Bộ truyền động nhiều pha (hơn 3) đang
dần được áp dụng trong nhiều ứng dụng đặc
biệt dẫn đến sự cần thiết trong việc phát triển
các giải thuật điều khiển của các bộ truyền
động này. Bài báo này trình bày lý thuyết
“Generalized Vectorial Formalism” để điều
khiển hai động cơ điện đồng bộ nhiều pha
mắc nối tiếp. Hai động cơ đồng bộ được cung
cấp bằng một bộ biến tần trong đó số nhánh
của bộ biến tần này bằng với số pha của mỗi
động cơ. Theo lý thuyết “Generalized
Vectorial Formalism”, một máy điện nhiều
pha tương ứng, bằng mô hình toán học, với
một vài máy điện ảo (hai pha và một pha). Số
lượng máy điện ảo phụ thuộc số pha và cách
đấu dây giữa các pha của máy điện nhiều
pha. Dựa trên lý thuyết này, một giải thuật đã
được đề ra để điều khiển một cách hoàn toàn
độc lập (vận tốc và moment xoắn) hai động
cơ đồng bộ nhiều pha mắc nối tiếp với duy
nhất một bộ biến tần. Kết quả thực nghiệm
với hai máy điện 5 pha cho thấy sự đúng đắn
của giải pháp điều khiển này.

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SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015
Generalized Vectorial Formalism – based
multiphase series-connected motors control
. Eric Semail
. Ngac Ky Nguyen
. Xavier Kestelyn
. Tiago Dos Santos Moraes
L2EP Laboratory, Arts et Métiers ParisTech, France
(Manuscript Received on July 15, 2015, Manuscript Revised August 30, 2015)
ABSTRACT
Multiphase drives are more and more magnet synchronous motors (PMSM) fed by
used in specific applications leading to a one voltage source inverter (VSI). Based on
necessity of control strategy development. a decomposition of multiphase machine, a
This paper presents the Generalized proposed control strategy has been
Vectorial Formalism (GVF) theory to control achieved. Some experimental results are
multiphase series-connected permanent given to illustrate this control method.
Keywords: Multiphase drives, Generalized Vectorial Formalism, Multiphase series-
connected machines.
1. INTRODUCTION
Higher reliability of classical 3-phase drives The multiphase theory has been developed
can be achieved by oversizing the converter- since 10 years ago [3]-[6] in objective to
machine set but this solution increases the cost of understand deeply and allow using simple
the whole system. Even if this oversizing is regulators for current and speed. Based on this
chosen, in case of an open circuit fault appears in theory, a multiphase machine can be
one or two phases of the drive, the system cannot decomposited to some equivalent fictitious ones.
ensure a functioning even at reduced power. Classically, a three-phase machine in the Park
Using multiphase drives instead of three- reference frame is the simplest case where we
phase drives, makes possible to increase the have a d-q diphase machine and a homopolar one.
power density, fault-tolerance capability and to If the machine is star connected, the latter one is
reduce torque pulsations at low frequencies [1, 2]. not considered. Indeed, the Park (or Concordia)
In fault mode, this kind of drives is able to work transformation is a linear mathematical
at a reduced power with satisfactory application where machines will be modelled into
performances. This aspect is very important in the eigen space and represented by the eigen
systems which are designed for specific vectors of the matrix inductance. In a general case
applications, such as offshore energy harvesting where a multiphase machine is considered, the
or electrical vehicles. generalized Concordia (or Park) transformation is
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TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015
needed. After the transformation, the multiphase machine. Then we consider that the stator
machine is modeled by some diphase machines inductance matrix is the characterization, in a
and some homopolar machines according to the natural base, of a linear application also called an
shape of the back-electromotive force (back- endomorphism.
EMF). In a general way, to control a diphase In the next paragraphs, we give some of its
machine, two independent currents are required. properties a n-phase machine and particularly a 5-
That is why a 3-phase Voltage Source Inverter phase machine.
(VSI) is needed to supply a 3-phase machine and 2.1 Endomorphism and stator inductance
a n-phase VSI is required to supply a n-phase matrix
machine. In the case where multiphase machines
Let us consider the stator inductance matrix
are series-connected, the model of each machine
L n of a multiphase machine. We consider it as
is not changed but the control is more complex s
the matrix of an endomorphism in an
since phase currents across all machines. An s
orthonormal base classified as “natural”. This
independent functionning (speed and torque) of n
endomorphism has properties independent of
each machine requires a decoupled strategy of s
control and imposes some contraints to machine the choice of the studying base: eigenvalues,
design in term of the back-EMF. eigenvectors and eigenspace. To get them we have
only to examine L n .
This paper presents firstly the theory of s
mutliphase machine based on the Generalized As mutual inductance between two windings
Vectorial Formalism. A two series-connected 5- j and k are identical (Mjk=Mkj) then the matrix is
phase machines supplied by one VSI structure is symmetrical. This symmetry implies the existence
presented thereafter. Some experimental results of a base of eigenvectors. Moreover the
will be given to confirm the feasibility of the
eigenspaces of s are orthogonal each other and
proposed structure.
the dimension of an eigenspace Es is equal to the
2. GENERALIZED VECTORIAL
multiplicity order of the associated eigenvalue
FORMALISM
. For example, if the order of multiplicity is one
In order to show that a polyphase machine is (respectively two), the eigenspace is a vectorial
equivalent to a set of 1-phase and 2-phase line (respectively vectorial plan). To obtain an
machines, we have to bring out vectorial orthonormal base of eigenvectors we have only to
properties of the stator self inductance matrix. The choose in each eigenspace an orthonormal base.
analysis of its properties enables the The classic transformation matrixes of Park or
generalization of the transformation concept. Of Concordia are nothing else than tables that allow
course, the matrices which generalize Park’s and us to find the coordinates of these eigenvectors.
Concordia’s transformations for a n-phase
If the order of multiplicity of all eigenvalues
machine have already been defined and used in
is one, then there is only one orthonormal base of
particular cases. Our formalism defines a larger
eigenvectors. Consequently, only one
class of systems for which these transformations
transformation that keeps the power invariant can
can be used. This is possible because, in the
then be elaborated.
vectorial approach, transformations are only the
On the other hand, if the order of multiplicity
expression of vectorial properties linked to the
is not one for one eigenvalue, then there is an
stator inductance matrix. At first, an Euclidian
infinity of orthonormal bases of eigenvectors.
vector n-space is associated with an n-phase
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SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015
d d d , the matrix of
Consequently an infinity of transformations d x1,,, x 2 x n
keeping invariant the power can be defined. This
endomorphism is given by:
property explains the great number of s
transformations that have been proposed in the 1 0 0
0 0
past. Ld 2 (6)
s 0
2.2 Applying to a n-phase machine
0 0 0 n
In an orthogonal base composed of the
n Each eigenvalue is associated to an
vectors xn,,, x n x n it can be defined the
1 2 n eigenspace whose the dimension depends on the
voltage, current and stator flux linkage vectors as multiplicity of the eigenvalue. For example, if
follows:
there are two solutions equal to 1 , then the 1
n n n
v vx1 1 v 2 x 2 vn x n (1) is belong to a 2-dimensional eigenspace.
n n n It can be noticed that all subspaces (the
i ix1 1 i 2 x 2 in x n (2)
eigenspaces) are orthogonal and it can be defined
n n n as a set of fictitious magnetically independent
1x 1 2 x 2 n x n (3)
systems.
These vectors are linked by:
For each subspace, the relationship between
d d d
v Ri Ri ss sf (4) the voltage and the current is given by:
dt dt dt
d di
v Ri j Ri j e (7)
where depends on i and is due to j jdt j j dt j
ss sf
magnets on the rotor.
where ej is the back-electromotive force.
The relationship between the current vector
Based on equation (7), we can consider that
and the flux vector is given by the endormorphism
in the new base, a multiphase can be decomposed
represented by the inductance matrix n . In
s L s into several fictitious machines where each
the natural frame, L n is expressed as: machine is characterized by a resistor R, an
s
inductance and a vector of the back-EMF e
LMMM11 12 13 1n j j
LLMM21 22 23 2n . According to the dimension of the eigenspace
Ln (5)
s
where the eigenvalue j is belong to, the
LLLLn1 n 2 n 3 nn
fictitious machine can be monophase or diphase.
where Lkk is the self-inductance of the phase Let us take an example in the case of a 3-phase
k and Ljk is the mutual inductance between the machine. The new base is defined by the
phases j and k. Concordia transformation where it exists a double
As mentionned above, the symetrical root and a single root of eigenvalues by solving
inductance matrix leads to a base of eigenvectors 3
the determination detLIs 3 0. The
whose corresponding eigenvalues are given by
double root corresponds to the d-q machine and
detLIn 0. In this new base defined by
s n the single root is linked to the homopolar one.
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TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015
2.3 Applying to a 5-phase machine 0 ia
i i
m b
T
imachim C5 i c (9)
i
s id
i
s ie
0 vaN
v v
m bN
T (10)
vvm C v
mach 5 cN
v
s vdN
Fig. 1. A 5-phase drive having star connection.
vs veN
For a 5-phase PMSM, after a generalized
Currents and voltages obtained using this
Concordia transformation (given in equation (8))
transformation can be decomposed into two
which brings electrical variables of machine to the
subsystems associated with the main and
eigenspace, we obtain thus: one single eigenvalue,
secondary machines:
two double eigenvalues. It means there are one 1-
dimensional fictitious machine and two 2-
v v v t v v v t
dimensional ones. They are called respectively m m m s s s
;
homopolar machine, main machine and secondary t t (11)
i i i i i i
machine. It is not always the case where these m m m s s s
machines exist. Their presence depend on the As the machine is supplied by a VSI,
shape of the back-EMF. Indeed, the homopolar
vmach vvsi and consequently:
machine is created by harmonics 5*k. The main
machine is issue from harmonics 1, 9, 11,, 5*k
vm v mvsi ; v s v svsi (12)
± 1 and the secondary is formed by harmonics 3,
Equations (13) - (16) give the mathematical
7, 13,, 5*k ± 2 by considering only odd
model of the two fictitious machines in the new
harmonics [3]. Thus the Main Machine (MM)
reference frame:
(resp. the Secondary Machine (SM)) produces Tm
d
(resp. Ts) torque mainly thanks to the first
vm R m i m L m i m e m (13)
harmonic (resp. third harmonic) of the back-EMF. dt
Relationships between actual phase variables d
vs R s i s L s i s e s (14)
(denoted with subscripts a, b, c, d and e) and dt
values of fictitious machines are then defined by:
TTTtot m s (15)
1
1 0 1 0
2 where :
1 2 2 4 4
cos sin cos sin
5 5 5 5
2 em. i m T m and e s . i s T s (16)
2 1 4 4 8 8
C cos sin cos sin
5 5 5 5 5 Equation (15) means that the total torque of
5 2
1 6 6 12 12 the 5-phase machine is the sum of the torque
cos sin cos sin
5 5 5 5
2 created by the two fictitious machine. We can
1 8 8 16 16
cos sin cos sin consider that there is a fictitious mechanical
2 5 5 5 5
coupling between them.
(8)
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SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015
Controlling a 5-phase machine leads to The special connection between the two
finally control two equivalent fictitious diphase machines can be expressed by:
ones after the Concordia transformation. Each T
i2 i 2a i 2 b i 2 c i 2 d i 2 e
ficitious machine is linked to some harmonics of T (17)
L5 i 1a i 1 b i 1 c i 1 d i 1 e
the back-EMF as discussed previously in section
L i
2.3. There are many works in the literature 5 1
presenting the control of 5-phase machines under 1 0 0 0 0
healthy and fault modes [6]-[12]. In both 0 0 0 1 0
with: (18)
operating modes, physical limits related to L5 0 1 0 0 0
voltages and currents of the drive have been also 0 0 0 0 1
studied [13]-[16]. This paper focuses to control 0 0 1 0 0
two 5-phase machine having a series connection
The vectors i1 and i 2 represent the current
and they are fed by two isolated DC-buses.
vectors of the 1st machine M1 and the second one
3. TWO SERIES-CONNECTED 5- M2 respectively.
PHASE PMSM
Using equations (9), (17) and the property
1 T
The studied structure is shown in Fig. 2. The [][]CC5 5 , a relation between the current vectors
objective is to control independently these two of the fictitious machines is given as follows:
machines. Normally, to do this, only one VSI is
required but a higher functional reliability can be
0 i2a i 1 a
achieved by adding another VSI [19], specially in
i i i
m 2 2b 1 b
short-circuit inverter switch fault. TT
imach2im 2 C 5 i 2 c C 5 L 5 i 1 c
Controlling independently two 5-phase i
s 2 i2d i 1 d
machines leads to a necessity of 8 independently is 2 i2e i 1 e
currents as degree of freedoms (DOF). In the 0
present structure, there are only 4 currents can be i
m1
TT
freely controlled. That is why a specific C5 L 5 C 5 im1 C 5 L 5 C 5 imach 1
connection between two machines is needed [20]. is1
i
3.1 Modelling s1
(19)
Fig. 2. Structure of two 5-phase open-end winding machines under study
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TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015
where: main machine is almost linked to the 1st
1 0 0 0 0 (fundamental) harmonic while the second
rd
0 0 0 1 0 machine is affected to the 3 harmonic.
rd
T Generally, the 3 harmonic can contribute to the
CLC5 5 5 0 0 0 0 1 (20)
machine torque but when the machine is operated
0 1 0 0 0
at high speed (constraint of voltage has to be
0 0 1 0 0
applied), the current calculation becomes more
Equations (19) and (20) lead to: complex, especially as in degraded mode where
TT some degrees of freedom are lost. That is why
is1 i s 1 i s 1 i m 2 i m 2
(21) until now, there are no analytical solutions for a
TT
i i i i i multiphase machine operating under limit of
m1 m 1 m 1 s 2 s 2
voltages and currents. For some specific
or:
applications as electric vehicle where the drive is
is1 i m 2 almost operated at high speed thanks to the flux
(22)
* weakening operation, a multiphase having
im1 i s 2
sinusoidal back-EMF is preferred.
Expression (22) lead to two significant
In this study, two 5-phase PMSM are
things:
considered sinusoidal. As a consequence, the
Thanks to the swapping connection
back-EMF es1 and es 2 does not exist leading to a
between two machines M1 and M2, the currents
very simple control strategy. The vector current im1
of the main machine MM1 (second SM1
controls the torque (and/or the speed) of the
respectively) of the M1 are linked to the ones of
the secondary SM2 (main MM2 respectively) machine M1 and the vector current is1 is
machine of the M2. This allows to control dedicated to the torque (and/or the speed) of the
independently two machines M1 and M2 with machine M2.
only 4 free currents. Let’s talk about the voltages limit that two
In point of view of electromagnetic machines can be fed.
r r r r
torque, i will create a torque not only for the v= v - v + v (23)
m1 å VSI1 VSI 2 N1 N 2
main machine of M1 but also for the second with:
r r r
machine of M2. In the same manner, is1 will v= v + v
å MM1 2 (24)
contribute to the torque created in the two
fictitious machines SM1 and MM2. This leads to is the sum of the phase voltages of the two
a more complex strategy of control to decouple machines M1 and M2 and
the previous mentioned interaction in order to T
vVSI1 v 1111111111 aN v bN v cN v dN v eN
obtain an independence of the two machines M1
(25)
and M2.
T
This second conclusion is obvious to take into vVSI2 v 2 aN 2 v 2'2 bN v 2'2 cN v 2' dN 2 v 2'2 eN
account for multiphase drives. As mentionned (26)
above, a multiphase machine can bedecomposed T
into some fictitious ones representing by vNNNNNNNNNNNN12 v 12 v 12 v 12 v 12 v 12
corresponding harmonics of the back-EMF. The (27)
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SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015
Fig. 3. Speed control scheme.
are the voltages of the two VSI and the voltage M2. A simplified strategy has been implemented
between the two negative points of two DC buses, r r1 r
in this work by using v= v = v .
respectively. MM1 2 2 å
Voltages delivered by two VSI can be 3.2 Control scheme
obtained by some techniques. A Space Vector The control scheme is reported in Fig. 3. The
Modulation has been proposed in [21] to exploit two machines M1 and M2 are required to rorate
better the two DC-buses. In [22], some simple
as Ω1ref and Ω2ref. PI controllers are used for speeds
modulation strategies have been presented for and currents tracking. It can be noticed that only
varying the machine voltages according to the the phase currents of M1 are measured since they
rotor speed without flux weanking operation. In
are the same ones acrossing the M2 by the L5
this paper, a bipolar modulation is chosen to transformation.
maximize the voltage of machines.
4. EXPERIMENTAL RESULTS
It’s interesting to discuss about the voltage
In order to validate the proposed control
sharing between the two machines. Indeed, by
strategy, some experimental results have been
observing the equation (24), voltages of M1 and
carried out. Fig. 4. shows the platform for
M2 can be shared in the optimal way while
experimental verification. Tab. I. gives some
respecting the voltage limit fed by two VSI based
details of the drive parameters.
on the bipolar modulation technique. As the M1
and M2 are independently controlled, an optimal A 5-phase VSI is communicated to a dSPACE
strategy could be employed according to technical 1005 board through an I/O interface. A DC-
specifications of the applications in term of programmable source is used to feed the drive
torques and speeds of the two machines M1 and during motor mode and recover energy during
braking.
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TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015
Fig. 4. Experimental test-bed.
Table 1. Drive parameters
Parameters 5-phase M1 5-phase M2
Phase resistance Rs = 2.24 [Ω] Rs = 9.1 [mΩ]
Fig. 5. Experimental results in three studied
Phase inductance Ls = 2.7 [mH] Ls = 0.09 [mH]
cases: a) Ω2ref = 0; b) Ω2ref = 40 rad/s and Ω1ref is
M = 0.02
Mutual induc. 1 M = 0.25 [mH] 1
1 [mH] varying; c) Both machine’s speeds are varying.
Mutual induc. 2 M1 = -0.75 [mH] M2 = -0.01[mH] Fig. 5 gives the experimental verification.
st
Pole pair number p = 2 p = 7 Three study cases have been considered. The 1
one consists to keep the machine M2 at stand-still
back-EMF constant
En 0.51 En 0.1358 and the M1 is trained following a speed profile.
The second test has been realized by keeping the
Max. RMS current 15 [A] 147 [A]
rotor speed of M2 at 40 rad/s and the M1 is tuned
Maximum speed 1500 [rpm] 16000 [rpm]
to track a speed profile. For the last case, both
Maximum torque 20 [N.m] 50 [N.m] machines M1 and M2 are operated at two different
Maximum power 3.1 [kW] 15 [kW] profiles of speed. Based on the results given in
For experimentation tests, the two machines Fig. 5, we can conclude that the proposed control
M1 and M2 are operated under speed control. strategy has been verified.
It should be highlighted that there was no
strategy for voltage sharing between M1 and M2.
The speed profiles were chosen in the way that
DC-buses are able to delivery enough voltages for
two machines M1 and M2
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SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015
5. CONCLUSION sinusoidal has been made to simplify to control
In this paper, a generalized vectorial strategy.
formalism has been presented and applied for This specific structure is suitable for
multiphase drives. Based on this approach, two 5- applications where we have the constraint for
phase PMSM having series connection (through compacity and weight of the whole systems.
windings) can be independently controlled.
However, one assumption that all machines are
Điều khiển động cơ điện nhiều pha mắc nối
tiếp dựa trên lý thuyết “Generalized
Vectorial Formalism”
. Eric Semail
. Ngac Ky Nguyen
. Xavier Kestelyn
. Tiago Dos Santos Moraes
L2EP Laboratory, Arts et Métiers ParisTech, France
TÓM TẮT
Bộ truyền động nhiều pha (hơn 3) đang pha tương ứng, bằng mô hình toán học, với
dần được áp dụng trong nhiều ứng dụng đặc một vài máy điện ảo (hai pha và một pha). Số
biệt dẫn đến sự cần thiết trong việc phát triển lượng máy điện ảo phụ thuộc số pha và cách
các giải thuật điều khiển của các bộ truyền đấu dây giữa các pha của máy điện nhiều
động này. Bài báo này trình bày lý thuyết pha. Dựa trên lý thuyết này, một giải thuật đã
“Generalized Vectorial Formalism” để điều được đề ra để điều khiển một cách hoàn toàn
khiển hai động cơ điện đồng bộ nhiều pha độc lập (vận tốc và moment xoắn) hai động
mắc nối tiếp. Hai động cơ đồng bộ được cung cơ đồng bộ nhiều pha mắc nối tiếp với duy
cấp bằng một bộ biến tần trong đó số nhánh nhất một bộ biến tần. Kết quả thực nghiệm
của bộ biến tần này bằng với số pha của mỗi với hai máy điện 5 pha cho thấy sự đúng đắn
động cơ. Theo lý thuyết “Generalized của giải pháp điều khiển này.
Vectorial Formalism”, một máy điện nhiều
Từ khóa: Bộ truyền động nhiều pha, Generalized Vectorial Formalism, Máy điện nhiều pha
mắc nối tiếp.
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TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015
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