The Brushless Doubly-Fed Machine Vector Model in the rotor flux oriented reference frame

Abstract

The paper presents the vector model of the brushless doubly-fed machine (BDFM) in the rotor flux oriented reference frame. The rotor flux oriented reference frame is well known in the standard AC machines analysis and control. Similar benefits can be sought by employing this method for the BDFM. The vector model is implemented in MATLAB/SIMULINK to simulate the BDFM dynamic performance under different operating conditions. The predictions from the vector model are compared to those from the coupled circuit model in simulation. The results are shown for the cascade mode of operation.

Full text

The Brushless Doubly-Fed Machine Vector Model in
the Rotor Flux Oriented Reference Frame
(1)Farhad Barati,(1)Hashem Oraee,(2)Ehsan Abdi,(2)Shiyi Shao and (2)Richard McMahon
(1) Electrical Engineering Department, Sharif University of Technology, Tehran, Iran
(2) Electrical Engineering Division, University of Cambridge, Cambridge, UK
Corresponding author: barati@ee.sharif.edu
Abstract-The paper presents the vector model of the
Brushless Doubly-Fed Machine (BDFM) in the rotor flux
oriented reference frame. The rotor flux oriented reference
frame is well known in the standard AC machines analysis and
control. Similar benefits can be sought by employing this
method for the BDFM. The vector model is implemented in
MATLAB/SIMULINK to simulate the BDFM dynamic
performance under different operating conditions. The
predictions from the vector model are compared to those from
the coupled circuit model in simulation. The results are shown
for the cascade mode of operation.
Keywords: Brushless Doubly-Fed Machine, Vector Model,
Rotor Flux Oriented Reference Frame
NOMENCLATURE
111 ,, cba
ν
ν
ν
stator1 phase voltages
222 ,, cba
ν
ν
ν
stator2 phase voltages
21 ,ss RR phase resistances of stator1 and stator2
windings
omi RRR ,, inner, middle and outer loops resistances
21 ,lsls LL leakage inductances of stator1 and stator2
windings
21 ,ss LL self inductances of stator1 and stator2
windings
lolmli LLL ,, inner, middle and outer loops leakage
inductances
omi LLL ,, inner, middle and outer loops self
inductances
iomoim LLL ,, mutual inductances between loops of a nest
oommii MMM ,, mutual inductance between identical loops
of 2 nests
iomoim MMM ,, mutual inductance between non-identical
loops of 2 nests
omi rsrsrs MMM 111 ,,
mutual inductance between stator1 phase
winding and inner, middle and outer loops
in each nest
omi rsrsrs MMM 222 ,,
mutual inductance between stator2 phase
winding and inner, middle and outer loops
in each nest
BJ , rotor moment of inertia and friction
coefficient
p
dtd operator
21 ,PP pole pairs of stator1 and stator2 windings
rss
ϕ
θ
θ
,, 21 arbitrary functions of time in stator1,
stator2 and rotor transformations
rss ',, 21
ϕ
ω
ω
time-derivatives of rss
ϕ
θ
θ
,, 21
111 ,, sss IV
λ
stator1 voltage, current and flux in the
vector model
222 ,, sss IV
λ
stator2 voltage, current and flux in the
vector model
rrr IV
λ
,, rotor voltage, current and flux in the vector
model
dr
λ
real part of r
λ
{
}
{
}
imgreal , real and imaginary parts of a complex
number
*
Z
complex conjugate of
Z
t transpose of a matrix or vector
I. INTRODUCTION
The Brushless Doubly-Fed Machine (BDFM) has the
potential to be employed as a motor in Adjustable Speed
Drive applications (ASD). It can also be an alternative to the
Doubly-Fed Induction Generators (DIFG) in wind power
applications [1].
The BDFM consists of two independent non-mutually
coupled balanced three phase windings wound on the same
core in the stator and a special rotor which couples both
fields of the stator. The nested-loop type is the most well
known rotor for the BDFM [1, 2]. The nested loop rotor
consists of nests which are equally spaced around the
circumference. The number of nests is equal to the sum of the
stator windings pole pairs. Fig. 1 shows the nested loop rotor
for the prototype machine at Cambridge University. The
machine has 4/8 - pole stator windings. Therefore, there exist
6 nests in the rotor cage. In the Cambridge prototype, there
are 3 loops in each nest. All the loops on the rotor are
connected via a common end ring on one end of the rotor [2].
For the purpose of machine study, a suitable mathematical
model for the BDFM is required to describe the machine
k,(((
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dynamic and steady-state performance in different operating
conditions.
Fig.1: Nested loop rotor of Cambridge BDFM
Several research activities have been carried out on the
BDFM modeling and analysis. These include the derivation
of the machine mathematical model based on the real rotor
structure, development of d-q models for the machine, vector
representation of d-q models equations and analysis of the
machine performance in different modes of operation.
A team at Oregon State University developed a detailed
mathematical model for a prototype BDFM for the first time.
Based on this model, a d-q model was derived for the
machine. The d-q model was then employed for the machine
operation analysis [3, 4, 5]. Later, work was done at
Cambridge University on BDFM modeling and analysis by
Williamson [6, 7]. A significant contribution was then made
by Roberts [8]. He developed a generalized framework for a
coherent and rigorous derivation of models for a wide class
of BDFMs, of which machines with nested-loop design
rotors are a subset. This framework was used to derive
coupled-circuit, d-q axes, sequence components and then
equivalent circuit models for the class of machines. The
coherence between the different models allows parameters
calculated for the coupled-circuit model to provide
parameters values for the other models. Poza in [10]
developed a vector model for the BDFM with the nested loop
rotor with one loop per nest.
The authors have recently developed a vector model for
the BDFM [11]. This vector model is a generic model, it is
not specified to any reference frame, and is derived for a
machine with 3 loops per nest. The approach presented in
[11] can be easily generalized to machines with any number
of loops per nest.
This paper describes the generic vector model in the rotor
flux oriented reference frame. At first the generic vector
model equations are presented in section II which is then
followed by introducing the rotor flux orientation concept in
section III. The vector model is implemented in
MATLAB/Simulink and is employed to simulate the
machine operation in the cascade mode. The predictions
obtained from the vector model are compared to those from
the coupled circuit model and are shown in section IV. Since
the coupled circuit model for the BDFM has been
experimentally verified in [2, 8], it can be used as a
benchmark for verification of the vector model in simulation.
II. GENERIC VECTOR MODEL OF THE PROTOTYPE BDFM
The mathematical models of AC machines can be
converted to d-q or vector models by employing appropriate
transformations [12, 13]. For the first time, a d-q model was
developed for the BDFM by Li et al at Oregon State
University [3]. Poza in [10] also presented a d-q model for
their BDFM which has one loop per nest.
Based on the mathematical model derived for the BDFM
by Roberts and Abdi, the authors have developed a vector
model for the machine by utilising appropriate
transformations [11]. The vector model has 3 free parameters
associated with 3 transformations employed in the derivation
of the vector model equations for stator 1, stator 2 and the
rotor circuit. They are 21 ,ss
θ
θ
and r
ϕ
. Although any values
can be assigned to these parameters, depending on the
application of the model appropriate assignment of these free
parameters will simplify the model.
In this section, the vector model equations are presented in
the generic form. The free parameters of the model will be
specified appropriately in the next section. The equations for
the voltage, flux and torque are as follows:
111111 ssssss jpIRV
λωλ
−+= (1-a)
222222 ssssss jpIRV
λωλ
−+= (1-b)
r
r
j
r
p
r
I
r
r
r
V
λϕλ
'
2.0 −+== (1-c)
Where
{
}
o
R
m
R
i
Rdiag
r
r,,
=
is the rotor circuit resistance
matrix.
The above are the voltage equations for stator 1, stator 2
and the rotor circuit in the BDFM vector model. As it can be
seen from equation (1-c), the rotor circuit voltage vector is
set to zero. This is due to the fact that the rotor loops are
shorted in the nested loop rotor.
The flux equations of the vector model are:
r
j
rssslss IeMILL
η
λ
−
++= 11111 3)
2
3
( (1-d)
*
22222 3)
2
3
(r
j
rssslss IeMILL
γ
λ
++= (1-e)
*
2211 2
3
2
3
)( s
j
t
rss
j
t
rsrNNNr IeMIeMIML
γη
λ
++−= (1-f)
Where ],,[ 1111 o
rs
m
rs
i
rsrs MMMM = is the stator1-rotor mutual
inductance vector, ],,[ 2222 o
rs
m
rs
i
rsrs MMMM = is the stator2-
rotor mutual inductance vector,
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α
β
1
d
1
q
1s
θ
−
2
d
2
q
2s
θ
−
αβ
λ
r
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
+
+
=
olomoio
momlmim
ioimili
N
LLLL
LLLL
LLLL
L is the nest self inductance
matrix,
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
oomoio
mommim
ioimii
NN
MMM
MMM
MMM
M is the nest-nest mutual
inductance matrix, rsr
θ
θ
ϕ
η
22 1
−
−
=
and )(42 2
ξ
θ
θ
ϕ
γ
−++= rsr .
r
θ
is the rotor position angle and
ζ
is the displacement angle
between the two stator windings in the stator core.
The machine torque for a 4/8-pole BDFM can be calculated
from the vector model quantities as:
}..{18}..{.9 )
2
(
*
)
2
(
*
2211
γ
π
η
π
+− += j
srrs
j
srrse eIIrealMeIIrealMT (1-g)
rrlrre pTBJpT
θ
ω
ω
ω
=++= , (1-h)
where T
e and Tl are the machine and load torques
respectively and r
ω
is the rotor shaft speed.
The above equations form the BDFM vector model. This
vector model is an appropriate tool for the analysis of the
machine operation both in dynamic and steady state regimes
and for development of vector control systems for the
machine. In order to employ the vector model for the
machine study or control purposes, the free parameters of the
model must be specified appropriately.
A suitable approach for specifying the free parameters of the
model is to use the rotor flux oriented reference frame
concept. This approach has yielded satisfactory results when
utilised for standard AC machines.
The method is to choose the rotor free parameter in the
vector model, r
ϕ
, such that the rotor flux vector which is in
general a complex vector becomes real.
The selection of the stator1 and stator2 free parameters in the
vector model, 1
s
θ
and 2
s
θ
, is then performed by assuming
0==
γ
η
in the vector model.
III. ROTOR FLUX ORIENTED REFERENCE FRAME VECTOR
MODEL
The rotor flux oriented reference frame has been employed
for AC machines vector models yielding satisfactory results
in the analysis and control of these machines [12, 13].
Similar benefits may be sought by employing the same
approach for the BDFM.
Therefore, the free parameters of the vector model are
specified according to the rotor flux orientation concept. By
setting 0==
γ
η
then:
rrs
θ
ϕ
θ
22
1−= (2-a)
)(42
2
ζ
θ
ϕ
θ
−−−= rrs (2-b)
The remaining free parameter r
ϕ
is selected according to the
rotor flux orientation concept. The method is that the free
parameter of the rotor transformation is selected such that the
rotor flux vector which is generally a complex vector
becomes real. Therefore:
0
2.r
r
j
re
λλ
ϕ
= (2-c)
where ).(
3
16
1
)1(
3
2
0∑
=
−
=
k
kj
rkr e
π
λλ
(2-d)
rk
λ
is the flux vector of the rotor kth nest which consists of
the rotor loops fluxes. In the prototype BDFM, the nested
loop rotor has three loops in each nest which are called the
inner, middle and outer loops. Hence, rk
λ
has three elements
corresponding to the three loops in the kth nest.
If r
ϕ
is specified such that:
2
0r
r
λ
ϕ
∠
−= (2-e)
then the rotor flux vector becomes real, that is:
drrr
λλλ
== * (2-f)
The machine vector model equations in the rotor flux
oriented reference frame can be arranged by assuming
0
=
=
γ
η
in the vector model and drrr
λλλ
== *.
Fig.2: Reference frame representation of the vector transformations
In Fig.2 reference frame representation of the vector
transformations is shown. In this figure, 1
s
θ
and 2
s
θ
are
according to Eq. (2-a, b).
αβ
is assumed to be the stationary
reference frame. 1dq and 2dq are other two reference frames
shown in the figure. It can be shown that the stator1 and rotor
quantities in the vector model belong to the 1dq reference
frame while the stator2 vector quantities are in the 2dq
reference frame. It can also be seen from the figure that 1
d
axis is aligned with the
αβ
λ
r which is the rotor flux vector
with respect to the stationary reference frame. In other words,
1dq reference frame is the rotor flux oriented reference
frame.
The vector model in the rotor flux oriented reference frame
can be utilised for the machine simulation, analysis and
control. In order to employ it to simulate the machine
behavior, the followings should be performed:
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21
21
PP
sync +
+
=
ω
ω
ω
- Determining the machine inputs. The machine has a
mechanical input, Tl , and in general two
independent electrical inputs. Depending whether
the power supplies to the machine are voltage or
current source supplies, the electrical inputs may be
voltages, currents or a combination of those.
- Arranging the machine state equations. In general,
the machine states include rrss
θλλλ
,,, 21 and r
ω
. If
the machine is fed from two voltage source supplies,
then the full state equations are used. For a machine
with one current source and one voltage source
supplies, the stator flux corresponding to the
winding supplied with the current source supply will
be omitted from the machine states. If the machine
is fed from 2 current sources, both of the stator
fluxes will be removed from the machine states and
the rotor flux, rotor position and rotor speed would
make the machine state vector.
Based on the above, the machine vector model equations in
the rotor flux reference frame are implemented in
MATLAB/Simulink to simulate the machine behavior under
different operating conditions.
By employing the rotor flux oriented reference frame vector
model for the analysis of BDFM, the vector model quantities
such as voltages, currents and fluxes become constant when
the machine is in the steady state. Therefore, all the
derivatives in the vector model are equal to zero and a set of
algebraic equations is obtained.
The BDFM vector model is derived assuming linear
characteristic for the machine core and fundamental
components for the rotor–stator mutual inductances. Such
assumptions are made to simplify the derivation of the vector
model equations. In practice, the machine core characteristic
is of hysteresis type rather than linear and also there are
harmonic components in the rotor-stator mutual inductances.
However, in general, these effects will not be significant in
commercially available machines and under normal operating
conditions. Therefore, the vector model is expected to give
predictions which are reasonably close to practice.
By arranging the equations of the rotor flux oriented
reference frame vector model for the rotor flux vector, then:
)..().(
2
3
.).( 2211
11
ds
t
rsds
t
rsrdrrdr iMiMMNNLNrMNNLNrp +−=−+ −−
λλ
(3)
It can be seen from Eq. (3) that the d components of the
stator1 and stator2 currents determine the rotor flux and
therefore by controlling one or both of these, the rotor flux
can respectively be partially or fully controlled.
The prototype machine torque equation in the rotor flux
oriented reference frame vector model is:
)4()...(.).(
2
81
..).(18..).(9
12212
1
1
2
1
21
1
1
dsqsdsqs
t
rsrs
qsdrrsqsdrrse
iiiiMMNNLNM
iMNNLNMiMNNLNMT
+−−
−+−=
−
−−
λλ
In this equation, the machine torque is described based on the
rotor flux and stator1 and stator2 currents d and q
components.
Equations (3) and (4) form the basis of the control of the
rotor flux and the machine torque.
In the BDFM, one of the stator windings, i.e. the control
winding, is often supplied by a controlled supply, while the
other winding, i.e. the power winding, is connected directly
to the grid. Therefore, the control inputs in the machine
model are the d and q components of the control winding
currents (or voltages). These control inputs must be well
regulated in order to achieve a high performance control
system.
Based on the above discussions, it can be concluded that the
main advantage of the machine vector model is its simplicity
of implementation for the machine study and control while
retaining reasonable accuracy in predicting the machine
behavior.
IV. VECTOR MODEL PERFORMANCE STUDY IN SIMULATION
The BDFM has several modes of operation including the
doubly-fed synchronous mode, the cascade mode and the
simple induction mode. In the two latter modes, the machine
has asynchronous operation. In the doubly-fed mode, both
windings are supplied and the BDFM has a synchronous
speed which only depends on the frequencies of the power
supplies.
(5)
where 1
ω
and 2
ω
are the stator1 and stator2 angular
frequencies respectively. The synchronous mode is the main
operating mode of the BDFM.
In the cascade mode, one of the stator windings is short
circuited while the other supplied from a three phase
balanced source. This mode has benefits in the machine study
and operation. For example, the machine parameters may be
extracted by performing suitable curve fitting methods on the
results from the cascade operation [9]. Further, it is shown in
[14] that the cascade mode can be used in starting the
machine as a motor for certain applications.
In this section, the predictions from the vector model are
compared to those obtained from the coupled circuit model.
The machine is operated in the cascade mode. The coupled
circuit model for the BDFM has been verified against
experimental tests in, for example, [2, 8]. Therefore, it is
used as a benchmark for verifying the vector model presented
in this paper.
Two operating conditions are simulated including:
A) The 8-pole winding is excited at 220V, 50Hz and the 4-
pole winding is shorted;
B) The 4-pole winding is excited at 220V, 50Hz and the 8-
pole winding is shorted.
In both cases the machine is run at no load.
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(3-a)
(3-b)
(3-c)
(3-d)
Figure 3: Machine quantities in the 8-pole excitation cascade mode
a- rotor speed b- machine torque c- phase “a” current of stator1
d- phase “a” current of stator2
solid line: vector model , dotted line: coupled circuit model
(4-a)
(4-b)
(4-c)
(4-d)
Figure 4: Machine quantities in the 4-pole excitation cascade mode
a- rotor speed b- machine torque c- phase “a” current of stator1
d- phase “a” current of stator2
solid line: vector model , dotted line: coupled circuit model
k,(((
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A. 8-Pole Excitation
Figure 3 shows the machine quantities during the 8-pole
excitation cascade mode. It can be seen from Fig. (3-a, b) that
the machine reaches from standstill to its steady state in
about 0.5s. In Fig. (3-c), the 4-pole winding current is shown.
In steady state, the stator2 winding currents are balanced.
The phase “a” current of the stator2 winding is shown in Fig.
(3-d). The machine torque is settled to the load torque i.e.
‘zero’ in steady state.
B. 4-Pole Excitation
The machine quantities for the 4-pole excitation cascade
mode are shown in Fig. 4. Similar behavior to that of the 8-
pole excitation can be seen in the 4-pole excitation.
From Fig. (3) and (4), it can be seen that the vector model
and the coupled circuit model predictions are in close
agreement. In the results from the couple circuit model, high
frequency ripples can be seen in the machine currents and
torque. But the vector model results do not show these
effects. This is most likely due to the harmonic components
of the stator-rotor mutual inductances being neglected in the
vector model.
V. CONCLUSIONS
The rotor flux oriented reference frame vector model for
the BDFM is presented in this paper. The model is an
appropriate tool for the machine analysis and control. The
derived vector model equations have been presented and the
free parameters have been specified to represent the model in
the rotor flux reference frame. The resulting model has been
implemented in MATLAB/Simulink and simulation results
for the machine operation in the cascade mode are provided.
The results have been compared to the predictions from the
coupled circuit model and satisfactory agreement has been
achieved. Although the comparison is made for the cascade
mode, we expect similar behaviors of the models in the
synchronous mode, which is the main operating mode of the
machine, as well.
The vector model approach offers a simple method of
modelling the BDFM operation while giving good accuracy.
Further, it can be used in developing appropriate control
algorithms for the machine.
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