6-scaleeffects

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Short Course on Wind EnergyShort Course on Wind EnergyShort Course on Wind EnergyShort Course on Wind Energy---- Scale Effects in Wind TurbinesScale Effects in Wind TurbinesScale Effects in Wind TurbinesScale Effects in Wind Turbines ----

Carlo L.Carlo L.Carlo L.Carlo L. BottassoBottassoBottassoBottassoPolitecnico di MilanoPolitecnico di MilanoPolitecnico di MilanoPolitecnico di Milano

NovemberNovemberNovemberNovember 2011201120112011

Short Course on Wind EnergyShort Course on Wind EnergyShort Course on Wind EnergyShort Course on Wind Energy

---- Scale Effects in Wind TurbinesScale Effects in Wind TurbinesScale Effects in Wind TurbinesScale Effects in Wind Turbines ----

Carlo L.Carlo L.Carlo L.Carlo L. BottassoBottassoBottassoBottassoPolitecnico di MilanoPolitecnico di MilanoPolitecnico di MilanoPolitecnico di Milano

NovemberNovemberNovemberNovember 2011201120112011


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Scale EffectsScale EffectsScale EffectsScale EffectsScale EffectsScale EffectsScale EffectsScale Effects


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POLITECNICO di MILANOPOLITECNICO di MILANOPOLITECNICO di MILANOPOLITECNICO di MILANO POLIPOLIPOLIPOLI----Wind Research LabWind Research LabWind Research LabWind Research Lab

Applications:Applications:Applications:Applications:

Understanding scaling effects in smallUnderstanding scaling effects in smallUnderstanding scaling effects in smallUnderstanding scaling effects in small----large wind turbineslarge wind turbineslarge wind turbineslarge wind turbines Scaling laws for wind tunnel modelsScaling laws for wind tunnel modelsScaling laws for wind tunnel modelsScaling laws for wind tunnel models

Tools:Tools:Tools:Tools:

BuckingamBuckingamBuckingamBuckingam ----Theorem (dimensional analysis)Theorem (dimensional analysis)Theorem (dimensional analysis)Theorem (dimensional analysis)

Scale EffectsScale EffectsScale EffectsScale EffectsScale EffectsScale EffectsScale EffectsScale Effects


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BuckinghamBuckinghamBuckinghamBuckinghamBuckinghamBuckinghamBuckinghamBuckingham --------TheoremTheoremTheoremTheoremTheoremTheoremTheoremTheorem::::

Given the governing equation for a physical system defined by nGiven the governing equation for a physical system defined by nGiven the governing equation for a physical system defined by nGiven the governing equation for a physical system defined by nphysical variables, which are expressible in terms ofphysical variables, which are expressible in terms ofphysical variables, which are expressible in terms ofphysical variables, which are expressible in terms ofkkkk independent fundamental quantities, one can construct anindependent fundamental quantities, one can construct anindependent fundamental quantities, one can construct anindependent fundamental quantities, one can construct anequivalent equation involving a set of m = nequivalent equation involving a set of m = nequivalent equation involving a set of m = nequivalent equation involving a set of m = nkkkk dimensionlessdimensionlessdimensionlessdimensionless

variables constructed from the original variablesvariables constructed from the original variablesvariables constructed from the original variablesvariables constructed from the original variables

The dimensionless parameters provide relations which define theThe dimensionless parameters provide relations which define theThe dimensionless parameters provide relations which define theThe dimensionless parameters provide relations which define the

scaling lawsscaling lawsscaling lawsscaling laws

Dimensional Analysis: BuckinghamDimensional Analysis: BuckinghamDimensional Analysis: BuckinghamDimensional Analysis: Buckingham ----TheoremTheoremTheoremTheorem



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Consider the generic model:Consider the generic model:Consider the generic model:Consider the generic model:

TheTheTheThe iiii----thththth physical parameterphysical parameterphysical parameterphysical parameterphysical parameterphysical parameterphysical parameterphysical parameter can be expressed in terms ofcan be expressed in terms ofcan be expressed in terms ofcan be expressed in terms ofkkkkindependent fundamental quantitiesindependent fundamental quantitiesindependent fundamental quantitiesindependent fundamental quantitiesindependent fundamental quantitiesindependent fundamental quantitiesindependent fundamental quantitiesindependent fundamental quantities::::

= dimensions of= dimensions of= dimensions of= dimensions of wrtwrtwrtwrt fundamental quantitiesfundamental quantitiesfundamental quantitiesfundamental quantities

TheTheTheThe Dimensional MatrixDimensional MatrixDimensional MatrixDimensional MatrixDimensional MatrixDimensional MatrixDimensional MatrixDimensional Matrix contains as elements thecontains as elements thecontains as elements thecontains as elements therrrr----thththth dimension of thedimension of thedimension of thedimension of the iiii----thththth parameterparameterparameterparameter

PhysicalPhysicalPhysicalPhysicalparametersparametersparametersparameters FundamentalFundamentalFundamentalFundamentalquantitiesquantitiesquantitiesquantities



f(p1;:::;pn; a1;:::;ak) = 0

di1; : : : ; dik

pi = ad1i1

: : : adkik

pi

D 2 Rk Rn dri


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Equivalent nonEquivalent nonEquivalent nonEquivalent nonEquivalent nonEquivalent nonEquivalent nonEquivalent non--------dimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional modeldimensional model::::

jjjj----thththth nonnonnonnon----dimensional parameter:dimensional parameter:dimensional parameter:dimensional parameter:

The exponents are the components of matrixThe exponents are the components of matrixThe exponents are the components of matrixThe exponents are the components of matrix

found from the solution of the following system of equations:found from the solution of the following system of equations:found from the solution of the following system of equations:found from the solution of the following system of equations:

RemarksRemarksRemarksRemarksRemarksRemarksRemarksRemarks::::

Set ofSet ofSet ofSet ofmmmm====nnnn----kkkkparameters is not uniqueparameters is not uniqueparameters is not uniqueparameters is not unique

Selection of proper/representative nonSelection of proper/representative nonSelection of proper/representative nonSelection of proper/representative non----dimensional parametersdimensional parametersdimensional parametersdimensional parametersneeds to be guided by physical considerations based on the modelneeds to be guided by physical considerations based on the modelneeds to be guided by physical considerations based on the modelneeds to be guided by physical considerations based on the model

under considerationunder considerationunder considerationunder consideration

i.e.:i.e.:i.e.:i.e.:



(1; : : : ; m) = 0, m = n k

j = pm1j1

: : : pmnjn

M 2 Rn Rm

D M = 0 M = null(D)


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Consider two systems:Consider two systems:Consider two systems:Consider two systems:

==== Physical systemPhysical systemPhysical systemPhysical systemPhysical systemPhysical systemPhysical systemPhysical system (full scale)(full scale)(full scale)(full scale)

==== ModelModelModelModelModelModelModelModel (reduced scale)(reduced scale)(reduced scale)(reduced scale)with governing equations:with governing equations:with governing equations:with governing equations:

and equivalent nonand equivalent nonand equivalent nonand equivalent non----dimensional relations obtained throughdimensional relations obtained throughdimensional relations obtained throughdimensional relations obtained through

BuckinghamBuckinghamBuckinghamBuckingham ----theorem:theorem:theorem:theorem:

SimilaritySimilaritySimilaritySimilaritySimilaritySimilaritySimilaritySimilarity

P

M

f(p1M; : : : ; pnM; a1M; : : : ; akM) = 0

f(p1P; : : : ; pnP; a1P; : : : ; akP) = 0

(1P; : : : ; mP) = 0

(1M; : : : ; mM) = 0


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isisisis similarsimilarsimilarsimilarsimilarsimilarsimilarsimilar to if:to if:to if:to if:

which provides a set of scaling relations:which provides a set of scaling relations:which provides a set of scaling relations:which provides a set of scaling relations:

GenericGenericGenericGeneric scaling relationscaling relationscaling relationscaling relationscaling relationscaling relationscaling relationscaling relation for a model parameter:for a model parameter:for a model parameter:for a model parameter:

SimilaritySimilaritySimilaritySimilaritySimilaritySimilaritySimilaritySimilarity

P M

jP = jM, for j = 1;::;m

pm1j1P

: : : pmnjnP = p

m1j1M

: : : pmnjnM

p1M = p1Pp2P

p2Mm2jm1j

pnP

pnMmnjm1j


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Typically, scaling is based on the assignment of aTypically, scaling is based on the assignment of aTypically, scaling is based on the assignment of aTypically, scaling is based on the assignment of a scaling parameterscaling parameterscaling parameterscaling parameterscaling parameterscaling parameterscaling parameterscaling parameter::::

Accordingly, the general scaling relation can be expressed as:Accordingly, the general scaling relation can be expressed as:Accordingly, the general scaling relation can be expressed as:Accordingly, the general scaling relation can be expressed as:

The scaling process reduces to aThe scaling process reduces to aThe scaling process reduces to aThe scaling process reduces to a linear transformation of parameterslinear transformation of parameterslinear transformation of parameterslinear transformation of parameterslinear transformation of parameterslinear transformation of parameterslinear transformation of parameterslinear transformation of parameters::::

Scaling matrixScaling matrixScaling matrixScaling matrixScaling matrixScaling matrixScaling matrixScaling matrix::::

Similarity and ScalingSimilarity and ScalingSimilarity and ScalingSimilarity and ScalingSimilarity and ScalingSimilarity and ScalingSimilarity and ScalingSimilarity and Scaling

Characteristic length of the modelCharacteristic length of the modelCharacteristic length of the modelCharacteristic length of the model

Characteristic length of full scale systemCharacteristic length of full scale systemCharacteristic length of full scale systemCharacteristic length of full scale system

pM = SpP

n =lMlP

p1M = p1Pn12 : : : n1n p1M = p1Pn

1

S = diag(n1 ; : : : ; nn)


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Wind Turbine Dimensional AnalysisWind Turbine Dimensional AnalysisWind Turbine Dimensional AnalysisWind Turbine Dimensional AnalysisWind Turbine Dimensional AnalysisWind Turbine Dimensional AnalysisWind Turbine Dimensional AnalysisWind Turbine Dimensional Analysis

StatesStatesStatesStatesStatesStatesStatesStates::::

Flap anglesFlap anglesFlap anglesFlap angles

Rotor azimuthRotor azimuthRotor azimuthRotor azimuth

Shaft torsionShaft torsionShaft torsionShaft torsion

ForeForeForeFore----aft angleaft angleaft angleaft angle

(and their rates)(and their rates)(and their rates)(and their rates)

InputsInputsInputsInputsInputsInputsInputsInputs::::

Pitch anglesPitch anglesPitch anglesPitch angles

Electrical torqueElectrical torqueElectrical torqueElectrical torque

ExampleExampleExampleExampleExampleExampleExampleExample: 6 state: 6 state: 6 state: 6 state 4 input4 input4 input4 inputindividualindividualindividualindividual----pitch model, withpitch model, withpitch model, withpitch model, withstiffness modeled usingstiffness modeled usingstiffness modeled usingstiffness modeled usingequivalent springsequivalent springsequivalent springsequivalent springs

Telc

c1c2

c3 K

K

K

K

K

1; 2; 3

Telc

c1 ; c2 ; c3

Readily generalized to more complex modelsReadily generalized to more complex modelsReadily generalized to more complex modelsReadily generalized to more complex models


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Derive equations of motion, apply BuckinghamDerive equations of motion, apply BuckinghamDerive equations of motion, apply BuckinghamDerive equations of motion, apply Buckingham ----theorem using astheorem using astheorem using astheorem using asfundamental quantitiesfundamental quantitiesfundamental quantitiesfundamental quantities

MassMassMassMassMassMassMassMass,,,, LengthLengthLengthLengthLengthLengthLengthLength,,,, TimeTimeTimeTimeTimeTimeTimeTime

NonNonNonNon----dimensional equations of motion:dimensional equations of motion:dimensional equations of motion:dimensional equations of motion:

States:States:States:States:

Inputs:Inputs:Inputs:Inputs:

NonNonNonNon----dimensional parametersdimensional parametersdimensional parametersdimensional parameters

x = (; i; ; )T

u =

i; eTelcT eTelc = Telc1=2ARV2

=

Ma; Re; Fr; Lo;;;

!-

;!i!

;!!

T


(x; _x; x;u;) = 0


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NonNonNonNonNonNonNonNon--------dimensional parametersdimensional parametersdimensional parametersdimensional parametersdimensional parametersdimensional parametersdimensional parametersdimensional parameters of physical relevance resulting fromof physical relevance resulting fromof physical relevance resulting fromof physical relevance resulting fromdimensional analysis:dimensional analysis:dimensional analysis:dimensional analysis:

Mach:Mach:Mach:Mach:

Reynolds:Reynolds:Reynolds:Reynolds:

Froude:Froude:Froude:Froude:

Lock:Lock:Lock:Lock:

TSR:TSR:TSR:TSR:

NonNonNonNon----dim. time:dim. time:dim. time:dim. time:

NonNonNonNon----dim. tower freq.:dim. tower freq.:dim. tower freq.:dim. tower freq.:

Flapping freq. placement:Flapping freq. placement:Flapping freq. placement:Flapping freq. placement:

Shaft freq. placement:Shaft freq. placement:Shaft freq. placement:Shaft freq. placement:

= -t

Re = V c=

Fr = V2=gR

= -R=V

Ma = V =a

Lo = CLcR4=J

Inertial/viscous aerodynamic force ratioInertial/viscous aerodynamic force ratioInertial/viscous aerodynamic force ratioInertial/viscous aerodynamic force ratio

Aerodynamic/gravitation force ratioAerodynamic/gravitation force ratioAerodynamic/gravitation force ratioAerodynamic/gravitation force ratio

Aerodynamic/inertial force ratioAerodynamic/inertial force ratioAerodynamic/inertial force ratioAerodynamic/inertial force ratio


Effect typically negligible forEffect typically negligible forEffect typically negligible forEffect typically negligible for WTsWTsWTsWTs

!i=!; !i =pKi=Jie

! = !=-; ! = pK=J!=!; ! =

pK=J


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Consider two wind turbines and with scale ratioConsider two wind turbines and with scale ratioConsider two wind turbines and with scale ratioConsider two wind turbines and with scale ratio

operating in theoperating in theoperating in theoperating in the same windsame windsame windsame windsame windsame windsame windsame wind

at theat theat theat the same TSRsame TSRsame TSRsame TSRsame TSRsame TSRsame TSRsame TSR

Size Effects on Wind TurbinesSize Effects on Wind TurbinesSize Effects on Wind TurbinesSize Effects on Wind TurbinesSize Effects on Wind TurbinesSize Effects on Wind TurbinesSize Effects on Wind TurbinesSize Effects on Wind Turbines

n = RM=RP

M P

VM = VP

M = P


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General scaling procedureGeneral scaling procedureGeneral scaling procedureGeneral scaling procedureGeneral scaling procedureGeneral scaling procedureGeneral scaling procedureGeneral scaling procedure::::

GivenGivenGivenGiven scale factorscale factorscale factorscale factorscale factorscale factorscale factorscale factor

FindFindFindFind velocityvelocityvelocityvelocityvelocityvelocityvelocityvelocity andandandand timetimetimetimetimetimetimetime scalingsscalingsscalingsscalings

EnforceEnforceEnforceEnforceEnforceEnforceEnforceEnforce::::

Time:Time:Time:Time:

TipTipTipTip----speedspeedspeedspeed----ratio:ratio:ratio:ratio:

Lock:Lock:Lock:Lock:

Remark: Lock can always be fixed with material densityRemark: Lock can always be fixed with material densityRemark: Lock can always be fixed with material densityRemark: Lock can always be fixed with material density

Scaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine Models

nV = VM=VP nt = tM=tP

M = P n- = -M=-P = 1=nt

M = P nV = n-n = n=nt(-R=V)M = (-R=V)P

(-t)M = (-t)P

LoM = LoP (CLcR4=J)M = (CLcR

4=J)P mM=mP = 1

n = RM=RP


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EnforceEnforceEnforceEnforceEnforceEnforceEnforceEnforce (continued):(continued):(continued):(continued):

Frequency placement:Frequency placement:Frequency placement:Frequency placement:

Remark: frequency placement can always be fixed with stiffnesRemark: frequency placement can always be fixed with stiffnesRemark: frequency placement can always be fixed with stiffnesRemark: frequency placement can always be fixed with stiffnesssss

Resulting errorsResulting errorsResulting errorsResulting errorsResulting errorsResulting errorsResulting errorsResulting errors::::

ReynoldsReynoldsReynoldsReynolds

FroudeFroudeFroudeFroude

MachMachMachMach

Important remarkImportant remarkImportant remarkImportant remarkImportant remarkImportant remarkImportant remarkImportant remark: only unknown left is time scaling: only unknown left is time scaling: only unknown left is time scaling: only unknown left is time scaling

Scaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine ModelsScaling Laws for Wind Turbine Models

ReM=ReP = n2=nt

FrM=FrP = n=n2t

MaM=MaP = n=n2

t

e!M = e!P (EJ=(mL4-4))M = (EJ=(mL4-4))P(EJ)M=(EJ)P = n

6=n2t

nt


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ApproachApproachApproachApproachApproachApproachApproachApproach::::

Choose comparison metrics (e.g., fatigue damage index in turbuleChoose comparison metrics (e.g., fatigue damage index in turbuleChoose comparison metrics (e.g., fatigue damage index in turbuleChoose comparison metrics (e.g., fatigue damage index in turbulentntntntwinds, load peak values for gust response, etc.)winds, load peak values for gust response, etc.)winds, load peak values for gust response, etc.)winds, load peak values for gust response, etc.)

Simulate response of scaled and fullSimulate response of scaled and fullSimulate response of scaled and fullSimulate response of scaled and full----scale modelsscale modelsscale modelsscale models

Compare responses upon backCompare responses upon backCompare responses upon backCompare responses upon back----scaling of scaled results using metricsscaling of scaled results using metricsscaling of scaled results using metricsscaling of scaled results using metrics

Testing of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsTesting of Scaling Laws

InputInputInputInputParametersParametersParametersParameters

InputInputInputInputParametersParametersParametersParameters

AeroelasticAeroelasticAeroelasticAeroelasticModelModelModelModel



AeroelasticAeroelasticAeroelasticAeroelastic

ModelModelModelModel

ScalingScalingScalingScalingLawsLawsLawsLaws

ScalingScalingScalingScalingLawsLawsLawsLaws

InverseInverseInverseInverseScaling LawsScaling LawsScaling LawsScaling Laws

InverseInverseInverseInverseScaling LawsScaling LawsScaling LawsScaling Laws

PerformancePerformancePerformancePerformancePerformancePerformancePerformancePerformance


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Testing of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsTesting of Scaling LawsExampleExampleExampleExampleExampleExampleExampleExample: performance comparison of two control laws on full scale and: performance comparison of two control laws on full scale and: performance comparison of two control laws on full scale and: performance comparison of two control laws on full scale andMachMachMachMach----mismatchedmismatchedmismatchedmismatched----scaled modelscaled modelscaled modelscaled model

Turbulent 16Turbulent 16Turbulent 16Turbulent 16 m/sm/sm/sm/s windwindwindwindRe full scale 5.25e+6Re full scale 5.25e+6Re full scale 5.25e+6Re full scale 5.25e+6

Re scaled 4.6e+5Re scaled 4.6e+5Re scaled 4.6e+5Re scaled 4.6e+5

GoodnessGoodnessGoodnessGoodness of one controllerof one controllerof one controllerof one controller wrtwrtwrtwrt thethethetheother is practically the same whenother is practically the same whenother is practically the same whenother is practically the same whentested on full scale and scaled modeltested on full scale and scaled modeltested on full scale and scaled modeltested on full scale and scaled model

HenceHenceHenceHenceHenceHenceHenceHence::::

Scaled model is appropriate forScaled model is appropriate forScaled model is appropriate forScaled model is appropriate forconducting control law comparisonsconducting control law comparisonsconducting control law comparisonsconducting control law comparisons

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