Probabilistic Wind Turbine System Models in Three Courses Composite Materials

Aerodynamics Grid Integration

Dr Curran Crawford

Department of Mechanical Engineering University of Victoria

4th Workshop on Systems Engineering for Wind Energy DTU Vindenergi September 13 2017

1 26

Outline

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

2 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

3 26

Wind energy systems inherently involve variability

Wind input

High f turbulence Fat tail distributions (extremes) Seasonalannual mean wind speed variation Decadal-scale variations

Wave loading offshore

Blade erosion amp soiling

Large-scale (manual) manufacturing

Limit amp fatigue strengths Stiffness variations

Mechanical amp electrical component reliability

Aero-structural response to these inputs

Controller actions Fatigue amp extreme loads Power output

4 26

System analysis models must handle this variability to be trusted and explore full design space

But we already do this donrsquot we Monte Carlo analyses

IEC load sets + statistical extrapolation Combined windwave conditions Decomposed MDO frameworks

Quite expensive even for low fidelity BEM-type models

My grouprsquos research goals Medium fidelity models viable for system optimization Intrinsically probabilistic models to quantifymitigate risk Design studies on advanced concepts airborne offshore Examine system economics amp grid integration

5 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

6 26

Micromechanics models use matrix amp fibre properties avoiding coupon tests for alternative layups

Relate micro σj and macro σ stresses σj = [SAF ]σ Elements of SAF computed via FEM simulations

Failure criteria applied at points j based on material strengths

7 26

Markov Chain Monte Carlo (MCMC) methods used to evaluate Bayesrsquos theorem estimates

f (ytest |Θ)f (Θ) L(Θ)f (Θ)f (Θ|ytest ) = = f (ytest ) f (ytest )

8 26

Statistical strengths calibrated using a limited set of experimental results

Uniform prior (unbiased)

Likelyhood function

Assume normal distribution of properties

Histogram algorithm output Kernel density estimator on MCMC results

Fibre and matrix strengths

Compressive tensile shear Ultimate and fatigue

9 26

Forward statistical analysis of candidate layups is then possible (ultimate amp fatigue)

10 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Outline

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

2 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

3 26

Wind energy systems inherently involve variability

Wind input

High f turbulence Fat tail distributions (extremes) Seasonalannual mean wind speed variation Decadal-scale variations

Wave loading offshore

Blade erosion amp soiling

Large-scale (manual) manufacturing

Limit amp fatigue strengths Stiffness variations

Mechanical amp electrical component reliability

Aero-structural response to these inputs

Controller actions Fatigue amp extreme loads Power output

4 26

System analysis models must handle this variability to be trusted and explore full design space

But we already do this donrsquot we Monte Carlo analyses

IEC load sets + statistical extrapolation Combined windwave conditions Decomposed MDO frameworks

Quite expensive even for low fidelity BEM-type models

My grouprsquos research goals Medium fidelity models viable for system optimization Intrinsically probabilistic models to quantifymitigate risk Design studies on advanced concepts airborne offshore Examine system economics amp grid integration

5 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

6 26

Micromechanics models use matrix amp fibre properties avoiding coupon tests for alternative layups

Relate micro σj and macro σ stresses σj = [SAF ]σ Elements of SAF computed via FEM simulations

Failure criteria applied at points j based on material strengths

7 26

Markov Chain Monte Carlo (MCMC) methods used to evaluate Bayesrsquos theorem estimates

f (ytest |Θ)f (Θ) L(Θ)f (Θ)f (Θ|ytest ) = = f (ytest ) f (ytest )

8 26

Statistical strengths calibrated using a limited set of experimental results

Uniform prior (unbiased)

Likelyhood function

Assume normal distribution of properties

Histogram algorithm output Kernel density estimator on MCMC results

Fibre and matrix strengths

Compressive tensile shear Ultimate and fatigue

9 26

Forward statistical analysis of candidate layups is then possible (ultimate amp fatigue)

10 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

3 26

Wind energy systems inherently involve variability

Wind input

High f turbulence Fat tail distributions (extremes) Seasonalannual mean wind speed variation Decadal-scale variations

Wave loading offshore

Blade erosion amp soiling

Large-scale (manual) manufacturing

Limit amp fatigue strengths Stiffness variations

Mechanical amp electrical component reliability

Aero-structural response to these inputs

Controller actions Fatigue amp extreme loads Power output

4 26

System analysis models must handle this variability to be trusted and explore full design space

But we already do this donrsquot we Monte Carlo analyses

IEC load sets + statistical extrapolation Combined windwave conditions Decomposed MDO frameworks

Quite expensive even for low fidelity BEM-type models

My grouprsquos research goals Medium fidelity models viable for system optimization Intrinsically probabilistic models to quantifymitigate risk Design studies on advanced concepts airborne offshore Examine system economics amp grid integration

5 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

6 26

Micromechanics models use matrix amp fibre properties avoiding coupon tests for alternative layups

Relate micro σj and macro σ stresses σj = [SAF ]σ Elements of SAF computed via FEM simulations

Failure criteria applied at points j based on material strengths

7 26

Markov Chain Monte Carlo (MCMC) methods used to evaluate Bayesrsquos theorem estimates

f (ytest |Θ)f (Θ) L(Θ)f (Θ)f (Θ|ytest ) = = f (ytest ) f (ytest )

8 26

Statistical strengths calibrated using a limited set of experimental results

Uniform prior (unbiased)

Likelyhood function

Assume normal distribution of properties

Histogram algorithm output Kernel density estimator on MCMC results

Fibre and matrix strengths

Compressive tensile shear Ultimate and fatigue

9 26

Forward statistical analysis of candidate layups is then possible (ultimate amp fatigue)

10 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Wind energy systems inherently involve variability

Wind input

High f turbulence Fat tail distributions (extremes) Seasonalannual mean wind speed variation Decadal-scale variations

Wave loading offshore

Blade erosion amp soiling

Large-scale (manual) manufacturing

Limit amp fatigue strengths Stiffness variations

Mechanical amp electrical component reliability

Aero-structural response to these inputs

Controller actions Fatigue amp extreme loads Power output

4 26

System analysis models must handle this variability to be trusted and explore full design space

But we already do this donrsquot we Monte Carlo analyses

IEC load sets + statistical extrapolation Combined windwave conditions Decomposed MDO frameworks

Quite expensive even for low fidelity BEM-type models

My grouprsquos research goals Medium fidelity models viable for system optimization Intrinsically probabilistic models to quantifymitigate risk Design studies on advanced concepts airborne offshore Examine system economics amp grid integration

5 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

6 26

Micromechanics models use matrix amp fibre properties avoiding coupon tests for alternative layups

Relate micro σj and macro σ stresses σj = [SAF ]σ Elements of SAF computed via FEM simulations

Failure criteria applied at points j based on material strengths

7 26

Markov Chain Monte Carlo (MCMC) methods used to evaluate Bayesrsquos theorem estimates

f (ytest |Θ)f (Θ) L(Θ)f (Θ)f (Θ|ytest ) = = f (ytest ) f (ytest )

8 26

Statistical strengths calibrated using a limited set of experimental results

Uniform prior (unbiased)

Likelyhood function

Assume normal distribution of properties

Histogram algorithm output Kernel density estimator on MCMC results

Fibre and matrix strengths

Compressive tensile shear Ultimate and fatigue

9 26

Forward statistical analysis of candidate layups is then possible (ultimate amp fatigue)

10 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

System analysis models must handle this variability to be trusted and explore full design space

But we already do this donrsquot we Monte Carlo analyses

IEC load sets + statistical extrapolation Combined windwave conditions Decomposed MDO frameworks

Quite expensive even for low fidelity BEM-type models

My grouprsquos research goals Medium fidelity models viable for system optimization Intrinsically probabilistic models to quantifymitigate risk Design studies on advanced concepts airborne offshore Examine system economics amp grid integration

5 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

6 26

Micromechanics models use matrix amp fibre properties avoiding coupon tests for alternative layups

Relate micro σj and macro σ stresses σj = [SAF ]σ Elements of SAF computed via FEM simulations

Failure criteria applied at points j based on material strengths

7 26

Markov Chain Monte Carlo (MCMC) methods used to evaluate Bayesrsquos theorem estimates

f (ytest |Θ)f (Θ) L(Θ)f (Θ)f (Θ|ytest ) = = f (ytest ) f (ytest )

8 26

Statistical strengths calibrated using a limited set of experimental results

Uniform prior (unbiased)

Likelyhood function

Assume normal distribution of properties

Histogram algorithm output Kernel density estimator on MCMC results

Fibre and matrix strengths

Compressive tensile shear Ultimate and fatigue

9 26

Forward statistical analysis of candidate layups is then possible (ultimate amp fatigue)

10 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

6 26

Micromechanics models use matrix amp fibre properties avoiding coupon tests for alternative layups

Relate micro σj and macro σ stresses σj = [SAF ]σ Elements of SAF computed via FEM simulations

Failure criteria applied at points j based on material strengths

7 26

Markov Chain Monte Carlo (MCMC) methods used to evaluate Bayesrsquos theorem estimates

f (ytest |Θ)f (Θ) L(Θ)f (Θ)f (Θ|ytest ) = = f (ytest ) f (ytest )

8 26

Statistical strengths calibrated using a limited set of experimental results

Uniform prior (unbiased)

Likelyhood function

Assume normal distribution of properties

Histogram algorithm output Kernel density estimator on MCMC results

Fibre and matrix strengths

Compressive tensile shear Ultimate and fatigue

9 26

Forward statistical analysis of candidate layups is then possible (ultimate amp fatigue)

10 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Micromechanics models use matrix amp fibre properties avoiding coupon tests for alternative layups

Relate micro σj and macro σ stresses σj = [SAF ]σ Elements of SAF computed via FEM simulations

Failure criteria applied at points j based on material strengths

7 26

Markov Chain Monte Carlo (MCMC) methods used to evaluate Bayesrsquos theorem estimates

f (ytest |Θ)f (Θ) L(Θ)f (Θ)f (Θ|ytest ) = = f (ytest ) f (ytest )

8 26

Statistical strengths calibrated using a limited set of experimental results

Uniform prior (unbiased)

Likelyhood function

Assume normal distribution of properties

Histogram algorithm output Kernel density estimator on MCMC results

Fibre and matrix strengths

Compressive tensile shear Ultimate and fatigue

9 26

Forward statistical analysis of candidate layups is then possible (ultimate amp fatigue)

10 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Markov Chain Monte Carlo (MCMC) methods used to evaluate Bayesrsquos theorem estimates

f (ytest |Θ)f (Θ) L(Θ)f (Θ)f (Θ|ytest ) = = f (ytest ) f (ytest )

8 26

Statistical strengths calibrated using a limited set of experimental results

Uniform prior (unbiased)

Likelyhood function

Assume normal distribution of properties

Histogram algorithm output Kernel density estimator on MCMC results

Fibre and matrix strengths

Compressive tensile shear Ultimate and fatigue

9 26

Forward statistical analysis of candidate layups is then possible (ultimate amp fatigue)

10 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Statistical strengths calibrated using a limited set of experimental results

Uniform prior (unbiased)

Likelyhood function

Assume normal distribution of properties

Histogram algorithm output Kernel density estimator on MCMC results

Fibre and matrix strengths

Compressive tensile shear Ultimate and fatigue

9 26

Forward statistical analysis of candidate layups is then possible (ultimate amp fatigue)

10 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Forward statistical analysis of candidate layups is then possible (ultimate amp fatigue)

10 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

11 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

To explore new (multi-MW scale floating) concepts need to get beyond BEM

Physics not captured by BEM

Non-linear motion amp deflections Sweptwinglet blades

Unsteady aerodynamics

CFD still to costly for unsteady design

Vortex-based methods appropriate

But costly for unsteady design

Also want gradients for unsteady performance

12 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

stochastic input process solve system

equations

several realizations(only if needed)

analyze results

Γξ i(t)=f (uξ i(t))

Γ(t ξ)=f (u(t ξ))

several realizations

one stochastic solution

several deterministic solutions

Fundamentally we want to replace many time-domain simulations with a single solution in the stochastic space

13 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

We have adopted intrusive chaos (polynomial) methods to analyze in the stochastic domain

Expand wind inflow model with random phase angles ξ isin [minusπ π] as

Rminus1R uinfin(tn ξ) = ur (tn)Ψr (ξ)

r=0

Lots of groundwork on describing correlated wind fields with minimum number of ξ phase angles Expand circulate strength Γ (lift) as

Rminus1R Γ(tn ξ) = gr (tn)Ψr (ξ)

r =0

Functionals Ψr (ξ) define polynomials (exponentials) of random variable ξ

Choose form based on PDF of ξ Legendre polynomials for uniform distributions

14 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Solution procedure is equivalent to one time-domain solve but obtain results for all possible ξ values

Use inner product (D Ψr (ξ)) lsquostochastic Galerkinrsquo projection

Orthogonality property of Ψr (ξ) when chosen correctly Equations for coefficients in time fall out

ur (tn) gr (tn)

Note that these solutions are functions of time

Not a Fourier transform Can handle non-linear time dependent effects

Output quantity calculations

Time-domain for specific ξ realization

Validation with time dependent code

Directly compute statistical moments from gr (tn) PDF reconstruction methods

All possible wind input fields evaluated simultaneously

15 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

tra

iling

fila

men

ts

Γ j(k

+1

)

k =

1

k =

M

i = N

+1

j= N

j= 1

i = 1

c)

xx

xx

x

r ijk

Γikshed

elements

x

x xxx

xxx x x x

x

xx

xx

xx

xxx

xxx

xX

ik

X(i+

1)k

x

x xxx

x

y

γ20

γ10

γ50 γN b N t

γ12Γ1(t n)

Γ2(t n)

Γ3(tn)

Γ4(tn)

Γ5(tn)

Γs 11

Γs 21

Γs 42H 3(s 51)

G3(s 23)

Γt 13

Γt 51

Nt elements

Nb e

lem

ents

γ31

16 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Wersquove applied the stochastic Galerkin method to stationary bladeswings (lifting line)

span position [m]-25 -15 -5 5 15 25

circula

tion [m

2s

]

5

10

15

steady state stochastic at t=[68101214] s

wind speed [ms]8 10 12 8 10 12 8 10 12 8 10 12 8 10 12 8 10 12

tim

e [s]

0

5

10

15

wind bound circulation (deterministic) bound circulation (stochstic)

circulation [m2s]

9 11 13 9 11 13 9 11 13 9 11 13 9 11 13 9 11 13

17 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

and simplified BEM compared to 100 deterministic sample wind inputs

3 6 9 12 15 18

time [s]

0

10

20

30

element at rR=014

element at rR=086

18 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Table of Contents

Why Probabilistic Models

Composite Structures

Turbulent Aero(structural) dynamics

(Smart) Grid Integration

19 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Wind energyrsquos value depends on the grid-delivered product hopefully greater than the costs to provide it

Various components in levelized cost of energy Base materials ndash capital costs Aerostructural performance ndash power captureloads Variability in system costs captured with previous approaches

But value is different than costs Levelized avoided cost of energy (LACE)

LCOE estimates revenue requirements LACE estimates revenues available

Firming services costs

How to model variable grid system performance

20 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

The interconnected grid is large and complicated requiring efficient solutions methods

BPAT

BCHAAESO

WAUWNWMT

IPCO

AVA

PGE

PACW

PACE

WACM

PSCO

PNM

EPE

WALC

SRP

AZPS

IID

PSEISCL

TPWR

DOPDCHPD

DEAAGRMA

HGMATEPC

LDWP

BANC

TIDC

CFE

CISO

GCPDWWA

GRIF

GWA

NEVP

GRID

Boundaries are approximate and for illustrative purposes only

Western InterconnectionBalancing Authorities (38)

AESO - Alberta Electric System OperatorAZPS - Arizona Public Service CompanyAVA - Avista CorporationBANC - Balancing Authority of Northern CaliforniaBPAT - Bonneville Power Administration - TransmissionBCHA - British Columbia Hydro AuthorityCISO - California Independent System OperatorCFE - Comision Federal de ElectricidadDEAA - Arlington Valley LLCEPE - El Paso Electric CompanyGRMA - Gila River Power LPGRID - GridforceGRIF - Grith Energy LLCIPCO - Idaho Power Company

IID - Imperial Irrigation DistrictLDWP - Los Angeles Department of Water and PowerGWA - NaturEner Power Watch LLCNEVP - Nevada Power CompanyHGMA - New Harquahala Generating Company LLCNWMT - NorthWestern EnergyPACE - PaciCorp EastPACW - PaciCorp WestPGE - Portland General Electric CompanyPSCO - Public Service Company of ColoradoPNM - Public Service Company of New MexicoCHPD - PUD No 1 of Chelan CountyDOPD - PUD No 1 of Douglas CountyGCPD - PUD No 2 of Grant County

PSEI - Puget Sound EnergySRP - Salt River ProjectSCL - Seattle City LightTPWR - City of Tacoma Department of Public Utilities TEPC - Tucson Electric Power CompanyTIDC - Turlock Irrigation DistrictWACM - Western Area Power Administration Colorado-Missouri RegionWALC - Western Area Power Administration Lower Colorado RegionWAUW - Western Area Power Administration Upper Great Plains WestWWA - NaturEner Wind Watch LLC

011915hr

21 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

The power grid is modeled by a set of (non-)linear power flow equations

Distribution and transmission grid governing equations

[Pi Qi ] = f (Vi δi ) [Pik Qik ] = f (Vi Vk δik )

22 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Cumulant-based analysis methods to handle stochastic generation and loads on each bus

Input definition Real Pi and imaginary Qi power injections represent generation and loads on each bus

Compute moments microv of the distributions Convert moments to cumulants κv of those distributions

Analysis method

Basic cumulant arithmetic for Y = AX κY v = Av κX v

We extended to cumulant tensors for correlated variables and polynomial functionals

Linearize polar form of equations amp truncate Rectangular form of power flow equation expansion rArr exact quadratic equation

Post-process results

[Vi δi Pik Qik ] outputs impacting costs etc Maximum-entropy PDF reconstruction from κv

23 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

IEEE 14 bus example with wind power generation and plug-in vehicle loads

24 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26

Thanks for listening

Dr Curran Crawford

E-mail currancuvicca

Website wwwssdluvicca

Twitter SSDLab

Students

Composites Ghulam Mustafa

Aero Manuel Fluck Rad Haghi

Grid Trevor Williams Pouya Amid

25 26