Sensorless Control of Permanent Magnet Synchronous Generators in Variable-Speed Wind Turbine Systems Mohamed Abdelrahem
Christoph Hackl
Zhenbin Zhang, Ralph Kennel
Student Member, IEEE Member, IEEE Student Member, Senior Member, IEEE Institute for Electrical Drive Systems Munich School of Engineering Research Group Institute for Electrical Drive Systems and Power Electronics “Control of Renewable Energy Systems (CRES)” and Power Electronics Technische Universität München (TUM) Technische Universität München (TUM) Technische Universität München (TUM) Munich, Germany Munich, Germany Munich, Germany Email:
[email protected] Email:
[email protected] Email: james.cheung,
[email protected]
Abstract—This paper proposes a sensorless control strategy for permanent magnet synchronous generators (PMSGs) in variablespeed wind turbine systems (WTSs). The proposed scheme uses an extended Kalman filter (EKF) for estimation of rotor speed and position. For EKF design, the nonlinear state space model of the PMSG is derived. Estimation and control performance of the proposed sensorless control method are illustrated by simulation results for all operation conditions. Moreover, the performance of the proposed EKF is tested under variations of the PMSG parameters.
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 stator (s) or filter (f ) quantity, i.e. z ∈ {s, f }. The space vector is expressed in either phase abc-, stationary s-, or arbitrarily rotating k-coordinate system, i.e. y ∈ {abc, s, 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.
certain ride through capability [3]. Operation above and below the synchronous speed is feasible. However, this situation has changed in the recent years with the development of WECSs with larger power capacity, lower cost/kW, increased power density, and the need for higher reliability. More and more attention has been paid to direct-drive gearless WECS concepts. Among different types of generators, the permanent magnet synchronous generator (PMSG), see Fig 2, has been found to be superior owing to their advantages of higher efficiency, higher power density, lower maintenance cost, and better grid compatibility [1]. Increased reliability and high performance make direct-drive PMSG-based WECSs more attractive in multi-Megawatt offshore applications, where the WECSs are installed in harsh and less-accessible environments. Currently, there is a variety of commercial direct-drive PMSG-based WECSs in the market with power ratings ranging from hundreds of watts to 6 MW [1].
Gear box Grid DFIG
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 various wind energy conversion system (WECS) configurations, the doubly-fed induction generator (DFIG)-based variable-speed WECSs have been the dominant technology in the market since late 1990s [1]-[3], see Fig 1. DFIGs can supply active and reactive power, operate with a partial-scale power converter (around 30% of the machine rating), and achieve a
Wind Turbine
Back-back converter
filter
Figure 1: DFIG topology for variable-speed wind turbine systems. Vector control has – so far – proven to be the most popular control technique for PMSGs in variable-speed WTSs [4]. This method allows for a decoupled control of active and reactive power of WTSs via regulating the direct and quadrature components of the stator current vector independently. Vector control requires accurate knowledge of rotor speed and rotor position [4]. Recently, the interest in sensorless methods (see,
Wind Turbine
Back-back converter filter Grid PMSG
Figure 2: PMSG topology for variable-speed wind turbine systems.
e.g., [5] 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 [4]-[7]. Many position sensorless control schemes have been developed for permanent magnet synchronous machines (PMSMs) used in applications such as electric vehicles, home appliances, and industrial drives [5]. Although little work has been reported on position sensorless vector control for PMSGbased WECSs, the methods developed for other industrial sensorless PMSM drives can be adapted to PMSG-based WECS applications. Position sensorless vector control for PMSGs used in direct-drive WECSs could be easier than those in other industrial applications because of several factors. First, the difference between the d- and q-axis inductances of the PMSGs used in direct-drive WECSs is usually small (Ld ≈ Lq ). Sensorless control of a nonsalient-pole PMSG is much easier than that of a PMSM with large saliency in the medium- and high-speed range. Second, the operation speed of PMSGs used in WECSs are relatively limited and rarely reach the flux-weakening region. Generally the rotor speed/position estimation schemes applicable for PMSGs can be grouped into two categories: 1) open-loop calculation (such as flux-based method, inductance based method, etc...) and 2) closed loop observers (such as model reference adaptive system (MRAS), sliding mode observer (SMO), etc..) [5]. The extended Kalman filter (EKF) is an optimal estimator in the least-square sense for estimating the states of dynamic nonlinear systems [8]. EKF has already been used for sensorless control and estimation of the electrical parameters of AC machines [8]-[15]. In [13], an EKF was designed in the stationary reference frame s = (α, β) for PMSG speed and position estimation. However, this paper neglected the mechanical system
dynamics as the author assumes dω dt = 0. A sensorless control of distributed power generators based on derivative free Kalman filter has been proposed in [14]. For the proposed method, the generator model is first subject to a linearization based on differential flatness and next state estimation is performed by applying the standard Kalman filter recursion to the linearized model. In [15], an EKF was designed in the rotating reference frame k = (d, q) for PMSG speed and position estimation. However, this paper neglected the mechanical system dynamics as the author also assumes dω dt = 0. Neglecting the mechanical system dynamics worsens the estimation performance of the EKF (see results in [15]) and will not represent the real physical model of the system. The observability of the linearized PMSG model is not checked in all previous works [12]-[15]. In this paper, an extended Kalman filter is proposed for the estimation of speed and position of the PMSG rotor. State, input and measurement variables are used in the rotating reference frame k = (d, q), which reduces the complexity of the 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 via simulation results. The results highlight the ability of the EKF in tracking the PMSG rotor speed and position. II. M ODELING AND C ONTROL OF THE WTS WITH PMSG The block diagram of the vector control problem of WTS with PMSG is shown in Fig. 3. It consists of a permanent magnet synchronous machine mechanically coupled to the wind turbine directly via a stiff shaft. The stator windings of the PMSG are connected via a back-to-back full-scale voltage source converter (VSC), a filter and a transformer to the grid. The transformer will be neglected in the upcoming modeling. The machine side converter (MSC) 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 [16]. The stator voltage equation of the PMSG is given by [16]: d usabc (t) = Rs isabc (t) + ψsabc (t), ψsabc (0) = 03 (1) dt where d abc d ψ (t) = Ls isabc (t) + eabc (2) s (t) dt s dt Here usabc = (usa , usb , usc )> [V], isabc = (isa , isb , isc )> [A], ψsabc = (ψsa , ψsb , ψsc )> [Vs], and eabc = (eas , ebs , ecs )> [V] are s the stator voltages, currents, fluxes, and back electromotive forces respectively, all in the abc-reference frame (three-phase system). Rs [Ω] and Ls [Vs/A] are the stator resistance and inductance. Note that the PMSG 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
Wind turbine Encoder
r & r PMSG
r Ls isq
d s,ref
isabc
i
r udc,ref
udc
e abc/dq
C
r (L i d s s
eL f i
i df ,ref i df uoabc
PI
uod
e abc/dq
i
u oq
q f
i qf ,ref
MSC
pm
)
DC Link
q f
Cdc
udc
Drive
PLL
PI
PWM
i abc f
PI
isq,ref
dq/abc
u
abc o
i PI
q s
r
Drive
r ,ref
abc/dq
PI PWM
r
isd
dq/abc
isabc
usabc
GSC
i abc f
Rf
PI
e L f i df
e
uoabc
Lf Grid
Figure 3: PMSG control structure for variable-speed wind turbine systems.
and the rotor reference frame is shifted by the rotor angle Z t φr (t) = ωr (τ )dτ + φ0r , φ0r ∈ R (3) 0
with respect to the stator reference frame (φ0r is the initial rotor angle). Equation (1) can be expressed in the stationary reference frame as follows xk = TP (φ)−1 xs = TP (φ)−1
TC xabc | {z }
xs =(xα ,xβ )>
by using the Clarke and Park transformation (see, e.g., [16]), 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
=:TP (φ)−1
(4) wherepγ = 23 for an amplitude-invariant transformation (or γ = 2/3 for a power-invariant transformation). Therefore,
(1) can be rewritten in the stationary reference frame s = (α, β) as follows uss (t) = Rs iss (t) +
d s ψ (t), dt s
ψss (0) = 02
(5)
Equation (5) can be written in the rotating reference frame k = (d, q) as usk (t) = Rs isk (t) +
d k ψ (t) + ωr J ψsk (t), dt s
ψsk (0) = 02 (6)
where [16] J := TP (π/2) =
0 1
−1 . 0
Assuming Lds = Lqs =: Ls (no anisotropy), the PMSG flux can be expressed by d d ψpm ψs is k ψs = = Ls q + (7) 0 ψsq is
and the dynamics of the mechanicals of the (stiff) wind turbine system are given by d 1 0 ωm = me − mm , ωm (0) = ωm ∈R (8) dt Θ
control is a non-trivial task due to the possible non-minimumphase behavior for a power flow from the grid to the DClink [16], [17], [18]. More details on controller design, phaselocked loop or, alternatively, virtual flux estimation and pulsewidth modulation (PWM) are given in, e.g., [4], [19], [16].
where
3 pt np ψpm isq and mm = (9) 2 ωm are the electro-magnetic machine torque (moment) and the mechanical wind turbine torque, respectively. The mechanical torque mm depends on the wind turbine power pt [W] (see Sec. III) and the mechanical angular speed ωm [rad/s]. Θ [kg/m2 ] is the rotor inertia and np [1] is the pole pair number. me =
A. Overall nonlinear model of the PMSG
d x = g(x, u), x(0) = x0 ∈ R4 and y = h(x), (10) dt is required. Therefore, introduce the state vector x, the output (measurement) vector y and the input vector u as follows > x = isd isq ωr φr ∈ R4 , > (11) y = isd isq ∈ R2 , q > d 2 ∈R . u = us us d k Inserting (7) into (6), solving for dt is and inserting (9) into (8) d and solving for dt ωr yields the nonlinear model (17) with −Rs d 1 d q Ls is + ωr is + Ls us −Rs iq − ωr id − ωr ψpm + 1 uq s Ls s Ls Ls s g(x, u) = (12) np 3 q n ψ i − m ] p pm m s Θ 2 ωr
1 h(x) = 0
0 1
0 0
0 x. 0
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 [20], [16], [21]: 1 3 pt = cp (λ, β) ρπrt2 vw 2 | {z }
(14)
wind power
For the design of the EKF, the derivation of a compact (nonlinear) state space model of the PMSG of the form
and
III. M AXIMUM POWER POINT TRACKING (MPPT)
(13)
Note that it is assumed that the mechanical torque mm as in (9) is known (at least roughly using the wind power as in (14) and the power coefficient as in (16)). B. Overall control system of the WTS The complete control block diagram of the PMSG in field oriented control is depicted in Fig. 3. For the machine-side converter (MSC), the q-axis current is used to control the PMSG stator active power in order to harvest the maximally available wind power (i.e., maximum power point tracking, see Sec. III), whereas the d-axis current is used to control the reactive power flow in the PMSG [4], [20]. For the grid-side converter (GSC), the stator voltage orientation is used [4], [16], 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 power flow. DC-link voltage
where ρ > 0 [kg/m3 ] is the air density, rt > 0 [m] is the radius of the wind turbine rotor (πrt2 is the area swept by the turbine), 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] (15) 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 [20], [21]. Many different (data-fitted) approximations for cp have been reported in the literature. This paper uses the power coefficient from [21], i.e. −21 116 cp (λ, β) = 0.5176 − 0.4β − 5 e λi + 0.0068λ λi 1 0.035 1 − 3 . (16) := 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 value c?p := cp (λ? , 0) = maxλ cp (λ, 0) as shown in Fig. 4. Only then, the WTS can extract the maximally available turbine 3 power p?t := c?p 21 ρπrt2 vw [16]. Maximum power point tracking is achieved by a speed controller, Fig. 3, which assures that the generator angular frequency ωm is adjusted to the actual wind speed vw such that ωm rt ! ? vw = λ holds. Therefore, the optimum generator angular frequency ωm,ref can be calculated and then it is compared with the actual mechanical speed ωm , which is estimated by the EKF, as shown in Fig. 3. Based on the difference ωm,ref − ωm the underlying PI controller generates the q-axis q stator reference current is,ref .
Algorithm 1: Extended Kalman filter
c
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, C[k] := ∂h(x) ∂x −
* p
*
ˆ [k] x
Figure 4: Typical power coefficient curve for β = 0. IV. E XTENDED K ALMAN F ILTER AND O BSERVABILITY 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 [22]. For discretization the (simple) forward Euler method with sampling time Ts [s] is applied to the timecontinuous model (10) with (11), (12) and (13). For sufficiently small Ts 1, the following holds x[k] := x(kTs ) ≈ x(t) d x(t) = x[k+1]−x[k] for all t ∈ [kTs , (k + 1)Ts ) and and dt Ts k ∈ N∪{0}. Hence, the nonlinear discrete model of the PMSG can be written as =:f (x[k],u[k]) z }| { x[k + 1] = x[k] + Ts g(x[k], u[k]) +w[k], (17) y[k] = h(x[k]) + v[k], x[0] = x0 where the random variables w[k] := (w1 [k], . . . , w4 [k])> ∈ R4 and v[k] := (v1 [k], v2 [k])> ∈ R2 are included to model system uncertainties and measurement noise, respectively. Both are assumed to be independent (i.e., E{w[k]v[j]> } = O 4×2 for all k, j ∈ N) and white noise (i.e., E{w[k]} = 04 and E{v[k]} = 02 for all k ∈ N) with normal probability 2 i }) distribution (i.e., p(αi ) = σ 1√2π exp −(αi −E{α with 2σ 2 αi
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. (18) Note that Q and R must be chosen positive semi-definite and positive definite, respectively. 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 (19) 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 [22]. The EKF achieves an optimal state estimation by minimizing the covariance of the estimation error for each time instant k ≥ 1.
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 − ˆ [k] x
Step III: Verification of (local) observability k ≥ 1: no [k] := rank S o [k] with S o [k] as in (21) 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 x− [k])) (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]).
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 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 [23] general guide lines are given how to select the values of Q and R. Following these guide lines, for this paper the
r [rad/ s]
4 2 0
r [rad ]
0
0.2
8 6 4 2 0
0.4 time (sec)
0.6
0.8
r
0.1
0.2 0.3 time (sec)
0.4
ˆ
r
0.5
Figure 5: Estimation performance of the proposed EKF (from top): wind speed vw , rotor speed ωr , Observability matrix rank, and rotor angle φr . following values have been selected Q =
diag{0.5, 0.5, 1 · 10−6 , 5 · 10−3 }
R
=
diag{1, 1}
P0
=
diag{1, 1, 0.001, 10}
x0
=
(5, 0.1, 0, 0)>
r
100 0 8 6 4 2 0
ˆ r
r
0
0.1
0.2 time (sec)
0.3
ˆr
0.4
Figure 6: Estimation performance of the proposed EKF at 50% step change in the PMSG resistance Rs (from top): wind speed vw , PMSG stator resistance Rs , rotor speed ωr , and rotor angle φr .
V. S IMULATION R ESULTS AND D ISCUSSION (20)
B. Observability The observability of a linear system can be verified by computing the observability matrix and its rank. For nonlinear systems, it is possible to analyze the observability “locally” by analyzing the linearized model around an operating point [24]. The observability matrix of the linearized model of the considered PMSG as in (17) is given by C[k] C[k]A[k] 8×4 S o [k] := (21) C[k]A[k]2 ∈ R , C[k]A[k]3 where A[k] and C[k] are computed numerically1 for each sampling instant k ≥ 0 (see Algorithm 1). The pair {A[k], C[k]} (i.e., the linearized model of the PMSG) is locally observable if and only if the observability matrix S o [k] has full rank, i.e., rank S o [k] = 4 for the considered PMSG as in (17). 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. 1 Future
25 20 15 10 5 0 0.4 0.3 0.2 0.1 0 400 300 200
r [rad]
vw[m/ s]
vw[m/ s] ˆ r
Rs [ ]
r
100 0
rank [1]
r [rad / s]
25 20 15 10 5 0 400 300 200
work will derive analytical conditions for local observability.
A simulation model of a 16 kW WTS with PMSG is implemented in Matlab/Simulink. The system parameters are listed in Table I. The implementation is as in Fig. 3. For more details on the implementation of e.g. back-to-back converter, PWM, current controller design, see [16]. The simulation results are shown in Figs. 5-9. The estimation performances of EKF are compared with the actual values for different wind speed and parameter uncertainties in Rs and Ls . Fig. 5 shows the simulation results for variable wind speeds. The presented wind speeds cover almost the complete speed range of the PMSG-based WTS (from low to high speed). Fig. 5 illustrates the tracking capability of the EKF of rotor speed and rotor position at low and high speeds. The EKF shows a high estimation accuracy as the estimation error is very small, see Table. II. The observability of the linearized PMSG model has been tested online under variable wind speeds, see Fig. 5: The linearized system is observable even if the PMSG operates at low speed, i.e., the stator frequency is almost zero (as shown in Fig. 5). The observability matrix has full rank for all times, i.e. rank S o [k] = 4 for all k ≥ 0. In order to check the robustness of the EKF under (unknown) parameter variations of the PMSG, the value of the stator resistance Rs is increased by 50% (e.g. due to warming or aging). For this scenario, Fig. 6 shows the estimation performances of the proposed EKF. The simulated wind speed profile is depicted in Fig. 6 (top). It is clear that
vw[m/ s]
m [rpm]
12 8 400 300 200
m,ref
m
i sq, ref i sq i sd, ref i sd
r
100 0
ˆ r
[1 ]
8.2 8.1 8.0
r
ˆr
0.49
c p [1]
8 6 4 2 0
0.48 0.47
0
0.1
0.2 time (sec)
0.3
0.4
Figure 7: Estimation performance of the proposed EKF at 25% step change in the PMSG inductance Ls (from top): wind speed vw , PMSG inductance Ls , rotor speed ωr , and rotor angle φr . Table I: Parameters of the PMSG-based WTS. Name Wind turbine rated power Wind turbine radius Rated wind speed Optimal tip speed ration PMSG rated power PMSG rated voltage (line-line] Number of pair poles Stator resistance Stator inductance Permanent magnet flux PMSG moment of inertia DC capacitor DC-link voltage Grid line-line voltage Grid normal frequency Grid resistance Grid inductance Filter resistance Filter inductance Sampling time Simulation step
Nomenclature pt rt vwrated λ? pnom urms s np Rs Ls ψpm Θ Cdc udc uo fe Rg Lg Rf Lf Ts Tsim
Value 16 kW 1.6 m 20 m/sec 8.11 16 kW 400 V 3 0.2[Ω] 15 mH 0.85 0.02[kg/m2 ] 3[mF] 700[V] 400[V] 50[Hz] 0.036[Ω] 2[mH] 0.12[Ω] 8[mH] 40[µs] 1[µs]
Table II: Estimation errors of the proposed EKF. Simulation case Normal conditions Rs increased by 50% Ls increased by 25%
ωr 0.9% 1.15% 1.8%
φr 1.2% 1.4% 2.6%
P[kW ] & Q
r [rad/ s]
16
r [rad]
25 20 15 10 5 0 1200 900 600 300 0 80 60 40 20 0
isd & isq [ A]
25 20 15 10 5 0 20
Ls [mH]
vw[m/ s]
the EKF is robust against parameter uncertainties in Rs . It estimates rotor speed and rotor position with very small errors (see Tab. II).
P Pmax Q
20 15 10 5 0 0.2
0.4
0.6 0.8 time [sec]
1.0
1.2
1.4
Figure 8: Performance of the MSC control system (from top): wind speed vw , rotor mechanical speed ωm , d & q currents ids & iqs , tip speed ratio λ, power coefficient cp , and output active and reactive power P & Q.
Finally, the robustness with respect to changes (due to magnetic saturation) in the stator inductance Ls is investigated. Therefore, Ls is increased by 25%. Fig. 7 shows the simulation results of the proposed EKF for this scenario. The used wind speed profile is depicted in Fig. 7 (top). Again, the EKF shows an accurate estimation performance and robustness against parameter uncertainties (see Tab. II). The final simulation results are shown in Fig. 8 and Fig. 9. Fig. 8 illustrates the control performance of the machine side converter under variable wind speed (top). It is clear that the MSC guarantees tracking of the maximum power from the wind turbine. The actual rotor speed w ˆm of the PMSG is following the reference value wm,ref , which ensures the MPPT capability. 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? ≈ 8.11 is reached. Moreover, the generated power from the PMSG is almost equal the maximum power as shown in Fig. 8 (bottom). Also, the GSC control system ensures practical tracking of the constant reference DC
udc udc,ref
700
i df & i qf [ A]
u dc [V ]
725
675 50 40 30 20 10 0
q i df i df ,ref i f i qf ,ref
0.2
0.4
0.6 0.8 time [sec]
1.0
1.2
1.4
Figure 9: Performance of the GSC control system (from top): DC-link voltage udc , and d & q currents idf & iqf . voltage (with some small deviations) and of the references d & q currents as shown in Fig. 9. VI. C ONCLUSION This paper proposed a sensorless vector control method for variable-speed wind turbine systems (WTSs) with permanent magnet synchronous generator (PMSG). The method uses an extended Kalman filter (EKF) for estimating the PMSG rotor speed and position. For the design of the EKF, a nonlinear state space model of the PMSG has been derived. The design procedure of the EKF has been presented in detail. The sensorless control scheme of the WTS with PMSG has been illustrated by simulation results and its performance has been compared with the real values of the rotor speed and position. The results have shown that the EKF tracks rotor speed and position with high accuracy. Moreover, the EKF is robust to variations in the PMSG stator resistance and inductance. R EFERENCES [1] 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. [2] A. Hansen, F. Iov, F. Blaabjerg, and L. Hansen, “Review of contemporary wind turbine concepts and their market penetration”, Journal of Wind Engergy, Vol. 58, No. 4, pp. 1081–1095, Apr. 2011. [3] 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. [4] M. Chinchilla, S. Arnaltes, J.C. Burgos, “Control of permanent magnet generators applied to variable-speed wind-energy systems connected to the grid”, IEEE Transactions on Energy Conversion, Vol. 21, No. 1, pp. 130–135, March 2006. [5] Y. Zhao, C. Wei, Z. Zhang, W. Qiao, “A Review on Position/Speed Sensorless Control for Permanent-Magnet Synchronous Machine-Based Wind Energy Conversion Systems”, IEEE Journal of Emerging and Selected Topics in Power Electronics, Vol. 1, No. 4, pp. 203–216, Dec. 2013. [6] M. Abdel-Salam, A. Ahmed, and M. Abdel-Sater, “Harmonic Mitigation, Maximum Power Point Tracking and Dynamic Performance of Variable Speed Grid Connected Wind Turbine”, Journal of Electric Power Component and Systems, Vol. 39, pp. 176–190, 2011.
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