ALGORITHMES DE COMMANDE NUMÉRIQUE OPTIMALE DES TURBINES ÉOLIENNES...
Transcript of ALGORITHMES DE COMMANDE NUMÉRIQUE OPTIMALE DES TURBINES ÉOLIENNES...
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.
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WIND TURBINE MATHEMATICAL MODEL
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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 :
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( ) [ ]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:
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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:
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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:
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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