PERFORMANCE COMPARISON OF CONTROL SCHEMES FOR VARIABLE-SPEED WIND TURBINES C.L. Bottasso, A. Croce, B. Savini Politecnico di Milano Milano, Italy TWIND 2007 DTU, Lyngby, August 28-31, 2007

Outline Wind turbine models: Aero-servo-elastic model Reduced model Observers: Reduced model state observer Wind observer Control laws: Wind-Scheduled optimal PID MIMO Nonlinear-wind LQR RAPC, Reference Augmented Predictive Control Results Conclusions and outlook

Outline Wind turbine models: Aero-servo-elastic model Reduced model Observers: Reduced model state observer Wind observer Control laws: Wind-Scheduled optimal PID MIMO Nonlinear-wind LQR RAPC, Reference Augmented Predictive Control Results Conclusions and outlook

Aero-servo-elastic Models ▶ FEM multibody code, extensively validated for rotorcraft applications: (Bauchau, Bottasso, Nikishkov, MCM 2001) ▶ Wind-energy version: CpLambda (Code for Performance, Loads and Aeroelasticity by Multi-Body Dynamic Analysis)

Aero-servo-elastic Models Examples of standard and teetering wind turbines in CpLambda:

Aero-servo-elastic Models CpLambda structural element library: ▶ Beams: Geometrically exact, composite-ready beams Curved and twisted NURBS reference lines Fully populated 6x6 stiffness (aeroelastic couplings) ▶ Joints: Enforced by Lagrange multipliers (DAE formulation) Spring, damper, backlash and friction models in all joints Flexible joints (contact beam-cylindrical, prismatic, screw) Unilateral joints (contact-impact analysis) ▶ Actuators: first and second order linear and rotational models ▶ Sensors and control elements

Aero-servo-elastic Models ▶ Aerodynamic model: Lifting lines (two-dimensional strip theory) Tip losses, radial & unsteady flow, dynamic stall Inflow models (Dynamic Pitt-Peters & Peters-He) Generic interface to external CFD or free wake Tower shadow Wind models (according to IEC 61400-1): Deterministic gusts (EOG1, ECG) 3D stochastic turbolent wind Wind shear (exponential and logarithmic) ▶ Analysis types: Static analysis Eigenanalysis Dynamic analysis Stability analysis (implicit Floquet or by excitation)

Aero-servo-elastic Models CpLambda time integrations schemes: ▶ Geometric integrators for DAEs: Exact treatment of geometric non-linearities Exact satisfaction of constraints (no drift) Scaling for improved numerical conditioning (Bottasso et al. 2007) Non-linear unconditional stability (Bottasso et al. 2003): bound on total energy of deformable bodies + vanishing of work of constraint forces + conservation of momenta ▶ Energy preserving/decaying scheme: High frequency modes artifacts of discretization Energy decaying scheme damps unresolved modes Improved robustness for large non-linear FEM models Solutions that satisfy the constraints Drifting solution System manifold Solutions that satisfy the invariants Constraint manifold Manifold of the invariants Energy preserving solution E D ¸ Dissipated total energy within a time step Energy manifold Energy decaying solution

Reduced Model for Model-Based Controllers ¯ e Equations: Drive-train shaft dynamics Elastic tower fore-aft motion Blade pitch actuator dynamics Electrical generator dynamics States: Inputs: ¯ c F a d J G d ; _ ­ ¯ e T l T e l c T e l T l T a ­ ¯ c ; T e l J R M T ; C K ▶ Remarks: Collective pitch only Individual-blade non-linear elastic model based on SymDyn also available (but not covered in this paper)

Reduced Model for Model-Based Controllers Equations of motion: Tip speed ratio: Wind: (mean wind + turbulence) ( J R + G ) _ ­ T l e ¡ a ; ¯ V w d m = M Ä C K F 2 » ! c 1 ¿ ¸ = ­ R ( V w ¡ _ d ) V w = m + t

Reduced Model for Model-Based Controllers Rotor force and moment coefficients: computed off-line with CpLambda aero-servo-elastic model, averaging periodic response over one rotor rev Stored in look-up tables T a = 1 2 ½ ¼ R 3 C P e ( ¸ ; ¯ V m ) w ¡ _ d F C F e ( ¸ ; ¯ V m ) P ▶ Dependence of and on mean wind accounts for deformability of tower and blades under high winds: C F e ( ¸ ; ¯ V m ) C P e ( ¸ ; ¯ V m ) V m

Outline Wind turbine models: Aero-servo-elastic model Reduced model Observers: Reduced model state observer Wind observer Control laws: Wind-Scheduled optimal PID MIMO Nonlinear-wind LQR RAPC, Reference Augmented Predictive Control Results Conclusions and outlook

Tower State Observer ▶ Remarks: ½ _ q = v ( © ) a + n c = © q ¡ n q © Kalman modal-based tower observer: Accelerations: Curvatures: Unknown modal amplitudes: Modal bases: Process & measurement noise: ▶ Remarks: Fore-aft and side-side identification Multiple modal ampl. (sensor number and position for observability) Formulation applicable also to identification of flap-lag blade states ½ _ q = v ( © T ) ¡ 1 a + n w Accelerometer c = © q ¡ n v q © n w ; v Strain gage

Tower State Observer ½ _ x = A + B u W n y C D V x = ( q ; v ) u a y c State space form: with Optimal Kalman state estimate: Filter gain matrix Propagated states and outputs based on accelerometric reading: Curvature reading: ½ _ x = A + B u W n w y C D V v x = ( q T ; v ) u a y c A = · I ¸ B ª C £ © ¤ D W V x k = ¡ + K ( ^ y ) K k x ¡ k ; y ^ y k

Tower State Observer Tower tip velocity estimation: Filter warm-up

Wind Observer _ V = n y = ( J + ) _ ­ T ¡ ; ¯ V d n ½ _ x = f ( ; u n Extended Kalman wind observer: Wind equation: Output measurement torque-balance equation: Non-linear state-space form: with Extended Kalman estimate with measured output to enforce torque-balance equation Mean wind reconstructed with moving average on 10 sec window _ V w = n y = ( J R + G ) _ ­ T l e ¡ a ; ¯ V w d m n v ½ _ x = f ( ; u n w ) y h v x = V w u ( _ ­ ; ¯ e d m ) T x k = ¡ + K ( ^ y ) ^ y k = V m

Wind Observer ▼ Turbulent wind ( m/sec) V = 1 5 ▲ EOG1-13 case Hub wind estimation: ▼ Turbulent wind ( m/sec) V m = 1 5 ▲ EOG1-13 case

Outline Wind turbine models: Aero-servo-elastic model Reduced model Observers: Reduced model state observer Wind observer Control laws: Wind-Scheduled optimal PID MIMO Nonlinear-wind LQR RAPC, Reference Augmented Predictive Control Results Conclusions and outlook

Control Laws: Virtual Testing Environment Virtual plant Sensor models CpLambda aero-servo-elastic model Wind generator Process noise Supervisor Choice of operating condition: Power production Emergency shut-down Start up … Controller Feedback controller PID MIMO LQR RAPC Adaptive reduced model Linux real-time environment Kalman filtering State estimation Wind estimation Measurement noise

Control Laws: Optimal PID Optimal wind-scheduled PID: Tabulated electrical torque ▶ Optimization of gains based on aeroelastic analyses in CpLambda ¯ c = K p ( V m ) ­ ¡ ¤ + i Z t T d ¿ _ e l K p ( V m ) ; i d

Control Laws: Optimal PID Gain optimization procedure: For each mean wind in region 3, define cost function Equivalent fatigue loads for tower and blades based on rain-flow analysis (ASTM E 1049-85): Tunable weighting factors: V m J ( V m ) = M e q T + B i 1 ; 3 Z 6 s c ¡ w _ ¯ 2 Ä d ­ ¤ P ¢ t M e q T ; B i M e q = Ã X i ¢ m f ; N t o ! 1 w _ ¯ ; Ä d ­ P

Control Laws: Optimal PID PID gain optimization procedure (continued): For each mean wind : ▶ Regard cost as sole function of unknown gains ▶ Minimize cost (using Noesis Optimus™): Evaluate cost with CpLambda aero-servo-elastic model Global optimization (GA) Local refinement (Response Surface + gradient based minimization) V m J ( V m ) = K p ; i d Optimizer Global & local algorithms Functional approximators CpLambda Aeroelastic response in turbulent wind for given gains K p ( V m ) ; i d J ( V m ) (possible constraints)

Control Laws: MIMO NonLinear-Wind LQR Wind-scheduled MIMO LQR: ▶ Reduced model in compact form: where ▶ Wind parameterized linear model: Remarks: Model linearized about current mean wind estimate Non-linear dependence on instantaneous turbulent wind Wind not treated as linear disturbance (as commonly done) _ x = f ( ; u V w m ) x = ( d ; _ ­ ¯ e T l ) u = ( ¯ c ; T e l ) ¢ _ x = A ( V w ; m ) + B u ¢ x = ¡ ¤ ( V m ) ¢ u = ¡ ¤ ( V m ) V m V w

Control Laws: MIMO NonLinear-Wind LQR Wind-scheduled MIMO LQR (continued): ▶ Regulation cost: where ▶ MIMO formulation: tracking quantities for reg. 2 & 3: J = 1 2 Z ¡ ¢ x T Q + u R d t ¢ x = ¡ ¤ ( V m ) ; u x ¤ ( V m ) ; u

Control Laws: MIMO NonLinear-Wind LQR Wind-scheduled MIMO LQR (continued): ▶ Closed loop controller: with Kalman estimated states and wind Periodic LQR (SymDyn model) available, but not covered in this paper u = ¡ K ( V w ; m ) x ¤

Control Laws: NonLinear Adaptive Ctrl. Design controller which: Can handle non-linearities of plant Is adaptive: Can adjust to off-design conditions (e.g. ice accretion, specifics of installation, hot-cold air variations, etc.) Can correct for unmodeled or unresolved physics and modeling errors Can handle constraints (e.g. max loads in blades or tower) Can be implemented in real-time (no iterative scheme, fixed number of operations per activation) ▶ Non-linear model-adaptive predictive control

Control Laws: NonLinear Adaptive Ctrl. Non-linear Model Predictive Control (NMPC): Find the control action which minimizes an index of performance, by predicting the future behavior of the plant using a non-linear reduced model. - Reduced model: - Initial conditions: - Output definition: Cost: with desired goal outputs and controls. Stability results: Findeisen et al. 2003, Grimm et al. 2005. m i n u ; x y J = Z t + T p L ( ) d s . : f _ 2 [ ] g L ( y ; u ) = ¡ ¤ T Q + R ( ¢ ) ¤

Control Laws: NonLinear Adaptive Ctrl. G o a l r e s p n x ¤ ( t ) t P r e d i c t s p o n x ( ) P a s t F u t r e P r e d i c t o n S t a e r c k i n g o P l a n t r e s p o x ( ) x G o a l c n t r u ¤ ( ) C o m p u t e d c n r l ( ) C o n t r l a c k i g e S t e r i n g w d o t + T s t P r e d i c t o n w t + T p

Control Laws: NonLinear Adaptive Ctrl. Predictive model-adaptive control: Prediction window Tracking cost Tracking cost Prediction window Prediction window Tracking cost Goal response Prediction error Prediction error Steering window Steering window Prediction error Plant response Past Future Past Future Past Future Past Future Steering window Past Future Predictive solutions 1. Tracking problem 2. Steering problem 3. Reduced model update Reduced model adaption: Predict plant response with minimum error (same outputs when same inputs) Self-adaptive (learning) model adjusts to varying operating conditions (ice, air density, terrain, etc.)

RAPC: Motivation For any given problem: wealth of knowledge and legacy methods which perform reasonably well Quest for better performance/improved capabilities: undesirable and wasteful to neglect valuable existing knowledge Reference Augmented Predictive Control (RAPC): exploit available legacy methods, embedding them in a non-linear model predictive adaptive control framework Specifically: Model: augment reduced models to account for unresolved or unmodeled physics Control: design a non-linear controller augmenting linear ones (MIMO Nonlinear-Wind LQR) which are known to provide a minimum level of performance about certain linearized operating conditions

RAPC: Motivation Approach: Choose a reference model / reference control law Augment the reference using an adaptive parametric function Adjust the function parameters to ensure good approximation of the actual system / optimal control law (parameter identification) Reasons for using a reference model / control: Reasonable predictions / controls even before any learning has taken place (otherwise would need extensive pre-training) Easier and faster adaption: the defect is typically a small quantity, if the reference solution is well chosen

RAPC: Reduced Model Identification The principle of reference model augmentation: Same wind, same inputs Same wind, same inputs u ; V w u ; V w u ; V w + Neural Network e x Trained on-line to minimize mismatch Augmented reduced model Reduced model Plant Dissimilaroutputs Similar outputs e x x x

RAPC: Reduced Model Identification Neural augmented reference model: reference (problem dependent) analytical model, Remark: reference model will not, in general, ensure adequate predictions, i.e. when = system states/controls, = model states/controls. Augmented reference model: where is the unknown reference model defect that ensures when i.e.: Hence, if we knew , we would have perfect prediction capabilities. Reference reduced model f r e ( _ x ; u ) = e x 6 = e u = ; e x ; u x ; u f r e ( _ x ; u ) = d d e x = e u = f r e ( _ x ; u ) ¡ d = d

RAPC: Reduced Model Identification Approximate with single-hidden-layer neural networks: where and = functional reconstruction error; = matrices of synaptic weights and biases; = sigmoid activation functions; = network input. The reduced model parameters are identified on-line using an Extended Kalman Filter. d d ( x ; u ) = p m + " d p ( x ; u m ) = W T ¾ V i + a b " W m ; V a b ¾ ( Á ) = 1 ; : N n T i = ( x T ; u ) p m = ( : ; W i k V a b ) T

RAPC: Reduced Model Identification Tower-tip velocity for multibody, reference, and neural-augmented reference with same prescribed inputs: Fast adaption Red: reference model Black: CpLambda multibody model Blue: reference model +neural network

RAPC: Reduced Model Identification Defect and remaining reconstruction error after adaption: d i " i Red: defect Blue: remaining reconstruction error

RAPC: Neural Control t A u g m e n t d s o l . x ( ) + N P a s t F u t The principle of neural-augmented reference control: t A u g m e n t d s o l . x ( r f ) + N P a s t F u t r e G o a l r e s p n x ¤ ( t ) O p t i m a l s o u n x ( N M P C ) x ( t ) ; < S u b - o p t i m a l s n x ( r e f ) x A u g m e n t d s o l . r f ( ) + N G o a l c n t r u ¤ ( ) O p t i m a l s o u n N M P C ( ) S u b - o p t i m a l s n r e f ( ) u ( t ) ; < t P r e d i c t o n w t + T p

RAPC: Neural Control m i n J = Z L ( ) d s . : f _ 2 [ ] g f ( _ x ; u Prediction problem: Enforcing optimality, we get: m i n u ; x y J = Z t + T p L ( ) d s . : f _ 2 [ ] g Model equations: f ( _ x ; u p m ) = t 2 [ + T ] ¡ d ¸ y L : State initial conditions: Adjoint equations: Co-state final conditions: Transversality conditions:

RAPC: Neural Control u ( t ) =  ¡ x ; y ¢ G o a l r e s p n x ( t ) t It can be shown that minimizing control is (Bottasso et al. 2007) u ( t ) =  ¡ x ; y ¤ ¢ G o a l r e s p n x ¤ ( t ) t P a s t F u t r e  ( ¢ ; ) x ( t ) ; < u ( t ) x G o a l c n t r u ¤ ( ) O p t i m a l c o n r u ( ) u ( t ) ; < t P r e d i c t o n w t + T p

RAPC: Neural Control u ( t ) = + À ¡ x ; y ¢ À ( ¢ ; ) À ( ¢ ; ) À ( ¢ Reference augmented form: where is the unknown control defect. Remark: if one knew , the optimal control would be available without having to solve the open-loop optimal control problem. Idea: Approximate using an adaptive parametric element: Identify on-line, i.e. find the parameters which minimize the reconstruction error . u ( t ) = r e f + À ¡ x ; y ¤ ¢ À ( ¢ ; ) À ( ¢ ; ) À ( ¢ ; ) À ¡ x ; y ¤ ( t ) u ¢ = p c + " À p ( ¢ ; ) p c "

RAPC: Neural Control u p f ( _ x ; u p ) = t 2 [ + T ] ¡ d ( f ¸ ) t + Iterative procedure to solve the problem in real-time: Integrate reduced model equations forward in time over the prediction window, using and the latest available parameters (state prediction): Integrate adjoint equations backward in time (co-state prediction): Correct control law parameters , e.g. using steepest descent: u r e f p c f ( _ x ; u p m ) = t 2 [ + T ] ¡ d ( f T ; _ x ¸ ) t + u y L = 2 [ p ] p c _ p c = ¡ ´ ^ J ; ! n e w o l d

RAPC: Neural Control _ p = ¡ ´ ^ J Z À ( L f ¸ ) d = Remark: the parameter correction step seeks to enforce the transversality condition Once this is satisfied, the control is optimal, since the state and co-state equations and the boundary conditions are satisfied. _ p c = ¡ ´ ^ J ; Z t + T p À ; c ( L u f ¸ ) d =

RAPC: Neural Control x ( t ) ¸ ( t ) u ( t ) _ p = ¡ ´ ^ J Past Future Past Future Target x ( t ) Tracking cost Prediction error ¸ ( t ) State u ( t ) _ p c = ¡ ´ ^ J ; Control Optimal control Steering window Prediction horizon Predict control action Predict state forward Repeat Predict co-state backwards Update estimate of control action, based on transversality violation Advance plant Update model, based on prediction error

RAPC: Neural Control À ¡ x ; y ( t ) u ¢ ¼ À ¡ x ; y ( t ) u ¿ ¢ ¼ 1 » Drop dependence on time history of goal quantities: Approximate temporal dependence using shape functions: Associate each nodal value with the output of a single-hidden-layer feed-forward neural network, one for each component: where Output: Input: Control parameters: À p ¡ x ; y ¤ ( t ) u c ¢ ¼ À p ¡ x ; y ¤ ( t ) u ¿ c ¢ ¼ 1 » k + o c = W T ¾ ( V i + a ) b o c = ( À T p ; 1 : M ¡ ) i c = ¡ x T ; ¤ ( t ) u ¢ p c = ( : ; W i j V a b ) T

RAPC: Neural Control t P a s t F u t r e x ( t ) x ( t ) N x ( t ) ; P a s t F u t r e x ¤ ( t ) x ¤ ( t ) N x ( t ) ; < À p k x u ¤ ( t ) u ¤ ( t ) u ( t ) ; < t P r e d i c t o n w t + T p

RAPC RAPC can handle constraints on inputs and outputs (not covered in this paper) Present results: Reference model: collective-only, Reference controller: MIMO Nonlinear-Wind LQR Work in progress: Reference model: SymDyn (individual blade pitch, flap dynamics) Reference controller: periodic MIMO Nonlinear-Wind LQR Constraints on inputs and outputs x = ( d ; _ ­ ¯ e T l )

Outline Wind turbine models: Aero-servo-elastic model Reduced model Observers: Reduced model state observer Wind observer Control laws: Wind-Scheduled optimal PID MIMO Nonlinear-wind LQR RAPC, Reference Augmented Predictive Control Results Conclusions and outlook

Results Two consecutive EOG1-13 in nominal conditions:

Results Normalized total regulation error in 600 sec turbulent wind Cold air & ice accretion (degraded airfoil performance):

Results Observations: Significant advantage of model-based (especially non-linear and adaptive) controllers in - Turbulent off-design conditions - Strong gusts It appears that adaptive element is able to correct deficiencies of reference reduced model, even in the presence of large errors In nominal conditions, and for the collective pitch case: - Differences in turbulent response of PID, LQR and RAPC are less pronounced - It appears difficult to very significantly outperform a well tuned simple controller (PID)

Outline Wind turbine models: Aero-servo-elastic model Reduced model Observers: Reduced model state observer Wind observer Control laws: Wind-Scheduled optimal PID MIMO Nonlinear-wind LQR RAPC, Reference Augmented Predictive Control Results Conclusions and outlook

Conclusions and Outlook Infrastructure for active control testing and development: ▶ Virtual testing environment: Developed a real-time environment for high-fidelity virtual testing of control schemes, based on CpLambda aero-servo-elastic models Ported all software in real-time, full software and hardware-in-the-loop capabilities in place ▶ Field testing: Hardware: pc-104, running Linux real-time OS Computing power (~ Pentium III) capable of supporting supervision, state and wind estimators, sophisticated active control laws, integrated diagnostics ▲ pc-104 hardware for supervision, active control and diagnostics Installed and fully integrated on-board an instrumented 1.5 MW wind turbine Tests in the field already started (PID, LQR only for now)

Conclusions and Outlook ▶ Analysis and development of control laws: Developed modifications of simple existing schemes (PID, LQR) Developed non-linear predictive adaptive controller: - Seems able to adapt quickly and correct even large model errors - No pre-training, fast convergence, thanks to reference elements - Double adaption level (model and control) Tests show importance of adaptive controllers, but primarily for gust response and turbulent off-design operations Work in progress: Implementation of RAPC with individual blade control based on SymDyn and periodic MIMO LQR reference controller Implementation of input and output constraints, will test load envelope-protection constraints on tower and blades