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Sensorless Control of Doubly-Fed Induction
Generators in Variable-Speed Wind Turbine Systems
Mohamed Abdelrahem
Christoph Hackl
Ralph Kennel
Student Member, IEEE
Institute for Electrical Drive Systems
and Power Electronics
Technische Universität München (TUM)
Munich, Germany
Email: [email protected]
Member, IEEE
Munich School of Engineering Research Group
“Control of Renewable Energy Systems (CRES)”
Technische Universität München (TUM)
Munich, Germany
Email: [email protected]
Senior Member, IEEE
Institute for Electrical Drive Systems
and Power Electronics
Technische Universität München (TUM)
Munich, Germany
Email: [email protected]
Abstract—This paper proposes a sensorless control strategy
for doubly-fed induction generators (DFIGs) in variable-speed
wind turbine systems (WTS). The proposed scheme uses an
extended Kalman filter (EKF) for the estimation of rotor speed
and rotor position. Moreover, the EKF is used to estimate the
mechanical torque of the generator to allow for maximum power
point tracking control for wind speeds below the nominal wind
speed. For EKF design, the nonlinear state space model of the
DFIG is derived. Estimation and control performance of the
proposed sensorless control method are illustrated by simulation
results at low, high, and synchronous speed. The designed EKF is
robust to machine parameter variations within reasonable limits.
Finally, the performances of the EKF and a model reference
adaptive system (MRAS) observer are compared for time-varying
wind speeds.
Keywords—DFIG, MPPT, Kalman filter, MRAS observer
N OTATION
N, R, C are the sets of natural, real and complex numbers.
x ∈ R or x ∈ C is a real or complex scalar. x ∈ Rn (bold)
is a real√valued vector with n ∈ N. x> is the transpose and
kxk = x> x is the Euclidean norm of x. 0n = (0, . . . , 0)>
is the n-th dimensional zero vector. X ∈ Rn×m (capital bold)
is a real valued matrix with n ∈ N rows and m ∈ N columns.
O ∈ Rn×m is the zero matrix. xyz ∈ R2 is a space vector of
a rotor (r) or stator (s) quantity, i.e. z ∈ {r, s}. The space
vector is expressed in either phase abc-, stator fixed s-, rotor
fixed r-, or arbitrarily rotating k-coordinate system, i.e. y ∈
{abc, s, r, k}, and may represent voltage u, flux linkage ψ or
current i, i.e. x ∈ {u, ψ, i}. E{x} or E{X} is the expectation
value of x or X, resp.
I.
I NTRODUCTION
The electrical power generation by renewable energy
sources (such as e.g. wind) has increased significantly during
the last years contributing to the reduction of carbon dioxide emissions and to a lower environmental pollution [1].
This increase will continue as countries are extending their
renewable action plans. Therefore, the share of wind power
generation will increase further worldwide. Among the various
types of wind turbine generators, the DFIG is the most commonly used generator in on-shore and off-shore applications,
accounting for more than 50% of the installed wind turbine
nominal capacity worldwide [1]. DFIGs can supply active and
reactive power, operate with a partial-scale power converter
(around 30% of the machine rating), and achieve a certain
ride through capability [2]. Operation above and below the
synchronous speed is feasible. Due to their wide use in WTS,
the development of advanced and reliable control techniques
for DFIGs has received significant attention during the last
years [2]. Examples of these control techniques are e.g. vector
control, direct torque control, and direct power control [2].
Vector control has – so far – proven to be the most popular
control technique for DFIGs in variable-speed WTS [2]. This
method allows for a decoupled control of active and reactive
power of WTS via regulating the quadrature components of
the rotor current vector independently. Vector control requires
accurate knowledge of rotor speed and rotor position [2].
Recently, the interest in sensorless methods (see, e.g., [5],
[2] and references therein) is increasing due to cost effectiveness/robustness, which implies that the vector controllers
must operate without the information of mechanical sensors
(such as position encoders or speed transducers) mounted on
the shaft. The required rotor signals must be estimated via the
information provided by electrical (e.g. current) sensors which
are cheap and easier to install than mechanical sensors. Furthermore, mechanical sensors reduce the system reliability due
to their high failure rate, which implies shorter maintenance
intervals and, so, higher costs.
Sensorless control methods for doubly-fed induction machines/generators have been proposed by several researchers.
The proposed approach in [6] uses the magnetizing currents
supplied from the rotor and stator to estimate the rotor position
and speed; however, observer design and its dynamics were
not addressed/analyzed. In [7], a rotor-flux-based sensorless
scheme is reported, where the rotor flux is obtained by the integration of the rotor back-electromotive force. This approach
might suffer from integration problems with poor performance
during operation close to synchronous speed, because the rotor
is excited with low frequency voltages. The sensorless methods
presented in [8]-[12] are open-loop and rely on rotor current
estimators in which the estimated and measured currents are
compared to obtain the rotor position. The rotational speed is
obtained by numerical differentiation which is very sensitive
to noise. None of these methods addresses the design of the
rotor position estimator bandwidth and the effect of parameter
uncertainties on the estimation accuracy.
Wind
turbine
sl (Lr irq  Lmisq )
mm
mm
r
PI
s
i rq
Q s , ref
PI
Qs
u dc , ref
d
f
i qf
i qf , ref
m
C dc
udc
PI
abc
s
u
 s abc/dq
u
d
s
u sq
isabc
Pdc
Qs
i abc
f
Ps
Rf L f
PI
s L f i df
gear
box
filter
Drive
 s abc/dq
PI
Qf
i df ,ref
i
i abc
f
Q f ,ref
DC
Link
s L i
PI
PLL
sl (L i  L i )
d
ms
q
f f
udc
Encoder
DFIG
C
d
r r
PWM
u
PI
irq,ref
dq/abc
abc
s
sl
Drive
i
sl abc/dq
PWM
irabc
d
r
t
irabc
PI
dq/abc
Lookup table r
RSC
ird,ref
GSC
s
Pf & Q f
Tr.
Grid
Figure 1: DFIG topology and control structure for the variable-speed wind turbine system.
The application of model reference adaptive system
(MRAS) observers for sensorless control for DFIGs has been
reported in [13], where MRAS observers are diversified with
various error variables, e.g. stator and rotor currents and
fluxes. Moreover, the respective estimation performances are
compared. The common disadvantage of the presented MRAS
observers is the DC-offset drift problem caused by the pure
integral action in the stator-flux estimation.
The sensorless control approach in [14] relies on signal
injection. The main advantage of this method is its high robustness against variations in the machine parameters. However,
the injection of high-frequency signals in the DFIG rotor is
not easy for large machines (> 1 MW) such those in modern
WTS. Another alternative is the use of an extended Kalman
filter (EKF) which has already been used for sensorless control
and the estimation of the electrical parameters of induction
machines and permanent magnet synchronous machines [15],
[16]. An EKF was used for DFIG speed and position estimation in [17], however the authors use state variables in
the rotating reference frame, whereas input and measurement
variables are in the stationary reference frame and are directly incorporated into the EKF design. This increases the
complexity of state, input and measurement signals, since the
Park transformation (from the stationary reference frame to
the rotating reference frame) has to be considered at each
sampling instant during time and measurement update and
computation of the Kalman gain of the EKF. This results in
high computational loads during real-time application.
In this paper, an extended Kalman filter is proposed to
estimate speed and position of the rotor and the mechanical
torque of the DFIG. State, input and measurement variables
are selected in the rotating reference frame, which reduces
the complexity of state, input and measurement matrices and,
hence, the computational time for real-time implementation.
The EKF performance and its robustness against parameter
variations are illustrated by simulation results. The results
highlight the ability of the EKF in tracking the DFIG rotor
speed and position. The results are compared with those of a
MRAS observer.
II.
M ODELING AND C ONTROL OF THE WTS WITH DFIG
The block diagram of the vector control problem of WTS
with DFIG is shown in Fig. 1. It consists of a wound rotor
induction machine mechanically coupled to the wind turbine
via a shaft and gear box with ratio gr ≥ 1 [1]. The stator
windings of the DFIG are directly connected to the grid via a
transformer, whereas the rotor winding is connected via a backto-back partial-scale voltage source converter (VSC), a filter
and a transformer to the grid. The transformer will be neglected
in the upcoming modeling. The rotor side converter (RSC) and
the grid side converter (GSC) share a common DC-link with
capacitance Cdc [As/V] with DC-link voltage udc [V]. Detailed
models of these components can be found in [18]. The stator
and rotor voltage equations of the DFIG are given by [19]:
usabc (t)
urabc (t)
d abc
ψ (t),
dt s
d
= Rr irabc (t) + ψrabc (t),
dt
= Rs isabc (t) +
ψsabc (0) = 03 (1)
ψrabc (0) = 03 (2)
|
{z
}
initial values
where (assuming linear flux linkage relations)
ψsabc (t) = Ls isabc (t) + Lm irabc (t)
ψrabc (t) = Lr irabc (t) + Lm isabc (t).
(3)
(4)
Here usabc = (usa , usb , usc )> [V], urabc = (ura , urb , urc )> [V],
isabc = (isa , isb , isc )> [A], irabc = (ira , irb , irc )> [A], ψsabc =
(ψsa , ψsb , ψsc )> [Vs], and ψrabc = (ψra , ψrb , ψrc )> [Vs] are the
stator and rotor voltages, currents and fluxes, respectively, all in
the abc-reference frame (three-phase system). Stator Ls [Vs/A]
and rotor Lr [Vs/A] inductance can be expressed by
Ls = Lm + Lsσ
and
Lr = Lm + Lrσ
(5)
where Lsσ and Lrσ are the stator and rotor leakage inductances
and Lm is the mutual inductance. Rs [Ω] and Rr [Ω] are stator
and rotor winding resistances. Note that the DFIG rotor rotates
with mechanical angular frequency ωm [rad/s]. Hence, for a
machine with pole pair number np [1], the electrical angular
frequency of the rotor is given by
ωr = np ωm
0
to the voltage equations (8) yields the description in the
rotating reference frame (neglecting initial values)

d
ψsk (t) + ωs J ψsk (t),
usk (t) = Rs isk (t) + dt


k
k
k
d
k
ur (t) = Rr ir (t) + dt ψr (t) + (ωs − ωr (t))J ψr (t),
(9)

| {z }

=:ωsl (t)
and the rotor reference frame is shifted by the rotor angle
Z t
ωr (τ )dτ + φ0r , φ0r ∈ R
(6)
φr (t) =
0
with respect to the stator reference frame (φ0r is the initial rotor
angle).
A. Model in stator (stationary) reference frame
The equations (1) and (2) can be expressed in the stationary
reference frame as follows
xs = (xα , xβ )> = TC xabc
by using the Clarke and Park transformation (see, e.g., [18]),
respectively, given by (neglecting the zero sequence)
1
1 −
−√12
cos(φ) sin(φ) s
abc
k
√2
xs = γ
x
x
&
x
=
3
− sin(φ) cos(φ)
0
− 23
2
{z
}
|
|
{z
}
=:TC
frequency f0 > 0, it holds that ωs = 2πf0 rad
s is constant).
Applying the (inverse) Park transformation with TP (φs )−1 as
in (7) with
Z t
φs (t) =
ωs (τ )dτ + φ0s , φ0s ∈ R
=:TP (φ)−1
(7)
wherepγ =
for an amplitude-invariant transformation (or
γ = 2/3 for a power-invariant transformation). Expressing
the rotor voltage equation (2) also with respect to the stationary
reference frame (i.e. urs = TP (φr )−1 TC urabc ), the voltage
equations (1) and (2) can be rewritten as
d
uss (t) = Rs iss (t) + dt
ψss (t), ψss (0) = 02
d
urs (t) = Rr irs (t) + dt
ψrs (t) − ωr (t)J ψrs (t), ψrs (0) = 02
(8)
where [18]
0 −1
J := TP (π/2) =
.
1 0
2
3
where usk = (usd , usq )> , urk = (urd , urq )> , isk = (isd , isq )> , irk =
(ird , irq )> , ψsk = (ψsd , ψsq )> , ψrk = (ψrd , ψrq )> , are the stator
and rotor voltages, currents and fluxes in the rotating reference
frame (k-coordinate system with axes d and q), respectively.
ωsl := ωs − ωr
is the slip angular frequency. Since, e.g., ψsk = TP (φs )−1 ψss =
TP (φs )−1 TC ψsabc , the flux linkages are given by
ψsk = Ls isk + Lm irk
(10)
ψrk = Lr irk + Lm isk .
C. Dynamics of the mechanical system
For a stiff shaft and a step-up gear with ratio gr ≥ 1, the
dynamics of the mechanical system are given by
d
1
mt
0
ωm =
me −
, ωm (0) = ωm
∈R
(11)
dt
Θ
gr
|{z}
=:mm
where
3
np is (t)> J ψss (t)
2 s
3
= np Lm isq (t)ird (t) − isd (t)irq (t) .
(12)
2
is the electro-magnetic machine torque (moment), mt [Nm] is
the turbine torque produced by the wind (see Sec. III) and
t
mm = m
gr [Nm] is the mechanical torque acting on the DFIG
shaft. Θ [kgm2 ] is the rotor inertia and np [1] is the pole pair
number.
me (t) =
D. Overall nonlinear model of the DFIG
B. Model in stator voltage orientation
An essential characteristic of the DFIG control strategy is
that the generated active and reactive power shall be controlled
independently. It is common to use an air-gap flux orientation [20] or a stator flux orientation [21]-[23] for the vector
control schemes. However, it has been shown that the stator
flux orientation can cause instability under certain operating
conditions [24]. Therefore, following the ideas in [19], [25],
in this paper, a stator (grid) voltage orientation for the vector
control scheme is used.
The stator voltage orientation is achieved by aligning the
d-axis of the synchronous (rotating) reference frame with the
stator voltage vector uss which rotates with the stator (grid)
angular frequency ωs (under ideal conditions, i.e. constant grid
For the design of the EKF, the derivation of a compact
(nonlinear) state space model of the DFIG of the form
d
x = g(x, u), x(0) = x0 ∈ R7 and y = h(x), (13)
dt
is required. Therefore, introduce the state vector x, the output
(measurement) vector y and the input vector u as follows:

>
x = isd isq ird irq ωr φr mm ∈ R7 , 

4
d
q
d
q >
(14)
y = is is ir ir
∈R ,


4
d
q
d
q >
u = us us ur ur
∈R .
Note that the mechanical torque mm is considered as an
additional virtual (constant) state. Combining the subsystems
of the DFIG as in (9), (10), (11) and (12), inserting (10) into (9)

1
d
2
q
d
q
d
d
σLs Lr (−Rs Lr is + (ωr Lm + ωs σLs Lr )is + Rr Lm ir + ωr Lm Lr ir + Lr us − Lm ur )
 1 ((−ωr L2 − ωs σLs Lr )isd − Rs Lr iq − ωr Lm Lr id + Rr Lm iq + Lr uq − Lm uq ) 
m
s
r
r
s
r 
 σLs Lr
 1 (Rs Lm id − ωr Ls Lm iq − Rr Ls id + (−ωr Lr Ls + ωs σLs Lr )iq − Lm ud + Ls ud )
s
s
r
r
s
r 
 σLs Lr

1
d
q
d
q
q
q 
 σLs Lr (ωr Ls Lm is + Rs Lm is + (ωr Lr Ls − ωs σLs Lr )ir − Rr Ls ir − Lm us + Ls ur ) 


np 3
q d
d q


Θ 2 np Lm (is ir − is ir ) − mm ]


ωr

g(x, u) =
(15)
0
d k
d k
and solving for dt
is and dt
ir yields the nonlinear model (13)
L2
with g(x, u) as in (15), σ := 1 − Lsm
Lr and
1 0 0 0 0 0 0
h(x) = 00 10 01 00 00 00 00 x.
(16)
0
0
0
1
0
0
0
|
{z
}
=:C=[I 4 , O 4×3 ]∈R4×7
E. Overall control system of the WTS
The complete control block diagram of the DFIG in stator
voltage orientation is depicted in Fig. 1. For the rotor-side
converter (RSC), the d-axis current is used to control the DFIG
stator active power (i.e., proportional to the electro-magnetic
torque) in order to harvest the maximally available wind power
(i.e., maximum power point tracking, see Sec. III), whereas
the q-axis current is used to control the reactive power flow
of the DFIG to the grid. For the grid-side converter (GSC),
also stator voltage orientation is used [25], [18], which allows
for independent control of active (d-axis current) and reactive
power (q-axis current) flow between grid and GSC. The main
control objective of the GSC is to assure an (almost) constant
DC-link voltage regardless of magnitude and direction of the
rotor power flow. DC-link voltage control is a non-trivial task
due to the possible non-minimum-phase behavior for a power
flow from the grid to the DC-link [18], [4]. More details on
controller design, phase-locked loop or, alternatively, virtual
flux estimation and pulse-width modulation (PWM) are given
in, e.g., [25], [3], [18].
III.
M AXIMUM POWER POINT TRACKING (MPPT)
Wind turbines convert wind energy into mechanical energy
and, via a generator, into electrical energy. The mechanical
(turbine) power of a WTS is given by [19], [18], [26]:
1
3
(17)
pt = cp (λ, β) ρπrt2 vw
2
| {z }
wind power
where ρ > 0 [kg/m3 ] is the air density, rt > 0 [m] is the
radius of the wind turbine rotor (πrt2 is the turbine swept area),
cp ≥ 0 [1] is the power coefficient, and vw ≥ 0 [m/s] is the
wind speed. The power coefficient cp is a measure for the
“efficiency” of the WTS. It is a nonlinear function of the tip
speed ratio
ωm rt
λ=
≥0
[1]
(18)
gr vw
and the pitch angle β ≥ 0 [◦ ] of the rotor blades. The Betz limit
cp,Betz = 16/27 ≈ 0.59 is an upper (theoretical) limit of the
power coefficient, i.e. cp (λ, β) ≤ cp,Betz for all (λ, β) ∈ R×R.
For typical WTS, the power coefficient ranges from 0.4 to 0.48
[19], [26]. Many different (data-fitted) approximations for cp
have been reported in the literature. This paper uses the power
coefficient cp from [26], i.e.
−21
116
− 0.4β − 5 e λi + 0.0068λ
cp (λ, β) = 0.5176
λi
1
1
0.035
− 3
.
(19)
:=
λi
λ + 0.08β
β +1
For wind speeds below the nominal wind speed of the WTS,
maximum power tracking is the desired control objective.
Here, the pitch angle is held constant at β = 0 and the
WTS must operate at its optimal tip speed ratio λ? (a given
constant) where the power coefficient has its maximum c?p :=
cp (λ? , 0) = maxλ cp (λ, 0). Only then, the WTS can extract
3
[18].
the maximally available turbine power p?t := c?p 12 ρπrt2 vw
Maximum power point tracking is achieved by the nonlinear
speed controller
2
m?m = −kp? ωm
≈ mm ,
kp? :=
ρπrt5 c?p
2gr (λ? )3
(20)
which assures that the generator angular frequency ωm is
!
adjusted to the actual wind speed vw such that ωgrmvrwt = λ?
holds. According to (20) the optimum torque m?m can be
calculated from the (estimated) shaft speed ωm = ωr /np and
then it is compared with the actual mechanical torque mm ,
which is estimated by the EKF, as shown in Fig. 1. Based on
the difference m?m − mm the underlying torque PI controller1
d
generates the rotor reference current ir,ref
.
Remark: For wind speeds above the nominal wind speed, the
WTS changes to nominal operation, i.e. m?m = mm,nom , where
mm,nom is the nominal/rated generator torque. Speed control
is achieved by (individual) pitch control such that the rated
power mm,nom ωm,nom of the WTS is generated.
IV.
E XTENDED K ALMAN F ILTER AND MRAS O BSERVER
A. Extended Kalman Filter (EKF)
The EKF is a nonlinear extension of the Kalman filter for
linear systems and is designed based on a discrete nonlinear
system model [27]. For discretization the (simple) forward
Euler method with sampling time Ts [s] is applied to the timecontinuous model (13) with (14), (15) and (16). For sufficiently
small Ts 1, the following holds x[k] := x(kTs ) ≈ x(t)
d
and dt
x(t) = x[k+1]−x[k]
for all t ∈ [kTs , (k + 1)Ts ) and
Ts
k ∈ N ∪ {0}. Hence, the nonlinear discrete model of the DFIG
1 The torque PI control loop still requires a thorough stability analysis which
is not considered in this paper.
can be written as
=:f (x[k],u[k])
z
}|
{
x[k + 1] = x[k] + Ts g(x[k], u[k]) +w[k],
y[k] = h(x[k]) + v[k],
x[0] = x0



(21)


where the random variables w[k] := (w1 [k], . . . , w7 [k])> ∈
R7 and v[k] := (v1 [k], . . . , v4 [k])> ∈ R4 are included to model system uncertainties and measurement
noise, respectively. Both are assumed to be independent
(i.e., E{w[k]v[j]> } = O 7×4 for all k, j ∈ N), while
(i.e., E{w[k]} = 07 and E{v[k]} = 04 for all k ∈
N) and with normal probability
distributions (i.e., p(αi ) =
−(αi −E{αi })2
1√
exp
with σα2 i := E{(αi − E{αi })2 }
2
2σα
σαi 2π
and αi ∈ {wi , vi }). For simplicity, it is assumed that the
covariance matrices are constant, i.e., for all k ∈ N:
Q := E{w[k]w[k]> } ≥ 0 and R := E{v[k]v[k]> } > 0.
(22)
Note that Q and R must be chosen positive semi-definite and
positive definite, resp.
Since system uncertainties and measurement noise are not
known a priori, the EKF is implemented as follows
x̂[k + 1] = f (x̂[k], u[k]) − K[k] y[k] − ŷ[k] ,
(23)
ŷ[k] = h(x̂[k]) = C x̂[k].
where K[k] is the Kalman gain (to be specified below) and x̂
and ŷ are the estimated state and output vector, respectively.
The recursive algorithm of the EKF implementation is listed
in Algorithm 1 [27]. The EKF achieves an optimal state
estimation by minimizing the covariance of the estimation error
for each time instant k ≥ 1.
Algorithm 1: Extended Kalman filter
Step I: Initialization for k = 0
x̂[0] = E{x0 },
P 0 := P [0] = E{(x0 − x̂[0])(x0 − x̂[0])> },
−1
K 0 := K[0] = P [0]C > CP [0]C > + R
Step II: Time update (“a priori prediction”) for k ≥ 1
(a) State prediction
x̂− [k] = f (x̂[k − 1], u[k − 1])
(b) Error covariance matrix prediction
P − [k] = A[k]P [k − 1]A[k]> + Q
where
(x,u) A[k] = ∂f ∂x
−
matrix P 0 represents the covariances (or mean-squared errors)
based on the initial conditions (often P 0 is chosen to be a
diagonal matrix) and determines the initial amplitude of the
transient behavior of the estimation process, while duration
of the transient behavior and steady state performance are
not affected. The matrix Q describes the confidence with the
system model. Large values in Q indicate a low confidence
with the system model, i.e. large parameter uncertainties are
to be expected, and will likewise increase the Kalman gain to
give a better/faster measurement update. However, too large
elements of Q may be lead to oscillations or even instability
of the state estimation. On the other hand, low values in
Q indicate a high confidence in the system model and may
therefore lead to weak (slow) measurement corrections.
The matrix R is related to the measurement noise characteristics. Increasing the values of R indicates that the measured
signals are heavily affected by noise and, therefore, are of
little confidence. Consequently, the Kalman gain will decrease
yielding a poorer (slower) transient response.
In [29] general guide lines are given how to select the
values of Q and R. Following these guide lines, for this paper
the following values have been selected
Q
R
P0
x0
Remark on the observability of the nonlinear DFIG model:
For nonlinear systems, it is possible to analyze observability
locally by analyzing the linearized model around an operating
point [28]. The observability of the linearized DFIG model has
been tested around several operating point (e.g. at low, high and
synchronous speed). The analysis shows that the observability
of the system is affected by the rotor current. When the DFIG
operates exactly at its synchronous speed, the rotor current
is zero and observability is lost at this (singular) point. For
operation close to synchronous speed, the DFIG is (locally)
observable (see also [17]).
B. MRAS Observer
The MRAS observer is based on two models [13]: a
reference model and an adaptive model, see Fig. 2.
iss
Rs
x̂ [k]
Step III: Computation of Kalman gain for k ≥ 1
−1
K[k] = P − [k]C > CP − [k]C > + R
Step IV: Measurement update (“correction”) for k ≥ 1
(a) Estimation update with measurement
x̂[k] = x̂− [k] + K[k](y[k] − h(x̂− [k]))
(b) Error covariance matrix update
P [k] = P − [k] − K[k]C > P − [k]
Step V: Go back to Step II.
A crucial step during the design of the EKF is the choice
of the matrices P 0 , Q and R, which affect the performance
and the convergence of the EKF. The initial error covariance
= diag{0.03, 0.03, 0.03, 0.03, 3 · 10−5 , 10−6 , 6 · 10−5 }
= diag{1, 1, 1, 1}
(24)
= diag{0.02, 0.02, 0.02, 0.02, 2 · 10−5 , 5 · 10−5 , 10−4 }
= (0, 0, 0, 0, 1, 0, 0.5)>
u ss
I
 ss
I
ˆr
Ls
1/Lm
iˆrs
TP(ˆr)1
e
iˆrr
i
r
r
PI
̂r
Figure 2: MRAS observer to estimate rotor position and speed.
For this paper, the reference model (see left part in Fig. 2)
is fed by the measured stator current iss and the measured stator
(grid) voltage uss . From the reference model (based on (10))
the rotor current iˆrs is estimated via
1
iˆrs (t) =
ψ s (t) − Ls iss (t)
(25)
Lm s
t
uss (τ ) − Rs iss (τ ) dτ.
(26)
r
r
r
r
r
r
r
S IMULATION R ESULTS AND D ISCUSSION
A simulation model of a 2 MW WTS with DFIG is
implemented in Matlab/Simulink. The system parameters are
listed in the Appendix. The implementation is as in Fig. 1.
For more details on the implementation of e.g. back-to-back
converter, PWM, current controller design, see [18]. The
simulation results are shown in Figures 3-6. The estimation
performances of MRAS observer and EKF are compared for
different wind speed and parameter uncertainties in Rs , Rr
and Lm .
Estimation results: Fig. 3a and Fig. 3b show the simulation
results for MRAS observer and EKF when the wind speed
changes from 7 ms to 11 ms and, then, to 9 ms (see top of Fig. 3a).
This wind speed range covers almost the complete speed
range of the DFIG (i.e. ±25% around the synchronous speed).
Fig. 3b illustrates the tracking capability of the EKF of rotor
speed and rotor position at low and high speeds, and close
to synchronous speed. For comparison, Fig. 3a shows the
estimation performance of the MRAS observer. Tab. I lists
the estimation errors of MRAS observer and EKF: the EKF
shows a (slightly) higher estimation accuracy than the MRAS
observer. An additional advantage of the EKF is its capability
of estimating the mechanical torque as shown in Fig. 3b.
Table I: Estimation errors of MRAS observer and EKF.
Observer
Estimated state
Normal conditions
Rs and Rr increase by 50%
Lm increases by 10%
MRAS
ωr
φr
1.4%
1.8%
2.2%
3.4%
4%
4.8%
ωr
1%
1.4%
2%
10
8
6
1.4
1.2
1.0
0.8
0.6
r
̂r
0
0.2 0.4 0.6 0.8
EKF
φr
1.2%
1.8%
3%
mm
2.3%
3.6%
6%
In order to check the robustness of the EKF under (unknown) parameter variations of the DFIG, the values of the
stator resistance Rs and the rotor resistance Rr are increased
by ±50% (e.g. due to warming or aging). For this scenario,
Fig. 4a and Fig. 4b show the estimation performances of
the MRAS observer and the EKF. The simulated wind speed
profile is depicted in Fig. 4a (top). The EKF is more robust
1.0
1.2 1.4 1.6 1.8
8
ˆr
r
5
0
0
0.05
r
The PI controller drives this error to zero by adjusting ω̂r . Its
output is the estimated speed ω̂r which is integrated to obtain
the estimated rotor angle φ̂r . For more details see [13].
V.
vw[m/ s]
The adaptive model (see right part in Fig. 2) is fed by the
estimated rotor current iˆrs and the measured rotor current irr
in the rotor reference frame which has been proven to be
the best option among all possible implementations of MRAS
observers [13]. The goal of the adaptive model (which is essentially a phase-locked loop) is to estimate rotor position φ̂r and
rotor speed ω̂r . To achieve that the estimated and the measured
rotor current must be compared; to do so, the estimated rotor
current iˆrs (in the stator reference frame) must be expressed in
the rotor reference frame, i.e. iˆrr = TP (φ̂r )−1 iˆrs . The “error”
between estimated iˆrr and measured rotor current irr is defined
as
e := iˆr J ir = iˆr J ir = kiˆs k kis k sin ∠(iˆs , is ) .
12
 r [ pu ]
0
r [rad/ S]
=
0.1
0.15
time (sec)
0.2
0.25
(a) Results of the MRAS observer (top: wind speed vw ).
m m [ pu ]
Z
1.51.5
mm
m̂ m
1.0 1
0.50.5
0
 r [ pu ]
ψss (t)
1.4
1.21.2
1.0 1
0.80.8
0.60.6
00
8
r [rad/ S]
where
0.2
0.4
0.6
0.8
1
1.4
1.6
r
̂r
0.2 0.4
0.4 0.6
0.6 0.8
0.8
0.2
1
1.0
1.2 1.4
1.4 1.6
1.6 1.8
1.2
r
5 5
0 0
00
1.2
0.05
0.05
0.1
0.15
0.1
0.15
time (sec)
0.2
0.2
ˆr
0.25
0.25
(b) Results of the proposed EKF (top: eletro-mechanical torque
mm and its estimation m̂m ).
Figure 3: Estimation performance of the MRAS observer and
the proposed EKF: Estimation of rotor speed ωr and rotor
angle φr .
than the MRAS observer under parameter uncertainties in Rs
and Rr . It estimates rotor speed and rotor position with smaller
errors than the MRAS observer (see Tab. I). In addition, the
EKF estimation accuracy of the mechanical torque is still
acceptable (see Fig. 4b).
Finally, the robustness with respect to changes (due to
magnetic saturation) in the mutual inductance Lm is investigated. Therefore, Lm is increased by 10%. Fig. 5a and Fig. 5b
show the simulation results of MRAS observer and EKF for
this scenario. The used wind speed profile is depicted in
Fig. 5a (top). Again, the EKF shows a more robust and more
accurate estimation performance than the MRAS observer
under variations in Lm (see Tab. I).
The final simulation results are shown in Fig. 6 and
illustrate the control performance of the maximum power point
tracking (MPPT) algorithm under wind condition as shown in
12
r [rad/ S]
vw[m/ s]
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6
r
̂r
0.2
0.2 0.4
0.4 0.6
0.6 0.8
0.8
1.0
1
1.2
1.2 1.4
1.4 1.6
1.6 1.8
r
55
00
00
0.05
0.05
0.1
0.15
0.1
0.15
time (sec)
0.2
0.2
ˆr
8
6
1.4
1.2
1.0
0.8
0.6
r
̂r
0
0.2 0.4 0.6 0.8
1.0
1.2 1.4 1.6 1.8
8
ˆr
r
5
0
0.25
0.25
0
0.05
0.1
0.15
time (sec)
0.2
0.25
(a) Results of the MRAS observer (top: wind speed vw ).
1.51.5
1.5
1
mm
r [rad/ S]
1.4
1.2
1.2
1.0 1
0.80.8
0.60.6
00
8
5
5
0
00
0
0.2 0.4 0.6 0.8
1
r
̂r
0.2 0.4
0.4 0.6
0.6 0.8
0.8
0.2
1
1.0
1.2 1.4
1.4 1.6
1.6 1.8
1.2
r
0.05
0.05
0.1
0.15
0.1
0.15
time (sec)
0.2
0.2
ˆr
0.25
0.25
(b) Results of the proposed EKF (top: eletro-mechanical torque
mm and its estimation m̂m ).
Figure 4: Robustness results of the MRAS observer and the
proposed EKF: Estimation of rotor speed ωr and rotor angle
φr for an 50% increase in Rs and Rr .
mm
m̂ m
1.0
0.5
1.2 1.4 1.6
 r [ pu ]
0.5
0
m̂ m
r
̂r
1.4
1.2
1.0
0.8
0.6
0
r [rad/ S]
1.0
m m [ pu ]
(a) Results of the MRAS observer (top: wind speed vw ).
0.5
 r [ pu ]
 r [ pu ]
1.4
1.21.2
1.0 1
0.80.8
0.60.6
0
0
8
r [rad/ S]
vw[m/ s]
0
 r [ pu ]
10
88
66
m m [ pu ]
12
12
1010
0.2 0.4 0.6 0.8
1.0
1.2 1.4 1.6 1.8
8
r
5
0
0
0.05
0.1
0.15
time (sec)
0.2
ˆr
0.25
(b) Results of the proposed EKF (top: eletro-mechanical torque
mm and its estimation m̂m ).
Figure 5: Robustness results of the MRAS observer and the
proposed EKF: Estimation of rotor speed ωr and rotor angle
φr for 10% increase in Lm .
0.5
VI.
0.45
cp
Fig. 3a (top). The estimation of mechanical torque by the EKF
is sufficiently accurate to achieve MPPT. The power coefficient
cp (λ, 0) is kept close to its maximal (optimal) value c?p = 0.48
when the optimal tip speed ratio λ = λ? is reached. Since the
tip speed ratio λ as in (18) is a function of the wind speed
vw and the mechanical angular velocity ωm it cannot change
immediately with the wind speed and, so during the transient
phase of the speed control loop, λ deviates from λ? which
results in deviations of the power coefficient cp and its optimal
value c?p .
cp
0.4
cp
0.35
0.3
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
time (sec)
Figure 6: Maximum power point tracking of the WTS: Evolution of the power coefficient cp .
C ONCLUSION
This paper proposed a sensorless vector control method
for variable-speed wind turbine systems (WTS) with doublyfed induction generator (DFIG). The method uses an extended
Kalman filter for state estimation. The EKF estimates position
and speed of the rotor and the mechanical torque of the
generator. For the design of the EKF, a nonlinear state space
model of the DFIG has been derived. The design procedure
of the EKF has been presented in detail. The sensorless
control scheme of the WTS with DFIG has been illustrated
by simulation results and its performance has been compared
with a MRAS observer. The results have shown that the EKF
tracks rotor speed and rotor position and the mechanical torque
with higher accuracy than the MRAS observer. Moreover, the
EKF is more robust to parameters variations than the MRAS
observer.
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A PPENDIX
The simulation parameters are given in Tab. II. Note that
the rotor parameters (resistance and inductance) are converted
to the stator of the DFIG.
Table II: DFIG parameters
Name
DFIG rated power (base power)
Stator voltage (base voltage)
Rotor voltage (base voltage)
Grid frequency (base frequency)
Number of pair poles
Stator resistance
Rotor resistance
Stator inductance
Rotor inductance
Mutual inductance
Nomenclature
pnom
urms
s
urms
r
s
f0 = ω
2π
np
Rs
Rr
Ls
Lr
Lm
Value
2 MW
690 V
2070 V
50 Hz
2
2.6 mΩ
2.9 mΩ
2.627 mH
2.633 mH
2.55 mH