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Nonlinear Dual-Mode Control of Variable-SpeedWind Turbines with Doubly Fed Induction

GeneratorsChoon Yik Tang,Member, IEEE,Yi Guo, Student Member, IEEE,and John N. Jiang,Senior Member, IEEE

Abstract—This paper presents a feedback/feedforward non-linear controller for variable-speed wind turbines with doublyfed induction generators. By appropriately adjusting the rotorvoltages and the blade pitch angle, the controller simultaneouslyenables: (a) control of the active power in both themaximumpower tracking and power regulation modes, (b) seamless switch-ing between the two modes, and (c) control of the reactive powerso that a desirable power factor is maintained. Unlike manyexisting designs, the controller is developed based on original,nonlinear, electromechanically-coupled models of wind turbines,without attempting approximate linearization. Its developmentconsists of three steps: (i) employ feedback linearizationto exactlycancel some of the nonlinearities and perform arbitrary poleplacement, (ii) design a speed controller that makes the rotorangular velocity track a desired reference whenever possible, and(iii) introduce a Lyapunov-like function and present a gradient-based approach for minimizing this function. The effectivenessof the controller is demonstrated through simulation of a windturbine operating under several scenarios.

Index Terms—Wind energy, wind turbine, active power, reac-tive power, maximum power tracking, power regulation, nonlin-ear control.

I. I NTRODUCTION

W IND power is gaining ever-increasing attention inrecent years as a clean, safe, and renewable energy

source. With the fast growth of wind generation in powersystems, wind power is becoming a significant portion of thegeneration portfolio in the United States as well as many coun-tries in Europe and Asia [1]. Indeed, wind power penetrationis planned to surpass 20% of the United States’ total energyproduction by 2030—a figure that is way beyond the currentlevel of less than 5% [2]. Hence, to realize this vision, it isnecessary to develop large-scale wind farms that effectivelyproduce electric power from wind, and integrate them withthe power systems.

The integration of large-scale wind farms into a powersystem, however, changes the fundamental principle of itsoperation, which is to maintain reliability by balancing loadvariation with “controllable” generation resources. Whenaportion of these resources comes from “uncontrollable” windgeneration, that portion of the resources can hardly be guar-anteed due to the intermittency of wind. As a result, thepower system may fail to achieve the required balance. When

The authors are with the School of Electrical and Computer En-gineering, University of Oklahoma, Norman, OK 73019 USA (e-mail:{cytang,yi.guo,jnjiang}@ou.edu).

This work was supported by the National Science Foundation under grantsECCS-0926038 and ECCS-0955265.

the level of wind power penetration is small, this issue maybe safely neglected. However, with the anticipated increasein penetration, the issue becomes critical for power systemreliability. As a case in point, half of the European gridsexperienced a severe difficulty in 2006 because several largewind farms, operating in themaximum power tracking(MPT)mode, produced excessive power that destabilized the grids.This event took place even though the wind penetration level,at that time, was low at only 7% [3].

Such an experience suggests that it is not advisable—and may be even disastrous—for a large-scale wind farmconnected to a grid to always operate in the MPT mode,making wind turbines in the farm harvest as much wind energyas they possibly could, following the “let it be when the windblows” philosophy of operation. Instead, it is highly desirablethat the wind farm can also operate in thepower regulation(PR) mode, whereby its total power output from the windturbines is closely regulated at some desired setpoint, despitethe fluctuating wind conditions.

The ability to operate in the PR mode in addition to theMPT mode, as well as the ability toseamlesslyswitch betweenthe two, offers many important advantages: not only does thePR mode provide a cushion to absorb the impact of windfluctuations on total power output through power regulation, italso enables a power system to effectively respond to changesin reliability conditions and economic signals. For instance,when a sudden drop in load occurs, the power system mayask the wind farm to switch from the MPT to the PR modeand generate less power, rather than rely on expensive down-regulation generation. As another example, the PR mode, whenproperly designed, allows the power output of a wind farm tosmoothly and accurately follow system dispatch requests, thusreducing its reliance on ancillary services such as reliabilityreserves.

To enable large-scale wind farms to operate well in thesetwo modes and switch seamlessly between them, numer-ous challenges must be overcome. This paper is devoted toaddressing a subset of these challenges, by presenting anintegrated framework for controlling the rotor voltages andthe blade pitch angle of variable-speed wind turbines withdoubly fed induction generators (DFIGs). The paper presents afeedback/feedforward nonlinear controller developed based onoriginal, nonlinear, and electromechanically-coupled modelsof wind turbines, without attempting approximate lineariza-tion. The controller simultaneously enables: (a) control of theactive power in both the MPT and PR modes, (b) seamless

2

switching between the two modes, and (c) control of thereactive power so that a desirable power factor is ensured.Its development consists of three steps. First, we show that,although dynamics of a wind turbine are highly nonlinearand electromechanically coupled, they offer a structure, whichmakes the electrical part feedback linearizable, so that arbitrarypole placement can be carried out. Second, we show thatbecause the electrical dynamics can be made very fast, it ispossible to perform model order reduction, so that only thefirst-order mechanical dynamics remain to be considered. Forthis reduced first-order model, a speed controller is designed,which enables the rotor angular velocity to track a desired ref-erence whenever possible. Finally, we introduce a Lyapunov-like function that measures the difference between the actualand desired powers and present a gradient-based approach forminimizing this function. The effectiveness of the controller isdemonstrated through simulation of a wind turbine operatingunder a changing wind speed, changing desired power outputs,modeling errors, and noisy measurements.

To date, a significant amount of research has been performedon the control of variable-speed wind turbines [4]–[37]. Theexisting publications, however, are substantially different fromour work in the following aspects:

(i) The mechanical and electrical parts of the wind turbinesare considered separately in most of the current litera-ture: [4]–[21] considered only the mechanical part, while[22]–[36] considered only the electrical part, focusingmostly on the DFIGs. In contrast, in this paper we con-siderboth the mechanical and electrical parts. Although[37] also considered both these parts, its controllerwas designed to maximize wind energy conversion, asopposed to achieving power regulation (i.e., only operatein the MPT mode). In comparison, our controller canoperate inboth the MPT and PR modes as well asseamlessly switch between the two.

(ii) For those references [4]–[21] considering only the me-chanical part, control of the active power has been themain focus: [4], [5] maximized the wind power captureat low to medium wind speeds by adjusting both thegenerator torque and the blade pitch angle, [6]–[9] didthe same by adjusting only the generator torque, and[13]–[21] aimed to maintain the rated rotor speed andlimit the power production at high wind speeds bycontrolling the blade pitch angle. Although the resultsare interesting, there is a lack of discussion on reactivepower control, which is sometimes crucial to powersystem reliability. On the other hand, for those refer-ences [22]–[36] considering only the electrical part, [22],[23] considered only the active power control, while[24]–[27] dealt with the reactive power. In contrast,we provide an integrated solution to the more difficultproblem of simultaneously controllingboth the activeand reactive powers by appropriately adjustingboth therotor voltages and the blade pitch angle.

(iii) From a modeling point of view, the wind turbine modelwe consider is one of the most comprehensive. Specifi-cally, [12]–[14] assumed a linearized model of the me-

chanical torque from wind turbines, while [4]–[11], [15]considered a nonlinear one. In addition, [23]–[32], [35]–[37] considered a reduced-order DFIG model, while[33], [34] considered a full-order one. In contrast, weconsider a fifth-order, nonlinear, electromechanically-coupled model and attempt no linearization around someoperating points.

(iv) Finally, from a controls point of view, the controltechniques we use are different from those adopted in[4]–[37]. For control of the mechanical part of the windturbines, several techniques have been considered, in-cluding proportional-integral-derivative (PID) [12]–[14],fuzzy control [4], [8]–[10], [21], adaptive control [6],[7], robust nonlinear control [5], [15], sliding modecontrol [11], and wind-model-based predictive control[17]. Similarly, for control of the electrical part, variousways of controlling the DFIGs have been proposed,including vector/decoupling control with or without aposition encoder [23]–[25], [31], [32], [34]–[37], directpower control [28], power error vector control [30],robust coordinated control [26], linear quadratic regu-lator [29], nonlinear inverse system method [27], andpassivity-based control [33]. In comparison, the con-troller we propose here consists of a mixture of severallinear/nonlinear control techniques, including feedbacklinearization, pole placement, and gradient-based opti-mization and involving three time scales. The reasonour controller is more complex is that the problem weaddress is inherently more challenging (see (i) and (ii))and the model is more comprehensive (see (i) and (iii)).

As it follows from the above brief review of the currentliterature, this paper contributes significantly to the state ofthe art on the control of wind turbines.

The remainder of the paper is organized as follows:Section II describes a model of variable-speed wind tur-bines with DFIGs. Section III introduces the proposed feed-back/feedforward nonlinear controller. Simulation results areshown in Section IV. Finally, Section V concludes the paper.

II. M ODELING

Consider a variable-speed wind turbine consisting of adoubly fed induction generator (DFIG) and a power electronicsconverter, as shown in Figure 1. The DFIG may be regarded asa slip-ring induction machine, whose stator winding is directlyconnected to the grid, and whose rotor winding is connectedto the grid through a bidirectional frequency converter usingback-to-back PWM voltage-source converters.

The dynamics of the electrical part of the wind turbine arerepresented by a fourth-order state space model, constructedusing the synchronously rotating reference frame (dq-frame),where the relation between the three phase quantities and thedq components is defined by Park’s transformation [38]. Thevoltage equations are [39]

vds = Rsids − ωsϕqs +d

dtϕds, (1)

vqs = Rsiqs + ωsϕds +d

dtϕqs, (2)

3

Wind

Wind Gear

box Grid

Stator power

Transformer

Filter DC/AC AC/DC

Rotor

turbine power Grid- side

converter

Rotor- side

converter

Doubly fed induction generator (DFIG)

Fig. 1. Schematic of a typical variable-speed wind turbine with a DFIG.

vdr = Rridr − (ωs − ωr)ϕqr +d

dtϕdr, (3)

vqr = Rriqr + (ωs − ωr)ϕdr +d

dtϕqr , (4)

wherevds, vqs, vdr, vqr are thed- andq-axis of the stator androtor voltages;ids, iqs, idr, iqr are thed- and q-axis of thestator and rotor currents;ϕds, ϕqs, ϕdr, ϕqr are thed- andq-axis of the stator and rotor fluxes;ωs is the constant angularvelocity of the synchronously rotating reference frame;ωr isthe rotor angular velocity; andRs, Rr are the stator and rotorresistances. The flux equations are [39]

ϕds = Lsids + Lmidr, (5)

ϕqs = Lsiqs + Lmiqr, (6)

ϕdr = Lmids + Lridr, (7)

ϕqr = Lmiqs + Lriqr, (8)

where Ls, Lr, and Lm are the stator, rotor, and mutualinductances, respectively, satisfyingLs > Lm andLr > Lm.From (5)–(8), the current equations can be written as

ids =1

σLsϕds −

Lm

σLsLrϕdr, (9)

iqs =1

σLsϕqs −

Lm

σLsLrϕqr , (10)

idr = − Lm

σLsLrϕds +

1

σLrϕdr, (11)

iqr = − Lm

σLsLrϕqs +

1

σLrϕqr, (12)

whereσ = (1 − L2

m

LsLr) is the leak coefficient. Selecting the

fluxes as state variables and substituting (9)–(12) into (1)–(4),the electrical dynamics in state space form can be written as

d

dtϕds = − Rs

σLsϕds + ωsϕqs +

RsLm

σLsLrϕdr + vds, (13)

d

dtϕqs = −ωsϕds −

Rs

σLsϕqs +

RsLm

σLsLrϕqr + vqs, (14)

d

dtϕdr =

RrLm

σLsLrϕds −

Rr

σLrϕdr + (ωs − ωr)ϕqr + vdr,

(15)d

dtϕqr =

RrLm

σLsLrϕqs − (ωs − ωr)ϕdr −

Rr

σLrϕqr + vqr.

(16)

Neglecting power losses associated with the stator and rotorresistances, the active and reactive stator and rotor powers are

given by [40]

Ps = −vdsids − vqsiqs, (17)

Qs = −vqsids + vdsiqs, (18)

Pr = −vdridr − vqriqr, (19)

Qr = −vqridr + vdriqr, (20)

and the total active and reactive powers of the turbine are

P = Ps + Pr, (21)

Q = Qs +Qr, (22)

where positive (negative) values ofP and Q mean that theturbine injects power into (draws power from) the grid.

The dynamics of the mechanical part of the wind turbineare represented by a first-order model

Jd

dtωr = Tm − Te − Cfωr, (23)

where Cf is the friction coefficient,Tm is the mechanicaltorque generated, andTe is the electromagnetic torque givenby [40]

Te = ϕqsids − ϕdsiqs, (24)

where positive (negative) values mean the turbine acts as agenerator (motor). The mechanical power captured by the windturbine is given by [41]

Pm = Tmωr =1

2ρACp(λ, β)V

3w , (25)

whereρ is the air density;A = πR2 is the area swept by therotor blades of radiusR; Vw is the wind speed; andCp(λ, β) isthe performance coefficient of the wind turbine, whose valueis a function [41] of the tip speed ratioλ, defined as

λ =ωrR

Vw, (26)

as well as the blade pitch angleβ, assumed to lie within somemechanical limitsβmin and βmax. This function is typicallyprovided by turbine manufacturers and may vary greatly fromone turbine to another [41]. Therefore, to make the results ofthis paper broadly applicable to a wide variety of turbines,no specific expression ofCp(λ, β) will be assumed, until it isabsolutely necessary in Section IV, to carry out simulations.Instead,Cp(λ, β) will only be assumed to satisfy the followingmild conditions:

(A1) FunctionCp(λ, β) is continuously differentiable in bothλ andβ overλ ∈ (0,∞) andβ ∈ [βmin, βmax].

(A2) There existsc ∈ (0,∞) such that for allλ ∈ (0,∞)and β ∈ [βmin, βmax], we haveCp(λ, β) ≤ cλ. Thiscondition is mild because it is equivalent to saying thatthe mechanical torqueTm is bounded from above, sinceTm ∝ Cp(λ,β)

λ according to (25) and (26).(A3) For each fixedβ ∈ [βmin, βmax], there existsλ1 ∈

(0,∞) such that for allλ ∈ (0, λ1), we haveCp(λ, β) >0. This condition is also mild because turbines aredesigned to capture wind power over a wide range ofλ, including times whenλ is small.

4

For the purpose of simulation in Section IV, the followingmodel ofCp(λ, β) will be assumed, which is presented in [42]and also adopted in MATLAB/Simulink R2007a:

Cp(λ, β) = c1

( c2λi

− c3β − c4

)

e−c5λi + c6λ, (27)

where

1

λi=

1

λ+ 0.08β− 0.035

β3 + 1, (28)

and the coefficients arec1 = 0.5176, c2 = 116, c3 = 0.4,c4 = 5, c5 = 21, andc6 = 0.0068.

As it follows from the above, the wind turbine studied in thispaper is described by a fifth-order nonlinear dynamical systemwith states[ϕds ϕqs ϕdr ϕqr ωr]

T , controls [vdr vqr β]T ,outputs[P Q]T , “disturbance”Vw, nonlinear state equations(13)–(16) and (23), and nonlinear output equations (17)–(22).Notice that the system dynamics are strongly coupled: the“mechanical” state variableωr affects the electrical dynam-ics bilinearly via (15) and (16), while the “electrical” statevariables[ϕds ϕqs ϕdr ϕqr ]

T affect the mechanical dynamicsquadratically via (9)–(12), (24) and (23). Since the statorwind-ing of the DFIG is directly connected to the grid, for reliabilityreasons[vds vqs]

T are assumed to be fixed, i.e., not to becontrolled, in the rest of this paper. Moreover, since (9)–(12)represent a bijective mapping between[ϕds ϕqs ϕdr ϕqr ]

T

and [ids iqs idr iqr]T and since the currents[ids iqs idr iqr]T ,the rotor angular velocityωr, and the wind speedVw can all bemeasured, a controller for this system has access to its entirestates (i.e., full state feedback is available) and its disturbance(i.e., the wind speedVw). A block diagram of this system isshown on the right-hand side of Figure 2.

III. C ONTROLLER DESIGN

In this section, a feedback/feedforward nonlinear controllerof the form depicted on the left-hand side of Figure 2 ispresented. By adjusting the rotor voltagesvdr andvqr and theblade pitch angleβ, the controller attempts to make the activeand reactive powersP andQ track, as closely as possible—limited only by wind strength—some desired, time-varyingreferencesPd andQd, presumably provided by a wind farmoperator. WhenPd is set to sufficiently large, i.e., larger thanwhat the turbine can possibly convert from wind, it meansthe operator wants the turbine to operate in the MPT mode;otherwise, the PR mode is sought. The value ofQd, along withthat ofPd, reflects a desired power factor PFd = Pd√

P 2

d+Q2

d

the

operator wants the turbine to also maintain.The controller development consists of three steps, which

are described in Sections III-A–III-C, respectively.

A. Feedback Linearization and Pole Placement

For convenience, let us introduce the variablex =[x1 x2 x3 x4]

T = [ϕds ϕqs ϕdr ϕqr]T and rewrite (9)–(12)

and (13)–(16) in matrix forms as follows:

x =

− Rs

σLsωs

RsLm

σLsLr0

−ωs − Rs

σLs0 RsLm

σLsLrRrLm

σLsLr0 − Rr

σLrωs

0 RrLm

σLsLr−ωs − Rr

σLr

︸ ︷︷ ︸

A

x

+

vdsvqs

vdr − ωrx4

vqr + ωrx3

, (29)

idsiqsidriqr

=

1σLs

0 − Lm

σLsLr0

0 1σLs

0 − Lm

σLsLr

− Lm

σLsLr0 1

σLr0

0 − Lm

σLsLr0 1

σLr

x,

(30)

whereA is a constant matrix and, as was pointed out at theend of Section II, bothvds and vqs are constants not to becontrolled. Note that the only nonlinearities in (29) are thetwo products of the state variables, i.e.,−ωrx4 and ωrx3.Also note that these nonlinearities appear on the same rows asthe control variablesvdr andvqr . Thus,feedback linearization[43] may be used to cancel them and subsequently performarbitrarypole placement[44], i.e., let

vdr = ωrx4 −KT1 x+ u1, (31)

vqr = −ωrx3 −KT2 x+ u2, (32)

whereK1,K2 ∈ R4, the first terms on the right-hand side

of (31) and (32) are intended to cancel the nonlinearities, thesecond terms are for pole placement, and the third are newcontrol variablesu1 andu2, to be designed later.

To implement (31) and (32), full state feedback on the fluxesx and the rotor angular velocityωr are needed. While the latteris relatively easy to measure, the former is not. Fortunately,this difficulty can be circumvented by first measuring thecurrents—which is feasible—and then calculating the fluxesfrom (5)–(8). This explains the fourth input of thenonlinearcontroller block in Figure 2.

Substituting (31) and (32) into (29) yields

x = (A−BK)x+[vds vqs u1 u2

]T, (33)

whereB = [02×2 I2×2]T andK = [K1 K2]

T is the statefeedback gain matrix. Since the electrical elements in theDFIG are physically allowed to have much faster responsesthan their mechanical counterparts,K in (33) may be chosenso thatA−BK is asymptotically stable with very fast eigen-values. With this choice ofK and with relatively slow-varyingu1 andu2 (recall thatvds andvqs are constants), the fourth-order linear differential equation (33) may be approximatedby the following static, linear equation:

x = −(A−BK)−1[vds vqs u1 u2

]T. (34)

As a result, the fifth-order model described in (13)–(16) and(23) may be approximated by the first-order model describedin (23) along with (34). As will be shown next, this approxi-mation greatly simplifies the design ofu1 andu2. Therefore,

5

Nonlinear

controller

Desired active/ Nonlinear

wind turbine

with DFIG

Rotor voltages vdr, vqr

Blade pitch angle β

Active/reactive

Current i and rotor

Wind speed Vw

reactive powers Pd, Qd powers P, Q

angular velocity ωr

Fig. 2. Structure of the multivariable, feedback/feedforward nonlinear controller, developed based on original, nonlinear dynamics of the wind turbine.

we will assume, in the sequel, thatK is chosen so that theelectrical dynamics (33) are asymptotically stable and so fastthat they may be approximated by (34).

B. Tracking of Desired Angular Velocity

The second step of the controller development involves con-structing a speed controller that ensures the angular velocityof the rotor,ωr, tracks a desired, time-varying reference,ωrd,whenever possible. The construction may be divided into foursubsteps as described below.

Substep 1.First, we show that the electromagnetic torqueTe defined in (24) may be expressed as aquadratic functionof the new control variablesu1 andu2. From (24) and (30),

Te = ϕqsids − ϕdsiqs =[x2 −x1

][idsiqs

]

= xT

0 − 1σLs

0 Lm

σLsLr1

σLs0 − Lm

σLsLr0

0 0 0 00 0 0 0

x. (35)

Equation (35) suggests thatTe is a quadratic function ofx,while (34) suggests thatx, in turn, is an affine function ofu1

andu2, sincevds andvqs in (34) are assumed to be constants.Hence,Te must be a quadratic function ofu1 andu2. Indeed,an explicit expression can be obtained as follows: sinceA −BK is asymptotically stable and thus nonsingular, it may bewritten as

(A−BK)−1 =

d11 d12 d13 d14d21 d22 d23 d24d31 d32 d33 d34d41 d42 d43 d44

, (36)

where eachdij depends onA, B, andK. From (34)–(36),

Te =Lm

σLsLr[(d11vds + d12vqs + d13u1 + d14u2)

(d41vds + d42vqs + d43u1 + d44u2)

− (d21vds + d22vqs + d23u1 + d24u2)

(d31vds + d32vqs + d33u1 + d34u2)]

=[u1 u2

][q1 q2q2 q3

] [u1

u2

]

+[b1 b2

][u1

u2

]

+ a, (37)

whereq1, q2, q3, b1, b2, anda are constants defined as

q1 =Lm

σLsLr(d13d43 − d23d33), (38)

q2 =1

2

Lm

σLsLr(d13d44 + d14d43 − d23d34 − d24d33), (39)

q3 =Lm

σLsLr(d14d44 − d24d34), (40)

b1=Lm

σLsLr

((d11vds+d12vqs)d43+d13(d41vds+d42vqs)

− (d21vds + d22vqs)d33 − d23(d31vds + d32vqs)),(41)

b2=Lm

σLsLr

((d11vds+d12vqs)d44+d14(d41vds+d42vqs)

− (d21vds + d22vqs)d34 − d24(d31vds + d32vqs)),(42)

a =Lm

σLsLr

((d11vds + d12vqs)(d41vds + d42vqs)

− (d21vds + d22vqs)(d31vds + d32vqs)). (43)

Substep 2.Next, we show that the quadratic function (37)relatingu1 andu2 to Te has a desirable feature: its associatedHessian matrix[ q1 q2

q2 q3 ] is alwayspositive definite, regardless ofthe parameters of the electrical part of the DFIG, as well asthe choice of the state feedback gain matrixK. The followinglemma formally states and proves this assertion:

Lemma 1. The Hessian matrix[ q1 q2q2 q3 ] in (37) is positive

definite.

Proof: From (29), (33), (36), (38), (39), and (40), thedeterminant ofA − BK and the leading principal minorsq1andq1q3 − q22 of [ q1 q2

q2 q3 ] can be written as

|A−BK| = ∆

(LsLr − L2m)2

, (44)

q1 =RsL

2m(∆2

1 +∆22)

∆2, (45)

q1q3 − q22 =R2

sL4m

∆2, (46)

where

∆ =(−RsLmk12 + (LsLr − L2

m)k13 −RsLrk14

+RrLs +RsLr

)∆1

+(RsLmk11 +RsLrk13 + (LsLr − L2

m)k14

+RsRr − LsLr + L2m

)∆2, (47)

∆1 = RsLmk21 +RsLrk23 + (LsLr − L2m)k24

+RrLs +RsLr,

6

∆2 = RsLmk22 + (−LsLr + L2m)k23 +RsLrk24

+ RsRr − LsLr + L2m,

andkij is the ij entry of K. SinceA − BK is nonsingular,Ls > Lm, andLr > Lm, it follows from (44) that∆ 6= 0.Since∆ 6= 0, it follows from (46) thatq1q3 − q22 > 0 andfrom (47) that∆1 and∆2 cannot be zero simultaneously. Thelatter, along with (45), implies thatq1 > 0. Sinceq1 > 0 andq1q3 − q22 > 0, [ q1 q2

q2 q3 ] in (37) must be positive definite.Substep 3. Next, we show that there is aredundancy

in the control variablesu1 and u2, which may be exposedvia a coordinate change. Observe from (23) that the first-order dynamics ofωr are driven byTe. Also observe fromSubsteps 1 and 2 thatTe is a convex quadratic function ofu1 and u2. Thus, we havetwo coupled control inputs (i.e.,u1 and u2) affecting one state variable (i.e.,ωr), implyingthat there is a redundancy in the control inputs, which maybe exploited elsewhere (to be discussed in Section III-C). Toexpose this redundancy, first notice that because the Hessianmatrix [ q1 q2

q2 q3 ] is positive definite, it can be diagonalized, i.e.,there exist an orthogonal matrixM containing its eigenvectorsand a diagonal matrixD containing its eigenvalues, such that

MT

[q1 q2q2 q3

]

M = D. (48)

Indeed,

M =

q2√q22+(λ1−q1)2

λ2−q3√(λ2−q3)2+q2

2

λ1−q1√q22+(λ1−q1)2

q2√(λ2−q3)2+q2

2

, D =

[λ1 00 λ2

]

,

where

λ1,2 =q1 + q3 ±

√

(q1 + q3)2 − 4(q1q3 − q22)

2.

Next, consider the following coordinate change, which trans-forms u1 ∈ R andu2 ∈ R in a Cartesian coordinate systeminto r ≥ 0 andθ ∈ [0, 2π) in a polar coordinate system:

r =√zT z, θ = tan−1

(z2z1

), (49)

where

z =

[z1z2

]

= D1/2MT

[u1

u2

]

+1

2D−1/2MT

[b1b2

]

. (50)

In terms of the new coordinatesr andθ, it follows from (37)and (48)–(50) that

Te = r2 + a′, (51)

where

a′ = a− 1

4

[b1 b2

][q1 q2q2 q3

]−1 [

b1b2

]

.

Using (38)–(43),a′ may be simplified to

a′ = −v2ds + v2qs4ωsRs

, (52)

implying that it is always negative. Comparing (51) with (37)shows that the coordinate change (49) and (50) allows us to de-couple the control variables, so that in the new coordinates, r isresponsible for driving the first-order dynamics ofωr through

Te of (51), while θ does not at all affectωr (and, hence, isredundant as far as the dynamics ofωr are concerned). Thedesign ofr andθ will be discussed in Substep 4 and SectionIII-C, respectively.

Substep 4.Finally, a speed controlleris presented, whichensures that the rotor angular velocityωr tracks a desiredtime-varying referenceωrd, to be determined in Section III-C,provided thatωrd is not exceedingly large. Combining (23)and (51) yields

Jωr = Tm(ωr, β, Vw)− r2 − a′ − Cfωr, (53)

where, according to (25),

Tm(ωr, β, Vw) =12ρACp(λ, β)V

3w

ωr. (54)

Here, Tm is written asTm(ωr, β, Vw) to emphasize its de-pendence onωr, β, and Vw. Observe from (53) that, if thecontrol inputr2 werereal-valuedinstead of being nonnegative,feedback linearization may be applied to cancel all the termson the right-hand side of (53) and insert linear dynamicsα(ωr − ωrd), i.e., we may let

r2 = Tm(ωr, β, Vw)− a′ − Cfωr + α(ωr − ωrd), (55)

so that

Jωr = −α(ωr − ωrd). (56)

By letting the controller parameterα be positive, (56) impliesthatωr always attempts to go toωrd. Unfortunately, becauser2 cannot be negative, the speed controller (55)—and, hence,the linear dynamics (56)—cannot be realized whenever theright-hand side of (55) is negative. To alleviate this issue, (55)is slightly modified by settingr2 to zero whenever that occurs,i.e.,

r2=max{Tm(ωr, β, Vw)−a′−Cfωr+α(ωr−ωrd), 0}.(57)

Notice that (57) contains a feedforward action involving the“disturbance”, i.e., the wind speedVw. This explains the firstinput of thenonlinear controllerblock in Figure 2.

To analyze the behavior of the speed controller (57), sup-poseωrd, β, andVw are constants and consider the functiong, defined as

g(ωr, β, Vw) = Tm(ωr, β, Vw)− a′ − Cfωr. (58)

The following lemma says thatg(ωr, β, Vw), when viewedas a function ofωr, has a positive rootω(1)

r , below whichg(ωr, β, Vw) is positive:

Lemma 2. For each fixedβ ∈ [βmin, βmax] and Vw > 0,there existsω(1)

r ∈ (0,∞) such thatg(ω(1)r , β, Vw) = 0 and

g(ωr, β, Vw) > 0 for all ωr ∈ (0, ω(1)r ).

Proof: Due to the fact thata′ in (52) is negative, thereexistsωr,1 such that−a′ − Cfωr > 0 for all ωr ∈ (0, ωr,1).Due to Assumption (A3) of Section II, (26), and (54), thereexistsωr,2 such thatTm(ωr, β, Vw) > 0 for all ωr ∈ (0, ωr,2).Hence, from (58), we haveg(ωr, β, Vw) > 0 for all ωr ∈(0,min{ωr,1, ωr,2}). In addition, due to Assumption (A2),

7

(26), (54), (58), anda′ < 0, there existsωr,3, sufficientlylarge, such thatg(ωr,3, β, Vw) < 0. These two properties ofg, along with Assumption (A1) and the Intermediate ValueTheorem, imply that there exists at least one positive rootωr

satisfyingg(ωr, β, Vw) = 0. Letting ω(1)r be the first of such

roots completes the proof.The following theorem, derived based on Lemma 2, says

that as long as the desired rotor angular velocityωrd is notexceedingly large, i.e., does not exceed the first rootω

(1)r of

g(ωr, β, Vw), the closed-loop dynamics (53) and (57) have anasymptotically stable equilibrium point atωrd:

Theorem 1. Consider the first-order dynamics(53) and thespeed controller(57). Supposeωrd, β, andVw are constants,with ωrd satisfying0 < ωrd < ω

(1)r . Then, for allωr(0) > 0,

limt→∞ ωr(t) = ωrd.

Proof: Substituting (57) into (53) and using (58) yield

Jωr = min{α(ωrd − ωr), g(ωr, β, Vw)}. (59)

Suppose0 < ωrd < ω(1)r . We first show thatωr = ωrd is the

unique equilibrium point of (59). Supposeωr = ωrd. Then,α(ωrd − ωr) in (59) is zero, whereasg(ωr, β, Vw) in (59)is positive, due to Lemma 2. Thus,ωr = 0, implying thatωrd is an equilibrium point. Next, suppose0 < ωr < ωrd.Then, α(ωrd − ωr) is positive, and so isg(ωr, β, Vw), dueagain to Lemma 2. Hence,ωr > 0, implying that thereis no equilibrium point to the left ofωrd. Finally, supposeωr > ωrd. Then,α(ωrd − ωr) is negative. Therefore,ωr < 0,implying that there is no equilibrium point to the right ofωrd. From the above analysis, we see thatωr = ωrd is theunique equilibrium point of (59). Next, we show that theequilibrium pointωr = ωrd is asymptotically stable in thatfor all ωr(0) > 0, limt→∞ ωr(t) = ωrd. Consider a quadraticLyapunov function candidateV : (0,∞) → R, defined as

V (ωr) =1

2(ωr − ωrd)

2, (60)

which is positive definite with respect to the shifted originωr = ωrd. From (59) and (60),

V (ωr) =1

J(ωr − ωrd)min{α(ωrd − ωr), g(ωr, β, Vw)}.

(61)

Note that whenever0 < ωr < ωrd, α(ωrd − ωr) > 0 andg(ωr, β, Vw) > 0, so thatV (ωr) < 0 according to (61). Onthe other hand, wheneverωr > ωrd, α(ωrd−ωr) < 0, so thatV (ωr) < 0. Finally, whenωr = ωrd, V (ωr) = 0. Therefore,V (ωr) is negative definite with respect to the shifted originωr = ωrd. It follows from [43] thatωr = ωrd is asymptoticallystable, i.e., for allωr(0) > 0, limt→∞ ωr(t) = ωrd.

Theorem 1 says that the first rootω(1)r is a critical root,

for which ωrd should never exceed, if we wantωr(t) to goto ωrd regardless ofωr(0). Figure 3 shows, for the MAT-LAB/Simulink R2007a model ofCp(λ, β) given in (27) and(28), how the critical rootω(1)

r depends onβ andVw. Noticefrom the figure thatω(1)

r is insensitive toβ but proportionalto Vw, meaning that the larger the wind speed, the higher the“ceiling” on the desired rotor angular velocity. Also notice that

0 2 4 6 8 10 12 14 16 18 20 22 243540

3545

3550

3555

3560

3565

Wind speed Vw (m/s)

Per

unit

Critical root ω(1)r

0 2 4 6 8 10 12 14 16 18 20 22 247080

7090

7100

7110

7120

7130

rad/s

β = 0β = 10β = 20

Fig. 3. Critical rootω(1)r as a function of blade pitch angleβ and wind

speedVw.

ω(1)r of more than 3500 in the per-unit system is extremely

large, meaning that for this particular turbine there is no needto be concerned aboutωrd exceedingω(1)

r .

C. Lyapunov-like Function and Gradient-based Approach

The third and final step of the controller development in-volves introducing a Lyapunov-like function, which measuresthe difference between the actual and desired powers, andutilizing a gradient-based approach, which minimizes thisfunction.

Recall from the beginning of Section III that the objectiveof the controller is to make the active and reactive powers,PandQ, track some desired references,Pd andQd, as closelyas possible. In the MPT mode, where the goal is to generate asmuch active power as possible while maintaining an acceptablepower factor,Pd is set to a value that far exceeds what thewind turbine can possibly produce (e.g., in the per-unit system,Pd > 1), while Qd is set to a value representing the desiredpower factor PFd = Pd√

P 2

d+Q2

d

. In this mode, makingP and

Q approachPd andQd is equivalent to maximizing the activepower output while preserving the power factor. In the PRmode, where the goal is to regulate the powers, bothPd andQd are set to values representing power demands from thegrid. In this mode, makingP and Q approachPd and Qd

amounts to achieving power regulation. Hence, the values ofPd andQd reflect the mode the wind farm operator wants thewind turbine to operate in. However, as far as the controlleris concerned, it does not distinguish between the two modes;all it does is try its best to driveP andQ to Pd andQd.

To mathematically describe the aforementioned controllerobjective, consider the following positive definite, quadraticLyapunov-like functionV of the differencesP −Pd andQ−Qd:

V =1

2

[P − Pd Q−Qd

][wp wpq

wpq wq

]

︸ ︷︷ ︸

>0

[P − Pd

Q −Qd

]

, (62)

wherewp, wq, andwpq are design parameters that allow oneto specify how the differencesP −Pd andQ−Qd, as well as

8

their correlation(P −Pd)(Q−Qd), should be penalized. Withthis V , the above controller objective can be restated simplyas:makeV go to zero, because when this happens,P andQmust both go toPd andQd. Since it is not always possibleto achieve this objective—due to the fact that the wind maynot always be strong enough—below we will attempt insteadto makeV as small as possible by minimizing it.

To minimizeV , we first show thatV is a function ofωrd,θ, β, Vw, Pd, andQd, i.e.,

V = f(ωrd, θ, β, Vw , Pd, Qd) (63)

for somef . Note from (62) thatV depends onP , Q, Pd, andQd. Also note from (17)–(20), (21), and (22) thatP andQ,in turn, depend oni, vdr, andvqr (recall thatvds andvqs areconstants). Thus,

V = f1(i, vdr, vqr, Pd, Qd) (64)

for somef1. Next, note from (9)–(12) thati depends onx;from (31) and (32) thatvdr andvqr depend onx, ωr, u1, andu2; and from (34) thatx further depends onu1 andu2. Hence,

(i, vdr, vqr) = f2(ωr, u1, u2) (65)

for somef2. Furthermore, note from (49) and (50) thatu1 andu2 depend onr andθ, wherer, in turn, depends onωr, ωrd,β, andVw through (57). Therefore,

(u1, u2) = f3(ωr, ωrd, θ, β, Vw) (66)

for somef3. Finally, assuming thatωrd does not exceed thefirst root ω(1)

r and assuming thatωrd, β, andVw are all rel-atively slow-varying (see below for a discussion), Theorem1says thatωr goes toωrd. Thus, after a short transient,

ωr ≈ ωrd. (67)

Combining (64)–(67), (63) is obtained as claimed.Now observe that the first three variables(ωrd, θ, β) in

(63) are yet to be determined, while the last three variables(Vw, Pd, Qd) are exogenous but known. Therefore, for eachgiven (Vw, Pd, Qd), (ωrd, θ, β) can be chosen correspond-ingly in order to minimizeV . This defines a mapping from(Vw, Pd, Qd) to (ωrd, θ, β), i.e.,

(ωrd, θ, β) = F (Vw, Pd, Qd)

, argmin(x1,x2,x3) f(x1,x2,x3,Vw,Pd,Qd). (68)

In principle, the mappingF in (68) may be constructedeitheranalytically, by setting the gradient off(·) to zero andsolving for the minimizer(ωrd, θ, β) in terms of(Vw , Pd, Qd),or numerically, by means of a three-dimensional lookup table.Unfortunately, the former is difficult to carry out, sincef ,being composed of several nonlinear transformations (64)–(67), has a rather complex expression. On the other hand, thelatter is costly to generate and can easily become obsolete dueto variations in system parameters. More important, selecting(ωrd, θ, β) as a static function of (Vw, Pd, Qd) as in (68)may lead to steep jumps in(ωrd, θ, β) becauseVw is ever-changing and may change dramatically, and bothPd andQd

from the wind farm operator may experience step changes.Such steep jumps are undesirable because large fluctuations

P,Q

Pd, Qd

Vw

ωr

V

λ Tm r u1, u2 vdr, vqr

β β

θ

ωrd

i x

(62)

(26)

(9)–(12)

(54) (57) (49),(50)(31),(32)

(71)

(70)

(69)

Fig. 4. Internal structure of the proposed nonlinear controller.

in ωrd may preventωr from tracking it, while discontinuouschanges inβ may be mechanically impossible to realize, causeintolerable vibrations, and substantially cut short the lifetimeof the turbine blades.

To alleviate the aforementioned deficiencies of selecting(ωrd, θ, β) according to (68), a gradient-based approach isconsidered for updating(ωrd, θ, β):

ωrd = −ǫ1∂f

∂ωrd, (69)

θ = −ǫ2∂f

∂θ, (70)

β = −ǫ3∂f

∂β, (71)

whereǫ1, ǫ2, ǫ3 > 0 are design parameters, which are meantto be relatively small, especiallyǫ1 and ǫ3, in order to avoidsteep changes inωrd andβ. The partial derivatives∂f

∂ωrd, ∂f

∂θ ,and ∂f

∂β in (69)–(71) can be calculated in a straightforwardmanner using (64)–(67), but are omitted from this paper dueto space limitations. These partial derivatives are practicallyimplementable since, likef , they depend onωrd, θ, β, Vw,Pd, andQd, all of which are known. With this gradient-basedapproach,(ωrd, θ, β) is guaranteed to asymptotically convergeto a local minimum when(Vw, Pd, Qd) is constant, and tracka local minimum when(Vw, Pd, Qd) varies.

To help the readers better understand the proposed nonlinearcontroller depicted in Figure 2 and described in this section,the internal structure of this controller is revealed in Figure 4.Observe that each arrow in this figure represents a signal,whereas each tiny box represents equations relating the signals.

IV. SIMULATION STUDIES

To demonstrate the effectiveness of the controller presentedabove, MATLAB simulations have been carried out, in whichthe controller is applied to a1.5MW GE turbine with575Vbase voltage and60Hz base frequency. To describe settingsand results of the simulations, both the per-unit system andthephysical unit system will be used, given that they are popularin the literature.

The simulation settings are as follows: All values ofthe wind turbine parameters are taken from the Wind Tur-bine Block of the Distributed Resources Library in MAT-LAB/Simulink R2007a. Specifically, the values are:ωs(pu) =1, Rs(pu) = 0.00706, Rr(pu) = 0.005, Ls(pu) = 3.071,Lr(pu) = 3.056, Lm(pu) = 2.9, J(pu) = 10.08, Cf (pu) =0.01, λnom = 8.1, and Cp nom = 0.48. In addition, the

9

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Vw

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Cp

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Pd

P

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PFd

PF

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vdrvqr

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ree

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β

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rad/s

ωrdωr

1000 1100 1200 1300 14000.5

1

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unit

Angular velocity (zoom in)

1000 1100 1200 1300 14001

2

3

rad/s

ωrdωr

Fig. 5. Scenario 1 illustrating the maximum power tracking (MPT) mode.

base wind speed isVw base= 12m/s. In MATLAB/SimulinkR2007a, the mechanical power captured is given by

Pm(pu) =Pwind basePnom

Pelec baseCp(pu)Vw(pu)

3, (72)

wherePwind base = 1.5MW, Pnom = 0.73, and Pelec base =1.5 × 106/0.9VA. From (72), it can be seen that at thebase wind speedVw(pu) = 1, Pm(pu) is capped at0.657.Furthermore, without loss of generality, the constant statorvoltages are set tovds(pu) = 1 andvqs(pu) = 0.

For the proposed controller, we let the desired poles of theelectrical dynamics (33) be located at−15, −5, and−10 ±5j, so that the corresponding state feedback gain matrixK,calculated using MATLAB’splace() function, is

K =

[5135.9 259.2 20.3 1.9−2676.7 4289.9 −1.3 19.7

]

.

In addition, we letwp = 10, wq = 1, andwpq = 0, implyingthat we penalize the difference betweenP andPd much morethan we doQ and Qd. Finally, we choose the rest of thecontroller parameters as follows:α = 10, ǫ1 = 4 × 10−3,ǫ2 = 1× 10−4, andǫ3 = 2.

Based on the above wind turbine and controller parameters,simulations have been carried out for four different scenarios.Description of each scenario, along with the simulation result,is given below:

Scenario 1: Maximum power tracking (MPT) mode. In thisscenario, we simulate the situation where the wind speedVw experiences step changes between12m/s and 7.2m/s,while the desired powersPd and Qd are kept constant at1.5MW and 0.15MW, so that the desired power factor isPFd = 0.995. SincePm cannot exceed0.657 × 1.5MW atthe base wind speedVw base = 12m/s, the wind turbine isexpected to operate in the MPT mode. Figure 5 shows thesimulation result for this scenario, where the key signals areplotted as functions of time in both the per-unit and physicalunit systems wherever applicable. Observe that, after a shorttransient, the wind turbine converts as much wind energyto electric energy as it possibly could, as indicated byCp

approaching its maximum value of0.48 in subplot 2 (whichtranslates intoP approaching its maximum possible value insubplot 3). Also observe that, whenVw goes from12m/s to7.2m/s and from7.2m/s back to12m/s, Cp drops sharplybut quickly returns to its maximum value. Note from subplot

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Vw

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Cp

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Pd

P

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PFd

PF

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vdrvqr

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ree

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β

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rad/s

ωrdωr

1000 1100 1200 1300 14000.6

0.65

0.7

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unit

Angular velocity (zoom in)

1000 1100 1200 1300 14001.2

1.3

1.4

rad/s

ωrdωr

Fig. 6. Scenario 2 illustrating the power regulation (PR) mode.

4 that, regardless ofVw, the power factor PF is maintainednear the desired level of0.995. Moreover, note from subplots5 and 6 that the angular velocityωr tracks the desired time-varying referenceωrd closely (subplot 6 is a zoom-in versionof subplot 5). Finally, the control inputsvdr, vqr , andβ areshown in subplots 7 and 8, respectively. Note that, to maximizeCp, β is kept at its minimum valueβmin = 0deg.

Scenario 2: Power regulation (PR) mode. In this scenario,we simulate the situation whereVw is kept constant at thebase value of12m/s, while Pd experiences step changes from0.45MW to 0.3MW and then to0.6MW, andQd is such thatPFd = 0.995. SincePd is always less than0.657×1.5MW atthe base wind speed of12m/s, the wind turbine is expected tooperate in the PR mode with different setpointsPd. Figure 6shows the simulation result for this scenario. Observe fromsubplot 2 thatCp is less than its maximum value of0.48.This suggests that the wind turbine attempts to capture lesspower than what it possibly could from wind, sincePd isrelatively small. Indeed, as can be seen from subplots 3 and4, the turbine produces just enough active and reactive powers,makingP trackPd closely while maintaining PF at PFd. Alsoobserve from subplots 5 and 6 thatωr closely followsωrd, asdesired. Finally, note from subplot 8 thatβ increases slightly

in order to capture less power between 1200s and 2400s, whenPd is smallest.

Scenario 3: Seamless switching between the MPT and PRmodes. In this scenario, we simulate the situation wherePd ex-periences large step changes between1.5MW and0.75MW,Qd again is such that PFd is 0.995, and an actual wind profilefrom a wind farm located in northwest Oklahoma is used todefineVw. The actual wind profile consists of 145 samples,taken at the rate of one sample per 10 minutes, over a 24-hour period. In order to use this wind profile in a 1-hoursimulation (as in Scenarios 1 and 2), we compress the timescale, assuming that the samples were taken over a 1-hourperiod. Note that compressing the time scale in this way makesthe problem more challenging because the wind speed variesfaster than it actually does. Figure 7 shows the simulationresult for this scenario, with subplot 1 displaying the windprofile. Observe from subplots 2 and 3 that, for the first 1200seconds during whichPd is 1.5MW, the turbine operates inthe MPT mode, grabbing as much wind energy as it possiblycould, by drivingCp to 0.48 and maximizingP . At time1200s whenPd abruptly drops from1.5MW to 0.75MW, theturbine seamlessly switches from the MPT mode to the PRmode, quickly reducingCp, accurately regulatingP around

11

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PR

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P

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ωrdωr

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1000 1100 1200 1300 14001.7

1.9

2.1

rad/s

ωrdωr

Fig. 7. Scenario 3 illustrating the seamless switching between the MPT and PR modes under an actual wind profile from a windfarm located in northwestOklahoma.

Pd, and effectively rejecting the “disturbance”Vw. Note thatbetween 2100s and 2400s, the wind is not strong enough tosustain the PR mode. As a result, the MPT mode resumesseamlessly, as indicated byCp returning immediately to itsmaximum value of0.48. Finally, at time 2400s whenPd goesfrom 0.75MW back to1.5MW, the turbine keeps working inthe MPT mode, continuing to maximize bothCp andP . Noticefrom subplots 4–6 that, over the course of the simulation, bothPF andωr are maintained at PFd andωrd, respectively, despitethe random wind fluctuations. Also notice from subplot 8 thatβ increases somewhat during the PR mode in order to helpcapture less power.

Scenario 4: Robustness of the proposed controller. In thisscenario, we simulate the exact same situation as that ofScenario 3 (i.e., with the sameVw, Pd, and Qd) but withmodeling errors and measurement noise. That is, we allowfor modeling errors in the friction coefficientCf and theperformance coefficientCp (due, for example, to changingweather conditions, blade erosions, and aging) as well asmeasurement noise in the wind speedVw (sinceVw is usuallymeasured by an anemometer located on the nacelle behindthe blades of a wind turbine). Specifically, we assume that

2 4 6 8 10 12 140

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Tip speed ratio λ

Bla

de

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angle

β(d

eg) Nominal Cp(λ, β) used by the controller

0.48

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5

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0

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0.3

0.3

0.35

Tip speed ratio λ

Bla

de

pitch

angle

β(d

eg) Actual Cp(λ, β) used in the simulation

0.39

Fig. 8. Contour plots of the nominal and actualCp(λ, β) for Scenario 4.

the nominalCf used by the controller is0.01(pu), whereasthe actualCf used in the simulation is0.012(pu), so that

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2

Time (s)

Per

unit

Angular velocity

0 1000 2000 30000

1

2

3

4

rad/s

ωrdωr

1000 1100 1200 1300 14000.85

0.95

1.05

Time (s)

Per

unit

Angular velocity (zoom in)

1000 1100 1200 1300 14001.7

1.9

2.1

rad/s

ωrdωr

Fig. 9. Scenario 4 illustrating the robustness of the proposed controller to modeling errors inCf andCp and noisy measurements inVw.

Cf has a 20% modeling error. Moreover, we assume that thenominalCp used by the controller is given by (27) and (28)with c1 = 0.5176, c2 = 116, c3 = 0.4, c4 = 5, c5 = 21, andc6 = 0.0068, whereas the actualCp used in the simulation isalso given by (27) and (28) but withc1 = 0.45, c2 = 115,c3 = 0.5, c4 = 4.5, c5 = 22, and c6 = 0.003. Figure 8displays the contour plots of the nominal and actualCp(λ, β)for λ ∈ [2, 15] andβ ∈ [0, 15], showing thatCp has noticeablemodeling errors. In particular, the nominalCp attains itsmaximum of0.48 at (λ, β) = (8.1, 0), whereas the actualCp

attains its maximum of0.39 at (λ, β) = (8.45, 0). Finally, weassume that the measuredVw used by the controller, denotedasVw meas, is related to the actualVw used in the simulationvia

Vw meas(t) = Vw(t) + 0.5 + 0.5 sin(0.5t) + 0.25 cos(t),

where the second term on the right-hand side representsa constant measurement bias, while the third and fourthrepresent measurement noises with different amplitudes andfrequencies. Figure 9 shows the simulation result for thisscenario. Comparing this figure with Figure 7, the followingobservations can be made: first,Cp in Figure 9 attains itsmaximum value of0.39 in the MPT mode, as opposed to the

0.48 attained byCp in Figure 7. Second, PF in Figure 9 has alarger fluctuation compared to PF in Figure 7, but nonethelessis maintained around PFd. Third,ωr in Figure 9 does not trackωrd as closely asωr in Figure 7 does. Nevertheless, despite thewind fluctuations, modeling errors, and noisy measurements,the controller performs reasonably well, as evident by howcloseCp is to its maximum value of0.39 in the MPT mode,how closeP is to Pd in the PR mode, and how close PF isto PFd throughout the simulation. Therefore, the controller isfairly robust.

As it follows from Figures 5–9 and the above discussions,the proposed controller exhibits excellent performance. Specif-ically, the controller works well in both the MPT mode understep changes in the wind speed (Scenario 1) and the PR modeunder step changes in the power commands (Scenario 2). Inaddition, it is capable of seamlessly switching between thetwo modes in the presence of changing power commandsand a realistic, fluctuating wind profile (Scenario 3). Finally,the controller is robust to small modeling errors and noisymeasurements commonly encountered in practice (Scenario 4).

13

V. CONCLUSION

In this paper, we have developed a feedback/feedforwardnonlinear controller, which accounts for the nonlinearities ofvariable-speed wind turbines with doubly fed induction gen-erators, and bypasses the need for approximate linearization.Its development is based on applying a mixture of linearand nonlinear control design techniques on three time scales,including feedback linearization, pole placement, and gradient-based minimization of a Lyapunov-like potential function.Simulation results have shown that the proposed scheme notonly effectively controls the active and reactive powers inboththe MPT and PR modes, it also ensures seamless switchingbetween the two modes. Therefore, the proposed controllermay be recommended as a candidate for future wind turbinecontrol.

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Choon Yik Tang (S’97–M’04) received the B.S. and M.S. degrees inmechanical engineering from Oklahoma State University, Stillwater, in 1996and 1997, respectively, and the Ph.D. degree in electrical engineering fromthe University of Michigan, Ann Arbor, in 2003. From 2003 to 2004,he was a Postdoctoral Research Fellow in the Department of ElectricalEngineering and Computer Science at the University of Michigan. From 2004to 2006, he was a Research Scientist at Honeywell Labs, Minneapolis. Since2006, he has been an Assistant Professor in the School of Electrical andComputer Engineering at the University of Oklahoma, Norman. His currentresearch interests include systems and control theory, distributed algorithmsfor computation and optimization over networks, and control and operationof wind farms.

Yi Guo (S’08) received the B.S. degree from Tianjin Polytechnic University,Tianjin, China, and the M.S. degree from Tianjin University, Tianjin, China, in2002 and 2005, respectively. He is currently working towardhis Ph.D. degreein the School of Electrical and Computer Engineering at the University ofOklahoma, Norman. His current research interests include control theory andapplications, with an emphasis on control of wind turbines and wind farms.

John N. Jiang (SM’07) is an Assistant Professor in the Power System Groupin the School of Electrical and Computer Engineering at the University ofOklahoma, Norman. He holds M.S. and Ph.D. degrees from the University ofTexas at Austin. He has been involved in a number of wind energy relatedprojects since 1989 in design, installation of stand-alonewind generationsystems, the market impact of wind generation in Texas, and recent large-scale wind farms development in Oklahoma.