Encoderless Model Predictive Control 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 ofEngineering Research Group “Control of Renewable Energy Systems (CRES)”.Technical University of Munich (TUM), Munich, Germany.† Corresponding author.

E-mail: mohamed.abdelrahem@tum.de, christoph.hackl@tum.de, ralph.kennel@tum.de

Abstract. In this paper, an encoderless finite-control-set model predictive control (FCS-MPC)strategy for doubly-fed induction generators (DFIGs) based on variable-speed wind turbinesystems (WTSs) is proposed. According to the FCS-MPC concept, the discrete states of thepower converter are taken into account and the future converter performance is predictedfor each sampling period. Subsequently, the voltage vector that minimizes a predefined costfunction is selected to be applied in the next sampling instant. Furthermore, a model referenceadaptive system (MRAS) observer is used to estimate the rotor speed and position of the DFIG.Estimation and control performance of the proposed encoderless control method are validatedby simulation results for all operation conditions. Moreover, the performance of the MRASobserver is tested under variations of the DFIG parameters.

NotationN,R,C are the sets of natural, real and complex numbers. x ∈ R or x ∈ C is a real or complex scalar. x ∈ Rn

(bold) is a real valued vector with n ∈ N. x> is the transpose and ‖x‖ =√x>x is the Euclidean norm of x.

0n = (0, . . . , 0)> is the n-th dimensional zero vector. X ∈ Rn×m (capital bold) is a real valued matrix with n ∈ Nrows and m ∈ N columns. On×m ∈ Rn×m is the zero matrix. xy

z ∈ R2 is a space vector of a rotor (r), stator (s) or

filter (f) quantity, i.e. z ∈ r, s, f. 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.

1. IntroductionThe electrical power generation by variable speed wind turbines (WTs) has increasedsignificantly during the last years contributing to the reduction of carbon dioxide emissions andto a lower environmental pollution [1]. Among various wind energy conversion systems (WECSs),WECSs with doubly-fed induction generator (DFIG) have been the dominant technology in themarket since the late 1990s [1]. DFIGs can supply active and reactive power, operate with apartial-scale power converter (around 30% of the generator rating), and fulfill a certain ridethrough capability [2]. Operation above and below synchronous speed is possible [2].Fig. 1 shows a DFIG mechanically coupled to the wind turbine via a shaft and gear box with

DFIG dcCdcu

αβsi

αβsu

MPPT

*

em

abc

rs

abc

ri

Gear Box

RSC

DC-link

GSC

Grid

Cost

Function

abc

fs

mω

rωrφ

dq

ri

abc

si

Prediction

Model

d

refri ,

d

smp

ss

uLn

L

3

2ω

q

refri ,

m

d

s

s

Lu

L

3

2

refsQ,

ms

d

s

L

u

ωMRAS

Observer

αβri

dcu

αβsi

7

abc

oufR fL

abc

fiCost

Function

sφPrediction

Model7

dq

fi

d

reffi ,

q

reffi ,

PI

refdcu,

dcu

dcu

abc

ouabc

fi

dq

fidq

oudqabc/ sφ

sφ

gImI

sωsω

Figure 1: Proposed encoderless FCS-MPC for DFIG in variable speed WECSs.

ratio gr ≥ 1. The stator windings of the DFIG are directly tied to the grid, whereas the rotorwinding is tied via a back-to-back partial-scale voltage source converter (VSC), and a filter tothe grid. The grid side converter (GSC) and the rotor side converter (RSC) share a commonDC-link. Currently, field oriented control (FOC) and direct torque control (DTC) methodsdominate both academic and industrial applications for RSC [3]. For GSC, voltage orientedcontrol (VOC) and direct power control (DPC) are two popular methods [3]. However, withthe development of faster and more powerful digital signal processors, the implementation ofnew control strategies such as fuzzy logic and predictive control is possible [4]. One of the mostpromising predictive controllers in power converters and electric drives is the finite-control-setmodel predictive control (FCS-MPC) [4]–[6], which exploits the finite number of switching statesof the power converter for solving an optimization problem.Lately, the interest in encoderless control techniques is increasing due to cost effectivenessand robustness, which means that the controllers must run without the information ofmechanical sensors (such as encoders or transducers) mounted on the shaft [3]. Therequired rotor speed/position must be estimated via the information supplied by electrical(e.g. current/voltage) sensors which are cheap and easier to install than mechanical sensors.Furthermore, mechanical sensors decrease the drive system reliability due to their high failurerate, which means shorter maintenance periods and, consequently, higher costs [3].Encoderless vector control techniques for DFIGs have been proposed by several researchers [7]-[10]. The encoderless method presented in [7] is open-loop and relies on rotor current estimatorin which the estimated and measured currents are compared to get the rotor position. Theapplication of model reference adaptive system (MRAS) observers for encoderless control ofDFIGs has been reported in [8], where MRAS observers are diversified with different errorvariables, e.g. stator and rotor currents and fluxes. The encoderless control approach in [9]relies on signal injection. Another alternative is the use of an extended Kalman filter (EKF) [10].However, encoderless FCS-MPC is rarely presented in the literature [11].In this paper, a FCS-MPC strategy for DFIGs based on variable-speed WTSs is proposed.The proposed control system uses a MRAS observer for estimation of the DFIG rotor speedand position. Estimation and control behavior of the proposed encoderless control method areillustrated by simulation results for all operation conditions. Moreover, the behavior of theproposed MRAS observer is investigated under variations of the DFIG parameters.

2. Modeling of the WECS with DFIGThe block diagram of WECS with DFIG is shown in Fig. 1. The RSC and the GSC share acommon DC-link with capacitance Cdc [As/V] with DC-link voltage udc [V].

2.1. Wind turbine (WT)The output mechanical power of a WT is given by [10, 12]

pt(t) = cp(λ, β)12ρπr

2t v

3w(t) (1)

where ρ > 0 [kg/m3] is the air density, rt > 0 [m] is the radius of the wind turbine rotor (πr2t

is the turbine swept area), cp ≥ 0 [1] is the power coefficient, and vw(t) ≥ 0 [m/s] is the windspeed. The power coefficient cp is an indication for the “efficiency” of the WT. It is a nonlinearfunction of the tip speed ratio

λ = ωm(t)rtgrvw(t) ≥ 0 [1] (2)

and the pitch angle β ≥ 0 [] of the rotor blades. In reality, the power coefficient ranges from0.4 to 0.48 [10, 12].

2.2. Doubly-Fed induction generator (DFIG)The stator and rotor voltage equations of the DFIG can be written as follows [2]:

uabcs (t) = Rsiabcs (t) + d

dtψabcs (t) and uabcr (t) = Rri

abcr (t) + d

dtψabcr (t) (3)

where (considering linear flux linkage relations)

ψabcs (t) = Lsiabcs (t) + Lmi

abcr (t) and ψabcr (t) = Lri

abcr (t) + Lmi

abcs (t). (4)

Here uabcs = (uas , ubs , u

cs )> [V], uabcr = (uar , u

br , u

cr )> [V], iabcs = (ias , i

bs , i

cs )> [A], iabcr =

(iar , ibr , i

cr )> [A], ψabcs = (ψas , ψ

bs, ψ

cs)> [Vs], and ψabcr = (ψar , ψ

br , ψ

cr )> [Vs] are the stator and rotor

voltages, currents and flux linkages, respectively, all in the abc-reference frame (three-phasesystem). Ls, Lr, and Lm [Vs/A] are the stator, rotor and mutual inductances. Rs [Ω] and Rr [Ω]are stator and rotor winding resistances. The DFIG rotor rotates with mechanical angularfrequency ωm [rad/s]. Hence, for a machine with pole pair number np [1], the electrical angularfrequency of the rotor is given by ωr = npωm and the rotor reference frame is shifted by the

rotor angle φr(t) =∫ t

0 ωr(τ)dτ + φ0r , φ

0r ∈ R, with respect to the stator reference frame (φ0

r isthe initial rotor angle). Equation (3) can be written in the stationary/rotating reference frameas follows xk = TP (φ)−1xs = TP (φ)−1TCx

abc by using the Clarke and Park transformation (see,e.g., [12]), respectively, given by (neglecting the zero sequence)

xs= 23

[1 −1

2 −12

0√

32 −

√3

2

]︸ ︷︷ ︸

=:TC

xabc and xk=

[cos(φ) sin(φ)− sin(φ) cos(φ)

]︸ ︷︷ ︸

=:TP (φ)−1

xs (5)

where xk = (xd, xq)>, and xs = (xα, xβ)>. The rotor voltage equation (3) with respect to thestationary reference frame (i.e. usr = TP (φr)

−1TCuabcr ) can be expressed as

uss (t) = Rsiss (t) + d

dtψss (t), and usr (t) = Rri

sr (t) + d

dtψsr (t)− ωr(t)Jψsr (t), (6)

where J := TP (π/2) =

[0 −11 0

][12]. The stator voltage orientation (SVO) is realized 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. constantgrid frequency f0 > 0, it holds that ωs = 2πf0 is constant). Applying the (inverse) Park

transformation with TP (φs)−1 as in (5) with φs(t) =

∫ t0 ωs(τ)dτ + φ0

s, φ0s ∈ R, to the voltage

equations (6) yields the description in the rotating reference frame

uks (t) = Rsiks (t) + d

dtψks (t) +ωsJψ

ks (t), and ukr (t) = Rri

kr (t) + d

dtψkr (t) +ωsl(t)Jψ

kr (t), (7)

where ωsl(t) := ωs − ωr(t) is the slip angular frequency. Since, e.g., ψks = TP (φs)−1ψss =

TP (φs)−1TCψ

abcs , the flux linkages are given by

ψks (t) = Lsiks (t) + Lmi

kr (t) and ψkr (t) = Lri

kr (t) + Lmi

ks (t). (8)

For a stiff shaft and a step-up gear with ratio gr ≥ 1, the dynamics of the mechanical systemare given by

ddtωm(t) = 1

Θ

(me(t)− mt(t)

gr︸ ︷︷ ︸=:mm(t)

), ωm(0) = ω0

m ∈ R (9)

whereme(t) = 3

2npiss (t)>Jψss (t) = 3

2npLm(iqs (t)idr (t)− ids (t)iqr (t)

). (10)

is the electro-magnetic machine torque (moment), mt [Nm] is the turbine torque produced bythe wind (see Sec. 3) and mm = mt

gr[Nm] is the mechanical torque acting on the DFIG shaft.

Θ [kg/m2] is the rotor inertia and np [1] is the pole pair number.

2.3. Back-to-back converter and DC-LinkAs shown in Fig. 1, a balanced generator and grid are assumed in this paper. The output voltageof the RSC and GSC can be calculated as follows [12]:

uabcr (t) = 13udc(t)T

abcsabcr (t) and uabcf (t) = 13udc(t)T

abcsabcf (t) (11)

where sabcr = (sar , sbr, s

cr)> ∈ 0, 1 and sabcf = (saf , s

bf , s

cf )> ∈ 0, 1 are the switching state

vectors of the RSC and GSC, respectively, and T abc is the transformation matrix [12]

T abc =

2 −1 −1−1 2 −1−1 −1 2

(12)

describing the relation between switching state vector and output phase voltage vector of theconverter. Considering all the possible combinations of the switching state vector sabcr or sabcf ,eight switching states, and consequently, eight voltage vectors are obtained. Note that twodifferent zero voltage vectors are available, see Fig. 2. The DC-link dynamics are given by [12](neglecting resistive losses)

ddtudc(t) = 1

Cdc(Ig(t)− Im(t)), udc(0) = 0 ∈ R (13)

whereIm(t) = iabcs (t)>sabcr (t) and Ig(t) = iabcf (t)>sabcf (t). (14)

are the rotor and grid side DC-link currents (see Fig. 1).

100011

110010

001 101

000

111

s

u0

s

u7

s

u1

s

u2

s

u3

s

u4

s

u5

s

u6

Figure 2: Different switching combination for2-level converter.

s

si

s

su

s

sψ s

ri

rω

rφ

r

ri

r

ri

e

sL

ImL

11−

PT PI

I

sR

Figure 3: MRAS observer for estimating therotor speed and position of the DFIG.

2.4. Filter and gridFig. 1 shows a grid-connected voltage source converter, which is connected to the grid via an RL-filter with resistance Rf [Ω] and inductance Lf [Vs/A]. The grid is considered as ideal voltage

source with grid voltage uabco = (uao , ubo , u

co )> [V]. The currents iabcf = (iaf , i

bf , i

cf )> [A] flow from

the grid to the GSC. Invoking Kirchhoff’s voltage law at the AC side of the GSC [12] gives

uabco (t) = Rf iabcf (t) + Lf

ddti

abcf (t) + uabcf (t), iabcf (0) = 03. (15)

Here uabcf = (uaf , ubf , u

cf )> [V] is the output voltage of the GSC. The voltage equation (15) can

be expressed in the rotating reference frame (grid voltage orientation) as follows

uko (t) = Rf ikf (t) + Lf

ddti

kf (t) + ωsJLf i

kf (t) + ukf (t). (16)

3. Maximum power point tracking (MPPT)For wind speeds below the rated wind speed of the WT, maximum power tracking is therequired control objective. Consequently, the pitch angle is kept constant at β = 0 and theWT must operate at its optimal tip speed ratio λ? (a constant) where the power coefficient hasits maximum c?p := cp(λ

?, 0) = maxλ cp(λ, 0). Thus, the WT can extract the maximally available

power p?t := c?p12ρπr

2t v

3w [12]. Maximum power point tracking is realized by the nonlinear speed

controllerme(t) ≈ m?

e(t) = −k?pωm(t)2 with k?p :=ρπr5t2gr

c?p(λ?)3

(17)

which guarantees that the generator angular frequency ωm(t) is adjusted to the actual wind

speed vw(t) such that ωm(t)rtgrvw(t)

!= λ? holds. According to (17) the optimum torque m?

e(t) can be

calculated from the (estimated) shaft speed ωm(t) = ωr(t)/np.

4. Proposed FCS-MPC4.1. FCS-MPC for RSCThe stator (grid) voltage orientation (SVO) is achieved by aligning the d-axis of the synchronousreference frame with the stator voltage vector uss (t). The resultant stator dq-axis voltages areuds (t) = ‖uss (t)‖ and uqs (t) = 0 [2]. By substituting the value of ψkr (t) from (8) in (7), the rotorvoltage ukr (t) can be written as

ukr (t) = Rrikr (t) + Lr

ddti

kr (t) + Lm

ddti

ks (t) + ωsl(t)LrJi

kr (t) + ωsl(t)LmJi

ks (t) (18)

Invoking (8), the stator current iks (t) can be expressed as

iks (t) = 1Lsψks (t)− Lm

Lsikr (t) (19)

and substituting (19) in (18) gives

ukr (t) = Rrikr (t) + σLr

ddti

kr (t) + Lm

Ls

ddtψ

ks (t) + ωsl(t)σLrJi

kr (t) + ωsl(t)

LmLsJψks (t) (20)

where σ = 1− L2m

LsLr. Substituting d

dtψks (t) from (7) and ψks (t) from (8) in (20) gives

ukr (t) = Rri

kr (t) +σLr

ddti

kr (t) + (ωsl(t)Lr−ωs(t)

L2m

Ls)Jikr (t)− (Rs

Lm

Ls+ωr(t)LmJ)iks (t) + Lm

Lsuks (t). (21)

Solving (21) for ddti

kr (and writing out both components) yields

ddt i

dr (t) = 1

σLsLr[−RrLsidr (t) + (ωsl(t)LrLs − ωs(t)L2

m)iqr (t) +RsLmids (t)

−ωr(t)LmLsiqs (t) + Lsudr (t)− Lmuds (t)]

ddt i

qr (t) = 1

σLsLr[−RrLsiqr (t)− (ωsl(t)LrLs − ωs(t)L2

m)idr (t) +RsLmiqs (t)

+ωr(t)LmLsids (t) + Lsu

qr (t)− Lmuqs (t)].

(22)

The FCS-MPC approach uses a discrete-time model for the prediction of the currents at a futuresample period. For discretization the forward Euler method with sampling time Ts [s] is appliedto the time-continuous model (22). The discrete model of the DFIG can be written as

idr [k + 1] = idr [k] + TsσLsLr

[−RrLsidr [k] + (ωsl[k]LrLs − ωs[k]L2

m)iqr [k] +RsLmids [k]

−ωr[k]LmLsiqs [k] + Lsu

dr [k]− Lmuds [k]

]iqr [k + 1] = iqr [k] + Ts

σLsLr

[−RrLsiqr [k]− (ωsl[k]LrLs − ωs[k]L2

m)idr [k] +RsLmiqs [k]

+ωr[k]LmLsids [k] + Lsu

qr [k]− Lmuqs [k]

] (23)

In this paper, for the RSC, the chosen cost function is defined by

gRSC =∣∣idr,ref [k + 1]− idr [k + 1]

∣∣+∣∣iqr,ref [k + 1]− iqr [k + 1]

∣∣ (24)

where idr,ref [k + 1] and iqr,ref [k + 1] are the reference values of the d− & q-axis currents.

For the prediction algorithm, the cost function (24) (and, hence, (23)) is calculated for eachof the seven voltage vectors, producing seven different current predictions. The voltage vectorwhose current prediction is minimizing the cost function (24) is applied at the next samplingperiod. However, the future reference current ikr,ref [k + 1] value is unknown. Therefore, ithas to be predicted from present and previous values of the current reference using Lagrangeextrapolation as follows [5]:

ikr,ref [k + 1] = 3ikr,ref [k]− 3ikr,ref [k − 1] + ikr,ref [k − 2]. (25)

The value of the reference current idr,ref [k] is calculated from the optimum torque m?e[k] and the

value of iqr,ref [k] is calculated from the reference stator reactive power Qs,ref [k] as follows [2]

idr,ref [k] = 2ωs[k]Ls

3npLmuds [k]m?e[k] and iqr,ref [k] = 2Ls

3Lmuds [k]Qs,ref [k]− uds [k]

ωs[k]Lm. (26)

4.2. FCS-MPC for GSCAgain, applying the forward Euler method to (16), the discrete model of the grid side filter canbe written as follows

idf [k + 1] = (1− TsRf

Lf)idf [k] + ωeTsi

qf [k] + Ts

Lf(udo [k]− udf [k])

iqf [k + 1] = (1− TsRf

Lf)iqf [k]− ωeTsidf [k] + Ts

Lf(uqo [k]− uqf [k]).

(27)

For the GSC, the cost function is defined by

gGSC =∣∣idf,ref [k + 1]− idf [k + 1]

∣∣+∣∣iqf,ref [k + 1]− iqf [k + 1]

∣∣ (28)

where idf,ref [k + 1] and iqf,ref [k + 1] are the reference values of the d- & q-axis currents.

Again, (27) is calculated for each of the seven voltage vectors, yielding seven different currentpredictions. The voltage vector whose current prediction is minimizing the cost function (28) isapplied at the next sampling interval. The future reference current ikf,ref [k+1] value is calculatedalso using Lagrange extrapolation as explained before.The value of the d-axis reference current idf,ref [k] is obtained from an outer DC-link voltage

control loop. The measured DC-link voltage udc[k] is compared with a constant reference valueudc,ref and the error is processed by a PI controller producing the d-axis reference currentidf,ref [k], see Fig. 1.

5. MRAS observerThe MRAS observer is consist of two models [8]: a reference model and an adaptive model, seeFig. 3. In this paper, the reference model (see left part in Fig. 3) is fed by the measured statorcurrent iss (t) and the measured stator (grid) voltage uss (t). From the reference model (based

on (8)) the rotor current isr (t) is estimated via

isr (t) = 1Lm

(ψss (t)− Lsiss (t)

)where ψss (t) =

∫ t

0

(uss (τ)−Rsiss (τ)

)dτ +ψss (0). (29)

The adaptive model (see right part in Fig. 3) is fed by the estimated rotor current isr (t)and the measured rotor current irr (t) in the rotor reference frame. The objective of the

adaptive model is to estimate rotor position φr(t) and rotor speed ωr(t). To achieve thatthe estimated and the measured rotor current must be compared; to do so, the estimated rotorcurrent isr (t) (in the stator reference frame) must be expressed in the rotor reference frame,

i.e. irr (t) = TP (φr(t))−1isr (t). The “error” between estimated irr (t) and measured rotor current

irr (t) is defined as

e(t) := irr (t)Jirr (t) = ‖irr (t)‖ ‖irr (t)‖ sin(∠(irr (t), irr (t))

).

The PI controller forces this error to zero by adjusting ωr(t). Its output is the estimated speed

ωr(t) which is integrated to obtain the estimated rotor angle φr(t). For more details see [8].

6. Simulation Results and DiscussionA simulation model of a 50kW WECS with DFIG is implemented in Matlab/Simulink. Thesystem parameters are listed in Table 1. The implementation is shown in Fig. 1. The simulationresults are shown in Figs. 4–8. The estimation performances of MRAS observer are comparedwith the actual values for different wind speeds and parameter uncertainties in Rs, Rr and Lm.

Table 1: Parameters of the WECS with DFIG.

Name Nom. Value Name Nom. Value

WT rated power pt 50kW Rotor inductance Lr 82.12mHWT radius rt 4.5m Mutual inductance Lm 77.2mH

Rated wind speed vwrated 14ms DFIG moment of inertia Θ 0.1 kg

m2

Optimal tip speed ratio λ? 8.036 DC-link capacitor Cdc 3mFDFIG rated power pnom 50 kW DC-link voltage udc 700VDFIG line-line voltage urms

s 400V Grid line-line voltage uo 400VNumber of pair poles np 2 Grid normal frequency fe 50HzStator resistance Rs 0.2448 Ω Filter resistance Rf 0.16ΩRotor resistance Rr 0.4847 Ω Filter inductance Lf 12mHStator inductance Ls 80.76mH Sampling time Ts 40µs

ω r[rad

/s]

150

300

450ωr ωr

φr[rad

/s]

0

7φr φr

id/q

r[A

]

0

50

100

idr idr,ref iqr iqr,ref

λ

5

10λ∗ λ

Time [s]0 0.25 0.5 0.75 1 1.25 1.5

c p

0.3

0.6 c∗p cp

12 [m/s]10 [m/s]

14 [m/s]11 [m/s]

8 [m/s] 8 [m/s]

Figure 4: Estimation and control performance of the proposed encoderless FCS-MPC (from

top): estimated and actual rotor speed (ωr, ωr), estimated and actual rotor position (φr, φr),actual and reference d− & q− axis current of the rotor (idr , i

dr,ref , iqr, i

qr,ref ), Optimal and actual

tip speed ratio (λ∗, λ), optimal and actual power coefficient (c∗p, cp).

The estimation performance of the proposed MRAS observer under variable wind speeds isshown in Fig. 4. This wind speed range covers almost the complete speed range of the DFIG(i.e. ±30% around the synchronous speed). Fig. 4 illustrates the tracking ability of the MRASobserver of the rotor speed and position at low and high speeds, and close to synchronous speed.An acceptably high estimation accuracy is achieved as the estimation error is very small. TheFCS-MPC performance of the rotor side converter (RSC) under variable wind speed is illustratedin Fig. 4. It is clear that the RSC control system ensures tracking of the maximum power from

the wind turbine. The actual rotor currents id/qr of the DFIG is following the reference value

ωr[rad

/s]

300

350

400ωr ωr

φr[rad

]

0

7φr φr

Time [s]0.2 0.3 0.4 0.5 0.6

id r[A

]

68

70

72idr idr,ref

Rs = 1.25Rs & Rr = 1.25Rr

12 [m/s]

Figure 5: Estimation and control perfor-mance of the proposed encoderless FCS-MPCat 25% step change in Rs and Rr (from top):estimated and actual rotor speed (ωr, ωr), es-

timated and actual rotor position (φr, φr), ac-tual and reference d− axis current of the rotor(idr , i

dr,ref ).

ωr[rad

/s]

300

350

400ωr ωr

φr[rad

]

0

7φr φr

Time [s]0.2 0.3 0.4 0.5 0.6

id r[A

]

68

70

72idr idr,ref

Lm = 1.1Lm vw = 12 [m/s]

Figure 6: Estimation and control perfor-mance of the proposed encoderless FCS-MPCat 10% step change in Lm (from top): esti-mated and actual rotor speed (ωr, ωr), esti-

mated and actual rotor position (φr, φr), ac-tual and reference d− axis current of the rotor(idr , i

dr,ref ).

id/qr,ref . Thus, the MPPT capability is ensured. The tip speed ratio λ is following the optimal

value λ∗. Subsequently, the power coefficient cp(λ) is kept close to its maximal (optimal) valuec?p ≈ 0.48.In order to check the robustness of MRAS observer and FCS-MPC under (unknown) parameter

variations of the DFIG, the value of the stator resistance Rs and rotor resistance Rr is increasedby 25% (e.g. due to warming or aging) at t = 0.3s. For this scenario, Fig. 5 shows the estimationand control performances of the proposed MRAS observer and FCS-MPC under wind speedvw = 12m

s . It is clear that MRAS observer and FCS-MPC are robust against parameteruncertainties in Rs and Rr. The MRAS observer is still capable of estimating the rotor speedand position with good accuracy and FCS-MPC is still able to track the reference current, seeFig. 5. Moreover, the robustness with respect to changes (due to magnetic saturation) in themutual inductance Lm is investigated. Therefore, Lm is increased by 10% at t = 0.3s. Fig. 6shows the simulation results of the proposed MRAS observer and FCS-MPC for this scenariounder wind speed vw = 12m

s . Again, MRAS observer and FCS-MPC show a good estimationand control performance and are robust against parameter uncertainties.

Figure 7 illustrates the FCS-MPC performance of the GSC under the same wind speed asshown in Fig. 4. Again, the GSC control system guarantees tracking of the constant referenceDC-link voltage and of the reference d & q currents as shown in Fig. 7. Finally, the robustness ofthe proposed FCS-MPC with respect to changes in the filter resistance Rf and inductance Lf isinvestigated. Therefore, Rf and Lf are increased by 25% at t = 0.3s. Fig. 8 shows the simulationresults of the proposed FCS-MPC for this scenario under wind speed vw = 10m

s . Again, theproposed FCS-MPC for GSC shows a good control performance and is robust against parametervariations of the filter.

7. ConclusionThis paper proposed an encoderless FCS-MPC method for variable-speed WECSs with DFIG.A MRAS observer for estimation of the DFIG rotor speed and position is utilized. The resultshave shown that the MRAS observer tracks rotor speed and position with high accuracy evenunder variations of the DFIG parameters. Also, the results show that the proposed FCS-MPC

udc

[V]

690

700

710udc udc,ref

Time [s]0 0.25 0.5 0.75 1 1.25 1.5

id/q

f[A

]

-20

0

20idf idf,ref iqf iqf,ref

Figure 7: Performance of the proposed FCS-MPC for GSC under variable wind speed(from top): actual and reference DC-linkvoltage (udc, udc,ref ), actual and reference d−& q− axis current of the filter (i

d/qf , i

d/qf,ref ).

udc[V

]

698

700

702udc udc,ref

Time [s]0.2 0.3 0.4 0.5 0.6

id/q

f[A

]

0

6

12

idf idf,ref iqf iqf,ref

Rf = 1.25Rf

Lf = 1.25Lf

vw = 10 [m/s]

Figure 8: Performance of the proposed FCS-MPC for GSC at 25% step change in Rf andLf (from top): actual and reference DC-linkvoltage (udc, udc,ref ), actual and reference d−& q− axis current of the filter (i

d/qf , i

d/qf,ref ).

for RSC tracks the reference currents for all the operation conditions and is robust againstparameter variation of the DFIG. Thus, tracking of the maximum power from the wind turbineis guaranteed. Moreover, the proposed FCS-MPC for the GSC tracks the reference currentsfor all the operation conditions and is robust against parameter variation of the output filter.Therefore, a constant DC-link voltage is ensured.

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