Application of EKF to parameter estimation paper final

Application of Extended Kalman Filter to Parameter Estimation of Doubly-Fed Induction Generators in Variable-Speed Wind Turbine Systems Mohamed Abdelrahem∗ , Christoph Hackl∗∗ , Ralph Kennel∗ ∗

∗∗

Institute for Electrical Drive Systems and Power Electronics Munich School of Engineering Research Group “Control of Renewable Energy Systems (CRES)” Technische Universität München (TUM), Munich (Germany) Email: [email protected], [email protected], [email protected]

competitive, therefore the share of wind power is likely to 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 around 50% of the installed wind turbines nominal capacity worldwide [1]. DFIGs can supply active and reactive power, operate with only a partial-scale power converter (around 30% of the machine rating), and achieve a certain ride through capability [2]. Operation above and below synchronous speed is feasible. Due to their wide use in the 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, direct power control, and model predictive control [2]-[5].

Abstract—This paper proposes a parameter estimation method for doubly-fed induction generators (DFIGs) in variable-speed wind turbine systems (WTS). The proposed method employs an extended Kalman filter (EKF) for estimation of all electrical parameters of the DFIG, i.e., the stator and rotor resistances, the leakage inductances of stator and rotor, and the mutual inductance. The nonlinear state space model of the DFIG is derived and the design procedure of the EKF is described. The observability matrix of the linearized DFIG model is computed and the observability is checked online for different operation conditions. The estimation performance of the EKF is illustrated by simulation results. The estimated parameters are plotted against their actual values. The estimation performance of the EKF is also tested under variations of the DFIG parameters to investigate the estimation accuracy for changing parameters. Keywords—DFIG, parameter estimation, extended Kalman filter, vector control, wind turbine systems

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 the active and reactive power of WTS via regulating the quadrature components of the rotor current vector independently. However, vector control relies on the accurate knowledge of the (electrical) parameters of the DFIG. If the control parameters do not match the actual values, the DFIG might not work properly: the closedloop system may be deteriorated or even become unstable. The electrical parameters of DFIG are sensitive to temperature changes, magnetic saturation and eddy currents [6]. Therefore, in most cases, the use of constant parameters does not allow to display the real dynamical behavior of the DFIG accurately. Online parameter estimation is necessary.

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. vector with n ∈ N. x is x ∈ Rn (bold) is a real valued √ 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 n×m ∈ 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, respectively. I.

Although there are several parameter estimation techniques available for induction machines [6], only few results are published on the parameter estimation of DFIG in variable-speed wind turbine systems [7]-[11]. In [7], two online parameter identification methods for DFIGs are proposed based on model reference adaptive systems (MRAS). The first method requires a testing signal to excite (all) systems eigenvalues, which is not feasible/reasonable when the machine is connected to the grid. The second method does not rely on excitation. However, this method neglects the stator and rotor resis-

I NTRODUCTION

The electrical power generation by renewable energy systems (RES; such as e.g. wind turbine systems) has increased during the last years and, so, RES significantly contribute to the reduction of carbon dioxide emission and therefore to a lower environmental pollution [1]. The increase of electrical power generation of RES will continue as countries are extending their renewable action plans. Moreover, wind power is already today economically

978-1-4799-8704-7/15/$31.00 ©2015 IEEE

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Wind turbine

Zsl (Lrirq  Lmisq )

me

m e

Zr

PI

i rq

Q s , ref

PI Qs

u dc , ref

irq,ref

PI

i

Q f ,ref

i PI

Qf

i

q f

q f , ref

udc

PI

usabc I s abc/dq

u

d s

u

q s

Pdc

Qs

i abc f

Ps

filter

Rf L f

PI Zs L f i df

gear box

isabc

Drive

I s abc/dq

C dc

DC Link

PWM

PLL

Zsl (L i  L i ) d ms

dq/abc

i

i abc f

d f

Zm C

d r r

Zs L i d f ,ref

Encoder

DFIG

PI

q f f

udc

Drive

Isl

Isl abc/dq

Is

Zt

irabc PWM

i

i

d r

PI dq/abc

abc r

Lookup table Ir

usabc

RSC

ird,ref

GSC

Is

Pf & Q f

Tr.

Grid

Figure 1:

DFIG topology and control structure for the variable-speed wind turbine system.

tances and is not capable of estimating the rotor leakage inductance. In [8], a search-based algorithm for parameter identification of DFIGs in wind turbine systems is proposed. However, if the initial values of the parameters are not properly selected (as mentioned in [8]), the objective function might converge solely to a local optimum instead of a global optimum. Moreover, the electrical parameters of the DFIG are not estimated separately: Combinations of the parameters such as the ratio of rotor resistance and inductance or the ratio of mutual and rotor inductances are estimated. In [9], an adaptive estimation algorithm is used for estimating the DFIG rotor resistance. However, all remaining DFIG parameters are assumed to be known. In [10], a Levenberg-Marquardt-Fletcher method for parameter estimation is used. However, solely stator resistance, stator inductance and mutual inductance are estimated. Rotor resistance and inductance are assumed to be constant and known. In [11] an algorithm is developed which allows for online estimation of stator, rotor and mutual inductances. However, stator and rotor resistances are assumed to be constant and known.

proposed for parameter estimation of DFIGs in wind turbine systems: The estimated parameters are the stator and rotor resistances, the leakage inductances of stator and rotor, and the mutual inductance. The nonlinear state space model of the DFIG is derived for designing the proposed EKF. The observability matrix of the linearized DFIG model is computed and the observability is checked online for different operation conditions. The estimation performance of the EKF is tested under variations of the DFIG parameters to investigate the estimation accuracy for changing parameters. The estimation performance of the EKF is illustrated by simulation results. II.

M ODELING AND C ONTROL OF THE WTS DFIG

WITH

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 back-to-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] and DC-link voltage udc [V]. Detailed models of these components can be found in [15]. The stator and rotor voltage equations of the DFIG are given by [16]:

Since the late 1960s, the Kalman filter has received huge attention from various fields in industry and academia and played a key role in many engineering disciplines for trajectory planing, state and parameter estimation, signal processing, etc. [12], [13]. In [14] the behavior of two estimation techniques based on the Kalman filter is analyzed. The methods utilize an Extended Kalman Filter (EKF) and an Unscented Kalman Filter (UKF) for parameter estimation of DFIGs in wind turbine systems. However, the observability of the (linearized) DFIG model is not addressed and the two Kalman filters are not tested under parameter variations. Moreover, the two filters were not optimally tuned (parameter estimation converges after 6 s) and the used nonlinear state space model depends on state, input and output (which is not admissible from a theoretical point of view).

usabc (t)

=

urabc (t)

=

d abc ψ (t), dt s d Rr irabc (t) + ψrabc (t), dt Rs isabc (t) +

ψsabc (0) = 0(1) 3 ψrabc (0) = 0(2) 3    initial values

where (assuming linear flux linkage relations) ψsabc (t) ψrabc (t)

In this paper, an Extended Kalman filter (EKF) is

227

= =

Ls isabc (t) + Lm irabc (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

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 with constant grid 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  t φs (t) = ωs (τ )dτ + φ0s , φ0s ∈ R

(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 ⎪ ⎬ d urk (t) = Rr irk (t) + dt ψrk (t) + (ωs − ωr (t))J ψrk (t), ⎪    ⎭ =:ωsl (t)

(9) 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.

and the rotor reference frame is shifted by the rotor angle  t ωr (τ )dτ + φ0r , φ0r ∈ R (6) φr (t) = 0

with respect to the stator reference frame (φ0r is the initial electrical rotor angle).

ωsl := ωs − ωr

A. Model in stator (stationary) reference frame

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 .

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., [15]), respectively, given by (neglecting the zero Moreover, if sequence)     L L + Lm Lsσ + Lrσ Lsσ L2 (5) 1 −√12 1 − cos(φ) sin(φ) s σ := 1− m = 2 m rσ = 0, s abc k 2 √ x & x = x x =γ L L L 3 3 r s m + Lm Lrσ + Lm Lsσ + Lrσ Lsσ − sin(φ) cos(φ) 0 − 2 2       one may re-write the currents as functions of the stator =:TP (φ)−1 =:TC and rotor flux linkages as follows [23, Sec. 5.4] (7)  where γ = 23 for an amplitude-invariant transformation 1 ψsk − σLLsmLr ψrk isk = σL 2/3 for a power-invariant transformation). (or γ = s (11) 1 irk = σL ψrk − σLLsmLr ψsk . Expressing the rotor voltage equation (2) also with r respect to the stationary reference frame (i.e. urs = TP (φr )−1 TC urabc ), the voltage equations (1) and (2) can C. Dynamics of the mechanical system be rewritten as

d For a stiff shaft and a step-up gear with ratio gr ≥ 1, ψss (t), ψss (0) = 02 uss (t) = Rs iss (t) + dt the dynamics of the mechanical system are given by s s s s d urs (t) = Rr ir (t) + dt ψr (t) − ωr (t)J ψr (t), ψr (0) = 02   (8) d 1 mt 0 ω m = − ∈ R (12) , ωm (0) = ωm m e where [15] dt Θ g r    0 −1 =:mm . J := TP (π/2) = 1 0 where [23, Sec. 5.4] B. Model in stator voltage orientation 3 3 me (t) = np iss (t) J ψss (t) = np isk (t) J ψsk (t) An essential characteristic of the DFIG control strat2 2 Lm 3 egy is that the generated active and reactive power shall (11) k  = − np ψ (t) J ψsk (t). (13) be controlled independently. It is common to use an air2 σLs Lr r gap flux orientation [17] or a stator flux orientation [18]is the electro-magnetic machine torque (moment), [20] for the vector control schemes. However, it has been mt [Nm] is the turbine torque produced by the wind (see shown that the stator flux orientation can cause instability t Sec. III) and mm = m under certain operating conditions [21]. Therefore, folgr [Nm] is the mechanical torque acting on the DFIG shaft. Θ [kg/m2 ] is the rotor inertia lowing the ideas in [16], [22], in this paper, a stator (grid) voltage orientation for the vector control scheme is used. and np [1] is the pole pair number.

228

D. Overall nonlinear model of the DFIG

mechanical (turbine) power of a WTS is given by [16], [15], [24]: 1 3 pt = cp (λ, β) ρπrt2 vw (18) 2  

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 ∈ R10 and y = h(x), dt (14) is required. Therefore, introduce the state vector x, the output (measurement) vector y and the input vector u as follows: ⎫   ⎪ ψsd , ψsq , ψrd , ψrq , ωr , 10 ⎪ ∈R , ⎪ x = ⎬ Rs , Rr , Lsσ , Lrσ , Lm  d q d q ⎪ y = is is ir ir ωr ∈ R5 , ⎪ ⎪   ⎭ u = usd usq urd urq ∈ R4 . (15)

wind power 3

where ρ [kg/m ] is the air density, rt [m] is the radius of the wind turbine rotor (πrt2 is the turbine swept area), cp [1] is the power coefficient, and vw [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 ω m rt [1] (19) λ= gr vw and the pitch angle β [◦ ] 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 [16], [24]. Many different (datafitted) approximations for cp have been reported in the literature. This paper uses the power coefficient cp from [24], i.e.   −21 116 − 0.4β − 5 e λi + 0.0068λ cp (λ, β) = 0.5176 λi 1 0.035 1 − 3 . (20) := 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 cp := cp (λ , 0) = maxλ cp (λ, 0). Only then, the WTS can extract the maximally available turbine 3 . Maximum power point tracking power pt = cp 21 ρπrt2 vw is achieved by the nonlinear speed controller [15] ρπrt5 cp 2 me = kp ωm ≈ mm , kp := (21) 2gr (λ )3 which assures that the generator angular frequency ωm is ! adjusted to the actual wind speed vw such that ωgrmvrwt = λ  holds. According to (21) the optimum torque me can be calculated from the shaft speed ωm = ωr /np and then it is compared with the actual electro-magnetic torque me , as shown in Fig. 1. Based on the difference me − me the underlying torque PI controller1 generates the rotor d . reference currents ir,ref

Combining the subsystems of the DFIG as in (9), (10), d d d d ψsk , dt ψrk , and dt ωr = np dt ωm (12) and solving for dt yields the nonlinear model (14) with ⎡ ⎤  ψd  usd − Rs σLs s − σLLsmLr ψrd + ωs ψsq q   ⎢ ⎥ ψ usq − Rs σLs s − σLLsmLr ψrq − ωs ψsd ⎢ ⎥ ⎢ ⎥ d   ⎢ud − R ψr − Lm ψ d + (ω − ω )ψ q ⎥ r σLr s r r⎥ ⎢ r s σL L s r  ψq  ⎢ q ⎥ r ⎢ur − Rr σL − σLLsmLr ψsq − (ωs − ωr )ψrd ⎥ r ⎢ np  3np Lm ⎥ k k  g(x, u) = ⎢ ⎥ ⎢ Θ − 2σLs Lr ψr (t) J ψs (t) − mm (ωr ) ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ 0 ⎣ ⎦ 0 0 (17) and h(x) as in (16). Note that, in h(x), the parameters Rs = x6 , Rr = x7 , Lsσ = x8 , Lrσ = x9 , and Lm = x10 are also state variables and mm (ωr ) will be approximated in Sec. III as a function of ωr = np ωm . 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 [22], [15], 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 the magnitude and direction of the rotor power flow. DC-link voltage control is a non-trivial task due to the possible non-minimumphase behavior for a power flow from the grid to the DClink [15], [4]. More details on controller design, phaselocked loop and pulse-width modulation (PWM) are given in, e.g., [22], [15]. III.

Remark: For wind speeds above the nominal wind speed, the WTS changes to nominal operation, i.e. me = me,nom , where me,nom is the nominal/rated generator torque. Speed control is achieved by (individual) pitch control such that the rated power me,nom ωm,nom of the WTS is generated. IV.

E XTENDED K ALMAN F ILTER AND O BSERVABILITY

A. Extended Kalman Filter The EKF is a nonlinear extension of the Kalman filter for linear systems. Its design is based on a discrete non-

M AXIMUM POWER POINT TRACKING (MPPT)

Wind turbines convert wind energy into mechanical energy and, via a generator, into electrical energy. The

1 The torque PI control loop still requires a thorough stability analysis which is not considered in this paper.

229

⎡⎡ h(x) =

1 Lm (Lsσ + Lrσ ) + Lsσ Lrσ

(Lm + Lrσ ) 0 ⎢⎢ ⎣⎣ −L m 0

0 (Lm + Lrσ ) 0 −Lm

⎤ 0 −Lm ⎥ ⎦, 0 (Lm + Lsσ )

⎤ ⎥ O 4×6 ⎦ x (16)

Algorithm 1: Extended Kalman filter

linear system model [25]. For discretization the (simple) forward Euler method with sampling time Ts [s] is applied to the time-continuous model (14) with (15), (17) and (16). For sufficiently small Ts 1, the following holds d x(t) = x[k+1]−x[k] for x[k] := x(kTs ) ≈ x(t) and dt Ts all t ∈ [kTs , (k + 1)Ts ) and k ∈ N ∪ {0}. Hence, the nonlinear discrete model of the DFIG can be written as ⎫ =:f (x[k],u[k]) ⎪ ⎬    (22) x[k + 1] = x[k] + Ts g(x[k], u[k]) +w[k], ⎪ ⎭ y[k] = h(x[k]) + v[k], x[0] = x0

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[0] C[0]P [0]C[0] + R where, for k ≥ 0,   ∂h(x)  C[k] := ∂x  ˆ − [k] x

Step II: Time update (“a priori prediction”) for k ≥ 1: (a) State prediction ˆ − [k] = f (ˆ x[k − 1], u[k − 1]) x (b) Error covariance matrix prediction P − [k] = A[k]P [k − 1]A[k] + Q where   ∂f (x,u)  A[k] := ∂x 

where the random variables w[k] := ∈ R10 and v[k] := (w1 [k], . . . , w10 [k]) (v1 [k], . . . , v5 [k]) ∈ R5 are included to model system uncertainties and measurement noise, respectively. Both are assumed to be independent (i.e., E{w[k]v[j] } = O 10×5 for all k, j ∈ N), while (i.e., E{w[k]} = 010 and E{v[k]} = 05 for all k ∈ N) and with normal probability distributions   −(αi −E{αi })2 1√ (i.e., p(αi ) = exp with 2σ 2 σ 2π αi

−Lm 0 (Lm + Lsσ ) 0

ˆ − [k] x

α

σα2 i := E{(αi − E{αi })2 } 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. (23) Note that Q and R must be chosen positive semi-definite and positive definite, respectively.

Step III: Verification observability k ≥ 1:  of (local)  no [k] := rank S o [k] with S o [k] as in (26) Step IV: Computation of  Kalman gain for k ≥ 1−1 K[k] = P − [k]C[k] C[k]P − [k]C[k] + R Step V: Measurement update (“correction”) for k ≥ 1: (a) Estimation update with measurement ˆ [k] = x ˆ − [k] + K[k](y[k] − h(ˆ x− [k])) x (b) Error covariance matrix update P [k] = P − [k] − K[k]C[k]P − [k] Step V: Go back to Step II (use C[k]).

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 ˆ [k] = h(ˆ y x[k]) (24) where K[k] is the Kalman gain (to be specified below) ˆ and y ˆ are the estimated state and output vector, and x respectively. The recursive algorithm of the EKF implementation is listed in Algorithm 1 [25]. The EKF achieves an optimal state estimation by minimizing the covariance of the estimation error for each time instant k ≥ 1.

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.

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 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

In [26] 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

The matrix R is related to the measurement noise characteristics. Increasing the values of R indicates that 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.

Q = 10−6 diag{2, 2, 2, 2, 500, 0.04, 0.04, 0.002, 0.002, 0.5} R = diag{1, 1, 1, 1, 1} (25) P 0 = 10−3 diag{5, 5, 5, 5, 3, 0.02, 0.02, 0.004, 0.004, 0.3} x0 = 10−3 {0, 0, 0, 0, 0, 1.4, 0.4, 0.04, 0.04, 0.5} . B. Observability The observability of a linear system can be verified by computing the observability matrix and its rank. For

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12

4.0

Rs [ m : ]

vw[m/ s]

10 8.0

4.0

Rr [m:]

1.0 0.8 0

0.5

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2.0

1.0

Overall simulation scenario: Wind speed profile vw and corresponding rotor speed ωr .

100

LsV LÖ sV

75 50 25 100

L rV [ P H ]

1.5

i rabc [ pu ]

RÖ r

2.0

2.5

Figure 2:

0

Lm [mH ]

-1.5 10

Rank

Rr

3.0

L sV [ P H ]

Zr [pu]

0

1.2

5.0 0

0.5

1.0 1.5 time [s]

2.0

L rV LÖ rV

75 50 25 4.0

Lm LÖ m

2.0 0

0

2.5

0.1

0.2 0.3 time [s]

0.4

0.5

Figure 4:

Estimation performance of the proposed EKF: Estimation of stator resistance Rs , rotor resistance Rr , stator leakage inductance Lsσ , rotor leakage inductance Lrσ , and mutual inductance Lm for large initial parameter errors.

Figure 3: Overall simulation scenario: Three-phase rotor currents irabc of the DFIG and rank of the observability matrix S o [k] as in (26).

nonlinear systems, it is possible to analyze the observability “locally” by analyzing the linearized model around an operating point [27]. The observability matrix of the linearized model of the considered DFIG as in (22) is given by ⎤ ⎡ C[k] ⎢ C[k]A[k] ⎥ ⎥ ⎢ ⎢C[k]A[k]2 ⎥ ⎥ ⎢ . (26) S o [k] := ⎢ ⎥ ∈ R50×10 , ⎥ ⎢ . ⎥ ⎢ ⎣ ⎦ . 9 C[k]A[k]

as in Fig. 1. For more details on the implementation of e.g. back-to-back converter, PWM, current controller design, see [15]. The simulation results are shown in Figures 2-6. The estimation performances of the EKF are illustrated for normal operation conditions and for parameter variations in Rs , Rr and Lm . Fig. 2 and 3 show the overall simulation scenario over time: Wind speed profile vw , corresponding rotor speed ωr , three-phase rotor currents irabc and rank of the observability matrix S o [k] as in (26). The observability of the linearized DFIG model has been tested online for several operating points (e.g. at low, high and synchronous speed, see Fig. 2): The linearized system is observable even if the DFIG operates close to synchronous speed, i.e., the rotor frequency is almost zero (as shown in Fig. 3). The observability matrix has full rank for all   times, i.e. rank S o [k] = 10 for all k ≥ 0.

where A[k] and C[k] are computed online for each sampling instant k ≥ 0 (see Algorithm 1). The pair {A[k], C[k]} (i.e., the linearized model of the DFIG) is observable if and only if matrix S o [k]   the observability has full rank, i.e., rank S o [k] = 10 for the considered DFIG as in (22). To check “local” observability, the rank of the observability matrix S o [k] is computed numerically for each sampling instant k ≥ 0 in Step III of Algorithm 1. V.

RÖ s

2.0

6.0

0

Rs

Fig. 4 shows the simulation results of the proposed EKF under normal operating conditions. Despite large initial errors in all electrical parameters, the parameter estimation is fast; in particular, for the mutual inductance. The steady state estimation errors of the EKF are small (see Tab. I). These results illustrate the capability of the EKF in estimating all electrical parameters of the DFIG.

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

In order, to illustrate the capability of the EKF to track the electrical parameters of the DFIG under parameter

231

3.0

Steady state estimation errors of the EKF.

Estimated state Normal conditions Rs and Rr increase by 10% Lm increases by 10%

Rs 0.5% 0.9%

Rr 0.4% 0.7%

Lsσ 0.2% 0.8%

Lrσ 0.2% 1%

Lm 0.4% 1.1%

1.4%

1.6%

1.5%

1.5%

1.6%

Rs [ m : ]

Table I:

Rs

RÖ s

2.5 2.0

Rs [ m : ]

3.0

Rr [m:]

3.5

Rs

RÖ s

2.5

100

L sV [ P H ]

Rr

RÖ r

3.0 2.5

LsV LÖ sV

80

Lm [mH ]

L sV [ P H ]

100

60 100

L rV [ P H ]

LsV LÖ sV

80 60 100

L rV [ P H ]

Rr [m:]

3.5

Lm [mH ]

RÖ r

3.0 2.5

2.0

60 3.0

m

0.6

0.7 0.8 time [s]

0.9

60 3.2

Lm LÖ m

2.6 0.5

0.6

0.7 0.8 time [s]

0.9

1.0

Figure 6: Robustness results of the proposed EKF: Estimation of stator resistance Rs , rotor resistance Rr , stator leakage inductance Lsσ , rotor leakage inductance Lrσ , and mutual inductance Lm for an 10% increase in Lm .

Lm LÖ

0.5

L rV LÖ rV

80

2.0

L rV LÖ rV

80

2.0

Rr

1.0

VI.

C ONCLUSION

This paper proposed a method for estimating all electrical parameters of doubly-fed induction generators (DFIGs) in wind turbine systems (WTS). The method utilizes an Extended Kalman filter (EKF). For implementation of the EKF, the nonlinear state space model of the DFIG has been derived to estimate all electrical parameters of the DFIG, i.e.. the stator and rotor resistances, the leakage inductances of stator and rotor, and the mutual inductance. The EKF design has been presented in detail and the observability of the linearized (fundamental) model of the DFIG has been tested online for different wind speeds. The results illustrate that the linearized model is “locally” observable for all considered operating conditions. The estimation performance of the EKF has been illustrated by simulation results and compared to the actual parameter values. The results showed that the EKF tracks all electrical parameters of the DFIG with high accuracy. Moreover, the EKF is also capable to estimate the electrical parameters under parameter variations due to temperature changes or magnetic saturation.

Figure 5:

Robustness results of the proposed EKF: Estimation of stator resistance Rs , rotor resistance Rr , stator leakage inductance Lsσ , rotor leakage inductance Lrσ , and mutual inductance Lm for an 10% increase in Rs and Rr .

variations, the values of stator Rs and rotor Rr resistances are increased by 10% (e.g. due to warming or aging). For this scenario, Fig. 5 shows the estimation performance of the proposed EKF. The EKF still achieves a high estimation accuracy. Moreover, the influence on the estimation accuracy of stator and rotor leakage inductances and mutual inductance is small (see last three subplots in Fig. 5). The steady state estimation error is still small (see Tab. I). Finally, the estimation performance of the EKF under a variation of the mutual inductance Lm (e.g., due to magnetic saturation) is investigated. Therefore, Lm is increased step-like by 10%. Fig. 6 shows the simulation results of the EKF for this scenario. Again, the EKF is able to track the change in Lm and the estimation of the other parameters is almost not affected (see Tab. I and Fig. 6).

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232

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A PPENDIX

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The simulation parameters are given in Tab. II. Note that the rotor parameters (resistance and inductance) are converted to the stator of the DFIG.

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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

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233

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