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Short Course on Wind Turbine
Modeling and Control- Part II: Control -
C.L. BottassoPolitecnico di Milano
Milano, Italy
Korea Institute of Machinery and Materials
&Kangwon National UniversityOctober 18-19, 2007
Short Course on Wind Turbine
Modeling and Control- Part II: Control -
C.L. BottassoPolitecnico di Milano
Milano, Italy
Korea Institute of Machinery and Materials
&Kangwon National UniversityOctober 18-19, 2007
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Control System Architectureontrol System Architecture
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Control System Architectureontrol System Architecture
Windindturbineurbine
SensorsPositions, speeds,accelerations, stresses,strains, temperature,electrical & fluidcharacteristics, etc.
SensorsPositions, speeds,accelerations, stresses,strains, temperature,electrical & fluidcharacteristics, etc.
SupervisorChoice of operating condition: Start up Power production Emergency shut-down
SupervisorChoice of operating condition:
Start up Power production Emergency shut-down Active control
systemControl strategy
Active controlsystemControl strategy
ObserversWind, tower & blades
ObserversWind, tower & blades
Wind farmsupervisorWind farmsupervisor
Communication and reportingommunication and reporting
ActuatorsActuator controlsystem
ActuatorsActuator controlsystem
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Supervisory Control Systemupervisory Control System
Main tasksain tasks : Operational managing and monitoring Diagnostics, safety
Communication, reporting and data logging
Operational statesperational states : Idling Start Up Normal power production Normal shut down
Emergency shut down
Main input dataain input data : Wind speed Rotor speed
Blade pitch Electrical power Temperatures in critical area Accelerations
but also Stresses, strains (blades, tower) Position, speed (yaw, blade, actuators, teeteringangle, rotor tilt, )
Fluid properties and levels Electrical systems (voltages, grid characteristics, ) Icing conditions, humidity, lighting,
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Supervisory Control Systemupervisory Control System
Idlingdling Power
production
Power
productiontart uptart up
Normal shut downormal shut down
Emergency
shut down
Emergencyshut down
V > V cut-inV > V cut-in RPM > cut-in
RPM > cut-in
V > V cut-offV > V cut-off
V < V cut-inV < V cut-in
Failures Overspeed & high rotor accel.
Vibrations
Failures Overspeed & high rotor accel.
Vibrations
Representative operational state monitoring logic:
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Basicasic wind turbine control strategies and power curvesower curves : Constant TSR strategy Constant rotor speed strategy Below and above rated speed control Variable speed pitch-torque regulated wind turbine Stall and yaw/tilt control
Control Strategiesontrol Strategies
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Control Strategiesontrol Strategies
C P
Power coefficient:
Tip speed ratio (TSR):
C P =P
1/ 2AV 3
= RV
C P = C P ( , ,Re,M )
C P
= const.
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C P
Control Strategiesontrol Strategies
Constant rotor speedonstant rotor speed : = const. =
V = R
V
C
P
V = R/
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C P
max
= const.
Direct grid connection:Generator provides whatever torque requiredto operate at or near given angular speed
V aero cut in =
R/ max < V cut in
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Control Strategiesontrol Strategies
Constant TSRonstant TSR : = const. = = V
R
P =12
AV 3C P
=
V
Constant rotorspeed strategy
Constant TSR strategy
= const. =
V
V r ( const. )
Indirect grid connection: Through power electronic converter Allows for rapid control of generator torque
V, C P ( const. ) C P ( const. )
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Control Strategiesontrol Strategies
V
P
P r
= Constant TSR strategy (cubic)
= Constant rotor speed strategy
= const.
Powerower -wind speed curveind speed curve :
V aero cut in ( const. ) V aero cut in ( const. )
Power deficit for constantspeed wrt constant TSR
V r ( const. )
V r ( const. )
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Control Strategiesontrol Strategies
P
P P
V V
V
1 2
=
= =
=
1
= 2
Constant rotor speed strategy 2 vs.onstant rotor speed strategy 2 vs.strategy 1trategy 1 : Higher cut in speed
Lower wind speed to reach rated power Smaller power deficit
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Control Strategiesontrol Strategies
P P
V V
= =
= 1
= 2
Annual energy yield: E = Y Z V
out
V inP (V )f w dV
f w f w
Weibull distribution
E
V
= 2
= 1Constant rotor speed strategy 2vs. strategy 1:
Smaller power deficit wrt toconstant TSR, but at improbablewind speeds Higher energy yield
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Control Strategiesontrol Strategies
Control above rated speedontrol above rated speed :
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V > V r
C P
C P
Constant power constant rotor speed curve(cubic)
C P = C
P
3
P = P r = 12AV 3C P ( , ) = const.
= = const.
T =P = const.
= R
V
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V
Control Strategiesontrol Strategies
P
V r
P r
=
V r
V V r
V V r
= V /RRegion 2 - below ratedspeed: constant TSRstrategy
Region 3 - above ratedspeed: constant powerstrategy
Below rated speed:torque control
Above rated speed:pitch control
Variableariable -speed pitch/torquepeed pitch/torqueregulated wind turbineegulated wind turbine :
T = 1 / 2ARV 2C P
/ T T = P r /
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C P
V cut outV cut in
Regio
1
No torque to promoterotor acceleration
V
Often, smoothing for mildertransition between regions
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V
Control Strategiesontrol Strategies
P
V r
P r =
Below rated speed:constant TSR strategy
Above rated speed:(roughly) constantpower strategy
Variableariable -speed passivepeed passive -stall/torquetall/torqueregulated wind turbineegulated wind turbine :
Stall region, highdispersion
V r
V V r
V V r
= V /R
Below rated speed:torque control
Above rated speed:torque-stall control
T T = P r /
V
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V
Control Strategiesontrol Strategies
P
V r
P r =
Below rated speed:
constant rotorspeed strategy
Above rated speed:
(roughly) constantpower strategy
Constant-speed passive-regulation wind turbine:
=
Stall region, highdispersion
V
P
V r
P r =
Below rated speed:
constant rotorspeed strategy
Above rated speed:
yaw or tilt rotor toreduce effective wind
=
Stall regulation Yaw/tilt out-of-the-wind
regulation
V
V cos
Rotor disk
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Further wind turbine control goalsurther wind turbine control goals : Fatigue damage reduction in turbulent wind Gust load alleviation Disturbance rejection Resonance avoidance
Actuator duty cycle reduction Periodic disturbance reduction (gravity, wind shear, tower shadow, )
Usually, these goals should be achieved together with the basic controlasic controlstrategiestrategies deriving from the power curves, i.e. Region 2: maximize energy capture Region 3: limit output power to rated value
Control Strategiesontrol Strategies
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Yaw Controlaw ControlOrient the rotor in line with the windn line with the wind field to increase powerNote: some small wind turbine will also yaw out of the wind to reduce loads in high winds
Passiveassive or free yaw, used in small wind turbines:
Activective yaw:- If V < V cut in = no action- If V > V cut in :
- Compute yaw error averaging over window (typically tens of sec.s) to reduce duty cycle- Region 2 = realign if yaw error > yaw threshold 2 (typically ~15 deg)- Region 3 = realign if yaw error > yaw threshold 3 (typically ~8 deg)- Realign at low yaw rate to reduce gyroscopic loads- If yaw error < small threshold (typically a fraction of a deg), engage yaw brake to eliminatebacklash between drive pinion and bull gear
Downwind rotorail fin
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R d d M d l f M d l B dReduced Models for Model Based
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Reduced Models for Model-BasedControllers
Reduced Models for Model-BasedControllers
Equationsquations : Drive-train shaft dynamics Elastic tower fore-aft motion Blade pitch actuator dynamics Electrical generator dynamics
Statestates :
Inputsnputs :
T el eT el c T l
J G
J R
c
e
T a
F a
M T , C T , K T
Non-linear collective-only reduced model:
c , T el c
d
d, d, , e , e , T el e
Reduced Models for Model BasedReduced Models for Model Based
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Reduced Models for Model-BasedControllers
Reduced Models for Model-BasedControllers
Equations of motionquations of motion :
Tip speed ratio: Wind: (mean wind + turbulence)
(J R + J G ) + T l ( ) + T el e T a ( , e , V w d, V m ) = 0M T d + C T d + K T d F a ( , e , V w d, V m ) = 0
e + 2 e + 2( e c) = 0
T el e + 1 (T el e T el c ) = 0
= R/ (V w d)
V w = V m + V t
Reduced Models for Model BasedReduced Models for Model Based
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Reduced Models for Model-BasedControllers
Reduced Models for Model-BasedControllers
Rotor force and moment coefficientsotor force and moment coefficients :
computed off-line with CpLambdapLambda aero-servo-elastic model, averaging periodic response over one rotor rev
Stored in look-up tables
T a =12
R3C P e ( , e , V m )
(V w d)2
F a =12 R2C F e ( , e , V m )(V w d)2
V m Dependence of and
on mean windaccounts for deformabilityeformability of towerand blades under high winds:
C P e ( , e , V m )C F e ( , e , V m )
C F e ( , e , V m ), C P e ( , e , V m )
Reduced Models for Model BasedReduced Models for Model Based
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Reduced Models for Model-BasedControllers
Reduced Models for Model-BasedControllers
Rigid body Beam Revolute joint Actuator Boundary condition
Blade
Generator
Nacelle inertia
Yaw
actuatorPitchactuator
Torqueactuator
Tower
Equivalenttowerstiffnesses
Equiv
en
sh
s
fne
Equivalent flap hingeand spring
Statestates : 3 flap angles (or blade modal amplitudes) Rotor azimuth Shaft torsion
3 tower angles (fore-aft, side-side, torsion)(or tower modal amplitudes)
Yaw angle
(and their rates)
Inputsnputs :
Examplexample : individual-pitch modelT el c
c1 c2
c3
c1 , c2 , c3 , T el c
Reduced Models for Model BasedReduced Models for Model Based
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Reduced Models for Model-BasedControllers
Reduced Models for Model-BasedControllers
Model linearizationodel linearization : needed for implementation of controllers(e.g. LQR) and model-based observers (e.g. Kalman filter)
Possible approaches: Analytical Automated (e.g. Maple, or directly from software usingAutomatic Differentiation tools like ADOL-C, ADIC, etc.) Numerical, by finite differences
P T
V V V VLinearization trim points
x = f (x , u , p) x = A (x , u , p ) x + B (x , u , p ) u u = u u x = x x
x u p
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Tower State Observerower State Observer
q = vv = ( T ) 1 T (a + n w )
Accelerometer
Strain gage
c = 00 q n v
q
n w , n v
Kalmanalman modalodal -based tower observerased tower observer :
Accelerations:
Curvatures: Unknown modal amplitudes: Modal bases: Process & measurement noise:
Remarksemarks : 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
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Tower State Observerower State Observer
State space form:
with
Optimal Kalman state estimate:
Filter gain matrix Propagated states and outputs based on accelerometric reading: Curvature reading:
x = Ax + Bu + W n wy = Cx + Du + V n v
x k = x
k + K k (y k y
k )
x = ( q T , v T )T u = a y = c
A = 0 I 0 0 B = 0 C = 00 0 D = 0 W =
0
V = I K k
x k , y
k
y k
b
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Tower State Observerower State Observer
Filter warm-up
Tower tip velocity estimation:
S Ob
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Tower State Observerower State Observer
Accelerometers
Strain gages
Kalmanalman modalodal -based tower and bladeased tower and bladestate observertate observer :
Compute or measure modal bases for
blades and tower
Integrate tower kinematic equationsfrom accelerations Correct with tower strain gage curvaturereadings Integrate blade kinematic equations
from blade and tower accelerations Correct with blade strain gage curvaturereadings
i d Obi d Ob
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Anemometer: Cup, but also laser, ultrasonic, etc. Measurements highly inaccurate because of
Rotor wake Wake turbulence Nacelle disturbance
Sufficient accuracy for supervision tasks and yaw alignment Not sufficient for sophisticated control law implementation
Need ways to reconstructeconstruct wind blowing on rotor from reliableeliablemeasurementseasurements (pitch setting, rotor speed, etc.)
Wind Observerind Observer
Wi d Obi d Ob
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Wind Observerind Observer
Extendedxtended Kalmanalman wind observerind 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 = nw
y = ( J R + J G ) + T l ( ) + T el e T a ( , e , V w d, V m ) + nv
x = f (x, u , n w )y = h(x, u , n v )x = V w
u= (
,
,
e ,
d, V m )T
xk = x
k + K k (yk y
k )
yk = 0
V m
Wi d Obi d Ob
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Wind Observerind Observer
Hub wind estimation:
Turbulent wind ( m/sec)V m = 15
EOG1-13 case
Wi d Obind Obser er
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Wind Observerind Observer
Simple mean hub wind reconstruction from torque balance equation
More in general:The rotor system is a sensorensor which responds to temporalemporal as well asspatialpatial wind variations
Model-based interpretation of response can be used for reconstructingvertical and horizontal wind shear for improved rotor control
Example: introduce spatial assumed modes and wind states
V (t, ) = V 0 (t) + V s (t)sin + V c(t)cos
Rotor disk
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Simulation Environmentimulation Environment
Control Laws: Virtual TestingControl Laws: Virtual Testing
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POLITECNICO di MILANO DIA
gEnvironmentEnvironment
Virtual plantirtual plant
Sensorensormodelsodels
CpLambdaCpLambda
aero-servo-elastic model
Windgenerator Processnoise
SupervisorupervisorChoice of operating condition:
Start up Power production Normal shut-down Emergency shut-down
Controller
Feedback controllereedback controller PID MIMO LQR RAPC Adaptive reduced model
Linux real-time environment
Kalmanalman filteringilteringWind & tower/blade state estimation
Measurementnoise
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P O L I
P O L I
d i
d i M I M I t e
c n
i c o
t e c n
i c o
l a
n o
l a
n o
Control Lawsontrol Laws
Control Laws: Three Case Studiesontrol Laws: Three Case Studies
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Case studiesase studies : PID: gain optimization and wind scheduling LQR: handling region 2-3 transition and wind scheduling Adaptive non-linear predictive control A simple LQR approach to cyclic pitch control
Control Laws: Three Case Studiesontrol Laws: Three Case Studies
Control Laws: Optimal PIDontrol Laws: Optimal PID
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Control Laws: Optimal PIDontrol Laws: Optimal PID
Optimal windptimal wind -scheduled PIDcheduled PID :
Tabulated electrical torque
Optimization of gains
based on aeroelastic analyses inCpLambda
c = K p(V m )( ) + K i (V m )
Z t
t
T i
( )d + K d (V m )
T el c = T el c ( )
K p(V m ), K i (V m ), K d (V m )
Control Laws: Optimal PIDontrol Laws: Optimal PID
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Control Laws: Optimal PIDontrol Laws: Optimal PID
Gain optimization procedureain 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
M eq T
, M eq Bi
J (V m ) = M eq T + M eq Bi =(1 , 3) +Z 600 sec0 w 2e + wd d2 + w ( )2 + wP (P P )2dt
w , wd , w , wP
M eq =
Xi
M mf,i N i /N tot
!1/m
Control Laws: Optimal PIDontrol Laws: Optimal PID
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Control Laws: Optimal PIDontrol Laws: Optimal PID
PID gain optimization procedureID gain optimization procedure (continued):
For each mean wind :
Regard cost as sole function of unknown gains
Minimize cost (using Noesis Optimus): Evaluate cost with CpLambdapLambda aero-servo-elastic model Global optimization (GA) Local refinement (Response Surface + gradient based minimization)
V m
J (V m ) = J (K p(V m ), K i (V m ), K d (V m ))
Optimizerptimizer Global & local algorithms Functional approximators
CpLambdapLambdaAeroelasticresponse inturbulent windfor given gains
K p (V m ) , K i (V m ) , K d (V m )
J (V m )(possible constraints)
Control Laws: MIMO NonLinear-Wind LQRontrol Laws: MIMO NonLinear-Wind LQR
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Control Laws: MIMO NonLinear Wind LQRontrol Laws: MIMO NonLinear Wind LQR
Windind -scheduled MIMO LQRcheduled MIMO LQR : Reduced model in compact form:
where Wind parameterized linear model:
where
Remarksemarks : 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 (x , u , V w , V m )
x = ( d,
d, , e ,
e , T el e )T
u = ( c , T el c )T
x = A (V w , V m ) x + B (V w , V m ) u
x = x x (V m ) u = u u (V m )
V mV w
Control Laws: MIMO NonLinear-Wind LQRontrol Laws: MIMO NonLinear-Wind LQR
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POLITECNICO di MILANO DIA
Control Laws: MIMO NonLinear Wind LQRontrol Laws: MIMO NonLinear Wind LQR
Windind -scheduled MIMO LQRcheduled MIMO LQR (continued):
Regulation cost:
where
MIMO formulation: tracking quantities for reg. 2 & 3:
J =12
Z
0
x T Q x + u T R u
dt
x (V m ), u (V m )
x = x x (V m ), u = u u (V m )
Control Laws: MIMO NonLinear-Wind LQRontrol Laws: MIMO NonLinear-Wind LQR
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Control Laws: MIMO NonLinear Wind LQRontrol Laws: MIMO NonLinear Wind LQR
Windind -scheduled MIMO LQRcheduled MIMO LQR (continued):
Closed loop controller:
with Kalman estimated states and wind
u = K (V w , V m )(x x (V m ))
Control Laws: NonLinear Adaptive Ctrl.ontrol Laws: NonLinear Adaptive Ctrl.
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Control Laws: NonLinear Adaptive Ctrl.ontrol Laws: NonLinear Adaptive Ctrl.
Design controller which: Can handle nonon -linearitiesinearities of plant Is adaptivedaptive :
- 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 constraintsonstraints (e.g. max loads in blades or tower) Can be implemented in realeal -timeime (no iterative scheme, fixednumber of operations per activation)
Nonon -linear modelinear model -adaptive predictive controldaptive predictive control
Control Laws: NonLinear Adaptive Ctrl.ontrol Laws: NonLinear Adaptive Ctrl.
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Control Laws: NonLinear Adaptive Ctrl.pNonon -linear Model Predictive Controlinear Model Predictive Control (NMPC):
Find the control action which minimizes an index of performance,by predicting the future behavior of the plant using a nonon -linearinearreduced modeleduced model .
- Reduced model:
- Initial conditions:- Output definition:
Cost:
with desired goal outputs and controls.
Stability resultstability results : Findeisen et al. 2003, Grimm et al. 2005.
L(y , u ) = ( y y
)T
Q (y y
) + ( u u
)T
R (u u
)
minu ,x ,y
J = Z t 0 + T p
t 0L(y , u ) d t
s.t.: f (x , x , u ) = 0 t [t0 , t 0 + T p]
x (t0) = x 0y = g (x ) t [t0 , t 0 + T p]
()
Control Laws: NonLinear Adaptive Ctrl.ontrol Laws: NonLinear Adaptive Ctrl.
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FuturePast
Steering window
Prediction window
Prediction errorState tracking error
Control tracking error
t0
t0 t0 + T p
t0 + T s
Computed control u (t)Goal control u (t)
: p .p
Goal response x (t)Predicted response x (t)
Plant response
ex (t)
x 0
Control Laws: NonLinear Adaptive Ctrl.ontrol Laws: NonLinear Adaptive Ctrl.
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pp
1. Tracking problem
Plant response
3. Reduced model update
Predictive solutions
2. Steering problem
Prediction window
Steering window
Tracking cost
Prediction error
Prediction window
Tracking cost
Steering windowPrediction error
Tracking costPrediction window
Steering window
Prediction error
Goal
response
Predictive modelredictive model -adaptive controldaptive control :
Reduced modeleduced model adaptiondaption : 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.)
Past Futureast Futureast Future
Past Future
Past Future
RAPC: MotivationAPC: Motivation
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POLITECNICO di MILANO DIA
For any given problem: wealth of knowledgenowledge and legacyegacy methodswhich perform reasonably well Quest for better performance/improved capabilities: undesirablendesirable
and wastefulasteful to neglect valuable existing knowledgeReference Augmented Predictive Controleference Augmented Predictive Control (RAPCAPC): exploit availablelegacy methods, embedding them in a non-linear model predictiveadaptive control framework
Specifically: Modelodel : augment reduced models to account for unresolved orunmodeled physics Controlontrol : design a non-linear controller augmenting linear ones(MIMO Nonlinear-Wind LQR) which are known to provide a minimumlevel of performance about certain linearized operating conditions
RAPC: MotivationAPC: Motivation
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Approach:pproach: Choose a reference model / reference control laweference model / reference control law Augment the reference using an adaptivedaptive parametric functionarametric function Adjustdjust the function parameters to ensure good approximation ofood approximation ofthe actual system / optimal control lawhe actual system / optimal control law (parameter identification)
Reasons for using a reference model / controleasons for using a reference model / control: Reasonable predictions / controls even before any learningven before any learning hastaken place (otherwise would need extensive pre-training) Easier and faster adaption: the defect is typically a small quantitymall quantity ,if the reference solution is well chosen
RAPC: Reduced Model IdentificationAPC: Reduced Model Identification
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Plantlant
u , V w u , V w
Reduced modeleduced modelAugmentedugmentedreduced modeleduced model
+ Neural NetworkNeural Network
Dissimilaroutputs
Same wind,same inputs u , V w
Same wind,same inputs
Similaroutputs
Trained on-lineto minimizemismatch
x
ex x
ex
The principle of reference model augmentationeference model augmentation :
RAPC: Reduced Model IdentificationAPC: Reduced Model Identification
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Neural augmented reference modeleural augmented reference model:reference (problem dependent) analytical model,
Remarkemark : reference model will notot , in general, ensure adequatedequatepredictions, i.e.
when = system states/controls,
= model states/controls.Augmented reference model:ugmented reference model:
where is the unknownnknown reference model defectefect that ensureswhen i.e.:
Hence, if we knewf we knew , we would have perfect predictionerfect prediction capabilities.
d
d
eu = u ,
ex 6= x
x , uex ,
eu
ex = x
f ref (x , x , u ) = 0
f ref (x , x , u ) = d (x , u )
eu = u
T el eT el eT el cT el c T lT l
J GJ G
J RJ R
c c
e e
T aT a
F aF a
M T , C T , K T M T , C T , K T
dd
Referencereduced model
f ref (
ex ,
ex ,
eu ) d (
ex ,
eu ) = 0
RAPC: Reduced Model IdentificationAPC: Reduced Model Identification
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Approximate withsingleingle
-hiddenidden
-layer neural networksayer neural networks :
where
and
= functional reconstruction error;
= matrices of synaptic weights and biases;= sigmoid activation functions;
= network input.
The reduced model parameterseduced model parameters
are identified on-line using an Extendedxtended Kalmanalman Filterilter .
d
( ) = ( (1), . . . , (N n ))T
W m , V m , a m , bm
i = ( x T , u T )T
pm = ( . . . , W m ik , V m ik , a m i , bm i , . . . )T
d (x , u ) = d p(x , u , pm ) +
d p(x , u , pm ) = W mT
(V mT
i + a m ) + bm
RAPC: Reduced Model IdentificationAPC: Reduced Model Identification
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Tower-tip velocity for multibody, reference, and neural-augmentedreference with same prescribed inputs:
Black: CpLambdamultibody model
Red: reference model
Blue: reference model+neural network
Fastast adaptiondaption
RAPC: Reduced Model IdentificationAPC: Reduced Model Identification
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Defect and remaining reconstruction error after adaption:di i
Red: defect
Blue: remainingreconstruction error
RAPC: Neural ControlAPC: Neural Control
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FuturePast
Prediction window
x 0
t0
t0
t0
+ T p
Goal response x
(t)
Goal control u (t)
x (t), t < t 0
u (t), t < t 0
The principle of neuraleural -augmented reference controlugmented reference control :
Optimal solution u NMPC (t)
Optimal solution x (u NMPC (t))
Sub-optimal solution u ref (t)
Sub-optimal solution x (u ref (t))
Augmented sol. x (u ref (t) + NN )
Augmented sol. u ref (t) + NN
RAPC: Neural ControlAPC: Neural Control
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Prediction problemrediction problem :
Enforcing optimalitynforcing optimality , we get:
minu ,x ,y
J = Z t 0 + T p
t 0L(y , u ) d t
s.t.: f (x , x , u ) = 0 t [t0 , t 0 + T p]
x (t0) = x 0y = g (x ) t [t0 , t0 + T p]
f (x , x , u , pm ) = 0 , t [t0 , t 0 + T p],x (t0) = x 0 ,
d( f T
,x )
dt + f T ,x + y T ,x L ,y = 0 , t [t0 , t 0 + T p], (t0 + T p) = 0 ,
L ,u + f T ,u = 0 , t [t0 , t 0 + T p].
Model equations:
Adjoint equations:
Transversality conditions:
State initial conditions:
Co-state final conditions:
RAPC: Neural ControlAPC: Neural Control
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FuturePast
Prediction window
x 0
t0
t0
t0
+ T p
Goal response x (t)
Goal control u (t)
u (t)
Optimal control u (t)
x (t), t < t 0
u (t), t < t 0
(, , , )
It can be shown that minimizing controlinimizing control is(Bottasso et al. 2007)
u (t) =
x 0 , y
(t), u (t), t
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Reference augmented form:eference augmented form:
where is the unknown control defect.
Remarkemark : if one knew , the optimal control would be availablewithout having to solve the open-loop optimal control problem.
Ideadea :- Approximatepproximate using an adaptive parametric element:
- Identifydentify on-line, i.e. find the parameterswhich minimize the reconstruction error .
pc
u (t) = u ref (t) + x 0 , y (t), u (t), t (, , , ) (, , , )
(, , , )
x 0 , y
(t), u (t), t
= p
x 0 , y
(t), u (t), t, p c
+ c
p(, , , )
RAPC: Neural ControlAPC: Neural Control
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Iterative procedureterative procedure to solve the problem in real-time: Integrate reduced model equations forwardorward in time over theprediction window, using and the latest available parameters(state prediction):state prediction):
Integrate adjoint equations backwardackward in time (coco -state prediction):tate prediction):
Correctorrect control law parameters , e.g. using steepest descent: pc
u ref p c
p c = J ,p c p newc = p oldc J ,p c
d( f T , x )
dt+ ( f ,x + u T ,x f ,u )
T + y T ,x L ,y + uT ,x L ,u = 0 t [t0 , t 0 + T p]
(t0 + T p) = 0
f (x , x , u , pm ) = 0 t [t0 , t0 + T p]x (t0) = x 0
RAPC: Neural ControlAPC: Neural Control
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Remarkemark : the parameter correction step
seeks to enforce the transversalityransversality conditionondition
Once this is satisfied, the control is optimalptimal , since the state and co-state equations and the boundary conditions are satisfiedatisfied .
pc = J ,p c
Z t 0 + T p
t0
T ,p c (L ,u + f T ,u ) d t = 0
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Tr ac k i n g c o s t
Fu tu re Targe t
Predictredict state forward
Predictredict co-state backwards
Predictredict control action
p c = J ,p c
Updatepdate estimate of control action, based on transversality violation Advancedvance plant Updatepdate model, based on prediction error
Past
Op t i m a l c o n t r o l
Pr e d i c t i o n e r r o r
Repeatepeat
Futu rePast
Pred ic t ion hor i zonSteer i ng w indow
S t a t e
Con t ro l
x (t)
(t)
u (t)
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- Drop dependence on time history of goal quantities:
- Approximate temporal dependence using shape functions:
- Associate each nodal value with the output of a singleingle -hiddenidden -layerayerfeed-forward neural networkeural network , one for each component:
where
Output:
Input:
Control parameters:
px 0 , y (t), u (t), t, pc px 0 , y (t0), u (t0), t, pc px
0 , y
(t0), u
(t0), , pc
(1 ) pkx 0 , y (t0), u (t0 ), pc+ pk +1 x 0 , y (t0), u (t0), p co c = W T c (V
T c i c + a c) + bc
o c = ( T p0, T p
1, . . . , T p
M
1)T
i c = x T 0 , x T
(t0), u T
(t0)T
pc = ( . . . , W c ij , . . . , V c ij , . . . , a c i , . . . , bc i , . . . )T
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FuturePast
Prediction window
x 0
t0
t0 t0 + T p
x (t), t < t 0
u (t), t < t 0
u (t0)
x (t0)NN pk
x (t)
u (t)
RAPCAPC
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RAPC can handle constraints on inputs and outputs (not covered inthis paper)
Present resultsresent results : Reference model: collective-only, Reference controller: MIMO Nonlinear-Wind LQR
Work in progressork in progress : Reference model with individual blade pitch, flap dynamics
Reference controller: periodic MIMO Nonlinear-Wind LQR Constraints on inputs and outputs
x = ( d, d, , e , e , T el e )T
Resultsesults
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Two consecutive EOG 1-13 in nominal conditions:
ResultsesultsNormalized total regulation error in 600 sec turbulent wind
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Normalized total regulation error in 600 sec turbulent wind Cold air & ice accretion (degraded airfoil performance):
Resultsesults
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Observationsbservations : Significant advantage of model-based (especially non-linear andadaptive) 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 lesspronounced
- It appears difficult to very significantly outperform a well tunedsimple controller (PID)
Cyclic Pitch Controlyclic Pitch Control
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Case studyase study : a simple LQR approach to cyclic pitch controlConsider individual-pitch model
where
= rotor azimuth
= all other states
Model linearization:
Remarkemark : azimuth dependent coefficient matrices
x = ( , x T )T
x
x = f (x , u , p) = f ( , x , u , p)
x = A ( , x , u , p ) x + B ( , x , u , p ) u
Cyclic Pitch Controlyclic Pitch Control
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Possible approachesossible approaches : Full state feedback:
a) Integrate Riccati eq. until periodic solution to obtain
optimal periodic feedback gain matrixb) Solve steady Riccati eq. for severalthen interpolate resulting gain matrices
c) Average periodic coefficient matrices over one revolution
solve steady Riccati eq. to get averaged gain matrix Output feedback: a), b) or c), but governing eq. more complexthan Riccati eq., approach a) complicated
K ( , x
, p
) i , 0 i 2
K ( i , x , p )
bA (x , p ) = (1 / 2 )Z
2
0A ( , x , p ) d
bB (x , p ) = (1 / 2 )
Z 2
0
B ( , x , p ) d
cK (x , p )
Full state feedback collective pitch vs. individual pitch LQR
Cyclic Pitch Controlyclic Pitch Control
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p p Q
Steady wind, wind shear, tower shadow, rotor up-tilt
Observationsbservations : Very similar behavior for a) and b) strategies, c) slightly worst Significant peak-to-peak reduction for cyclic control, at the cost ofincreased duty cycle
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P
O L I
P
O L I
d i
d i M I M I t
e c n i c
o
t e c n i c
o
l a n o
l a n o
Hardware Implementationardware Implementation
Control System Hardwareontrol System HardwareDecentralized PC/PLC based architecture
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Main controller
Remote visualization www access
Decentralizedcontrol module
Pitch regulator
Ethernet Wireless,ADSL
Realtime fiber optic network(FAST-Bus, Profibus, Ethernet)
CAN-Bus, RS485
Ethernet
Remote visualization
Decentralized PC/PLC based architecture
Control panel
Slip-ring orwireless bridge
RIO = Reconfigurable I/O PLC = Programmable Logic Controller PROFIBUS = Process Field Bus CAN BUS = Controller Area Network RS485 = Serial communication
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PoliMi Control Research PlatformoliMi Control Research PlatformPLC-based decentralizedcontrol module cabinet
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Leitwind 1.2 MW Wind Turbine Hub height 65m Rotor radius 38m
control module cabinet
PC/104 architecture, Pentium M 1.6 GHz Linux real-time operative system
Hardwareardware for supporting researchesearch and fieldieldtestingesting on advanced control laws, state andwind estimators, integrated diagnostics
V l i Ch t h PC104 SBC ith I t l
PoliMi Control Research PlatformoliMi Control Research Platform
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Data acquisition module 16-bit A/D
Versalogic Cheetah PC104 SBC with IntelPentium M 1.6 GHz and Extreme Graphics 2Video (-40 to +60C), 2 configurable serialports, 1 Ethernet interface, 2 usb ports
HE104 High Efficiency PowerSupply 50 Watt, +5V@10A,+12V@2A, -40 to +85C
Hard disk 44 pin (replaceablewith a solid state disk)
Internal communicationPC/104 bus
To servos:From sensors:
PoliMi Control Research PlatformoliMi Control Research Platform
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Torque Rotor speed Azimuth Blade pitch angle Wind
Serial communication RS485 @1Hz
Pitch control Torque control
Pitch, yaw, torquesetpoints
Anemometer, inverter,pitch regulator, yaw
Analog inputs:
Tower accelerations andstrain gauges
Collect data, interface withservos, compute yaw control
Controller and observer algorithms,interface with on-board industrialcontroller
Complete compatibility with andminimum impact on existing on-boardsystem Substantial computing power On-board system can give control toand regain control from researchplatform at any time
Referenceseferences
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Wind Turbines Part 1: Design Requirements, IEC 61400-1, 2005
Manwell J.F., McGowan J.G., and Rogers A.L., Wind Energy Exp lained: Theory, Design andApplication , John Wiley & Sons, New York, NY, 2002
Burton T., Sharpe D., Jenkins N., and Bossanyi E., Wind Energy Handbook , John Wiley & Sons, New
York, NY, 2001Stol K.A., and Fingersh L.J., Wind Turbine Field Testing of State-Space Control Designs, NREL/SR-500-35061, 2003
Findeisen R., Imland L., Allgower F., and Foss B., State and Output Feedback Nonlinear ModelPredictive Control: An Overview, European Journal of Control , 9:190206, 2003
Fausett L., Fundamentals of Neural Networks , Prentice-Hall, New York, 1994