Energy and Power Engineering, 2013, 5, 429-433
doi:10.4236/epe.2013.54B083 Published Online July 2013 (http://www.scirp.org/journal/epe)
Small Signal Stability Analysis for a DFIG-Based Offshore
W ind Farms Collected Thr ough VSC-HVDC
Transmission System
Kai Liao, Zhengyou He, Bin Sun, Yong Jia
School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China
Received April, 2013
ABSTRACT
This paper modeled a doubly fed induction generator (DFIG) - based offshore wind farm integrated through a voltage
source converter –based high voltage direct current (VSC-HVDC) transmission system, which is collected with infin ite
bus for small signal stability analysis. The control system of HVDC system is considered for the stability analysis. The
impact of the VSC control parameters on the network stability is studied. The lineared dynamic model is employed to
do small signal stability analysis by th e eigenvalue analysis. The locus of th e eigenvalue, which is corresponding to the
oscillation model is studied. Time domain si mulations conducted in Matlab/Simulink are used to validate the small sig-
nal stability analysis.
Keywords: DFIG; Offshore Wind Farm; VSC-HVDC; Small Signal Stability
1. Introduction
Because the advantages in speed control, reduced flicker,
and four-quadrant active and reactive power capabilities,
are primarily achieved via control of a rotor side con-
verter. Many large offshore wind farms based on DFIG
(Doubly fed induction generator) have been planned
around the world wide [1, 2]. The High-voltage dc (HVDC)
transmission is emerging as the prospective technology
to address the challenges associated with the integration
of future offshore wind power [3].
In [4] and [5], a VSC transmission system was used to
connect a 6-MWwind farm to the grid. VSC transmission
systems were also proposed in [6], for transmitting off-
shore wind power equipped fixed speed generators to the
grid. However, many large wind farms under develop-
ment will employ DFIG-based wind turbines whose op-
eration and response to net-work disturbances are sig-
nificantly different from other types of generators. In [7],
line commutated HVDC systems were used to connect a
large DFIG-based offshore wind farm into the grid. In [li
xue] described the use of VSC-HVDC transmission sys-
tem technology for connecting large DFIG-based wind
farms over long distance. New control strategies for
normal and grid fault conditions are proposed. To obtain
smooth operation, the wind farm side VSC is controlled
as an infinite voltage source that automatically absorbs
power generated by the wind farm and maintains a stable
local ac network.
The dynamic behavior of the doubly fed induction ge-
nerator (DFIG) has been investigated in many papers.
The majority of these studies are based on time-domain
simulations to show the impact on power syste m dynam-
ics [8,9], the performance of decoupled control and
maximum power tracking, the response to grid distur-
bances, the control methods to make the DFIG behave
like a synchronous generator [10-13], etc. Time-domain
studies offer a direct appreciation of the dynamic behav-
ior in terms of visual clarity. However, the time domain
simulation can’t observe all oscillation modals [14 ].
In [15], a model suitable for small-signal stability
analysis and control design of multi-terminal dc networks
is presented. A generic test network that combines con-
ventional synchronous and offshore wind generation
connected to shore via a dc network is used to illustrate
the design of enhanced voltage source converter (VSC)
controllers. The impact of VSC control parameters on
network stability is discussed and the overall network
dynamic performance assessed in the event of small and
large pert urbations .
In this paper, the grid-connected DFIG via VSC-HVDC
system is studied. The single-machine infinite-bus
(SMIB) approach is followed. The paper is organized as
follows. In Section II, the mathematical model is ob-
served. In Section III, modal analysis method and the
eigenvalue locus under different controller parameters is

K. LIAO ET AL.
430
studied. Sectio n IV p rese nt s the conclusion.
2. Modeling of Study System
The study system is shown in Figure 1 where a DFIG-
based wind farm (100MW form aggregation of 2 MW
units) is connected to a VSC-HVDC link.
2.1. DFIG Generator
The one-phase equivalent electric circuit of the DFIG is
shown in Figure 2. The dynamic equations of DFIG are
usually described by transforming the machine ‘abc’
voltage equations into a synchronously rotating frame,
referred to as the ‘d-q’ frame.
For stability analysis, the generato rs are modeled as an
equivalent voltage source based on transient impedance.
The DFIG is modeled as
122
qs elel rel
qsel dsqsds
ss ss rss
elmrr el
qs qr
ss ss
di wRww w
iwi ee
dtw LwL TwL
wKw
vv
wL wL

 

(1)
122 dselel rel
el qsdsdsqs
ss ss rss
elmrr el
ds dr
ss ss
diw Rw ww
wi iee
dtw LwL TwL
wKw
vv
wL wL

 

(2)
2()
qs els r
el dsqselds
rs s
mrrel dr
de www
wRie we
dtTww
Kwv




(3)
2()
dselr s
el dsdselqs
rs s
mrrel qr
deww w
wRiewe
dtT ww
Kwv


 

(4)
Figure 1. The studied system structure.
Figure 2. The equivalent electric circuit of the DFIG.
where ds e

and ds e

are the equivalent internal q- and
d-axis voltages, respectively; ids and iqs are the stator q-
and d- axis currents, respectively. All the parameters are
converted to p.u.
2.2. Converter
Figure 3 illustrated a converter circuit diagram in ‘abc’
frame.
The dynamic model of the converter in ‘abc’ frame
can be modeled as
caaasa
cbbbsb
cccc sc
uii
d
uLiRiu
dt
uii
u
u












(5)
Use the Park transform as shown in (6), the dynamic
model in ‘d-q’ frame is shown in (7).
sinsin(120 )sin(120 )
2
3coscos(120 )cos(120 )
wt wtwt
Pwt wtwt











(6)
11
0
0
mdcd mdsd
mqcq mqsq
md
mq
iviu
dR
iviu
dtL L L
i
i

















(7)
where, Vc is the converter voltage and Vs is the grid volt-
age. Under PWM control, the amplitude of the converter
output fundamental voltage is controlled by the modula-
tion index as
/2
cdc
VMV

 (8)
In ideal condition, the dc side transmission syste m can
be expressed as
13
()
4
dc dcd mdmq
dV
I
Mi Mqi
dt CC
  (9)
where, the imd and imq are the d-and q- axis converter
current, respectively. M is the modulation index. Vdc is
the DC voltage and Idc is the DC current.
dl
I
dc
I
2C


2C
d
u
s
a
u
s
b
u
s
c
u
ca
u
cb
u
cc
u
m
I
R
L
Figure 3. Converter circuit diagram.

K. LIAO ET AL. 431
2.3. Controller of Wind Farm Side Converter
The aim of the wind farm side converter controller sys-
tem is to maintain the wind farm network work at con-
stant frequency and voltage. One of the primary require-
ments for the WFVSC is to collect energy from the wind
farm. The output power of DFIG’s is contro lled by pow-
er electronic converters and MPPT system. The network
frequency variations have little influence on the power
generation. Therefore, to simplify the control system
design, the control strategy adopted here is to control the
wind farm side converter to resemble an infinite voltage
source with constant frequency, voltage amplitude, and
phase angle. The control block diagram for the wind
farm side converter is shown as Thus, as in the case
when a wind farm is connected to an infinite ac system,
the power generated by the wind farm is automatically
absorbed by the source resembled by the wind farm side
converters and then transmitted to the grid via the DC
lines. The main tasks for the WFVSC are then to collect
energy from the wind farm and to control the ac voltage
and frequency of the local wind farm network.
2.4. Controller of Grid Side Converter
The wind farm side converter collects energy from wind
farm and then transmit it to the power grid via the dc
transmission and grid side converter. For the normal op-
eration of a VSC transmission system, its dc link voltage
must be maintained at a constant value under all condi-
tions. A constant dc voltage indicates balanced active
power flow between the two sides. Abnormal dc link
voltage can cause the system to trip and disrupt its nor-
mal operation. Furthermore, to achieve this balance, the
grid side converter is assigned to control the dc voltage,
to ensure the energy collected by the wind farm side
converter is transmitted to the grid network. The control
system of the grid side conv erter is shown as Figure 5.
The current control loop is designed as:
11
()()
sd
dpsd sdisd sd
di
ukiikii
dt

 
dt
(10)
Figure 4. Control block diagr am of the wind farm side con-
verter.
11
()()
sq
qp sqsqisqsq
di
ukiikii
dt

 
dt
(11)
s
d
i

is designed as the dc voltage control loop and
s
q
i

is
designed as the connection bus voltage control loop.
33
()(
sdpdc dcidc dc
ikVVkVVd
 
 
)t
t
(12)
22
()()
sqpGGiGG
ikVVkVVd
 
 
 (13)
According to (9), the modulation index M in dq frame
are give n as (14) and (15).
21
()
ddsdssq
dc
LR sd
M
ui iv
VL L
  (14)
21
()
qqsqssd
dc
LR sq
M
ui i v
VL L
  (15)
2.5. DFIG Control
The DFIG control is comprised by two PWM modulation
inverters connected back to back via a dc link. The con-
trol system is to ensure the stator frequency of DFIG
operates at a constant value, and constraint for maximum
power capture. So, the rotor side converter operates as a
controlled voltage source since it injects an ac voltage
with varying frequency to the rotor to keep the stator
voltage frequency be a constant value under varying
wind speed. The ac voltage of the rotor-side converter
depends on the control objectives. For grid-connected
-
dc ref
V
PI
-
dref
I
dc
V
-
ac ref
V
PI
-
qref
I
ac
V
r
I
i
I
PLL
()cos ref
s( )in ref
d
I
q
I
md
P
mq
P
PI
PI
mr
P
mi
P
Inverter-side Controller
Current Controller
Voltage Controller
Figure 5. Control block diagram of the grid side converter.
dc,ref
V
2
2s
i
p
k
k
s
d
i

dc
V
d
u
G,ref
V
q
u
G
V
3
3
s
i
p
k
k
1
1
s
i
p
k
k
1
1
s
i
p
k
k
s
d
i
s
q
i

s
q
i
Figure 6. The control system of the grid side converter.

K. LIAO ET AL.
432
WECS applications, a sensible choice is to impose a con-
straint for maximum power capture (equivalent to air gap
power, electromagnetic torque, or speed constraint) and
another for the voltage control (reactive power con-
straint). These two objectives determine the DFIG rotor
voltage.
For the grid-side converter, the control has to be coor-
dinated so that the dc-link voltage is constant and the
desired sharing of reactive power with stator is achieved.
Usually, for minimum the converter rating, the reactive
power delivered to the grid only from the stator. So, the
there is no reactive power delivered to the grid and the
rotor side converter works at unity power factor.
In this paper the dynamic of the DFIG rotor is ignored.
The rotor side voltage, electromagnetic torque, rotor side
current and reactive and active power is constant.
3. Modal Analysis
The most direct way to assess small-signal stability is via
eigenvalue analysis of a model of the power system. In
this case, the “small-signal” disturbances are considered
sufficiently small to permit the equations representing the
system to be linearized and expressed in state-space form.
The model of a power system can be expressed as a set
of DAE. The linearized model of the test system can be
expressed in state-space form as
 xxB
u (16)
where x is the where is the state vector, u is the input
vector, A is the state matrix, and B is the input or control
matrix. The eigenvalue of the state matrix provide the
necessary information about the small-signal stability of
the system.
The purpose of th is study is to observe the influence of
VSC-HVDC control parameters on the small signal sta-
bility of the studied system. Vary the value of Kp1,
which is employed in the DC line voltage control in
VSC-HVDC system. The corresponding eigenvalue dis-
placements are shown in Figure 7. With a larger Kp, the
observations indicate that better damped for the oscilla-
tion modal.
Varying the control parameters of Ki, which can also
employed to DC voltage regulations. The corresponding
eigenvalue displacements are shown in Figure 8. The
analysis result indicates that the small Ki are better
damped and their oscillation frequencies are lover com-
pared to with large Ki controller.
The time domain simulation result is shown in Figure
9. Under a single phase fault in ac transmission line, the
transmission power of the bus, which collected the VSC
-HVDC system and wind farm operate at different con-
trol parameters is shown in the figure. The result of the
time domain simulation validates the results obtained
from the small-signal stability analysis above.
Figure 7. The eigenvalue locus under varying Kp.
Figure 8. The eigenvalue locus under varying Ki.
Figure 9. The time domain simulation result.
4. Conclusions
A lineared mathematical model for small signal stability
analysis of VSC-HVDC transmission system collected
with a DFIG based wind farm has been presented in this
paper. The lineared model is based the state-space mod-
els. The state matrix is employed to investigate the small
signal stability performance of the studied system
through the eigenvalue analysis. The eigenvalue locus
under different HVDC system control parameters is ob-
served. It was validated that using the small-signal stabil-
ity model, it was possible to design improved controllers
for the VSCs of the multi-terminal dc network, which
ensure stable network operation and enhanced dynamic

K. LIAO ET AL.
433
performance. The time domain simulation also validated
the analysis results.
REFERENCES
[1] R. Pena, J. C. Clare and G. M. Asher, “Double Fed In-
duction Generator using Back-to-back PWMconverter
and its Application to Variable Speed Wind Energy Gen-
eration,” Proceedings Institution Electric Engineering,
Electric Power Applications, Vol. 143, No. 3, 1996, pp.
231-241. doi:10.1049/ip-epa:19960288
[2] T. Ackermann, “Transmission Systems for Offshore
Wind Farms,” IEEE Power Engineering Review, Vol. 22,
No. 12, 2002, pp. 23-27.
doi:10.1109/MPER.2002.1098040
[3] W. Lu and B. T. Ooi, “Optimal Acquisition and Aggrega-
tion of Offshore Wind Power by Multiterminal Volt-
age-source HVDC,” IEEE Transactions Power Delivery,
Vol. 18, No. 1, 2003, pp. 201-206.
doi:10.1109/TPWRD.2002.803826
[4] K. H. Sobrink, P. L. Sorensen, P. Christensen, N. San-
dersen, K. Eriksson, and P. Holmberg, “DC Feeder for
Connection of a Wind Farm,” In Proceedings Cigre
Symp., Malaysia, 1999.
[5] X. I. Koutiva, T. D. Vrionis, N. A. Vovos, and G. B.
Giannakopoulos, “Optimal Integration of an Offshore
Wind Farm to a Weak Grid,” IEEE Transactions Power
Delivery, Vol. 21, No. 2, 2006, pp. 987-994.
doi:10.1109/TPWRD.2005.859275
[6] D. Xiang, L. Ran, J. R. Bumby, P. Tavner, and S. Yang,
“Coordinated Control of an HVDC Link and Doubly Fed
Induction Generators in a Large Offshore Wind Farm,”
IEEE Transactions Power Delivery, Vol. 21, No. 1, 2006,
pp.463-471.doi:10.1109/TPWRD.2005.858785
[7] L. Xu, L. Z. Yao, Christian Sasse. “Grid Integration of
Large DFIG-based Wind Farms using VSC Transmis-
sion,” IEEE Transactions Power System, Vol. 22, No. 3,
pp 2007, 976-984. doi:10.1109/TPWRS.2007.901306
[8] J. G. Slootweg, “Wind Power Modelling and Impact on
Power Systems Dynamics,” Ph.D. Dissertation, Delft
Univ. Technology, Delft, The Netherlands, 2003.
[9] V. Akhmatov, “Analysis of Dynamic Behaviour of Elec-
tric Power Systems with Large Amount of Wind Power,”
Ph.D. dissertation, Technology University Denmark,
Lyngby, Denmark, 2003.
[10] D. Alvira, J. Arevalo, C. Bermudez, R. Granadino, A.
Granda, C. Pincella, G.P. Stigliano, and R. Vailati, “Fea-
sibility Studies of the HVDC Submarine Interconnection
between the Spanish Peninsula and the Balearic Island of
Mallorca,” Presented at the 41th CIGREGen. Session,
Paris, France, 2006, Paper B4-104.
doi:10.1109/TEC.2004.832078
[11] M. V. A. Nunes, J. A. P. Lopes, H. H. Zurn, U. H. Bez-
erra, and R. G. Almeida, “Influence of the Variable-speed
wind Generators in Transient Stability Margin of the
Conventional Generators Integrated in Electrical Grids,”
IEEE Transctions Energy Conversion, Vol. 19, No. 4,
2004, pp. 692-701. doi:10.1109/TPWRS.2005.857275
[12] F. M. Hughes, O. Anaya-Lara, N. Jenkins, and G. Strbac,
“Control of DFIG-based Wind Generation for Power
Network Support,” IEEE Transactions Power System,
Vol. 20, No. 4, 2005, pp. 1958-1966.
[13] R. S. Pena, “Vector Control Strategies for a Doubly-fed
induction Generator Driven by a Wind Turbine,” Ph.D.
Dissertation, University Nottingham, Nottingham, U.K.,
1996.
[14] F. Mei and B. Pal, “Modal Analysis of Grid-connected
doubly-fed Induction Generators,” IEEE Transactions
Energy Conversions, Vol. 22, No. 3, 2007, pp. 728-736.
doi:10.1109/TEC.2006.881080
[15] G. O. Kalcon, G. P. Adam, Olimpo Anaya-Lara,
Stephen Lo, and Kjetil Uhlen. “Small-Signal Stabil-
ity Analysis of Multi-Terminal VSC-Based DC
Transmission Systems,” IEEE Transactions Power
System, Vol. 27, No. 4, 2012, pp.1818-1830.
doi:10.1109/TPWRS.2012.2190531