Sensorless Control of Doubly-Fed Induction Generators in Variable ...

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 ˆ [k + 1] = f (ˆ ˆ [k] , x x[k], u[k]) − K[k] y[k] − y (23) ˆ [k] = h(ˆ ˆ [k]. y x[k]) = C x ˆ where K[k] is the Kalman gain (to be specified below) and x ˆ are the estimated state and output vector, respectively. and y 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 ˆ [0] = E{x0 }, x ˆ [0])(x0 − x ˆ [0])> }, P 0 := P [0] = E{(x0 − x −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 ˆ − [k] = f (ˆ x x[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

ˆ [k] x

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 ˆ [k] = x ˆ − [k] + K[k](y[k] − h(ˆ x x− [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

ˆ 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 ˆ 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 ω ˆ r . Its output is the estimated speed ω ˆ r which is integrated to obtain the estimated rotor angle φˆr . 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 φˆr and rotor speed ω ˆ r . 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 (φˆr )−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

mˆ 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 ˆ 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 m ˆ m ).

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 ˆ 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 ˆ 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


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 ˆ 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


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

mˆ m

1.0 0.5

1.2 1.4 1.6  r [ pu ]

0.5 0

mˆ m

r ˆ r

1.4 1.2 1.0 0.8 0.6 0

r [rad/ S]

1.0

m m [ pu ]


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


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. R EFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

M. Liserre, R. Cardenas, M. Molinas, and J. Rodriguez, “Overview of Multi-MW Wind Turbines and Wind Parks”, IEEE Transactions on Industrial Electronics, Vol. 58, No. 4, pp. 1081-1095, April 2011. R. Cardenas, R. Pena, S. Alepuz, and G. Asher, “Overview of Control Systems for the Operation of DFIGs in Wind Energy Applications”, IEEE Transactions on Industrial Electronics, Vol. 60, No. 7, pp. 2776-2798, July 2013. 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-back power converters”, IEEE Transactions on Industrial Electronics, vol. PP, no. 99, p. 1 (IEEE Early Access Article), 2015. C. Dirscherl, C. Hackl, and K. Schechner, "Explicit model predictive control with disturbance observer for grid-connected voltage source power converters", in to be published in the Proceedings of the 2015 IEEE International Conference on Industrial Technology, 2015. Z. Zhang, C. Hackl, F. Wang, Z. Chen, and R. Kennel, "Encoderless model predictive control of back-to-back converter direct-drive permanent-magnet synchronous generator wind turbine systems", in Proceedings of 15th European Conference on Power Electronics and Applications, 2013, pp. 1–10. O. Mohammed, Z. Liu, and S. Liu, "A novel sensorless control strategy of Doubly Fed induction motor and its examination with the physical modeling of machines", IEEE Transactions on Magnetics, Vol. 41, No. 5, pp. 1852-1855, May 2005. L. Xu and W. Cheng, "Torque and reactive power control of a doubly fed induction machine by position sensorless scheme", IEEE Transactions on Industrial Applications, Vol. 31, No. 3, pp. 636-641, May/June 1995. M. Abolhassani, P. Niazi, H. Toliyat, and P. Enjeti, "A sensorless integrated Doubly-Fed electric alternator/active filter (IDEA) for variable speed wind energy system", Conference Record of the 38th Industrial Application Society Annual Meeting, Salt Lake City, UT, Vol. 1, pp. 507514, 12-16 October 2003. R. Datta and V. Ranganathan, A simple position sensorless algorithm for rotor side field oriented control of wound rotor induction machine, IEEE Transactions on Industrial Electronics, Vol. 48, No. 4, pp. 786-793, August 2001. L. Morel, H. Godfroid, A. Mirzaian, and J. Kauffmann, Double-Fed induction machine: Converter optimisation and field oriented control without position sensor, IEE Proceedings on Electric Power Applications, Vol. 145, No. 4, pp. 360-368, July 1998. E. Bogalecka and Z. Krzeminski, Sensorless control of a double-fed machine for wind power generators, in Proceeding of Power Electronics and Motion Control, Dubrovnik, Croatia, 2002. B. Hopfensperger, D. Atkinson, and R. Lakin, Stator-flux oriented control of a doubly-fed induction machine with and without position encoder, IEE Proceedings on Electric Power Applications, Vol. 147, No. 4, pp. 241-250, July 2000. R. Cardenas, R. Pena, J. Clare, G. Asher, and J. Proboste, MRAS observers for sensorless control of doubly-fed induction generators, IEEE Transactions on Power Electronics, Vol. 23, No. 3, pp. 1075-1084, May 2008. D. Reigosa, F. Briz, C. Charro, A Di Gioia, P. Garcia, and J. Guerrero, Sensorless Control of Doubly Fed Induction Generators Based on Rotor High-Frequency Signal Injection, IEEE Transactions on Industry Applications, Vol. 49, No. 6, pp. 2593-2601, November 2013. F. Auger, M. Hilairet, J. Guerrero, E. Monmasson, T. OrlowskaKowalska, 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]

[17]

[18]

[19] [20]

[21]

[22]

[23]

[24]

[25]

[26] [27]

[28]

[29]

M. Abdelrahem, C. Hackl, and R. Kennel, "Application of Extended Kalman Filter to Parameter Estimation of Doubly-Fed Induction Generators in Variable-Speed Wind Turbine Systems", to be published in the Proceedings of the 5th International Conference on Clean Electrical Power, Taormina, Italy, 2015. I. Perez, J. Silva, E Yuz, and R. Carrasco, Experimental sensorless vector control performance of a DFIG based on an extended Kalman filter, 38th IEEE Annual Conference on Industrial Electronics Society (IECON), pp. 1786-1792, 25-28 October 2012. C. Dirscherl, C. Hackl, and K. Schechner, “Modellierung und Regelung von modernen Windkraftanlagen: Eine Einführung (available at the authors upon request),” Chapter 24 in Elektrische Antriebe – Regelung von Antriebssystemen, D. Schröder (Ed.), Springer-Verlag, 2015. B. Wu, Y. Lang, N. Zargari, and S. Kouro, Power conversion and control of wind energy systems, Wiley-IEEE Press, 2011. M. Yameamoto and o. Motoyoshi, "Active and reactive power control for doubly-fed wound rotor induction generator", IEEE Transactions on Power Electronics, Vol. 6, No. 4, pp. 624-629, October 1991. R. Pena, J. Clare, and G. Asher, Doubly fed induction generator using back-to-back PWM converters and its application to variable-speed wind-energy generation, IEE Proceedings on Electric Power Applications, Vol. 143, No. 3, pp. 231-241, May 1996. A. Hansen, P. Sorensen, F. Lov, and F. Blaabjerg, Control of variable speed wind turbines with doubly-fed induction generators, Wind Energy, Vol. 28, No. 4, pp. 411-434, June 2004. R. Fadaeinedjad, M. Moallem, and G. Moschopoulos, Simulation of a wind turbine with doubly fed induction generator by FAST and Simulink, IEEE Transaction on Energy Conversions, Vol. 23, No. 2, pp. 690-700, June 2008. A. Petersson, L. Harnefors, and T. Thiringer, Comparison between stator-flux and grid-flux-oriented rotor current control of doubly-fed induction generators, proceeding IEEE 35th Annual Power Electronics Specialist Conference, Vol. 1, pp. 482-486, 20-25 June 2004. S. Müller, M Deicke, and R. De Doncker, Doubly Fed Induction Generator Systems for Wind Turbines, IEEE Industrial Applications Magazine, Vol. 8, No. 3, pp. 26-33, May/June 2002. Siegfried Heier, Grid Integration of Wind Energy Conversion Systems, John Wiley & Sons Ltd, 1998. G. Bishop, and G. Welch, An introduction to the Kalman filter, Technical report TR 95-041, Department of Computer Science, University of North Carolina at Chapel Hill, 2006. C. De Wit, A. Youssef, J. Barbot, P. Martin, F. Malrait, "Observability conditions of induction motors at low frequencies," Proceedings of the 39th IEEE Conference on Decision and Control, Vol. 3, pp. 2044-2049, 2000. S. Bolognani, L. Tubiana, and M. Zigliotto, "Extended Kalman filter tuning in sensorless PMSM drives", IEEE Transactions on Industry Applications, Vol. 39, No. 6, pp. 1741-1747, November 2003.

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