514
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 2, JUNE 2010
Transient Performance Improvement of Wind
Turbines With Doubly Fed Induction Generators
Using Nonlinear Control Strategy
Mohsen Rahimi and Mostafa Parniani, Senior Member, IEEE
Abstract—This paper first discusses dynamic characteristics of
wind turbines with doubly fed induction generator (DFIG). Rotor
back electromotive force (EMF) voltages in DFIG reflect the effects
of stator dynamics on rotor current dynamics, and have an important role on rotor inrush current during the generator voltage dip.
Compensation of these voltages can improve DFIG ride-through
capability and limit the rotor current transients. It is found that
the electrical dynamics of the DFIG are in nonminimum phase for
certain operating conditions. Also, it is shown that the dynamics of
DFIG, under compensation of rotor back EMF and grid voltages,
behave as a partially linearizable system containing internal and
external dynamics. The internal and external dynamics of DFIG
include stator and rotor dynamics, respectively. It is found that under certain operating conditions, the internal dynamics, and thus,
the entire DFIG system becomes unstable. This phenomenon deteriorates the DFIG postfault behavior. Since the DFIG electrical
dynamics are nonlinear; the linear control scheme cannot properly
work under large voltage dips. We address this problem by means
of a nonlinear controller. The proposed approach stabilizes the internal dynamics through rotor voltage control, and improves the
dynamic behavior of the DFIG after clearing the fault.
Index Terms—Doubly fed induction generator (DFIG), internal
dynamics, nonlinear control, transient performance, wind turbine
(WT).
I. INTRODUCTION
MONG the different alternatives to obtain variable speed
wind turbines (WTs), the system based on doubly fed induction generator (DFIG) has become the most popular [1]. The
stator of DFIG is directly connected to the power grid, and the
rotor windings are supplied from a back-to-back voltage source
converter (VSC) via slip rings. Fig. 1 shows the schematic diagram of WT with DFIG connected to an infinite bus through
the equivalent grid impedance Re + jXe . A common feature in
most DFIG-related papers is the field-oriented control (FOC),
which enables decoupled control of real and reactive powers.
FOC has been implemented in two ways. One way is to control
the DFIG with stator flux orientation [2], the other is with air gap
flux orientation [3]. This paper deals with the analysis and improvement of transient performance in the DFIG modeled with
the stator flux orientation. Transient performance improvement
is realized by Lyapunov-based nonlinear control design.
A
Manuscript received March 15, 2009; revised July 5, 2009; accepted August
23, 2009. Date of publication October 30, 2009; date of current version May
21, 2010. Paper no. TEC-00109-2009.
The authors are with the Department of Electrical Engineering, Sharif University of Technology, Tehran 11155-9363, Iran (e-mail: m_rahimi@ee.sharif.edu;
parniani@sharif.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEC.2009.2032169
Fig. 1.
WT with DFIG connected to the infinite bus.
As the penetration of wind power in electrical power system increases, the behavior of WT under faults, voltage dips,
and disturbances becomes more important. From this point of
view, power system operation is divided into three operating
phases: prefault, fault-on, and postfault. It is desired that WTs
remain connected, and actively contribute to the system stability during and after faults and disturbances. The ability of WT
to stay connected to the grid during faults and voltage dips is
stated as low-voltage ride-through capability [4]. Considerable
research has been done on ride-through capability and dynamic
behavior investigation of DFIG-based WTs during faults and
voltage dips [5]–[9]. Two main problems must be overcome in
achieving the ride-through requirements of DFIGs during the
voltage dip. The first one is the peak rotor fault current that may
exceed its limit, and the second one is the dc-link overvoltage.
Compensation of rotor back EMF voltages is one of the efficient
methods used in [10] to limit the rotor inrush current during the
fault. In fact, by using this control strategy, the rotor dynamics
will improve and become independent of stator dynamics. However, we show that this approach can weaken the other system
dynamics and deteriorate the DFIG postfault behavior. The behavior of turbine generator after clearing the fault is in the
domain of postfault transient studies. These studies determine
whether the postfault system will converge toward an acceptable
steady state as time increases. Little work has been published for
assessing and improving postfault transient behavior of DFIG.
There are several papers about dynamic and transient behaviors
of DFIGs [5], [6], [9], [11]–[15], but none of them discusses the
nature of instability, neither they present an analytical method
for stability evaluation. Instead, the operation of DFIG in these
literatures has been studied by means of simulations.
This paper develops a theoretical basis for analysis and improvement of DFIG transient behavior after clearing the fault.
0885-8969/$26.00 © 2009 IEEE
RAHIMI AND PARNIANI: TRANSIENT PERFORMANCE IMPROVEMENT OF WIND TURBINES WITH DOUBLY FED INDUCTION GENERATORS
Modal analysis, eigenvalue tracking, and Lyapunov-based nonlinear controller design are used to identify the nature of instability and improve the generator postfault performance. It is found
that electrical dynamics of the DFIG are in nonminimum phase
for certain operating conditions, and thus, have an inherent limitation on the achievable dynamic response. Also, it is shown
that the dynamics of DFIG, under compensation of rotor back
EMF and grid voltages, behave as a partially linearizable system
containing internal and external dynamics. It is found that under
certain operating conditions, the internal dynamics, and thus, the
entire DFIG system becomes unstable. Since the DFIG electrical dynamics are nonlinear, the linear control scheme cannot
properly work under large voltage dips. We address this problem by means of a nonlinear controller. The proposed approach
is a combination of proportional–integral (PI) and Lyapunovbased auxiliary control, which stabilizes the internal dynamics
and improves the DFIG postfault behavior through rotor control voltage. The structure of the paper is as follows. Following
this introduction, the dynamic model of DFIG will be derived.
Then, stability of the system is discussed using modal analysis. The analysis includes stator, rotor, and grid filter dynamics
and controllers. It is shown that stator dynamics contain poorly
damped modes. Next, a Lyapunov-based nonlinear controller is
proposed to improve and stabilize the DFIG transient behavior. At the end, the results of theoretical analysis are verified
by time-domain simulations. Simulation results also consider
the situations in which there is uncertainty in knowledge of the
system parameters, such as stator resistance and magnetizing
inductance.
The purpose of this section is to present the dynamic model
of single-machine infinite bus (SMIB) system of Fig. 1 in d–q
reference frame with the stator flux orientation. The generalized
machine model is developed based on the following conditions
and assumptions.
1) Positive direction for the stator and rotor currents is assumed toward the generator, and for the grid-side filter, it
is toward the grid-side converter (see Fig. 1).
2) All system parameters and variables are in per unit and
referred to the stator side of DFIG.
The following base equations are used to model the DFIG
generator [16]:
1 dψsdq
ωb dt
vr dq = Rr ir dq + jω2 ψr dq +
1 dψr dq
ωb dt
Rotor current control loops.
the base angular frequency, and ω is the speed of d–q reference
frame, coinciding with the stator flux. Also, Rs and Rr are the
stator and rotor resistances. Electromechanical torque Te and
reactive power injected to the grid by the stator windings Qs are
given by
Te =
Lm
(ψsq ir d − ψsd ir q )
Ls
Qs = vsd isq − vsq isd .
(5)
(6)
A. Rotor Modeling
From (2)–(4), the rotor dynamics is described in terms of
rotor current and stator flux, as follows:
II. DFIG-BASED WT MODELING IN STATOR
FLUX ORIENTATION
vsdq = Rs isdq + jωψsdq +
Fig. 2.
515
(1)
(2)
ψs = Ls is + Lm ir
(3)
ψr = Lm is + Lr ir
(4)
where, ψ, v, and i represent flux, voltage, and current, respectively, subscripts s and r denote the stator and rotor quantities,
respectively, Ls and Lr are the stator and rotor self-inductances,
Lm is the mutual inductance, ω2 is the rotor slip frequency, ωb is
Lr dir dq
= −Rr ir dq − jω2 Lr ir dq − edq + vr dq
ωb dt
(7)
where Lr = Lr − (L2m /Ls ), Rr = Rr + (Lm /Ls )2 Rs , and
Lm
Rs
edq =
ψsdq
vsdq − jωr ψsdq −
(8)
Ls
Ls
where ωr in (8) is the rotor speed and is equal to ωr = ω − ω2 .
The variables ed and eq in (7) are functions of stator flux and
stator voltage. These terms, called rotor back EMF voltages,
reflect the effects of stator dynamics on rotor current dynamics
and have an important role in DFIG transient performance. By
compensating the cross-coupling terms ω2 Lr ir q and ω2 Lr ir d
using d–q rotor current controllers, the d and q rotor current
control loops will be decoupled. The rotor d–q current control
loops, under compensation of cross-coupling terms, are shown
in Fig. 2. The back EMF voltages ed and eq are represented as
disturbance in current control loops of Fig. 2. The superscriptˆ
in the figures denotes the measured or calculated variables used
as control inputs.
In order to decrease tracking error, the back EMF voltages can
be compensated by rotor current controllers using feedforward
terms.
Considering the rotor controllers to be PI, KI dq (s) =
kp idq + (ki idq /s). Also, with the control structure of Fig. 2,
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 2, JUNE 2010
converter voltage could be stated as
vg dq (t) = −kp g (ig dq
ref (t) − ig dq (t)) − ki
g
(ig dq
− ig dq (t))dt − jωLg ig dq + vsdq (t)
ref (t)
(11)
where ig dq ref denotes the d–q components of grid filter reference current.
C. Stator Modeling
In stator flux orientation, ψs = ψsd and ψsq = 0. Then, from
(1), (3), and (4), and according to Fig. 1, the stator is described by
the following state equations as a function of rotor and grid-side
filter currents, stator flux, and infinite bus voltage:
1 Ls + Le dψsd
Rs + Re
Rs + Re
=−
ψsd +
Lm ir d − Re ig d
ωb Ls
dt
Ls
Ls
Fig. 3.
Grid filter current control loops.
under compensation of cross-coupling terms, the d–q rotor voltage could be stated as
vr dq (t) = kp idq (ir dq ref (t) − ir dq (t)) + ki idq (ir dq ref (t)
− ir dq (t))dt + jω2 Lr ir dq + kcom edq (t)
(9)
where ir dq ref represents the d–q components of rotor reference
current. In (9), kcom is either 0 or 1. kcom = 1 means that back
EMF voltages are compensated by rotor current controllers, and
kcom = 0 means that they are not compensated. Considering
Fig. 2, the open-loop bandwidth of current control in per unit is
αs = Rr /Lr , which is relatively small.
B. Grid-Side Filter Modeling
The grid-side filter, as shown in Fig. 1, consists of an inductance Lg and resistance Rg , and its dynamics are described
by
Lg dig dq
= −Rg ig dq − jωLg ig dq − vg dq + vsdq
ωb dt
(10)
where the subscript g denotes the grid filter quantities, and vsdq ,
ig dq , and vg dq are the d–q components of the generator terminal
voltage, and grid-side filter current and voltage, respectively. vg
is supplied from the grid-side converter. The d–q grid filter current control loops, under compensation of cross-coupling terms,
are shown in Fig. 3. In this figure, the grid voltages vsd and vsq
are represented as disturbance. In order to decrease the tracking
error, these voltages are compensated using feedforward terms,
as shown in Fig. 3. Considering the grid filter current controllers
to be PI, KG d q (s) = kp g + (ki g /s), and under compensation
of cross-coupling terms and grid voltages, the d–q grid-side
ω=
−
Le Lm
Le Lm 1 dir d
ωir q + Le ωig q +
Ls
Ls ωb dt
−
Le dig d
+ V∞ cos γ
ωb dt
(12)
dγ
= ωb (ωs − ω)
(13)
dt
and (14) shown at the bottom of this page, where ωs is the synchronous frequency and is equal to 1 p.u., ω is the speed of d–q
reference frame, in p.u., and is equal to stator flux frequency.
Also, ωs = (1/ωb ) (dθs /dt) and ω = (1/ωb ) (dθ/dt). The variables θs and θ are the infinite bus voltage angle and stator flux
angle in stationary reference frame, respectively. Also, γ is the
difference between θs and θ, and V∞ is the infinite bus voltage.
D. Drive Train Model and Speed Controller
The drive train comprises turbine, gear box, shafts, and other
mechanical components of WT. The two mass drive train models
of DFIG are given by [17]
Te + Ks β + D(ωt − ωr )
dωr
=
dt
2Hr
(15)
dωt
Tm − Ks β − D(ωt − ωr )
=
dt
2Ht
(16)
dβ
= ωb (ωt − ωr )
(17)
dt
where ωt and ωr are the turbine and generator speeds (in per
unit), β is the shaft twist angle (in radians), Hr and Ht are
the inertia constants of turbine and generator (in seconds), respectively, ks is the shaft stiffness coefficient (in per unit per
electrical radian), D is the damping coefficient (in per unit), and
Te and Tm are the generator electrical torque and the turbine
mechanical torque, respectively, (in per unit). With stator flux
orientation, the rotor speed is controlled by the q-components
of rotor voltage and current, vr q and ir q [1]. The control scheme
((Rs + Re )/Ls )Lm ir q − Re ig q + ((Le Lm )/Ls )(1/ωb )(dir q /dt) − (Le /ωb )(dig q /dt) + V∞ sin γ
ψsd (1 + (Le /Ls )) − (Le /Ls )Lm ir d + Le ig d
(14)
RAHIMI AND PARNIANI: TRANSIENT PERFORMANCE IMPROVEMENT OF WIND TURBINES WITH DOUBLY FED INDUCTION GENERATORS
517
TABLE II
THE SAMPLE DFIG SYSTEM MODES AND PARTICIPATION FACTORS
Fig. 4.
Speed control loop.
TABLE I
STATOR POWER FACTOR AS A FUNCTION OF d-AXIS ROTOR CURRENT
Fig. 5.
III. SMALL-SIGNAL ANALYSIS
Reactive power control loop.
used for speed control is shown in Fig. 4. In this figure, αq
is the bandwidth of the q-axis rotor current control loop, and
Tm = ks β + Dωt . Employing a PI controller for the speed
controller, Kω (s) = kpω + (kI ω /s), state equation of the speed
controller is
dx7
= kI ω (ωr
dt
− ωr ).
Equations (7)–(9), (10)–(18), and (20) describe the dynamics
of turbine generator with its rotor speed and reactive power
controllers. The dynamic model of the system may be rewritten
in the form of state equations, and summarized as
x• = f (x, z, h)
0 = g(x, z, h)
(21)
(18)
where x, z, and h are the vectors of the system state variables,
reference inputs, and exogenous inputs, respectively, i.e.,
An active damping term can be used with the controller to
increase the open-loop bandwidth (−(D/2Hr )) in Fig. 4, and
thus to improve the dynamic response of the rotor speed.
x = [ψsd , γ, ir d , ir q , x5 , x6 , x7 , x8 , ωr , β, ωt , ig d , ig q , x14 , x15]T ,
ref
E. Reactive Power Control
With stator flux orientation, terminal voltage and reactive
power exchange between the generator and the grid can be
controlled by the d-components of rotor voltage and current [1].
Considering (6), with stator flux orientation, the reactive power
injected to the grid by the stator can be written as
Qs =
1
ψsd (Lm ir d − ψsd ).
Ls ωs
(19)
This equation shows the direct relation between d-axis rotor
current and stator reactive power, and the generator power factor.
Table I displays the stator power factor with different d-axis rotor
currents, for the study system described later.
Thus, the d-axis rotor reference current is determined by
reactive power controller, as shown in Fig. 5. In this figure, αd
is the bandwidth of the d-axis rotor current control loop. Using
a PI controller as Kpf (s) = kp pf + (kI pf /s), state equation
of the reactive power controller is
dx8
= kI
dt
pq (Qs ref
− Qs ).
(20)
z = [ωr
T
ref , Qs ref ]
,
h = [V∞ , Tm ]T
and the state variables x5 , x6 and x14 , x15 correspond to the
integral terms in (9) and (11), respectively.
Linearizing and rearranging (21) yields the linearized model
of the DFIG as follows:
∆x• = A∆x.
(22)
To investigate the system dynamics, a 1.76-MVA, 575-V,
60-Hz DFIG is considered. Appendix A gives the generator
parameters. The study is done under operating conditions, as
in Appendix B, and unity power factor at the stator terminal.
The PI controller parameters are shown in Appendix C. These
parameters correspond to the rotor and grid-side filter current
bandwidths of 2 p.u. (754 rad/s), speed control loop bandwidth
of 4.4 rad/s (0.7 Hz), including active damping, and reactive
power control loop bandwidth of 4.4 rad/s (0.7 Hz).
As mentioned in Section II, it is possible to include a feedforward compensating term in the control law that will compensate
for the tracking error caused by variations in the rotor back EMF
and grid voltages. After compensating these voltages, i.e., with
kcom = 1 in (9), the system modes and the corresponding state
variables with the highest participation factors are obtained as
in Table II. Using the participation factors [18], the degree of
contribution of each state variable in the system modes and the
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 2, JUNE 2010
physical nature of dynamic modes can be detected. Examining
the results in Table II, the following key points are found.
1) The dynamics of DFIG contain poorly damped modes
(λ1,2 = −0.43 ± 375.9j) with a corresponding natural
frequency near the network frequency. Stator variables
ψsd and γ have the highest contributions in these modes;
thus, we call these modes as stator modes. As we will
see later, these modes have significant impact on transient performance of DFIG, and under special operating
conditions, may become unstable.
2) The modes λ3 = −758.4 and λ4 = −745.1 are the d–q
rotor current modes, and ir d and ir q have the highest
participations in these modes. These modes are very fast
and their damping is nearly equal to the rotor current
control bandwidth (2 p.u. or 754 rad/s). Therefore, the
larger the rotor current control bandwidth, the larger the
damping of rotor current modes.
3) The modes λ5,6 = −3.89 ± 13.11j are the electromechanical modes. The mechanical variables ωr and β have
the highest contributions in these modes. The corresponding natural frequency is approximately 2 Hz.
4) The real mode λ7 = −4.4 is associated with state variable
x8 and is equal to the bandwidth of reactive power control
loop.
5) The modes λ8,9 = −0.56 ± 1.56j are the mechanical
modes associated with the state variables x7 and ωt . These
modes are weakly damped, and are dependent on the speed
control bandwidth and damping.
6) The modes λ10,11 are both equal to −13.41, and are the rotor electrical modes associated with state variables x5 and
x6 . These modes are equal to the d–q rotor current openloop bandwidth, αs = (Rr /Lr )ωb . By actively or passively increasing the bandwidth, damping of these modes
will increase.
7) The modes λ12,13 are both equal to −753.98, and are
the grid-side filter current modes associated with state
variables ig d and ig q . These modes are very fast and their
damping is nearly equal to the grid filter current control
bandwidth (2 p.u. or 754 rad/s).
8) The modes λ14,15 = −3.77 are also the grid-side filter
modes associated with state variables x14 and x15 . These
modes are equal to the grid filter current open-loop bandwidth, αg = (Rg /Lg )ωb .
As stated before, the stator modes are weakly damped, e.g.,
the damping ratio of stator modes in Table II is ξ = 0.0012. Usually in the literature, the stator dynamics are neglected. However,
these modes could have significant effects on DFIG transient behavior. Considering the dependency between stator, rotor, and
grid filter dynamics, and to clarify the effects of stator dynamics
on the DFIG dynamic performance, in the following, stability
analysis of stator, rotor, and grid filter dynamics is presented.
IV. STABILITY ANALYSIS OF NONLINEAR
ELECTRICAL DYNAMICS
The effects of stator dynamics on the stability of the system
are further investigated in this section, based on the theory of
partially feedback linearizable systems. In this theory, the input–
output relations of the system are linearized, while the state
equations may only be partially linearized [19]. Thus, the system
dynamics are divided into two subsystems called the internal and
external dynamics.
By selecting the d–q components of the rotor, and grid-side
filter control voltages vr dq (t) and vg dq (t) in (7) and (10), as
vr dq (t) = jω2 Lr ir dq + edq (t) + vr dq
crl (t)
vg dq (t) = −jωLg ig dq + vsdq (t) − vg dq
crl (t)
(23)
the back EMF voltages and cross-coupling terms of rotor and
grid filter dynamics will be compensated. Then, by substituting
(23) into (7) and (10), the rotor and grid filter dynamics can be
described by
Lr dir dq
= −Rr ir dq + vr dq
ωb dt
crl (t)
(24)
Lg dig dq
= −Rg ig dq + vg dq
ωb dt
crl (t).
(25)
The control law of vr dq crl (t) and vg dq crl (t), in (24) and (25)
is chosen such that
1) it stabilizes the rotor and grid filter dynamics, i.e., (24)
and (25);
2) rotor and grid filter currents track their reference values
without tracking error.
Then, vr dq crl (t) and vg dq crl (t) will be the functions of rotor and grid filter currents, respectively. Accordingly, the terms
dir dq /dt and dig dq /dt in stator dynamics (12)–(14) will be the
functions of only rotor and grid filter currents, respectively.
In other words, the dynamics of stator, rotor, and grid filter,
described by (12)–(14), (24), and (25), behave as a partially
feedback linearizable system in the following general form:
η • = f (η, z)
z • = A1 z + Bv(t)
(26)
where η = [ψsd , γ]T , z = [ir d , ir q , ig d , ig q ]T , v(t) = [vr d crl (t),
vr q crl (t), vg d crl (t), vg q crl (t)]T , and B = I4×4 .
The dynamics of z, i.e., rotor and grid filter dynamics, are
called the external dynamics and the ones of η, stator dynamics,
are the internal dynamics [19]. The operating points of these
dynamic variables are z 0 and η0 . If
η • = f (η, z0 )
(27)
which are usually referred to as zero dynamics, are unstable, the
system (26) is said to be in nonminimum phase [19]. In this case,
a control law that stabilizes the external dynamics, but does not
take into account the internal dynamics, may result in a closedloop unstable system. The zero dynamics of DFIG system can
be obtained by setting the rotor and grid filter currents to their
desired reference values in (12)–(14), i.e., ir dq (t) = ir dq 0 (t) =
ir dq ref (t) and ig dq (t) = ig dq 0 (t) = ig dq ref (t). Thus, the zero
dynamics is given in (28) and (29) shown at the bottom of the
next page.
The equilibrium points of (28) and (29), under normal conditions, are γ0 and ψsd0 ≈ 1 p.u.. The linearized dynamic model
RAHIMI AND PARNIANI: TRANSIENT PERFORMANCE IMPROVEMENT OF WIND TURBINES WITH DOUBLY FED INDUCTION GENERATORS
of (28) and (29) around the operating point is obtained as
•
∆ψsd
a11 a12
∆ψsd
= ωb
(30)
∆γ •
a21 a22
∆γ
in which a11 to a22 are given in Appendix D. After some manipulation, the simplified characteristic polynomial is approximately found as
Rs + Re
λ2 + ωb
Ls + Le
((Rs +Re )/Ls)ψsd0 − ((Rs + Re)/Ls)Lm ir d0 + Re ig d0
+
ψsd0 (1 + (Le /Ls )) − (Le Lm /Ls)ir d0 + Le ig d0
+ ωb2 = 0.
≺
where Er dq and Eg dq are the regulation errors in the d–q axes
of rotor and grid filter currents. By substituting (33) into (7) and
(10), the rotor and grid filter dynamics can be described by
Lr dEr d
L dir d ref
= −Rr Er d + ω2 Lr ir q − ed − r
ωb dt
ωb dt
− Rr ir d
2ψsd0
.
Lm
(32)
If inequality (32) is satisfied, the nonlinear stator dynamics
and the zero dynamics are locally asymptotically stable. Considering (32), it is clear that the stability of zero dynamics depends
on the operating conditions ir d0 and ig d0 , and consequently, on
stator power factor. Also, it depends on the network parameters
such as Le and Re . Under voltage dip, ψsd0 is lower than its
value at normal conditions. Thus, the stability margin of the zero
dynamics under voltage dip decreases.
V. IMPROVEMENT OF DFIG TRANSIENT PERFORMANCE BY
NONLINEAR CONTROLLER
As discussed previously, a control strategy that compensates
the rotor back EMF and grid voltages can improve and stabilize the rotor and grid filter dynamics by removing the effects
of stator dynamics on the rotor and grid-side filter. However,
it may weaken the stator dynamics, and under certain operating conditions, the stator dynamics may become unstable. The
purpose of this section is to design a nonlinear controller for
stabilizing both the internal dynamics (stator dynamics) and
the external dynamics of DFIG system. To achieve the desired
control objectives, the following error functions are considered:
Er dq = ir dq − ir dq
Eg dq = ig dq − ig dq
ref
ref
(33)
ref
+ vr d
Lr dEr q
L dir q ref
= −Rr Er q − ω2 Lr ir d − eq − r
ωb dt
ωb dt
− Rr ir q
ref
+ vr q
Lg dEg d
= −Rg Eg d + ωLg ig q + vsd − Rg ig d
ωb dt
(31)
Considering (31), it is clear that the stator-mode natural frequency is near the line frequency, i.e., 1 p.u. For maintaining
stability of the zero dynamics, (28) and (29), it is required that
Le
Re
Le
Ls
ig d0
+
ir d0 1 +
−
Ls + Le
Lm
Ls + Le
Rs + Re
519
−
(34)
ref
Lg dig d ref
− vg d
ωb
dt
Lg dEg q
= −Rg Eg q − ωLg ig d + vsq − Rg ig q
ωb dt
ref
Lg dig q ref
− vg q .
(35)
ωb dt
Considering the dependency of the stator dynamics stability
on ir d , as in (32), a nonlinear control strategy is proposed that
stabilizes the entire system [(12)–(14), (34), and (35)], through
d-axis rotor voltage vr d . For this purpose, the control inputs vr dq
and vg dq in (34) and (35) are selected as follows:
vr d = vr d crl + vr d auxilary
−
v r q = vr q
crl
vg d = −vg d
crl
vg q = −vg q
crl
(36)
where vr d auxilary is the proposed nonlinear control that will be
described later. Each of the other terms, vr d crl , vr q crl , vg d crl ,
and vg q crl , consists of a PI control, back EMF, and crosscoupling compensations, and feedforward terms as follows:
vr d
crl (t)
= kp
id (ir d ref (t)
+ ki
(ir d
id
+ Rr ir d
vr q
crl (t)
= kp
ref
+
ref (t)
+ ki
iq
+ Rr ir q
(ir q
ref
+
− ir d (t))dt
Lr dir d ref
− ω2 Lr ir q + ed
ωb dt
iq (ir q ref (t)
− ir d (t))
− ir q (t))
ref (t)
− ir q (t))dt
Lr dir q ref
+ ω2 Lr ir d + eq
ωb dt
(37)
Rs + Re
Le Lm 1 dir d0 Le dig d0
1 Ls + Le dψsd
Rs + Re
Le Lm
= −
+ V∞ cos γ
ψsd +
Lm ir d0 − Re ig d0 −
ωir q 0 + Le ωig q 0 +
−
ωb Ls
dt
Ls
Ls
Ls
Ls ωb dt
ωb dt
dγ
= ωb
dt
ωs −
(28)
((Rs + Re )/Ls )Lm ir q 0 − Re ig q 0 + (Le Lm /Ls )(1/ωb )(dir q 0/dt) − (Le/ωb )(dig q 0/dt) + V∞ sin γ
.
ψsd (1 + (Le/Ls )) − (Le Lm /Ls )ir d0 + Le ig d0
(29)
520
vg d
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 2, JUNE 2010
crl (t)
= kp g (ig d ref (t) − ig d (t))
+ ki g (ig d ref (t) − ig d (t))dt
+ Rg ig d
vg q
crl (t)
ref
+
function of state variables as
vr d
Lg dig d ref
− ωLg ig q − vsd
ωb
dt
= kp g (ig q ref (t) − ig q (t))
+ ki g (ig q ref (t) − ig q (t))dt
+ Rg ig q
ref
+
Lg dig q ref
+ ωLg ig d − vsq . (38)
ωb dt
Replacing (36)–(38) in (34) and (35) yields the rotor and grid
filter dynamics as
Lr dEr d
= −(Rr + kp
ωb dt
id )Er d
+ x5 + vr d
Lr dEr q
= −(Rr + kp
ωb dt
iq )Er q
+ x6
dx5
= −ki id Er d
dt
dx6
= −ki iq Er q
dt
Lg dEg d
= −(Rg + kp g )Eg d + x14
ωb dt
auxilary
= −K
∆Xint
+ vn l
∆Xext
(42)
where ∆Xint = Xint − Xint 0 and ∆Xext = Xext − Xext 0 .
Subscript 0 denotes the operation point, and Xint 0 = [ψsd 0 γ0 ]T
and Xext 0 = 08×1 . From (41) and (42), we obtain
•
Xint
∆Xint
•
= (A − BK)
X =
Xext 10×1
∆Xext 10×1


02×1
fext (X)


+
+  vn l 
(43)
08×1
10×1
07×1
where B = [02×1 1 07×1 ]T . If K is chosen so that A − BK is
Hurwitz, then ∀Q > 0, ∃P > 0 such that [19]
auxilary
(A − BK)T P + P (A − BK) = −Q.
Thus,
V (Xint , Xext ) =
dx14
= −ki g Eg d
dt
dx15
= −ki g Eg q .
(40)
dt
In the control law of (37) and (38), the PI control terms
are used to improve and stabilize the d–q current dynamics of
rotor and grid-side filter. Also, the rotor auxiliary control input
vr d auxilary is used to stabilize the internal dynamics, and thus
to improve the entire dynamics of the system after clearing
the fault. Considering (12)–(14), (39), and (40), we rewrite the
electrical dynamics, stator, rotor, and grid filter dynamics, by
separating their linear parts from nonlinear parts, as follows:
•
Xint
∆Xint
fext (X)
•
=A
+
X =
Xext 10×1
∆Xext 10×1
08×1
10×1


02×1


+  vr d auxilary 
(41)
07×1
10×1
where Xint denotes the internal dynamics containing stator
dynamics with order of 2 [see (12) and (13)] and Xext denotes the external dynamics including rotor and grid filter dynamics with order of 8 [see (39) and (40)]. Also, fext (X) =
[fext 1 (X)fext 2 (X)]T is a 2 × 1 vector function of electrical
state variables and A is a 10 × 10 matrix. The proposed auxiliary control consists of a state feedback control and a nonlinear
T
P
∆Xext
(39)
Lg dEg q
= −(Rg + kp g )Eg q + x15
ωb dt
∆Xint
∆Xint
(44)
∆Xext
(45)
can be a Lyapunov candidate function for the closed-loop system. The derivative of V (Xint , Xext ) is given by
T ∆X
∆X
int
int
Q
V• =−
∆Xext
∆Xext


+ 2
fext (X)
vn l
T

 P
∆Xint
.
∆Xext
07×1
If vn l in (46) is selected such that

T
fext (X)
∆Xint


=0
 vn l
 P
∆Xext
07×1
or, if
2
i=1
fext i (X)Pi
vn l = −
P3
Then,
•
V =−
∆Xint
∆Xext
∆Xint
(46)
(47)
∆Xint
∆Xext
.
(48)
∆Xext
T
Q
∆Xint
∆Xext
≺ 0.
(49)
Thus, the closed-loop system becomes stable with the control law of (48). The matrix P is determined by solving the
Lyapunov equation (44). Pi in (48) is the ith row vector of matrix P and fext i (X) is the ith element of fext (X). The nonlinear
parts of stator dynamics, fext 1 (X) and fext 2 (X), are given in
RAHIMI AND PARNIANI: TRANSIENT PERFORMANCE IMPROVEMENT OF WIND TURBINES WITH DOUBLY FED INDUCTION GENERATORS
Fig. 7.
Fig. 6.
521
Evolution of stator modes in the instability case.
Block diagram of the proposed nonlinear controller.
Appendix E. We can show that in (43), fext (X) → 0 when
(Xint , Xext ) → (Xint 0 , Xext 0 ). Thereafter, it can be found that
in (48), vn l → 0 when (Xint , Xext ) → (Xint 0 , Xext 0 ) or when
the system state trajectories reach their stable operating point.
Fig. 6 shows block diagram of the d-axis rotor current control,
including the proposed nonlinear auxiliary control vr d-auxilary .
The q-axis current control is the same as in Fig. 2, and therefore,
is not repeated here.
VI. TRANSIENT PERFORMANCE EVALUATION
A. Eigenvalue Tracking
For identifying the nature of instability during transient states,
the method of eigenvalue tracking is introduced [20].
In this method, the system is repeatedly linearized by building
the state Jacobian matrix at selected time instants during the
simulation, and the system eigenvalues are computed at each
snapshot. The information provided by this online linearization
eigenvalues is not a strictly rigorous indication of stability, but
it is quite useful, in practice, to characterize the nature of a
possible instability, as will be done in the following.
B. Simulation Results
The studies are done on the SMIB system shown in Fig. 1,
with the DFIG parameters given in Appendix A. The rotor and
grid filter controllers are PI controllers introduced in Section II.
To evaluate transient performance of the system, a 70% voltage
dip with duration of 300 ms is imposed on the high-voltage
side of DFIG transformer at t = 20 s. At the moment of voltage dip, the generator slip and power are s0 = −0.095 and
pe0 = 0.6 p.u., respectively, and the generator is operated at
unity power factor. To take into account the converter limits,
rotor control voltage is limited to 0.4 p.u. Simulation results
show that in this case, the transient behavior of DFIG with con-
ventional PI controls is unstable. Fig. 7 shows the evolution
of the real part of the first critical eigenvalue in this instability
case. The examined eigenvalues correspond to stator modes of
the DFIG, after the fault has being cleared. The instability, in
Fig. 7, starts at t = 20.55. After this time, undamped oscillations
in electrical torque, terminal voltage, and generator speed are
observed. Online linearization at time instants after clearing the
fault shows that the stator modes move to unstable state, and the
DFIG loses its equilibrium.
As stated in Section IV and considering (32), the stability
of zero dynamics (stator dynamics) is closely related to the
network parameters Le and Re . To demonstrate the prominent
role of stator dynamics in the instability, the same voltage dip is
imposed at t = 0.9 s to the system with two different network
impedances. In the first case, Re +jXe = 0.05 + 0.3j p.u., and
in the second case, Re +jXe = 0.05 + 0.5j p.u. In both cases,
the slip and real power of the system in equilibrium point are
−0.095 and 0.6 p.u.
Fig. 8 shows that in case 2 with higher network reactance, the
stator flux at the time of clearing the fault is out of the stator
domain of attraction, and consequently, stator state trajectories
do not reach their stable postfault operating point, and the transient behavior of the DFIG is unstable. The final outcome of
instability is large oscillations in the terminal voltage, electrical
torque, and rotor speed of DFIG.
In WT generators, the nature of instability is different from
the rotor angle instability of conventional synchronous generators. In the WTs with the DFIG, the generator speed range
is approximately ±30% around the synchronous speed. The
upper limit of the generator speed is determined by the backto-back converter capacity rating. If the generator speed after
clearing the fault is higher than the limit, the converter cannot
handle the slip power and the generator may become unstable.
For the DFIG system studied in this paper, the operating speed
before the fault is 1.095 p.u. and the generator speed at the
moment of clearing the fault (at t = 1.2 s) is 1.11 p.u. Thus,
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 2, JUNE 2010
Fig. 8. DFIG transient response under 300 ms voltage dip, with two different values of network reactance. (a) Stator flux (ψ s d ). (b) Terminal voltage.
(c) Electrical torque. (d) Rotor speed.
Fig. 9. DFIG transient response under 300 ms voltage dip, with linear and
nonlinear controller (R e + jX e = 0.05 + j0.5 p.u.). (a) Stator flux (ψ s d ).
(b) Terminal voltage. (c) Electrical torque. (d) Rotor speed.
the generator slip after the fault is within the allowable range
and the back-to-back converter is able to handle the slip power.
Moreover, growth of the generator speed during the fault is relatively low. Therefore, the system does not face angle/speed
instability. However, the electrical dynamics of the DFIG, as
explained in Section IV, are in nonminimum phase, and under
compensation of back EMF voltages, the electrical dynamics
behave as a partially linearizable system containing internal and
external dynamics. This phenomenon can deteriorate the DFIG
postfault behavior. In this case, then, in spite of admissible slip,
the system may experience instability that roots in the stator
dynamics.
Next, the d-axis PI rotor controller is replaced with the nonlinear controller proposed in Section V, and transient behavior
of the DFIG is simulated with the same 70% voltage dip and
Re + jXe = 0.05 + 0.5j p.u. The results are shown in Fig. 9.
It is clear that in this case, transient behavior of DFIG with the
proposed nonlinear controller is stable and well damped.
RAHIMI AND PARNIANI: TRANSIENT PERFORMANCE IMPROVEMENT OF WIND TURBINES WITH DOUBLY FED INDUCTION GENERATORS
Fig. 10.
523
DFIG-based WT with crowbar protection and resistive chopper.
In ideal conditions, without rotor converter and dc-link constraints, we can completely compensate the rotor back EMF
voltages, and consequently remove the rotor inrush current and
dc-link voltage fluctuations. In fact, by full compensation of
rotor back EMF voltages, the rotor current dynamics will be independent of stator dynamics, and there will be no rotor inrush
current and dc-link overvoltage. However, in actual conditions,
due to limited capacity of the rotor-side converter, full compensation of back EMF voltages may not be possible during large
voltage dips. Hence, by using the proposed nonlinear control
applied to the rotor converter with limited capacity, the transients and fluctuations of the rotor current and dc-link voltage
are not completely removed.
Typically, the tolerable limit of the rotor current during the
network fault is 2 p.u., and that of the dc-link voltage is 1.2 times
its nominal value [8], [21]. Thus, a crowbar is used to protect the
rotor-side converter and power switches against the excessive
rotor current. It is also common to add a resistive chopper to
the dc-link voltage to limit the dc-link voltage by dissipating
the excess energy, as shown in Fig. 10.
By activating the resistive chopper, the dc-link voltage begins to fall. When the dc-link voltage falls below the minimum
threshold value (usually nominal dc-link voltage), the switch is
opened, and the resistive chopper is deactivated.
Further, the first step to limit the dc-link voltage fluctuations
before triggering the resistive chopper is reducing the changes
of rotor instantaneous power fed to the dc-link capacitor. This
step to a large extent is realized by compensation of rotor back
EMF voltages. The second step is the efficient control of the dclink voltage through grid-side converter control, which is not
dealt with in this paper.
Fig. 11 shows the rotor current, and dc-link voltage using
the proposed nonlinear control of Section V, for the same 70%
voltage dip of Fig. 9. As in the previous simulations, the converter voltage is limited to 0.4 p.u. Considering Fig. 11, the peak
value of the rotor current does not exceed its tolerable limit, i.e.,
2 p.u., during and after clearance of the fault. Also, with the
nominal dc-link voltage of 1200 V, the peak value of the dc-link
voltage during the fault is about 1300 V and does not exceed
its allowable limit. Thus, by using the proposed nonlinear control, with limited rotor and dc-link rated voltages, not only the
dynamic performance of the DFIG improves, but also the peak
values of the rotor current and dc-link voltage do not exceed
their acceptable limits.
Fig. 11. DFIG transient response under 70% voltage dip, using nonlinear
controller (R e + jX e = 0.05 + j0.5 p.u.). (a) Rotor current. (b) DC-link
voltage.
The reader might have noted that realization of the nonlinear
control law (48) requires the knowledge of the DFIG parameters such as rotor, and stator resistances and inductances. In
practical applications, however, situations may arise in which
these parameters are not exactly known. To examine robust performance of the controller against parameter uncertainties, in
the following, we simulate the system transient performance as
rotor/stator resistances and magnetizing inductance are severely
altered.
Fig. 12 shows the transient behavior of the DFIG system,
for the same 70% voltage dip of Fig. 9, under uncertainties in rotor/stator resistances and magnetizing inductance. In
Fig. 12(a) and (b), the rotor/stator resistances are unknown
and two cases are considered. In the first case, 60% underestimation, i.e., R̂s = 0.4Rs , R̂r = 0.4Rr , and in the second
case, 100% overestimation, R̂s = 2Rs , R̂r = 2Rr , are considered. In Fig. 12(c) and (d), ±40% error in estimation of Lm
is considered, i.e., L̂m = 0.6Lm , L̂m = 1.4Lm . Considering
Fig. 12, it is clear that the transient behavior of the system with
proposed nonlinear controller is stable for −60% to +100%
error in estimation of rotor/stator resistances. Also, the nonlinear controller shows good results for ±40% error in estimation
of magnetizing inductance. Thus, the performance of the proposed nonlinear controller, under a wide range of uncertainty in
524
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 2, JUNE 2010
the effects of stator dynamics on the rotor current dynamics
through rotor control voltage can improve the rotor dynamics
and enhance the DFIG ride-through capability during the fault.
However, it was shown that it can weaken the stator dynamics
and deteriorate the DFIG transient behavior after clearing the
fault. It was found that the electrical dynamics of the DFIG
are in nonminimum phase for certain operating conditions, and
thus, have an inherent limitation on the achievable transient
response.
Also, it was shown that the dynamics of the DFIG, under
compensation of rotor back EMF and grid voltages, behave as
a partially linearizable system containing internal and external
dynamics. The internal dynamics comprises stator dynamics and
have an important role on the DFIG transient behavior. They
can move the DFIG to unstable behavior after clearing the fault.
Since the DFIG electrical dynamics are nonlinear, the linear
control scheme cannot properly work under large voltage dips.
We addressed this problem by means of a nonlinear controller.
The proposed approach is a combination of PI and Lyapunovbased auxiliary control, which stabilizes the internal dynamics
and improves the DFIG postfault behavior through rotor control
voltage.
Simulation results showed that the proposed control method
is robust under uncertainties of generator parameters.
APPENDIX
A. Parameters of the 1.76-MVA, 575-V, 60-Hz DFIG WT
Vbase = 575 V,
Sbase = 1.76 MVA,
fbase = 60 Hz
ωb = 2πfb = 377 rad/s, Rs = 0.00706 p.u.
Rr = 0.005 p.u.,
Ls = 3.07 p.u.,
Lr = 3.056 p.u.
Lr = 3.056 p.u.,
Lm = 2.9 p.u.,
Lg = 0.3 p.u.
Rg = 0.003 p.u.,
Hr = 0.75 s,
Ht = 4.3 s
ks = 0.6 p.u./elec. rad, ωs = 1 p.u.
D = 1.2 p.u.,
B. Operating Conditions Used for Modal Analysis
in Section III-A
Fig. 12. DFIG transient response under 300 ms voltage dip, with nonlinear
controller, under uncertainties in rotor/stator resistances and magnetizing inductance. (a) Stator flux (ψ s d ). (b) Electrical torque. (c) Stator flux. (d) Electrical
torque.
s0 = −0.21,
Pe0 = 0.9 p.u.,
ψsd0 = 1.005 p.u.,
ψsq 0 = 0,
vsd0 = 0,
vsq 0 = 1 p.u.
ir d0 = 0.345 p.u.
ir q 0 = 0.7874.
estimation of rotor/stator resistances and magnetizing inductance, is robust.
C. Controller Parameters Used for Modal Analysis
in Section III-A
VII. CONCLUSION
This paper discussed the dynamic characteristics and improvement of transient performance in WTs with DFIGs. Modal
analysis and eigenvalue tracking were used to identify the nature of instability, and Lyapunov-based nonlinear controller
was used for improving the transient performance. Removing
kp
idq
= 0.633,
kpω = 6.98,
kI
pf
= 4.656
ki
idq
= 8.5,
kI ω = 0.04656,
kp
g
kp
= 0.6,
pf
ki
g
= 0.01235
= 2.26
RAHIMI AND PARNIANI: TRANSIENT PERFORMANCE IMPROVEMENT OF WIND TURBINES WITH DOUBLY FED INDUCTION GENERATORS
D. Elements a11 , a12 , a21 , and a22 of State Matrix Related
to Zero Dynamics (30)
a11
a12
Rs + Re
Le
Lm
=−
−
ir q 0
ig q 0 −
Ls + Le
D
Ls
Ls
v∞d0 Le (ig q 0 − (Lm /Ls ) ir q 0 )
=−
v∞q 0 −
Ls + Le
D
1 + (Le /Ls )
D
v∞d0
a22 = −
D
−(Rs + Re /Ls )ψsd0 + (Rs + Re /Ls )Lm ir d0 − Re ig d0
=
D
Le
Lm
+
ir q 0
ig q 0 −
D
Ls
Le
Le Lm
D = ψsd0 1 +
ir d0 + Le ig d0 .
(50)
−
Ls
Ls
a21 =
E. Nonlinear Parts of Stator Dynamics fext 1 and fext 2
With Le = 0
Rs
ψsd 0 + V∞ (γ − γ0 )Sin γ0 + V∞ Cos γ
Ls
Rs
Lm ir d ref
+
Ls
(Rs /Ls ) Lm (ir q ref − eq ) + V∞ Sin γ
= ωb ωs −
ψsd
fext 1 = ωb
fext 2
b21
−
− ωb (b21 (ψsd − ψsd 0 ) + b24 eq + b22 (γ − γ0 ))
(Rs Lm /Ls ) ir q ref + V∞ Sin γ0
=
,
2
ψsd
0
b22 = −
V∞ Cos γ0
,
ψsd 0
b24 =
Rs Lm
.
Ls ψsd 0
(51)
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Mohsen Rahimi received the B.Sc. degree in electrical engineering in 2001 from Isfahan University
of Technology, Isfhan, Iran, and the M.Sc. degree in
2003 from Sharif University of Technology, Tehran,
Iran, where he is currently working toward the Ph.D.
degree at the Department of Electrical Engineering.
His current research interests include modeling
and control of power system dynamics with particular interest in control of grid connected wind turbines
and renewable energy sources.
Mostafa Parniani (S’93–M’95–SM’06) received the
B.Sc. degree in electrical power engineering from
Amirkabir University of Technology, Tehran, Iran, in
1987, and the M.Sc. degree in electrical power engineering from Sharif University of Technology (SUT),
Tehran, in 1989, and the Ph.D. degree in electrical
engineering from the University of Toronto, Toronto,
ON, Canada, in 1995.
From 1988 to 1990, he was with GhodsNiroo Consulting Engineers Corporation and Electric
Power Research Center, Tehran. Since 1995, he has
been an Assistant Professor in the Department of Electrical Engineering, SUT.
From 2005 to 2006, he was a Visiting Scholar at Rensselaer Polytechnic Institute, Troy, NY. His current research interests include power system dynamics
and control, and applications of power electronics in power systems.