Effect of Steady-State Wind Turbine Generator Models on Power
Flow Convergence and Voltage Stability Limit
Genevieve Coath
Mechatronics Research Group
Department of Mechanical Engineering
The University of Melbourne, Australia
Email: gcoath@ieee.org
Majid Al-Dabbagh
Electrical Energy and Control Systems
School of Electrical and Computer Engineering
RMIT University, Australia
Email: majid@rmit.edu.au
A BSTRACT
Environmental concerns over the use of fossil fuels as energy sources for electric power generation have resulted in
the rapid increase in the amount of wind-generated power
connected to electrical power networks worldwide. As such,
the inclusion of accurate wind turbine generator models in
power system simulation software has become increasingly
important. Steady-state and dynamic models of wind turbine
generators are required for different types of simulation
studies. Steady-state fixed-speed and variable-speed models
are examined in this paper in terms of their effect on the
power flow convergence, voltage stability limits, and their
ability to be used to provide good initial values in dynamic
models.
1. I NTRODUCTION
In recent years, there has been a rapid increase in the
amount of wind-generated power on electrical power grids
worldwide. Subsequently, the inclusion of accurate models
of wind turbine generators (WTGs) in power system analysis
software is important. Steady-state WTG models are required
for voltage stability analysis and to initialise dynamic models; dynamic simulation models require correct initialisation
otherwise time will be wasted as the models try to find a
steady-state operating point as the initial condition [1]. Thus,
load flow data can provide dynamic models with an initial
starting point [1]. The way in which the WTG is modelled
in load flow studies can influence the load flow convergence
and the system’s voltage stability limit (VSL). Variablespeed WTGs have seen a recent increase in popularity as the
problems associated with fixed-speed WTGs become more
prominent with the increase in penetration level of windgenerated power into electrical power grids.
2. W IND T URBINE G ENERATORS
2.1. S QUIRREL -C AGE I NDUCTION G ENERATORS
Squirrel-cage induction generators (SCIGs) are fixed-speed
machines that are directly grid connected, as shown in
Figure 1. The slip and the rotor speed of a SCIG vary with
the generated power; these variations in rotor speed are so
small that SCIGs are described as fixed-speed [2]. They are
widely employed in WTGs today, however their inability to
generate reactive power is a limiting factor in their future
use.
Fig. 1. Squirrel cage induction generator [2]
extraction from the wind [3]. This is achieved by feeding
the rotor circuit with real and reactive power from the rotorside converter, as shown in Figure 2. The converter circuit
allows the production or consumption of reactive power,
making it different to the SCIG, which can only consume
reactive power. Hence, DFIGs do not cause the same voltage
instability problems as do SCIGs.
Fig. 2. Doubly-fed induction generator [2]
2.3. D IRECT-D RIVE S YNCHRONOUS G ENERATORS
Figure 3 shows a direct-drive synchronous generator
(DDSG) which is studied in this paper, and has a wound
rotor, and no gear box due to its operation at low speeds and
its many poles. It is connected to the grid via a back-to-back
voltage source converter [1].
2.2. D OUBLY-F ED I NDUCTION G ENERATORS
Doubly-fed induction generators (DFIGs) have seen a recent
surge in popularity for wind turbine applications for several
reasons [2]. The primary reason for this is their ability to
vary their operating speed in order to gain optimum power
Fig. 3. Direct drive synchronous generator [2]
3. W IND T URBINE G ENERATOR M ODELLING
3.1. S QUIRREL C AGE I NDUCTION G ENERATOR M ODEL
The SCIG is usually modelled as a conventional PQ bus, with
the real power generated and reactive power demand specified. However, the reactive power demand can be expressed
as a funtion of the bus voltage if the SCIG is modelled as
an improved PQ bus by using the representation introduced
in [4] and later used in [5], as shown below:
Q≈V2
Xc − X m
X
+ 2P2
Xc Xm
V
(1)
where Q is the generator’s reactive power consumption, and
is calculated taking into account the capacitive reactance Xc ,
magnetising reactance Xm , sum of stator and rotor reactances
X, terminal voltage V , and real power P of the generator.
In this paper it was assumed that the capacitor was there
primarily to compensate the no-load reactive power consumption, hence Xc = Xm . The improved PQ bus representation
involves updating the specified reactive power at every load
flow iteration [4]; a full derivation of the approximation can
be found in [4].
3.2. D OUBLY-F ED I NDUCTION G ENERATOR M ODELS
DFIGs can be included in load flow studies as PQ or PV
buses, as they can operate in either power factor controlled
or voltage-controlled modes. When modelling a DFIG as a
PQ bus, it is assumed that the DFIG is operating in power
factor controlled mode, meaning that the specified reactive
power is zero. In voltage controlled mode, the DFIG can be
represented as a PV bus with Q limits applied.
3.3. D IRECT-D RIVE S YNCHRONOUS G ENERATOR M ODEL
The DDSG can be modelled as a PV bus in the load flow
study, with or without its reactive power limits enforced.
When enforcing these limits, if the limit is reached the PV
bus is converted to a PQ bus.
attached to PQ bus 30 of the 57-bus network. The DFIG was
then modelled in voltage controlled mode as a PV bus with
reactive power generator limits enforced, assuming a power
factor of 0.85. The DDSG was modelled as a PV bus with
and without reactive power generator limits enforced. All PV
models were attached to PV bus 6 of the 57-bus network.
All load flow studies were performed using the BX version
of the fast-decoupled load flow which was initialised to
a flat start. A continuation power flow was implemented
using a secant predictor and a bus voltage magnitude as
the continuation parameter, as in [7], to trace the PV curve
of the weakest bus in the system under study. Hence, the
continuation power flow simulated a load increase on bus
31 of the 57-bus system for all PV curves traced. The load
increase was performed using a loading factor, λ, whose
value was obtained as part of the continuation power flow.
The MATPOWER software package was used for all
simulations due to the ease with which it can be modified
to include additional generation which is entirely specified
by the user, and also for its ease of use with continuation
power flow analysis. It is a single-phase load flow package
with many features including both ac and dc power flows.
It also provides a variety of solvers for optimal power
flow problems which make use of some of the features in
MATLAB’s Optimization Toolbox. It was designed to be
highly modifiable for use in power system analysis research.
6. R ESULTS
6.1. E FFECT OF M ODELS ON P OWER F LOW C ONVERGENCE
Each of the WTG model types were implemented into
the systems as aggregated wind farms with only one type
of WTG modelled within the wind farm, i.e. all PQ, or all
improved PQ, etc. The PQ-based WFs were in turn added to
PQ bus 30 of the 57-bus system and the PV-based WFs were
in turn added to PV bus 6. The effect of the models on the
power flow convergence in terms of P θ and QV iterations
at the bus at which they were applied was assessed. From
4. DYNAMIC M ODEL I NITIALISATION
5. S IMULATION S TUDIES
Studies were carried out using the IEEE 57-bus test network, which was modified to incorporate the wind farms.
In all cases, a wind farm capable of producing 30MW was
assumed. The wind farm was comprised of 15 aggregated
WTGs, each rated at 2MW. The WTG base values were
as follows [6]: Vbase = 690V , Sbase = 2M W , ωbase =
2πfbase and fbase = 50Hz. The WTGs’ parameters were
[6]: stator resistance (Rs ) = 0.0048pu, rotor resistance
(Rr ) = 0.00549pu, stator reactance (Xls ) = 0.09241pu, rotor reactance (Xlr ) = 0.09955pu and magnetising reactance
(Xm ) = 3.95279pu.
The SCIG was modelled as a conventional PQ bus and as
an improved PQ bus, as described in Section 3.1. The DFIG
was modelled as a PQ bus with a constant power factor,
the assumption being that the DFIG has local power factor
control. Hence, a real power was assumed (nominal power
output of each WTG within the wind farm), and the reactive
power consumption was zero. All PQ models studied were
−3
6
x 10
P mismatch
Q mismatch
4
2
Real and reactive power mismatch
The use of dynamic models of wind turbine generators in
power system dynamic simulation software is becoming of
increasing relevance as the amount of wind-generated electrical power on electrical grids worldwide increases. Dynamic
models must be initialised with a valid steady-state operating
point, otherwise during dynamic simulations time is wasted
as the models try to find a valid operating point [1], [6].
0
−2
−4
−6
−8
−10
−12
1
2
3
4
5
6
Number of iterations
7
8
9
10
Fig. 4. Convergence of Pθ iterations for PQ model of
SCIG on bus 30 of 57 bus system
Figures 4-7, it can be seen that using the improved PQ bus
model for the SCIG results in a higher number of P and
Q iterations than the conventional PQ model of the SCIG.
This can be attributed to the calculation of the specified
reactive power as a function of bus voltage, which is therefore
updated within the complex power injection at each load
flow iteration. The PQ model of the DFIG had a similar
convergence behaviour to the conventional PQ model of the
SCIG, as shown in Figures 8 and 9. Figures 10 and 11 show
−3
0.005
P mismatch
Q mismatch
0
4
2
−0.005
Real and reactive power mismatch
Real and reactive power mismatch
x 10
6
P mismatch
Q mismatch
−0.01
−0.015
−0.02
0
−2
−4
−6
−8
−0.025
−10
−0.03
1
2
3
4
5
6
Number of iterations
7
8
9
−12
1
2
Fig. 5. Convergence of QV iterations for PQ model of
SCIG on bus 30 of 57 bus system
3
4
5
6
Number of iterations
7
8
9
Fig. 8. Convergence of Pθ iterations for PQ model of
DFIG on bus 30 of 57 bus system
−3
6
x 10
0.005
P mismatch
Q mismatch
P mismatch
Q mismatch
4
0
2
Real and reactive power mismatch
Real and reactive power mismatch
−0.005
0
−2
−4
−6
−0.01
−0.015
−0.02
−0.025
−8
−0.03
−10
−12
−0.035
0
5
10
15
1
2
3
4
Number of iterations
5
6
Number of iterations
7
8
9
Fig. 9. Convergence of QV iterations for PQ model of
DFIG on bus 30 of 57 bus system
Fig. 6. Convergence of Pθ iterations for improved PQ
model of SCIG on bus 30 of 57 bus system
1.4
0.005
P mismatch
Q mismatch
P mismatch
Q mismatch
1.2
0
Real and reactive power mismatch
Real and reactive power mismatch
1
−0.005
−0.01
−0.015
−0.02
0.6
0.4
0.2
−0.025
−0.03
0.8
0
−0.2
0
2
4
6
8
Number of iterations
10
12
14
Fig. 7. Convergence of QV iterations for improved PQ
model of SCIG on bus 30 of 57 bus system
that the PV model with reactive power limits applied had
approximately the same convergence as the PQ model for
the DFIG and SCIG. Figures 12-15 show the convergence
characteristic of the load flow calculation for the DDSG
models. There was no effect on convergence of applying
the reactive power generation limits; PV buses whose limits
are reached are converted into PQ buses which doesn’t
affect the load flow convergence. It should be mentioned,
however, that during this process several attempts at the load
flow calculation itself may be made in order to satisfy all
generators’ reactive power generation limits. A summary of
1
2
3
4
5
6
Number of iterations
7
8
9
10
Fig. 10. Convergence of Pθ iterations for PV model
with Q limits of DFIG on bus 6 of 57 bus system
the results of the load flow convergence is shown in Tables
1 and 2.
6.2. E FFECT OF M ODELS ON VOLTAGE S TABILITY L IMIT
The effect of the models on the VSL of bus 31 of the the
57-bus system can be seen in Figures 16-21. Figures 1619 illustrate the impact of the different induction generator
models on the voltage stability limit (VSL) of the 57-bus
system. The conventional PQ model of the SCIG gives a VSL
of 4.4, as does the improved PQ model. Hence, the increased
accuracy of the reactive power consumption of the machine
−3
0.01
1
P mismatch
Q mismatch
P mismatch
Q mismatch
0.5
0
0
Real and reactive power mismatch
Real and reactive power mismatch
x 10
−0.01
−0.02
−0.03
−0.5
−1
−1.5
−2
−0.04
−2.5
−0.05
1
2
3
4
5
6
Number of iterations
7
8
9
−3
1
Fig. 11. Convergence of QV iterations for PV model
with Q limits of DFIG on bus 6 of 57 bus system
2
3
4
5
6
Number of iterations
7
8
9
Fig. 14. Convergence of Pθ iterations for PV model
with Q limits of DDSG on bus 6 of 57 bus system
−3
14
x 10
0.09
P mismatch
Q mismatch
P mismatch
Q mismatch
0.08
12
0.07
Real and reactive power mismatch
Real and reactive power mismatch
10
8
6
4
0.06
0.05
0.04
0.03
0.02
2
0.01
0
−2
0
−0.01
1
2
3
4
5
6
Number of iterations
7
8
9
2
3
4
5
Number of iterations
6
7
8
Fig. 15. Convergence of QV iterations for PV model
with Q limits of DDSG on bus 6 of 57 bus system
Fig. 12. Convergence of Pθ iterations for PV model of
DDSG on bus 6 of 57 bus system
0.95
−3
1
1
x 10
0.9
P mismatch
Q mismatch
0.85
0
−1
Voltage (p.u.)
Real and reactive power mismatch
0.8
−2
0.75
0.7
0.65
0.6
−3
0.55
0.5
−4
0.45
−5
1
2
3
4
5
Number of iterations
6
7
1
1.5
2
2.5
3
Loading factor, λ
3.5
4
4.5
8
Fig. 13. Convergence of QV iterations for PV model
of DDSG on bus 6 of 57 bus system
in the improved PQ model makes no difference to the VSL
obtained. As the DFIG operating in power factor controlled
mode doesn’t consume reactive power, it yields a higher
VSL, at 4.92, and also has a less steep gradient over the first
few incremental load increases than the SCIG models. The
decline in bus voltage in this region is particularly important,
as a large decrease may often mean that the voltage is
below its minimum allowable value. The PV representation
of the DFIG gives a lower voltage stability margin at bus
31 than any of the PQ models; its placement in the network
Fig. 16. PV curve for PQ model of SCIG on 57 bus
system
is electrically distant from bus 31, hence having little effect
on its VSL, but causing a more rapid decline in bus voltage
than any of the other models. The results are summarised in
Table 3. The VSLs for the two DDSG models differed by
only 0.07, as the effect of applying the reactive power limits
had virtually no impact at bus 31, again due to the distance
between it and the wind farm bus; the results are shown in
Table 4.
6.3. DYNAMIC M ODEL I NITIALISATION
DATA
FROM
L OAD F LOW
Model
# of Iterations for
P V Convergence
10
15
9
10
PQ (SCIG)
Improved PQ
PQ (DFIG)
PV with Q limits
# of Iterations for
Qθ Convergence
9
14
9
9
1
0.9
0.8
Model
# of Iterations for
P V Convergence
9
9
PV
PV with Q limits
Voltage (p.u.)
Table 1. Comparison of convergence characteristics for
induction generator models
# of Iterations for
Qθ Convergence
8
8
0.7
0.6
0.5
Table 2. Comparison of convergence characteristics for
DDSG models
0.4
1
1.5
2
2.5
3
Loading factor, λ
3.5
4
4.5
0.95
Fig. 19. PV curve for PV model with Q limits of DFIG
on 57 bus system
0.9
0.85
Model
PQ (SCIG)
Improved PQ
PQ (DFIG)
PV with Q limits
Voltage (p.u.)
0.8
0.75
0.7
Critical Point
4.4
4.4
4.92
4.02
Table 3. Comparison of voltage stability limits for
induction generator models
0.65
0.6
0.55
1
0.5
0.45
1
1.5
2
2.5
3
Loading factor, λ
3.5
4
4.5
0.9
Fig. 17. PV curve for improved PQ model of SCIG on
57 bus system
Voltage (p.u.)
0.8
1
0.9
0.6
0.8
Voltage (p.u.)
0.7
0.5
0.7
0.4
1
1.5
2
2.5
3
Loading factor, λ
3.5
4
4.5
0.6
Fig. 20. PV curve for PV model of DDSG on 57 bus
system
0.5
1
0.4
1
1.5
2
2.5
3
Loading factor, λ
3.5
4
4.5
5
0.9
Fig. 18. PV curve for PQ model of DFIG on 57 bus
system
6.3.1. S QUIRREL C AGE I NDUCTION G ENERATOR : The SCIG
is an induction generator with a zero rotor voltage. The
voltage equations of the machine can be written as [1]:
vds
= −Rs ids + ωs ((Lsσ + Lm )iqs + Lm iqr )
0 = −Rr iqr − sωs ((Lrσ + Lm )idr + Lm ids )
The real power generated, P , and the reactive power consumed, Q, are [1]:
= vds ids + vqs iqs
Q = vqs ids − vds iqs
0.7
0.6
0.5
vqs = −Rs iqs − ωs ((Lsσ + Lm )ids + Lm idr ) (2)
0 = −Rr idr + sωs ((Lrσ + Lm )iqr + Lm iqs )
P
Voltage (p.u.)
0.8
(3)
0.4
1
1.5
2
2.5
3
Loading factor, λ
3.5
4
4.5
Fig. 21. PV curve for PV model with Q limits of DDSG
on 57 bus system
where v is voltage, i is current, R is resistance, L is
inductance, ω is frequency and s is slip. The equations are
Model
PV
PV with Q limits
Critical Point
4.08
4.01
different to the current. Hence, the necessity of applying
these limits is evident, otherwise unrealistic values would be
obtained to initialise the dynamic models.
Table 4. Comparison of voltage stability limits for
DDSG models
7. C ONCLUSIONS
expressed in terms of their direct (d) and quadrature (q)
components. The subscripts m, s and r stand for mutual,
stator and rotor, respectively. The symbol σ symbolises
leakage.
6.3.2. D OUBLY-F ED I NDUCTION G ENERATOR : The voltage
equations required to initialise the reduced order dynamic
DFIG are [1]:
vds
vqs
= −Rs ids + ωs ((Lsσ + Lm )iqs + Lm iqr )
= −Rs iqs − ωs ((Lsσ + Lm )ids + Lm idr )
vdr
vqr
= −Rr idr + sωs ((Lrσ + Lm )iqr + Lm iqs )
= −Rr iqr − sωs ((Lrσ + Lm )idr + Lm ids )
(4)
and the real and reactive power of the generator are given by
[1]:
P = Ps + Pr = vds ids + vqs iqs + vds idc + vqs iqc
Q = vqs ids − vds iqs
(5)
(6)
where the subscript c means converter. All other symbols are
as described earlier.
6.3.3. D IRECT-D RIVE S YNCHRONOUS G ENERATOR : As seen
from the power grid, the DDSG’s real and reactive power
generated is that provided by the voltage source converter,
as in Figure 3, [1]:
Pc
Qc
= vdc idc + vqc iqc
= vqc idc − vdc iqc
(7)
and the voltage equations are given by:
q
2 + v2
vg =
vds
qs
vds
= ωm (Lqm + Lsσ )iqs − Rs ids
vqs
vf d
= ωm (−(Ldm + Lsσ )ids + Ldm if d ) − Rs iqs
= R f d if d
(8)
The real and reactive power generated is expressed as [1]:
Pg
= vds ids + vqs iqs
Qg
= vqs ids − vds iqs
(9)
Using the load flow data obtained for each of the models,
the initial conditions of stator voltages and stator currents
in the d-q reference frame were calculated. It can be seen
in Table 5 that depending on the model used, a different
initial condition is obtained. The SCIG models gave similar
The effect of steady-state models of fixed-speed and variablespeed wind turbine generators (WTGs) on power flow convergence, voltage stability limit and suitability for dynamic
model initialisation was studied in this paper. The squirrelcage induction generator (SCIG) was modelled as a conventional PQ bus and as an improved PQ bus, where the
dependence of reactive power on nodal voltage was taken
into account during the load flow iterations. The doubly-fed
induction generator (DFIG) was modelled in power factor
controlled mode as a conventional PQ bus, and in voltage
controlled mode as a PV bus with reactive power generator limits enforced. The direct-drive synchronous generator
(DDSG) was modelled as a PV bus with and without reactive
power limits enforced. If its reactive power generation was
at a limit, it was converted into a PQ bus.
The improved PQ model required more load flow iterations to converge, however it yielded an identical voltage
stability limit (VSL) to the conventional PQ model, meaning
that its costly computation provided no advantage over the
conventional PQ model when calculating the VSL. Its use
would be more suited to conventional load flow studies,
where an accurate calculation of an induction generator’s
reactive power consumption is required. The PQ model of
the DFIG gave a higher VSL than the other PQ models, as
the nature of the DFIG operating in power factor controlled
mode means that it does consume any reactive power, and
hence doesn’t negatively affect the VSL. It also provided the
slowest decline in bus voltage for the continuation power flow
study, meaning a higer bus voltage was maintained over the
load increase.
The effect of enforcing reactive power limits for the PV
models for the DFIG and DDSG made no difference to either
the convergence of the load flow iterations or to the VSL. If
the wind farm were placed closer to the weak load bus being
monitored, a decrease in that bus’s VSL would be expected.
It can therefore be concluded that the placement of such wind
farms is preferably some distance from weak buses.
The importance of correct initialisation of dynamic models
was highlighted in this paper, and the results of each of
the models were used to calculate the initial conditions of
some sample SCIG, DFIG and DDSG dynamic models.
The importance of applying generator reactive power limits
was evident; without doing so the dynamic models could
be initialised with unrealistic or impossible values. Future
directions of this research include dynamic model validation
and verification.
A PPENDIX
Generator Type
SCIG
SCIG
DFIG
DFIG
DDSG
DDSG
Model
PQ
Improved PQ
PQ
PV with Q limits
PV
PV with Q limits
vds , vqs (pu)
0.98, -0.067
0.95, -0.058
1.03, -0.09
0.97, -0.11
1.03, -0.14
0.97, -0.11
ids , iqs (pu)
0.26, -0.95
0.27, -0.12
0.25, -0.39
0.29, -0.10
0.43, 1.08
0.29, -0.10
Table 5. Comparison of initial conditions for all models
results for the stator voltages and currents, whilst the PQ
model of the DFIG showed a higher value for the voltage.
The application of generator reactive power limits to the PV
model made a difference to the voltage and a significant
The equivalent circuits and equations for the wind turbine
generator models used in this paper are presented below. The
following equations represent the SCIG:
Zin
= Rs + jX1 + (jXm||((
Rr
) + jX2 ))
s
(10)
Va
Vs
Zin
Va
=
Z2
= Vs + Is (Rs + jX1 )
(13)
P
= 3|Vs ||Is |cosθ
(14)
Is
Ir
=
(11)
(12)
ACKNOWLEDGEMENT
The authors would like to thank the A.E. Rowden
White Foundation for their generous, ongoing financial support of this research. The use of the MATPOWER software package is gratefully acknowledged:
http://www.pserc.cornell.edu/matpower/.
R EFERENCES
Fig. 22. Per-phase equivalent circuit of SCIG
Q = 3|Vs ||Is |sinθ
(15)
where Zin is the input impedance, Rs is the stator resistance,
Rr is the rotor resistance, X1 is the stator reactance, X2 is
the rotor reactance, s is the slip, Is is the stator current, Ir
is the rotor current, Z2 is the rotor impedance and Va is the
voltage across the magnetizing reactance. P and Q are the
real and reactive power, respectively.
Fig. 23. Per-phase equivalent circuit of DFIG
The following equations represent the DFIG:
Vs = Rs Is + jX1 Is + jXm (Is + Ir )
Vr
Rr
=
Ir + jX2 Ir + jXm (Is + Ir )
s
s
(16)
(17)
The power generated is the sum of the stator and rotor
generated powers:
Ps + jQs = 3Vs Is∗
Pr + jQr = 3Vr Ir∗
(18)
(19)
Pe = P s + P r
(20)
where the subscripts are as described earlier. The following
Fig. 24. Per-phase equivalent circuit of DDSG
equations represent the DDSG:
Vφ = EA − jXs IA − RA IA
(21)
where Vφ is the per-phase terminal voltage, EA is the internal
generated voltage, XS is the synchronous reactance, IA is the
current, and RA is the stator resistance. The power generated
can be expressed as:
P = 3Vφ IA cosθ
Q = 3Vφ IA sinθ
(22)
(23)
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