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
Abdulrahman Kalbat Columbia University 3
10/03/2013 /19
Linear Quadra+c Gaussian (LQG) Control
• LQG block diagram
Abdulrahman Kalbat Columbia University 4
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
10/03/2013 /19
LQG General Form
• The state space equaMons of the open loop plant for a standard LQG problem:
where
Abdulrahman Kalbat Columbia University 5
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
10/03/2013 /19
LQG General Form
• The state space equaMons of the Kalman Filter:
where
Abdulrahman Kalbat Columbia University 6
˙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
10/03/2013 /19
LQG General Form
• AssumpMons:
Abdulrahman Kalbat Columbia University 7
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
10/03/2013 /19
LQG Design Procedure
Step 1: Check opMmal gain existance criteria:
Step 2: OpMmal state esMmaMon gain calculaMon:
Abdulrahman Kalbat Columbia University 8
(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)
10/03/2013 /19
LQG Design Procedure
Step 3
• OpMmal State Feedback Gain CalculaMon:
• Weighing Matrices SelecMon:
Abdulrahman Kalbat Columbia University 9
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
LQG Design Procedure
Step 4: Linear QuadraMc Gaussian Regulator by combining
OpMmal State EsMmaMon and OpMmal State Feedback:
where
Abdulrahman Kalbat Columbia University 10
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)
10/03/2013 /19
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
Abdulrahman Kalbat Columbia University 11
10/03/2013 /19
Controls Advanced Research Turbine (CART)
Abdulrahman Kalbat Columbia University 12
10/03/2013 /19
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.
Abdulrahman Kalbat Columbia University 13
10/03/2013 /19
Numerical Example using MATLAB
• CART’s state space model:
Abdulrahman Kalbat Columbia University 14
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)
10/03/2013 /19
Results
• Wind turbine’s response to different wind speeds
Abdulrahman Kalbat Columbia University 15
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
10/03/2013 /19
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)
Abdulrahman Kalbat Columbia University 16
10/03/2013 /19
Results
• Poles/Zeros plot of the closed loop system
Abdulrahman Kalbat Columbia University 17
−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
10/03/2013 /19
Results
• Bode plot of the open and closed loop systems
Abdulrahman Kalbat Columbia University 18
−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
10/03/2013 /19
Conclusion
• LQG general formulaMon, assumpMons and design procedure
were stated.
• LQG regulator for CART was simulated
• Robustness problem of LQG was shown
Abdulrahman Kalbat Columbia University 19