Electric Power Systems Research 210 (2022) 108108
Contents lists available at ScienceDirect
Electric Power Systems Research
journal homepage: www.elsevier.com/locate/epsr
Short-circuit calculation method for unbalanced distribution networks with
doubly fed induction generators
Fan Xiao a, *, Yongjun Xia a, Kanjun Zhang a, Zhe Zhang b, Xianggen Yin b
a
b
Electric Power Research Institute of Hubei Electric Power Co., Ltd., Wuhan 430077, China;
State Key Laboratory of Advanced Electromagnetic Engineering and Technology (Huazhong University of Science and Technology), Wuhan 430074, China
A R T I C L E I N F O
A B S T R A C T
Keywords:
New energy source
Wind turbines
Photovoltaic power
Unbalanced distribution network
Short circuit calculation
Phase component
In an unbalanced distribution network, transforming three-phase unbalanced components into components with
positive, negative, and zero sequences is difficult. Moreover, the fault currents of doubly fed induction generators
(DFIGs) exhibit a complex nonlinear relationship. The fault characteristics of DFIGs with an activated crowbar
differ from those with a non-activated crowbar. Moreover, distinguishing between these two crowbar conditions
based on calculations is difficult. Therefore, the traditional symmetric component method is unsuitable for
unbalanced distribution networks with DFIGs owing to its low calculation accuracy. In this work, the equivalent
calculation phase component models of DFIGs are analyzed using low-voltage control strategies. In particular,
the activation criterion of crowbar protection in the time domain is proposed. A novel fault analysis method for
unbalanced distribution networks with DFIGs is also proposed. The action condition of crowbar protection is
assessed using the traversal method, and the coupling relationship among DFIGs is determined using an iterative
calculation method. Finally, the effectiveness of the proposed fault analysis method is verified by simulation.
1. Introduction
In power systems, accurate calculation of short-circuit currents is
vital for the selection of electrical equipment, calculation of relay pro­
tection settings, and operation as well as control of grids. Traditional
power grid fault analysis methods are primarily employed for symmet­
rical components. However, in unbalanced distribution networks, such
methods are incapable of decoupling three-phase asymmetrical com­
ponents into positive-component, negative-component, and zero se­
quences. More specifically, in unbalanced distribution networks, the use
of such methods can result in considerable errors between the calculated
and actual fault values. Moreover, the fault currents of different types of
new energy sources exhibit a complex nonlinear relationship, and it is
difficult to calculate the short-circuit currents in power grids. Further­
more, under grid fault conditions, the crowbar of doubly fed induction
generators (DFIGs) could either be activated or non-activated; therefore,
it is difficult to accurately distinguish fault conditions using traditional
fault analysis methods. An incorrect assessment of the crowbar condi­
tion leads to a significant deviation between the calculated and actual
results. Therefore, it can be preliminary deduced that the traditional
symmetrical component calculation method is unsuitable for computing
the fault values of unbalanced distribution networks with DFIGs.
Accurate fault calculation models of new energy power sources are
the bases of fault analysis in power grids. Accordingly, the fault calcu­
lation model of an inverter-interfaced distributed generator (IIDG) can
be considered equivalent to the current source model [1]. Fault calcu­
lation models used for DFIGs vary depending on the status of the
crowbar, i.e., whether the crowbar is activated or non-activated. Under
activated crowbar protection conditions, the fault calculation model for
DFIGs is equivalent to the asynchronous motor model proposed in [2].
Under non-activated crowbar protection conditions, the low-voltage
ride-through control strategies (LVRT) of DFIGs under symmetrical or
unsymmetrical fault conditions have been implemented to enhance the
safety of DFIGs [3–6]. Accordingly, the short-circuit current model of
the DFIG under symmetrical fault conditions was developed [7].
Moreover, the transient response of DFIG under unsymmetrical fault
conditions was also investigated [8]. Thus far, the short-circuit calcu­
lation model of DFIGs under unsymmetrical fault conditions has not
been formulated. In particular, there are considerable differences in the
fault currents of DFIGs under activated and non-activated crowbars;
however, no assessment method for distinguishing the DFIG fault cur­
rents between the two crowbars has been formulated.
For the fault analysis method, several studies have focused on the
* Corresponding author.
E-mail address: xiao103fan@163.com (F. Xiao).
https://doi.org/10.1016/j.epsr.2022.108108
Received 16 March 2021; Received in revised form 8 May 2022; Accepted 15 May 2022
Available online 20 May 2022
0378-7796/© 2022 Elsevier B.V. All rights reserved.
F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
A list of symbols
DFIG
IIDG
LVRT
RSC
IDFIG,1
γDG,1
i+
sd
i+
sq
δU,1
UDFIGre,1
UDFIGim,1
IDFIG,2
γDG,2
−
Isd
isq
δU,2
UDFIGre,2
UDFIGim,2
ZDFIG,1
ZDFIG,2
Lr σ
R ;r
Lm
Lsσ
s
i∗rf
R
′
double-fed induction generator
inverter interfaced distributed generator
low voltage ride through control
rotor-side converter grid-side converter
amplitude of the positive-sequence component in the fault
current of the DFIG
phase angle of the positive-sequence component in the
fault current of the DFIG
d-axis components of the stator current in (dq)+
synchronous rotating reference frames
q-axis components of the stator current in (dq)+
synchronous rotating reference frames
phase angle of positive-sequence component of the stator
voltage
real parts of the positive-sequence component in the stator
voltage
imaginary parts of the positive-sequence component in the
stator voltage
amplitude of the negative-sequence component in the fault
current of the DFIG
phase angle of the negative-sequence component in the
fault current of the DFIG
d-axis components of the stator current in (dq)synchronous rotating reference frames
d-axis components of the stator current in (dq)synchronous rotating reference frames
phase angle of negative-sequence component of the stator
voltage
real part of the negative-sequence component of the stator
voltage
imaginary part of the negative-sequence component of the
stator voltage
positive-sequence equivalent impedance in sequence
networks
negative-sequence equivalent impedance in sequence
networks
leakage reactance of the rotor
ω1
i∗r
udc0
C
t0
Pg
Pr
n
ZDFIGi,1
ZDFIGi.2
I˙DFIGi,a
I˙DFIGi,b
I˙DFIGi,c
I˙DFIGi,1
I˙DFIGi,2
Uk
Ik
UDGm
IDGm
h
Ik,a,sc
short-circuit current calculation method of power grids with new energy
sources. The IIDG in a power grid can be considered equivalent to a
model with variable impedance and constant voltage source in series,
PQ node, or PI node [9–11]. For power grids with IIDGs, a calculation
method has been proposed for the short-circuit current contribution of
current control inverter-based distributed generation sources [12].
Moreover, several improved iterative calculation methods based on
symmetrical component calculation have also been proposed [13–15].
However, existing research has not considered unbalanced distribution
networks and the activation of the DFIG crowbar. For power grids with
DFIGs, the short-circuit calculation model of DFIGs based on the voltage
of a grid-connected point is derived according to the LVRT strategy.
Moreover, short-circuit calculation methods based on symmetrical
component calculation have been proposed [16–17]. However, if the
power grid structure is unbalanced, the calculation results obtained via
traditional symmetrical component methods are most likely to deviate
significantly from the actual results. In [16], the feeder was modified to
be balanced in the symmetrical pre-fault state using line section pa­
rameters. In [17], the fault analysis of balanced power grids with
distributed generators was based on the symmetrical component
method. Existing fault calculation methods cannot satisfy the fault
analysis requirements of unbalanced distribution networks.
equivalent resistance of the rotor winding
mutual inductance between the stator and rotor
leakage reactance of the stator
slip
reference signal of the DFIG rotor current
resistance
synchronous angular velocity
setting value of the rotor current under the condition of
activated crowbar
rated voltage of the DC bus
capacitor of the DC bus
time of failure
output power of the GSC
active power of the RSC
number of grid nodes
positive-sequence equivalent impedance of DFIG at i − th
node under condition of activated crowbar
negative-sequence equivalent impedance of DFIG at i − th
node under the condition of activated crowbar
short-circuit current injected by the DFIG at i − th node
into the a-phase circuit
short-circuit current injected by the DFIG at i − th node
into the b-phase circuit
short-circuit current injected by the DFIG at i − th node
into the c-phase circuit
short-circuit current injected by the DFIG at i − th node in
the positive-sequence networks
short-circuit current injected by the DFIG at i − th node in
the negative-sequence networks
three-phase voltage at the fault pointkafter modification of
the phase component node admittance matrix
three-phase current at the fault pointkafter modification of
the phase component node admittance matrix
three-phase voltage of the DFIG at the m − th node
three-phase injection current of the DFIG at the m − th
node
h-th iteration
ground current at node k
Furthermore, a considerable difference exists in the DFIG fault current
under activated and non-activated crowbar conditions, leading to a
substantial deviation between calculated and actual results. Further­
more, existing fault calculation methods cannot determine whether the
crowbar is activated. Hence, the above-mentioned short-circuit calcu­
lation methods cannot satisfy the fault analysis requirements of unbal­
anced distribution networks with DFIGs.
In this study, a novel fault analysis method is proposed for unbal­
anced distribution networks with new energy sources. Existing fault
analysis methods of power systems are based on the symmetrical
component method. Consequently, they are unable to meet the fault
analysis requirements of an unbalanced distribution network with new
energy sources and cannot determine whether the crowbar of DFIGs is
activated. Thus, the use of these methods can result in considerable
errors between the calculated and actual fault values in unbalanced
distribution networks with DFIGs.
This article is organized as follows. Section 2 presents the phase
component short-circuit calculation models of DFIGs under nonactivated and activated crowbar conditions. Moreover, the activation
criterion of the crowbar protection in the time domain is also discussed.
Section 3 elaborates on an improved phase component method for un­
balanced distribution networks with DFIGs. It also presents the
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F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
Fig. 1. DFIG structure.
components in the output current of the DFIG through the coordi­
nated control of RSC and GSC.
2) The LVRT strategy for the DFIG has a balanced stator current. The
control target of this strategy involves eliminating components with
a negative sequence in the output current of the DFIG.
3) The LVRT strategy for the DFIG has a constant electromagnetic tor­
que. The control target of this strategy involves eliminating doublefrequency components in the electromagnetic torque of the DFIG,
thereby reducing the mechanical torsional vibration in the shaft
system.
calculation results obtained via an iterative method. The action condi­
tion of crowbar protection assessed by the traversal method is eluci­
dated. In Section 4, the results of the proposed method are compared
with those of other methods. PSCAD/EMTDC is used to simulate and
validate the effectiveness of the proposed fault analysis method. The
results are discussed in this section.
2. Short-circuit calculation model of distribution network with
DFIGs
Each type of new energy source has a unique structure, LVRT strat­
egy, and protection device (such as a crowbar). Therefore, the shortcircuit current of different new energy sources can vary considerably.
Therefore, it is necessary to formulate an equivalent short-circuit
calculation model for new energy sources. The inverter power supply,
such as the photovoltaic (PV) power supply and direct-drive wind tur­
bine, has the same short-circuit characteristics as those of a DFIG
without crowbar action. Hence, the DFIG is used as an example for
analysis in this work.
The crowbar and rotor-side converter (RSC) have a significant in­
fluence on the fault current characteristics of the DFIG. Therefore, the
short-circuit calculation models of the DFIG are established under acti­
vated and non-activated crowbar conditions.
Fig. 1
Under the three-phase asymmetrical fault condition, the equivalent
calculation model of the DFIG is identical to the controlled current
source model in [3,4][3,4]. Accordingly, the positive-sequence compo­
nent of short-circuit current in the DFIG can be characterized as follows:
⎧
⎪
I˙DFIG,1 = IDFIG,1 ∠γDG,1
⎪
⎪
⎪
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
⎨
( + )2 ( + )2
(1)
I
=
isd + isq
DFIG,1
⎪
⎪
⎪
⎪
+
⎩γ
DG,1 = θI + δU,1
where IDFIG,1 and γDG,1 denote the amplitude and phase angle of the
positive-sequence component in the DFIG fault current, respectively,
+
+
+
+
and θ+
I = arctan[isq /isd ], where isd and isq represent the d-axis and q-axis
components of the stator current in (dq)+ synchronous rotating refer­
ence frames, respectively. Moreover,δU,1 = arctan(UDFIGim,1 /UDFIGre,1 )
indicates the phase angle of the positive-sequence component of stator
voltage. Finally, UDFIGre,1 and UDFIGim,1 represent the real and imaginary
parts of the positive-sequence component of the stator voltage,
respectively.
Similarly, the negative-sequence component of the short-circuit
current in the DFIG under the non-activated crowbar condition is
characterized as follows:
⎧
⎪
I˙DFIG,2 = IDFIG,2 ∠γDG,2
⎪
⎪
⎪
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
⎨
( − )2 ( − )2
(2)
I
=
isd + isq
DFIG,2
⎪
⎪
⎪
⎪
−
⎩γ
DG,2 = θI + δU,2
2.1. Calculation model of DFIG under non-activated crowbar condition
Under the non-activated crowbar condition, the fault current char­
acteristics of the DFIG are related to the LVRT strategy of the RSC and
grid-side converter (GSC). According to the technical regulations of the
wind farm power system reported in [18] and [19], the reactive power
of the DFIG must be specified to ensure that the voltage requirement of
the power grid is satisfied. Under asymmetric fault conditions, unbal­
anced heating, torque ripple, and output power oscillation could occur
in the stator winding of the DFIG [18]. The LVRT strategy of DFIG was
studied [5–8] to ensure the safety of the DFIG. The typical LVRT stra­
tegies for DFIGs are as follows.
1) The LVRT strategy for the DFIG has constant active power. The
control target of this strategy involves eliminating double-frequency
where IDFIG,2 and γDG,2 denote the amplitude and phase angle of the
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F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
is turned off when the rotor current flows through the crowbar. Under
this condition, the fault current of the DFIG under the activated crowbar
condition significantly differs from that under the non-activated
crowbar condition. Accordingly, a novel equivalent short-circuit calcu­
lation model of the DFIG needs to be established.
As reported in [20], the steady-state fundamental frequency
component of the DFIG can be considered equivalent to an asynchronous
motor model. Accordingly, the equivalent impedance of the DFIG in the
positive-sequence and negative-sequence networks can be obtained as
follows:
(
⎧
′)
⎪
⎪ ZDFIG,1 = jω1 Lsσ + jω1 Lrσ + Rr ‖ jω1 Lm
⎪
⎨
s
(3)
(
)
′
⎪
⎪
⎪
⎩ ZDFIG,2 = jω1 Lsσ + jω1 Lrσ + Rr ‖ jω1 Lm
2− s
where ZDFIG,1 and ZDFIG,2 denote the positive-sequence and negativesequence
equivalent
impedances
in
sequence
networks,
′
respectively.Lrσ symbolizes the rotor leakage reactance, Rr denotes the
equivalent resistance of rotor winding, Lm represents the mutual
inductance between stator and rotor, Lsσ denotes the stator leakage
reactance, andsis the slip. The equivalent DFIG circuit in the sequence
networks under an activated crowbar condition is shown in Fig. 3.
2.3. Activation criterion of crowbar protection
Considering that the crowbar activation is based on the overcurrent
of the RSC or the overvoltage of the direct current (DC) bus, the acti­
vation criteria of the crowbar under the two conditions are investigated.
Fig. 2. Equivalent
crowbar condition.
circuit
of
the
DFIG
under
the
2.3.1. RSC Over-current
According to Mohammadi et al. [3,4], the DFIG rotor current can be
ascertained using the following relationships:
⎧
′
⎨ i+ (t) = i+∗ + Kd ⋅K+ e− Rs t/L s sin(ω1 t + θ1 )
rd+
rdf +
(4)
⎩ i+ (t) = i+∗ + K ⋅K e− Rs t/L′ s cos(ω t + θ )
d
+
1
1
rq+
rqf +
non-activated
negative-sequence component in the DFIG fault current, respectively,
and θ−I = arctan[i−sq /i−sd ], where isd and isd represent the d-axis and q-axis
components of the stator current in (dq)− synchronous rotating refer­
ence frames, respectively. Moreover,δU,2 = arctan(UDFIGre,2 /UDFIGim,2 )
indicates the phase angle of the negative-sequence component of stator
voltage. Finally, UDFIGre,2 and UDFIGim,2 represent the real and imaginary
parts of the negative-sequence component of stator voltage,
respectively.
+ −
i
Under fault conditions, i+
sd , isq , Isd , and Isq are related to the reference
rotor current value, which can be calculated according to the LVRT
strategy for the DFIG. Therefore, the equivalent DFIG model under
asymmetric fault conditions can be derived by substituting the d-axis
and q-axis components of the DFIG’s short-circuit current into Eqs. (1)
and (2). Under symmetric fault conditions, there is no negative-sequence
component in the short-circuit current of the DFIG. In other words, the
amplitude of the negative-sequence component in Eq. (2) is zero, and the
controlled negative-sequence current source model is disconnected from
the negative-sequence equivalent circuit. The equivalent DFIG model in
the sequence networks of the power grid is depicted in Fig. 2.
The fault current characteristics of the IIDG are the same as those of
the DFIG under the non-activated crowbar condition. Therefore, the
IIDG can be considered to be equivalent to a controlled current source
model. The corresponding equivalent model of short-circuit calculation
can be formulated according to the LVRT scheme of the IIDG.
⎧
⎨ i− (t) = i− ∗
rd−
rdf −
+ Kd ⋅K− e−
⎩ i− (t) = i− ∗ − K ⋅K e−
d
−
rq−
rqf −
Rs t/L
Rs t/L
′
′
s
s
sin(ω1 t + θ2 )
cos(ω1 t + θ2 )
(5)
where Kd = Lm /(σ Ls Lr ); “∗” indicates the reference signal, i∗rf denotes the
reference signal of the DFIG’s rotor current, and R denotes the resis­
tance. Subscripts r and s indicate the rotor and stator, respectively; ω1
symbolizes
the
synchronous
angular
velocity;K+ =
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
√̅̅̅̅̅̅̅̅̅̅̅
(9Us+ − 17Usn )2 + 72U2s+ /(17ω1 ); K− = 3Us− /( 17ω1 ); θ1 =
√̅̅̅
√̅̅̅
′
arctan[6 2Us+ /(17Usn − 9Us+ )]; θ2 = arctan (2 2 /3); and Ls = Ls −
2
Lm /Lr .
When the rotor current exceeds the activation value of the crowbar,
crowbar resistance is activated. Therefore, the crowbar is activated if the
following condition is satisfied:
+2
− 2
− 2
∗2
i2r (t) =i+2
rd+ (t)+irq+ (t)+ird− (t)+irq− (t) ≥ ir
(6)
where i∗r indicates the setting value of the rotor current under the acti­
vated crowbar condition.
2.3.2. DC capacitor of over-voltage
According to the RSC circuit, GSC circuit, and capacitor, the voltage
equation of the capacitance can be expressed as follows:
/
∫t
(
)
u2dc (t) = u2dc0 + 2
Pr − Pg dt
C
(7)
2.2. Calculation model of DFIG under activated crowbar condition
Under fault conditions, the switching device of the crowbar is usually
turned on, and the crowbar can be maintained in a short-circuited state
to ensure the safe operation of the RSC. The switching device of the RSC
t0
where udc0 and C are the rated voltage and capacitance of the DC bus,
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Electric Power Systems Research 210 (2022) 108108
Fig. 3. Equivalent circuit of DFIG considering the influence of crowbar.
Fig. 4. Outer voltage control loop of the GSC.
respectively, t0 denotes the time of failure, Pg is the GSC output power,
and Pr denotes the RSC active power.
If Δudc is assumed as the fault component of the DC bus voltage, then
udc = udc0 + Δudc exists under fault conditions. Generally, the DC bus
voltage does not exceed 1.1 times the rated voltage, therefore, Δu2dc can
be neglected. According to Eq. (7), Δudc can be derived as follows.
/
∫t
(
Pr − Pg )dt Cudc0
Δudc (t) =
(8)
Thus, we obtain the following.
p2 Δudc (t)+
The outer voltage control loop of the GSC is depicted in Fig. 4.
The output current of the GSC can fast-track the reference value.
Hence, the current value of the GSC can be considered approximately
equal to the reference value of the GSC. Accordingly, Δudc can be
rewritten as follows:
]
[∫ t
/
∫t
∫t
Δudc (t) =
(9)
Pr dt −
ugd ki
Δudc (t)dt)dt + kp Δudc (t) Cudc0
t0
(10)
By assuming that the GSC output current is continuous when the
failure occurs, i.e., igd (t0− ) = igd (t0+ ), the definite solution of (10) can be
obtained as follows:
{
(
( Δu)dc (t(0 ) =)/0
(11)
pΔudc (t0 ) = Pr − ugd t0+ igd t0+ Cudc0 = Us+ Pr /CUs udc0
t0
t0
kp ugd
ki ugd
pΔudc (t)+
Δudc (t) = 0
Cudc0
Cudc0
where Us indicates the rated voltage of the DFIG.
Eq. (10) is a second-order homogeneous differential equation with
constant coefficients; hence, it can have two conjugate complex roots
and two mutually different real roots. In this study, both cases are
considered.
(1) Eq. (10) has two distinct roots.
0
5
F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
Fig. 5. Flowchart of short-circuit current calculation method for unbalanced distribution network with DFIGs.
Under this condition, Δudc can be derived by combining Eqs. (10)
and (11).
Δudc (t) = N1 e−
λ1 t
− N1 e −
λ2 t
Under this condition, Δudc can be obtained according to Eqs. (10)
and (11).
(12)
Δudc (t) = N2 e− αt sin(βt)
(13)
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
where N2 = 2Us+ Pr /CUs udc0 4ki σ − k2p σ2 ; α = kp σ /2; and β =
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
4ki σ − k2p σ 2 /2.
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
where λ1 = (kp σ −
k2p σ 2 − 4ki σ ) /2; σ = ugd /Cudc0 ; λ2 = (kp σ +
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
k2p σ2 − 4ki σ) /2; and N1 = 2Us+ Pr /CUs udc0 k2p σ 2 − 4ki σ.
According to the conservation principle of power, the sum of the
output active powers of RSCPr and statorPs is equal to the output wind
(2) Eq. (10) also has two conjugate complex roots.
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F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
Fig. 6. Single-line diagram of IEEE 13 node feeder with DFIGs.
TABLE 1
Branch currents when a three-phase fault occurs at f.
Branch
650-632
632-633
633-634
632-645
645-646
632-671
671-680
671-692
692-675
671-684
684-652
684-611
DG1
DG2
Measured values
Phase A
Mag.
Ang.
(kA)
(◦ )
4.42
-66
0.076
-35
0.66
-35
/
/
/
/
4.49
-65
4.6
-68.5
0.28
69
0.068
-23
0.013
-36
0.013
-35.8
/
/
1.2
-55
1.54
-126
Phase B
Mag.
(kA)
4.38
0.059
0.51
0.117
0.039
4.36
4.5
0.25
0.012
/
/
/
1.2
1.52
Ang.
(◦ )
161
-158
-158
-145.5
-120
162
161
-30
-167
/
/
/
-175
114
Phase C
Mag.
(kA)
4.38
0.059
0.51
/
/
3.74
3.85
0.24
0.052
0.02
/
0.02
1.2
1.53
Ang.
(◦ )
49
82
83
/
/
50
47
-176
82
118
/
118
65
-5.8
Calculated values
Phase A
Mag.
Ang.
(kA)
(◦ )
4.31
-65
0.076
-36.5
0.65
-36
/
/
/
/
4.4
-64
4.5
-67.5
0.28
64
0.07
-28
0.01
-41
0.01
-41
/
/
1.19
-57
1.53
-126
Phase B
Mag.
(kA)
4.29
0.06
0.5
0.116
0.04
4.3
4.4
0.24
0.01
/
/
/
1.19
1.53
Ang.
(◦ )
163.5
-160
-160
-146.5
-120.5
163.5
162
-36
-172
/
/
/
-177
114
Phase C
Mag.
(kA)
4.29
0.06
0.5
/
/
3.67
3.76
0.24
0.05
0.02
/
0.02
1.19
1.53
Ang.
(◦ )
50.5
81
81
/
/
51.5
48
-177.5
78.5
114
/
114.5
63
-6
Calculated values
Phase A
Mag.
Ang.
(kV)
(◦ )
2.4
0
1.48
-0.7
1.48
-0.6
0.165
-1.6
/
/
/
/
0.5
-7.5
0
/
0.5
-7.3
0.5
-7.6
0.5
-7.2
0.5
-7.2
/
/
Phase B
Mag.
(kV)
2.4
1.47
1.485
0.166
1.45
1.45
0.53
0
0.53
0.53
/
/
/
Ang.
(◦ )
-120
-120.7
-121.6
-121.6
-120
-120
-129.5
/
-129.5
-130
/
/
/
Phase C
Mag.
(kV)
2.4
1.47
1.485
0.166
1.48
1.48
0.53
0
0.54
0.54
0.535
/
0.53
Ang.
(◦ )
120
119.3
118.4
118.4
120
120
114.5
/
114.4
114.5
114.3
/
114.2
TABLE 2
Node voltages when a three-phase fault occurs at f.
Node
650
632
633
634
645
646
671
680
692
675
684
652
611
Measured values
Phase A
Mag.
Ang.
(kV)
(◦ )
2.4
0
1.45
2.3
1.497
0.3
0.166
-0.7
/
/
/
/
0.49
-2
0
/
0.48
-2
0.48
-2.3
0.48
-2
0.485
-1.95
/
/
Phase B
Mag.
(kV)
2.4
1.44
1.497
0.167
1.46
1.465
0.52
0
0.52
0.52
/
/
/
Ang.
(◦ )
-120
-119
-120
-121
-121
-121
-125
/
-125
-125.5
/
/
/
Phase C
Mag.
(kV)
2.4
1.44
1.497
0.167
1.5
1.5
0.52
0
0.53
0.53
0.53
/
0.52
Ang.
(◦ )
120
119
120
119
120
120
118
/
118.5
118.6
118.5
/
118
turbine active powerP0 . Thus, the output active power of RSC can be
expressed as follows:
/
/
Ls − Lm i−rdf∗ − Us−
Ls
Pr = P0 − Ps = P0 − Lm i+∗
(14)
rdf + Us+
where u∗dc denotes the threshold value of voltage in the DC bus when the
crowbar is activated. The crowbar is activated if the following condition
is satisfied.
7
F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
Fig. 7. Deviation between the simulated and calculated values of branch currents (under three-phase fault conditions).
udc0 + Δudc t ≥ u∗dc
should be developed for unbalanced distribution networks with DFIGs.
In the present study, an improved short-circuit calculation method
based on the traditional phase component calculation is proposed for
unbalanced distribution networks with DFIGs. The principle steps of the
traditional phase component calculation method are as follows. First,
the phase component node admittance matrix is established according
to the power system network under normal conditions and then modi­
fied according to the various types of grid faults. Second, the phase
component node voltage equation is established such that the threephase node voltage and three-phase current of each branch can be
solved analytically. However, the short-circuit current characteristics of
DFIGs are more complex than those of the traditional synchronous
generators. In particular, the LVRT strategy has a significant impact on
the short-circuit current when the crowbar is activated. A non-linear
relationship between output current and stator voltage exists. There­
fore, the traditional phase component short-circuit calculation method
cannot satisfy the requirements for the analysis of unbalanced distri­
bution networks with DFIGs.
(15)
3. Short-circuit calculation method for unbalanced distribution
networks with DFIGs
For unbalanced power grids with DFIGs, the improved short-circuit
calculation method based on symmetrical component calculation can
satisfy the short-circuit calculation requirement of the power system.
For an asymmetrical power grid with DFIGs, the distribution network
may exhibit unbalanced phase load, unbalanced line impedance and
admittance, and possible asymmetrical structures of some transformers
in. A complex coupling relationship may also exist among the networks
with positive, negative, and zero sequences. Consequently, it is difficult
to accurately decouple the three-phase power components into positivesequence, negative-sequence, and zero-sequence coordinates of the
power grid. A considerable deviation between the calculated and actual
values is expected if an improved calculation method based on the
traditional symmetrical component analysis is employed to compute the
short-circuit current of the unbalanced distribution network with DFIGs.
It has a substantial impact on the protection setting, grid operation, and
equipment selection. Therefore, a new short-circuit calculation method
8
F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
Fig. 8. Deviation between the simulated and calculated values of node voltages (under three-phase fault conditions).
TABLE 3
Branch currents under the condition that two-phase fault occurs at f.
Branch
650-632
632-633
633-634
632-645
645-646
632-671
671-680
671-692
692-675
671-684
684-652
684-611
DG1
DG2
Measured values
Phase A
Mag.
Ang.
(kA)
(◦ )
0.308
-31.5
0.086
-34.8
0.72
-34
/
/
/
/
0.39
-16.2
0
/
0.115
24.2
0.22
-21
0.06
-33
0.06
-33
/
/
1.3
-34
1.17
-53
Phase B
Mag.
(kA)
3.38
0.067
0.6
0.125
0.038
3.33
3.42
0.2
0.03
/
/
/
1.3
1.17
Ang.
(◦ )
-167
-170
-169
-155
-119
-165
-163
31
159
/
/
/
-154
-173
Phase C
Mag.
(kA)
3.38
0.067
0.6
/
/
3.17
3.42
0.16
0.12
0.04
/
0.045
1.3
1.17
Ang.
(◦ )
17.7
95
96
/
/
18.88
17
-147
122
159
/
158.8
86
67
3.1. Phase component node admittance matrix under normal condition
According to the phase component model of each power component,
the phase component node admittance matrix under normal conditions
can be established as follows:
9
Calculated values
Phase A
Mag.
Ang.
(kA)
(◦ )
0.3
-29.5
0.085
-35
0.73
-35
/
/
/
/
0.395
-16
0
/
0.12
24.5
0.23
-21
0.06
-33
0.06
-33
/
/
1.285
-36
1.14
-55
Phase B
Mag.
(kA)
3.3
0.07
0.595
0.127
0.038
3.25
3.34
0.2
0.03
/
/
/
1.285
1.14
Ang.
(◦ )
-165.7
-170
-170
-156
-120
-163
-161
30
159
/
/
/
-156
-175
Phase C
Mag.
(kA)
3.17
0.07
0.595
/
/
3.1
3.34
0.165
0.115
0.04
/
0.04
1.285
1.14
Ang.
(◦ )
19.5
95
95
59.5
/
21
19
-148
121
157
/
157
84
65
F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
TABLE 4
Node voltages under the condition that two-phase fault occurs at f.
Node
650
632
633
634
645
646
671
680
692
675
684
652
611
Measured values
Phase A
Mag.
Ang.
(kV)
(◦ )
2.4
0
2.394
0.819
2.396
1.044
0.2696
0.342
/
/
/
/
2.316
0.66
2.3
1.86
2.316
0.65
2.3
0.42
2.31
0.65
2.29
0.76
/
/
Phase B
Mag.
(kV)
2.4
1.74
1.748
0.1968
1.722
1.72
1.26
1.18
1.26
1.268
/
/
/
Ang.
(◦ )
-120
-132.8
-132.1
-132.9
-133
-133.1
-159.2
179.8
-159
-159
/
/
/
Phase C
Mag.
(kV)
2.4
1.74
1.748
0.1968
1.758
1.755
1.26
1.15
1.24
1.24
1.24
/
1.236
Ang.
(◦ )
120
133.1
133.5
132.8
133
133.2
159
179.6
159
159
158
/
158
Calculated values
Phase A
Mag.
Ang.
(kV)
(◦ )
2.4
0
2.36
0.7
2.36
0.9
0.266
0.19
/
/
/
/
2.28
0.4
2.26
1.6
2.28
0.4
2.26
0.2
2.27
0.5
2.26
0.65
/
/
Phase B
Mag.
(kV)
2.4
1.73
1.74
0.196
1.68
1.68
1.29
1.14
1.3
1.3
/
/
/
Ang.
(◦ )
-120
-133
-133
-133
-133
-133
-159
179
-159
-159.5
/
/
/
Fig. 9. Deviation between the simulated and calculated values of branch currents (under phase-to-phase fault conditions).
10
Phase C
Mag.
(kV)
2.4
1.74
1.74
0.196
1.72
1.72
1.18
1.14
1.18
1.18
1.18
/
1.18
Ang.
(◦ )
120
132
133
133
133
133
157
179
157
157
157
/
157
F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
Fig. 10. Deviation between the simulated and calculated values of node voltages (under phase-to-phase fault conditions).
⎧
⎪
⎪
⎡
⎪
⎪
Y11 Y12
⎪
⎪
⎪
⎪
⎢Y
⎪
⎪
Y22
21
⎢
⎪
⎪
Y=⎢
⎪
⎪
⎣ ... ...
⎪
⎪
⎪
⎪
⎨
Yn1 Yn2
⎡
⎪
⎪
ij
ij
⎪
⎪
⎪
⎢ Yaa Yab
⎪
⎪
⎢
ij
ij
⎪
⎪
Y
Ybb
⎪ Yij = ⎢
⎪
⎣ ba
⎪
ij
ij
⎪
⎪
Yca Ycb
⎪
⎪
⎪
⎪
⎩
Y1n
...
Y2n ⎥
⎥
⎥
... ⎦
...
...
ij
Yac
ij
Ybc
ij
Ycc
at the ith node; ZDFIGi,1 and ZDFIGi.2 represent the positive and negativesequence equivalent impedances of DFIG at the ith node under the
activated crowbar condition, respectively.
Under the non-activated crowbar condition, the short-circuit current
of DFIG is related to the LVRT strategy. Combining Eqs. (1) and (2), the
phase component short-circuit calculation model of the DFIG becomes
equivalent to the controlled current source phase component model,
which is expressed as follows.
⎡
⎤
⎤
⎡
I˙DFIGi,a
I˙DFIGi,1
−
1
⎣ I˙DFIGi,b ⎦ = S ⎣ I˙DFIGi,2 ⎦
(18)
I˙DFIGi,c
0
⎤
...
Ynn
⎤
(16)
⎥
⎥
⎥
⎦
where n denotes the number of grid nodes, and i and j are node indices.
where I˙DFIGi,a , I˙DFIGi,b , and I˙DFIGi,c represent the three-phase short-circuit
currents injected by the DFIG at the ith node into the circuits of Phases A,
B, and C, respectively. I˙DFIGi,1 and I˙DFIG,2 represent the short-circuit
current injected by the DFIG at the ith node in the positive-sequence
and negative-sequence networks, respectively.
If the LVRT strategy for a DFIG is adopted to eliminate the negativesequence components in the fault current, then the components in the
DFIG fault current can be reduced to zero. The short-circuit calculation
model for the phase components of the DFIG can be expressed as
IDFIGi,120 = [ I˙DFIGi,1 0 0 ]T .
3.2. Phase component model of DFIG
Under the activated crowbar condition, the equivalent DFIG model
in the power system can be considered equivalent to an asynchronous
motor model. The phase component short-circuit calculation model of
the DFIG can be expressed as follows:
⎡
⎤
⎡
⎤
ZDFIGi,a
ZDFIGi,1
⎣ ZDFIGi,b ⎦ = S− 1 ⎣ ZDFIGi,2 ⎦
(17)
ZDFIGi,c
0
⎡
⎤
1 1 1
∘
∘
2
⎣
where S = a
a 1 ⎦, a = ej120 , and a2 = ej240 ; ZDFIGi,a , ZDFIGi,b ,
2
a a 1
and ZDFIGi,c represent the three-phase equivalent impedances of the DFIG
3.3. Node voltage equation based on fault types
− 1
Under the non-activated crowbar condition, the short-circuit current
of other branches increases when the short-circuit current of DFIG is
11
F. Xiao et al.
TABLE 5
Branch currents when the two-phase grounding fault occurs at f.
Branch
12
Phase A
Mag.
(kA)
1.6
Ang.
(◦ )
-145
Phase B
Mag.
(kA)
1.79
Ang.
(◦ )
100
Phase C
Mag.
(kA)
1.1
Ang.
(◦ )
-29
Proposed method
Considering the activation of crowbar
(The crowbar of DFIG 2 is activated)
Phase A
Phase B
Mag.
Ang.
Mag.
Ang.
(kA)
(◦ )
(kA)
(◦ )
1.6
-145
1.78
100
0.162
162
0.178
42
0.159
-78
0.162
162
0.178
42
0.16
-78
0.149
160
0.169
38.3
0.159
-80.7
0.062
164
0.065
44
0.081
-72
0.064
164
0.066
44
0.082
-72
0.083
-74
0.066
166.6
0.065
45.5
/
/
0.135
136.7
/
/
/
/
0.136
136.8
/
/
/
/
0.135
136.4
/
/
/
/
0.063
132
/
/
/
/
0.065
133
/
/
/
/
0.063
132.3
/
/
0.129
-127
0.102
109
0.146
-8.44
0.131
-128
0.106
108
0.148
-8.2
0.387
-125.7
0.375
109
0.327
-3
0.515
-148
0.574
127
0.01
32
0.512
-146
0.58
128
0.01
33
0.742
-177
0.887
109
0.015
8.5
0.213
-10.3
0.241
-22
0.33
-10.6
0.213
-10
0.242
-21
0.33
-10.8
0.586
-24.7
0.465
-71.9
0.295
-4.76
0.0005
-156
0.00015
-164
0.02
60
0.0005
-156
0.0001
-164
0.02
60
0.0005
0
0.00025
171
0.0294
31.3
0.11
-132
/
/
0.047
25
0.11
-132
/
/
0.046
24
0.0027
-174
/
/
0.0161
-174
/
/
/
/
0.049
-149
/
/
/
/
0.048
-150
0.0161
-173.6
/
/
/
/
/
/
/
/
0.01
26.3
/
/
/
/
0.01
26.2
/
/
/
/
0.0155
0.57
0.213
0.207
170
-39
0.24
0.208
158
-153
0.291
0.204
159
83
0.214
0.207
170
-38
0.24
0.207
158
-158
0.29
0.207
158
82
0.207
0.207
156
-36
0.207
0.207
36
-156
0.207
0.207
-84
84
Without considering the activation of crowbar (Balanced stator current strategy of DFIGs)
Phase C
Mag.
(kA)
1.1
Ang.
(◦ )
-29
Phase A
Mag.
(kA)
0.46
Ang.
(◦ )
-143
Phase B
Mag.
(kA)
0.56
Ang.
(◦ )
89.5
Phase C
Mag.
(kA)
0.397
Ang.
(◦ )
25.2
Electric Power Systems Research 210 (2022) 108108
650632
632633
633634
632645
645646
632671
671680
671692
692675
671684
684652
684611
DG1
DG2
Measured values
F. Xiao et al.
added to the grid. Under the activated crowbar condition, the DFIG is
equivalent to an asynchronous motor model, and the short-circuit cur­
rent of other branches decreases. Therefore, the phase component node
admittance matrix is established based on the non-activated crowbar
condition. Note that the crowbar condition has to be determined. If the
crowbar is activated, then the phase component node admittance matrix
must be modified according to Eq. (3). If the crowbar is not activated,
then the phase component node admittance matrix under normal con­
ditions can be used directly.
The phase component nodal admittance matrixY is established ac­
cording to the fault types of the unbalanced distribution network. The
voltage equation of the phase component node under fault conditions is
expressed as follows:
⎡
⎤ ⎡
⎤
U1
I1
⎢ U 2 ⎥ ⎢ I2 ⎥
⎢
⎥ ⎢
⎥
⎢ ... ⎥ ⎢ ... ⎥
⎢
⎥ ⎢
⎥
⎢
⎥
⎢
′
U k ⎥ ⎢ Ik ⎥
⎥
=
(19)
Y⎢
⎢ ... ⎥ ⎢ ... ⎥
⎢
⎥ ⎢
⎥
⎢ UDGm ⎥ ⎢ IDGm ⎥
⎢
⎥ ⎢
⎥
⎣ ... ⎦ ⎣ ... ⎦
UN
IN
Ang.
(◦ )
-90
-63.9
-59.3
-60.3
/
/
-173
-174
-173
-173
-93.7
-96
/
Phase B
Mag.
(kV)
2.85
2.69
2.47
0.293
1.87
2.97
0.008
0.0088
0.009
0.009
/
/
/
Ang.
(◦ )
150
177
-176
-177
176
0
109
108
109
106
/
/
/
Phase C
Mag.
(kV)
2.85
2.75
2.59
0.292
2.8
1.36
0.61
0.62
0.67
0.608
0.49
/
0.503
Ang.
(◦ )
29
58.4
2.55
62.4
59
58
42
39
42
34
23.7
/
26.1
′
Phase A
Mag.
(kV)
2.85
2.72
2.45
0.288
/
/
0.007
0.0067
0.007
0.007
0.018
0.019
/
]T
]T
[
[
where Ui = U̇i,a U̇i,b U̇i,c
and Ii = I˙i,a I˙i,b I˙i,c ; Uk and Ik
represent the three-phase voltage and current at the fault pointk after
modifying the phase component node admittance matrix, respectively.
UDGm and IDGm represent the three-phase voltage and injection current of
the DFIG at the mth node, respectively.
Due to the nonlinear relationship between the DFIG short-circuit
current and stator voltage, the traditional phase component method
cannot be used to solve the problem of an unbalanced distribution
network with DFIGs. Accordingly, an improved phase component shortcircuit calculation method for unbalanced distribution networks with
DFIGs is proposed. The calculation flowchart of the improved method is
shown in Fig. 5. The key steps of the proposed method are as follows.
Step 1) The phase component node admittance matrix and phase
component node voltage equation are established according to the
actual structure and parameters of the distribution network.
Step 2) The DFIG crowbar is not activated and the output current of
the DFIG under normal conditions is considered as the first iteration
value of its equivalent current source model.
Step 3) The phase component node admittance matrix and the cor­
responding phase component node voltage equation are modified ac­
cording to the different fault types; here, a modification method is
proposed.
Proposed method
Consider the activation of crowbar
(The crowbar of DFIG 2 is activated)
Phase A
Phase B
Mag.
Ang.
Mag.
Ang.
(kV)
(◦ )
(kV)
(◦ )
2.85
-90
2.85
150
2.72
-60
2.70
178
2.52
-38.5
2.53
-156
0.28
-30.8
0.291
-156.8
/
/
1.86
-156.8
/
/
1.36
9.6
0.0052
-148
0.0052
128
0.0052
-148
0.0054
128
0.0052
-148
0.0052
128
0.0052
-148
0.0053
128
0.011
-68.4
/
/
0.011
-68.4
/
/
/
/
/
/
Phase C
Mag.
(kV)
2.85
2.74
2.55
0.293
2.6
2.68
0.4
0.4
0.4
0.4
0.33
/
0.33
Ang.
(◦ )
30
60
84
83.3
59
56.7
63
62.5
62.5
62.5
52
/
52
Without consider the activation of crowbar (Balanced stator current strategy of DFIGs)
Electric Power Systems Research 210 (2022) 108108
Phase B
Mag.
(kV)
2.85
2.69
2.55
0.293
1.87
1.37
0.005
0.0051
0.0057
0.0057
/
/
/
Ang.
(◦ )
150
179
-156
-156.8
-156.8
9.73
128
128
128
128
/
/
/
Phase C
Mag.
(kV)
2.85
2.73
2.53
0.291
2.598
2.67
0.4
0.405
0.405
0.405
0.33
/
0.33
If an A-phase grounding fault occurs at node k, then the A-phase
potential of node k is zero, U̇k,a = 0. Ik,a,sc is the ground current at node k.
Accordingly, the current source connected to other nodes is related to
the system or node potential. Moreover, the output value of corre­
sponding current sources can be obtained according to the potential of
′
′
′
the access node. Then, I = Ii ; IDGm = IDGm ; Ik = Ik + Ika,sc ; and U̇k,a = 0.
The phase component node voltage equation can be rewritten as follows:
Phase A
Mag.
(kV)
2.85
2.72
2.52
0.286
/
/
0.005
0.0055
0.0057
0.0051
0.012
0.012
/
650
632
633
634
645
646
671
680
692
675
684
652
611
Ang.
(◦ )
-90
-60
-38.3
-30.6
/
/
-148
-148
-148.7
-148
-68.7
-68.7
/
Measured values
Node
TABLE 6
Node voltages when the two-phase grounding fault occurs at f.
Ang.
(◦ )
29.9
59.9
84.2
83.5
59.3
58.3
63
62.7
62.8
63
51.6
/
51.6
(1) Single-phase grounding fault
13
F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
Fig. 11. Amplitude error among calculated values of branch currents under Phase B-to-Phase C grounding fault condition (crowbar activation considered
or neglected).
Fig. 12. Phase angle error among calculated values of branch currents under Phase B-to-Phase C grounding fault condition (crowbar activation considered
or neglected).
⎡
⎤
⎡
⎤
]
′
′ T
Ik,b Ik,c . In Eq. (20), the injection current source includes the in­
jection current source before and after the fault as well as the injection
current source affected by the terminal voltage. The voltage value of
each phase at each node can be solved iteratively by substituting the
output current of the DFIG under normal conditions into Eq. (20).
[
⎢ U′ ⎥ ⎢ I ′ ⎥
⎢ 1 ⎥ ⎢ 1 ⎥
⎢ ′ ⎥ ⎢ ′ ⎥
⎢ U 2 ⎥ ⎢ I2 ⎥
⎢
⎥ ⎢
⎥
⎢ ... ⎥ ⎢ ... ⎥
⎢
⎥ ⎢
⎥
⎢ ′ ⎥ ⎢ ′ ⎥
⎢
⎥ ⎢ Ik ⎥
′′ ⎢ Uk ⎥
⎢
⎥
Y ⎢
⎥=⎢
⎥
⎢ ... ⎥ ⎢ ... ⎥
⎢ ′ ⎥ ⎢ ′
⎥
⎢U
⎥ ⎢I
⎥
⎢ DGm ⎥ ⎢ DGm ⎥
⎢
⎥ ⎢
⎥
⎢ ... ⎥ ⎢ ... ⎥
⎢ ′ ⎥ ⎢ ′ ⎥
⎣ U ⎦ ⎣ I ⎦
N
N
(20)
(1) Phase-to-phase fault
By assuming that an interphase short-circuit fault occurs at Phases B
and C of node k, these phases can be injected with the short-circuit
′
′
currents, I˙sc and I˙ = I˙k,b + I˙sc , I˙ = I˙k,c − I˙sc . The voltages of Phases
k,b
[
where U k = 0
′
Uk,b
′
Uk,c
′
]T
and I k =
′
[
Ik,a + Ik,a,sc
′
Ik,b
′
Ik,c
′
]T
k,c
′
′
′
B and C at node k are U̇k,b = U̇k,c = U̇k . The Phase B element at the kth
node is replaced with the sum of the Phase B and Phase C elements to
eliminate I˙sc . Accordingly, the rows and columns of Phase C at the kth
node in the pre-fault phase component node admittance matrix are
.
When Uk,a = 0, the rows and columns of Phase A of this node are deleted
′
without affecting the calculation of Eq. (20). Subsequently, Ik,a,sc is
]
[
′
′
′
′
′ T
eliminated. Therefore, Y becomes Y ; U k = Uk,b Uk,c ; and I k =
14
F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
Fig. 13. Amplitude error among calculated values of node voltages under Phase B-to-Phase C grounding fault condition (crowbar activation considered or neglected).
Fig. 14. Phase angle error among calculated values of node voltages under Phase B-to-Phase C grounding fault condition (crowbar activation considered
or neglected).
′
]T
[ ′
′
′
′
′
deleted; thus, Y becomes Y . Moreover, Uk = Uk,a Uk,b + Uk,c and Ik
]T
[ ′
′
′
= Ik,a Ik,b + Ik,c , and one equation is removed from the expression
where I˙k,ba = − I˙k,ab , I˙k,ac = − I˙k,ca , and I˙k,cb = − I˙k,bc . The phase voltage
at node k is U̇k,a = U̇k,b = U̇k,c = U̇k . Therefore, the simplified method
under the three-phase fault condition is equivalent to the simplified
method under the phase-to-phase fault condition. Rows a, b, and c of the
kth node are added, and I˙k,ba , I˙k,ac , and I˙k,cb are eliminated. Accordingly,
the fault currents of the unbalanced distribution networks with DFIGs
can be solved iteratively.
Step 4) According to the node voltage equation under the fault
condition and based on the phase component calculation model of the
DFIG ((1) and (2)), the current along each branch and the voltage on
each node are solved iteratively.
Step 5) If the new energy source is a DFIG, then the crowbar con­
dition is evaluated according to Criterion 1. If the crowbar is activated,
then the phase component node admittance matrix and phase compo­
nent node voltage equation should be modified according to the
equivalent asynchronous motor model of the DFIG under the activated
crowbar condition. Then, return to Step 3); if the crowbar is not acti­
vated, then proceed to Step 6).
in (20). Under this condition, the injection current source only includes
such sources before and after the fault as well as the injection current
source affected by the terminal voltage. Accordingly, the fault currents
of unbalanced distribution networks with DFIGs can be solved
iteratively.
(1) Three-phase fault
Assuming that a three-phase fault occurs at node k, the injected
current of each phase at node k can be expressed as follows:
⎡
⎤
⎡
⎤
I˙k,a + I˙k,ba + I˙k,ca
⎢ ′ ⎥
⎢ I˙ ⎥ = ⎣ I˙k,b + I˙k,ab + I˙k,cb ⎦
⎣ k,b ⎦
′
I˙k,c + I˙k,ac + I˙k,bc
I˙k,c
′
I˙k,a
(21)
15
F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
Eqs. (6) and (15) indicate that the relationship between the rotor
current ir (t) and DC bus voltage udc (t), which are time-varying functions.
The amplitudes of the rotor current and DC bus voltage are affected by
the terminal voltage and time (t) after failure. Therefore, the resulting
amplitudes of the rotor current and DC bus voltage are ergodic. In other
words, the aforementioned amplitudes can be calculated at every time
point.
Let ihm,r = {ihm,r (t), t > 0} and uhm,dc = {uhm,dc (t), t ≥ 0} denote the rotor
fundamentally ignored. Therefore, the transient DFIG process for the
relay protection setting and equipment selection in power systems can
be ignored.
In this study, the voltage level of the test system is 4.16 kV, and the
system frequency is 60 Hz. Parameters T1 and T2 are the same. The rated
capacity is 1.6 MVA, the turns ratio is 0.69 kV/4.16 kV, the winding type
is Y/D, and the leakage reactance is 0.0622 pu. The parameters of the
IEEE 13 Node Test Feeder are described in [21]. The rated capacities of
the two DFIGs are 1.5 MW, Usn = 690 V, Lm = 2.1767 pu, and Ls = Lr
=0.1245 pu; the rated rotor speed is 1.2 pu.
The faults of node f are assumed to be a three-phase balanced fault
and a Phase B-to-Phase C fault. The calculated and measured values of
the phase fault current and phase node voltage when a three-phase fault
occurs at node f are summarized in Tables 5 and 6, respectively. Tables 7
and 8 list the calculated and measured values of the phase fault current
and phase node voltage of the Phase B-to-Phase C fault occurrence at
node f, respectively.
current and DC bus voltage, respectively. Let the DFIG values at the mth
node be in the finite set S = {0,1,2,...,Z}, where 0, 1, 2,..., Z indicates the
state at different times after failure, and Z = 100 000. Accordingly, the
following is derived:
⎧
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
⎪
⎨ ih (t) = i+h2 (t) + i+h2 (t) + i− h2 (t) + i− h2 (t)
m,r
m,rd+
m,rq+
m,rd−
m,rq−
(22)
⎪
⎩
uhm,dc (t) = um,dc0 + Δuhm,dc (t)
where h indicates the h-th iteration.
The corresponding amplitude vectors of rotor current and DC bus
voltage are defined as follows:
[
]
⎧
⎨ Ihm,r = ihm,r0 (t), ihm,r1 (t),ihm,r2 (t),..., ihm,rZ (t)
[
]
(23)
⎩ Uh = uh (t), uh (t),uh (t),..., uh (t)
1
1
Z
m,dc
m,dc
m,dc
m,dc
2
4.1. Three-phase fault
Three-phase short-circuit faults are simulated at different nodes of
the IEEE 13 node model to verify the effectiveness of the proposed
method. In this study, a balanced three-phase fault occurs at node f. The
calculated and measured values of the phase fault current and phase
node voltage are summarized in Tables 1 and 2, respectively. The de­
viation between the simulated and calculated values of the branch
current is illustrated in Fig. 7. Moreover, the deviation between the
simulated and calculated values of the node voltage is shown in Fig. 8.
Table 2 and Fig. 8 indicate that the maximum deviation between the
calculated and measured values of node voltage is 4.16%, while the
maximum angle error between the calculated and measured values is
5.5◦ . As indicated in Table 1 and Fig. 7, the Phase A load connected to
node 652 is minimal; thus, the load current is also considerably small.
Therefore, a certain deviation exists between the calculated and
measured values of the branch currents in nodes 671–684 and 684–652.
The maximum deviation between the calculated and measured values of
currents of the other branches is 2.5%, and the angular error is 5◦ . In
other words, the proposed phase component short-circuit current
calculation method satisfies the requirements of short-circuit calculation
for unbalanced distribution networks with DFIGs.
m,dc
where ihm,r0 (t) =ihm,r0 (0) and ihm,rZ (t) = ihm,rZ− 1 (t + 10− 6 ). Moreover,
uhm,dc 0 (t) = uhm,dc 0 (0) and uhm,dc Z (t) = uhm,dc Z− 1 (t + 10− 6 ). Therefore, the
activation criterion of the crowbar protection of the DFIG at the m-th
node in the h-th iteration can be obtained as follows.
Criterion 1 :
h
h
Im,r
> i∗m,r or Um,dc
> u∗m,dc
(24)
Step 6) If Criterion 2 is satisfied, then the iterative process is
completed; proceed to Step 7). If Criterion 2 is not satisfied, then the
last iteration value of the node voltage can be substituted into the
equivalent phase component calculation model of the DFIG (i.e.,
Eqs. (1) and (2)) as the initial value of the DFIG’s injection current in
the next iteration; return to Step 5). Criterion 2 is given as follows:
⃒ ⃒
⃒ ⃒
⃒)
(⃒ h
h− 1 ⃒ ⃒ h
h− 1 ⃒ ⃒ h
h− 1 ⃒
⃒
Criterion 2 : max ⃒V̇ i,a − V̇ i,a ⃒, ⃒V̇ i,b − V̇ i,b ⃒, ⃒V̇ i,c − V̇ i,c ⃒ ≤ ε (25)
4.2. Asymmetric fault
The simulation results and theoretical values are comparatively
analyzed by considering the Phase B-to-Phase C fault at node f as an
example. The calculated and measured values of the branch currents and
node voltages are summarized in Tables 3 and 4, respectively, and their
deviations are shown in Figs. 9 and 10, respectively.
Table 3 and Fig. 9 indicate that the maximum deviations between the
calculated and measured values of the branch current and angular error
are 4.5% and 2◦ , respectively. From Table 4 and Fig. 10, it is evident that
the maximum deviation values between the calculated and measured
values of node voltage and angular error are 6.7% and 2%, respectively.
For an unbalanced distribution network, regardless of the symmetry
or asymmetry of fault conditions, the maximum magnitude and angle
errors between the calculated and measured values of the branch current
and node voltage are extremely small. In other words, the proposed fault
calculation method satisfies the requirements of short-circuit calculation
for unbalanced distribution networks with DFIGs.
where h represents the h-th iteration, irepresents the ith node, andε in­
dicates the threshold value, which is typically 0.02 pu.
Step 7) Based on the impedance relationship of the phases of each
branch, the current in the branch is calculated according to the last
iterative voltage value of the corresponding node.
Step 8) End.
4. Simulation
A simulation model of IEEE 13 Node Test Feeder with DFIGs was
formulated to verify the effectiveness of the proposed short-circuit
current calculation method for unbalanced distribution networks with
DFIGs, as shown in Fig. 6. Many LVRT strategies can be used for DFIGs.
For instance, the balanced stator current strategy is applied to DFIG1
and DFIG2. The phase component short-circuit current calculation
method was tested under various fault conditions. In this study, two
DFIGs that are connected to nodes 633 and 692 are considered as ex­
amples. The effectiveness of the proposed method was verified. Ac­
cording to Mohammadi et al. [3] and [4], the DFIG short-circuit current
undergoes a slight change during the power grid fault, which can be
4.3. Comparison of methods
Generally, crowbar activation is not considered in the traditional
fault analysis method. In this study, crowbar activation can be assessed
using the proposed method whose calculation results were found to be
more accurate. The fault at node f is assumed to be a Phase B-to-Phase C
16
F. Xiao et al.
Electric Power Systems Research 210 (2022) 108108
grounding fault. Under this fault condition, the crowbar of DFIG2 shown
in Fig. 6 is activated. Tables 5 and 6 summarize the results of the pro­
posed method with or without the activation of the crowbar. The
calculation results in the second column of these tables are based on the
activated condition of the crowbar of DFIG2, whereas those in the third
column are based on the balanced stator current strategy of DFIG2.
The amplitude and phase angle errors of the proposed method with
or without the activation of the crowbar are presented in Figs. 11–14,
where CB denotes the crowbar. In these figures, “considered CB” means
that crowbar activation can be considered, whereas “not considered CB”
indicates that this activation is neglected. For DFIG2, the action condi­
tion of the crowbar is assessed using the proposed method. Additionally,
the fault calculation model of DFIG2 is consistently based on the
balanced stator current control strategy.
As shown in Figs. 11 and 12, when crowbar activation is neglected,
the maximum amplitude and angular errors among the calculated values
of branch currents are 267% and 157◦ , respectively. However, when
crowbar activation is considered, these errors are only 3.23% and 5◦ ,
respectively.
As shown in Figs. 13 and 14, when crowbar activation is neglected,
the maximum amplitude and angular errors among the calculated node
voltages are 116.8% and − 81◦ , respectively. However, when crowbar
activation is considered, these errors decrease to 2.1% and − 1.6◦ ,
respectively. The result indicates that the effect of crowbar activation
must be considered in the short-circuit calculation method for unbal­
anced distribution networks with DFIGs.
Therefore, the difference between the measured values and calcu­
lation results when crowbar activation is neglected is considerable.
However, when crowbar activation is considered, the measured and
calculated values are essentially the same. Therefore, under activated
crowbar conditions, the DFIG cannot be considered equivalent to a
current source model based on the previously developed LVRT strategy,
and crowbar activation must be correctly evaluated using the failure
analysis method. The proposed method is also applicable to unbalanced
distribution networks with IIDGs, such as photovoltaic power systems
and direct drive wind turbines.
Xianggen Yin: Investigation, Conceptualization.
Jinyu Wen:Conceptualization, Methodology,
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgments
The research is supported by the State Grid Headquarters Science
and Technology Project (No. 5400-202122573A-0-5-SF).
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5. Conclusions
The symmetrical component method cannot satisfy the requirements
of short-circuit calculation for unbalanced distribution networks with
DFIGs. Therefore, a novel short-circuit calculation method was proposed
in this study. The short-circuit calculation equivalent models based on
the phase components of DFIGs were established based on the LVRT
strategy. In particular, the activation criteria of crowbar protection were
proposed. Subsequently, an improved fault analysis method for unbal­
anced distribution networks with DFIGs was proposed based on
traversal and iteration. Finally, the efficacy of the proposed short-circuit
calculation method was verified by simulations. The simulation results
indicate that the proposed short-circuit calculation method is highly
accurate. This study has considerable significance for short-circuit
calculation, operation control, and protection setting of unbalanced
distribution networks with DFIGs.
Credit author statement
Fan Xiao: Conceptualization, Methodology, Validation, WritingOriginal draft preparation.
Yongjun Xia: Data curation, Validation.
Kanjun Zhang: Validation, Writing-Original draft preparation.
Zhe Zhang: Investigation, Conceptualization, Writing-Reviewing and
Editing.
17