julio bravo's master graduation project

Optimal Control Tracking Problem Applied to a Wind PowerSystem

May 4, 2015

Julio [email protected]

Abstract

We consider a wind power system with the goal of track a desired trajectory ofits states using an extension of the Linear Quadratic Regulator method. The nonlinearsystem is mathematically modeled and then it is linearized for Tracking Problem designpurposes. We mathematically design the controller and compare the system states withthe desired trajectories.

1. Introduction

Power generation using wind turbines has become an important source of clean energy duringthe last decades. The necessity of control these systems has been the focus of many researcherswith the goal to improve efficiency considering fatigue loads and life time of the systemcomponents as described by [1] and [2].

In Section 2 we begin showing the nonlinear model of a wind power system developed by [3]and [4] then we show the linearization of the system choosing the appropriate operating points.

In Section 3 we describe the analytical analysis of the tracking problem applied to this linearsystem explained by [5], [6] and [7].

In Section 4 we simulate all the system states separately and compare them against theirrespective desired trajectories.

2. Model

From [3] we have a wind power energy conversion system (WECS) where ”the aerodynamicsblock converts the power from wind into mechanical power (rotations) and these rotations areincreased and transmitted to the generator block by the drive train block. Finally, the mechanicalpower is transformed into electrical power by the generator block”. A WECS block diagram isshown in Fig.1.

Figure 1: WECS Block Diagram

For the control purpose, only aerodynamics, drive train dynamics, and generator dynamics aretaken into account. All the definitions, values, and units of the variables and constants used inthe following deduction of the system model are tabulated in Table 1.

Parameter Definition Value and UnitsLd = Lq d and q components of the stator inductance 0.04156 mHRs Stator Resistance 3.3 Ωρ Standard Air Density 1.2041 kg/m3

Jg Generator Inertia 0.22 kgm2

Bg Damping Coefficient of the Generator Shaft 0.3i Gearbox Ratio 6Kg Stiffness Coefficient of the Generator Shaft 75γ Defined in the Text -0.51215CQ Torque Coefficient 0.3422

λ = λopt Tip-speed Ratio 7.36218Jr Wind Rotor Inertia 7.1062 kgm2

η Gearbox Efficiency 0.405288R Radius of the Wind Rotor Plane 2 mp Number of Pole Pairs 2φm Flux 0.6573 Wba0 λ - Polynomial Coefficient 0.0084948a1 λ - Polynomial Coefficient 0.05186a2 λ - Polynomial Coefficient -0.022818a3 λ - Polynomial Coefficient 0.01191a4 λ - Polynomial Coefficient -0.0017641a5 λ - Polynomial Coefficient 7.48 × 10−5

ωr Wind Rotor Speed 36.8109 rad/sωg Generator Speed 220.8654 rad/sTH Internal Torque 329.49 Nmid d Component of Stator Current 6.68 Aiq q Component of Stator Current 5.25705 A

Table 1: Definition and Units of System Variables

The inputs of the aerodynamics block are V and ωr. The output is Tr which can be expressedas

Tr(λ, t) =1

2ρπR3CQ(λ)V (t)2 (1)

where λ and CQ are defined as

λ(t) =ωr(t)R

V (t)(2)

and

CQ(λ) = a0 + a1λ+ a2λ2 + a3λ

3 + a4λ4 + a5λ

5 (3)

The drive train can be represented by a rigid or flexible model as described in [2]. A flexibledrive train block is shown in Fig.2

Figure 2: Drive Train Block Diagram

From [3] a flexible drive train has the model

ωr(t) = − i

ηJrTH(t) +

1

JrTr(t) (4)

ωg(t) =1

JgTH(t) − 1

JgTg(t) (5)

˙TH(t) = iKgωr(t) −Kgωg(t) −Bg(1

Jg+

i2

ηJr)TH(t) +

iBg

JrTr(t) +

Bg

JgTg(t) (6)

”Essentially, asynchronous and synchronous generators are two primary types of generatorwhich have been used in WECSs. Three popular generators are Squirrel Cage Induction Generator(SCIG), Doubly Fed Induction Generator (DFIG), and Permanent Magnet Synchronous Generator(PMSG)” [3]. This study is focused on the PMSG which, from [8], has the model in (d,q) axes

id(t) = −Rs

Ldid(t) −

pLqLd

iqωg(t) −1

Ldud(t) (7)

iq(t) = −Rs

Lqiq(t) −

p

Lq(Ldid(t) − φm)ωg(t) −

1

Lquq(t) (8)

where Tg is defined as Tg(t) = pφmiq(t) and can be used in the model of the drive trainexplained above.

The complete nonlinear model of a PMSG-based WECS is obtained by combining equations4 to 8.CQ(λ) has its maximum value at λopt. Maintaining λopt constant, regardless the fluctuations

of wind speed, we can expect that a WECS operates optimally. To keep λ(t) at the λopt valueas V (t) changes, according to [3], ωr(t) must be adjusted by controlling Tg(t).

The nonlinear system has the state vector x(t) = [ωr(t);ωg(t);TH(t); id(t); iq(t)]T and the

input vector u(t) = [ud(t);uq(t);V (t)]T .Using Taylor’s Series expansion around the operating points given byx(t) = [ωr(t)ωg(t); TH(t); id(t); iq(t)]

T and u(t) = [ud(t); uq(t); V (t)]T , we obtain a model asfollows

δx(t) = Aδx(t) +Bδu(t) (9)

where δx(t) = x(t) − x(t) and δu(t) = u(t) − u(t) are variations of the variables in theneighborhood of the operating points and A and B are the Jacobian of the system defined as

A =

∂ωr

∂ωr

∂ωr

∂ωg

∂ωr

∂TH

∂ωr

∂id

∂ωr

∂iq∂ωg

∂ωr

∂ωg

∂ωg

∂ωg

∂TH

∂ωg

∂id

∂ωg

∂iq∂TH∂ωr

∂TH∂ωg

∂TH∂TH

∂TH∂id

∂TH∂iq

∂id∂ωr

∂id∂ωg

∂id∂TH

∂id∂id

∂id∂iq

∂iq∂ωr

∂iq∂ωg

∂iq∂TH

∂iq∂id

∂iq∂iq

(10)

and

B =

∂ωr

∂ud

∂ωr

∂uq∂ωr

∂V∂ωg

∂ud

∂ωg

∂uq

∂ωg

∂V∂TH∂ud

∂TH∂uq

∂TH∂V

∂id∂ud

∂id∂uq

∂id∂V

∂iq∂ud

∂iq∂uq

∂iq∂V

(11)

Evaluating the Jacobian at the operating points, we have the system and control matrices givenas

A =

12Jrωr

ρπR3CQ(λ)γV 2 0 − iηJr

0 0

0 0 1Jg

0 −pφmJG

iKg + iBg

2JrωrρπR3CQ(λ)γV 2 −Kg −Bg(

1Jg

+ i2

ηJr) 0 Bgpφm

Jg

0 pLq

Ldiq 0 −Rs

Ld

pLqωg

Ld

0 pLq

(Ldid − φm 0 −pLdωg

Lq−Rs

Ld

(12)

and

B =

0 0 2−γ

2JrρπR3CQ(λ)V

0 0 0

0 0 2−γ2Jr

iBgρπR3CQ(λ)V

− 1Ld

0 0

0 − 1Lq

0

(13)

where γ =λC′

Q(λ)

CQ(λ).

Assuming a wind velocity V = 10 m/s, and evaluating the Jacobian in the numerical valuesof the operating points, the system and control matrices are given as bellow

A =

−1.0133 0 −2.0833 0 0

0 0 4.5455 0 −5.9755

448.176 −75 −5.1136 0 1.7926

0 10.5141 0 −79.4033 441.7308

0 18.2715 0 −441.7308 −79.4033

(14)

and

B =

0 0 18.3058

0 0 0

0 0 32.9504

−24.0616 0 0

0 −24.0616 0

(15)

The linear system have five eigenvalues λ1=-79.45+441.86j, λ2=-79.45-441.86j, λ3=-0.2, λ4=-2.92+35.63j, and λ5=-2.92-35.63j. The negative real part of all eigenvalues demonstrate that thesystem is stable.

3. Tracking Problem

The goal of Optimal Control is to find a control u(t) which causes the systemx(t) = f(x(t), u(t), t) to follow a trajectory x(t) that minimizes the cost function

J = φ(x(tf ), tf ) +

∫ tf

t0

L(x(t), u(t), t)dt (16)

The purpose of this analysis is to maintain the system state x(t) as close as possible to thedesired state xd(t) in the interval [t0, tf ] and because is specially important, that the states beclose to their desired values at the final time. To satisfy the goal of our analysis, we can choosea cost function as

J =1

2< x(tf )− xd(tf ), S(x(tf )− xd(tf ) > +

1

2

∫ tf

t0

[< x(t)− xd(t), Q(x(t)− xd(t)) > + < u(t), Ru(t) >]dt

(17)

where the operator < ·, · > represents the dot product of vectors. Q is a real symmetricn×n matrix that is positive semi-definite. The elements of Q are selected to weight the relativeimportance of the different components of the state vector and to normalize the numerical valuesof the deviations. R is a real symmetric positive definite m×m and S is a real symmetric positivesemi-definite n× n matrix.

Because our system has the linear form

x(t) = Ax(t) +Bu(t) (18)

with initial conditions x(t0) = x0 the above cost function leads to an optimal control trackingproblem.

Defining the Hamiltonian we have

H(x, ψ, u, t) =1

2< x(t)− xd(t), Q(x(t)− xd(t)) > +

1

2< u(t), Ru(t) > + < ψ(t), Ax(t) +Bu(t) > (19)

where ψ(t) are the costates which are deducted from the minimization problem solved usingthe approach of Calculus of Variations. They represent the Lagrange multipliers used to solvesuch problem.

Using the Pontryagin’s minimum principal we find the adjoint equation

ψ(t) = −Qx(t) − ATψ(t) +Qxd(t) (20)

and the algebraic relations that must be satisfied are given by

Ru(t) +BTψ = 0 (21)

Now the problem became a two point boundary problem as shown below

x = Ax(t) +Bu(t) (22)

with initial conditions x(t0) = x0 and

ψ(t) = −Qx(t) − ATψ(t) +Qxd(t) (23)

with ”initial” conditions ψ(tf )=S(x(tf ) − xd(tf )).The problem described above can be written in a state space form as[

x

ψ

]=

[A −BR−1BT

−Q −AT

][x

ψ

]+

[0

Q

]xd (24)

Now let ψ(t) = P (t)x(t) − η(t), where P (t) is a real symmetric n × n matrix and η(t) is an× 1 matrix.

Differentiating we have ψ(t) = P (t)x(t) + P (t)x(t) − η(t).Replacing we obtain

−Qx(t)−AT [P (t)x(t)− η(t)] +Qxd(t) = P (t)x(t) + P (t)(Ax(t)−BR−1BT (P (t)x(t)− η(t)))− η(t) (25)

expanding and regrouping we find

(−Q−ATP (t)− P (t)−P (t)A+P (t)BR−1BTP (t))x(t)+(AT −P (t)BR−1BT )η(t)+ η(t)+Qxd(t) = 0 (26)

To satisfy the above equation we need

P (t) + P (t)A+ ATP (t) − P (t)BR−1BTP (t) +Q = 0 (27)

the above equation is the Riccati differential equation which has to be solved backwards usingP (tf ) = S as ”initial” conditions.

We also have

η(t) − (AT − P (t)BR−1BT )η(t) −Qxd(t) (28)

which has to be solved backwards with ”initial” conditions η(tf ) = Sxd(tf ).Since we know that u(t) = −R−1BTψ(t), once we solve the above equations for P (t) and

η(t) we can find the feedback control law which is

u(x(t), t) = −R−1BTP (t)x(t) +R−1BTη(t) (29)

and finally the closed loop system will be

x(t) = (A−BR−1BTP (t))x(t) +BR−1BTη(t) (30)

4. Simulations

Using a code written in Matlab , we simulated and found the results for different desired tra-jectories. In our case we chose trajectories of the type: constant, linear, quadratic, and sinusoidal.

To solve Riccati system of differential equations we used Runge-Kutta 4th order with stepsize h = 0.0001. We used the same numerical method to solve the extra system of differentialequations that involve η and the state space equations.

For simulation we used the following weight matrices:S is a matrix of dimensions 5 by 5 with all its elements equals to zero. These values were

used in all cases.R is a matrix of dimesions 1 by 1 with its only element equals to one. These values were

used in all cases.The initial conditions used for the state space simulation were x0 = [0; 0; 0; 0; 0]T . These

values were used in all cases.The matrix Q varies depending on what kind of desired trajectory we want to follow. Dif-

ferent Qs, for constant trajectory, linear trajectory, quadratic trajectory, and sinusoidal trajectoryrespectively are presented below

Qcons =

500 0 0 0 0

0 200 0 0 0

0 0 200 0 0

0 0 0 20000 0

0 0 0 0 20000

(31)

QLinear =

1000 0 0 0 0

0 200 0 0 0

0 0 200 0 0

0 0 0 20000 0

0 0 0 0 20000

(32)

Qquad =

1000 0 0 0 0

0 1000 0 0 0

0 0 1000 0 0

0 0 0 20000 0

0 0 0 0 20000

(33)

Qsin =

2000 0 0 0 0

0 200 0 0 0

0 0 0 0 0

0 0 0 20000 0

0 0 0 0 20000

(34)

The desired trajectories of the states were chosen based on the relation that exists among thestates. For easier calculations purpose we chose:

for constant desired trajectory ωrd(t) = 4 rad/s, ωgd(t) = 25 rad/s, THd(t) = 13 Nm,idd(t) = 3 A, and iqd(t) = 10 A;for linear desired trajectory ωrd(t) = t rad/s, ωgd(t) = 6t rad/s, THd(t) = 14 Nm,idd(t) = 3 A, and iqd(t) = 10 A;for quadratic desired trajectory ωrd(t) = 0.85t2 rad/s, ωgd(t) = 5t2 rad/s,THd(t) = 18 Nm, idd(t) = 3 A, and iqd(t) = 10 A;for sinusoidal desired trajectory ωrd(t) = 0.8sint rad/s, ωgd(t) = 5sint rad/s,THd(t) = 12.5 Nm, idd(t) = 3 A, and iqd(t) = 10 A.Where the subscript d stands for desired.The solutions of the State Space system of differential equations for different desired trajec-

tories are shown from Fig.3 to Fig.22.

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

ωr

0

1

2

3

4

5

6

7

8

9

10Desired Trajectory and Tracking Problem ω

r

desired ωr

controlled ωr

Figure 3: Rotor Speed Constant

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

ωg

-5

0

5

10

15

20


g

desired ωg

controlled ωg

Figure 4: Generator Speed Constant

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

TH

0

5

10

15

20

25

30

35

40Desired Trajectory and Tracking Problem T

H

desired TH

controlled TH

Figure 5: Internal Torque Constant

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

i d

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5Desired Trajectory and Tracking Problem i

d

desired id

controlled id

Figure 6: d current Constant, id

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

i q

0

1

2

3

4

5

6

7

8

9

10Desired Trajectory and Tracking Problem i

q

desired iq

controlled iq

Figure 7: q current Constant, iq

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ωr

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5


r

desired ωr

controlled ωr

Figure 8: Rotor Speed Linear

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ωg

-5

0

5

10

15

20

25


g

desired ωg

controlled ωg

Figure 9: Generator Speed Linear

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

TH

0

5

10


H

desired TH

controlled TH

Figure 10: Internal Torque Linear

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

i d

0

0.5

1

1.5

2

2.5

3

3.5

4


d

desired id

controlled id

Figure 11: d current Linear, id

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

i q

0

1

2

3

4

5

6

7

8

9


q

desired iq

controlled iq

Figure 12: q current Linear, iq

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ωr

0

5

10

15

20


r

desired ωr

controlled ωr

Figure 13: Rotor Speed Quadratic

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ωg

-20

0

20

40

60

80

100

120


g

desired ωg

controlled ωg

Figure 14: Generator Speed Quadratic

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

TH

0

5

10

15

20


H

desired TH

controlled TH

Figure 15: Internal Torque Quadratic

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

i d

0

0.5

1

1.5

2

2.5

3

3.5

4


d

desired id

controlled id

Figure 16: d current Quadratic, id

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

i q

0

2

4

6

8

10


q

desired iq

controlled iq

Figure 17: q current Quadratic, iq

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ωr

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8


r

desired ωr

controlled ωr

Figure 18: Rotor Speed Sinusoidal

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ωg

-5

-4

-3

-2

-1

0

1

2

3

4


g

desired ωg

controlled ωg

Figure 19: Generator Speed Sinusoidal

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

TH

0

2

4

6

8

10

12

14

16


H

desired TH

controlled TH

Figure 20: Internal Torque Sinusoidal

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

i d

0

0.5

1

1.5

2

2.5

3

3.5

4


d

desired id

controlled id

Figure 21: d current Sinusoidal, id

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

i q

0

1

2

3

4

5

6

7

8

9


q

desired iq

controlled iq

Figure 22: q current Sinusoidal, iq

The Control Law for different desired trajctories are shown from Fig.23 to Fig.30.

Time0 0.5 1 1.5

ud

0

100

200

300

400

500

600

700

800Control Law, u

d

Figure 23: Control Law for Constant Trajectory, ud

Time0 0.5 1 1.5

uq

0

500

1000

1500

2000

2500

3000Control Law, u

q

Figure 24: Control Law for Constant Trajectory, uq

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ud

0

100

200

300

400

500

600

700

800Control Law, u

d

Figure 25: Control Law for Linear Trajectory, ud

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

uq

0

500

1000

1500

2000

2500

3000Control Law, u

q

Figure 26: Control Law for Linear Trajectory, uq

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ud

0

100

200

300

400

500

600

700

800

900

1000Control Law, u

d

Figure 27: Control Law for Quadratic Trajectory, ud

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

uq

0

500

1000

1500

2000

2500

3000Control Law, u

q

Figure 28: Control Law for Quadratic Trajectory, uq

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ud

0

100

200

300

400

500

600

700

800Control Law, u

d

Figure 29: Control Law for Sinusoidal Trajectory, ud

Time0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

uq

0

500

1000

1500

2000

2500

3000Control Law, u

q

Figure 30: Control Law for Sinusoidal Trajectory, uq

The solutions of Riccati system of differential equations for the case when the desired trajectoryis a constant are shown from Fig.31 to Fig.38.

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

P

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4Riccati equation solutions

p11p12

Figure 31: Riccati Differential Equation Solution

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

P

-0.25

-0.2

-0.15

-0.1

-0.05

0


p13p14


Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

P

-10

0

10

20

30

40

50Riccati equation solutions

p15p22


Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

P

-0.2

-0.1

0

0.1

0.2

0.3

0.4


p23p24


Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

P

-0.1

0

0.1

0.2

0.3

0.4

0.5


p25p33


Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

P

×10-4

-5

-4

-3

-2

-1

0

1

2

3

4


p24p35


Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

P

-1

0

1

2

3

4

5


p45p55


Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

p44

0

1

2

3

4

5

6Riccati equation solution p

44


The solutions of the extra system of differential equations that involve η for the case whenthe desired trajectory is a constant are shown in Fig.39 and Fig.40.

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

η

-70

-60

-50

-40

-30

-20

-10

0

10Extra differential equation η

1,η

3,η

4,η

5

η1

η3

η4

η5

Figure 39: η Differential Equation Solution

Time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

η

-1200

-1000

-800

-600

-400

-200

0Extra differential equation η

2

Figure 40: η Differential Equation Solution

Because of space reasons the solutions of Riccati system of differential equations and thesolutions of the extra system of differential equations for the cases where the desired trajectoryis a function of time are not shown in this report but they can be provided upon request.

5. Conclusions

After apply the Tracking Problem Theory to this Wind Power System we can appreciate thefollowing:

a) It is very important to take the time, use the correct criterion, and try many times withdifferent options before choose the appropriate desired states trajectories and weight matrices.

Changes in any desired trajectory will affect the trajectory of the other states since the statesare related. It is important to balance the remain trajectories and also be careful tuning the weightmatrices when we want to follow some trajectory for a specific state.

b)The Tracking Problem analysis can be expanded to Time Variant Systems and also can beapplied when the desired trajectory is a function of time.

In our case of study we noticed that when we chose a desired trajectory that is a function oftime the perturbation on the other states is more critical when the function is not linear and inour specific case the more affected state is the internal torque, TH .

c)The time of response of the states is different depending on the desired trajectory chosen.In our case the constant trajectory has the fastest response. That is the reason why we plot theother cases for an interval of 5 seconds rather than 2 seconds as we did it in the case of constanttrajectory.

d)As an extra conclusion, I would like to mention that in order to solve the tracking problem wehad to solve three different systems of differential equations, the Riccati system which actuallyhave 25 equations but because it is symmetric we only have to consider 15 equations, the systemthat involve η with 5 equations, and the state space system with 5 equations. All the systemswere solved independently but at the same time they were related, the form in which the codewas written is a kind of cascade where one system is solved and its solutions are inputs of thenext system. Because of this way to write the code I used the same step size for all the systemsbut I noticed that if I used step values greater than 0.0001 then the numerical method blows up.A possible reason that could cause the instability of the numerical method when we use biggerstep values could be the nonlinearity and the order 15 of the Riccati Equations. The problemwith the optimal control analysis of this report is that we need to solve first the Riccati Equationsand then use its results as input for the controlled space state system as I mention before.

6. Matlab Code

Matlab codes were too long to be included in this report. This could be provided upon request.

References

[1] H.Battista F. Bianchi and R. Mantz, “Wind turbine control system: Principles, modelling, and gain scheduledesign,” 2007.

[2] N. Cutuluslis I. Munteanu, A. Bratcu and E. Ceanga, “Optimal control of wind energy systems: Toward aglobal aproach,” 2008.

[3] Yang Zhang, D. Naidu, and Chenxiao Cai, “Time scale analysis and sysnthesis for model predictive controlunder stochastic enviroments,” Control Theory and Applications, IEE Proceedings -, nov.

[4] Naidu Nguyen, “Time scale analysis and control of wind energy conversion system,” Energy Conversion, IEEETransactions on, 2012.

[5] Saroj Biswas, “Lecture notes of optimal control,” .

[6] Doald Kirk, “Optimal control theory: an introduction,” 2004.

[7] Frank Lewis, “Optimal control,” 2012.

[8] H. Li and Z. Chen, “Overview of different wind generator systems and their comparisons,” Renewable PowerGeneration, vol. 2, no. 2, pp. 123–138, June 2008.

julio bravo's master graduation project

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