ALGORITHMES DE COMMANDE NUMÉRIQUE OPTIMALE DES TURBINES ÉOLIENNES
Eng. Raluca MATEESCUDr.Eng Andreea PINTEA
Prof.Dr.Eng. Nikolai CHRISTOVProf.Dr.Eng. Dan STEFANOIU
AERT 2013 [CA'NTI 19]
CONTENT
IntroductionWind Turbine Mathematical Model LQG Controller DesignMPC Controller DesignResults & Conclusions
Eng. Raluca MATEESCU
INTRODUCTION – WIND ENERGY
Electrical energy production with minimum environment damage.Romania in 2012 – first place in energy production from wind energy in Central and Eastern Europe.Romania in 2012 – 3.300 MW from facilities connected to the grid.
Eng. Raluca MATEESCU
INTRODUCTION – EFFICIENCY
Requirement – keep constant the electrical power despite wind speed variations thus the need for a dedicated controller.Discrete-time controllers in order to use it on a wind turbine.
Eng. Raluca MATEESCU
WIND TURBINE MATHEMATICAL MODEL
Eng. Raluca MATEESCU
Placement Nacelle axis orientation Rotor speed
Onshore Horizontal Fixed
Offshore Vertical Variable
Above rated regime goal: Power Limitation and Mechanical Structure protection!Solution: Pitch Control!
WIND TURBINE MATHEMATICAL MODEL
Eng. Raluca MATEESCU
QqE
qE
qE
qE
dtd
i
P
i
d
i
c
i
c =++−δδ
δδ
δδ
δδ
)(
The motion equation:
( )( ) ( ) ( ) , ,t t t t+ + =Mq Cq Kq Q q q
where M, C and K are the mass, damping and the stiffness matrices and Q is the vector of the forces acting on the system, depending the vector of generalized coordinates, q.
Lagrange equation:
[ ]1 2T
T G Ty= θ θ ζ ζq
,1 ,2 2T
aero em aero aero aeroC C F F F⎡ ⎤= −⎣ ⎦Q
WIND TURBINE MATHEMATICAL MODEL
Eng. Raluca MATEESCU
The resulting model is a highly nonlinear 8 order model.
After linearization the system was put into the general form:
[ ]ely P=
[ ], Temu C= β
( , , , , , , , )TT G T T G Tx y y= θ − θ ζ ω ω ζ β
2 1( ) ( ) ( ) ( )( ) ( ) ( ) ( )
wt t t v tt t t t= + +⎧
⎨ = + +⎩
x Fx G u Gy Hx Mu w
WIND TURBINE MATHEMATICAL MODEL
Eng. Raluca MATEESCU
Aerodynamic Block
Mechanical Block
Pitch Control Block
Generator Block
Wind Turbine
Wind Speed Block
Energy Production ProcessLinearizationOperating Point
8 Order Linear State Space Model
ofWind Turbine
LQG CONTROLLER DESIGN
A discrete-time Linear Quadratic Gaussian with integral action controller is proposed for horizontal wind turbines.The control objective – keep the output power constant, despite the wind variation, and reduce the fatigue on turbine components.Command vector: Output :
Eng. Raluca MATEESCU
( ) [ ]Temt C= βu
elP
CONTROLLER DESIGN – LQG BASICS
Stochastic system:
Find the control law u*(t) that minimizes the quadratic cost function:
u*(t) is computed based on the optimal state vector estimation obtained using the continuous-time Kalman filter.
Eng. Raluca MATEESCU
( ) ( ) ( ) ( ),
( ) ( ) ( ) ( )t t t v t
tt t t t +
= + +⎧∈⎨ = + +⎩
x Ax Buy Cx Du w
( )1 10
1( ) limT
T T
TJ E dt
T→∞
⎧ ⎫⎪ ⎪= +⎨ ⎬⎪ ⎪⎩ ⎭∫u y Q y u R u
ˆ( ) ( )t t∗ = −u Kx
ˆ ˆ ˆ( ) ( ) ( ) ( ( ) ( ))ft t t t t= + + −x Ax Bu K y Cx
DISCRETE TIME AUGMENTED MODEL
Discrete augmented model of the wind turbine:
The corresponding quadratic cost function :
Eng. Raluca MATEESCU
[ ][ 1] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] , 1, , 4
t s
i
n n n v n v nn n n n y n i+ = + + +⎧⎪
⎨ = + + = =⎪⎩
z Az Bu E Ey Cz Du w …
}
1 10
0
1( [ ]) lim ( [ ] [ ] [ ] [ ] [ ] [ ])
1lim ( [ ] [ ] [ ] [ ] [ ] [ ]
2 [ ] [ ] [ ]) ,
NT T
N
NT T
N
T
J n E n n n n n nN
E n n n n n nN
n n n
→∞
→∞
⎧ ⎫= + =⎨ ⎬
⎩ ⎭⎧
= +⎨⎩
+
∑
∑
u y Q y u R u
z Q z u R u
z S u
1, 1[ ] [ ] [ ] and [ ] [ ]TT
refn n n n y y n⎡ ⎤= ε ε = −⎣ ⎦z xwhere:
CONTROLLER DESIGN – CONTROL LAW
The discrete-time LQG control law is :
where is the optimal estimation of , which is obtained by the Kalman filter:
The gain matrix is computed as:
The gain matrix of the Kalman filter:
Eng. Raluca MATEESCU
ˆ[ ] [ ] [ ],d in n K n∗ = − + εu K xˆ[ ]nx [ ]nx
( )ˆ ˆ ˆ[ 1] [ ] [ ] [ ] [ ]ˆ ˆ[ ] [ ] [ ]
fn n n n n
n n n
⎧ + = + + −⎪⎨
= +⎪⎩
x Ax Bu K y y
y Cx Du
[ ]d iK= −K K
( ) ( )1T T T−= + +K R B PB B PA S
( ) 1T Tf f f
−= +K AP C W CP C
LQG STRATEGY – RESULTS & CONCLUSIONS
Simulation Environment – Matlab SIMULINK
Wind speed profile:
Eng. Raluca MATEESCU
Output power of the wind turbine obtained using the designed discrete-time LQG controller.
Eng. Raluca MATEESCULQG STRATEGY – RESULTS & CONCLUSIONS
MPC CONTROLLER DESIGN
A discrete-time model predictive control (MPC) strategy is proposed for horizontal axis wind turbines.The control objective – keep the output power constant, despite the wind variation, and reduce the fatigue on turbine components.Command vector: Output :
Eng. Raluca MATEESCU
( ) [ ]Temt C= βu
elP
DISCRETE TIME AUGMENTED MODELAugmented state-space model of the HAWT:
where:
Eng. Raluca MATEESCU
( 1) ( ) ( ) ( )( ) ( )
e e
e
x k x k u k ky t x k
ε+ = ⋅ + ⋅∆ + ⋅ε⎧⎨ = ⋅⎩
A B BC
[ ]( 1) ( 1) ( 1) Tmx k x k y k+ = ∆ + +
( ) ( ) ( 1)u k u k u k∆ = − −
kε( ) is the input disturbance – wind speed variation.
Step 1: Calculate the predicted plant output with the future control signal as the adjustable variable. This prediction is described within an optimization window
.The future control trajectory is denoted by:
The future state variables are denoted as:
The output and command vectors are defined as:
Eng. Raluca MATEESCU
PN
( ), ( 1), ..., ( 1),i i i Cu k u k u k N∆ ∆ + ∆ + − is control horizon.CN
( 1 | ), ( 2 | ), ...., ( | ), ..., ( | )i i i i i i i P ix k k x k k x k m k x k N k+ + + +
[ ]
( ) ( 1) ... ( 1)
( 1 | ) y( 2 | ).. ( | )
TT T Ti i i C
Ti i i i i P i
U u k u k u k N
Y y k k k k x k N k
⎡ ⎤∆ = ∆ ∆ + ∆ + −⎣ ⎦
= + + +
CONTROLLER DESIGN – MPC
The future state variables are calculated sequentially using the set of future control parameters as follows:
Effectively, we have:
With:
Eng. Raluca MATEESCU
CONTROLLER DESIGN – MPC
1 2
1
2
( | ) ( ) ( ) ( 1)
( 1)
( 1 | ) ... 1 |
p P P
pP C
P
N N Ni P i i i i
NN Ni C i
Ni i i P i
x k N k A x k A B u k A B u k
A B u k N A B k
A B k k B k N k
− −
−−ε
−ε ε
+ = + ∆ + ∆ +
+ ∆ + − + ε( )
+ ε + + + ε( + − )
( )iY Fx k U= +Φ∆
2
3 2
1 2
0 00
; 0
P CP P P N NN N N
CA CBCA CAB CB
F CA CA B CAB
CA CA B CA B CA B−− −
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= Φ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
For a given set-point signal at sample time the objective of the predictive control system is to bring the predicted output as close as possible to the set-point signal . This objective is then translated into a design to find the ‘best’control parameter vector such that an error function between the set-point and the predicted output is minimized.
The set point signal is defined as:
Eng. Raluca MATEESCU
CONTROLLER DESIGN – MPC
( ) ( ) ( ) ( )1 2
T
i i i q ir k r k r k r k⎡ ⎤= ⎣ ⎦…
Assuming that the data vector that contains the set-point information is:
the cost function that reflects the control objective is:
Control vector ∆U is linked to the set-point signal r(ki) and the state variable x(ki) via the following equation:
CONTROLLER DESIGN – MPCEng. Raluca MATEESCU
( ) ( ) ( )( )1
s i iU R R r k Fx k−Τ Τ Τ∆ = Φ Φ + Φ −Φ
[ ]1 1 ... 1 ( )PN
TS iR r k=
Step 3: Receding horizon control - Applying the receding horizon control principle, the first m elements in ∆U are taken to form the incremental optimal control:
With:
Eng. Raluca MATEESCU
CONTROLLER DESIGN – MPC
( ) [ ]( )( ) ( )( )
( ) ( )
1
y
0 0
=K
CN
i m m m
s i i
i mpc i
u k I R
R r k Fx k
r k K x k
−Τ
Τ Τ
∆ = Φ Φ +
× Φ −Φ
−
…
( )
( )
1
11... (first row of the matrix)0
y T S
mpc T
K R R
K R F
− Τ
− Τ
= Φ Φ + Φ
⎡ ⎤⎢ ⎥= Φ Φ + Φ⎢ ⎥⎢ ⎥⎣ ⎦
Step 4: Building the Observer for State estimation –Considering that the given information considered for MPC design x(ki) is not measurable an observer is needed. The observer is constructed using the equation:
Where Kob is the observer gain matrix, and Am and Bm
correspond to the discrete-time state-space model of the plant. Kob was computed using the Matlab ‘place’function, based on the augmented state space model.
Eng. Raluca MATEESCU
CONTROLLER DESIGN – MPC
( ) ( ) ( ) ( )( )correction termmodel
ˆ ˆ ˆ( 1)m m m m ob m mx k A x k B u k K y k C x k+ = + + −
MPC – RESULTS & CONCLUSIONS
Simulation Environment – Matlab SIMULINK
Wind speed profile:
Eng. Raluca MATEESCU
Output power of the wind turbine obtained using the designed discrete-time MPC.
Eng. Raluca MATEESCU
MPC – RESULTS & CONCLUSIONS
Q&A
En vous remerciant de votre attention, je vous souhaite
une agreable journée!
Eng. Raluca MATEESCU