8th PVPMC Workshop, May 9-10 2017

1. Calibrating Global Diode Models from I-V Curve
Measurement Matrices without Short-Circuit
Temperature Coefficients
Mark Campanelli 1 Behrang Hamadani 2
1Intelligent Measurement Systems LLC, Bozeman, MT, USA
mark.campanelli@gmail.com
www.pv-fit.com
2National Institute of Standards and Technology, Gaithersburg, MD, USA
8th PV Performance Modeling and Monitoring Workshop
Albuquerque, New Mexico, USA
9 May 2017
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2. Agenda and Take Home Message
1. Reparameterizing diode models in terms of effective irradiance
ratio F and effective temperature ratio H alleviates model
calibration issues related to irradiance and temperature effects.
2. A “typical” single diode model is readily calibrated with
collections of V-I-F-H curve measurements using a reference
PV device for F. H can be harder to define/measure.
3. PV devices hold potential for stochastic tuning of satellite
data when sufficiently accurate performance models are well
calibrated in terms of F and H.
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3. Irradiance and Temperature Effects on I-V Curves
Well recognized issues with—
1. Measuring irradiance: spectral and angular distribution effects
2. Measuring temperature: cell-junction temperature(s) vs.
back-of-module vs. dry-bulb + insolation + wind, etc.
3. Well-defined and measured temperature coefficients
4. Correcting I-V data to constant irradiance and temperature
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4. Effective Temperature Ratio H
Some preliminary definitions1—
TD test device (cell/module/array)
RD reference device (cell/module/array)
OC operating conditions (spectral & total irradiance, temp, ...)
RC reference conditions (e.g., Standard Test Conditions)
Unitless effective temperature ratio—
H :=
T
T0
,
← cell junction temperature at OC
← cell junction temperature at RC
where definition includes the temperature measurement technique.
Ideally, OC for I-V curve measurements “matches” OC in the field,
e.g., continuous illumination to establish temperature gradients.
1
also see Symbol Legend slide at end
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5. Effective Irradiance Ratio F
Unitless effective irradiance ratio—
F :=
Isc
Isc0
← TD’s short-circuit current at OC
← TD’s short-circuit current at RC
= M
Isc,r
Isc,r0
,
← RD’s short-circuit current at OCr
← RD’s short-circuit current at RC
with spectral correction factor M given by the function2—
M = fM (T, Tr, T0
, ¯S, ¯Sr, ¯E, ¯Er, ¯E0
)
=
Isc
∞
λ=0
¯S(λ, T) ¯E(λ) dλ
Isc,r0
∞
λ=0
¯Sr(λ, T0
) ¯E0
(λ) dλ
∞
λ=0
¯S(λ, T0
) ¯E0
(λ) dλ
Isc0
∞
λ=0
¯Sr(λ, Tr) ¯Er(λ) dλ
Isc,r
.
2
assumes linearity in short-circuit currents w.r.t. normally incident irradiance
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6. From Local to Global Single Diode Model
7-parameter global single diode model (SDM-G) is given by3—
Iph − Irs e
V +I Rs
n∗
− 1 − Gp(V + I Rs) − I = 0, (SDM-L)
with 5 auxiliary equations (AE)—
Iph = Isc0 F + Irs e
Isc0 F Rs
n∗
− 1 + Gp Isc0 F Rs,
Irs = Irs0 H3
e
E∗
g0
(1− 1
H ) 1−α∗
Eg0 ,
n∗
= n∗
0
H, n∗
0
:= Ns n kBT0 q
Rs = Rs0 , E∗
g0
:= Eg0 (kBT0 )
Gp = Gp0 . α∗
Eg0
:= αEg0
T0
NOTE: F appears in AE for Iph instead of Isc temp coefficient.
3
7 model parameters at RC in blue, 4 observables in orange (as data)
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7. I-V Curve Matrices as V-I-F-H Datasets
Figure: Level surfaces of constant temperature for optimal fit using ODR
Search for the 7 SDM-G parameter values that minimize the
sum-of-squared orthogonal distances of the V-I-F-H data
points to the 4-dimensional manifold
Orthogonal distance regression (ODR) uses implicit SDM-G
model =⇒ easier to code and computationally efficient
Python module scipy.odr (but no parameter constraints)
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8. Verification: Synthetic I-V Curve Matrix Data for Module
275 W, 72-cell, multi-Si module
Mean F’s: {0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 1.1}
Mean H’s: {0.966, 1.000, 1.083, 1.168}
101 noisy V-I-F-H points per curve
100 SDM-G fits to V-I-F-H matrices
Parameter True Mean C.V.
Value Value (%)
Isc0 [A] 8.30 8.30 0.0053
Irs0 [10−10 A] 3.30 3.31 3.4
n∗
0
[V] 1.880 1.880 0.149
Rs0 [Ω] 0.527 0.527 0.22
Gp0 [10−3 ] 2.59 2.59 0.46
E∗
g0
[·] 43.6 43.6 0.27
α∗
Eg0
[·] -0.0798 -0.0798 3.4
Pmp0
[W] 275 275 0.0183
NOTE: Local calibration of Rs0
and Gp0
using SDM-L shows significantly
higher coefficients of variation (C.V.) depending on (F, H) combination.
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9. Validation: Real I-V Curve Matrix Data for Cell (1)
2 cm×2 cm mono-Si test cell & ref. cell
I-V-F curves and T-dependent spectral
responses measured at NIST
Cell junction temperatures measured under
steady-state continuous illumination
69 points in 60 ms with same T throughout
Mean F’s:
1.13033045 1.14452734 1.15250381
0.93853937 0.94416165 0.95768569
0.72473101 0.73131190 0.73878091
0.32117537 0.32109446 0.32604454
0.15503686 0.15674316 0.15675615
0.12015046 0.12215738 0.12309654
Mean T’s (test and reference cells):
15, 25, 50 ±1◦
C
Isc0
Irs0
n∗
0
Rs0
Gp0
E∗
g0
α∗
Eg0
Pmp0
0.1125 A 7.21×10−9
A 0.0355 V 0.1202 Ω 3.94×10−3
29.2 -0.0781 0.0494 W
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10. Validation: Real I-V Curve Matrix Data for Cell (2)
SDM-L fits over matrix suggest model discrepancy in SDM-G
Seek better SDM-G AEs or try double-diode model?
Isc0
Irs0
n∗
0
Rs0
Gp0
E∗
g0
α∗
Eg0
Pmp0
0.1125 A 7.21×10−9
A 0.0355 V 0.1202 Ω 3.94×10−3
29.2 -0.0781 0.0494 W
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11. Potential for Stochastic Tuning of Satellite Datasets (1)
Calibrated PV device + I-V curve tracer = F-H weather station
After model calibration, re-task SDM-G4—
Iph − Irs e
V +I Rs
n∗
− 1 − Gp(V + I Rs) − I = 0, (SDM-L)
with five auxiliary equations (AE)—
Iph = Isc0 F + Irs e
Isc0 F Rs
n∗
− 1 + Gp Isc0 F Rs,
Irs = Irs0 H3
e
E∗
g0
(1− 1
H ) 1−α∗
Eg0 ,
n∗
= n∗
0
H, n∗
0
:= Ns n kBT0 q
Rs = Rs0 , E∗
g0
:= Eg0 (kBT0 )
Gp = Gp0 . α∗
Eg0
:= αEg0
T0
4
2 fit parameters at OC in blue, 2 observables in orange (as data)
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12. Potential for Stochastic Tuning of Satellite Datasets (2)
Preliminary work suggests the feasibility of
F-H measurements using I-V curves
What about model discrepancy in SDM-G?
Second diode?
Cell inhomogeneity?
Device degradation?
Transient (F, H) conditions?
Does elimination of pyranometers, reference cell, etc. more
than offset cost of I-V curve tracer plus calibrated module(s)?
. . . and deployed at what angle(s)?
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13. Potential for Stochastic Tuning of Satellite Datasets (3)
Over time, a measured (F, H) value would correspond to multiple
satellite dataset values due to inherent non-determinism—
How to model the resulting stochastic process for (F, H) as a
function of satellite dataset value?
Does knowledge of this process significantly improve the
quantification of performance prediction uncertainty?
How much (F, H) and satellite data are sufficient?
We welcome feedback/partners in tackling this opportunity!
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14. Cloud-Based Computational Service
RESTful API
# Python requests package POST with numpy vector data
response =
requests.post(‘https://api.pv-fit.com/beta/SDMfit’,
headers = {‘Content-Type’:‘application/json’},
json = {‘v V data’:v V data.tolist(),
‘i A data’:i A data.tolist()})
# Assuming successful call, unpack response
response json dict = json.loads(response.json())
# Print maximum power
print(response json dict[‘derived params’][‘p mp W’])
Web client at http://www.pv-fit.com
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15. References
M. B. Campanelli and C. R. Osterwald, “Effective Irradiance Ratios to Improve I-V Curve Measurements
and Diode Modeling Over a Range of Temperature and Spectral and Total Irradiance,” IEEE Journal of
Photovoltaics, vol. 6, pp. 48–55, 2016.
C. R. Osterwald, M. Campanelli, T. Moriarty, K. A. Emery, and R. Williams, “Temperature-Dependent
Spectral Mismatch Corrections,” IEEE Journal of Photovoltaics, vol. 5, pp. 1692–1697, 2015.
C. R. Osterwald, M. Campanelli, G. J. Kelly, and R. Williams, “On the Reliability of Photovoltaic
Short-Circuit Current Temperature Coefficient Measurements,” in Proceedings of the 42nd
Photovoltaic
Specialists Conference. IEEE, 2015.
B. Zaharatos, M. Campanelli, and L. Tenorio, “On the Estimability of the PV Single-Diode Model
Parameters,” Statistical Analysis and Data Mining, vol. 8, pp. 329–339, October/December 2015.
K. Roberts, “A Robust Approximation to a Lambert-Type Function,” arXiv:1504.01964v1, April 2015.
J. Brynjarsd´ottir and A. O’Hagan, “Learning about physical parameters: the importance of model
discrepancy,” Inverse Problems, vol. 30, 2014.
B. Paviet-Salomon, J. Levrat, V. Fakhfouri, Y. Pelet, N. Rebeaud, M. Despeisse, and C. Ballif, “New
guidelines for a more accurate extraction of solar cells and modules key data from their current-voltage
curves,” Progress in Photovoltaics: Research and Applications, 2017.
P. T. Boggs and J. E. Rogers, “Orthogonal Distance Regression,” NISTIR, vol. 89–4197, November 1989,
revised July 1990.
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16. Symbol Legend
T Cell junction temperature [K]
Isc Short-circuit current [A]
Iph Photocurrent [A]
Irs Reverse saturation current [A]
n∗ Modified ideality factor [V]
Rs Series resistance [Ω]
Gp Parallel conductance [ ]
E∗
g Modified material band gap [·],
α∗
Eg
Modified temperature coefficient of
material band gap [·]
Pmp Maximum power [W]
¯S Absolute spectral response [
A W/m2
nm
]
¯E Absolute spectral irradiance [W/m2]
Eg Material band gap [J],
αEg Temperature coefficient of material
band gap [1/K],
Ns Number of cells in series per string
n Ideality factor
kB Boltzmann constant
[1.3806488 × 10−23 J/K]
q electron charge
[1.602176565 × 10−19 C]
An additional subscript of—
0 indicates a value at reference conditions (RC)
r indicates a value for the reference device (RD)
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