energies

Article

Feedback Linearization Controller for a Wind EnergyPower System

Muthana Alrifai *, Mohamed Zribi and Mohamed Rayan

Department of Electrical Engineering, Kuwait University, P. O. Box 5969, Safat 13060, Kuwait;mohamed.zribi@ku.edu.kw (M.Z.); mmsrayan@gmail.com (M.R.)* Correspondence: m.alrifai@ku.edu.kw; Tel.: +965-2498-7381

Academic Editor: Paolo MercorelliReceived: 14 June 2016; Accepted: 19 September 2016; Published: 23 September 2016

Abstract: This paper deals with the control of a doubly-fed induction generator (DFIG)-basedvariable speed wind turbine power system. A system of eight ordinary differential equations isused to model the wind energy conversion system. The generator has a wound rotor type withback-to-back three-phase power converter bridges between its rotor and the grid; it is modeled usingthe direct-quadrature rotating reference frame with aligned stator flux. An input-state feedbacklinearization controller is proposed for the wind energy power system. The controller guaranteesthat the states of the system track the desired states. Simulation results are presented to validatethe proposed control scheme. Moreover, further simulation results are shown to investigate therobustness of the proposed control scheme to changes in some of the parameters of the system.

Keywords: wind energy; doubly-fed induction generator (DFIG); feedback linearization

1. Introduction

There has been a global interest in renewable energy resources due to the worldwide increasein power demand and the limitation of fossil fuels and their harmful impact on the environment.Renewable energy resources, such as wind energy, are naturally available, clean and have a much lessharmful impact on the environment than fossil fuels.

Wind energy is the fastest growing renewable energy resource. Globally, the annual cumulativeinstalled wind energy capacity has increased rapidly during the period from 1997–2014 [1]. In addition,the advancement on the design of the components of wind energy power systems that include powerelectronics inverters, electric generators and drive train systems has also contributed to the fast growthand high demand of wind energy conversion systems (WECs). The increasing market share of variablespeed wind energy conversion systems has led to further investigations of wind turbines’ controltechnology. Furthermore, researchers have recognized that appropriate control algorithms can greatlyimprove the efficiency of wind power conversion systems.

Among the various types of available wind turbines, the doubly-fed induction generator-basedwind turbines are widely used for variable speed wind turbine systems because of their simplestructures, their reliable operations, their high power densities and their energy efficiency.Moreover, doubly-fed induction generators-based energy systems have other advantages, such asthe reduction of mechanical stress, the flexibility in controlling active and reactive powers and theability to track maximum power using different control techniques.

Many research articles dealing with the design of different types of control schemes forDFIG-based variable speed wind turbine systems were published in the last few years; for example,the reader can refer to the works in [2–16] for a detailed review of wind turbine systems and theircontrol. Some of these control techniques are highlighted below.

Energies 2016, 9, 771; doi:10.3390/en9100771 www.mdpi.com/journal/energies

Energies 2016, 9, 771 2 of 23

Researchers have used different control techniques to control DFIG-based wind turbine energysystems. For example, PID controllers and several modified PID control schemes were used to controlwind energy conversion systems [17–19]. These types of schemes are believed to be the easiestand simplest design methods. However, DFIG-based wind energy systems are highly nonlinearsystems, which are characterized by strong couplings between the different variables of the systems.Thus, nonlinear control schemes need to be designed to control wind energy power systems so thatthe desired performances are attained.

The work in [20] proposed a method for obtaining the maximum power output of a DFIG-based windenergy system. The proposed control technique was based on the usage of an improved maximumpower point tracking (MPPT) curve method and a Lyapunov function. In [21], an artificial neuralnetwork technique based on Markov scheme approaches was developed to optimize the output of a windturbine system. An artificial neural network, which is based on the Jordan re-current concept, was used toestimate the reference tracking speed of the rotor in [22]. A direct and an indirect control structureusing a quantum neural network to enhance the efficiency of the system and to optimize the output of a windenergy power system was proposed in [23]. Furthermore, the combination of fuzzy logic control andsliding mode control was investigated in the literature; for example, refer to the works in [24–28].

Investigations to determine the best nonlinear control technique that can be used to improvethe performance of DFIG-based WEC systems are still ongoing [29]. The control of both the gridside converter (GSC) and the rotor side converter (RCS) in one control loop was reported in [30].In [31], a direct active power and reactive power controller based on the estimation of the stator fluxwas proposed, and a basic hysteresis controller was used. Vector control and direct power controlwere proposed in [32] to regulate the active power and reactive power generated by a DFIG-basedwind energy system. In [33], two distinct control strategies were used to control a DFIG-basedWEC system; backstepping and sliding mode control strategies were first used to control the rotorside converter (RCS); then, the same strategies were used to control the grid side converter (GSC).A perturbation observer-based control scheme for the control of DFIG in a multi-machine powersystem was introduced in [34]. The controller is achieved with a four-loop perturbation observer-basedcontrol configuration, and the controlled dynamics were investigated through simulation studies.

Several researchers investigated the usage of the feedback linearization technique for the controlof DFIG-based WEC systems. Some of these works are highlighted below. An input-output feedbacklinearization controller for a DFIG-based system connected to an infinite bus was discussed in [35];the authors differentiated the electromagnetic torque, the stator power and the grid side converterpower outputs of the DFIG to obtain a linear relationship with the rotor voltages, which serve ascontrol inputs. Then, decoupled torque control and decoupled power control were developed. In [36],an input-output linearizing and decoupling control strategy was proposed for a doubly-fed inductiongenerator system. The developed controller was verified by using a 7.5 kW wind power test rig.A mathematical model of the DFIG based on the so-called stator magnetizing current field orientationwas given in [37]; then, an input-output linearizing controller is proposed. The controller resultsin decoupled control and the tracking of the generated active and reactive powers of the system.Direct control of the torque and direct control of the power outputs of the DFIG were establishedbased on the input-output feedback linearization technique in [38,39]. In [40], an adaptive nonlinearcontrol strategy based on a feedback linearization scheme was proposed for a DFIG-based WECsystem; a disturbance observer was added to estimate the uncertainties in the parameters of the system.The performance of the system is checked in the presence of nearby faults. Moreover, an exact feedbacklinearization scheme was proposed in [41] for a DFIG-based WEC system. The wind turbine with DFIGis represented by a third-order model. This model is exactly linearized, and a linear quadratic regulatoris used to design an optimal controller for the linearized system. Simulations were performed on asingle machine infinite bus system and on a four-machine system; the simulation results indicate thatthe proposed control scheme improves the transient stability of the power system, and it enhances thedamping of the system.

Energies 2016, 9, 771 3 of 23

The contribution of this paper involves the design of an input-state feedback linearizationcontroller for a wind energy conversion system. Previous works on feedback linearization of WECsystems, generally, used fifth order models or sixth order models. Our work uses an eighth ordermodel of the WEC system; the state space representation of the eighth order model under considerationuses four electrical states and four mechanical states. The proposed controller controls the rotor sideconverter (RCS) of a DFIG. It guarantees the stability of the closed loop system and enables the trackingof the desired states of the system. In addition, the proposed controller maximizes the power extractedfrom the wind. The proposed theoretical work is verified through MATLAB simulations; moreover,the validity of the proposed controller was tested when some of the parameters of the system change.

The rest of the paper is organized as follows. Section 2 presents the model of a DFIG-basedwind energy power system; the system consists of eight nonlinear ordinary differential equations.In Section 3, the computations of the values of the desired states of the system, as well as the referenceinputs are discussed in detail. Section 4 presents the design of an input-state feedback linearizationcontroller for the wind energy conversion system. To show the effectiveness of the proposed controller,simulation studies are presented in Section 5. In addition, some simulation results are shown toinvestigate the robustness of the controller to changes in some of the parameters of the system.Finally, the conclusion is drawn in Section 6.

2. Model of the Wind Energy Conversion System

The model of a DFIG-based wind energy conversion power system is presented in this section.A block diagram depiction of the system is shown in Figure 1. The system consists of a wind turbineconnected through a gear box to a doubly-fed induction generator. The stator windings of the standardwound induction generator are directly connected to the grid. The rotor windings are connected to thegrid through a voltage source and power electronics converters (AC/DC/AC). The wind energy iscaptured by the motor blades and transferred to the motor hub. The hub has a gearbox, which attachesthe low-speed shaft to the high-speed shaft that drives the DFIG, which converts the mechanical energyto electrical energy. This electrical energy is delivered to the grid.

Figure 1. A block diagram representation of the variable speed DFIG-based wind energyconversion system.

Note that Ωr, Ωg, Th and Tg depicted in Figure 1 represent the rotational speed of the turbine onthe low-speed side of the gearbox, the mechanical speed of the generator, the high-speed shaft torqueand the generator torque, respectively. Furthermore Ps and Qs are the stator active power and thestator reactive power.

The mechanical power extracted from the wind can be expressed as [42],

Pm =12

Cp(λ, β)ρπR2V3 (1)

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where:

Pm: the captured wind power (the mechanical power),Cp(λ, β): the power coefficient,λ: the tip speed ratio (TSR),β: the pitch angle,ρ: the air density,R: the rotor-plane radius,V: the speed of the wind.

The tip speed ratio can be expressed as,

λ = ΩrR/V (2)

where Ωr is the rotational speed of the turbine on the low-speed side of the gearbox.The power coefficient Cp versus the tip speed ratio λ for different values of the pitch angle β

is depicted in Figure 2. It can be seen from the figure that when β = 0, the maximum value of Cp

is achieved when Cp(max) = 0.48 at an optimal value of the tip speed ratio λopt = 8. To capture themaximum power from the wind, a variable speed wind turbine normally follows the Cp(max) up to therated speed by varying the rotor speed to keep the tip speed ratio at its optimum value λopt.

Tip speed ratio (λ)

0 2 4 6 8 10 12 14 16

Pow

er c

oeffi

cien

t Cp (β

, λ

)

-0.1

0

0.1

0.2

0.3

0.4

0.5

β=0

β=5

β=10

β=15

β=20

Figure 2. The power coefficient Cp versus the tip speed ratio λ.

Using Equation (1), the maximum captured wind power, Pm(max), can be written as,

Pm(max) =12

Cp(max)(λopt, β)ρπR2V3. (3)

The aerodynamic torque, Tr, can be expressed as follows,

Tr = Pm/Ωr =12

Cp(λ, β)ρπR2V3/Ωr. (4)

The wind turbine varies its speed by following the maximum value of the power coefficientCp(max) so that maximum power is captured. This is done by changing the speed of the rotor tomaintain the tip speed ratio at its optimum value λopt [42]. Using Equations (2)–(4), the aerodynamictorque, Tr, can be written as follows,

Tr = KoptΩ2r (5)

Energies 2016, 9, 771 5 of 23

where Kopt = 0.5ρπR5Cp(max)/λ3opt.

To model the wind energy power system, we first need to define the state variables of the system.The system consists of an electrical part and a mechanical part. The state variables of the electricalpart of the system are the stator currents in the d and q axes (isd and isq) and the rotor currents in thed and q axes (ird and irq). The state variables of the mechanical part of the system are the rotationalspeed of the turbine on the low-speed side of the gearbox (Ωr), the mechanical speed of the generator(Ωg), the high-speed shaft torque (Th) and the generator torque (Tg). The inputs for the system arethe stator and rotor voltages and the required generator torque (Tg,r). Note that the voltage usd is thed-axis stator voltage, and usq is the q-axis stator voltage. The voltage urd is the d-axis rotor voltage, andurq is the q-axis rotor voltage.

The rotational speed of the generator Ωg is related to the rotor angular frequency ωr, such that,

Ωg =1

npωr (6)

where np is the number of pole pairs of the generator.The stator angular frequency ωs is related to the stator frequency, fs, such that,

ωs = 2π fs. (7)

Furthermore, the leakage coefficient can be written, such that,

ξ =1 − L2

mLsLr

(8)

where Lm is the magnetizing inductance, Lr is the rotor leakage inductance and Ls is the statorleakage inductance.

The model of the wind energy power system consists of eight first order ordinary differentialequations (odes). The first four odes are related to the electrical part of the system; the second fourodes are related to the mechanical part of the system. Hence, the wind turbine energy conversionsystem can be described using the following set of nonlinear ordinary differential equations, suchthat [42,43],

disddt

=−Rs

ξLsisd + ωsisq +

1 − ξ

ξnpΩgisq +

RrLm

ξLrLsird +

Lm

ξLsnpirqΩg −

1ξLs

usd +Lm

ξLrLsurd

disq

dt= −ωsisd −

1 − ξ

ξnpΩgisd +

−Rs

ξLsisq −

Lm

ξLsnpΩgird +

RrLm

ξLrLsirq −

1ξLs

usq +Lm

ξLrLsurq

dirddt

=RsLm

ξLrLsisd −

Lm

ξLrnpΩgisq −

Rr

ξLrird + ωsirq −

1ξ

npΩgirq +Lm

ξLrLsusd −

1ξLr

urd

dirq

dt=

Lm

ξLrnpΩgisd +

RsLm

ξLrLsisq − ωsird +

1ξ

npΩgird −Rr

ξLrirq +

Lm

ξLrLsusq −

1ξLr

urq

dΩr

dt= −Dr

JrΩr +

Kopt

JrΩ2

r −nbJr

Th (9)

dΩg

dt= −

Dg

JgΩg +

1Jg

Th −1Jg

Tg

dThdt

=1nb

(Kls −DrDls

Jr)Ωr +

1nb

(DlsKopt

Jr)Ω2

r −1n2

b(Kls −

DgDls

Jg)Ωg

−Dls(1Jr+

1n2

b Jg)Th +

Dls

n2b Jg

Tg

dTg

dt= − 1

τgTg +

1τg

Tg,r

Energies 2016, 9, 771 6 of 23

where:

Dg: the damping constants for the generator,Dls: the damping constants for the the equivalent low-speed shaft,Dr: the damping constants for the rotor,im: the magnetizing current,Jg: the moment of inertia of the generator,Jr: the moment of inertia of the rotor,Kls: the equivalent torsional stiffness of the low-speed shaft,nb: the gearbox ratio (the gearbox is considered as a lossless device for this model),Rr: the rotor resistance,Rs: the stator resistance,τg: the time constant of the model,Vs: the stator voltage magnitude.

For the ease of presentation, we define the following parameters for the wind energy conversionsystem, p1 = −Rs

ξLs, p2 = ωs, p3 = 1−ξ

ξ np, p4 = Rr LmξLr Ls

, p5 = LmξLs

np, p6 = − 1ξLs

,

p7 = LmξLr Ls

, p8 = − LmξLr

np, p9 = − RrξLr

, p10 = − 1ξ np, p11 = − 1

ξLr, p12 = Rs Lm

ξLr Ls, p13 = −Dr

Jr,

p14 =Kopt

Jr, p15 = − nb

Jr, p16 = −Dg

Jg, p17 = 1

Jg, p18 = 1

nb(Kls − Dr Dls

Jr), p19 = 1

nb(

DlsKoptJr

),

p20 = − 1n2

b(Kls −

DgDlsJg

), p21 = −Dls(1Jr+ 1

n2b Jg

), p22 = Dlsn2

b Jg, p23 = − 1

τg.

Moreover, we define the state variable vector x, which contains the states x1–x8, such that,

x = [x1 x2 x3 x4 x5 x6 x7 x8]T = [isd isq ird irq Ωr Ωg Th Tg]

T . (10)

Therefore, the model of the wind energy power system in (10) can be written in compact formas follows,

x1 = p1x1 + p2x2 + p3x6x2 + p4x3 + p5x4x6 + p6usd + p7urd

x2 = −p2x1 − p3x6x1 + p1x2 − p5x6x3 + p4x4 + p6usq + p7urq

x3 = p12x1 + p8x6x2 + p9x3 + p2x4 + p10x6x4 + p7usd + p11urd

x4 = −p8x6x1 + p12x2 − p2x3 − p10x6x3 + p9x4 + p7usq + p11urq (11)

x5 = p13x5 + p14x25 + p15x7

x6 = p16x6 + p17x7 − p17x8

x7 = p18x5 + p19x25 + p20x6 + p21x7 + p22x8

x8 = p23x8 − p23Tg,r

Let the desired state vector xd be such that xd = [x1d x2d x3d x4d x5d x6d x7d x8d]T , where

xid (i = 1, 2, ..., 8) are the desired values of the states xi (i = 1, 2, ..., 8) of the power system. Since thedesired states have to be an operating point of the power system, then they must satisfy the equationsof the model of the wind energy power system in (12). Therefore, the desired states of the powersystem are governed by the following set of differential equations:

x1d = p1x1d + p2x2d + p3x6dx2d + p4x3d + p5x4dx6d + p6usd + p7urd (12)

x2d = −p2x1d − p3x6dx1d + p1x2d − p5x6dx3d + p4x4d + p6usq + p7urq (13)

x3d = p12x1d + p8x6dx2d + p9x3d + p2x4d + p10x6dx4d + p7usd + p11urd (14)

x4d = −p8x6dx1d + p12x2d − p2x3d − p10x6dx3d + p9x4d + p7usq + p11urq (15)

x5d = p13x5d + p14x25d + p15x7d (16)

x6d = p16x6d + p17x7d − p17x8d (17)

x7d = p18x5d + p19x25d + p20x6d + p21x7d + p22x8d (18)

x8d = p23x8d − p23Tg,r (19)

Energies 2016, 9, 771 7 of 23

Note that usd, usq, urd, urq and Tg,r in (12)–(19) represent the reference inputs of the wind energyconversion system. The computation of these reference inputs will be discussed in the next section.

Define the errors ei (i = 1, 2, ..., 8), such that:

e1 = x1 − x1d

e2 = x2 − x2d

e3 = x3 − x3d

e4 = x4 − x4d (20)

e5 = x5 − x5d

e6 = x6 − x6d

e7 = x7 − x7d

e8 = x8 − x8d

Using Equations (12)–(21), the dynamic model of the error system can be written, such that:

e1 = p1e1 + p2e2 + p3(x6x2 − x6dx2d) + p4e3 + p5(x4x6 − x4dx6d) + v1

e2 = −p2e1 − p3(x6x1 − x6dx1d) + p1e2 − p5(x6x3 − x6dx3d) + p4e4 + v2

e3 = p12e1 + p8(x6x2 − x6dx2d) + p9e3 + p2e4 + p10(x6x4 − x6dx4d) + v3

e4 = −p8(x6x1 − x6dx1d) + p12e2 − p2e3 − p10(x6x3 − x6dx3d) + p9e4 + v4 (21)

e5 = p13e5 + p14(x25 − x2

5d) + p15e7

e6 = p16e6 + p17e7 − p17e8

e7 = p18e5 + p19(x25 − x2

5d) + p20e6 + p21e7 + p22e8

e8 = p23e8 + vg

where the inputs v1, v2, v3, v4 and vg are defined, such that,

v1 = p6(usd − usd) + p7(urd − urd)

v2 = p6(usq − usq) + p7(urq − urq)

v3 = p7(usd − usd) + p11(urd − urd) (22)

v4 = p7(usq − usq) + p11(urq − urq)

vg = −p23(Tg,r − Tg,r).

After some manipulations, the error dynamics in (22) can be written as follows,

e1 = p1e1 + p2e2 + p3(e6e2 + x6de2 + x2de6) + p4e3 + p5(e4e6 + x4de6 + x6de4) + v1 (23)

e2 = −p2e1 − p3(e6e1 + x6de1 + x1de6) + p1e2 − p5(e6e3 + x6de3 + x3de6) + p4e4 + v2 (24)

e3 = p12e1 + p8(e6e2 + x6de2 + x2de6) + p9e3 + p2e4 + p10(e4e6 + x4de6 + x6de4) + v3 (25)

e4 = −p8(e6e1 + x6de1 + x1de6) + p12e2 − p2e3 − p10(e6e3 + x6de3 + x3de6) + v4 (26)

e5 = p13e5 + p14(e25 + 2x5de5) + p15e7 (27)

e6 = p16e6 + p17e7 − p17e8 (28)

e7 = p18e5 + p19(e25 + 2x5de5) + p20e6 + p21e7 + p22e8 (29)

e8 = p23e8 + vg (30)

Define the matrix Mu and the vectors u, v and ur, such that,

Energies 2016, 9, 771 8 of 23

Mu =

p6 p7 0 0 00 0 p6 p7 0p7 p11 0 0 00 0 p7 p11 00 0 0 0 −p23

, u =

usdurdusq

urq

Tg,r

, v =

v1

v2

v3

v4

vg

, ur =

usdurdusq

urq

Tg,r

. (31)

Remark 1. The controller u =[

usd urd usq urq Tg,r

]Tof the wind energy conversion system model

in (12) is computed using the controller v =[

v1 v2 v3 v4 vg

]Tand the reference inputs usd, usq, urd, urq

and Tg,r, such that,

usdurdusq

urq

Tg,r

=

p6 p7 0 0 00 0 p6 p7 0p7 p11 0 0 00 0 p7 p11 00 0 0 0 −p23

−1

v1

v2

v3

v4

vg

+

usdurdusq

urq

Tg,r

(32)

Equation (32) can be written in compact form, such that,

u = M−1u v + ur. (33)

Note that the determinant of the matrix Mu equals p23(p27 − p6 p11)

2, which is different from zero.Hence, the matrix Mu is nonsingular, and its inverse always exists.

3. Computations of the Desired States and the Reference Inputs of the WEC System

This section deals with the computations of the values of the desired states of the wind energyconversion system, as well as the values of the reference inputs of the system.

3.1. Computation of the Desired States of the WEC System

The flux linkages of the stator and the rotor of the generator can be expressed as,

ψsd = Lsisd + Lmird (34)

ψsq = Lsisq + Lmirq (35)

ψrd = Lrird + Lmisd (36)

ψrq = Lrirq + Lmisq (37)

where ψsd is the stator d-axis flux linkage and ψsq is the stator q-axis flux linkage. The flux ψrd is therotor d-axis flux linkage, and ψrq is the rotor q-axis flux linkage.

The stator active power, Ps, and the stator reactive power, Qs, can be expressed as follows,

Ps =32(usdisd + usqisq) (38)

Qs =32(usqisd − usdisq) (39)

where isd and isq are the stator currents; usd and usq are the stator voltages.In the stator voltage oriented reference frame, the q-axis is aligned with the supply voltage Vs;

then, the reference inputs usd and usq are such that usd = 0 and usq = Vs.Using Equations (38)–(39), the desired stator active and reactive powers can be written as follows,

Energies 2016, 9, 771 9 of 23

Psd =32(usdx1d + usqx2d) =

32

x2dVs (40)

Qsd =32(usqx1d − usdx2d) =

32

x1dVs (41)

where x1d is the desired value of isd and x2d is the desired value of isq.Moreover, the stator flux ψs is set to be aligned with the d-axis; hence, the stator d-axis and q-axis

flux linkages are such that ψsd = ψs = Vs/ωs and ψsq = 0. In addition, we choose to set the desiredstator active power Psd to be equal to the maximum power that can be captured from the wind and thedesired stator reactive power Qsd to be zero, then Psd = Pm(max) and Qsd = 0.

Remark 2. Note that the desired reactive power is selected to ensure a unity power factor.Hence, the desired states x1d, x2d, x3d, x4d, x5d, x6d, x7d and x8d are computed as follows.At first, using Equations (40) and (41), we can compute x1d and x2d as follows,

x1d =23

QsdVs

= 0 (42)

x2d =23

PsdVs

=23

Pm(max)

Vs. (43)

Then, using Equations (34) and (35) with ψsd = ψs = Vs/ωs and ψsq = 0, we can compute x3d and x4d,which are the desired values of ird and irq, as follows,

x3d =1

Lm

Vs

ωs(44)

x4d = −23

Ls

Lm

Pm(max)

Vs. (45)

Furthermore, the desired value of generator torque x8d is chosen to provide an appropriate opposing torque.Using Equations (2) and (4), x8d can be written as follows,

x8d = −12

nbρπR3Cp(max)V2/λopt. (46)

Finally, the remaining desired states x5d, x6d and x7d, which are the desired values of Ωr, Ωg, Th, aredetermined by solving the ordinary differential Equations (16)–(18). Recall that these equations are as follows,

x5d = p13x5d + p14x25d + p15x7d (47)

x6d = p16x6d + p17x7d − p17x8d (48)

x7d = p18x5d + p19x25d + p20x6d + p21x7d + p22x8d (49)

3.2. Computation of the Reference Inputs of the WEC System

Recall that in the stator voltage-oriented reference frame, it is assumed that the q-axis is alignedwith the supply voltage Vs. Hence, the reference inputs usd and usq are such that usd = 0 and usq = Vs.On the other hand, the reference inputs urd, urq and Tg,r are computed as follows. First, we define theterms c1, c2, c3, c4 and c5, such that,

c1 = p1x1d + p2x2d + p3x6dx2d + p4x3d + p5x4dx6d (50)

c2 = −p2x1d − p3x6dx1d + p1x2d − p5x6dx3d + p4x4d (51)

c3 = p23x8d (52)

c4 = x2d = Cp(max)(λopt, β)ρπR2V2V (53)

c5 = x8d = − nbλopt

Cp(max)(λopt, β)ρπR2V2V (54)

Energies 2016, 9, 771 10 of 23

where V represents the derivative with respect to time of the wind speed model.Then, using Equations (12), (13) and (19), the reference inputs are obtained, such that,

usd = 0 (55)

urd = − 1p7

c1 (56)

usq = Vs (57)

urq =1p7

(c4 − c2 − p6Vs) (58)

Tg,r =1

p23(p23x8d − c5) (59)

4. A Feedback Linearization Controller for the Wind Energy Conversion System

This section deals with the design of a feedback linearization control law for the wind energyconversion system represented by the nonlinear system of odes given by (12).

We start with defining the variables z1, z2 and z3, such that,

z1 = e5 (60)

z2 = p13e5 + p14(e25 + 2x5de5) + p15e7 (61)

z3 = (p13 + 2p14e5 + 2p14x5d)[p13e5 + p14(e25 + 2x5de5) + p15e7]

+2p14 x5de5 + p15[p18e5 + p19(e25 + 2x5de5) + p20e6 + p21e7 + p22e8]. (62)

Furthermore, define the functions f1 and f2, such that:

f1 = (p13 + 2p14e5 + 2p14x5d)[p13e5 + p14(e25 + 2x5de5) + p15e7]

+2p14 x5de5 + p15[p18e5 + p19(e25 + 2x5de5) + p20e6 + p21e7] (63)

f2 = 2p14(p13e5 + p14(e25 + 2x5de5) + p15e7 + 2x5d)z2

+(p13 + 2p14e5 + 2p14x5d)z3 + 2p14 x5de5

+p15[p18z2 + 2p19(e5 + x5d)z2 + 2p19 x5de5] + p15 p20(p16e6 + p17e7 − p17e8)

+p15 p21(p18e5 + p19(e25 + 2x5de5) + p20e6 + p21e7 + p22e8) (64)

Let the control parameters α1, α2, α3 and α4 be positive scalars and choose the parameters β1,β2 and β3 to be positive scalars, such that the polynomial P1(s) = s3 + β3s2 + β2s + β1 is Hurwitz.

The following proposition gives the feedback linearization controller.

Proposition 1. The feedback linearization controller,

u = M−1u v + ur (65)

with Mu and ur given by (31) and (55)–(59) and v = [v1 v1 v1 v1 vg]T , such that,

v1 = −p2e2 − p3(e6e2 + x6de2 + x2de6)− p4e3 − p5(e4e6 + x4de6 + x6de4)− α1e1 (66)

v2 = p2e1 + p3(e6e1 + x6de1 + x1de6) + p5(e6e3 + x6de3 + x3de6)− p4e4 − α2e2 (67)

v3 = −p12e1 − p8(e6e2 + x6de2 + x2de6)− p2e4 − p10(e4e6 + x4de6 + x6de4)− α3e3 (68)

v4 = p8(e6e1 + x6de1 + x1de6)− p12e2 + p2e3 + p10(e6e3 + x6de3 + x3de6)− α4e4 (69)

vg =1

p15 p22(− f2 − p15 p22 p23e8 − β1z1 − β2z2 − β3z3) (70)

Energies 2016, 9, 771 11 of 23

when applied to the model of the wind energy power system given by (12) guarantees the asymptotic convergenceof the states of the system to their desired values.

Note that the block diagram representation of the DFIG-based WEC system with the proposed controller isdepicted in Figure 3.

Figure 3. A block diagram representation of the variable speed DFIG-based wind energy conversionsystem with the proposed controller. GSC, grid side converter.

Proof. Applying the controllers given by (66)–(69) to the first four differential equations of the errorsystem given by (23)–(26), we obtain:

e1 = −(α1 − p1)e1

e2 = −(α2 − p1)e2 (71)

e3 = −(α3 − p9)e3

e4 = −α4e4.

Because the αi’s (i = 1, ..., 4) are chosen to be positive scalars and since the parameters p1 = − RsξLs

and p9 = − RrξLr

are negative, then it can be concluded from the system of odes in (72) that the errorse1, e2, e3 and e4 asymptotically converge to zero as t tends to infinity.

The second part of the proof involves proving the asymptotic convergence to zero of the errors e5,e6, e7 and e8. Taking the time derivative of the variables z1, z2 and z3 defined in Equations (60)–(62)and using the equations of the error system given by (23)–(30), we obtain:

z1 = z2

z2 = z3 (72)

z3 = f2 + p15 p22(p23e8 + vg)

The application of the controller vg in (70) to the system of odes in (73), yields,

z1 = z2

z2 = z3 (73)

z3 = −β1z1 − β2z2 − β3z3

Energies 2016, 9, 771 12 of 23

The above system can be written in compact form, such that:

z = Mzz (74)

with the matrix Mz and the vector z being such that,

Mz =

0 1 00 0 1

−β1 −β2 −β3

, z =

z1

z2

z3

.

Since β1, β2 and β3 are positive scalars, such that the polynomial P1(s) = s3 + β3s2 + β2s + β1 isHurwitz, then the matrix Mz is a stable matrix (i.e., its eigenvalues are located in the left half of thes-plane). The solution of the Equation (74) is z(t) = exp(Mzt)z(0), where z(0) is the value of z(t) whent = 0. Therefore, the asymptotic convergence of z1, z2 and z3 to zero as t tends to infinity is guaranteedbecause Mz is a stable matrix.

The zero dynamics [44] is defined as the internal dynamics of the system when the output iskept identically zero by a suitable input function. For the third order system given by (73), the zerodynamics is analyzed by studying Equations (27)–(30) when z1 = 0, z2 = 0 and z3 = 0.

Using Equations (60) and (61), it is clear that the asymptotic convergence of z1 and z2 to zeroimplies the asymptotic convergence of e5 and e7 to zero as t tends to infinity.

Moreover, since z1, z2, z3, e5 and e7 converge to zero as t tends to infinity, then the equation givenby (62) yields,

e8 = − p20

p22e6. (75)

As t tends to infinity, the differential Equation (28) of the error system given by (23)–(30), whichrepresents the zero dynamics of the system, reduces to,

e6 = p16e6 − p17e8

= (p16 +p17 p20

p22)e6 = aee6. (76)

The constant ae = p16 +p17 p20

p22in (76) is such that,

ae = p16 +p17 p20

p22

= −Dg

Jg− 1

Jg

1n2

b(Kls −

DgDls

Jg)

n2b Jg

Dls(77)

= −Dg

Jg− (

KlsDls

−Dg

Jg) = − Kls

Dls.

Therefore, since ae = − KlsDls

is always negative, Equation (76) guarantees the asymptoticconvergence of the error e6 to zero as t tends to infinity. Moreover, Equation (75) implies the asymptoticconvergence of the error e8 to zero as t tends to infinity.

The asymptotic convergence to zero of the errors ei (i = 1, ..., 8) as t tends to infinity implies theasymptotic convergence of the states of the wind energy power system given by (12) to their desiredvalues as t tends to infinity.

Energies 2016, 9, 771 13 of 23

5. Simulation Results of the Controlled Wind Energy Power System

The performance of the wind energy conversion system controlled using the proposed feedbacklinearization controller was simulated using the MATLAB software. The parameters of the systemused for the simulation studies are given in Table 1.

Table 1. The parameters of the DFIG-based wind turbine power system.

Parameter Value

Rated power 1.5 MWRated apparent power 1.5 MVARated voltage (line to line) 575 vFrequency/angular speed 2π60 rad/sNominal system frequency 60 HzStator resistance 0.0014 ΩStator leakage inductance 89.98 mHRotor resistance 0.99187 mΩRotor leakage inductance 82.088 mHMagnetizing inductance 1.526 mHInertia of the generator 53.036 kg·m2

Pole pairs 3Wind turbine with a rotor diameter 70 mAir density 0.55 kg/m3

Cut-in wind speed 4 m/sCut-out wind speed 25 m/sRated wind speed 12 m/sRated rotor speed 19.7 rpmDrive-train torsion damper 1.0 × 107 Nm/sDrive-train torsion spring 5.6 × 109 Nm/radGearbox ratio 75.7098Inertia of the rotor 34.6 × 103 kg·m2

The wind speed model used for simulation purposes is given by the following formula:

V = 12 + 0.55[sin(0.0625w)− 0.875 sin(0.1875w) + 0.75 sin(0.3125w)

− 0.625 sin(0.625w) + 0.5 sin(1.875w) + 0.25 sin(3.125w) + 0.125 sin(6.25w)] (78)

with w = 2π10 t. The wind speed profile versus time generated using Equation (78) is depicted in

Figure 4.

Time (Sec)

0 20 40 60 80 100 120 140 160 180

Win

d S

peed (

m/s

)

8

9

10

11

12

13

14

Figure 4. The wind speed profile versus time.

Energies 2016, 9, 771 14 of 23

The wind energy conversion system was modeled using Equation (12) and controlled using theproposed feedback linearization control scheme given by Equations (65)–(70). The desired values ofthe states of the wind energy conversion system are obtained using Equations (42)–(49). The gains ofthe controller are taken to be: α1 = 1e4, α2 = 3e5, α3 = 2e6, α4 = 3.5e8, β1 = 3.15e10, β2 = 6.35e8 andβ3 = 9.7e4.

5.1. Simulation Studies of the WEC System When Using the Nominal Parameters of the System

The simulation results of the controlled system when using the nominal parameters are presentedin Figures 5–13. Figure 5 shows the trajectory of the rotor speed Ωr versus time, and Figure 6 depictsthe trajectory of the generator speed Ωg versus time. The high speed shaft torque and the generatortorque versus time are shown in Figures 7 and 8. The errors between the actual and the desired valuesof the eight states of the system are shown in in Figures 9 and 10. Figure 9 shows the errors in theelectrical state variables of the system versus time. It is clear from this figure that the errors e1, e3 ande4 converge to zero. However, the error e2 shows some fluctuations around zero; these fluctuationsare due to the fact that e2, which is equal to x2 − x2d, where x2d is proportional to V3 (V is the speedof the wind). Figure 10 shows the errors of the mechanical states variables of the system. It is clearfrom this figure that the errors e5 and e6 converge to zero. However, the errors e7 and e8 show somefluctuations around zero; these fluctuations are due to the fact that their desired values are dependenton the wind speed. Note that the average value of the generated torque Tg = x8 is about −2.5e7 Nm.Therefore, it can be concluded that the states of the wind energy power system track the desired states.Figures 11 and 12 present the trajectories of the stator active and reactive powers Ps and Qs versustime. The average of the stator active power Ps is about −0.6 MW. The average of the stator reactivepower Qs is about −0.002 MVAR. Figure 13 depicts the power coefficient Cp versus time; it is clear thatthe maximum value of the power coefficient is achieved when Cp(max) ≈ 0.48. It is clear from thesefigures that all of the closed loop system signals are bounded and that all of the states of the powersystem converge to their desired values.

Time (sec)

0 20 40 60 80 100 120 140 160 180

Ωr (

rpm

)

20

22

24

26

28

30

32

34

Figure 5. The turbine rotational speed on the low-speed side of the gearbox, Ωr, versus time.

Energies 2016, 9, 771 15 of 23

Time (sec)

0 20 40 60 80 100 120 140 160 180

Ωg (

rpm

)

0

500

1000

1500

2000

2500

3000

Figure 6. The mechanical generator speed Ωg versus time.

Time (sec)

0 20 40 60 80 100 120 140 160 180

Th (

Nm

)

×105

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Figure 7. The high-speed shaft torque Th versus time.

Time (sec)

0 20 40 60 80 100 120 140 160 180

Tg (

Nm

)

×107

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Figure 8. The generator torque Tg versus time.

Energies 2016, 9, 771 16 of 23

0 20 40 60 80 100 120 140 160 180

e1 (

Am

p)

-10

0

10

0 20 40 60 80 100 120 140 160 180

e2 (

Am

p)-10

0

10

0 20 40 60 80 100 120 140 160 180

e3 (

Am

p)

-10

0

10

Time (sec)

0 20 40 60 80 100 120 140 160 180

e4 (

Am

p)

-10

0

10

Figure 9. The errors e1, e2, e3, e4 of the electrical state variables of the WEC system versus time.

0 20 40 60 80 100 120 140 160 180

e5 (

rpm

)

-10

0

10

0 20 40 60 80 100 120 140 160 180

e6 (

rpm

)

-10

0

10

0 20 40 60 80 100 120 140 160 180

e7 (

Nm

)

-10

0

10

Time (sec)

0 20 40 60 80 100 120 140 160 180

e8 (

Nm

)

-1000

0

1000

Figure 10. The errors e5, e6, e7, e8 of the mechanical state variables of the WEC system versus time.

Time (sec)

0 20 40 60 80 100 120 140 160 180

Ps (

MW

)

-1

-0.5

0

0.5

Figure 11. The stator active power Ps versus time.

Energies 2016, 9, 771 17 of 23

Time (sec)

0 20 40 60 80 100 120 140 160 180

Qs (

MV

AR

)

×10-3

-1

0

1

2

3

4

5

Figure 12. The stator reactive power Qs versus time.

Time (sec)

0 20 40 60 80 100 120 140 160 180

Pow

er c

oeffi

cien

t Cp

0.4

0.42

0.44

0.46

0.48

0.5

0.52

Figure 13. The power Coefficient Cp versus time.

Therefore, it can be concluded that the simulation results indicate that the wind energy conversionsystem controlled using the proposed feedback linearization controller shows good performance.

5.2. Robustness of the Proposed Control Scheme

Simulation studies were carried out to investigate the robustness of the proposed control schemeto changes in some of the parameters of the wind energy conversion system. At first, the effects of thechanges in the stator resistance Rs, the rotor resistance Rr and the rotor inductance Lr are investigated.We simulated the performance of the closed loop system when the stator resistance Rs, the rotorresistance Rr and the rotor inductance Lr are increased by 30% of their nominal values. The responsesof the system are not shown because of space limitations. However, the steady state performance ofthe system with the changed parameters is very similar to the steady state performance of the systemwith the nominal parameters. Hence, the change in some of the electrical parameters of the system didnot affect the steady state performance of the system.

In addition, simulation studies were carried out to investigate the robustness of the proposedcontrol scheme to changes in some of the electrical and some of the mechanical parameters of the windenergy conversion system. The effects of the changes in the stator resistance Rs, the rotor resistanceRr, the rotor inductance Lr, the moment of inertia of the rotor Jr and the moment of inertia of thegenerator Jg are investigated. Figures 14–22 show the system responses when the stator resistance

Energies 2016, 9, 771 18 of 23

Rs, the rotor resistance Rr and the rotor inductance Lr are increased by 30%; and when the momentof inertia of the rotor Jr and the moment of inertia of the generator Jg are increased by 15% of theirnominal values. Figures 14 and 15 show the rotor and the generator speed responses versus time.The high speed shaft torque and the generator torque versus time are shown in Figures 16 and 17.The errors between the actual and the desired values of the eight states of the system are shown inin Figures 18 and 19. Figure 18 shows the errors of the electrical state variables of the system versustime. It is clear from this figure that the errors e1, e3 and e4 converge to zero. However, the error e2

shows some fluctuations around zero. Figure 19 shows the errors of the mechanical states variablesof the system. It is clear from this figure that the error e5 converges to zero. However, the errors e6,e7 and e8 show some fluctuations around zero. Note that in this case, the error e7 fluctuates between−10 and 10 Nm. The error e8 fluctuates between −5e3 and 5e3 Nm. The fluctuations in the error e8

seems to be high. However, it should be kept in mind that the average value of Tg is a bout −2.5e7Nm. Even though the error is high in absolute value, it represents less than one percent of the value ofTg. Figures 20 and 21 depict the stator active and reactive powers versus time. Figure 22 depicts thepower coefficient Cp. In this case, the reader can see the differences in the steady state performances ofthe system with the nominal parameters and the system with the changed parameters.

Time (sec)

0 20 40 60 80 100 120 140 160 180

Ωr (

rpm

)

20

22

24

26

28

30

32

34

36

Figure 14. The turbine rotational speed on the low-speed side of the gearbox, Ωr, versus time when Rs,RL, Lr, Jr and Jg are different from their nominal values.

Time (sec)

0 20 40 60 80 100 120 140 160 180

Ωg (

rpm

)

500

1000

1500

2000

2500

Figure 15. The mechanical generator speed Ωg versus time when Rs, RL, Lr, Jr and Jg are differentfrom their nominal values.

Energies 2016, 9, 771 19 of 23

Time (sec)

0 20 40 60 80 100 120 140 160 180

Th (

Nm

)

×105

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Figure 16. The high-speed shaft torque Th versus time when Rs, RL, Lr, Jr and Jg are different fromtheir nominal values.

Time (sec)

0 20 40 60 80 100 120 140 160 180

Tg (

Nm

)

×107

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Figure 17. The generator torque Tg versus time when Rs, RL, Lr, Jr and Jg are different from theirnominal values.

0 20 40 60 80 100 120 140 160 180

e1 (

Am

p)

-10

0

10

0 20 40 60 80 100 120 140 160 180

e2 (

Am

p)

-10

0

10

0 20 40 60 80 100 120 140 160 180

e3 (

Am

p)

-10

0

10

Time (sec)

0 20 40 60 80 100 120 140 160 180

e4 (

Am

p)

-10

0

10

Figure 18. The errors e1, e2, e3, e4 of the electrical state variables of the WEC system versus timewhen Rs, RL, Lr, Jr and Jg are different from their nominal values.

Energies 2016, 9, 771 20 of 23

0 20 40 60 80 100 120 140 160 180

e5 (

rpm

)

-10

0

10

0 20 40 60 80 100 120 140 160 180e

6 (

rpm

)-10

0

10

0 20 40 60 80 100 120 140 160 180

e7 (

Nm

)

-100

0

100

Time (sec)

0 20 40 60 80 100 120 140 160 180

e8 (

Nm

) ×104

-1

0

1

Figure 19. The errors e5, e6, e7, e8 of the mechanical state variables of the WEC system versus timewhen Rs, RL, Lr, Jr and Jg are different from their nominal values.

Time (sec)

0 20 40 60 80 100 120 140 160 180

Ps (

MW

)

-1

-0.5

0

0.5

Figure 20. The stator active power Ps versus time when Rs, RL, Lr, Jr and Jg are different from theirnominal values.

Time (sec)

0 20 40 60 80 100 120 140 160 180

Qs (

MV

AR

)

×10-3

-1

0

1

2

3

4

5

Figure 21. The stator reactive power Qs versus time when Rs, RL, Lr, Jr and Jg are different from theirnominal values.

Energies 2016, 9, 771 21 of 23

Time (sec)

0 20 40 60 80 100 120 140 160 180

Pow

er c

oeffi

cien

t Cp

0.4

0.42

0.44

0.46

0.48

0.5

0.52

Figure 22. The power Coefficient Cp versus time when Rs, RL, Lr, Jr and Jg are different from theirnominal values.

Hence, it can be concluded that the simulation results show that the proposed controller isrobust to changes in some of the electrical parameters of the wind turbine conversion system.However, the simulation results indicate that the proposed controller is sensitive to changes in someparameters of the mechanical system.

6. Conclusions

In this paper, the control of a DFIG-based wind energy power system is investigated. The powersystem is modeled using a system of eight nonlinear ordinary differential equations. The proposedcontroller is based on the feedback linearization technique. It is proven that the control schemeguarantees the asymptotic convergence of the states of the power system to their desired values.Simulation results are presented and discussed to show the effectiveness of the proposed scheme.In addition, the effects of the change of some of the parameters of the power system are studiedthrough simulations.

Future work will address the design of other types of controllers, such as observer-basedcontrollers and sliding mode controllers for DFIG-based wind energy power systems.

Acknowledgments: The authors would like to thank Kuwait University for covering the costs to publish thispaper in open access.

Author Contributions: The first two authors contributed equally to this work. The third author contributed tothe simulations of the performance of the system. All authors approved the final manuscript.

Conflicts of Interest: The authors declare no conflict of interest.

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