figure 13 shows the comparisonwith the traditional two-vector SVPWM algorithm and thesimulation results As can be seen from Figure 13 the THD
8 Mathematical Problems in Engineering
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(a) Overmodulation region I (m=0974)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(b) Overmodulation region II(m=0988)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(c) Overmodulation region III(m=1)
Figure 10 The voltage vector of 120572-120573 and x-y subspaces in different modulation regions
Mathematical Problems in Engineering 9
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus1
Times (s)
SVPW
M w
avef
orm
02018016014012010080060040020
Figure 11 The modulated wave with different modulation ratio
08
06
04
02
0
minus02
minus04
minus06
minus08
Times (s)08070605040302010
Phas
e vol
tage
(pu
)
Figure 12 The modulated wave in different modulation regions
091 092 093 094 095 096 097 098 099 1modulation ratio m
0
10
20
30
40
50
THD
The method in the paperTwo-vector methodThe method in Ref[1]
Figure 13The comparisonwith the traditional two-vector SVPWMalgorithm
content of the proposed control strategy is relatively smallThus it is verified that the proposed algorithm has bettercontrol performance
5 Experimental Results Analysis
To demonstrate the effectiveness of the proposed a novelSVPWM algorithm in full modulation region for six-phase
3MWConverter system
2MW IPMSM system
Figure 14 The overall experiment setup
voltage source inverter the drive operation is examined atexperiment platform shown in [19] Figures 14 and 15 illus-trate a 2MW IPMSMdriving system and detailed experimentsetup a high-power back-to-back converter system is used tofeed the IPMSM system and the parameters of IPMSM arepresent in Table 1 All three-phase currents and voltages aremeasured using magnetic current and voltage transducers
The system control algorithms are developed in Mat-labSimulink followed by implementation on an OPAL RT-Lab (Real-time Digital Simulator) controller board In thispaper the method of changing the amplitude of a given signalis used to change the modulation ratio m value The motor iscontrolled byVF and the dead time is 10usThe phase voltagewaveforms under different modulation degrees are observed
10 Mathematical Problems in Engineering
High-power inverter
Communication Circuit board
RT-Lab system
Figure 15 The detailed experiment setup
500V
div
Times (50msdiv)
(a) m=09
500V
div
Times (50msdiv)
(b) m=098
500V
div
Times (50msdiv)
(c) m=1
Figure 16 Phase A voltage experimental waveforms in different modulation index
Table 1 Parameters of IPMSM prototype
Rate power [kW] 2180Rate speed [rmin] 17Rate frequency [Hz] 85Rotor inertia [kgsdotm2] 16000Rate phase to phase voltage [V(rms)] 690Rate current [A] 1960Number of pole pairs 30Stator resistance per phase(R) [Ω] 00192d-axis inductance (Ld) [mH] 4q-axis inductance (Lq) [mH] 5Permanent magnet flux (Ψ119891) [Wb] 10228
by an oscilloscope as shown in Figure 16 It can be seen thatwith different modulation ratios along with the increasing
modulation the modulated wave in the PWM also graduallychanges from a sine wave to a square wave signal This isthe same as the simulation result shown in Figure 9 andthe phase voltage also transitions from the linear modulationregion to the overmodulation region119880A and 119880U phase voltage reconstruction waveforms areshown in Figure 17 It can be seen that the experimentalwaveform is consistent with the simulation waveform shownin Figure 10 Phase voltage UU lags UA 30 electrical anglesAs the modulation index m increases the voltage utilizationrate increases gradually and the phase voltage graduallytransitions from a sine wave to a twelve-step wave
6 Conclusion
To improve the utilization ratio ofDCbus voltage and restrainthe stator current harmonics and torque ripple a novel
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 5
Out
put p
hase
volta
ge (V
)
08
06
04
02
0
minus02
minus04
minus06
minus08
Modulation ratio (M)09709609509409309209109
Figure 5 The relationship between fundamental amplitude of output phase voltage and m in mode I
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)
0988098609840982098097809760974
Figure 6 The relationship between fundamental amplitude of output phase voltage and m in mode II
vector the amplitude or phase of Vr must be adjusted so thatthe average value of output voltage for the entire period isequal to the reference voltageTheovermodulation coefficientof over modulation II is defined as
1198962 = 119898 minus 11989821198983 minus 1198982 (15)
wherem3=0988 In addition the value of k2 is set to 0 whenvoltage vector is located in overmodulation I and the valueof value of k2 is set to 1 when voltage vector rotates along theregular dodecagon
According to (8) and (9) the expression of Vlowast in sector 2can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = 2 (radic3 + 1)2 11988111988911988824 (1 minus 1198962) + 2 (radic3 + 1)
2 11988111988911988824 cos (1205876 minus 120593)1198962 (16)
In addition when the reference voltage vector is locatedoutside the regular dodecagon an unreasonable situation thatthe dwell time of the zero vector is less than 0 will happen So(3) is modified by
119905119894 = 1199051198941199051 + 1199052 + 1199053 + 1199054 119894 = 1 2 3 41199050 = 0
(17)
Figure 6 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0974 to0988 As can be seen from the diagram with the gradualincrease of the modulation ratio m the distortion of theoutput phase voltage is more and more obvious when m isclose to 0988 the output voltage vector will rotate along thetrack of the regular dodecagon
323 Overmodulation Region III (0988 lt m le 1) Theovermodulation coefficient of overmodulation III is definedas
1198963 = 119898 minus 11989831 minus 1198983 (18)
where the value of k3 is set to 0 when voltage vector is locatedin overmodulation II and the value of k3 is set to 1 whenvoltage vector rotates along the twelve-step wave
Like overmodulation II region to illustrate the processof adjusting the reference voltage vector sector 2 shown inFigure 7 will be taken as an example and 119881119889119900119889 and 119881119905119908119890 willbe used to reconstruct the reference voltage vector Vlowast ie
Vlowast = 119881119889119900119889 (1 minus 1198963) + 1198811199051199081198901198963 (19)
Although the introduced 119881119905119908119890 vector makes the phaseof reconstructed voltage vector different from the reference
6 Mathematical Problems in Engineering
o
o
B
lowast
A
B
k3Vtwe
(1 minus k3)Vdod
r
Figure 7 The principle diagram reference voltage vector in overmodulation section III
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)109980996099409920990988
Figure 8 The relationship between fundamental amplitude of output phase voltage and m in mode III
voltage vector it increases the output amplitude of voltagevector According to (9) and (10) the magnitude of thereconstructed voltage vector on the 120572-120573 axis can be obtainedas
1003816100381610038161003816Vlowast1205721003816100381610038161003816 = 2 (radic3 + 1)2cos120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) cos (1205874) sdot 1198811198891198886 1198963(20)
10038161003816100381610038161003816Vlowast12057310038161003816100381610038161003816 = 2 (radic3 + 1)2 sin 120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) sin (1205874) sdot 1198811198891198886 1198963(21)
The amplitude and phase of the reconstructed voltagevector can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = radic1003816100381610038161003816Vlowast12057210038161003816100381610038162 + 10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038162
120574 = arctan(10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038161003816100381610038161003816Vlowast1205721003816100381610038161003816)
(22)
Figure 8 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0987 to 1As can be seen from the diagram with the gradual increaseof the modulation ratio m the distortion of the output phasevoltage is more and more obvious when m is close to 1 theoutput voltage vector will rotate along the track of the regulardodecagon vertex
4 Simulation Results Analysis
To verify the validity of the proposed SVPWM algorithm ofsix-phase VSI with wide modulation range the simulationmodel is built in MatlabSimulink environment Figure 9shows the phase voltage and phase voltage harmonic spectrain different modulation regions The voltage is per unitvalue When m=0906 the harmonic voltages vx and vyare zero in x-y subspaces and the phase voltage is sinewave THD=0 As the modulation ratio increases the THDof phase voltage gradually increases and the low-orderharmonic components are mainly from 120572-120573 subspace and x-ysubspace
Figure 10 shows the voltage vector of the different mod-ulation ratios in the 120572-120573 and x-y subspaces Among themthe left side of Figures 9(a) 9(b) and 9(c) is 120572-120573 subspaceand the right side is x-y subspace It can be seen from
Mathematical Problems in Engineering 7
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 0577 THD= 000
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
1 2 3 4 5 6 7 8 9 10 110 1312Harmonic order
(a) m=0906
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 06048 THD= 1050
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
Harmonic order43210 98765 13121110
5
(b) m=095
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u) Fundamental (5Hz) = 06288 THD= 1640
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(c) m=098
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u)
Fundamental (5Hz) = 0637 THD= 3138
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(d) m=1
Figure 9 Phase voltage and phase voltage harmonic spectra in different modulation regions
Figure 9 with the increase of the modulation ratio m thefundamental amplitude of voltage vector on 120572-120573 subspaceincreases gradually and the modulation ratio ism=0988 theoutput voltage vector tracks along the regular dodecagon andthe output voltage vector trajectory rotates along the trackof the regular dodecagon vertex with the modulation ratioof m=1 which verify the correctness of the above theoryanalysis
In addition although the harmonic component in the x-y subspace is gradually increasing the utilization ratio of DCvoltage is increased to a certain extent and the distortion rateand harmonic content of phase voltage increase graduallywith the increase of modulation ratio According to theabove analysis when the modulation ratio is higher thanthe linearmodulation region (mgt0907)modulation strategycompletely degenerates into two-vector SVPWM algorithmthe distortion rate of harmonic distribution is applicable to
all the two-largest-vector synthesis and only the harmonicamplitude is different
Figure 11 shows the modulation curve of SVPWMmethod with different modulation ratio it can be seen thatwhen the modulation ratio is m=1 the modulation wavebecomes a square wave signal
To verify the validity of the proposed SVPWM algorithmachieves the smooth transitions from linear to overmodula-tion region Figure 12 shows the curve of phase voltage whenthe modulation ratio is m increased from 0907 to 1 It canbe seen that as the modulation ratio increases stepwise thephase voltage waveform achieves a smooth transition fromlinear modulation region to overmodulation region
In order to illustrate the superiority of the control algo-rithmproposed in this paper Figure 13 shows the comparisonwith the traditional two-vector SVPWM algorithm and thesimulation results As can be seen from Figure 13 the THD
8 Mathematical Problems in Engineering
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(a) Overmodulation region I (m=0974)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(b) Overmodulation region II(m=0988)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(c) Overmodulation region III(m=1)
Figure 10 The voltage vector of 120572-120573 and x-y subspaces in different modulation regions
Mathematical Problems in Engineering 9
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus1
Times (s)
SVPW
M w
avef
orm
02018016014012010080060040020
Figure 11 The modulated wave with different modulation ratio
08
06
04
02
0
minus02
minus04
minus06
minus08
Times (s)08070605040302010
Phas
e vol
tage
(pu
)
Figure 12 The modulated wave in different modulation regions
091 092 093 094 095 096 097 098 099 1modulation ratio m
0
10
20
30
40
50
THD
The method in the paperTwo-vector methodThe method in Ref[1]
Figure 13The comparisonwith the traditional two-vector SVPWMalgorithm
content of the proposed control strategy is relatively smallThus it is verified that the proposed algorithm has bettercontrol performance
5 Experimental Results Analysis
To demonstrate the effectiveness of the proposed a novelSVPWM algorithm in full modulation region for six-phase
3MWConverter system
2MW IPMSM system
Figure 14 The overall experiment setup
voltage source inverter the drive operation is examined atexperiment platform shown in [19] Figures 14 and 15 illus-trate a 2MW IPMSMdriving system and detailed experimentsetup a high-power back-to-back converter system is used tofeed the IPMSM system and the parameters of IPMSM arepresent in Table 1 All three-phase currents and voltages aremeasured using magnetic current and voltage transducers
The system control algorithms are developed in Mat-labSimulink followed by implementation on an OPAL RT-Lab (Real-time Digital Simulator) controller board In thispaper the method of changing the amplitude of a given signalis used to change the modulation ratio m value The motor iscontrolled byVF and the dead time is 10usThe phase voltagewaveforms under different modulation degrees are observed
10 Mathematical Problems in Engineering
High-power inverter
Communication Circuit board
RT-Lab system
Figure 15 The detailed experiment setup
500V
div
Times (50msdiv)
(a) m=09
500V
div
Times (50msdiv)
(b) m=098
500V
div
Times (50msdiv)
(c) m=1
Figure 16 Phase A voltage experimental waveforms in different modulation index
Table 1 Parameters of IPMSM prototype
Rate power [kW] 2180Rate speed [rmin] 17Rate frequency [Hz] 85Rotor inertia [kgsdotm2] 16000Rate phase to phase voltage [V(rms)] 690Rate current [A] 1960Number of pole pairs 30Stator resistance per phase(R) [Ω] 00192d-axis inductance (Ld) [mH] 4q-axis inductance (Lq) [mH] 5Permanent magnet flux (Ψ119891) [Wb] 10228
by an oscilloscope as shown in Figure 16 It can be seen thatwith different modulation ratios along with the increasing
modulation the modulated wave in the PWM also graduallychanges from a sine wave to a square wave signal This isthe same as the simulation result shown in Figure 9 andthe phase voltage also transitions from the linear modulationregion to the overmodulation region119880A and 119880U phase voltage reconstruction waveforms areshown in Figure 17 It can be seen that the experimentalwaveform is consistent with the simulation waveform shownin Figure 10 Phase voltage UU lags UA 30 electrical anglesAs the modulation index m increases the voltage utilizationrate increases gradually and the phase voltage graduallytransitions from a sine wave to a twelve-step wave
6 Conclusion
To improve the utilization ratio ofDCbus voltage and restrainthe stator current harmonics and torque ripple a novel
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Mathematical Problems in Engineering
o
o
B
lowast
A
B
k3Vtwe
(1 minus k3)Vdod
r
Figure 7 The principle diagram reference voltage vector in overmodulation section III
Out
put p
hase
volta
ge (V
)
1
05
0
minus05
minus1
Modulation ratio (M)109980996099409920990988
Figure 8 The relationship between fundamental amplitude of output phase voltage and m in mode III
voltage vector it increases the output amplitude of voltagevector According to (9) and (10) the magnitude of thereconstructed voltage vector on the 120572-120573 axis can be obtainedas
1003816100381610038161003816Vlowast1205721003816100381610038161003816 = 2 (radic3 + 1)2cos120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) cos (1205874) sdot 1198811198891198886 1198963(20)
10038161003816100381610038161003816Vlowast12057310038161003816100381610038161003816 = 2 (radic3 + 1)2 sin 120593 sdot 11988111988911988824 cos (1205876 minus 120593) (1 minus 1198963)
+ radic2 (radic3 + 1) sin (1205874) sdot 1198811198891198886 1198963(21)
The amplitude and phase of the reconstructed voltagevector can be obtained by
1003816100381610038161003816Vlowast1003816100381610038161003816 = radic1003816100381610038161003816Vlowast12057210038161003816100381610038162 + 10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038162
120574 = arctan(10038161003816100381610038161003816Vlowast120573100381610038161003816100381610038161003816100381610038161003816Vlowast1205721003816100381610038161003816)
(22)
Figure 8 shows the A phase voltage curve in whichVdc=1V and modulation ratio m increased from 0987 to 1As can be seen from the diagram with the gradual increaseof the modulation ratio m the distortion of the output phasevoltage is more and more obvious when m is close to 1 theoutput voltage vector will rotate along the track of the regulardodecagon vertex
4 Simulation Results Analysis
To verify the validity of the proposed SVPWM algorithm ofsix-phase VSI with wide modulation range the simulationmodel is built in MatlabSimulink environment Figure 9shows the phase voltage and phase voltage harmonic spectrain different modulation regions The voltage is per unitvalue When m=0906 the harmonic voltages vx and vyare zero in x-y subspaces and the phase voltage is sinewave THD=0 As the modulation ratio increases the THDof phase voltage gradually increases and the low-orderharmonic components are mainly from 120572-120573 subspace and x-ysubspace
Figure 10 shows the voltage vector of the different mod-ulation ratios in the 120572-120573 and x-y subspaces Among themthe left side of Figures 9(a) 9(b) and 9(c) is 120572-120573 subspaceand the right side is x-y subspace It can be seen from
Mathematical Problems in Engineering 7
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 0577 THD= 000
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
1 2 3 4 5 6 7 8 9 10 110 1312Harmonic order
(a) m=0906
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 06048 THD= 1050
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
Harmonic order43210 98765 13121110
5
(b) m=095
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u) Fundamental (5Hz) = 06288 THD= 1640
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(c) m=098
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u)
Fundamental (5Hz) = 0637 THD= 3138
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(d) m=1
Figure 9 Phase voltage and phase voltage harmonic spectra in different modulation regions
Figure 9 with the increase of the modulation ratio m thefundamental amplitude of voltage vector on 120572-120573 subspaceincreases gradually and the modulation ratio ism=0988 theoutput voltage vector tracks along the regular dodecagon andthe output voltage vector trajectory rotates along the trackof the regular dodecagon vertex with the modulation ratioof m=1 which verify the correctness of the above theoryanalysis
In addition although the harmonic component in the x-y subspace is gradually increasing the utilization ratio of DCvoltage is increased to a certain extent and the distortion rateand harmonic content of phase voltage increase graduallywith the increase of modulation ratio According to theabove analysis when the modulation ratio is higher thanthe linearmodulation region (mgt0907)modulation strategycompletely degenerates into two-vector SVPWM algorithmthe distortion rate of harmonic distribution is applicable to
all the two-largest-vector synthesis and only the harmonicamplitude is different
Figure 11 shows the modulation curve of SVPWMmethod with different modulation ratio it can be seen thatwhen the modulation ratio is m=1 the modulation wavebecomes a square wave signal
To verify the validity of the proposed SVPWM algorithmachieves the smooth transitions from linear to overmodula-tion region Figure 12 shows the curve of phase voltage whenthe modulation ratio is m increased from 0907 to 1 It canbe seen that as the modulation ratio increases stepwise thephase voltage waveform achieves a smooth transition fromlinear modulation region to overmodulation region
In order to illustrate the superiority of the control algo-rithmproposed in this paper Figure 13 shows the comparisonwith the traditional two-vector SVPWM algorithm and thesimulation results As can be seen from Figure 13 the THD
8 Mathematical Problems in Engineering
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(a) Overmodulation region I (m=0974)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(b) Overmodulation region II(m=0988)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(c) Overmodulation region III(m=1)
Figure 10 The voltage vector of 120572-120573 and x-y subspaces in different modulation regions
Mathematical Problems in Engineering 9
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus1
Times (s)
SVPW
M w
avef
orm
02018016014012010080060040020
Figure 11 The modulated wave with different modulation ratio
08
06
04
02
0
minus02
minus04
minus06
minus08
Times (s)08070605040302010
Phas
e vol
tage
(pu
)
Figure 12 The modulated wave in different modulation regions
091 092 093 094 095 096 097 098 099 1modulation ratio m
0
10
20
30
40
50
THD
The method in the paperTwo-vector methodThe method in Ref[1]
Figure 13The comparisonwith the traditional two-vector SVPWMalgorithm
content of the proposed control strategy is relatively smallThus it is verified that the proposed algorithm has bettercontrol performance
5 Experimental Results Analysis
To demonstrate the effectiveness of the proposed a novelSVPWM algorithm in full modulation region for six-phase
3MWConverter system
2MW IPMSM system
Figure 14 The overall experiment setup
voltage source inverter the drive operation is examined atexperiment platform shown in [19] Figures 14 and 15 illus-trate a 2MW IPMSMdriving system and detailed experimentsetup a high-power back-to-back converter system is used tofeed the IPMSM system and the parameters of IPMSM arepresent in Table 1 All three-phase currents and voltages aremeasured using magnetic current and voltage transducers
The system control algorithms are developed in Mat-labSimulink followed by implementation on an OPAL RT-Lab (Real-time Digital Simulator) controller board In thispaper the method of changing the amplitude of a given signalis used to change the modulation ratio m value The motor iscontrolled byVF and the dead time is 10usThe phase voltagewaveforms under different modulation degrees are observed
10 Mathematical Problems in Engineering
High-power inverter
Communication Circuit board
RT-Lab system
Figure 15 The detailed experiment setup
500V
div
Times (50msdiv)
(a) m=09
500V
div
Times (50msdiv)
(b) m=098
500V
div
Times (50msdiv)
(c) m=1
Figure 16 Phase A voltage experimental waveforms in different modulation index
Table 1 Parameters of IPMSM prototype
Rate power [kW] 2180Rate speed [rmin] 17Rate frequency [Hz] 85Rotor inertia [kgsdotm2] 16000Rate phase to phase voltage [V(rms)] 690Rate current [A] 1960Number of pole pairs 30Stator resistance per phase(R) [Ω] 00192d-axis inductance (Ld) [mH] 4q-axis inductance (Lq) [mH] 5Permanent magnet flux (Ψ119891) [Wb] 10228
by an oscilloscope as shown in Figure 16 It can be seen thatwith different modulation ratios along with the increasing
modulation the modulated wave in the PWM also graduallychanges from a sine wave to a square wave signal This isthe same as the simulation result shown in Figure 9 andthe phase voltage also transitions from the linear modulationregion to the overmodulation region119880A and 119880U phase voltage reconstruction waveforms areshown in Figure 17 It can be seen that the experimentalwaveform is consistent with the simulation waveform shownin Figure 10 Phase voltage UU lags UA 30 electrical anglesAs the modulation index m increases the voltage utilizationrate increases gradually and the phase voltage graduallytransitions from a sine wave to a twelve-step wave
6 Conclusion
To improve the utilization ratio ofDCbus voltage and restrainthe stator current harmonics and torque ripple a novel
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 7
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 0577 THD= 000
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
1 2 3 4 5 6 7 8 9 10 110 1312Harmonic order
(a) m=0906
Phas
e vol
tage
(pu
)A
mpl
itude
(pu
) Fundamental (5Hz) = 06048 THD= 1050
05
0
minus05
1
06
08
04
02
0
0201501Time (s)
0050
Harmonic order43210 98765 13121110
5
(b) m=095
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u) Fundamental (5Hz) = 06288 THD= 1640
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(c) m=098
Phas
e vol
tage
(pu
) 05
0
minus05
0201501Time (s)
0050
Am
plitu
de (p
u)
Fundamental (5Hz) = 0637 THD= 3138
1
06
08
04
02
0
Harmonic order43210 98765 13121110
(d) m=1
Figure 9 Phase voltage and phase voltage harmonic spectra in different modulation regions
Figure 9 with the increase of the modulation ratio m thefundamental amplitude of voltage vector on 120572-120573 subspaceincreases gradually and the modulation ratio ism=0988 theoutput voltage vector tracks along the regular dodecagon andthe output voltage vector trajectory rotates along the trackof the regular dodecagon vertex with the modulation ratioof m=1 which verify the correctness of the above theoryanalysis
In addition although the harmonic component in the x-y subspace is gradually increasing the utilization ratio of DCvoltage is increased to a certain extent and the distortion rateand harmonic content of phase voltage increase graduallywith the increase of modulation ratio According to theabove analysis when the modulation ratio is higher thanthe linearmodulation region (mgt0907)modulation strategycompletely degenerates into two-vector SVPWM algorithmthe distortion rate of harmonic distribution is applicable to
all the two-largest-vector synthesis and only the harmonicamplitude is different
Figure 11 shows the modulation curve of SVPWMmethod with different modulation ratio it can be seen thatwhen the modulation ratio is m=1 the modulation wavebecomes a square wave signal
To verify the validity of the proposed SVPWM algorithmachieves the smooth transitions from linear to overmodula-tion region Figure 12 shows the curve of phase voltage whenthe modulation ratio is m increased from 0907 to 1 It canbe seen that as the modulation ratio increases stepwise thephase voltage waveform achieves a smooth transition fromlinear modulation region to overmodulation region
In order to illustrate the superiority of the control algo-rithmproposed in this paper Figure 13 shows the comparisonwith the traditional two-vector SVPWM algorithm and thesimulation results As can be seen from Figure 13 the THD
8 Mathematical Problems in Engineering
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(a) Overmodulation region I (m=0974)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(b) Overmodulation region II(m=0988)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(c) Overmodulation region III(m=1)
Figure 10 The voltage vector of 120572-120573 and x-y subspaces in different modulation regions
Mathematical Problems in Engineering 9
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus1
Times (s)
SVPW
M w
avef
orm
02018016014012010080060040020
Figure 11 The modulated wave with different modulation ratio
08
06
04
02
0
minus02
minus04
minus06
minus08
Times (s)08070605040302010
Phas
e vol
tage
(pu
)
Figure 12 The modulated wave in different modulation regions
091 092 093 094 095 096 097 098 099 1modulation ratio m
0
10
20
30
40
50
THD
The method in the paperTwo-vector methodThe method in Ref[1]
Figure 13The comparisonwith the traditional two-vector SVPWMalgorithm
content of the proposed control strategy is relatively smallThus it is verified that the proposed algorithm has bettercontrol performance
5 Experimental Results Analysis
To demonstrate the effectiveness of the proposed a novelSVPWM algorithm in full modulation region for six-phase
3MWConverter system
2MW IPMSM system
Figure 14 The overall experiment setup
voltage source inverter the drive operation is examined atexperiment platform shown in [19] Figures 14 and 15 illus-trate a 2MW IPMSMdriving system and detailed experimentsetup a high-power back-to-back converter system is used tofeed the IPMSM system and the parameters of IPMSM arepresent in Table 1 All three-phase currents and voltages aremeasured using magnetic current and voltage transducers
The system control algorithms are developed in Mat-labSimulink followed by implementation on an OPAL RT-Lab (Real-time Digital Simulator) controller board In thispaper the method of changing the amplitude of a given signalis used to change the modulation ratio m value The motor iscontrolled byVF and the dead time is 10usThe phase voltagewaveforms under different modulation degrees are observed
10 Mathematical Problems in Engineering
High-power inverter
Communication Circuit board
RT-Lab system
Figure 15 The detailed experiment setup
500V
div
Times (50msdiv)
(a) m=09
500V
div
Times (50msdiv)
(b) m=098
500V
div
Times (50msdiv)
(c) m=1
Figure 16 Phase A voltage experimental waveforms in different modulation index
Table 1 Parameters of IPMSM prototype
Rate power [kW] 2180Rate speed [rmin] 17Rate frequency [Hz] 85Rotor inertia [kgsdotm2] 16000Rate phase to phase voltage [V(rms)] 690Rate current [A] 1960Number of pole pairs 30Stator resistance per phase(R) [Ω] 00192d-axis inductance (Ld) [mH] 4q-axis inductance (Lq) [mH] 5Permanent magnet flux (Ψ119891) [Wb] 10228
by an oscilloscope as shown in Figure 16 It can be seen thatwith different modulation ratios along with the increasing
modulation the modulated wave in the PWM also graduallychanges from a sine wave to a square wave signal This isthe same as the simulation result shown in Figure 9 andthe phase voltage also transitions from the linear modulationregion to the overmodulation region119880A and 119880U phase voltage reconstruction waveforms areshown in Figure 17 It can be seen that the experimentalwaveform is consistent with the simulation waveform shownin Figure 10 Phase voltage UU lags UA 30 electrical anglesAs the modulation index m increases the voltage utilizationrate increases gradually and the phase voltage graduallytransitions from a sine wave to a twelve-step wave
6 Conclusion
To improve the utilization ratio ofDCbus voltage and restrainthe stator current harmonics and torque ripple a novel
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Mathematical Problems in Engineering
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(a) Overmodulation region I (m=0974)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(b) Overmodulation region II(m=0988)
06
04
02
0
minus02
minus04
minus06
V(V
)
V(V)
Vy(V
)
Vx(V)
015
01
005
0
minus005
minus01
minus015
0604020minus02minus04minus06 015010050minus005minus01minus015
(c) Overmodulation region III(m=1)
Figure 10 The voltage vector of 120572-120573 and x-y subspaces in different modulation regions
Mathematical Problems in Engineering 9
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus1
Times (s)
SVPW
M w
avef
orm
02018016014012010080060040020
Figure 11 The modulated wave with different modulation ratio
08
06
04
02
0
minus02
minus04
minus06
minus08
Times (s)08070605040302010
Phas
e vol
tage
(pu
)
Figure 12 The modulated wave in different modulation regions
091 092 093 094 095 096 097 098 099 1modulation ratio m
0
10
20
30
40
50
THD
The method in the paperTwo-vector methodThe method in Ref[1]
Figure 13The comparisonwith the traditional two-vector SVPWMalgorithm
content of the proposed control strategy is relatively smallThus it is verified that the proposed algorithm has bettercontrol performance
5 Experimental Results Analysis
To demonstrate the effectiveness of the proposed a novelSVPWM algorithm in full modulation region for six-phase
3MWConverter system
2MW IPMSM system
Figure 14 The overall experiment setup
voltage source inverter the drive operation is examined atexperiment platform shown in [19] Figures 14 and 15 illus-trate a 2MW IPMSMdriving system and detailed experimentsetup a high-power back-to-back converter system is used tofeed the IPMSM system and the parameters of IPMSM arepresent in Table 1 All three-phase currents and voltages aremeasured using magnetic current and voltage transducers
The system control algorithms are developed in Mat-labSimulink followed by implementation on an OPAL RT-Lab (Real-time Digital Simulator) controller board In thispaper the method of changing the amplitude of a given signalis used to change the modulation ratio m value The motor iscontrolled byVF and the dead time is 10usThe phase voltagewaveforms under different modulation degrees are observed
10 Mathematical Problems in Engineering
High-power inverter
Communication Circuit board
RT-Lab system
Figure 15 The detailed experiment setup
500V
div
Times (50msdiv)
(a) m=09
500V
div
Times (50msdiv)
(b) m=098
500V
div
Times (50msdiv)
(c) m=1
Figure 16 Phase A voltage experimental waveforms in different modulation index
Table 1 Parameters of IPMSM prototype
Rate power [kW] 2180Rate speed [rmin] 17Rate frequency [Hz] 85Rotor inertia [kgsdotm2] 16000Rate phase to phase voltage [V(rms)] 690Rate current [A] 1960Number of pole pairs 30Stator resistance per phase(R) [Ω] 00192d-axis inductance (Ld) [mH] 4q-axis inductance (Lq) [mH] 5Permanent magnet flux (Ψ119891) [Wb] 10228
by an oscilloscope as shown in Figure 16 It can be seen thatwith different modulation ratios along with the increasing
modulation the modulated wave in the PWM also graduallychanges from a sine wave to a square wave signal This isthe same as the simulation result shown in Figure 9 andthe phase voltage also transitions from the linear modulationregion to the overmodulation region119880A and 119880U phase voltage reconstruction waveforms areshown in Figure 17 It can be seen that the experimentalwaveform is consistent with the simulation waveform shownin Figure 10 Phase voltage UU lags UA 30 electrical anglesAs the modulation index m increases the voltage utilizationrate increases gradually and the phase voltage graduallytransitions from a sine wave to a twelve-step wave
6 Conclusion
To improve the utilization ratio ofDCbus voltage and restrainthe stator current harmonics and torque ripple a novel
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 9
1
08
06
04
02
0
minus02
minus04
minus06
minus08
minus1
Times (s)
SVPW
M w
avef
orm
02018016014012010080060040020
Figure 11 The modulated wave with different modulation ratio
08
06
04
02
0
minus02
minus04
minus06
minus08
Times (s)08070605040302010
Phas
e vol
tage
(pu
)
Figure 12 The modulated wave in different modulation regions
091 092 093 094 095 096 097 098 099 1modulation ratio m
0
10
20
30
40
50
THD
The method in the paperTwo-vector methodThe method in Ref[1]
Figure 13The comparisonwith the traditional two-vector SVPWMalgorithm
content of the proposed control strategy is relatively smallThus it is verified that the proposed algorithm has bettercontrol performance
5 Experimental Results Analysis
To demonstrate the effectiveness of the proposed a novelSVPWM algorithm in full modulation region for six-phase
3MWConverter system
2MW IPMSM system
Figure 14 The overall experiment setup
voltage source inverter the drive operation is examined atexperiment platform shown in [19] Figures 14 and 15 illus-trate a 2MW IPMSMdriving system and detailed experimentsetup a high-power back-to-back converter system is used tofeed the IPMSM system and the parameters of IPMSM arepresent in Table 1 All three-phase currents and voltages aremeasured using magnetic current and voltage transducers
The system control algorithms are developed in Mat-labSimulink followed by implementation on an OPAL RT-Lab (Real-time Digital Simulator) controller board In thispaper the method of changing the amplitude of a given signalis used to change the modulation ratio m value The motor iscontrolled byVF and the dead time is 10usThe phase voltagewaveforms under different modulation degrees are observed
10 Mathematical Problems in Engineering
High-power inverter
Communication Circuit board
RT-Lab system
Figure 15 The detailed experiment setup
500V
div
Times (50msdiv)
(a) m=09
500V
div
Times (50msdiv)
(b) m=098
500V
div
Times (50msdiv)
(c) m=1
Figure 16 Phase A voltage experimental waveforms in different modulation index
Table 1 Parameters of IPMSM prototype
Rate power [kW] 2180Rate speed [rmin] 17Rate frequency [Hz] 85Rotor inertia [kgsdotm2] 16000Rate phase to phase voltage [V(rms)] 690Rate current [A] 1960Number of pole pairs 30Stator resistance per phase(R) [Ω] 00192d-axis inductance (Ld) [mH] 4q-axis inductance (Lq) [mH] 5Permanent magnet flux (Ψ119891) [Wb] 10228
by an oscilloscope as shown in Figure 16 It can be seen thatwith different modulation ratios along with the increasing
modulation the modulated wave in the PWM also graduallychanges from a sine wave to a square wave signal This isthe same as the simulation result shown in Figure 9 andthe phase voltage also transitions from the linear modulationregion to the overmodulation region119880A and 119880U phase voltage reconstruction waveforms areshown in Figure 17 It can be seen that the experimentalwaveform is consistent with the simulation waveform shownin Figure 10 Phase voltage UU lags UA 30 electrical anglesAs the modulation index m increases the voltage utilizationrate increases gradually and the phase voltage graduallytransitions from a sine wave to a twelve-step wave
6 Conclusion
To improve the utilization ratio ofDCbus voltage and restrainthe stator current harmonics and torque ripple a novel
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Mathematical Problems in Engineering
High-power inverter
Communication Circuit board
RT-Lab system
Figure 15 The detailed experiment setup
500V
div
Times (50msdiv)
(a) m=09
500V
div
Times (50msdiv)
(b) m=098
500V
div
Times (50msdiv)
(c) m=1
Figure 16 Phase A voltage experimental waveforms in different modulation index
Table 1 Parameters of IPMSM prototype
Rate power [kW] 2180Rate speed [rmin] 17Rate frequency [Hz] 85Rotor inertia [kgsdotm2] 16000Rate phase to phase voltage [V(rms)] 690Rate current [A] 1960Number of pole pairs 30Stator resistance per phase(R) [Ω] 00192d-axis inductance (Ld) [mH] 4q-axis inductance (Lq) [mH] 5Permanent magnet flux (Ψ119891) [Wb] 10228
by an oscilloscope as shown in Figure 16 It can be seen thatwith different modulation ratios along with the increasing
modulation the modulated wave in the PWM also graduallychanges from a sine wave to a square wave signal This isthe same as the simulation result shown in Figure 9 andthe phase voltage also transitions from the linear modulationregion to the overmodulation region119880A and 119880U phase voltage reconstruction waveforms areshown in Figure 17 It can be seen that the experimentalwaveform is consistent with the simulation waveform shownin Figure 10 Phase voltage UU lags UA 30 electrical anglesAs the modulation index m increases the voltage utilizationrate increases gradually and the phase voltage graduallytransitions from a sine wave to a twelve-step wave
6 Conclusion
To improve the utilization ratio ofDCbus voltage and restrainthe stator current harmonics and torque ripple a novel
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 11
200V
div
Times (50msdiv)
UU
UA
(a) m=09
200V
div
Times (50msdiv)
UU
UA
(b) m=09820
0Vd
iv
Times (50msdiv)
UU
UA
(c) m=1
Figure 17 Phases A and U voltage experimental waveforms in different modulation index
SVPWM modulation strategy based on vector space decou-pling transform approach and vector weighting method isproposed in this paper The overmodulation of the six-phaseinverter is divided into 3 regions such as overmodulation Iovermodulation II and overmodulation III and the dwelltime of the SVPWM algorithm based on vector weightingmethod is deduced Simulation and experimental analysesdemonstrate the effectiveness and feasibility of the proposedstrategy
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
The authors declare no conflict of interest
Authorsrsquo Contributions
All authors contributed to this article Peng Wu and Lei Yuandesigned the study and put forward the articlersquos methodsThey also participated in programming Zhen Zuo collected agreat deal of materials to form the articlersquos structure as well asreviewing the manuscript Junyu Wei carried out the simula-tions for the proposedmethods and handled the experimentsAll authors read and approved the final manuscript
Acknowledgments
This work is supported by National Nature ScienceFoundation of China (under Grant 51507188) and Nature
Science Foundation of HuNan province China (under Grant2019JJ50121)
References
[1] C Zhou G Yang and J Su ldquoPWM strategy with minimumharmonic distortion for dual three-phase permanent-magnetsynchronous motor drives operating in the overmodulationregionrdquo IEEE Transactions on Power Electronics vol 31 no 2pp 1367ndash1380 2016
[2] XGuoMHe andY Yang ldquoOvermodulation strategy of powerconverters A reviewrdquo IEEE Access 2018
[3] E E M Mohamed and M A Sayed ldquoMatrix convert-ers and three-phase inverters fed linear induction motordrivesmdashPerformance comparerdquoAin Shams Engineering Journalvol 9 no 3 pp 329ndash340 2018
[4] Q Luo J Zheng Y Sun and L Yang ldquoOptimal modeled six-phase space vector pulse width modulation method for statorvoltage harmonic suppressionrdquo Energies vol 11 no 10 articleno 2598 2018
[5] X Tang C Lai Z Liu andM Zhang ldquoA SVPWM to eliminatecommon-mode voltage for multilevel invertersrdquo Energies vol10 no 5 p 715 2017
[6] M Moranchel F Huerta I Sanz E Bueno and F J RodrıguezldquoA comparison of modulation techniques for modular multi-level convertersrdquo Energies vol 9 no 12 p 1091 2016
[7] L Yuan M-L Chen J-Q Shen and F Xiao ldquoCurrent harmon-ics elimination control method for six-phase PM synchronousmotor drivesrdquo ISA Transactions vol 59 pp 443ndash449 2015
[8] L Yuan B X Hu K Y We et al ldquoA novel current vectordecomposition controller design for six-phase PMSMrdquo Journalof Central South University vol 23 pp 443ndash449 2016
[9] KHatua andVTRanganathan ldquoDirect torque control schemesfor split-phase inductionmachinerdquo IEEETransactions on Indus-try Applications vol 41 no 5 pp 1243ndash1254 2005
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
12 Mathematical Problems in Engineering
[10] Y Zhao and T A Lipo ldquoSpace vector PWM control of dualthree-phase induction machine using vector space decomposi-tionrdquo IEEE Transactions on Industry Applications vol 31 no 5pp 1100ndash1109 1995
[11] D Dujic M Jones and E Levi ldquoAnalysis of output currentripple rms in multiphase drives using space vector approachrdquoIEEE Transactions on Power Electronics vol 24 no 8 pp 1926ndash1938 2009
[12] D Dujic M Jones and E Levi ldquoAnalysis of output current-ripple RMS inmultiphase drives using polygon approachrdquo IEEETransactions on Power Electronics vol 25 no 7 pp 1838ndash18492010
[13] A Iqbal and S Moinuddin ldquoComprehensive relationshipbetween carrier-based PWM and space vector PWM in a five-phase VSIrdquo IEEE Transactions on Power Electronics vol 24 no10 pp 2379ndash2390 2009
[14] K Marouani L Baghli D Hadiouche A Kheloui and A Rez-zoug ldquoA new PWM strategy based on a 24-sector vector spacedecomposition for a six-phase VSI-Fed dual stator inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no5 pp 1910ndash1920 2008
[15] D Hadiouche L Baghli and A Rezzoug ldquoSpace-vector PWMtechniques for dual three-phase ac machine Analysis perfor-mance evaluation and DSP implementationrdquo IEEE Transac-tions on Industry Applications vol 42 no 4 pp 1112ndash1122 2006
[16] S Bolognani and M Zigliotto ldquoSpace vector Fourier analysisof SVM inverters in the overmodulation rangerdquo in Proceedingsof the International Conference on Power Electronics Drives andEnergy Systems for Industrial Growth pp 319ndash324 New DelhiIndia 1996
[17] D Yazdani S Ali Khajehoddin A Bakhshai and G Joos ldquoFullutilization of the inverter in split-phase drives by means of adual three-phase space vector classification algorithmrdquo IEEETransactions on Industrial Electronics vol 56 no 1 pp 120ndash1292009
[18] G Grandi G Serra and A Tani ldquoSpace Vector Modulationof a Six-Phase VSI based on three-phase decompositionrdquo inProceedings of the 2008 International Symposium on Power Elec-tronics Electrical Drives Automation and Motion (SPEEDAM)pp 674ndash679 Ischia Italy June 2008
[19] L Yuan F Xiao J-Q Shen M-L Chen Q-M Shi and LQuan-feng ldquoSensorless control of high-power interior perma-nentmagnet synchronous motor drives at very low speedrdquo IETElectric Power Applications vol 7 no 3 pp 199ndash206 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom