Linear’Quadra+c’Gaussian’Control’of’Wind’...

Linear Quadra+c Gaussian Control of Wind Turbines

Abdulrahman Kalbat Columbia University in the City of New York

(Teaching Assistant at UAE University “Currently on leave”)

October 3rd, 2013

10/03/2013 /19

Paper Structure

•  Linear Quadra+c Gaussian (LQG) Control –  General problem formulaMon

–  AssumpMons

–  Design procedures

•  Real system simula+on –  Controls Advanced Research Turbine (CART)

–  State space model of CART

•  Results and Discussion –  SimulaMon results

–  LimitaMon of LQG

Abdulrahman Kalbat Columbia University 2

10/03/2013 /19

Linear Quadra+c Gaussian (LQG) Control

•  LQG is rooted in opMmal stochasMc control theory

•  It is simply a combinaMon of:

–  Linear Quadra+c Regulator (LQR): for full state feedback

–  Kalman Filter: for state esMmaMon

•  Linear QuadraMc Gaussian (LQG) –  Linear system model

–  Quadra+c cost funcMon

–  Gaussian noises


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Linear Quadra+c Gaussian (LQG) Control

•  LQG block diagram


u(t)B

G

w(t)

R

A

C

v(t)

++ x(t) x(t)+

y(t)

+

+

BR

A

C

Kk

Kf

+

+˙x(t) x(t) y(t)

�

+

+

y(t)� y(t)

+

Kalman-Bucy Filter

Noisy System

Determinisric Optimal Controller

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LQG General Form

•  The state space equaMons of the open loop plant for a standard LQG problem:

where


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

y(t) = Cx(t) + v(t)

x(t) : state vector

u(t) : control input vector

y(t) : measured output vector

w(t) : process noise

v(t) : measurement noise

A : state matrix

B : control input gain matrix

G : plant noise gain matrix

C : measured state matrix

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LQG General Form

•  The state space equaMons of the Kalman Filter:

where


˙x(t) = (A�KkC)x(t) +Bu(t) +Kky(t)

y(t) = Cx(t)

x(t) : estimated state vector

y(t) : estimated output vector

Kk : optimal state estimation gain vector

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LQG General Form

•  AssumpMons:


E[w(t)w(⌧)T ] =

⇢W, if t = ⌧

0, if t 6= ⌧

E[v(t)v(⌧)T ] =

⇢V, if t = ⌧

0, if t 6= ⌧

E[w(t)v(⌧)T ] =

⇢R12, if t = ⌧

0, if t 6= ⌧

E[x(0)] = x

o

E[w(t)] = 0

E[v(t)] = 0

E[x(0)w(t)T ] = 0

E[x(0)v(t)T ] = 0

White Gaussian noises

Covariance

UncorrelaMon of iniMal states with the noises

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LQG Design Procedure

Step 1: Check opMmal gain existance criteria:

Step 2: OpMmal state esMmaMon gain calculaMon:


(A,B) is Controllable

(A,C) is Observable

Jk = E

�(x� x)T (x� x)

APk + PkAT +GWG

T � PkCTV

�1CPk = 0

Kk = PkCTV

�1

Filter Algebraic RiccaM EquaMon (FARE)

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Step 3

•  OpMmal State Feedback Gain CalculaMon:

•  Weighing Matrices SelecMon:


Jf =

Z T

0(zTQfz + uTRfu)dt

ATPf + PfA� PfBR�1f BTPf +Qf = 0

Kf = R�1f BTPf

Q = CTC

R = ⇢I

Qii =1

Max (x2ii)

Rii =1

Max (u2ii)

or (Bryson’s Rule)

Control Algebraic RiccaM EquaMon (CARE)

10/03/2013 /19


Step 4: Linear QuadraMc Gaussian Regulator by combining

OpMmal State EsMmaMon and OpMmal State Feedback:

where


x(t)e(t)

�=

A�BKf BKf

0 A�KkC

� x(t)e(t)

�

+

G 0G �Kk

� w(t)v(t)

�

y(t) =⇥C 0

⇤ x(t)e(t)

�+

⇥0 1

⇤ w(t)v(t)

�

e(t) = x(t)� x(t) (state estimation error)

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Controls Advanced Research Turbine (CART)

•  Used by the NaMonal Wind Technology Center (NWTC) and

operated by the NaMonal Renewable Energy Laboratory

(NREL) and located at Boulder, Colorado

•  Used for: –  Exploring potenMal control innovaMons

–  Field test advanced control systems.

•  Very flexible: –  More than 80 sensors

–  CollecMve Blade Pitching + Individual Blade Pitching


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Controls Advanced Research Turbine (CART)


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Numerical Example using MATLAB

•  The model was created by linearizing the moMon equaMons at

a control design point of:

–  18 m/s for wind speed

–  12 degress for rotor collecMve pitch

–  42 RPM for rotor speed.

•  The main objecMve is to operate the machine as a variable

speed wind turbine in region 3

–  by applying constant torque to the generator

–  by maintaining a constant rotor speed

–  through the collecMve rotor blade pitching.


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Numerical Example using MATLAB

•  CART’s state space model:


x(t) =

2

4x1(t)x2(t)x3(t)

3

5where

8<

:

x1(t) : is rotor speed

x2(t) : is drive train torsion

x3(t) : is generator speed

u(t) =

w(t)v(t)

�where

⇢w(t) : is turbine system noise

v(t) : is measurement noise

A =

2

4�1.4454x10�1 �3.1078x10�6 0.02.6910x107 0.0 �2.6910x107

0.0 1.5601x10�5 0.0

3

5

B =⇥�3.4559 0.0 0.0

⇤T

C =⇥0 0 1

⇤

G =⇥7.8938x10�2 0.0 0.0

⇤T

W = E[w(t)w(⌧)T ] = 0.1 (Turbine system noise covariance)

V = E[v(t)v(⌧)T ] = 0.1 (Measurement noise covariance)

Q =

2

41 0 00 1x10�13 00 0 1

3

5 R = 1

Kk =⇥7.6282x10�3 1.2663x102 6.2859x10�2

⇤T

Kf =⇥�2.0336 �2.1225x10�7 6.6055x10�1

⇤

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

y(t) = Cx(t) + v(t)

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Results

•  Wind turbine’s response to different wind speeds


0 10 20 30 40 50 60 70

15

20

Wind Speed (m/sec)

0 10 20 30 40 50 60 7020304050

Generator Speed (RPM)

0 10 20 30 40 50 60 705

10

15Pitch Angle (degrees)

Time sec

Opera+on point

18 m/s

12 degress

42 RPM

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Discussion

•  The simulated control system is Not Robust

–  robust control system works not only for the linear system which

serves as the plant model but it also works for the real physical system

with minor performance degradaMon

•  LQR à (Robust)

•  Kalman Filter à (Robust)

•  LQR + Kalman Filter à (Robustness not guaranteed)

•  To recover the robustness of LQR and Kalman Filter:

–  Loop Transfer Recovery (LTR)


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Results

•  Poles/Zeros plot of the closed loop system


−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−25

−20

−15

−10

−5

0

5

10

15

20

25

Pole−Zero Map

Real Axis

Imag

inar

y Ax

is

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Results

•  Bode plot of the open and closed loop systems


−150

−100

−50

0

50From: In(1)

To: O

ut(1

)

100

−270

−180

−90

0

90

180

To: O

ut(1

)From: In(2)

100

Closed loop)

Frequency (rad/sec)

Mag

nitu

de (d

B) ;

Phas

e (d

eg)

Openclosed

−150

−100

−50

0

50From: In(1)

To: O

ut(1

)

100

−270

−180

−90

0

90

180

To: O

ut(1

)

From: In(2)

100

Closed loop)

Frequency (rad/sec)

Mag

nitu

de (d

B) ;

Phas

e (d

eg)

Openclosed−150

−100

−50

0

50From: In(1)

To: O

ut(1

)

100

−270

−180

−90

0

90

180

To: O

ut(1

)

From: In(2)

100

Closed loop)

Frequency (rad/sec)

Mag

nitu

de (d

B) ;

Phas

e (d

eg)

Openclosed

−150

−100

−50

0

50From: In(1)

To: O

ut(1

)

100

−270

−180

−90

0

90

180

To: O

ut(1

)

From: In(2)

100

Closed loop)

Frequency (rad/sec)

Mag

nitu

de (d

B) ;

Phas

e (d

eg)

Openclosed

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Conclusion

•  LQG general formulaMon, assumpMons and design procedure

were stated.

•  LQG regulator for CART was simulated

•  Robustness problem of LQG was shown


Thank You

Contact Informa+on Abdulrahman Kalbat

[email protected]

[email protected]

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