1
Comparative Stability Analysis of DFIG-based
Wind Farms and Conventional Synchronous
Generators
J. C. Muñoz, and C. A. Cañizares, Fellow, IEEE
Abstract—This paper presents a comparative stability analysis
of conventional synchronous generators and wind farms based on
double feed induction generators (DFIG). Based on an
appropriate DFIG wind generator model, PV curves, modal
analysis and time domain simulations are used to study the effect
on system stability of replacing conventional generation by
DFIG-based wind generation on the IEEE 14-bus benchmark
system, for both fixed power factor and voltage control
operation. The results show that the oscillatory behavior
associated with the dominant mode of the synchronous generator
is improved when the DFIG-based wind turbine is connected to
the system; this improvement in the damping ratios is more
evident when the wind turbines are operated with terminal
voltage control.
Index Terms— Power system stability, Wind power generation,
Synchronous generators.
I. INTRODUCTION
N
OWDAYS, wind power energy is increasingly penetrating
ele electrical grids. This penetration is mainly driven by
policies, global warming concerns and better wind
technologies.
The control capabilities of these new
technologies are continuously improving to satisfy grid code
requirements, ensuring a safe operation under normal and
fault conditions. Double feed induction generators (DFIGs) is
one of the most commonly used technologies nowadays, as
these offer advantages such as the decoupled control of active
and reactive powers and maximum power tracking. These
capabilities are possible due to the power electronic converters
used in this type of generator.
When the penetration of wind generation is high, it is
important to keep these generators on line as much as possible
during grid disturbances as per grid code requirements.
Therefore, there is a significant interest in investigating the
dynamic performance and characteristics of the system under
high penetration of wind generation.
Various studies have been carried out regarding modeling of
DFIG for stability analysis. In [2]-[6], different models of
DFIG-based wind generator farms are discussed and
simulations are performed. The tuning of the parameters of the
This work has been supported in part by MITACS and NSERC, Canada;
and the Universidad de Los Andes, Merida-Venezuela.
J.C. Muñoz and C. A. Cañizares are with the Department of Electrical and
Computer Engineering, University of Waterloo, Waterloo, ON, Canada,
N2L3G1 (jcmunozg@uwaterloo.ca, ccanizar@uwaterloo.ca).
DFIG controllers is also addressed in various papers. Thus, in
[11], a tuning method to optimise the parameters of the DFIG
controllers is proposed to improve small-and large-disturbance
stability performance. In [12], a methodology to tune damping
controllers based on eigenvalue sensitivities is presented.
Reference [13] studies the increase in system transient
stability margins when DFIG generators are introduced instead
of cage generators. A complete analysis of transient stability
considering the point of connection of the DFIG at
transmission, subtransmission and distribution levels is
presented in [14]. In [15], the impact of the increased
penetration of DFIG-based wind turbines on small and
transient stability is assessed by replacing the DFIG by
synchronous generators and evaluating the sensitivity of the
eigenvalues with respect to inertia. This methodology
identifies inter-area modes that are worsen, and
electromechanical modes whose damping is increased by the
penetration of DFIG based wind turbines. In [16], the steady
state voltage stability of power systems with high penetration
of wind turbines is studied using time-series ac power flow
techniques. The methodology used in [16] incorporates
resource and system assessment for wind power, unit
commitment and economic dispatch; the historical data of
loading and wind power output are time synchronized, and the
worst operating point is identified as the point when wind
generation feeds the largest portion of load. According to this
paper, the voltage control capabilities of the DFIG wind
turbines improve the voltage stability margin at distribution
and transmission levels. Also, the eigenvalues trajectories as a
function of the load for a power system containing DFIGs are
computed in [17]. In this paper, the authors conclude that the
DFIGs do not participate in the unstable modes associated
with oscillatory instability; a sensitivity analysis of the
eigenvalues with respect to the parameters of the active and
reactive power controllers is also carried out. Finally, in [18],
a four generator test system is used to study the effect of
replacing one synchronous generator by a wind farm on the
oscillation modes of the system, resulting in an increase of the
stability of the observed modes when wind farms are
connected.
The present paper presents a comparative study of the effect
on stability of DFIG-based wind turbines vis-a-vis
conventional synchronous generators. PV curves are used for
analyzing static load margins, and the effect on the damping
ratio of the dominant mode of oscillation for the DFIG
operating at fixed power factor and terminal voltage control.
2
, : Stator and rotor resistances;
: Stator self-reactance;
: Rotor self-reactance;
: Mutual reactance;
ω : Rotor speed.
The wind turbine, generator shaft, and the gearbox is
modeled in [19] as a lumped inertia ; therefore, the motion
equation can be represented by:
(5)
Fig.1. DFIG overall scheme
The presented studies are based on the IEEE 14-bus
benchmark system. Thus, in this system, one of the
synchronous generators is replaced by an aggregated DFIGbased wind turbine of equivalent size. Simulations are carried
out using the Matlab-based toolbox (PSAT) [19], which
includes power flow, optimal power flow, continuation power
flow, small-disturbance stability and time domain simulation
tools.
The rest of this paper is organized as follows: Section II
presents a brief description of the PSAT DFIG model and
associated controls. In Section III the study methodologies
used in this paper are briefly discussed. Section IV presents
and discusses the obtained simulations results. Finally, in
Section V the main conclusions of the present work are
discussed.
II. DFIG MODEL
The overall scheme for a wind farm based on DFIG is
depicted in Fig. 1. Thus, it is composed by two voltage fed
PWM converters in back-to-back configuration. These
converters allow the decoupled control of the active and
reactive power flow between the DFIG and the ac network by
adjusting the switching of the IGBTs. For this structure, the
equations of the double feed induction generator in terms of
the d and q axes and neglecting the stator and rotor flux
transients can be written as [19]:
•For the stator circuit:
(1)
where
: Mechanical torque;
: Electromagnetic torque.
This simplification in the inertia is valid only if it is assumed
that the controllers associated to the DFIG(s) are able to
quickly minimize the shaft oscillations [19]. The
electromagnetic torque is represented by:
(6)
Vector control schemes decouple the control of active and
reactive power in the rotor. Thus, the active power P derived
ω is
from the wind turbine power-speed characteristic
associated with the rotor current in the q axis as follows:
ω
(7)
whereas the reactive power Q is associated with the rotor
current in the d axis trough the following voltage control
equation:
(8)
where
: Actual terminal voltage;
: Desired terminal voltage.
This controller uses the current rotor speed to optimize the
energy extracted from the wind. Furthermore, for rotor speeds
greater than 1 p.u., the power is set to 1 p.u. and for rotor
speeds lower than 0.5 p.u. the power is set to zero. The limits
for the rotor currents are then computed in PSAT as follows:
(9)
(2)
(10)
•For the rotor circuit:
where
,
,
,
1
ω
(3)
1
ω
(4)
: d and q axes stator voltages;
: d and q axes stator currents;
: d and q axes rotor currents;
(11)
(12)
These limits are carefully selected to ensure a proper dynamic
and steady state operation of the model.
3
Fig.3. DFIG Collector system (PSAT).
loading level slowly changes. Here, the damping ratios are
used to identify proximity to these bifurcations, which is
defined for the i-th eigenvalue of the state matrix α
σ
ω as follows:
γ
Fig. 2. IEEE 14 bus system (PSAT).
Four state variables can be identified in the DFIG model used
, , , and β. Where β is the pitch angle. The
in [19]:
pitch angle control only operates for super-synchronous
speeds, and for sub-synchronous speeds the pitch control is
locked. For the speed ranges used here, this pitch angle is
inactive and hence, not considered in this paper.
It should be mentioned that the dynamics of the converter
are fast and are neglected. Therefore, the converter is
represented as a current source.
The wind is modeled by using the Weibull distribution
available in [19], with a shape factor equal to two, which
results in a Rayleigh Distribution.
III. STUDY METHODOLOGY
The theoretical static load margin is computed in this paper
by using PV curves. These curves are obtained in PSAT by
means of continuation power flows; this method uses
predictor-corrector steps to ensure convergence of the
nonlinear algebraic equations that describe the power system,
avoiding the singularity of the Jacobian matrix near the
maximum loading point.
The eigenvalues from the linearization of the differential
algebraic equations that describe the dynamic operation of the
power system are used to perform stability studies around the
equilibrium points of the PV curves [20]. This paper focuses
in small oscillatory phenomena, which can be associated with
Hopf Bifurcations (HB). These types of bifurcations are
identified by the presence of a complex pair of eigenvalues
crossing the imaginary axes of the complex plane when the
σ
√ σ
ω
(13)
Moreover, by using participation factors, a better
understanding of the states that influence the dominant or
critical modes can be achieved [20].
Time domain simulations are also carried out using PSAT
with the aim of studying the system dynamic behaviour under
contingency (large-disturbance) operation. These simulations
are based on the numerical integration of the differentialalgebraic equations that describe the dynamic operation of the
system, and also allow to study the effect on all system
variables of wind speed variations.
IV. RESULTS
The following three study cases are addressed here:
• Case A corresponds to the IEEE 14-bus system with
synchronous generators, as depicted in Fig. 2. This
benchmark system, described in detail in [21], is comprised
of two synchronous generators providing active and reactive
power connected at Buses 1 and 2, and three synchronous
condensers connected at Buses 3, 6 and 8. Automatic voltage
regulators (AVR) Type II are incorporated in each machine.
The model for the synchronous generator connected at Bus 1
is a 5th order model, and the models for the generator
connected at Bus 2 and all the synchronous condensers are
6th order models. The system base load is 259 MW and
81.4 MVAR. In all simulations, the load are represented
using exponential recovery dynamic models.
• In Case B, the 60 MVA synchronous generator located at
Bus 2 in Case A is replaced by an aggregated DFIG-based
wind turbine of equivalent size and limits, operating at unity
power factor. The corresponding collector system is shown
in Fig. 3; two transformation stages are modeled: one from
480 V to 25 kV and the other one from 25kV to 69kV.
Detailed data for the DFIG-based wind turbine, wind model
and collector system can be found in the Appendix.
• In Case C, the DFIG is assumed to operate under terminal
voltage control.
4
Fig. 4. PV curves for normal operation.
Fig. 5. PV curves for contingency operation.
Normal and single contingency (Line 2-4 trip) operation are
considered for each study case.
A. Case A
Figures 4 and 5 depict PV curves for normal and
contingency operation respectively. Observe that the static
load margin in this case under normal operating condition is
435 MW, while for contingency operation this margin is
reduced to 391 MW. Moreover, HBs are identified for a
loading level of 341 MW and 329 MW for normal and
contingency operation, respectively. The participation factors
associate the AVR of the generator connected at Bus 1 with
this oscillatory instability, which has a frequency of 1.32 Hz
for normal operating condition and 1.31 Hz for contingency
operation.
Time domain simulations for a Line 2-4 trip at 1s at a
loading level of 332 MW is shown in Fig. 6. Notice from this
figure the oscillatory instability predicted by the eigenvalue
analysis.
Fig. 6. Bus 14 voltage.
B. Case B
The static load margin for Case B under normal operation is
depicted in Fig. 4. This margin is 424 MW, which is 2.53%
lower than the static load margin for Case A. This reduction is
mainly due to the DFIG unity power factor operation, which
does not allow for the control of the terminal voltage. As it can
be seen from Fig. 5, a similar observation can be made for the
system under contingency conditions, where the static load
margin becomes 382 MW, which represents a 2.30%
reduction with respect to Case A for the same operating
conditions.
The eigenvalue analysis for Case B shows an important
improvement in the dominant mode damping ratios associated
with the generator connected at Bus 1. Indeed, for the same
loading levels corresponding to the HBs in Case A (341 Mw
and 329 MW), the damping ratios become 1.91% and 1.82%
for normal and contingency operation, respectively, at similar
frequencies as before. The participation factors do not link the
DFIG state variables with the oscillatory modes, which is
consistent with the observations reported in [17].
If the loading level is increased, an HB can be observed at a
386 MW total load level in normal operating condition. This
corresponds to a 25% increase in the dynamic load margin
with respect to Case A. Moreover, HBs are not observed for
contingency conditions.
The above eigenvalue discussion is consistent with the time
domain simulations depicted in Fig. 6. Thus, observe that the
oscillations in this case are completely damped after 25 s.
C. Case C
The DFIG-based wind turbine with terminal voltage control
operation delivers the reactive power required to keep the
voltage at terminals constant at 1.09 p.u. The limits for this
reactive power are set so that they are similar to the replaced
synchronous generator limits. As can be seen in Fig. 4, the
static load margin for Case C under normal operating
condition is 431 MW; this value is slightly lower than the
Case A static load margin, but 1.6% greater than the margin
observed for the DFIG operating at unity power factor.
Moreover, in contingency operation, the static load margin for
5
TABLE I
DOMINANT- MODE DAMPING AS A FUNCTION OF THE DFIG VOLTAGE
CONTROLLER GAIN (NORMAL OPERATION CONDITION AT 341 MW
LOADING)
Dominant-mode Damping
Ratio (%)
10
1.67
11
1.86
12
2.05
13
2.23
14
2.41
15*
2.59
16
2.76
17
2.92
18
3.08
19
3.24
20
3.39
* Base DFIG voltage controller gain
Fig.7. DFIGs output power.
TABLE II
DOMINANT- MODE DAMPING AS A FUNCTION OF THE WIND POWER
PENETRATION (NORMAL OPERATION CONDITION AT 341 MW LOADING)
Wind Power
Dominant-mode
Penetration (MW)
Damping Ratio (%)
5
0.89
10
1.14
15
1.39
20
1.64
25
1.88
30
2.12
35
2.36
40*
2.59
45
2.81
50
3.04
* Base synchronous generator power output
Fig. 8. Wind speed.
Fig. 9. Reactive power output.
Case C becomes 387 MW, which is 1.3% greater than the
corresponding margin for unity power factor operation.
The normal and contingency operation dominant mode
damping ratios for the same loading levels associated with the
HBs in Case A are 2.59% and 2.50%, respectively. These
values are roughly 74% greater than those obtained for the
DFIG with unity power factor operation. As a result,
oscillatory instabilities are not observed in Case C.
Figure 6 further demonstrates the better damping response in
this case by means of time domain simulations. Observe that
oscillations are damped faster than in Case B.
Figure 7 illustrates the power output variations associated
with wind speed changes for the DFIG operating at terminal
voltage control. The output power oscillation at 1s is mostly
the result of the voltage drop when Line 2-4 is tripped; after
this oscillation is damped, the output power is controlled to
optimize the energy extracted from the wind speed shown in
Fig. 8. Figure 9 depicts the DFIG reactive power support;
notice that the reactive power changes to regulate the terminal
voltage and thus increase system security.
Results of a sensitivity study of the dominant-mode damping
ratios with respect to the DFIG voltage controller gain
can
be seen in Table I. Observe that there is a linear correlation
and the oscillatory mode damping ratios.
between the gain
This correlation suggests that by properly tuning the voltage
controller gain, DFIGs equipped with voltage control
6
TABLE IV
WIND MODEL PARAMETERS
capabilities can properly damp the oscillatory modes,
eliminating the occurrence of oscillatory instability associated
with HBs.
Table II shows the sensitivity of the dominant mode
damping ratio with respect to the wind power penetration.
These results suggest that as wind power penetration
increases, the damping ratio of the dominant mode improves.
For this study, the wind power output connected at Bus 2 was
gradually increased from 5 MW to 50 MW at a 342 MW
loading level. These wind power output levels range between
1.33 to 15% of the total system generated power.
Wind model type
Average wind speed vωA (m/s)
Air density ρ(kg/m3)
Filter time constant τ(s)
Sample time for wind measurements Δt
(s)
Scale factor for Weibull distribution c
Shape factor for Weibull distribution k
Frequency step Δf (Hz)
Weibull
Distribution
14.50
1.225
4
0.1
20
2
0.2
TABLE IV
COLLECTOR SYSTEM PARAMETERS
IV. CONCLUSIONS
A comparative stability analysis based on PV curves, modal
analysis and time domain simulations of DFIG-based wind
generators replacing synchronous generators has been carried
out for the IEEE-14 bus system. The obtained results show
that the oscillatory behaviour associated with the dominant
mode of the synchronous generator is improved when the
DFIG-based wind turbine is connected to the system, which is
consistent with similar observations by other authors. This
improvement in the damping ratio is more evident for DFIG
wind turbines operating with terminal voltage control.
Moreover, the static load margins are not significantly affected
when the DFIG-based wind turbine with voltage control
operation replaces an equivalent synchronous generator; the
small differences could be accredited to the impedances of the
collector system. However, when the DFIG is operated with
unity power factor, static load margins are reduced; thus,
negatively affecting the system security.
First transformation stage
Voltage ratio (kV/kV)
0.480/25
Resistance (p.u.)
0.00
Rectance (p.u.)
0.1
Fixed tap ratio (p.u./p.u.)
1.00
Second transformation stage
Voltage ratio (kV/kV)
25/69
Resistance (p.u.)
0.00
Rectance (p.u.)
0.1
Fixed tap ratio (p.u./p.u.)
1.00
Transmission line
Length of Line (km)
0*
Resistance (p.u.)
0.035
Reactance (p.u.)
0.017
Susceptance (p.u.)
1e-3
* Zero indicates to PSAT that the line parameters
are given in p.u.
REFERENCES
APPENDIX
[1]
TABLE III
DFIG PARAMETERS
Power rating Sn (MVA)
Voltage rating Vn (kV)
Frequency rating fn (Hz)
Stator resistance Rs (p.u.)
Stator reactance Xs (p.u.)
Rotor resistance Rr (p.u.)
Rotor reactance Xr (p.u.)
Magnetization reactance Xm (p.u.)
Initial constant Hm (kWs/kVA)
Pitch control gain Kp (p.u.)
Pitch control time constant Tp (s)
Voltage control gain KV (p.u)
Power control time constant Te (s)
Rotor radius R (m)
Number of poles p
Number of Blades nb
Gear box ratio ηGB
Pmax (p.u.)
Pmin (p.u.)
Qmax (p.u.)
Qmin (p.u.)
60.0
0.480
60.0
0.01
0.10
0.01
0.08
3.00
3.00
10
3
15
0.01
75
4
3
1/89
0.9
0.0
0.35
-0.219
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