Control 15MW Turbine

Wind turbine control

Control of a 1.5MW wind turbine for load reduction

B.P. Lemmen BSc.CST 2010.047

Eindhoven, June 22, 2010

2

Contents

1 Linearization 71.1 Floquet analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 PID control 112.1 Angular velocity high speed shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Tower acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Individual pitch control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Power control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Simulations and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.1 Wind input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.2 Sensor noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Kalman filter 273.1 Linearizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Appending states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 LQG controller 334.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Conclusions and recommendations 39

A Coleman transformations 41

3

4

Introduction

In this report the goal is to design different ways to control a 1.5 MW wind turbine and comparethe performance of these different controllers. All of the controllers will be used to control thewind turbine in above rated wind conditions. In above rated wind conditions the generatortorque is kept constant while the blades of the rotor are pitched to assure a constant angu-lar velocity of the rotor. This way the power output of the generator should be constant. Inthese conditions increasing the power output further could potentially damage the wind turbine.

A non-linear model of a wind turbine is used for simulations. This wind turbine model is a3 blade, 1.5 MW wind turbine with a hub height of 85 meters and blades with a length of 35meters. In above rated wind conditions the desired angular velocity of the rotor is 20.48 rpmfor this turbine. Maintaining this angular velocity is the first requirement for the controllers.The second requirement is based on load reduction. Load reduction should, if possible, be fo-cussed on reducing the loadings on the tower and blades. To do this the blades can be pitchedindividually and small changes can be made to the generator torque. For the load reductionthe tower accelerations in both for-aft and side-side directions are measured as well as the bladeedgewise and flapwise bending moments at the root of the blade. Here the edgewise direction isthe bending in the rotational plane of the rotor and the flapwise direction is the bending out ofthis rotational plane.

The controllers that will be designed first are several PID controllers and secondly a Kalmanfilter and LQG controller. For all of these controllers it is necessary to linearize the non-linearmodel equations. This will be discussed in the first chapter where also a structural analysis isperformed. In the second part the design and performance of the PID controllers is discussed.Controllers will be designed to control the angular velocity of the rotor, to reduce the toweracceleration and to reduce the flapwise bending moment on the blades. Also a multi inputsingle output controller will be designed to control the power by varying both the torque andthe pitch of the blades. Next the design of a Kalman filter is discussed and the performance ofthis filter is assessed. The state estimates from the Kalman filter are than used together with aLQG controller in the last part of this report. Finally the results from the PID controllers andthe LQG controller are compared.

5

6

Chapter 1

Linearization

Before a controller can be designed a linearized model of the plant is needed. In this chapterthe linearization of the plant model will be discussed. For the simulations the NWTC (NationalWind Technology Center) Design Code FAST [1] is used. FAST has the ability to linearizethe nonlinear model. First a simulation is started and FAST will simulate until a steady statesolution, over one rotation, of the wind turbine is reached. After the steady state has beenreached the model will be linearized at a number of different azimuth angles. FAST thendetermines system matrices, either for a first order model, see equation 1.1, or a second ordermodel, see equation 1.2.

x = Ax+Bu+Bdud

y = Cx+Du+Ddud (1.1)

Mq + Cq +Kq = Fu+ Fdud (1.2)

The linearization will be performed for four different wind speeds, 12, 16, 20 and 25 meters persecond. Each solution will be determined at 36 different azimuth angles. This results in foursets of 36 linearizations. For these linearizations the following parameters are used:

Inputs:

• Torque

• Individual pitch of the blades

Degrees of freedom:

• First flapwise blade mode

• Second flapwise blade mode

• First edgewise blade mode

• Generator (angle)

• First fore-aft tower bending-mode

• Second fore-aft tower bending-mode

• First side-to-side tower bending-mode

• Second side-to-side tower bending-mode

7

Outputs:

• Tower for-aft and side-side acceleration

• Low speed shaft angle and speed

• High speed shaft speed

• Yaw angle

• Blade root edgewise bending moments for individual blades

• Blade root flapwise bending moments for individual blades

• Individual pitch angles

In the model that results from these computations some of the variables are in a non-rotatingcoordinate frame while others are defined in a rotating coordinate frame. To bring all of thevariables in a non-rotating frame the Coleman transformation is used. These transformationsare discussed in appendix A. After this transformation the found system matrices can be usedfor both structural analysis and determination of transfer functions.

1.1 Floquet analysis

A structural analysis can be performed using the results from the linearizations and the sub-sequent Coleman transformations. This analysis consists of four different steps, as describedin [7]:

1. Compute the transition matrix for one rotation of the rotor. The solution after one periodis given by:

x(T ) = Φ(T, 0)x(0)

Where Φ is the transition matrix. This matrix can be determined by integration fromt = 0 until t = T with Φ(0, 0) = I, the identity matrix.

2. Determination of the characteristic multipliers σi by eigenvalue analysis of Φ(T, 0). Theeigenvalues of the matrix Φ(T, 0) are the characteristic multipliers of the system.

3. Using the following two equations the modal damping coefficient and the modal frequencycan be computed.

ζi =1Tln|σi|

ωi =1T

(tan−1 Im(σi)

Re(σi)+ 2πk

)If all the ζi < 0 the system is stable.

4. With the eigenvectors of the eigenanalysis from step 2 the modeshapes of the system canbe computed.

In the figures 1.1 until 1.4 the characteristic exponents λi = ζi+ jωi, with j =√−1, are plotted.

For 20 and 25 meters per second wind speeds there is one characteristic exponent which is largerthan zero. This suggests that for these wind speeds the structure is unstable.

8

−12 −10 −8 −6 −4 −2 0−1.5

−1

−0.5

0

0.5

1

1.5Characteristic exponents

Real [rad/s]

Ima

gin

ary

[ra

d/s

]

Figure 1.1: Characteristic exponents Floquet analysis for 12 mps winds.

−12 −10 −8 −6 −4 −2 0−1.5

−1

−0.5

0

0.5

1


Real [rad/s]

Ima

gin

ary

[ra

d/s

]


9

−10 −8 −6 −4 −2 0 2−1.5

−1

−0.5

0

0.5

1


Real [rad/s]

Ima

gin

ary

[ra

d/s

]


−12 −10 −8 −6 −4 −2 0 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Characteristic exponents

Real [rad/s]

Ima

gin

ary

[ra

d/s

]


10

Chapter 2

PID control

System matrices obtained from the Coleman transformations can be used to determine transferfunctions for different input output combinations. These can be used in the design of a controller.In this chapter PID control will be used for two main different approaches. First only the angularvelocity of the generator will be used as a control variable. Next both the angular velocity andthe torque of the generator will be controlled, this could potentially reduce the loads on thewind turbine. As a final step individual pitch control will be introduced for load reduction onthe blades.

2.1 Angular velocity high speed shaft

In this section the angular velocity of the generator side of the turbine is used as a controlledvariable. With the transformed linearization obtained in the previous chapter the transfer be-tween the pitch angle as an input and the rotational speed can be determined. The 36 differentmodels for one wind speed are first averaged. Using equation 2.1 the transfer function can bedetermined. So now there are four transfer functions at different wind speeds. These transferfunctions now describe the system over one rotation.

G(s) = C(sI −A)−1B +D; With I the identity matrix (2.1)

By extracting different columns and rows from B and C respectively different transfer functionscan be computed. The derived model can now be verified by means of the step responses. Infigure 2.1 the simulation results for different wind speeds are shown. In these simulations thepitch angle is stepped up by one degree after 100 seconds. Several system properties can bederived. First is it clear that for each wind speed the transfer is stable. Also when the pitchangle is stepped up the rotational speed decreases. In the right figure the step responses areshown zoomed, it can be seen that there are several right half plane zeros for these systems.The models that where derived using the linearizations show for some wind speeds unstablebehavior, i.e. open loop right half plane poles. These systems are corrected to make sure thatthe step response of the model is close to the actual step response from the simulations. Thestep response of these corrected models are compared to the simulation results in figure 2.2. Allof the system properties mentioned above have to be considered in the design of a controller.The bode diagrams of the models are shown in figure 2.3

11

100 150 2001600

1620

1640

1660

1680

1700

1720

1740

1760

1780

1800

Time [s]

Hig

h sp

eed

shaf

t ang

ular

vel

ocity

99 100 101 102 1031750

1755

1760

1765

1770

1775

1780

1785

1790

1795

1800

Time [s]H

igh

spee

d sh

aft a

ngul

ar v

eloc

ity

12 mps16 mps20 mps25 mps

Figure 2.1: Step responses from simulations. Right figure is zoomed in on left figure.

100 150 2001600

1650

1700

1750

1800

Time [s]

Hss

ang

ular

vel

ocity

[rpm

]

FASTModel

100 150 2001600

1650

1700

1750

1800

Time [s]

Hss

ang

ular

vel

ocity

[rpm

]

FASTModel

100 150 2001600

1650

1700

1750

1800

Time [s]

Hss

ang

ular

vel

ocity

[rpm

]

FASTModel

100 150 2001600

1650

1700

1750

1800

Time [s]

Hss

ang

ular

vel

ocity

[rpm

]

FASTModel

Figure 2.2: Step responses from simulations and models. Top left: 12 mps. Top right: 16 mps.Lower left: 20 mps. Lower right: 25 mps.

12

0

20

40

60

80

Mag

nitu

de (

dB)

10−5

100

−900

−720

−540

−360

−180

0

Pha

se (

deg)

Bode Diagram

Frequency (Hz)

12mps16mps20mps25mps

Figure 2.3: Bode diagram transfer from pitch to HSS angular velocity.

To control this plant first two integrators are added. Next a lead filter was used to create thenecessary phase lead at the crossover frequencies. In figure 2.3 it can be seen that for lowfrequencies the slope of the bode diagram is zero while the phase is −180. This is because ofthe inverse reaction of the angular velocity which was noted in the step responses. To correct forthis the gain of the controller is multiplied by -1. In figure 2.4 the resulting open loop transferfunctions for all four different systems are shown. It is clear that for wind speeds near 12 metersper second the phase and magnitude margins, 16.7 and 10.8 dB respectively, are the smallestbut quickly increase for higher wind speeds. Figure 2.5 shows the sensitivity functions. Herethe sensitivity function for 12 meters per second winds has the highest peak at 11.2 dB, this isexpected because of the low robustness margins. Since all of the systems are open loop stablethe simplified Nyquist stability criterium can be used to assess the stability of the controlledsystems. This is why the Nyquist plots in figure 2.6 only show the positive frequencies. TheNyquist diagram shows that all of the systems should be stable. This controller will determine acollective pitch angle for all three blades. To assess the performance of this controller simulationswill be performed. The results of the simulation will be discussed in section 2.5.

13

−300

−200

−100

0

100

200

300

Mag

nitu

de (

dB)

10−5

100

−900

−720

−540

−360

−180

0P

hase

(de

g)

Bode Diagram

Frequency (Hz)


Figure 2.4: Open loop transfer functions for controlled systems.

10−2

10−1

−40

−30

−20

−10

0

10

20

Mag

nitu

de (

dB)

Bode Diagram

Frequency (Hz)


Figure 2.5: Sensitivity functions for controlled systems.

14

−6 −5 −4 −3 −2 −1 0 1−6

−5

−4

−3

−2

−1

0

1

Nyquist Diagram

Real Axis

Imag

inar

y A

xis


Figure 2.6: Nyquist diagrams for controlled systems.

2.2 Tower acceleration

Now that a controller has been designed to control the angular velocity additional controllers canbe added in order to reduce the loadings on the blades and tower of the wind turbine. The firstadditional controller for this purpose will be a controller to reduce the tower for-aft acceleration.For the design of this controller the same strategy is used as for the angular velocity controller.Starting again with the open loop bode diagram from figure 2.7.

−100

−50

0

50

Mag

nitu

de (

dB)

10−6

10−4

10−2

100

102

−900

−720

−540

−360

−180

Pha

se (

deg)

Bode Diagram

Frequency (Hz)


Figure 2.7: Open loop transfer functions.

15

To control this system the a weak and a normal integrator is added. Next a lead filter arounda frequency of 1 · 10−4 Hz. Last a second order low-pass filter with the poles at 0.032 Hz. Thisresults in the bode diagram in figure 2.8. With this controller the minimal stability margins,for a wind speed of 20 mps, are a gain margin of 9.15 dB, a phase margin of 15.3 degrees anda modulus margin of 0.25. For different wind speeds these margins increase. In figures 2.9and 2.10 the sensitivity plots and Nyquist diagrams for the controlled systems are shown.

−400

−200

0

200

Mag

nitu

de (

dB)

10−8

10−6

10−4

10−2

100

102

−540

−450

−360

−270

−180

−90

0

Pha

se (

deg)

Bode Diagram

Frequency (Hz)



10−6

10−4

10−2

−40

−30

−20

−10

0

10

Mag

nitu

de (

dB)

Bode Diagram

Frequency (Hz)



16

−5 −4 −3 −2 −1 0 1−6

−5

−4

−3

−2

−1

0

1

Nyquist Diagram

Real Axis

Imag

inar

y A

xis



2.3 Individual pitch control

For the purpose of load reduction individual pitch control can be used, as described in the fourpapers [3]-[6]. In [5] the use of a PI-based controller is discussed to reduce the flapwise loadingson the blades. For this controller the measured flapwise bending moments, at the root of theblades, are transformed using the inverse of the transformation matrix as defined in equation 2.2.This signal is than used for the controller. For this loop a simple PI controller is sufficient. Thetuning of this PI controller is done using time simulations and tuning the P-action and I-actionto reduce bending moments as much as possible. After the controller the resulting two controlsignals have to be transformed again using the transformation matrix defined in equation 2.3.The total control scheme is depicted in figure 2.11. Results for the control without individualpitch control (IPC) and with are also discussed in the simulation section 2.5.

T =23

[sin(φ1) sin(φ2) sin(φ3)cos(φ1) cos(φ2) cos(φ3)

](2.2)

T =

sin(φ1) cos(φ1)sin(φ2) cos(φ2)sin(φ3) cos(φ3)

(2.3)

Blade root bending moments

Inverse Coleman transformations

PIPI

Coleman trans-formations

Individual pitch angles

Figure 2.11: Control setup for individual pitch control.

17

2.4 Power control

Until now only the pitch was used to control the rotational speed. But if also the torque is usedas a control input this could help to further reduce the loads on the structure. When both pitchand torque are used as control inputs the power is now chosen as the controlled output. Thepower is simply the product between rotational speed and torque.Again to determine transfer function linearizations have to be performed. This will result in aMISO (Multi Input Single Output) system. The bode diagrams are shown in figure 2.12.

−30

−25

−20

−15

−10

−5

Mag

nitu

de (

dB)

10−5

100

−135

−90

−45

0

45

Pha

se (

deg)

Bode Diagram

Frequency (Hz)

20

40

60

80

100

Mag

nitu

de (

dB)

10−5

100

−900

−720

−540

−360

−180

0

Pha

se (

deg)

Bode Diagram

Frequency (Hz)


Figure 2.12: Bode diagram for MISO system, torque to power (left), pitch to power (right).

An extra consideration for the tuning of this controller is that preferably the torque has to bekept constant. To ensure this requirement the gain in the torque controller will be kept low.The torque controller consists of a double integrator and a lead filter. In figures 2.13 until 2.15the bode diagrams, sensitivity plots and Nyquist diagrams for the controlled system are shown.

−200

−100

0

100

200

Mag

nitu

de (

dB)

10−6

10−4

10−2

100

102

−315

−270

−225

−180

−135

−90

Pha

se (

deg)

Bode Diagram

Frequency (Hz)



18

10−4

10−3

10−2

10−1

−50

−40

−30

−20

−10

0

Mag

nitu

de (

dB)

Bode Diagram

Frequency (Hz)



−6 −5 −4 −3 −2 −1 0 1−5

−4

−3

−2

−1

0

1

Nyquist Diagram

Real Axis

Imag

inar

y A

xis



It can be seen that with this controller the stability and robustness margins for this system areabout the same for all the wind speeds. The controller for control of the pitch also contains adouble integrator and lead filter. But as can be seen in the open loop bode diagram in figure 2.12the phase is −180 degrees at low frequencies where the slope is 0. To correct this a negative gainis used in the controller. The resulting bode diagram, sensitivity plots and Nyquist diagramsfor the controlled system are shown in figure 2.16 until 2.18.

19

−200

0

200

400

Mag

nitu

de (

dB)

10−5

100

−900

−720

−540

−360

−180

0P

hase

(de

g)

Bode Diagram

Frequency (Hz)



Bode Diagram

Frequency (Hz)10

−410

−310

−2

−60

−50

−40

−30

−20

−10

0

10

Mag

nitu

de (

dB)



20

Nyquist Diagram

Real Axis

Imag

inar

y A

xis

−6 −5 −4 −3 −2 −1 0 1−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5



In this case the stability and robustness margins are smallest for wind speeds of 12 mps. Themagnitude margin in this case is 42 dB, the phase margin is 34.3 degrees and the maximum ofthe sensitivity function is 5.44. So all of these margins are within the standard guidelines.

Now that all of these different controller are designed they can be used in a simulation. Af-ter the simulations the results can be compared.

2.5 Simulations and results

In this section simulations will be performed using the various PID controllers from the previoussections. For these and later simulations it is important that simulation conditions are identicalfor a objective comparison between different controllers and control strategies. So before actuallystating the simulations and discussing the results first the wind input will be discussed. Secondlythe sensor noise which is used in all of the simulations is presented.

2.5.1 Wind input

For the wind used in the simulations a two dimensional profile in the wind is desired. This meansthat wind speeds will only vary according to the height and not in the horizontal direction. Nowa wind input file for Aerodyn will be computed. In Aerodyn there are various parameters thatdetermine the wind profile. A full description can be found in the Aerodyn manual [8]. Hereonly the parameters that are used will be discussed.

The profile of the wind in these simulations will be a combination between linear vertical shearand vertical power law shear. These components are described by the functions 2.4 and 2.5.

Vlin = α

(h−H

2R

)(2.4)

Vpower =(h

H

)βVmean (2.5)

21

Where α is the linear vertical shear parameter, β the vertical power law parameter, H the rotorhub height, R the rotor radius (blade length), Vmean the mean hub height wind speed and h theheight. This height only defined over the rotorplane. So from the hub height minus the bladelength till the hub height plus the blade length. By choosing β = 0.14 and varying the α thisresults in the wind profiles shown in the left side of figure 2.19. It can be seen that all of theseprofiles cross at the hub height. To get the required profiles a correction is added to the meanwind speed. The equation for this correction velocity is in equation 2.6. Now the profiles arelike the ones depicted on the right of figure 2.19.

10 12 14 16 18 2040

50

60

70

80

90

100

110

120

Wind speed [m/s]

Hei

ght [

m]

10 15 20 2540

50

60

70

80

90

100

110

120

Wind speed [m/s]

Hei

ght [

m]

Figure 2.19: Wind profiles, (left) uncorrected, (right) corrected.

Vcorr = 10α (2.6)

Now that the wind profile is as desired it needs to be put into a format suitable for simulation.This would mean adding noise, since the wind basically is a disturbance on the system. Butsince wind is not white noise the noise needs to be colored. This is done by filtering white noisewith a second order filter. This filter is based on a wind turbulence model [9]. The transferfunction for this filter is in equation 2.7.

Hf (s) = KvsTva1 + 1

(sTv + 1)(sTva2 + 1)(2.7)

Kv =

√2Tv(1− a2

2)(a2

1

a22

− a2 + 1− a21

)−1

(2.8)

Where a1 = 0.4 and a2 = 0.25. The filtered noise is now used to determine the linear verticalshear parameter at each simulation step. After this the mean wind speed is adjusted using thecorrection velocity is using equation 2.6. This data can now be written into a simple data fileand be used in a simulation.

2.5.2 Sensor noise

For an accurate model description sensor noise is added to the measured outputs. The amountof noise for each of these measurements is based on [10]. The noise added is Gaussian whitenoise with a zero mean and a standard deviation as defined in table 2.1.

22

Measured variable Standard deviationGenerator speed 0.0158 rad/sGenerator torque 45 NmPitch rotor angle 0.2

Rotor speed 0.025 rad/sTower accelerations 0.5 m/s2

Table 2.1: Table with standard deviations for sensor noise.

2.5.3 Simulations

Now the simulations can be performed. Each of the simulations will be 600 seconds long. Intotal four different control setups will be compared. First only the PID controller for the angularvelocity of the rotor. Second this PID controller but now together with the controller for thetower acceleration and the individual pitch controllers. Third is the MISO controller. Andfinally the MISO with again the controller for the tower acceleration and the individual pitchcontrollers. Comparison will be based on two main criteria. In the above rated wind conditionsconsidered the rotational speed of the rotor should be kept constant. The generator used in thesesimulations should have a rotational speed of 20.48 rpm. Next to this the controller should tryto reduce loads on both the tower and the blades. For this the root mean square values of thetower accelerations in the two directions and the average and standard deviation of the bendingmoments in the two directions are compared. Also the frequency content of the tower for-aft,edgewise bending moment and flapwise bending moment are compared.In figure 2.20 the angular velocity of the rotor over time is shown for the four different controlsetups. From this figure it can be concluded that all of the controllers are able to keep theangular velocity around the desired value of 20.48 rpm. So if any of the controllers are able toreduce loadings this should not have a significant effect on the generated power.

0 100 200 300 400 500 60019

19.5

20

20.5

21

21.5

22

22.5

23

23.5

24

Time [s]

Rot

or a

ngul

ar v

eloc

ity [r

pm]

HSS PIDHSS PID+IPC+TAMISOMISO+IPC+TA

Figure 2.20: Angular velocity over time for different control setups.

As stated above also the frequency content of various signals can be used as basis for the com-parison. In figure 2.21 the frequency content of three signals is shown. First the tower for-aftacceleration, second the edgewise bending moment and third the flapwise bending moment.What can be noted in these figures is that, instead of using normal frequencies. periodic compo-nents are used. These periodic components are based on an N-times per revolution component

23

HSS PID HSS PID + IPC + TA MISO MISO + PID + IPCRMS Acc. for-aft [m/s2] 1e−3 1e−3 1e−3 1e−3

RMS Acc. side-side [m/s2] 2e−4 3e−4 3e−4 4e−4

RMS Pitch rate [/s] 1e−5 0.03 8e−6 0.03Avg. edgewise bending moment 43 39 43 40Avg. flapwise bending moment 657 664 623 630Std edgewise bending moment 45 43 45 44Std flapwise bending moment 185 121 183 120

Table 2.2: Results simulations PID control setups.

in the signal. For instance the tower shadow will have an effect on each of the three blades ina single revolution. This would then show up as a 3p component in the tower acceleration butas a 1p component on the blade bending moments. For the results the 1p components of thebending moments are considered and the 3p component of the tower acceleration.

0.9 0.95 1 1.05 1.10

10

20PSD edgewise bending moment

Periodic component

|Y(f

)|

0.9 0.95 1 1.05 1.10

50

PSD flapwise bending moment

Periodic component

|Y(f

)|

2.9 2.95 3 3.05 3.10

0.05PSD For−Aft Tower Acceleration

Periodic component

|Y(f

)|

HSS PIDHSS PID+IPC+TAMISOMISO+IPC+TA

Figure 2.21: Frequency content for different signals.

Clearly this result shows that the addition of the tower acceleration and IPC controllers has thedesired effect. In all three signals a reduction in the content of the periodic component can beseen.Finally the RMS values, averages and standard deviation of some signals are in table 2.2. Inthis table also the RMS values of the pitch rate are added. These show that the more extensivecontrol schemes do increase the pitch rate. For the rest of the values it can be concluded that thecontrol setups with the IPC and tower acceleration controller are able to decrease the averagesand standard deviations on the bending moments in both directions of the blades. But thestandard deviation of the side-side tower acceleration is slightly increased as a consequence.From these results it shows that the designed controllers perform as expected. They decreasemost of the loadings on the tower and blades. But the blade pitch rate is increased as aconsequence. In the next part of the report the focus will be on the second control setup.

24

This will involve a combination of a Kalman filter, for state estimation, together with a LQGcontroller. After the design of these elements simulation results will be compared to the resultsfrom this part.

25

26

Chapter 3

Kalman filter

In this chapter the design and performance of a Kalman filter for the wind turbine model willbe discussed. The Kalman filter will be used to estimate the states of the model to be able touse them with a LQG controller.

3.1 Linearizations

The model used for the Kalman filter will be based on linearizations from FAST. These lin-earizations will be computed for different wind speeds and different azimuth positions. In thesimulations a linear combination of the systems matrices resulting from these linearizations willbe used to get a model for the Kalman filter. Since this is a non-linear system it will need tobe verified that there are enough linearizations to accurately describe the system dynamics. Ifthis is not the case the Kalman filter will not be able to estimate the states properly.

In total there will be linearizations for 11 different wind speeds ranging from 12 until 25 mps.For each wind speed there will be 24 linearizations at equally spaced azimuth positions. Using24 here will ensure that each blade is linearized at exactly the same position and therefor shouldlead to repeating models for each one third azimuth rotation. After the computations the resultscan be examined to assess the use of linear combinations for the determination of the systemmatrices. In figure 3.1 the system matrix A1 is shown in a mesh plot, this is for a azimuthangle of 0. Figure 3.2 shows also the A2 matrix but now for a azimuth angle of 15. Now thelinear combination of 0.5(A1 + A2) is computed, the result is shown in figure 3.3 Comparingthese three figure is can be concluded that the mesh plots for the three different matrices arevery similar. This suggest that using a linear combination as an estimate of the system matrixis allowed. Similar results can be shown for different azimuth positions, wind speeds as well asthe other matrices B, C and D.

27

0

10

20

30

0

10

20

30−6000

−4000

−2000

0

2000

4000

6000

Figure 3.1: Mesh plot for A matrix at 0 azimuth position.

0

10

20

30

0

10

20

30−6000

−4000

−2000

0

2000

4000

6000

Figure 3.2: Mesh plot for A matrix at 15 azimuth position.

28

0

10

20

30

0

10

20

30−6000

−4000

−2000

0

2000

4000

6000

Figure 3.3: Mesh plot for linear combination for A matrix at 7.5 azimuth position.

3.2 Appending states

For a linear plant in equation 3.1 a Kalman filter is able to estimate the states of the system bythe inputs and measurements. In this equation w and v are the process and measurement noiserespectively. A requirement for these signals is that they are white noise with a zero mean. Asdiscussed in section 2.5.2 the sensor noise is white noise. But for the signal w the mean hubheight wind speed and the linear vertical shear parameter will have to be used. But as discussedin section 2.5.1 these parameters are colored noise. And a second order filter was used to colorwhite noise. So the Kalman filter will have to be adjusted to correct for this. This is simplydone by addition of states to the system. These state can be computed from the second orderfilter that was used to color the white noise. By converting the transfer function to a secondorder state space system in equation 3.2.

x = Ax+Bu+Gw

y = Cx+Du+ v (3.1)

xf = Afxf +Bfm

yf = Cfxf +Dfm (3.2)

Now by simply appending the computed states xf to the state vector x the xA from equation 3.3is used in the Kalman filter. The system dynamics can now be written into the form of equa-tion 3.4, where m is now white noise for the two signals for mean wind speed and linear verticalshear.

xA = [x xf ]T (3.3)

xA =[A G[10Cf Cf ]T

0 Af

]xA +

[B0

]u+

[0 0Bf Bf

]m

y = Cx+Du+ v (3.4)

29

The factor 10Cf in the AA matrix comes from the correction velocity used in the computationof the wind input. This correction velocity was ten times the linear vertical shear parameter.With these appended states the Kalman filter can now be computed. These computations willresult in a gain L which is used to produce state estimate ˙x using equation 3.5 The gain L isdetermined by solving an algebraic Riccati equation and will ensure stable estimations, i.e. theestimated state should converge to the real states over time.

˙x = Ax+Bu+ L(y − Cx−Du) (3.5)

3.3 Performance

Now that a Kalman filter is designed the quality of the estimations can be assessed in a simu-lation. In this simulation the measured outputs are compared with the estimated outputs. Asmentioned in the previous section linear combination as a function of both mean wind speed andazimuth position is used to determine the system matrices which are then used for the estima-tions. Initial simulations suggested that the Kalman filter did not function. In these simulationsthe mean wind speed is a very noisy signal. This also means that the dynamics in the Kalmanfilter change rapidly. Because the mean wind speed is used in the computation of a linear combi-nation to determine the model at each time instance. When adding a weighted moving averageto the mean wind speed signal the changes are less rapidly. When using a weighting over just0.12 second the results improve significantly. The results are shown in figures 3.4 and 3.5.From these results it can be concluded that the Kalman filter is not able to correctly estimateall of the outputs. For some of the outputs like the bending moments and generator torquethe estimation is sufficiently close to the measured value. But for the low and high speed shaftangular velocity a steady state offset is visible. This could be because the linearization algorithmof FAST only searches for a steady state between certain bounds. When the differences in thestates in one rotation are within these bounds the algorithm assumes steady state has beenreach. Furthermore as discussed in the PID control design the linearized model errors can evencause a stable system to seem unstable. As for the tower for-aft acceleration the estimationsdiffer between good and bad. This could be because of the amount of linearization point whichare used. The total estimation is build up from two distinct parts. First there is the nominaltrajectory which is computed in the linearization. And second there is the estimator part. If thesteady state in the simulation is about the same as in the linearization the estimation part shouldbe relatively small. This would suggest that the biggest part of the estimation comes from thenominal trajectory. But what can be noted in the figure for the tower for-aft acceleration is thatin one rotation (about 3 seconds), there are a total of nine oscillations. These oscillations haveto be described for the biggest part by the nominal trajectory. But in one rotation the nominaltrajectory only consists of 24 points. So 24 point are available to describe nine oscillations. Thiscould also lead to incorrect estimations. Unfortunately because of the amount of linearizationsit is not possible to correct each of these models like in the PID case. The amount of pointsof the nominal trajectory could be solved by just performing new linearizations with a moredetailed description of this nominal trajectory.Still the Kalman filter works sufficiently and can be used in union with a LQG controller. Inthe next chapter the design of this controller will be discussed as well as the simulation of thesystem with the Kalman filter and LQG controller.

30

300 301 302 303 304 305 306

−1

−0.5

0

0.5

1

Time [s]

Tow

er fo

r−af

t acc

eler

atio

n [m

/s2 ]

Measured outputEstimated output

300 301 302 303 304 305 306

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time [s]

Tow

er s

ide−

side

acc

eler

atio

n [m

/s2 ]


300 301 302 303 304 305 30620.1

20.2

20.3

20.4

20.5

20.6

20.7

20.8

20.9

21

Time [s]

Low

spe

ed s

haft

angu

lar

velo

city

[rpm

]


300 301 302 303 304 305 3061770

1780

1790

1800

1810

1820

1830

1840

Time [s]

Hig

h sp

eed

shaf

t ang

ular

vel

ocity

[rpm

]


Figure 3.4: Measured and estimated outputs.

31

300 301 302 303 304 305 306−50

0

50

100

150

200

250

300

350

400

Time [s]

Azi

mut

h an

gle

[ °]


300 301 302 303 304 305 3066.5

7

7.5

8

8.5

9

9.5

10

Time [s]

Gen

erat

or to

rque


300 301 302 303 304 305 306−50

0

50

100

150

200

Time [s]

Edg

ewis

e be

ndin

g m

omen

t [kN

m]


300 301 302 303 304 305 306

200

400

600

800

1000

1200

Time [s]

Fla

pwis

e be

ndin

g m

omen

t [kN

m]


Figure 3.5: Measured and estimated outputs.

32

Chapter 4

LQG controller

In this chapter the design and performance of a linear quadratic gain controller is discussed.This controller will use the state estimations from the Kalman filter to control each of the bladesindividually. For this controller the choice is made to use an output weighting instead of anstate weighting. With this the quadratic cost function that needs to be minimized changes toequation 4.1. The choice to use an output weighting is because of the fact that all rotating statesand outputs are transformed using the Coleman transformations. After these transformationsthe states change and it is much more difficult to find a proper weighing matrix Q. But for theoutputs a simple simulation where the outputs are transformed is enough to give a good view ofhow the weighting have to be adjusted. The same adjustment has to be made to the weightingon the pitch inputs. Since these inputs are also in a rotating frame and are transformed. Nextto tuning the weights on in- and outputs the reference that the LQG-controller should try tofollow can be tuned for better performance. After this tuning the controller can be used in asimulation. In the next section results are discussed and compared with the results from thePID controllers.

J =∫ ∞

0(yTQy + uTRu)dt (4.1)

4.1 Results

As with the results from the PID the first priority is the angular velocity of the rotor. Infigure 4.1 the angular velocity is shown over time. It shows that the LQG controller is faster toget to desired the steady state value of 20.48 rpm.

Now the frequency content can be compared. This is done in figure 4.2. What can be notedis that here no improvement is shown for the LQG-controller compared to the simplest PIDcontroller for just the angular velocity of the rotor. Also when looking at the data in table 4.1the same conclusion can be made. So based on this data it seem like the LQG-controller is notable to improve on any the load reduction compared to PID controllers. When looking at thetower accelerations in both for-aft and side-side direction there is an improvement to be seen.In figures 4.3 and 4.4 it can be seen that the for-aft acceleration is reduced by 0.5 m/s2 and theside-side acceleration by about 0.1 m/s2. This while not increasing the bending moments whichare plotted in figures 4.5 and 4.6.

33

0 100 200 300 400 500 60019.5

20

20.5

21

21.5

22

22.5

23

Time [s]

Ang

ular

vel

ocity

LS

S [r

pm]

LQGMISO + IPC + TA

Figure 4.1: Angular velocity over time for different control setups.

0.9 0.95 1 1.05 1.10

20

40PSD edgewise bending moment

Periodic component

|Y(f

)|

0.9 0.95 1 1.05 1.10

50

100PSD flapwise bending moment

Periodic component

|Y(f

)|

2.9 2.95 3 3.05 3.10

0.05PSD For−Aft Tower Acceleration

Periodic component

|Y(f

)|

LQGHSS PIDMISO+IPC+TA

Figure 4.2: Periodic frequency content for different control setups.

34

0 100 200 300 400 500 600−1.5

−1

−0.5

0

0.5

1

1.5

MISO + IPC + TALQG

Figure 4.3: Tower for-aft acceleration for PID controller and LQG controller.

0 100 200 300 400 500 600

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

MISO + IPC + TALQG

Figure 4.4: Tower side-side acceleration for PID controller and LQG controller.

35

HSS PID LQGRMS Acc. for-aft [m/s2] 1e−3 1e−3

RMS Acc. side-side [m/s2] 2e−4 2e−4

RMS Pitch rate [/s] 1e−5 0.057Avg. edgewise bending moment 43 43Avg. flapwise bending moment 657 656Std edgewise bending moment 45 47Std flapwise bending moment 185 193

Table 4.1: Results simulation LQG control setup.

200 220 240 260 280 300 320 340 360−150

−100

−50

0

50

100

150

200

MISO + IPC + TALQG

Figure 4.5: Edgewise bending moments for PID controller and LQG controller.

200 220 240 260 280 300 320 340 360−200

0

200

400

600

800

1000

1200

1400

Figure 4.6: Flapwise bending moments for PID controller and LQG controller.

36

As was concluded in the section about the Kalman filter design there is a lot of improvementpossible in the state estimations. Because the performance of a LQG-controller is heavily depen-dent on the quality of the state estimates the best way to improve is to get a better estimator.Since the FAST model is now a ’black box’ no non-linear model information is available. Whena non-linear model would be available an extended Kalman filter could be used for better esti-mations.

37

38

Chapter 5

Conclusions and recommendations

In this report a comparison was made between PID controllers and a LQG controller for thecontrol of a 1.5 MW wind turbine. The control was focused on keeping a steady angular velocitywhile reducing loads on tower and blades. From the results several conclusions can be made. Forthe PID controllers it was shown that a single PID controller could control the non-linear modelfor all considered wind speeds. Gain scheduling could potentially improve the performance ofthese controllers further and is dependent on the specifications for the controlled system. Whencomparing results for the different PID control setups it was shown that the most elaborate con-troller was able to achieve the biggest load reduction on the system. This controller used bothindividual pitches and the generator torque to control the power while reducing tower for-aftacceleration and the flapwise bending moments on the blades.

Next a Kalman filter was designed. The model in the filter was based on linearizations ofthe system for different winds speeds and azimuth angles. It was shown that is was allowed totake linear combinations of the linearizations for the determination of the system dynamics inthe simulations. By using a weighted moving average on the wind speed signal the Kalman filterwas able to estimate the outputs of the system. The performance of the filter is dependent ofthe linearizations which, as was already shown in the PID controller part, can contain errors. Abig improvement for the performance of the Kalman filter can be potentially made by using anon-linear model in the Kalman filter and using an extended Kalman filter.

Finally a LQG controller was designed to use the state estimates from the Kalman filter tocontrol the system. Comparing results with the PID controllers it could be seen that the per-formance of the LQG controller was, for the biggest part, comparable with the high speed shaftangular velocity PID controller. The LQG controller was able to reduce tower acceleration bya considerable amount. Because the performance of this controller is dependent on the esti-mations from the Kalman filter improving the Kalman filter could also the performance of theLQG controller. When comparing the tuning of the LQG controller with the PID controllersit is much easier to adjust the LQG controller and the weighting filters make it easier to seethe connection between different signals. With this the potential of the use of a LQG controllershows.

39

40

Appendix A

Coleman transformations

Before the linearization can be used for structural analysis and controller design a coordinatetransformation has to be performed. In this section the Coleman transformations are explained.The Coleman transformation can be used to transform the variables in a rotating frame intovariables in the fixed, non-rotating frame. This transformation process is described in [2]. Belowthe process will be described briefly for a three blade wind turbine.

A transfer matrix t is defined in equation A.1. This matrix describes the relation between threeDOF’s in a rotating and a non-rotating frame. To transform three DOF’s (qrot1 ,qrot2 ,qrot3 ) in arotating frame to a non-rotating frame (qnr1 ,qnr2 ,qnr3 ) the transformation in equation A.2 can beused. It can be noticed that the transformation is dependent on the azimuth angles (ψ1,ψ2,ψ3)of the individual blades.

t =

1 cos(ψ1) sin(ψ1)1 cos(ψ2) sin(ψ2)1 cos(ψ3) sin(ψ3)

(A.1)

qnr1

qnr2

qnr1

= t−1

qrot1

qrot2

qrot3

(A.2)

The transformation of the systems matrices, determined by the linearization, is dependent onthe number of non-rotating and rotating variables. For the setup chosen the following division,given in the table below, can be made between non-rotating and rotating variables.

Non-rotating RotatingGenerator (angle) First flapwise blade modeFirst fore-aft tower bending-mode Second flapwise blade modeSecond fore-aft tower bending-mode First edgewise blade modeFirst side-side tower bending-mode Blade root flapwise bending momentsSecond side-side tower bending-mode Blade root edgewise bending momentsIndividual pitch of the bladesTorqueIndividual pitch anglesTower for-aft and side-side accelerationLow speed shaft angle and speedHigh speed shaft speedYaw angle

Transformation for the system matrices, both for the first and second order model, are nowgiven in equation A.3.

41

MNR = MT1

CNR = 2ΩMR2 + CT1

KNR = Ω2MT3 + ΩMT2 + ΩCT2 +KT1

FNR = FTlc

Fd NR = Fd

Cv NR = T−1lo CvT1

Cd NR = T−1lo (ΩCvT2 + CdT1)

DNR = T−1lo DTlc (A.3)

Dd NR = T−1lo Dd

ANR =[T−1

1 00 T−1

1

](A

[T1 0

ΩT2 T1

]−[

ΩT2 0Ω2T3 + ΩT2 2ΩT2

])BNR =

[T−1

1 00 T−1

1

]BTlc

Bd NR = Bd

CNR = T−1lo C

[T1 0

ΩT2 T1

]DNR = T−1

lo DTlc

Dd NR = T−1lo Dd

Where Ω and Ω are the angular velocity and acceleration of the rotor shaft. The transformationsmatrices: T1, T2, T3, Tlo and Tlc are defined in equation A.4.

T1 =

InF×nF

ttt

T2 =

0nF×nF

t2t2

t2

T3 =

0nF×nF

t3t3

t3

(A.4)

Tlo =

IFo×Fo

ttt

Tlc =

IFc×Fc

ttt

Here nF is the number of fixed DOF, Fo is the number of fixed outputs and Fc is the numberof fixed control inputs. The two transformations matrices t2 and t3 are given in equations A.5and A.6.

42

t2 =

0 −sin(ψ1) cos(ψ1)0 −sin(ψ2) cos(ψ2)0 −sin(ψ3) cos(ψ3)

(A.5)

t3 =

0 −cos(ψ1) −sin(ψ1)0 −cos(ψ2) −sin(ψ2)0 −cos(ψ3) −sin(ψ3)

(A.6)

43

44

Bibliography

[1] NWTC Design Codes (FAST by Jason Jonkman). http://wind.nrel.gov/designcodes/simulators/fast/.Last modified 12-August-2005; accessed 12-August-2005.

[2] Bir G., Multiblade Coordinate Transformation and Its Application to Wind Turbine Analysis,NREL, January 2008

[3] Bossanyi E.A., The Design of Closed Loop Controllers fo Wind Turbines, Garrad Hassanand Partners Ltd, St Vincent’s Works, Bristol, UK, Wind Energy 2000; 3(143-163)

[4] Bossanyi E.A., Wind Turbine Control for Load Reduction, Garrad Hassan and Partners Ltd,St Vincent’s Works, Bristol, UK, Wind Energy 2003; 6(229-244)

[5] Bossanyi E.A., Individual Blade Pitch Control for Load Reduction, Garrad Hassan and Part-ners Ltd, St Vincent’s Works, Bristol, UK, Wind Energy 2003; 6(119-128)

[6] Bossanyi E.A., Further Load Reductions with Individual Pitch Control, Garrad Hassan andPartners Ltd, St Vincent’s Works, Bristol, UK, Wind Energy 2005; 8(481-485)

[7] Stol K.A., Dynamics Modeling and Periodic Control of Horizontal-axis Wind Turbines, Uni-versity of Colorado, December 2001

[8] Laino D.J., Hansen A.C., USER’S GUIDE to the Wind Turbine Aerodynamics ComputerSoftware AeroDyn,Windward Engineering, LC, Salt Lake City, AeroDyn 12.50, 24 December2002

[9] Fernando D. Bianchi, Ricardo J. Mantz and Hernn De Battista, Wind Turbine ControlSystems, Springer London 2007; 3(29-48)

[10] Thomas Esbensen, Christoffer Sloth, Fault Diagnosis and Fault-Tolerant Control of WindTurbines, Aalborg University, Department of Electronic Systems, Section for Automationand Control, June 2009

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