Model Predictive Control of a Wave Energy Converter

101 Pages·2007·2.78 MB·English

Model Predictive Control of a Wave Energy
Converter:
Archimedes Wave Swing
Abstract:
Waveenergyisapromisingsourceofcleanandrenewableenergy. Inorder
totapthisenergyvariousdesignsforwave-energyconvertersarecurrently
underdevelopment. One ofthesedevicesis theArchimedes Wave Swing.
It is basically a submerged air vessel consisting of a floater and a stator.
The floater is free to move vertically under varying wave pressure, while
the statoris anchored to the seabed. Energyis extractedfrom the relative
motionofthetwoparts.
Thecontrolobjectiveistooptimizethepowerproducedwhileensuring
thatthemotionofthefloaterremainswithincertainprescribedlimits. The
main difficulty is due to the high degree of irregularity of ocean waves.
Two consecutive waves can have significantly different heights and peri-
ods,andthecontrollershouldbeabletodealwiththat.
The controllers are designed based on two different principles. The first
methodcontrolsthetrajectorysothatthevelocityofthefloaterremainsin
phasewiththeexcitationforce. Modelpredictivecontrolisusedinorderto
beabletoeacttotheconstraintsaheadoftime. Thesecondmethodmakes
use of a prediction of the excitation force in order to calculate the control
forcewhichmaximizestheenergyproduced.
PaulGieske
1020897
Contents
1 Introduction 1
2 ControlProblemDescription 2
2.1 TheAWSDesign . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Oceanwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 ControlProblem . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 ControlStrategy . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Modelling 8
3.1 AWSmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 LinearizingtheAWSmodel . . . . . . . . . . . . . . . . . . . 12
3.3 PredictingtheExcitationForce . . . . . . . . . . . . . . . . . 15
3.4 EstimatingtheExcitationForce . . . . . . . . . . . . . . . . . 16
4 ControllerDesign 19
4.1 ReferenceTrackingforPhaseControl . . . . . . . . . . . . . . 19
4.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.2 Controllerdesign . . . . . . . . . . . . . . . . . . . . . 20
4.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 DirectEnergyMaximization . . . . . . . . . . . . . . . . . . . 24
4.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.2 ControllerDesign . . . . . . . . . . . . . . . . . . . . . 25
4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 SwitchingbetweentheControllers . . . . . . . . . . . . . . . 29
4.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3.2 ControllerDesign . . . . . . . . . . . . . . . . . . . . . 30
4.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 ApproximatingtheEnergyFunctionforFasterOptimization 31
4.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Evaluation 36
5.1 ComparisonoftheControllers . . . . . . . . . . . . . . . . . 36
5.2 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
i
5.3 ConstraintHandling . . . . . . . . . . . . . . . . . . . . . . . 41
5.4 EffectofAssumptions . . . . . . . . . . . . . . . . . . . . . . 42
5.4.1 ExcitationEstimation . . . . . . . . . . . . . . . . . . . 43
5.4.2 ClosedLoopPerformance . . . . . . . . . . . . . . . . 46
6 Conclusions 51
7 Reccomendations 53
A ListofSymbols 55
B MatlabCode 59
B.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
B.1.1 Centralm-file . . . . . . . . . . . . . . . . . . . . . . . 60
B.1.2 Parameters. . . . . . . . . . . . . . . . . . . . . . . . . 63
B.1.3 Wavemodel . . . . . . . . . . . . . . . . . . . . . . . . 64
B.1.4 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . 65
B.1.5 AWSdynamics . . . . . . . . . . . . . . . . . . . . . . 69
B.2 Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
B.2.1 ReferenceTrackingController . . . . . . . . . . . . . . 72
B.2.2 EnergyMaximizing Controller . . . . . . . . . . . . . 75
B.2.3 SwitchingController . . . . . . . . . . . . . . . . . . . 78
B.2.4 2-stepController . . . . . . . . . . . . . . . . . . . . . 82
B.3 Calculating thehydrodynamicloads . . . . . . . . . . . . . . 85
C ApproximatingtheRadiationForce 88
C.1 AnAnalyticalExpressionfortheRaditaionForce . . . . . . . 88
C.2 ApproximatingtheMemoryTermasaDamping . . . . . . . 89
C.3 ApproximatingtheMemoryTermasaLinearSystem . . . . 90
D ModelPredictiveControl 91
D.1 BasicPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
D.2 UsingLinearModelstomakePredictions . . . . . . . . . . . 91
D.3 ReformulatingtotheQuadraticProgrammingProblem . . . 93
E RandomSearchResults 95
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Chapter 1
Introduction
A rapidly increasing oil price and increased awareness of global climate
change have led to a search for clean and renewable energy. One option
is harnessing the energy in ocean waves. A number of wave energy con-
verters (WECs) are currently being developed and have reached the pre-
commercial stage of development. One of these is the Archimedes Wave
Swing(AWS).TheAWSisacylindricalunderwaterair-reservoirconsisting
ofafloaterwhichisfreetomovevertically, andastatorwhichisanchored
to theseabed. The positionofthe floater varies with thechanging hydro-
dynamicpressureandenergyisextractedfromthemotion.
The AWS is subject to some constraints: the maximum floater position
andvelocityareconstrainedandthegeneratorforceislimitedto1MN.Fur-
thermore, ocean waves are highly irregular, meaning that there is a large
variation in the wave periods and wave heights for the same sea condi-
tions. Themotionofthefloatermustbecontrolledinordertooptimizethe
waveenergyconvertedwhilesatisfyingthegivenconstraints.
Thecombinationoftheirregularityofthewavesandtheimportanceofthe
constraintsobediencemakesapredictivecontrolstrategyasuitablecandi-
date. Predictivecontrolcanreactinadvancetotheincomingwaves,allow-
ingittosafelyoperateclosertotheconstraints. Forexample,thecontroller
providesanextrabrakingforceonlyforparticularlylargewaveswherethe
constraintsareendangered.
The report is structured as follows. Chapter 2 describes the control prob-
lem in more detail. Chapter 3 describesthe modelling doneto designand
evaluatethecontrollers. Itincludesmodelsofthesystemaswellasmodels
used to estimate and predict the wave force. Chapter 4 describes the con-
trollerdesign. Chapter5evaluatesthecontrollers. Finally,chapters6and7
containtheconclusionsandrecomendationsrespectively.
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Chapter 2
Control Problem Description
This chapter briefly describes the control problem. First the AWS is intro-
duced in section 2.1. A more complete overview of the systemis given in
chapter 2 of the literature study report. Section 2.2 briefly describes the
wave profiles used to generate the results presented in this report. Next
the basics of the control problem are described in section 2.3. Finally the
controlstrategytobeusedisintroducedinsection2.4.
2.1 The AWS Design
Figure 2.1 shows the working principle of the AWS. The AWS is a sub-
mergedair-vessel consistingof two parts. The toppart (the floater)is free
tomovevertically whilethebottompart(thestator)is anchoredtothesea
bed. When the trough of the wave passes over the AWS the water pres-
sure acting on it is especially low and the air inside expands, pushing the
floaterupwards. WhenthepeakofthewavepassesovertheAWSthewa-
terpressureactingonitisespeciallyhigh,pushingthefloaterdownwards.
Ageneratorextractsenergyfromtherelativemotionofthetwoparts.
Figure 2.2 shows a simple schematic of the AWS. The motion of the
AWS is constrained. If the vertical displacement floater exceeds a certain
limit a setof water brakes will activate providing an extradamping force.
Additionally,therearerubberendstopsatthefurthestallowablelimit. The
velocityofthefloaterisalsobelimitedbecauselargevelocitiesinducelarge
electricalcurrents.
The design also incorporates a pump which can pump water into and
out of the AWS, changing the air pressure. This is done in order to tune
the natural frequency of the AWS to match it with the average excitation
periodatthattime.
The generator is an experimental permanent magnet linear generator.
The main difference to a conventional generator is that the motion of the
rotorislinear. Thegeneratorprovidesadampingforcetothefloaterwhen
2
Figure2.1: WorkingPrincipleoftheAWS
Figure2.2: SchematicoftheAWS
3
itextractsenergyfromthemotion. Itisalsocapableextractingenergyfrom
the electrical grid to providing a control force in the same direction as the
velocity.
There may be a floating buoy present in the neighbourhood of the de-
vice. Ifsomeasurementsofthesurfacewavesarealsoavailable tothecon-
troller.
In 2004 a prototype of the AWS was implemented off of the coast of
Portugal. The floater has a diameter of 9.5 m and weighs 0.4106 kg. The
·
totalheightwith the floater at mid positionis 38 m, with an air volume of
3000m3. Atotalof1500m3ofwatercanbepumpedintotheAWSallowing
ittobetunedtobetween7and13secondwaveperiods.
2.2 Ocean waves
Ocean waves are irregular on the short term but vary on the long term as
well depending on weather conditions. The average characteristics of the
wavesoverintervalsof30minutesisreferedtoastheseastate.
Inthisreportseastatesarecharacterizedbythesignificantwaveheight
(H ) and the average upwards zero-crossing period (T ). The significant
s z
wave heightis definedas the average ofthe one third highestwaves. The
average upwards zero-crossing period is defined as the average interval
betweenupwardscrossingsofthemeanwaterlevel.
Thisreportmakesuseofsyntheticallygeneratedwaves. Thewavepro-
files are generated by filtering white noise such that the resulting signal
has thedesiredpowerspectrum. Thedesiredpowerspectrumis basedon
theJONSWAPpowerspectrum. Thespectrumismodifiedtoresemblethe
spectraattainedfromthemeasurementsatthetestsite. Themeasurements
weretakenat theAWSprototypetestsite,offofthecoastofPortugal. The
JONSWAPspectrumisgivenas:
5.23π3H2 π3
J(H,T,γ,ω) = s exp 32.39 γp, (2.1)
T4ω5 − T4ω4 ·
(cid:20) (cid:21)
where
0.191ωT 1
p = exp − 2 ,
− √2σ
(cid:18) (cid:18) (cid:19) (cid:19)
σ = 0.07forω 5.24/T (2.2)
≤
σ = 0.09forω > 5.24/T
Thefollowingspectrumisusedtogeneratethewavedata:
J = 4J(H,0.9T ,2.8,ω) (2.3)
mod z
Chapter3.1oftheliteraturestudydiscussesoceanwavesmoredeeply. Ref-
erences[11]and[12]andchapter3.1.4oftheliteraturestudyreportdescribe
theprocedureforgeneratingthewaveprofiles.
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Parameter Value
c 250kW
1
c 20kW
2
c 10kW
3
F 1MN
rated
v 2.2m/s
rated
Table2.1: Generatorparameters
2.3 Control Problem
The AWS prototype made use to an experimental linear generator. The
generator is capable of producing a 1 MN control force (F ) parallel to
gen
thefloatermotion. Thegeneratorforceisaresultofextractingenergyfrom
themotionandcanalsobeusedtocontrolthemotionofthefloater.
The power produced by the generator is given by the following equa-
tion:
F 2 z˙
gen
P = F z˙ c c c (2.4)
gen 1 2 3
− − F − v −
(cid:18) rated(cid:19) (cid:12) rated(cid:12)
(cid:12) (cid:12)
Thefirsttermexpressesthepowerextractedbyt(cid:12)hegen(cid:12)eratorfromthemo-
(cid:12) (cid:12)
tion. Note that when the control force and the velocity are the same sign,
the power produced is negative, i.e. it costs energy to increase the speed
of the floater. The second term represents the resistive losses, also known
as copperlosses, and is approximated as proportional to the square of the
generator force. The third term represents the iron losses and is approxi-
matedasproportionaltothespeedofthefloater. Thefinaltermrepresents
thepowerusedbytheAWSformiscellaneousfunctionssuchas,forexam-
ple,coolingtheconverter. Table2.3liststhevaluesfortheparametergiven
inequation2.4. Reference[4]givesadescriptionofthegenerator.
The control objective is to maximize the energy produced while satis-
fyingtheconstraintsonthemotionaswellasthelimitationsofthecontrol
force. Note that maximizing the energy output is not equivalent to max-
imizing the power produced at every point in time. The generator can
extract power from the grid to facilitate the motion of the AWS, thereby
increasingtheoverallamountofenergyproduced.
The control problem is made more challenging by the nature of ocean
waves. Thesurfacewavesarehighlyirregular, meaningthatthereisasig-
nificant difference between the periods and heights of consecutive waves,
whichthesystemmustbeabletohandle. References[2],[3],[8],[9]and[10]
discussvariousmannersofusingcontroltodealwiththewaveirregularity.
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2.4 Control Strategy
The significance of the constraints in combination with the irregularity of
theexcitationforcemakemodelpredictivecontrolapromisingcontrolmethod-
ology. For example, if an unexpectedly large wave approaches the AWS,
endangering the constraints, a predictive controller can react to it in ad-
vance, thereby allowing it to safely operate close to the constraints. An
example of a model predictive control technique used to handle irregular
waves is discussedin reference [3], where a predictionis usedtocalculate
theoptimallatchingrelasetime.
Basically,twokindsofcontrollersaredesigned:
Trajectory control: A controller of this kind controls the velocity of
•
the floater to ensure that it is in phase with the excitation force. The
magnitudeisdefinedasaconstantscalingoftheexcitationforce. The
scaling factor is based on an equation for the optimal velocity tra-
jectory, as derived in chapter 4.3.2 of the literature study report and
chapter 6 of Ocean Waves and Oscillating Systems [6]. The expression
isonlyvalidwhenthegeneratorlossesandconstraintsareneglected
and the motion is unconstrained. The true optimal trajectoryis very
difficulttocalculate.
Energy maximization: An alternative is to design a controller which
•
attempts to optimize the energy produced directly. The difficulty
with this kind of controller is in formulating an optimization prob-
lemwhichaccurately describestheenergyproducedasafunctionof
the predicted excitation force, the initial state and the future control
force.
In addition, two other possibilities are suggested. The possibility of
switchingbetweenthereferencetrackingandenergymaximizingcontrollers
is discussed. Also a method for expressingthe electrical energyproduced
bythegeneratorapproximatelyinordertoallowtheoptimizationproblem
tobesolvedwithlesscalculationsisdiscussed.
In order to design the controllers it is necessary to have an estimate of
thefutureexcitationforce. Theexcitationforcecanbepredictedbyextrap-
olating it from past values. This is be done using an autoregressive (AR)
model,asisdiscussedinsection3.3.
To predict the excitation force past values are needed. These can be
providedbyuseofapressuresensor. Includingapressuresensorhowever,
wouldinfluencetheoverallcostandreliability ofthesystem. Incasethere
excitation force cannot be measured, the excitation force can be estimated
fromthemotionoftheAWS.Thisisdiscussedinsection3.4.
Inordertoinvestigatetheeffectsofthepredictionandestimationonthe
performance, the code is written to be able to handle the following three
cases:
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The excitation force is known a priori. This is an ideal case. Un-
•
lessotherwisestatedtheresultspresentedin this reportmake useof
knowledgeofthefutureexcitationforce.
The excitation force is known for past and present, and the future
•
excitation force mustbe predicted. Inthis case it isassumedthat the
AWSisequippedwithanidealpressuresensor.
Theexcitationforceisnotknownatall. Itmustfirstbeestimatedand
•
thenextrapolated.
By comparing the performance for each of the above cases the need for
improvedestimationorpredictioncanbeevaluated.
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