# Modelling and Optimal Control of Tethered Airfoils for Wind by murplelake83

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```									Modelling and Optimal Control of Tethered
Airfoils for Wind Power Generation

Moritz Diehl
Optimization in Engineering Center (OPTEC) & ESAT
K.U. Leuven, Belgium

Symposium on Control and Modelling of Alternative Energy Systems,
Urbana-Champaign, April 2-3, 2009
Optimization in Engineering Center OPTEC
Center of Excellence of K.U. Leuven, since 2005
About 25 professors, 10 postdocs, and 50 PhD students involved in OPTEC
research

Promoted by four departments:
Electrical Engineering
Mechanical Engineering
Chemical Engineering
Computer Science

Many real world applications at OPTEC...
Aim: Connect Optimization Methods and Applications
Applications: Smart
problem formulations allow
efficient solution (e.g.
convexity)

Methods: New developments
are inspired and driven by
application needs
Overview

Idea: Wind Power Generation by Fast Flying Kites

Optimal Periodic Orbits

Stabilization by Nonlinear Model Predictive Control (NMPC)

Do Open-Loop Stable Periodic Orbits exist?

Status of Real-World Implementation
Conventional Wind Turbines

Due to high speed, wing tips are most
efficient part of wing
High torques at wings and mast limit size
and height of wind turbines
But best winds are in high altitudes!

Could we construct a wind turbine
with only wing tips and generator?
Conventional Wind Turbines

Due to high speed, wing tips are most
efficient part of wing
High torques at wings and mast limit size
and height of wind turbines
But best winds are in high altitudes!

Could we construct a wind turbine
with only wing tips and generator?
Crosswind Kite Power (Loyd 1980)

Fly kite fast in crosswind direction
Very strong force (100x higher than static
kite)

But where could a generator be driven?
Variant 1: On-Board Generator
attach small wind turbines to kite
due to high relative wind speed, can be
small
optimally increase system drag by 50%
lines must transmit power

Pros: fast spinning generators
Cons: heavy, moving parts on board,
electric cables, conversion losses
Variant 2: New Pumping Cycle
New cycle consists of two phases:
Power generation phase:
• Fly kite fast, have high force
• unwind cable (!!)
• generate power

Retraction phase:
• Slow down kite, reduce force
• pull back line

Cycle produces same power as (gigantic)
windmill of same wing area
Variant 2: New Pumping Cycle
New cycle consists of two phases:
Power generation phase:
• Fly kite fast, have high force
• unwind cable (!!)
• generate power

Retraction phase:
• Slow down kite, reduce force
• pull back line

Pros: all heavy and electric parts on ground

Can generate same power as (gigantic) wind
turbine of same wing area
Can stack kites, can use on sea
High Altitude Wind Power investigated by many groups

TU Delft: Wubbo Ockels, …
U Sussex: Allister Furrey
K.U. Leuven: MD, J. Swevers, …

AmpyxPower: R. Ruiterkamp, B. Lansdorp
KiteGen: M. Milanese, M. Ippolito, …
SkySails
NTS
makani
JobyEnergy
…
Many Important Questions:

Cable length? 100 m, 1 km, 10 km ?

Cable strength? (drag vs. elastic losses)

Shall kites fly high, or close to ground?

How long should each phase be?

How fast to roll in and out cable?

What are the optimal flight figures?

How to control?                             (old illustration of TU Delft‘s „laddermill“)
Five Claims [Kite Workshop, January 2007]

The future kite power plants will look as follows:

Kite lines will be far from vertical, kites fly at low angles
Lift control will play crucial role
Kites will be “pumping” rather than turning a carousel
Plants will be built rather on sea than on land
Connection to ground by only one line, not two or more
Overview

Idea: Wind Power Generation by Fast Flying Kites

Optimal Periodic Orbits

Stabilization by Nonlinear Model Predictive Control (NMPC)

Do Open-Loop Stable Periodic Orbits exist?

Status of Real-World Implementation
Kite Model for Optimal Control (Boris Houska)
Have to regard also cable elasticity

ODE Model with 12 states and 3 controls

forces at kite (here:                Control inputs:
500 m2)                             line length
roll angle (as for toy kites)
lift coefficient (pitch angle)
Some Kite Parameters
e.g. 10 m x 50 m,
like Boeing wing,
but much lighter
material

standard wind velocity for
nominal power of wind
turbines
Periodic Optimization Problem
Maximize mean power production:

by varying line thickness, period duration,
controls,    subject to periodicity and other
constraints:
Solve by Bock’s Direct Multiple Shooting Method [Bock&Plitt,’84]
implemented in MUSCOD-II[Leineweber‘99] or ACADO[Ferreau, Houska’09]
Solution of Periodic Optimization Problem
Maximize mean power production:

by varying line thickness, period duration,
controls,    subject to periodicity and other
constraints:

Cable length 1.3km, thickness 7 cm
Periodic Orbit: 5 MW mean power production
2 x 500 m2 airfoils
Optimal ‚Dancing‘ Kites: 14 MW possible

2 x 500 m2 airfoils
kevlar line 1500 m, diameter 8 cm
wind speed 10 m/s
Overview

Idea: Wind Power Generation by Fast Flying Kites

Optimal Periodic Orbits

Stabilization by Nonlinear Model Predictive Control (NMPC)

Do Open-Loop Stable Periodic Orbits exist?

Status of Real-World Implementation
Problem: kite orbits unstable. What to do?
Feedback by Nonlinear Model Predictive Control (NMPC)

Always look a bit into the future!

Controller predicts and optimizes:
e.g. slow down before curve
Principle of Optimal Feedback Model Predictive Control (MPC)
Computations in Control / Nonlinear MPC:

x0

u0

x0

u0

Main challenge for MPC: fast and reliable real-time optimization
Fast MPC Group Leuven -
Software and Applications
Our aim: make fastest possible online optimization codes

Our two open source MPC “products”:

qpOASES for linear MPC
qpOASES: Tailored QP Solver

Solve p-QP via „Online Active Set Strategy“:

go on straight line in parameter space
from old to new problem data

solve each QP on path exactly (keep
primal-dual feasibility)

Update matrix factorization at boundaries
of critical regions

Up to 10 x faster than standard QP

qpOASES: open source C++ code by Hans Joachim Ferreau
The Predictive Prefilter: a 100 Hz Application
Quarter car: oscillating spring
damper system
MPC Aim: settle at any new
setpoint in in minimal time
Re-Formulate with L1-slacks as
p-QP:
6 online data
30 variables
180 constraints (in-&output)
use qpOASES on dSPACE
CPU time: 0.5 – 2.9 ms

Lieboud Van den Broeck in front of
quarter car experiment
Slow Motion Video of qpOASES content
qpOASES running on Industrial Control Hardware (20 ms)

Project manager (Dec. 2008): “…we had NO problem at all
with the qpOASES code. Your Software has throughout the
whole project shown reliable and robust performance.”

A Toolkit for „Automatic Control and Dynamic Optimization“

C++ code along with user-friendly Matlab interfaces
Open-source software (LGPL 3)
Since mid 2008 developed at OPTEC by
Boris Houska and Hans Joachim Ferreau

Problem Classes:                 Discretization Methods:
• Optimal control                 • Single shooting
• State & parameter estimation    • Multiple shooting
• Robust optimization             • Collocation
• Model predictive control
Integrators:
• RKF and BDF methods
Dynamic Optimization:
• Efficient sensitivity generation
• Linear and Nonlinear
• Second order sensitivities
• ODE and DAE
• Continuous and discrete time   NLP solvers:
• Automatic differentiation       • Adjoint-based SQP
• Convexity detection             • Interior point methods
NMPC of Wind Power Generating Kites
(with Boris Houska, A. Ilzhoefer, K. Evers, B. Struyven...)
NMPC of Large Kite Systems (Andreas Ilzhoefer)

Track Reference. Predict 20 secs. Sample time: 1 sec
Use real-time iteration scheme within MUSCOD-II
Combine with Moving Horizon Estimator (MHE)
Computation always < 1s:            Closed-loop simulation:
Kite NMPC Problem solved with ACADO

9 states, 3 controls
Penalize deviation from “lying eight”
Predict half period
zero terminal constraint
10 multiple shooting intervals

Solve with SQP real-time iterations
Kite NMPC: CPU Time per RTI below 50 ms

Initial-Value Embedding            : 0.03 ms
QP solution (qpOASES)               : 2.23 ms
-----------------------------------------------------------
Feedback Phase:                               3 ms
(QP after condensing: 30 vars. / 240 constr.)

Expansion of the QP                : 0.10 ms
Simulation and Sensitivities : 44.17 ms
Condensing (Phase I)               : 2.83 ms
-----------------------------------------------------------
Preparation Phase:                         47 ms

(on Intel Core 2 Duo CPU T7250, 2 GHz)
NMPC Control with large turn of wind direction
Overview

Idea: Wind Power Generation by Fast Flying Kites

Optimal Periodic Orbits

Stabilization by Nonlinear Model Predictive Control (NMPC)

Do Open-Loop Stable Periodic Orbits exist?

Status of Real-World Implementation
Question: Could kite also fly WITHOUT feedback?

Stability just by smart choice of open-loop controls?
Monodromy Matrix determines stability

„Monodromy matrix“ = Jacobian of Poincare map.

Stability   Spectral radius of monodromy matrix smaller one

nonsmooth criterion, difficult for optimization
constraints and uncertainty of parameters not taken into account
Alternative Approach (Houska, D. 2007)

Regard linearized propagation of noise:

Compute covariance matrix P by
Lyapunov Equation:

Infinitely long time: covariance blows up, or
becomes periodic

THEOREM: If periodic Lyapunov solution
exists (with     ), system is stable.
New robust stability optimization problem

Allows to robustly satisfy inequality constraints!

Numerical challenges: higher order sensitivities, large scale ODEs,
non-smooth constraints, relaxations for infeasible initializations, ...
Orbit optimized for power and robust stability

Kite never touches
ground: inequalities
robustly satisfied

We have GENERATED a stable attractor!
Only possible due to nonlinearity
Overview

Idea: Wind Power Generation by Fast Flying Kites

Optimal Periodic Orbits

Stabilization by Nonlinear Model Predictive Control (NMPC)

Do Open-Loop Stable Periodic Orbits exist?

Status of Real-World Implementation
Prototypes built in Hamburg, Delft, Torino, California, …
The groups known to the speaker

TU Delft: Wubbo Ockels, …
U Sussex: Allister Furrey
K.U. Leuven: MD, J. Swevers, …

AmpyxPower: R. Ruiterkamp, B. Lansdorp
KiteGen: M. Milanese, M. Ippolito, …
SkySails
NTS
makani
JobyEnergy
…
Leuven Kite Power Group

Build toy prototypes to validate
mathematical predictions
Investigate rather rigid than flexible
airfoils (kite power grows
Involved groups: Diehl, Swevers,
Vandepitte
Good contacts with AmpyxPower,
TU Delft, KiteGen
Makani power

Financed by google with 15 MDollar
In “stealth” mode, but seems to bet on “generator on board” variant
Like everyone else, first builds pumping kite wind generator

Nice recent video (google “TED kite griffith”)
KiteGen

Financed by small investors and local government subsidies
Believes in “carousel” variant
Like everyone, first builds pumping kite wind generator (two lines)
KiteGen Test Flight in 2006
TU Delft Team headed by Wubbo Ockels

Financed by sponsors and local companies (port of rotterdam)
Group working longest experimentally on kite power
Has built pumping wind generator with single line
Kite Demo in Groningen August 28, 2007
AmpyxPower, Delft

Founded by ex-members of TU Delft team
Financed by seed investors and subsidies
Is building pumping wind generator with rigid airfoils
AmpyxPower Test Flight in Dec 2008
AmpyxPower Test Flight in Dec 2008
Summary: Optimal Control for Kites

Open-Loop Optimization
(finds optimal designs, unstable, non-robust)

Model Predictive Control          Open-Loop Robustness and Stability
(powerful feedback by fast        (no sensor feedback needed, simple)
online optimization)
Job Announcements – Join the OPTEC team!

Join the OPTEC team!

Open PhD positions:
•   Optimal Control of Power Generating Kite Systems
•   Distributed Optimization for MPC
•   Nonlinear State and Parameter Estimation
Thank you!

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