Control and disturbances compensation for doubly-fed induction generators using the Derivative-Free Nonlinear Kalman Filter

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Control and Disturbances Compensation for DoublyFed Induction Generators Using the Derivative-Free

Nonlinear Kalman FilterGerasimos Rigatos, Member, IEEE, Pierluigi Siano, Member, IEEE, Nikolaos Zervos,

and Carlo Cecati, Fellow, IEEE

Abstract—The paper studies differential flatness properties andan input–output linearization procedure for doubly fed inductiongenerators (DFIGs). By defining flat outputs which are associatedwith the rotor’s turn angle and the magnetic flux of the stator, anequivalent DFIG description in the Brunovksy (canonical) form isobtained. For the linearized canonical model of the generator, afeedback controller is designed. Moreover, a comparison of the dif-ferential flatness theory-based control method against Lie algebra-based control is provided. At the second stage, a novel KalmanFiltering method (Derivative-free nonlinear Kalman Filtering) isintroduced. The proposed Kalman Filter is redesigned as distur-bance observer for estimating additive input disturbances to theDFIG model. These estimated disturbance terms are finally usedby a feedback controller that enables the generator’s state vari-ables to track desirable setpoints. The efficiency of the proposedstate estimation-based control scheme is tested through simulationexperiments.

Index Terms—Derivative-free nonlinear Kalman filtering, dif-ferential flatness theory, disturbance estimator, doubly fed in-duction generator (DFIG), input–output linearization, nonlinearcontrol.

I. INTRODUCTION

DOUBLY fed induction generators (DFIG) have beenwidely used in variable-speed fixed frequency hydro-

power generation systems, wind-power generation systems, andturbine engine power generation systems [1]–[3]. DFIGs haveproven to be more efficient than squirrel-cage induction gener-ator systems and the synchronous generator systems in terms ofcost and losses of the associated power electronics converters.DFIG systems can operate either in grid-connected mode or instand-alone mode [4]–[9]. Results on the reliable connection ofDFIGs to the electricity grid have been presented in [10]–[12].Moreover, several field-oriented control schemes have been pro-posed for both operation modes. Additionally, to control elec-tric power generators and the power electronics that enable their

Manuscript received February 25, 2014; revised August 24, 2014; acceptedOctober 28, 2014. Recommended for publication by Associate Editor Z. Chen.

G. Rigatos and N. Zervos are with the Unit of Industrial Automation, In-dustrial Systems Institute, 26504 Rion Patras, Greece (e-mail: [email protected];[email protected]).

P. Siano is with the Department of Industrial Engineering, University ofSalerno, 84084 Fisciano, Italy (e-mail: [email protected]).

C. Cecati is with the Department of Informatics, University of L’ Aquila,67100 L’ Aquila, Italy (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2014.2369412

connection to the grid, feedback linearization approaches havebeen developed [13]–[14]. In parallel, several results have beenpublished on sensorless control of DFIG [15]–[19]. Taking intoaccount that the installation and maintenance of sensors for mea-suring several parameters of the generator’s state vector can betechnically difficult or costly, the need for developing sensor-less control schemes for DFIG becomes apparent. In this paper,a novel sensorless control scheme is developed using flatness-based control theory and a state estimation method that is basedon Kalman Filtering.

Using the electric equations of the stator and rotor, a dynamicmodel for the DFIG is derived. The DFIG is analogous to theinduction motor. In an induction motor, the stator voltage playsthe role of an input variable, while the rotor voltage is a constant.In case of the doubly fed induction machine it is quite similarbut the other way round, with a dual analogy to hold betweenthe stator and rotor parameters of the generator and the motor.This means that the rotor voltage now acts as an input, while thestator voltage is a constant parameter. The stator’s and rotor’svoltages, currents, and magnetic flux are represented as vectorsin a rotating orthogonal axis frame. The complete sixth-ordermodel of the DFIG captures efficiently transients at both thestator and the rotor side.

In this paper, differential flatness theory has been proposed forthe control of the DFIG. Differential flatness theory is currentlya main direction in nonlinear dynamical systems and enableslinearization and control for a wide class of systems, in a moreefficient manner than Lie-algebra methods [20]–[23]. To find outif a dynamical system is differentially flat, the following shouldbe examined: 1) the existence of the so-called flat output, i.e.,a new variable which is expressed as a function of the system’sstate variables. The flat output and its derivatives should not becoupled in the form of an ordinary differential equation (ODE),2) the components of the system (i.e., state variables and controlinput) should be expressed as functions of the flat output andits derivatives [24]–[29]. In certain cases, differential flatnesstheory enables transformation to a linearized form (canonicalBrunovsky form) for which the design of the controller becomeseasier. In other cases by showing that a system is a differentiallyflat one can easily design a reference trajectory as a function ofthe so-called flat output and can find a control law that assurestracking of this desirable trajectory [25], [26].

This paper is concerned with proving differential flatnessof the model of the DFIG and its resulting description in theBrunovksy (canonical) form [20], [21]. By defining flat outputs

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which are associated with the rotor’s angle and with the mag-netic flux of the stator, an equivalent DFIG description in theBrunovksy (linear canonical) form is obtained. It is shown thatfor the linearized DFIG’s model, it is possible to design a feed-back controller. At the second stage, a novel Kalman Filteringmethod, the Derivative-free nonlinear Kalman Filter, is pro-posed for estimating the state vector elements of the linearizedsystem which are not directly measurable. With the redesignof the proposed Kalman filter as a disturbance observer, it be-comes possible to estimate also disturbance terms affecting theDFIG model and to use these terms in the feedback controller.By avoiding linearization approximations, the proposed filter-ing method improves the accuracy of estimation and resultsin smooth control signal variations and in minimization of thetracking error of the associated control loop [30]–[32].

The structure of the paper is as follows. In Section II, themodel of the DFIG is analyzed. and the associated state-spaceequations are formulated. In Section III, input–output lineariza-tion for the DFIG model is performed using Lie algebra theory.In Section IV, differential flatness for nonlinear dynamical sys-tems is analyzed. Conditions, which are based on differentialflatness theory, are provided for transforming MIMO dynami-cal systems into the linear canonical form. In Section V, input–output linearization of the DFIG is performed using differentialflatness theory. In Section VI, the design of a Kalman Filter-based disturbance observer for the DFIG model is explained.In Section VII, simulation tests are carried out to evaluate theperformance of the DFIG control scheme that is based on dif-ferential flatness theory. Finally, in Section VIII, concludingremarks are given.

II. MODEL OF THE DFIG

A. Complete Sixth-Order Model of the Induction Generator

The DFIG is not only the most widely used technology inwind turbines due to its good performance, but it is also usedin many other fields such as hydropower generation, pumpedstorage plants and flywheel energy storage systems. The DFIGmodel is derived from the voltage equations of the stator androtor. It is assumed that the stator and rotor windings are sym-metrical and symmetrically fed. Usually, the saturation of theinductances, iron losses, skin effect, and bearing friction is ne-glected. Moreover, the winding resistance is considered to beconstant.

This type of wound-rotor machine is connected to the grid byboth the rotor and stator side. The DFIG stator can be directlyconnected to the electric power grid while the rotor is interfacedthrough back-to-back converters (see Fig. 1). By decoupling thepower system’s electrical frequency and the rotor mechanicalfrequency, the converter allows a variable speed operation of thewind turbine. The DFIG is analogous to the induction motor. Inan induction motor, the stator voltage plays the role of an inputvariable, while the rotor voltage is a constant (it is usually zero).In case of the doubly fed induction machine, it is very similarbut the other way round, with a dual analogy to hold betweenthe stator and rotor parameters of the generator and the motor.This means that the rotor voltage now acts as an input, while

Fig. 1. Configuration of a DFIG unit in the power grid.

the stator voltage depends on the voltage at the bus to which theDFIG is connected, and in the dq reference frame is a constantparameter [33]–[35].

In a compact form, the DFIG can be described by the follow-ing set of equations in the dq reference frame that rotates at anarbitrary speed denoted as ωdq [4]

dψsq

dt= − 1

τsψsq

− ωdqψsd+

M

τsirq

+ vsq(1)

dψsd

dt= ωdqψsq

− 1τs

ψsd+

M

τsird

+ vsd(2)

dirq

dt=

β

τsψsq

+ βωrψsd− γ2irq

−(ωdq − ωr )ird− βvsq

+1

σLrvrq

(3)

dird

dt= −βωrψsq

+β

τsψsd

+ (ωdq − ωr )irq

− γ2ird− βvsd

+1

σLrvrd

(4)

where ψsq, ψsd

, irq, ird

are the stator flux and the rotor currents,vsq

, vsd, vrq

, vrdare the stator and rotor voltages, Ls and Lr

are the stator and rotor inductances, ωr is the rotor’s angularvelocity, M is the magnetizing inductance. Moreover, denotingas Rs and Rr the stator and rotor resistances the followingparameters are defined

σ = 1 − M 2

LrLs, β = 1−σ

M σ , τs =Ls

Rs(5)

τr =Lr

Rr, γ2 =

(1−σστs

+ 1στr

).

The angle of the vectors that describe the magnetic flux ψsα andψsb is first defined for the stator, i.e.,

ρ = tan−1(

ψsb

ψsa

). (6)

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RIGATOS et al.: CONTROL AND DISTURBANCES COMPENSATION FOR DOUBLY FED INDUCTION GENERATORS 3

The angle between the inertial reference frame and the rotatingreference frame is taken to be equal to ρ.

Moreover, it holds that cos(ρ) = ψs a

||ψ || , sin(ρ) = ψs b

||ψ || , and

||ψ|| =√

ψ2sα

+ ψ2sb

. Therefore, in the rotating dq frame ofthe generator, and under the condition of field orientation, therewill be only one nonzero component of the magnetic flux ψsd

,while the component of the flux along the q-axis equals 0.

The dynamic model of the DFIG can be also written in state-space equations form by defining the following state variables:x1 = θ, x2 = ωr , x3 = ψsd

, x4 = ψsq, x5 = ird

, and x6 = irq.

It holds that

x1 = x2 (7)

x2 = − Km

Jx2 −

Tm

J+

η

J(irq

x3 − irdx4) (8)

x3 = − 1τs

x3 + ωdqx4 +M

τsx5 + vsd

(9)

x4 = −ωdqx3 −1τs

x4 +M

τsx6 + vsq

(10)

x5 = −βx2x4 +β

τsx3 + (ωdq − x2)x6 − γ2x5

+1

σLrvrd

− βvsd(11)

x6 =β

τsx4 + βx2x3 − (ωdq − x2)x5

− γ2x6 +1

σLrvrq

− βvsq. (12)

In the above set of equations, J is the moment of inertia of therotor, Tm is the externally applied mechanical torque that makesthe turbine rotate, Km is the friction coefficient, η is a variablethat is associated to the number of poles and to the mutualinductance M . Variable η in turn determines the electrical torqueTe which is associated with rotor currents and stator magneticflux. Equation (7)–(12) can be written also in the form

x = f(x) + ga(x)vrd+ gb(x)vrq

(13)

where x = [x1 , x2 , x3 , x4 , x5 , x6 ]T and

f(x) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x2

−Km

J x2 − Tm

J + nJ (irq

x3 − irdx4)

− 1τs

x3 + ωdqx4 + Mτs

x5 + vsd

−ωdqx3 − 1τs

x4 + Mτs

x6 + vsq

−βx2x4 + βτs

x3 + (ωdq − x2)x6 − γ2x5 − βvsd

βτs

x4 + βx2x3 − (ωdq − x2)x5 − γ2x6 − βvsq

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ga(x) = (0 0 0 01

σLr0)T

gb(x) = (0 0 0 0 01

σLr)T . (14)

The active and reactive power delivered by the DFIG stator areassociated with the real and imaginary part of the power at thestator’s terminals, i.e.

Ps = Re{UsI∗s} = vsd

isd+ vsq

isq(15)

Qs = Im{UsI∗s} = vsd

isq− vsq

isd. (16)

III. INPUT–OUTPUT LINEARIZATION OF THE DFIG USING LIE

ALGEBRA THEORY

A. Input–Output Linearization of the DFIG Model

The following variables are defined:

h1(x) = x1 = θ

h2(x) = x23 + x2

4 = ψ2sd

+ ψ2sq

. (17)

Next, based on h1 and h2 , the following transformed state vari-ables are defined :

z1 = h1(x) = θ (18)

z2 = Lf h1(x)⇒ (19)

z2 = f1⇒z2 = x2⇒z2 = ω.

Similarly, one has

z3 = L2f h1(x) = Lf z2 ⇒ (20)

z3 = f2⇒z3 = −Km

Jx2 −

Tm

J+

η

J(irq

x3 − irqx4) ⇒

z3 = −Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4).

For the transformed state variable z4 one has

z4 = h2(x) = ψ2sd

+ ψ2sq

= x23 + x2

4 (21)

and

z5 = Lf h2(x)⇒z5 = 2x3f3 + 2x4f4⇒ (22)

z5 = 2x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+ 2x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

].

After the change of the state variables, it holds (the completeproof is given in Appendix I)

z1 = z2

z2 = z3

z3 = L3f h1(x) + (Lga

L2f h1(x))u1 + (Lgb

L2f h1(x))u2

z4 = z5

z5 = L2f h2(x) + (Lga

Lf h2)(x)u1 + (LgbLf h2(x))u2 . (23)

The inputs of the above linearized and decoupled DFIG modelare u1 = urd

and u2 = urq. The system of (24) can be written

in the input–output linearized form(

z(3)1

z4

)= fa + Mu (24)

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where

fa(x)=

(L3

f h1(x)

L2f h2(x)

), M =

(Lga

L2f h1(x) Lgb

L2f h1(x)

LgaLf h2(x) Lgb

Lf h2(x)

)

(25)

or equivalently, one has the system’s description in the MIMOcanonical form⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎝

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎝

0 00 01 00 00 1

⎞⎟⎟⎟⎟⎠

(v1v2

)

(26)where

v1 = L3f h1(x) + (Lga

(L2f h1(x))u1 + (Lgb

L2f h1(x))u2

v2 = L2f h2(x) + (Lga

Lf h2(x))u1 + (LgbLf h2(x))u2 . (27)

Returning to the compact form of (26), one has

(z

(3)1

z4

)=

(v1v2

)(28)

and the control signal that assures convergence of the z1 and z4to the reference setpoints zd

1 and zd4 is given by

v1 = zd1

(3) − k(1)1 (z1 − zd

1 ) − k(1)2 (z1 − zd

1 ) − k(1)3 (z1 − zd

1 )

v2 = zd4 − k

(2)1 (z4 − zd

4 ) − k(2)2 (z4 − zd

4 ). (29)

B. State Estimation-Based Control

For the implementation of the aforementioned control law,there is a need to obtain measurements of all elements of theDFIG’s state vector. The rotor’s turn angle can be measured di-rectly with the use of an encoder [36], [37]. Knowing the rotor’sangle and with the use of the decoupled induction machine’smodel of (28), it is possible to estimate the rotor’s angular speed.Similarly, after obtaining measurements of the magnetic flux atthe stator and with the use of the decoupled induction machine’smodel of (28), it is possible to estimate the derivatives of themagnetic flux. Due to the fact that the magnetic flux of the statorψs cannot be measured directly, equations that provide indirectmeasurements of the flux (computed through measurements ofthe stator and rotor currents) will be used, that is

ψsd= Lsisd

+ Mird

ψsq= Lsisq

+ Mirq. (30)

It is noted that the currents are measured in the ab referenceframe, and their computation in the dq reference frame requiresthe application of the associated reference frame transformation.

Using the model of (28), the state estimator for the DFIG isgiven by

˙z = Az + Bv + K(zmeas − Cz) (31)

where the estimator’s gain K∈R5×2

A =

⎛⎜⎜⎜⎜⎝

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

⎞⎟⎟⎟⎟⎠

, B =

⎛⎜⎜⎜⎜⎝

0 00 01 00 00 1

⎞⎟⎟⎟⎟⎠

C =(

1 0 0 0 00 0 0 1 0

)(32)

IV. DIFFERENTIAL FLATNESS FOR NONLINEAR

DYNAMICAL SYSTEMS

A. Definition of Differentially Flat Systems

Differential flatness is a structural property of a class of non-linear systems, denoting that all system variables (such as statevector elements and control inputs) can be written in terms ofa set of specific variables (the so-called flat outputs) and theirderivatives. The following nonlinear system is considered:

x(t) = f(x(t), u(t)). (33)

The time is t∈R, the state vector is x(t)∈Rn with initial condi-tions x(0) = x0 , and the input is u(t)∈Rm . Next, the propertiesof differentially flat systems are given [20]–[29].

The finite dimensional system of (37) can be written inthe general form of an ODE, i.e., Si(w, w, w, . . . , w(i)), i =1, 2, . . . , q. The term w is a generic notation for the systemvariables (these variables are for instance the elements of thesystem’s state vector x(t) and the elements of the control inputu(t)) while w(i) , i = 1, 2, . . . , q are the associated derivatives.Such a system is differentially flat if there are m functionsy = (y1 , . . . , ym ) of the system variables and of their time-derivatives, i.e., yi = φ(w, w, w, . . . , w(αi )), i = 1, . . . , m sat-isfying the following two conditions [23]–[27].

1) There does not exist any differential relation of the formR(y, y, . . . , y(β )) = 0 which implies that the derivativesof the flat output are not coupled in the sense of an ODE,or equivalently, it can be said that the flat output is differ-entially independent.

2) All system variables (i.e., the elements of the system’sstate vector w and the control input) can be expressedusing only the flat output y and its time derivatives wi =ψi(y, y, . . . , y(γi )), i = 1, . . . , s. An equivalent definitionof differentially flat systems is as follows.

Definition: The system x = f(x, u), x∈Rn , u∈Rm is differ-entially flat if there exist relations

h : Rn×(Rm )r+1→Rm

φ : (Rm )r→Rn and

ψ : (Rm )r+1→Rm (34)

such that

y = h(x, u, u , . . . , u(r))

x = φ(y, y , . . . , y(r−1)), and

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u = ψ(y, y, . . . , y(r−1) , y(r)). (35)

This means that all system dynamics can be expressed as afunction of the flat output and its derivatives; therefore, the statevector and the control input can be written as

x(t) = φ(y(t), y(t), . . . , y(r)(t)), and

u(t) = ψ(y(t), y(t), . . . , y(r+1)(t)). (36)

It is noted that for linear systems the property of differentialflatness is equivalent to that of controllability.

B. Conditions for Applying Differential Flatness Theory

The generic class of nonlinear systems x = f(x, u) is con-sidered. Such a system can be transformed to the form of anaffine in the input system by adding an integrator to each input[28]

x = f(x) +m∑

i=1

gi(x)ui. (37)

If the system of (41) can be linearized by a diffeomorphismz = φ(x) and a static state feedback u = α(x) + β(x)v intothe following form:

zi,j = zi+1,j for 1≤j≤m and 1≤ i≤ vj − 1

zvi , j= vj (38)

with∑m

j=1vj = n, then yj = z1,j for 1≤ j ≤m are the 0-flatoutputs which can be written as functions of only the elementsof the state vector x. To define conditions for transforming thesystem of (41) into the canonical form described in (42), thefollowing theorem holds [28].

Theorem: For the nonlinear systems described by (41), thefollowing variables are defined: 1) G0 = span[g1 , . . . , gm ],2) G1 = span[g1 , . . . , gm , adf g1 , . . . , adf gm ], · · · (k) Gk =span{adj

f gi for 0≤ j ≤ k, 1≤ i≤m}. Then, the linearizationproblem for the system of (41) can be solved if and only if: 1)The dimension of Gi, i = 1, . . . , k is constant for x∈X⊆Rn

and for 1≤ i≤n − 1, 2) the dimension of Gn−1 is of order n,3) the distribution Gk is involutive for each 1≤ k≤n − 2.

C. Transformation of MIMO Nonlinear Systems Into theBrunovsky Form

It is assumed now that after defining the flat outputs of theinitial MIMO nonlinear system, and after expressing the systemstate variables and control inputs as functions of the flat outputand of the associated derivatives, the system can be transformedin the Brunovsky canonical form:

x1 = x2

· · ·xr1 −1 = xr1

xr1 = f1(x) +p∑

j=1

g1j(x)uj + d1

xr1 +1 = xr1 +2

· · ·xp−1 = xp

xp = fp(x) +p∑

j=1

gpj(x)uj + dp

y1 = x1

y2 = x2

· · ·yp = xn−rp +1 (39)

where x = [x1 , . . . , xn ]T is the state vector of the transformedsystem (according to the differential flatness formulation),u = [u1 , . . . , up ]T is the set of control inputs, y = [y1 , . . . , yp ]T

is the output vector, fi are the drift functions, and gi,j , i, j =1, 2 , . . . , p are smooth functions corresponding to the controlinput gains, while dj is a variable associated to external distur-bances. In holds that r1 + r2 + · · · + rp = n. Having writtenthe initial nonlinear system into the canonical (Brunovsky) formit holds

y(ri )i = fi(x) +

p∑j=1

gij (x)uj + dj . (40)

Next the following vectors and matrices can be defined f(x) =[f1(x), . . . , fn (x)]T , g(x) = [g1(x), . . . , gn (x)]T with gi(x) =[g1i(x), . . . , gpi(x)]T , and also A = diag[A1 , . . . , Ap ], B =diag[B1 , . . . , Bp ], C = diag[C1 , . . . , Cp ], d = [d1 , . . . , dp ]T ,where matrix A has the MIMO canonical form, i.e., with block-diagonal elements

Ai =

⎛⎜⎜⎜⎜⎜⎝

0 1 · · · 00 0 · · · 0...

... · · ·...

0 0 · · · 10 0 · · · 0

⎞⎟⎟⎟⎟⎟⎠

ri ×ri

(41)

BTi = (0 0 · · · 0 1)1×ri

Ci = (1 0 · · · 0 0)1×ri.

Thus, (44) can be written in state-space form

x = Ax + Bv + Bd

y = Cx (42)

where the control input is written as v = f(x) + g(x)u.

V. INPUT–OUTPUT LINEARIZATION OF THE DFIG USING

DIFFERENTIAL FLATNESS THEORY

A. Differential Flatness Properties of the DFIG

The flat outputs of the system are defined as

y1 = θ or y = x1

y2 = ψ2sd

+ ψ2sq

or y2 = x23 + x2

4 . (43)

It holds that

y1 = ω or y1 = x2 ⇒

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y1 = ω = −Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4) ⇒

y1 = ω = −Km

Jy1 −

Tm

J+

η

J(x6x3 − x5x4). (44)

Deriving the last row of (44) with respect to time, one obtains

y(3)1 = −Km

Jy1 +

η

J(x6x3 + x6 x3 − x5x4 − x5 x4) ⇒

y(3)1 = −Km

Jy1 +

η

Jx3

{[β

τsx4 + βx2x3 + (ωdq − x2)x5

−γ2x6− βvsq

]+

1σLr

u1

}+

η

Jx6

[− 1

τsx3 + ωdqx4 +

M

τsx5

+ vsd

]− η

Jx4

{[− βx2x4 +

β

τsx3 + (ωdq − x2)x6 − γ2x5

− βvsd

]+

1σLr

u2

}− η

Jx5

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

].

(45)

Moreover, about the second flat output, it holds

y2 = 2x3 x3 + 2x4 x4⇒

y2 = 2x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+ 2x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]⇒ (46)

Consequently, it holds

y2 = 2x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+ 2x3

[− 1

τsx3 + ωdq x4 +

M

τsx5

]

+ 2x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]

+ 2x4

[−ωdq x3 −

1τs

x4 +M

τsx6

](47)

or equivalently,

y2 = 2[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]2

− 2τs

x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

− 2ωdqx3

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]

+2M

τsx3

{[− βx2x4 +

β

τsx3 + (ωdq − x2)x6

− γ2x5 − βvsd

]+

1σLr

u1

}

+ 2[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]2

− 2ωdqx4

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

− 2τs

x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]

2x4M

τs

{[β

τsx4 + βx2x3 + (ωdq − x2)x5 −

− γ2x6 − βvsq

]+

1σLr

u2

}. (48)

It holds that x1 = y1 , x2 = y1 . From the second row of (43) andconsidering that the field orientation condition requires x4 =ψsq

= 0, one obtains that x3 =√

y2 . Moreover, from (44) itholds

y1 = −Km

Jy1 −

Tm

J+

η

J

√y2x6⇒

x6 =y1 + Km

J y1 + Tm

JηJ

√y2

, y2 = 0. (49)

From (46), one obtains

y2 = − 2τs

x23 +

2M

τsx3x5 + 2vsd

x3 ⇒

y2 +(

2τs

x3 − 2vsd

)x3 =

2M

τsx3x5 ⇒

x5 =y2 +

(2τs

√y2 − 2vsd

)√y2

2Mτs

√y2

y2 = 0. (50)

Therefore, x5 is also a function of the flat output and of itsderivatives. Additionally, by solving the system of (49) and (52)with respect to the control inputs u1 and u2 , one obtains that thecontrol inputs are functions of the flat output and its derivatives.Therefore, the model of the DFIG is a differentially flat one.

Next, to design the flatness-based controller for the DFIG,the following transformation of the state variables is introduced:z1 = y1 , z2 = y1 , z3 = y1 , z4 = y2 , z5 = y2 for which holds

z1 = z2

z2 = z3

z3 = L3f h1(x) + (Lga

L2f h1(x))u1 + (Lgb

L2f h1(x))u2

z4 = z5

z5 = L2f h2(x) + (Lga

Lf h2(x))u1 + (LgbLf h2(x))u2 . (51)

Therefore, one obtains the decoupled and linearized representa-tion of the system(

z(3)1

z4

)=

(L3

f h1(x)

L2f h2(x)

)+

+

(Lga

L2f h1(x) Lgb

L2f h1(x)

LgaLf h2(x) Lgb

Lf h2(x)

)(u1u2

)(52)

or equivalently,(

z(3)1

z4

)= fa + Mu (53)

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where

fa =

(L3

f h1(x)

L2f h2(x)

), M =

(Lga

L2f h1(x) Lgb

L2f h1(x)

LgaLf h2(x) Lgb

Lf h2(x)

).

(54)

By defining the control inputs v1 = L3f h1(x) +

(LgaL2

f h1(x))u1 + (LgbL2

f h1(x))u2 and v2 = L2f h2(x) +

(LgaLf h2(x))u1 + (Lgb

Lf h2(x))u2 , one can also have thedescription in the MIMO canonical form⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎝

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎝

0 00 01 00 00 1

⎞⎟⎟⎟⎟⎠

(v1v2

).

(55)The control input for the linearized and decoupled model of theDFIG is chosen as follows:

v1 = zd1

(3) − k(1)1 (z1 − zd

1 ) − k(1)2 (z1 − zd

1 ) − k(1)3 (z1 − zd

1 )

v2 = zd4 − k

(2)1 (z4 − zd

4 ) − k(2)2 (z4 − zd

4 ) (56)

and finally, the control input that is applied to the system is

u = M−1(−fa + v). (57)

The proposed control scheme can work with the use of mea-surements from a small number of sensors. That is, there isneed to obtain measurements of only y1 = θ which is theturn angle of the generator’s rotor, and of the magnetic fluxy2 = ψ2

s = ψ2sd

+ ψ2sq

, or due to the orientation of the magneticfield y2 = ψ2

s = ψ2sd

. The stator flux (ψs) cannot be measureddirectly from a sensor (e.g., the use of Hall sensor in an electricmachine with a rotating part would not be efficient); however(32) that relates stator flux and stator and rotor currents can beused to calculate ψs . Thus one has

ψsd= Lsisd

+ Mird

ψsq= 0 (58)

which means that by measuring stator and rotor currents onecan obtain an indirect measurement of the stator’s magneticflux ψsd

. Next, one can compute the dynamics of the magneticflux, jointly with the dynamics of the rotor’s motion throughthe use of the Derivative-free Nonlinear Kalman Filter. Thisestimation method is based on the application of the KalmanFilter recursion to the linearized equivalent of the generator’smodel which is given by (59). Actually, (59) can be written inthe state-space form

z = Az + Bv

zmeas = Cz (59)

where

A =

⎛⎜⎜⎜⎜⎝

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

⎞⎟⎟⎟⎟⎠

, B =

⎛⎜⎜⎜⎜⎝

0 00 01 00 00 1

⎞⎟⎟⎟⎟⎠

C =(

1 0 0 0 00 0 0 1 0

). (60)

The estimator’s dynamics is

˙z = A·z + B·v + K(zmeas − Cz) (61)

where K∈R5×2 is the state estimator’s gain. Defining as Ad ,Bd , and Cd , the discrete-time equivalents of matrices A, B, andC, respectively, the associated Kalman Filter-based estimator isgiven by [38]–[42]

measurement update:

K(k) = P−(k)CTd [Cd ·P−(k)CT

d + R]−1

z(k) = z−(k) + K(k)[zmeas(k) − Cdz−(k)]

P (k) = P−(k) − K(k)CdP−(k) (62)

time update:

P−(k + 1) = Ad(k)P (k)ATd (k) + Q(k)

z−(k + 1) = Ad(k)z(k) + Bd(k)v(k). (63)

Remark 1: The first linearization approach followed inSection III was based on differential geometry and the compu-tation of Lie Derivatives. The second linearization approach fol-lowed in Section V was based on differential flatness theory. Formulti-input systems which admit static feedback linearization,the differential flatness theory-based approach is equivalent tolinearization based on Lie algebra. As it can be confirmed from(28) and (59), the two linearization methods provided the samelinearized model of the DFIG. The differential flatness theorycan be also extended to MIMO systems that admit only dynamicfeedback linearization. In the latter case, an extended state vec-tor of the controlled system is defined containing as additionalstate variables the derivatives of the control input. In dynamicfeedback linearization, the control input that is finally applied tothe system contains integral terms of the error’s state vector. Interms of computation, the differential flatness theory-based lin-earization is simpler because it does not require the calculationof Lie derivatives. Moreover, by expressing all state variablesas functions of the flat output and its derivatives, the differen-tial flatness theory-based linearization enables to perform stateestimation and to reconstruct the state variables of the initialnonlinear system. This is not possible for the Lie algebra-basedapproach, where to perform filtering it is necessary to computeand invert the Jacobian matrix of the transformed state vector[43].

Remark 2: Regarding comparison to existing results, it isnoted that in other control approaches for DFIGs, e.g., controlof the rotor’s speed and of the stator’s magnetic flux in cascadingloops analyzed in [33], one has to use again measurements ofthe rotor currents. One can estimate the rest of the DFIG statevariables with filtering methods that make use of the initialnonlinear model of the system, such as the Extended or theUnscented Kalman Filter. Being based on an exact linearizationmethod, the control and state estimation approach for the DFIGthat is presented in Section V has the advantage of using areduced number of sensors while at the same time remaining

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robust to modeling uncertainties and external perturbations andavoiding numerical approximation errors.

Remark 3: It has been explained that the concept of sensor-less control is to reduce the number of sensors needed for theimplementation of feedback control. As explained in Section IIIand in Section V, the DFIG MIMO nonlinear model is trans-formed into two decoupled systems in the canonical linear form.The first system has as output the turn angle of the rotor whichcan be measured with the use of an encoder. The second sys-tem has as output the magnetic flux of the stator, where due tothe field orientation condition only the d-axis flux componentis nonzero. Using measurements of the stator’s and rotor’s cur-rents, one can obtain a measurement of the stator’s magneticflux too. By considering as measurable outputs the rotor’s turnangle and the stator’s magnetic flux, the observability of thelinearized DFIG model is assured. Thus, it is possible to per-form state estimation for the nonmeasurable state variables andto develop sensorless control, using the observers of (33) and(61).

VI. KALMAN FILTER-BASED DISTURBANCE OBSERVER FOR

THE DFIG MODEL

A. Application of a Disturbance Observer to the DFIG Model

Next, it will be considered that additive input disturbances(e.g., due to load variations) affect the DFIG model. The si-multaneous estimation of the nonmeasurable elements of theDFIG state vector as well as the estimation of additive distur-bance terms affecting the generator is possible with the use of adisturbance estimator [44]–[47].

It is assumed that the third and fifth row of the state-spaceequations of the DFIG of (55) include a disturbance term

z3 = L3f h1(x) + Lga

(L2f h1(x))u1

+ Lgb(L2

f h2(x))u2 + d1


(Lf h2(x))u1

+ Lgb(Lf h2(x))u2 + d2 . (64)

Without loss of generality, the dynamics of the disturbanceterms is described by their second-order derivatives and the

associated initial conditions, i.e., ¨d1 = fa(x) and ¨

d2 = fb(x).Next, an extended state-space model of the system is definedthat comprises as additional state variables the disturbance

terms z6 = d1 , z7 = ˙d1 , while z8 = d2 , and z9 = ˙

d2 . Thus,the extended state-space model is written as z1 = z2 , z2 = z3 ,z3 = v1 + z6 , z4 = z5 , and z5 = v2 + z8 , z6 = z7 , z7 = fa ,z8 = z9 and z9 = fb , or in matrix form one has

˙z = Az + Bv

zmeas = Cz (65)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

z1z2z3z4z5z6z7z8z9

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

z1z2z3z4z5z6z7z8z9

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 00 0 0 01 0 0 00 0 0 00 1 0 00 0 0 00 0 1 00 0 0 00 0 0 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎝

v1v2

fa

fb

⎞⎟⎟⎠

(zmeas

1zmeas

4

)=

(1 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0

)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

z1z2z3z4z5z6z7z8z9

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (66)

The associated state estimator is

ˆz = Ao z + Bo v1 + Ko(zmeas − Cz) (67)

where

Ao =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Bo =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 00 0 0 01 0 0 00 0 0 00 1 0 00 0 0 00 0 0 00 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Co =(

1 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0

)(68)

while the estimator’s gain Ko∈R9×2 is obtained from the stan-dard Kalman Filter recursion [38]–[42].

Defining as Ad , Bd , and Cd , the discrete-time equivalents ofmatrices Ao , Bo , and Co , respectively, a Derivative-free non-linear Kalman Filter can be designed for the aforementionedrepresentation of the system dynamics [23], [31]. The associ-ated Kalman Filter-based disturbance estimator is given by

measurement update:


d + R]−1

ˆz(k) = z−(k) + K(k)[Cd z(k) − Cdˆz−(k)]

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P (k) = P−(k) − K(k)CdP−(k) (69)

time update:

P−(k + 1) = Ad(k)P (k)ATd (k) + Q(k)

ˆz−(k + 1) = Ad(k)ˆz(k) + Bd(k)v(k). (70)

Remark 4: The advantages of the proposed nonlinear feed-back control method for DFIGs (that is based on differential flat-ness theory and on the Derivative-free nonlinear Kalman Filter)against PID-type control (included in vector control loops) areobvious. In most cases, the application of PID control to electricmachines is based on heuristic parameters tuning, has no stabil-ity proof, and has limited robustness to the change of operatingpoints or to the effects of external perturbations. Moreover, inthe case of multivariable systems such as DFIGs the applicationof PID control is known to have questionable performance. Thefirst vector control approaches for asynchronous machines (e.g.[35]) made use of multiple PID loops which were implementedin a cascaded manner (for controlling separately the magneticflux and the rotation angle of the machine). Such methods werebased on the assumption that the flux and the rotation speedbecome finally decoupled at steady state. However, there is noproof about that (it cannot be always assured that transients willbe eliminated and the machine will reach a steady state), andtherefore, the performance of the control loop is not alwaysguaranteed (see attached paper). Consequently, although PIDcontrol is met in some cases in asynchronous machines, it is notthe recommended solution.

Remark 5: Field-oriented (vector) control has been for manyyears a common approach for the control of DFIGs. However,comparing to the flatness-based control approach developed inthis paper, vector control exhibits several weaknesses whichmake its performance be questionable [50], [51]. As it is shownin detail in Appendix II, the implementation of vector control re-quires measurement or estimation of the stator’s magnetic flux.Therefore, one comes against the observer or Kalman Filter de-sign problem that was solved in a conclusive manner in Section 6of this manuscript. Moreover, vector control for DFIGs requiresthe tuning of the several PID and PI controllers, and this limits itsreliability only round local operating points. Consequently, thestability and robustness properties of the field-oriented controlfor DFIGs are doubtful.

Remark 6: It is confirmed that the linearized equivalentmodel of the DFIG, after application of the pole placement tech-nique has poles, exclusively in the left complex semiplane. Be-sides the inclusion of the additional control input that compen-sates for the estimated additive disturbance terms improves therobustness features of this control loop. It is also noted that thelinearized equivalent model of the DFIG exhibits multiple polesat the origin. This particular form implies an infinite gain mar-gin and a sufficiently large phase margin. Finally, it is noted thatthe stability and robustness features of the control scheme whichcomprises also estimation and compensation of the disturbancesare similar to those of LQG control. According to the above, thepaper justifies sufficiently the stability and disturbance rejec-tion capability of the proposed feedback control scheme. On the

Fig. 2. Control loop of the DFIG comprising a flatness-based control elementand an estimator for disturbances compensation.

TABLE IRATINGS OF THE MODELED DFIG

Rated power 15.5 kW

Number of Pole pairs 4Stator Resistance 0.58 ΩStator Inductance 13 mHRotor Resistance 1.30 ΩRotor Inductance 3 mHMutual Inductance 10 mHRotor’s inertia 20.0 kg · m2

other hand, the presented simulation experiments demonstratedthe efficiency of the control method in tracking rapidly chang-ing reference setpoints while also succeeding good transients.The disturbances appearing in the simulation experiments couldbe met in adverse operating conditions of the power generator.Even for the latter case, the good performance of the controlloop is confirmed.

VII. SIMULATION TESTS

The structure of the proposed control scheme is depictedin Fig. 2. The control scheme comprises 1) the flatness-basedcontrol part which computes the control signal for the system’sequivalent model that is transformed to the linear canonicalform, 2) a Kalman Filter-based disturbances estimator whichprovides estimates for the elements of the state vector of theDFIG, such as rotor’s speed , magnetic flux at the stator aswell as disturbances affecting the generator’s model. Indicativenumerical values for the parameters of the considered DFIGmodel are given in Table I.

Simulation tests were carried out for two different setpointsof the turn speed of the generator’s rotor. The values of the gen-erator’s state vector elements are actually measured in SI units;however, in the simulation results, they are expressed in the perunit (p.u.) system. The results obtained for the first setpoint aredepicted in Figs. 3 and 4. Similarly, the results obtained for thesecond setpoint are depicted in Figs. 5 and 6. It can be observedthat the proposed control scheme assures that the rotor’s turn

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Fig. 3. DFIG setpoint 1: (a) Control of state variable x2 = ω. (b) Control ofstate variable x3 = ψsd

.

Fig. 4. DFIG setpoint 1: (a) Control of state variable x5 = irdand of state

variable x6 = ir q . (b) Estimation of disturbance inputs di , i = 1, 2 and of theirderivatives.


.

speed follows a specific setpoint, while tracking of referencesetpoints is succeeded for the components of the magnetic fluxand for the rotor’s currents. Several reference setpoints havebeen defined for the DFIG state variables, i.e., rotor’s angularspeed ω, rotor currents ird

, irq, and the magnetic flux ψsd

, and asit can be observed from the associated diagrams, the proposedcontrol scheme resulted in fast and accurate convergence tothese setpoints. The disturbance observer that was based on the



Fig. 7. Convergence of the stator’s magnetic flux ψsdto the reference setpoint

(a) without using the disturbance observer, (b) when using the disturbanceobserver.

Derivative-free nonlinear Kalman Filter was capable of estimat-ing the unknown and time-varying input disturbances affectingthe DFIG model. The selection of the magnetic flux setpointsappearing in Figs. 3 and 5 did not aim to be restricted only to thecase that the DFIG is connected to a grid, which is characterizedby constant voltage amplitude and frequency. The purpose of thesimulation experiments was to show that the proposed nonlinearcontrol scheme succeeds convergence to time-varying magneticflux setpoints (e.g., piecewise constant ones). Of course, the an-alyzed control method for the DFIG enables also convergenceof the stator’s flux to constant setpoints, but this is a subcase ofwhat has already been presented.

The improvement in the performance of the control loopthat is due to the use of a disturbance observer based on theDerivative-free nonlinear Kalman Filter is explained as follows:1) compensation of the disturbance terms which are generatedby parametric uncertainty or unknown external inputs, 2) moreaccurate estimation of the disturbance terms because the filter-ing procedure is based on an exact linearization of the system’sdynamics and does not introduce numerical errors (as for exam-ple in the case of the Extended Kalman Filter). This is shown inFig. 7.

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Remark 7: The implementation of the control scheme withthe use of a digital processor does not exhibit any difficulty.The application of Kalman Filter-based control loops is a com-mon practice in other complicated and demanding cases (e.g., inautonomous navigation of aircrafts, in robots, etc., [23]). There-fore, the method can be also applied, through a programmabledigital controller, in the case of asynchronous electric machinestoo. As explained, the use of the Derivative-free nonlinearKalman Filter as a disturbance observer enables identificationand compensation of external perturbations in real time (such asdisturbances due to grid faults). Therefore, the proposed controlscheme exhibits improved robustness.

Remark 8: The generator’s speed can be efficiently con-trolled, and the associated rotation speed setpoints can bereached by applying the proposed control scheme. It is pos-sible to operate the generator at variable speed, thus also chang-ing the levels of the generated power. Moreover, as shown inSection V, all currents and voltages defining the active and re-active power of the generator, according to (16) and (17), can bewritten explicitly or implicitly, as functions of the flat outputs ofthe DFIG model. This explains why these variables are finallyassociated with the turn speed of the rotor, and consequently,why the produced power of the generator is determined by therotor’s angular velocity.

Remark 9: The differential flatness properties of the DFIGdynamic model are initially proven, considering that the exter-nal mechanical torque is constant or piecewise constant. Equiv-alently, it can be considered that Tm stands for an unknowntime-varying disturbance term to the DFIG model [48], [49]. Insuch a case, variable Tm is omitted from the DFIG model, whichat a second stage is transformed into the linear canonical anddecoupled form using the diffeomorphism provided by differ-ential flatness theory. For the linearized equivalent of the DFIGmodel, a Kalman Filter-based disturbance observer is designedfollowing the method of Section VI. The Kalman Filter-baseddisturbance estimator can identify in real time the aggregate dis-turbance term which incorporates the time-varying torque Tm .Therefore, the proposed nonlinear control scheme, that is basedon disturbances estimation with the use of the Derivative-freenonlinear Kalman Filter, can work well even if the mechanicaltorque Tm causing the turbine’s rotation is completely unknownand time-varying.

Remark 10: Variables d1 and d2 appearing in (64) are aggre-gate disturbance terms which include any type of perturbationsthat may be due to load variations and change of the statorcurrents, change of the mechanical torque, voltage fluctuationand faults in the grid (vsd

and vsqnon constant) or modeling

uncertainty, and changes in the numerical values of the param-eters appearing in the DFIG model. Representing the aggregatedisturbances effects as in (64) enables the design of a distur-bances estimator and compensator based on the Derivative-freenonlinear Kalman Filter.

VIII. CONCLUSION

The paper has proposed a nonlinear control scheme forDFIGs. Estimation of disturbance terms affecting the DFIG

model has been performed with the use of a new nonlinear fil-tering approach, the so-called derivative-free nonlinear KalmanFilter. First, it was proven that the dynamic model of the DFIGis a differentially flat one, and this enabled its description inthe Brunovksy (linear canonical) form. It has been shown thatfor the linearized DFIG model, it is possible to design a statefeedback controller. At the second stage, a novel Kalman Fil-tering method, the Derivative-free nonlinear Kalman Filter, hasbeen proposed for estimating the nonmeasurable elements of thedynamic model of the DFIG. It has been shown that by avoid-ing linearization approximations, the proposed filtering method,improves the accuracy of estimation and results in smooth con-trol signal variations and in minimization of the tracking errorof the DFIG control loop. Moreover, with the redesign of theproposed Kalman Filter as a disturbance observer, it becamepossible to obtain estimates of disturbance terms affecting theDFIG model. The DFIG’s control input was generated by in-cluding in the state-feedback control law an input that is basedon the estimate of the disturbance terms. Simulation tests havebeen provided to evaluate the performance of the nonlinear con-trol scheme.

APPENDIX I

Input–output linearization of the DFIG model with use of Liealgebra

The following variables have been defined:

h1(x) = x1 = θ

h2(x) = x23 + x2

4 = ψ2sd

+ ψ2sq

. (71)

Next, based on h1 , h2 , the following transformed state variablesare defined

z1 = h1(x) = θ (72)

z2 = Lf h1(x)⇒z2 = f1⇒z2 = x2⇒z2 = ω. (73)

Similarly, one has

z3 = L2f h1(x) = Lf z2 ⇒

z3 = f2⇒z3 = −Km

Jx2 −

Tm

J+

η

J(irq

x3 − irqx4) ⇒

z3 = −Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4). (74)

Moreover, one has

z3 = L3f h1(x) + (Lga

L2f h1(x))u1 + (Lgb

L2f h2(x))u2 . (75)

It holds that

L3f h1(x) = Lf z3 (76)

L3f h1(x) = −Km

J

[−Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4)

]

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+η

Jx6

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

− η

Jx5

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]

− η

Jx4

[−βx2x4 +

β

τsx3 + (ωdq − x2)x6 − γ2x5 − βvsd

]

+η

Jx3

[β

τsx4 + βx2x3 + (ωdq − x2)x5 − γ2x6 − βvsq

].

(77)

Equivalently, one has

Lga(L2

f h1(x)) = Lgaz3⇒

Lga(L2

f h1(x)) = − η

J

1σLr

x4 (78)

and similarly,

Lgb(L2

f h1(x)) = Lgbz3⇒

Lgb(L2

f h1(x)) =η

J

1σLr

x3 . (79)


z4 = h2(x) = ψ2sd

+ ψ2sq

= x23 + x2

4 (80)

and

z5 = Lf h2(x)⇒z5 = 2x3f3 + 2x4f4⇒

z5 = 2x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+ 2x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

](81)

and equivalently, one has


(Lf h2(x))u1 + Lgb(Lf h2(x))u2 . (82)

It holds that

L2f h2(x) =

(− 4

τsx3−

2M

τsx5 + 2vsd

)[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+(− 4

τsx4 +

2M

τsx6 + 2vsq

)[−ωdqx3−

1τs

x4 +M

τsx6 + vsq

]

+(

2M

τsx3

)[−βx2x4 +

β

τsx3 +(ωdq − x2)x6− γ2x5−βvsd

]

+(

2M

τsx3

) [β

τsx4 + βx2x3 +(ωdq − x2)x5 − γ2x6 − βvsq

].

(83)

Moreover, it holds that

Lga(Lf h2(x)) =

2M

τsx3ga5 ⇒Lga

(Lf h2(x)) =2M

τs

1σLs

x3

(84)

and in a similar manner

Lgb(Lf h2(x)) =

2M

τsx4ga6 ⇒Lga

(Lf h2(x)) =2M

τs

1σLs

x4 .

(85)Next, it is confirmed that after change of the state variables, itholds

z1 = z2

z2 = z3


(L2f )h1(x)u1 + Lgb

(L2f )h1(x)u2

z4 = z5


(Lf )h2(x)u1 + Lgb(Lf )h2(x)u2 . (86)

It holds that z1 = θ, z1 = ω = z2 , z2 = ω = f2(x) + ga2 u1 +gb2 u2⇒z2 = f2(x) + 0u1 + 0u2 which finally gives z2 =f2(x). Moreover, it has been proven that z3 = f2 , therefore,it holds z2 = z3 . Moreover, it holds that

z3 =∂z3

∂x1x1 +

∂z3

∂x2x2 +

∂z3

∂x3x3 +

∂z3

∂x4x4 +

∂z3

∂x5x5 +

∂z3

∂x6x6

(87)which in turn gives

z3 =∂z3

∂x1f1 +

∂z3

∂x2f2 +

∂z3

∂x3f3 +

∂z3

∂x4f4

+∂z3

∂x5

(f5 +

1σLr

u1

)+

∂z3

∂x6

(f6 +

1σLr

u2

)(88)

that is also written as


(L2f h1(x))u1 + Lgb

(L2f h2(x))u2 . (89)

Similarly, one has

z4 = x23 + x2

4⇒z4 = 2x3 x3 + 2x4 x4⇒z4 = 2x3f3 + 2x4f4

z4 = 2x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+ 2x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]⇒z4 = z5 . (90)

Additionally, it holds

z5 =∂z5

∂x1x1 +

∂z5

∂x2x2 +

∂z5

∂x3x3 +

∂z5

∂x4x4 +

∂z5

∂x5x5 +

∂z6

∂x6

x6


z5 =∂z5

∂x1f1 +

∂z5

∂x2f2 +

∂z5

∂x3f3 +

∂z5

∂x4f4

+∂z5

∂x5

(f5 +

1σLr

u1

)+

∂z5

∂x6

(f6 +

1σLr

u2

)(92)

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which subsequently gives


(Lf h2(x))u1 + Lgb(Lf h2(x))u2 (93)

which is the anticipated relation about z5 . Consequently, (86)is confirmed to hold. The system of (86) can be written in theinput–output linearized form

(z

(3)1

z4

)= fa + Mu (94)

where

fa(x) =

(L3

f h1(x)

L2f h2(x)

)

M =

(Lga

L2f h1(x) Lgb

L2f h2(x)

LgaLf h1(x) Lgb

Lf h2(x)

)(95)

or equivalently, one has the system’s description in the MIMOcanonical form

⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎝

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎝

0 00 01 00 00 1

⎞⎟⎟⎟⎟⎠

(v1v2

)

(96)where

v1 = L3f h1(x) + Lga

(L2f h1(x))u1 + Lgb

(L2f h2(x))u2


(Lf h1(x))u1 + Lgb(Lf h2(x))u2 . (97)

APPENDIX II

Field-oriented control of the DFIG The classical method forinduction machines control was introduced by Blaschke (1971)and in the DFIG case is based on a transformation of the rotor’scurrents (ir α ) and (ir b ) and of the magnetic fluxes of the stator(ψsα and ψsb ) to the reference frame dq which rotates togetherwith the rotor [52]. Thus, the controller’s design uses the currentsir d and ir q and the fluxes ψsd and ψsq [35]. The angle of thevectors that describe the magnetic fluxes ψsα and ψsb is firstdefined, i.e.,

ρ = tan−1(

ψsb

ψsa

). (98)

The angle between the inertial reference frame of the stator andthe rotating reference frame of the rotor is taken to be equal toρ. The transition from (ir α , ir b) to (ir d , ir q ) is given by

(ird

ir q

)=

(cos(ρ) sin(ρ)−sin(ρ) cos(ρ)

)(irα

ir b

). (99)

The transition from (ψsα , ψr b) to (ψsd, ψsq ) is given by

(ψsd

ψsq

)=


) (ψsα

ψsb

). (100)


||ψ || , sin(ρ) = ψs b

||ψ || , and ||ψ|| =√ψ2

sα+ ψ2

sb. Using the above transformation ones obtains

ir d =ψsαir α + ψsbir b

||ψ|| ψsd = ||ψ||(101)

ir q =ψsαir b − ψsbir α

||ψ|| ψsq = 0.

Therefore, in the rotating frame dq of the generator, there willbe only one nonzero component of the magnetic flux ψsd

, whilethe component of the flux along the d-axis equals 0. The newinputs of the system are considered to be vr d , vr q , which areconnected to vr a , vr b according to the relation

(vrα

vr b

)= ||ψ||·

(ψsa ψsb

ψsb ψsa

)−1 (vrd

vr q

). (102)

In the new coordinates, the induction generator model has beendescribed in (7) to (12). The state-space model of the inductiongenerator has been defined in (13) and (14). Using the statevariables notation, the DFIG model was written in the form

x1 = x2

x2 = −Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4)

x3 = − 1τs

x3 + ωdqx4 +M

τsx5 + vsd


x4 +M

τsx6 + vsq

x5 = −βx2x4 +β

τsx3 + (ωdq − x2)x6 − γ2x5

+1

σLrvrd

− βvsd

x6 = − β

τsx4 + βx2x3 − (ωdq − x2)x5 − γ2x6

+1

σLrvrq

− βvsq. (103)

Next, the following nonlinear feedback control law is defined(

vrd

vr q

)=

σLr

(βx2x4 − β

τsx3 − (ωdq − x2)x6 + βvsd

+ βv1βτs

x4 − βx2x3 + (ωdq − x2)x5 + βvsq+ βv2

).(104)

The terms in (104) have been selected so as to linearize the fifthand sixth row of the state-space model of the induction generatorin (103) and to produce first-order linear differential equations.The control signal in the fixed coordinates system a − b will be

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(vr α

vr b

)= ||ψ||σLr

(ψsα

ψsb

−ψsb ψsα

)−1

·(

βx2x4 − βτs

x3 − (ωdq − x2)x6 + βvsd+ βv1

βτs


). (105)

Substituting (104) into (103), one obtains:

x1 = x2 (106)

x2 = −Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4) (107)

x3 = − 1τs

x3 + ωdqx4 +M

τsx5 + vsd

(108)


x4 +M

τsx6 + vsq

(109)

x5 = −γ2x5 + βv1 (110)

x6 = −γ2x6 + βv2 (111)

The system of (106)–(111) comprises two linear subsystems,where the first one has as output the magnetic flux x3 = ψsd

and the second has as output the rotation speed x2 = ω [35].Thus, from (108) and (110), one obtains

x3 = − 1τs

x3 +M

τsx5 + vsd

(112)

x5 = −γ2x5 + βv1 (113)

while from (107) and (111), one obtains

x2 = −Km

Jx2 −

Tm

J+

η

Jx3x6 (114)

x6 = −γ2x6 + βv2 . (115)

For x3 = ψsd, it holds that if ψsd→ψs

refd , i.e., the transient

phenomena for ψsd have been eliminated, and therefore, ψsd

has converged to a steady-state value, then the two subsystemsdescribed by (112)–(113) and (114)–(115) are decoupled.

The subsystem that is described by (112) and (113) is lin-ear with control input v1 and can be controlled using methodsof linear control, such as optimal control, or PID control. Forinstance, the following PI controller has been proposed for thecontrol of the magnetic flux [35]

v1(t) = −kd1(ψsd − ψsrefd ) − kd2

∫ t

0(ψsd(τ) − ψsd

ref (τ)dτ.

(116)Thus, if (116) is applied to the subsystem that is described by(112) and (113), one anticipates to succeed ψsd(t)→ψs

refd (t).

Now, the subsystem that consists of (114) and (115) is ex-amined. The term T = η

J x6x3 denotes the torque developed inthe rotor. After succeeding ψsd→ψs

refd , one can also control the

generator’s speed ω, using linear feedback control algorithms.A first approach to the control of the speed ω is to use nested PI

loops, i.e.,

v2 = −Kq 1(T − Tref ) − Kq 2

∫ t

0(T (t) − Tref (t))dτ

Tref = −Kq 3(ω − ωref )− Kq 4

∫ t

0(ω(t)− ωref (t))dτ. (117)

From the above analysis, it becomes clear that a remaining prob-lem in the implementation of field-oriented control for DFIGs ishow to measure efficiently x3 = ψsd(t). Therefore, one comesagainst the need for applying a state observer or Kalman Filter-ing. Besides, the tuning of the multiple PID and PI controllersthat constitute the field-oriented control scheme, as described in(116) and (117), remains valid only round local operating points,and thus, the stability and robustness of the field-oriented controlfor DFIGs cannot be assured.

REFERENCES

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[14] X. Bao, F. Zhuo, Y. Tian, and P. Tan, “Simplified feedback linearizationcontrol of three-phase photovoltaic inverter with an LCL filter,” IEEETrans. Power Electron., vol. 28, no. 6, pp. 2739–2752, Jun. 2013.

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[16] P. Aguglia, P. Voiarouge, R. Wamkeue, and J. Cros, “Determination offault operation dynamical constraints for the design of wind turbine DFIGdrives,” Math. Comput. Simul., vol. 81, pp. 252–262, 2010.

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[17] F. Wu, X. P. Zhang, P. Ju, and M. J. H. Sterling, “Decentralized nonlinearcontrol of wind turbine with doubly-fed induction generator,” IEEE Trans.Power Syst., vol. 23, no. 2, pp. 613–621, May 2008.

[18] S. Peresada, A. Tilli, and A. Tonielli, “Power control of a doubly fedinduction machine via output feedback,” Control Eng. Pract. ,vol. 12, pp.41–57, 2004.

[19] V. Calderaro, V. Galdi, A. Piccolo, and P. Siano, “A fuzzy controller formaximum energy extraction from variable speed wind power generationsystems,” Electric Power Syst. Res., vol. 78, pp. 1109–1118, 2008.

[20] J. Rudolph, Flatness Based Control of Distributed Parameter Systems.Aachen, GermanyX: Shaker, 2003.

[21] H. Sira-Ramirez and S. Agrawal, Differentially Flat Systems. New York,NY, USA: Marcel Dekker, 2004.

[22] F. Wu, X. P. Zhang, P. Ju, and M. J. Sterling, “Decentralized nonlinearcontrol of wind turbine with doubly fed induction generator,” IEEE Trans.Power Syst., vol. 23, no. 2, pp. 613–621, May 2008.

[23] G. G. Rigatos, “Modelling and control for intelligent industrial systems:Adaptive algorithms in robotcs and industrial engineering,” New York,NY, USA: Springer, 2011.

[24] J. Levine, “On necessary and sufficient conditions for differential flatness,”Appl. Algebra Eng., Commun. Comput., vol. 22, no. 1, pp. 47–90, 2011.

[25] M. Fliess and H. Mounier, “Tracking control and π-freeness of infinitedimensional linear systems,” in Dynamical Systems, Control, Coding andComputer Vision, vol. 258, G. Picci and D. S. Gilliam Eds. Basel, Switzer-land: Birkhauser, 1999, pp. 41–68.

[26] J. Villagra, B. d’Andrea-Novel, H. Mounier, and M. Pengov, “Flatness-based vehicle steering control strategy with SDRE feedback gains tunedvia a sensitivity approach,” IEEE Trans. Control Syst. Technol., vol. 15,no. 3, pp. 554–565, May 2007.

[27] P. Martin and P. Rouchon, “Two remarks on induction motors,” in Proc.Comput. Eng. Syst. Appl., Lille, France, 1996, vol. 1, pp. 76–79.

[28] S. Bououden, D. Boutat, G. Zheng, J. P. Barbot, and F. Kratz, “A triangularcanonical form for a class of 0-flat nonlinear systems,” Int. J. Control, vol.84, no. 2, pp. 261–269, 2011.

[29] H. Sira-Ramirez and M. Fliess, “On the output feedback control of asynchronous generator,” in Proc. 43rd IEEE Conf. Decision Control, Dec.2004, pp. 4459–4464.

[30] G. Rigatos, P. Siano, and N. Zervos, “PMSG sensorless control with theuse of the derivative-free nonlinear Kalman Filter,” presented at the IEEEInt. Conf. Clean Electr. Power, Alghero, Italy, Jun. 2013.

[31] G. G. Rigatos, “A derivative-free Kalman Filtering approach to stateestimation-based control of nonlinear dynamical systems,” IEEE Trans.Ind. Electron., vol. 59, no. 10, pp. 3987–3997, Oct. 2012.

[32] G.Rigatos, P. Siano, and N. Zervos, “Sensorless control of distributedpower generators with the derivative-free nonlinear Kalman Filter,” IEEETrans. Ind. Electron., vol. 61, no. 11, pp. 6369–6382, Nov. 2014.

[33] G. Rigatos and P. Siano, “DFIG control using differential flatness theoryand extended Kalman filtering,” presented at the 14th IFAC Int. Conf.Inform. Control Problems Manuf., Bucharest, Romania, May 2012.

[34] G. Rigatos and P. Siano, “A nonlinear H-infinity feedback control approachfor asynchronous generators IEEE ICCEP 2015,” presented at the 5th Int.Conf. Clean Electr. Power, Taormina, Italy, Jun., 2015.

[35] G. G. Rigatos, “Adaptive fuzzy control for field-oriented induction motordrives,” Neural Comput. Appl., vol. 21, no. 1, pp. 9–23, 2011.

[36] E. Barrera-Cardiel and N. Pastor-Gomez, “Microcontroller-based power-angle instrument for a power system laboratory,” in Proc. IEEE PoweringEng. Soc. Summer Meet., 1999, pp. 1008–1012.

[37] R. M. Kennel, “Why do incremental encoders do a reasonably good jobin electrical drives with digital control?” in Proc. IEEE 41st IAS Annu.Meeting. Conf. Rec. Ind. Appl. Conf., 2006, pp. 925–930.

[38] E. W. Kamen and J. K. Su, Introduction to Optimal Estimation. New York,NY USA: Springer, 1999.

[39] M. Basseville and I. Nikiforov, Detection of Abrupt Changes: Theory andApplications. Englewood Cliffs, NJ, USA: Prentice-Hall, 1993.

[40] J. Xiong, An Introduction to Stochastic Filtering Theory. London, U.K.:Oxford Univ. Press, 2008.

[41] G.G. Rigatos and S.G. Tzafestas, “Extended Kalman filtering for fuzzymodelling and multi-sensor fusion, ” in, Mathematical and ComputerModelling of Dynamical Systems. New York, NY, USA: Taylor & Francis,vol. 13, pp. 251–266, 2007.

[42] G. Rigatos and Q. Zhang, “Fuzzy model validation using the local statis-tical approach,” Fuzzy Sets Syst., vol. 60, no. 7, pp. 882–904, 2009.

[43] G. Rigatos, P. Siano, and N. Zervos, “Derivative-free nonlinear Kalmanfiltering for PMSG sensorless control,” in Mechatronics Engineering:

Research Development and Education, M. Habibf[ , Ed. New York, NY,USA: Wiley, 2012.

[44] R. Miklosovich, A. Radke, and Z. Gao, “Discrete implmentation andgeneralization of the Extended State Observer,” in Proc. IEEE Amer.Control Conf., Minneapolis, MN, USA, Jun. 2006.

[45] R. Cortesao, “On Kalman Active Observers, ” Journal of Intelligent andRobotic Systems. New York, NY, USA: Springer, vol. 48, no. 2, pp. 131–155, 2006.

[46] A. Gupta and M. K. O. Malley, “Disturbance-observer-based force esti-mation for haptic feedback,” ASME J. Dyn. Syst. Meas. Control, vol. 133,no. 1, art. no. 14505, 2011.

[47] S. Kwon and W.K. Chung, “Combined synthesis of state estimator andperturbation observer,” J. Dyn. Syst. Meas. Control, vol. 125, pp. 19–26,2003

[48] G. Rigatos, P. Siano, and C. Cecati, “An H-infinity feedback controlapproach for three-phase voltage source converters,” presented at the IEEEInd. Electron. Soc., Dallas, TX, USA, Oct. 2014.

[49] G. Rigatos, P. Siano, and N. Zervos, “PMSG sensorless control with thederivative-free nonlinear Kalman Filter for distributed generation units,”presented at the 11th IFAC Int. Workshop Adapt. Learn. Control SignalProcess., Caen, France, Jul. 2013.

[50] G. D. Marques and D. M. Sousa, “Stator flux active damping methodsfor field-oriented doubly fed induction generator,” IEEE Trans. EnergyConvers., vol. 27, no. 3, pp. 799–806, Sep. 2012.

[51] J. Yao, Hui Li, Z. Chen, X. Xia, X. Chen, Q. Li, and Y. Liao, “Enhancedcontrol of a DFIG-based wind-power generation system with series grid-side converter under unbalanced grid voltage conditions,” IEEE Trans.Power Electron., vol. 28, no. 7, pp. 3167–3181, Jul. 2013.

[52] F. Blaschke, “The principle of field orientation applied to the new transvec-tor closed-loop control system for rotating field machines,” Siemens-Rev.,vol. 39, pp. 217–220, 1971.

Gerasimos Rigatos (M’98) received the Diploma de-gree in electrical engineering and the Ph.D. degreein control systems both from the National Techni-cal University of Athens (NTUA), Athens, Greece,in 1995 and 2000, respectively.

He is currently a Researcher (Grade B’) at the In-dustrial Systems Institute, in Rion Patras, Greece. Hehas held research and teaching positions at INRIA-IRISA France in 2001, Universite Paris XI in 2007,Harper-Adams University College, U.K. from 20112012) and in Greek universities. His research interests

include the areas of control and robotics, mechatronics, electric power systems,computational intelligence, fault diagnosis and optimization.

Dr. Rigatos is an Editor-in-Chief of the Journal of Intelligent Industrial Sys-tems (Springer) and a Member of the IET and IMACS.

Pierluigi Siano (M’09–SM’14) received the M.Sc.degree in electronic engineering and the Ph.D. de-gree in information and electrical engineering fromthe University of Salerno, Salerno, Italy, in 2001 and2006, respectively.

He is an Aggregate Professor of electrical energyengineering with the Department of Industrial Engi-neering, University of Salerno. In 2013, he receivedthe Italian National Scientific Qualification as a FullProfessor in the competition sector electrical energyengineering. His research interests include the inte-

gration of distributed energy resources in smart distribution systems and onplanning and management of power systems. He has coauthored more than 160papers including more than 70 international journals.

Dr. Siano is an Associate Editor of the IEEE TRANSACTIONS ON INDUSTRIAL

INFORMATICS, and an Editor-in-Chief of the Journal of Intelligent IndustrialSystems (Springer).

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Nikolaos Zervos received the Ph.D. degree in elec-trical engineering from the University of Toronto,Toronto, ON, Canada, the M.Sc. degree in systemsand computing science from Carleton University, Ot-tawa, ON, and the Diploma degree in electrical andmechanical engineering from the National TechnicalUniversity of Athens, Athens, Greece.

He is currently a Researcher (Grade A’) at theIndustrial Systems Institute, Rion Patras, Greece. Hewas at Bell Laboratories first with AT&T and thenwith Lucent Technologies, as Acting Technical Man-

ager of Multimedia Access Communication Networks. He is one of the world’sexperts in bandwidth-efficient digital transmission and author of several patentsin the areas of data transmission and digital signal processing.

Carlo Cecati (M’90-SM’03–F’06) received the Dr.Ing. degree in electrotechnics from the University ofLAquila, LAquila, Italy, in 1983.

Since 1983, he has been with the Department ofElectrical and Information Engineering, University ofLAquila, where he is currently a Professor of indus-trial electronics and drives and is a Rectors Delegate.He is the Founder and the Coordinator of the Ph.D.courses on management of renewable energies andsustainable building at the University of LAquila. In2007, he founded DigiPower Ltd., LAquila, which is

a spin-off dealing with industrial electronics and renewable energies. His re-search and technical interests include several aspects of power electronics andelectrical drives.

Dr. Cecati is an Editor-in-Chief of the IEEE TRANSACTIONS ON INDUSTRIAL

ELECTRONICS, and has been also a Technical Editor of the IEEE/ASME TRANS-ACTIONS ON MECHATRONICS. He is a Member of IEEE IES Committees onRenewable Energy Systems and on Power Electronics.

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IEEE TRANSACTIONS ON POWER ELECTRONICS 1

Control and Disturbances Compensation for DoublyFed Induction Generators Using the Derivative-Free

Nonlinear Kalman FilterGerasimos Rigatos, Member, IEEE, Pierluigi Siano, Member, IEEE, Nikolaos Zervos,

and Carlo Cecati, Fellow, IEEE

Abstract—The paper studies differential flatness properties andan input–output linearization procedure for doubly fed inductiongenerators (DFIGs). By defining flat outputs which are associatedwith the rotor’s turn angle and the magnetic flux of the stator, anequivalent DFIG description in the Brunovksy (canonical) form isobtained. For the linearized canonical model of the generator, afeedback controller is designed. Moreover, a comparison of the dif-ferential flatness theory-based control method against Lie algebra-based control is provided. At the second stage, a novel KalmanFiltering method (Derivative-free nonlinear Kalman Filtering) isintroduced. The proposed Kalman Filter is redesigned as distur-bance observer for estimating additive input disturbances to theDFIG model. These estimated disturbance terms are finally usedby a feedback controller that enables the generator’s state vari-ables to track desirable setpoints. The efficiency of the proposedstate estimation-based control scheme is tested through simulationexperiments.

Index Terms—Derivative-free nonlinear Kalman filtering, dif-ferential flatness theory, disturbance estimator, doubly fed in-duction generator (DFIG), input–output linearization, nonlinearcontrol.

I. INTRODUCTION

DOUBLY fed induction generators (DFIG) have beenwidely used in variable-speed fixed frequency hydro-

power generation systems, wind-power generation systems, andturbine engine power generation systems [1]–[3]. DFIGs haveproven to be more efficient than squirrel-cage induction gener-ator systems and the synchronous generator systems in terms ofcost and losses of the associated power electronics converters.DFIG systems can operate either in grid-connected mode or instand-alone mode [4]–[9]. Results on the reliable connection ofDFIGs to the electricity grid have been presented in [10]–[12].Moreover, several field-oriented control schemes have been pro-posed for both operation modes. Additionally, to control elec-tric power generators and the power electronics that enable their

Manuscript received February 25, 2014; revised August 24, 2014; acceptedOctober 28, 2014. Recommended for publication by Associate Editor Z. Chen.

G. Rigatos and N. Zervos are with the Unit of Industrial Automation, In-dustrial Systems Institute, 26504 Rion Patras, Greece (e-mail: [email protected];[email protected]).

P. Siano is with the Department of Industrial Engineering, University ofSalerno, 84084 Fisciano, Italy (e-mail: [email protected]).

C. Cecati is with the Department of Informatics, University of L’ Aquila,67100 L’ Aquila, Italy (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2014.2369412

connection to the grid, feedback linearization approaches havebeen developed [13]–[14]. In parallel, several results have beenpublished on sensorless control of DFIG [15]–[19]. Taking intoaccount that the installation and maintenance of sensors for mea-suring several parameters of the generator’s state vector can betechnically difficult or costly, the need for developing sensor-less control schemes for DFIG becomes apparent. In this paper,a novel sensorless control scheme is developed using flatness-based control theory and a state estimation method that is basedon Kalman Filtering.

Using the electric equations of the stator and rotor, a dynamicmodel for the DFIG is derived. The DFIG is analogous to theinduction motor. In an induction motor, the stator voltage playsthe role of an input variable, while the rotor voltage is a constant.In case of the doubly fed induction machine it is quite similarbut the other way round, with a dual analogy to hold betweenthe stator and rotor parameters of the generator and the motor.This means that the rotor voltage now acts as an input, while thestator voltage is a constant parameter. The stator’s and rotor’svoltages, currents, and magnetic flux are represented as vectorsin a rotating orthogonal axis frame. The complete sixth-ordermodel of the DFIG captures efficiently transients at both thestator and the rotor side.

In this paper, differential flatness theory has been proposed forthe control of the DFIG. Differential flatness theory is currentlya main direction in nonlinear dynamical systems and enableslinearization and control for a wide class of systems, in a moreefficient manner than Lie-algebra methods [20]–[23]. To find outif a dynamical system is differentially flat, the following shouldbe examined: 1) the existence of the so-called flat output, i.e.,a new variable which is expressed as a function of the system’sstate variables. The flat output and its derivatives should not becoupled in the form of an ordinary differential equation (ODE),2) the components of the system (i.e., state variables and controlinput) should be expressed as functions of the flat output andits derivatives [24]–[29]. In certain cases, differential flatnesstheory enables transformation to a linearized form (canonicalBrunovsky form) for which the design of the controller becomeseasier. In other cases by showing that a system is a differentiallyflat one can easily design a reference trajectory as a function ofthe so-called flat output and can find a control law that assurestracking of this desirable trajectory [25], [26].

This paper is concerned with proving differential flatnessof the model of the DFIG and its resulting description in theBrunovksy (canonical) form [20], [21]. By defining flat outputs

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which are associated with the rotor’s angle and with the mag-netic flux of the stator, an equivalent DFIG description in theBrunovksy (linear canonical) form is obtained. It is shown thatfor the linearized DFIG’s model, it is possible to design a feed-back controller. At the second stage, a novel Kalman Filteringmethod, the Derivative-free nonlinear Kalman Filter, is pro-posed for estimating the state vector elements of the linearizedsystem which are not directly measurable. With the redesignof the proposed Kalman filter as a disturbance observer, it be-comes possible to estimate also disturbance terms affecting theDFIG model and to use these terms in the feedback controller.By avoiding linearization approximations, the proposed filter-ing method improves the accuracy of estimation and resultsin smooth control signal variations and in minimization of thetracking error of the associated control loop [30]–[32].

The structure of the paper is as follows. In Section II, themodel of the DFIG is analyzed. and the associated state-spaceequations are formulated. In Section III, input–output lineariza-tion for the DFIG model is performed using Lie algebra theory.In Section IV, differential flatness for nonlinear dynamical sys-tems is analyzed. Conditions, which are based on differentialflatness theory, are provided for transforming MIMO dynami-cal systems into the linear canonical form. In Section V, input–output linearization of the DFIG is performed using differentialflatness theory. In Section VI, the design of a Kalman Filter-based disturbance observer for the DFIG model is explained.In Section VII, simulation tests are carried out to evaluate theperformance of the DFIG control scheme that is based on dif-ferential flatness theory. Finally, in Section VIII, concludingremarks are given.

II. MODEL OF THE DFIG

A. Complete Sixth-Order Model of the Induction Generator

The DFIG is not only the most widely used technology inwind turbines due to its good performance, but it is also usedin many other fields such as hydropower generation, pumpedstorage plants and flywheel energy storage systems. The DFIGmodel is derived from the voltage equations of the stator androtor. It is assumed that the stator and rotor windings are sym-metrical and symmetrically fed. Usually, the saturation of theinductances, iron losses, skin effect, and bearing friction is ne-glected. Moreover, the winding resistance is considered to beconstant.

This type of wound-rotor machine is connected to the grid byboth the rotor and stator side. The DFIG stator can be directlyconnected to the electric power grid while the rotor is interfacedthrough back-to-back converters (see Fig. 1). By decoupling thepower system’s electrical frequency and the rotor mechanicalfrequency, the converter allows a variable speed operation of thewind turbine. The DFIG is analogous to the induction motor. Inan induction motor, the stator voltage plays the role of an inputvariable, while the rotor voltage is a constant (it is usually zero).In case of the doubly fed induction machine, it is very similarbut the other way round, with a dual analogy to hold betweenthe stator and rotor parameters of the generator and the motor.This means that the rotor voltage now acts as an input, while

Fig. 1. Configuration of a DFIG unit in the power grid.

the stator voltage depends on the voltage at the bus to which theDFIG is connected, and in the dq reference frame is a constantparameter [33]–[35].

In a compact form, the DFIG can be described by the follow-ing set of equations in the dq reference frame that rotates at anarbitrary speed denoted as ωdq [4]

dψsq

dt= − 1

τsψsq

− ωdqψsd+

M

τsirq

+ vsq(1)

dψsd

dt= ωdqψsq

− 1τs

ψsd+

M

τsird

+ vsd(2)

dirq

dt=

β

τsψsq

+ βωrψsd− γ2irq

−(ωdq − ωr )ird− βvsq

+1

σLrvrq

(3)

dird

dt= −βωrψsq

+β

τsψsd

+ (ωdq − ωr )irq

− γ2ird− βvsd

+1

σLrvrd

(4)

where ψsq, ψsd

, irq, ird

are the stator flux and the rotor currents,vsq

, vsd, vrq

, vrdare the stator and rotor voltages, Ls and Lr

are the stator and rotor inductances, ωr is the rotor’s angularvelocity, M is the magnetizing inductance. Moreover, denotingas Rs and Rr the stator and rotor resistances the followingparameters are defined

σ = 1 − M 2

LrLs, β = 1−σ

M σ , τs =Ls

Rs(5)

τr =Lr

Rr, γ2 =

(1−σστs

+ 1στr

).

The angle of the vectors that describe the magnetic flux ψsα andψsb is first defined for the stator, i.e.,

ρ = tan−1(

ψsb

ψsa

). (6)

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The angle between the inertial reference frame and the rotatingreference frame is taken to be equal to ρ.


||ψ || , sin(ρ) = ψs b

||ψ || , and

||ψ|| =√

ψ2sα

+ ψ2sb

. Therefore, in the rotating dq frame ofthe generator, and under the condition of field orientation, therewill be only one nonzero component of the magnetic flux ψsd

,while the component of the flux along the q-axis equals 0.

The dynamic model of the DFIG can be also written in state-space equations form by defining the following state variables:x1 = θ, x2 = ωr , x3 = ψsd

, x4 = ψsq, x5 = ird

, and x6 = irq.

It holds that

x1 = x2 (7)

x2 = − Km

Jx2 −

Tm

J+

η

J(irq

x3 − irdx4) (8)

x3 = − 1τs

x3 + ωdqx4 +M

τsx5 + vsd

(9)


x4 +M

τsx6 + vsq

(10)

x5 = −βx2x4 +β

τsx3 + (ωdq − x2)x6 − γ2x5

+1

σLrvrd

− βvsd(11)

x6 =β

τsx4 + βx2x3 − (ωdq − x2)x5

− γ2x6 +1

σLrvrq

− βvsq. (12)

In the above set of equations, J is the moment of inertia of therotor, Tm is the externally applied mechanical torque that makesthe turbine rotate, Km is the friction coefficient, η is a variablethat is associated to the number of poles and to the mutualinductance M . Variable η in turn determines the electrical torqueTe which is associated with rotor currents and stator magneticflux. Equation (7)–(12) can be written also in the form

x = f(x) + ga(x)vrd+ gb(x)vrq

(13)

where x = [x1 , x2 , x3 , x4 , x5 , x6 ]T and

f(x) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x2

−Km

J x2 − Tm

J + nJ (irq

x3 − irdx4)

− 1τs

x3 + ωdqx4 + Mτs

x5 + vsd

−ωdqx3 − 1τs

x4 + Mτs

x6 + vsq

−βx2x4 + βτs

x3 + (ωdq − x2)x6 − γ2x5 − βvsd

βτs

x4 + βx2x3 − (ωdq − x2)x5 − γ2x6 − βvsq

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ga(x) = (0 0 0 01

σLr0)T

gb(x) = (0 0 0 0 01

σLr)T . (14)

The active and reactive power delivered by the DFIG stator areassociated with the real and imaginary part of the power at thestator’s terminals, i.e.

Ps = Re{UsI∗s} = vsd

isd+ vsq

isq(15)

Qs = Im{UsI∗s} = vsd

isq− vsq

isd. (16)

III. INPUT–OUTPUT LINEARIZATION OF THE DFIG USING LIE

ALGEBRA THEORY

A. Input–Output Linearization of the DFIG Model

The following variables are defined:

h1(x) = x1 = θ

h2(x) = x23 + x2

4 = ψ2sd

+ ψ2sq

. (17)

Next, based on h1 and h2 , the following transformed state vari-ables are defined :

z1 = h1(x) = θ (18)

z2 = Lf h1(x)⇒ (19)

z2 = f1⇒z2 = x2⇒z2 = ω.

Similarly, one has

z3 = L2f h1(x) = Lf z2 ⇒ (20)

z3 = f2⇒z3 = −Km

Jx2 −

Tm

J+

η

J(irq

x3 − irqx4) ⇒

z3 = −Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4).


z4 = h2(x) = ψ2sd

+ ψ2sq

= x23 + x2

4 (21)

and

z5 = Lf h2(x)⇒z5 = 2x3f3 + 2x4f4⇒ (22)

z5 = 2x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+ 2x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

].

After the change of the state variables, it holds (the completeproof is given in Appendix I)

z1 = z2

z2 = z3

z3 = L3f h1(x) + (Lga

L2f h1(x))u1 + (Lgb

L2f h1(x))u2

z4 = z5

z5 = L2f h2(x) + (Lga

Lf h2)(x)u1 + (LgbLf h2(x))u2 . (23)

The inputs of the above linearized and decoupled DFIG modelare u1 = urd

and u2 = urq. The system of (24) can be written

in the input–output linearized form(

z(3)1

z4

)= fa + Mu (24)

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where

fa(x)=

(L3

f h1(x)

L2f h2(x)

), M =

(Lga

L2f h1(x) Lgb

L2f h1(x)

LgaLf h2(x) Lgb

Lf h2(x)

)

(25)

or equivalently, one has the system’s description in the MIMOcanonical form⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎝

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎝

0 00 01 00 00 1

⎞⎟⎟⎟⎟⎠

(v1v2

)

(26)where

v1 = L3f h1(x) + (Lga

(L2f h1(x))u1 + (Lgb

L2f h1(x))u2

v2 = L2f h2(x) + (Lga

Lf h2(x))u1 + (LgbLf h2(x))u2 . (27)

Returning to the compact form of (26), one has

(z

(3)1

z4

)=

(v1v2

)(28)

and the control signal that assures convergence of the z1 and z4to the reference setpoints zd

1 and zd4 is given by

v1 = zd1

(3) − k(1)1 (z1 − zd

1 ) − k(1)2 (z1 − zd

1 ) − k(1)3 (z1 − zd

1 )

v2 = zd4 − k

(2)1 (z4 − zd

4 ) − k(2)2 (z4 − zd

4 ). (29)

B. State Estimation-Based Control

For the implementation of the aforementioned control law,there is a need to obtain measurements of all elements of theDFIG’s state vector. The rotor’s turn angle can be measured di-rectly with the use of an encoder [36], [37]. Knowing the rotor’sangle and with the use of the decoupled induction machine’smodel of (28), it is possible to estimate the rotor’s angular speed.Similarly, after obtaining measurements of the magnetic flux atthe stator and with the use of the decoupled induction machine’smodel of (28), it is possible to estimate the derivatives of themagnetic flux. Due to the fact that the magnetic flux of the statorψs cannot be measured directly, equations that provide indirectmeasurements of the flux (computed through measurements ofthe stator and rotor currents) will be used, that is

ψsd= Lsisd

+ Mird

ψsq= Lsisq

+ Mirq. (30)

It is noted that the currents are measured in the ab referenceframe, and their computation in the dq reference frame requiresthe application of the associated reference frame transformation.

Using the model of (28), the state estimator for the DFIG isgiven by

˙z = Az + Bv + K(zmeas − Cz) (31)

where the estimator’s gain K∈R5×2

A =

⎛⎜⎜⎜⎜⎝

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

⎞⎟⎟⎟⎟⎠

, B =

⎛⎜⎜⎜⎜⎝

0 00 01 00 00 1

⎞⎟⎟⎟⎟⎠

C =(

1 0 0 0 00 0 0 1 0

)(32)

IV. DIFFERENTIAL FLATNESS FOR NONLINEAR

DYNAMICAL SYSTEMS

A. Definition of Differentially Flat Systems

Differential flatness is a structural property of a class of non-linear systems, denoting that all system variables (such as statevector elements and control inputs) can be written in terms ofa set of specific variables (the so-called flat outputs) and theirderivatives. The following nonlinear system is considered:

x(t) = f(x(t), u(t)). (33)

The time is t∈R, the state vector is x(t)∈Rn with initial condi-tions x(0) = x0 , and the input is u(t)∈Rm . Next, the propertiesof differentially flat systems are given [20]–[29].

The finite dimensional system of (37) can be written inthe general form of an ODE, i.e., Si(w, w, w, . . . , w(i)), i =1, 2, . . . , q. The term w is a generic notation for the systemvariables (these variables are for instance the elements of thesystem’s state vector x(t) and the elements of the control inputu(t)) while w(i) , i = 1, 2, . . . , q are the associated derivatives.Such a system is differentially flat if there are m functionsy = (y1 , . . . , ym ) of the system variables and of their time-derivatives, i.e., yi = φ(w, w, w, . . . , w(αi )), i = 1, . . . , m sat-isfying the following two conditions [23]–[27].

1) There does not exist any differential relation of the formR(y, y, . . . , y(β )) = 0 which implies that the derivativesof the flat output are not coupled in the sense of an ODE,or equivalently, it can be said that the flat output is differ-entially independent.

2) All system variables (i.e., the elements of the system’sstate vector w and the control input) can be expressedusing only the flat output y and its time derivatives wi =ψi(y, y, . . . , y(γi )), i = 1, . . . , s. An equivalent definitionof differentially flat systems is as follows.

Definition: The system x = f(x, u), x∈Rn , u∈Rm is differ-entially flat if there exist relations

h : Rn×(Rm )r+1→Rm

φ : (Rm )r→Rn and

ψ : (Rm )r+1→Rm (34)

such that

y = h(x, u, u , . . . , u(r))

x = φ(y, y , . . . , y(r−1)), and

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u = ψ(y, y, . . . , y(r−1) , y(r)). (35)

This means that all system dynamics can be expressed as afunction of the flat output and its derivatives; therefore, the statevector and the control input can be written as

x(t) = φ(y(t), y(t), . . . , y(r)(t)), and

u(t) = ψ(y(t), y(t), . . . , y(r+1)(t)). (36)

It is noted that for linear systems the property of differentialflatness is equivalent to that of controllability.

B. Conditions for Applying Differential Flatness Theory

The generic class of nonlinear systems x = f(x, u) is con-sidered. Such a system can be transformed to the form of anaffine in the input system by adding an integrator to each input[28]

x = f(x) +m∑

i=1

gi(x)ui. (37)

If the system of (41) can be linearized by a diffeomorphismz = φ(x) and a static state feedback u = α(x) + β(x)v intothe following form:

zi,j = zi+1,j for 1≤j≤m and 1≤ i≤ vj − 1

zvi , j= vj (38)

with∑m

j=1vj = n, then yj = z1,j for 1≤ j ≤m are the 0-flatoutputs which can be written as functions of only the elementsof the state vector x. To define conditions for transforming thesystem of (41) into the canonical form described in (42), thefollowing theorem holds [28].

Theorem: For the nonlinear systems described by (41), thefollowing variables are defined: 1) G0 = span[g1 , . . . , gm ],2) G1 = span[g1 , . . . , gm , adf g1 , . . . , adf gm ], · · · (k) Gk =span{adj

f gi for 0≤ j ≤ k, 1≤ i≤m}. Then, the linearizationproblem for the system of (41) can be solved if and only if: 1)The dimension of Gi, i = 1, . . . , k is constant for x∈X⊆Rn

and for 1≤ i≤n − 1, 2) the dimension of Gn−1 is of order n,3) the distribution Gk is involutive for each 1≤ k≤n − 2.

C. Transformation of MIMO Nonlinear Systems Into theBrunovsky Form

It is assumed now that after defining the flat outputs of theinitial MIMO nonlinear system, and after expressing the systemstate variables and control inputs as functions of the flat outputand of the associated derivatives, the system can be transformedin the Brunovsky canonical form:

x1 = x2

· · ·xr1 −1 = xr1

xr1 = f1(x) +p∑

j=1

g1j(x)uj + d1

xr1 +1 = xr1 +2

· · ·xp−1 = xp

xp = fp(x) +p∑

j=1

gpj(x)uj + dp

y1 = x1

y2 = x2

· · ·yp = xn−rp +1 (39)

where x = [x1 , . . . , xn ]T is the state vector of the transformedsystem (according to the differential flatness formulation),u = [u1 , . . . , up ]T is the set of control inputs, y = [y1 , . . . , yp ]T

is the output vector, fi are the drift functions, and gi,j , i, j =1, 2 , . . . , p are smooth functions corresponding to the controlinput gains, while dj is a variable associated to external distur-bances. In holds that r1 + r2 + · · · + rp = n. Having writtenthe initial nonlinear system into the canonical (Brunovsky) formit holds

y(ri )i = fi(x) +

p∑j=1

gij (x)uj + dj . (40)

Next the following vectors and matrices can be defined f(x) =[f1(x), . . . , fn (x)]T , g(x) = [g1(x), . . . , gn (x)]T with gi(x) =[g1i(x), . . . , gpi(x)]T , and also A = diag[A1 , . . . , Ap ], B =diag[B1 , . . . , Bp ], C = diag[C1 , . . . , Cp ], d = [d1 , . . . , dp ]T ,where matrix A has the MIMO canonical form, i.e., with block-diagonal elements

Ai =

⎛⎜⎜⎜⎜⎜⎝

0 1 · · · 00 0 · · · 0...

... · · ·...

0 0 · · · 10 0 · · · 0

⎞⎟⎟⎟⎟⎟⎠

ri ×ri

(41)

BTi = (0 0 · · · 0 1)1×ri

Ci = (1 0 · · · 0 0)1×ri.

Thus, (44) can be written in state-space form

x = Ax + Bv + Bd

y = Cx (42)

where the control input is written as v = f(x) + g(x)u.

V. INPUT–OUTPUT LINEARIZATION OF THE DFIG USING

DIFFERENTIAL FLATNESS THEORY

A. Differential Flatness Properties of the DFIG

The flat outputs of the system are defined as

y1 = θ or y = x1

y2 = ψ2sd

+ ψ2sq

or y2 = x23 + x2

4 . (43)

It holds that

y1 = ω or y1 = x2 ⇒

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y1 = ω = −Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4) ⇒

y1 = ω = −Km

Jy1 −

Tm

J+

η

J(x6x3 − x5x4). (44)

Deriving the last row of (44) with respect to time, one obtains

y(3)1 = −Km

Jy1 +

η

J(x6x3 + x6 x3 − x5x4 − x5 x4) ⇒

y(3)1 = −Km

Jy1 +

η

Jx3

{[β

τsx4 + βx2x3 + (ωdq − x2)x5

−γ2x6− βvsq

]+

1σLr

u1

}+

η

Jx6

[− 1

τsx3 + ωdqx4 +

M

τsx5

+ vsd

]− η

Jx4

{[− βx2x4 +

β

τsx3 + (ωdq − x2)x6 − γ2x5

− βvsd

]+

1σLr

u2

}− η

Jx5

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

].

(45)

Moreover, about the second flat output, it holds

y2 = 2x3 x3 + 2x4 x4⇒

y2 = 2x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+ 2x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]⇒ (46)

Consequently, it holds

y2 = 2x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+ 2x3

[− 1

τsx3 + ωdq x4 +

M

τsx5

]

+ 2x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]

+ 2x4

[−ωdq x3 −

1τs

x4 +M

τsx6

](47)

or equivalently,

y2 = 2[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]2

− 2τs

x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

− 2ωdqx3

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]

+2M

τsx3

{[− βx2x4 +

β

τsx3 + (ωdq − x2)x6

− γ2x5 − βvsd

]+

1σLr

u1

}

+ 2[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]2

− 2ωdqx4

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

− 2τs

x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]

2x4M

τs

{[β

τsx4 + βx2x3 + (ωdq − x2)x5 −

− γ2x6 − βvsq

]+

1σLr

u2

}. (48)

It holds that x1 = y1 , x2 = y1 . From the second row of (43) andconsidering that the field orientation condition requires x4 =ψsq

= 0, one obtains that x3 =√

y2 . Moreover, from (44) itholds

y1 = −Km

Jy1 −

Tm

J+

η

J

√y2x6⇒

x6 =y1 + Km

J y1 + Tm

JηJ

√y2

, y2 = 0. (49)

From (46), one obtains

y2 = − 2τs

x23 +

2M

τsx3x5 + 2vsd

x3 ⇒

y2 +(

2τs

x3 − 2vsd

)x3 =

2M

τsx3x5 ⇒

x5 =y2 +

(2τs

√y2 − 2vsd

)√y2

2Mτs

√y2

y2 = 0. (50)

Therefore, x5 is also a function of the flat output and of itsderivatives. Additionally, by solving the system of (49) and (52)with respect to the control inputs u1 and u2 , one obtains that thecontrol inputs are functions of the flat output and its derivatives.Therefore, the model of the DFIG is a differentially flat one.

Next, to design the flatness-based controller for the DFIG,the following transformation of the state variables is introduced:z1 = y1 , z2 = y1 , z3 = y1 , z4 = y2 , z5 = y2 for which holds

z1 = z2

z2 = z3

z3 = L3f h1(x) + (Lga

L2f h1(x))u1 + (Lgb

L2f h1(x))u2

z4 = z5

z5 = L2f h2(x) + (Lga

Lf h2(x))u1 + (LgbLf h2(x))u2 . (51)

Therefore, one obtains the decoupled and linearized representa-tion of the system(

z(3)1

z4

)=

(L3

f h1(x)

L2f h2(x)

)+

+

(Lga

L2f h1(x) Lgb

L2f h1(x)

LgaLf h2(x) Lgb

Lf h2(x)

)(u1u2

)(52)

or equivalently,(

z(3)1

z4

)= fa + Mu (53)

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where

fa =

(L3

f h1(x)

L2f h2(x)

), M =

(Lga

L2f h1(x) Lgb

L2f h1(x)

LgaLf h2(x) Lgb

Lf h2(x)

).

(54)

By defining the control inputs v1 = L3f h1(x) +

(LgaL2

f h1(x))u1 + (LgbL2

f h1(x))u2 and v2 = L2f h2(x) +

(LgaLf h2(x))u1 + (Lgb

Lf h2(x))u2 , one can also have thedescription in the MIMO canonical form⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎝

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎝

0 00 01 00 00 1

⎞⎟⎟⎟⎟⎠

(v1v2

).

(55)The control input for the linearized and decoupled model of theDFIG is chosen as follows:

v1 = zd1

(3) − k(1)1 (z1 − zd

1 ) − k(1)2 (z1 − zd

1 ) − k(1)3 (z1 − zd

1 )

v2 = zd4 − k

(2)1 (z4 − zd

4 ) − k(2)2 (z4 − zd

4 ) (56)

and finally, the control input that is applied to the system is

u = M−1(−fa + v). (57)

The proposed control scheme can work with the use of mea-surements from a small number of sensors. That is, there isneed to obtain measurements of only y1 = θ which is theturn angle of the generator’s rotor, and of the magnetic fluxy2 = ψ2

s = ψ2sd

+ ψ2sq

, or due to the orientation of the magneticfield y2 = ψ2

s = ψ2sd

. The stator flux (ψs) cannot be measureddirectly from a sensor (e.g., the use of Hall sensor in an electricmachine with a rotating part would not be efficient); however(32) that relates stator flux and stator and rotor currents can beused to calculate ψs . Thus one has

ψsd= Lsisd

+ Mird

ψsq= 0 (58)

which means that by measuring stator and rotor currents onecan obtain an indirect measurement of the stator’s magneticflux ψsd

. Next, one can compute the dynamics of the magneticflux, jointly with the dynamics of the rotor’s motion throughthe use of the Derivative-free Nonlinear Kalman Filter. Thisestimation method is based on the application of the KalmanFilter recursion to the linearized equivalent of the generator’smodel which is given by (59). Actually, (59) can be written inthe state-space form

z = Az + Bv

zmeas = Cz (59)

where

A =

⎛⎜⎜⎜⎜⎝

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

⎞⎟⎟⎟⎟⎠

, B =

⎛⎜⎜⎜⎜⎝

0 00 01 00 00 1

⎞⎟⎟⎟⎟⎠

C =(

1 0 0 0 00 0 0 1 0

). (60)

The estimator’s dynamics is

˙z = A·z + B·v + K(zmeas − Cz) (61)

where K∈R5×2 is the state estimator’s gain. Defining as Ad ,Bd , and Cd , the discrete-time equivalents of matrices A, B, andC, respectively, the associated Kalman Filter-based estimator isgiven by [38]–[42]

measurement update:


d + R]−1

z(k) = z−(k) + K(k)[zmeas(k) − Cdz−(k)]

P (k) = P−(k) − K(k)CdP−(k) (62)

time update:

P−(k + 1) = Ad(k)P (k)ATd (k) + Q(k)

z−(k + 1) = Ad(k)z(k) + Bd(k)v(k). (63)

Remark 1: The first linearization approach followed inSection III was based on differential geometry and the compu-tation of Lie Derivatives. The second linearization approach fol-lowed in Section V was based on differential flatness theory. Formulti-input systems which admit static feedback linearization,the differential flatness theory-based approach is equivalent tolinearization based on Lie algebra. As it can be confirmed from(28) and (59), the two linearization methods provided the samelinearized model of the DFIG. The differential flatness theorycan be also extended to MIMO systems that admit only dynamicfeedback linearization. In the latter case, an extended state vec-tor of the controlled system is defined containing as additionalstate variables the derivatives of the control input. In dynamicfeedback linearization, the control input that is finally applied tothe system contains integral terms of the error’s state vector. Interms of computation, the differential flatness theory-based lin-earization is simpler because it does not require the calculationof Lie derivatives. Moreover, by expressing all state variablesas functions of the flat output and its derivatives, the differen-tial flatness theory-based linearization enables to perform stateestimation and to reconstruct the state variables of the initialnonlinear system. This is not possible for the Lie algebra-basedapproach, where to perform filtering it is necessary to computeand invert the Jacobian matrix of the transformed state vector[43].

Remark 2: Regarding comparison to existing results, it isnoted that in other control approaches for DFIGs, e.g., controlof the rotor’s speed and of the stator’s magnetic flux in cascadingloops analyzed in [33], one has to use again measurements ofthe rotor currents. One can estimate the rest of the DFIG statevariables with filtering methods that make use of the initialnonlinear model of the system, such as the Extended or theUnscented Kalman Filter. Being based on an exact linearizationmethod, the control and state estimation approach for the DFIGthat is presented in Section V has the advantage of using areduced number of sensors while at the same time remaining

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robust to modeling uncertainties and external perturbations andavoiding numerical approximation errors.

Remark 3: It has been explained that the concept of sensor-less control is to reduce the number of sensors needed for theimplementation of feedback control. As explained in Section IIIand in Section V, the DFIG MIMO nonlinear model is trans-formed into two decoupled systems in the canonical linear form.The first system has as output the turn angle of the rotor whichcan be measured with the use of an encoder. The second sys-tem has as output the magnetic flux of the stator, where due tothe field orientation condition only the d-axis flux componentis nonzero. Using measurements of the stator’s and rotor’s cur-rents, one can obtain a measurement of the stator’s magneticflux too. By considering as measurable outputs the rotor’s turnangle and the stator’s magnetic flux, the observability of thelinearized DFIG model is assured. Thus, it is possible to per-form state estimation for the nonmeasurable state variables andto develop sensorless control, using the observers of (33) and(61).

VI. KALMAN FILTER-BASED DISTURBANCE OBSERVER FOR

THE DFIG MODEL

A. Application of a Disturbance Observer to the DFIG Model

Next, it will be considered that additive input disturbances(e.g., due to load variations) affect the DFIG model. The si-multaneous estimation of the nonmeasurable elements of theDFIG state vector as well as the estimation of additive distur-bance terms affecting the generator is possible with the use of adisturbance estimator [44]–[47].

It is assumed that the third and fifth row of the state-spaceequations of the DFIG of (55) include a disturbance term


(L2f h1(x))u1

+ Lgb(L2

f h2(x))u2 + d1


(Lf h2(x))u1

+ Lgb(Lf h2(x))u2 + d2 . (64)

Without loss of generality, the dynamics of the disturbanceterms is described by their second-order derivatives and the

associated initial conditions, i.e., ¨d1 = fa(x) and ¨

d2 = fb(x).Next, an extended state-space model of the system is definedthat comprises as additional state variables the disturbance

terms z6 = d1 , z7 = ˙d1 , while z8 = d2 , and z9 = ˙

d2 . Thus,the extended state-space model is written as z1 = z2 , z2 = z3 ,z3 = v1 + z6 , z4 = z5 , and z5 = v2 + z8 , z6 = z7 , z7 = fa ,z8 = z9 and z9 = fb , or in matrix form one has

˙z = Az + Bv

zmeas = Cz (65)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

z1z2z3z4z5z6z7z8z9

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

z1z2z3z4z5z6z7z8z9

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 00 0 0 01 0 0 00 0 0 00 1 0 00 0 0 00 0 1 00 0 0 00 0 0 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎝

v1v2

fa

fb

⎞⎟⎟⎠

(zmeas

1zmeas

4

)=

(1 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0

)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

z1z2z3z4z5z6z7z8z9

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (66)

The associated state estimator is

ˆz = Ao z + Bo v1 + Ko(zmeas − Cz) (67)

where

Ao =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Bo =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 00 0 0 01 0 0 00 0 0 00 1 0 00 0 0 00 0 0 00 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Co =(

1 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0

)(68)

while the estimator’s gain Ko∈R9×2 is obtained from the stan-dard Kalman Filter recursion [38]–[42].

Defining as Ad , Bd , and Cd , the discrete-time equivalents ofmatrices Ao , Bo , and Co , respectively, a Derivative-free non-linear Kalman Filter can be designed for the aforementionedrepresentation of the system dynamics [23], [31]. The associ-ated Kalman Filter-based disturbance estimator is given by

measurement update:


d + R]−1

ˆz(k) = z−(k) + K(k)[Cd z(k) − Cdˆz−(k)]

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P (k) = P−(k) − K(k)CdP−(k) (69)

time update:

P−(k + 1) = Ad(k)P (k)ATd (k) + Q(k)

ˆz−(k + 1) = Ad(k)ˆz(k) + Bd(k)v(k). (70)

Remark 4: The advantages of the proposed nonlinear feed-back control method for DFIGs (that is based on differential flat-ness theory and on the Derivative-free nonlinear Kalman Filter)against PID-type control (included in vector control loops) areobvious. In most cases, the application of PID control to electricmachines is based on heuristic parameters tuning, has no stabil-ity proof, and has limited robustness to the change of operatingpoints or to the effects of external perturbations. Moreover, inthe case of multivariable systems such as DFIGs the applicationof PID control is known to have questionable performance. Thefirst vector control approaches for asynchronous machines (e.g.[35]) made use of multiple PID loops which were implementedin a cascaded manner (for controlling separately the magneticflux and the rotation angle of the machine). Such methods werebased on the assumption that the flux and the rotation speedbecome finally decoupled at steady state. However, there is noproof about that (it cannot be always assured that transients willbe eliminated and the machine will reach a steady state), andtherefore, the performance of the control loop is not alwaysguaranteed (see attached paper). Consequently, although PIDcontrol is met in some cases in asynchronous machines, it is notthe recommended solution.

Remark 5: Field-oriented (vector) control has been for manyyears a common approach for the control of DFIGs. However,comparing to the flatness-based control approach developed inthis paper, vector control exhibits several weaknesses whichmake its performance be questionable [50], [51]. As it is shownin detail in Appendix II, the implementation of vector control re-quires measurement or estimation of the stator’s magnetic flux.Therefore, one comes against the observer or Kalman Filter de-sign problem that was solved in a conclusive manner in Section 6of this manuscript. Moreover, vector control for DFIGs requiresthe tuning of the several PID and PI controllers, and this limits itsreliability only round local operating points. Consequently, thestability and robustness properties of the field-oriented controlfor DFIGs are doubtful.

Remark 6: It is confirmed that the linearized equivalentmodel of the DFIG, after application of the pole placement tech-nique has poles, exclusively in the left complex semiplane. Be-sides the inclusion of the additional control input that compen-sates for the estimated additive disturbance terms improves therobustness features of this control loop. It is also noted that thelinearized equivalent model of the DFIG exhibits multiple polesat the origin. This particular form implies an infinite gain mar-gin and a sufficiently large phase margin. Finally, it is noted thatthe stability and robustness features of the control scheme whichcomprises also estimation and compensation of the disturbancesare similar to those of LQG control. According to the above, thepaper justifies sufficiently the stability and disturbance rejec-tion capability of the proposed feedback control scheme. On the

Fig. 2. Control loop of the DFIG comprising a flatness-based control elementand an estimator for disturbances compensation.

TABLE IRATINGS OF THE MODELED DFIG

Rated power 15.5 kW

Number of Pole pairs 4Stator Resistance 0.58 ΩStator Inductance 13 mHRotor Resistance 1.30 ΩRotor Inductance 3 mHMutual Inductance 10 mHRotor’s inertia 20.0 kg · m2

other hand, the presented simulation experiments demonstratedthe efficiency of the control method in tracking rapidly chang-ing reference setpoints while also succeeding good transients.The disturbances appearing in the simulation experiments couldbe met in adverse operating conditions of the power generator.Even for the latter case, the good performance of the controlloop is confirmed.

VII. SIMULATION TESTS

The structure of the proposed control scheme is depictedin Fig. 2. The control scheme comprises 1) the flatness-basedcontrol part which computes the control signal for the system’sequivalent model that is transformed to the linear canonicalform, 2) a Kalman Filter-based disturbances estimator whichprovides estimates for the elements of the state vector of theDFIG, such as rotor’s speed , magnetic flux at the stator aswell as disturbances affecting the generator’s model. Indicativenumerical values for the parameters of the considered DFIGmodel are given in Table I.

Simulation tests were carried out for two different setpointsof the turn speed of the generator’s rotor. The values of the gen-erator’s state vector elements are actually measured in SI units;however, in the simulation results, they are expressed in the perunit (p.u.) system. The results obtained for the first setpoint aredepicted in Figs. 3 and 4. Similarly, the results obtained for thesecond setpoint are depicted in Figs. 5 and 6. It can be observedthat the proposed control scheme assures that the rotor’s turn

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.




.

speed follows a specific setpoint, while tracking of referencesetpoints is succeeded for the components of the magnetic fluxand for the rotor’s currents. Several reference setpoints havebeen defined for the DFIG state variables, i.e., rotor’s angularspeed ω, rotor currents ird

, irq, and the magnetic flux ψsd

, and asit can be observed from the associated diagrams, the proposedcontrol scheme resulted in fast and accurate convergence tothese setpoints. The disturbance observer that was based on the



Fig. 7. Convergence of the stator’s magnetic flux ψsdto the reference setpoint

(a) without using the disturbance observer, (b) when using the disturbanceobserver.

Derivative-free nonlinear Kalman Filter was capable of estimat-ing the unknown and time-varying input disturbances affectingthe DFIG model. The selection of the magnetic flux setpointsappearing in Figs. 3 and 5 did not aim to be restricted only to thecase that the DFIG is connected to a grid, which is characterizedby constant voltage amplitude and frequency. The purpose of thesimulation experiments was to show that the proposed nonlinearcontrol scheme succeeds convergence to time-varying magneticflux setpoints (e.g., piecewise constant ones). Of course, the an-alyzed control method for the DFIG enables also convergenceof the stator’s flux to constant setpoints, but this is a subcase ofwhat has already been presented.

The improvement in the performance of the control loopthat is due to the use of a disturbance observer based on theDerivative-free nonlinear Kalman Filter is explained as follows:1) compensation of the disturbance terms which are generatedby parametric uncertainty or unknown external inputs, 2) moreaccurate estimation of the disturbance terms because the filter-ing procedure is based on an exact linearization of the system’sdynamics and does not introduce numerical errors (as for exam-ple in the case of the Extended Kalman Filter). This is shown inFig. 7.

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Remark 7: The implementation of the control scheme withthe use of a digital processor does not exhibit any difficulty.The application of Kalman Filter-based control loops is a com-mon practice in other complicated and demanding cases (e.g., inautonomous navigation of aircrafts, in robots, etc., [23]). There-fore, the method can be also applied, through a programmabledigital controller, in the case of asynchronous electric machinestoo. As explained, the use of the Derivative-free nonlinearKalman Filter as a disturbance observer enables identificationand compensation of external perturbations in real time (such asdisturbances due to grid faults). Therefore, the proposed controlscheme exhibits improved robustness.

Remark 8: The generator’s speed can be efficiently con-trolled, and the associated rotation speed setpoints can bereached by applying the proposed control scheme. It is pos-sible to operate the generator at variable speed, thus also chang-ing the levels of the generated power. Moreover, as shown inSection V, all currents and voltages defining the active and re-active power of the generator, according to (16) and (17), can bewritten explicitly or implicitly, as functions of the flat outputs ofthe DFIG model. This explains why these variables are finallyassociated with the turn speed of the rotor, and consequently,why the produced power of the generator is determined by therotor’s angular velocity.

Remark 9: The differential flatness properties of the DFIGdynamic model are initially proven, considering that the exter-nal mechanical torque is constant or piecewise constant. Equiv-alently, it can be considered that Tm stands for an unknowntime-varying disturbance term to the DFIG model [48], [49]. Insuch a case, variable Tm is omitted from the DFIG model, whichat a second stage is transformed into the linear canonical anddecoupled form using the diffeomorphism provided by differ-ential flatness theory. For the linearized equivalent of the DFIGmodel, a Kalman Filter-based disturbance observer is designedfollowing the method of Section VI. The Kalman Filter-baseddisturbance estimator can identify in real time the aggregate dis-turbance term which incorporates the time-varying torque Tm .Therefore, the proposed nonlinear control scheme, that is basedon disturbances estimation with the use of the Derivative-freenonlinear Kalman Filter, can work well even if the mechanicaltorque Tm causing the turbine’s rotation is completely unknownand time-varying.

Remark 10: Variables d1 and d2 appearing in (64) are aggre-gate disturbance terms which include any type of perturbationsthat may be due to load variations and change of the statorcurrents, change of the mechanical torque, voltage fluctuationand faults in the grid (vsd

and vsqnon constant) or modeling

uncertainty, and changes in the numerical values of the param-eters appearing in the DFIG model. Representing the aggregatedisturbances effects as in (64) enables the design of a distur-bances estimator and compensator based on the Derivative-freenonlinear Kalman Filter.

VIII. CONCLUSION

The paper has proposed a nonlinear control scheme forDFIGs. Estimation of disturbance terms affecting the DFIG

model has been performed with the use of a new nonlinear fil-tering approach, the so-called derivative-free nonlinear KalmanFilter. First, it was proven that the dynamic model of the DFIGis a differentially flat one, and this enabled its description inthe Brunovksy (linear canonical) form. It has been shown thatfor the linearized DFIG model, it is possible to design a statefeedback controller. At the second stage, a novel Kalman Fil-tering method, the Derivative-free nonlinear Kalman Filter, hasbeen proposed for estimating the nonmeasurable elements of thedynamic model of the DFIG. It has been shown that by avoid-ing linearization approximations, the proposed filtering method,improves the accuracy of estimation and results in smooth con-trol signal variations and in minimization of the tracking errorof the DFIG control loop. Moreover, with the redesign of theproposed Kalman Filter as a disturbance observer, it becamepossible to obtain estimates of disturbance terms affecting theDFIG model. The DFIG’s control input was generated by in-cluding in the state-feedback control law an input that is basedon the estimate of the disturbance terms. Simulation tests havebeen provided to evaluate the performance of the nonlinear con-trol scheme.

APPENDIX I

Input–output linearization of the DFIG model with use of Liealgebra

The following variables have been defined:

h1(x) = x1 = θ

h2(x) = x23 + x2

4 = ψ2sd

+ ψ2sq

. (71)

Next, based on h1 , h2 , the following transformed state variablesare defined

z1 = h1(x) = θ (72)

z2 = Lf h1(x)⇒z2 = f1⇒z2 = x2⇒z2 = ω. (73)

Similarly, one has

z3 = L2f h1(x) = Lf z2 ⇒

z3 = f2⇒z3 = −Km

Jx2 −

Tm

J+

η

J(irq

x3 − irqx4) ⇒

z3 = −Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4). (74)

Moreover, one has

z3 = L3f h1(x) + (Lga

L2f h1(x))u1 + (Lgb

L2f h2(x))u2 . (75)

It holds that

L3f h1(x) = Lf z3 (76)

L3f h1(x) = −Km

J

[−Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4)

]

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+η

Jx6

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

− η

Jx5

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]

− η

Jx4

[−βx2x4 +

β

τsx3 + (ωdq − x2)x6 − γ2x5 − βvsd

]

+η

Jx3

[β

τsx4 + βx2x3 + (ωdq − x2)x5 − γ2x6 − βvsq

].

(77)

Equivalently, one has

Lga(L2

f h1(x)) = Lgaz3⇒

Lga(L2

f h1(x)) = − η

J

1σLr

x4 (78)

and similarly,

Lgb(L2

f h1(x)) = Lgbz3⇒

Lgb(L2

f h1(x)) =η

J

1σLr

x3 . (79)


z4 = h2(x) = ψ2sd

+ ψ2sq

= x23 + x2

4 (80)

and

z5 = Lf h2(x)⇒z5 = 2x3f3 + 2x4f4⇒

z5 = 2x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+ 2x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

](81)

and equivalently, one has


(Lf h2(x))u1 + Lgb(Lf h2(x))u2 . (82)

It holds that

L2f h2(x) =

(− 4

τsx3−

2M

τsx5 + 2vsd

)[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+(− 4

τsx4 +

2M

τsx6 + 2vsq

)[−ωdqx3−

1τs

x4 +M

τsx6 + vsq

]

+(

2M

τsx3

)[−βx2x4 +

β

τsx3 +(ωdq − x2)x6− γ2x5−βvsd

]

+(

2M

τsx3

) [β

τsx4 + βx2x3 +(ωdq − x2)x5 − γ2x6 − βvsq

].

(83)

Moreover, it holds that

Lga(Lf h2(x)) =

2M

τsx3ga5 ⇒Lga

(Lf h2(x)) =2M

τs

1σLs

x3

(84)

and in a similar manner

Lgb(Lf h2(x)) =

2M

τsx4ga6 ⇒Lga

(Lf h2(x)) =2M

τs

1σLs

x4 .

(85)Next, it is confirmed that after change of the state variables, itholds

z1 = z2

z2 = z3


(L2f )h1(x)u1 + Lgb

(L2f )h1(x)u2

z4 = z5


(Lf )h2(x)u1 + Lgb(Lf )h2(x)u2 . (86)

It holds that z1 = θ, z1 = ω = z2 , z2 = ω = f2(x) + ga2 u1 +gb2 u2⇒z2 = f2(x) + 0u1 + 0u2 which finally gives z2 =f2(x). Moreover, it has been proven that z3 = f2 , therefore,it holds z2 = z3 . Moreover, it holds that

z3 =∂z3

∂x1x1 +

∂z3

∂x2x2 +

∂z3

∂x3x3 +

∂z3

∂x4x4 +

∂z3

∂x5x5 +

∂z3

∂x6x6


z3 =∂z3

∂x1f1 +

∂z3

∂x2f2 +

∂z3

∂x3f3 +

∂z3

∂x4f4

+∂z3

∂x5

(f5 +

1σLr

u1

)+

∂z3

∂x6

(f6 +

1σLr

u2

)(88)

that is also written as


(L2f h1(x))u1 + Lgb

(L2f h2(x))u2 . (89)

Similarly, one has

z4 = x23 + x2

4⇒z4 = 2x3 x3 + 2x4 x4⇒z4 = 2x3f3 + 2x4f4

z4 = 2x3

[− 1

τsx3 + ωdqx4 +

M

τsx5 + vsd

]

+ 2x4

[−ωdqx3 −

1τs

x4 +M

τsx6 + vsq

]⇒z4 = z5 . (90)

Additionally, it holds

z5 =∂z5

∂x1x1 +

∂z5

∂x2x2 +

∂z5

∂x3x3 +

∂z5

∂x4x4 +

∂z5

∂x5x5 +

∂z6

∂x6

x6


z5 =∂z5

∂x1f1 +

∂z5

∂x2f2 +

∂z5

∂x3f3 +

∂z5

∂x4f4

+∂z5

∂x5

(f5 +

1σLr

u1

)+

∂z5

∂x6

(f6 +

1σLr

u2

)(92)

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which subsequently gives


(Lf h2(x))u1 + Lgb(Lf h2(x))u2 (93)

which is the anticipated relation about z5 . Consequently, (86)is confirmed to hold. The system of (86) can be written in theinput–output linearized form

(z

(3)1

z4

)= fa + Mu (94)

where

fa(x) =

(L3

f h1(x)

L2f h2(x)

)

M =

(Lga

L2f h1(x) Lgb

L2f h2(x)

LgaLf h1(x) Lgb

Lf h2(x)

)(95)

or equivalently, one has the system’s description in the MIMOcanonical form

⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎝

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 10 0 0 0 0

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

z1z2z3z4z5

⎞⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎝

0 00 01 00 00 1

⎞⎟⎟⎟⎟⎠

(v1v2

)

(96)where


(L2f h1(x))u1 + Lgb

(L2f h2(x))u2


(Lf h1(x))u1 + Lgb(Lf h2(x))u2 . (97)

APPENDIX II

Field-oriented control of the DFIG The classical method forinduction machines control was introduced by Blaschke (1971)and in the DFIG case is based on a transformation of the rotor’scurrents (ir α ) and (ir b ) and of the magnetic fluxes of the stator(ψsα and ψsb ) to the reference frame dq which rotates togetherwith the rotor [52]. Thus, the controller’s design uses the currentsir d and ir q and the fluxes ψsd and ψsq [35]. The angle of thevectors that describe the magnetic fluxes ψsα and ψsb is firstdefined, i.e.,

ρ = tan−1(

ψsb

ψsa

). (98)

The angle between the inertial reference frame of the stator andthe rotating reference frame of the rotor is taken to be equal toρ. The transition from (ir α , ir b) to (ir d , ir q ) is given by

(ird

ir q

)=


)(irα

ir b

). (99)

The transition from (ψsα , ψr b) to (ψsd, ψsq ) is given by

(ψsd

ψsq

)=


) (ψsα

ψsb

). (100)


||ψ || , sin(ρ) = ψs b

||ψ || , and ||ψ|| =√ψ2

sα+ ψ2

sb. Using the above transformation ones obtains

ir d =ψsαir α + ψsbir b

||ψ|| ψsd = ||ψ||(101)

ir q =ψsαir b − ψsbir α

||ψ|| ψsq = 0.

Therefore, in the rotating frame dq of the generator, there willbe only one nonzero component of the magnetic flux ψsd

, whilethe component of the flux along the d-axis equals 0. The newinputs of the system are considered to be vr d , vr q , which areconnected to vr a , vr b according to the relation

(vrα

vr b

)= ||ψ||·

(ψsa ψsb

ψsb ψsa

)−1 (vrd

vr q

). (102)

In the new coordinates, the induction generator model has beendescribed in (7) to (12). The state-space model of the inductiongenerator has been defined in (13) and (14). Using the statevariables notation, the DFIG model was written in the form

x1 = x2

x2 = −Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4)

x3 = − 1τs

x3 + ωdqx4 +M

τsx5 + vsd


x4 +M

τsx6 + vsq

x5 = −βx2x4 +β

τsx3 + (ωdq − x2)x6 − γ2x5

+1

σLrvrd

− βvsd

x6 = − β

τsx4 + βx2x3 − (ωdq − x2)x5 − γ2x6

+1

σLrvrq

− βvsq. (103)

Next, the following nonlinear feedback control law is defined(

vrd

vr q

)=

σLr

(βx2x4 − β

τsx3 − (ωdq − x2)x6 + βvsd

+ βv1βτs


).(104)

The terms in (104) have been selected so as to linearize the fifthand sixth row of the state-space model of the induction generatorin (103) and to produce first-order linear differential equations.The control signal in the fixed coordinates system a − b will be

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(vr α

vr b

)= ||ψ||σLr

(ψsα

ψsb

−ψsb ψsα

)−1

·(

βx2x4 − βτs

x3 − (ωdq − x2)x6 + βvsd+ βv1

βτs


). (105)

Substituting (104) into (103), one obtains:

x1 = x2 (106)

x2 = −Km

Jx2 −

Tm

J+

η

J(x6x3 − x5x4) (107)

x3 = − 1τs

x3 + ωdqx4 +M

τsx5 + vsd

(108)


x4 +M

τsx6 + vsq

(109)

x5 = −γ2x5 + βv1 (110)

x6 = −γ2x6 + βv2 (111)

The system of (106)–(111) comprises two linear subsystems,where the first one has as output the magnetic flux x3 = ψsd

and the second has as output the rotation speed x2 = ω [35].Thus, from (108) and (110), one obtains

x3 = − 1τs

x3 +M

τsx5 + vsd

(112)

x5 = −γ2x5 + βv1 (113)

while from (107) and (111), one obtains

x2 = −Km

Jx2 −

Tm

J+

η

Jx3x6 (114)

x6 = −γ2x6 + βv2 . (115)

For x3 = ψsd, it holds that if ψsd→ψs

refd , i.e., the transient

phenomena for ψsd have been eliminated, and therefore, ψsd

has converged to a steady-state value, then the two subsystemsdescribed by (112)–(113) and (114)–(115) are decoupled.

The subsystem that is described by (112) and (113) is lin-ear with control input v1 and can be controlled using methodsof linear control, such as optimal control, or PID control. Forinstance, the following PI controller has been proposed for thecontrol of the magnetic flux [35]

v1(t) = −kd1(ψsd − ψsrefd ) − kd2

∫ t

0(ψsd(τ) − ψsd

ref (τ)dτ.

(116)Thus, if (116) is applied to the subsystem that is described by(112) and (113), one anticipates to succeed ψsd(t)→ψs

refd (t).

Now, the subsystem that consists of (114) and (115) is ex-amined. The term T = η

J x6x3 denotes the torque developed inthe rotor. After succeeding ψsd→ψs

refd , one can also control the

generator’s speed ω, using linear feedback control algorithms.A first approach to the control of the speed ω is to use nested PI

loops, i.e.,

v2 = −Kq 1(T − Tref ) − Kq 2

∫ t

0(T (t) − Tref (t))dτ

Tref = −Kq 3(ω − ωref )− Kq 4

∫ t

0(ω(t)− ωref (t))dτ. (117)

From the above analysis, it becomes clear that a remaining prob-lem in the implementation of field-oriented control for DFIGs ishow to measure efficiently x3 = ψsd(t). Therefore, one comesagainst the need for applying a state observer or Kalman Filter-ing. Besides, the tuning of the multiple PID and PI controllersthat constitute the field-oriented control scheme, as described in(116) and (117), remains valid only round local operating points,and thus, the stability and robustness of the field-oriented controlfor DFIGs cannot be assured.

REFERENCES

[1] H. Xu, J. Hu, and Y. He, “Operation of wind-turbine-driven DFIG sys-tems under distorted grid voltage conditions: Analysis and experimentalvalidations,” IEEE Trans. Power Electron., vol. 27, no. 5, pp. 2354–2366,May 2012.

[2] M. Mohseni and S. M. Islam, “Transient control of DFIG-based windpower plants in compliance with the australian grid code,” IEEE Trans.Power Electron., vol. 27, no. 6, pp. 2813–2824, Jun. 2012.

[3] J. Yao, H. Li, Z. Chen, X. Xia, X. Q. Li, and Y. Liao, “Enhanced controlof a DFIG-based wind-power generation system with series grid-sideconverter under unbalanced grid voltage conditions,” IEEE Trans. PowerElectron., vol. 28, no. 7, pp. 3167–3181, Jul. 2013.

[4] D. G. Forchetti, J. A. Solsona, G. O. Garcia, and M. I. Valla, “A controlstrategy for stand-alone wound rotor induction machine,” Electric PowerSyst. Res., vol. 77, pp. 163–169, 2007

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Gerasimos Rigatos (M’98) received the Diploma de-gree in electrical engineering and the Ph.D. degreein control systems both from the National Techni-cal University of Athens (NTUA), Athens, Greece,in 1995 and 2000, respectively.

He is currently a Researcher (Grade B’) at the In-dustrial Systems Institute, in Rion Patras, Greece. Hehas held research and teaching positions at INRIA-IRISA France in 2001, Universite Paris XI in 2007,Harper-Adams University College, U.K. from 20112012) and in Greek universities. His research interests

include the areas of control and robotics, mechatronics, electric power systems,computational intelligence, fault diagnosis and optimization.

Dr. Rigatos is an Editor-in-Chief of the Journal of Intelligent Industrial Sys-tems (Springer) and a Member of the IET and IMACS.

Pierluigi Siano (M’09–SM’14) received the M.Sc.degree in electronic engineering and the Ph.D. de-gree in information and electrical engineering fromthe University of Salerno, Salerno, Italy, in 2001 and2006, respectively.

He is an Aggregate Professor of electrical energyengineering with the Department of Industrial Engi-neering, University of Salerno. In 2013, he receivedthe Italian National Scientific Qualification as a FullProfessor in the competition sector electrical energyengineering. His research interests include the inte-

gration of distributed energy resources in smart distribution systems and onplanning and management of power systems. He has coauthored more than 160papers including more than 70 international journals.

Dr. Siano is an Associate Editor of the IEEE TRANSACTIONS ON INDUSTRIAL

INFORMATICS, and an Editor-in-Chief of the Journal of Intelligent IndustrialSystems (Springer).

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Nikolaos Zervos received the Ph.D. degree in elec-trical engineering from the University of Toronto,Toronto, ON, Canada, the M.Sc. degree in systemsand computing science from Carleton University, Ot-tawa, ON, and the Diploma degree in electrical andmechanical engineering from the National TechnicalUniversity of Athens, Athens, Greece.

He is currently a Researcher (Grade A’) at theIndustrial Systems Institute, Rion Patras, Greece. Hewas at Bell Laboratories first with AT&T and thenwith Lucent Technologies, as Acting Technical Man-

ager of Multimedia Access Communication Networks. He is one of the world’sexperts in bandwidth-efficient digital transmission and author of several patentsin the areas of data transmission and digital signal processing.

Carlo Cecati (M’90-SM’03–F’06) received the Dr.Ing. degree in electrotechnics from the University ofLAquila, LAquila, Italy, in 1983.

Since 1983, he has been with the Department ofElectrical and Information Engineering, University ofLAquila, where he is currently a Professor of indus-trial electronics and drives and is a Rectors Delegate.He is the Founder and the Coordinator of the Ph.D.courses on management of renewable energies andsustainable building at the University of LAquila. In2007, he founded DigiPower Ltd., LAquila, which is

a spin-off dealing with industrial electronics and renewable energies. His re-search and technical interests include several aspects of power electronics andelectrical drives.

Dr. Cecati is an Editor-in-Chief of the IEEE TRANSACTIONS ON INDUSTRIAL

ELECTRONICS, and has been also a Technical Editor of the IEEE/ASME TRANS-ACTIONS ON MECHATRONICS. He is a Member of IEEE IES Committees onRenewable Energy Systems and on Power Electronics.

Documents

Control and disturbances compensation for doubly-fed induction generators using the Derivative-Free Nonlinear Kalman Filter