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.


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.


WIND TURBINE MATHEMATICAL MODEL


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!



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



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



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 :


( ) [ ]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.


( ) ( ) ( ) ( ),

( ) ( ) ( ) ( )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 :


[ ][ 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:


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:


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 :


( ) [ ]Temt C= βu

elP

DISCRETE TIME AUGMENTED MODELAugmented state-space model of the HAWT:

where:


( 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:


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:



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:



( ) ( ) ( ) ( )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:



( ) [ ]( )( ) ( )( )

( ) ( )

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.



( ) ( ) ( ) ( )( )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:


Output power of the wind turbine obtained using the designed discrete-time MPC.


MPC – RESULTS & CONCLUSIONS

Q&A

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