IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. XX, NO. X, XXXXXX 20XX 1
Vo onoi Mul i-phase P edic i e Cu en Con ol
wi h Va iable Applica ion Times
Manuel R. A ahal, Fede ico Ba e o, Manuel G. Sa u´
e and F ancisco Colod o
Abs ac —P edic i e S a o Cu en Con ol (PSCC) is a
lexible echnique o d i es o di e en ypes. Fo he mul i-
phase case, PSCC mus deal wi h an inc eased numbe o
a ailable con ol op ions ( he ol age ec o s) and cope wi h he
di e en cu en spaces: o que p oducing (α−β) and ha monic
planes (x−y). In his pape , a Vo onoi egion based scheme is
designed o cope wi h bo h issues while main aining an a o dable
commu a ion equency. The p oposal is enhanced by using
a iable applica ion imes wi h ine esolu ion. The me hod is
compa ed wi h s a e o he a dead-bea mul i- ec o app oach.
The compa ison is made using a eal, labo a o y se up designed
o expe imen a ion based on a i e-phase induc ion machine.
The compa ison shows enhanced con ol esul s oge he wi h a
educ ion in ha monic con en , wi hou comp omising he swi ch-
ing equency limi s o he powe con e e s and main aining
lexibili y due o he cos unc ion.
Index Te ms—Mul i-phase d i es, P edic i e con ol, Va iable
applica ion ime, Vo onoi egions.
I. INTRODUCTION
VARIABLE speed d i es equi e p ecise dynamic egu-
la ion ha can be achie ed h ough adequa e con ol o
s a o cu en s. Con ol me hods conside ing con inuous ol -
age mus use some modula ion [1]. P edic i e S a o Cu en
Con ol (PSCC), howe e , a oids he modula ion s age and
is able o deal wi h a ious s a o cu en planes; α−β,
and x−y[2]. PSCC is cha ac e ized by i s high lexibili y
o accommoda e di e en objec i es in a cos unc ion. Fo
ins ance, managing he con lic ing c i e ia o o que p oduc ion
s. x−yplane con en . In addi ion i ea u es buil -in aul
ole ance [3].
The numbe o epo ed applica ions o PSCC is la ge,
encompassing di e en ypes o d i es based on Induc ion
Machines (IM) [4], [5], pe manen magne machines [6], [7],
and o he s. Howe e , se e al p oblems ha e been epo ed
equi ing new solu ions. Fo ins ance, he compu a ional cos
associa ed wi h he minimiza ion o he cos unc ion [8].
Se e al schemes ha e been pu o wa d o diminish he
compu a ional load, such as: educ ion o he con ol se [9]
(i.e. he se o allowed Vol age Vec o s (VV)), wi hin sample
This wo k is pa o p ojec I+D+i / PID2021-125189OB-I00, unded by
MCIN/AEI/10.13039/501100011033/FEDER, UE ”ERDF A way o mak-
ing Eu ope”. Manuel R. A ahal and Manuel G. Sa u´
e a e wi h he Sys-
ems Enginee ing and Au oma ion Depa men o he Uni e si y o Se ille,
Se ille, 41092, Spain (e-mail: [email p o ec ed]; [email p o ec ed]). Fede ico
Ba e o and F ancisco Colod o a e wi h he Elec onic Enginee ing De-
pa men o he Uni e si y o Se ille, Se ille, 41092, Spain (e-mail: ba -
[email p o ec ed]; [email p o ec ed]). (Co esponding au ho Fede ico Ba e o, phone:
+34 954481304; ax: +34 954487372; e-mail: [email p o ec ed]).
Manusc ip ecei ed Mon h xx, 2xxx; e ised Mon h xx, xxxx; accep ed
Mon h x, xxxx.
modula ion (o mul i- ec o app oach) [10], simpli ica ion o
he combina o ial sea ch [11], e c. The wo ks di ec ly ela ed
o he p oposal a e c i ically e iewed in wha ollows.
A. Li e a u e e iew
Mul i- ec o app oaches use a dead-bea concep aiming
a compu ing a desi ed ol age V∗ ha mus be syn hesized
using di e en VV du ing one sampling pe iod [12], [13]. This
app oach, e e ed o as DB-PSCC in wha ollows, is as
because compu ing V∗does no need i e a ions. DB-PSCC has
been ex ensi ely epo ed o he h ee-phase case [14], and e-
cen ly o mul i-phase d i es [15]–[17]. The me hod, howe e ,
mus swi ch he Vol age Sou ce In e e (VSI) con igu a ion
mo e o en han single ec o app oaches. This is easy o
unde s and since he ec o s in he mul i- ec o scheme ha e
di e en VSI con igu a ions ha a e accessible only h ough
commu a ion. This is a p oblem ha has no been widely
epo ed because as me hods o single- ec o PSCC did no
exis un il ecen ly.
Ano he p oblem o DB-PSCC is ha he dead-bea ob-
jec i e equi es he exac cancella ion o con ol e o in
one sho . This canno be achie ed in p ac ice due o: un-
modelled dynamics, pa ame e excu sions and he ac ha
V∗mus be syn hesized wi h a ious VV, so ha an a e age
e ec is ob ained. This has been c i icized in [18] as he
”a e age decep ion p oblem.” In sho , his occu s because he
op imal ol age is no ac ing du ing he whole sampling pe iod
bu a he is an a e age alue esul ing om he successi e
applica ion o basic VV. This is specially impo an in me hods
ha conside x−ycomponen s in open loop, elying on an
a e age null exci a ion o he x−yplane, bu wi hou eedback.
In ela ion wi h his las aspec , he ade-o s be ween
α−β acking and x−y egula ion a e los in DB-PSCC
me hods. These ade-o s ha e been i s epo ed in [19].
Thei impo ance has been ecognized in [20] and [21], whe e
he possibili y o balancing he ade-o is seen as an ex a
deg ee o eedom. This is impo an o ene gy e iciency
conside a ions as x−ycu en s p oduce losses.
Ano he c i icism ha DB-PSCC me hods ace is ha o
DC bus u iliza ion. This p oblem a ises as combina ions o
basic VV do no g an he same ol age span as he basic
VV hemsel es as poin ed ou in [18]. Finally, igonome ic
unc ions a e used by DB-PSCC. These unc ions a e known
o be ime-consuming and migh be no a ailable in low-end
ha dwa e.
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On he o he hand, exis ing single- ec o p edic i e me hods
(SV-PSCC) ha e a numbe o p oblems o hei own. P e ious
wo ks ha e been hinde ed by he lack o as me hods o VV
compu a ion. The p oblem is specially acu e in mul i-phase
d i es since he numbe o VV g ows exponen ially wi h he
numbe o phases [8]. The se o a emp s o diminish he
compu a ional load include: educ ion o he con ol se [22],
simpli ica ion o he combina o ial sea ch [11], use o spe-
cialized ha dwa e [23], and he use o model- ee app oaches
[24]. None o hese achie e he le els o compu a ional speed
o DB-PSCC. As a esul hey mus eso o highe sampling
pe iods, deg ading he con ol loop (unless special ha dwa e
is used).
Mo e ecen ly, a egion-based me hod o mul i-phase d i es
has been p esen ed [25]. This cons i u es an ex ension o he
mul i-phase a iable-speed case o p e ious wo ks [26]–[31].
Wi h his echnique i is possible o compe e wi h mul i- ec o
app oaches.
Ano he p oblem o SV-PSCC me hods in gene al, is ha
he applica ion ime o he VV (Ts) always co e s he whole
sampling pe iod (Ts). This p oduces a ha monic dis o ion
in s a o cu en s ha is mo e no iceable ha hose ound in
adi ional app oaches using modula ion. The idea o a iable
sampling ime has been p oposed in [32] o PSCC. The
me hod di ides he sampling pe iod in o sub-in e als. This
allows o op imize bo h, he VV o be applied, and i s applica-
ion ime. Howe e , combina o ial sea ch is in bo h cases. This
en ails a la ge compu a ional bu den. This bu den p e en s he
me hod om using ine esolu ion o he applica ion imes.
The idea o a iable sampling imes has been applied o mul i-
phase sys ems be o e wi h he help o cu en obse e s in
[33]. Howe e , he me hod s ill uses exhaus i e sea ch o he
VV compu a ion, being unable o compe e wi h DB-PSCC in
e ms o sampling ime educ ion.
B. Con ibu ions and no el y
As opposed o p e ious me hods, he p oposal uses educed
and non uni o m sampling imes o he applica ion o a single
basic VV. The p oposal achie es a numbe o goals ha a e
ele an (as he p e ious analysis o he s a e o he a shows).
The con ibu ions o he pape can be summa ized as ollows.
1) Fas Vo onoi based compu a ion o a single VV (i.e.
p o iding less commu a ions pe sampling pe iod).
2) Closed loop ea men o x−ycu en s conside ing he
whole x−yplane, wi h lexibili y o balance α−β
acking and x−y egula ion.
3) Reduced con ol pe iod ha educes he eac ion lag.
4) No educ ion in DC bus ol age u iliza ion.
The es o he pape is o ganized as ollows. Sec ion II
e isi s he PSCC con ol s uc u e using a i e phase induc ion
machine o he p esen a ion and expe imen al esul s. The
p oposal is de ailed in Sec ion III. Expe imen al esul s on a
eal induc ion machine a e p o ided o compa e he p oposal
wi h p e ious app oaches. The las sec ion p esen s he main
conclusions o he wo k.
*
iq
ωm
*
*
id
ωm
PI θa PSCC
*
iαβ u
ii
D
ωm
ωm
VSC
5
VSC
1
Fig. 1. Diag am o PSCC o a i e phase IM d i e.
II. MULTI-PHASE PREDICTIVE CURRENT CONTROL
The P edic i e S a o Cu en Con ol PSCC uses he Cla ke
ans o ma ion o α−βand xj−yjplanes. Fo a i e phase
IM, jus one x−yplane is p oduced. Cu en s in α−βspace
mus ollow a e e ence ha is ob ained om he speed con ol
loop. The scheme is p esen ed in Fig. 1, whe e lux and o que
a e independen ly egula ed. The lux se poin is p o ided
by i∗
dwhe eas e e ence i∗
qis used o he elec ical o que.
These e e ences a e p ojec ed o he α−βspace using he
Pa k ans o ma ion, ob aining a e e ence o s a o cu en in
α−βplane as I∗
α−β=Di∗
d, i∗
q⊺, whe e ma ix Dis gi en
by
D=cos θasin θa
−sin θacos θa(1)
The lux angle θais ob ained as θa=Rωed . As a
esul , he se poin o s a o cu en acking i∗(k)has an
ampli ude I∗=qi∗2
d+i∗2
q. Finally, he α−β e e ences
can be exp essed as i∗
α( ) = I∗sin ωe ,i∗
β( ) = I∗cos ωe ,
i∗
x( ) = 0,i∗
y( ) = 0.
In single- ec o PSCC, he con ol ac ion is he s a e o
he VSI u= (Ka, Kb,· · · , Ke)⊤, whe e he alues Kj
indica e he s a e o he co esponding VSI swi ch o each
phase. Fo a i e phase IM The e a e 32 possible VSI con-
igu a ions each p oducing a VV as shown in Fig. 2. The
s a o ol ages p o ided by each VSI s a e can be ound as
V(k) = VDC TMu(k), whe e VDC is he DC-link ol age
and
T=1
5
4−1−1−1−1
−1 4 −1−1−1
−1−1 4 −1−1
−1−1−1 4 −1
−1−1−1−1 4
,(2)
M=2
5
1γc
1γc
2γc
3γc
4
0γs
1γs
2γs
3γs
4
1γc
2γc
4cϑγc
3
0γs
2γs
4γs
1γs
3
1/2 1/2 1/2 1/2 1/2
.(3)
whe e γc
h= cos hϑ,γs
h= sin hϑ,ϑ= 2π/5.
The PSCC uses a p edic i e model o link u u e s a o
cu en s o ac ual ol ages. This model is o en a se o
disc e e- ime s a e-space equa ions as ollows:
ˆ
i(k+ 1) = (C(ω) + G)i(k) + BV (k)(4)
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31
1
2
3
4
5
6
7
8
9
10
11
12
14
15 16
17
19
20
21
22
23
24
25
26
27
28
29
30
0
β
α
13
18
Fig. 2. Dis ibu ion o he 32 basic VV o a i e phase VSI in α−βplane.
whe e icon ains he α,β,xand ys a o cu en s, ˆ
iis he
p edic ion o iand ωis he angula speed. Ma ices Cand B
a e ob ained om i s p inciples applying ime disc e iza ion
wi h sampling ime Ts. This gi es C= (I+AcTs), and B=
TsBc, whe e
Ac=
a2a40 0
a4a20 0
0 0 a30
000a3
, Bc=
c20 0 0
0c20 0
0 0 c30
0 0 0 c3
(5)
In he p e ious exp essions he coe icien s used a e: c1=
LsL −L2
M,c2=L /c1,c3= 1/Lls,c4=LM/c1,
a2=−Rsc2,a3=−Rsc3,a4=LMc4ω ,a= 4/5,
b=−1/5,c=−b,d= 2/5,a 4=R c4,al4=L c4ω . The
measu ed alues o he elec ical pa ame e s co esponding o
he machine used in he expe imen s a e shown in Table II.
To compensa e o he ime needed o compu e he con ol
ac ion, a delay compensa ion scheme is used. This is he
s anda d p ac ice o bo h single ec o and mul i ec o PSCC
me hods. This amoun s o le go a whole sampling pe iod and
op imize o k+ 2 du ing ha ime. Then, he wo-s ep ahead
p edic ion o he s a o cu en s mus be conside ed. F om
equa ion (4) he p edic ion is ound o be
ˆ
i(k+ 2) = A(ω)ˆ
i(k+ 1) + BV (k+ 1).(6)
yielding he p edic ed con ol e o as
ˆe(k+ 2) = i∗(k+ 2) −A(ω)ˆ
i(k+ 1) + BV (k+ 1),(7)
his p edic ion is used o de i e he con ol ac ion as will be
shown in wha ollows.
Vol age ec o de e mina ion
Con ol ac ion compu a ion in PSCC can be pe o med in
a a ie y o ways. Fo his pape i is ele an o conside
a) he case o mul i- ec o app oxima ion o he dead-bea
solu ion and b) he case o single- ec o minimiza ion o a
cos unc ion.
In case a), and using equa ion (7), one can de i e he ol age
ha , applied du ing he k+ 1 in e al, would d i e he e o
o ze o a k+ 2. Deno ing i by V∗i u ns ou o be
2h
3h
1h
Fig. 3. Regions esul ing om h ee adjacen VV de e mined by h ee planes.
V∗=B−1i∗(k+ 2) −A(ω)ˆ
i(k+ 1).(8)
Vol age V∗migh no be one o he 32 VV ha he VSI can
p oduce. Then i mus be syn hesized using a ious basic VV
applied a di e en imes and o di e en sub-pe iods du ing
he whole sampling pe iod. The basic VV ha need be applied
can be ound using inonome ic ules and look-up ables. To
u he simpli y ma e s, hese algo i hms suppose open-loop
x−ycu en elimina ion, so hey can wo k wi h jus he α−β
plane.
In case b), jus a single basic VV is applied. The dead-bea
concep canno be applied. The usual way o ind he con ol
signal is by minimiza ion o he ollowing cos unc ion
J(k+ 2) = ∥ˆe(k+ 2)∥2
w,(9)
whe e ∥.∥w ep esen s a weigh ed no m de ined as
∥e∥2
w=e2
α+e2
β+λxye2
x+λxye2
y,(10)
whe e λxy is he Weigh ing Fac o (WF) o he CF ha allows
balancing α−β acking and x−y egula ion.
III. PSCC WITH VARIABLE APPLICATION TIMES
The p oposal compu es he solu ion ha minimizes he
CF de ined in (9). The single basic VV is hen applied
du ing a sampling ime o a iable du a ion. This is he co e
idea o he me hod. The expec ed bene i s ha e been al eady
commen ed in he in oduc ion sec ion. In wha ollows, hese
wo ing edien s a e explained wi h echnical de ail.
A. Vo onoi egion o op imal ol age ec o
To unde s and he p oposal i is impo an o no ice ha ,
minimiza ion o Jamoun s o minimizing ∥V∗−Vh∥2
w, whe e
index hgoes om 1 o he numbe o a ailable VV. This can be
checked no ing ha ˆe=i∗−ˆ
iwi h ˆ
i=A(ω)ˆ
i(k+1)+BV h o
some basic VV o index h. As a esul he ollowing iden i y
is ound
B−1ˆe=B−1i∗−A(ω)ˆ
i(k+ 1)−Vh,(11)
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whe e he igh hand side is exac ly V∗−Vhby de ini ion. So,
he e m ˆecan be eplaced by B(V∗−Vh)in he exp ession o
he cos unc ion. Finally, since Bdoes no depend on he VV
selec ed, minimizing Jand minimizing ∥V∗−Vh∥2
wyields
he same esul .
Ano he impo an s ep in he de i a ion o he p oposal
is unde s anding ha minimiza ion o ∥V∗−Vh∥wcan be
made using he idea o Vo onoi egions. This will allow he
a oidance o he cumbe some exhaus i e explo a ion. Vo onoi
egions appea because ∥V∗−Vh∥wde ines a me ic in he
ma hema ical sense [34]. Then, o compu e he VV minimizing
he cos unc ion one jus needs o ind in which Vo onoi egion
is V∗placed.
The Vo onoi egion con aining V∗is easily ound since
he egions a e bounded by planes as shown in Fig. 3. The
planes in ques ion a e he pe pendicula bisec o s o each pai
o adjacen Vh. The planes a e comple ely de ined by hei
no mal ec o s, ound as nij =Vj−Vi, whe e indices i
and j ep esen wo adjacen egions, co esponding o ol age
ec o s Viand Vj.
I is well known ha , gi en a plane Πand a poin V, hei
ela i e posi ion can be ound as he scala p oduc
p=V⊺·Π,(12)
i p > 0 hen poin Vlies in a subspace wi h ela ion o Π
co esponding o egion i. Howe e , i p < 0 hen poin Vlies
in he o he subspace, co esponding o egion j. The case in
which p= 0 co esponds o Vbeing pa o plane Π. In his
case he wo adjacen VV p oduce he same alue o he CF.
Gi en he ac ha a la ge numbe o basic VV a e p o ided
by a mul i-phase VSI, many scala p oduc s and compa ison
would be needed o es ablish he egion o V∗. Howe e , i
he quad an o V∗is known, hen jus h ee compa isons a e
needed as shown in Fig. 3. This is in e es ing because he
quad an Q∗can be ound easily ei he using some nes ed
i -else s a emen s o in he ollowing way
Q∗= 1 + S∗·(8,4,2,1)⊺,(13)
whe e S∗is a one-by- ou ec o o signs compu ed as S∗=
sgn(V∗), and sgn() is he sign unc ion.
To suppo he me hod conside Table I whe e he indices o
he basic VV lying on each quad an Qa e shown. Jus h ee
VVs a e ound o each quad an and iden i ied by hei indices
h1,h2, and h3. Please no ice ha VV lying on he on ie s o
quad an s a e coun ed as belonging o bo h adjacen quad an s;
in his way, hey a e no le ou o he pe inen compa isons.
Wi h his in mind, he de e mina ion o he egion con aining
V∗is done conside ing jus h ee p oduc s:
TABLE I
INDICES hOF NON-ZERO VV BELONGING TO EACH QUADRANT Q.
Q h1h2h3Q h1h2h3
01 16 24 26 09 06 22 30
02 16 20 28 10 04 06 22
03 08 09 25 11 10 14 15
04 09 25 29 12 12 13 15
05 16 18 19 13 02 06 22
06 16 17 21 14 06 22 23
07 09 25 27 15 03 11 15
08 01 09 25 16 05 07 15
POWER ELECTRONIC
CONVERTERS
DSP
DC MOTOR 5-PHASE IM
Phase Cu en s
Swi ching Signals
Posi ion Encode
a b c d e
Fig. 4. Diag am and pho og aphs o he labo a o y se up used in he
expe imen s.
p1=V∗⊺·nh1,h2(14)
p2=V∗⊺·nh1,h3(15)
p3=V∗⊺·nh2,h3,(16)
whe e h1,h2,h3a e e ie ed om Table I om he ow
indica ed by Q∗.
Finally, he index o he egion con aining V∗is ound as
h∗=h1i p1<0,p2<0,h∗=h2i p1>0and p3<0,
h∗=h3i p2>0,p2=p3>0.
I is qui e appa en ha he compu a ional bu den is small,
equi ing a ew scala p oduc s and compa isons. This allows
he compu a ion o he op imal single VV wi hou i e a ions
and conside ing he o iginal cos unc ion. In his way he x−y
plane is conside ed p ope ly and in ull. Also, he weigh ing
ac o can e ained in he cos unc ion. This cons i u es a
deg ee o eedom o balance α−β acking wi h x−ycon en .
B. Non uni o m sampling ime
The sampling ime can be modi ied om one pe iod o
ano he in o de o make he selec ed VV ha e an op imal
du a ion. In his way, he applica ion ime Tsbecomes a deg ee
o eedom ha can be used o ob ain be e esul s. The idea
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o he non uni o m sampling ime is as ollows. Ins ead o
he ypical ead-compu e-hold-wai scheme o a digi al con ol
sys em, he a iable sampling ime uses an addi ional s ep in
which he sampling ime is compu ed. In his way, he wai ing
pa o he digi al con ol me hod has a a iable du a ion.
Wi h his idea in mind, one would like o use he selec ed
VV and selec he sampling ime (Ts) o minimize he de ia-
ion Fde ined as
F=i∗( 1+Ts)−ˆ
ih∗( 1+Ts),(17)
whe e 1is he ime co esponding o he p e ious sampling
pe iod, h∗is he index o he op imal VV compu ed as
explained in he p eceding sec ion and Tsis he applica ion
ime being op imized. Using he con inuous- ime model o he
IM i is possible o w i e he ime de i a i e o ˆ
ih∗as
D∗=Acˆ
i( 1) + BcVh∗,(18)
whe e ˆ
i( 1) = ˆ
i(k+ 1) is he one-s ep ahead p edic ion and
Vh∗is he basic VV selec ed in he op imiza ion phase.
The alue D∗o equa ion (18) ep esen s he di ec ion o
a line (in cu en space) ha is ans e sed as ime p og esses
om 1 o 1+Ts. The poin in he line ha is closes o he
e e ence alue is ound when Fis pe pendicula o said line.
This p o ides he ollowing equa ion
F⊺·Acˆ
i( 1) + BcVh∗= 0.(19)
The only unknown is Ts, appea ing only in he F ac o .
Thus, he solu ion can be ound as
Ts=i∗( 1+Ts)−ˆ
i( 1)⊺
·D∗/∥D∗∥2,(20)
his can be checked subs i u ing (20) in o (19). This solu ion
has ine esolu ion, unlike p e ious me hods ha elied on
disc e iza ion o he applica ion ime.
IV. EXPERIMENTAL RESULTS
The p oposal is pu o es using a es -bench ha is
desc ibed below. The pe o mance indica o s used o he
compa isons a e p esen ed la e on.
The compa ison is d awn be ween he p oposed egion-
based PSCC wi h a iable applica ion imes and a s a e o
he a mul i- ec o app oach p esen ed in [13], e e ed o
as DB-PSCC. Said con ende me hod uses Vi ual Vol age
Vec o s (VVV) ob ained wi hou combina o ial sea ch, hus
p o iding e y low compu a ion ime. The applied ol age
is u he op imized by combining he op imal VVV wi h a
ze o con igu a ion o a non- ixed ime op . Fu he mo e, o
a oid a iable swi ching equency he mul i- ec o me hod
conside s a modula ion p ocess in which he basic VV a e
p oduced in a pa icula sequence (see Fig. 7 in [13]). In said
DB-PSCC, he ha monic plane con en is conside ed in open-
loop, since he VVV a e designed o p o ide (on a e age)
ze o con en in he hi d-ha monic. The x−yplane is hus
TABLE II
PARAMETERS OF THE EXPERIMENTAL FIVE PHASE IM
Pa ame e Value Uni
S a o esis ance, Rs12.85 Ω
Ro o esis ance, R 4.80 Ω
S a o leakage induc ance, Lls 79.93 mH
Ro o leakage induc ance, Ll 79.93 mH
Mu ual induc ance, LM681.7 mH
Ro a ional ine ia, Jm0.02 kg m2
Numbe o pai s o poles, P3 -
TABLE III
SUMMARY OF RESULTS FOR DB MULTI-VECTOR (TOP)AND THE
PROPOSAL (BOTTOM).
Case Eα−βEx−yASF T HD
(mA) (mA) (kHz) (%)
A) 73.27 81.87 9.50 9.3
B) 84.25 100.1 9.50 3.9
C) 107.7 145.0 9.50 4.1
D) 101.2 136.3 9.50 6.5
E) 81.96 101.0 9.50 5.5
A g. 89.70 112.9 9.50 5.5
Case Eα−βEx−yASF T HD
(mA) (mA) (kHz) (%)
A) 69.06 80.43 9.46 7.2
B) 74.22 85.16 7.30 3.4
C) 102.6 109.5 6.34 4.1
D) 99.69 106.5 8.48 5.8
E) 79.40 88.27 7.62 5.4
A g. 85.00 94.01 7.84 4.6
no conside ed in ull. Finally, he me hod makes use o a
igonome ic unc ion (see Eq. (28) in [13]) o de i e he
sec o o he e e ence ol age V∗. The me hod is he e adap ed
o he i e phase IM used in he expe imen s.
A. Tes acili y
A es bench is used o he expe imen s o app aise
he p oposal. Fig. 4 shows pho og aphs o i s main le-
men s: a i e phase IM, a powe con e e wi h SEMIKRON
SKS 22F modules and a 300V DC powe supply. Fo
he con ol, he ollowing elemen s a e used: a MSK28335
boa d including a TMS320F28335 digi al signal p ocesso , a
GHM510296R/2500 inc emen al posi ion encode , and Hall
e ec senso s (LH25-NP) o measu e he s a o phase cu en s.
Finally, a DC mo o p o ides an opposing o que o he es s.
Table II p esen s he pa ame e s o he sys em.
B. Figu es o Me i
Se e al igu es o me i a e conside ed: speed ipple, To al
Ha monic Dis o ion (THD), s a o cu en e o in he o que
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p oducing plane, x−ycu en con en , and swi ching e-
quency a he VSI. The swi ching equency o mos PSCC
me hods is no cons an . Howe e i s alue is impo an o
ha dwa e selec ion (e.g., s anda d IGBTs o SiC-based powe
swi ches) and o e iciency conside a ions (VSI commu a ion
losses). Thus, an a e age alue, e e ed o as A e age Swi ch-
ing F equency (ASF), is usually conside ed. Ha ing his in
mind, he ollowing e iciency ac o s a e conside ed:
Eα−β=
u
u
1
N
k1+N
X
k=k1
e2
αβ(k)(21)
Ex−y=
u
u
1
N
k1+N
X
k=k1
e2
xy(k)(22)
ASF =1/(5 ·2)
(k1+N)− (k1)
k1+N
X
k=k1
∆S(k)(23)
THD =100
I1
u
u
∞
X
i=2
I2
i(24)
whe e ∆S(k) = P5
i=1 |ui(k+ 1) −ui(k)|is he numbe o
swi ch changes p oduced a he VSI when con igu a ion u(k)
is changed o u(k+ 1), and Iiis he ampli ude o he i- h
ha monic componen o s a o cu en s. These quan i ies a e
de ined o e a empo al ho izon o Nsamples, de ined by he
disc e e- ime index k1 o k1+N. Please no ice ha , in he case
o he p oposal, and due o he a iable sampling, imes (k1)
and (k1+N)a e no , in gene al, exac mul iples o a base
pe iod Ts. In o he app oaches on simply ge s (k) = k·Ts
as Tsis a cons an .
C. S eady S a e Analysis
The easibili y o he p oposal is i s es ed on he expe -
imen al benchma k o a ed condi ions. The esul s can be
obse ed in Fig. 5, whe e he α−βs a o cu en s, and phase
cu en s a e shown as wa e o ms. T acking o sinusoidal α−β
e e ence can be clea ly seen.
The igu es o me i o his es a e: Eα−β= 0.1(A),
Ex−y= 0.08 (A), ASF = 8240 (Hz), and THD = 4.1(%).
These esul s imp o e p e ious schemes as will be shown la e .
I is in e es ing o no e ha he ASF is low o he con e e
used ha can handle up o 15 (kHz). This means ha he
p oposal is compe i i e wi hou eso ing o high commu a ion
a es.
A compa ison wi h DB-PSCC is p esen ed in he nex
sec ion o di e en ope a ing egimes o he a iable-speed
d i e.
D. Ope a ing Regime Analysis
A se ies o s eady s a e es s on di e en ope a ing poin s
ha e been pe o med o compa e he pe o mance o he p o-
posal wi h DB-PSCC. Bo h con olle s use he same ha dwa e
and o he esou ces. In pa icula , he sampling pe iods a e
adjus ed so ha bo h me hods use simila swi ching equency.
0 0.01 0.02 0.03 0.04 0.05
−2
−1
0
1
2
Time (s)
Cu en s (A)
0 0.01 0.02 0.03 0.04 0.05
−2
−1
0
1
2
Time (s)
Fig. 5. S eady s a e esul s o he p oposal in a ed condi ions. Top plo is
o α−βs a o cu en s, and bo om plo o phase cu en s.
Di e en ope a ing poin s a e used o he compa ison. The
ope a ing poin s ea u e a pe cen age o a ed mechanical
speed and opposing load p o ided by he DC machine. The
pai s (speed, opposing load) o ope a ing poin s A) o F) a e:
A) (20 %, 10 %), B) (80 %, 50 %), C) (100 %, 100 %), D)
(20 %, 100 %), and E) (50 %, 30 %). This co e s many cases
whe e he beha io o he IM is di e en . This co e age allows
o a ho ough e alua ion on di e e en ope a ing egimes o
he a iable-speed d i e.
Table III summa izes he esul s ob ained. Please no ice
ha , in o de o make he compa ison ai bo h app oaches
use app oxima ely he same swi ching equency. In ac , he
p oposal ea u es a lowe ASF, so he con ende is in a be e
posi ion. I can be seen, in said Table III, ha he p oposal
ob ains he bes esul s o all igu es o me i s and o all
cases conside ed. In pa icula he Ex−yo he DB-PSCC a e
much la ge . This is a e lec ion o he ac ha he x−y
componen s ha e no been conside ed, jus he hi d ha monic.
This has some ele ance as x−ycu en s p oduce losses. I
is also in e es ing o see ha , he lowe con ol e o s o he
p oposal a e ob ained wi h sligh ly less ASF han he DB-
PSCC app oach. The a iable swi ching equency is hus he
only pa ame e ha emains sub-pa in SV-PSCC wi h espec
o o he app oaches.
The p oposal also has an in e es ing ea u e ha canno be
achie ed wi h he DB app oaches: he x−ycon en can be
aded wi h Eα−β o sui he pa icula applica ion a hand (see
[19]). Fo ins ance, i one is conce ned mo e wi h x−y ela ed
losses and a bi less wi h acking, hen one can inc ease he
alue o λxy. To demons a e his, conside again he esul s
o he p oposal in Table III (whe e λxy = 0.05 is used) and
compa e i wi h Table IV (whe e λxy = 0.08 is used). I can
be seen ha in he la e case, he x−ycon en has been
diminished, ye he a e age alues o he o he igu es o
me i s ill emain lowe han hose o he con ende me hod
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0 0.005 0.01 0.015 0.02 0.025 0.03
−2
−1
0
1
2
3
Cu en s (A)
i* , iα , iβ , ix , iy
0 0.005 0.01 0.015 0.02 0.025 0.03
−2
−1
0
1
2
3
Time (s)
Cu en s (A)
Fig. 6. S eady s a e esul s o he p oposal in ope a ing poin A o he
p oposal ( op plo ) and o DB-PSCC (bo om plo ).
DB-PSCC. The mo e impo an bi , howe e , is ha he single
ec o app oach has lexibili y whe eas DB-PSCC has no .
TABLE IV
EXPERIMENTAL RESULTS FOR THE PROPOSAL WITH REDUCED x−y
CONTENT
Ope a ing Eα−βEx−yASF T HD
Poin (mA) (mA) (kHz) (%)
A) 70.90 58.80 10.41 7.6
B) 75.68 62.25 7.88 3.8
C) 104.4 80.11 8.24 4.1
D) 100.5 77.88 9.07 6.2
E) 79.71 64.52 8.76 6.5
A g. 86.25 68.71 8.87 5.2
In addi ion o he p e ious ables, a se ies o g aphs a e
used o p esen he expe imen al ajec o ies o s a o cu en s
in α−βplane and x−yplane o he p oposal and o
he DB-PSCC me hod. These a e shown in Figs. 6 o 9. The
di e en ope a ing poin s a e indica ed in he cap ion o each
igu e. The co ec beha io o he p oposal is clea ly seen.
Acco ding o [19], in he single- ec o app oach, he x−y
con en is aded o wi h he o he igu es o me i . In his way
he me hod can be uned o he speci ic applica ion unlike he
mul i- ec o app oach. Also, due o he use o all 32 basic VV
he p oposal achie es be e esul s. Recall ha he DB-PSCC
can only use he speci ic VVV, his es ic s he possibili ies
o cope wi h di e en si ua ions.
0 0.005 0.01 0.015 0.02 0.025
−2
−1
0
1
2
3
Cu en s (A)
i* , iα , iβ , ix , iy
0 0.005 0.01 0.015 0.02 0.025
−2
−1
0
1
2
3
Time (s)
Cu en s (A)
Fig. 7. S eady s a e esul s o he p oposal in ope a ing poin B o he
p oposal ( op plo ) and o DB-PSCC (bo om plo ).
0 0.005 0.01 0.015 0.02 0.025 0.03
−2
−1
0
1
2
3
Cu en s (A)
i* , iα , iβ , ix , iy
0 0.005 0.01 0.015 0.02 0.025 0.03
−2
−1
0
1
2
3
Time (s)
Cu en s (A)
Fig. 8. S eady s a e esul s o he p oposal in ope a ing poin C o he
p oposal ( op plo ) and o DB-PSCC (bo om plo ).
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0 0.005 0.01 0.015 0.02 0.025 0.03
−2
−1
0
1
2
3
Cu en s (A)
i* , iα , iβ , ix , iy
0 0.005 0.01 0.015 0.02 0.025 0.03
−2
−1
0
1
2
3
Time (s)
Cu en s (A)
Fig. 9. S eady s a e esul s o he p oposal in ope a ing poin D o he
p oposal ( op plo ) and o DB-PSCC (bo om plo ).
E. T ansien Response Tes s
In addi ion o he s eady s a e esul s, he capabili ies o he
p oposal ha e been es ed o ansien condi ions. A speed
e e sal expe imen has been pe o med whe e he desi ed
speed goes om 500 ( pm) o -500 ( pm). The esul s can be
seen in Fig. 10, whe e he ajec o ies o he measu ed speed
a e compa ed o he p oposal and o he mul i- ec o me hod
(DB-PSCC). Please no ice ha he esul s a e simila . This is
due o he ac ha he same PI is used o bo h me hods. This
PI is la gely esponsible o he pe o mance in mechanical
e ms. Please no ice ha his is no a nega i e esul since
he same mechanical pe o mance has been ob ained o he
p oposal wi h be e elec ical indica o s, hus, in a mo e
ene gy e icien way (i.e. less commu a ions and less ha monic
con en ).
F. Robus ness and s abili y
One p oblem o model-based app oaches is ha misma ched
pa ame e s can cause deg ada ion in con ol esul s. This is
he case o PSCC because he p edic i e model con ains
pa ame e s ha can di e om he eal ones. This si ua ion
can appea as a esul o he iden i ica ion p ocedu e o om
pa ame e excu sions. Excu sions can appea o ins ance due
o empe a u e a ia ions.
Robus beha io is achie ed when he in luence o pa ame e
de uning is accep able. In he case o PSCC, pa ame e s Rs,
R ,Lls,Ll , and LMplay a ole in he p oduc ion o
p edic ions ha a e la e used o de i e he con ol ac ion.
Analysis o PSCC obus ness ha e been ca ied ou in he
li e a u e conside ing de uned pa ame e s.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−600
−400
−200
0
200
400
600
ω ( pm)
ω* ,
P oposal,
DB−PSCC
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−3.0
−2.0
−1.0
0.0
1.0
Time (s)
is q (A)
Fig. 10. Resul s ob ained in a e e sal es using he p oposal and he mul i-
ec o DB app oach.
Fo he analysis i is assumed ha he pa ame e s in Table
II co espond o he uned sys em. A se ies o es is hen pe -
o med using a de uned p edic i e model. The pa ame e s used
by he model will be: ˆ
Rs=ϕ1Rs,ˆ
R =ϕ2R ,ˆ
Lls =ϕ3Lls,
ˆ
Ll =ϕ4Ll , and ˆ
LM=ϕ5LM. The ϕcoe icien s allow
o conside a ious si ua ions wi h misma ched pa ame e s.
Please no e ha o ϕ= 0.5 he model is using a pa ame e
wi h a alue hal he co ec one. Simila ly, ϕ= 1 indica es no
de uning and ϕ= 2 implies ha he model uses a pa ame e
alue double o he co ec one.
In Table V, he e ec caused by he misma ched pa ame e s
is indica ed as a deg ada ion ac o δ, de ined as δ=Ed/E .
The deg ada ion is conside ed o a pe o mance indica o
E. The supe sc ip dindica es de uning and he supe sc ip
indica es pe ec uning (i.e. ϕi= 1 o all i). In said Table
V he indica o s a e Eα−βand Ex−y.
TABLE V
ROBUSTNESS ANALYSIS FOR DETUNED MODELS.
De uning Con en ional P oposal
Fac o δα−βδx−yδα−βδx−y
ϕ1= 0.51.09 1.02 1.08 1.02
ϕ1= 2.01.01 1.01 1.01 1.01
ϕ2= 0.51.46 1.14 1.45 1.14
ϕ2= 2.01.41 1.05 1.41 1.06
ϕ3= 0.51.18 1.03 1.18 1.03
ϕ3= 2.01.31 1.04 1.30 1.03
ϕ4= 0.51.00 1.01 1.01 1.00
ϕ4= 2.01.01 1.01 1.01 1.01
ϕ5= 0.51.00 1.00 1.00 1.00
ϕ5= 2.01.95 1.22 1.90 1.22
F om he esul s o he analysis, i is clea ha he p oposed
me hod has a simila obus ness as ha o he con en ional
PSCC.
Simila esul s a e ob ained ega ding s abili y. Please no e
ha s abili y p oo o PSCC conside Lyapuno me hods ha
use he cos unc ion as s epping s one. In Sec ion III-A i
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con en may change p io o inal publica ion. Ci a ion in o ma ion: DOI 10.1109/TEC.2024.3497212
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has been shown ha he minimiza ion due o Vo onoi egion
iden i ica ion is equi alen o minimiza ion o J. As a esul
he s abili y p oo would o he p oposal would ollow he
same s eps as exis ing p oo s o PSCC and a e, hus, omi ed.
V. CONCLUSION
Model based p edic i e me hods ha e become an impo an
esea ch a ea in mode n elec omechanical d i es, p o id-
ing a lexible solu ion o mul i a iable-cons ained con ol
p oblems. One o i s main d awbacks, is he compu a ional
bu den ha limi s i s applicabili y in low-end and medium
DSP solu ions. This esul s in high applica ion imes p oducing
high THD alues. Bo h p oblems a e sol ed wi h he p oposal
by deli e ing a as , Vo onoi-based compu a ion me hod and
a iable applica ion imes.
The expe imen al esul s show ha he p oposal e ains he
abili y o PSCC o deal wi h he ade-o be ween o que
p oducing acking and x−ycon en . Compa ed wi h a s a e o
he a mul i- ec o solu ion, he p oposal imp o es all igu es
o me i e en using less swi ching equency. The p oposal,
hus, pu s back single- ec o app oaches in he on ie o
esea ch making i compa able wi h mul i- ec o app oaches.
ACKNOWLEDGMENT
This wo k is pa o p ojec I+D+i / PID2021-125189OB-
I00, unded by MCIN/AEI/10.13039/501100011033/FEDER,
UE ”ERDF A way o making Eu ope”.
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