Modeling of a Twelve-phase Synchronous Machine
Using Matlab/SimPowerSystems
Liu Sheng
Cheng Yinzhong
College of Automation
Harbin Engineering University
Harbin Heilongjiang
liu.sch@163.com
College of Automation
Harbin Engineering University
Harbin Heilongjiang
cyzelvis@163.com
Abstract—Electric propulsion is now gaining more and more
interests in both civil and military ships. Multiphase machine,
with many inherent advantages, is supposed to be a proper choice
of the propulsion motor for electric propulsion systems. In this
paper, mathematical model of twelve-phase synchronous
machine with four Y-connected three-phase symmetrical
windings displaced in turn by 15 degree is obtained. Then
simulation model of the machine is established in Malabo using
SimPowerSystems toolbox. In order to validate the simulation
model, a simulation case is constructed and implemented, and the
simulation results show that the simulation model of the twelvephase synchronous machine is usable.
Keywords-twelve-phase synchronous
eletric propulsion; model validation
I.
machine;
modeling;
INTRODUCTION
Electric propulsion is now gaining more and more interests
in both civil and military ships. Ships with Electric propulsion
will be powered through a combined propulsion and electrical
power distribution system. This revolutionary change will
bring a lot of benefits for future ships [1].
1) Unlocking of propulsion power, releasing the prime
movers from ship propulsion;
2) Lower fuel cost, significant fuel savings can be realized
through the use of electric propulsion;
3) Reducing manpower cost, reducing the amount of
installed equipment reduces manpower requirements and
therefore cost;
4) Design flexibility, reduced number of prime movers
and auxiliary equipment brings an enormous amount of ship
volume;
5) Improve the survivability and reliability.
Therefore, significant effort is made to develop electric
propulsion system all over the world.
Electric propulsion presents important opportunities for
overall ship improvement, but they are accompanied by
unique challenges as well [2], such as increased system
complexity, increased construction cost, increased requirement
on components’ power density. One of the major challenges is
the choice of the propulsion motor. Multiphase machine has
many inherent advantages including ability to obtain high
power in lower voltage level, fault tolerance, reducing the
This paper is funded by the International Exchange Program of Harbin
Engineering University for Innovation-oriented Talents Cultivation
978-1-4577-0321-8/11/$26.00 ©2011 IEEE
2131
torque ripple and so on, which make it be a proper choice for
electric propulsion systems.
The roots of multiphase variable speed drives can be traced
back to the late 1960s. The pace of research started
accelerating in the second half of the 1990s, predominantly
due to the developments in the area of electric ship propulsion,
which remains nowadays one of the main application areas for
multiphase variable-speed drives. In view of the development
of manufacturing twelve-phase synchronous machine is
chosen to be the research object in this paper. In [3] Zhou
presents the vector control of the twelve-phase synchronous
machine, but the mathematical model given is just a part. Lin
gives the whole mathematical model of the twelve-phase
synchronous machine [4], but the model is not validated.
In this paper, mathematical model of twelve-phase
synchronous machine with four Y-connected three-phase
symmetrical windings displaced in turn by 15 degree is
obtained by deduction. Then simulation model of the twelvephase synchronous machine is established using
SimPowerSystems toolbox in Matlab. A simulation case is
implemented to validate the simulation model of the twelvephase synchronous machine.
II.
MATHEMATICAL MODEL
The twelve-phase synchronous machine in this paper has
four Y-connected three-phase symmetrical windings displaced
in turn by 15 degree. The stator winding arrangement [5] of
the twelve-phase synchronous machine is shown in Fig. 1.
With some assumptions that the four three-phase windings are
identical to each other and the structure of the rotor is
symmetrical in d-q axis. Another assumption, also leading to a
negligible error, is to set the permeability of the iron core to
in¿nity and to ignore the parasitic interaction between the
different winding systems.
In order to model the machine, the first step is to transform
the stator variables to a reference frame by Park’s
transformation. In this paper, the rotor reference frame is
chosen to be the reference frame [6]. Use the subscript “i” to
represent four windings respectively. So we have: i=1, 2, 3, 4.
The transformation equation can be written as:
K S = [ K1 , K 2 , K 3 , K 4 ]
(1)
­id 1 = id 2 = id 3 = id 4
®
¯iq1 = iq 2 = iq 3 = iq 4
(5)
­λd 1 = λd 2 = λd 3 = λd 4
®
¯λq1 = λq 2 = λq 3 = λq 4
(6)
Where “i” stands for the current and Ȝ represents the flux
linkage.
All the variables are transformed to the rotor reference
frame by using the transformation equation, We can obtain the
voltage equations in d-q axis:
­udi = − rs idi − ωr λqi + pλdi
°u = − r i + ω λ + pλ
s qi
r di
qi
°° qi
®u fd = rfd i fd + pλ fd
°u = r i + pλ
kd
° kd kd kd
°̄ukq = rkq ikq + pλkq
Figure 1. Stator winding arrangement of twelve-phase synchronous machine
is:
For the “i”th winding, the voltage in rotor reference frame
Where
ªu Ai º
ªudi º
« »
=
K
i u Bi
«uqi »
«u »
¬ ¼
¬ Bi ¼
π
ª
« cos[θ − ( i − 1) 12 ]
«
2
π
2π
K i = « cos[θ −
− ( i − 1) ]
«
3
12
3
«
π
2π
« cos[θ +
− ( i − 1) ]
3
12
¬
Where rs is the stator resistance, Ȧe is the rotor electrical
angular velocity, the subscript “fd”, “kd” and “kq” denote the
field winding and the d-q axis damper windings respectively.
And the expressions for the flux linkages are:
(2)
º
»
12
»
π
− ( i − 1) ] »
12 »
»
π
− ( i − 1) ] »
12 ¼
− sin[θ − ( i − 1)
− sin[θ −
− sin[θ +
2π
3
2π
3
π
T
­ λdi = − Lls idi + Lmd ( −4idi + i fd + ikd )
°
° λqi = − Lls iqi + Lmq ( −4iqi + ikq )
°
® λfd = Llfd i fd + Lmd ( −4idi + i fd + ikd )
°λ = L i + L ( −4 i + i + i )
lkd kd
md
di
fd
kd
° kd
°̄ λkq = − Llkq ikq + Lmq ( −4iqi + ikq )
]
The voltage of each stator winding can be written as:
π
­
°u Ai = 2U cos(ωt − (i − 1) 12 )
°
2π
π
°
®u Bi = 2U cos(ωt − − (i − 1) )
3
12
°
°u = 2U cos(ωt + 2π − (i − 1) π )
°̄ Ci
3
12
(8)
Where the Lls is the stator leakage inductance and Lmd and
Lmq are the magnetizing inductances of the stator windings in
d-q axis, respectively. Llfd and Llkd are the leakage inductance
of the field winding and the damper winding in d axis
respectively. Llkq is the leakage inductance of the damper
winding in q axis.
(3)
Then we can obtain the expression of the electromagnetic
torque:
Applying (3) into (2) we have the relation of the voltage in
d-q axis:
­ud 1 = ud 2 = ud 3 = ud 4
®
¯uq1 = uq 2 = uq 3 = uq 4
(7)
3
2
p
2
4
Te = ( )( ) ¦ (λdi iqi − λqi idi )
i =1
(9)
The torque and the rotor speed are relatived by:
(4)
2 d ωr
+ Tl
p dt
Te = − J ( )
Similarly, we can obtain the currents and flux linkages
relations of four windings in d-q axis:
2132
(10)
Where P is the pole pairs of the machine, J is the inertia,
and Tl is the load torque on the shaft.
III.
SIMULATION MODEL
It is often convenient to express machine parameters and
variables as per unit quantities. In order to obtain the
simulation model of the twelve-phase synchronous machine,
the equations of the machine are per unitized, and then the
equations convenient for simulating the twelve-phase
synchronous machine are established by solving the flux
linkage per second equations for the current. The simulation
model of the machine can be expressed as follows:
­
rs
ωb
°ψ di = p [ u di + ω rψ qi + L (ψ md − ψ di )]
ls
°
rs
ωb
°
°ψ qi = p [ u qi − ω rψ di + L (ψ mq − ψ qi )]
ls
°
r
°
ωb
fd
[ u fd +
(ψ md − ψ fd )]
®ψ fd =
p
L
lfd
°
°
r
ω
°ψ kd = b [u kd + kd (ψ md − ψ kd )]
p
Llkd
°
°
r
ωb
[ u kq + kd (ψ mq − ψ kq )]
°ψ kq =
p
Llkd
¯
(11)
­
1 4
°°Te = 4 i¦=1 ( iqiψ di − idiψ qi )
®
°ω r = − ω b (Te − Tl )
2 Hp
°̄
(14)
Where H is the inertia constant, Ȧb is the base electrical
angular velocity, p is the differential operator.
According to the equations convenient for simulating the
machine, the simulation model of the twelve-phase
synchronous machine is constructed in Matlab software using
Simpowersystems toolbox. Fig. 2 shows the configuration of
the simulation model of the twelve-phase synchronous
machine, where the controlled-current source blocks and the
voltage measurement blocks are used in the “interface”
subsystem to interface the Simulink components with the
Simpowersystems components of the simulation model [7].
Where p is the differential operator, ȥ denotes the flux
linkage per second, and ȥmdand ȥmqare the magnetizing flux
linkage per second of d-axis and q-axis respectively. The
current can be expressed as:
­
1
(ψ di − ψ md )
°idi = −
L
°
ls
°
1
(ψ qi − ψ mq )
°iqi = −
L
ls
°
°
1
(ψ fd − ψ md )
®i fd = −
L
lfd
°
°
1
(ψ kd − ψ md )
°ikd = −
Llkd
°
°
1
(ψ kq − ψ mq )
°ikq = −
L
lkq
¯
(12)
Figure 2. Configuration of the simulation model
IV.
When constructing and using a simulation model, it is
important to consider its validity and how the validity can be
evaluated. In this section, a simulation case is configured and
implemented to validate the simulation model.
In the above equations, we have:
­
4ψ di ψ fd
ψ
+
+ kd )
°ψ m d = L a d (
L
L
L lkd
ls
lfd
°
°
4ψ q i ψ kq
)
+
°ψ m q = L aq (
L ls
L lkq
°
®
° L = ( 1 + 4 + 1 + 1 ) −1
° ad
Lm d
L ls
L lfd
L lkd
°
° L = ( 1 + 4 + 1 ) −1
° ad
Lm q
L ls
L lkq
¯
MODEL VALIDATION
This case investigates the starting of the synchronous
machine. The configuration of the simulation case is shown in
Fig. 3. The machine is powered by four three-phase
programmable voltage source blocks. The rated power of the
machine in the simulation is 45MW, other parameters we have
p=6, UN=3300V, fN=18Hz, H=0.8. The simulation results of
the rotor electrical angular velocity, the electromagnetic
torque and the current in q-axis are shown in Fig. 4, Fig. 5 and
Fig. 6 respectively.
(13)
The electromagnetic torque equation and the relation
between the torque and rotor speed are:
2133
From these figures, it can be observed that the simulation
results accord with the actual, which proves the validity of the
simulation model.
6
4
q - a x is c u r re n t / p u
2
0
-2
-4
-6
-8
0
2
4
6
8
10
12
14
16
18
t/s
20
Figure 6. Simulation results of the current in q-axis
Figure 3. Configuration of the simulation case
V.
In this paper, mathematical model of a twelve-phase
synchronous machine with four Y-connected three-phase
symmetrical windings displaced in turn by 15 degree is
obtained. By using Simpowersystems toolbox of Matlab
software, the simulation model of the twelve-phase machine is
established. In order to validate the simulation model, a
simulation case of the twelve-phase synchronous machine is
constructed and implemented. The simulation results show
that the simulation model of the twelve-phase synchronous
machine is usable.
1.2
1
0.8
a n g u l a r v e lo c it y / p u
CONCLUSION
0.6
0.4
0.2
ACKNOWLEDGMENT
0
-0.2
0
2
4
6
8
10
12
14
16
18
This paper is funded by the International Exchange
Program of Harbin Engineering University for Innovationoriented Talents Cultivation.
20
t/s
Figure 4. Simulation results of the rotor electrical angular velocity
REFERENCES
3
[1]
T o rq u e / p u
2
1
[2]
0
[3]
-1
[4]
-2
[5]
-3
0
2
4
6
8
10
12
14
16
18
20
t/s
[6]
Figure 5. Simulation results of the electromagnetic torque
[7]
2134
M. Benatmane, LCDR T. McCoy, T. Dalton and T.L. Cooper, “Electric
power generation and propulsion motor development for U.S. Navy
surface ships” Proceedings All Electric Ship: Developing Benefits for
Maritime Applications, The Institute of Marine Engineers, (IMarE)
Conference, Vol. 110, (2), pp 53-61, 29-30 September 1998.
Kyle Schmitt, “Modeling and Simulation of an All-Electric Ship in
Random Seas” Master thesis of MIT, 2010.
Zhou Dejia, Zhao Zhengming, Wang Shanming and CHAI Jianyun,
“Vector Control for a Twelve-Phase Synchronous Motor” Proceeding of
International Conference on Electrical Machines and Systems 2007, Oct.
8~11, Seoul, Korea.
Lin Hua and Zou Yunping, “Mathematical Models of 12-phase
Synchronous Machine with Air Gap Flux-oriented Control” S&M
Electrical Machines. 2004.
Zhang Jingnan, Cong Wang and Dai Ke, “Research on a mathematical
model of a multi_phase Y-coil synchronous motor” Journal of Harbin
Engineering University, July 2007.
Paul C. Krause, Oleg Wasynczuk and Scott D. Sudhoff, Analysis Of
Electric Machinery And Drive Systems Second Edition. IEEE Press
Power Engineering Series 2002.
The Mathworks, Simulink User’s Guide. 2010, http://www.mathwor
ks.com/help/toolbox/simulink/.