Predictive Control Strategy for DFIG WindTurbines with Maximum Power Point Tracking

Using Multilevel ConvertersJosé Sayritupac∗, Eduardo Albánez∗, Johnny Rengifo∗, José M. Aller∗†‡ and José Restrepo∗†‡

∗Universidad Simón Bolívar, Caracas–VenezuelaEmail: {ealbanez,jwrengifo}@usb.ve†PROMETEO-SENESCYT, Ecuador

‡Universidad Politécnica Salesiana, Cuenca–EcuadorEmail: jaller@ups.edu.ec

Abstract—This paper proposes a control scheme for a windturbine using a DFIG electromechanical converter, implementedthrough an NPC three-level back to back converter, that keeps aunitary power factor injection to the grid and maximizes theenergy harvested from wind, using a maximum power pointtracking algorithm (MPPT). A predictive direct power control(DPC) strategy drives the grid-side converter, to maintain theDC bus reference voltage. Whereas, a predictive direct torquecontrol (DTC) strategy drives the machine-rotor-side converter,to control the power extraction, the power factor and balancing ofthe DC bus capacitors. The developed model allows to study thewind energy conversion system (WECS) for energy harvesting,rotor dynamics and power quality analysis. Simulation resultsendorse the effectiveness of the advanced control techniques forthe whole wind spectrum, including the pitch angle control. Asensitivity analysis shows that the predictive DTC control strategyis robust with an uncertainty up to 20% of the induction machineparameters.

Index Terms—WECS, DFIG, DTC, DPC, Predictive control

NOMENCLATURE

θe Electrical angle among rotor & stator α-axisθp,0 Blade pitch angleρ Air densityρT , ρλ, ρv DTC cost function weightsρp, ρq DPC cost function weightsϕ Angle of relative windΛ Tip speed ratioΛh Tip speed ratio at the hubΛr Tip speed ratio at a distance r from the hubωm Generator shaft angular speedωt Turbine rotor angular speedωe Electric system synchronous speeda, a′ Axial and angular induction factorsagb Gear box transformation ratioA Area swept by the turbine rotorC Capacitance of each DC bus capacitorsCD Airfoil drag coefficientCL Airfoil lift coefficientCp Turbine power coefficient

ira, irb, irc Phase rotor currentsiga, igb, igc Phase grid side currentsJ Moment of inertiaLs, Lr, Lsr Stator, rotor and mutual inductancesnp Number of pole pairsp Derivative operator d/dtR Turbine rotor radiusRs, Rr Stator and rotor resistancesRg, Lg Transformer resistance and inductanceSa, Sb, Sc Bridge leg statesSw Switching state vectorTe, Tm Electrical and mechanical torqueTs Sampling time~vs, ~is Stator voltage and current space vectors

I. INTRODUCTION

There is a great interest in the scientific community onresearching wind energy conversion systems (WECS), due toits sustained growing inclusion in electric power systems in thelast decades [1]. Doubly fed induction generators (DFIG) arethe most installed technology used to exploit the wind energyresources [2], mainly because the rated power of the electronicconverter is a third of the induction generator capacity.

Several strategies based on direct torque control (DTC) [3]have been proposed in recent years, including predictive con-trol and multilevel converters [4]–[6]. Some DFIG applicationsare reported in [1], [7]–[9]. Maximum power point tracking(MPPT) algorithms have been presented [10], [11] in order tooptimize the energy extraction from the wind.

This paper presents a WECS based on a DFIG topologycontrolled by a back to back power converter, build upon twothree-level neutral point clamped (NPC) bridges linked by aDC bus. The machine-side converter is controlled with the pro-posed DTC technique, and the grid-side converter use a directpower control (DPC) strategy proposed by [12]. Both controlsminimize cost functions based on the control variables errorsat every control cycle to select the switching states of eachconverter. The DPC references are set to keep constant the DCbus voltage, whereas the DTC references are set to maximize

the power extraction from the wind, control the stator powerfactor and guarantee the balance of DC bus capacitors. Thetorque reference is established by a maximum power pointtracking (MPPT) algorithm, that takes into account the bladepitch angle. The WECS model developed allows to study therotor dynamics, power quality analysis, and control strategiesfor different wind turbine models.

II. WIND ENERGY CONVERSION SYSTEM

The wind energy conversion system studied on this paperis shown in Fig. 1.

A. Wind speed

A wind speed measurement that includes wind gust and awind ramp was obtained from [13], and provides a globalscenario suitable to analyze the energy conversion systemdynamics and the performance of the control strategies.

B. Wind turbine

The torque developed by the turbine rotor is [14], [15],

Tm =Pmωt

=ρAV 3

2ωtCp (Λ, θp,0) (1)

where the power coefficient in accordance with the BladeElement Momentum Theory is expressed as [16],

Cp =8

Λ

ˆ Λ

Λh

Λ3ra′ (1− a)

(1− CD

CLcotϕ

)dΛr (2)

The power coefficient surface, as a function of the the tipspeed ratio and the pitch angle, for the data shown in appendixA, is depicted in Fig. 2.

05

1015

2025

0

5

10

0

0.1

0.2

0.3

0.4

θp,0Λ

Cp

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Figure 2: Cp vs λ curves.

C. Doubly fed induction generator

The fixed stator reference frame αβ space vector model ofthe induction machine is [17],

~vs = Rs~is + Lsp~is + Lsrp~isr (3)

~vsr = Rr~isr + Lsrp~is + Lrp~i

sr − jnpωm

(Lsr~is + Lr~i

sr

)(4)

J pωm = Te − Tm = npLsr

(~isr ×~is

)− Tm (5)

Figure 3: Back to back NPC three level converter

pθe = npωm (6)

The space-vector transformation used in this paper is,

~x =

√2

3

(xa + xbe

j 2π3 + xce

j 4π3

)= xα + jxβ (7)

and the transformation to refer to the rotor variables to theframe of reference fixed in the stator is,

~xsr = ~xrejθe (8)

D. Multilevel back to back converter

The back to back converter is shown in Fig. 3. Eachinverter generates 33 = 27 different valid voltage space vectorsin the αβ reference frame. The switching variable Sw =[Sa, Sb, Sc]

t denotes the converter switching position for eachleg, these integer variables are -1, 0 or 1 corresponding to thephase voltages −Vdc2 , 0, Vdc2 .

The NPC topology requires to keep the voltage on eachcapacitor equal, but these voltages are affected by the connec-tivity states when Sa, Sb or Sc are zero [18]. The strategy forbalance operation consists in controlling the neutral voltage vnaround zero volts. The dynamic response of vn is describedby [19],

pvn =1

C

∑m={a,b,c}

|Sgm| igm − |Srm| irm (9)

III. CONTROL STRATEGIES

A. Predictive direct power control

The predictive DPC presented in [12], [20] proposes analgorithm for computing the optimum voltage space vectorthat satisfies the active and reactive power references [21].This formulation is based on computing the optimum trajec-tory using Lagrange operators. The system voltage may beexpressed as a function of the grid-side converter voltage as,

~vsys = Rg~ig + Lgp~ig + ~vg (10)

Using a first order approximation, an expression for theincrement of the apparent power for each control cycle k is,

∆~sk = ∆pk + j∆qk (11)

∆~sk = ∆~s0,k −TsLg

(~vsys,k+1 ~v

∗g,k

)(12)

m g

Figure 1: WECS topology and system control block diagrams.

∆~s0,k = ∆~vsys,k~i∗g + ~vsys,k+1

TsLg

(~vsys,k −Rg~ig,k

)∗(13)

where the electric system voltage ~vsys,k is sinusoidal, andthe estimated voltage for the next control cycle ~vsys,k+1 isobtained by rotating ~vsys,k

~vsys,k+1 = ~vsys,kejωeTs (14)

∆~vsys,k = ~vsys,k(ejωeTs − 1

)Given an active and reactive power references, the errors

are defined as,

εk = εp,k+jεq,k = (pref,k+1 − pk)+j (qref,k+1 − qk) (15)

The cost function Ψ1 follows as,

Ψ1 = ρp (εp,k −∆pk)2

+ ρq (εq,k −∆qk)2 (16)

B. Predictive direct torque control

The predictive strategy is based on applying the optimalrotor voltage space vector ~vr that best accomplishes thefollowing targets:

1) Maximize the energy harvested from wind, through thetorque reference obtained from the MPPT algorithm.

2) Control the stator power factor, using a closed-loop PIcontroller to set the flux reference.

3) Balance each capacitor voltage of the DC bus, bycomputing the ~vr using redundant switching states ofthe back to back converter when possible.

The selection of ~vr is performed by minimizing a cost functionthat takes into account the electric torque, the rotor fluxmagnitude and the neutral point voltage of the DC bus, atthe next control cycle.

The discrete-time form of the estimated torque is,

Te,k+1 = Te,k + ∆Te,k (17)

Differentiating the electric torque in (5) leads to,

∆Te = TsnpLsr

(p~ir,k ×~i rs,k +~ir,k × p~i rs,k

)(18)

Replacing p~irs and p~ir from (3) and (4) respectively, in (18),the following expression for the change of the electric torqueis obtained,

∆Te,~vs = −TsnpLsr

LsLr − L2sr

[~v rs,k ×

(Lsr~i

rs,k + Lr~ir,k

)+

(LsRr + LrRs)(~ir,k ×~i rs,k

)+

ωmLsr

(Ls|~i rs,k|2 + Lr|~ir,k|2

)+

ωm(LsLr + L2

sr

)<e{~ir,k~i

rs,k

}](19)

∆Te,~vr = TsnpLsr

LsLr − L2sr

[~vr,k ×

(Ls~i

rs,k + Lsr~ir,k

)](20)

with,

∆Te = ∆Te,~vs + ∆Te,~vr (21)

The discrete-time form of the estimated rotor flux amplitudeis, ∣∣∣~λr,k+1

∣∣∣ =∣∣∣~λr,k∣∣∣+ ∆

∣∣∣~λr,k∣∣∣ (22)

the rotor voltage equation is,

p~λr = ~vr −Rr~ir (23)

p∣∣∣~λr,k∣∣∣ =

<e(~λr,k

(~vr,k −Rr~ir,k

))∣∣∣~λr,k∣∣∣

then,

∆λr,k = Ts<e(~λr,k

(~vr,k −Rr~ir,k

))∣∣∣~λr,k∣∣∣ (24)

The change of the neutral voltage of the DC bus from (9)is,

∆vn,k =TsC

∑m={a,b,c}

|Sgm,k| igm,k − |Srm,k| irm,k (25)

The electric torque, rotor flux and neutral point voltageerrors are defined as,

εT = Tref−Te,k ; ελ = λref−∣∣∣~λr,k∣∣∣ ; εv = vref−vn,k (26)

The cost function Ψ2 used in this work is,

Ψ2 = ρT (εT,k −∆Te,k)2

+ ρλ (ελ,k −∆λr,k)2...

+ ρv (εv,k −∆vn,k)2 (27)

C. Maximum power point tracking

The wind turbine has an optimum operating point(Cp−opt, λopt) for a given pitch angle θp,0 and wind speed V ,as depicted in Fig. 4, where the the power Pm−opt capturedfrom the wind is maximum,

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

ωt (pu)

Pt(pu)

5 m/s7 m/s9 m/s11 m/s13 m/sMPPT

Figure 4: Wind turbine power extraction vs angular mechani-cal speed, for a fixed pitch angle.

Pm−opt =1

2ρAV 3Cp−opt (28)

ωt−opt =V

Rλopt (29)

then the torque reference for the DTC is,

Tref =Pm−optagbωt−opt

(30)

D. Pitch control

For wind speeds exceeding the turbine rated value, theblades pitch angle can be controlled to maintain the generatorpower at its rated capacity. The pitch angle model can beexpressed as [22], [23],

pθp,0 =θref − θp,0

τθ(31)

where τθ represents the mechanical servomotor time delay.The angle θref is set to zero when the wind speed is belowits rated value, otherwise the reference is obtained from a PIcontroller to keep the generator operating at rated power.

IV. SIMULATION RESULTS

A detailed analysis of the WECS to evaluate the per-formance of the proposed control strategy is presented, thesimulation data is given in Appendix A. All per units resultspresented in this section are based on the generator ratedvalues.

A. Case of study

The system dynamics are studied within a 50 s window ofincident wind speed. The power delivered to the electric gridvaries accordingly with the wind fluctuations as shown in Fig.5a. The turbine power is effectively limited to its rated valuewhen the wind gust (12 s < t < 17 s) and the wind ramp(t > 39 s) take place, as a result of the active blade pitchangle control. When the wind speed is under its rated value thepitch angle is set to zero, and the DTC shows a fast responseto the MPPT reference, to keep the optimum turbine powercoefficient of 0.42. As the wind speed exceeds its rated valueof 12m/s, the pitch angle grows reducing the Cp accordinglyto Fig. 5b. When the power coefficient differs from zero, thetrajectory followed by it, is the crest of the surface depictedin Fig. 2.

The DPC holds the DC bus voltage at its reference valueof 3 pu, whereas the DTC maintains the capacitors voltagebalanced at 1.5 pu each, as shown in Fig. 6. The rotor fluxlinkage was set to keep a lagging power factor of 0.95 at theelectric generator terminals. Table I synthesizes these results.The variation of the turbine mechanical speed is presentedin Fig. 7, which varies slowly due to its large inertial timeconstant. Lastly, the phase current detail and its frequencyspectrum is depicted Fig. 8, in which the signal harmoniccontent is kept 40 decades below the fundamental frequencyuntil the 49th harmonic.

Table I: Power factor results

Mean Median Standard deviation

0.9455 0.9568 0.0454

0 10 20 30 40 500

0.25

0.5

0.75

1

1.25Psys(pu)

0 10 20 30 40 500

4

8

12

16

20

time (s)

V(m

/s)

incident windrated speed

(a) Active power delivered and incident wind speed

0 10 20 30 40 500

0.25

0.5

0.75

1

1.25

Pt(pu)

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

time (s)C

p

(b) Turbine power and power coefficient

Figure 5: WECS performance during the study case

0 10 20 30 40 500

1

vC1

0 10 20 30 40 500

1

vC2

0 10 20 30 40 50−0.1

0

0.1

time (s)

vn

Figure 6: Capacitors and neutral DC voltages

B. Sensitivity analysis

The proposed DTC strategy requires knowledge of themachine parameters to predict the optimum operating pointat the next control cycle. Therefore, is of interest to assessthe robustness of the control under parameter detuning condi-tions. To this purpose, the algorithm was tested varying eachparameter over a range of ±20%. The variable σT is define in(32) to quantify the torque ripple. The results of the sensitivityanalysis are presented in Table II, where the control schemeexhibits a primary dependency with Lsr. For the worst casescenario considered, the torque ripple is kept below 3.5%.

σT = 1001

Ni

Ni∑k=1

1

Tref,k

√(Tref,k − Te,k)

2 (32)

V. CONCLUSION

A DFIG-based wind energy conversion system has beensimulated, using a detailed model of the turbine operating

Table II: Torque ripple under parameters uncertainty

Rs Rr Lσs Lσr Lsr

+20% 1.5476 1.5476 1.7532 1.7699 3.54310% 1.5474−20% 1.5483 1.5434 1.5229 1.5287 3.2035

0 10 20 30 40 500.95

1

1.05

1.1

1.15

time (s)

ωm(pu)

Figure 7: Angular mechanical speed

40 40.02 40.04 40.06 40.08 40.1−1.5

−1

−0.5

0

0.5

1

1.5

time (s)

i s(pu)

(a) Detail of stator phase current

0 500 1000 1500 2000 2500 3000

−80

−60

−40

−20

0

f (Hz)

|Is(jω)|(dB)

(b) Frequency spectrum of the stator current

Figure 8: Analysis of the stator phase current

under demanding wind speed conditions. The proposed DTCstrategy is applied using a three-level NPC converter to controlthe electric generator has proven to be effective, with a fasttracking MPPT reference that maximizes the harvesting of thewind energy; greatly reducing the reactive power requirementsby keeping the power factor close to unity; and balancing thevoltage of the back to back converter DC bus capacitors.

The proposed control technique was tested for machine-parameters detuning, revealing a main dependence with themutual inductance. In spite of this, the sensitivity analysisshowed the robustness of the DTC under this operating con-

dition.

ACKNOWLEDGMENT

The authors gratefully acknowledge the support of PrometeoProject, Secretaría de Educación Superior, Ciencia, Tecnologíae Innovación, and Universidad Politécnica Salesiana, bothin República del Ecuador, as well as Universidad SimónBolívar and FONACIT Research Project #2011000970 bothin Venezuela.

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APPENDIX AAERODYNAMIC DATA

Chord

α

φ θp

θθp

θ : Section twist angleθ: Section pitch angle

p

Plane of blade rotation

V(1-a)Urel

U : Relative wind velocityrel

Figure 9: Airfoil aerodynamics..

Table III: Simulation data

Specification Variable Value Unit

Turbine rated power - 1 MWRated wind speed - 12 m/sInertia time constant Ht 4 sTurbine rotor radius R 26 mGear box transformation ratio agb 65 -Servomotor time delay τθ 0.25 sNumber of blades 3Blade model LM-26

Generator rated power 1 MWStator resistance Re 0.0108 puRotor resistance Rr 0.0121 puStator inductance Ls 3.4640 puRotor inductance Lr 3.4720 puMutual inductance Ler 3.3620 puStator rated voltage - 575 VRotor rated voltage - 1975 VCapacitance of each capacitor C 2 puReference DC voltage - 3 puConverter switching frequency - 10 kHz

Transformer resistance Rg 0.01 puTransformer leakage inductace Lg 0.2 pu