Electrical Power and Energy Systems 70 (2015) 70–82
Contents lists available at ScienceDirect
Electrical Power and Energy Systems
journal homepage: www.elsevier.com/locate/ijepes
Small signal stability enhancement of DFIG based wind power system
using optimized controllers parameters
Bhinal Mehta a,⇑, Praghnesh Bhatt a, Vivek Pandya b
a
b
Department of Electrical Engineering, C.S. Patel Institute of Technology, CHARUSAT, Changa, Gujarat, India
Department of Electrical Engineering, School of Technology, PDPU, Gandhinagar, Gujarat, India
a r t i c l e
i n f o
Article history:
Received 19 August 2014
Received in revised form 30 December 2014
Accepted 31 January 2015
Available online 21 February 2015
Keywords:
Wind turbine generators
Doubly fed induction generator
Particle swam optimization
Eigen value analysis
Low voltage ride through
a b s t r a c t
This paper presents the state space modelling of doubly fed induction generator (DFIG) for small signal
stability assessment. The gains of PI controller in torque and voltage control loop of rotor-side converter
(RSC) are optimized by particle swarm optimization (PSO) to improve the dynamic performance of DFIG.
These optimized parameters results in improved damping of DFIG and minimizes the oscillations in rotor
currents and electromagnetic torque. The nature of modes of oscillations for DFIG integrated to infinite
bus are analysed under different operating conditions such as varying wind speed and grid strength.
The transient analysis with optimized parameters shows the enhancement in LVRT capability during
voltage sag and three phase fault as desired by grid codes.
Ó 2015 Elsevier Ltd. All rights reserved.
Introduction
The use of wind power has increased significantly over recent
decades and its integration with the power system is now an
important topic of study. India ranks fifth amongst the wind energy producing countries of the world after USA, China, Germany and
Spain. The installed capacity of wind power in India has reached
about 20 GW by 2013. Estimated potential is around 49,130 MW
at 50 m above ground level and 102,788 MW at 80 m above ground
level [1].
The wind energy conversion system (WECS) could be
operationally classified into fixed speed and variable speed wind
turbine generating system (WTGS). In the early stage of wind power generations, most wind farms were equipped with fixed speed
induction generators (FSIG). The operation of FSIG is fairly simple
but it is unable to extract maximum power at varying wind speed
as its slip can be varied in a very small range. The development in
technology has encouraged to switch from the fixed speed WTGS
to variable speed WTGS mainly due to its advantages such as
improved efficiency for wider range of wind speeds, independent
control of active and reactive power, better fault ride through capability, etc. Currently the most common variable-speed wind turbine configurations are DFIG wind turbine and fully rated
converter (FRC) wind turbine based on a synchronous or induction
⇑ Corresponding author. Mobile: +91 9427045058.
E-mail address: bhinalmehta.ee@charusat.ac.in (B. Mehta).
http://dx.doi.org/10.1016/j.ijepes.2015.01.039
0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
Downloaded from http://www.elearnica.ir
generator. Amongst many variable speed concepts, the DFIG
equipped wind turbine is very popular as it has many advantages
over others like improved power quality, higher efficiency, the
power converter rating can be kept fairly low, approximately 25%
of the total machine power, more economical than a series configuration with a fully rated converter, the controllability of reactive power and thus it help DFIG equipped wind turbines play a
similar role to that of synchronous generators [2–5].
PI controllers are the most common controllers as they have
simple structure. Their performance greatly depends on an optimal
tuning of the gains. The tuning of the PI gains is very important
task and even more vital to have optimized performance for varying operating conditions. The sound knowledge of the dynamic
modelling of DFIG integrated to power system is required to adjust
the PI gains in order to have optimized performance of DFIG in normal operating conditions as well as under severe disturbance on
the system [6].
One of the major advantages of DFIG is to have decoupled
control of active and reactive powers with the use of different
control strategies for grid side and rotor side convertors. The
decoupled control of DFIG has the following controllers, namely
Pref; V sref ; V dcref , and qcref . These controllers are required to maintain
maximum power tracking, stator terminal voltage, dc voltage level,
and GSC reactive power level respectively. The coordinated tuning
of these controllers may or may not result into optimized performance of DFIG while adopting hit and trail method for tuning of
gains [7]. The coordinated tuning of the controllers to improve
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B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
the damping of the electromechanical mode in the DFIG was presented in [8] by replacing the DC link capacitor with battery energy
storage (BES) system which eliminated the oscillatory mode corresponding to DC link.
With increasing penetration level of DFIG type wind turbines
into the grid, it is very important to investigate the impacts of wind
turbine generating units on the power system stability [4,9]. The
wind farms are generally located far from demand centres where
the network is relatively weak and congested. Therefore, if the
integration and penetration of wind energy are not properly
assessed for the given network, it is difficult to maintain the reliability and stability [10]. In order to protect the security and operation of the transmission system, it is imperative to investigate the
impact of wind at various penetration levels.
A detailed investigation to analyse the small signal behaviour of
squirrel cage induction generator (SCIG), DFIG and permanent
magnet synchronous generator (PMSG) based wind turbines are
carried out in [11] to see how each turbine technology affects the
local, inter area, torsional and control modes of the system. The
comprehensive studies regarding the modelling of DFIG and to
identify their interaction with the power system have been reported in [12]. [13] shows the presence of DFIG can alter the local and
inter-area mode shapes and shows the improvement in dynamic
behaviour of multi machine power system.
Large voltage dip occurs as a result of a large network disturbance, such as a short circuit, and it may trigger a sequence of
other events in the network. Most of the countries have introduced
and implemented the gird code regulations to fulfil the fault ride
through requirements as the penetration level of wind power generation in the power system has increased drastically. In [14], the
importance of the correct design of the control system is discussed
where, an adequate adjustment of the PI gains helps to limit the
generator currents during a fault. Hence the operations of the
crowbar can be avoided and the convertors continuously remain
in operation.
PSO is an evolutionary computation technique, motivated by
the simulation of social behaviour. In searching the optimal solution of a problem, information of the best position of each individual particle and the best position among the whole swarm are
used to direct the searching. Hence, in comparison with GA, PSO
is quite immune to local optima and is reasonably efficient in solving problems with complex hyperspace [15].
This paper presents the state space model of DFIG to study its
small signal and transient performance. The dynamic performance
of DFIG under different wind speeds and system disturbances for
strong and weak grid are presented in the paper. Therefore, the
objectives of the paper are as follows:
1. To formulate the state space model of DFIG connected to infinite bus for small signal and transient stability assessment.
2. To optimize the controllers gains by PSO for dynamic performance improvement of DFIG.
3. To investigate the impact of optimized controllers’ gains on the
nature of modes of oscillations with varying wind conditions
and with different strength of transmission network.
4. To investigate the fault ride through capability of DFIG with
optimized controllers gains under fault and voltage sag
conditions.
The paper is organized in six sections. Section ‘Mathematical
modelling of DFIG’ presents the modelling concepts of wind turbine
generating system associated with DFIG along with its control
strategies. Interfacing of DFIG with infinite bus is discussed in Section ‘Interfacing of DFIG with infinite bus’. Section ‘Particle swarm
optimization’ defines the objective function and describes the PSO
algorithm for the optimization of the controller gains. Section
‘Results and discussions’ details the approach to analyse the impact
of DFIG on small signal and transient stability along with results and
discussions of different scenarios. The conclusion is drawn in
Section ‘Conclusion’.
Mathematical modelling of DFIG
Fig. 1 shows the schematic of a DFIG connected to an infinite
bus through transmission line and transformer. The DFIG is wound
rotor induction generator whose stator is directly connected to a
power grid. The rotor of DFIG is connected to the power grid
through IGBT based controlled back-to-back voltage source converters. The rotor-side converter (RSC) controls the injected rotor
voltage that allows the control of the electromagnetic torque,
which must follow the reference speed provided by the control
system. It can also provide reactive power control and voltage control or power factor control of the machine. This ensures the variable speed operation of DFIG with maximum power point tracking
characteristics. The grid side converter (GSC) is connected to the
grid through a grid-side filter and is used to control dc-link voltage
and reactive power exchange with the grid. Thus GSC represents a
shunt power converter. As RSC can provide reactive power control,
GSC may offer additional voltage support capabilities in conditions
of excessive speed ranges or in transient operations. In this work
for the proposed model only the control of RSC is discussed and
it was assumed that the dc link voltage between the converters
is kept constant by converter. Thus in order to evaluate the performance of the DFIG based scheme proposed to control the RSC,
operational aspects concerning to the GSC control will not be
depicted in detail because it is not the main objective of this work.
Induction generator model
Assumptions:
The following assumptions are made while modelling the
induction generator.
(1) Stator current is negative when flowing toward the machine,
i.e. generator convection is used.
(2) Equations are derived in the synchronous reference frame.
(3) q-axis is 90° ahead of the d-axis.
The stator of the induction machine carries three-phase windings. The windings produce a rotating magnetic field which rotates
at synchronous speed. The dynamic equations for stator and rotor
in d–q reference frame rotating at synchronous speed [3,16,17] are
described in (1)–(3).
Stator voltage equations:
"
v ds ids
uqs
1 d uds
¼ Rs
þ xs
þ
xb dt uqs
v qs
iqs
uds
#
ð1Þ
Rotor voltage equations:
"
v dr idr
uqr
1 d udr
¼ Rr
þ s xs
þ
xb dt uqr
v qr
iqr
udr
#
ð2Þ
Flux equations:
uds ¼ X ss ids þ X m idr
uqs ¼ X ss iqs þ X m iqr
udr ¼ X rr idr X m ids
uqr ¼ X rr iqr X m iqs
ð3Þ
The expression for the stator and rotor currents as the state
variables are obtained by substituting the flux Eqs. (3), into the stator and rotor voltage Eqs. (1) and (2), respectively.
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B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
Three Winding
Transformer
MV
AC Side
Slip
Rings
V∞
HV
Re
Xe
LV
Gear
Box
Infinite Bus
AC low voltage and
Variable frequency
DC
PWM Converter I
(Rotor Side
Converter)
(RSC)
PWM Converter II
(Grid Side
Converter)
(GSC)
Fig. 1. Schematic of DFIG integrated with infinite bus.
and the rated value, the rotor speed reference can be obtained by
substituting k into (4) as follows:
Wind turbine model for DFIG
To complete the induction generator state model, it is necessary
to combine the equations that describe electrical voltage and current components of the machine with swing equation that provides rotor speed as state variable. In power system studies,
drive trains are modelled as a series of rigid disks connected via
mass less shafts.
Maximum Power Point Tracking (MPPT)
The aim of the DFIG wind turbine is to extract maximum power
from the wind. The mechanical power Pw extracted from the wind
is given by (4) [3,18].
Pw ¼
Tm ¼
q
cp ðk; bÞAr v 3w
2
Pw
xm
ð4Þ
ð5Þ
where q is the air density, v w is the wind speed, b is the pitch angle,
Ar is the area swept by the rotor, k is the blade tip speed ratio and b
is the blade pitch angle.
The power coefficient and the tip-speed ratio describe the performance of a wind turbine rotor. In fact, the maximum power
coefficient is only achieved at a single tip speed ratio kopt and if
can
the turbine is operated at variable speed, xr a maximum cmax
p
be achieved over a range of wind speeds [3,18].
The performance co-efficient or the power co-efficient cp is represented as follows:
c2
C p ¼ c1
c3 b c4 ec5 ki þ c6 k
ki
ð6Þ
where
ki ¼
1
0:035
b3 þ1
1
kþ0:08b
ð7Þ
C p ðk; bÞ has a maximum cmax
for a particular tip speed ratio kopt and
p
pitch angle b = 0. A typical wind turbine characteristics and maximum power point extraction curve is shown in Fig. 2(a). The speed
control of the DFIG is achieved by driving the generator speed along
the optimum power-speed characteristic curve (intersect cmax
for
p
each wind speed) shown in Fig. 2(a), which corresponds to the maximum energy capture from the wind. The complete generator torque speed characteristics in shown in Fig. 2(b). When generator
speed is less than the low limit or higher than the rated value, the
reference speeds is set to the minimal value or rated value,
respectively. When generator speed is between the lower limit
k¼
vt
2Rxr
¼g
v w GB pv w
ð8Þ
where v t is the blade tip speed, v w the wind speed, gGB is the gear
box ratio, p is the number of poles of the induction generator and R
is the wind turbine blade radius.
Lumped mass model
For small signal stability analysis of synchronous generators in
conventional power plants, the one mass or lumped mass model is
used because the drive train behaves as a single equivalent mass.
The participations of all inertias, which include the rotating masses
of turbine and generator rotor, are nearly equal. Hence the mode of
interest is non-torsional [16]. Similarly in case of WTGS, drive train
can be represented by the lumped mass system, where there is
only a single inertia which is equivalent to the sum of the generator rotor and the wind turbine. The mathematical equation of
a one-mass model is given by (9).
dxr
1
¼
ðT m T e Þ
2Htot
dt
ð9Þ
where Htot is the total inertia of generator rotor Hg and wind turbine
Ht ; T e is the electromagnetic torque and T m is the mechanical
torque, T e and T m are given by (10) and (11).
T e in terms of the state variables
T e ¼ X m ðidr iqs iqr ids Þ
Tm ¼
C pðpuÞ V 3wðpuÞ
xrðpuÞ
Vw
V w base
Cp
¼
C p nom
ð10Þ
ð11Þ
V wðpuÞ ¼
ð12Þ
C pðpuÞ
ð13Þ
Control strategies for a DFIG
The PVdq control scheme is employed for the control of DFIG.
The rotor current is split into two orthogonal components, d-axis
and q-axis. The q-axis component of the current is used to regulate
the torque and the d-axis component is used to regulate power factor or terminal voltage of DFIG [3].
Torque control scheme
The purpose of the torque controller is to modify the electromagnetic torque of the generator according to wind speed
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B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
Rated power
Generator power
v = 10 m/s
v = 8 m/s
v = 6 m/s
v = 4 m/s
v = 2 m/s
cut-in speed
speed
limit
Generator speed
Generator speed
(a)
(b)
Tsp
ωr
v = 12 m/s
Generator torque
Maximum
power curve
Rated
torque
Fig
2(b)
xss
v qr '
x1
iqr ,ref
+
xmωs vs
∫
K i1
−
shutdown
speed
+
vqr
+
+
+
Te Vs Wr
Characteristics
K p1
iqr
s
idr
vs
2
⎡⎛
xm vs ⎤
x ⎞
s ⎢⎜ xrr − m ⎟ idr +
⎥
xss ⎠
xssωs ⎦
⎣⎝
(c)
vs
x3
ref
+
vs
−
∫
K p3
x2
idr ,ref
+
+
+
∫
Ki 2
−
vdr
+
+
'
vdr
+
−
K p2
1
xmωs
idr
s
(d)
s
iqr
⎡⎛
2
xm ⎞ ⎤
−
x
⎜ rr
⎟ iqr
xss ⎠ ⎦
⎣⎝
⎢
⎥
Fig. 2. Control strategies for DFIG: (a) characteristic for Maximum Power Point Tracking (MPPT) and (b) generator torque speed characteristics (c) torque control scheme (d)
voltage control scheme.
variations and drive the system to the optimal operating point reference. Given a rotor speed measurement, the reference torque T sp
is provided by the wind turbine characteristic for maximum power
extraction as shown in Fig. 2(b). With this computed value of T sp , a
reference rotor current in the q axis i.e. iqr;ref can be found as shown
in Fig. 2(c). The rotor voltage v qr required to operate DFIG at the
reference torque set point T sp is obtained through a summation
of the term obtained by PI controller and the compensation term.
This compensation term is required to minimize cross-coupling
between speed and voltage control loops.
Neglecting the stator resistance and stator transients from (1)
and use of uqs and uds from (3) into (1), we get d-axis and q-axis
stator voltages as follows:
v ds ¼ xs ðX ss iqs þ X m iqr Þ
v qs ¼ xs ðX ss ids þ X m idr Þ
ð14Þ
ð15Þ
With the use of (14) and (15) and q-axis and d-axis stator current
are obtained as (16) and (17), respectively
Xm
iqr
X ss
1
X
¼
v þ mi
xs X ss qs X ss dr
iqs ¼
ids
1
xs X ss
v ds þ
ð16Þ
ð17Þ
Eq. (16) can be represented by (18) after considering d-axis component of stator voltage v ds ¼ 0 as stator flux oriented reference frame
is used.
iqs ¼
Xm
iqr
X ss
ð18Þ
With the use of (17) and (18) in (10), the electromagnetic magnetic
torque of DFIG is derived in (19).
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B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
Xm
1
X
T e ¼ X m idr
iqr iqr v qs þ m idr
X ss
xs X ss
X ss
ð19Þ
With the help of the reference torque set point T sp , the reference qaxis current iqr;ref is given by (20).
iqr;ref ¼
xs X ss
T sp
X m v qs
ð20Þ
The complete block diagram of torque control scheme is shown in
Fig. 2(c). All the variables shown in the block diagram are in per
unit.
The use of udr from (3) and ids from (17) in (2), the final
equation of q-axis rotor voltage v qr can be obtained as (21) after
neglecting the transient term from (2).
!
v qr ¼ Rr iqr þ sxs
X rr X 2m
Xm
idr v
X ss
xs X ss qs
ð21Þ
From Fig. 2(c), (22) can be obtained as follows
v 0qr ¼ x1 þ K p1 ðiqr;ref iqr Þ
idr;ref ¼ x3 þ
X m xs
x3 ¼ K p3 ðv s;ref v s Þ
v ds ¼ v d1 X T iqs þ RT ids
v qs ¼ v d1 þ X T ids þ RT iqs
ð31Þ
ð32Þ
where
X T ¼ X TR þ X e
ð33Þ
RT ¼ Re þ RS
ð34Þ
where v q1 and v d1 are q and d axis components of infinite bus
voltage.
State variables for control loops of DFIG shown in Fig. 2(c) and
(d) are represented by (35)–(37)
x_ 1 ¼ K i1 iqr þ ðK i1 X ss =xs X m Þ
ð22Þ
Voltage control scheme
The voltage control or power factor control at the terminal of
DFIG is achieved through the rotor side converter. The terminal
voltage will increase or decrease with the change in reactive power
delivered to the grid. The voltage controller should fulfil the following requirements in such a situation: (i) the reactive power consumed by the DFIG should be compensated and (ii) if the terminal
voltage is too low or too high compared with the reference value
then idr;ref , for d axis rotor current should be adjusted appropriately.
The required d-axis rotor voltage v dr is obtained through the output
of a PI controller, v 0dr minus a compensation term to eliminate
cross-coupling between control loops [3]. The complete block diagram of the DFIG terminal voltage controller is shown in Fig. 2(d).
All the variables shown in the block diagram are in per unit.
v 0dr ¼ x2 þ K p2 ðidr;ref idr Þ
vs
System matrix A, Control matrix B, Output matrix C and Feed
forward matrix D for the induction machine are represented in
Appendix C.
In order to consider the integration of DFIG to the transmission
network, the stator voltage equations are represented in (31) and
(32).
ð23Þ
ð24Þ
ð25Þ
T sp
ð35Þ
vs
x_ 2 ¼ K i2 x_ 3 K i2 idr þ ðK i2 =xs X m Þv s
ð36Þ
x_ 3 ¼ K p3 v s þ K p3 v sref
ð37Þ
The state variables and control inputs for DFIG after the integration
to transmission network are given in (38) and (39), respectively.
T
x_ ¼ ½ids iqs idr iqr xr x2 x1 x3 u1 ¼ ½idr iqr
v s v s;ref
T sp ð38Þ
T
ð39Þ
The complete system matrix Asys of DFIG connected to the infinite
bus for small signal stability analysis is represented in (40). All
the elements of system matrix Asys are listed in Appendix D.
2
Asys
3
A11
A12
A13
A14
A15
A16
A17
A18
6 A21
6
6
6 A31
6
6A
6 41
¼6
6 A51
6
6A
6 61
6
4 A71
A22
A23
A24
A25
A26
A27
A32
A33
A34
A35
A36
A37
A42
A43
A44
A45
A46
A47
A52
A53
A54
A55
A56
A57
A62
A63
A64
A65
A66
A67
A72
A73
A74
A75
A76
A77
A28 7
7
7
A38 7
7
A48 7
7
7
A58 7
7
A68 7
7
7
A78 5
A81
A82
A83
A84
A85
A86
A87
A88
ð40Þ
Interfacing of DFIG with infinite bus
Particle swarm optimization
The test system for the analysis is shown in Fig. 1 where DFIG is
integrated to the infinite bus through the transmission line. The
infinite bus is considered as voltage source of constant voltage
and constant frequency. To carry out the small signal stability analysis of the system shown in Fig. 1, the linearization of the induction machine equations given in (1) and (2) and the rotor
mechanical equation given in (9) are presented in (26) and (27).
The system state space representation can be described by (26)
and (27). The system state matrix after the integration of DFIG to
infinite bus is represented in (40) and its elements are listed
Appendix D. The eigenvalues of this state matrix Asys is used to
design the objective function.
Design of the objective function
x_ ¼ Ax þ Bu
ð26Þ
y ¼ Cx þ Du
ð27Þ
where
T
x_ ¼ ½ids iqs idr iqr xr ð28Þ
In general, input and output vectors for the system under considerations are defined as follows:
u ¼ ½v ds
v qs v dr v qr T
y ¼ ½idr iqr T
ð29Þ
ð30Þ
The PI gains of the RSC controller are to be optimized in such a
manner that some degree of relative stability and damping of electromechanical modes of oscillations can be achieved. Therefore, a
multi-objective optimization function, OF (Figure of Demerit) is
designed as given in (41) to have minimized undershoot, minimized overshoot and faster settling time of oscillations in the transient responses.
P
OF1 ¼ i ðr0 ri Þ2 , if ri P 0, r0 = 2.0, ri is the real part of the
th
i eigenvalue. The relative stability is determined by r0 . By optimizing OF1, closed loop system poles are thus consistently pushed
further left of jx axis with simultaneous reduction in imaginary
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B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
i
0
Table 2
Original and optimized controller parameters.
j
0
K p1
K i1
K p2
K i2
K p3
0
Original parameters
Optimized parameters
0.05
10
0.05
10
7
0.7
11.3367
0.5761
13.4624
10
Particle swarm optimization
0
The PSO was first introduced by Eberhart and Shi [22]. It is an
evolutionary optimization technique based on swarm intelligence.
The PSO is a population-based optimization technique, where the
population is called ‘swarm’. Based on PSO concept, mathematical
equations for the searching process are:
Velocity updating equation:
Fig. 3. D-shaped sector in the negative half of s – plane.
v kþ1
¼ v ki þ c1 r1 i
parts also, thus increasing the damping ratio above f0 (0.3). Finally,
all closed loop system poles should lie within a D-shaped sector
shown in Fig. 3.
P
OF2 = ðf0 fi Þ2 , if (imaginary part of the ith eigenvalue) > 0.0,
i
fi is the damping ratio of the ith eigenvalue and fi < f0 . Minimum
damping ratio considered, f0 = 0.3. Minimization of this objective
function will minimize maximum overshoot.
P
OF3 = i (imaginary part of ith eigenvalue), if ri P 1:0. High
value of imaginary part of ith eigenvalue to the right of vertical line
r0 = 2.0 is to be prevented. OF3 will be high if imaginary part of
ith eigenvalue is large.
OF4 = an arbitrarily chosen very high fixed value (say, 106),
which will indicate some ri values P 0.0. This means unstable
oscillation occurs for the transient responses.
So, multi-objective optimization function (Figure of Demerit),
OF ¼ 10 OF1 þ 10 OF2 þ 0:01 OF3 þ OF4
ð41Þ
pBesti xki þ c2 r2 gBest xki
ð42Þ
Position updating equation:
xkþ1
¼ xki þ v kþ1
i
i
ð43Þ
The following modifications help to enhance the global search ability of PSO algorithm.
Position and velocity updating
In (42), the second term on the right hand side represents the
personal behavior whereas the third term represents the social
behavior of the particles. As numbers r1 and r2 are generated randomly, they could be too large or too small. In such cases personal
and social experiences will be over used or not fully utilized.
Hence, to strike a balance between two, the velocity and position
equations are modified as follow.
v kþ1
¼ r2 signðr3Þ v ki þ ð1 r2Þ c1 r1
i
pBesti xki þ ð1 r2Þ c2 ð1 r1Þ
gBest xki
The weighting factors ‘10’ and ‘0.01’ are chosen to impart more
weight to OF1 and OF2 and to reduce high value of OF3, to make
them mutually competitive during optimization. All the closed
loop poles lie in the negative half plane of jx axis for which
ri 2:0, fi 0:3.
In (44), sign(r3) may be defined as,
Table 1
Initial condition for test system.
signðr3Þ ¼
1 ðr3 6 0:05Þ
ðr3 > 0:05Þ
1
ð44Þ
ð45Þ
Initial condition for test system
Scenario
Scenario 1 (40 MVA)
Scenario 2 (16 MVA)
Case
a
b
C
a
b
c
wr0
ids0
iqs0
idr0
iqr0
vds0
vqs0
vdr0
vqr0
vdsinf
vqsinf
Dvds
Dvqs
Te
vs0
Dvs
iqrref
0.8
0.024
0.35
0.229
0.367
0.035
0.999
0.02
0.206
0
1
0.0351
1.001
0.3516
0.9996
1.0016
0.3676
1.1
0.055
0.661
0.197
0.693
0.066
0.998
0.02
0.098
0
1
0.0664
1.0010
0.6653
1.0001
1.0032
0.6954
1.29
0.097
0.977
0.154
1.024
0.098
0.995
0.085
0.286
0
1
0.0984
0.9999
0.9873
0.9998
1.0047
0.9586
0.8
0.035
0.343
0.217
0.367
0.06
0.998
0.025
0.206
0
1
0.0604
0.9998
0.3449
0.9998
1.0016
0.3675
1.1
0.084
0.649
0.166
0.693
0.114
0.993
0.025
0.097
0
1
0.1146
0.9965
0.6559
0.9995
1.0031
0.6962
1.29
0.151
0.959
0.096
1.024
0.1169
0.986
0.105
0.28
0
1
0.1698
0.9902
0.9751
0.9929
1.0033
0.9706
The steps of proposed PSO based algorithms as implemented for
optimization of PI gains of RSC controller are listed below:
Generation of population.
Evaluation of Figure of Demerit of each particle as per (41).
Search for individual minimum Figure of Demerit and corresponding individual best particle (pBest).
Search for global minimum Figure of Demerit and its corresponding global best particle (gBest).
Velocity and position updating.
Searching for Individual best position updating and global best
position updating.
Iteration updating and stopping criteria.
The parameters used are c1 = 2.15, c2 = 2.05. The chosen maximum population size np = 50, maximum allowed iteration
cycles = 500.
76
B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
Table 3
Results of small signal stability analysis of DFIG using original controller parameters for Scenario 1 at different wind speed.
Strong connection (VAsc = 40 MVA)
Mode No.
Frequency of oscillation in Hz
Damping ratio
Most influential states in the control of the Mode with their % participation
CASE (a) Wr = 0.8 p.u.
k1, k2
53.42 ± 485.82i
k3, k4
29 ± 131.56i
k5, k6
24.15 ± 76.38i
k7
0.1
k8
0.68
Eigen values
77.24
20.92
12.14
0.00
0.00
0.11
0.22
0.30
1.00
1.00
ids = 26, iqs = 25.8, idr = 24.1, iqr = 24
ids = 23.5, iqs = 23.8, idr = 25.3, iqr = 25.7, x2 = 0.7, x1 = 0.8
ids = 22.5, iqs = 21, idr = 24.7, iqr = 23, x2 = 4.3, x1 = 4.1
wr = 99, x1 = 0.9
x3 = 99.7
CASE (b) Wr = 1.1 p.u.
k1, k2
46.42 ± 484.29i
k3, k4
39.11 ± 121.04i
k5, k6
20.77 ± 82.03i
k7
0.1
k8
0.67
76.99
19.24
13.04
0.00
0.00
0.10
0.31
0.25
1.00
1.00
ids = 26, iqs = 25.6, idr = 24.3, iqr = 24
ids = 23.2, iqs = 23.4, idr = 25, iqr = 25.3, x2 = 1.4, x1 = 1.5
ids = 23.6, iqs = 23, idr = 25.2, iqr = 24.5, x2 = 1.8, x1 = 1.8
wr = 99.5
x3 = 99.6
CASE (c) Wr = 1.29 p.u.
k1, k2
43.18 ± 483.58i
k3, k4
47.73 ± 160.93i
k5, k6
15.13 ± 61.96i
k7
0.07
k8
0.66
76.88
25.59
9.85
0.00
0.00
0.09
0.28
0.24
1.00
1.00
ids = 26, iqs = 25.5, idr = 24.4, iqr = 23.9
ids = 24.1, iqs = 23.2, idr = 25.9, iqr = 25, x2 = 0.8, x1 = 0.8
ids = 22.7, iqs = 22.8, idr = 24.2, iqr = 24.2, x2 = 2.8, x1 = 2.8
wr = 98.9
x3 = 99.4
Table 4
Results of small signal stability analysis of DFIG using original controller parameters for Scenario 2 at different wind speed.
Weak connection (VAsc = 16 MVA)
Mode No.
Frequency of oscillation in Hz
Damping ratio
Most influential states in the control of the mode with their % participation
CASE (a) Wr = 0.8 p.u.
k1, k2
75.91 ± 610.31i
k3, k4
22.8 ± 120.97i
k5, k6
19 ± 67.53i
k7
0.1
k8
1.17
Eigen values
97.03
19.23
10.74
0.00
0.00
0.12
0.19
0.27
1.00
1.00
ids = 26.3, iqs = 26.1, idr = 23.8, iqr = 23.6
ids = 23.2, iqs = 23.4, idr = 25.5, iqr = 26, x2 = 1, x1 = 1
ids = 22.1, iqs = 20.2, idr = 24.6, iqr = 22.4, x2 = 5.4, x1 = 5.1
wr = 98.9, x1 = 0.9
x3 = 99.3
CASE (b) Wr = 1.1 p.u.
k1, k2
68.83 ± 607.92i
k3, k4
32.3 ± 112.52i
k5, k6
15.95 ± 71.46i
k7
0.11
k8
1.14
96.65
17.89
11.36
0.00
0.00
0.11
0.28
0.22
1.00
1.00
ids = 26.2, iqs = 25.8, idr = 24.1, iqr = 23.7
ids = 23, iqs = 23.1, idr = 25.1, iqr = 25.3, x2 = 1.6, x1 = 1.7
ids = 23, iqs = 22.3, idr = 25.3, iqr = 24.3, x2 = 2.5, x1 = 2.4
wr = 99.3
x3 = 99.1
CASE (c) Wr = 1.29 p.u.
k1, k2
65.34 ± 606.6i
k3, k4
39.84 ± 153.62i
k5, k6
11.53 ± 52.35i
k7
0.08
k8
1.13
96.44
24.42
8.32
0.00
0.00
0.11
0.25
0.22
1.00
1.00
ids = 26.3, iqs = 25.6, idr = 24.3, iqr = 23.6
ids = 24, iqs = 23, idr = 26, iqr = 25, x2 = 0.8, x1 = 0.9
ids = 21.9, iqs = 22, idr = 23.8, iqr = 23.9, x2 = 4.1, x1 = 4.1
wr = 98.8, x1 = 0.5
x3 = 98.7
Results and discussions
Case a. rotor speed = 0.8 p.u. Case b. rotor speed = 1.1 p.u.
Case c. rotor speed = 1.29 p.u.
Test scenario
Small signal stability analysis
The impacts of DFIG on dynamic behaviour of power system
under different wind conditions have been evaluated for the test
system shown in Fig. 1. It has been observed that the dynamic performance of the power system is also affected by the strength of
the transmission network to which the wind farms are connected.
Hence, the effect of strong and weak transmission network are
considered with short circuit level of 40 MVA and 16 MVA, respectively. The DFIG model along with torque and voltage control
strategies as discussed in Section ‘Interfacing of DFIG with infinite
bus’ is implemented in MATLAB/Simulink [19]. The parameters
used for simulation are given in Appendices A and B. The following
two scenarios are simulated for the analysis.
Scenario1: Strong grid with short circuit level of 40 MVA.
Scenario2: Weak grid with short circuit level of 16 MVA.
The dynamic behaviour of the DFIG is also analysed for the following rotor speeds considering above both scenarios.
Modal analysis or small signal analysis has been popularly used
in power system for identification of low frequency oscillation.
Small signal stability studies are based on linearization of system
equations around the operating point and modes of oscillations of
system response can be derived from the eigenvalues of the system
state matrix. The analysis of the Eigen properties of system state
matrix provides valuable information regarding the stability characteristics of the system [16]. The states associated in the eigenvalue
analysis for above scenarios are given in (38). The dynamic performance of DFIG is evaluated under different operating conditions
such as varying wind speed and varying network strength. The
steady state initial operating points for these varying operating conditions are tabulated in Table 1. Table 2 shows the original controller
parameters [18] and optimized controller parameters obtained after
minimizing the objective function given in (41) with the help of PSO.
The results of eigenvalue analysis including frequency of oscillation, damping ratio and percentage participation of all the states
B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
77
Table 5
Results of small signal stability analysis of DFIG using optimized controller parameters for Scenario 1 at different wind speed.
Strong connection (VAsc = 40 MVA)
Mode No.
Frequency of oscillation in Hz
Damping ratio
Most influential states in the control of the mode with their % participation
CASE (a) Wr = 0.8 p.u.
k1, k2
1017.82 ± 195.21i
k3, k4
40.41 ± 319.21i
k5
23.04
k6
17.83
k7
0.24
k8
0.19
Eigen values
31.04
50.75
0.00
0.00
0.00
0.00
0.98
0.13
1.00
1.00
1.00
1.00
ids = 23.2, iqs = 25.9, idr = 21.3, iqr = 29.3
ids = 28, iqs = 22.6, idr = 27.2, iqr = 22
ids = 20.6, iqs = 1.4, idr = 22.5, iqr = 1.6, x2 = 50.6, x1 = 2.8
ids = 0.4, iqs = 13.6, idr = 0.5, iqr = 14.8, wr = 3, x2 = 3.3, x1 = 64
wr = 94.9, x1 = 4.4
x3 = 99.1, wr = 0.6
CASE (b) Wr = 1.1 p.u.
k1, k2
1002.23 ± 47.64i
k3, k4
55.56 ± 323.75i
k5
23.94
k6
17.62
k7
0.19
k8
0.36
7.57
51.47
0.00
0.00
0.00
0.00
1.00
0.17
1.00
1.00
1.00
1.00
ids = 25, iqs = 21.8, idr = 23.9, iqr = 29
ids = 28.2, iqs = 22.6, idr = 27.4, iqr = 21.6
ids = 26.5, idr = 28.3, x2 = 44.1
iqs = 14.1, iqr = 15.1, wr = 7.7, x1 = 62.4
wr = 88.1, x1 = 11
x3 = 98.9, wr = 0.7
CASE (c) Wr = 1.29 p.u.
k1
1092.94
k2
893.80
k3, k4
64.60 ± 328.32i
k5
22.30
k6
18.74
k7
0.19
k8
0.43
0.00
0.00
52.20
0.00
0.00
0.00
0.00
1.00
1.00
0.19
1.00
1.00
1.00
1.00
ids = 33, iqs = 8.4, idr = 33.1, iqr = 25
ids = 16.3, iqs = 27.8, idr = 27.8
ids = 28.2, iqs = 22.6, idr = 27.4, iqr = 21.5
ids = 25.7, iqs = 2, idr = 27.1, iqr = 2.2, wr = 0.5, x2 = 39.3, x1 = 2.9
ids = 0.9, iqs = 13.2, idr = 0.8, iqr = 14.2, wr = 10, x2 = 2.7, x1 = 57.9
wr = 84.5, x1 = 14.4, x3 = 0.5
x3 = 99, wr = 0.6
Table 6
Results of small signal stability analysis of DFIG using optimized controller parameters for Scenario 2 at different wind speed.
Weak connection (VAsc = 16 MVA)
Mode No.
Frequency of oscillation in Hz
Damping ratio
Most influential states in the control of the mode with their % participation
CASE (a) Wr = 0.8 p.u.
k1, k2
995.37 ± 334.99i
k3, k4
73.39 ± 313.83i
k5
22.60
k6
18.21
k7
0.24
k8
0.33
Eigen values
53.26
49.89
0.00
0.00
0.00
0.00
0.95
0.23
1.00
1.00
1.00
1.00
ids = 25, iqs = 24.4, idr = 23.7, iqr = 26.6
ids = 28, iqs = 22.1, idr = 27.8, iqr = 21.9
ids = 20, iqs = 2.1, idr = 22.2, iqr = 2.3, x2 = 48.2, x1 = 4.7
ids = 1, iqs = 13.2, idr = 1.1, iqr = 14.7, wr = 2.9, x2 = 5.6, x1 = 61
wr = 94.8, x1 = 4.4
x3 = 98.5, wr = 0.7
CASE (b) Wr = 1.1 p.u.
k1, k2
963.21 ± 220.86i
k3, k4
104.85 ± 319.01i
k5
24.13
k6
17.77
k7
0.34
k8
0.34
35.11
50.72
0.00
0.00
0.00
0.00
0.97
0.31
1.00
1.00
1.00
1.00
ids = 29.5, iqs = 20, idr = 27.1, iqr = 23
ids = 28.3, iqs = 22.1, idr = 28, iqr = 21.3
ids = 26.7, iqs = 0.4, idr = 29, iqr = 0.5, x2 = 42.5, x1 = 0.6
ids = 0.2, iqs = 14.1, idr = 0.2, iqr = 15.5, wr = 7.7, x1 = 61.4
wr = 87.6, x1 = 11.1, x3 = 1
x3 = 97.8, wr = 1.1
CASE (c) Wr = 1.29 p.u.
k1
1051.1
k2
760.1
k3, k4
129.5 ± 345i
k5
22.2
k6
13.8
k7
2.1
k8
1.7
0.00
0.00
54.85
0.00
0.00
0.00
0.00
1.00
1.00
0.35
1.00
1.00
1.00
1.00
ids = 36.5, iqs = 10.1, idr = 36.5, iqr = 16.7
ids = 24, iqs = 33.8, idr = 8.9, iqr = 32.7, wr = 0.5
ids = 28.4, iqs = 22.3, idr = 27.8, iqr = 21.2
ids = 23.6, iqs = 3.8, idr = 25.3, iqr = 4.1, wr = 1, x2 = 37, x1 = 4.8
ids = 1.4, iqs = 13.3, idr = 1.3, iqr = 14.5, wr = 10, x2 = 4.3, x1 = 54.8
wr = 83.6, x1 = 14.9, x3 = 1
x3 = 97.6, wr = 1
for two scenarios and all three cases under considerations are
tabulated in Tables 3–6. The results shown in Tables 3–6 reveal
that system exhibit the stable behaviour for all the scenarios with
varying wind speeds. Tables 3 and 4 shows the results with using
original parameters of PI gains and Tables 5 and 6 with optimized
parameters of controllers. As per Tables 3 and 4, for scenario 1 and
2, respectively, five stable modes have been identified for each
wind speed, two of which are non-oscillating modes. The physical
nature of the modes can be identified by observing the participation factors: Mode 1 (k1;2 Þ and mode 2 (k3;4 Þ are oscillating modes
associated with the stator and the rotor electrical dynamics,
respectively. Mode 3 (k5;6 Þ is also oscillating mode associated with
rotor electrical and mechanical dynamics (rotor currents and gen-
erator speed) and therefore it is referred as electromechanical
mode. Mode 4 (k7 Þ is non oscillating mode associated with rotor
speed. k8 is also the non oscillating mode associated with voltage
controller. The stator mode (k1;2 Þ has a large real part magnitude
and the much higher frequency of oscillations which results in
lowest damping ratio. As the wind speed increases, damping ratio
of mode 1 and mode 3 is slightly reduced while that of mode 2
increases up to the synchronous speed and then again reduces.
Tables 5 and 6 show the results obtained with optimized controller
parameters for both the scenarios in which different rotor speeds
as represented in cases a–c above are considered. The results
revealed that for speed below the synchronous and just above
the synchronous speed, only two oscillating modes exist, i.e. mode
B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
Comparison of Damping Performance for Scenario 1
λ1,2
λ3,4
λ5,6
λ7
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
λ1,2
λ8
Case (a) Wr=0.8 p.u Original Parameters
Case (a) Wr=0.8 p.u Optimized Parameters
λ3,4
λ5,6
λ7
λ8
Case (a) Wr=0.8 p.u Original Parameters
Case (a) Wr=0.8 p.u Optimized Parameters
λ3,4
λ5,6
λ7
λ8
Comparison of Damping Peformance for Scenario 2
1
Damping Ratio
Damping Ratio
λ1,2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
λ1,2
Case (b) Wr=1.1 p.u Original Parameters
Case (b) Wr=1.1 p.u Optimized Parameters
Comparison of Damping Peformance for Scenario 2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Comparison of Damping Performance for Scenario 1
Damping Ratio
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Damping Ratio
Damping Ratio
Comparison of Damping Performance for Scenario 1
0.8
0.6
0.4
0.2
0
λ1,2
λ3,4
λ5,6
λ7
λ8
Case (b) Wr=1.1 p.u Original Parameters
Case (b) Wr=1.1 p.u Optimized Parameters
λ3,4
λ5,6
λ7
λ8
Case (c) Wr=1.29 p.u Original Parameters
Case (c) Wr=1.29 p.u Optimized Parameters
Comparison of Damping Peformance for Scenario 2
Damping Ratio
78
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
λ1,2
λ3,4
λ5,6
λ7
λ8
Case (c) Wr=1.29 p.u Original Parameters
Case (c) Wr=1.29 p.u Optimized Parameters
Fig. 4. Comparison of damping performance of all three cases for Scenario 1 and Scenario 2.
1 and mode 2. For the rotor speeds much above the synchronous
speed results in more stabilized operations for both scenarios
and left with only one oscillating mode.
Fig. 4 show the comparative performance of the damping ratio
of all three cases for scenarios 1 and 2, respectively, with and
without optimized controller parameters. The improvement in
the damping performance with use of optimized parameter can
be clearly noticed from Fig. 4.
Transient response of DFIG for short circuit
As the penetration of wind power in electrical power system
increases, the behaviour of wind turbine (WT) under faults, voltage
dips and disturbances becomes more important, especially for
those with power electronic converters, such as DFIGs. The grid
voltage dips imposed at the connection point of the DFIG due to
short circuit results in high rotor current. This high rotor current
can damage the RSC and may cause large increases in the dc-link
voltage. Such large rotor current, dc-link over voltage and torque
oscillations occurring due to grid faults are quite harmful for the
DFIG-based WTs. In these conditions either the DFIG may be disconnected from the grid, or the rotor-side converter may be deactivated using the crowbar resistors. A sudden loss of wind power
during grid faults results in rapid rate of change of frequency
(ROCOF) in the system. In addition DFIG will behave as squirrel
cage induction generator after the deactivation of RSC. Thus DFIG
consumes more reactive power and caused the voltage instability
problem. Thus, it is desired that the wind turbines must remain
connected and actively contribute to the system stability during
and after the faults and disturbances. The ability of WT to stay
connected to the grid during the faults and voltage dips is termed
as low voltage ride through (LVRT) capability [20]. Nowadays, in
order to ensure the security of power system, most of the countries
have introduced and implemented their grid codes for LVRT
capability while integrating WTGs into the utility grid. The grid
code for LVRT capability of DFIG as demanded by UK TSO is given
in [18,21].
Three phase fault
The transient performance of DFIG connected to infinite bus has
been analyzed for three phase balanced fault applied at the terminal
of DFIG. The fault is applied at 5 s which persists for 140 ms and after
that the normal operation is restored. The comparative transient
responses, of stator currents ðids; iqs Þ , rotor currents ðidr; iqr Þ and electromagnetic torque (T e Þ for both the scenarios are shown in Figs. 5
and 6, for original controller parameters and optimized controller
parameters. It is clearly depicted in Figs. 5 and 6 that the currents
and electromagnetic torque were operating at their initial settled
value as indicated in Table 1 before the application of fault. The sudden application of three phase fault causes the significant excursions in transient responses of currents and electromagnetic
torque. The optimized parameters of the controllers successfully
limit the peak values in these excursions and suppress them very
quickly as per the grid code requirements. The systems regain its
original operating points after the removal of the fault and all the
parameters are restored back to their initial steady state value.
Voltage sag
The LVRT capability specified in grid codes also requires the
WTGs to operate at reduced voltage for a few hundreds of ms to
several seconds. It can be seen, from the requirement specified
by the TSO of UK [18,21] that wind turbines should ride through
a 50% fault for 710 ms. This condition is also investigated, for both
the scenarios by reducing the terminal voltage of DFIG from its
nominal value to 50% of its nominal value for the duration of
710 ms. The comparative transient responses shown in Figs. 7
79
B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
6
4
original
original
optimized
Te
Te
optimized
2
original
4
optimized
Te
2
2
0
0
5.2
5.3
5.4
5.5
5.6
4.9
5.7
4
4
2
2
0
-2
5.1
5.2
5.3
5.5
5.4
5.6
4.9
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5
0
0
-2
-4
-5
-4
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
4.9
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
4.9
5.7
4
4
2
4
0
2
iqs
2
iqs
iqs
0
5
ids
5.1
5
ids
ids
-2
4.9
0
0
-2
-4
-2
5
5.1
5.2
5.3
5.4
5.5
5.6
4.9
5.7
4
2
2
idr
4
0
5
5.1
5.2
5.3
5.4
5.5
5.6
4.9
5.7
5
idr
4.9
idr
-2
0
0
-2
-2
-4
-4
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
4.9
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
-5
4.9
5.7
4
4
4
2
0
2
iqr
iqr
iqr
2
0
0
-2
-2
-2
-4
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
4.9
5.7
5
5.1
Time in Seconds
5.2
5.3
5.4
5.5
5.6
4.9
5.7
Time in Seconds
Time in Seconds
Fig. 5. Comparative transient response of all three cases for Scenario 1 considering three phase fault.
3
original
Te
Te
optimized
1
3
original
2
2
optimized
Te
2
1
0
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
-1
5
5.1
5.2
5.3
5.4
5.5
5.6
5.1
5.2
5.3
5.4
5.5
5.6
5.7
-4
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5
5.1
5.2
5.3
5.4
2
2
ids
2
-4
4.9
0
5
5.1
5.2
5.3
5.4
5.5
5.6
-4
4.9
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
3
2
0
iqs
iqs
2
1
0
0
-1
-2
4.9
0
-2
-2
2
iqs
5
4
-2
5
5.1
5.2
5.3
5.4
5.5
5.6
4.9
5.7
5
5.1
5.2
5.3
5.4
5.5
5.6
-2
4.9
5.7
4
4
2
2
2
0
0
-2
-2
4.9
idr
4
idr
idr
4.9
5.7
4
0
optimized
1
4
ids
ids
4.9
original
optimized
0
0
-1
-1
4.9
original
5
5.1
5.2
5.3
5.4
5.5
5.6
-4
4.9
5.7
0
-2
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
4.9
3
2
0
2
iqr
iqr
iqr
2
1
0
0
-1
-2
4.9
5
5.1
5.2
5.3
5.4
Time in Seconds
5.5
5.6
5.7
4.9
5
5.1
5.2
5.3
5.4
Time in Seconds
5.5
5.6
5.7
-2
4.9
Time in Seconds
Fig. 6. Comparative transient responses of all three cases for Scenario 2 considering three phase fault.
5.5
5.6
5.7
B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
optimized
optimized
2
original
optimized
0.5
3
2
original
1
1
Te
Te
1.5
Te
80
0
0
0
-0.5
-1
5
5.2
5.4
5.6
5.8
6
6.2
5
5.4
5.6
5.8
6
6.2
0
-2
0
5.2
5.4
5.6
5.8
6
5
6.2
2
5.2
5.4
5.6
5.8
6
6.2
5
5.2
5.4
5.6
5.8
6
6.2
5
5.2
5.4
5.6
5.8
6
6.2
5
5.2
5.4
5.6
5.8
6
6.2
5
5.2
5.4
5.6
5.8
6
6.2
0
-2
-2
5
5
2
ids
ids
ids
5.2
2
2
5.2
5.4
5.6
5.8
6
6.2
3
2
2
0
0
iqs
iqs
1
-1
5
5.2
5.4
5.6
5.8
6
5
6.2
5.2
5.4
5.6
5.8
6
-1
6.2
2
2
idr
idr
2
1
0
-2
idr
iqs
original
1
0
0
0
-2
-2
5
5.2
5.4
5.6
5.8
6
5
6.2
2
5.2
5.4
5.6
5.8
6
-2
6.2
3
2
0
2
1
iqr
iqr
iqr
1
0
1
0
-1
-1
5
5.2
5.4
5.6
5.8
6
5
6.2
5.2
5.4
5.6
5.8
6
6.2
Time in Seconds
Time in Seconds
Time in Seconds
Fig. 7. Comparative transient responses of all three cases for Scenario 1 considering voltage sag.
1.5
1.5
optimized
Te
0
5
5.2
5.4
5.6
5.8
6
5
5.2
5.4
5.6
5.8
6
5.6
5.8
6
6.2
5
5.2
5.4
5.6
5.8
6
iqs
iqs
0
5.2
5.4
5.6
5.8
6
1
6.2
5.4
5.6
5.8
6
6.2
5
5.2
5.4
5.6
5.8
6
6.2
5
5.2
5.4
5.6
5.8
6
6.2
5
5.2
5.4
5.6
5.8
6
6.2
5
5.2
5.4
5.6
5.8
6
6.2
2
0
5
5.2
0
-2
6.2
2
1
5
2
0
-2
6.2
2
-1
5.4
ids
0
-2
5.2
2
ids
ids
2
1
0
5
6.2
iqs
Te
0.5
0
-0.5
original
original
1
original
0.5
optimized
2
optimized
Te
1
1
0
5
5.2
5.4
5.6
5.8
6
6.2
2
2
idr
0
idr
idr
2
0
0
-2
5
5.2
5.4
5.6
5.8
6
-2
6.2
2
5
5.2
5.4
5.6
5.8
6
-2
6.2
2
2
iqr
iqr
iqr
1
1
1
0
0
-1
5
5.2
5.4
5.6
5.8
Time in Seconds
6
6.2
0
5
5.2
5.4
5.6
5.8
6
6.2
Time in Seconds
Fig. 8. Comparative transient responses of all three cases for Scenario 2 considering voltage sag.
Time in Seconds
81
B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
and 8 demonstrate that the DFIG exhibits the superior performance
with the optimized controller parameters for the voltage sag
considerations.
Conclusion
With increasing wind penetration in power systems, grid codes
demand complete dynamic models of WTGs and its simulation
studies under different operating conditions to prevent any detrimental impact of these energy sources on the network to which
it is connected. In this paper, a dynamic model of DFIG and its associated controllers with the reduced order representation is presented, which is suitable to capture its impact on small signal
and transient stability of power system. The RSC of DFIG is controlled by q-axis and d-axis rotor currents through torque and voltage controller loop respectively. It is observed from the eigenvalue
analysis that the dynamic behaviour of DFIG has been significantly
improved by optimizing the gains of torque and voltage control
loops for different network strength and also over a wide range
of rotor speed variations. The transient analysis of DFIG for three
phase fault and voltage sag reveals that the optimized gains plays
a vital role to improve the LVRT capability of DFIG.
Appendix A. Parameter of DFIG (in p.u. otherwise specified)
Dsh = 0.01, K sh = 10, Htot = 3.5, Hg = 0.5, Ht = 3, V w base = 9 m/s,
k = 8.1, cp = 0.48, Pnom = 2 MVA, P mec = 2 MVA, Pnom1 = 2.2222 MVA,
P elec base = 2.2222 MVA,
P wind base = 1,
c1 = 0.5176,
c2 = 116,
c3 = 0.4, c4 = 5, c5 = 21, c6 = 0.0068, pitch_rate = 2, pitch_max = 45,
K opt = 0.56, K p = 5, K i = 25, V b = 690 V, Sb = 2 MVA, F b = 50, Ws = 1,
W b ¼ 2 pi F b ,
X tr = 0.05,
Rs = 0.00488,
X ls = 0.09241,
Rr = 0.00549, X lr = 0.09955, X m = 3.95279, X rm = 0.02, W s = 1,
X ss = 4.0452 (X ss ¼ X ls þ X m Þ, X rr ¼ X lr þ X m , vdsinf = 0, vqsinf = 1.
2
Rs
6 x X
6
e
ss
6
6
0
F ¼ 6
6
6 s x X
e
m
4
X m iqr0
VAsc
X/R
Ze
Re
Xe
Rt
Xt
Scenario 2 weak grid
40 MVA
10
0.05
0.0050
0.0498
0.0099
0.0998
16 MVA
10
0.125
0.0124
0.1244
0.0173
0.1744
0
0
0
X rr
Xm
0
0
0
0
X rr
0
0
0
2Hxb
3
7
7
7
7
7
7
5
X m iqs0
X m ids0
Feed forward matrix D;
3
7
7
7
7
7
xb
7
X m ids0 X rr idr0 7
5
xb
0
X rr iqr0 X m iqs0
1 0 0
0 1 0
0 0 0
0 0 0
Appendix D. Elements of system matrix Asys
The elements of system matrix are as follows:
A11 ¼
A12 ¼
xb
X rr X ss X 2m
xb
X rr X ss X 2m
fRT X rr þ ðK p2 =v s0 xs ÞðRT v ds0 þ X T v qs0 Þg
f X rr X ss s0 X 2m xs þ X T X rr þ ðK p2 =v s0 xs ÞðRT v qs0
X T v ds0 Þg
A13 ¼
A14 ¼
A21 ¼
xb
X rr X ss X 2m
xb
X rr X ss X 2m
xb
X rr X ss X 2m
xb X m
X rr X ss X 2m
xb
X rr X ss X 2m
fX m ðRr þ K p2 Þg;
fðX m X rr þ s0 X m X rr Þxs g
fðX m iqs0 X rr iqr0ÞX m g;
A17 ¼ 0;
;
A18 ¼
xb X m K p2
X rr X ss X 2m
f X rr X ss þ s0 X 2m xs X T X rr
þ ðK p1 K opt X ss xs x2r0 =v 3s0 ÞðRT v ds0 þ X T v qs0 Þg
A22 ¼
xb
X rr X ss X 2m
fRT X rr þ ðK p1 K opt X ss xs x2r0 =v 3s0 ÞðRT v qs0
X T v ds0 Þg
A31 ¼
0
Rr
X m idr0
Output matrix C;
A32 ¼
A24 ¼
xb
X rr X ss X 2m
A26 ¼ 0;
0
s xe X rr
B ¼ ðEÞ1
0 0
C¼
0 0
0 0
D¼
0 0
A25 ¼
Xm
s xe X rr
Control matrix B;
Appendix C. Derivation of A, B, C and D matrices
d
x ¼ Fx þ u
dt
2
X ss
0
6
X ss
6 0
1 6
6 X m
E¼
0
xb 6
6
X m
4 0
Rr
0
A ¼ ðEÞ1 F
A23 ¼ A14 ;
E
s xe X m
System matrix A;
A16 ¼
Scenario 1 strong grid
Rs
0
0
y ¼ Cx þ Du
A15 ¼
Type of grid
xe X m
0
0
xe X m
x_ ¼ Ax þ Bu
Appendix B. System parameters
Parameters
xe X ss
xb
X rr X ss X 2m
fðX m ids0 þ X rr idr0ÞX m g
A27 ¼ A16 ;
xb
X rr X ss X 2m
xb
X rr X ss X 2m
fX m ðRr þ K p1 Þg;
A28 ¼ 0;
fRT X m þ ðK p2 X ss =X m v s0 xs ÞðRT v ds0 þ X T v qs0 Þg
fðX m X ss s0 X m X ss Þxs þ X T X rr
þ ðK p2 X ss =X m v s0 xs ÞðRT v qs0 X T v ds0 Þg
A33 ¼
A34 ¼
xb
X rr X ss X 2m
xb
X rr X ss X 2m
fX ss ðRr þ K p2 Þg;
f X 2m þ s0 X ss X rr xs g;
82
B. Mehta et al. / Electrical Power and Energy Systems 70 (2015) 70–82
A35 ¼
A36 ¼
A41 ¼
xb
X rr X ss X 2m
xb X ss
X rr X ss X 2m
xb
X rr X ss X 2m
References
fðX m iqs0 X rr iqr0ÞX ss g;
;
A37 ¼ 0;
A38 ¼
xb X ss K p2
X rr X ss X 2m
fðX m X ss þ s0 X m X ss Þxs X T X rr
þ ðK p1 X ss K opt X ss xs x2r0 =v 3s0 X m ÞðRT v ds0 þ X T v qs0 Þg
A42 ¼
xb
X rr X ss X 2m
fRT X m þ ðK p1 X ss K opt X ss xs x2r0 =v 3s0 X m ÞðRT v qs0
X T v ds0 Þg
A43 ¼
A44 ¼
A45 ¼
xb
X rr X ss X 2m
xb
X rr X ss X 2m
xb
X rr X ss X 2m
A46 ¼ 0;
n
X 2m s0 X ss X rr xs g;
)
fX ss ðRr þ K p1 Þ
fðX m ids0 þ X rr idr0ÞX ss þ 2ðK p1 X ss =X m Þg;
A47 ¼ A36 ;
A48 ¼ 0;
A51 ¼
X m iqr0
2H
X m idr0
X m iqs0
X m ids0
; A53 ¼
; A54 ¼
;
2H
2H
2H
¼ 0; A56 ¼ 0; A57 ¼ 0; A58 ¼ 0;
A52 ¼
A55
A61 ¼ ðK i2 =v s0 xs X m ÞðRT v ds0 þ X T v qs0 Þ;
A62 ¼ ðK i2 =v s0 xs X m ÞðRT v qs0 X T v ds0 Þ;
A65 ¼ 0;
A66 ¼ 0; A67 ¼ 0;
A71 ¼ ðK i1 K opt X ss xs x
A63 ¼ K i2 ;
A68 ¼ K i2 ;
v
2
3
r0 = s0 X m ÞðRT
v ds0 þ X T v qs0 Þ
A72 ¼ ðK i1 K opt X ss xs x2r0 =v 3s0 X m ÞðRT v qs0 X T v ds0 Þ;
A73 ¼ 0;
A74 ¼ K i1 ;
A75 ¼ ð2K i1 K opt X ss xs xr0 =X m v s0 Þ;
A78 ¼ 0;
A76 ¼ 0;
A77 ¼ 0;
A81 ¼ ðK p3 =v s0 ÞðRT v ds0 þ X T v qs0 Þ
A82 ¼ ðK p3 =v s0 ÞðRT v qs0 X T v ds0 Þ;
A85 ¼ 0;
A86 ¼ 0;
A87 ¼ 0;
A64 ¼ 0;
A83 ¼ 0;
A88 ¼ 0
A84 ¼ 0;
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