sensorless control of doubly-fed induction generators in ... · pdf filesensorless control of...
TRANSCRIPT
Sensorless Control of Doubly-Fed InductionGenerators in Variable-Speed Wind Turbine Systems
Mohamed AbdelrahemStudent Member, IEEE
Institute for Electrical Drive Systemsand Power Electronics
Technische Universität München (TUM)Munich, Germany
Email: [email protected]
Christoph HacklMember, IEEE
Munich School of Engineering Research Group“Control of Renewable Energy Systems (CRES)”
Technische Universität München (TUM)Munich, Germany
Email: [email protected]
Ralph KennelSenior Member, IEEE
Institute for Electrical Drive Systemsand Power Electronics
Technische Universität München (TUM)Munich, Germany
Email: [email protected]
Abstract—This paper proposes a sensorless control strategyfor doubly-fed induction generators (DFIGs) in variable-speedwind turbine systems (WTS). The proposed scheme uses anextended Kalman filter (EKF) for the estimation of rotor speedand rotor position. Moreover, the EKF is used to estimate themechanical torque of the generator to allow for maximum powerpoint tracking control for wind speeds below the nominal windspeed. For EKF design, the nonlinear state space model of theDFIG is derived. Estimation and control performance of theproposed sensorless control method are illustrated by simulationresults at low, high, and synchronous speed. The designed EKF isrobust to machine parameter variations within reasonable limits.Finally, the performances of the EKF and a model referenceadaptive system (MRAS) observer are compared for time-varyingwind speeds.
Keywords—DFIG, MPPT, Kalman filter, MRAS observer
NOTATION
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‖x‖ =
√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 ofa rotor (r) or stator (s) quantity, i.e. z ∈ r, s. The spacevector is expressed in either phase abc-, stator fixed s-, rotorfixed r-, or arbitrarily rotating k-coordinate system, i.e. y ∈abc, s, r, k, and may represent voltage u, flux linkage ψ orcurrent i, i.e. x ∈ u,ψ, i. Ex or EX is the expectationvalue of x or X , resp.
I. INTRODUCTION
The electrical power generation by renewable energysources (such as e.g. wind) has increased significantly duringthe last years contributing to the reduction of carbon diox-ide emissions and to a lower environmental pollution [1].This increase will continue as countries are extending theirrenewable action plans. Therefore, the share of wind powergeneration will increase further worldwide. Among the varioustypes of wind turbine generators, the DFIG is the most com-monly used generator in on-shore and off-shore applications,accounting for more than 50% of the installed wind turbinenominal 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 certainride through capability [2]. Operation above and below thesynchronous speed is feasible. Due to their wide use in WTS,the development of advanced and reliable control techniquesfor DFIGs has received significant attention during the lastyears [2]. Examples of these control techniques are e.g. vectorcontrol, direct torque control, and direct power control [2].
Vector control has – so far – proven to be the most popularcontrol technique for DFIGs in variable-speed WTS [2]. Thismethod allows for a decoupled control of active and reactivepower of WTS via regulating the quadrature components ofthe rotor current vector independently. Vector control requiresaccurate 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 effec-tiveness/robustness, which implies that the vector controllersmust operate without the information of mechanical sensors(such as position encoders or speed transducers) mounted onthe shaft. The required rotor signals must be estimated via theinformation provided by electrical (e.g. current) sensors whichare cheap and easier to install than mechanical sensors. Fur-thermore, mechanical sensors reduce the system reliability dueto their high failure rate, which implies shorter maintenanceintervals and, so, higher costs.
Sensorless control methods for doubly-fed induction ma-chines/generators have been proposed by several researchers.The proposed approach in [6] uses the magnetizing currentssupplied from the rotor and stator to estimate the rotor positionand speed; however, observer design and its dynamics werenot addressed/analyzed. In [7], a rotor-flux-based sensorlessscheme is reported, where the rotor flux is obtained by the in-tegration of the rotor back-electromotive force. This approachmight suffer from integration problems with poor performanceduring operation close to synchronous speed, because the rotoris excited with low frequency voltages. The sensorless methodspresented in [8]-[12] are open-loop and rely on rotor currentestimators in which the estimated and measured currents arecompared to obtain the rotor position. The rotational speed isobtained by numerical differentiation which is very sensitiveto noise. None of these methods addresses the design of therotor position estimator bandwidth and the effect of parameteruncertainties on the estimation accuracy.
C
PI
abc/dqPLL
dq/a
bc
PW
M
Drive
PI
abc/dq
PI
PI
RSC
PI
dq/a
bc
PW
M
Drive
PI
abc/dq
PI
PI
GSC
DC
Link
Wind
turbine
Grid
Lookup tableDFIG
Encoder
mmmm
abc
ri
sr
d
ri
q
ri
d
refri ,
q
refri ,
sl
)( q
sm
q
rrsl iLiL
sl
s
)( d
sm
d
rrsl iLiL sQ
refsQ ,
abc
ri
abc
si
sQ
sP
gear
box
filter
abc
fi
Tr.ff QP &
dcu
q
fi
d
fi
sabc
su
d
reffi ,
q
reffi ,
dcurefdcu ,
abc
fi
r
reffQ ,
fQ
s
abc
sud
su
q
su
q
ffs iL
d
ffs iL
fLfR
t
m
dcPdcC
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 beenreported in [13], where MRAS observers are diversified withvarious error variables, e.g. stator and rotor currents andfluxes. Moreover, the respective estimation performances arecompared. The common disadvantage of the presented MRASobservers is the DC-offset drift problem caused by the pureintegral action in the stator-flux estimation.
The sensorless control approach in [14] relies on signalinjection. The main advantage of this method is its high robust-ness against variations in the machine parameters. However,the injection of high-frequency signals in the DFIG rotor isnot easy for large machines (> 1 MW) such those in modernWTS. Another alternative is the use of an extended Kalmanfilter (EKF) which has already been used for sensorless controland the estimation of the electrical parameters of inductionmachines and permanent magnet synchronous machines [15],[16]. An EKF was used for DFIG speed and position es-timation in [17], however the authors use state variables inthe rotating reference frame, whereas input and measurementvariables are in the stationary reference frame and are di-rectly incorporated into the EKF design. This increases thecomplexity of state, input and measurement signals, since thePark transformation (from the stationary reference frame tothe rotating reference frame) has to be considered at eachsampling instant during time and measurement update andcomputation of the Kalman gain of the EKF. This results inhigh computational loads during real-time application.
In this paper, an extended Kalman filter is proposed toestimate speed and position of the rotor and the mechanicaltorque of the DFIG. State, input and measurement variablesare selected in the rotating reference frame, which reducesthe complexity of state, input and measurement matrices and,hence, the computational time for real-time implementation.
The EKF performance and its robustness against parametervariations are illustrated by simulation results. The resultshighlight the ability of the EKF in tracking the DFIG rotorspeed and position. The results are compared with those of aMRAS observer.
II. MODELING AND CONTROL OF THE WTS WITH DFIG
The block diagram of the vector control problem of WTSwith DFIG is shown in Fig. 1. It consists of a wound rotorinduction machine mechanically coupled to the wind turbinevia a shaft and gear box with ratio gr ≥ 1 [1]. The statorwindings of the DFIG are directly connected to the grid via atransformer, whereas the rotor winding is connected via a back-to-back partial-scale voltage source converter (VSC), a filterand a transformer to the grid. The transformer will be neglectedin the upcoming modeling. The rotor side converter (RSC) andthe grid side converter (GSC) share a common DC-link withcapacitance Cdc [As/V] with DC-link voltage udc [V]. Detailedmodels of these components can be found in [18]. The statorand rotor voltage equations of the DFIG are given by [19]:
uabcs (t) = Rsiabcs (t) +
d
dtψabcs (t), ψabcs (0) = 03 (1)
uabcr (t) = Rriabcr (t) +
d
dtψabcr (t), ψabcr (0) = 03︸ ︷︷ ︸
initial values
(2)
where (assuming linear flux linkage relations)
ψabcs (t) = Lsiabcs (t) + Lmi
abcr (t) (3)
ψabcr (t) = Lriabcr (t) + Lmi
abcs (t). (4)
Here uabcs = (uas , ubs , u
cs )> [V], uabcr = (uar , u
br , u
cr )> [V],
iabcs = (ias , ibs , i
cs )> [A], iabcr = (iar , i
br , i
cr )> [A], ψabcs =
(ψas , ψbs, ψ
cs)> [Vs], and ψabcr = (ψar , ψ
br , ψ
cr )> [Vs] are the
stator and rotor voltages, currents and fluxes, respectively, all inthe 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 inductancesand Lm is the mutual inductance. Rs [Ω] and Rr [Ω] are statorand rotor winding resistances. Note that the DFIG rotor rotateswith mechanical angular frequency ωm [rad/s]. Hence, for amachine with pole pair number np [1], the electrical angularfrequency of the rotor is given by
ωr = npωm
and the rotor reference frame is shifted by the rotor angle
φr(t) =
∫ t
0
ωr(τ)dτ + φ0r, φ0
r ∈ R (6)
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 stationaryreference frame as follows
xs = (xα, xβ)> = TCxabc
by using the Clarke and Park transformation (see, e.g., [18]),respectively, given by (neglecting the zero sequence)
xs=γ
[1 − 1
2 − 12
0√
32 −
√3
2
]︸ ︷︷ ︸
=:TC
xabc & xk=
[cos(φ) sin(φ)− sin(φ) cos(φ)
]︸ ︷︷ ︸
=:TP (φ)−1
xs
(7)where γ = 2
3 for an amplitude-invariant transformation (orγ =
√2/3 for a power-invariant transformation). Expressing
the rotor voltage equation (2) also with respect to the stationaryreference frame (i.e. usr = TP (φr)
−1TCuabcr ), the voltage
equations (1) and (2) can be rewritten as
uss (t) = Rsiss (t) + d
dtψss (t), ψss (0) = 02
usr (t) = Rrisr (t) + d
dtψsr (t)− ωr(t)Jψsr (t), ψsr (0) = 02
(8)
where [18]
J := TP (π/2) =
[0 −11 0
].
B. Model in stator voltage orientation
An essential characteristic of the DFIG control strategy isthat the generated active and reactive power shall be controlledindependently. It is common to use an air-gap flux orienta-tion [20] or a stator flux orientation [21]-[23] for the vectorcontrol schemes. However, it has been shown that the statorflux orientation can cause instability under certain operatingconditions [24]. Therefore, following the ideas in [19], [25],in this paper, a stator (grid) voltage orientation for the vectorcontrol scheme is used.
The stator voltage orientation is achieved by aligning thed-axis of the synchronous (rotating) reference frame with thestator voltage vector uss which rotates with the stator (grid)angular frequency ωs (under ideal conditions, i.e. constant grid
frequency f0 > 0, it holds that ωs = 2πf0rads is constant).
Applying the (inverse) Park transformation with TP (φs)−1 as
in (7) with
φs(t) =
∫ t
0
ωs(τ)dτ + φ0s, φ0
s ∈ R
to the voltage equations (8) yields the description in therotating reference frame (neglecting initial values)
uks (t) = Rsiks (t) + d
dtψks (t) + ωsJψ
ks (t),
ukr (t) = Rrikr (t) + d
dtψkr (t) + (ωs − ωr(t)︸ ︷︷ ︸
=:ωsl(t)
)Jψkr (t),
(9)
where uks = (uds , uqs )>, ukr = (udr , u
qr )>, iks = (ids , i
qs )>, ikr =
(idr , iqr )>, ψks = (ψds , ψ
qs )>, ψkr = (ψdr , ψ
qr )>, are the stator
and rotor voltages, currents and fluxes in the rotating referenceframe (k-coordinate system with axes d and q), respectively.
ωsl := ωs − ωris the slip angular frequency. Since, e.g., ψks = TP (φs)
−1ψss =TP (φs)
−1TCψabcs , the flux linkages are given by
ψks = Lsiks + Lmi
kr
ψkr = Lrikr + Lmi
ks .
(10)
C. Dynamics of the mechanical system
For a stiff shaft and a step-up gear with ratio gr ≥ 1, thedynamics of the mechanical system are given by
d
dtωm =
1
Θ
(me −
mt
gr︸︷︷︸=:mm
), ωm(0) = ω0
m ∈ R (11)
where
me(t) =3
2npi
ss (t)>Jψss (t)
=3
2npLm
(iqs (t)idr (t)− ids (t)iqr (t)
). (12)
is the electro-magnetic machine torque (moment), mt [Nm] isthe turbine torque produced by the wind (see Sec. III) andmm = mt
gr[Nm] is the mechanical torque acting on the DFIG
shaft. Θ [kgm2] is the rotor inertia and np [1] is the pole pairnumber.
D. Overall nonlinear model of the DFIG
For the design of the EKF, the derivation of a compact(nonlinear) state space model of the DFIG of the form
d
dtx = g(x,u), x(0) = x0 ∈ R7 and y = h(x), (13)
is required. Therefore, introduce the state vector x, the output(measurement) vector y and the input vector u as follows:
x =(ids iqs idr iqr ωr φr mm
)> ∈ R7,
y =(ids iqs idr iqr
)> ∈ R4,
u =(uds uqs udr uqr
)> ∈ R4.
(14)
Note that the mechanical torque mm is considered as anadditional virtual (constant) state. Combining the subsystemsof the DFIG as in (9), (10), (11) and (12), inserting (10) into (9)
g(x,u) =
1σLsLr
(−RsLrids + (ωrL2m + ωsσLsLr)i
qs +RrLmi
dr + ωrLmLri
qr + Lru
ds − Lmudr )
1σLsLr
((−ωrL2m − ωsσLsLr)isd −RsLriqs − ωrLmLridr +RrLmi
qr + Lru
qs − Lmuqr )
1σLsLr
(RsLmids − ωrLsLmiqs −RrLsidr + (−ωrLrLs + ωsσLsLr)i
qr − Lmuds + Lsu
dr )
1σLsLr
(ωrLsLmids +RsLmi
qs + (ωrLrLs − ωsσLsLr)idr −RrLsiqr − Lmuqs + Lsu
qr )
npΘ
[32npLm(iqs i
dr − ids iqr )−mm]ωr0
(15)
and solving for ddti
ks and d
dtikr yields the nonlinear model (13)
with g(x,u) as in (15), σ := 1− L2m
LsLrand
h(x) =
[1 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 0
]︸ ︷︷ ︸
=:C=[I4,O4×3]∈R4×7
x. (16)
E. Overall control system of the WTS
The complete control block diagram of the DFIG in statorvoltage orientation is depicted in Fig. 1. For the rotor-sideconverter (RSC), the d-axis current is used to control the DFIGstator active power (i.e., proportional to the electro-magnetictorque) in order to harvest the maximally available wind power(i.e., maximum power point tracking, see Sec. III), whereasthe q-axis current is used to control the reactive power flowof the DFIG to the grid. For the grid-side converter (GSC),also stator voltage orientation is used [25], [18], which allowsfor independent control of active (d-axis current) and reactivepower (q-axis current) flow between grid and GSC. The maincontrol objective of the GSC is to assure an (almost) constantDC-link voltage regardless of magnitude and direction of therotor power flow. DC-link voltage control is a non-trivial taskdue to the possible non-minimum-phase behavior for a powerflow from the grid to the DC-link [18], [4]. More details oncontroller design, phase-locked loop or, alternatively, virtualflux estimation and pulse-width modulation (PWM) are givenin, e.g., [25], [3], [18].
III. MAXIMUM POWER POINT TRACKING (MPPT)
Wind turbines convert wind energy into mechanical energyand, via a generator, into electrical energy. The mechanical(turbine) power of a WTS is given by [19], [18], [26]:
pt = cp(λ, β)1
2ρπr2
t v3w︸ ︷︷ ︸
wind power
(17)
where ρ > 0 [kg/m3] is the air density, rt > 0 [m] is theradius of the wind turbine rotor (πr2
t is the turbine swept area),cp ≥ 0 [1] is the power coefficient, and vw ≥ 0 [m/s] is thewind speed. The power coefficient cp is a measure for the“efficiency” of the WTS. It is a nonlinear function of the tipspeed ratio
λ =ωmrtgrvw
≥ 0 [1] (18)
and the pitch angle β ≥ 0 [] of the rotor blades. The Betz limitcp,Betz = 16/27 ≈ 0.59 is an upper (theoretical) limit of thepower 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 powercoefficient cp from [26], i.e.
cp(λ, β) = 0.5176
(116
λi− 0.4β − 5
)e
−21λi + 0.0068λ
1
λi:=
1
λ+ 0.08β− 0.035
β3 + 1. (19)
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 theWTS must operate at its optimal tip speed ratio λ? (a givenconstant) where the power coefficient has its maximum c?p :=cp(λ
?, 0) = maxλ cp(λ, 0). Only then, the WTS can extractthe maximally available turbine power p?t := c?p
12ρπr
2t v
3w [18].
Maximum power point tracking is achieved by the nonlinearspeed controller
m?m = −k?pω2
m ≈ mm, k?p :=ρπr5
t
2gr
c?p(λ?)3
(20)
which assures that the generator angular frequency ωm isadjusted to the actual wind speed vw such that ωmrt
grvw
!= λ?
holds. According to (20) the optimum torque m?m can be
calculated from the (estimated) shaft speed ωm = ωr/np andthen it is compared with the actual mechanical torque mm,which is estimated by the EKF, as shown in Fig. 1. Based onthe difference m?
m −mm the underlying torque PI controller1
generates the rotor reference current idr,ref .
Remark: For wind speeds above the nominal wind speed, theWTS changes to nominal operation, i.e. m?
m = mm,nom, wheremm,nom is the nominal/rated generator torque. Speed controlis achieved by (individual) pitch control such that the ratedpower mm,nomωm,nom of the WTS is generated.
IV. EXTENDED KALMAN FILTER AND MRAS OBSERVER
A. Extended Kalman Filter (EKF)
The EKF is a nonlinear extension of the Kalman filter forlinear systems and is designed based on a discrete nonlinearsystem model [27]. For discretization the (simple) forwardEuler method with sampling time Ts [s] is applied to the time-continuous model (13) with (14), (15) and (16). For sufficientlysmall Ts 1, the following holds x[k] := x(kTs) ≈ x(t)
and ddtx(t) = x[k+1]−x[k]
Tsfor all t ∈ [kTs, (k + 1)Ts) and
k ∈ N∪0. Hence, the nonlinear discrete model of the DFIG
1The torque PI control loop still requires a thorough stability analysis whichis not considered in this paper.
can be written as
x[k + 1] =
=:f(x[k],u[k])︷ ︸︸ ︷x[k] + Tsg(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 in-cluded to model system uncertainties and measurementnoise, respectively. Both are assumed to be independent(i.e., Ew[k]v[j]> = O7×4 for all k, j ∈ N), while(i.e., Ew[k] = 07 and Ev[k] = 04 for all k ∈N) and with normal probability distributions (i.e., p(αi) =
1σαi√
2πexp
(−(αi−Eαi)2
2σ2α
)with σ2
αi := E(αi − Eαi)2and αi ∈ wi, vi). For simplicity, it is assumed that thecovariance matrices are constant, i.e., for all k ∈ N:
Q := Ew[k]w[k]> ≥ 0 and R := Ev[k]v[k]> > 0.(22)
Note that Q and R must be chosen positive semi-definite andpositive definite, resp.
Since system uncertainties and measurement noise are notknown a priori, the EKF is implemented as follows
x[k + 1] = f(x[k],u[k])−K[k](y[k]− y[k]
),
y[k] = h(x[k]) = Cx[k].
(23)
where K[k] is the Kalman gain (to be specified below) and xand y are the estimated state and output vector, respectively.The recursive algorithm of the EKF implementation is listedin Algorithm 1 [27]. The EKF achieves an optimal stateestimation by minimizing the covariance of the estimation errorfor each time instant k ≥ 1.
Algorithm 1: Extended Kalman filter
Step I: Initialization for k = 0x[0] = Ex0,
P 0 := P [0] = E(x0 − x[0])(x0 − x[0])>,K0 := K[0] = P [0]C>
(CP [0]C> +R
)−1
Step II: Time update (“a priori prediction”) for k ≥ 1(a) State prediction
x−[k] = f(x[k − 1],u[k − 1])(b) Error covariance matrix prediction
P−[k] = A[k]P [k − 1]A[k]> +Qwhere
A[k] = ∂f(x,u)∂x
∣∣∣∣x−[k]
Step III: Computation of Kalman gain for k ≥ 1
K[k] = P−[k]C>(CP−[k]C> +R
)−1
Step IV: Measurement update (“correction”) for k ≥ 1(a) Estimation update with measurement
x[k] = x−[k] +K[k](y[k]− h(x−[k]))
(b) Error covariance matrix updateP [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 choiceof the matrices P 0, Q and R, which affect the performanceand the convergence of the EKF. The initial error covariance
matrix P 0 represents the covariances (or mean-squared errors)based on the initial conditions (often P 0 is chosen to be adiagonal matrix) and determines the initial amplitude of thetransient behavior of the estimation process, while durationof the transient behavior and steady state performance arenot affected. The matrix Q describes the confidence with thesystem model. Large values in Q indicate a low confidencewith the system model, i.e. large parameter uncertainties areto be expected, and will likewise increase the Kalman gain togive a better/faster measurement update. However, too largeelements of Q may be lead to oscillations or even instabilityof the state estimation. On the other hand, low values inQ indicate a high confidence in the system model and maytherefore lead to weak (slow) measurement corrections.
The matrix R is related to the measurement noise charac-teristics. Increasing the values ofR indicates that the measuredsignals are heavily affected by noise and, therefore, are oflittle confidence. Consequently, the Kalman gain will decreaseyielding a poorer (slower) transient response.
In [29] general guide lines are given how to select thevalues of Q and R. Following these guide lines, for this paperthe following values have been selected
Q = diag0.03, 0.03, 0.03, 0.03, 3 · 10−5, 10−6, 6 · 10−5R = diag1, 1, 1, 1 (24)P 0 = diag0.02, 0.02, 0.02, 0.02, 2 · 10−5, 5 · 10−5, 10−4x0 = (0, 0, 0, 0, 1, 0, 0.5)>
Remark on the observability of the nonlinear DFIG model:For nonlinear systems, it is possible to analyze observabilitylocally by analyzing the linearized model around an operatingpoint [28]. The observability of the linearized DFIG model hasbeen tested around several operating point (e.g. at low, high andsynchronous speed). The analysis shows that the observabilityof the system is affected by the rotor current. When the DFIGoperates exactly at its synchronous speed, the rotor currentis zero and observability is lost at this (singular) point. Foroperation close to synchronous speed, the DFIG is (locally)observable (see also [17]).
B. MRAS Observer
The MRAS observer is based on two models [13]: areference model and an adaptive model, see Fig. 2.
I
I
s
si
s
sus
ss
ri1/Lm
r
r
r
riPI
r
ri
e1)ˆ(
rPT
Rs Ls
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 isr is estimated via
isr (t) =1
Lm
(ψss (t)− Lsiss (t)
)(25)
where
ψss (t) =
∫ t
0
(uss (τ)−Rsiss (τ)
)dτ. (26)
The adaptive model (see right part in Fig. 2) is fed by theestimated rotor current isr and the measured rotor current irrin the rotor reference frame which has been proven to bethe best option among all possible implementations of MRASobservers [13]. The goal of the adaptive model (which is essen-tially a phase-locked loop) is to estimate rotor position φr androtor speed ωr. To achieve that the estimated and the measuredrotor current must be compared; to do so, the estimated rotorcurrent isr (in the stator reference frame) must be expressed inthe rotor reference frame, i.e. irr = TP (φr)
−1isr . The “error”between estimated irr and measured rotor current irr is definedas
e := irr Jirr = irr Ji
rr = ‖isr ‖ ‖i
sr ‖ sin
(∠(isr , i
sr )).
The PI controller drives this error to zero by adjusting ωr. Itsoutput is the estimated speed ωr which is integrated to obtainthe estimated rotor angle φr. For more details see [13].
V. SIMULATION RESULTS AND DISCUSSION
A simulation model of a 2 MW WTS with DFIG isimplemented in Matlab/Simulink. The system parameters arelisted in the Appendix. The implementation is as in Fig. 1.For more details on the implementation of e.g. back-to-backconverter, PWM, current controller design, see [18]. Thesimulation results are shown in Figures 3-6. The estimationperformances of MRAS observer and EKF are compared fordifferent wind speed and parameter uncertainties in Rs, Rrand Lm.
Estimation results: Fig. 3a and Fig. 3b show the simulationresults for MRAS observer and EKF when the wind speedchanges from 7 m
s to 11 ms and, then, to 9 m
s (see top of Fig. 3a).This wind speed range covers almost the complete speedrange of the DFIG (i.e. ±25% around the synchronous speed).Fig. 3b illustrates the tracking capability of the EKF of rotorspeed and rotor position at low and high speeds, and closeto synchronous speed. For comparison, Fig. 3a shows theestimation performance of the MRAS observer. Tab. I liststhe estimation errors of MRAS observer and EKF: the EKFshows a (slightly) higher estimation accuracy than the MRASobserver. An additional advantage of the EKF is its capabilityof estimating the mechanical torque as shown in Fig. 3b.
Table I: Estimation errors of MRAS observer and EKF.
Observer MRAS EKFEstimated state ωr φr ωr φr mmNormal conditions 1.4% 1.8% 1% 1.2% 2.3%Rs and Rr increase by 50% 2.2% 3.4% 1.4% 1.8% 3.6%Lm increases by 10% 4% 4.8% 2% 3% 6%
In order to check the robustness of the EKF under (un-known) parameter variations of the DFIG, the values of thestator resistance Rs and the rotor resistance Rr are increasedby ±50% (e.g. due to warming or aging). For this scenario,Fig. 4a and Fig. 4b show the estimation performances ofthe MRAS observer and the EKF. The simulated wind speedprofile is depicted in Fig. 4a (top). The EKF is more robust
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0 0.250.20.05 0.1 0.15
time (sec)
6
8
10
12
]/
[s
mv w
0.6
0.8
1.0
1.2
1.4
][pu
r
0
5
8
rr
r r
]/
[S
rad
r
(a) Results of the MRAS observer (top: wind speed vw).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.5
1
1.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.6
0.8
1
1.2
0 0.05 0.1 0.15 0.2 0.250
5
rr
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0 0.250.20.05 0.1 0.15
time (sec)
0
5
8]/
[S
rad
r
0.6
1.0
1.4
0.5
1.0
1.5
0.8
1.2][pu
r
][pu
mm
r r
mm mm
(b) Results of the proposed EKF (top: eletro-mechanical torquemm and its estimation mm).
Figure 3: Estimation performance of the MRAS observer andthe proposed EKF: Estimation of rotor speed ωr and rotorangle φr.
than the MRAS observer under parameter uncertainties in Rsand Rr. It estimates rotor speed and rotor position with smallererrors than the MRAS observer (see Tab. I). In addition, theEKF estimation accuracy of the mechanical torque is stillacceptable (see Fig. 4b).
Finally, the robustness with respect to changes (due tomagnetic saturation) in the mutual inductance Lm is investi-gated. Therefore, Lm is increased by 10%. Fig. 5a and Fig. 5bshow the simulation results of MRAS observer and EKF forthis scenario. The used wind speed profile is depicted inFig. 5a (top). Again, the EKF shows a more robust and moreaccurate estimation performance than the MRAS observerunder variations in Lm (see Tab. I).
The final simulation results are shown in Fig. 6 andillustrate the control performance of the maximum power pointtracking (MPPT) algorithm under wind condition as shown in
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.66
8
1012
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.60.8
11.2
0 0.05 0.1 0.15 0.2 0.250
5
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0 0.250.20.05 0.1 0.15
time (sec)
6
8
10
12
]/
[s
mv w
0.6
0.8
1.0
1.2
1.4
][pu
r
0
5
8
rr
r r
]/
[S
rad
r
(a) Results of the MRAS observer (top: wind speed vw).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.5
1
1.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.60.8
11.2
0 0.05 0.1 0.15 0.2 0.250
5
rr
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0 0.250.20.05 0.1 0.15
time (sec)
0
5
8]/
[S
rad
r
0.6
1.0
1.4
0.5
1.0
1.5
0.8
1.2][pu
r
][pu
mm
r r
mm mm
(b) Results of the proposed EKF (top: eletro-mechanical torquemm and its estimation mm).
Figure 4: Robustness results of the MRAS observer and theproposed EKF: Estimation of rotor speed ωr and rotor angleφr for an 50% increase in Rs and Rr.
Fig. 3a (top). The estimation of mechanical torque by the EKFis sufficiently accurate to achieve MPPT. The power coefficientcp(λ, 0) is kept close to its maximal (optimal) value c?p = 0.48when the optimal tip speed ratio λ = λ? is reached. Since thetip speed ratio λ as in (18) is a function of the wind speedvw and the mechanical angular velocity ωm it cannot changeimmediately with the wind speed and, so during the transientphase of the speed control loop, λ deviates from λ? whichresults in deviations of the power coefficient cp and its optimalvalue c?p.
VI. CONCLUSION
This paper proposed a sensorless vector control methodfor variable-speed wind turbine systems (WTS) with doubly-fed induction generator (DFIG). The method uses an extendedKalman filter for state estimation. The EKF estimates position
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0 0.250.20.05 0.1 0.15
time (sec)
6
8
10
12
]/
[s
mv w
0.6
0.8
1.0
1.2
1.4
][pu
r
0
5
8
rr
r r
]/
[S
rad
r
(a) Results of the MRAS observer (top: wind speed vw).
rr
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0 0.250.20.05 0.1 0.15
time (sec)
0
5
8]/
[S
rad
r
0.6
1.0
1.4
0.5
1.0
1.5
0.8
1.2][pu
r
][pu
mm
r r
mm mm
(b) Results of the proposed EKF (top: eletro-mechanical torquemm and its estimation mm).
Figure 5: Robustness results of the MRAS observer and theproposed EKF: Estimation of rotor speed ωr and rotor angleφr for 10% increase in Lm .
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8time (sec)
0.35
0.4
0.45
0.5
pc
0.30
pc pc
Figure 6: Maximum power point tracking of the WTS: Evolu-tion of the power coefficient cp.
and speed of the rotor and the mechanical torque of thegenerator. For the design of the EKF, a nonlinear state spacemodel of the DFIG has been derived. The design procedureof the EKF has been presented in detail. The sensorless
control scheme of the WTS with DFIG has been illustratedby simulation results and its performance has been comparedwith a MRAS observer. The results have shown that the EKFtracks rotor speed and rotor position and the mechanical torquewith higher accuracy than the MRAS observer. Moreover, theEKF is more robust to parameters variations than the MRASobserver.
REFERENCES
[1] M. Liserre, R. Cardenas, M. Molinas, and J. Rodriguez, “Overviewof Multi-MW Wind Turbines and Wind Parks”, IEEE Transactions onIndustrial Electronics, Vol. 58, No. 4, pp. 1081-1095, April 2011.
[2] R. Cardenas, R. Pena, S. Alepuz, and G. Asher, “Overview of ControlSystems for the Operation of DFIGs in Wind Energy Applications”, IEEETransactions on Industrial Electronics, Vol. 60, No. 7, pp. 2776-2798,July 2013.
[3] Z. Zhang, H. Xu, M. Xue, Z. Chen, T. Sun, R. Kennel, and C. Hackl,“Predictive control with novel virtual flux estimation for back-to-backpower converters”, IEEE Transactions on Industrial Electronics, vol. PP,no. 99, p. 1 (IEEE Early Access Article), 2015.
[4] C. Dirscherl, C. Hackl, and K. Schechner, "Explicit model predictivecontrol with disturbance observer for grid-connected voltage sourcepower converters", in to be published in the Proceedings of the 2015IEEE International Conference on Industrial Technology, 2015.
[5] Z. Zhang, C. Hackl, F. Wang, Z. Chen, and R. Kennel, "Encoder-less model predictive control of back-to-back converter direct-drivepermanent-magnet synchronous generator wind turbine systems", inProceedings of 15th European Conference on Power Electronics andApplications, 2013, pp. 1–10.
[6] O. Mohammed, Z. Liu, and S. Liu, "A novel sensorless control strategyof Doubly Fed induction motor and its examination with the physicalmodeling of machines", IEEE Transactions on Magnetics, Vol. 41, No. 5,pp. 1852-1855, May 2005.
[7] L. Xu and W. Cheng, "Torque and reactive power control of a doubly fedinduction machine by position sensorless scheme", IEEE Transactions onIndustrial Applications, Vol. 31, No. 3, pp. 636-641, May/June 1995.
[8] M. Abolhassani, P. Niazi, H. Toliyat, and P. Enjeti, "A sensorlessintegrated Doubly-Fed electric alternator/active filter (IDEA) for variablespeed wind energy system", Conference Record of the 38th IndustrialApplication Society Annual Meeting, Salt Lake City, UT, Vol. 1, pp. 507-514, 12-16 October 2003.
[9] R. Datta and V. Ranganathan, A simple position sensorless algorithmfor rotor side field oriented control of wound rotor induction machine,IEEE Transactions on Industrial Electronics, Vol. 48, No. 4, pp. 786-793,August 2001.
[10] L. Morel, H. Godfroid, A. Mirzaian, and J. Kauffmann, Double-Fedinduction machine: Converter optimisation and field oriented controlwithout position sensor, IEE Proceedings on Electric Power Applications,Vol. 145, No. 4, pp. 360-368, July 1998.
[11] E. Bogalecka and Z. Krzeminski, Sensorless control of a double-fedmachine for wind power generators, in Proceeding of Power Electronicsand Motion Control, Dubrovnik, Croatia, 2002.
[12] B. Hopfensperger, D. Atkinson, and R. Lakin, Stator-flux orientedcontrol of a doubly-fed induction machine with and without positionencoder, IEE Proceedings on Electric Power Applications, Vol. 147,No. 4, pp. 241-250, July 2000.
[13] R. Cardenas, R. Pena, J. Clare, G. Asher, and J. Proboste, MRASobservers for sensorless control of doubly-fed induction generators, IEEETransactions on Power Electronics, Vol. 23, No. 3, pp. 1075-1084, May2008.
[14] D. Reigosa, F. Briz, C. Charro, A Di Gioia, P. Garcia, and J. Guer-rero, Sensorless Control of Doubly Fed Induction Generators Based onRotor High-Frequency Signal Injection, IEEE Transactions on IndustryApplications, Vol. 49, No. 6, pp. 2593-2601, November 2013.
[15] F. Auger, M. Hilairet, J. Guerrero, E. Monmasson, T. Orlowska-Kowalska, S. Katsura, "Industrial Applications of the Kalman Filter:A Review", IEEE Transaction on Industrial Electronics, vol.60, no.12,pp.5458-5471, Dec. 2013.
[16] M. Abdelrahem, C. Hackl, and R. Kennel, "Application of ExtendedKalman Filter to Parameter Estimation of Doubly-Fed Induction Gen-erators in Variable-Speed Wind Turbine Systems", to be published inthe Proceedings of the 5th International Conference on Clean ElectricalPower, Taormina, Italy, 2015.
[17] I. Perez, J. Silva, E Yuz, and R. Carrasco, Experimental sensorlessvector control performance of a DFIG based on an extended Kalmanfilter, 38th IEEE Annual Conference on Industrial Electronics Society(IECON), pp. 1786-1792, 25-28 October 2012.
[18] C. Dirscherl, C. Hackl, and K. Schechner, “Modellierung und Regelungvon modernen Windkraftanlagen: Eine Einführung (available at theauthors upon request),” Chapter 24 in Elektrische Antriebe – Regelungvon Antriebssystemen, D. Schröder (Ed.), Springer-Verlag, 2015.
[19] B. Wu, Y. Lang, N. Zargari, and S. Kouro, Power conversion and controlof wind energy systems, Wiley-IEEE Press, 2011.
[20] M. Yameamoto and o. Motoyoshi, "Active and reactive power controlfor doubly-fed wound rotor induction generator", IEEE Transactions onPower Electronics, Vol. 6, No. 4, pp. 624-629, October 1991.
[21] R. Pena, J. Clare, and G. Asher, Doubly fed induction generator usingback-to-back PWM converters and its application to variable-speedwind-energy generation, IEE Proceedings on Electric Power Applica-tions, Vol. 143, No. 3, pp. 231-241, May 1996.
[22] A. Hansen, P. Sorensen, F. Lov, and F. Blaabjerg, Control of variablespeed wind turbines with doubly-fed induction generators, Wind Energy,Vol. 28, No. 4, pp. 411-434, June 2004.
[23] R. Fadaeinedjad, M. Moallem, and G. Moschopoulos, Simulation of awind turbine with doubly fed induction generator by FAST and Simulink,IEEE Transaction on Energy Conversions, Vol. 23, No. 2, pp. 690-700,June 2008.
[24] A. Petersson, L. Harnefors, and T. Thiringer, Comparison betweenstator-flux and grid-flux-oriented rotor current control of doubly-fedinduction generators, proceeding IEEE 35th Annual Power ElectronicsSpecialist Conference, Vol. 1, pp. 482-486, 20-25 June 2004.
[25] S. Müller, M Deicke, and R. De Doncker, Doubly Fed InductionGenerator Systems for Wind Turbines, IEEE Industrial ApplicationsMagazine, Vol. 8, No. 3, pp. 26-33, May/June 2002.
[26] Siegfried Heier, Grid Integration of Wind Energy Conversion Systems,John Wiley & Sons Ltd, 1998.
[27] G. Bishop, and G. Welch, An introduction to the Kalman filter, Tech-nical report TR 95-041, Department of Computer Science, University ofNorth Carolina at Chapel Hill, 2006.
[28] C. De Wit, A. Youssef, J. Barbot, P. Martin, F. Malrait, "Observabilityconditions of induction motors at low frequencies," Proceedings of the39th IEEE Conference on Decision and Control, Vol. 3, pp. 2044-2049,2000.
[29] S. Bolognani, L. Tubiana, and M. Zigliotto, "Extended Kalman filtertuning in sensorless PMSM drives", IEEE Transactions on IndustryApplications, Vol. 39, No. 6, pp. 1741-1747, November 2003.
APPENDIX
The simulation parameters are given in Tab. II. Note thatthe rotor parameters (resistance and inductance) are convertedto the stator of the DFIG.
Table II: DFIG parameters
Name Nomenclature ValueDFIG rated power (base power) pnom 2 MWStator voltage (base voltage) urms
s 690 VRotor voltage (base voltage) urms
r 2070 VGrid frequency (base frequency) f0 = ωs
2π 50 HzNumber of pair poles np 2Stator resistance Rs 2.6 mΩRotor resistance Rr 2.9 mΩStator inductance Ls 2.627 mHRotor inductance Lr 2.633 mHMutual inductance Lm 2.55 mH