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Sliding Mod Control of Torque Ripple in Wind Energy Conversion Systems withe Slip Power Recovery

e m b De Battista

d fluctuations, wind shear and tower

i. NTRODUCTION

nal speed is such that the ratioand wind speed, called tip speed

any time. Therefore,

cted directly to grid by stator andby rotor (Fig. 1). The features

of conversion over

static converter.

generators or synchronous ones are

0-7803-4503-7/98~$10.00 1998IEEE 65 1

Ricardo J u l i b MantzNational University of La Plata- CICpBA

CC 91, (1900) La Plata, Argentinamantz@,venus. isica.unb.edu .ar

Wind turbine generato rs are characterized by a lowfrequency mode of oscillation, which is a consequence ofhigh turbine inertia and low effective shaft stiffnessbetween turbine and generator[4,5]. This mode can beexcited by different disturbances:- Random wind fluctuations [6].

- Tower shadow : Wh en a blade passes in fro nt of the tower,turbulence effects reduce the aerodyn amic orque[7,8].- Wind she ar: The wind through the turbine is non-constantand depends on the height. T hat is, the power captured byeach blade depends on its angular position [5].

These disturbances produce a periodical fluctuation inthe aerodynamic torque. Its frequency is given by theproduct of the number of blades and the rotational speed.When the natural frequency falls into the range of winddisturbance frequencies, large torque oscillations may betransmitted through the d rive train, leading in electricpower fluctuations. Torque oscillations have harmfuleffects on the fatigu e life of drive train components[7]. Inaddition, when the WECS is connected to a weak grid,fluctuation of the generated electric power may cause

flicker [8].External damping at the natural frequency is a way of

reducing torque oscillations [6,9]. Based on this idea, asliding mode control of the WECS with double-outputinduction generator is proposed. The f i gangle of theconverter is controlled to track the optimum speed whiledamping torque oscillations. This control strategy providesrobustness to unc ertain ties in electrical parameters of thegenerator and grid voltage disturban ces.

The advantages of the proposed control strategy, i.e.torque ripple attenuation and good efficiency ofconversio n, are verified by digital simulation.

11. MOD EL OF THE WIND ENERGY CONVERSIONSYSTEM

The powe r captured from the w ind is given by

wherep is the air density,R is the blades leng th andf& isthe turbine angular spee d. Given a wind speed, this poweris maximum at the o ptimum tip speed ratio,Iopt(Fig. 2) :

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with

Fig. 1. Wind enera conversion systemwith double-output inductiongenerator.

ct

3.075

0 . 0 6 0

0.045

0.030

0.015

0

Fig.2. Power (Cp)and torque (G) coefficients as fmction sof the tip speedratio x.

From (l) , the following expression of the aerodynamictorque can be obtained:

(3)

Usually, the prime objectiveof control is maximizingthe efficiency of conversion, i.e. forcing the tip speed ratiod to equalize its optimum valuejlopr. This can be achievedadjusting the firing of the converter.

Fig. 3 shows the investigated plant. Wind shear andtower shadow effects are modeled as fluctuations of thewind speedw. he aerodynamic torque model is describedby (3). The generator torqueTG s function of the generatorspeed aG nd the fiing angle of the convertera. Itsexpression can be obtained from the equivalent circuitofthe generator shown inFig. 4 [3]:

where

and

Rs , RR and RF are the resistance of stator, rotor and dc-link, respectively.Lds and L d R are the leakage inductan ce ofstator and rotor windings.0 s and COS are the mechanicaland electrical synchronous speed, ands is the generatorslip. Vs is the stator voltage. Theturn atio of the generatorand transformer aren l nd 122, respectively.

A third order physical model of the WECS is considered[9], which is shown in Fig. 5. The resonance lies in themost flexible part of the rotational system. Therefore,JTand JG do not correspond to the actual turbine andgenerator inertia but refer to the bodies of the modelcoupled by the effective flexible shaft. The effective shaftcompliance and damping coefficients areKs and Bsrespectively.Tis the shaft torque.

The model of the WECS can be described by thefollowing differential equations:

For grid-linked squirrel cage induction generators, theconnection to the synchronous speed is very stiff.

l - d y p y + 1!2G

Fig. 3.Blockdiagram of the WECS.

Fig. 4. Equivalent circuit of the generator.

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TG---+

Fig. 5.Phys cal modelof turbine and generator.iand the shaft compliance

Similar problems of resonance may occur inWECSwithgenerators connecte to grid via a static converter. In fact,a control action thatImposes0, =a, ! (7)at every time, wi the reference speedOR qual to theoptimum one for m ximizing the efficiency of conversion,causes a lightly da ped resonance at frequencywR. On theother hand, a contro action that forces the generator torqueto be equalto its reienceT, = TR ~

also causes resonanle but at a frequencyWAR:

(9)

I

that reduces the peak ofd by means of generator speed

(10)

of stator and rotor

The discontinu

robust to uncertaintiesin the mentioned electrical variablesand parameters.

The system state( 5 )is expanded with the state variable

z = c o s a . (11)

Its derivative is the input signalU :

A. Switching surface

The input signal takes the valueU + or U * epending onthe sign of a sw itching functions of the system statex

u + ( x ) if s(x)20

u - ( x ) if s ( x ) < O, u + ( X ) f U - ( X ) ' (13)

where

and

The reference torque is given by

where Top,s the aerodynamic torque at,Iopt and wind speedw. This reference forces the system to follow wind speedvariations.

B. Sliding mode dynamics

As a result of the control policy, and if the existenceconditions are satisfied, a sliding regime e xists on theswitching surface[10,111

s(x) = 0. (17)

That is, the state trajectory is constrained to the surface.Thus, the closed-loop system dynamics is order-reduced,and can be written as

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The nonlinear expression of the generator torque doesnot appear in the resultant dynamics, and then, thisdynamics is completely robust to uncertainties in gridvoltage, frequency, and generator and dc-link resistancesand inductances.

C .Existence conditions

The equivalent control ueq(x) is defiied as thecontinuous control action that forces the system state toslide on the switchin g surface. Its expression is obtainedfrom the invariance condition[101

s ( x ) = 0J (19)

and results

In order to establish a sliding regime on the switchingsurface, U + and U - must satisfy the neces sary and sufficientcondition:

A =

0

B -B GKs-- -Bs* +- -K,+-S__

- I2 -JT JT

Bs'BTT (j, jG) JG0 __

3JG JG

r -T

B w = [ $ 0]ij

AT, = T, - TT o- BG Do)

and

AT,,, = 3 . BT " P A . Aw .

The poles of the system are the roots of thecharacteristicpolynomialP(s),

U+ (x, t ) < uq (x ) < U - (x, t ) (21)

whereD. urbine torque linearizatio n

If the aerodynamic torque model is linearized at meanwind speedwoand optimum tip speed ratioA it can bewritten as

where

m

TTO is the aerodyna mic torque atAopl nd mean wind speedwo,and no s the optimum speed atwo. rom (2) and(3), itfollows thatBTis proportional to the mean wind speedwo.The resultant dynamic model is linear:

and

JT + J G +z.1 =-Be BT K S

The intrinsic turbine speed feedbackBT is weak [8], andits influenceon the pole placementis negligible over theentire range of operation.

Fig. 6 shows the wind disturbance rejection responsesfor different values of the feedback gainBG. Lightlydamped resonance occur whenBG=O and BG+oo. There isa set of values ofBG, from (24), that provides adequatedamping of the torsional mode. Torque ripple due to wind

shear and tower shadow is therefore almost eliminated.With regards to wind speed variations, the better theoptimum speed tracking, the larger the torque fluctuations.Fig. 7 shows the refe rence tracking responses for differentvalues of the feedback gainBG.

Hence, a tradeoff exists in the design ofBG betweentorque ripple elimination and tracking of the optimumspeed.

0-7803-4503-7/98/$10.00 1998IEEE 654

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*

-g o Lv)

-20

I . . .

10-9

Fig. 6 .Frequency respo ses. (a) From ATw to AT. (b) From ATw o ATG.f

. . . . , . . . . . I . . ilo-' IO 0

Frequew (Hz)

101. ,

1 lV.EXAMPLE

(Fig. 8a, with dotted line) arque in case A. This torquecomponents and may cause

a also shows (with solid line)

B) that the system operates insystem state is constrained to a

aveforms of the generator torque, ase switchingsurface.

~

/$10.00 1998IEEE 65 5

5 10 15 20 25 30Time (s)

E 2 0 '2.

3w, . I , , , , , , , ,

2 2.1 2.2 2.3 2.4 2.5 2.8 2.7 2.8 2.9 31801

" " "4 ' I

T h e 9)

Fig. 9. Generator torque with control of case A (dotted line), and caseB(solid line).

the wind disturbances due to wind shear and tower shadowcoincides with the natural frequencywR f the system.

The robustness of the proposed control strategy tochanges in grid voltage, frequency and various parametersof the generator has been verified by simulation. Thewaveformsof speed and torque obtained under the samewind conditions, but changing the mentioned variables andparameters, d o not differ from those shown in Figs. 8 and9. Obviously, this robustness property is verified whenevercondition(21) holds.

V. CONCLUSIONS

Converter-fed wind turbine generators canbe controlledto rotate at the optimum speedto maximize the conversionefficiency. The serious problems of resonance that mayoccur due to the stiff connection to the synchronous speedcan be solved by partial state feedback. The control lawproposed in this paper establishes a sliding regimeon aswitching surface in the state space. It provides damping atthe torque resonance frequency and allows the system totrack wind speed fluctuations. The sliding mode dynamics

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is completely robust to grid voltage disturbances andgenerator parameters uncertainties.

REFERENCES

[11 R.D. Richardson, and G.M . McNerney, Wind EnergySystems,IEEE Proceed ings, vol. 81, no. 3, Mar. 93,

[2] M. Ermis, H.D. Ertan, M. Demirekler, B.M. S aribatir,Y . Uctug, M.E. Sezer and I. Cardici, Variousinduction generator schemes for wind-electricitygeneration,Electric Power Systems Research,vol.

[3] P.F. Puleston, Control of wind energy conversionsystem using double-output induction generator, PhDThesis, Univ. La Plata, Argentina, Nov. 1997 (InSpanish).

[4] E.N. Hinrichsen, and P.J. Nolan, Dynamics andStability of Wind Turbine Generators,IEEE Trans.on Power Apparatus and Systems,vol. PAS-101, no.

8, Aug. 1982, pp. 2640-2648.

pp. 378-389.

23, 1992, pp. 71-83.

0-~803-4503-7/98/$10.00 1998IEEE

[5] 0. Wasynczuk, D.T. Man, and J.P. Sullivan, DynamicBehavior of a Class of Wind Turbine GeneratorsDuring Random Wind Fluctuations,IEEE Trans. onPower Apparatus and Systems, vol. PAS-100, no. 6,Jun. 1981, pp.2837-2845.

[6] T. Ekelund,Modeling and Linear Quadratic OptimalControl of Wind Turbines, PhD Thesis, ChalmersUniv. of Technology, Goteborg, Sweden, Mar. 1997.

[7] L. Dessaint, H. Nakra, and D. Mukhedkar,Propagation and Elimination of Torque Ripplein aWind Energy Conversion System:IEEE Trans. onEnergy Conversion,vol. EC-1, no.2, Jun. 1986, pp.

[8] T. Thiringer, Power Quality MeasurementsPerformed on a Low-Voltage Grid Equipped with TwoWind Turbines,IEEE Trans. on Energy Conversion,vol. EC-11, no. 3, Sep. 1996, pp. 601-606.

[9] P Novak,T. Ekelund, I. Jovik, and B. Schmidtbauer,Modeling and Control of a Variable-Speed Wind-Turbine Drive-System Dynamics,IEEE ControlSystems Magazine,vol. 15, no. 4, 199 5, pp. 28-38.

[101H. Sira-Ramirez, Differential geometric methods invariable structure control,Int. Journal of Control,vol. 48, no. 4, 1988, pp.1359-1390.

[ll]V.I. Utkin, Sliding Modes in Control andOptimization,Springer-Verlag,Berlin; 1992.

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