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Abstract— Substantial advantages of brushless doubly-fed induction machine (BDFIM) bring up it as an interesting alternative for conventional solutions. In order to better analyse of BDFIM performance, it is required to employ a more precise model of BDFIM. In this paper, an approach, which completely relies on dynamic modelling in arbitrary reference frame, is proposed to consider both iron loss and main flux saturation effects in two-axis model of BDFIM. In this regard, the dynamic and static inductances are derived by using the magnetizing flux curve. If the cross-saturation is ignored, the time derivative of magnetizing inductances will yield incorrect simulation results. Therefore, to increase the accuracy, the proposed model contains terms which describe the cross-saturation effects. Due to the close relation between torque and power, the torque expression is also achieved in the presence of iron loss and magnetic saturation using expression of electrical input power. The simulation and experimental results are finally given to show the performance of the proposed model.
Index Terms—Brushless doubly-fed induction machine, iron loss, magnetic saturation, two-axis model.
NOMENCLATURE
, 𝐼 , 𝜆 Voltage, current, flux vectors
𝑇𝑒 Electromagnetic torque
𝑃 Power
𝑅 Winding resistance
𝐿𝑙 Leakage inductance
𝐿𝑚 Magnetizing inductance
𝐿𝑀𝑝𝐷 𝐷-axis component of power winding (PW)
magnetizing inductance
𝐿𝑀𝑝𝑄 𝑄-axis component of PW magnetizing inductance
𝐿𝑀𝑐𝐷 𝐷-axis component of control winding (CW)
magnetizing inductance
𝐿𝑀𝑐𝑄 𝑄-axis component of CW magnetizing inductance
𝐿𝑝𝐷𝑄 Cross-coupling inductance between 𝐷- and 𝑄-
axes of PW
𝐿𝑐𝐷𝑄 Cross-coupling inductance between 𝐷- and 𝑄-
axes of CW
𝐿𝑝𝐷 𝐷- axis component of PW self-inductance
𝐿𝑝𝑄 𝑄- axis component of PW self-inductance
𝐿𝑐𝐷 𝐷- axis component of CW self-inductance
𝐿𝑐𝑄 𝑄- axis component of CW self-inductance
𝑝 Number of pole pair
𝑁𝑟 Number of rotor loops (nests)
𝑓 Frequency
𝜔𝑟 Synchronous rotor speed
𝜔𝑎 Arbitrary angular speed
𝜔𝑛 Natural synchronous speed
𝑠 Derivative operator
Subscripts
𝑝 , 𝑐, 𝑟 PW, CW, rotor
𝐷 , 𝑄 Arbitrary frame axis
𝑑 , 𝑞 Stationary frame axis
𝑚 Magnetizing
𝑐𝑢 Copper loss
𝑖𝑟𝑜𝑛 Iron loss
𝑠𝑙𝑙 Stray load loss
𝑓&𝑤 Friction and windage losses
I. INTRODUCTION
ynamic modelling is necessary for analyzing the
performance of brushless doubly-fed induction machine
(BDFIM); moreover, it is helpful to realize the control
techniques. The coupled-circuit model was first suggested by
Wallace et al. [1] which makes it possible to evaluate the
dynamic and the steady-state behavior of BDFIM. In order to
replace the position-dependent inductances with speed-
dependent ones, an approach was proposed in [2], which
transformed the coupled-circuit model into the two-axis form.
Considering the BDFIM as two interconnected induction
machines (IMs), an equivalent circuit was introduced in [3]
which is valid for all modes of operation. To reduce the order
of this model, a simplified model has been developed in [4],
excluding the iron loss effect. This model is a much better
approximation to represent the behavior of BDFIM compared
to the core model.
For the purpose of understanding the performance of BDFIM,
it is required to derive its precise dynamic model considering
the iron loss and magnetic saturation effect. They are possible
sources to deteriorate the electrical machine performance when
the close-loop control strategies are employed [5]. The
combined impact of flux saturation and iron loss has been
studied on operation of vector controlled IM by Levi et al. [6].
It has been shown detuned operation necessarily will occur if
the iron loss and parameter variations are neglected.
A novel modelling method has been introduced for BDFIM
in [7] based on magnetic equivalent circuit (MEC). The
operating characteristics are evaluated using the proposed
model considering saturation. The impact of magnetic
saturation and iron loss on the performance prediction of
BDFIM have been studied in [8] using time-stepping finite
element analysis. By providing the steady-state conditions, the
proposed finite-element model can be used for optimization of
BDFIM design. In [9], the steady-state model of the cascade
doubly-fed induction machine (CDFIM) is presented that
accounts for the magnetic saturation and iron loss. Using this
model, a general performance study of the CDFIM has been
Generalized Analysis of Brushless Doubly-Fed Induction Machine Taking Magnetic Saturation and Iron Loss into Account
Hamidreza Mosaddegh Hesar, Mohammad Ali Salahmanesh, Hossein Abootorabi Zarchi
D
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carried out. A dynamic saturated model is proposed for dual-
stator brushless doubly-fed induction generator (DS-BDFIG)
by Zeng et al. [10] and it is confirmed through experimental
tests which the proposed model is more accurate than the model
without saturation effect. In this research work, the measured
magnetizing inductances are updated in each cycle from fitting
formulas and then placed in the electrical equations of DS-
BDFIG.
According to literature review, although few research works
have been published to date on modelling of types of the
brushless doubly-fed machines taking the both iron loss and
flux saturation into account, this is the first time that they are
considered in electric equivalent circuit (EEC) model of
BDFIM. This paper attempts to fill this void by achieving the
following purposes:
1) The two-axis model is a simple but precise mathematical
method that is useful for studying the electrical machines.
Meanwhile, the arbitrary reference frame provides a direct
means to achieve the voltage equations in all other reference
frames. Accordingly, the dynamic model of the BDFIM is
proposed considering both PW and CW sides iron loss
resistances in the arbitrary reference frame. Furthermore, by
introducing the full nonlinear BDFIM model, the cross-
saturation effect is also considered. If the cross-saturation is
ignored, the time derivative of magnetizing inductances will
yield incorrect simulation results. Generally, there are two
approaches to consider the effect of saturation in the electrical
machines model. In first which is known as simplified model,
the saturation effect is taken into account by substituting a
nonlinear function of magnetizing inductance into the voltage
equations of the unsaturated model [10]. The second approach
is a full nonlinear model in which the time variation of
magnetizing inductance leads to different voltage equations in
comparison with the unsaturated model. Therefore, new terms
arise as dynamic cross-saturation effects in the full nonlinear
model which are neglected in the simplified form. The full
nonlinear model is obviously better than the simplified model
due to the benefits such as high accuracy [11, 12] and fast
response [13].
2) The torque expression is derived in the presence of iron
loss and magnetic saturation using expression of electrical input
power. This power includes the power that is being converted
to mechanical energy, the power that is stored in the field of
machine, and the power for losses. It should be noted that the
various expressions for the electromagnetic torque will not be
derived for the saturated machine, since saturation does not
introduce new terms into the torque expression. Of course, as a
result of saturation, the saturation-dependent machine
parameters which are present in the different expressions for the
torque, e.g. the magnetizing inductance, or the rotor self-
inductance or the stator self-inductance, will be different to
their unsaturated values and are variables which depend on the
machine currents.
3) Due to the complex structure of BDFIM, the iron loss
resistance cannot be determined using the conventional
approaches proposed for standard IM. In order to identify the
BDFIM iron loss resistance, an approach is presented in which
the speed-dependent iron loss resistances are obtained through
some effective experiments. The acceptable accuracy of this
method is validated by comparing the calculated iron loss
resistances of BDFIM to the measured data.
This paper is organized as follows; in Section II the
dynamic modelling principle of BDFIM is introduced which
includes the magnetic saturation and iron loss modelling, as
well as derivation of two-axis model of BDFIM in the arbitrary
reference frame. In Section III the performance of proposed
model is confirmed by simulation and experiments. Section IV
contains the concluding remarks. The torque expression
achieved using active input power is also given in Appendix.
II. MODELLING OF BDFIM
A. Two-Axis Electrical Equations
The two-axis equations of an ideal BDFIM in the arbitrary
reference frame are: pQampDpDlppDppD dtddtdiLiRV (1)
pDampQpQlppQppQ dtddtdiLiRV (2)
cQrramcDcDlccDccD NdtddtdiLiRV (3)
cDrramcQcQlccQccQ NdtddtdiLiRV (4)
rQrpa
mcDmpDrDlrrDrrD
p
dtddtddtdiLiRV
(5)
rDrpa
mcQmpQrQlrrQrrQ
p
dtddtddtdiLiRV
(6)
mpDpDlppD iL , mpQpQlppQ iL (7)
mcDcDlccD iL , mcQcQlccQ iL (8)
mcDmpDrDlrrD iL , mcQmpQrQlrrQ iL (9)
where 𝜆𝑚𝑝𝐷,𝑄 = 𝐿𝑚𝑝𝑖𝑚𝑝𝐷,𝑄 and 𝜆𝑚𝑐𝐷,𝑄 = 𝐿𝑚𝑐𝑖𝑚𝑐𝐷,𝑄 are the two-
axis components of power winding (PW) and control winding
(CW) magnetizing flux linkages.
The orthogonal 𝐷 − 𝑄 axis equivalent circuit of the BDFIM
obtaining from (1)-(9) has been depicted in Fig. 1. This circuit
is valid for all operation modes of BDFIM including
synchronous mode. In this model, the magnetizing inductances
are constant and the iron loss resistances has not been included.
Later in this paper, the BDFIM model is developed taking into
consideration the iron loss and magnetic saturation.
Fig. 1. The 𝐷 − 𝑄 axis dynamic model of BDFIM in arbitrary reference frame
B. BDFIM Model Including Magnetic Saturation
In this section, the effect of iron saturation on the electrical
equations are discussed. As a result of saturation, the
magnetizing inductances are not constant, and vary with
saturation. The variation of the magnetizing inductances will be
incorporated into the voltage equations. In the two-axis model
cRlrLpDi lpL
lcLpRrR
rDi
pDv
pQa rQrpa p )( cQrra N )( cDi
cDv
mpDi mcDi
cRlrLlpL
lcLpRrR
pQv
pQi pDa
mpQi rQi
rDrpa p )(
mcQi
cDrra N )(
cQv
cQi
mpL
mpL
mcL
mcL
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introduced in (1)-(6), the saturation effect is neglected and
parameters such as magnetizing inductances, stator and rotor
self-inductances are considered constant. Tacking the magnetic
saturation into account, the voltage equations considerably
differ from those obtained in the ideal form. In order to derive
the dynamic electrical equations, we start with the PW-side.
There is similar procedure for the CW. It is required to compute
the first time derivative of 𝐷- and 𝑄-axis components of PW
magnetizing flux as: dtdLidtdiLdtd mpmpDmpDmpmpD (10)
dtdLidtdiLdtd mpmpQmpQmpmpQ (11)
In the saturation conditions, the first time derivative of
magnetizing inductance can be calculated as follows [14]:
dtidiLLdtdL mpmpmppmp
(12)
where 𝐿𝑝 and 𝐿𝑚𝑝 are called dynamic and static the time variant
inductances of PW, respectively. These inductances are defined
as follows:
mp
mpp
id
dL
, mp
mpmp
iL
(13)
In (12), the first time derivative of magnetizing current
amplitude (|𝑖 𝑚𝑝|) is also obtained as:
dtdidtdidtid mpQpmpDpmp sincos
(14)
where angle 𝜇𝑝 is the angular displacement of 𝑖 𝑚𝑝 with respect
to the direct axis of arbitrary reference frame. By substituting
(12) into (10) and (11) and through straightforward
computation, the first time derivative of 𝐷 and 𝑄 components
of PW magnetizing flux can be rewritten as: dtdiLdtdiLdtd mpQpDQmpDMpDmpD (15)
dtdiLdtdiLdtd mpDpDQmpQMpQmpQ (16)
In a similar way, we have for CW: dtdiLdtdiLdtd mcQcDQmcDMcDmcD (17)
dtdiLdtdiLdtd mcDcDQmcQMcQmcQ (18)
where 𝐿𝑀𝑝𝐷, 𝐿𝑀𝑝𝑄, 𝐿𝑀𝑐𝐷, 𝐿𝑀𝑐𝑄, 𝐿𝑝𝐷𝑄, and 𝐿𝑐𝐷𝑄 are achieved as:
pmpppMpD LLL 22 sincos (19)
pmpppMpQ LLL 22 cossin (20)
cmcccMcD LLL 22 sincos (21)
cmcccMcQ LLL 22 cossin (22)
ppmpppDQ LLL cossin (23)
ccmcccDQ LLL cossin (24)
As observed, (19)-(24) depend on both static and dynamic
inductances. Under linear magnetic conditions, the static
inductances are equal to the dynamic ones. As a result, it can be
simply attained 𝐿𝑀𝑝𝐷 = 𝐿𝑀𝑝𝑄 = 𝐿𝑚𝑝, 𝐿𝑀𝑐𝐷 = 𝐿𝑀𝑐𝑄 = 𝐿𝑚𝑐 ,
and 𝐿𝑝𝐷𝑄 = 𝐿𝑐𝐷𝑄 = 0 .
According to above equations, the two-axis saturated model
of BDFIM is obtained in arbitrary reference frame without rotor
equations as follows:
rQmpapQmplparQpQpDQ
rDMpDpDpDpDppD
iLiLLdtdidtdiL
dtdiLdtdiLiRV
(25)
rDmpapDmplparDpDpDQ
rQMpQpQpQpQppQ
iLiLLdtdidtdiL
dtdiLdtdiLiRV
(26)
rQmcrra
cQmclcrrarQcQcDQ
rDMcDcDcDcDccD
iLN
iLLNdtdidtdiL
dtdiLdtdiLiRV
.
(27)
rDmcrra
cDmclcrrarDcDcDQ
rQMcQcQcQcQccQ
iLN
iLLNdtdidtdiL
dtdiLdtdiLiRV
.
(28)
where 𝐿𝑝𝐷,𝑄 = 𝐿𝑙𝑝 + 𝐿𝑀𝑝𝐷,𝑄, and 𝐿𝑐𝐷,𝑄 = 𝐿𝑙𝑐 + 𝐿𝑀𝑐𝐷,𝑄.
After derivation of PW and CW voltage equations in
saturation condition, the same procedure is applied to achieve
the rotor voltage equations. The first time derivative of 𝐷 and
𝑄 components of rotor flux are obtained as:
dtdLidtdiL
dtdLidtdiLdtdiLdtd
mcmcDmcDmc
mpmpDmpDmprDlrrD
(29)
dtdLidtdiL
dtdLidtdiLdtdiLdtd
mcmcQmcQmc
mpmpQmpQmprQlrrQ
(30)
According to the described process from (10)-(18), (29) and
(30) are restated as:
dtdiLdtdiL
dtdiLdtdiLdtdiLdtd
mcQcDQmcDMcD
mpQpDQmpDMpDrDlrrD
(31)
dtdiLdtdiL
dtdiLdtdiLdtdiLdtd
mcDcDQmcQMcQ
mpDpDQmpQMpQrQlrrQ
(32)
In the next step, the rotor voltage equations are derived by
substituting (31) and (32) into (5) and (6) as:
cQmcpQmprpa
rQmcmplrrparQcQcDQ
rDcDMcDrQpQpDQ
rDpDMpDrDlrrDr
iLiLp
iLLLpdtdidtdiL
dtdidtdiLdtdidtdiL
dtdidtdiLdtdiLiR
.
.
0
(33)
cDmcpDmprpa
rDmcmplrrparDcDcDQ
rQcQMcQrDpDpDQ
rQpQMpQrQlrrQr
iLiLp
iLLLpdtdidtdiL
dtdidtdiLdtdidtdiL
dtdidtdiLdtdiLiR
.
.
0
(34)
The two-axis model of the BDFIM will attain in the arbitrary
reference frame if equations (33) and (34) are combined with
equations (25)-(28), as the matrix form (35) (see top of next
page). As observed in this matrix, not only all windings are
coupled owing to the cross-saturation, but also all inductances
are modified in saturated model. In order to get the magnetizing
curves, the experimental procedure described in [10] is fulfilled
and the results are used to find the best fit of PW and CW
magnetizing curves. The approximation of these curves are
obtained as an exponential function: 208928.1.27009.0
.9551414.0880384.0 mpimp e
(36)
56675.1.18937486.0.8291694.0879062.0 mcimc e (37)
Fig. 2 shows the magnetizing test results (star points) and the
fitted curves (solid lines). Considering (13), the static
inductances of PW and CW are obtained from (36) and (37),
respectively, as follows:
mpi
mp ieL mp
208928.1.27009.0.9551414.0880384.0 (38)
mci
mc ieL mc
56675.1.18937.0.8291694.0879062.0 (39)
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TrQrDcQcDpQpD
McQMpQlrrmcmplrrpa
cDQpDQMcQmcrpacDQMpQmprpapDQ
mcmplrrpa
cDQpDQMcDMpDlrrmcrpacDQMcDmprpapDQMpD
McQmcrracDQcQcmclcrracDQ
mcrracDQMcDmclcrracDQcDc
MpQmpapDQpQpmplpapDQ
mpapDQMpDmplpapDQpDp
TrQrDcQcDpQpD
iiiiii
sLLLRLLLp
sLLsLLpsLsLLpsL
LLLp
sLLsLLLRLpsLsLLpsLsL
sLLNsLsLRLLNsL
LNsLsLLLNsLsLR
sLLsLsLRLLsL
LsLsLLLsLsLR
VVVVVV
.00
00
00
00
(35)
The first order derivatives of (36) and (37) are also given the
dynamic inductances corresponding to PW and CW: 208928.1.27009.0208928.0 ..3118723.0 mpi
mpp eiL
(40)
56675.1.18937.056675.0 ..246016.0 mcimcc eiL
(41)
The dynamic and static inductances are illustrated in Fig. 3,
in terms of the magnetizing currents. It should be noted that 𝐿𝑚𝑝
(𝐿𝑚𝑐) is chord (static) slop and 𝐿𝑝 (𝐿𝑐) is tangent slop of PW
(CW) magnetizing curve. The tangent-slop inductance is not
zero under saturated conditions, in contrast to linear magnetic
condition in which it is equal to chord-slope inductance.
(a)
(b)
Fig. 2. Magnetizing curves of BDFIM; (a) PW, (b) CW
(a)
(b)
Fig. 3. Static and dynamic inductance curves; (a) PW, (b) CW
C. Full Nonlinear Model of BDFIM with Incorporating Iron Losses
The rotor speed of BDFIM is not close to the synchronous
speeds of stator windings. Hence, the rotor frequency is high
and the rotor iron loss is significant. Moreover, the existence of
two rotating flux densities in the stator increases the stator
hysteresis loss [15]. Therefore, the iron loss of BDFIM is higher
than the conventional IM with the same capacity. The complete
model of BDFIM is now derived including the iron losses. In
the stationary reference frame, the iron loss resistance can be
simply paralleled with the magnetizing branch in the dynamic
equivalent circuit [16]. Since we have previously defined
𝜆𝑚𝑝 = 𝐿𝑚𝑝𝑖𝑚𝑝, we can write KVL (Kirchhoff’s voltage law) as
follows:
dtiLdiRdtdiR smpmp
siipmp
siip
pp
(42)
where 𝑅𝑖𝑝 is the PW-side iron loss resistance. The “s”
superscript is to emphasize that the frame is stationary. In order
to convert any frame to another frame a general expression is: jexx 12
(43)
where 𝑥 1 and 𝑥 2 are space vectors in old and new frames,
respectively, and 𝜃 is difference between new and old frame
angles (𝜃 = 𝜃2 − 𝜃1). In a particular frame conversion, the
stationary reference frame is converted to an arbitrary rotating
frame. Hence, 𝜃1 = 0 and 𝜃2 = 𝜃𝑎, therefore 𝜃 = 𝜃𝑎. In this
regard, (42) is rewritten as follows:
dtdLei
iLejdtideL
dteiLdeiR
mpj
mp
mpmpj
ampj
mp
jmpmp
jiip
a
aa
aap
(44)
Considering (12), we get the following:
dtidiLLi
iLjdtidLiR
mpmpmppmp
mpmpampmpiip p
.
(45)
The D-axis component of (45) is determined as:
dtdidtdiiLLi
iLdtdiLiR
mpQpmpDpmpmppmpD
mpQmpampDmpiip pD
.sin.cos.
(46)
By knowing 𝑖𝑚𝑝𝐷 = |𝑖 𝑚𝑝|. cos 𝜇𝑝 and after some
mathematical manipulations, (46) is rewritten as follows:
mpQmpampQpDQmpDMpDiip iLdtdiLdtdiLiRpD
.. (47)
2 4 6 8 10 120
0.25
0.5
0.75
1
PW Magnetizing Current (A)
P
W M
ag
ne
tizin
g
Flu
x L
ink
ag
e (
Wb
)
Fitting Curve
Experiment
2 4 6 8 100
0.25
0.5
0.75
1
CW Magnetizing Current (A)
C
W M
ag
ne
tizin
g
Flu
x L
ink
ag
e (
Wb
)
0 2 4 6 8 10 120
0.25
0.5
0.75
1
PW Magnetizing Current (A)
PW
Ma
gn
eti
zin
g F
lux
(W
b)
Fitting Curve
Experiment
2 4 6 8 10 120
0.05
0.1
0.15
0.2
PW Magnetizing Current (A)
PW
In
du
cta
nc
e (
H)
Static Ind.
Dynamic Ind.
2 4 6 8 100
0.05
0.1
0.15
0.2
CW Magnetizing Current (A)
CW
In
du
cta
nc
e (
H)
Static Ind.
Dynamic Ind.
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The same procedure proposed for obtaining (47) is now
carried out to derive the following equations:
mpDmpampDpDQmpQMpQiip iLdtdiLdtdiLiRpQ
.. (48)
mcQmcrramcQcDQmcDMcDieqi iLNdtdiLdtdiLiRcD
.., (49)
mcDmcrramcDcDQmcQMcQieqi iLNdtdiLdtdiLiRcQ
.., (50)
where 𝑅𝑖,𝑒𝑞 is the CW-side iron loss resistance. Expressions of
(47) – (50) are then considered in the proposed equivalent
circuit. It should be noted that 𝑅𝑖,𝑒𝑞 is introduced for equivalent
resistance of CW iron loss (𝑃𝑖𝑟𝑜𝑛𝐶𝑊 ) and rotor iron loss (𝑃𝑖𝑟𝑜𝑛
𝑟 ).
These power loss components are obtained from power diagram
of BDFIM as:
sllwfrcu
PWiron
CWcu
PWculoadno
riron
CWiron
PPPPPPP
PP
&
(51)
where 𝑃𝑛𝑜−𝑙𝑜𝑎𝑑 is total no-load input power. According to (51),
the calculation process of resistance of CW-side iron loss (sum
of the CW and rotor iron losses) consists of four steps:
Step 1) The no-load power is calculated at the synchronous
mode for rotor speed varied in the range of ±30%
around the natural synchronous speed.
Step 2) Through DC measurements and considering skin effect
and temperature, the PW and the CW copper losses are
determined.
Step 3) The PW iron loss is obtained by proposed approach in
[17] for conventional IM. This step is fulfilled in
simple induction mode.
Step 4) The stray load loss, the friction and windage losses,
and the rotor copper loss are calculated according to
[18], [19], and [20], respectively.
Since the PW is fed by a constant frequency supply, we have
a constant resistance at the PW side. At the rated voltage and
frequency, the PW-side iron loss resistance (𝑅𝑖𝑝) is nearly 1025
Ω. The speed-dependent resistance of CW-side iron loss (𝑅𝑖,𝑒𝑞)
is achieved by supplying the PW through an autotransformer.
In order to keep the CW magnitude flux at a specific level, the
CW is controlled using 𝑉 𝑓⁄ (voltage-versus-frequency) control
approach. The CW-side iron loss has been reported in Table I
for the speed range of 350-650 rpm. The finite element analysis
(FEA) is also presented to verify the experimental results.
According to the proposed results in Table I, values of 𝑅𝑖,𝑒𝑞 are
therefore tabulated in Table II. The experimental and FEA
results have a close agreement which confirm the accuracy of
iron loss calculation algorithm. TABLE I
CW-SIDE IRON LOSS (EXPERIMENTAL AND FEA RESULTS) Rotor speed (rpm)
350 400 450 500 550 600 650
𝑷𝒊𝒓
𝒐𝒏
𝑪𝑾
+𝑷
𝒊𝒓𝒐𝒏
𝒓 (𝑾
)
Exp. 64 60 53 36 51 52.5 53.9
FEM 63 59.3 52.1 35.9 50.01 51.07 53.1
Relative error
between Exp.
and FEM (%)
1.58 1.18 1.72 0.27 1.98 2.8 1.5
TABLE II CW-SIDE IRON LOSS RESISTANCE OBTAINED BY EXPERIMENT AND FEA
Rotor speed (rpm)
350 400 450 500 550 600 650
𝑹𝒊,𝒆𝒒 (𝛀
) Exp. 506.3 540 611.3 900 635.3 617.1 604.5
FEM 514.3 546.4 621.9 902.5 647.7 634.4 610.2
Relative error
between Exp.
and FEM (%)
1.55 1.17 1.7 0.28 1.91 2.72 0.93
Based on the mentioned points in section B and C, the 𝐷 − 𝑄
full saturated model of BDFIM in arbitrary reference frame is
finally derived tacking iron loss into account. In this model
which is illustrated in Fig. 4, the orthogonal 𝐷 − 𝑄 axes are
coupled through cross-coupling inductances (𝐿𝑝𝐷𝑄 and 𝐿𝑐𝐷𝑄).
It is interesting to note that the state-space model will not
comprise clear terms that express the mutual influence of direct
axis on quadrature one and vice versa if the flux linkage are
selected as state variables.
Fig. 4. Two-axis full saturated model of BDFIM including iron loss effect in arbitrary reference frame
cRlrL
eqiR ,
pDi lpLlcLpR
rR
rDi
.pDv
pQa
MpDL
rQrpa p )(
McDL
. mcQrra N )(
cQrra N )( cDi
cDv
pDii mpDi mcDicDii
ipR
cRlrL
eqiR ,
lpLlcLpR
rR
.
mpDa .
ipR
pDQL cDQL
pQv
pQi pDa
pQii
mpQa
mpQi
MpQL
rQi
rDrpa p )(
mcQicQii
mcDrra N )(
cDrra N )(
cQv
cQi
McQL
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III. RESULTS AND DISCUSSION
The accuracy of proposed dynamic model is confirmed by
means of simulation and experiment. The main parameters of
BDFIM are shown in Table III. Due to the complex structure of
machine, most of its parameters cannot be determined using the
no-load and locked-rotor tests, unlike the standard IM. The
machine parameters except the iron loss resistances were
therefore extracted from D132s prototype BDFIM using the
procedure described in [3]. Accordingly, the parameters were
obtained from experimentally determined torque-speed
characteristics. Finding the best fit to experimental data was
done using non-linear least-squares optimization. The stator
winding resistances are also obtained from DC measurements.
The laboratory setup illustrated in Fig. 5 consists of: a voltage
source inverter with corresponding driver board, a sensor board
and a TMS320F28335 signal processor board designed with
Texas Instrument Co. The stator phase currents are measured
using Hall-effect current sensors and the line-to-line voltages
are detected by voltage sensors. To measure the torque, a
separated excitation DC generator with an external rheostat in
the armature terminal as a load is connected to the shaft of the
BDFIM. The DC generator specifications are given in Table IV.
TABLE III D132 PROTOTYPE BDFIM PARAMETERS
Symbol Parameter Value
𝑝𝑝/𝑝𝑐 PW/CW pole-pairs 2/4
𝑉𝑝 PW rated voltage (rms) 180 (V)
𝑉𝑐 CW rated voltage (rms) 180 (V)
𝐼𝑝 PW rated current (rms) 10 (A)
𝐼𝑐 CW rated current (rms) 4.5 (A)
𝑇𝑒 Rated torque 20 (N.m)
𝑅𝑝 PW resistance 1.3012 (Ω)
𝑅𝑐 CW resistance 3.7171 (Ω)
𝑅𝑟 Rotor resistance 1.1237 (Ω)
𝐿𝑙𝑝 PW leakage inductance 0.0047 (H)
𝐿𝑙𝑐 CW leakage inductance 0.0053 (H)
𝐿𝑙𝑟 Rotor leakage inductance 0.0206 (H)
TABLE IV DC GENERATOR SPECIFICATIONS
Symbol Parameter Value
𝑃𝐷𝐶 Power 4.8 (kW)
𝑉𝐷𝐶 Rated voltage 230 (V)
𝐼𝐷𝐶 Rated current 21 (A)
𝜔𝑟 Rotor speed 157 (𝑟𝑎𝑑 𝑠⁄ )
Fig. 5. Experimental setup
(a)
(b)
(c)
(d)
Fig. 6. BDFIM variables during free acceleration; (a) Rotor speed, (b) Electromagnetic torque, (c) PW current, (d) CW current.
To assess the dynamic performance of proposed model, the
machine variables including rotor speed, torque, as well as PW
and CW currents (Phase A) are obtained during free
acceleration from stall. If the IGBTs is turned off in the high-
DC Link
Capacitors
3 phase Inverter &
IGBT Driver Board
Voltage
Sensors
TMS
Board
Analog FilterCurrent
Sensors
Resistive
load
BDFIMDC
Generator
Encoder
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
150
300
450
600
time (s)
Ro
tor
sp
ee
d (
rpm
)
0 0.1 0.2 0.3 0.4 0.5-15
-10
-5
0
5
10
15
time (s)
CW
cu
rren
t (A
)
Without iron loss and
magnetic saturation
Proposed model
Experimental
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
7
14
21
28
35
time (s)
To
ruq
e (
N.m
)
0 0.1 0.2 0.3 0.4 0.5-15
-10
-5
0
5
10
15
time (s)
CW
cu
rren
t (A
)
Without iron loss and
magnetic saturation
Proposed model
Experimental
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
-10
0
10
20
time (s)
PW
cu
rre
nt
(A)
0.25 0.26 0.27 0.28 0.29 0.3-8
-4
0
4
8
0 0.1 0.2 0.3 0.4 0.5-15
-10
-5
0
5
10
15
time (s)
CW
cu
rren
t (A
)
Without iron loss and
magnetic saturation
Proposed model
Experimental
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
-7.5
0
7.5
15
time (s)
CW
cu
rre
nt
(A)
0 0.1 0.2 0.3 0.4 0.5-15
-10
-5
0
5
10
15
time (s)
CW
cu
rren
t (A
)
Without iron loss and
magnetic saturation
Proposed model
Experimental
0.2 0.21 0.22 0.23 0.24 0.25-6
-3
0
3
6
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side (or low-side) of machine-side inverter (MSI), the CW will
be short circuit. In this condition, the BDFIM operates similar
to the self-cascaded machine and it accelerates from zero to
natural synchronous speed. According to Fig. 6, the simulation
and experimental results verify the accuracy of the proposed
model. Hence, it can be understood that the BDFIM dynamic
model including iron loss and magnetic saturation is
sufficiently accurate for purpose of control.
For a specific operating point (magnetizing current), the PW
magnetizing flux and the static and dynamic inductances are
experimentally obtained (Figs. 2a and 3a). In order to more
assess the proposed dynamic model, among the experimental
voltage levels supplied to the PW, the voltages with amplitude
of 120 V, 200 V and 240 V are selected to apply to the PW in
the cascade mode of operation in the simulation. The accuracy
of values obtained for PW flux linkage, static inductance, and
dynamic inductance are validated through the PW magnetizing
curve (Fig. 2a) and PW inductances curve (Fig. 3a). As
observed in Fig. 7, for a same PW voltage, these variables
which are derived from proposed model are nearly equal to
those attained from no-load test.
For additional study, a three-phase terminal fault is applied
in steady-state by setting the power supply voltage to zero at
𝑡 = 0.5 𝑠. The system configuration for three-phase fault is
demonstrated in Fig. 8. In this scenario, the load torque is 50%
of rated torque and the machine is operating at synchronous
mode. The stepping of the terminal voltage to zero at the time
of fault causes transients in the both PW and CW currents.
These transients are damped before the fault is removed and
thereafter the machine currents equal to zero. After 25 cycles
(grid frequency ꞊ 50 Hz) the fault is cleared by closing the
circuit breaker and the source voltage is reapplied to the
BDFIM. Therefore, the machine reestablishes its original
operating condition. As shown in Fig. 9, the simulation results
of the proposed model are very close to the experimental ones.
The error between theoretical and experimental tests is due to
the stray load loss, and the mechanical losses which are not
considered in the BDFIM modelling.
(a)
(b)
Fig. 7. Analysis of saturation effect for three levels of PW voltage (Cascade mode); (a) PW voltage and magnetizing flux linkage, (b) Static and dynamic inductances of PW.
Fig. 8. System configuration for three-phase voltage fault (Synchronous mode).
(a)
(b)
Fig. 9. PW and CW currents during a terminal fault (Phase A); (a) PW current, (b) CW current. (𝑓𝑃𝑊 = 50 𝐻𝑧 and 𝑓𝐶𝑊 = 20 𝐻𝑧).
The accuracy of proposed model is also evaluated under
loading condition. In this regard, the dynamic behavior of
BDFIM during step load change is demonstrated in Fig. 10. As
observed, the load torque is first stepped up to 50% of rated
torque. The load torque is then stepped down and the BDFIM
returns to its initial operating condition. A little disagreement is
observed between the proposed model and experimental results.
We change the load torque standardly with connecting/
disconnecting the load resistor instead of connecting/
disconnecting the DC generator field supply which results in
the sluggish response.
It is worth noting that the transient behavior during the
terminal fault and loading has not been shown in Figs. 9 and 10.
0 0.2 0.4 0.6 0.8 1-400
-200
0
200
400
time (s)
PW
Mag
neti
zin
g f
lux l
inkag
e (
Wb
)
0 0.5 1 1.5 2
-200
-100
0
100
200
time (s)
PW
Vo
lta
ge
(V
)
0
0.5
1
PW
flu
x l
inkag
e (
Wb
)
1.5
0.621 Wb 0.755 Wb0.372 Wb
1
0
-1
-1.5
Voltage
Flux linkage
0 0.2 0.4 0.6 0.8 10
0.1
0.2
time (s)
PW
In
du
cta
nc
e (
H)
Static Ind.
Dynamic Ind.
0.172 H
0.112 H
0.144 H
0.061 H
0.181 H
0.191 H
BDFIM
AC
DC
DC
AC
Grid
Con
trol
Win
din
g
Machine-Side
Inverter
Grid-Side
Inverter
Frequency Converter
Power Winding
DC
Gen.
Circuit
Breaker
Auto-
transformer
0 0.25 0.5 0.75 1 1.25 1.5-20
-10
0
10
20
time (s)
PW
cu
rre
nt
(A)
0.25 0.26 0.27 0.28 0.29 0.3-10
-5
0
5
10
0 0.1 0.2 0.3 0.4 0.5-15
-10
-5
0
5
10
15
time (s)
CW
cu
rren
t (A
)
Without iron loss and
magnetic saturation
Proposed model
Experimental
0 0.25 0.5 0.75 1 1.25 1.5-20
-10
0
10
20
time (s)
CW
cu
rre
nt
(A)
0 0.1 0.2 0.3 0.4 0.5-15
-10
-5
0
5
10
15
time (s)
CW
cu
rren
t (A
)
Without iron loss and
magnetic saturation
Proposed model
Experimental
0.5 0.52 0.54 0.56 0.58 0.6-15
-10
-5
0
5
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Fig. 10. Dynamic performance of BDFIM during step change in load torque (Phase A); (a) PW current, (b) CW current. (𝑓𝑃𝑊 = 50 𝐻𝑧 and 𝑓𝐶𝑊 = 20 𝐻𝑧).
(a)
(b)
Fig. 11. Comparison of two magnetic saturation models during starting
transient, (a) 𝑑 − 𝑞 axes PW currents, (b) 𝑑 − 𝑞 axes CW currents
Comparison of two magnetic saturation models of BDFIM
has been given in Fig. 11. One of the models accounts for
dynamic cross-saturation effects (full nonlinear model),
whereas the other neglects them (simplified model). It is
evident that the full nonlinear model is more accurate than the
simplified one ignoring the cross-coupling inductances. The
difference between the models is visible through showing the
two-axis stator currents in stationary reference frame during
free acceleration.
It is worthwhile to provide a comparison of the proposed
model and existing methods in the literature for modeling the
some types of brushless doubly-fed machines. In this regard,
the most significant features of them have been summarized in
Table V.
IV. CONCLUSION
The two-axis theory of the three-phase electrical machines is
well-developed and it was used as a starting point for
development of the model of BDFIM tacking the iron loss and
saturation into account. In this regard, the two-axis full
saturated model of BDFIM was derived in arbitrary reference
frame including iron loss which covers the cross-saturation
effect. The experimental and simulation results for cascade and
synchronous modes were shown the accuracy of dynamic
model during free acceleration, three-phase fault, and step
change in the load torque. Moreover, an experimental process
was presented to determine the fundamental iron loss of
BDFIM and the equivalent iron loss resistances were obtained
from these data. According to the results, the proposed dynamic
model can be used to implement the various control strategies
of the BDFIM as a possible substitute for adjustable speed
drives.
APPENDIX
DERIVATION OF TORQUE EXPRESSION
The active input power can be written for the BDFIM in space
0 0.5 1 1.5 2 2.5 3-10
-5
0
5
10
PW
cu
rre
nt
(A)
time (s)
1.19 1.21 1.48 1.25-10
-5
0
5
10
Without iron loss and magnetic saturation Proposed model Experimental
0 0.5 1 1.5 2 2.5 3-5
0
5
time (s)
CW
cu
rre
nt
(A)
2.8 2.825 2.85 2.875 2.9
-3
0
3
(a)
(b)
0.2 0.21 0.22 0.23 0.24
-10
-5
0
5
10
0.25 0.26 0.27 0.28 0.29 0.3-8
-4
0
4
80 0.05 0.1 0.15 0.2 0.25 0.3 0.35
-10
0
10
20
time (s)
PW
cu
rre
nt
(A)
0.25 0.3 0.35 0.4 0.45
-5
0
5
time (s)
CW
cu
rren
t (A
)
Simplified model
Proposed model
(Full nonlinear model)
Experimental
0.25 0.26 0.27 0.28 0.29 0.3-8
-4
0
4
8
0.08 0.09 0.1 0.11 0.12-10
-5
0
5
10
15
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-20
-10
0
10
20
time (s)
CW
cu
rre
nt
(A)
0.25 0.3 0.35 0.4 0.45
-5
0
5
time (s)
CW
cu
rren
t (A
)
Simplified model
Proposed model
(Full nonlinear model)
Experimental
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TABLE V
COMPARISON OF MODELS PROPOSED IN THE LITERATURE
Ref. Type of
machine
Has iron loss
been considered?
Has magnetic saturation
been considered?
Type of
model
Dynamic or
steady-state
modelling?
Model accounting for
magnetic saturation
[1] BDFIM No No EEC1 Dynamic -
[2] BDFIM No No EEC Dynamic -
[3] BDFIM No No EEC Steady-state -
[7] BDFIM No Yes MEC2 - -
[8] BDFIM Yes Yes MEC - -
[9] CDFIM Yes Yes EEC Steady-state Simplified model
[10] DS-BDFIG No Yes EEC Dynamic Simplified model
[20] BDFIM Yes No EEC Steady-state -
Proposed model BDFIM Yes Yes EEC Dynamic Full nonlinear model
1 Electric equivalent circuit (EEC) 2 Magnetic equivalent circuit (MEC)
vector form as:
ccpp IVIVP
Re2
33 (52)
This active input power includes the power for losses, the
power that is stored in the three-phase field of the machine, and
the power that is being converted to mechanical energy. The
power stored in the field is constant in steady state (there is a
constant amplitude three-phase flux density waveform in the
air-gap of the machine), and therefore there is no energy stored
in this, after the initial transient. On the other hand, the
individual phases have reactive power transferred to and from
the machine. These reactive powers add instantaneously to be
zero over the three phases of the machine.
Applying KVL to loop between magnetizing and iron loss
branches at the PW (CW) side, we have respectively:
pDpQpQpDp impQimpDai
mpQi
mpDiip iii
dt
di
dt
dIR ._...
2
(53)
cDcQcQcDc imcQimcDrrai
mcQi
mcDieqi iiNi
dt
di
dt
dIR ._....
2,
(54)
In a similar way, KVL is written for the rotor loop:
0._....
..2
rDrQrQrDrrarQrQ
lrrDrD
lr
rQmcQmpQ
rDmcDmpD
rr
iiNidt
diLi
dt
diL
idt
d
dt
di
dt
d
dt
dIR
(55)
Substituting (53)-(55) into (52) and after a small amount of
manipulation, the expression of active input power is derived as
follows:
rDrQrQrDrpa
imcDimcQrracDcQcQcDrra
impDimpQapDpQpQpDa
rQrQ
lrrDrD
lrmcQmcQ
mcDmcD
cQcQ
lc
cDcD
lcmpQmpQ
mpDmpD
pQpQ
lppDpD
lp
rrieqiiipccpp
iip
iiNiiN
iiii
idt
diLi
dt
diLi
dt
di
dt
di
dt
diL
idt
diLi
dt
di
dt
di
dt
diLi
dt
diL
IRIRIRIRIRP
cQcD
pQpD
cp
...
......
....
22,
222
23
(56)
This expression includes following components:
Resistive losses ꞊
22
,
222
2
3rrieqiiipccpp IRIRIRIRIR
cp
(57)
Changes in Field energy ꞊
rQrQ
lrrDrD
lr
mcQmcQ
mcDmcD
cQcQ
lccDcD
lc
mpQmpQ
mpDmpD
pQpQ
lppDpD
lp
idt
diLi
dt
diL
idt
di
dt
di
dt
diLi
dt
diL
idt
di
dt
di
dt
diLi
dt
diL
2
3
(58)
Power converted to mechanical energy ꞊ rDrQrQrDp
imcDimcQrcDcQcQcDrr
iip
iiNiiNcQcD
..
....
(59)
The electromagnetic torque is therefore obtained from
rotational power (𝑃𝑟𝑜𝑡) as:
** .Im
2
3.Im
2
3rmcmccrmpmpp
r
rote IILpIILp
PT
(60)
REFERENCES
[1] A. K. Wallace, R. Spee, and H. K. Lauw, “Dynamic Modelling of
Brushless Doubly-Fed Machines,” in proc. IEEE Ind. Appl. Society Annu. Meeting, San Diego, CA, USA, 1989, pp. 329-334.
[2] R. Li, A. K. Wallace, and R. Spee, “Two-Axis Model Development of
Cage-Rotor Brushless Doubly-Fed Machines,” IEEE Trans. Energy Convers., vol. 6, no. 3, pp. 453–460, Sep. 1991.
[3] P. C. Roberts, R. A. McMahon, P. J. Tavner, J. M. Maciejowski and T. J.
Flack, “Equivalent Circuit for the Brushless Doubly-Fed Machine (BDFM) including Parameter Estimation and Experimental Verification,”
IEE Elect. Power Appl., vol. 152, no. 4, pp. 933-942, July 2005.
[4] S. Tohidi, “Analysis and Simplified Modelling of Brushless Doubly-Fed Induction Machine in Synchronous Mode of Operation”, IET Elect.
Power Appl., vol. 10, no. 2, pp. 110-116, Feb. 2016.
[5] E. Levi, and V. Vuckovic, “Rotor Flux Computation in Saturated Field-Oriented Induction Machines,” Electric Machines & Power Systems, vol.
21, no. 6, pp. 741-754, Oct. 1992.
[6] M. Sokola, and E. Levi, “Combined Impact of Iron Loss and Main Flux Saturation on Operation of Vector Controlled Induction Machines,” in
proc. Int. Conf. Power Electron. & Variable Speed Drives, Nottingham,
UK, 1996, pp. 36-41. [7] H. Gorginpour, H. Oraee and R. A. McMahon, “A Novel Modeling
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Hamidreza Mosaddegh Hesar received the
B.Sc. and M.Sc. degrees from Ferdowsi
University of Mashhad, Iran, in 2011 and
2014, respectively, from Ferdowsi University of Mashhad, Iran, where he is currently
working toward the Ph.D. degree.
His current interests and activities include control of high-performance drives, nonlinear
control, and modelling of electrical machines.
Mohammad Ali Salahmanesh received the
B.Sc. degree in electrical engineering from the Sajjad University of Technology, Iran, in 2017,
and the M.Sc. degree from Ferdowsi
University of Mashhad, Iran, in 2019 (both with honors).
His research interests and activities include control of electric drives, nonlinear control, modeling of electrical machines and power electronics.
Hossein Abootorabi Zarchi received the M.S. and Ph.D. degrees from the Isfahan University of Technology, Isfahan, Iran, in 2004 and 2010, respectively. He was a Visiting Ph.D. Student with the Control and Automation Group, Denmark Technical University, Denmark, from May 2009 to February 2010. He is currently an Assistant Professor in the Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.
His research interests include electrical machines, applied nonlinear control in electrical drives, and renewable energies.
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