Automatic Control and Computer Science_2011

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Buletinul Institutului Politehnic din Iaşi

Tome LVII (LXI)

Fasc. 3, 2011

AUTOMATIC CONTROL and COMPUTER SCIENCE Section

CONTENTS

Multi-Agent System for Monitoring and Analysis Prahova Hydrographical

Basin

Alexandra Maria Matei

9 - 19

An Alghoritm Designed to Determine the Optimum Number ofCommunication Channels in a Modular Simulator

Lucian-Florentin Bărbulescu

21 - 32

Unsupervised Color - Based Image Recognition Using a LAB Feature

Extraction Technique

Tudor Barbu, Adrian Ciobanu and Mihaela Costin

33 - 41

Algorithmic Solution for Design and Optimisation of Multi-Phase Pulse

Generators

Aleodor Daniel Ioan

43 - 59

Parallel Tools and Techniques for Biological Cells Modelling

Salvatore Cuomo, Pasquale De Michele and Marta Chinnici 61 - 75

Nonlinear H ∞ Control of Variable Speed Wind Turbines for Power

Regulation and Load Reduction

Jován Oseas Mérida Rubio and Luis Tupak Aguilar Bustos

77 - 94

Achieve Faster Spanning Tree Convergence

Roxana St ănică and Emil Petre 95 - 111

Singular Perturbation Approach to RBNN Adaptive Control of UnknownFlexible Joint Manipulators

Bahram Karimi and Morteza Ghateei

113 - 127

2-Connected Synchronizing Networks

Eduardo Canale, Pablo Monzon and Franco Robledo 129 - 141

Implementing a Public-Key Infrastructure for the Academic Environment

Marius Marian and Andrei Pîrvan 143 - 155



BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞIPublicat de

Universitatea Tehnică „Gheorghe Asachi” din IaşiTomul LVII (LXI), Fasc. 3, 2011

SecţiaAUTOMATICĂ şi CALCULATOARE

NONLINEAR H ∞ CONTROL OF VARIABLE SPEED WIND

TURBINES FOR POWER REGULATION AND LOADREDUCTION

BY

JOVÁN OSEAS MÉRIDA RUBIO∗∗∗∗ and LUIS TUPAK AGUILAR BUSTOS

Instituto Politécnico Nacional, Avenida del parque 1310,Mesa de Otay, Tijuana 22510, México

Received: August 3, 2011Accepted for publication: September 16, 2011

Abstract. A control strategy is realized which solve the problem of output power regulation of variable speed wind energy conversion systems bycombining a linear control for blade pitch angle with a nonlinear H ∞ torquecontrol which mitigate the effects of external disturbances that occur at the inputand output of the system. The controller exhibits better power and speedregulation when compared to classic linear controllers. We assume that theeffective wind speed and acceleration are available from measurements on thewind turbine. In order to validate the mathematical model and evaluate the

performance of proposed controller, we used the National Renewable EnergyLaboratory (NREL) aerolastic wind turbine simulator FAST. Simulation andvalidation results show that the proposed control strategy is effective in terms of

power and speed regulation.Key words: nonlinear control, power and speed regulation, wind turbine

simulator.

2000 Mathematics Subject Classification: 34H05, 93A30.

∗Corresponding author; e-mail : [email protected]



Jován Oseas Mérida Rubio and Luis Tupak Aguilar Bustos78

1. Introduction

As a result of increasing environmental concern, more and moreelectricity is being generated from renewable sources. Wind energy has provedto be an important source of clean and renewable energy in order to produceelectrical energy. Nowadays, wind energy is by far the fastest-growingrenewable energy technology, between 2000 and 2009, wind energy generationworldwide increased by a factor of almost 9 (Gelman et al., 2010). However,the performance of wind turbine must be improved. There are two primarytypes of horizontal-axis wind turbines: fixed speed and variable speed(Ofualagba & Ubeku, 2008). In this work we choose the variable speed becausealthough the fixed speed system is easy to build and operate, does not have theability that the variable speed system has in energy extraction, up to a 20%increase over fixed speed (Ofualagba & Ubeku, 2008; Burton et al., 2001).Moreover, the variable speed system is much more complex to control.Advanced control plays an important role in the performance of large windturbines. This allows better use of resources of the turbine, increasing thelifetime of mechanical and electrical components, earning higher returns. On theother hand, the new technological advancements improve the prospects of wind

power allowing the design of cost-effective of wind turbines. Wind turbinecontrollers objectives depend on the operation area (Pao & Johnson, 2009; Lakset al., 2009). Variable speed wind turbine operation can be divided into three

operating regions:− Region I: Below cut-in wind speed.− Region II: Between cut-in wind speed and rated wind speed.− Region III: Between rated wind speed and cut-out wind speed.In region I wind turbines do not run, because power available in wind is

low compared to losses in turbine system. Region II is an operational modewhere it is desirable that the turbine capture as much power as possible from thewind, this because wind energy extraction rates are low, and the structural loadsare relatively small. Generator torque provides the control input to vary therotor speed while the blade pitch is held constant. Region III is encounteredwhen the wind speeds are high enough that the turbine must limit the fraction ofthe wind power captured such that safe electrical, and mechanical loads are not

exceeded. If the wind speeds exceed the contained ones in the region III, thesystem will make a forced stop the machine, protecting it from aerodynamicloads excessively high. Generally the rated rotor speed and power output aremaintain by the blade pitch control with the generator torque constant at itsrated value. Region III is considered in the present work where the poweravailable in the wind exceeds the limit for which the turbine mechanics has

been designed. Much of the research work in the wind energy conversionsystem control have used classical controllers. This, for several reasons. First,linear control theory is a well-developed topic while nonlinear control theory is






controlled, pt y R ∈)( − the only available measurement on the system. The

following assumptions are assumed to hold:

(A1) The functions f ( x), g 1( x), g 2( x), h1( x), h2( x), k 12( x), and k 21( x) aretwice continuously differentiable in x locally around x = 0.

(A2) (0) = 0 f , 1(0)=0h , and 2 (0)=0h .

(A3) ,=)()(0,=)()( 1212121 I xk xk xk xh T T

.=)()(0,=)()( 2121121 I xk xk x g xk T T

These assumptions are inherited from the standard nonlinear H ∞ control

theory (Doyle et al ., 1989; Isidori & Astolfi, 1992) and they are made fortechnical reasons. Assumption (A1) guarantees the well-posedness of the abovedynamic system, while being enforced by integrable exogenous inputs.Assumption (A2) ensures that the origin is an equilibrium point of the non-driven (u = 0) disturbance-free (w = 0) dynamic system (1). Assumption (A3) isa simplifying assumption inherited from the standard H ∞ -control problem.

A causal dynamic feedback compensator

)(= xu K (2)

is said to be globally (locally) admissible controller if the closed-loop system(1)−(2) is globally asymptotically stable when = 0w .

Given a real number > 0γ , it is said that system (1), (2) has 2L -gainless than γ if the response z , resulting from w for initial state (0)=0 x , satisfies

dt t wdt t z T T 2

0

22

0)(<)( ∫∫ γ (3)

for all T > 0 and all piecewise continuous functions w(t ).The H ∞ -control problem is to find a globally admissible controller (2)

such that 2L -gain of the closed-loop system (1)−(2) is less than γ . In turn, alocally admissible controller (2) is said to be a local solution of the H ∞ -control

problem if there exists a neighborhood U of the equilibrium such that inequality(3) is satisfied for all > 0T and all piecewise continuous functions )(t w forwhich the state trajectory of the closed-loop system starting from the initial

point (0) = 0 x remains in U for all [0, ]t T ∈ .

2.1. Local State-Space Solution

Assumptions (A1)-(A3) allow one to linearize the correspondingHamilton-Jacobi-Isaacs inequalities from that arise in the state feedback and



Bul. Inst. Polit. Iaşi, t. LVII (LXI), f. 3, 2011 81

output-injection design thereby yielding a local solution of the time-invariant H ∞ -control problem. The subsequent local analysis involves the linear time-invariant H ∞ -control problem for the system

w D xC y

u D xC z

u Bw B Ax x

212

121

21

=

=

=

+

+

++ɺ

(4)

where

(0).=)((0),=

(0),=(0),=

(0),=(0),=(0),=

21212

2

12121

1

2211

k t Dh

C

k D x

hC

g B g B x

f A

∂

∂∂

∂∂

∂

(5)

Such a problem is now well-understood if the linear system (4) isstabilizable and detectable from u and y, respectively. Under these assumptions,the following conditions are necessary and sufficient for a solution to exist (see,

e.g ., (Doyle et al., 1989)):C1) There exists a positive semidefinite symmetric solution of theequation

01

2211211 =

−++= P B B B B P C C P A PA T T T T

γ (6)

such that the matrix has all eigenvalues with negative real part.

P B B B B A T T )( 112

22−−− γ

According to the strict bounded real lemma (Anderson &Vreugdenhil,1973), condition C1) ensures that there exists a positive constant ε0 such that thesystem of tehe perturbed Riccati equation

01

22112

11

=

−+

+++

E

T T

T T

P B B B B P

I C C P A A P

γ

ε

ε

ε ε

(7)




has a unique positive definite symmetric solution ( , ) P Z ε ε for each 0(0, )ε ε ∈

where ε γ P B B A AT

112=

~ −+ .Eq. (7) is subsequently utilized to derive a local solution of the

nonlinear H ∞ -control problem for (1). The following result is extracted from.

Theorem 1. Let condition C1) be satisfied and let ),( ε ε Z P be the

corresponding positive solution of (7) under some > 0ε . Then the output

feedback

x P x g u T ε )(= 2− (8)

is a local solution of the H ∞ -control problem.

3. Wind Turbine Model and Problem Statement

The aerodynamic power captured by the rotor is given by the nonlinearexpression (Burton et al., 2001)

32 ),(2

1= vC R P pa β λ ρπ (9)

where: v is the wind speed, ρ − the air density, and R − the rotor radius. The

efficiency of the rotor blades is denoted as pC , which depends on the blade

pitch angle β , or the angle of attack of the rotor blades, and the tip speed ratio

λ , the ratio of the blade tip linear speed to the wind speed. The parameters β

and λ affect the efficiency of the system. The coefficient pC is specific for

each wind turbine. The relationship of tip speed ratio is given by

.=v

R r λ (10)

The turbine estimated pC λ β − − surface, derived from simulation is

illustrated in Fig. 1. This surface was created with the modeling softwareWTPerf (Marshall, 2009), which uses blade-element-momentum theory to

predict the performance of wind turbines (Burton et al., 2001). The WTPerfsimulation was performed to obtain the operating parameters for the CART(Controls Advanced Research Turbine).




Fig. 1 − Power coefficient curve.

Fig. 1 indicates that there is one specific λ at which the turbine is mostefficient. From (9) and (10), one can note that the rotor efficiency is highlynonlinear and makes the entire system a nonlinear system. The efficiency of

power capture is a function of the tip speed ratio and the blade pitch. The powercaptured from the wind follows the relationship

r aa T P ω = (11)

where:

3 2( , )1=

2 p

a

C T R v

λ β ρπ

λ (12)

is the aerodynamic torque which depends nonlinearly upon the tip speed ratio.A variable speed wind turbine generally consists of an aeroturbine, a gearbox,and generator, as shown in Fig. 2.




Fig. 2 − Two-mass wind turbine model.

The wind turns the blades generating an aerodynamic torque aT , which

spin a shaft at the speed r . The low speed torque lsT acts as a braking torque

on the rotor. The gearbox, which increases the rotor speed by the ratio g n to

obtain the generator speed g ω and decreases the high speed torque hsT . Thegenerator is driven by the high speed torque hsT and braked by the generator

electromagnetic torque emT (Boukhezzar et al., 2007). The mathematical model

of the two mass wind turbine, can be described as follows:

r r lsr lslsr lsr ar r D D K vT J ω ω ω θ θ β ω ω −−−−− )()(),,(=ɺ

g g g lsr lslsr ls g em g g g n D D K nT n J ω ω ω θ θ ω −−+−+− )()(=ɺ (13)

where: lsω is the low shaft speed, r θ − the rotor side angular deviation, lsθ −

the gearbox side angular deviation, r J − the rotor inertia, g J − the generatorinertia, r D − the rotor external damping, g D − the generator external damping,

ls D − the low speed shaft damping, and ls K − the low speed shaft stiffness. The

gearbox ratio g n is:

.==hs

ls

ls

g

g T

T n

ω

ω (14)




If a perfectly rigid low speed shaft is assumed, a single mass model ofthe turbine may be then considered, upon using (14) and (13), one gets:

,),,(= g r t r ar t T DvT J −− ω β ω ω ɺ (15)

where: 2=t r g g J J n J + , 2=t r g g D D n D+ , and = g g emT n T are the turbine total

inertia, turbine total external damping, and generator torque in the rotor side,respectively. The parameters of the model are given in Table 1.

Table 1One-Mass Model Parameters

Notation Numerical

valueUnits

R 21.65 m ρ

1.308 kg/m3 J t

3.92x105 kg m2

Dt

400 N m/rad/s H 36.6 m P e

600 kW

n g

43.165

Those parameters are based on the CART which is a two-bladed,

teetered, active-yaw, upwind, variable speed, variable pitch, horizontal axiswind turbine. The nominal power is 600 kW, the rated wind speed of 13 m/s, acut out wind speed of 26 m/s, and it has a maximum power coefficient max pC =

0.3659. The rated rotor speed is 41.7 rpm. The required constraints for torqueand rotor speed are 162 kN-m and 58 rpm respectively (Fingersh & Johnson,2002). The gearbox is connected to an induction generator via the high speedshaft, and the generator is connected to the grid via power electronics.

Generator power will be controled in region III when the wind speedsare high enough that the turbine must limit the fraction of the wind powercaptured so that safe electrical and mechanical loads are not exceeded. Then itoperates at the rated power with power regulation during high wind periods by

active control of the blade pitch angle or passive regulation based onaerodynamic stall (Burton et al., 2001). The objective control in this region is tofind a control law g T and β to achieve the best tracking of rated of power

while r ω follows d ω , as well as to reject fast wind speed variations and

avoiding significant control efforts that induce undesirable torques and forceson the wind turbine structure. For variable speed wind turbine we design acontroller using blade pitch and generator torque as control inputs.

Our objective is to design a regulator of the form




−−−− uT DT

J

T C T g r t a

t

g

p

r

g )(1

= 0 ω ε ω

ɺ (16)

where: g r e T P ω = is the electrical power, e p P P −nom=ε , and 0C − a positive

constant while the rotor speed regulation is partly guaranteed by the pitchcontroller.

We confine our investigation to the velocity regulation problem where1) The output to be controlled is given by

nom

0 1

0=

0r

y

e

w z u

P P ρ

+ −

(17)

with a positive weight coefficient y ρ , and

2) The measurements

y g

p

r

wT

y +

ε

ω

= (18)

corrupted by the vector 3 yw ∈ R , are only available.

4. Controller Synthesis

4.1. H ∞ Synthesis

To begin with, let us introduce the error state vector

),,(=),,(= nom321 g er T P P x x x x −ω . After that, let us rewrite the state Eq. (15) as

−−−−

+−

+−−

u x x x DT J

xc x

x

u xc x

T J

x J

x J

D x

t a

t

a

t t t

t

331201

3

202

311

)(11

=

=

11=

ɺ

ɺ

ɺ

(19)




Since the right-hand of the Eqs. (19) are twice continuously

differentiable in x locally around the origin 3= 0 x ∈ R , the above H ∞ control problem is nothing else than the earlier theoretical approach to the nonlinear H ∞ control problem for the system (1) specified as follows:

[ ]1 3

0 2

23 1 3 3

0 21

1

( ) = ,

1

a t

a t

t t t

T D x x Jt

f x c x

T x D x x xc x x J J J

− − −

− + +

1 2

1

0 0 0 0

( ) = 0 0 0 , ( ) = 1 ,

0 0 0 1

g x g x

−

1 1 12

2

0 1

( ) = , ( ) = 0 ,

0

h x x k x

x

1

2 2 21

3

1 0 0

( ) = , ( ) = 0 1 0

0 0 1

x

h x x k x

x

.

(20)

The subsequent local analysis involves the linear H ∞ -control problemfor the system (4), (5), the state vector x contains the desviation from the

operating point

1

1 0

031 33

1

1 10

= 0 0 ,

at

t t

op

T D

J x J

A c

C A A

x

∂ −− ∂

−




1 2

1

0 0 0 0

= 0 0 0 , = 1 ,

0 0 0 1

op

B B

x

−

1 12

0 0 0 1

= 1 0 0 , = 0 ,0 1 0 0

C D

2 21

1 0 0 1 0 0

= 0 1 0 , = 0 1 0

0 0 1 0 0 1

C D

where

213op 0 2op 3op

31 2 21 1op 1op

33 op 1op 3op

( )=

1= 2 .

a

t t

a t

t

x c x xT x A

J x x J x

A T D x x J

−∂− − −

∂

− + +

Now by applying Theorem 1 to system (1) thus specified, we derive alocal solution of the H ∞ regulation problem.

4.2. Pitch Controller

In order to regulate the rotor speed and reduce generator torqueoscillation, the torque control is aided by the pitch action. The pitch controlallows to maintain the rotor speed around its nominal value. To achieve this,

proportional action is used




w p K ε β =∆ (21)

where: r w ω ω ε −nom= is the rotor speed tracking error and > 0 p K .

5. Simulation Results with Turbulent Wind

Fig. 3 − Wind speed profile of 20 m/s mean value.

The proposed control approach has been simulated on based on theCART. This turbine was modelled with the mathematical model on Matlab-Simulink as well as in FAST aeroelastic simulator for validation. The windspeed is described as a slowly varying average wind speed superimposed by arapidly varying turbulent wind speed. The model of the wind speed v at themeasured point is

t m vvv += (22)




where: mv is the mean value and t v − the turbulent component. The hub-

reference wind field was generated using Turbsim (Kelley & Jonkman, 2009).The wind data consist of 600 sec in = 20mv m/s with 16% turbulence

intensity, via the Kaimal turbulence model. Fig. 3 shows the profile of windspeed. In order to improve power regulation, rotor speed, and reduce themechanical loads, we design H ∞ -regulation controller (8), (20) with = 12γ

and = 0.001ε .

First, the proposed controller is evaluated using the simplifiedmathematical model and then is evaluated with FAST simulator. In Figs. 4and 5, simulation results are presented. The nominal values for regulation

problem are shown in Figs. 4 a and 4 b; it is observed that the proposedcontrol is able to achieve precise speed and power regulation. In Fig. 4 a therotor speed is well regulated close to its nominal value. Fig. 4 b shows thatthe electrical power P e follows the nominal power. This value is almostequal to the nominal power P nom. In this region pitch control alters the pitchof the blade, thereby changing the airflow around the blades resulting in thereduced torque capture of wind turbine rotor. Because of the pitch control,

the control torque is reduced as shown in Fig. 5 a; if variation of g T are

large can be result in loads over the wind turbine affecting its behavior, butin this case its value goes up to 138.25 kNm, which is under the maximumone 162 kNm. These result in the reduction of the drive train mechanicalstresses and output power fluctuations. In Fig. 5 b we see that the collective

pitch action did not exceed its limit. It was observed that a good tracking ofa power reference and regulation of the rotor speed near of its nominal valuecan be successfully enforced with the controller used.

To verify the results obtained from the mathematical model, the proposed controller strategy performance has been tested for validation

using FAST simulator. In this study, two degree of freedom are simulated:the variable generator and rotor speed. The performance increases incomparison with the mathematical model getting a better regulation speedand power tracking (Figs. 4 a and 4 b). The high speed shaft torsionalmoment with the mathematical model is greater then the one obtained withthe simulator as well as the pitch angle (Figs. 5 a and 5 b). Checking outthe results, one should conclude that the wind turbine control issatisfactory.




Fig. 4 − Closed-loop system responses: a − rotor speed andb − generator power.




Fig. 5 − Closed-loop system responses:a − generator torque and b − pitch angle.




6. Conclusions

Variable speed operation of wind turbine is necessary to increase powergeneration efficiency. As explained above, the wind turbine output power islimited to the rated power by the pitch and torque controller. We havedeveloped the structure for nonlinear H ∞ control system in conjunction with alinear control strategy whose design procedure shown to be acceptable tosolving the tracking of a power problem and regulation of the rotor speed nearof its nominal value. To evaluate the performance the proposed strategy controlapproach has been simulated on a 600 kW two-blade wind turbine. Then, it has

been validated using the wind turbine simulator FAST. Simulation results showthat the proposed method is able to achieve the power and speed regulationwhile the load is limited.

The states of the system were supposed available.

REFERENCES

Anderson B., Vreugdenhil R., Network Analysis and Synthesis. Prentice Hall,Englewood Cliffs NJ, 1973.

Bianchi F.D., de Battista H., Mantz R.J., Wind Turbine Control Systems: Principles,

Modelling and Gain Scheduling Design of Advances in Industrial Control .

Springer-Verlag, London Limited, 1st Edition, 2007.Boukhezzar B., Lupu L., Siguerdidjane H., Hand M., Multivariable Control Strategy for

Variable Speed, Variable Pitch Wind Turbines. Renewable Energy, 32, 8,1273−1287, 2007.

Burton T., Sharpe D., Jenkins N., Bossanyi E., Wind Energy Handbook . Wiley, 1stEdition, 2001.

Doyle J., Glover K., Khargonekar P., Francis B., State-Space Solutions to Standard H 2

and H ∞ Control Problems. IEEE Trans. Autom. Control, 34, 8, 831−847, 1989.Fingersh L., Johnson K., Controls Advanced Research Turbine (CART) Comissioning

and Baseline Data Collection. Technical Report, NREL, 2002.Gelman R., Kubik M., Buchanan S., 2009 Renewable Energy Data Book. [Online].

Available: http://www.eere.energy.gov/maps_data/pdfs/eere_databook.pdf, 2010.Grimble M., Horizontal Axis Wind Turbine Control: Comparison of Classical, LQG

and H ∞ Designs. Dynamics and Control, 6, 2, 143−161, 1996.Isidori A., Astolfi A., Disturbance Attenuation and H ∞ -Control via Measumerent

Feedback in Nonlinear System. IEEE Trans. Autom. Control, 37, 9,1283−1293, 1992.

Jonkman J., Buhl M. Jr., FAST User’s Guide. Technical report, NREL/EL-500-38230, National Renewable Energy Laboratory, Golden, CO, 2005.

Kelley N., Jonkman B., TurbSim User's Guide: Version 1.50., Technical Report, NREL/TP-500-46198, National Renewable Energy Laboratory (NREL), 2009.

Laks J.H., Pao L.Y., Wright A.D., Control of Wind Turbines: Past, Present, and Future. IEEE Proc. Amer. Control Conf., 2096−2103, St. Louis, MO, 2009.




Marshall Buhl., NWTC Design Codes WTPerf. [Online]. Available:http://wind.nrel.gov/designcodes/simulators/wtperf/, 2009.

Ofualagba G., Ubeku E.U., Wind Energy Conversion System- Wind Turbine Modeling.

IEEE Power and Energy Society General Meeting - Conversion and Deliveryof Electrical Energy in the 21st Century, 1−8, 2008.

Pao L.Y., Johnson K.E., A Tutorial on the Dynamics and Control of Wind Turbines andWind Farms. IEEE Proc. Amer. Control Conf., 2076−2089, St. Louis, MO,2009.

Thomsen S., Nonlinear Control of a Wind Turbine. ME Thesis, Lyngby: Informatik ogMatematisk Modellering, Danmarks Tekniske Universitet, 2006.

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NREL/TP-500-42437, NREL, 2008.

CONTROLUL H ∞ NELINIAR AL TURBINELOR EOLIENECU VITEZĂ VARIABILĂ

(Rezumat)

Este prezentată o strategie de control care rezolvă problema reglării puterii laieşirea sistemului eolian de conversie a energiei la viteză a vântului variabilăcombinând un sistem de control liniar al unghiului elicei cu un un sistem de control H ∞

neliniar al cuplului, ce diminuează efectul perturbaţiilor externe ce apar la intrarea şi laieşirea sistemului. Sistemul de control propus prezintă o mai bună reglare a puterii laieşire şi a vitezei de rotaţie în comparaţie cu regulatoarele liniare clasice. Se presupunecă valori reale ale vitezei şi acceleraţiei vântului sunt disponibile din măsurătoriefectuate pe o turbină eoliană. Pentru a valida modelul matematic şi pentru a evalua

performanţele sistemului de control propus a fost folosit simulatorul unei turbineeoliene FAST de la National Renewable Energy Laboratory (NREL). Rezultateleobţinute prin simulare arată eficacitatea strategiei de control propusă pentru reglareavitezei şi a puterii la ieşirea turbinei eoliene.

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