International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011
Modal Analysis of DFIG-based Wind Farms
Considering Converter Controllers
Milad Yadollahi, Amir Abbas Shayegani Akmal, and Abbas Eskandari
problems have to be investigated.
Several types of WTGs are commercially available today
including: squirrel cage induction generators (SCIG), doublyfed induction generators (DFIG) and permanent magnet
synchronous generators (PMSG). Two more general
categories are fixed-speed and variable-speed turbines. The
dominant technology today, however, is the DFIG.
Some research work has been conducted to study the
stability issues corresponding to the WFs. In [2], an approach
for sensitivity analysis of electromechanical modes (EMM) to
the inertia of the generators is introduced and the results have
shown both detrimental and beneficial impacts of increased
DFIG penetration into the power system. Inter-area and local
oscillations have been investigated in [3] in a test system; the
results are reported for both the variable and fixed-speed
turbines replacing the conventional generation units. It is
shown that the damping of EMMs is improved as the level of
penetration is increased. However, in the case of large
amounts of installed WFs, there is a possibility for the interarea modes to be adversely affected.
Different control strategies for DFIGs and their impacts on
the system stability were investigated in [4]. It is shown that
DFIGs operating in frequency control mode and the SCIGs
could increase the eigenvalues damping.
Different control modes for DFIGs including VAR control
mode, power factor control mode and voltage control mode
were extensively analyzed in [5]. The impacts of DFIGs on
the EMMs are demonstrated to be highly dependent on their
control strategies. The authors in [6] have reported the
likelihood of the contribution of DFIGs to the system damping
with application of appropriate controllers.
Besides the mentioned studies, other related studies are also
available in the literature with similar conclusions [7], [8]. All
the aforementioned research was conducted with the
assumption that wind speed is fixed. In other words, the
simulations are performed for different WF capacities. It is,
however, known that a WF output power may significantly
alternate throughout a day; thus, affecting power system
variables such as voltages and tie-line flows. Therefore, in this
paper, the effects of changing WF output power (due to
changing in wind speed) on system small signal stability are
examined.
Modal analysis is carried out in [9] for DFIG-based WF and
the impacts of system parameters, operating points and grid
strengths are studied. However, the controllers for rotor-side
and grid-side converters are not modeled which highly affect
Abstract— Wind power generations are now being
installed widely in the power systems throughout the world.
Due to the differences between conventional synchronous
generators and doubly-fed induction generator (DFIG) –based
wind farms (WFs), the impacts of WFs on power system
stability should be investigated. In this paper, the impacts of
DFIG-based WF on power system small signal stability are
analyzed. The converter controllers are modeled and the
effects of controller parameters on system dominant modes are
studied in a single machine infinite bus (SMIB) system. The
appropriate values for controller parameters are decided
according to the eigenvalue analysis results. Besides, the
impacts of shaft model and its parameters on dominant modes
are evaluated.
Keywords— Doubly-fed induction generator, Power system
small signal stability, Eigenvalue analysis.
I. INTRODUCTION
T
HE document rate of rise of installed wind power
generation capacity has been substantially increased
worldwide. Compared to the other renewables, wind energy
has received more attention with higher growth. The
remarkable advancements which have been reached during the
past decade in wind power generation technology along with
the environmental issues pertaining to the fuel-consuming
power plants have made the wind power one of the most
economical resources for replacing the conventional power
plants.
Wind power integration, in turn, will introduce new issues
to the existing power systems in several aspects, especially
when the level of penetration is significant. Among these
aspects, the system stability is from crucial importance.
Conventional generators are generally synchronous machines
with high inertia, coupled to the steam or hydro turbines. The
reciprocal effects of these types of machines were broadly
studied in detail [1]. Since the nature of the wind turbine
generators (WTG) is basically different from the synchronous
generators due to the power electronic interfaces and
induction generators application, the well-known stability
Milad Yadollahi, Amir Abbas Shayegani Akmal and Abbas Eskandari are
with the Department of Electrical and Computer Engineering, University of
Tehran, Tehran, Iran (phone: +98 911 320 0647; e-mails:
m.yadollahi@ut.ac.ir, shayegani@ut.ac.ir, abbasskandari@yahoo.com).
161
International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011
θt
the results. The mentioned shortcoming is addressed in
present study.
This paper is organized as follows: Section II describes the
modeling procedure of WTG and introduces the test systems
used in this work. Section III presents the results of small
signal analysis in a SMIB system. The main findings of this
paper are reported in Section IV.
θg
Ktg
Jg
Jt
Te
Tm
II. WTG MODELING PROCEDURE
A WTG consists of blades, rotor shaft, gearbox, the
induction generator and the interface with the network. The
blades and gearbox are modeled as the mechanical part, which
converts the wind energy into the mechanical torque applied
to the induction generator. The power extracted from wind
speed by the turbine is given by [10]:
Pm = c p (λ , β )
ρA
2
Dtg
Fig. 1 Two-mass model of the windmill.
Rs
Vds
v w3
 c2

− c3 β − c 4  e λ + c 6λ ,
 λi

c p ( λ , β )= c1 
i
1
=
λi
L′ls
Lls
1
λ − 0.08β
−
0.035
β 3 +1
i′dr
J gθg + Dtgθg + K tgθ g =
−Te + Dtgθt + K tgθt
(4
)
ωλds
L′ls (ω-ωr)λ′dr
Lls
V′dr
R′r
i′qr
iqs
Lm
Vqs
V′qr
Fig. 2 Equivalent d-q circuits for the induction machine.
Te =
Values of the parameters c 1 to c 6 are given in [10]. A
typical turbine speed–output power characteristic, which is
used in this study, is given in [11].
The generated power is then given to the generator through
the shaft as the prime mover torque (T m ). It has been shown
that the single mass model for the shaft is not suitable for
stability studies in case of squirrel cage induction generator
[12]. Therefore, the two-mass model shown in Fig. 1 is used.
Considering this model, the following equations are derived
for the motion of rotating mass:
(3
)
R′r
Lm
Rs
(2
)
J tθt + Dtgθt + K tgθt =
Tm + Dtgθg + K tgθ g
(ω-ωr)λ′qr
ids
(1)
where P m is the mechanical output power of the turbine (W),
c p is the performance coefficient of the turbine, ρ is the air
density (kg/m3), A is the turbine swap area (m2), v w is the
wind speed (m/s), λ is the ratio of the rotor blade tip speed to
wind speed and β is the blade pitch angle (deg). c p is related to
λ and β as:
c5
ωλqs
3 p
( ) Lm (iqs idr′ − ids iq′r )
2 2
(5
)
where p is the number of pole pairs.
B. DFIG Controllers
Figure 3 portraits the general structure of DFIG-based
WTG. Two converters, namely the rotor-side converter (RSC)
and the grid-side converter (GSC), are coupled through a DC
capacitor.
With the aid of RSC, the frequency and magnitude of the
injected voltage to the rotor windings are determined. The
block diagram representation of the RSC and GSC controllers
are shown in Fig. 4. The subscript gc refers to the GSC. All
the regulators are proportional-integral (PI) controllers with
constants K p and K i . The switch shown in Fig. 4 for RSC is
used to select between the VCM and VARCM. Details about
the control system are given in [11], [14].
The GSC is able to absorb or generate reactive power and
keep the DC bus voltage constant. Parameters used for
aforementioned controllers are reported in Appendix.
in which J t and J g are the turbine and generator moment of
inertias; θ t and θ g are the angular displacement of turbine and
generator; K tg and D tg are the shaft stiffness and mutual
damping, respectively; and T m and T e are the mechanical and
electrical torques, respectively.
Gearbox
A. Induction Machine Modeling
The asynchronous machine model is obtained from [13]. A
4th order model is used for the electrical part with equivalent
circuit shown in Fig. 2. The electrical torque is calculated as:
GSC
Grid
RSC
Controller
Fig. 3 General structure of DFIG-based WTG.
162
International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011
The time constants of pitch angle controller are large
compared to power system transient period and the wind
speed is kept between the allowable values. Therefore, this
controller is not modeled here.
C. Test System Description
A single machine infinite bus (SMIB) model is employed to
study the WF regardless of the network. This system is shown
in Fig. 5. All parameters are given in the figure and the WF
parameters are given in Appendix. For simplicity, the
aggregated model of small wind turbines in a WF is used here
which does not affect the results [15]. The corresponding data
for each WTG is given in Appendix.
Fig. 5 Single machine-infinite bus test system
TABLE I
OPERATING POINT FOR DFIG IN SMIB TEST SYSTEM
ωg (p.u.) P (MW) Q (MVAr) Wind speed (m/s) V (p.u.)
0.965
3.3
Small signal analysis is performed here using the concepts
of linearizing the system equations at the operating point and
eigenvalue analysis. The system equations are described in the
following general form:
(7)
Iq_GSC (4.15%), RLq (28.3), RLd (67.5%)
λ2 = -2066.28
Id_GSC (4.19%), RLq (57.9%), RLd (24.5%),
λ4 = -344.53
λ5 = -180.07 ± 30.03i
(8)
where A is the system state matrix. This matrix is then used
for calculating the system eigenvalues.
V
Voltage
Reg.
Droop
Xs
VAR
Meas.
I
(Vdr, Vqr)
Qref
Current
Meas.
Ir
ωr
Idr
Current
Reg.
ωr
Is&Ir
Igc
Power
Meas.
Power
Reg.
Iqr_r
Power
Losses
DC
Vol.Reg.
VDC
Idgc_ref
VDC_ref
Igc
Current
Meas.
Idg
Current
Reg.
Id_GSC (6.13%)
ψqr (28.9%), ψdr (33.5%), ψqs (10.3%),
Id_RSC (5.6%), Iq_RSC (6.1%)
CDC Bus (5.97%), VDC (2.09%), Id_GSC
(88.11%)
ψqr (6.86), ωg (42.9%), Preg (39.95%),
Iq_RSC (5.31%)
λ9 = -27.96 ± 0.12i
CDC Bus (5.28%), Iq_RSC (44.37%), VDC
λ10 = -30.85
ψqr (2.48%), ψdr (11%), Id_RSC (85.74%)
λ11 = -0.12 ± 3.01i
ωt (49.9%), θt (49.2%)
(44.55%)
In the present work, the Linearization block in
MATLAB/SIMULINK is used for calculating this matrix at
the operating point. Considering the model described above
for DFIG, the resulted state space model for the SMIB test
case is an 18-variable system.
A 9 MW WF consisted of 6×1.5 MW turbines is assumed
in the SMIB system. At the operating point given in Table 1,
small signal analysis is carried out and the results are reported
in Table 2. In this table, the following notations are used:
ψ dr , ψ qr , ψ ds , ψ qs : Rotor and stator fluxes on the d and q
axis
I d_GSC , I q_GSC : d and q axis components of grid-side
current regulator
I d_RSC , I q_RSC : d and q axis components of rotor-side
current regulator
RL d , RL q : d and q axis components of coupling inductor
C DC Bus , V DC : DC bus capacitor and voltage regulator
P reg : Active power regulator
ω g , ω t , θ t : generator and turbine rotational speeds and
Iqr
Power
Tracking
V
I
ψqr (6.34%), RLq (11%), CDC Bus (62.6%),
λ7 = -96.51
Idr_ref
VAR Reg.
(38.8%)
RLd (2.87%), RLq (1.27%), Iq_GSC (95.82%)
λ8 = -6.68 ± 46.96i
VARCM /
VCM
CDC Bus (13%)
ψdr (9.46%), ψqr (9.1%), ψqs (39.5%), ψds
λ6 = -104.32
Vref
I
1.02
λ1 = -2409.4
λ3 = -176.8 ± 558.2i
where x, z and u are vectors of state variables, control output
variables and input variables, respectively. After performing
the linearization procedure around the current operating point,
the following relation is derived [1]:
Voltage
Meas.
10
TABLE II
EIGENVALUES AND CORRESPONDING PARTICIPATION FACTORS FOR THE DFIG
IN SMIB SYSTEM
III. SMALL SIGNAL STABILITY ANALYSIS
V
0.5
Vgc
Iqg
Iqr_ref
Fig. 4 Up: Rotor-side converter controller; Down: Grid-side
converter controller.
163
International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011
Imaginary part (rad/s)
angular displacement
There are two dominant modes in this system, namely the
mechanical mode (λ 11 ) and the electromechanical mode (λ 8 ).
The impacts of the control system parameters on the system
modes are studied in the next section.
A. Wind Speed
Variation of wind speed would change the generated power
and thus the operating point of the system. Figure 7 shows the
variation of the dominant eigenvalues by increasing the wind
speed from 8 to 14 m/s. The electromechanical mode heads to
the right, while the mechanical mode is not considerably
affected.
100
λ
8
80
60
40
-80
-70
-60
-50
-40
-30
-20
-10
0
10
Imaginary part (rad/s)
Real part
3.0045
λ
3.004
11
3.0035
3.003
3.0025
3.002
-0.126
-0.125
-0.124
-0.123
-0.122
-0.121
-0.12
-0.119
-0.118
Real part
B. Active Power Regulation
The active power regulator is a PI controller which
determines the power delivered into the grid. The proportional
gain is increased from 0.3 to 4.5 and the resulted eigenvalue
trajectory is depicted in Fig. 8. The integral gain is also
increased from 100 to 220 and Fig. 9 shows the results. It can
be seen that increasing K p has beneficial impacts while
increasing K i has detrimental effects.
Fig. 8 Dominant mode trajectories when K p for power regulator is
increased from 0.3 to 4.5.
Imaginary part (rad/s)
60
50
45
-7
Imaginary part (rad/s)
46.5
-6.4
-6.2
-6
-5.8
-5.6
λ
3.0026
0
1
λ
11
3
-0.121
-0.12
-0.119 -0.118 -0.117 -0.116 -0.115 -0.114 -0.113 -0.112 -0.111
λ
55
8
50
45
-9
-8
-7
-6
Real part
-5
-4
-3
-2
72
11
data2
λ8
70
68
3.0024
Imaginary part (rad/s)
Imaginary part (rad/s)
3.0028
-1
Fig. 10 Electromechanical mode trajectory when K p for rotor-side
current regulator is increased from 0.1 to 7.
Real part
3.003
-2
3.01
40
-10
-6.6
-3
60
47
-6.8
-4
Real part
data2
-7
-5
3.02
λ8
-7.2
-6
3.03
Fig. 9 Dominant mode trajectories when K i for power regulator is
increased from 100 to 220.
47.5
46
λ8
55
Imaginary part (rad/s)
Imaginary part (rad/s)
65
Real part
C. Rotor-Side Current Regulator
Both the K p and K i for rotor-side current regulator are
varied and only the electromechanical mode was influenced.
This is depicted in Figs. 10 and 11. It is obvious that
increasing K p from 0.1 to 7 is of interest. Also, increasing Ki
from 5 to 12 is beneficial. No significant impact on
mechanical mode was observed.
48
70
3.0022
3.002
3.0018
3.0016
66
64
62
60
58
56
3.0014
3.0012
54
-0.13
-0.125
-0.12
-0.115
52
-20
-0.11
-15
-5
-10
0
5
Real part
Real part
Fig. 11 Electromechanical mode trajectory when K i for rotor-side
current regulator is increased from 5 to 12.
Fig. 7 Dominant mode trajectories when wind speed is increased
from 8 to 14 m/s.
164
International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011
(a)
4.32
Imaginary part (rad/s)
4
λ11
3.5
1.5
Terminal Voltage Regulator
3
Ki
KP
X droop
300
1.25
0.02
Active Power Regulator
Ki
KP
100
1
R
2.5
2
1.5
-0.18
-0.14
-0.16
-0.04
-0.06
-0.08
-0.1
-0.12
80.27
R
R
Real part
R
R
0.685
1200
60
0.0015 0.15
RSC Current
DC Bus Voltage Regulator
Regulator
Ki
KP
Ki
KP
8
0.3
0.05
0.002
Grid-Side Converter Current Regulator
Ki
KP
100
1
R
R
R
R
R
R
(b)
Imaginary part (rad/s)
6
λ
5
REFERENCES
11
4
[1]
[2]
3
2
1
-0.17
-0.1
-0.11
-0.12
-0.13
-0.14
-0.15
-0.16
Real part
Imaginary part (rad/s)
Fig. 12 Mechanical mode trajectory when (a) H t is increased from 3
to 13 s; (b) K tg is increased from 20 to 280.
[3]
3.005
[4]
3
2.995
2.99
[5]
λ11
2.985
2.98
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
[6]
-0.05
Imaginary part (rad/s)
Real part
47.05
47
[7]
46.95
46.9
λ8
46.85
46.8
-8.5
-8
-7.5
-7
[8]
-6
-6.5
Real part
[9]
Fig. 13 Dominant mode trajectories (D tg is increased from 0.5 to 6).
D. Mechanical Parameters
The mechanical part of the turbine is shown in Fig. 1.
Increasing H t would displace the mechanical mode toward
lower stability and increasing Hg would do the same for
electromechanical mode. On the other hand, increasing K tg
would make the opposite impact, as shown in Fig. 12.
Mutual damping (D tg ) increment has made beneficial
impacts on both mechanical and electromechanical modes, as
depicted in Fig. 13.
[10]
[11]
[12]
[13]
[14]
IV. CONCLUSION
[15]
Small disturbance stability of DFIG-based WFs was
investigated. Modeling procedure for converter controllers
was described. The impacts of changes in wind speed and
influential controller parameters on dominant modes of DFIG
were studied. The results could be used for tuning the PI
controllers. In addition, the effects of shaft parameters were
studied. The results can be used in manufacturing process to
select the mechanical parameters suitable for power system
stability.
[16]
[17]
[18]
[19]
APPENDIX
P (MW)
1.5
V (V)
575
R
R
D tg
R
K tg
R
Hg
R
L' lr
Lm
0.156 0.92
Coupling
DC Bus
Inductor
(V) C DC (mF) R GS
L GS
R
Turbine Shaft
Ht
[20]
Induction Generator
f (Hz)
Rs
L ls
R' r
60 0.00706
0.171 0.005
V DC
R
R
R
R
R
R
R
R
[21]
R
165
P. Kundur, Power system stability and control, McGraw-Hill, 1994.
D. Guatam, V. Vittal, T. Harbour, Impact of increased penetration of
DFIG-based wind turbine generators on transient and small signal
stability of power systems, IEEE Trans. Power Syst., vol. 24, no. 3, pp.
1426-1434, 2009.
J J.G. Slootweg, W.L. Kling, The impact of large scale wind power
generation on power system oscillations, Electr. Power Syst. Res., vol.
67 no. 1, pp. 9-20, 2003.
R.D. Fernandez, R.J. Mantz, P.E. Battaiotto, Impact of wind farms on a
power system: An eigenvalue analysis approach, Renew. Energy, vol. 32
no. 10, pp. 1676–1688, 2007.
G. Tsourakis, B.M. Nomikos, C.D. Vournas, Effect of wind parks with
doubly fed asynchronous generators on small-signal stability, Electr.
Power Syst. Res., vol. 79, pp. 190–200, 2009.
G. Tsourakis, B.M. Nomikos, C.D. Vournas, Contribution of doubly fed
wind generators to oscillation damping, IEEE Trans. Power Syst., vol.
24, no. 3, pp. 783–791, 2009.
E. Hagstrøm, I. Norheim, K. Uhlen, Large-scale wind power integration
in Norway and impact on damping in the Nordic grid, Wind Energy, vol.
8, no. 3, pp. 375–384, 2005.
L. Fan, Z. Miao, and D. Osborn, Impact of doubly fed wind turbine
generation on inter-area oscillation damping, Proc. IEEE Power Eng.
Soc. Gen. Meeting, pp. 1–8, 2008.
F. Mei and B. Pal, Modal analysis of grid-connected doubly fed
induction generators, IEEE Trans. Energy Convers., vol. 22, no. 3, pp.
728-736, 2007.
T. Ackerman, Wind power in power systems, John Wiley & Sons,
London, UK, 2005.
MATLAB Help Tutorial, The Math Works, Inc. Version 7.8.0.347,
2009.
S. K. Salman and A. L. J. Teo, Windmill modeling consideration and
factors influencing the stability of a grid-connected wind power-based
embedded generator, IEEE Trans. Power Syst., vol. 18, no. 2, pp. 793802, 2003.
P.C. Krause, O. Wasynczuk, and S.D. Sudhoff, Analysis of electric
machinery, IEEE Press, 2002.
N. W. Miller, W. W. Price, J. J. Sanchez-Gasca, Dynamic modeling of
GE 1.5 and 3.6 wind turbine-generators, GE Power Systems, Ver. 3
2003.
L.M. Fernández, C.A. García, J.R. Saenz, F. Jurado, Equivalent models
of wind farms by using aggregated wind turbines and equivalent winds,
Energy Conv. Man., vol. 50, no. 3, pp. 691-704, 2009.
Z. Miao, L. Fan, D. Osborn, S. Yuvarajan, Control of DFIG-based wind
generation to improve interarea oscillation damping, IEEE Trans. Energy
Convers., vol. 24, no. 2, pp. 415-422, 2009.
Y. Mishra, S. Mishra, M. Tripathy, N. Senroy, Z. Y. Dong, Improving
stability of a DFIG-based wind power system with tuned damping
controller, IEEE Trans. Energy Convers., vol. 24, no. 3, pp. 650-660,
2009.
B. Badrzadeh, S. K. Salman, Critical clearing time of induction
generator, Power Tech Con., Russia, pp. 1-7, 2005.
H. Müller, M. Pöller, Ch. Eping, J. Stenzel, Impact of large scale wind
power on power system stability, Fifth International Workshop on
Large-Scale Integration of Wind Power and Transmission Networks for
Offshore Wind Farms, Glasgow, Scotland, April 2005.
F. Wu, X.-P. Zhang, P. Ju, Small signal stability analysis and control of
the wind turbine with the direct-drive permanent magnet generator
integrated to the grid, Electr. Power Syst. Res., vol. 79, pp. 1661-1667,
2009.
Siegfried Heier, Grid integration of wind energy conversion systems,
John Wiley & Sons Ltd, 1998.