Scheduled Model Predictive Scheduled Model Predictive Control of Wind turbines in Control of Wind turbines in

Above Rated WindAbove Rated Wind

Avishek Kumar

Dr Karl Stol

Department of Mechanical Engineering

2

OverviewOverview

3

BACKGROUNDBACKGROUND

4

Horizontal Axis Wind TurbinesHorizontal Axis Wind Turbines

Source: US Department of Energy

5

Control ObjectivesControl Objectives

Speed controlSpeed control Maintain rated rotor speed in above rated Maintain rated rotor speed in above rated

windswinds

Load controlLoad control Oscillations occur in the Low Speed Shaft Oscillations occur in the Low Speed Shaft

(LSS)(LSS) Reduce loads in LSSReduce loads in LSS

6

Turbine NonlinearitiesTurbine Nonlinearities

),(2

14

32 xCVRP pwrr

w

rr

V

R

7

Model Predictive ControlModel Predictive Control

Choose the control input trajectory that will Choose the control input trajectory that will minimize a cost function over the minimize a cost function over the prediction horizon prediction horizon HHpp

Example:Example:

maxmin

maxmin

:subject to

min

uuu

xxx

uuxxu

RQJ TH

k

Tp

8

Why MPC?Why MPC?

Accommodate disturbancesAccommodate disturbances

MIMOMIMO

ConstraintsConstraints

Many cost functionsMany cost functions

Can extend to nonlinear systemsCan extend to nonlinear systems

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Current State of MPC for Current State of MPC for Wind TurbinesWind Turbines

MPC using linear models of turbine (LMPC)MPC using linear models of turbine (LMPC) Lacks ability to deal with system nonlinearitiesLacks ability to deal with system nonlinearities

MPC using nonlinear models of turbineMPC using nonlinear models of turbine Difficult to increase order of model as explicit Difficult to increase order of model as explicit

nonlinear equations become very complexnonlinear equations become very complex Computationally expensiveComputationally expensive

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Bridging the GapBridging the Gap

Scheduled MPC (SMPC)Scheduled MPC (SMPC)

Uses a network of linear models easily obtained Uses a network of linear models easily obtained from linearization codes (FAST)from linearization codes (FAST)

Optimization remains convex for each controllerOptimization remains convex for each controller

Controllers can be specifically tuned at various Controllers can be specifically tuned at various operating points to operate with different aimsoperating points to operate with different aims

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ObjectivesObjectives

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MPC OVERVIEWMPC OVERVIEW

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Constrained Linear Constrained Linear Quadratic RegulatorQuadratic Regulator

Up till now, MPC has been posed as a Up till now, MPC has been posed as a finite horizonfinite horizon problem problem

For better performance set up MPC as an For better performance set up MPC as an infinite infinite horizon problemhorizon problem

This allows LQR control with constraintsThis allows LQR control with constraints

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Infinite Horizon Cost Function for Infinite Horizon Cost Function for CLQRCLQR

ki

Tki

ikik

Tkik uRuxQxJ |1|1

0|1|1

kiHkTkiHk

ikiHk

TkiHk

kikTkik

H

ikik

Tkik

pppp

p

uRuxQxJ

uRuxQxJ

JJJ

||0

|1|12

||

1

0|1|11

21

pp HkHk

T xPxJ 2

pp

p

HkHkT

ikTik

H

iik

Tik xPxuRuxQxJ

1

011

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Constrained Linear Constrained Linear Quadratic RegulatorQuadratic Regulator

Design a LQR for the linear system giving Design a LQR for the linear system giving predictions:predictions:

)(

|1|

|1|

|1|

kikkik

kikkik

kikkik

xx

xBKAx

xKu

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Constrained Linear Constrained Linear Quadratic RegulatorQuadratic Regulator

Create a MPC to calculate perturbations Create a MPC to calculate perturbations cc about control input given by the LQR about control input given by the LQR onlyonly over over HHp p so constraints are met so constraints are met

|1|

|1|1|

|1|1|

kikkik

kikkikkik

kikkikkik

xx

cBxx

cxKu

p

p

p

Hi

Hi

Hi

...2 ,1

...2 ,1

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CLQR MinimizationCLQR Minimization

maxmin

maxmin

maxmin

1

011

:subject to

min

uuu

uuu

uuu

c

pp

p

HkHkT

ikTik

H

iik

Tik xPxuRuxQxJ

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CLQR Block DiagramCLQR Block Diagram

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Scheduled MPCScheduled MPC

Create a network of MPCs at enough Create a network of MPCs at enough operating points to capture nonlinearities operating points to capture nonlinearities of systemof system

Tune each controller for the region it Tune each controller for the region it operates inoperates in

Weight the outputs of each controller Weight the outputs of each controller based on scheduling variablebased on scheduling variable

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SMPC Block DiagramSMPC Block Diagram

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ModelModel

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Linear Model for Control Linear Model for Control Design/Disturbance EstimationDesign/Disturbance Estimation

op

op

uuu

xxx

uBxAx

Speed Wind

errorpower Integral

ratepitch Bladepitch BladeTorqueGenerator

rate twist DrivetrainspeedRotor

twistDrivetrain

positionazimuth Rotor

VerrorP

T

r

r

xg

pitch Blade Commanded

torqueGenerator Commanded

,,

c

Tu cg

23

Nonlinear Model for EKFNonlinear Model for EKF(7)

where

0

1

1

),,(

)(

5

4

21

532

21

321

1

41

x

x

N

xx

J

x

JN

Kx

JN

Dx

NJ

DxJ

Kx

NJ

Dx

J

Dx

Jx

VxxP

xf

T

g

ggg

s

gg

s

gg

s

r

s

gr

s

r

s

r

wr

uxgxfx )()(

T

xg

1

0

01

00

00

00

)(

cg

c

Tu

,

V

T

x

g

g

r

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WIND TURBINE CONTROL WIND TURBINE CONTROL DESIGNDESIGN

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Baseline ControllersBaseline ControllersGSPIGSPI

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Baseline ControllersBaseline ControllersCLQRCLQR

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Scheduled MPCScheduled MPC

Linearization Point

1 2 3

Wind Speed (Vi0)

14ms-1 18ms-1 22ms-1

Blade Pitch 2.2° 11.1° 16.1°

Generator Torque

3524Nm 3524Nm 3524Nm

Rotor Speed 41.7rpm 41.7rpm 41.7rpm

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Scheduled MPCScheduled MPC

kCLQRk

kCLQRkCLQRk

kCLQRkCLQRk

kCLQRk

uu

VV

uuu

VV

uuu

uu

,3

02

03

,3,2

01

02

,2,1

,1

4/)(

)1(

4/)(

)1(

V

V

V

V

1

11

11

1

ms22

ms22ms18

ms18ms14

ms14

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Scheduled MPCScheduled MPC

V̂

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SimulationsSimulations

Simulations conducted in MATLAB/Simulink with Simulations conducted in MATLAB/Simulink with FAST modelFAST model

Active DOFActive DOF Blade flap (modes 1 and 2)Blade flap (modes 1 and 2) Blade EdgewiseBlade Edgewise TeeterTeeter Tower fore-aft (mode 1 and 2)Tower fore-aft (mode 1 and 2) DrivetrainDrivetrain GeneratorGenerator Tower side-sideTower side-side

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Wind InputsWind Inputs

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Performance CriteriaPerformance Criteria

Rotor Speed RMS ErrorRotor Speed RMS Error

Low Speed Shaft Damage Equivalent LoadLow Speed Shaft Damage Equivalent Load

RMS Pitch AccelerationRMS Pitch Acceleration

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TuningTuning

Each SMPC controller tuned to have same Each SMPC controller tuned to have same speed control as GSPI in respective low speed control as GSPI in respective low turbulence windturbulence wind

Each SMPC controller tuned to have same Each SMPC controller tuned to have same LSS load control as CLQR in respective LSS load control as CLQR in respective low turbulence windlow turbulence wind

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RESULTSRESULTS

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ConstraintsConstraints

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Speed ControlSpeed Control

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LSS DELLSS DEL

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Pitch AccelerationPitch Acceleration

39

ConclusionsConclusions

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Future WorkFuture Work

Questions?Questions?

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Nonlinear ModelNonlinear Model(7)

where

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Extended Kalman FilterExtended Kalman Filter

FL design needs FL design needs accurate accurate wind speed wind speed estimateestimate

Extended Kalman Filter (EKF) is a Extended Kalman Filter (EKF) is a nonlinear state estimatornonlinear state estimator

Sub optimalSub optimal

Linearizes the system model each time Linearizes the system model each time step, then estimates states like a linear step, then estimates states like a linear Kalman FilterKalman Filter

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Choosing HpChoosing Hp