Control Design Methodologies for Extenuation of Vibration on Wind Turbine Systems

7/27/2019 Control Design Methodologies for Extenuation of Vibration on Wind Turbine Systems

1/16

Control Design Methodologies for Extenuation ofVibration on Wind Turbine Systems

Ade-Bello, Abdul-JeliliDepartment of Electrical and Computer Engineering

University of New-MexicoAlbuquerque,New-Mexico

United State.

Abstract

Developing and harnessing alternative sources of energy is an important challenge with criticalconcerns for both present and the future of our fast growing society. Therefore, to harvest and

maximize the under-developed power present in wind turbine will be of great importance towards thereductions of carbon dioxide emission generated from other sources. Examining and overcoming any

deficiencies that arise from the design of a wind turbine will help these alternative energy sources to

compete economically with the traditional energy suppliers. This paper addresses the control

methodologies for vibration mitigation in a wind turbine that disrupt the structure rectitude andsystem performance. The turbine system is modelled as a flexible structure operating in the presence of

turbulent wind disturbances and the control law is based on a linearization of the system around a

selected set of operating points. The goal is to identify different processes of multiple sensor input thatcan be used to construct state-space models which are compatible with MIMO modern control

techniques such as LQR, LQG ,

and robustness (LQG/LTR) design to reduce the torque variations by

using speed control with collective blade pitch adjustments. Simulation results show effects and

performances of the optimal control structures evaluated over optimization criterion variables.

1. Introduction

For the past recent years there has been an intense research activity in alternative electricity

generation due to depletion in fossil fuel and generation of bio-hazards that are harmful to the

environment. The world's energy consumption from the 18th century till now has increased in analarming rate also the large part of this energy has been generated from sources that have negative

impacts on the environment. Therefore, a more sustainable and eco-friendly energy production

methods are emphasised among researchers and environment protections communities. For thisreason renewable energies particularly, wind power, have now become an essential part of the energy

programs for most countries and futuristic companies around the world. Collaboration among wind

power and other renewable power production methods has begin to play a more significant role in the

future of electricity supply [1]&[12].Over the years, a closer look and study of the potential of windresources has proven to be rewarding and sustainable with great expectation and results. One of the

most comprehensive studies on this topic [2] found the potential of wind power on land and near-shore

to be 72TW, which alone could have provided over five times the worlds current energy use in all

forms averaged over a year. In [3] the World Wind Energy Association (WWEA) estimates the windpower investment worldwide to expand from approximately 273GW installed capacity at the end of

2012 to 1900 GW installed capacity by 2020.Economical advantage of installing larger wind turbines with typical size of utility-scale has

grown dramatically over the past few years. In addition to the increasing turbine-sizes, cost reductiondemands imply use of lighter and hence more flexible structures. If the energy-price from wind turbines


2/16

in the future is to be competitive with other power production methods, an optimal balance must be

made between maximum power capture on one side, and load-reduction capability on the other side. To

be able to obtain this, a well defined advanced control-design is needed to improve energy capture andreduce dynamic loads. The combination of the difficult and expensive maintenance of Wind Turbines

further increases the need for a reliable control system for fatigue and load reduction [4].

Structurally, wind turbines often consist of a tower on which a nacelle is placed with a generator

connected to it. The generator is driven by a main shaft at which the turbine's blades are fixed with aconnecting gearbox enabling the possibility of different rotational speeds. The most important

problems of a wind turbine is vibration which causes inefficient generation of the electricity, caused by

variable wind forces which are need to be solved in order to get more efficient performance. By using a

good technique, rotating of each blade around its axel which has been designed in different shapes onits sides will change the pushing force from wind to rotate the blades around the origin, and then the

speed of rotation will be changed. The structures are subject to wind and wave loadings. These

disturbances cause vibrations that produce fatigue in the slender structure and decrease the

performance of the mechanical system in the nacelle that generates the electricity [5].The operationregion of a wind turbine is often divided into four regions as shown in Fig.1

Fig.1 Operating region of a typical wind turbine

In region I, the wind speed is lower than the cut-in wind speed and no power can be produced. In

region II, the pitch is usually kept constant while the generator torque is the controlling variable. Inregion III, the main concern is to keep the rated power and to limit loads on critical parts of the

structure by pitching the blades. In region IV, the wind speed is too high, and the turbine is shut down.

In this paper we will focus on the above rated wind speed scenario (region III).The optimality of thewhole system is defined in relation with the trade-off between the wind energy conversion

maximization and the minimization of the asynchronous generator torque variation that is responsible

for the frequency fluctuations (vibration).This is achieved by using a combined optimization criterion,

resulting in a LQ tracking problem with an infinite horizon and a measurable exogenous variable (windspeed)

2. Wind Turbine Dynamic Modelling

There are different methods available for modelling purposes. Large multi-body dynamic codes,

as reported in [6] divide the structure into numerous rigid body masses and connect these parts with

springs and dampers. This approach leads to dynamic models with hundreds or thousands of degrees of

freedom (DOFs). Hence, the order of these models must be greatly reduced to make them practical forcontrol design [1]. This section presents a simplified control-oriented model. In this approach, a state

space representation of the dynamic system is derived from a quite simple mechanical description of

the wind turbine. This state space model is totally non-linear due to the aerodynamics involved, and will

thus be linearized around a specific operation point.


3/16

Fig. 2 WT subsystems with corresponding models

The Wind Turbine is a complicated mechanical system with many interconnecting DOF. Thenonlinearities of a wind turbine system, for instance due to the aerodynamics, may bring along

challenges when it comes to the control design. Since the control input gains of a pitch control usually isthe partial derivative of the rotor aerodynamic torque with respect to blade pitch angle variations, these

input gains will depend on the operating condition, described by a specific wind and rotor speed. A

method which bypasses the challenges of directly involving the nonlinear equations is by using a linear

time invariant system (LTI) on state space form. Such a system relates the control input vector u andoutput of the plant y using first-order vector ordinary differential equation of the form

= + + , = + (1)where x is the system states and matrices A, B, G, C and D are the state, input, disturbance, output and

feed through matrix, respectively, and is the disturbance input vector. The wind turbine drive-trainmodelled with its high and low speed shaft separated by a gearbox is shown in Figure 3. As its seen, thedrive train is modelled as a simple spring-damper configuration with the constants and denotingthe spring stiffness and damping in the rotor shaft, and similarly; and as representing the springstiffness and damping in the generator shaft. Figure 2 also shows the inertia, torque, rotor speed and

displacement of the rotor and generator shafts. The parameters named as1 , 1,1, 1, 1 are thetorque, speed, displacement, number of teeth, and inertia of gear 1, and similarly for gear 2.

Fig.3 Model of the drive train with the high and low speed shafts.


4/16

The model in Figure 3 results in the following equation of motion for the rotor torque

= + ( 1) + 1+ 1 + 11 (2)where the factor ( 1) + 1is the reaction torque in the low speed shaft.Equivalently, the equation for generator motion is as follows

= + 2+ 2+ 2 + 2 2 (3)where the factor 2+ 2 is the reaction torque at the high speed shaft. Therelationship between1and 2 is derived based on the equation describing a constrained motionbetween two gears in the following way

= 1 = 2 (4)From Eqn. (3) we find

2 = + 2 2+ 22 (5)then, the following equation for the rotor rotation holds

= + ( 1) + 1 + ( + 2 2+ 2 2 + 11 (6)

Since the goal of the modelling is to use it for control design, this equation will be simplified in the

following. First, it can be assumed that the high speed shaft is stiff. This will imply that =2 , = 2 and so on. Secondly, the gearbox can be assumed lossless, hence the terms involving 1 and 2can be omitted. This reduces eqn. (6) to the following equation

=

+

(

1) +

1+

(

) (7)

The model is now of three degrees of freedom ; rotor speed, generator speed, and a DOF describing the

torsional spring stiffness of the drive train. These DOFs correspond with the three states shown Figure

4.

Fig. 4 Illustration of the 3-state model used in the control design having Kd and Cd as drive train torsional stiffness

and damping constants, respectively.


5/16

The states in Figure 3 will in the following be regarded as perturbations from any steady-state

equilibrium point (operating point), around which the linearization is done. Hence the states are

assigned with the notation and describes the following DOFs

1 = (perturbed rotor speed)

2=

( perturbed drive train torsional spring stiffness)

3 = (perturbed generator speed)where ( = ) and ( = ).Now, from the Newtons second law, the following relation holds

= (8)where the left hand side expresses the difference between the aerodynamic torque on the rotor causedby the wind force, and the reaction torque in the shaft. This reaction torque can be expressed according

to equation (7) as

=

+

= + (1 3) + ( ) (9)This equation can be expressed in terms of deviations from the steady state operation as:

= + + (10)Hence the BEM theory provides a way to calculate the power coefficient CP based on the combination of

a momentum balance and a empirical study of how the lift and drag coefficients depend on the

collective pitch angle, , and tip speed ratio, . In this way an expression of the aerodynamic torque can

be found to be;

(, ,) = 12 (,) 2 (11)where is the air density, is the rotor radius, and is the wind speed. Let us now assume anoperating point at(0,0 ,0)such that Eqn. (11) can be written as

= (0,0 ,0) + (12)where Tr is deviations in the torque from the equilibrium point (= ,0) and consists of partialderivatives of the torque with respect to the different variables, i.e. Taylor series expansion [7]and [8]inthe following way

=

+

+

(13)

where = 0, = 0and = 0 By assigning , and to denote the partialderivatives of the torque at the chosen operating point (0,0 ,0) equation(12) becomes = (0,0 ,0) + () + () + () (14)If the above expression is put into eqn.(8), it follows that

= (0,0 ,0) + ,0 (15)At the operation point is (0,0 ,0) = ,0 since this is a steady state situation. This reduces eqn.(15) to

= () + () + () 2 (1 3) (16)


6/16

when substituting with the corresponding state equations. The following expressions for the

derivatives of the state variables can now be set up

1 = ( )1 2 + 3 + () + () (17)

2=

=

(

1 3) (18)

3 = + (19)Note that in the derivative of the generator speed state3 is used such that = 3 = = when assuming a constant generator torque. The dynamic system can now be represented in astate space system on the form as described by equation (1) yielding

12

3 =

() 1 0 1

123

+ 00

+ 00

(20)where the disturbance input vector w from the general form in Eqns.(1) now is given as , which isthe perturbed wind disturbance (i.e deviations from the operating point, = 0) and the controlinput vector u from eqns. (1) now given as (i.e perturbed (collective) pitch angle, = 0).Theparameters will be assigned when coming to Section 3. It is worth no notice that the measured outputsignal here is the generator speed. Optimally speaking to have a power production which ensures that

speed, torque and power are within acceptable limits for the different wind speed regions it is

necessary to control the wind turbine[13]. This requirement can be further crystallized to the following

properties which the control system should possess; (1) Good closed loop performance in terms ofstability, disturbance rejection, and reference tracking at an acceptable level of control effort, (2) Low

dynamic order (because of hardware constraints), (3) Good robustness.

The linear quadratic regulator (LQR)

The objective of this controller was to alleviate blade loads due to wind shear, gravity, and tower

deflection using individual blade pitch control. The results will include the reduction of blade and tower

cyclic responses [9].The practical application of this technique will be limited by the challenges ofobtaining accurate measurements of the states needed in the controller. The LQR problem rests upon

the following three assumptions;

1. All the states are available for feedback, i.e. it can be measured by sensors etc.

2. The system is stabilizable which means that all of its unstable modes are controllable,

3. The system is detectable having all its unstable modes observable.

This regulator provides an optimal control law for a linear system with quadratic performance index

yielding a cost function on the form [11]

= ( 0 ()() + ()()) (21)where and are weighting parameters that penalize the states and the control effort, respectively.These matrices are therefore controller tuning parameters. It is crucial that Q must be chosen in

accordance to the emphasis we want to give the response of certain states, or in other words; how wewill penalize the states. Likewise, the chosen value(s) of R will penalize the control effort . As an


7/16

example, if Q is increased while keeping R at the same value, the settling time will be reduced as the

states approach zero at a faster rate. This means that more importance is being placed on keeping the

states small at the expense of increased control effort. On the other side, if R is very large relative to Q,the control energy is penalized very heavily. Hence, in an optimal control problem the control system

seeks to maximize the return from the system with minimum cost. In a LQR design, because of the

quadratic performance index of the cost function, the system has a mathematical solution that yields an

optimal control law given as

() = () , = 1 (22)where is the control input andis the state feedback gain matrix given as found by solving thefollowing algebraic Riccati

0 = + + 1 (23)the control designer still needs to specify the weighting factors and compare the results with the

specified design goals. This means that controller synthesis will often tend to be an iterative processwhere the designed "optimal" controllers are tested through simulations and will be adjusted according

to the specified design goals.

Disturbance Rejection

This section will explain the various principles of analysing a system with an assumed wind disturbance

utilized together with the above LQR. This vector (G) describes the magnitude of the disturbance, while

the input signal w is the disturbance quantity (which in our case is the wind speed perturbation)from

equation(1). By using a proportional-plus-Integral (PI) control we can drive the noise to zero providedthe disturbance is constant through-out the entire time. In a steady state condition to control the

disturbance we will have to define new states and variables for LQR optimization in order to derive ourobjective. From equation (1)

=

(perturbed wind disturbance, (

), if this perturbed wind

disturbance was an unknown factor but assumed to be constant we can use PI to drive the states to zeroasymptotically. Minimizing the below equation

= (()()0 + ) (24)Subject to

= 0 0 + 0 (25)Therefore the final control law

() = 1() 2 ()0 (26)will converge to cancel the disturbance (which will be verified with simulations in section 4) this turnsout that with proper time-varying Q and R and considering the above cost function the system is stable.

The LGQ Controller

A well known method for handling stochastic noise in a system is linear quadratic Gaussian (LQG)

controller, which is simply the combination of a Kalman -Bucy filter added to a LQR controller. Theseparation principle can be designed and computed independently and still assure a well define

controller for the above mentioned noise. The application to linear time-variant systems enables the

design of linear feedback controllers for non-linear uncertain systems, which is the case for the wind


8/16

turbine system. With the Stochastic Hamilton Jacobi equation, optimality conditions are derived by

finding the expected value of

= lim { + } (27)Since the state variablesand the disturbance variable cannot be measured directly. Observers willbe utilized to estimate these. The mathematical description of the observer will be solved using the

optimal LQG problem by separately solving the optimal-estimation problem and the deterministiccertainty-equivalent control problem. Therefore, the dynamic equation depends on the filter gain and not the control input. The estimated variables can then be fed into the controller to minimize the

effect of the disturbances. The result of this is that the state feedback also includes the feedback of

disturbance observer yielding a "new" feedback control law. The LQR gain and is computed inMATLAB, and the full state numerical values will be given in Section 3.

Robustness Design

This is design in order to balance the trade-off between maximization of the wind energy conservation

and minimization of the asynchronous generator torque variation (vibration). Several methods areavailable for this but we will make use of LQG/LTR design in this paper since the LQR system is a

minimum phase system. This rectify or gives an alternative to the fact that LQG solution do have poorstability margins while LQR solution have a sharp infinite increasing gain margin. This method will

asymptotically recover the robustness properties of optimal LQR state feedback when optimal LQG

state-estimate feedback is used and also LQG/LTR allows the approach of any LTF of the form( )1 giving good recovery to even non-minimum phase plants provided the zeros aresufficiently located outside the loop pass band. All these will be shown in the simulation in section 3.

3. Control Design and Simulations

We will investigate the application of modern control theories such as LQR and LQG on the wind turbine

system and see how the speed can be made with these theories to reduce torque variations due to the

wind disturbances of different natures. In addition, a further idea is to investigate if a better loadreduction can be obtained by using a disturbance rejection technique called proportional-plus-Integral

control and whether factors as gain will affect the result.

Linearization

As was described in section 2 the wind turbine compose of a complex non-linear relationship between

factors such as the aerodynamic properties of the blades, pitch angle, and wind speeds. The linearizedstate space model, which will be used in the control design, is based on what is presented in [4]and The

operation point, around which the linearization is done, is given as

0 = 18m/s, 0= 42rpm, 0 = 12 degreesThe corresponding linearization with reference to (Wright, 2004) around this point yields the followingstate space system.

123 =

0.145 3.108 106 0.024452.691 106 0 2.691 106

0.1229 1.56 105 0.1229 123+

3.45600

+ 100

= [0 0 1] 123 (28)


9/16

where the factors and are calculated by using the known relation to the other parameters.

Recalling from Section 2 that vector is given as /Ir, where describes the relationship betweentorque variations and wind variations,

. This means that if is increased, it will follow that thetorque is more sensitive to wind variations, which is surely a negative effect when the matter of fatigue

is concerned. The torque is given as a non-linear function which shows the dependency on the windspeed, rotor speed and blade pitch, and will be fixed at the chosen operating point. The transfer function

(TF) of the open loop system formed by the linearized state space system equation is

() = 0.42471451+0.2679+503.4+50.61 (29)the above transfer function as all it zeros and poles in the LHP leading to a characteristic equation withall it coefficients been positive. This explains the characteristics of the response in figure below graph.The three poles of the system are located at(0.1005,0.0837 22.4371).

Fig.5 Perturbations in rotor speed without any control

Simulation with LQR

This design will be experimented without any noise in the system(assuming =0) . After checking thethree above assumptions of stabilizability, detectability and available feedback state. The initial

parameters of the weighting matrices Q and R were chosen arbitrarily. A simulation to check whether

the results correspond with the expected performance while iteratively changing the values of Q and Rwas carried out .The calculated controllability matrix and observability is defined by;

= 0.000 107 0.0000 107 0.0000 1070 9.3001 107 2.4915 1070 0.000 107 0.0001 107 (30)

= 0 0 1.00000.1229 0.00000 0.1229419.7631 0.00000 419.7779 (31)

These matrices both have full ranks of 3, thus the system is stabilizable and detectable and the poles canbe placed arbitrarily. The objective is to place the generator speed poles further to the left which will be

0 10 20 30 40 50 60-30

-25

-20

-15

-10

-5

0

Step response of H(s)

Time (seconds)

perturbationinrotorspeed[rpm]


10/16

achieved with a gain corresponding to these pole locations which are been calculated for differentvalues of Q (1 = ,2 = 10). Applying the LQR control method for optimization of the wind turbinesystem[14]. The gains 1 and2 were calculauted respectively for Q1 and Q2.

= 0 0 00 0 00 0 1

1 = [1.5659 0.0000 0.6002]2 = [4.5930 0.0000 1.4654]

Note that initial condition of (0) = [42 0 0] and = [1] was used for simulation (i.e. Rotor speedwas initially at 42rpm with generator speed at zero)

Fig 6. Simulation of perturbed rotor speed, generator speed and torsional spring stifness for Q1

Fig7. Simulation of perturbed rotor speeds, generator speed and torsional spring stifness for Q2

0 0.5 1 1.5 2 2.5 3 3.5 4-10

0

10

20

30

40

50

Time

Perturbationinrotorspeed[rpm]forQ1

0 0.5 1 1.5 2 2.5 3 3.5 4-5

0

5x 10

7

Time

Perturbationintraint

orsionalstiffnessforQ1

0 0.5 1 1.5 2 2.5 3 3.5 4-10

0

10

20

30

40

50

60

Time

Pe

rturbationingeneratorspeed[rpm]forQ1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-10

0

10

20

30

40

50

Time

Per

turba

tion

inro

torspee

d[rpm

]for

Q2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-3

-2

-1

0

1

2

3x 10

7

Time

Perturb

ation

intra

intors

iona

ls

tiffness

for

Q2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-10

0

10

20

30

40

Time

Perturba

tion

ingenera

torspee

d[rpm

]for

Q2


11/16

Fig.8. Perturbations in rotor speed with LQR gain and without control

From the above simulation we could see an improved stability of the plant where the perturbations in

rotor speed moves closer to zero in the case of the LQR control method. We could also design for LQR

robustness and have control over the number of encirclement of the -1 point of the return-difference

matrix. By making use of kalman inequality with a degree of stability of alpha ( =4) and using the rootlocus to monitor the movement of the closed-loop eigenvalues about the origin we could achieve a

robust-stability conditions for the plant perturbations of the rotor and generator speeds. As shown in

the fig.9 and fig.10. The gain for this stability degree is

3 = [7.1492 0.0000 3.7249]From fig 10 the plant system remains a minimum phase system seen through the positions of the

singular values of the return-difference matrix (I + H(s)).

Fig.9 root locus plot of LQR control with degree of stability

0 10 20 30 40 50 60-30

-25

-20

-15

-10

-5

0


Time (seconds)

perturbationinrotorsp

eed[rpm]

w ith LQR gain

H

-1.5 -1 -0.5 0 0.5 1 1.5 2

x 104

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500root locus plot of H(s)

Real Axis (seconds-1)

ImaginaryAx

is(seconds

-1)


12/16

Fig .10 singular values of I + H(s)

The improvement of the control and robustness of the states properties can also been in figure 11

where the perturbation in rotor speed in the case of normal LQR is compared with that of the degree of

stability.

Fig.11 comparison of perturbation of rotor speed in LQR and DOS

Simulation for Disturbance Rejection/Tracking

Making use of the PI control as discussed earlier, the unknown constant noise can now be introduced

and simulation can reveal ways to reject this noise and give a good tracking property to the system. The

matrix for the disturbance is = [1 0 0] from the linearized equation.

10-3

10-2

10-1

100

101

102

103

0

5

10

15

20

25

30

singular values of I + H(s)

Frequency (rad/s)

Singu

lar

Va

lues

(dB)

0 0.5 1 1.5 2 2.5 3-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0


Time (seconds)

perturbationinrotorspeed[rpm

]

w ith LQR gain

w ith degree of s tability


13/16

Fig .12. Using PI control design

4 = [1.8187 0.000 0.5406 1.000]() = [1.8187 0.000 0.5406]1() + 2 ()0 (32)The output of the PI control asymptotically goes to zeros meaning the system is stable and the

performance objective was achieved. The chosen values in Q will result in a relatively large penalty of

the states 1 and3. This means that if1 or 3 is large; the large values in Q will amplify the effect of1 and 3 in the optimization problem. Since the optimization problem is to minimize, the optimalcontrolmust force the states x1 and x3 to be small (which make sense physically since x1 and x3represent the of the perturbations in rotor and generator speed, respectively). On the other hand, the

small R relative to the max values in Q involves very low penalty on the control effort u in the

minimization of V, and the optimal control u can then be large. For this small R, the gain K can then belarge resulting in a faster response the plots in figure shows clearly that the disturbance now is much

better attenuated while comparing with the first two scenarios. The reason why the speed have this

downwards undershoot is due to the low values of the gain in the LQR controller system seen fromFigure 13 the pitch angle increased and end up having a zero value. This is an expected situation since

it is now assuming a constant wind disturbance which must be accounted for by an increase in pitch

angle but stabilized with the design. The disturbance rejection simulations in this section shows that

this control method enables the mitigate of the disturbance due to the wind affecting the rotor state.

Fig.13 comparison of perturbation of pitch angle in LQR and PI design

0 10 20 30 40 50 60 70-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

7 perturbation of wind turbine states

Time(secs)

amplitudeofrotorspeed

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-10

0

10

20

30

40

50

60

70

80Comparison of Wind Turbine reaction

Time

perturba

tion

inp

itc

hang

le[deg

]

LQR

PI design


14/16

LQG simulation

The control objective for the LQR is the same as for the PI design; making the perturbations in the rotorspeed as small as possible in order to attenuate the wind disturbance. The wind disturbance is made by

calculation of the kalman gain used in the LQG controller as discussed in the theory in Section 3.When

applying the same weighting matrices as for the LQR in the PI approach and the weighting functions

representing the process and measurement noise covariances in the kalman function, comparing the

control efforts in both cases (perturbation in pitch angles).The corresponding load attenuation for thethree cases is shown in Figure 14 below

Fig.14 perturbation of rotor speed in LQG

Fig.15 comparison of control effect for all used control methods

0 0.5 1 1.5 2 2.5 3 3.5-4

-3

-2

-1

0

1

2

3

4x 10

7

Time

Perturbationinrotorspeed[rpm]withLQG

0 1 2 3 4-50

0

50

100

LQG

Timeperturbation

in

pitch

angle

[deg]

0 2 4 6-50

0

50

100

PI

Timeperturbation

in

pitch

angle

[deg]

0 1 2 3 4-50

0

50

100

LQR

Timeperturbation

in

pitch

angle

[de

g]


15/16

Fig. 16 bode plot of the LQG + () 1Simulation for Robustness using LQG/LTR

To improve the robustness properties of the LQG, the LTR approach was introduced as a way to ensure

good robustness in spite of uncertainties in the plant model. Designing a compensator that recovers aphase margin of 30 degrees with added gain and a simulation for the nyquist plot to monitor the

corresponding crossover frequencies of the new model system.

A minimum gain margin of 4.0029 with a new phase margin of -70.1403degrees and a corresponding

crossover frequencies of 7.6567 and 20.1182 was achieved by making use of and of values 10 eachto improve the stability of the LQG and nominal optimality the plant achieved by the LQR.

-400

-300

-200

-100

0

100From: y1 To: Out(1)

Magn

itude(dB)

10-2

100

102

104

-360

-180

0

180

Phase(deg)

Bode Diagram

Frequency (rad/s)

-5 0 5 10 15 20 25 30-20

-15

-10

-5

0

5

10

15

20

Nyquist Diagram

Real Ax is

ImaginaryAxis


16/16

4. Conclusion

The purpose of this project has been to reduce the loads on a wind turbine for above rated wind speeds(Region III) by speed control when applying different controlling methods. This was first performed by

pitch regulation with the PI design. This regulation policy shows to have some regulation capacity, but

also resulted in some bias in the control signal. It was found when utilizing the LQG controller thatalthough the wind disturbance was well extenuated, the settling time was relatively longer and a

reduction in the rotor speed occurred due to the extenuation of the wind disturbances. It could havebeen interesting to extend the work also to include torque regulation in below rated wind speeds. This

regulation policy would aim at extenuation of speed and torque variations due to wind disturbances inRegion II. The LQG regulator showed to give well speed attenuation, but since PI and LQG are quite

different approaches were a comparison between them not possible. It would have been of major

interest to extend the work to also considering pitch actuator constraints to see how this would have

affected the results, especially the control input signal.

References

[1]Connor, Iyer, Leithead, and Grimble. Control of a horizontal axis wind turbine using H-infinity control. In FirstIEEE Conference on Control Applications, pages 117122, 1992.

[2]Archer, C. L. & Jacobson, M. Z. (2005). Evaluation of global wind power, Journal of Geophysical Research Vol.

110: p. 17.

[3]WWEA (2012), "Half-Year Report", Worldwide Wind Energy Statistics, World Wind Energy Association

(WWEA)

[4]T. Pan, Z. Ji, and Z. Jiang, Maximum Power Power Point Tracking of Wind Energy Conversion Systems Basedon Sliding Mode Extremum Seeking Control, Proc. IEEE Energy Conf., pp. 15, Nov. 2008.National RenewableEnergy Laboratory.

[5] S. Colwell and B. Basu, Tuned liquid column dampers in offshore wind turbines for structural control,Engineering Structures, vol. 31,2009, pp 358-368.

[6]Elliot, A. S. & Wright, A. D. (2004). Adams/wt: An industry-specific interactive modelling interface for wind

turbine analysis, Wind Energy.

[7]Henriksen, L. C. (2007). Model Predictive Control of a Wind Turbine, Informatics and Mathematical Modelling -

Technical University of Denmark.

[8]Wright, A. D. (2004). Modern control design for flexible wind turbines, Technical report, National Renewable

Energy Laboratory

[9]Liebst, B. (1985). A pitch control system for the KaMeWa Wind Turbine, Journal of Dynamic Systems and

Control Vol. 107(No.1): pp.4652.

[10]Balas, M., Lee, Y. & Kendall, L. (n.d.). Disturbance tracking control theory with application to horizontal axis

wind turbines, Proceeding of the 1998 ASME Wind Energy Symposium, Reno, Nevada, 12-15 January pp. 9599.

[11]Burns, R. S. (2001). Advanced Control Engineering, Butterworth Heinemann.

[12]Lee, J. & Kim, S. (2010). Wind power generations impact on peak time demand and on future power mix,

Green Energy and Technology 3: 108112.

[13]2010 IEEE International Conference on Control Applications Part of 2010 IEEE Multi-Conference on Systems

and Control Yokohama, Japan, September 8-10, 2010.

[14]D. D. Moerder, A. J. Calise, Convergence of a numerical algorithm for calculating optimal output feedback

gains, IEEE Transactions on Automatic Control, Vol. 30, pp. 900903, September 1985.

Documents

Control Design Methodologies for Extenuation of Vibration on Wind Turbine Systems