Towards Pitch-Scheduled Drive Train Dampingin Variable-Speed, Horizontal-Axis Large Wind Turbines

Amit Dixit and Shashikanth Suryanarayanan

AbstractGiven the prohibitive costs of replacement ofdamaged gearboxes, damping drive train oscillations is ofimmense signicance in large wind turbine control design. TheDrive Train Damper (DTD) is an important constituent ofthe power production control routine in variable-speed, pitch-regulated large wind turbines. The DTD is used to mitigatefatigue loading of drive train components. This paper motivatesthe need for scheduling the parameters of the damper based onblade pitch angle. It is shown that due to the strong couplingbetween blade edgewise motion and the drive train torsion,the fundamental frequency associated with the drive trainstorsional motion can vary signicantly across the range of bladepitch angles observed in practice. If left unaccommodated, thisvariation in fundamental frequency is shown to adversely affectthe performance of the DTD.

I. INTRODUCTION

Over the last couple of decades, development ofenvironment-friendly alternate power production technolo-gies has attracted considerable attention. Wind power has,thus far, proved to be the most promising green poweroption. Today, several wind farms produce power at a Cost-of-Energy (CoE) comparable to that of coal and natural gasbased power plants [1].For economic reasons, much of the work on wind turbine

development has focused on large wind turbines. Typicallarge wind turbines of today (Fig. 1) are massive 3-bladed,horizontal-axis structures with enormous blade spans (70-100m in diameter), tall towers (60-100m in height) andpower ratings in the 1-5 MW range. Most of these machinessupport variable-speed operation i.e., the rotor speed is notrigidly coupled to the grid frequency. Variable speed actionenables realization of higher power capture efciencies.Amongst the technologies that enable realization of these

machines, advanced control plays a pivotal role. Modernlarge wind turbines are endowed with sophisticated controlsystems which are organized to support several modes ofoperation such as start-up. shut-down, power production etc.In this paper, we look at a component of the control routineused in the power production mode of variable-speed largewind turbines known as the Drive Train Damper (DTD).The drive-train in horizontal-axis large wind turbines con-

sists of a low speed shaft coupled to the rotor, a step-up gearbox and a high speed shaft that is coupled to the generator.Damping drive train oscillations becomes critical given theprohibitive costs of replacement of failed gearboxes. Drive

A. Dixit and S. Suryanarayanan are with the Department of Me-chanical Engineering, Indian Institute of Technology Bombay, Powai,Mumbai 400076, India. Email: : adixit@me.iitb.ac.in,shashisn@iitb.ac.in

Fig. 1. Horizontal-Axis Large Wind Turbine (Courtesy: GE Energy)

train damping solutions proposed in the past have revolvedaround the idea of utilizing ltered versions of generatorspeed information to damp oscillations of the resonant modeof the drive train [2], [3]. In the absence of precise estimatesof the fundamental frequency of interest, xed parametersolutions tend to be conservative. In this paper, we proposea modeling approach to predict the frequency associatedwith that of the drive trains fundamental torsional mode.Specically, we assert that the fundamental frequency inquestion varies with the blade pitch angle. This result gainssignicance since drive train oscillations/fatigue loading aremost signicant in high wind speed operating conditionsduring which the pitch control loop is active [4].

The paper is organized as follows. Section II discussesthe state-of-the-art of drive train damping solutions andmotivates the need for an adaptive DTD structure. Thissection also includes a tutorial that illustrates the effect ofcoupling in mechanical systems. Section III details the effectof coupling between blade edge and drive train torsionaloscillations that results in variation of the fundamentalfrequency associated with the drive train torsional mode.Section IV discusses simulation results performed on FAST -

Proceedings of the44th IEEE Conference on Decision and Control, andthe European Control Conference 2005Seville, Spain, December 12-15, 2005

MoIB18.6

0-7803-9568-9/05/$20.00 2005 IEEE 1295

a horizontal-axis wind turbine simulator developed at NREL1

that corroborate analytical predictions of DTD performance.The paper concludes with a summary and a discussion onfuture problems of interest.The primary contribution of the paper is the result that the

fundamental frequency associated with the torsional mode ofvibration of the drive train in horizontal-axis wind turbinesvaries with pitch angle. It is shown that since the couplingbetween the blade edge and drive train torsion modes ofoscillation is pronounced, the fundamental frequency canvary signicantly. These results are used to motivate the needto incorporate pitch scheduling into the DTD architecture.

II. PRELIMINARIESA. Drive Train DampingLarge wind turbines of today are often equipped with

four actuators that include three blade pitch motors andthe generator (used as a torque control device) to controlrotor speed, power produced and mechanical loads on theturbine structure. Fig. 2 shows the typical power productionmode control architecture used in large wind turbine control.The torque control loop is active, primarily, in the lowerwind speed region of operation during which the controlobjective is to maximize energy capture. On the other hand,the pitch control loop is active in higher wind speed regionsof operation where the goal is to minimize fatigue damageof the turbine structure. See [5], [4] for more details.

Torque Controller

Wind Disturbance

Pitch Controller Wind TurbineAerodynamics

DamperDrive Train

DynamicsDriveTrain

OtherLoads

GeneratorSpeed

DTLoads

Plant

T

T

i

aero

gen

Fig. 2. Typical Control Architecture

In this paper, we concern ourselves with the drive traindamper. As illustrated in Fig. 2, the drive trains torsionalmode is excited by the rotor aerodynamic torque input(Taero) produced by the interaction of the turbines bladeswith a turbulent wind eld incident upon the rotor plane. Inthe past, drive train dynamics have been modeled [2] usinga mass-spring model as described below.

J(t) + K(t) = Taero Tgen (1)where, J and K are lumped parameters representing theequivalent rotational inertia and torsional stiffness of thedrive train. The generator torque demand is denoted as Tgen.

1National Renewable Energy Labs, Boulder, USA

Natural damping is often neglected in the model structuresince the drive train components are largely made out ofsteel with no explicit interconnecting damping elements.Drive train damping solutions revolve around utilizing

the generator as a damping device. The essential idea isto introduce damping action by making a component ofTgen proportional to a ltered version of generator speed. Bossanyi [3] has proposed a DTD structure of the form:

B(s) =Ks(1 + s)

s2 + 2ns + 2n(2)

where n corresponds to the frequency of the resonant modeassociated with the torsional oscillations of the drive train.Such a structure is motivated by the following requirements:1) Large gain close to the resonant frequency n to

provide adequate damping.2) Small gains at low frequencies so that unwarranted

power uctuations are not introduced.3) DTD component of torque demand should have a phase

lag of 90o to that of the forcing input. It turns out thatfor the dynamics considered in Eqn. 1, the lter shouldadd close to zero phase at the resonant frequency n.

Fig. 3 shows the Bode plot of a candidate lter that uses theabove structure.

20

30

40

50

60

Mag

nitu

de (d

B)

100 101 102 10390

45

0

45

90

Phas

e (de

g)

Bode Diagram

Frequency (rad/sec)

Fig. 3. Bode Plot of Candidate Filter

If the resonant frequency n is precisely known, the lterparameter may be chosen small so that the requirementsabove are met. However, since an estimate of the resonantfrequency has been found to be difcult to obtain in practice,the damper parameter is chosen fairly large to realize robustperformance (damper peak has larger spread for larger valuesof ).In this paper, we provide a methodology to predict changes

in n with changes in blade pitch angle. The benets ofsuch a strategy gains signicance since drive train oscilla-tions/fatigue loading are most signicant in high wind speed

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operating conditions during which the pitch control loop isactive.

B. Coupling in Mechanical SystemsIn this subsection, we discuss the effects of coupling

between elements of an interconnected mechanical systemand its effect on system behavior. Consider an n degree-of-freedom system whose dynamics are described by thefollowing coupled set of differential equations.

[M ] x(t) + [C] x(t) + [K]x(t) = {f(t)} (3)where x(t), x(t) and x(t) respectively denote n 1 dis-placement, velocity and acceleration vectors, while [M ], [K]and [C] denote n n mass, damping and stiffness matricesrespectively. The forcing function is denoted by {f(t)}. Wewill refer to the off-diagonal elements in the mass, dampingand stiffness matrices respectively as the coupled mass,coupled damping and coupled stiffness terms.An important set of parameters which governs the behavior

of the system described in Eqn. 3 is the set of fundamental(natural) frequencies associated with the system. With theaid of a pedagogical example, we reiterate the well-knownresult that the natural frequencies of a coupled systemcan be signicantly different from those of its constituentuncoupled systems. Consider the dynamics of a 2 degree-of-freedom system represented in Fig. 4. The pendulumsystem consists of a mass m attached to one end of amassless link of length L. The other end of the link is hingedto the mass M . For small displacements, the dynamics maybe approximated as:

[m1 mcmc m2

] [x(t)(t)

]+[k1 00 k2

] [x(t)(t)

]=[F (t)(t)

]

where m1 = M + m, m2 = mL2,mc = mL, k1 = k andk2 = mgL. The natural frequencies of this system are givenby

1,2 =m1k2 + m2k1

(m1k2 m2k1)2 + 4m2ck1k2

2(m1m2 m2c)which are different from the uncoupled natural frequenciesuc1 =

k1m1 and uc2 =

k2m2

(t)

M

m

L

k xF(t)

Fig. 4. A pendulum system

In general, the coupled mass, coupled stiffness or coupleddamping terms are functions of the system geometry. In thependulum system considered above, if the plane of rotationof the simple pendulum were at an angle to the plane ofmotion of the mass M , it can be shown that the expressionfor the coupled mass changes to mc = mLcos, whileall the other quantities remain unchanged. Changes in thecoupled mass/stiffness terms result in changes in the naturalfrequencies associated with the the system. As shown inthe following section, such a situation arises in the studyof the dynamics of horizontal-axis wind turbines where thecoupled mass of interest is a function of the blade pitch an-gle. For different pitch angles, the natural frequencies of theturbine system and in particular the frequency correspondingto drive trains torsional oscillations can vary signicantly.As shown later in Section IV, such a change in the naturalfrequncy can have a signicant impact on the performanceof the DTD.

C. Fatigue Data Analysis

The general approach to fatigue life prediction of strucutralcomponents subjected to a random stresses is to relate thelife of the element to laboratory fatigue experiments ofsimple specimens subjected to constant amplitude uctuatingstresses. The so-called S-N data for a specimen relates themagnitude of the uctating stress with the number such stresscycles to failure. A typical S-N curve for steel is shown inFig. 5.To estimate the fatigue-life of a specimen subjected to a

combination of constant amplitude stress-reversals, a damagerule is assumed. A commonly used damage rule (due toPalmgren and Miner) states that if there are k different stressmagnitudes in a spectrum, Si(1 i k), each contributingni(Si) cycles, then if Ni(Si) is the number of cycles tofailure of a constant stress reversal Si (read from the S-Ncurve), failure occurs when

ki=1

niNi

= C (4)

For design purposes, C is often assumed to be equal to 1.

Fluc

tuat

ing

stre

ss m

agni

tude

"S"

Number of cycles to failure "N"

Fig. 5. Typical S-N curve for steel-like materials

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Rainow counting [6], [7] is a method used in the analysisof fatigue data to translate a spectrum of varying stressinto sets of reversing stress cycles of constant magnitude.Each such set (often called a bin) is characterized bythe magnitude of the uctuating stress. For estimating thenumber of cycles corresponding to each bin, the varyingstress spectrum is rst converted to a series of tensile andcompressive peaks. By observing the pattern the peaks, anestimate of the number of cycles corresponding to each binis made.Consider two time series of stress variation S1 and S2

in a structural element where S2 is a damped version ofS1 i.e., the magnitude of stress variation in S2 is lesser thanin S1. It is useful to note that in comparison to S2, S1 willcontribute more rainow counts to bins corresponding tolarger magnitudes of stress variation. However, a rainowcount excercise on S2 will yield larger counts for binsassociated with small magnitudes of stress variation sincelarge stress amplitude variations are moved to bins withsmaller stress amplitudes. As will be seen, this fact willbe useful in interpreting the load mitigating performance ofdrive train dampers.

III. COUPLING BETWEEN BLADE EDGE MOTIONAND DRIVE TRAIN TORSION

The dynamics of horizontal-axis wind turbines can beexpressed using a 6 degree-of-freedom model (Fig. 6). Themotion of the tower along the direction of wind is termedtower fore-aft, while the motion of the tower perpendicularto the wind direction is termed tower side-to-side. The mo-tion of blade in the plane of rotaion is termed blade edgewhile the out-of-plane motion of the blades is termed bladeap. The generator azimuth degree-of-freedom accounts forthe variable speed nature of the turbine, and essentiallyconstitutes a rigid body rotational mode. The oscillationsof the drive-train are governed by the drive-train torsionaldegree-of-freedom. It should be noted that this torsionalmotion of the drive train is strongly coupled to blade edge-wise motion.

Generatorazimuth

Wind direction

Blade flap

Tower foreaft Tower sidetoside

Blade edge

Drivetrain torsion

Fig. 6. 6-DOF model of a horizontal axis variable speed wind turbine

The structural components of wind turbines can be viewedas distributed parameter systems. The response of sucha system to an external excitation may be written as aweighted sum of an innite number of basis functions (called

modes). Associated with each mode is a mode shape - anon-dimensional function which describes the spatial varia-tion of the amplitude of vibration of the mode. The responseof distributed parameter systems can often be approximatedby superimposing responses of a nite number of dominantmodes of the system (the Assumed Modes method). Thenumber of modes to be considered for a good approximationof the response is found by ensuring that the norm of thedifference between the exact response and the approximatedresponse for a predened input is less than a critical value.See [8] for more details.We now derive an expression for how the blade edge and

drive-train torsion degrees-o-freedom are coupled. We usethe Assumed Modes method to model blade deections.In the interests of simplicity, only the rst dominant modeof the blade edge-wise deection is used. Let (r) denotethe mode shape function associated with the dominant bladeedgewise deection mode at a distance r from the axis of theturbine. Then, by denition, the blade edge-wise deectionis given by xe(r) = (r)q, where q denotes the generalizeddisplacement of the rst blade edge-wise mode.Fig. 7 shows a blade section at radius r. It can be seen that

due to pitching of the blades, the edge-wise motion of theblade section is not entirely in the plane of rotation. The in-plane component of the blade edge-wise deection is givenby

xp(r) = ((r)q)cos (5)

The coupled mass between drive-train and the blade edge-wise degrees of freedom is the torque exerted on the drive-train due to produce unit acceleration along the blade-edgedegree-of-freedom. The inertial force associated with a bladesection at radius r (in-plane) is

f(r) = [(r)m(r)dr]cos (6)

where m(r) is the mass per unit length of the section anddr denotes the thickness of the section. This force exertsa torque on the drive-train. The total torque on the drive-train due to the in-plane acceleration of the entire blade iscomputed by summing up the torques exerted by all suchblade sections. By denition, this torque equals the coupledmass. Hence, the expression for the coupled mass may beexpressed as:

mc = R0

{r(r)m(r)dr}cos (7)

As is clear from Eqn. 7, the coupled mass is a functionof the blade pitch angle.Fig. 8 and Fig. 9 represent the power spectral density

functions of the simulated drivetrain torque for a typical 1.5MW wind turbine under two different operating conditions,denoted by OC1 and OC2. In turbines of this rating, the pitchangle can vary from a low of 02o below rated wind speedto around 25o in high wind speed conditions. It can be seenthat the drive-train frequency differs by approximately 4%between the two operating conditions.

1298

Zero pitch line

chord line

x (r)e

Fig. 7.

The change in the drive-train natural frequency with achange in the operating conditions is not accounted for inthe current xed-gain drive-train damper structures. Thissituation provides an opportunity to investigate methods ofincreasing the effectiveness of the drive train damper byscheduling its gains based on pitch angle. This idea is furtherdiscussed in the next section.

0 5 10 15 2060

50

40

30

20

10

0

10

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB/H

z)

Power Spectral Density Estimate via Welch

DT Frequency 2.64Hz

Fig. 8. OC1: mean wind speed = 12 m/s; mean pitch angle 80

IV. SIMULATION RESULTS

In this section, we present simulation results to demon-strate a case where improved DTD performance is achievedby accommodating change in drive-train fundamental fre-quency due to pitching. We focus on the above-rated windspeed region, where the goal is to minimize the structuralloads on the turbine. As mentioned earlier, the pitch controlloop is active in this region. The pitch controller regulatesthe generator speed to a desired value while the torquedemanded by the torque controller is kept constant (seeFig. 2). The drive train damper imposes an additional torquedemand on top of that demanded by the torque controller.A nonlinear model representing the dynamics of a typical

1.5MW horizontal-axis wind turbine was constructed usingthe wind turbine simulation code FAST [9]. A PI controlleracting on the generator speed was used to compute the bladepitch angles for regulating the generator speed.

0 2 4 6 8 10 12 14 16 18

50

40

30

20

10

0

10

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB/H

z)

Power Spectral Density Estimate via Welch

DT Frequency 2.54 Hz

Fig. 9. OC2: mean wind speed = 18 m/s; mean pitch angle 200

The model was subjected to two sets of wind inputsdenoted by OC1 and OC2 with mean wind speeds of 12m/sand 18m/s respectively. A turbulence intensity of 10%(small in comparison to IEC2A wind class) was used for thesimulations. Note that the mean pitch angles (1 = 8o, 2 =20o) required for speed regulation differ due to the differencein mean wind speeds. Based on the computational procedurediscussed in Section III, a correspondence can be establishedbetween each of the mean pitch angles i and the naturalfrequency ni associated with the torsional oscillations inthe drive-train.To provide corroboration for the need to accommodate

variation in the drive train natural frequency due to pitchaction, two lters Bi, (i = 1, 2) corresponding to meanpitch situation i were designed. The DTD structure usedis identical to that proposed by Bossanyi [3] (see Eqn. 2).

Bi(s) =Kis(1 + is)

s2 + 2inis + 2ni

The load/oscillation mitigation performance of the damperswas evaluated by using the Rainow Counting method (seeSec. II-C).Rainow counts of the drive-train torque for wind input

condition OC1 with and without damper B1 included arepresented in Table. I. Fig. 10 shows a time history of thedrive-train torque with and without the use of a damper.

TABLE IRAINFLOW CYCLE COUNTS; WIND CONDITION OC1

Torque [N.m] No Damper Damper B1300 18 75910 110 191520 112 112130 19 2

Table. II lists the rainow counts for the performanceof the damper B1 and B2 both simulated for wind input

1299

12 14 16 18 20 22 24 26 28 30

9

9.5

10

10.5

11

Time (sec)

DT

torq

ue (N

.m)

No damperWith B1

Fig. 10. Variation of Drive Train Torque with and without DTD

condition OC2. This comparison was made to highlight theill-effects of a xed-parameter damper (say B1).

TABLE IIRAINFLOW CYCLE COUNTS; WIND CONDITION OC2

Torque [N.m] No Damper DamperB1 DamperB2270 269 173 189810 67 60 401340 4 3 1

The results in Tables I, II and Fig. 10, indicate thefollowing:

The inclusion of a damper signicantly reduces drive-train oscillations

The DTD that accommodates natural frequency varia-tion displays better load mitigation performance

As discussed earlier, improved oscillation damping leadsto a reduction in large amplitude oscillations and conse-quently increased smaller amplitude oscillations. It shouldbe noted that the exponential nature of the S-N curve (seeSec. II-C) indicates that the number of cycles to failurereduces drastically with increasing stress cycle magnitude.Typically, the number cycles to failure reduce by around 150times when stress cycle amplitude increases 3 times. The im-plication is that an increased number of low amplitude stresscycles can be tolerated, if it is accompanied by a signicantdecrease in the number of cycles in high magnitude region.

V. SUMMARY AND FUTURE WORK

Active drive train damping solutions are deployed in largehorizontal-axis wind turbines to mitigate fatigue damagedue to drive-train oscillations. In this paper, we studied theeffect of blade pitch on the natural frequency associatedwith drive train torsional oscillations and its impact ondrive train damper design. It was shown that due to strongcoupling between the blade edge-wise motion and drive traintorsional oscillations, the drive train natural frequency varies

signicantly across the range of pitch angles encounteredduring operation of large wind turbines. Further, it wasshown that this variation, if unaccommodated, leads to sub-optimal drive train damper performance.The results discussed in this paper provides a platform

for the investigation of strategies for scheduling the gainsof the drive train damper based on the prevalent pitch angle.Given that in turbulent high wind speed conditions, signifcantvariations in pitch angle is not uncommon, it is expectedthat stability considerations would play a signicant rolein the design of scheduling algorithms. Addressing suchconsiderations is an issue of immediate interest.

REFERENCES[1] Danish Wind Industry Associationhttp://www.windpower.org.[2] P. Novak et al., Modeling and Control of Variable-Speed Wind Turbine

Drive-System Dynamics, IEEE Control Systems Magazine, pp. 28-38,August 1995.

[3] E. A. Bossanyi, The Design of Closed-loop Controllers for WindTurbines, Wind Energy, vol. 3, 2000, pp. 149-163.

[4] T. Burton, D. Sharpe, N. Jenkins and E. A. Bossanyi, Wind EnergyHandbook, John Wiley & Sons, Ltd, NY; 2001.

[5] S. Suryanarayanan and A. Dixit, On the Dynamics of the Pitch ControlLoop in Horizontal-Axis Large Wind Turbines, Proc. of the AmericanControl Conference, Portland, 2005, pp. 686-690.

[6] I. Rychlik, A New Denition of the Rainow Cycle Counting Method,Intl. Journal of Fatigue, vol. 9, pp. 119-121, 1987.

[7] N. E. Dowling, Fatigue Prediction for Complicated Stress-StrainHistories, Journal of Materials, vol. 7, pp. 71-87, 1972.

[8] L. Meirovich, Elements of Vibration Analysis, McGraw-Hill , NY;1986.

[9] FAST: An aero-elastic design code for horizontal axis wind turbineshttp://wind.nrel.gov/designcodes/simulators/fast

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