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PARAMETER SENSITIVITY ANALYSIS OF PHOTON RECYCLING IN GALIUM ARSENIDE SOLAR CELLS:
METHODOLOGICAL DEVELOPMENT
GRACE CAREY, ILYA KORSUNSKY, ARJUNEN KUTAYIAH, KATHLEEN MCGOVERN, LAUREN SWADDELL
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Outline
• Motivation: Environmental Impact• Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression)• Implementation• Results• Design Conclusions• Future Directions
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U.S. Energy Consumption and Production predictions
Source: U.S. Energy Information Administration, Annual Energy Outlook 2011, Early Release, December 16, 2011.
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U.S. energy consumption in 2009
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U.S. Primary Energy Flow
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Source: U.S. Department of Energy, Department of Fossil Fuels, 2011
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Carbon Dioxide Emissions
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Ice Core Data and The Keeling Curve
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Vostok Ice-Core Data
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Alternatives to fossil fuels?The suspense is terrible… I hope it’ll last
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Nuclear Energy• 400 nuclear plants in the world• 100 nuclear plants in the US alone• Powers ~15% of US energy needs• Relies on the use of uranium and
other fissible materials to generate electricity
• Uranium is a finite mineral resource• Cooling methods often employ the use
of local water systems endangering aquatic life
• Nuclear power plants in the US produce 2000 metric tons of radioactive waste
• Nuclear disasters can emit large amounts of radiation which can be lethal and detrimental to the environment
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Solar Energy• Sustainable and renewable resource
which does not emit greenhouse gases• ~1% of U.S. energy • Solar energy hitting the earth is
approximately 274 million giga-watt/year = 8.2 million quads of Btu/year
• Solar cells currently have an average efficiency of 15% 369 thousand quads of Btu/year can be collected if all land mass of earth had solar panels
• Total potential for solar energy is 444,000 TWh
• The world’s total energy consumption is 132,000 TWh
• The total annual energy consumption in the US is less than 0.5% the theoretical amount of sunlight received
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Solar Cell Efficiency Tables
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Solar Cell Efficiency Tables
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Solar Cell Efficiency Tables
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Solar Cell Efficiency Tables
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Outline
• Motivation: Environmental Impact• Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression)• Implementation• Results• Design Conclusions• Future Directions
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N type
P type
Space
Charged
Region
Electric field
N type
P type
N type
P type
Contacts
Photon
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Valence Band, Ev
Conducing Band, Ec
Band Gap, Eg = Ec - Ev
Radiative Recombination
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GaAs•Semiconductor •Direct Band Gap
• No energy is lost to phonons (lattice vibrations) as a result of radiative recombinations.
• Good for optical devices
Photon Recycling•Re-absorption of photons generated in a semiconductor device as a product of radiative recombinations. •Increases efficiency by 1-2%
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Outline
• Motivation: Environmental Impact• Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression)• Implementation• Results• Design Conclusions• Future Directions
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Modeling: Motivation
• Goal: create the best solar cell we can!– Efficacy – Cost – Environmental Impact
• Need some design guidelines• Computational model handles complexity
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The Model
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The Model
• Output: Photon Recycling Rate• Inputs:
– Temperature – Front Surface Reflection – Width– Angle of Incidence– Refractive index – Light Wavelength– Internal Surface Reflectance– Reflectance of Metal Grid– Front Internal Shadow Factor
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How do we use the model?
• Optimize Photon Recycling over the input parameters
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Dealing with Complexity
• 9 input parameters => 9 dimensional hypercube
• Are all the parameters important? • Sensitivity analysis gives importance of each
parameter • Cut down search space
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Outline
• Motivation: Environmental Impact• Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression)• Implementation• Results• Design Conclusions• Future Directions
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The simplest and most powerful relationship between independent and dependent variables is linear.
The dependent variable can be predicted from the independent variable by fitting the data to as
follows:
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The problem is almost always more complicated.
If the dependent variable is a function of multiple independent variables, we have:
This describes multilinear dependencies for only one sample; for n samples y can be written as a column vector and the values of x form the rows of matrix X:
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In multiple linear regression, the solution for the b vector take the form:
Can anyone see a potential problem here?
The formula for b depends on the invertability of the product matrix of the X row vector and the X matrix!
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Partial least squares (PLS) regression does not depend on the invertability of input data.
Assume a linear relationship between independent parameter matrix X and dependent output matrix
Y:
PLS regression uses a variation of the NIPALS algorithm to find the best approximation of this
linear relationship in the form of regression matrix, B.
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What does the PLS algorithm look like?
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The Model
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The previous complexity can be reduced to the following:
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The regression coefficients (Bpls) can give us the following information:
(1) Significance of independent parameters to output(s) of interest
(2) Prediction of dependent parameters from independent parameters (unlike PCA)
(3) Indication of parameters to be tested in future experiments
(4) Unreasonable results indicate that a mathematical model needs to be reevaluated in some regard
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Outline
• Motivation: Environmental Impact• Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression)• Implementation• Results• Design Conclusions• Future Directions
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The Model
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GPR (x) = 2π ∫d E ∫ dμ α b(E, x, μ)∞
EG
bn(E,x) = 2
h3c2
n2 E 2
(E -qφ(x)
) - 1kT
ˆΦ= exp
2αwμ
- RF RR
ΨOF = RR ∫bn exp(αx'
)μdx’ + exp(
2αw
)μ∫ bn exp(
- α x’
)μdx’
ΨOR = RF ∫bn exp(α x'
)μ dx’ +∫ bn exp(
- α x’
)μdx’
b(E,x, μ) = {+ exp x [ ΨOF + ∫ bn exp dx’ ] if 1 ≤ μ < 0
- exp x [ ΨOR + ∫ bn exp dx’ ] if 0 > μ ≥ -1
αμ (
- α x
)μ
RF
Φ (α x'
)μαμ (
- α x
)μ
RR
Φ (α x'
)μ
1
-1
∞
0
x
w
Photon recycling rate (GPR): function map
α =4 log(10) π κ
λ
RF = κF * FSF + ρF * (1 – FSF)
E = h*c
λ
μ = cosθ
Experimental variablesConstants
Functions
KEY
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Φ RF μ α ΨOF ΨOR bn
E b μ α
GPR
α μ RF bn μ α RF bn E
E E
Photon recycling rate (GPR): function dependency chart
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Photon recycling rate (GPR): code sample
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Outline
• Motivation: Environmental Impact• Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression)• Implementation• Results• Design Conclusions• Future Directions
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We can apply the PLS algorithm to our input and output data.
Parameters
GPR
Output
k l k F rF SF W q
n hat T
Tria
ls
Input
-1.5
0
1.5
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Results of PLS regression:
k l k F rF SF W q
n hat T
Input Matrix
GPR
Output
-1.5
0
1.5
GPR
BPLS
k
l
kF
r
FSF
W
q
nhat
T
* =
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Results of PLS Regression:
-1
-0.5
0
0.5
1
k l k F rF SF W q
n hat T
GPR Regression Coefficients
These give quantitative insight into how changing input parameters affects output.
Significant parameters include wavelength of light, temperature, and the front reflectance.
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Accuracy of regression
-2 -1 0 1 2-2
-1
0
1
2
True GPR
Pre
dic
ted G P
R
Predicted vs True GPR
Predicted GPR
True GPR
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Conclusions:
1. PLS regression is an accurate tool for both determining parameter sensitivities from our simulated data sets and predicting the output variable data of interest.
2. As conserving energy is of optimal interest to the environment, photon recycling is an important physical phenomenon to energy conservation and solar cell efficiency.
3. From our regression analysis, the parameters which should be maximized in future cell design are wavelength of light directed at the solar cell, temperature, and front reflectance.
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Future Directions
• Function for cost• Function for environmental impact • Convex optimization
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Questions
• Any?• No?• Thanks!
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Acknowledgements
• The Catalyst Scholarship Program!• Dr. Haydee Salmun• All of our wonderful advisors• Dr. Eric Sobie and Amrita Sarkar