AERT 2013 [CA'NTI 19] ALGORITHMES DE COMMANDE NUMÉRIQUE OPTIMALE DES TURBINES ÉOLIENNES

Transcription

1 AER 2013 [CA'NI 19] ALGORIHMES DE COMMANDE NUMÉRIQUE OPIMALE DES URBINES ÉOLIENNES Eng. Raluca MAEESCU Dr.Eng Andreea PINEA Prof.Dr.Eng. Nikolai CHRISOV Prof.Dr.Eng. Dan SEFANOIU

2 Eng. Raluca MAEESCU CONEN Introduction Wind urbine Mathematical Model LQG Controller Design MPC Controller Design Results & Conclusions

3 Eng. Raluca MAEESCU INRODUCION 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 MW from facilities connected to the grid.

4 Eng. Raluca MAEESCU INRODUCION 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.

5 Eng. Raluca MAEESCU WIND URBINE MAHEMAICAL 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!

6 Eng. Raluca MAEESCU WIND URBINE MAHEMAICAL MODEL he motion equation: ( ) Mq () t + Cq () t + Kq() t = Q q, q, t 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. Q = C C F F F aero em aero,1 aero,2 2 aero Lagrange equation: [ G 1 2 y] q = θ θ ζ ζ d dt δe ( δq c i ) δe δq c i + δe δq d i + δe δq P i = Q

7 Eng. Raluca MAEESCU WIND URBINE MAHEMAICAL MODEL he resulting model is a highly nonlinear 8 order model. x = ( θ θ, ζ, y, ω, ω, ζ, y, β) u =β [, C em ] y = [ P el ] G G After linearization the system was put into the general form: x () t = Fx() t + G2u() t + G1vw() t y() t = Hx() t + Mu() t + w() t

8 Eng. Raluca MAEESCU WIND URBINE MAHEMAICAL MODEL Aerodynamic Block Mechanical Block Pitch Control Block Wind urbine Generator Block 8 Order Linear State Space Model of Wind urbine Wind Speed Block Energy Production Process Linearization Operating Point

9 Eng. Raluca MAEESCU LQG CONROLLER DESIGN A discrete-time Linear Quadratic Gaussian with integral action controller is proposed for horizontal wind turbines. he control objective keep the output power constant, despite the wind variation, and reduce the fatigue on turbine components. Command vector: Output : u() t = [ β C ] em P el

10 CONROLLER DESIGN LQG BASICS Eng. Raluca MAEESCU Stochastic system: x () t = Ax() t + Bu() t + v() t, y() t = Cx() t + Du() t + w() t t + Find the control law u*(t) that minimizes the quadratic cost function: 1 J ( u) = lime ( ) dt y Q y u R u 0 u*(t) is computed based on the optimal state vector estimation obtained using the continuous-time Kalman filter. u () t = Kxˆ () t xˆ () t = Axˆ() t + Bu() t + K ( y() t Cxˆ()) t f

11 DISCREE IME AUGMENED MODEL Discrete augmented model of the wind turbine: he corresponding quadratic cost function : Eng. Raluca MAEESCU z[ n+ 1] = Az[ n] + Bu[ n] + Evt[ n] + Evs[ n] y[ n] = Cz[ n] + Du[ n] + w [ n] = [ yi[ n] ], i = 1,,4 N 1 J( u[ n]) = lim E ( y [ n] Q1[ n] y[ n] + u [ n] R1[ n] u[ n]) = N N 0 N 1 = lim E ( z [ n] Q[ n] z[ n] + u [ n] R[ n] u[ n] N N z [][][]), n S n u n } where: z[ n] = x [ n] ε[ n] and ε [ n] = y1, ref y1[ n]

12 CONROLLER DESIGN CONROL LAW he discrete-time LQG control law is : ˆ[ ] where x n is the optimal estimation of x[, nwhich ] is obtained by the Kalman filter: he gain matrix K = K K is computed as: he gain matrix of the Kalman filter: u [ n] = K xˆ [ n] + K ε[ n], d Eng. Raluca MAEESCU i ( ) xˆ[ n+ 1] = Axˆ[ n] + Bu[ n] + K [ ] ˆ f y n y[ n] yˆ[ n] = Cxˆ[ n] + Du[ n] [ ] d i ( 1 ) ( ) K = R+ B PB B PA+ S ( ) 1 K = APC W+ CPC f f f

13 LQG SRAEGY RESULS & CONCLUSIONS Eng. Raluca MAEESCU Simulation Environment Matlab SIMULINK Wind speed profile:

14 LQG SRAEGY RESULS & CONCLUSIONS Eng. Raluca MAEESCU Output power of the wind turbine obtained using the designed discrete-time LQG controller.

15 Eng. Raluca MAEESCU MPC CONROLLER DESIGN A discrete-time model predictive control (MPC) strategy is proposed for horizontal axis wind turbines. he control objective keep the output power constant, despite the wind variation, and reduce the fatigue on turbine components. Command vector: Output : u() t = [ β C ] em P el

16 DISCREE IME AUGMENED MODEL Augmented state-space model of the HAW: Eng. Raluca MAEESCU where: xk ( + 1) = Ae xk ( ) + Be uk ( ) + Bε ε( k) yt () = Ce xk ( ) [ ] xk ( + 1) = x ( k+ 1) yk ( + 1) uk ( ) = uk ( ) uk ( 1) m ε( k) is the input disturbance wind speed variation.

17 Eng. Raluca MAEESCU CONROLLER DESIGN MPC Step 1: Calculate the predicted plant output with the future control signal as the adjustable variable. his prediction is described within an optimization window N P. he future control trajectory is denoted by: uk ( ), uk ( + 1),..., uk ( + N 1), is control horizon. i i i C he future state variables are denoted as: xk ( + 1 k), xk ( + 2 k),..., xk ( + m k),..., xk ( + N k) i i i i i i i P i he output and command vectors are defined as: N C U = u( ki) u( ki + 1)... u( ki + NC 1) [ ( i 1 i) y( i 2 i).. ( i P i) ] Y = y k + k k + k x k + N k

18 Eng. Raluca MAEESCU CONROLLER DESIGN MPC he future state variables are calculated sequentially using the set of future control parameters as follows: N p NP 1 NP 2 ( i + P i) = ( i) + ( i) + ( i + 1) N N p 1 P N C + A B u( ki + NC 1) + A Bεε( ki) NP 2 + A Bεε ( ki + 1 ki) Bεε( ki + NP 1 ki) x k N k A x k A B u k A B u k Effectively, we have: With: Y = Fx( k ) +Φ U 3 2 F = CA Φ = CA B CAB i CA CB CA CAB CB 0 ; 0 NP NP 1 NP 2 N P N C CA CA B CA B CA B

19 CONROLLER DESIGN MPC Eng. Raluca MAEESCU 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. his 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. he set point signal is defined as: ( ) ( ) ( ) ( ) r ki = r1 ki r2 ki rq k i

20 CONROLLER DESIGN MPC Eng. Raluca MAEESCU Assuming that the data vector that contains the setpoint information is: N P R = [ ] rk ( ) the cost function that reflects the control objective is: S i Control vector U is linked to the set-point signal r(ki) and the state variable x(ki) via the following equation: ( ) 1 Τ ( Τ Τ ( ) ( )) s i i U = Φ Φ+ R Φ R r k Φ Fx k

21 Eng. Raluca MAEESCU CONROLLER DESIGN MPC 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: Τ ( i) [ m 0m 0m]( ) Τ Τ ( Φ Rr s ( ki) Φ Fx( ki) ) =K N C u k = I Φ Φ+ R y ( ) ( ) r k K x k i mpc i 1 ( ) 1 Τ y = ΦΦ+ Φ S K R R 1 1 Τ Kmpc =... ( Φ Φ+ R) Φ F (first row of the matrix) 0

22 Eng. Raluca MAEESCU CONROLLER DESIGN MPC 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. he observer is constructed using the equation: model correction term xˆ ˆ ( ) ( ) ( ( ) ˆ ( )) m( k+ 1) = Amxm k + Bmu k + Kob y k Cmxm k 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.

23 MPC RESULS & CONCLUSIONS Simulation Environment Matlab SIMULINK Eng. Raluca MAEESCU Wind speed profile:

24 MPC RESULS & CONCLUSIONS Eng. Raluca MAEESCU Output power of the wind turbine obtained using the designed discrete-time MPC.

25 Q&A En vous remerciant de votre attention, je vous souhaite une agreable journée! Eng. Raluca MAEESCU