The document presents two approaches - Monte Carlo simulation and convolution - for modeling the time evolution of photovoltaic module power distributions to evaluate performance warranty risks. It applies these approaches using realistic initial distributions for module power and degradation rates from literature. The results show the distribution of module powers broadening over time and shifting to lower powers. Warranty risks are calculated for different example warranty terms and shown to increase substantially for more aggressive terms. The impact of potential light-induced degradation is also demonstrated. The approaches allow flexible incorporation of updated input data and factors over time.

1. Modeling of PV Module PowerDegradation to EvaluatePerformance Warranty Risks
Thomas Roessler1andKenneth J. Sauer2
1Yingli Green Energy Europe GmbH
2Yingli Green Energy Americas, Inc.
27thEUPVSEC, Frankfurt, September 24-28, 2012

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Motivation
•Module manufacturers provide performance warranties
•Goal: Quantify associated financial risk
•Illustration: Data from SUPSI –ISAAC PV system built 1982
Warranty Limit
Warranty Risk
Model for time evolution of module power distributions is needed!

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Outline
•Motivation
•Context
•Annual Degradation Rate: A Fresh Look
•Systematic Approaches:
Monte Carlo
Convolution
•Application
•Summary and Outlook

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Context: Vázquez and Rey-Stolle
• Time evolution of a Gaussian distribution of module powers
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M. Vázquez and
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Prog. Photovolt.:
Res. Appl.
16 (2008) p. 419.
Extend to arbitrary, realistic module power distributions
Relate broadening to degradation rate distributions

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Context: Jordan and Kurtz
• Encompassing compilation of published measured annual
degradation rates rD for PV modules
• Example result: Distribution q(rD) for crystalline Si modules
D.C. Jordan and
S.R. Kurtz,
Prog. Photovolt.:
Res. Appl. (2011).
Reasonably good fit with gamma distribution

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Annual Degradation Rate: A Fresh Look
• Determined from power P measured at time t and t+Dt
• Repeating this for module population yields distribution q(rD)
• Two interpretations of q(rD):
0 2 4
Frequency
Degradation Rate [%/y]
250 255 260
Frequency
Module Power [W]
t
P t P t t
rD D
  D

[ ( ) ( )]
Probability that the
power of a given module
will degrade by some
amount DP after one
year ( Monte Carlo)
Result of evolution of an
initial Dirac delta function
distribution of module
powers after one year
( Convolution)
(with Dt in years)

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Systematic Approaches: Introduction
• Monte Carlo simulation (evolution from year n to year n+1
by sampling module power & degradation rate distributions):
• Convolution approach [used, e.g., to solve heat or diffusion
equation, kernel is “fundamental solution” for d(P-P0)]:
• Main new aspects:
Applicable to arbitrary initial module power distributions
Explicitly consider distributions of degradation rates q(rD)
( ) ( ' )~ ( ' ) ' 1 p P p P q P P dP n  n


   
( ) ( ) ~ ( ) 1 p P p P q P s s
n
s
n   D 
Kernel

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Systematic Approaches: Choice of Parameters
• Realistic initial module
power distribution:
Truncated Gaussian
Representing one
power class for given
module construction
• Degradation rate distribution:
Gamma distribution fit to data
for mono c-Si by Jordan & Kurtz
Maximum at 0.42%, median of 0.59%
230 240 250 260 270 280 290
Frequency
Module Power [W]
All 60 Cell Modules
260W Power Class
0 2 4
Frequency
Degradation Rate [%/y]

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0.7 0.8 0.9 1 1.1
Frequency
P/P0
Initial
5y
10y
15y
20y
25y
Systematic Approaches: Central Result
• Time evolution:
Warranty Limit,
e.g. 80%
Warranty Risk

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Application: Performance Warranties
• Three different examples:
75
80
85
90
95
100
105
0 5 10 15 20 25
Warranty Limit [%]
Module Age [years]
Stepwise Performance Warranty
Linear Performance Warranty
Agressive Linear Performance Warranty
Linear decrease:
0.71%
0.5%

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Application: Incremental Warranty Risk
• “Incremental warranty risk” = percentage of modules
underperforming for the first time at a given module age
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25
Aggressive Linear [%]
Incremental Warranty Risk
Stepwise and Linear [%]
Module Age [years]
Stepwise
Linear
Aggressive Linear
Total warranty
risk:
3.6%
3.6%
74%!!!
Higher cost of
replacement, tighter
control over system
performance

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Application: Influence of Light-Induced Degradation
•LID: Gaussian with average 1.3%, std. deviation 0.24%
Total warranty risk:
3.6%
10%
00.10.20.30.40.50.60.70.80.90510152025 Incremental Warranty Risk [%] Module Age [years] Linear, no LIDLinear, with LID

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Summary and Outlook
•Summary:
Two systematic and general approaches for time evolution of module power distributions presented
Simple application to comparison of incremental warranty risk for different warranty terms demonstrates basic utility
Effects of LID easily incorporated
•Outlook:
Use actual distributions for module power & degradation rates
Consider module volume from different production years
Consider time dependence of cost of module replacement
Time dependence of degradation rate distributions?
Thanks for your attention!