Interleaving Modulation Schemes in Asymmetrical Dual Three-Phase Machines for the DC-Link Stress Reduction

Ci a ion: DeMa cos, A.; Robles, E.;
Ugalde, U.; Ma inez de Aleg ia, I.;
And eu, J. In e lea ing Modula ion
Schemes in Asymme ical Dual
Th ee-Phase Machines o he
DC-Link S ess Reduc ion. Machines
2023,11, 267. h ps://doi.o g/
10.3390/machines11020267
Academic Edi o : K zysz o
Pienkowski
Recei ed: 13 Janua y 2023
Re ised: 1 Feb ua y 2023
Accep ed: 7 Feb ua y 2023
Published: 10 Feb ua y 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
machines
A icle
In e lea ing Modula ion Schemes in Asymme ical Dual
Th ee-Phase Machines o he DC-Link S ess Reduc ion
Ande DeMa cos * , Endika Robles , Unai Ugalde , Inigo Ma inez de Aleg ia and Jon And eu
Depa men o Elec onic Technology, Facul y o Enginee ing in Bilbao, Uni e si y o he Basque
Coun y (UPV/EHU), Plaza Ingenie o To es Que edo 1, 48013 Bilbao, Spain
*Co espondence: ande [email p o ec ed]; Tel.: +34-946-01-3915
Abs ac :
The DC-Link capaci o plays a c ucial ole as a as powe densi y and eliabili y a e
conce ned: i occupies app oxima ely 40% o he in e e , and causes app oxima ely 30% o i s
ailu es. Asymme ical dual h ee-phase (ADTP) mul iphase a angemen s a e gaining ele ance in
he au omo i e sec o o powe ain applica ions. This wo k ocuses on educing he impac ha he
widely used double ze o sequence injec ion (DZSI) amily o PWM echniques ha e on such a bulky
and ailu e-p one componen in an ADTP a angemen by means o in e lea ing echniques. By
using he double Fou ie in eg al o malism, he inpu cu en spec a and he o e all pe o mance
o hese PWM echniques ha e been de i ed, in e ms o cu en ms alue and ol age ipple in
he DC-Link capaci o . Simula ions ha e shown ha choosing an adequa e in e lea ing scheme and
angle conside ably elie es bo h cu en and ol age s esses on he DC-Link capaci o compa ed o
nonin e lea ed ope a ion. Reduc ions o 84% cu en ms and 86% ol age ipple ha e been achie ed
a s a ic ope a ing poin s. Finally, by a e aging he ms cu en o e WLTP s anda d d i ing cycle,
educ ions up o 26% ha e been ob ained unde mo e ealis ic condi ions. All his would enhance he
eliabili y and educe he size o he onboa d capaci o s in u u e elec ic ehicles.
Keywo ds:
mul iphase; in e lea ing; asymme ical dual h ee-phase; double ze o-sequence injec ion
(DZSI) PWM; DC-Link capaci o ; DC-Link cu en spec um
1. In oduc ion
The elec ic ehicle (EV) powe ain is expe imen ing a huge change. WBG semicon-
duc o s, mo o s wi hou dependence on a e ea h ma e ials and new con e e a chi ec-
u es a e being in oduced. Au omo i e manu ac u e s and in e na ional p og ams such
as Ho izon Eu ope, USCAR, DOE, and UN ESCAP a e ocusing on imp o ing speci ic
powe (kW/kg), powe densi y (kW/
`
), e iciency (%), and cos ($/kW) [
1
]. In his con ex ,
mul iphase p opulsion sys ems p o ide se e al ad an ages a an a o dable cos compa ed
wi h classic h ee-phase elec ic mo o -d i en sys ems. Such bene i s include powe spli -
ing be ween phases (lowe cu en s and powe losses o he same a ed ou pu powe ),
educ ion o he o que ipple (enhanced e iciency), o que densi y imp o emen using
ha monic cu en injec ion (in concen a ed winding machines), lowe DC-Link cu en
ipples (smalle DC-Link capaci o ), and in insic aul - ole an ope a ion [2–5].
In o de o bene i om he abo emen ioned ad an ages o mul iphase sys ems, he
ecen scien i ic li e a u e shows ha he dual h ee-phase opology (Figu e 1) is p obably
he mos widesp ead mul iphase solu ion [
6
–
10
]. Al hough odd phase numbe mul iphase
s a -connec ed a angemen s o e a be e ela ionship be ween he deg ees o eedom
and he phase numbe and semiconduc o de ice numbe , hey ha e lowe aul ole ance
ega ding sho ci cui and powe supply aul s (when dual h ee-phase a angemen s a e
supplied independen ly) and hei modula ion scheme is mo e complex [
11
]. The e o e,
mul iple h ee-phase winding machines a e p e e ed. Theo e ically, any numbe o h ee-
phase winding se s can cons i u e his kind o elec ic machine; he mos common is o
Machines 2023,11, 267. h ps://doi.o g/10.3390/machines11020267 h ps://www.mdpi.com/jou nal/machines
Machines 2023,11, 267 2 o 31
ind dual h ee-phase a angemen s wi h wo isola ed neu al poin s known as “dual
h ee-phase” [
12
]. These con igu a ions a e he mos in e es ing because (i) hey ep esen a
good adeo be ween pe o mance and complexi y; (ii) easy mig a ion om h ee-phase
echnologies is possible because mul iple gene ic and modula h ee-phase in e e s can
eed he wo h ee-phase winding se s independen ly [
13
]; and (iii) hey ha e e y good
pe o mance in e ms o aul ole ance (open and sho ci cui aul s as well as on he DC
powe supply) [
6
,
7
,
14
,
15
]. Gene ally, 0
◦
, 30
◦
, and 60
◦
a e he p e e ed angle displacemen s
be ween he wo se s. Howe e , he 30
◦
ype, which is commonly called asymme ical
six-phase o asymme ical dual h ee-phase (ADTP) machine (Figu e 1), p o ides highe
o que densi y and lowe o que ipple han he o he s [
16
] because i elimina es he six h
o que ha monic pulsa ion h ough he synch oniza ion o he wo winding se s [17,18].
iba iin
icap sa1 sb1 sc1
sa1 sb1 sc1
ia1 ib1 ic1
n1
VDC
a1b1c1
sa2 sb2 sc2
sa2 sb2 sc2
ia2 ib2 ic2
n2
a2b2c2
iin 1 VSI1
ADTP
VSI2
iin 2
iin 1a
iin 2a
iin 1b iin 1c
iin 2b iin 2c
30º
a1
a2
b1
n1
b2
c1c2
CDC n2
Figu e 1. Asymme ic dual h ee-phase elec ic machine wi h wo pa allel h ee-phase VSIs.
Thus, ADTP-speci ic ol age-sou ce in e e s (VSIs) ha e been de eloped, which
show g ea po en ial in sa e y-c i ical applica ions, and when high powe densi y is needed,
such as in elec ic ehicle (EV) d i e ains [
19
]. These VSIs (Figu e 1) can be con olled
by using ei he app op ia e space ec o (SV) [
20
–
24
] o ca ie -based (CB) [
25
–
30
] PWM
echniques. The e is a wide a ie y o SV-PWM echniques o he ADTP opology; some o
hem u ilize wo la ge adjacen ac i e ec o s o syn hesize he e e ence ol age ec o [
20
],
and o he s use ou la ge ac i e ec o s [
20
,
21
], h ee la ge ac i e ec o s plus one medium
ac i e ec o [22], o wo la ge and wo medium ac i e ec o s [23]. In con as , CB-PWM
s a egies a e usually ea ed as a dual h ee-phase s uc u e (VSI
1
and VSI
2
, Figu e 1), in-
s ead o a six-phase sys em. This implies an ad an age o e he SV-based app oach because
con en ional CB-PWM echniques can be exploi ed. Figu e 2shows how hese h ee-phase
CB-PWM echniques a e implemen ed, whe e
∗=Mcos(θ1)
is he modula ing signal,
0s
is he injec ed ze o-sequence componen ,
∗∗ = ∗+ 0s
is he modi ied modula ing signal,
θ1
is he modula ing signal’s angula posi ion and he modula ion index (
M
) is de ined as
M=ˆ
V1/(0.5·VDC)
[
31
], whe e
ˆ
V1
is he peak phase-neu al ol age and
VDC
is he DC-Link
ol age (Figu e 1).
Machines 2023,11, 267 3 o 31
++
++
++
+
a** sa
sc
sb
b**
c**
0s
c
a*
b*
c*1
0
1
0
1
0
−
+
−
+
−
(a) CB-PWM block diag am.
0
1
θ1 [ ad]
2π
0π
π
2
3π
2
** 0s
*
−1
(b) MINMAX-PWM.
0
1
−1
θ1 [ ad]
2π
0π
π
2
3π
2
** 0s
*
(c) THI-PWM.
0
1
θ1 [ ad]
2π
0π
π
2
3π
2
** 0s
*
−1
(d) D-PWMMIN.
0
1
θ1 [ ad]
2π
0π
π
2
3π
2
** 0s
*
−1
(e) D-PWMMAX.
θ1 [ ad]
0
1
2π
0π
π
2
3π
2
** 0s *
−1
( ) D-PWM0.
0
1
θ1 [ ad]
2π
0π
π
2
3π
2
**
0s
*
−1
(g) D-PWM1.
θ1 [ ad]
0
1
2π
0π
π
2
3π
2
**
0s
*
−1
(h) D-PWM2.
θ1 [ ad]
0
1
2π
0π
π
2
3π
2
** 0s
*
−1
(i) D-PWM3.
Figu e 2. CB-PWM block diag am as well as hei ol age e e ences and ze o sequence signals.
CB-PWM echniques applied in a “spli ” six-phase in e e a e commonly known as
double ze o-sequence injec ion (DZSI) PWM echniques because his implies injec ing one
ze o-sequence componen (
0s
) in o each h ee-phase s uc u e [
25
]. They can be classi ied
in o con inuous and discon inuous modula ion echniques [
31
]. Sinusoidal PWM (SPWM,
as he ze o sequence signal which is injec ed in SPWM is 0, some imes i is no conside ed
Machines 2023,11, 267 4 o 31
as DZSI-PWM echnique), hi d ha monic injec ion PWM (THI-PWM) and min-max PWM
me hod (MINMAX-PWM, some imes also called symme ical SV-PWM) a e known as
con inuous modula ions (C-PWM), in which all he in e e b anches swi ch con inuously.
D-PWMMIN, D-PWMMAX, D-PWM0, D-PWM1, D-PWM2, and D-PWM3 a e known as
discon inuous PWM (D-PWM) echniques, in which one b anch does no swi ch o e a
whole swi ching pe iod (while he modula ing signal is clamped o
±
1, Figu e 2). Thus,
he swi ching powe losses in he semiconduc o s o he VSI a e educed o discon inuous
PWM echniques because only wo ou o he h ee b anches a e ac ually swi ching and he
a e age equi alen swi ching equency is educed o
2/3 sw
. Finally, all hese ca ie -based
PWM echniques allow a maximum modula ion index
Mmax =
1.15 wo king in he linea
egion excep o SPWM echnique in which Mmax =1.
A ha dwa e le el, he DC-Link capaci o (
CDC
, Figu e 1) is a c ucial elemen o he
VSI. This capaci o is esponsible o educing he low- equency ol age ipple a he inpu
o he con e e , in bo h s eady and ansien s a es, as well as s o ing he necessa y ene gy
o allow an ins an aneous powe balance be ween he con e e inpu and ou pu . I mus
p o ide a low impedance pa h o high- equency cu en s in o de o decouple and educe
he cu en ipple om he ba e y. Mo e impo an ly, in ac ion applica ions, he DC-Link
capaci o is a bulky and expensi e componen because i amoun s o up o 40% o he
o al olume o he VSI [
32
–
35
]. In addi ion, DC-Link capaci o s a e also conside ed o
be one o he mos c i ical elemen s in powe elec onics because hey cause 30% o he
o al ailu es in powe elec onic in e e s [
36
–
38
]. Fo his eason, he eliabili y o hese
eac i e componen s has been discussed deeply du ing he las se e al yea s [39–42].
The selec ion o he DC-Link capaci o ( echnology, capaci ance, size, weigh , cos ,
e c.) is highly dependen on he DC ol age a ing o he applica ion whe e hey a e
in eg a ed [
40
]. To da e, ligh EV ba e ies anged om 250 o 450 V [
43
], whe eas o hea y
ehicles he a ed ol age is abou 800 V. In his con ex , a end change is aking place in
which elec ic mobili y manu ac u e s a e commi ed o o e ing mo e solu ions o hese
800-V sys ems, because his pe mi s he u iliza ion o ligh e wi ing [
44
] and also p oduces
smalle on-s a e powe losses, highe e iciency and powe densi y mo o s, and as e
cha ge o he ba e y pack [
45
]. A highe DC ol ages, he capaci ance o
CDC
dec eases o
he same size o encapsula ion [
46
], and i s li e ime is signi ican ly educed. Fo example,
AVX au omo i e ilm capaci o s o he FHC1 se ies has a li e ime o an o de o magni ude
o 10,000 h a 400 V, whe eas a 900 V he li e ime d ops o app oxima ely 1000 h [46].
In addi ion o he DC ol age a ing, he e a e o he impo an speci ica ions o selec
an app op ia e DC-Link capaci o : ol age ipple, which is in e sely p opo ional o he
capaci ance and he size o he DC-Link capaci o [
47
,
48
], and cu en ipple [
32
,
34
,
49
].
Any ol age ipple on he DC-Link p oduces an addi ional cu en ipple on he phase
cu en s, which wo sens o que ipple in he elec ic machine. In his con ex , he e is o en
a speci ica ion o he maximum allowable ol age ipple on he DC-Link ( ypically anging
om 5–10%).
The cu en ipple is condi ioned by he maximum ho -spo empe a u e o he DC-
Link capaci o . This in e nal empe a u e depends on he powe losses due o he equi alen
se ies esis ance (ESR) and is in e sely p opo ional o he li e ime o he componen . Thus,
manu ac u e s ypically speci y he maximum ms ipple cu en a ing a an ambien
empe a u e and a speci ic equency.
Because he DC-Link capaci o is a c i ical componen , signi ican e o s a e be-
ing made o enhance i s pe o mance. Some wo ks p opose o minimize he DC-Link
capaci o ’s cu en s ess by he synch oniza ion o pa allel-in e lea ed single-phase in-
e e s [
50
,
51
] and h ee-phase in e e s [
52
–
59
]. In ADTP machines, cons an in e lea ing
angles can be used o educe he DC-Link capaci o cu en o he MINMAX-PWM and
some discon inuous PWM echniques [
49
,
60
–
62
]. Re . [
63
] p oposes a dynamic in e lea ing
me hod o educe he DC-Link cu en ipple, which is only applicable o discon inuous
PWM echniques.
Machines 2023,11, 267 5 o 31
Howe e , i is no common o ind esea ch wo ks ha analyse in-dep h he e ec o
he in e lea ing angle on he inpu ha monic cancella ion and he ms cu en minimiza ion
o he ADTP a angemen . Fu he mo e, he ew exis ing con ibu ions usually ocus
on one o wo speci ic PWM echniques, which makes i ha d o compa e and quan i y
he pe o mance o he a ious DZSI-PWM echniques. As a esul , iden i ying he bes
in e lea ing angle o each DZSI-PWM echnique applied in an ADTP a angemen is also
missing in he scien i ic li e a u e.
This pape ocuses on educing he cu en and ol age s ess in he DC-Link capaci o
by using a mul iphase VSI and DZSI-PWM echniques sui able o ADTP a angemen .
Di e en aspec s a e discussed h oughou he pape . In Sec ion 2, DC-Link cu en spec a
a e analysed by using he double Fou ie in eg al me hod o DZSI-PWM echniques in
ADTP con e e s, which a e di ec ly ela ed o he main ol age and cu en s ess a iables
( ms cu en and peak- o-peak ol age ipple) o he DC-Link capaci o . The igu e-o -
me i conside ed in he scien i ic li e a u e is he ou pu cu en quali y, which is di ec ly
ela ed o he o al ha monic dis o ion (THD) o he lux ha monic dis o ion ac o (HDF).
Howe e , his igno es how he modula ion echnique a ec s he con e e inpu . Thus,
Sec ion 3ma hema ically analyses and simula es he e ec o he DZSI-PWM echniques
on he inpu cu en spec um and how i a ec s he DC-Link capaci o in an ADTP
a angemen . Due o he need o educe he ms alue o he DC-Link capaci o cu en ,
Sec ion 4exp esses di e en in e lea ing schemes and how he ela ionship be ween he
inpu cu en ha monic spec um and he cons an in e lea ing can be exploi ed in o de o
cancel ce ain dominan ha monics. Nex , in o de o display he impo ance o using his
ype o in e lea ing echniques, Sec ion 5shows bo h he ms cu en and he peak- o-peak
ol age o he DC-Link capaci o applying he op imal in e lea ing scheme o each DZSI-
PWM echnique. Finally, Sec ion 6d aws he co esponding conclusions, emphasizing ha
he pe o mance o he DC-Link capaci o can be signi ican ly imp o ed by applying a
sui able in e lea ing scheme.
2. DC-Link Capaci o Cu en and Vol age S ess in Asymme ical Dual
Th ee-Phase In e e s
Vol age ipple and DC-Link capaci o cu en (
icap
, Figu e 1) ms alue play a c ucial
ole in he selec ion o an app op ia e DC-Link capaci o . The cu en in he ADTP in e e
de e mines he capaci o cu en . Figu e 3shows he de ailed low cha ollowed in
his wo k in o de o ob ain he ha monic spec um o he DC-Link capaci o o he
desc ibed ADTP sys em. He e, wo pa allel p ocedu es ha e been implemen ed: (i) in
Ma lab execu ing he co esponding code o he ma hema ical equa ions syn hesized in
his sec ion; and (ii) in Ma lab–Simulink by using a highe -le el block en i onmen . The
cu en ha monic spec a ob ained om bo h p ocedu es ha e been compa ed o each o he
in o de o check he esul ma ching. In he nex lines, he ha monic spec um o he ADTP
in e e is s udied in de ail.
2.1. Cu en Spec um Theo e ical Basics o an ADTP In e e
2.1.1. Inpu Cu en o One B anch o VSI1(iin 1a)
The double Fou ie in eg al o mula ion cha ac e izes a double pe iodic unc ion in he
equency domain [
64
]. Such is he case o a gene ic analog PWM wa e o m
g[x( ),y( )]
,
whe e
x( )=
2
π sw
and
y( )=
2
π 1 =θ1
a e wo ime a iables, wi h
sw
he ca ie
equency and 1< sw he undamen al equency.
Thus, acco ding o ha o mula ion,

Machines 2023,11, 267 6 o 31
g(x,y)=A00
2
DC o se
+
∞
∑
n=1hA0ncos ny +B0nsin nyi
Fundamen al, and Baseband Ha monics
+
∞
∑
m=1hAm0cos mx +Bm0sin mxi
Ca ie Ha monics
(1)
+
∞
∑
n=−∞
bu n6=0
∞
∑
m=1hAmn cos(mx +ny)+Bmn sin(mx +ny)i
Sideband Ha monics
,
whe e
Amn =1
2π2
π
Z
−π
π
Z
−π
g(x,y)cos(mx +ny)dx dy, (2)
Bmn =1
2π2
π
Z
−π
π
Z
−π
g(x,y)sin(mx +ny)dx dy, (3)
o , in complex o m,
Cmn =Amn +jBmn =1
2π2
π
Z
−π
π
Z
−π
g(x,y)ej(mx+ny)dx dy. (4)
Thus,
|Cmn|=pA2
mn +B2
mn
ep esen s he spec al magni ude o each ha monic,
which a ises a equency alues equalling
h=m sw +n 1
, whe e
m
is he ca ie index
a iable and nis he baseband index a iable.
Figu es 2and 4show how he ol age pa e ns o a PWM-d i en in e e a e syn he-
sized as a unc ion o he e e ence and ca ie ol ages, i.e.,
∗∗
and
c
, espec i ely. As
is usual [
65
,
66
], le us assume ha
sw  1
; he e o e, he phase cu en s a e sinusoidal,
wi h ampli ude ˆ
Iou and phase lag φwi hou any high- equency ipple cu en , so
ia1=ˆ
Iou cos(2π 1 −φ)=ˆ
Iou cos(y−φ); (5)
hen, because
iin 1a
esul s om he sampling o
ia1
by ollowing he same ol age PWM
pa e n, Equa ions (1)–(4) can be applied wi h
g(x,y) = (0 when ∗∗ ≤ c ,
ˆ
Iou cos(y−φ)when ∗∗ > c .(6)
As Figu e 4shows, he limi s o he inne in eg al o he ising and alling po ions
o
g(x
,
y)
equal
x =−π
2[1+ ∗∗(y)]
and
x =π
2[1+ ∗∗(y)]
. The e o e, om
(1)
–
(4)
he
ha monic coe icien s o iin 1a esul [65,66] in
Cin 1a
mn =ˆ
Iou
2π2
2π
Z
0



[1+ ∗∗(y)]π
2
Z
−[1+ ∗∗(y)]π
2
cos(y−φ)·ej(mx+ny)dx

dy. (7)
Thus, Equa ion
(7)
quan i ies he spec um o he cu en
iin 1a
, emphasizing i s de-
pendance on he PWM echnique and modula ion index
M
h ough he limi s o he inne
in eg al ∗∗(y)as well as on he phase lag φ.
Machines 2023,11, 267 7 o 31
DZSI-PWM
echnique
selec ion
Cmn calcula ion (6)
in 1a
in 1
Cmn calcula ion (9)
icap, ms calcula ion
(12)-(14)
in
Cmn calcula ion:
- wi hou (11)
- wi h (31)
M alue selec ion
h alue selec ion
M ange
comple ed?
Yes
No h ange
comple ed?
Yes
No
Ma lab sc ip Simulink model
CDC ha monic analysis CDC s ess educ ion
DZSI-PWM
echnique
selec ion
Run
simula ion
End
End
θ1 [ ad]
0
1
-1
2π
0π
π
23π
2
** 0s
*
0
-0.5
-1
0.5
2Tsw
Tsw
2
3Tsw
0
2
Tsw
h
h ange
M ange
0.07
-0.07
-0.14
0.14
2Tsw
Tsw
2
3Tsw
0
0
2
Tsw
Δ cap,pp
m=1 m=2 m=3 m=4
0.1
0.2
0.3
0.4
0
n=-3 n=3 n=0
peak=0.765 n=0
Modula ing signal
cons uc ion ( )
**
Cmn calcula ion (6)
in 1a
Cmn calcula ion (6)
in 1
Cmn calcula ion (6)
in
a1b1c1
a2b2c2
in 1
in
h=n 1+m sw
()
m and n
de e mina ion
0
0.1
0.25
0.5
0.75
0.2 0.3 0.4 0.5 0.6 0.80.7
1
10.9
ζc= ad
π
2
ζ=0 ad ζd
ζop
m=1 m=2 m=3 m=4
0.1
0.2
0.3
0.4
0
n=-3 n=3 n=0
peak=0.765 n=0
0
0.1
0.25
0.5
0.75
0.2 0.3 0.4 0.5 0.6 0.80.7
1
10.9
ζc= ad
π
2
ζ=0 ad ζd
ζop
m=1
n=-3
Signal
measu emen
(iin , iin 1,
icap, cap...)
Plo iin FFT
0
0.1
0.2
0.4
0.6
0.2 0.3 0.4 0.5 0.6 0.80.7 10.9
ζd=ζc
ζop
ζ=0 ad
Plo ΔVcap,max
Plo icap, ms
Repea
simula ion
un il
M ange
comple ed
m0.1
0.5
1
0
0.5
1
1234567M
Ampli ude [p.u]
Cmn
Ini ializa ion
h ange,
M ange,
1, sw,
Io,
...
Cmn
in
icap, ms
in
Check esul
ma ching
Check esul
ma ching
icap, ms
Δ cap,max
Δ cap
iin FFT
icap
Figu e 3. CDC ha monic analysis and ms calcula ion low cha .
2.1.2. Inpu Cu en o VSI1(iin 1)
The Fou ie coe icien s o he consecu i e b anches ‘1b’ and ‘1c’ (
Ciin 1b
mn
and
Ciin 1c
mn
)
o he VSI
1
(Figu e 1) can be ob ained by aking (7) and eplacing
ny
wi h
n(y+2π
3)
and
n(y+4π
3)
, espec i ely ( he cu en pulses o his b anches a e phase shi ed
2π/3
ad
4π/3 ad conce ning he b anch ‘1a’). Likewise, i can be phase shi ed Ciin 1a
mn as
Ciin 1b
mn =Ciin 1a
mn ·ejn 2π
3, (8)
Ciin 1c
mn =Ciin 1a
mn ·ejn 4π
3. (9)
Howe e , because iin 1=iin 1a+iin 1b+iin 1c(Figu e 1),
Ciin 1
mn =Ciin 1a
mn ·h1+ejn 2π
3+ejn 4π
3i=Ciin 1a
mn ·1+2cosn2π
3, (10)
whe e, i can be in e ed ha
Ciin 1
mn =
0 o all alues o
n
excep o 0 and mul iples o 3.
This means ha a he ou pu phase cu en s o each h ee-phase VSI wi hin he ADTP,
whe e he ha monics mul iples o 3 imes he undamen al cancel each o he ou , a he
Machines 2023,11, 267 8 o 31
inpu , he opposi e happens. Thus, he e only exis high- equency ca ie and sideband
iplens ha monics a he con e e inpu because he PWM me hods elimina e all he
low-o de baseband ha monics.
+1
-1
-π+π
0
0x x
x=ωsw
x=ωsw
0
+1
-1
0
**(y)
g(x,y)
Tsw
c
c
-π
2π
+2
V
Block
diag am
**
c
**
Figu e 4. CB-PWM and in eg a ion limi s o (7).
2.1.3. Inpu Cu en o ADTP (iin )
The ADTP is composed o wo h ee-phase s a o windings displaced spa ially by
30 elec ical deg ees (
π/6
ad). Usually, each h ee-phase subsys em has i s own isola ed
neu al poin (n
1
and n
2
, Figu e 1). Thus, ollowing a simila analysis o (10), he Fou ie
coe icien s o he inpu cu en ha monics o he second h ee-phase s a o winding (
Ciin 2
mn
)
esul in
Ciin 2
mn =Ciin 1a
mn ·1+2cosn2π
3·ejn π
6, (11)
whe e
ejn π
6
co esponds o he 30
◦
displacemen o he second s a o winding wi h espec
o he i s one. F om (10)–(11), he Fou ie coe icien s o he inpu cu en o he ADTP
esul in
Ciin
mn =Ciin 1a
mn ·1+2cosn2π
3·h1+ej(nπ
6)i. (12)
The di e ence among he inpu cu en ha monics o a h ee-phase sys em (10) and
an asymme ical dual h ee-phase VSI (12) is he
h1+ej(nπ
6)i
e m, which equals 0 o
n=
6
(2k+1)∀k∈Z
. This leads o he cancella ion o some o he exis ing high- equency
ca ie and sideband iplens ha monics a he inpu o each o he ADTP con e e .
2.2. De ini ion o RMS Cu en and Vol age Ripple in DC-Link Capaci o
The wo s scena io o he DC-Link capaci o is when he whole inpu cu en ipple o
he ADTP comes om he DC-Link capaci o (
icap =iin ,AC
) and he ba e y only supplies
he a e age cu en o he in e e (
Iba =Iin ,a g
) [
53
,
67
–
70
]. Thus, he ms alue o he
DC-Link cu en (Figu e 1) can be exp essed as
Icap, ms =qI2
in , ms −I2
in ,a g , (13)
whe e he ms alue o he inpu cu en is
Iin , ms =
u
u
u
∞
∑
n=0
Ciin
0n
√2

2
+
∞
∑
m=1
∞
∑
n=−∞
Ciin
mn 
√2

2
, (14)
Machines 2023,11, 267 9 o 31
and he a e age alue o he inpu cu en o he ADTP is
Iin ,a g =6
4Mˆ
Iou cos φ. (15)
Assuming ha a he dominan cu en ha monic’s equency he capaci o has a
p edominan ly capaci i e eac ance, he RL impedance can be neglec ed, so he DC-Link
ol age a ia ion can be exp essed as
∆ cap( ) = 1
C
Z
0
icap d =1
C
Z
0
(iin −Iin ,a g)d . (16)
Conside ing his, Figu e 5a shows he cu en h ough he DC-Link capaci o and
Figu e 5b i s ol age ipple acco ding o (16). Applying a no maliza ion ac o
∆ base =ˆ
Iou ·Tsw/C
o he ol age ipple o (16), he peak- o-peak alue o he ol age
ipple in e e y swi ching pe iod (Tsw) can be ob ained by
∆ cap,ppTsw =max∆ cap( )Tsw −min∆ cap( )Tsw , (17)
which makes i possible o de e mine he maximum alue o he swi ching ol age ipple
o e
T1
(o
θ1=
2
π ad
). Because he inpu cu en o he ADTP is epea ed e e y
2π/6 ad
(Figu e 5c), i is su icien o pe o m he analysis du ing his in e al, so
∆ cap,max =max∆ cap,pp2π=max∆ cap,pp2π/6 . (18)
0
−0.5
−1
0.5
Tsw
4
3Tsw
0
icap
4
Tsw
2
Tsw
ime [s]
(a) Cu en wa e o m o e wo Tsw.
0
Tsw
4
3Tsw
4
Tsw
2
Tsw
ime [s]
0.07
−0.07
−0.14
0.14
0
Δ cap Δ cap,pp
(b) Vol age wa e o m o e wo Tsw.
0.2
0.1
0
0.3
Δ cap,pp
Δ cap,max
ime [s]
0T1
6
5T1
3
2T1
6
T1
2
T1
3
T1
(c) Vol age wa e o m o e a T1.
Figu e 5.
No malized cu en and ol age in he DC-Link capaci o wi h SPWM echnique,
M=
0.9
and cos φ= 1.
Machines 2023,11, 267 16 o 31
la cu en pulses depends on he ampli ude
ˆ
Iou
and he squa e oo o he du y cycle
(
Iin 1, ms =√D·ˆ
Iou
, Figu e 11). When an in e lea ing angle
ζ=
0 ad is applied be ween
wo in e e s, he cu en pulses o bo h VSIs a e o ally o e lapped, and he esul ing
pulses a e doubled in ampli ude, which leads o a double ms cu en alue o he in e -
lea ed in e e cu en (
Iin , ms =
2
·√D·ˆ
Iou
). As he in e lea ing angle is inc eased, he
o e lap is educed, and in u n, he ms alue o he esul ing wa e o m is educed. When
he in e lea ing angle be ween he pulses is la ge enough o hem no o o e lap (
ζ≥D
),
he ms cu en is minimized (
Iin , ms =√2·√D·ˆ
Iou
). This way, a educ ion o he ms
alue is achie ed wi h espec o he p e ious cases wi hou in e lea ing (Figu e 11).
In o de o analyse he concep o in e lea ing in dep h, h ee di e en schemes will
be in oduced: cons an in e lea ing, dynamic in e lea ing, and op imal in e lea ing.
c 2
iin 1
D
iin 2
iin
Iou
Iou
2Iou
Iin , ms=2· D·Iou
Iin , ms
c 1
**
x
Tsw
(a) No in e lea ing (ζ=0◦).
iin 2
c 1
iin 1
iin Iin , ms
D
ζ
Iou
Iou
2Iou Iou
c 2
x
**
Iin , ms= 2·ζ+4·(D-ζ)·Iou
(b) O e lapping (ζ<D).
iin 1
iin 2
iin
Iou
Iou
Iou
Iin , ms
D
ζ
Iin , ms= 2· D·Iou
x
c 1
c 2
**
(c) No o e lapping (ζ≥D).
Figu e 11. In e lea ing concep on ec angula cu en pulses.
4.1. Cons an In e lea ing: ζc
The concep o in e lea ing can be applied o he Fou ie double se ies o he inpu
cu en o he VSI
2
by adding an addi ional phase shi wi h espec o he inpu cu en o
he VSI
1
(Figu e 1). As Figu e 11 shows, because
x( )
is he a iable ela ed o he ca ie
wa e angle, he Fou ie coe icien s o b anch ‘a’ o he consecu i e in e e ‘2’ (
Ciin 2a
mn
) can
be calcula ed by aking (7) and eplacing
mx
wi h
m(x+ζc)
. This can be also
ep esen ed as
Ciin 2a
mn =Ciin 1a
mn ·ejmζc, (30)
whe e
ejmζc
co esponds o he shi ing caused by he in e lea ing o he second ca ie
signal ( c 2, in Figu e 11).
The in oduc ion o his new pa ame e a ec s
Ciin 2
mn
o (11) and
Ciin
mn
o (12), leading o
Ciin 2
mn =Ciin 1a
mn ·1+2cosn2π
3·ejn π
6·ejmζc, (31)
Ciin
mn =Ciin 1a
mn ·1+2cosn2π
3·h1+ej(nπ
6+mζc)i. (32)
In (32), i can be obse ed ha making he second e m in b acke s
h1+ej(nπ
6+mζc)i
equal o 0, an in e lea ing angle
ζc
ha cancels some speci ic inpu cu en ha monics (
Ciin
mn
,
(12)) is ob ained:
ζc=(2k+1)·π−nπ
6
m∀k∈Z+. (33)

Machines 2023,11, 267 17 o 31
In his sense, and because he dominan ha monic is he mos impo an componen
when i comes o compu ing he ms alue o he whole cu en spec um h ough he
capaci o , he in e lea ing angle (
ζc
) should elimina e his dominan ha monic. Table 2
summa izes he mos signi ican cons an in e lea ing angles and he ha monic o de s
cancelled. This able also poin s ou when he mos ele an inpu cu en ha monics
a e emo ed: le ing
ζc=π/2
ad e ases he (2,0) and (1,3) ha monics, whe eas le ing
ζc=π ad e ases he (1,0) ha monic.
As a colla e al e ec , some imes o he ha monics can sligh ly inc ease hei ampli ude.
The e o e, he angle which elimina es he dominan inpu cu en ha monic and he one
which minimizes
Icap, ms
can be di e en . Fo example, Figu e 12 shows he inpu cu en
spec um o he ADTP wi h SPWM echnique,
M=
0.9,
cos φ
= 1 and wo in e lea ing
angles (
ζc=
0 ad and
ζc=π/2
ad). Fo his case, i can be obse ed ha he dominan ha -
monic (2,0) has been cancelled ou , bu (1,
−
3) and (2,
±
6) ha e inc eased hei ampli ude.
The e o e, he selec ion o he in e lea ing angle has o be made ca e ully.
m=1
n=3
n=0
n=6
n=-6
n=-3
m=2
0.2
0.4
0.6
0.8
0
π
2
=0 ad
ζc= ad
ζc
Figu e 12.
Inpu cu en spec um o he ADTP wi h SPWM echnique,
M=
0.9,
cos φ
= 1 and wo
in e lea ing angles.
In iew o his, nume ical calcula ions ha e been ca ied ou in Ma lab ollowing
he p ocedu e explained in Sec ion 2 o each DZSI-PWM echnique, by sweeping he
modula ion index
M
and he in e lea ing angle
ζc
. All hese calcula ions show ha he
ζc=π/2
ad cons an angle minimizes
Icap, ms
ega dless o he alue o
M
o all he
analysed DZSI-PWM echniques excep o D-PWMMIN and D-PWMMAX, in which
ζc=π
ad is he bes choice o any alue o
M
. Howe e , om he nume ical analysis
ca ied ou , i has been seen ha he cons an angle applied o he en i e pe iod o he
undamen al o D-PWMMIN and D-PWMMAX
ζc=π
is he bes op ion only o
M⩽
0.75;
and o
M>
0.75 he in e lea ing angle which minimizes he
Icap, ms
is only cons an o
each undamen al pe iod bu no o es o he linea ange. This is explained be e in
Sec ion 4.3. This analysis coincides wi h he abo e pe o med inpu ha monic spec a o
Figu e 10, as well as wi h he emo al o he dominan ha monics obse ed in Table 2.
Table 2. Elimina ed (m,n) ha monics depending on ζcacco ding o (33).
k=0k=1k=2
ζc=π/2 ad
(2,0) (6,0) (10,0)
(1,3) (5,3) (9,3)
(3,−3) (7,−3) (11,−3)
ζc=π ad
(1,0) (3,0) (5,0)
(2,6) (4,6) (6,6)
(2,−6) (4,−6) (6,−6)
Finally, as an example o his analysis, Figu e 13 shows he case o he SPWM echnique.
He e, no e ha o
M=
0.35 (Figu e 13a),
Icap, ms
is a minimum when
ζc
lies be ween 1.08
and 2.06 ad e en hough he cu en spec a is no he same (Figu e 13c–e. Pe o ming
he same analysis in he whole linea egion o
M
, o 0
⩽M<
0.5, he e is no a single
ζc
which minimizes
Icap, ms
bu a ange o alues (Figu e 13b); howe e , o 0.5
⩽M⩽
1
Machines 2023,11, 267 18 o 31
he e is jus one
ζc
alue (
π/2
ad) ha is conside ed he bes choice
ζc
in he en i e ange o
M o he SPWM echnique.
in e lea ing angle ζ [ ad]
0.4
0.8
0.6
1
0
Icap, ms
π
4
π
2
π
4
3ππ
4
5π
4
7
π
2
32π
1.08 2.06
(
a
) No malized
Icap, ms
as a unc ion o he in e -
lea ing angle ζ o M=0.35.
modula ion index [p.u.]
π
10.90.8
0.70.60.50.40.30.20.1
ζ
π
4
3
π
2
π
4
0
1.08
2.06
ζc
(
b
) Cons an in e lea ing angle
ζc
o he whole
linea ange.
m=1 m=2 m=3 m=4
0.2
0.4
0.6
0
(
c
) Inpu cu en spec um o he ADTP wi h
ζ=1.08 ad and M=0.35.
m=1 m=2 m=3 m=4
0.2
0.4
0.6
0
(
d
) Inpu cu en spec um o he ADTP wi h
ζ=π/2 ad and M=0.35.
m=1 m=2 m=3 m=4
0.2
0.4
0.6
0
(
e
) Inpu cu en spec um o he ADTP wi h
ζ=2.06 ad and M=0.35.
Figu e 13. Inpu cu en unde cons an in e lea ing angle applying he SPWM echnique.
4.2. Dynamic In e lea ing Scheme o Discon inuous PWM Techniques: ζd
Unlike he cons an in e lea ing scheme discussed abo e, a dynamic in e lea ing
algo i hm
ζd
applicable only o discon inuous PWM has been ecen ly p oposed in [
63
] o
educe he DC-Link cu en ipple o he ADTP. In his case, he in e lea ing angle o he
p oposed me hod is no cons an . In some subin e als o he undamen al pe iod cycle,
ζ
is se o π ad whe eas he es o he ime i is 0 ad.
In any discon inuous PWM echnique, he peak- o-peak alue o he inpu cu en
o he VSI wi hou any in e lea ing ises signi ican ly when any wo nea by phases
(
Figu e 14a
) in he phaso diag am a e clamped o
±
1. As an example, Figu e 14 shows he
ope a ion basics o
ζd
o D-PWM1 and D-PWMMIN. Figu e 14b,c show he inpu cu en
o he VSI wi hou in e lea ing (
ζ=
0 ad) o hese discon inuous PWM echniques and
Figu e 14d,e hei espec i e ol age e e ences o he ADTP.
In he case o D-PWMMIN, wo nea by phases in he phaso diag am a e con inu-
ously clamped o
±
1 (Figu e 14e). This leads o a con inuous ac i a ion o he dynamic
in e lea ing algo i hm (Figu e 14g,
ζd
in blue). This means ha o his speci ic PWM
echnique, he dynamic in e lea ing scheme is exac ly he same as applying a cons an
angle o
ζc=π
ad. The e ec o applying his angle can be isualized in Figu e 14 ,g o
D-PWM1 and D-PWMMIN, espec i ely. The peak- o-peak alues o he inpu cu en s
a e educed signi ican ly compa ing o he
ζ=
0 ad scena io o Figu e 14b,c. The ech-
niques D-PWM0, D-PWM2 and D-PWM3 ollow he same pa e n as D-PWM1, whe eas
D-PWMMAX beha es as D-PWMMIN in e ms o ull- ime clamping. The e o e, o bo h
D-PWMMIN and D-PWMMAX, applying he dynamic in e lea ing scheme would be
exac ly like applying he scheme ζc=π ad explained in he p e ious subsec ion.
Machines 2023,11, 267 19 o 31
a1
a2
b1
n1
b2
c1c2
n2
(
a
) ADTP phaso diag am.
0
1.5
0.5
2
iin 1
θ1 [ ad]
2π
0π
π
2
3π
2
high peak- o-peak cu en
(
b
) Inpu cu en o he VSI wi hou in e lea -
ing (ζ=0 ad) o D-PWM1.
0
1.5
0.5
2
iin 1
θ1 [ ad]
2π
0π
π
2
3π
2
high peak- o-peak cu en
(
c
) Inpu cu en o he VSI wi hou in e lea -
ing (ζ=0 ad) o D-PWMMIN.
θ1 [ ad]
0
1
**
2π
0π
π
2
3π
2
b1
b2
0
1
**
−1
2π
0π
π
2
3π
2
wo nea by phases clamped
a2
a1
0
1
**
2π
0π
π
2
3π
2
c1
c2
−1
−1
(
d
) Vol age e e ences o he VSI o he ADTP
o D-PWM1.
θ1 [ ad]
0
1
**
A1B2
C1
C2
2π
0π
π
2
3π
2
A2
B1
a1
a2
c2
−1
0
1
**
A1B2
C1
C2
2π
0π
π
2
3π
2
A2
B1b2c2
c1
−1
0
1
**
A1B2
C1
C2
2π
0π
π
2
3π
2
A2
B1
−1
b1
b2
a2
(
e
) Vol age e e ences o he VSI o he ADTP
o D-PWMMIN.
0
1.5
0.5
2
iin 1
θ1 [ ad]
2π
0π
π
2
3π
2
peak- o-peak cu en educ ion
ζd
0
π
(
) Inpu cu en o he VSI and he ins an-
aneous in e lea ing angle applying
ζd
o
D-PWM1.
0
1.5
0.5
2
iin 1ζd
0
π
θ1 [ ad]
2π
0π
π
2
3π
2
peak- o-peak educ ion
(
g
) Inpu cu en o he VSI and he ins an-
aneous in e lea ing angle applying
ζd
o
D-PWMMIN.
Figu e 14. Dynamic in e lea ing me hod p oposed in [63] o D-PWM1 and D-PWMMIN.
As a coun e pa , his dynamic in e lea ing scheme (
ζd
) has some disad an ages
compa ed o he cons an in e lea ing (
ζc
); i.e., mo e compu a ional esou ces a e needed
Machines 2023,11, 267 20 o 31
in o de o de ec he ol age e e ence clampings be ween wo nea by phases in he phaso
diag am; in addi ion, i is only applicable o discon inuous PWM echniques.
4.3. Op imal In e lea ing Scheme o any DZSI-PWM: ζop
The op imal in e lea ing scheme (
ζop
) can be de ined as he one which minimizes
Icap, ms
o e he en i e linea egion, 0
⩽M⩽
1. In he case o con inuous echniques,
ζop =ζc
because a dynamic in e lea ing scheme canno be used. The e o e, he op imal
in e lea ing angle can be conside ed ζc=π/2 ad.
In he case o he D-PWMMIN and D-PWMMAX echniques, i is a bi mo e complex.
Fi s , i is wo h emembe ing ha as he dynamic in e lea ing is applied con inuously, i
is he same as applying a cons an in e lea ing o
ζc=π
ad. In addi ion, ollowing he
nume ical analysis desc ibed in Sec ion 4.1 and as Figu e 15 shows,
ζ=π
ad minimizes
Icap, ms
o 0
⩽M⩽
1. Howe e , o highe modula ion indexes, he in e lea ing angle
becomes smalle . This happens because in his ange o
M
, al hough he (1,0) ha monic is
no comple ely cancelled, he ha monics ha ha e g ea in luence on he ms alue o he
cu en , such as (1,0) and (2,0), a e conside ably a enua ed.
modula ion index [p.u.]
π
10.90.8
0.70.60.50.40.30.20.1
ζ
π
4
3
π
2
π
4
0
ζop (M)
D-PWMMIN
&
D-PWMMAX
ζd=ζc=π ad
Figu e 15.
Cons an and op imal in e lea ing angles o he whole linea ange o D-PWMMIN and
D-PWMMAX.
Finally, o he es o discon inuous echniques (D-PWM0, D-PWM1, D-PWM2 and
D-PWM3), he op imal in e lea ing scheme consis s o compa ing bo h in e lea ing algo-
i hms and selec ing he one which gi es he smalles Icap, ms o each modula ion index.
The e o e, he implemen a ion o his op imal in e lea ing scheme is pe o med ol-
lowing he indica ions in Figu e 16. The nex sec ion will show he esul s ob ained o all
he in e lea ing schemes.
DZSI-PWMs Modula ion index (M)
0-0.30 0.30-0.80 0.80-0.85 0.85-0.95 0.95-1
Con inuous
SPWM
THI-PWM
MINMAX-PWM
Discon inuous
D-PWMMIN
D-PWMMAX
D-PWM0
D-PWM1
D-PWM2
D-PWM3
π o
d
π/2
as a unc ion o M (Figu e 15)
o
π/2
π/2
π/2
π/2
d d
d
d
d
Figu e 16.
Op imal in e lea ing scheme as a unc ion o he selec ed DZSI-PWM echnique and
M
o cos φ=1.
5. In luence o In e lea ing o DZSI-PWM Techniques on he Cu en Ripple and
Vol age in he DC-Link Capaci o
The in luence o DZSI-PWM echniques on
Icap, ms
and
∆ pp,max
has been iden i ied
h ough he in e lea ing schemes desc ibed in Sec ion 4as a unc ion o
M
and o
cos φ=
1
by using he me hodology desc ibed in Figu e 3.
In con as o h ee-phase VSIs, whe e he ms cu en h ough he DC-Link capaci o
is independen o he chosen PWM echnique [
67
], in he ADTP wi h double ze o sequence
injec ion echnique i is s ongly dependen on i . This happens due o he 30
◦
shi ing
be ween he wo h ee-phase in e e s o ming he ADTP. The ms alue o he inpu
Machines 2023,11, 267 21 o 31
cu en o ei he he i s in e e o he second in e e do no depend on he modula ion
echnique bu hei sum does:
Iin , ms 6=Iin 1, ms +Iin 2, ms. (34)
In gene al e ms, in EV applica ions, he DC-Link capaci o is selec ed mainly consid-
e ing he ipple cu en h ough he capaci o . As he e a e no low- equency ha monic
componen s as in single-phase con e e s, he ol age ipple is lowe o a mul iphase ap-
plica ion wi h he same powe a ing. Fo his eason, he op imal in e lea ing angle mus
be selec ed based on he cu en p o ile and no on he basis o he ol age ipple. Howe e ,
in any case, i is also in e es ing o analyse whe he his op imal in e lea ing scheme which
minimizes
Icap, ms
also educes
∆ pp,max
. The
Icap, ms
esul s shown in he ollowing sec ion
we e ob ained by using he double Fou ie in eg al analysis o Sec ion 2and he esul s
ha co espond o
∆ pp,max
we e ob ained by a nume ical analysis in Ma lab–Simulink.
Finally, in Sec ion 5.3
Icap, ms
is simula ed o EV dynamic condi ions applying he WLTP
d i ing cycle.
5.1. RMS Value o he Cu en Th ough DC-Link Capaci o a S a ic Ope a ing Poin s
Figu e 17 shows he ms cu en cu es o he di e en DZSI-PWM echniques.
He e, i can be obse ed ha wi hou using in e lea ing algo i hm (in blue) D-PWM1, D-
PWM0, D-PWM2, and D-PWM3 p esen he lowes Icap, ms (in ha o de ) and con inuous
echniques, as well as D-PWMMIN and D-PWMMAX ha e he highes alues o
Icap, ms
.
Al hough he di e ences be ween hese wo main PWM g oups a e bigge o cen al alues
o
M
(0.4
<M<
0.7), when
M
ge s close o 1 (
M>
0.9), all analysed PWM echniques end
o equalise. These con inuous PWM echniques, as well as D-PWMMIN and D-PWMMAX,
p esen a maximum alue a M≈0.6.
When he p oposed op imal in e lea ing scheme is applied (Figu e 17, in black),
con inuous modula ions wi h
ζop =π/2
ad educe
Icap, ms
o up o 62% o SPWM, 84%
o MINMAX-PWM and 80% o THI-PWM. Fo discon inuous modula ions, educ ions
up o 80% o D-PWMMIN and D-PWMMAX and 78% o D-PWM0, D-PWM1, D-PWM2
and D-PWM3 a e ob ained. D-PWM1 and MINMAX-PWM p o ides he lowes alues o
Icap, ms (in ha o de ).
5.2. Vol age Ripple in he DC-Link Capaci o a S a ic Ope a ing Poin s
E en hough he main idea o he in e lea ing app oach is o educe he ms alue o
he cu en h ough he DC-Link capaci o , Figu e 18 shows ha i also educes he DC-Link
ol age ipple (∆ cap,max) o all DZSI-PWMs.
Unlike he ms alue o he cu en h ough he DC-Link capaci o , when no in e lea -
ing (ζ=0 ad) is applied, con inuous PWM echniques (MINMAX-PWM, THI-PWM and
SPWM, in ha o de ) p esen he lowes
∆ pp,max
, and discon inuous echniques p esen
he highes alues o
∆ cap,max
(Figu e 18). Speci ically, MINMAX-PWM p o ides he
smalles and D-PWM0 p o ides he highes ol age ipple among hese nonin e lea ed
DZSI-PWM echniques.
When he p oposed op imal in e lea ing scheme (
ζ=ζop
) is applied, he ollowing
maximum educ ions o
∆ pp,max
a e ob ained: up o 64% o SPWM, 86% o MINMAX-
PWM, 85% o THI-PWM, 90% o D-PWMMIN and D-PWMMAX, 88% o D-PWM0,
90% o D-PWM1, 90% o D-PWM2, and 91% o D-PWM3. The DZSI-PWM echnique
which p esen s he smalles
∆ pp,max
once he op imal in e lea ing scheme is applied
is MINMAX-PWM.

Machines 2023,11, 267 22 o 31
Icap, ms
0
0.1
0.25
0.5
0.75
0.2 0.3 0.4 0.5 0.6 0.80.7
1
M
10.9
ad
π
2
ζc=
ζop =
ζ=0 ad
62 %
(a) SPWM.
Icap, ms
0
0.1
0.25
0.5
0.75
0.2 0.3 0.4 0.5 0.6 0.80.7
1
M
10.9
ζ=0 ad
ad
π
2
ζc=
ζop =
84 %
(b) MINMAX-PWM.
Icap, ms
M
0
0.1
0.25
0.5
0.75
0.2 0.3 0.4 0.5 0.6 0.80.7
1
10.9
ζ=0 ad
ad
π
2
ζc=
ζop =
80 %
(c) THI-PWM.
Icap, ms
0
0.1
0.25
0.5
0.75
0.2 0.3 0.4 0.5 0.6 0.80.7
1
M
10.9
ζ=0 ad
ζop
ad
π
ζc=
ζd=
80 %
(d) D-PWMMIN and D-PWMMAX.
Icap, ms
0
0.1
0.25
0.5
0.75
0.2 0.3 0.4 0.5 0.6 0.80.7
1
M
10.9
ζc= ad
π
2
ζ=0 ad ζd
ζop
78 %
(e) D-PWM0.
Icap, ms
0
0.1
0.25
0.5
0.75
0.2 0.3 0.4 0.5 0.6 0.80.7
1
M
10.9
ζc= ad
π
2
ζ=0 ad
ζd
ζop =
78 %
( ) D-PWM1.
Icap, ms
0
0.1
0.25
0.5
0.75
0.2 0.3 0.4 0.5 0.6 0.80.7
1
M
10.9
ζc= ad
π
2
ζ=0 ad ζd
ζop
78 %
(g) D-PWM2.
Icap, ms
0
0.1
0.25
0.5
0.75
0.2 0.3 0.4 0.5 0.6 0.80.7
1
M
10.9
ζc= ad
π
2
ζ=0 ad ζd
ζop
78 %
(h) D-PWM3.
Figu e 17. No malized DC-Link ms cu en o DZSI-PWM echniques as a unc ion o M.
As has been seen in his sec ion, he combina ion o he op imal in e lea ing scheme
and DZSI-PWM echniques is a good al e na i e in o de o educe he s ess a iables on
he DC-Link capaci o s o he ADTP powe con e e s o elec ic ehicles. Howe e , i is
enough o apply a cons an in e lea ing scheme (
ζc
) in o de o educe he cu en s ess on
he DC-Link capaci o conside ably.
5.3. RMS Value o he Cu en h ough DC-Link Capaci o du ing S anda dized D i ing Cycles
Figu e 19 shows he block diag am o he ADTP pla o m and he simula ion model
used o ob ain he esul s du ing EV dynamic condi ions. The ehicle ba e y has been
modelled as a 400 V DC sou ce acco ding o he ypical alues o elec ic ehicles [
43
]. The
selec ed DC-Link capaci ance has been 600
µ
F. The mo o ha has been modelled is he
one indica ed in Table 1(Sec ion 3.1). A maximum mo o cu en o 25 A has been se and
he base mechanical speed o he mo o has been changed o 2400 pm. Las ly, he con ol
algo i hm used is explained in de ail in [
79
] and he swi ching equency o he powe
de ices has been se o 25 kHz.
Ma lab–Simulink has been used o execu e he simula ion models, and hey ha e
been embedded on an OPAL-RT OP4510 high-pe o mance, eal- ime pla o m, which has
accele a ed he simula ion ime conside ably. The simula ion s ep has been se o 1
µ
s.
In addi ion, in o de o isualize in a mo e didac ic way and compa e o he esul s wi h
Machines 2023,11, 267 23 o 31
and wi hou in e lea ing, he esul s ha e been pos p ocessed wi h a mo ing a e age o
2000 samples.
0
0.1
0.2
0.4
0.6
0.2 0.3 0.4 0.5 0.6 0.80.7
M
10.9
ζop =ζc= ad
π
2
ζ=0 ad
64 %
(a) SPWM.
ζ= ad
π
2
0
0.1
0.2
0.4
0.6
0.2 0.3 0.4 0.5 0.6 0.80.7
M
10.9
ζ=0 ad
86 %
ζop =ζc= ad
π
2
(b) MINMAX-PWM.
0
0.1
0.2
0.4
0.6
0.2 0.3 0.4 0.5 0.6 0.80.7
M
10.9
ζ=0 ad
ζop =ζc= ad
π
2
(c) THI-PWM.
0
0.1
0.2
0.4
0.6
0.2 0.3 0.4 0.5 0.6 0.80.7
M
10.9
ζd=ζc
ζop
ζ=0 ad
(d) D-PWMMIN and D-PWMMAX.
0
0.1
0.2
0.4
0.6
0.2 0.3 0.4 0.5 0.6 0.80.7
M
10.9
ζd
ζc= ad
π
2
ζ=0 ad
ζop
(e) D-PWM0.
0
0.1
0.2
0.4
0.6
0.2 0.3 0.4 0.5 0.6 0.80.7
M
10.9
ζc= ad
π
2
ζ=0 ad
ζop =ζd
( ) D-PWM1.
0
0.1
0.2
0.4
0.6
0.2 0.3 0.4 0.5 0.6 0.80.7
M
10.9
ζc= ad
π
2
ζ=0 ad
ζop ζd
(g) D-PWM2.
0
0.1
0.2
0.4
0.6
0.2 0.3 0.4 0.5 0.6 0.80.7
M
10.9
ζ=0 ad
ζ= ad
π
2
ζop ζd
(h) D-PWM3.
Figu e 18. No malized maximum swi ching ol age ipple wi hin one pe iod o he undamen al.
iba iin
icap sa1 sb1 sc1
sa1 sb1 sc1
ia1 ib1 ic1
n1
a1b1c1
sa2 sb2 sc2
sa2 sb2 sc2
ia2 ib2 ic2
n2
a2b2c2
iin 1 VSI1
VSI2
iin 2
iin a1
iin a2
iin b1 iin c1
iin b2 iin c2
CDC
GUI
OP4510 RTlab pla o m
Communica ions
bus
Lap op
Powe con e e (ADTP) IPMSM model (VDQ)
Resul s
G1-G12
wmech
θ,ω
Du y
cycles
Console
ehicula model
Field
O ien ed
Con ol
(FOC)
Vba
Tem
*
Vba
Ba e y
icap da a
Con ol da a
Modula ion
0.5
0.25
0.75
speed
1
0
ime [s]
200 180014001000600
WLTP d i ing cycle
2
1
3
4
0
ime [s]
200 180014001000600
Icap, ms
0
-0.5
0.5
o que
1
-1
ime [s]
200 180014001000600
i
6
Figu e 19.
RTlab OP4510 simula ion pla o m diag am o elec ic ehicle ADTP p opulsion sys ems.
Machines 2023,11, 267 24 o 31
Figu e 20 shows he esul s o
Icap, ms
in he whole WLTP d i ing cycle, wi h and
wi hou cons an in e lea ing angle o each DZSI-PWM echnique (
ζc=π/2
ad o
he echniques SPWM, MINMAX-PWM, THI-PWM, D-PWM0, D-PWM1, D-PWM2, D-
PWM3; and
ζc=π
ad o D-PWMIN and D-PWMMAX). He e, i can be obse ed ha
all he analysed PWM echniques educe
Icap, ms
when he in e lea ing scheme is applied.
Likewise, Table 3shows he mean alue o he ms cu en h oughou he en i e WLTP
d i ing cycle. These esul s con i m ha an adequa e cons an in e lea ing scheme educes
he cu en s ess on he DC-Link capaci o conside ably. These simula ions show ha
educ ions up o 26% can be achie ed. The la ges educ ions a e p o ided by con inuous
PWM echniques, and he lowes alues o
Icap, ms
a e ound o discon inuous PWM
echniques. Howe e , hese D-PWMs a e no so o en used in EV applica ion because hey
wo sen he quali y o he ou pu ol age wa e o m.
2
1
3
4
0
ime [s]
200 180014001000600
Icap, ms
ζ=0 ad ad
π
2
ζc=
(a) SPWM.
2
1
3
4
0
ime [s]
200 180014001000600
Icap, ms
ζ=0 ad ad
π
2
ζc=
(b) MINMAX-PWM.
2
1
3
4
0
ime [s]
200 180014001000600
Icap, ms
ζ=0 ad ad
π
2
ζc=
(c) THI-PWM.
2
1
3
4
0
ime [s]
200 180014001000600
Icap, ms
ζ=0 ad ad
π
ζc=
(d) D-PWMMIN and D-PWMMAX.
2
1
3
4
0
ime [s]
200 180014001000600
Icap, ms
ζ=0 ad ad
π
2
ζc=
(e) D-PWM0.
2
1
3
4
0
ime [s]
200 180014001000600
Icap, ms
ζ=0 ad ad
π
2
ζc=
( ) D-PWM1.
2
1
3
4
0
ime [s]
200 180014001000600
Icap, ms
ζ=0 ad ad
π
2
ζc=
(g) D-PWM2.
2
1
3
4
0
ime [s]
200 180014001000600
Icap, ms
ζ=0 ad ad
π
2
ζc=
(h) D-PWM3.
Figu e 20.
DC-Link RMS cu en o DZSI-PWM echniques in WLTP d i ing cycle wi h (in ed) and
wi hou (in blue) cons an in e lea ing schemes.
Machines 2023,11, 267 25 o 31
Table 3.
DC-Link ms cu en (
Icap, ms
) o nonin e lea ed DZSI-PWM echniques and using he
co esponden cons an in e lea ing angle ζc o he WLTP d i ing cycle.
Nonin e lea ed (A) In e lea ed (A) Reduc ion (%)
SPWM 1.06 0.82 22.94
MINMAX-PWM 1.13 0.82 27.48
THI-PWM 1.13 0.82 27.25
D-PWMMIN 0.82 0.64 21.23
D-PWMMAX 0.83 0.64 22.67
D-PWM0 0.78 0.65 16.01
D-PWM1 0.71 0.59 16.27
D-PWM2 0.72 0.61 16.01
D-PWM3 0.79 0.67 15.37
6. Conclusions
The ADTP has u ned ou o be one o he mos success ul mul iphase a angemen
in he sho e m o elec ic ac ion applica ions due o i s in insic ad an ages, such as
enhanced e iciency and aul - ole an ope a ion. In powe con e e s in gene al, and he e-
o e also in ADTP a chi ec u e, he DC-Link capaci o is a c i ical elemen ha ep esen s a
conside able ac ion o he olume and sou ce o ailu es because i is esponsible o up
o a 40% o he o al olume and 30% o he o al ailu es in powe elec onic in e e s.
This wo k ocuses on he DC-Link s ess educ ion in o de o bene i his capaci o s
by means o he ollowing igu es o me i : ms alue o he ipple cu en h ough he
DC-Link capaci o (
Icap, ms
) and he maximum peak- o-peak ol age ipple (
∆ cap,max
). Fo
ha pu pose, he inpu cu en spec a o he ADTP a angemen ha e been analysed by
using he double Fou ie in eg al me hod o he DZSI-PWM echniques. Al hough each
b anch o he mul iphase VSI p esen s ce ain inpu cu en ha monics, he in e ac ion
be ween he di e en b anches inhe en o he ADTP a chi ec u e cancels hem ou and
changes he ampli ude o some o hese ha monics.
All hese inpu cu en ha monics depend mainly on he selec ed DZSI-PWM ech-
nique, as well as on
M
and
cos φ
. Fo elec ic ehicle applica ions, he main ehicle s anda d
d i ing cycles NEDC and WLTP using a PMSM demons a e ha he alue o he powe
ac o is
cos φ>
0.97. Thus, he cu en spec a o hese DZSI-PWM echniques ha e
been ob ained as a unc ion o
M
and o
cos φ=
1. He e, i has been obse ed ha
con inuous PWM echniques (SPWM, MINMAX-PWM, and THI-PWM) ha e a p edomi-
nan ca ie wa e ha monics a 2
sw
; D-PWM0, D-PWM1, D-PWM2, and D-PWM3 ha e a
wide sideband ha monic ange a ound
sw
, e en hough hei ca ie wa e ha monics a
2
sw
canno be neglec ed; inally, D-PWMMIN and D-PWMMAX ha e hei p edominan
cu en ha monics a sw and 2 sw.
Due o lack o in-dep h esea ch in he scien i ic li e a u e abou he in e lea ing
schemes o his kind o ADTP a angemen s, his wo k has analy ically de i ed he ela-
ionship be ween he inpu cu en ha monic spec um and he cons an in e lea ing angle
(
ζc
), as well as how his can be exploi ed in o de o cancel ce ain dominan ha monics
inhe en o hese DZSI-PWM echniques. As a esul , he ms alue o he ipple cu en
h ough he DC-Link capaci o (
Icap, ms
) and he maximum peak- o-peak ol age ipple
(
∆ cap,max
) has been educed. This con i ms ha he elimina ion o he dominan inpu
cu en ha monics is di ec ly ela ed o he minimiza ion o Icap, ms.
I has been concluded ha o all con inuous PWM echniques he op imal in e lea ing
angle is
ζc=π/2
ad because i elimina es he dominan ca ie ha monic a 2
sw
. Fo
discon inuous PWMs, a combina ion be ween he dynamic (only applicable o discon-
inuous PWM echniques) and he cons an in e lea ing schemes is gene ally p e e ed.
Du ing he subin e al in which he cons an in e lea ing scheme is applied, o D-PWM0,
D-PWM1, D-PWM2, and D-PWM3
ζc=π/2
ad is p e e ed because hei 2
sw
ca ie wa e
ha monic and
sw +
3
1
sideband ha monic a e cancelled ou . Howe e , o D-PWMMIN