Model P edic i e Con ol wi h Cons an Swi ching
F equency Using a Disc e e Space Vec o
Modula ion wi h Vi ual S a e Vec o s
Abs ac — Fini e S a es Model P edic i e Con ol (FS-MPC)
appea s as a p omising con ol echnique o be applied o
powe con e e s in he indus y. Howe e , he FS-MPC p esen s
some d awbacks as non cons an swi ching equency and high
sampling equency. This wo k p oposes a FS-MPC wi h cons an
swi ching equency and low sampling equency applying a
Disc e e Space Vec o Modula ion (DSVM) echnique. The eal
s a e ec o s o he con e e a e used oge he wi h new i ual
s a e ec o s o ming swi ching sequences each sampling pe iod.
The ad an ages and disad an ages o he p oposed FS-MPC
wi h he DSVM a e analyzed using a wo-le el h ee-phase
in e e connec ed o he g id as se up o in oduce he p oposed
echnique. Simula ion esul s a e p esen ed, showing ha using
he p oposed echnique he swi ching equency is ixed and he
sampling equency can be lowe ed wi hou educing he quali y
o he con e e beha io .
I. INTRODUCTION
SINCE powe con e e s appea he de elopmen o di -
e en applica ions and hei associa ed con ol s a egies
ha e been a challenge in he powe elec onics ield. Se e al
con ol echniques ha e been applied o he powe con e e s.
Linea con olle s and non linea con olle s including adap i e
con ol, epe i i e con ol, uzzy con ol and p edic i e con ol
ha e been widely used in di e en applica ions as ec i ie s,
in e e s, ac i e powe il e s, unin e up ible powe supplies
and mo o s con ol [1]–[4].
Among he p edic i e echniques, deadbea con olle s and
Model P edic i e Con ol (MPC) ha e s and ou due o hei
good p ope ies [5]–[8]. The implemen a ion o MPC o
powe con e e s can be di icul due o he la ge amoun o
calcula ions needed o sol e online he op imiza ion p oblem.
To o e come his, wo solu ions has been p oposed: o line
calcula ion o he op imal solu ions [9], and calcula ion o
he op imal solu ion by e alua ion o all possible swi ching
s a es. This las solu ion has been called Fini e S a es Model
P edic i e Con ol (FS-MPC), because i akes in o accoun
only he ini e numbe o possible swi ching s a es, and can
be easily implemen ed using s anda d con ol ha dwa e. In he
pa icula case o FS-MPC, se e al applica ions as ec i ie
[10], in e e [11], [12], unin e up ible powe supply [13]
and mo o con ol [14], ha e shown ha his ype o con ol
p esen s high pe o mance wi hou he necessi y o adjus
con olle pa ame e s o ob ain an op imized beha io o he
o e all sys em. In despi e o he good ea u es shown, he
FS-MPC has wo main d awbacks ha makes e y di icul i s
use in eal indus ial powe con e e s. The FS-MPC needs
a high sampling equency o achie e high pe o mance, his
en ails high compu a ional cos and he e o e expensi e DSP
o FPGA ha dwa e. Besides FS-MPC applica ions show non
cons an swi ching wi h a widesp ead spec um o he ou pu
cu en s and ol ages. This ac leads o he necessi y o
la ge induc o and capaci o s alues o he connec ion passi e
il e s, inc easing he weigh , olume and cos o he powe
con e e .
These d awbacks a e p esen in o he disc e e con olle s
as Di ec Powe Con ol (DPC) in ec i ie s applica ions and
Di ec To que Con ol (DTC) in mo o con ol [15], [16]. Bo h
con ol echniques a e based on choosing in a look up able
(LUT) he bes ou pu s a e o minimize he e o s be ween
he command e e ences and he ac ual alues o ins an aneous
ac i e and eac i e powe (DPC), o o que and lux (DTC).
To enhance he pe o mance o DTC he use o a Disc e e
Space Vec o Modula ion (DSVM) has been p oposed [17]–
[22], imp o ing he d i e pe o mance h ough he use o new
s a es wi h a p e ixed ime in e als.
In his pape his concep is ex ended o he FS-MPC and he
use o DSVM o ca y ou he FS-MPC is p oposed. This new
echnique is called MPC-DSVM. The p oposed me hod allows
o achie e he con ol objec i es wi h low sampling equency
and cons an swi ching equency. To assess he MPC-DSVM,
simula ions esul s a e p esen ed using a con en ional wo-
le el h ee-phase in e e connec ed o he g id. Finally he
ad an ages and disad an ages o he p oposed FS-MPC wi h
he DSVM a e analyzed.
II. FS-MPC
FS-MPC is based on he minimiza ion o a de ined cos
unc ion s udying he p edic ed esponse o each disc e e
ou pu s a e o a powe con e e . Each sampling ime (Ts), he
FS-MPC e alua es all possible ou pu s a es and chooses he
s a e ha minimizes a cos unc ion as he s a e o be applied o
he con e e . FS-MPC echnique p esen s he ad an ages and
disad an ages summa ized in Table I and Table II espec i ely.
Among he mos aluable cha ac e is ics o FS-MPC i
can be no iced ha high pe o mance con olle s can be
implemen ed s aigh o wa d. Howe e his equi es high sam-
pling equency, which de e mines he con ol ha dwa e o
TABLE I
ADVANTAGES OF FS-MPC CONTROLLERS
Easy implemen a ion wi h high pe o mance
No necessi y o cons an s con olle uning p ocedu es
Easy o include any pa ame e op imiza ion c i e ia in he cos unc ion
Takes in o accoun he disc e e na u e o he con e e
TABLE II
DRAWBACKS OF FS-MPC CONTROLLERS
Needs high sampling equency s o ob ain high pe o mance
P esen s non cons an swi ching equency sw , limi ed by s/2
Does no use he comple e con e e con ol egion
be used o con ol he powe con e e . Ano he no iceable
ea u es a e he no necessi y o cons an s con olle uning
p ocedu es and ease o include any cons ain o pa ame e
op imiza ion c i e ia in he cos unc ion. Finally, FS-MPC
akes in o accoun he disc e e na u e o he con e e . As a
consequence, FS-MPC does no use he comple e con e e
con ol egion, only he disc e e eal s a es o he con e e
(Fig. 1). This ac oge he wi h he possibili y o he cos
unc ion o choose se e al consecu i e imes he same disc e e
eal s a e as he op imum s a e, leads FS-MPC o p esen non
cons an swi ching equency( sw). Besides sw can be as high
as hal he sampling equency( s)due o he e is no es ic ion
in he swi ching sequence unless i is included in he cos
unc ion.
FS-MPC con olle s a e also cha ac e ized by he way hey
gene a e he powe swi ch i ing pulses. In compa ison wi h
ex e nal modula o based con olle s, FS-MPC does no need a
modula o o gene a e he i ing pulses. The powe swi ch ga e
signals can be ob ained om a LUT, whe e he co espondence
be ween he disc e e eal s a es o he con e e and he ga e
signals a e s o ed. Al hough i is no necessa y, i is possible o
gene a e he FS-MPC i ing pulses using an ex e nal modula o
as i is shown in Fig. 2. The use o an ex e nal modula o does
no complica e he ha dwa e used in he applica ion, bu opens
new possibili ies o FS-MPC as he comple e con e e con ol
egion is easily accessible.
III. MPC-DSVM
FS-MPC only conside s he con e e disc e e eal s a es o
p edic he sys em esponse and o e alua e he FS-MPC cos
unc ion. Al hough his leads o high dynamic pe o mance o
he sys em, he comple e con ol egion o he con e e is no
used and as a consequence some d awbacks appea as was
shown in sec ion II.
The idea o MPC-DSVM is o use no only he con e e
disc e e eal s a es bu in addi ion o he poin s o he con ol
egion called i ual ec o s [23]. These i ual ec o s can
be loca ed in any posi ion o he con e e con ol egion and
ha e o be syn hesized by a linea combina ion o he con e e
disc e e eal s a es. Fig. 3 shows he con ol egion o he
wo-le el h ee-phase con e e whe e he con e e disc e e
eal s a e ec o s ( ound ma ks) and wel e addi ional i ual
ec o s (squa e ma ks) ha e been ep esen ed. In gene al, any
numbe o i ual ec o s can be plo ed in he con ol egion.
The con e e i ual ec o s can be gene a ed as a linea
combina ion o he disc e e eal s a es o he con e e by
using a disc e e space ec o modula ion (DSVM). The DSVM
echnique is a space ec o modula ion based on he use o a
ini e numbe o i ual ec o s placed in he con ol egion
o he powe con e e ins ead o using he whole con e e
α
β
(100)
V1
(000)
V0
(111)
V7
(110)
V2
(010)
V3
(011)
V4
(001)
V5(101)
V6
Con e e con ol egion
Con e e eal s a es
Fig. 1. Con ol egion o a wo-le el h ee-phase con e e in he αβ ame.
Sa,k+1 = 0
Sb,k+1 = 0
Sc,k+1 = 1
Sa,k = 1
Sb,k = 0
Sc,k = 1
α
β
(100)
V1
(000)
V0
(111)
V7
(110)
V2
(010)
V3
(011)
V4
Con e e con ol egion
Con e e eal s a es
(001)
V5(101)
V6
a b
TsTs
Sa,k
Sa,k+1
PWM
TsTs
On O LUT
Ga e
Signal
Ga e
Signal
Fig. 2. Gene a ion o he powe swi ch i ing pulses wi h FS-MPC. a)
Th ough LUT. b) Th ough ex e nal PWM modula o .
α
β
(100)
V1
(000)
V0
(111)
V7
(110)
V2
(010)
V3
(011)
V4
(001)
V5(101)
V6
Con e e con ol egion
Con e e eal s a es
Con e e i ual s a es
(1;0.5;0)
Vk
Fig. 3. Con ol egion o a wo-le el h ee-phase con e e in he αβ ame
including wel e i ual s a es.
con ol egion. The concep o DSVM was in oduced in [17],
[18], and has been used o educe he o que and cu en ipple
in DTC applica ions [19]–[22].
Any i ual ec o i can be achie ed by a linea combi-
na ion o h ee eal ec o s V eal, whe e each eal ec o is
applied in a swi ching sequence a ce ain ime quan i y.
i =X
j=1,2,3
jV eal
j
1+ 2+ 3=Tsw (1)
V eal
j∈ {V0, V1, . . . , V7}
In (1), Tsw ep esen s he swi ching ime, i Tsw is ixed
equal o all he i ual ec o s hen Tsw is ans o med in he
swi ching pe iod and he swi ching equency o MPC-DSVM
is ixed, sol ing in his way one o he mayo issues o
con en ional FS-MPC. As in FS-MPC, he numbe o ec o s
o be e alua ed is ini e so i is possible o use a LUT o
s o e he swi ching ec o sequence and imes. Howe e when
he numbe o i ual ec o s inc eases a be e solu ion is o
use an ex e nal modula o o gene a e he powe swi ch ga e
signals as i is shown in Fig. 4.
IV. SIMULATIONS RESULTS
MPC-DSVM echnique can be employed in any powe con-
e e applica ion ha uses FS-MPC as con ol echnique. Fo
he shake o simplici y, o p esen he concep o MPC-DSVM
a con en ional wo-le el h ee-phase in e e connec ed o he
g id is used. The h ee-phase wo-le el powe con e e is
depic ed in Fig. 5, whe e he neu al poin is deno ed by n. The
sys em is connec ed o he g id h ough smoo hing induc o s
Land a cons an DC ol age sou ce Vdc is connec ed in he
DC e minals o he con e e . The sys em pa ame e s and
a iables a e desc ibed in Table III.
The equa ions ha desc ibe he inpu cu en s dynamics can
be de i ed om he sys em model in he s a iona y αβ ame.
The a iables in he s a iona y ame a e calcula ed as {·}T
αβ =
A{·}T
abc.
αβ =Ldiαβ
d +Vdc
2Sαβ (2)
A=q2
3"1−1
2−1
2
0√3
2−√3
2#(3)
The con ol objec i e o he h ee-phase powe in e e is
he inpu cu en acking owa ds hei e e ences, i∗
αand i∗
β
TABLE III
SYSTEM VARIABLES AND PARAMETERS
Va iable Desc ip ion
={ an bn cn}TPhase o neu al inpu ol age ec o
i={iaibic}TPhase inpu cu en ec o
S={SaSbSc}TCon ol inpu ec o
ωG id equency
LSmoo hing induc o
Vdc DC ol age sou ce alue
a b
TsTs
Sa,k
Sa,k+1
PWM
TsTs
On On/O /OnLUT
Ga e
Signal
Ga e
Signal
Sa,k+1 = 0.5
Sb,k+1 = 0
Sc,k+1 = 1
Sa,k = 1
Sb,k = 0
Sc,k = 1
α
β
(100)
V1
(000)
V0
(111)
V7
(110)
V2
(010)
V3
(011)
V4
Con e e con ol egion
Con e e eal s a es
Con e e i ual s a es
(001)
V5(101)
V6
Fig. 4. Gene a ion o he powe swi ch i ing pulses wi h MPC-DSVM. a)
Th ough LUT. b) Th ough ex e nal PWM modula o .
a
b
c
L
L
L
s
dc
V
a
i
c
i
b
i
an
cn
n
bn
Fig. 5. Th ee-phase wo le el powe in e e .
espec i ely. These e e ences can be calcula ed in such a way
ha ce ain alues o ac i e and eac i e powe a e injec ed o
consumed om he g id, howe e his is ou o scope o his
pape .
iα→i∗
α(4)
iβ→i∗
β(5)
In [11] a FS-MPC cu en con olle o he h ee-phase
wo-le el in e e is p oposed. This con ol algo i hm has
been modi ied as i is shown in Fig. 6, whe e he numbe
o disc e e ou pu s a es unde conce n is n , including he
e alua ion o he eal and i ual s a e ec o s. The inal
ha dwa e con igu a ion is p esen ed in Fig. 7 showing he
powe con e e diag am block, whe e an ex e nal modula o
has been used o gene a e he i ing pulses.
Se e al simula ions ha e been pe o med o analyze he
in luence on he powe con e e pe o mance when i ual
ec o s a e used diminishing he sampling equency. In Table
IV he pa ame e alues used in he simula ions a e sum-
ma ized. Fou di e en si ua ions ha e been conside ed. The
Fig. 6. Flow diag am o he MPC-DSVM con ol algo i hm.
L
PWM-SVM
Ga e Signals
Con e e
n
Fig. 7. Powe con e e diag am block using MPC-DSVM.
i s one is a FS-MPC e alua ing only he eigh eal s a e
ec o s using high sampling equency. Secondly he FS-MPC
pe o mance is e alua ed when he sampling equency is
i e imes slowe . The hi d expe imen uses he p oposed
MPC-DSVM wi h he same sampling equency as he second
si ua ion, bu e alua es hi y eigh ou pu ec o s a es (eigh
eal and hi y i ual s a es). In his way he compu a ional
e o has been kep cons an compa ed wi h expe imen 1.
Finally MPC-DSVM wi h a highe numbe o i ual ec o s
is conside ed bu he sampling equency is kep as in he
second and hi d expe imen .
In Table V a e shown he sampling equency and he
TABLE IV
SIMULATION PARAMETERS VALUES
Pa ame e Value
an, bn, cn 230V ms
i∗
a,peak, i∗
b,peak, i∗
c,peak 20A
ω50Hz
L2mH
Vdc 750V
TABLE V
GRID CURRENT THD VALUES
sn cu en THD
Expe imen 1 50 kHz 8 2.0%
Expe imen 2 10 kHz 8 23.1%
Expe imen 3 10 kHz 38 11.0%
TABLE VI
GRID CURRENT THD VALUES FOR HIGH n
s:10 kHz
n 62 332 1262 4922
cu en THD 7.2% 3.7% 2.0% 1.8%
numbe o disc e e ou pu ec o s (n ) used in expe imen s
1, 2 and 3, oge he wi h he cu en o al ha monic dis o ion
(THD) alue ob ained in each case. The cu en THD has
been calcula ed up o ha monic 49 h. I can be no iced ha
expe imen 1, co esponding wi h FS-MPC echnique, p esen s
he lowes ha monic con en . Expe imen 2 also co esponds
wi h FS-MPC, bu as he sampling equency has been educed
hen he pe o mance o he con olle is highly de e io a ed.
Finally, he p oposed MPC-DSVM is pe o med in expe imen
3. When compa ed wi h expe imen 2, he THD alue has
been educed in 50%, howe e his alue is highe han THD
ob ained wi h FS-MPC in expe imen 1.
Fig. 8, 9 and 10 show he g id cu en wa e o m o phase a,
i s e e ence and i s ha monic spec um o expe imen s 1, 2
and 3 espec i ely. I can be no iced in Fig. 8 ha cu en
ipple o FS-MPC wi h high sampling equency is e y
small. In addi ion, ha monic con en is small bu he spec um
is widesp ead. In Fig. 9 he cu en ipple has inc eased as
he sampling equency is lowe , besides he spec um is
widesp ead as FS-MPC is used. When MPC-DSVM is em-
ployed he cu en ipple achie es lowe alues ha FS-MPC
wi h he same sampling equency as i is shown in Fig.
10. Besides, in he ha monic spec um can be no iced ha
equency componen s a e less sp ead and a peak appea s in
he swi ching equency.
In expe imen 4 is s udied he pe o mance o MPC-DSVM
when he numbe o addi ional i ual ec o s is inc eased.
Table VI shows he cu en THD o di e en alues o n .
I can be no iced ha THD diminishes as he numbe o
addi ional i ual ec o inc eases, being possible o ob ain
e en lowe THD han wi h FS-MPC. In Fig. 11 he esul s
ob ained wi h n = 4922 a e shown, i can be obse ed ha
cu en ipple is almos iden ical o he FS-MPC con olle
wi h high sampling equency in expe imen 1, bu in con as
ha ing a ixed swi ching equency. To s udy a high numbe
o ou pu s a es p o ides high pe o mance o MPC-DSVM.
Howe e , MPC-DSVM has o e alua e all he disc e e ou pu
ec o s o minimize he cos unc ion. The con ol ha dwa e
de e mines he maximum numbe o disc e e ou pu s a es ha
can be s udied each sampling ime, limi ing in his way he
pe o mance o he sys em.
0 0.01 0.02 0.03 0.04
ÿ35
0
35
Time(s)
ia,i*
a(A)
0 5000 10000 15000
0
3.5
7
F equency(Hz)
THD(%)
Fig. 8. Simula ion esul s o s= 50kHz and n = 8. F om op o bo om:
a) Phase a cu en and e e ence. b) Phase a cu en ha monic spec um.
0 0.01 0.02 0.03 0.04
ÿ35
0
35
Time(s)
ia,i*
a(A)
0 5000 10000 15000
0
3.5
7
F equency(Hz)
THD(%)
Fig. 9. Simula ion esul s o s= 10kHz and n = 8. F om op o bo om:
a) Phase a cu en and e e ence. b) Phase a cu en ha monic spec um.
V. ANALYSIS OF MPC-DSVM
Tables VII and VIII summa ize he ad an ages and disad-
an ages o p oposed MPC-DSVM. This echnique has he
same ad an ages o FS-MPC and includes wo new powe -
ul ea u es, cons an swi ching equency and low sampling
equency. Bo h cha ac e is ics a e consequence o a be e
use o he con e e con ol egion. The cons an swi ching
equency makes easie he design o passi e il e s and he low
equency educes he con ol ha dwa e equi emen s. On he
o he side, when MPC-DSVM is employed new conce ns ap-
pea compa ed wi h FS-MPC. MPC-DSVM needs an ex e nal
modula o and his o ces a mo e complex con ol ha dwa e.
Mode n DSPs include build-in pe iphe als o do his ask o
i can be buil easily in a FPGA. Besides a new challenge
eme ges in low sampling equency MPC-DSVM. To ob ain
simila pe o mance o ha wi h high sampling equency
FS-MPC i is necessa y o e alua e a high numbe o disc e e
ou pu s a es. This issue could be sol ed wi h a sma choosing
o he i ual ec o s o be analyzed in each sampling ime.
0 0.01 0.02 0.03 0.04
ÿ35
0
35
Time(s)
ia,i*
a(A)
0 5000 10000 15000
0
3.5
7
F equency(Hz)
THD(%)
Fig. 10. Simula ion esul s o s= 10kHz and n = 38. F om op o bo om:
a) Phase a cu en and e e ence. b) Phase a cu en ha monic spec um.
0 0.01 0.02 0.03 0.04
ÿ35
0
35
Time(s)
ia,i*
a(A)
0 5000 10000 15000
0
3.5
7
F equency(Hz)
THD(%)
Fig. 11. Simula ion esul s o s=10kHz and n =4922. F om op o bo om:
a) Phase a cu en and e e ence. b) Phase a cu en ha monic spec um.
TABLE VII
ADVANTAGES OF MPC-DSVM CONTROLLERS
Easy implemen a ion wi h high pe o mance
No necessi y o cons an s con olle uning p ocedu es
Easy o include any pa ame e op imiza ion c i e ia in he cos unc ion
Takes in o accoun he disc e e na u e o he con e e
P esen s cons an swi ching equency sw
High pe o mance wi h low sampling equency s
Takes be e ad an age o he con e e con ol egion
TABLE VIII
DRAWBACKS OF MPC-DSVM CONTROLLERS
Needs a LUT o an ex e nal modula o o gene a e he i ing pulses
Low sinc eases he compu a ional e o o achie e high pe o mance
VI. CONCLUSIONS
This pape p esen s a Fini e S a es Model P edic i e Con ol
(FS-MPC) s a egy wi h cons an swi ching equency. To

achie e his ea u e, a Disc e e Space Vec o Modula ion
(DSVM) is added o he FS-MPC. The p oposed MPC-DSVM
is based on he e alua ion o new ou pu ec o s in he
con olle algo i hm. These ou pu ec o s, called i ual ec-
o s, a e no eal con e e s a e ec o s and ha e o be
syn hesized h ough a ex e nal modula o . I has been shown
ha MPC-DSVM p esen s he same ad an ages o FS-MPC
and addi ionally can wo k wi h low sampling equency and
ixed swi ching equency. Besides, he in luence o he num-
be o ou pu ec o s e alua ed ha e been analyzed. I has
been shown ha o low sampling equency wi h highe
pe o mance, MPC-DSVM equi es a la ge numbe o i ual
ec o s. Howe e e en be e pe o mance han FS-MPC can
be achie ed, so new challenges ha e been add essed.
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