New Con ollabili y C i e ia o 3-phase 4-wi e in e e s applied o Shun Ac i e
Powe Fil e s
Manuel A. Pe ales, Luis Te ón, Juan A. Sánchez, An onio de la To e, Juan M. Ca asco, Leopoldo
G. F anquelo
Dep . de Ingenie ía Elec ónica. Escuela Supe io de Ingenie os. Uni e sidad de Se illa
Camino de los Descub imien os s/n
Isla de la Ca uja. 41092-Se illa . SPAIN
[email p o ec ed]
Abs ac – In Shun Ac i e Fil e applica ions, he 3-phase 4-
wi e opology is equen ly used when dealing wi h unbalanced
loads con aining ze o sequence componen s. A new design
c i e ia o his opology is p esen ed, based on he well known
exis ing me hod o he 3-phase 3-wi e sys em. Simula ion and
expe imen al esul s con i ms he alidi y o his new c i e ia,
p o iding an easy me hod o he design o he eac i e elemen s
in ol ed in a Shun Ac i e Fil e .
I. INTRODUCTION
Ac i e Fil e s a e a as g owing ield in he powe
elec onics a ea, due o he new egula ions and s anda ds on
powe quali y. In pa icula , Shun Ac i e Fil e s a e he mos
common s uc u e, ha ing nowadays a lo o p ac ical
implemen a ions ins alled.
The mos common s uc u e in his ield is based on he
Cu en Con olled Vol age Sou ce In e e (CC-VSI). When
designing his in e e , a special ca e ha e o be aken in he
elec ion o he eac i e elemen s, ha is, he capaci ance and
nominal ol age o he DC-Link and he alue o he
smoo hing induc ance.
When he load o be compensa ed is a 3-phase load,
wi hou neu al connec ion, a 3-phase 3-wi e CC-VSI is
commonly used. Fo his kind o in e e s, a design c i e ia
was de eloped long ago [1] ha is e y use ul o selec he
co ec alues o his pa ame e s, using he well known α-β
ans o ma ion.
In his pape , an ex ension o his me hod is p esen ed,
co e ing he 3-phase 4-wi e opology, also e y employed in
he Shun Ac i e Fil e s ield [2] [3] [4]. In his case, he α-β
ans o ma ion is no so use ul, as i will be s a ed, because o
he necessi y o adding ano he componen , he 0 axis,
making he g aphical analysis oo much complica ed.
In Sec ion II, he 3-phase 4-wi e opology is b ie ly
p esen ed and analysed, showing he RST-αβ0
ans o ma ion o he in e e simpli ied equa ions. Then in
Sec ion III he con ollabili y condi ion is de ined, showing
he di icul y on he assessmen o his condi ion in αβ0 axis.
In Sec ion IV a new change o axis is s a ed, called IMC
ans o ma ion, ha g ea ly simpli ies he g aphical analysis
o he con ollabili y p oblem. A e ha , in Sec ion V
simula ion and expe imen al esul s a e p esen ed, ha
con i ms he alidi y o he p oposed me hod, and inally in
Sec ion VI some conclusions a e ex ac ed.
II. 3-PHASE 4-WIRE TOPOLOGY
The 3-phase 4-wi e opology is cha ac e ised by he
connec ion o he neu al o he middle poin o he DC-Link.
In his si ua ion, he h ee phases a e independen o each
o he , and he cu en lowing o each leg depends only on
he posi ion o he associa ed swi ches and i s phase ol age.
In ig. 1 a basic scheme o his opology is p esen ed. I
should be no ed ha he cu en is conside ed posi i e
lowing ou o he in e e .
L
L
L
DC
DC
R
S
T
i
FR
S
1
S
2
S
3
S
4
S
5
S
6
N
Load
i
FS
i
FT
Fig 1. Basic scheme o a 3-phase 4-wi e based Shun Ac i e Fil e
So, he equa ions desc ibing he beha iou o he phase
cu en s could be exp essed as (1)
DCF
FT
FS
FR
F
T
S
R
·(k)
d
d
·L
i
i
i
;
SiV;iV+−=










=










= (1)
Whe e S(k) is a ec o ial disc e e unc ion depending on
he s a e (k) o he in e e , wi h he ollowing alues:
TABLE I
VALUES OF S(K)
S a e (k) Swi ch. ON S(k)R S(k)S S(k)T
0 S2,S4,S6 -
1
-1 -1
1 S1,S4,S6 1 -1 -1
2 S1,S3,S6 1 1 -1
3 S2,S3,S6 -1 1 -1
4 S2,S3,S5 -1 1 1
5 S2,S4,S5 -1 -1 1
6 S1,S4,S5 1 -1 1
7 S1,S3,S5 1 1 1
As he h ee phase a e decoupled, i is no possible o
educe i o a wo componen sys em, i.e. using he αβ
ans o ma ion. Ins ead, a h ee componen e e ence sys em
mus be applied. I we use he αβ0 ans o ma ion, de ined
by he ma ix Mαβ0 (2), he equa ions can be ansla ed o (3):
















−
−−
=
2
1
2
1
2
12
3
2
3
0
21211
3
2
0
//
Mαβ
(2)
)k(VU
d
d
L+−=
F
i (3)
Whe e he ec o s iF, U and V(k) a e de i ed om (4):
DC
T
S
R
T
S
R
FT
FS
FR
F
F
F
V·
)k(S
)k(S
)k(S
M
)k(V
)k(V
)k(V
)k(V
;
M
U
U
U
U;
i
i
i
M
i
i
i










=










=










=










=










=










=
0
0
0
0
0
0
αββ
α
αββ
α
αββ
α
F
i
(4)
I is impo an o no e ha all ec o s V(k) ha e a
componen on he 0 axis, as i can be seen in (5). Tha leads
o wo conclusions:
• Vec o s 0 and 7 a e no in e changeable, as i
happened in he 3-wi e opology.
• The hexagon o med o he ec o s in he 3-wi e
opology is now a cube, o a ed in he αβ0
coo dina es.


























−
−−−
−
−−
−
−
=
























2
3
2
1
2
1
2
1
2
1
2
1
2
12
3
3
2
00
31
31
02
31
31
02
00
7
6
5
4
3
2
1
0
DC
V·
)(V
)(V
)(V
)(V
)(V
)(V
)(V
)(V
(5)
Fig 2. S a e ec o s in αβ0 coo dina es, gene a ing he con ollabili y cube
This cube is he h ee dimensional equi alen o he well
known wo-dimensional hexagon p oduced by he ec o s in
he αβ coo dina es and will play an impo an ole on he
de ini ion o he con ol capabili y o he in e e in
ollowing a ce ain cu en e e ence, so we called i
con ollabili y cube.
III. CONTROLLABILITY CRITERIA IN αβ0
COORDINATES
Using he same echnique as in [1] bu o he h ee
dimensional case, le us de ine i*F as he cu en e e ence o
be injec ed, and hen he acking e o (∆iF) could be
exp essed as:
FFF i*ii −=∆ (6)
The condi ion o assu e ha he e e ence can be acked is
ha he acking e o dec eases wi h he ime. So he
con ollabili y condi ion will be ha he ime de i a i e o
his e o is nega i e. Making his de i a i e, and using (3):
)k(VU
d
d
L
d
d
L−





+=
∆FF *ii (7)
I we name e0 he e m in pa en hesis, (7) can be ew i en
in he ollowing o m:
)k(V
d
d
L
U
d
d
L
−=
∆
+=
0
F
F
0
e
i
*i
e
(8)
Analysing (8) i is clea ha he con ollabili y depends on
which ec o is g ea e , e0 o V(k). To ensu e he
con ollabili y in he h ee phases simul aneously, he h ee
componen s o he ec o e0 should be smalle han hose o
he V(k) ec o .
I a Space Vec o modula ion is implemen ed, V(k) will
ep esen he ol age ec o o he in e e , and i will be
included on he p e iously de ined con ollabili y cube.
Then, he con ollabili y c i e ia could be exp essed as:
Fo gi en alues o VDC and L, he so de ined in e e will
be capable o acking a ce ain cu en e e ence only i he
ec o e0 de ined in (8) is con ained on he inside o he
con ollabili y cube desc ibed by he ec o s V(k).
I is impo an o no ice ha , e en in he case ha he load
has no ze o sequence componen , is no applicable he wo
dimensional c i e ia, because he in e sec ion o he
con ollabili y cube wi h he αβ plane is no equal o he
hexagon desc ibed by he p ojec ion o he s a e ec o s o e
he αβ plane. This p ojec ion is equal o he hexagon used in
[1], bu he in e sec ion o he cube wi h he αβ plane, whe e
should be loca ed e0 i i do no has ze o sequence o ensu e
con ollabili y, is an hexagon smalle han ha , wi h i s
e ices in he middle o he sides o he i s , as i is shown
in ig. 3.
α
β
1
2
3
4
5 6
0,7
Fig 3. Con ollabili y hexagon in he wo dimensional case (solid) and
in e sec ion o con ollabili y cube wi h he αβ plane(dashed)
IV. NEW CONTROLLABILITY CRITERIA IN IMC
COORDINATES
A. IMC ans o ma ion
The complexi y o he applica ion o his new
con ollabili y c i e ia lays on he di icul y o de e mine
whe he e0 is con ained o no in he con ollabili y cube, as i
is o a ed om he e e ence sys em.
To o e come his p oblem, a new ans o ma ion is
p oposed, called IMC ans o ma ion (whose name comes
om In e se o Ma ix o Con ollabili y), s a ed as:
• The new coo dina es will be de ined by he uni a y
ec o s I, M, C, in he di ec ion o V(0)-V(5), V(4)-V(5)
and V(6)-V(5) espec i ely.
• The cen e o he e e ence sys em will be placed on
V(5).
• The axis will be scaled down so he ec o s V(0)-V(5),
V(4)-V(5) and V(6)-V(5) will ha e uni a y modulus.
To make his coo dina es change, i is use ul o de ine he
Ma ix o Con ollabili y (MC), cons uc ed by columns wi h
he ec o s V(0)-V(5), V(4)-V(5) and V(6)-V(5) exp essed in
αβ0 coo dina es:










−
=
⇒







=−
=−
−=−
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
022
2
0256
254
250
DC
DC
DC
DC
VMC
),,(V)(V)(V
),,(V)(V)(V
),,(V)(V)(V
(9)
No e ha he columns o his ma ix ep esen s h ee
o hogonal ec o s placed in h ee sides o he con ollabili y
cube, and he h ee o hem ha e he same modulo: 2VDC.
Now, o exp ess a gene ic ec o Y
αβ0 in he IMC
e e ence sys em, i is only necessa y o sub ac om i he
V(5) ec o , in αβ0 coo dina es, and hen p e-mul iply his by
he in e se o he ma ix o con ollabili y, esul ing:
))(VY(MCYIMC 00
15αβαβ −= − (10)
Ope a ing wi h (10), (9) and (5), he change o coo dina es
om αβ0 o IMC can be s a ed inally as:






















−
−
−
−






















−
−
=










2
1
3
2
0
2
3
2
1
2
3
2
1
3
1
102
1
1
3
1DC
DC
C
M
I
V
Y
Y
Y
·
V
Y
Y
Y
β
α (11)
B. Con ollabili y cube on IMC axis
The i ue o his new coo dina e sys em is ha , on IMC
axis, he con ollabili y cube will be a uni a y cube (wi h all
i s sides measu ing 1) and wi h i s aces pa allel o he
coo dina e planes IM, IC and MC, as ep esen ed in ig.4
Fig 4. Con ollabili y cube ep esen ed in IMC axis.
In his si ua ion, he accomplishmen o he con ollabili y
c i e ia can be e y easily checked, only by es ing ha he
IMC componen s o he e0 ec o emains be ween 0 and 1
always. I hey do ha , he ec o will be con ained in he
con ollabili y cube.
C. Con ollabili y C i e ia in IMC coo dina es
Using he new coo dina es ans o ma ion, he
con ollabili y c i e ia de ined in III may be ew i en in his
way:
Fo gi en alues o VDC and L, he so de ined in e e will
be capable o acking a ce ain cu en e e ence only i he
IMC coo dina es o he ec o e0 de ined in (8) emains
be ween 0 and 1.
No e ha e0 depends no only on he cu en e e ence
de i a i es, bu also on he alue o he smoo hing
induc ance, and e en on he g id ol age. On he o he hand,
V(k) depends on he DC-Link ol age. Consequen ly, o a
gi en cu en e e ence and g id ol age, his wo pa ame e s
(VDC and L) could be adjus ed o comply wi h his
con ollabili y c i e ia.
V. SIMULATION AND EXPERIMENTAL RESULTS
To alida e he c i e ia s a ed be o e, many simula ion and
expe imen al esul s we e ob ained. Fi s a h ee phase
unbalanced load, wi h ze o sequence componen , we e used
as load o compensa e, doing ex ensi e simula ions o
de e mine he op imal alues o VDC and L. Then , a h ee
phase balanced load we e simula ed and compensa ed
expe imen ally, o e ing he posibili y o compa ing he
simula ion and expe imen al esul s.
A. Simula ion wi h an unbalanced load
A 3-phase 4-wi e load wi h unbalance, ze o sequence,
cu en ha monics and eac i e powe was modelled om
expe imen al measu es o a eal one. The wa e shapes o he
h ee phase cu en s and neu al cu en a e d awn in ig. 5. I
is no iceable he magni ude o he neu al cu en , ela ed o
he ze o sequence componen .
Fig 5. Unbalanced load simula ed. (a) Load cu en s in R-S-T phases. (b)
Neu al Cu en
Simula ions we e made calcula ing he ec o e0 needed o
compensa e adequa ely his load, o many alues o VDC and
L. In ig. 6, he h ee componen s o e0 in IMC coo dina es
a e plo ed o wo pa icula condi ions (VDC=250, L=1.5mH
and V
DC=300, L=1.2mH). I can be seen ha in he i s
si ua ion he con ollabili y c i e ia will no be ul illed, and i
will in he second case.
Nex , in ig. 7, he op imal alues o VDC depending on
he alue o L is d awn. I can be seen ha a quad a ic
ela ionship is almos ollowed (plo ed in solid) by he
calcula ed pai s (plo ed as ci cles).
Fig 6. plo o e0 in IMC coo dina es. (a) VDC=300V, L=2mH.
(b) VDC=300V, L=1mH
Fig 7. Op imal alues o VDC, o a gi en alue o L.
The cu e ob ained in ig. 7 gi es us a design c i e ia o a
Shun Ac i e Fil e ha migh ha e o compensa e his
pa icula load: ixing he DC-Link ol age, due o swi ching
o capaci o limi a ions, he induc ance equi ed can be
calcula ed by means o his quad a ic app oxima ion.
B. Simula ions wi h a load wi hou ze o sequence
Nex , simula ions o balanced load wi hou ze o sequence
we e pe o med, in o de o p o e ha , in his case i a 3-
phase ou -wi e in e e is employed, he con ollabili y
hexagon applicable is he one de ined in ig.3, and no he
bigge one s a ed in [1].
The load simula ed was modelled o m labo a o y
measu es, and is equal o he eal one ha will be
compensa ed using an expe imen al Shun Ac i e Fil e . The
wa e shape o he phase cu en is plo ed in ig. 8.
In ig. 9, he h ee componen s o e0 a e plo ed, o h ee
alues o VDC and he same alue o L, showing ha only he
hi d case (VDC=300V) complies wi h he con ollabili y
c i e ia. Then, in ig.10 he pola ep esen a ion o he α, β
componen s o e0 a e plo ed, o he same condi ions as
abo e.
Fig 8. Load cu en used in simula ions
No ice ha , as iL has no ze o sequence componen , nei he
i*F will, so supposing ha g id ol age has no ze o sequence
componen s, e0 will be loca ed on he αβ plane. Then, he
con ollabili y c i e ia could be simpli ied o ha eo(αβ)
emains in he hexagon de ined in ig. 3. I is impo an o
poin ou ha , i he c i e ia de ined o he 3-phase 3-wi e
case should be used, he si ua ion ep esen ed in (b) should
be conside ed as con ollable e oneously, aking he ou e
hexagon (dashed) ins ead o he inne hexagon (solid) as he
con ollabili y hexagon.
Fig 9. Componen s o e0 in IMC coo dina es. (a) VDC=225V, L=2.4mH. (b) VDC=250V, L=2.4mH. (c) VDC=300V, L=2.4mH.
Fig 10. Pola ep esen a ion o e0αα,e0ββ wi h he new con ollabili y hexagon (solid) and he 3-phase 3-wi e case hexagon (dashed), o he same alues as
abo e.

C. Expe imen al esul s
To alida e he con ollabili y c i e ia s a ed in his pape ,
expe imen s we e made, using he same load han in he
p e ious simula ions.
Fo his expe imen s, an Ac i e Powe Fil e de eloped by
he au ho s was employed [5]. The con ol algo i hm
implemen ed was based on Sel -Tuned Vec o Fil e s [6],
able o compensa e cu en ha monics e en in p esence o a
highly dis o ed ol age g id.
The mos impo an pa ame e s o he powe sys em a e
shown in Table I .
TABLE I. POWER SYSTEM PARAMETERS
Topology 3-phase 4-wi e
Vol age (phase-neu al) 150V
DC-Link capaci y 2 x 13.2 mF
DC-Link ol age (max) 2 x 350V
Smoo hing induc ances 2.4mH
Powe swi ches IGBT’s (SKM300 GB 123 D)
Commu a ion equency 20kHz
Nominal powe (load) 12kW
Finally, in ig. 11 expe imen al cu es o he compensa ed
cu en a e shown. I can be obse ed easily ha in he i s
and second cases a peak appea s on he compensa ed cu en ,
due o he loose o con ollabili y. This peak is g ea e in he
i s case (VDC=225V), and do no appea s in he hi d case
(VDC=300V).
These esul s ma ch p ecisely he p edic ed con ollabili y
o he sys em in e ms o complying he con ollabili y
c i e ia s a ed, alida ing his way he p oposed me hod.
VI. CONCLUSIONS
The 3-phase 4-wi e opology is widely used in he
implemen a ion o Shun Ac i e Fil e solu ions, ha ing he
abili y o compensa e unbalanced loads wi h ze o sequence
componen s. I is impo an he e o e o ha e a design
c i e ia, based on a con ollabili y c i e ia in he same way
han o he 3-phase 3-wi e opology.
In his pape , a new con ollabili y c i e ia is p esen ed, as
an ex ension o he wo dimensional case o a h ee
dimensional analysis.
By means o a ans o ma ion o coo dina es, he c i e ia
can be easily checked, allowing he use o i o es ima e he
co ec alues o he ol age o he DC-Link (VDC) and he
smoo hing induc ance (L).
The ma ch be ween heo e ical and expe imen al esul s
con i ms he alidi y o his echnique, ha should be used
when a 3-phase 4-wi e in e e is being designed o an
Ac i e Fil e applica ion, e en i he load o compensa e has
no ze o sequence componen s.
REFERENCES
[1]. H. Akagi, Y. Tsukamo o, A. Nabae. “Analysis and
design o an ac i e powe il e using quad-se ies ol age
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Fig 11. Expe imen al esul s. Compensa ed cu en o di e en alues o VDC, and L=2.4mH. (a) VDC=225V. (b) VDC=250V. (c) VDC=300V