Stability Analysis of a Brushless Doubly-Fed Machine under Closed Loop Scalar
Current Control
Izaskun Sarasola1
1
Javier Poza1
Estanis Oyarbide2
Miguel Ángel Rodríguez1
Faculty of Engineering, University of Mondragon, Loramendi 4, Aptdo. 23, 20500 Arrasate, Spain
isarasola@eps.mondragon.edu, jpoza@eps.mondragon.edu, marodriguez@eps.mondragon.edu
2
Aragón Institute for Engineering Research (I3A), University of Zaragoza, María de Luna 1, 50018 Zaragoza, Spain
eoyarbid@unizar.es
Abstract – This paper presents a stability analysis of the
Brushless Doubly-Fed Machine (BDFM) under Closed Loop
Scalar Current Control (CLSCC). The recently developed
unified reference-frame dq model leads to an invariant small
signal model reflecting the dynamic of the BDFM under
CLSCC. Theoretical stability analysis shows a stable behavior
along all the operation range. This result is confirmed by an
experimental BDFM set-up. It is concluded that CLSCC
becomes the simplest control alternative of any real BDFM
application.
I. NOMENCLATURE
Nr
p
ω
v
i
V
I
R
L
Lh
ρ
TL
Tem
J
D
θr
Kp
Ki
ψ
A
B
x
u
ϕ
τ
ωd
number of rotors bars
number of poles pair of the stator winding
angular speed
instantaneous voltage
instantaneous current
rms voltage value
rms current value
resistance
self inductance
coupling inductance
derivative symbol
load torque
electromagnetic torque
inertia coefficient of the mechanical system
friction coefficient of the mechanical system
rotor shaft displacement
regulator proportional coefficient
integration proportional coefficient
flux
state equation A matrix
state equation B matrix
state variables
state equation input vector
angular displacement of CW current
time constant
damping natural frequency
Subscripts
p, c, r
power winding, control winding, rotor
dq
PW voltage orientation reference frame
αβ
CW reference frame
0
steady state operation point
Superscripts
*
conjugate
⋅
for time derivatives
1-4244-0136-4/06/$20.00 '2006 IEEE
II. INTRODUCTION
Recent developments have revitalized research activity in
the area of doubly-fed machines [1]. The expression doublyfed applies generally to machines where electrical power can
be fed or extracted from two accessible three-phase
windings. The wound rotor induction machine is a good
example of that. Generally the stator winding (through which
most of the power flows) is connected directly to the grid and
the rotor winding is connected to a bi-directional power
converter. The power rating of the rotor winding, i.e. the
converter size, depends on the required speed range and the
reactive power requirements. This fact can be of especial
interest in systems where small speed variation range is
needed. This is the case of variable speed wind turbines and
adjustable speed drives, like pumps and fans. The main
problem is that the slip rings and wound rotor arrangement
render the rotor of a slip-ring doubly-fed machine more
vulnerable to faults than a cage induction machine. Among
other solutions, the use of the so-called Brushless DoublyFed Machine (BDFM) could overcome this problem.
The BDFM (which is also known as self-cascaded
machine) is composed of two three-phase windings in the
stator. One of them, the Power Winding (PW), is usually
branched to the power line, and the other one, the Control
Winding, (CW), is connected to a voltage source inverter
(VSI). A special rotor configuration leads to a cross-coupling
effect between the two stator windings, in such a way that the
VSI can control, through the CW, the instantaneous energyexchange of the PW [2;3].
The electromagnetic behavior of the BDFM was
exhaustively analyzed by Williamson et al in [3]. This paper
developed a generalized harmonic analysis of the BDFM,
obtaining an accurate mathematical model in the
synchronous steady-state operation mode. This work is a
powerful tool for the study of the BDFM, and it is especially
useful for static performance analysis as well as for machine
design tasks.
Contrary to most of standard electromagnetic machines,
the BDFM shows large unstable domains under Open Loop
Voltage Control (OLVC) strategy. This phenomenon has
been experimentally observed and theoretically explained.
The first theoretical stability analysis of the BDFM was
based on two wound-rotor induction machines with a
common shaft and cascaded rotors [4;5]. It is concluded that
an appropriate machine design may reduce but never remove
such unstable domains. Because of that, the BDFM must be
1527
preferably used under closed-loop control configuration [6].
Wallace et al developed a dynamic vector model of the
BDFM referred to the rotor’s shaft position [7]. This model
was first experimentally validated [8] and later on exploited
in the stability analysis of the BDFM [9].
The reference frame in which this model was written was
aligned with the rotor, implying a non-linear time-variant set
of dynamic equations. As these variations are periodic in
time, Wallace et al exploited the generalized theory of
Floquet, leading, after linearization, to a linear time-invariant
set of equations. This way small-signal stability analysis
could be performed.
Trying to simplify the analysis procedure, a simplest
vector model under a new generic dq reference frame was
presented [10], leading, in a straightforward way, to a timeinvariant small signal model. The resulting model is,
somehow, equivalent to those of classical AC machines, but
it is restricted to the nested-loop rotor type BDFM family.
Overcoming this restriction, a generalized “synchronous
reference frame” model was proposed by Roberts [11] for a
wide class of BDFM. This model has been derived in a
different way but, in the particular case of the nested-loop
rotor type BDFM, it becomes to be identical to the unified
reference frame model [10].
By means of the generic dq reference frame it becomes
easy to design and to analyze any control strategy, e.g. the
open loop voltage control ([11-13]), the scalar current control
[12], or the PW-flux oriented vector control [12].
This paper deals with the CLSCC. Though this control
does not offer the same dynamic behavior as the vector
control, it is robust and simple to implement, requiring only
some few sensors.
Traditionally, the stability margin of the CLSCC has been
obtained only by simulation or by experimental verification
[6]. Few years ago a theoretical stability study of the closedloop scalar current control has been carried out [12], but
unfortunately results are not experimentally validated.
The purpose of the work presented in this paper is to check
the validity of the theoretical stability analysis of the CLSCC
by an experimental BDFM set-up.
ωr =
ω p + ωc
(3)
p p + pc
But synchronous operation requires a minimum level of
CW current, as shown in Fig. 1. This figure draws the
possible steady state PW current values (or domains) in a idpiqp plane, (idp-active current and iqp-reactive current) [12]. This
current must match two operation constraints: the
electromagnetic torque (right-hand domains) and the CW
current level (elliptic domain). Fig. 1. shows four possible
operation points obtained from the intersection of four
different torque requirement domains with the elliptic-kind
CW steady current domain. As it can be observed, in this
particular case the CW current must be above 2.5Arms,
otherwise synchronism will be lost.
IV. CLOSED LOOP SCALAR CURRENT CONTROL
Once the synchronization procedure is achieved and
provided that the minimum required CW current is
maintained [6], the operation of the BDFM is governed by
equation (3). This way, rotor speed can be easily controlled
by an adequate frequency variation of the CW feeding
current. Based on this idea, a simple control scheme can be
implemented, see Fig. 2. As it can be observed, the
maximum required torque is used to set the CW current
Tem=1Nm
iqp
Tem=3 Nm
Tem=6 Nm
fc=-50 Hz
Tem=7.2 Nm
idp
|Ic|=2.5 A
Fig. 1.Theoretical steady-state operating domains of the PW
current under constant |Ic| and Tem
III. OPERATING PRINCIPLE
50 Hz
400 V
ωr TL
This work considers the simplest BDFM with a ‘nestedloop’ type rotor. The simplest configuration is obtained
selecting the number of symmetrical rotor loops (Nr) as:
N r = p p + pc
V=0÷400 V
f=-50÷50 Hz
(2)
In order to get the desired cross coupling effect, the
currents that both the PW and the CW induce at the rotor
bars must evolve with the same frequency. This operating
restriction leads to the so-called synchronous rotor speed,
which is equal to:
RECT
INV
(1)
Direct coupling phenomena between the two stator
windings must be prevented, so next supplementary
restriction must be adopted,
p p ≠ pc
BDFM
Qf
Vbus_ref
Converter
Control
vabc_c_ref
Machine Control
Γmax_ ref
ωr _ ref
f (Γmax_ref )
idC c _ ref
I c max ref
ωr _ ref ( p p + pc ) − ω p fc _ ref
2π
Current
references
idC c
PI
iqC c _ ref
iqC c
PI
vdC c _ ref
2→3
vqC c _ ref
θc
Fig. 2. Closed Loop Current Control
1528
Qf_ref
ρLhp − ω p Lhp 
0
0
v dp   R p + ρL p ω p L p
v   ω L
+
R
L
ρ
ω
ρLhp 
0
0
p p
p
p
p Lhp
 qp  
v dc  
− ϑc Lhc 
Rc + ρLc ϑc Lc
ρLhc
0
0
 =

ϑc Lc Rc + ρLc ϑc Lhc
ρLhc 
0
0
v qc  
v dr   ρLhp
− γ c Lhp
− γ c Lhc Rr + ρLr − γ c Lr 
ρLhc
  

ρLhp
γ c Lhc
ρLhc
γ c Lr Rr + ρLr 
v qr   γ c Lhp
amplitude reference, and the desired rotor speed defines the
CW current frequency reference.
Using these values the “current reference” block generates
instantaneous dq current references as well as the angular
position θc of the dcqc reference-frame. It has to be noted that
for control purposes we use a dcqc reference frame aligned
with the CW current instead of the above mentioned unified
generic dq reference frame. This way fast and precise
regulation behavior is obtained by conventional PI
controllers.
V. DYNAMIC NON-LINEAR MODEL
Next we are going to deal with the particular case of the
nested-loop rotor type BDFM with a single spire per nest. In
the case of multiple spires per nest an equivalent single-nest
system can be obtained by model reduction techniques, so
the next development can be easily extended to the family of
nested-loop rotor type BDFM. In order to get the simplest
possible model, the unified dq reference frame is aligned
with the PW voltage. Resulting BDFM voltage model is
shown in (4), [10], with ϑc=ωp-(pp+pc)ωr, γc=(ωp-pp ωr) and
ρ=d/dt.
The electromagnetic torque can be expressed as:
Tem =
(
)
(
3
3
p p Lhp − i dp i qr + i qp i dr + p c Lhc i dc i qr − i qc i dr
2
2
)
(5)
And the mechanical system dynamics is computed as
follows:
TL = Tem
dω r
−J
− Dω r
dt
(6)
The CW current controllers can be modeled as:
(
)
(
)
(7)
(
)
(
)
(8)
v dc c = K p id c c _ ref − id c c + K i ∫ i d c c _ ref − i d c c dt
v qc c = K p i qc c _ ref − i qc c + K i ∫ i qc c _ ref − i qc c dt
With dcqc the controller reference frame, aligned with the
CW current reference and located at θc from the static
reference. In order to get a unified description of the system,
R + ρL p
∆vdp   p
∆v   ω p Lp
 qp  
0
 ∆vdc  

 
0
 ∆vqc  = 
ρLhp
 ∆vdr  


γ c 0 Lhp
 ∆vqr  
 ∆T  − 3 p p Lhpiqr 0
 L  2
− ω p Lp
0
0
R p + ρL p
0
0
− γ c 0 Lhp
ρLhp
3
p p Lhpidr 0
2
0
Rc + ρLc
− ϑc 0 Lc
ρLhc
γ c 0 Lhc
3
pc Lhciqr 0
2
0
− ϑc 0 Lc
Rc + ρLc
− γ c 0 Lhc
ρLhc
3
− pc Lhcidr 0
2
idp 
i 
 qp 
idc 
 
iqc 
idr 
 
iqr 
(4)
(7) and (8) must be transformed from the local dcqc reference
frame to the unified dq reference frame. First, using
conventional Park’s transformation, equations are
transformed to the static αcβc reference frame:
xα c β c = e − jθ c x d c qc
(10)
Transformation from the static αcβc reference frame to the
synchronous dq PW reference frame is performed by [10]:
xdq = e
[
(
) ]
− j ω pt − p p + pc θ r
xα* c β c
(11)
Without loss of generality we can consider that the initial
position of the rotor and the angle between the PW and CW
reference axis are nulls. Using (11) it is straightforward to get
the new controller’s equations under the unified dq reference
frame:
ρv dc = ω T [v qc − K p e qc ] + K p ρedc + K i e dc
(12)
ρv qc = −ω T [v dc + K p edc ] + K p ρeqc + K i e qc
(13)
where, ωT=ωc+ωp-(pp+pc)ωr, edc=idc_ref-idc and eqc=iqc_ref-iqc. The
dq components of the PW line-voltage are vdp= 2 Vp and
vqp=0, whereas the CW current set-point remains as follows:
[
(
)
i dc _ ref = 2 I c _ ref sin ω c t + ω p t − p p + p c θ r + ϕ
[
(
)
]
i qc _ ref = 2 I c _ ref cos ω c t + ω p t − p p + p c θ r + ϕ
(14)
]
(15)
In steady-state synchronous operation [ωct+ωpt-(pp+pc)θr]
is zero, leading to constant dq CW current components. This
fact is the main advantage of the unified dq reference frame,
as it has been explained in the introduction.
VI. SMALL SIGNAL MODEL
A lineal model is required for stability analysis purposes.
As the dynamic model described in the previous section is
non-linear, a small signal model around a given equilibrium
point is computed. This is performed replacing all large
signal variables by their constant equilibrium component, x0,
and the new small signal ∆x variable (17).
ρLhp
ω p Lhp
ρLhc
ϑc 0 Lhc
Rr + ρLr
γ c 0 Lr
− ω p Lhp
ρLhp
− ϑc 0 Lhc
ρLhc
γ c 0 Lr
Rr + ρLr
3
( p p Lhpiqp 0 − pc Lhciqc0 ) 3 (− p p Lhpidp 0 + pc Lhcidc 0 )
2
2
1529
  ∆i 
  dp 
0
 ∆i
(Lciqc0 + Lhciqr 0 )( p p + pc )   ∆iqpdc 

− (Lcidc 0 + Lhcidr 0 )( p p + pc )  ⋅  ∆i 
(Lr iqr 0 + Lhpiqp 0 + Lhciqc 0 )p p   ∆iqcdr 

− (Lr idr 0 + Lhpidp 0 + Lhcidc 0 ) p p   ∆i 
  qr 
− ρJ − D
 ∆ωr 

0
(9)
 ρ∆v dc   ρK p + K i

=
 ρ∆v qc   ω T 0 K p
− ωT 0 K p
− ρK p − K i
ρK p + K i
− ωT 0 K p
ωT 0 K p
ρK p + K i
x = x 0 + ∆x
0
ωT 0 K p
− ωT 0 K p
0
(17)
The resulting small signal model of the machine is
presented in equation (9), with ϑc0=ωp-(pp+pc)ωr0 and γc0=(ωp ppωr0).
The non linear behavior of the cos/sin trigonometric
operators of controller’s reference equations (14)-(15) has to
be also linealized. They can be approximated by:
(
)
∆i dc _ ref ≈ −i qc 0 _ ref p p + p c ∆θ r
(
(18)
)
∆i qc _ ref ≈ i dc 0 _ ref p p + p c ∆θ r
(19)
And computing their derivatives we get:
ρ∆idc _ ref ≈ − iqc 0 _ ref ( p p + p c )ρ∆θ r
(20)
ρ∆i qc _ ref ≈ i dc 0 _ ref ( p p + p c )ρ∆θ r
(21)
Linealizing the controller’s model (12)-(13) and replacing
the small signal current reference values by (18) to (21), the
small signal model of the control is obtained (16), with
ωT0=[ωc+ωp-(pp+pc)ωr0].
Combining both machine’s and controller’s small signal
models, (9) and (16), into one equation we get the overall
system’s linear model:
x = Ax + Bu
(22)
where
[
x = ∆idp
[
∆iqp
u = ∆TL ∆u dp
∆idc
∆iqc
∆idr
∆iqr
∆vdc
∆vqc
∆ωr
∆θ r
]
T
]
T
(
(p
)[
)[
(
)]
)]
*
− p p + p c v qc 0 − K p i qc
0 − i qc 0
p
(
+ p c v dc 0 − K p i dc* 0 − i dc0
 ρ∆i dc 


 ρ∆i qc 
 ∆i 
  dc 
  ∆i qc 
 
∆v 
 dc 
 ∆v qc 


 ∆ω r 
system is then easily evaluated by inspection of the
eigenvalues of matrix A, i.e., the system’s poles.
VII. STABILITY ANALYSIS
Fig. 3 schematizes the stability analysis procedure. The
equilibrium point depends on the current reference, the rotor
speed, the load torque and the PW voltage. These variables
and steady state equations define equilibrium state variables
ip0, ic0, ir0 and vc0, required by the small signal model.
The small signal model has 10 conjugated complex poles,
each of which corresponds to a particular system: the power
winding, the control winding, the rotor, the mechanical
system or the controller.
The parameters of the BDFM involved in this analysis are
those of the prototype described in the next section. The
machine stability has been analyzed along all the speed range
(–50Hz to 50Hz) with TL=25Nm, |Vp(rms)|=220V (50Hz),
|Ic(rms)|=9.9A and D=0.1Kgm2/s. Fig. 4 shows the evolution of
the dominant poles, related to the mechanical system (p1,2)
and the controller (p3,4). The dominant pole is the mechanical
one, and it can be concluded that the overall system is stable
under closed loop scalar current control.
As the overall dynamic is mainly determined by the
mechanical system, it could be of interest the study of the
influence of different mechanical parameters. Fig. 5 shows
that the Closed Loop Scalar Current Control is stable for any
coherent inertia value, J.
VIII. BDFM PROTOTYPE
(23)
(24)
The small signal model requires the knowledge of the
equilibrium values, which can be obtained through the
steady-state model of the BDFM [10]. The stability of the
The goal of the prototype is to collect the maximum
experimental data needed in the validation of the theoretical
models and the control strategies, regardless of the
optimization of machine performance. The power ratings of
STABILITY ANALYSIS TOOL
|Vp|
|Ic|
ωr0
ΓL
Steady
state
model
p1,2
idqp0
Small
signal
model
idqc0
idqr0
vdqc0
D
p3,4
p5,6
p7,8
p9,10
(16)
poles
analysis
J
2
Fig. 3. Stability analysis tool for the BDFM under CLSCC
Fig. 4. Dominant poles (J=0.4 Kgm )
1530
Fig. 6. Test bench at the Power Electronics Laboratory of the Univeristy of
Mondragón
Fig. 5. Dominant pole evolution with J variations
the two stators windings are the same (220V/50Hz, 10A), in
such a way that the same machine can test two different PW
and CW configurations. One of the windings (the PW in our
tests) has two poles and the other (the CW for us) is
composed of six poles. There are four nests in the rotor, with
a single loop per nest (slots for additional two loops per nest
are available). Copper wire type coils have been employed in
the rotor loops.
The BDFM prototype is built around the core of an IEC180 frame four-pole wound rotor induction machine. The
core is 200mm long and the stator is composed of 36 slots.
The two stator windings have 23 turns per coil with a
2.75mm2 wire and each rotor coil is made up of 65 turns of a
1.77mm2 wire (rated for 10 A(rms)). The air gap is 0.6mm width.
Further constructional details can be found in [12].
The BDFM can operate from 0rpm to 2ωsyn. Saturation
problems are avoided by a conservative design of the
magnetic circuit [11]. Table I collects the most relevant
parameters of the prototype.
IV. EXPERIMENTAL RESULTS
The experimental test bench is located at the Power
Electronics Laboratory of the University of Mondragón, see
Fig. 6. The BDFM is coupled to a controlled reversible DC
motor, which emulates the mechanical behavior of the load.
The CW voltage is fed by a bi-directional IGBT-based power
converter. The PW is directly branched to the standard
European 400V-50Hz grid. The mechanical parameters are
J=0.4Kgm2 and D=0.1Kgm2/s. All control algorithms have
been implemented in a DSP-based Dspace DS1103 platform
with a sample time of 250µs.
Fig. 7 shows an example of the machine operation along a
wide speed reference variation. As we are dealing with an
open-loop control scheme a reference ramp limiter is
employed, minimizing this way the stable but fairly damped
oscillations.
In order to validate the stability analysis a pulse type load-
torque perturbation is applied to the machine’s shaft, which
leads to current, torque and speed oscillations. The first test
is carried out at N=450rpm, TL=4.2Nm and |Vp(rms)|=230V. As
speed is constant, CW voltage is kept at |Ic(rms)|=4.9A. The
load-torque perturbation is 7Nm depth and 0.5s long.
Resulting current transients show damped oscillations, see
Fig. 8. These current oscillations are transmitted to the
electromagnetic torque, which produces oscillations in the
rotor speed. Dynamic behavior of Fig. 8 can be characterized
by a time constant (τ) and a damped frequency (ωd). These
parameters are related to the dominant poles of the small
signal model (p1,2=-(1/τ)±jωd). Fig. 9 shows the real part of
this dominant pole for both experimental and theoretical
cases along all the operation range. Test parameters are:
TL=4.1Nm, |Vp(rms)|=220V (50Hz) and |Ic(rms)|=4.2 A.
As it can be observed the system is always stable.
Experimental and theoretical values match around |fc|≈25Hz.
When the feeding frequency is under this value (|fc|<25Hz)
theoretical system is more damped than the experimental
one. In the other hand, for high feeding frequencies
(|fc|>25Hz), the experimental damping is higher. This bias
could be originated by any frequency-dependent phenomena,
as iron losses: damping ratio depends on losses, but the
model assumes constant losses (identified around 25Hz),
which leads to the observed symmetrical deviation.
TABLE I
BDFM ELECTRICAL PARAMETERS
Resistance (Ω)
Self Inductance (mH)
Mutual Inductance (mH)
PW
Rp=1.732
Lp=714.8
Lhp=242.1
CW
Rc=1.079
Lc=121.7
Lhc=59.8
Rotor
Rr=0.473
Lr=132.6
Fig. 7. Speed reference variation and tracking
1531
XII. REFERENCES
[1]
[2]
[3]
[4]
[5]
Fig. 8. Amplitude of the stator currents (RMS) and rotor speed
(filtered values with flow pass=10Hz)
[6]
[7]
[8]
[9]
Fig. 9. Real part of dominant pole
[10]
X. CONCLUSIONS
A small signal model of the BDFM under scalar current
control has been proposed. This model can be useful in
stability margin analysis of BDFM drives.
Experimental damping data shows frequency-dependent
deviations from theoretical case. Improvements in the
modeling of iron-losses could probably minimize these
deviations.
Closed Loop Scalar Current Control renders the BDFM
behavior stable along all the operation range, which means
that it is the simplest control strategy for any practical BDFM
installation (contrary to the Open Loop Voltage Control).
This type of control is very simple, requires only a couple
of current sensors and it can be used on Adjustable Speed
Drives where high dynamic is not required, like pumps and
fans.
XI. ACKNOWLEDGMENT
[11]
[12]
[13]
This work has been partially supported by the programs:
Investigación Básica y Aplicada (PI 2003-11), Formación de
Investigadores
del
Departamento
de
Educación,
Universidades e Investigación from the Basque Government
and by ENE2005-09218-C02-00 from MCYT DGI/FEDER.
1532
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machines: classification and comparison," in 9th European
Conference on Power Electronics and Applications. EPE 2001
Graz, Austria: EPE Assoc, 2001, p. 17.
J. Poza, A. Foggia, E. Oyarbide, and D. Roye, "Brushless
doubly-fed machine model," in ICEM 2002. 15th International
Conference on Electrical Machines Brugges, Belgium:
Institute of Mechatronics and Information Systems, 2002, p. 6.
S. Williamson, A. C. Ferreira, and A. K. Wallace,
"Generalized theory of the brushless doubly-fed machine. Part
1: Analysis," IEE Proceedings: Electric Power Applications,
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